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1,116,691,498,415 | arxiv | \section{Introduction}
\vspace{5mm}
Ordinary $RO(G)$-graded equivariant homology and cohomology for a compact Lie group was
defined by Lewis, May, and McClure in \cite{LewisMayMcClureOrdinary}.
This theory is thus represented by a $G$-equivariant spectrum in the sense of \cite{LewisMaySteinbergerEquivariant}.
For $G$ finite, however, equivariant (co)homology is also deeply connected with Mackey functors \cite{DressMackeyFunctors, GreenleesMackeyFunctors} and can be
characterized entirely on chain level.
As a beginning of the story, ordinary equivariant (co)homology is a special case of Bredon cohomology \cite{Bredon}, but more needs to be said to capture
the whole Mackey functor structure.
The purpose of this paper is to examine this story in more detail and show how certain constructions on
$G$-equivariant spectra can be done on chain level.
While much of the material presented here is ``standard" (with the exception, perhaps, of the last section),
some of it may not be easy to find in the literature.
Given a $G$-CW-complex $X$, we have the equivariant coefficient system-valued cellular chain complex $C_G(X)$ (see Section
\ref{MackeyFunctorSection}). Equivariant cohomology
(resp. homology) of $X$ with respect to a coefficient system (resp. a co-coefficient system) $M$ is the cohomology (resp. homology) of
\beg{EquivariantCoHomologyDefn}{Hom_{\mathscr{O}_G} (C_G(X), M) \text{ (resp. } C_G (X) \otimes_{\mathscr{O}_G} M).}
Of course, both constructions apply when $M$ is a Mackey functor, which is our main case of interest.
(The more general case corresponds to $\Z$-graded Bredon equivariant cohomology and homology theory \cite{Bredon}.)
One point is to clarify how this structure behaves under products.
From a spectral point of view, constructions \rref{EquivariantCoHomologyDefn} correspond to
$$F(X_+, HM), \; \; X_+ \wedge HM,$$
so the discussion of products reduces to multiplicative properties of equivariant Eilenberg-Mac Lane spectra \cite{LewisMayMcClureOrdinary}.
On the chain level, there is a tensor product $\boxtimes$ of coefficient systems such that
$$C_G (X\times Y) \cong C_G (X) \boxtimes C_G (Y).$$
We discuss this in Section \ref{CoefficientSystemLevel}. On the level of spectra, equivariant Eilenberg-Mac Lane spectra with coefficients
in a Mackey functor \cite{LewisMayMcClureOrdinary} are (rigid) module spectra
over the $E_{\infty}$ ring spectrum $H\mathscr{A}$ where $\mathscr{A}$ is the universal (Burnside ring) Green functor.
Thus, for Mackey functors $M, N$ we have the smash product
\beg{SmashProductFromMackeyFunctors}{HM \wedge_{H\mathscr{A}} MN.}
This is reflected in Mackey functors by the box-product $\Box$ (see \cite{DressMackeyFunctors, LiBoxProduct} and Section \ref{MackeyFunctorSection} below), and in fact,
\rref{SmashProductFromMackeyFunctors} is equivalent to the total left derived functor of $H(M\Box N)$.
We shall also discuss duality.
Starting from $G$-CW-complexes, the coefficient system-valued chain complex $C_G(X)$ is not sufficient,
and we need to discuss what is the appropriate Mackey functor-valued chain complex $C_M (X)$.
The right construction, which also carries $\boxtimes$ to $\Box$,
turns out to be the left Kan extension from coefficient systems to Mackey functors, which we discuss in Section \ref{MackeyFunctorSection}.
Duality is discussed in Section \ref{DualitySection}. Mackey functors form a closed symmetric monoidal category
with an internal Hom functor $Hom_M$. For a $G$-CW-complex $X$,
$Hom_M (C_M X,\mathscr{A})$ is the right Kan extension from co-coefficient systems to Mackey functors applied to
$$C_G^* X = Hom (C_G X, \Z).$$
We conclude with Section \ref{AppendixMackeyChains}, where we discuss the role of modules over the constant Green functor $\underline{\Z}$.
An example is
the Mackey functor
\beg{IntroductionFixedPointsMackeyFunctor}{C_{\underline{\Z}} X: G/ H \mapsto (CX)^H}
for a $G$-CW complex $X$.
Other aspects of equivariant homology with constant coefficients were discussed in a previous paper \cite{GeomFixedPoints}.
The facts presented here are, however, different.
One reason for the significance of Section \ref{AppendixMackeyChains} in the present story
is that it clarifies why we cannot just work with the fixed points of the ordinary chain complex as a chain-level model
of general equivariant homology with Mackey functor coefficients.
However, Mackey $\underline{\Z}$-modules are of independent interest and in some sense, can be
considered as an alternative type of representation theory.
In fact, \rref{IntroductionFixedPointsMackeyFunctor} turns out to correspond to the spectrum $X_+ \wedge H\underline{\Z}$, i.e. the universal $\underline{\Z}$-module valued
Mackey chain complex on $X$, where $\underline{\Z}$ denotes the constant Green functor.
While the $H$-fixed points of a $\Z[G]$-module, for a subgroup $H\subseteq G$, always give a Mackey $\underline{\Z}$-module,
the converse is not always true (we give a characterization of Mackey $\underline{\Z}$-modules).
We discuss, however, a certain sense in which the converse is true on the derived level.
Finally, we briefly discuss cofixed points which give, in some sense, an equivalent theory
for $G$ cyclic, but not in general.
We give an example.
\vspace{5mm}
\section{Coefficient Systems and Products}\label{CoefficientSystemLevel}
\vspace{5mm}
For a CW-complex $X$, we denote by $C_*(X)$ the cellular chain complex of $X$. We denote by $\mathscr{O}_G$ the orbit category of $G$.
A $G$-coefficient system is a functor $\mathscr{O}_G^{Op} \rightarrow Ab $ (the category of abelian groups). Likewise, a $G$-co-coefficient
system is a functor $\mathscr{O}_G \rightarrow Ab$.
The category of finite $G$-sets and $G$-maps will be denoted by
``f.$G$-Sets." For a $G$-CW complex $X$, let
$$C_G (X) (G/H) := C_G (X^H)$$
denote the cellular coefficient-system-valued chain complex of $X$.
On the other hand, let
$$C^*_G(X)(G/H) : = Hom_{Ab}(C_G(X)(G/H), \Z)$$
denote the dual of $C_G(X)$. Since $Hom_{Ab}$ is contravariant in the first variable,
$C_G^*(X)$ is a co-coefficient system.
A {\em Mackey functor} is a pair consisting of a coefficient system and a co-coefficient system which agree on objects.
There is a compatibility condition which will be discussed in Section \ref{MackeyFunctorSection}.
In this section, we will define a ``tensor product" $A \boxtimes B$ of coefficient systems
$$A, B:\mathscr{O}_G^{Op}\rightarrow Ab$$
First, for any coefficient system $A$, we can define an extension
$$A: \text{f.}G \text{-Sets} \rightarrow Ab$$
by
$$A(\coprod G/H_i) = \oplus A (G/H_i)$$
We have the Cartesian product functor
\beg{CartesianFiniteGSets}{\times :\text{f.} G\text{-Sets}^{Op}\times \text{f.} G\text{-Sets}^{Op} \rightarrow \text{f.}G \text{-Sets}^{Op}.}
Recall that for a finite $G$-set $S$, the representable functor by $S$ is a coefficient system
$$F_S : G/H \mapsto \Z Map_G (G/H, S)$$
which is the value at $\Z$ of the left adjoint to evaluation at $S$ (the value of the left adjoint on any abelian group $A$ is obtained by tensoring with $A$,
i.e.
$$A \mapsto A\otimes F_S).$$
To see this adjunction, which is a variant of the Yoneda lemma,
first we have
$$A\otimes F_S (U) = A\{ \text{f.}G\text{-Sets} (U , S)\}$$
for a finite $G$-set $U$.
Then, for an additive functor
$$\Phi: \text{f.} G\text{-Sets}^{Op} \rightarrow Ab,$$
it is enough to show that a homomorphism on abelian groups
\beg{AssumingFunctor}{h:A \rightarrow \Phi(S)}
uniquely determines a natural transformation
\beg{DeterminedFunctor}{H:A\otimes F_S \rightarrow \Phi.}
We have
$$H(S)(a\otimes (Id:S\rightarrow S)) = h(a)$$
for $a\in A$.
Now for any $G$-set $U$ we must define
$$H(U): A\otimes F_S(U) = A \otimes \{U\rightarrow S\} \rightarrow \Phi (U)$$
by putting, for an $a\in A$, and a $f: U\rightarrow S$,
$$H(U) (a\otimes f) = (\Phi(f)) \circ H(S) (a\otimes Id_S) = \Phi(f)(h(a)).$$
Now, given coefficient systems $A, B : \text{f.} G\text{-Sets}^{Op}\rightarrow Ab$, first define
$$A\underline{\boxtimes} B : \text{f.} G\text{-Sets}^{Op}\times \text{f.} G\text{-Sets}^{Op} \rightarrow Ab$$
by $A\underline{\boxtimes} B (S, T) := A(S) \otimes B(T)$.
We define $A\boxtimes B : \mathscr{O}_G^{Op} \rightarrow Ab$ as the left Kan extension of $A\underline{\otimes} B$
by the Cartesian product \rref{CartesianFiniteGSets}.
\begin{lemma}\label{CoefficientSystemBoxtimesConnection}
\begin{enumerate}
\item\label{BoxtimesConnectionFinite}
For finite $G$-sets $S, T$ we have
$$F_S \boxtimes F_T = F_{S\times T}.$$
\item\label{BoxtimesConnectionCWComplexes}
For $G$-CW complexes $X, Y$, we have
$$C_G(X) \boxtimes C_G (Y) = C_G (X\times Y).$$
\end{enumerate}
\end{lemma}
\begin{proof}
To prove \rref{BoxtimesConnectionFinite}, the left Kan extension along \rref{CartesianFiniteGSets}
$$Funct(\text{f.} G\text{-Sets}^{Op} \times \text{f.} G\text{-Sets}^{Op}, Ab)\rightarrow Funct(\text{f.} G\text{-Sets}^{Op}, Ab)$$
is defined to be the left adjoint of $Funct(\times, Ab)$.
Then we shall study the composition of functors
\beg{LemmaCompositionABFUNCTS}{Ab \rightarrow Funct(\text{f.} G\text{-Sets}^{Op} \times \text{f.} G\text{-Sets}^{Op}, Ab)\rightarrow Funct(\text{f.} G\text{-Sets}^{Op}, Ab),}
where the first functor is the left adjoint to evaluation at some fixed $(S, T)$.
The composition \rref{LemmaCompositionABFUNCTS} is left adjoint to the functor that evaluates a functor in $Funct(\text{f.} G\text{-Sets}^{Op}, Ab)$ to $ S\times T$.
Thus it sends $\Z$ to $F_{S\times T}$. On the other hand, the first functor \rref{LemmaCompositionABFUNCTS} sends $\Z$ to $F_S \underline{\boxtimes} F_T$.
As above, this means that a homomorphism
$$h:A \rightarrow \Phi(S, T)$$
uniquely determines a natural transformation
$$H:A\otimes (F_S \underline{\boxtimes} F_T) \rightarrow \Phi.$$
By definition, we have
$$F_S \underline{\boxtimes} F_T (U, V) = \Z \{ U\rightarrow S\} \otimes \Z \{ V\rightarrow T\} = \Z \{ (U\rightarrow S, V\rightarrow T)\}.$$
Again, we have
$$H(S,T) (a\otimes (Id:S\rightarrow S, Id:T\rightarrow T)) = h(a)$$
for $a\in A$.
Now, for any finite $G$-set $U$, we must define
$$H(U,V) : A \otimes (F_S \underline{\boxtimes} F_T (U,V)) \rightarrow \Phi (U,V)$$
by putting, for $a\in A$, $f_1: U\rightarrow S$, $f_2: V\rightarrow T$,
$$H(U,V) (a\otimes (f_1 \otimes f_2))= (\Phi (f_1, f_2)) \circ H(S, T) (a \otimes (Id_S\otimes Id_T))=$$
$$= (\Phi (f_1, f_2))(h(a)).$$
This concludes the proof of \rref{BoxtimesConnectionFinite}.
Finally, \rref{BoxtimesConnectionCWComplexes} is an immediate consequence, since
$$C_G(X)_n = F_{I_n}$$
where $I_n$ is the set of $n$-cells of $X$.
\end{proof}
\vspace{5mm}
\section{Mackey Functors and Products}\label{MackeyFunctorSection}
\vspace{5mm}
Ordinary equivariant $RO(G)$-graded homology and cohomology, however, work on the level
of Mackey functors, not coefficient systems.
This becomes important when one studies duality.
Mackey functors have their own product, different from the tensor product $\boxtimes$ of coefficient systems.
It is called the {\em box product}, and we denote it by $\Box$ (see \cite{DressMackeyFunctors, LiBoxProduct}).
The purpose of this section is to relate the results of the last section to statements about $\Box$.
We begin with a more detailed treatment of Mackey functors.
\vspace{3mm}
Recall the Burnside category $\mathscr{B}_G$. The objects of $\mathscr{B}_G$ are finite $G$-sets.
Morphisms in $\mathscr{B}_G$ from $T$ to $U$
are given by the group completion of the set of isomorphism classes (in the $S$-coordinate) of diagrams
of finite $G$-sets
$$
\diagram
& S\drto\dlto & \\
T & & U\\
\enddiagram
$$
with respect to the operation of coproducts (again in the $S$-coordinate).
The composition of two diagrams is defined in the obvious way using pullbacks (for details, see \cite{DressMackeyFunctors}).
Then a Mackey functor is defined to be an additive functor
$$\mathscr{B}_G \rightarrow Ab.$$
The category of Mackey functors over $G$ is denoted by $Mackey_G$.
(Note that by definition, $\mathscr{B}_G$ is self-dual.)
There is an operation for Mackey functors similar to the operation for coefficient systems $\boxtimes$ defined in the previous section.
This ``tensor product" of Mackey functors is denoted by $\Box$ (c.f. \cite{DressMackeyFunctors, LiBoxProduct}).
We will review this construction.
First, as above in the previous section, given Mackey functors $M,N: \mathscr{B}_G\rightarrow Ab$,
define
\beg{UnderlinedBox}{M\underline{\Box} N : \mathscr{B}_G\times \mathscr{B}_G\rightarrow Ab}
by $M\underline{\Box} N (S, T) := M(S) \otimes N(T)$.
On the other hand, we also have a ``Cartesian product"
\beg{CartesianProductBurnside}{\times:\mathscr{B}_G \times \mathscr{B}_G \rightarrow \mathscr{B}_G.}
(Note that \rref{UnderlinedBox} and \rref{CartesianProductBurnside} are biadditive functors.
Also note that to define \rref{CartesianProductBurnside}, one must consider the group completion.)
Define $M\Box N$ to be the left Kan extension of $M\underline{\Box} N$ via \rref{CartesianProductBurnside}.
\vspace{5mm}
\noindent {\bf Comment:}
Commutative monoids $\mathscr{G}$ with respect to the box product
$$\mathscr{G}\Box \mathscr{G} \rightarrow \mathscr{G}$$
are called Green functors.
A module $\mathscr{M}$ over a Green functor $\mathscr{G}$ is defined by
$$\mathscr{M} \Box \mathscr{G} \rightarrow \mathscr{M}$$
with the usual axioms.
Of course, we also must consider the unit of the box product. This is the universal Green functor $\mathscr{A}$, i.e. the Burnside ring functor.
By the above method, it is the left Kan
extension of the functor
$$*\rightarrow Ab$$
$$* \mapsto \Z$$
via the inclusion
$$* \rightarrow \mathscr{B}_G$$
$$* \mapsto G/G.$$
Then $\mathscr{A}(G/H)$ is the value of the functor that sends
$$G/H \mapsto K \{ G/G \leftarrow S \rightarrow G/H\} = K \{ S\rightarrow G/H\},$$
where $K$ denotes the group completion.
This is is the Burnside ring of $H$, since the category finite $G$-sets over $G/H$ is equivalent to the category of finite $H$-sets via the functor
$$\{ S\rightarrow G/H\} \rightarrow H\text{-Sets}$$
$$f\mapsto f^{-1}(*).$$
\vspace{5mm}
\begin{lemma}\label{BoxTimesVSBoxConnection}
The left Kan extension
$$L: Funct(\mathscr{O}_G^{Op}, Ab) \rightarrow Mackey_G$$
takes $\boxtimes$ to $\Box$, i.e.
$$L(A\boxtimes B) = L(A) \Box L(B)$$
for coefficient systems $A, B$.
\end{lemma}
\begin{proof}
We have left Kan extensions
$$L: Funct(\mathscr{O}_G^{Op}, Ab) \rightarrow Mackey_G = Add(\mathscr{B}_G, Ab)$$
and
$$\mathscr{L}: Funct(\mathscr{O}_G^{Op}\times \mathscr{O}_G^{Op}, Ab) \rightarrow BiAdd(\mathscr{B}_G \times \mathscr{B}_G, Ab)$$
where $Add$ denotes the category of additive functors and $BiAdd$ denotes the category of biadditive functors.
Then we can form a diagram
$$
\diagram
Funct(\mathscr{O}_G^{Op}, Ab)\times Funct(\mathscr{O}_G^{Op}, Ab) \rto^(0.575){\underline{\boxtimes}}\dto_{L\times L} &Funct(\mathscr{O}_G^{Op}\times \mathscr{O}_G^{Op}, Ab)\dto^{\mathscr{L}}\\
Add(\mathscr{B}_G, Ab)\times Add(\mathscr{B}_G, Ab) \rto_{\otimes} & BiAdd(\mathscr{B}_G\times\mathscr{B}_G, Ab)\\
\enddiagram
$$
This diagram commutes, since for coefficient systems $A, B$,
$$\mathscr{L}(A\otimes B) = \mathscr{B}_G\times \mathscr{B}_G \otimes_{\mathscr{O}_G^{Op}\times\mathscr{O}_G^{Op}}A\otimes B=L(A)\otimes L(B).$$
On the other hand, the diagram
$$
\diagram
Add(\mathscr{B}_G, Ab) \times Add(\mathscr{B}_G, Ab)\rto^(0.55)\otimes \drto_{\Box} & BiAdd(\mathscr{B}_G\times \mathscr{B}_G, Ab)\dto^{\times_{\#}} \\
& Add(\mathscr{B}_G, Ab)
\enddiagram
$$
where $\times_{\#}$ the left Kan extension of $\times$, commutes since their right adjoints commute.
Also, we have a commutative diagram
$$
\diagram
Funct(\mathscr{O}_G^{Op} \times \mathscr{O}_G^{Op}, Ab) \rrto^(0.55){\times_{\#}}\dto_{\mathscr{L}} & &Funct(\mathscr{O}_G^{Op}, Ab)\dto^L \\
BiAdd(\mathscr{B}_G \times \mathscr{B}_G. Ab) \rrto_{\times_{\#}} & & Add(\mathscr{B}_G, Ab).\\
\enddiagram
$$
Also,
$$
\diagram
Funct(\mathscr{O}_G^{Op}, Ab)\times Funct(\mathscr{O}_G^{Op}, Ab) \dto_{\underline{\boxtimes}} \drto^{\boxtimes}& \\
Funct(\mathscr{O}_G^{Op}\times \mathscr{O}_G^{Op}, Ab)\rto_(0.55){\times_{\#}} & Funct (\mathscr{O}_G^{Op}, Ab)\\
\enddiagram
$$
commutes by definition.
Therefore, by combining the diagrams, we get that
$$
\diagram
Funct(\mathscr{O}_G^{Op}, Ab)\times Funct(\mathscr{O}_G^{Op}, Ab)\dto_{L\times L} \rto^(0.65){\boxtimes}& Funct(\mathscr{O}_G^{Op}, Ab)\dto^L \\
Add(\mathscr{B}_G, Ab)\times Add(\mathscr{B}_G, Ab) \rto^(0.6){\Box}&Add(\mathscr{B}_G, Ab) \\
\enddiagram
$$
commutes.
\end{proof}
\begin{definition}
For a $G$-CW-complex $X$, define
$$C_M(X)= L C_G (X).$$
For a based $G$-CW-complex $X$, its based Mackey complex $\widetilde{C}_M(X)$ is defined analogously using $\widetilde{C}_n(X)$.
\end{definition}
A simpler construction may come to mind: sending $H \mapsto (C(X))^H$ also forms a Mackey functor.
However, this turns out to be the wrong construction for the present purpose.
We clarify the role of this construction (``constant coefficients in a looser sense") in Section \ref{AppendixMackeyChains}.
\begin{lemma}
For $G$-CW-complexes $X, Y$,
$$C_M(X\times Y) = C_M(X) \Box C_M(Y).$$
For based $G$-CW-complexes $X, Y$,
$$\widetilde{C}_M(X\wedge Y) = \widetilde{C}_M(X) \Box \widetilde{C}_M(Y).$$
\end{lemma}
\begin{proof}
This follows immediately from Lemma \ref{CoefficientSystemBoxtimesConnection} and Lemma \ref{BoxTimesVSBoxConnection}.
The based case is analogous, omitting the base point cell.
\end{proof}
To give an example how these constructions are used, given an element $e_V$ of
$\widetilde{H}_{m|V|}^G (S^{mV};\mathscr{A})$, one can use it to construct a map, for a based $G$-CW-complex $X$,
$$\widetilde{H}_k (X;M) \rightarrow \widetilde{H}^G_{k+m|V|}(X\wedge S^{mV}, M).$$
This can be described on chain level as follows. We have maps:
\beg{VerticalDiagramForIsomorphismForBorel}{
\diagram
\widetilde{H}_k (X;M) = H_k((\widetilde{C}_M(X) \Box M)(G/G))\dto^{\otimes e_V}\\
H_k((\widetilde{C}_M (X)\Box M) (G/G) \otimes H_{m|V|} (\widetilde{C}_M(S^{mV})(G/G)\dto \\
H_{k+m|V|}((\widetilde{C}_M(X)\Box M)(G/G) \otimes \widetilde{C}_M(S^{mV})(G/G))\dto \\
H_{k+m|V|}( (\widetilde{C}_M(X) \Box M \Box \widetilde{C}_M(S^{mV}))(G/G)),\\
\enddiagram}
which is isomorphic to $\widetilde{H}^G_{k+m|V|}(X\wedge S^{mV}, M)$, since
$$(\widetilde{C}_M(X) \Box M \Box \widetilde{C}_M(S^{mV}))(G/G) \cong \widetilde{C}_M(X\wedge S^{mV}) \Box M (G/G).$$
\vspace{5mm}
\section{Mackey Functors and Duality}\label{DualitySection}
\vspace{5mm}
We can also use these constructions to treat the $G$-equivariant duality
between ordinary homology and cohomology on chain level.
\begin{lemma}
$Mackey_G$ is a closed category, i.e. there exists a functor
$$Hom_M : Mackey_{G}^{Op} \times Mackey_G \rightarrow Mackey_G$$
such that we have a natural isomorphism
$$Mackey_G( M \Box N, P) \cong Mackey_G(M, Hom_M(N, P)).$$
\end{lemma}
\begin{proof}
Let $M, N, P$ be Mackey functors.
Suppose we have a natural transformation
$$M\Box N \rightarrow P.$$
By definition, this consists of morphisms, for finite $G$-sets $S,T$,
$$M(S)\otimes N(T) \rightarrow P(S\times T)$$
with naturality diagrams, for $\mathscr{B}_G$-morphisms $f: S\rightarrow S'$ and $\varphi: T\rightarrow T'$,
of the form
\beg{DiagramBoxTimesDuality}{
\diagram
M(S) \otimes N(T) \rto \dto &P(S\times T)\dto\\
M(S')\otimes N(T')\rto & P(S' \times T').\\
\enddiagram
}
By adjunction, we can write this as
$$M(S) \rightarrow (T\mapsto Hom_{Ab}(N(T), P(S\times T)).$$
By diagram \rref{DiagramBoxTimesDuality}, the system $g_T\in Hom_{Ab} (N(T), P(S\times T))$ satisfies
$$g_{T'} \circ N(\varphi) = P(S\times \varphi) \circ g_T.$$
Thus we can define
$$Hom_M(N, P)(S)=\{ g_T: N(T) \rightarrow P(S\times T)\; \mid$$
$$\text{ for } \varphi:T\rightarrow T', \; g_{T'} \circ N(\varphi ) = P (S\times \varphi)\circ g_T\}$$
Also, from diagram \rref{DiagramBoxTimesDuality}, we get a diagram
$$
\diagram
M(S)\dto\rto & (T\mapsto Hom_{Ab}(N(T), P(S\times T)))\dto\\
M(S')\rto & (T'\mapsto Hom_{Ab}(N(T'), P(S'\times T'))).
\enddiagram
$$
Thus, $Hom_M(N,P)$ is made into a Mackey functor by
$$f(T\rightarrow g_T): T\mapsto P(f\times T) \circ g_T.$$
Clearly, these choices are forced and reversible, thus proving the natural isomorphism.
\end{proof}
\begin{lemma}
Suppose $X$ is a $G$-CW-complex, then
$$Hom_M(C_M(X),\mathscr{A}) \cong R^*(C^*_G(X)),$$
where $R^*$ is the right Kan extension from co-coefficient systems to Mackey functors.
\end{lemma}
\begin{proof}
We will prove
$$Hom_M(L F_U, \mathscr{A})(S) \cong R^*F_U^*$$
for a general finite $G$-set $U$, extend to the case of infinite $G$-sets, and then apply this to $U= I_n$, in which case
$$F_U = C_G(X)_n.$$
We have, for a $G$-set $S$,
\beg{DualityLemma}{Hom_M(LF_U, \mathscr{A})(S):T\mapsto g_T \in Hom_{Ab}(F_U(T), \mathscr{A}(S\times T)),}
where for $\varphi\in \mathscr{O}_G^{Op}(T,T')$,
$$g_{T'} \circ F_U(\varphi) = \mathscr{A}(S\times \varphi) \circ g_T.$$
Also, $F_U(T) = \mathscr{O}_G(T, U)$.
So, \rref{DualityLemma} is uniquely determined by
$$g_U(Id_U: U\rightarrow U) \in \mathscr{A}(S\times U).$$
So,
$$Hom_M(LF_U, \mathscr{A})(S) = \mathscr{A}(S\times U).$$
On the other hand, we will also show
$$R^* (Hom(F_U,\Z))(S)= \mathscr{A}(S\times U).$$
We have
$$Hom(F_U, \Z): T\mapsto Hom(\Z\mathscr{O}_G (T,U),\Z).$$
Denote $\Gamma:= R^* Hom (F_U, \Z )$.
Then we have, by definition,
$$\Gamma = Hom_{\mathscr{O}_G}(\mathscr{B}, Hom_{Ab} (F_U, \Z).$$
Thus, $\Gamma(T)$ is given by morphisms
$$\mathscr{B}_G(T,T')\otimes\Z \mathscr{O}_G(T',U) \rightarrow \Z$$
subject to the usual identification coming from $\mathscr{O}_G$-functors of $T'$.
Therefore
$$\Gamma(T) = Hom_{Ab}(\mathscr{B}_G(T,U),\Z)= Hom_{Ab}(\mathscr{A}(T\times U),\Z).$$
The proof is concluded by noting that the representable Mackey functor
$T\mapsto \mathscr{B}_G(U,T)=\mathscr{B}_G(T,U)=\mathscr{A}(T\times U)$ is isomorphic to $T\mapsto Hom_{Ab}(\mathscr{B}_G(U,T), \Z).$
This proves the claim for $U$ finite. For $U$ infinite, $C^*$, $R^*$ turn direct colimits into limits.
\end{proof}
\vspace{5mm}
\section{Mackey Chains with Constant Coefficients}\label{AppendixMackeyChains}
\vspace{5mm}
Since a chain complex is a type of a ``stable object," one could ask to what extent one can simply use chain complexes of $\Z[G]$-modules
(and their fixed points under subgroups) to generate our Mackey functors, instead of the more complicated structures treated earlier in this paper.
The answer is that a chain complex of $\Z[G]$-modules captures, essentially, information with constant Mackey functor coefficients,
rather than general Mackey coefficients.
The purpose of this section is to discuss this point.
\vspace{3mm}
We begin by characterizing Mackey functors which are modules over the Green functor $\underline{\Z}$.
(In this case, the Mackey functor structure determines the module structure.)
Next, we observe that fixed points and cofixed points of a $\Z[G]$-module give rise to a $\underline{\Z}$-module Mackey functor.
However, not all examples arise in this way.
On the other hand, a derived statement along such lines is true.
More precisely, we describe a notion of an equivalence on chain complexes of $\Z[G]$-modules based on quasi-isomorphisms of $H$-fixed points
(called fp-{\em equivalence})
and prove that the corresponding derived category is equivalent to the derived category of the abelian category of Mackey $\underline{\Z}$-modules.
There is also a seemingly symmetrical notion of equivalence based on quasi-isomorphisms on $H$-cofixed points (called cfp-{\em equivalence}) and
one may ask if it coincides with the equivalence based on fixed points.
We prove that this is in fact true for $G$ cyclic, but give an example showing that it is false for $G=\Z/2\times \Z/2$.
\vspace{5mm}
Our classification of $\underline{\Z}$-Mackey modules is given by the following
\begin{proposition}
The category of Mackey functor $\underline{\Z}$-modules is equivalent to the full subcategory of $Mackey_G$ on Mackey functors $M$
such that for $f: G/H \rightarrow G/K \in Mor(\mathscr{O}_G)$,
\beg{ConditionForModule}{f_* f^* = \frac{|K|}{|H|}.}
\end{proposition}
\begin{proof}
By definition, $M$ is a $\underline{\Z}$-module when, for a map
$$f: G/H \rightarrow G/K,$$
the diagrams
$$
\diagram
M(G/H) \otimes \Z \rto^(0.55)= & M(G/H)\\
M(G/K) \otimes \Z \uto^{f^* \otimes f^*}\rto_(0.55)= & M(G/K) \uto_{f^*}
\enddiagram
$$
$$
\diagram
&M(G/H) \otimes \Z \rto^= & M(G/H)\ddto^{f_*} \\
M(G/H) \otimes \Z \urto^{Id \otimes f^*}\drto_{f_*\otimes Id} & & \\
& M(G/K) \otimes \Z \rto_= & M(G/K)
\enddiagram
$$
$$
\diagram
&M(G/H) \otimes \Z \rto^= & M(G/H)\ddto^{f_*} \\
M(G/K) \otimes \Z \urto^{f^* \otimes Id}\drto_{Id \otimes f_*} & & \\
& M(G/K) \otimes \Z \rto_= & M(G/K)
\enddiagram
$$
commute (see \cite{LiBoxProduct}, Lemma 21).
The conditions that the first and second diagrams commute are trivial.
The third diagram gives the condition that
$$(f_* f^*: M(G/H) \rightarrow M(G/K)) = \frac{|K|}{|H|}.$$
\end{proof}
\vspace{5mm}
To describe the relation with $\Z[G]$-modules, note that for a $\Z[G]$-module $N$,
\beg{ExampleFixedPoints}{G/H \mapsto N^H}
\beg{ExampleDual}{G/H \mapsto N_H= N\otimes_{\Z[G]} \Z}
are Mackey functors satisfying \rref{ConditionForModule} where for \rref{ExampleFixedPoints}, restrictions are given by inclusions
and corestrictions are given by summing over cosets, and for \rref{ExampleDual}, vice versa. Thus, they give examples of $\underline{\Z}$-modules.
If $N=\Z$ (with trivial $G$-action), \rref{ExampleFixedPoints} gives $\underline{\Z}$ and \rref{ExampleDual} gives its dual.
This also shows that neither \rref{ExampleFixedPoints} nor \rref{ExampleDual} give all $\underline{\Z}$-modules.
\vspace{5mm}
On the other hand, we have the following
\begin{proposition}\label{MackeyAbelianConnection}
The functor $\Xi_{G/H}$ from the category of abelian groups to the category of Mackey functors defined by, for an object $A \in Ab$
and a subgroup $H\subseteq G$,
$$\Xi_{G/H}: A \mapsto (G/K \mapsto ( A((G/H)^K):= A\otimes \Z ((G/H)^K))),$$
(where restrictions are defined by restrictions of fixed points, and co-restrictions are given by sums over representatives of cosets)
is a universal $\underline{\Z}$-module on $F_{G/H}\otimes A$. Thus, $\Xi_{G/H}$ is left adjoint to evaluation of a $\underline{\Z}$-module
Mackey functor at $G/H$.
\end{proposition}
\begin{proof}
To simplify notation, we will just treat the case of $A= \Z$ (the general case is analogous).
We need to show that $\Xi_{G/H} (\Z)$ is the quotient $Q_{G/H}$ of the representable Mackey functor
$$G/K \mapsto \mathscr{B} (G/K, G/H)$$
modulo the relation \rref{ConditionForModule}.
To this end, note that $(\Z(G/H))^K$ is freely generated, as an abelian group, by
$$\sum_{\gamma\in K/(gHg^{-1})\cap K} 1\cdot (\gamma g H)$$
where $g$ runs through representatives of double cosets $K\backslash G / H$.
Thus $\Z [G/H]^K$ is the free abelian group on generators of the form
\beg{LeftAdjointsLemmaDiagram}{
\diagram
& G/(gHg^{-1} \cap K) \drto \dlto& \\
G/H & & G/K, \\
\enddiagram
}
or
\beg{FixedPointsAreFreeAbelianGroup}{\Xi_{G/H} (G/K) = (\Z(G/H))^K = \Z \{ G/H \leftarrow G/(gHg^{-1} \cap K) \rightarrow G/K\}.}
On the other hand,
$$\mathscr{B}(G/K, G/H) = \Z \{ G/H \leftarrow G/J \rightarrow G/K | J\subseteq G/gHg^{-1} \cap K\}.$$
This can be written as
$$
\diagram
& & S\dlto_q \drto^q & & \\
& G/ gHg^{-1} \cap K \dlto& & G/ gHg^{-1} \cap K\drto & \\
G/H & & & & G/K \\
\enddiagram
$$
for some unique $q$.
By \rref{ConditionForModule}, this is identified with a multiple of \rref{LeftAdjointsLemmaDiagram}.
Thus $Q_{G/H}$ is a quotient of $\Xi_{G/H} (G/K)$ (see \rref{FixedPointsAreFreeAbelianGroup}).
On the other hand, $\Xi_{G/H}$ is a $\underline{\Z}$-module by the previous comment, and thus, the quotient map is the identity.
\end{proof}
\begin{corollary}\label{CorollaryUnivComplex}
For a $G$-CW-complex $X$,
$$C_{\underline{\Z}} : G/K \mapsto (C(X))^K$$
is the universal complex of $\underline{\Z}$-modules on the complex of coefficient systems $C_G (X)$.
\end{corollary}
\begin{proof}
Apply Proposition \ref{MackeyAbelianConnection} to the orbit summands of the set $I_n$ of $n$-cells (and $A= \Z$), and take direct sum.
\end{proof}
\noindent {\bf Comment:}
\noindent Thus, $C_{\underline{\Z}} (X)$ corresponds to the $G$-equivariant spectrum $X_+ \wedge H\underline{\Z}$.
\vspace{5mm}
Proposition \ref{MackeyAbelianConnection} and Corollary \ref{CorollaryUnivComplex}
can be used to show that a certain derived category of the category of chain complexes of
$\Z[G]$-modules is equivalence to the derived category of $\underline{\Z}$-Mackey modules.
Denote by $\Z[G]\text{-Chain}$ the category of chain complexes of $\Z[G]$-modules.
\begin{definition}
A morphism $f:C\rightarrow D$ in $\Z[G]\text{-Chain}$ is called an {\em fp}-equivalence if for every subgroup $H\subseteq G$, the map induced on fixed points $f^H : C^H\rightarrow D^H$ is a quasi-isomorphism (i.e., induces an isomorphism in chain homology). Symmetrically, it is called
a {\em cfp}-equivalence if for every subgroup $H\subseteq G$, the map on cofixed points
$f_H :C_H \rightarrow D_H$
is a quasi-isomorphism.
\end{definition}
By Corollary \ref{CorollaryUnivComplex}, the homotopy category $h\Z[G]\text{-Chain}$ has colocalization with respect to fp-equivalences by cell chain complexes of $\Z[G]$-modules (see \cite{AGBook}, Section 5.2) where a cell chain complex of $\Z[G]$-modules $C$ is defined to be of the form
$$C=colim C_{(n)}$$
for some
$$C_{(-1)} \rightarrow C_{(0)} \rightarrow C_{(1)}\rightarrow \dots$$
where $C_{(-1)}=0$, and $C_{(n+1)}$ is the mapping cone of a chain map
$$P_{(n)} \rightarrow C_{(n)}$$
where $P_{(n)}$ is a direct sum of chain complexes of $\Z[G]$-modules of the form $\Z[G/H_i][n_i]$ and has 0 differential.
Thus we have proved
\begin{proposition}
The derived category of $\Z[G]\text{-Chain}$ with respect to fp-equivalences exists.
\end{proposition}
\qed
\begin{proposition}
The derived category $D_{fp}(G)$ of $\Z[G]\text{-Chain}$ with respect to fp-equivalences is equivalent to the
derived category $D\underline{\Z}\text{-Mod}$ of the abelian category $\underline{\Z}\text{-Mod}$ of Mackey modules over the constant Green functor
$\underline{\Z}$.
\end{proposition}
\begin{proof}
By construction, $D_{fg}(G)$ is a full subcategory of $D\underline{\Z}\text{-Mod}$.
The fully faithful functor is onto on isomorphism classes because every $D\underline{\Z}$-module
has a resolution in $D_{fp}(G)$ by Proposition \ref{MackeyAbelianConnection}.
\end{proof}
It is easy to see examples where a quasi-isomorphism of chain complex of $\Z[G]$-modules is not an fp-equivalence or a cfp-equivalence
(e.g., a free $\Z[\Z/2]$-resolution of $\Z$).
Because of the apparent symmetry, one can ask if there is a relationship between fp-equivalence and cfp-equivalence.
By the following proposition, they are actually the same when $G$ is cyclic.
\vspace{5mm}
\begin{proposition}
For a finite cyclic group $G$, a chain map of $\mathbb{Z}[G]$-modules is a quasi-isomorphism after taking $H$-fixed points for all subgroups
$H\subseteq G$ if and only if it is a quasi-isomorphism after taking $H$-cofixed points for all subgroups $H\subseteq G$.
\end{proposition}
\begin{proof}
First, a chain map
\beg{GeneralChainMapDiagram}{\diagram
\dots \rto & C_n \dto\rto & C_{n-1} \dto\rto & C_{n-2} \dto\rto & \dots\\
\dots \rto & D_n \rto & D_{n-1} \rto & D_{n-2} \rto & \dots\\
\enddiagram
}
is a quasi-isomorphism if and only if the totalization of \rref{GeneralChainMapDiagram} (considering it as a
double chain complex) is an exact sequence. Hence it suffices to show a sequence is long exact on the fixed points with respect to every subgroup
if and only if it is long exact on the cofixed points with respect to every subgroup.
Then, using the fact that fixed points (and cofixed points) are a left (right) exact functor on representations, this statement is equivalent to the statement
obtained by
replacing ``long exact" with ``short exact."
Now suppose we have a short exact sequence
$$0 \rightarrow K\rightarrow N \rightarrow M \rightarrow 0.$$
Then for every element $\alpha \in G$, we have
\vspace{5mm}
\hspace{15mm}\begin{tikzcd}
& 0\arrow[d]& 0\arrow[d] &0 \arrow[d] & \\
0\arrow[r] & K^{\langle \alpha\rangle}\arrow[r] \arrow[d]& N^{\langle \alpha\rangle}\arrow[r]\arrow[d] & M^{\langle \alpha\rangle}\arrow[d]\arrow[llddd, controls={+(2,-1) and +(-2,1)}]& \\
0 \arrow[r] & K \arrow[r]\arrow[d]{d}{1-\alpha} & N \arrow[r] \arrow[d]{d}{1-\alpha} & M\arrow[r]\arrow[d]{d}{1-\alpha} & 0\\
0 \arrow[r] & K \arrow[r]\arrow[d] & N \arrow[r] \arrow[d] & M\arrow[r]\arrow[d] & 0\\
& K_{\langle \alpha\rangle}\arrow[r] \arrow[d]& N_{\langle \alpha\rangle}\arrow[r]\arrow[d] & M_{\langle \alpha\rangle}\arrow[d]\arrow[r]& 0\\
& 0 & 0 & 0 & \\
\end{tikzcd}
\noindent and the Snake Lemma gives a connecting map
$\gamma: M^{\langle \alpha \rangle} \rightarrow K_{\langle \alpha \rangle}$
making a six-term long exact sequence.
Then it follows that we have a short exact sequence
$$0 \rightarrow K^{\langle \alpha\rangle} \rightarrow N^{\langle \alpha \rangle} \rightarrow M^{\langle \alpha \rangle}\rightarrow 0$$
if and only if $\gamma: M^{\langle \alpha \rangle} \rightarrow K_{\langle \alpha \rangle }$ is 0, which also happens if and only if we have a
short exact sequence
$$0 \rightarrow K_{\langle \alpha \rangle} \rightarrow N_{\langle \alpha \rangle} \rightarrow M_{\langle \alpha \rangle} \rightarrow 0.$$
This proves the statement for $H= \langle \alpha \rangle$ for every $\alpha \in G$, hence implying it for every subgroup of $G$, since they are all of that form.
\end{proof}
One might ask if a similar statement is true for general finite groups $G$. However, this is false.
For a counterexample, consider the $\Z[\Z/2 \times \Z/2]$-module
\beg{Z2Z2RepWithRelation}{M= \Z[\Z/2 \times \Z/2] / (\alpha h - \alpha + h -1)}
where $h, \alpha $ are the generators of the two copies of $\Z/2$.
\vspace{5mm}
\begin{proposition}
There exists a sequence
\beg{CounterExSequence}{0 \rightarrow K \rightarrow N \rightarrow M \rightarrow 0}
which is short exact after taking fixed points with respect to any subgroup $H \subseteq \Z /2 \times \Z/2$ while
\beg{CofixedNotInjective}{K_{\Z/2\times \Z/2}\rightarrow N_{\Z/2\times \Z/2}}
is not injective.
\end{proposition}
\begin{proof}
First note that the relation of \rref{Z2Z2RepWithRelation} is preserved up to sign by the $\Z/2 \times \Z/2$-action,
and thus $M$ is a free $\Z$-module with basis $\{1, h, \alpha\}$.
Additionally, rationally, it splits as a sum of a fixed representation generated by $1+ \alpha$, an $\alpha h$-fixed sign representation generated by
$h-1$, and an $h$-fixed sign representation generated by $\alpha h -1$.
It follows that
$$M^{\Z/2\times \Z/2} = M^{\langle \alpha \rangle} = \Z \{1+\alpha \}$$
$$M^{\langle \alpha \rangle} = \Z \{ \alpha h-1, 1+\alpha\}$$
$$M^{\langle \alpha h\rangle} = \Z \{ h-1 , 1+\alpha\}.$$
Also,
$$\Z[\Z/2 \times \Z/2]^{\Z/2 \times \Z/2} = \Z\{1+\alpha +h + \alpha h\} $$
$$\Z[\Z/2\times \Z/2] ^{\langle h \rangle} = \Z\{1+h , \alpha + \alpha h\} $$
$$\Z [ \Z/2 \times \Z/2]^{\langle \alpha h \rangle} = \Z\{ 1+ \alpha h, \alpha + h\}.$$
Considering the map $\epsilon : \Z [\Z/2 \times \Z/2]\rightarrow M$ given by $\epsilon (1) =1 $, we have
$$\Z[\Z/2 \times \Z/2]^{\Z/2 \times \Z/2} \ni 1+ \alpha + h + \alpha h \mapsto 2(1+ \alpha)$$
$$\Z [\Z/2\times \Z/2]^{\langle h \rangle} \ni \alpha h + \alpha \mapsto (\alpha h-1) + (1+\alpha) $$
$$\Z[\Z/2 \times \Z/2]^{\langle\alpha h\rangle} \ni \alpha = h \mapsto (\alpha +1 ) + (h-1).$$
Thus, putting $N = \Z [\Z/2 \times \Z/2] \oplus \Z$ (where $\Z$ is the trivial $\Z/2 \times \Z/2$-representation), the map
$$\lambda: N \rightarrow M$$
given by $\lambda = (\epsilon, 1+\alpha)$ is onto on $H$-fixed points for all
$H\subseteq \Z/2\times \Z/2$.
Let $K= \text{\em Ker}(\lambda)$.
Since fixed points are a left exact functor on representations, \rref{CounterExSequence} is short exact after taking fixed points with
respect to every subgroup $H\subseteq \Z/2 \times \Z/2$.
Now, by construction,
$$K = \langle (\alpha h-\alpha + h-1, 0) \rangle \oplus \langle (1+ \alpha + h + \alpha h, -2)\rangle\subseteq N$$
where $\Z/2 \times \Z/2$ acts by the sign representation fixing $\alpha$ on the first summand and by the trivial representation on the
second summand.
Hence,
$$K_{\Z/2 \times \Z/2} \cong \Z/2 \oplus \Z$$
coming from the first and second summand, respectively.
Meanwhile,
$$N_{\Z/2\times \Z/2} \cong \Z \oplus \Z,$$
and thus \rref{CofixedNotInjective} cannot be injective.
\end{proof}
|
1,116,691,498,416 | arxiv | \section{Introduction}
Frustration plays an important role in the conceptual understanding of the physical properties of novel magnetic materials ~\cite{diep}. Frustration can arise either due to the underlying geometry as in a triangular
lattice ~\cite{weihong992D,zhit962D} case or due to competing
interactions as in the quantum spin-$1/2$ Heisenberg antiferromagnet (AF) on a square
lattice~\cite{tassi75gen,dot942D,valeri992D,singh992D,watabe91,mam06}. There are several examples of
frustrated magnetic materials: spinels~\cite{tristan08,wiebe03},
all face-centered-cubic (FCC) AFs including type-I
systems (e.g. CeAs, CeSb, USb, NpBi)~\cite{furrer,bossy,hagen,jensen}, type-II systems (e.g. FeO, MnO, NiO, $\alpha$-MnS, CoO, EuTe, NiS$_{2}$)~\cite{yamamotoFCC}, type-III systems (e.g. Cd$_{1-x}$Mn$_x$Te for larger $x$) ~\cite{gieb86},
triangular stacked AF's~\cite{chubukov91triangle,chubukov94triangle,chern09triangle}, pyrochlore magnets~\cite{canals98Pyro,maged05Pyro}, kagome lattices~\cite{Fak08SL,harris92Kagome,subir92,chubu92Kagome,oleg08Kagome},
and fully frustrated cubic systems~\cite{diep85SC,viana,derrida80SC}.
The tendency of a magnetic system to support long range order is more pronounced in three dimensions (3D) than in two- or one- dimension. Recently, motivated by the results for the 2D lattices, some work has been done by analytical (non-linear spin wave theory) ~\cite{dattaBCC, dattaSC} and numerical techniques (exact diagonalization, and linked-cluster series expansions) ~\cite{oitmaa,Schmidt,viana} to understand the magnetic phase diagram of 3D quantum spin-$1/2$ Heisenberg AF on a body-centered-cubic (BCC) lattice and simple cubic (SC) lattice. There also exists a limited amount of work on the effects of local anisotropy, four-spin exchange interactions, and biquadratic interaction of spin-$1/2$ Heisenberg AF on 3D lattices ~\cite{oguchi,katanin08FCC,ader02FCC,oja}.
In this paper, we study the effects of quantum fluctations in a 3D quantum AF on a FCC lattice using linear spin wave theory (LSWT). We choose the Type IIA FCC structure (see Fig.~\ref{subfig:Type2A}) which is proven to be stable from among the (initially) degenerate ground states of the FCC AF ~\cite{HarrisFCC,henley}. We then perform a LSWT calculation for spins in a canted basis about the Type IIA ground state and obtain the dispersion including the effects of external magnetic field, single-ion anisotropy, and biquadratic interaction (refer Eq.~\ref{eq:dispersion}). We calculate the expected finite frequency neutron scattering intensity for the FCC AF (see
Figs.~\ref{fig:MnOneutron} and ~\ref{fig:CoOneutron}). We also compute the effect of quantum fluctuations on the sublattice magnetization (see Fig.~\ref{fig:deltaS}).
The motivation for considering the Type IIA FCC lattice is twofold. First, as mentioned earlier, there are several experimentally relevant Type IIA FCC materials. Second, to the best of our knowledge a systematic theoretical study of neutron scattering and the effects of magnetic field and single-ion anisotropy for the Type IIA FCC AF is missing. A knowledge of the neutron scattering pattern is crucial to determining information relevant for the quantum/classical dynamics of the system and to help understand the effects of frustration.The Type II FCC lattice had been an earlier topic of theoretical investigation where the authors studied the effect of frustration and quantum fluctuations ~\cite{HarrisFCC}. The Type I FCC AF has already been investigated in some detail theoretically ~\cite{ader01FCC,ader02FCC}.
This paper is organized as follows. In Section~\ref{sec:fccreview}
we begin with a brief description of the existing theoretical and
experimental understanding of the FCC AF system and the
properties of the lattice relevant to our calculations. In
Section~\ref{sec:lswt} we set-up the Hamiltonian and perform the
boson transformation for spins in a canted basis to obtain
the spin wave dispersion within LSWT. In
Section~\ref{subsec:dispersion} we discuss the effects of magnetic
field, single-ion anisotropy, and bi-quadratic interaction on the
dispersion relation. In Section~\ref{subsec:neutron} we present the
results of neutron scattering for MnO and CoO. In Section~ \ref{subsec:submag} we
show the effects of quantum fluctuations on the sublattice
magnetization. Finally, in Section~\ref{sec:discon} we
summarize the main results.
{\section{FCC review}\label{sec:fccreview}} The classical ground states of the FCC
AF have been investigated theoretically
~\cite{yamamotoFCC,oguchi}. Extending these studies to include
the effect of quantum fluctuations it was shown that the continuous
degeneracy of the classical ground states can be removed to favor a
collinear ground state ~\cite{shender82,henley}. In general there are 4 types or kinds of
collinear structure - Type I, Type IIA, Type IIB, Type III, and Type
IV. Quantum fluctuations are unable to remove the twofold structural
degeneracy between the inequivalent Type IIA and Type IIB structure
(see Fig.~\ref{fig:TypeAB} for spin arrangement). Classically
these two structures are stable for $|J^{'}|<2|J|$. As shown by
~\cite{HarrisFCC,henley} a spin wave theory up to order $(J^{'}/J)^4$ is
needed to lift the degeneracy to select the second kind of type A as
having the lower energy. In this paper we focus on this spin
arrangement. The spin wave gaps occur at the relative order of
$(J^{'}/J)^2$. The gap can be phenomenologically modeled by a
bi-quadratic interaction and microscopically justified through a spin
wave theory Hartree decoupling of quartic interaction terms ~\cite{HarrisFCC,henley}.
\begin{figure}[h]
\centering
\subfigure[FCC Type IIA]{\includegraphics[width=1.5in]{Type2A.jpg}\label{subfig:Type2A}} \subfigure[FCC Type IIB]{\includegraphics[width=1.5in]{Type2B.jpg}\label{subfig:Type2B}}
\caption{\label{fig:TypeAB} (Color online) Twofold structurally degenerate inequivalent Type IIA and Type IIB FCC structures. The four colors - \textcolor{black}{\bf black}, \textcolor{red}{{\bf red}}, \textcolor{blue}{\bf blue}, and \textcolor{green}{\bf green} represent the four sublattices respectively. The $+$ and $-$ denote the up and down collinear spin configurations respectively. Spin wave theory upto quartic order in the ratio of interaction strengths show that quantum fluctuations select the Type IIA structure as having the lower energy ~\cite{HarrisFCC}. We focus on this spin arrangement in this paper.}
\end{figure}
The Type II FCC AF system has also been investigated experimentally. The data in Table ~\ref{tab:FCCDatatable} documents the experimentally measured N\'{e}el transition temperature (T$_{N}$), nearest neighbor (nn) exchange interaction (J$^{'}$), next-nearest neighbor (nnn) exchange interaction (J), and the ratio of $J'/J$. The table shows that J $>$ J$^{'}$ for these materials. Under this condition the FCC lattice may be viewed as four interpenetrating SC AF sublattices in which the mean field on one sublattice due to any other vanishes. This fact forms the basis for writing our Hamiltonian, Eq.~\ref{eq:Hamiltonian}, in the sublattice formulation. In the next section, Sec.~\ref{sec:lswt}, we state the Hamiltonian and carry out the LSWT calculation in the canted spin basis about the FCC Type IIA AF ground state (see Fig.~\ref{subfig:Type2A}).
\begin{figure}[b]
\includegraphics[width=2.5in]{fccunitcell.jpg}
\caption{\label{fig:fcc} (Color online) Four sublattice formulation of the AF FCC lattice. The sublattices are indicated by - \textcolor{black}{\bf black} (1), \textcolor{red}{{\bf red}} (2), \textcolor{blue}{\bf blue} (3), and \textcolor{green}{\bf green} (4) colors. The corresponding sublattice number is indicated in the parenthesis. The intra-sublattice nearest neighbor vector is given by ${\bf \Delta}$ and inter-sublattice vectors are given by $\Vec{\delta}_{12},\Vec{ \delta}_{13}$ and $\Vec{\delta}_{14}$. The choice of coordinate axis is as shown in the figure.}
\end{figure}
\begin{table}[t]
\caption{\label{tab:FCCDatatable} Transition temperature (T$_{N}$), nearest neighbor (J$^{'}$), next-nearest neighbor $ (J)$, and relative coupling strength $\gamma$ for Type II FCC antiferromagnets. The data has been compiled from Yamamoto and Nagamiya ~\cite{yamamotoFCC}.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
Material & T$_N$ [K (meV)] & J$^{'}$ [K (meV)] &J[K (meV)] & $\gamma$=J$^{'}$/J \\ \hline
FeO & 198 (17.07) & 7.8 (0.67) & 8.2 (0.71) & 0.95\\
MnO & 117 (10.08) & 5 (0.43) & 5.5 (0.47) &0.91\\
NiO & 523 (45.09) & 50 (4.31)& 85 (7.33)& 0.59\\
$\alpha$-MnS & 147 (12.7) & 3.5 (0.30) & 6.25 (0.54)&0.56\\
CoO & 292 (25.2)& 6.9 (0.60) & 21.6 (1.86)&0.32\\
EuTe & 9.76 (0.84)& 0.07 (0.006) & 0.21 (0.02)&0.33
\end{tabular}
\end{ruledtabular}
\end{table}
{\section{Hamiltonian and LSWT}\label{sec:lswt}}
The model Hamiltonian that we study is given by,
\begin{eqnarray}
\mathcal{H}&=&J\sum_{\langle \alpha,i;\alpha,j\rangle}{{\bf S}_{\alpha,i}\cdot {\bf S}_{\alpha,j}}+J^{'}\sum_{\langle \alpha, i;\beta,j \rangle}{{\bf S}_{\alpha,i}\cdot {\bf S}_{\beta,j}}\nonumber \\&-&\lambda\sum_{ \alpha, i}( S^{z}_{\alpha,i})^{2}+g\mu_{B}H\sum_{\alpha, i}S^{y}_{\alpha, i}
\label{eq:Hamiltonian}
\end{eqnarray}
where {\bf S$_{\alpha,i}$} denotes the $i$th spin on a sublattice $\alpha$. The first term, with strength J is the interaction within the sublattices (which is a nn interaction based on the SC formulation). The second term with strength J$^{'}$ is the inter sublattice interaction. In Fig.~\ref{fig:fcc} we show the nn (${\bf \Delta}$) and nnn ($\Vec{\delta}_{12},\Vec{\delta}_{13}$ and $\Vec{\delta}_{14}$) vectors in the four sublattice SC formulation. The four colors - black (1), red (2), blue (3), and green (4) represent the four sublattices respectively. The corresponding sublattice number is indicated in the parenthesis. The third term is the single-ion anisotropy with strength $\lambda >0$, and the last term is the Zeeman energy due to the external magnetic field which has been applied along the negative y-direction ~\cite{syromaleyev}. The gyromagnetic ratio is given by the symbol g and the Bohr magneton by $\mu_{B}$.
In the presence of a magnetic field the sublattices are canted by an angle $\Theta$ and we apply the canted spin wave theory approach to calculate the LSWT dispersion ~\cite{zhitniku,zhitchern,syromaleyev,mourigal}. In this method the spin components are first represented in a rotating frame such that the local z$_{i}$-axis points in the direction of each magnetic sublattice. We then express the spin components in the rotated frame relative to the laboratory frame as,\begin{eqnarray}
&{\bf S}_{\alpha,i}&=\tilde{S^{x}}_{\alpha,i}{\bf \hat{x}}+\left(\tilde{S^{y}}_{\alpha,i}e^{i{\bf k_{o}}\cdot{\bf r}_{\alpha,i}}\cos\Theta+\tilde{S^{z}}_{\alpha,i}\sin\Theta\right){\bf \hat{y}}\nonumber\\
&+&\left(\tilde{S^{z}}_{\alpha,i}e^{i{\bf k_{o}}\cdot{\bf r}_{\alpha,i}}\cos\Theta-\tilde{S^{y}}_{\alpha,i}\sin\Theta\right){\bf \hat{y}}\label{eq:rotbasis}
\end{eqnarray} where {\bf k}$_{o}=(\frac{\pi}{a},\frac{\pi}{a},\frac{\pi}{a})$. The advantage of the rotated basis is that it allows us to perform the calculation using only one type of sublattice boson. In the second step of the process we obtain the boson representation (note that within LSWT there is no distinction between the Dyson-Maleev and the Holstein-Primakoff transformation) of the spins in the Hamiltonian by applying the transformation,
\begin{eqnarray}
\tilde{S}^{x}_{\alpha,i}&=&\sqrt{\frac{S}{2}}\left(a_{\alpha,i}+a^{\dag}_{\alpha,i}\right)\label{eq:sxdm}\\
\tilde{S}^{y}_{\alpha,i}&=&-i\sqrt{\frac{S}{2}}\left(a_{\alpha,i} - a^{\dag}_{\alpha,i}\right)\label{eq:sydm}\\
\tilde{S}^{z}_{\alpha,i}&=&S - a^{\dag}_{\alpha,i}a_{\alpha,i}\label{eq:szdm}
\end{eqnarray}
where S is the spin, $a^{\dag}_{\alpha, i}$ and $a_{\alpha,i}$ are the boson creation and annihilation operators for each site $i$ in a sublattice $\alpha$ respectively. Finally, we use the transformation Eqs.~\ref{eq:sxdm},~\ref{eq:sydm}, and~\ref{eq:szdm}, to write the Hamiltonian bilinear in the boson operators as,
\begin{eqnarray}
\mathcal{H}&=&-12NJS^{2}\cos2\Theta+4g\mu_{B}HNS\sin\Theta-4NS^{2}\lambda\cos^{2}\Theta\nonumber\\
&+&(-g\mu_{B}H\sin\Theta+2S\lambda\cos^{2}\Theta)\sum_{\alpha,i}a^{\dag}_{\alpha,i}a_{\alpha,i}\nonumber\\
&+&JS\sum_{\langle\alpha_{i},\alpha_{j}\rangle}\cos2\Theta(a^{\dag}_{\alpha,i}a_{\alpha,i}+a^{\dag}_{\alpha,j}a_{\alpha,j})\nonumber\\
&+&JS\sum_{\langle\alpha_{i},\alpha_{j}\rangle}\sin^{2}\Theta(a^{\dag}_{\alpha,i}a_{\alpha,j}+a_{\alpha,i}a^{\dag}_{\alpha,j})\nonumber\\
&+&JS\sum_{\langle\alpha_{i},\alpha_{j}\rangle}\cos^{2}\Theta(a^{\dag}_{\alpha,i}a^{\dag}_{\alpha,j}+a_{\alpha,i}a_{\alpha,j})\nonumber\\
&+&J^{'}S\sum_{\langle\alpha_{i},\beta_{j}\rangle}A^{'}_{\alpha_{i},\beta_{j}}(a^{\dag}_{\alpha,i}a_{\beta,j}+a_{\alpha,i}a^{\dag}_{\beta,j})\nonumber\\
&+&J^{'}S\sum_{\langle\alpha_{i},\beta_{j}\rangle}B^{'}_{\alpha_{i},\beta_{j}}(a^{\dag}_{\alpha,i}a^{\dag}_{\beta,j}+a_{\alpha,i}a_{\beta,j})
\label{eq:Hsite}
\end{eqnarray}
where $\Theta$ is the canting angle of the sublattice magnetization. In the above derivation the linear terms from \~{S}$^{y}_{\alpha,i}$ do not contribute. We also ignore interaction terms of order $\lambda\sin^{2}\Theta$ which are small in the presence of weak field and anistropy. These terms could be relevant in the presence of strong magnetic field, but, we do not consider that analysis here. Furthermore, in the presence of strong magnetic field the present derivation will break down and a more careful analysis is required ~\cite{mourigal}. The canting angle, $\Theta$, can be obtained by minimizing the classical energy (the first three terms of Eq.~\ref{eq:Hsite}) to obtain,
\begin{equation}
\sin\Theta=-\frac{g\mu_{B}H}{12JS+2S\lambda}
\label{eq:cant}
\end{equation}
We now Fourier Transform the above Hamiltonian using the following definition $a^{\dag}_{\alpha,i}=\frac{1}{\sqrt{N}}\sum_{{\bf q}}a^{\dag}_{\alpha}({\bf q})e^{i{\bf q}\cdot {\bf r}_{i}}$ where N is the number of sites in each of the four SC AF sublattice, {\bf q} is summed over N values in the interval $-\pi < aq_{j}< \pi$ (j = x,y,z). The Fourier Transformed Hamiltonian may be written as a sum of the classical energy $\mathcal{E}_{cl}$, the intra (same) sublattice interaction $\mathcal{H}_{intra}$, and inter (different) sublattice interaction $\mathcal{H}_{inter}$ as,
$\mathcal{H}=\mathcal{E}_{cl}+\mathcal{H}_{intra}+\mathcal{H}_{inter}$
where,\begin{equation}
\mathcal{E}_{cl}=-12NJS^{2}\cos2\Theta+4g\mu_{B}HNS\sin\Theta-4NS^{2}\lambda\cos^{2}\Theta
\label{eq:Ecl}
\end{equation}
\begin{eqnarray}
&\mathcal{H}_{intra}&= 6JS[\sum_{\alpha,{\bf q}}(1+\lambda/3J+\gamma_{\bf q}\sin^{2}\theta)a^{\dag}_{\alpha}({\bf q})a_{\alpha}({\bf q})\nonumber\\
&+& \frac{1}{2}\gamma_{\bf q}\cos^{2}\theta\sum_{\alpha,{\bf q}}(a^{\dag}_{\alpha}({\bf q})a^{\dag}_{\alpha}({-\bf q})+a_{\alpha}({\bf q})a_{\alpha}({-\bf q})]]\nonumber\\
\label{eq:hintra}
\end{eqnarray}
and $\gamma_{\bf q}$ is given by $\gamma_{\bf q}=\frac{1}{6}\sum_{\bf \Delta}e^{i{\bf q}\cdot{\bf \Delta}}$
with ${\bf \Delta}$ as the nn neighbor vector within a sublattice (see Fig.~\ref{fig:fcc}). Finally,
\begin{eqnarray}
&\mathcal{H}_{inter}&= J^{'}S[\sum_{\alpha,\beta,{\bf q}}A^{'}_{\alpha,\beta}({\bf q})[a^{\dag}_{\alpha}({\bf q})a_{\beta}({\bf q})+a_{\alpha}({\bf q})a^{\dag}_{\beta}({\bf q})]\nonumber\\
&+& B^{'}_{\alpha,\beta}({\bf q})\sum_{\alpha,\beta,{\bf q}}(a^{\dag}_{\alpha}({\bf q})a^{\dag}_{\beta}({-\bf q})+a_{\alpha}({\bf q})a_{\beta}({-\bf q})]]\nonumber\\
\label{eq:hinter}
\end{eqnarray}
The A$^{'}_{\alpha,\beta}({\bf q})$ and B$^{'}_{\alpha,\beta}({\bf q})$ coefficients are given by
\begin{eqnarray}
A^{'}_{\alpha,\beta}{\bf (q)}&=&\frac{1}{4}\sum_{{\Vec{\delta}}_{\alpha,\beta}}\left[1+\left(e^{i{\bf k_{o}}\cdot{\Vec{\delta}}_{\alpha,\beta}}\cos^{2}\Theta+\sin^{2}\Theta\right)\right]e^{-i{\bf q}\cdot{\Vec{\delta}}_{\alpha,\beta}}\nonumber\\\\
B^{'}_{\alpha,\beta}{\bf (q)}&=&\frac{1}{4}\sum_{{\Vec{\delta}}_{\alpha,\beta}}\left[1-\left(e^{i{\bf k_{o}}\cdot{\Vec{\delta}}_{\alpha,\beta}}\cos^{2}\Theta+\sin^{2}\Theta\right)\right]e^{-i{\bf q}\cdot{\Vec{\delta}}_{\alpha,\beta}}\nonumber\\
\label{eq:NewAB}
\end{eqnarray}
where ${\Vec{\delta}_{\alpha,\beta}}$ is summed over the four first-neighbor vectors which connect sublattices $\alpha$ and $\beta$ (see Fig.~\ref{fig:fcc}). With the above definitions the bilinear Hamiltonian, Eq.~\ref{eq:Hsite}, can be written as follows,
\begin{equation}
\mathcal{H}=\mathcal{E}_{cl}+\frac{1}{2}\sum_{{\bf q}}{\bf X}^{\dag}({\bf q}){\bf M}({\bf q}){\bf X}({\bf q})
\label{eq:Hdiag}
\end{equation}
with the following defintions,
\begin{eqnarray}
{\bf X}({\bf q})=\left(\begin{tabular}{c}
{\bf V}({\bf q}) \\
{\bf V}$^{\dag}$(-{\bf q})\\
\end{tabular}
\right)\\
{\bf V}({\bf q})=\left(\begin{tabular}{c}
a$_{1}$({\bf q}) \\
a$_{2}$({\bf q}) \\
a$_{3}$({\bf q}) \\
a$_{4}$({\bf q}) \\
\end{tabular}\right)\\
{\bf M}({\bf q})=\left(\begin{tabular}{c c}
{H}$_{1}$({\bf q}) &{H}$_{2}$({\bf q})\\
{H}$_{2}$({\bf q}) &{H}$_{1}$({\bf q})
\end{tabular}\right)
\end{eqnarray}
and H$_{1}$({\bf q}) and H$_{2}$({\bf q}) are defined as,
\begin{eqnarray}
H_{1}({\bf q})&=&6JS\{(1+\lambda/3J+\gamma_{\bf q}\sin^{2}\theta)+[J^{'}/(3J)]{A}^{'}({\bf q})\}\nonumber\\
\label{eq:eqhone}
\end{eqnarray}
\begin{eqnarray}
H_{2}({\bf q})&=&6JS\{\gamma_{\bf q}\cos^{2}\theta+[J^{'}/(3J)]{B}^{'}({\bf q})\}
\label{eq:eqhtwo}
\end{eqnarray}
Now in the above Hamiltonian one can include the biquadratic interaction defined by ~\cite{HarrisFCC},
\begin{equation}
\Delta\mathcal{H}_{Q}=-\frac{1}{2}Q\sum_{i,j}\Delta_{ij}[{\bf S}_{i}\cdot{\bf S}_{j}]^{2}/S^{3}
\label{eq:biquad}
\end{equation}
where Q is the strength of the biquadratic interaction, $\Delta_{ij}$ is unity if spins i and j are nn and is zero otherwise. Following the same strategy as outlined before we have the following \emph{new definitions} of H$_{1}({\bf q})$ and H$_{2}({\bf q})$ in the presence of biquadratic interaction,
\begin{eqnarray}
H_{1}({\bf q})&=& 6JS\left[1+\frac{\lambda}{3J}+\gamma_{{\bf q}}\sin^{2}\Theta+\frac{2Q}{JS}\right]\\
&+&6JS\left[2j - \frac{Q}{3JS}\right]h_{1}({\bf q})+6JS\left[2j\sin^2\Theta\right]h_{2}({\bf q})\nonumber\\
H_{2}({\bf q})&=&6JS\gamma_{{\bf q}}\cos^{2}\Theta +6JS\left[2j\cos^{2}\Theta+\frac{Q}{3JS}\right]h_{2}({\bf q})\nonumber\\
\label{eq:NewH1H2}
\end{eqnarray}
where h$_{1}$({\bf q}) and h$_{2}$({\bf q}) are now given by, \begin{eqnarray}
h_{1}({\bf q})&=&\cos[a(q_x-q_y)/2]
+\cos[a(q_y-q_z)/2]\nonumber\\&+&\cos[a(q_z-q_x)/2]\\
h_{2}({\bf q}) &=&\cos[a(q_x+q_y)/2]+\cos[a(q_y+q_z)/2]\nonumber\\
&+&\cos[a(q_z+q_x)/2]
\end{eqnarray} with j=$J^{'}/6J$.
We choose the fourth eigenvalues h$_{1}$({\bf q}) and h$_{2}$ ({\bf q}) because H$_{1}$({\bf q}) and H$_{2}$({\bf q}) commute and can be simultaneously diagonalized for the
Type IIA structure (see appendix A of Ref. [39] for more
details). The other three branches can be obtained by "folding" the single branch spectrum ~\cite{HarrisFCC}. The neutron scattering is nonzero only for the single mode appearing in this basis. Hence the choice of this representation is convenient to perform the neutron scattering calculation. The LSWT dispersion is given by,
\begin{equation}
\omega({\bf q})=\sqrt{[H_{1}({\bf q}) - H_{2}({\bf q})][H_{1}({\bf q}) + H_{2}({\bf q})]}
\label{eq:dispersion}
\end{equation}
\begin{figure}[t]
\centering
\subfigure[MnO]{\includegraphics[width=3.5in]{SWDMnO}\label{subfig:MnO}}
\subfigure[CoO]{\includegraphics[width=3.5in]{SWDCoO}\label{subfig:CoO}}
\caption{\label{fig:SWD}(Color online) Linear spin-wave theory dispersion for (a) MnO and (b) CoO. The x-axis and y-axis correspond to wave-vectors q$_{x}$ and q$_{y}$ respectively with the range (0,2$\pi$). The z-axis corresponds to energy in units of kelvin. The value of q$_z=\pi$. We compute the dispersions for the parameter set (S=5/2, J$^{'}$=5 K, J=5.5 K) for MnO and (S=3/2, J$^{'}$=6.9 K, J=21.6 K) with the ratios $\frac{g\mu_{B}H}{6J}$=0.05, $\frac{\lambda}{6J}=0.002$, and $\frac{Q}{J}=0.01$. For this choice of parameters the energy gap at the antiferromagnetic $(\pi,\pi,\pi)$ point $\Delta(\pi,\pi,\pi)$, indicated by a red arrow on the plot, is 10.2 K (0.88 meV) for MnO and 20.1 K (1.73 meV) for CoO. The dashed lines in the q$_x$-q$_y$ plane are a guide for the eye to locate the $(\pi,\pi,\pi)$ point. The spin of the compound is given by S, J$^{'}$ is the inter sublattice interaction, J is the interaction within same sublattice, $\lambda$ is the anisotropy parameter, Q is the bi-quadratic interaction, and H is the external magnetic field.}
\end{figure}
{\section{Results}\label{sec:results}}
{\subsection{Spin wave dispersion - effects of magnetic field,
anisotropy, and bi-quadratic interaction}\label{subsec:dispersion}}
The spin wave dispersion for MnO and CoO is displayed in Fig.~\ref{fig:SWD} for (q$_x$, q$_y$, $\pi$). The range for q$_{x}$ and q$_{y}$ is 0 to 2$\pi$ on the dispersion plot. The energy is measured in units of kelvin. We use
Eq.~\ref{eq:dispersion} to compute the energy profile for the parameter set (S, J$^{'}$, J) with the ratios $\frac{g\mu_{B}H}{6J}$=0.05, $\frac{\lambda}{6J}=0.002$, and $\frac{Q}{J}=0.01$. The spin of the compound is given by S, J$^{'}$ is the inter sublattice interaction, J is the interaction within same sublattice, $\lambda$ is the anisotropy parameter, Q is the bi-quadratic interaction, and H is the external magnetic field. From Table \ref{tab:FCCDatatable} and
Ref.~\onlinecite{HarrisFCC} we take the parameter set (5/2, 5 K (0.43 meV), 5.5 K(0.47 meV)) for MnO and (3/2, 6.9 K (0.60 meV), 21.6 K(1.86 meV)) for CoO. We find that for the given choice of parameters the AF gap at the $(\pi,\pi,\pi)$-point is 10.2 K (0.88 meV) for MnO and 20.1 K (1.73 meV) for CoO. The two compounds also have a different band top - 134 K (11.6 meV) for MnO and 239 K (20.6 meV) for CoO. The dispersion also shows that the energy range is different and there is an overall qualitative difference in the curvature of the dispersion.
In Figs.~\ref{fig:magfield} and ~\ref{fig:andis} we display the dispersion curves for MnO, $\omega$({\bf q})/15J, for different cuts along the Brillouin zone (BZ) - (2$\pi$, 0, 0), $\Gamma$, M, X, and then back to $\Gamma$. From Fig.~\ref{fig:magfield} we observe that in the absence of a magnetic field (solid black line) there are no gaps in the BZ. But, with the inclusion of a field (solid red line) an energy gap of 9.45 K (0.81 meV) opens up at the $\Gamma$-point as shown in the inset of Fig.~\ref{fig:magfield}. The value of this gap will increase if the strength of the field is increased (for reasons mentioned in Section \ref{sec:lswt} a very large value of the magnetic field should not be used). Gaps can also be created due to the presence of single-ion anisotropy (solid black line) and bi-quadratic interaction (solid red line) as shown in Fig.~\ref{fig:andis}. With single-ion anisotropy present only, a gap of 10.2 K (0.87 meV) opens up at the $\Gamma$ \& X point and 6.2 K (0.53 meV) at the (2$\pi$, 0, 0) point. However, with the biquadratic interaction a gap of 10.1 K (0.86 meV) is present \emph{only} at the (2$\pi$, 0, 0) point and this fact is in agreement with the work of Yildirim \emph{et.al.}~\cite{HarrisFCC}. The CoO dispersion cuts along the BZ (not shown here) also display a qualitatively similar behavior.
\begin{figure}[t]
\centering
\includegraphics[width=3.5in]{DispersionMagfield.jpg}
\caption{\label{fig:magfield} (Color online) MnO spin wave dispersion, $\omega$({\bf q})/15J, for zero external magnetic field (solid black line) and magnetic field $\frac{g\mu_{B}H}{6J}$=0.15 (solid red line). The field ratio is chosen to emphasize the effect of magnetic field on the spin wave dispersion. An energy gap of 9.45 K (0.81 meV) opens up at the $\Gamma$-point as displayed in the inset. Smaller values of the field produces a smaller gap. The Brillouin zone is traversed along (2$\pi$,0,0), $\Gamma$, X, M, and then back to $\Gamma$.The inset is displayed for $(\frac{\pi}{8},0,0) \rightarrow \Gamma \rightarrow (\frac{\pi}{8},\frac{\pi}{8},0)$.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=3.5in]{DispersionAnisBQ.jpg}
\caption{\label{fig:andis} (Color online) MnO spin wave dispersion, $\omega$({\bf q})/15J, with single-ion anisotropy parameter $\frac{\lambda}{6J}=0.002$ (solid red line) and bi-quadratic interaction parameter $\frac{Q}{J}=0.01$ (solid black line). In the presence of single-ion anisotropy only, multiple energy gaps are developed at the high symmetry points - 10.2 K (0.87 meV) at $\Gamma$ \& X and 6.2 K (0.53 meV) at $(2\pi,0,0)$. In the presence of bi-quadratic interaction only, a single energy gap of 10.1 K (0.86 meV) opens up at the $(2\pi, 0, 0)$-point in confirmation with ~\cite{HarrisFCC}. The Brillouin zone is traversed along (2$\pi$,0,0), $\Gamma$, X, M, and then back to $\Gamma$.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=3.0in]{MnOneutron.jpg}
\caption{\label{fig:MnOneutron} (Color online) Constant energy cuts of the inelastic neutron scattering pattern for MnO at {\bf q}=$(q_x,q_y,\pi)$. The x-axis and y-axis correspond to q$_{x}$ and q$_{y}$ respectively with the range (0, 2$\pi$). The left hand column is for an untwinned crystal and the right hand for a twinned crystal. The parameter set and the scaled values of magnetic field, anisotropy, and bi-quadratic interaction used to compute the pattern are the same as the MnO spin wave dispersion plot.}
\end{figure}
\begin{figure}[t]
\includegraphics[width=3.0in]{CoOneutron.jpg}
\caption{\label{fig:CoOneutron} (Color online) Constant energy cuts of the inelastic neutron scattering pattern for CoO at {\bf q}=$(q_x,q_y,\pi)$. The x-axis and y-axis correspond to q$_{x}$ and q$_{y}$ respectively with the range (0, 2$\pi$). The left hand column is for an untwinned crystal and the right hand for a twinned crystal. The parameter set and the scaled values of magnetic field, anisotropy, and bi-quadratic interaction used to compute the pattern are the same as the CoO spin wave dispersion plot.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=3.0in]{SwTotal.jpg}
\caption{\label{fig:swdisp} (Color online) Integrated structure factor S(E) for the full Brillouin zone. The computations were done for the same parameter set as in the neutron intensity calculation. The red line represents the structure factor for MnO and the blue line for CoO. The energy is in units of kelvin.}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=3.0in]{Qfluct.jpg}
\caption{\label{fig:deltaS}(Color online) Spin reduction, $\Delta S $, due to zero point quantum fluctuations for $J^{'}/J$ ratio in the absence of magnetic field, single-ion anisotropy, and bi-quadratic interaction. Various Type II FCC antiferromagnets stated in Table~\ref{tab:FCCDatatable} are indicated on the graph.}
\end{figure}
{\subsection{Inelastic neutron Scattering}\label{subsec:neutron}}
Neutron scattering is a very useful tool to detect magnetic order in
crystals ~\cite{boothroyd,tranquada,dai,tranaip}. Neutron
diffraction (elastic scattering) can be used to determine the spin
structure and magnetic moments. Inelastic neutron scattering can be
used to study spin dynamics including spin waves
~\cite{ewcneutron,yaoPRL,yaoPRB}. The technique has been used
successfully in magnetic materials, high T$_C$ superconductors, and
manganites.
The neutron scattering cross section is proportional to the dynamic
structure factor S({\bf q}, $\omega$) ~\cite{ewings}. In the linear
spin wave approximation the transverse parts contribute to the
structure factor and by symmetry we have,
\begin{eqnarray}
\frac{S^{\alpha\alpha}({\bf q},\omega)}{g^{2}\mu^{2}_{B}S_{eff}}
=\frac{H_{1}({\bf q})-H_{2}({\bf q})}{2\omega({\bf q})}[n(\omega)+1]\delta(\omega - \omega({\bf q}))\nonumber\\
\end{eqnarray} where $\alpha$=x,y and and n($\omega$) is the Bose occupation
factor. In Fig.~\ref{fig:MnOneutron} (MnO) and Fig.~\ref{fig:CoOneutron} (CoO) we show the expected neutron scattering intensity for constant energy cuts in {\bf q} space. The calculated spectra are predictions from LSWT. In either figure, the left hand column shows the expected neutron scattering intensity from a single domain of the magnetic order (untwinned cyrstal) and the right hand column shows the expected neutron scattering intensity from domains with both orientations of the magnetic order (twinned cyrstal). In real materials twinning occurs due to a finite correlation length, local disordered pinning, or crystal twinning.
We use the value of spin for the compound (S), J$^{'}$, J, H, $\lambda$, and Q. For MnO we have (5/2, 5 K (0.43 meV), 5.5 K (0.47 meV)) and for CoO (3/2, 6.9 K (0.60 meV), 21.6 K (1.86 meV)).
We choose $\frac{g\mu_{B}H}{6J}$=0.05, $\frac{\lambda}{6J}=0.002$, and
$\frac{Q}{J}=0.01$ as the scaled values of the magnetic field, single-ion anisotropy, and bi-quadratic interaction. The spectra are computed at {\bf q}=$(q_x,q_y,\pi)$ for both MnO and
CoO. The x-axis and y-axis correspond to q$_{x}$ and q$_{y}$
respectively with the range (0, 2$\pi$). The value of Q obtained
from Ref.~\onlinecite{HarrisFCC} is of the order of 0.001 meV. Such
a small value has negligible effect on the dispersion and the
neutron scattering plot. Therefore to highlight the effect of Q we
choose a slightly higher value. The choice of $\lambda$ is typical for AFs ~\cite{yao}.
For the untwinned case, at low energies the strongest diffraction peaks are centered around the ($\pi$,$\pi$) point and has an elliptical shape for MnO and a more circular shape for CoO. As the energy is increased the elliptical neutron pattern for MnO is stretched out and the circular pattern for CoO increases in size. Simultaneously the neutron intensity starts to non uniformly concentrate along the edges of the ellipse connecting the (0,0) and the (2$\pi$, 2$\pi$) line for MnO. For the CoO case the intensity starts to spread uniformly along the circular ring. At even higher energies the MnO spectra acquires a flattened elliptical shape. The CoO spectra on the other hand becomes a distorted circular shape. The neutron scattering patterns are determined mainly by the ratio of J$^{'}$/J. For the twinned case the intensity is located at particular points. As the energy is increased the high intensity patches simply grow in size.
The intensity patterns for the MnO and CoO case are different at high energy. This fact is reflected in the integrated structure factor which is given by,
\begin{equation}
S^{\alpha\alpha}(\omega)=\int\int\int_{BZ}dk_{x}dk_{y}dk_{z}S^{\alpha\alpha}({\bf q},\omega)\delta(\omega-\omega({\bf q}))
\end{equation}
where $\alpha=x,y$ and BZ means integrate over the full magnetic
Brillouin zone. Numerical results for the structure factor are presented in
Fig.~\ref{fig:swdisp}. The energy scale at which the peaks appear
are different. The MnO peaks at a lower energy $\approx$
80 K ( 6.9 meV) while the CoO at a higher energy $\approx$ 200 K (17.2 meV).
{\subsection{Sublattice Magnetization}\label{subsec:submag}}
Fig.~\ref{fig:deltaS} shows the spin reduction, $\Delta S$, due to zero point quantum fluctuations, \begin{equation}
\Delta S=-\frac{1}{2}+\frac{1}{8N}\sum_{{\bf q}}\frac{H_{1}({\bf q})}{\omega({\bf q})}
\label{eq:deltaS}
\end{equation}
For spin suppression the various Type II FCC AFs are also indicated
on the graph. The black line in Fig.~\ref{fig:deltaS} is the quantum
fluctuation in the absence of magnetic field, single-ion anisotropy,
and biquadratic interaction. The red line shows the effect of
single-ion anisotropy on quantum fluctuations only. With the anisotropy value that we choose there is not much of a change from when it is absent. Both curves lie on top of each other. On the
figure the various Type II FCC AFs are indicated.
{\section{Summary and Conclusion}\label{sec:discon}} In this paper we compute the
LSWT dispersion for a FCC AF in the presence of magnetic field,
single-ion anisotropy, and biquadratic interaction. We carry out the spin wave theory computation about the Type IIA FCC structure which is known to be stable. We highlight the effects of magnetic field, single-ion anisotropy, and
biquadratic interaction on the FCC dispersion and the energy gaps that can be created. Using MnO and CoO as typical Type II FCC materials we compute the predicted inelastic neutron scattering pattern. The two predicted patterns differ at high energies and in the ellipticity of their plots which is controlled by the ratio of J$^{'}$/J. The effects of quantum fluctations on sublattice magnetization at zero temperature is also explored for various ratios of nn and nnn interactions.
\begin{acknowledgments}
T.D. acknowledges the invitation, kind hospitality, and research
funding support from Sun Yat-sen University and Fundamental Research
Funds for the Central Universities. T. D. also thanks Augusta State
University Katherine Reese Pamplin College of Arts and Sciences for
partial research funding support. D. X. Y. is supported by the
NSFC-11074310, Sun Yat-sen University, and Fundamental Research
Funds for the Central Universities.
\end{acknowledgments}
|
1,116,691,498,417 | arxiv | \section{Maintaining high girth in graph packings}
We say that two $n$-vertex graphs $G_1$ and $G_2$
{\em pack} if
there exists an edge-disjoint
placement of them on the same set of $n$ vertices.
There is an extensive literature
dealing with sufficient conditions ensuring that two graphs
$G_1$ and $G_2$ on $n$ vertices pack. A well known open conjecture on the
subject is the one of Bollob\'as and Eldridge
\cite{BE} asserting that if the maximum degrees in $G_1$ and
$G_2$ are $d_1$ and $d_2$, respectively, and if
$(d_1+1)(d_2+1) \leq n+1$ then $G_1$ and $G_2$ pack.
Sauer and Spencer (\cite{SS}, see also Catlin \cite{Ca}),
proved that this is the case if
$2 d_1 d_2 <n$. For a survey of packing results including
extensions, variants and relevant references, see
\cite{KKY}.
A natural extension of the packing problem is that of requiring
a packing in which the girth of the combined graph whose edges
are those of the two packed graphs is large, assuming this is the
case for each of the individual graphs. Indeed, in the basic problem
the girth of each of the
packed graphs exceeds $2$, and the packing condition
is simply the requirement that in the combined graph the girth
exceeds $2$. Here we prove such an extension, observe that it implies
the old result of Erd\H{o}s and Sachs \cite{ES} about the existence of high-
girth regular graphs, and describe an application for obtaining an
explicit construction of high-girth directed expanders.
\begin{theo}
\label{t111}
Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$
be two $n$-vertex graphs, let $d_1$ be the
maximum degree of $G_1$ and let $d_2$ be the maximum degree of
$G_2$. Suppose the girth of each of the graphs $G_i$ is at least
$g >2$ and let $k$ be the largest integer satisfying
\begin{equation}
\label{e111}
1+(d_1+d_2)+(d_1+d_2)(d_1+d_2-1)+
\ldots + (d_1+d_2)(d_1+d_2-1)^{k-1} < n
\end{equation}
Then there is a packing of the two graphs so that the combined
graph has girth at least $\min\{g, k\}$.
\end{theo}
Note that for fixed $d_1+d_2 \geq 3$ and large $n$, the number $k$
above is $(1+o(1))\frac{\log n}{\log (d_1+d_2-1)}$.
\subsection{Proof}
Clearly we may assume that both $G_1$ and $G_2$ have edges, thus
$d_1$ and $d_2$ are positive. If $2d_1d_2 \geq n$
then $d_1+d_2 \geq \sqrt {2n}$ implying that
$1+(d_1+d_2)+(d_1+d_2)(d_1+d_2-1) \geq
1 + \sqrt {2n}+ \sqrt {2n} (\sqrt {2n} -1) =2n+1>n$, that is, the
largest $k$ satisfying (\ref{e111}) is at most $1$. In this case the
conclusion is trivial since $\min\{g,k\} \leq 1$ and any
placement of $G_1,G_2$ will do. We thus may and will assume that
$2d_1d_2<n$. Since any placement will do even if $k=2$ we assume
that $k \geq 3$.
By the result of \cite{SS}, $G_1,G_2$ pack.
Among all possible packings choose one in which the girth $m$ of the
combined graph is maximum, and the number of cycles of length $m$
in this combined graph is minimum (if the girth is infinite there
is nothing to prove). Suppose this packing is given by two
bijections $f_1:V_1 \mapsto V$ and $f_2:V_2 \mapsto V$
where $V$ is the fixed set of $n$ vertices of the
combined graph, which we denote by $H=(V,E)$. As $G_1$ and $G_2$ pack,
$m \geq 3$.
ref{e111}). Let
$v_1=f_1^{-1}(v)$ be the preimage of $v$ in $V_1$. Let
$f_1': V_1 \mapsto V$ be the bijection obtained from $f_1$ by
swapping the images of $u_1$ and $v_1$. Formally, $f'_1(u_1)=v,
f'_1(v_1)=u$, and $f'_1(w)=f_1(w)$ for all $w \in V_1-\{u_1,v_1\}$.
We claim that in the embedding of $G_1,G_2$ given by $f'_1,f_2$
the girth of the combined graph, call it $H'$, is at least $m$ and
the number of cycles of length $m$ in $H'$ is smaller than the
corresponding number in $H$, contradicting the minimality in the
choice of $f_1,f_2$. To prove this claim put $u_2=f_2^{-1}(u)$,
$v_2=f_2^{-1}(v)$.
Let $X_1$ denote the set of
images under $f_1$ of all the neighbors of $u_1$ in $G_1$, and let
$X_2$ denote the set of images under $f_2$ of all neighbors of
$u_2$ in $G_2$. Similarly, let
$Y_1$ be the set of
images under $f_1$ of all the neighbors of $v_1$ in $G_1$, and let
$Y_2$ be the set of images under $f_2$ of all neighbors of
$v_2$ in $G_2$. Note that since $m \geq 3$ and
$k+1 \geq 3$ all four sets $X_1,X_2,Y_1,Y_2$ are pairwise disjoint.
The cycles of length $m$ in $H$ and $H'$ that do
not contain any of the two vertices $u,v$ are exactly the same
cycles. On the other hand, the
cycle $C$ is of length $m$ and it exists in $H$ but not in $H'$, since
all edges of $H$ between $u$ and $X_1$ do not belong to
$H'$, and $C$ contains such an edge (as well as an edge from
$u$ to $X_2$). Any cycle $C'$ of $H'$ that is not a cycle of $H$ must
contain at least one edge either between $u$ and $Y_1$ or between
$v$ and $X_1$ (or both). Consider the following possible cases.
\vspace{0.2cm}
\noindent
{\bf Case 1a:}\, $C'$ contains $u$ but not $v$ and contains two
edges from $u$ to $Y_1$. In this case the cycle of $H$ obtained
from $C'$ by replacing $u$ by $v$ is of the same length as $C'$.
This is a one-to-one correspondence between cycles as above of length
$m$ in $H'$ and in $H$ (if there are any such cycles).
\vspace{0.2cm}
\noindent
{\bf Case 1b:}\, $C'$ contains $u$ but not $v$ and contains an edge
$uy_1$ from $u$ to $Y_1$ and an edge $ux_2$ from $u$ to $X_2$.
In this case the part of the cycle between $y_1$ and $x_2$ which
does not contain $u$ is a path in $H$ between $y_1$ and $x_2$. The
length of this path is at least $k-1$, since the distance in $H$
between $u$ and $v$ is at least $k+1$. Therefore, the length of
$C'$ is at least $(k-1)+2=k+1>m$.
\vspace{0.2cm}
\noindent
{\bf Case 1c:}\, $C'$ contains $v$ but not $u$: this is symmetric
to either Case 1a or Case 1b.
\vspace{0.2cm}
\noindent
{\bf Case 2a:}\, $C'$ contains both $u$ and $v$ and contains two
edges $uy_1,uy'_1$ from $u$ to $Y_1$. If both neighbors of $v$
in $C'$ belong to $Y_2$ then each of the parts of $C'$ connecting any of
them to $y_1$ or to $y'_1$ is of length at least
$m-2$, since the girth of $H$ is $m$, hence the total length
of $C'$ is at least $2(m-2)+4=2m>m$. If both neighbors of
$v$ in $C'$ are in $X_1$ then since the distance in $H$ between
$X_1$ and $Y_1$ is at least $k-1$, in this case the length of $C'$
is at least $2(k-1)+4=2k+2>m$. If the two neighbors of $v$
in $C'$ are $y_2 \in Y_2$ and $x_1 \in X_1$ then the cycle
$C'$ contains a path from $y_2$ to either $y_1$ or $y'_1$, whose
length is at least $m-2$, and a path from $x_1$ to either $y_1$ or
$y'_1$, of length at least $k-1$. Thus the total length of $C'$
is at least $(m-2)+(k-1)+4 >m$.
\vspace{0.2cm}
\noindent
{\bf Case 2b:}\, $C'$ contains both $u$ and $v$ and the two
neighbors of $u$ in $C'$ are $y_1 \in Y_1$ and $x_2 \in X_2$.
In this case the path in $C'$ from $v$ to $y_1$ is of length at
least $m-1$ if it does not pass through $X_1$, and at least
$k$ if it passes through $X_1$, and the path from $v$ to $x_2$ is
of length at least $k$ if it does not pass through $X_1$ and of
length at least $m-1$ if it does pass through $X_1$. In all these cases
the length of $C'$ is at least $2+2 \min\{m-1,k\}=2m >m$
(where here we used the assumption that $m<k$).
\vspace{0.2cm}
\noindent
{\bf Case 2c:}\, $C'$ contains both $u$ and $v$ and at least one
edge from $v$ to $X_1$. This is symmetric to either Case 2a or Case
2b.
It thus follows that the number of cycles of length $m$ in $H'$ is
smaller than that number in $H$, contradicting the minimality in
the choice of $H$ and implying that $m \geq \min \{g,k\}$.
This completes the proof of the theorem.
\hfill $\Box$
\vspace{0.2cm}
\noindent
{\bf Remark:}\, The above proof is constructive, that is, it
provides a polynomial algorithm to find a packing of given graphs
$G_1,G_2$ as above, with the asserted bound on the girth of the
combined graph. Indeed, as long as the girth is too small we can
find a shortest cycle $C$, take in it a vertex $u$ as in the
proof, find a vertex $v$ far from it and swap their roles in the
image of $G_1$. By the argument above this decreases the number of
short cycles by at least $1$. As the total number of such cycles is
less than $n$ by the choice of the parameters
and by (\ref{e111}), this process terminates in
polynomial time.
\subsection{Directed expanders}
By applying Theorem \ref{t111} repeatedly, starting with a cycle of
length $n$, it follows that for every
$d$ and all large $n$ there is a $2d$-regular graph on $n$ vertices
with girth at least $(1+o(1))\frac{\log n}{ \log (2d-1)}$ which can
be decomposed into $d$ Hamilton cycles. This is a (modest)
strengthening of the result of Erd\H{o}s and Sachs about the
existence of regular graphs of high girth. A more interesting
application of Theorem \ref{t111} is a strengthening of a result
proved in \cite{AMP} about the existence of high-girth directed
expanders.
\begin{theo}
\label{t131}
For every prime $p$ congruent to $1$ modulo $4$ and any $n>n_0(d)$
there is an explicit
construction of a $2d$-regular graph on $n$ vertices
with (undirected) girth
at least $(\frac{2}{3}-o(1)) \frac{\log n}{\log (d-1)}$
and an orientation of this graph
so that for every two sets of vertices $X,Y$ satisfying
\begin{equation}
\label{e131}
\frac{|X|}{n} \cdot \frac{|Y|}{n} \geq \frac{16}{d}
\end{equation}
there is a directed edge from $X$ to $Y$ and a directed edge from
$Y$ to $X$.
\end{theo}
This improves the estimate on the girth in the result proved in \cite{AMP}
by a factor of $3$, and also works for all large $n$. The proof
combines Theorem \ref{t111} with an argument from \cite{AMP} and
a recent result proved in \cite{Al4}. An explicit construction
here means that there is a polynomial time deterministic algorithm
for constructing the desired graphs.
\begin{proof}
An $(n,d,\lambda)$-graph is a $d$ regular graph on $n$ vertices in
which the absolute value of any nontrivial eigenvalue is at most
$\lambda$. The graph is Ramanujan if $\lambda=2 \sqrt {d-1}$.
Lubotzky, Phillips and Sarnak \cite{LPS}, and
independently Margulis \cite{Ma} gave,
for every prime $p$ congruent to $1$ modulo $4$, an explicit
construction of infinite
families
of $d=p+1$-regular Ramanujan graphs. The girth of these graphs
is at least $(1+o(1))\frac{2}{3}\log_{d-1} n'$, where $n'$ is the
number of vertices. In \cite{Al4} it is shown
how one can modify these graphs by deleting a set of
appropriately chosen $n'-n$ vertices and by adding edges
among their neighbors to get an $(n,d,2\sqrt{d-1}+o(1))$-graph with
exactly $n$ vertices keeping the girth essentially the same.
Fix such a graph $H$. By Theorem \ref{t111} we can pack two
copies of it $H_1,H_2$ keeping the girth of the combined graph at least
$$
\min \{(1+o(1))\frac{2}{3}\log_{d-1} n, (1+o(1)) \log_{2d-1} n\}
=(1+o(1))\frac{2}{3}\log_{d-1} n,
$$
where here we used the fact that for all admissible $d$,
$$
\frac{2}{3 \log (d-1)} \leq \frac{1}{\log (2d-1)}.
$$
Let $G$ be the combined graph. Number its vertices $1,2, \ldots ,n$
and orient every edge $ij$ with $i<j$ from $i$ to $j$ if it belongs
to the copy of $H_1$ and from $j$ to $i$ if it belongs to the
copy of $H_2$.
It is well known (c.f. \cite{AS}, Corollary 9.2.5) that if
$A,B$ are two subsets of an $(n,d,\lambda)$-graph and
$$
\frac{|A||B|}{n^2} > \frac{\lambda^2}{d^2}
$$
then there is an edge connecting $A$ and $B$. Let
$X$ and $Y$ be two sets of vertices satisfying (\ref{e131}).
Let $x$ be the median of $X$ (according to the numbering
of the vertices), $y$ the median of $Y$. Without loss of generality
assume that $x \leq y$. Let $A$ be the set of all vertices of $X$
which are smaller or equal to $x$, $B$ the set of all vertices of
$Y$ that are larger or equal to $y$. Then $|A| \geq |X|/2$ and
$|B| \geq |Y|/2$. Therefore
$$
\frac{|A||B|}{n^2} \geq \frac{|X||Y|}{4 n^2 } \geq
\frac{4}{d} > \frac{(2\sqrt{d-1}+o(1))^2}{d^2}.
$$
Therefore there is an edge of $H_1$ connecting $A$ and $B$
which, by construction, is oriented from $A$ to $B$.
Similarly there is an
edge of $H_2$ oriented from $B$ to $A$.
This completes the proof.
\end{proof}
\section{Nearly fair representation }
The approach described here was initiated in discussions with
Eli Berger and Paul Seymour \cite{BS}.
Let $G=(V,E)$ be a graph and let $P$ be an arbitrary partition
of its set of edges
into $m$ pairwise disjoint subsets $E_1, E_2, \ldots ,E_m$.
The sets $E_i$ will be called the color classes of the partition.
For any
subgraph $H'=(V',E')$ of $G$, let
$x(H',P)$ denote the vector $(x_1,x_2, \ldots ,x_m)$, where
$x_i=|E_i \cap E'|$ is the number of edges of $H'$ that lie in
$E_i$. Thus, in particular, $x(G,P)=(|E_1|, \ldots ,|E_m|).$
In a completely fair representation of the sets $E_i$ in $H'$,
each entry $x_i$ of the vector $x(H',P)$ should be equal
to $|E_i| \cdot \frac{|E'|}{|E|}$. Of course such equality can hold
only if all these numbers are integers. But
even when this is not the case the equality may hold up to a small
additive error.
In this section we are interested in results (and conjectures)
asserting that when $G$ is either the complete graph $K_n$ or the
complete bipartite graph $K_{n,n}$, then for certain graphs $H$
and for any partition $P$ of $E(G)$ into color classes $E_1,\ldots ,E_m$,
there is a subgraph $H'$ of $G$ which is isomorphic to $H$ so that
the vector $x(H',P)$ is close (or equal) to the vector
$x(G,P)\frac{|E(H')|}{|E(G)|}$. Stein
\cite{St} conjectured that if $G=K_{n,n}$ and $P$ is any partition
of the edges of $G$ into $n$ sets, each of size $n$, then
there is always a perfect matching $M$ in $G$ satisfying
$x(M,P)=\frac{1}{n} x(G,P)$, that is, a perfect matching containing
exactly one edge from each color class of $P$. This turned out to
be false, a clever counter-example has been given by Pokrovskiy and
Sudakov. In \cite{PS} they describe a partition of the edges of
$K_{n,n}$ into $n$ sets, each of size $n$, so that every perfect
matching misses at least $\Omega( \log n)$ color classes.
In \cite{AABCKLZ}
it is conjectured that when $G=K_{n,n}$, $P$ is arbitrary,
and $H$ is a matching of
size $n$, then there is always a copy $H'$ of $H$ (that is, a
perfect matching $H'$ in $G$), so that
$$
\|x(H',P)-\frac{1}{n} x(G,P)\|_{\infty} <2.
$$
This is proved in \cite{AABCKLZ} (in a slightly stronger form)
for partitions $P$
with $2$ or $3$ color classes. Here we first prove the following,
showing that when allowing a somewhat larger additive error
(which grows with the number of colors $m$ but is independent of $n$)
a similar result holds for partitions with any fixed number of
classes.
\begin{theo}
\label{t211}
For any partition $P$ of the edges of the complete bipartite graph
$K_{n,n}$ into $m$ color classes, there is a perfect matching $M$ so that
$$
\| x(M,P)-\frac{1}{n} x(K_{n,n},P) \|_{\infty} \leq
\| x(M,P)-\frac{1}{n} x(K_{n,n},P) \|_{2}
< (m-1)2^{(3m-2)/2}.
$$
\end{theo}
It is worth noting that a random perfect matching $M$ typically
satisfies
$$
\| x(M,P)-\frac{1}{n} x(K_{n,n},P) \|_{\infty} \leq O(\sqrt n).
$$
The main challenge addressed in the theorem is to get an upper
bound independent of $n$.
Theorem \ref{t211} is a special case of a general result which we
describe next, starting with the following definition.
\begin{definition}
\label{d221}
Let $G$ be a graph and let $H$ be a subgraph of it. Call a family of
graphs ${\cal H}$ (which may have repeated members)
a {\em uniform cover of width $s$ of the pair $(G,H)$}
if every member $H'$ of ${\cal H}$ is a subgraph of $G$ which is isomorphic
to $H$, the number of edges of each such $H'$ which are not edges
of $H$ is at most $s$, every edge of $H$ belongs to the same number
of members of ${\cal H}$, and every edge in $E(G)-E(H)$
belongs to the same positive number of members of ${\cal H}$.
\end{definition}
An example of a uniform cover of width $s=2$ for
$G=K_{n,n}$ and $H$ a perfect matching in it is the following.
Let the $n$ edges of $H$ be
$a_ib_i$ where $\{a_1,a_2, \ldots ,a_n\}$ and
$\{b_1, b_2, \ldots ,b_n\}$ are the vertex classes of $G$. Let
${\cal H}$ be the family of all perfect matchings of $G$ obtained from $H$
by omitting a pair of edges $a_ib_i$ and $a_jb_j$ and by adding
the edges $a_ib_j$ and $a_jb_i$. The width is $2$, every edge of
$H$ belongs to exactly ${n \choose 2}-(n-1)$ members of ${\cal H}$, and every
edge in $E(G)-E(H)$ belongs to exactly $1$ member of
${\cal H}$.
\begin{theo}
\label{t213}
Let $G$ be a graph with $g$ edges,
let $F$ be a
subgraph of it with $f$ edges,
and suppose there is a uniform cover of width $s$ of the
pair $(G,F)$. Then for any partition $P$ of the edges of $G$ into
$m$-subsets, there is a copy $H$ of $F$ in $G$ so that
$$
\| x(H,P)-\frac{f}{g} x(G,P) \|_{\infty} \leq
\| x(H,P)-\frac{f}{g} x(G,P) \|_{2} \leq (m-1)2^{(m-2)/2}s^m.
$$
\end{theo}
Theorem \ref{t211} is a simple consequence of Theorem \ref{t213}.
A similar simple consequence is the following.
\begin{prop}
\label{p214}
For any partition $P$ of the edges of the complete graph
$K_{n}$ into $m$ color classes, there is a Hamilton cycle $C$ so that
$$
\| x(C,P)-\frac{2}{n-1} x(K_{n},P) \|_{\infty} \leq
\| x(C,P)-\frac{2}{n-1} x(K_{n},P) \|_{2}
< (m-1)2^{(3m-2)/2}.
$$
\end{prop}
Similar statements follow, by the same reasoning,
for a Hamilton cycle in a complete bipartite
graph, or for a perfect matching in a complete graph on an even number
of vertices. We proceed to describe a more general application.
For a fixed graph $T$ whose number of vertices $t$ divides $n$,
a $T$-factor in $K_n$ is the graph consisting of $n/t$ pairwise
vertex disjoint copies of $T$. In particular, when $T=K_2$ this is
a perfect matching.
\begin{theo}
\label{t215}
For any fixed graph $T$ with $t$ vertices and $q$ edges
and any $m$ there is a
constant $c=c(t,q,m) \leq (m-1)2^{(m-2)/2}(qt)^m$
so that for any $n$ divisible by $t$ and for
any partition $P$ of the edges of the complete graph $K_n$ into $m$
subsets, there is a $T$-factor $H$ so that
$$
\| x(H,P)-\frac{2q}{(n-1)t} x(K_{n},P) \|_{\infty} \leq
\| x(H,P)-\frac{2q}{(n-1)t} x(K_{n},P) \|_{2} \leq c.
$$
\end{theo}
\subsection{Proofs}
We start with the proof of Theorem \ref{t213}.
\begin{proof}
Let $P$ be a partition of the edges of $G$ into $m$ color
classes $E_i$.
Put
$$
y=(y_1,y_2, \ldots ,y_m)=\frac{f}{g}x(G,P).
$$
Let $H$ be a copy of $F$ in $G$ for which the quantity
$\|y-x\|_2^2 = \sum_{j=1}^m (y_i-x_i)^2$ is minimum where
$x=(x_1, x_2, \ldots ,x_m)=x(H,P)$.
Let ${\cal H}$ be a uniform cover of width $s$ of the pair
$(G,H)$. Suppose each edge of $H$ belongs to $a$ members of ${\cal H}$
and each edge in $E(G)-E(H)$ belongs to $b>0$ such members.
For each member $H'$ of ${\cal H}$, let $v_{H'}$
denote the vector of length $m$ defined as follows. For each
$1 \leq i \leq m$, coordinate number
$i$ of $v_{H'}$ is the number of edges in $E(H')-E(H)$
colored $i$ minus the number of edges in $E(H)-E(H')$
colored $i$. Note that the $\ell_1$-norm of
this vector is at most $2s$ and its sum of coordinates is $0$.
Therefore, its $\ell_2$-norm is at most $\sqrt {2s^2}$.
Note also that $x(H',P)=x(H,P) +v_{H'}$.
We claim that the sum $S$ of all $|{\cal H}|$-vectors $v_{H'}$ for $H' \in
{\cal H}$ is a positive multiple of the vector $(y-x)$. Indeed, each
edge in $E(G)-E(H)$ is covered by $b$ members of ${\cal H}$, and each
edge of $E(H)$ is covered by $a$ members of ${\cal H}$. In the sum
$S$ above this contributes to the coordinate corresponding to color
number $i$, $b$ times the number of edges of color $i$ in
$E(G)-E(H)$
minus $(|{\cal H}|-a)$ times the number of edges of color
$i$ in $H$. Equivalently, this is
$b$ times the number of all edges of $G$ colored $i$ minus
$(|{\cal H}|+b-a)$ times the number of edges of $H$ colored $i$.
Since the sum of coordinates of each of the vectors
$v_{H'}$ is zero, so is the sum of coordinates of
$S$, implying that $bg=(|{\cal H}|+b-a) f$, that is,
$|{\cal H}|+b-a=\frac{g}{f}b$. Since $\frac{g}{f} y=x(G,P)$
this implies that
$S=\frac{bg}{f}(y-x)$, proving the claim.
Since the vector $y-x$ is a linear combination
with positive coefficients of the
vectors $v_{H'}$ it follows,
by Carath\'eodory's Theorem for cones,
that there exists a set $L$ of linearly independent vectors
$v_{H'}$ so that $y-x$ is
a linear combination with positive coefficients of them.
Indeed, starting
with the original expression of
$y-x$ mentioned above, as long as there is a linear dependence
among the vectors $v_{H'}$ participating in the combination
with nonzero (hence positive) coefficients, we can subtract an
appropriate multiple of this dependence and ensure that at least
one of the nonzero coefficients vanishes and all others stay
non-negative (positive, after omitting all the ones with coefficient
$0$). As each vector $v_{H'}$ has $m$ coordinates and their sum is
$0$, it follows that $|L| \leq m-1$.
We can now solve the system of linear equations $y-x=\sum z_{H'} v_{H'}$
with the variables $z_{H'}$ for $v_{H'} \in L$.
Note that it is enough to
consider any $|L|\leq m-1$ coordinates of $y-x$ and solve the system
corresponding to these coordinates.
By Cramer's rule applied to this system each $z_{H'}$ is a ratio of
two determinants. The denominator is a determinant of a nonsingular matrix
with integer coefficients, and its absolute value is thus at least $1$.
The numerator is also a determinant, and
by Hadamard's Inequality its absolute value is at most the product of
the $\ell_2$-norms of the columns of the corresponding matrix.
The norm of one column is at
most $\|y-x\|_2$ (this can be slightly improved by
selecting the $|L|$-coordinates with the smallest $\ell_2$-norm,
but we do not include this slight improvement here). Each other column
has norm at most $(2s^2)^{1/2}$. Therefore
each coefficient $z_{H'}$ satisfies
$0 \leq z_{H'} \leq \|y-x\|_2 (2s^2)^{(m-2)/2}.$
By taking the inner product with $y-x$ we get
$$
\|y-x\|_2^2 =\sum_{v_{H'} \in L} z_{H'} \langle y-x,v_{H'} \rangle
$$
$$
\leq \sum_{v_{H'} \in L, \langle y-x, v_{H'} \rangle >0}
z_{H'} \langle y-x,v_{H'} \rangle
$$
$$
\leq (m-1) \|y-x\|_2 (2s^2)^{(m-2)/2} \max \langle y-x,v_{H'} \rangle.
$$
Therefore, there is a $v_{H'}$ so that
$$
\frac{\|y-x\|_2}{(m-1) (2s^2)^{(m-2)/2} }=
\frac{\|y-x\|_2^2}{(m-1) (2s^2)^{(m-2)/2} \|y-x\|_2} \leq
\langle y-x,v_{H'} \rangle,
$$
that is,
\begin{equation}
\label{e221}
\|y-x\|_2 \leq (m-1) (2s^2)^{(m-2)/2} \langle y-x,v_{H'} \rangle
=(m-1)2^{(m-2)/2} s^{m-2} \langle y-x,v_{H'} \rangle.
\end{equation}
By the minimality of $\|y-x\|_2^2$
$$
\|x+v_{H'}-y\|_2^2 =\|x-y\|_2^2 -2 \langle y-x,v_{H'} \rangle
+\|v_{H'}\|_2^2 \geq \|x-y\|_2^2,
$$
implying that
$$
2 s^2 \geq \|v_{H'}\|_2^2 \geq 2 \langle y-x,v_{H'} \rangle.
$$
Plugging in (\ref{e221}) we get
$$
\|y-x\|_2 \leq (m-1)2^{(m-2)/2} s^{m},
$$
and the desired results follows since
$ \|y-x\|_{\infty} \leq \|y-x\|_2$.
\end{proof}
The assertions of Theorem \ref{t211} and Proposition \ref{p214}
follow easily from Theorem \ref{t213}. Indeed, as described above
there is a simple uniform cover of width $s=2$ for the pair
$(K_{n,n},M)$ where
$M$ is a perfect matching. There is also a similar uniform cover
${\cal H}$
of width $s=2$ for the pair $(K_n,C)$ where $C$ is a Hamilton
cycle. The $n(n-3)/2$ members of ${\cal H}$ are all Hamilton cycles
obtained from $C$ by omitting two nonadjacent edges of it and by
adding the two edges that connect the resulting pair of paths to
a cycle.
To prove Theorem \ref{t215}
we need the following simple lemma.
\begin{lemma}
\label{l221}
Let $T$ be a fixed graph with $t$ vertices and $q$ edges, suppose
$t$ divides $n$ and let $H$ be a $T$-factor in $K_n$. Then there
is a uniform cover of width at most $qt$ of the pair $(K_n,H)$.
\end{lemma}
\begin{proof}
Let $H$ be a fixed $T$-factor in $K_n$, it consists of
$p=n/t$ (not necessarily connected) vertex disjoint copies of $T$
which we denote by $T_1,T_2, \ldots ,T_p$. Let ${\cal H}_1$ be the set of
all copies $H'$ of the $T$-factor obtained from $H$ by replacing one
the copies $T_i$
by another copy of $T$ on the same set of vertices, in all
possible $t! $ ways. Note that if $T$ has a nontrivial automorphism
group some members of ${\cal H}_1$ are identical, and ${\cal H}_1$
is a multiset.
By symmetry it is clear that each edge
of $H$ belongs to the same number of members of ${\cal H}_1$. Similarly,
each edge connecting two vertices of the same $T_i$ which does not
belong to $H$ lies in the same positive number of members of ${\cal H}_1$. Beside
these two types of edges, no other edge of $K_n$ is covered by any
member of ${\cal H}_1$. Let ${\cal H}_2$ be the (multi)-set of all copies of
the $T$-factor obtained from $H$ by choosing, in all possible ways,
$t$ of the copies of $T$, say, $T_{i_1}, T_{i_2}, \ldots
,T_{i_t}$, removing them, and replacing them by all possible
placements of $t$ vertex disjoint copies of $T$
where each of the newly placed copies
contains exactly one vertex of each $T_{i_j}$. Again by symmetry it
is clear that each edge of $H$ belongs to the same number of
members of ${\cal H}_2$. In addition, each
edge of $K_n$ connecting vertices from
distinct copies of $T$ in $H$
belongs to the same (positive) number of members of ${\cal H}_2$. No
other edges of $K_n$ are covered by any $H' \in {\cal H}_2$.
It is now simple
to see that there are two integers $a$, $b$, so that the multiset
${\cal H}$ consisting of $a$ copies of each member of ${\cal H}_1$ and $b$
copies of each member of ${\cal H}_2$ is a uniform cover of the pair
$(K_n,H)$. The width of this cover is clearly $qt$, as every member
of ${\cal H}_2$ contains $qt$ edges not in $E(H)$, and every member of
${\cal H}_1$ contains at most $2q$ edges not in $E(H)$. This completes
the proof.
\end{proof}
The assertion of Theorem \ref{t215} clearly follows from
the last Lemma together with Theorem \ref{t213}.
\subsection{Concluding remarks and open problems}
\begin{itemize}
\item
The statement of Theorem \ref{t215} holds for any graph $H$
consisting of $n/t$ (not necessarily connected)
vertex disjoint components, each having $t$ vertices
and $q$ edges. The proof applies with no need to assume that all these
components are isomorphic.
\item
The proof of Theorem \ref{t213} is algorithmic in the sense that if
the cover ${\cal H}$ is given then one can find, in time polynomial
in $n$ and $|{\cal H}|$,
a copy $H$ of $F$ satisfying the conclusion. Indeed, the proof
implies that as long as we have a copy $H$ for which
the conclusion does not hold, there is a member $H' \in {\cal H}$ for
which $\| x(H',P) - \frac{f}{g} x(G,P) \|_2^2$ is strictly
smaller than $\| x(H,P) - \frac{f}{g} x(G,P) \|_2^2$.
By checking all members of ${\cal H}$ we can find an $H'$ for which
this holds.
As both these
quantities are non-negative rational numbers smaller than
$n^4$ with denominator
$g^2<n^4$, this process
terminates in a polynomial number of steps. We make no attempt to
optimize the number of steps here.
\item
The results can be extended to $r$-uniform hypergraphs by a
straightforward
modification of the proofs.
\item
There are graphs $H$ for which no result
like those proved above
holds when $G$ is either a complete or a complete bipartite
graph even if the number of colors is small. A simple example
is when $G=K_{2n}$, $H=K_{1,2n-1}$ and $m=3$. The edges of $K_{2n}$ can be
partitioned into two vertex disjoint
copies of $K_n$ and a complete bipartite graph
$K_{n,n}$. For this partition, every copy of the star $H$ misses
completely one of the color classes, although it's fair share in it
is roughly a quarter of its edges. More generally, let $H$ be any
graph with a vertex cover of size smaller than $m-1$ (that is, $H$
contains a set of less than $m-1$ vertices touching all its edges).
Consider a partition of the edges of the complete graph $K_n$ into
$m-1$ pairwise vertex disjoint copies of the complete graph on
$\lfloor n/(m-1) \rfloor$ vertices, and an additional class
containing all the remaining edges. Then any copy of $H$ in this
graph cannot contain edges of all those $m-1$ complete subgraphs,
as the edges of the copy can be covered by less than $m-1$ stars.
It is easy to see that a
similar example exists for $G=K_{n,n}$ as well.
\item
The discussion here suggests the following conjecture.
\begin{conj}
\label{c231}
For every $d$ there exists a $c(d)$ so that for any graph $H$ with
at most $n$ vertices and maximum degree at most $d$ and for any
partition $P$ of the edges of $K_n$ into $m$ color classes, there is
a copy $H'$ of $H$ in $K_n$ so that
$$
\| x(H',P)-\frac{|E(H)|}{E(K_n)|} x(K_{n},P) \|_{\infty} \leq
c(d).
$$
\end{conj}
The analogous conjecture for bipartite bounded-degree graphs $H$
with at most $n$ vertices in each color class and for partitions of
the edges of $K_{n,n}$ is also plausible. Note that the conjecture
asserts that the same error term $c(d)$ should hold for any number of
colors $m$. Note also that $c(d)$ must be at least $\Omega(d)$ as shown by
the example of a star $H=K_{1,d}$ and the edge-coloring of $K_{2n}$ with
$m=3$ colors described above.
\end{itemize}
\section{The choice number of complete multipartite graphs with
equal color classes}
The choice number of a graph $G$ is the
smallest integer $s$ so that for any
assignment of a list of $s$ colors to each vertex of $G$
there is a proper coloring of $G$ assigning to each vertex a color
from its list. This notion was introduced in \cite{Vi}, \cite{ERT}.
Let $K_{m*k}$ denote the complete $k$-partite graph with
$k$ color classes, each of size $m$. Several researchers
investigated the choice number
$ch(K_{m*k})$ of this graph. Trivially $ch(K_{1*k})=1$ as
$K_{1*k}$ is a $k$-clique. In \cite{ERT} it is proved that
$ch(K_{2*k})=k$. Kierstead \cite{Ki} proved that
$ch(K_{3*k})=\lceil (4k-1)/3 \rceil$ and
in \cite{KSW} it is proved that
$ch(K_{4*k})= \lceil (3k-1)/2 \rceil.$
In \cite{ERT} it is shown that as $m$ tends to infinity
$ch(K_{m*2})=(1+o(1))\log_2 m$. In \cite{Al} the author shows
that there are absolute constants $c_1,c_2>0$ so that
$c_1 k \ln m \leq ch(K_{m*k}) \leq c_2 k \ln m$ for all $m$ and
$k$. In \cite{GK} it is proved that for fixed $k$,
as $m$ tends to infinity, $ch(K_{m*k})=(1+o(1)) \frac{\ln m}{\ln
(k/(k-1)}$ and in \cite{Sh} it is proved that if both
$m$ and $k$ tend to infinity and $\ln k =o(\ln m)$ then
$ch(K_{m*k})=(1+o(1)) k \ln m$. Our first result here is that
the assumption
that $\ln k =o(\ln m)$ can be omitted, obtaining the asymptotics of
$ch(K_{m*k})$ when $m$ and $k$ tend to infinity (with no assumption
on the relation between them).
\begin{theo}
\label{t311}
If $m$ and $k$ tend to infinity then
$$
ch(K_{m*k})=(1+o(1))k \ln m.
$$
\end{theo}
The proof is probabilistic, similar to the one in \cite{Al},
where the main additional argument is in the proof of the upper
bound for values of $k$ which are much bigger than $m$.
Our second result is the following.
\begin{theo}
\label{t312}
For any fixed integer $m \geq 1$ the limit
$$
\lim_{k \rightarrow \infty} \frac{ch(K_{m*k})}{k}
$$
exists (and is $\Theta(\ln m)$).
\end{theo}
For $m \geq 1$, let $c(m)$ denote the above limit.
By the known results stated above
$c(1)=c(2)=1$, $c(3)=4/3$, $c(4)=3/2$ and
$c(m)=(1+o(1)) \ln m$. The problem of determining $c(m)$ precisely
for every $m$ seems very difficult.
We prove Theorem \ref{t311} without trying to optimize the error terms.
To simplify the presentation, we omit all floor and ceiling signs
whenever these are not crucial.
\subsection{The upper bound}
\begin{prop}
\label{p321}
For every $m,k \geq 2$
$$
ch(K_{m*k}) \leq k (\ln m + \ln \ln m+20).
$$
\end{prop}
\noindent
{\bf Proof:}\,
Since $\ln m + \ln \ln m +20 \geq 20$ for all $m \geq 2$
we may and will assume that $m > 20$. We consider two possible cases.
\vspace{0.2cm}
\noindent
{\bf Case 1:}\, $k \leq 10 \ln m$.
In this case we show that lists of size
$s=k(\ln m + \ln \ln m +3)$ suffice. Let $G=K_{m*k}=(V,E)$, and
suppose we assign a list $S_v$ of colors to each vertex $v \in
V$, where $|S_v|=s$ for all $v$. Let $S=\cup_{v \in V} S_v$
be the union of all lists. Let $S=T_1 \cup T_2 \ldots \cup T_k$ be
a random partition of all colors in $S$ into $k$ pairwise disjoint
subsets, where each color $x \in S$ is assigned, randomly, uniformly
and independently, to one of the subsets $T_j$. We obtain a proper
coloring of $G$ by coloring each vertex $v$ that lies in color
class number $j$ by a color from $S_v \cap T_j$. Clearly, if there is
indeed such a color for each vertex, then the resulting coloring
is proper. The probability that for a fixed vertex $v$ the
above fails is exactly
$$
(1-\frac{1}{k})^{|S_v|} \leq e^{-(\ln m + \ln \ln m +3)}
<\frac{1}{e^3 m \ln m } < \frac{1}{km}.
$$
As there are $mk$ vertices,
the probability that there is a vertex for which the
above fails is smaller than $1$, completing the proof in this
case.
\vspace{0.2cm}
\noindent
{\bf Case 2:}\ $ k > 10 \ln m$.
Note that since by assumption
$m>20$ this implies that $k \geq 30$.
In this case we show that lists of size $s=k(\ln m+20)$ suffice.
Let $G=K_{m*k}=(V,E)$, and
suppose we assign a list $S_v$ of $s$ colors to each vertex $v \in
V$. As before, let $S=\cup_{v \in V} S_v$
be the union of all lists. Our strategy now is to first define a
set of reserve colors $R$, these colors will be used to
assign colors to the vertices that will not be colored
by the procedure applied in Case 1. Let $R$ be a random subset of
$S$ obtained by picking each color in $S$ to lie in $R$
with probability $p=\frac{10}{\ln m + 20}$, where all choices are
independent. For a fixed vertex $v$, the random variable
$|S_v \cap R|$ is a Binomial random variable with expectation
$sp=10k$. By the standard estimates for Binomial distributions
(see, e.g., \cite{AS}, appendix A, Theorems A.1.11 and A.1.13),
the probability that this random variable is smaller than $k$ is
less than $e^{-10k/8}$ and the probability it is larger than
$20k$ is less than $e^{-10k/14}$. Thus the probability it is not
between $k$ and $20k$ is less than
$$
2e^{-10k/14}<2e^{-k/3} e^{-k/3} < 2\frac{1}{2k}\frac{1}{m^3}
< \frac{1}{mk},
$$
where here we used the fact that $k \geq 10 \ln m$ to conclude
that $e^{-k/3} < \frac{1}{m^3}$ and the fact that $k \geq 30$ to
conclude that $e^{-k/3} < \frac{1}{2k}.$
It follows that with positive probability
$k \leq |S_v \cap R| \leq 20 k$ for every vertex $v \in V$.
Fix a set of colors $R$ for which this holds. Now proceed as
in Case 1.
Let $S-R=T_1 \cup T_2 \ldots \cup T_k$ be
a random partition of all colors in $S-R$ into $k$ pairwise disjoint
subsets, where each color in $S-R$ is assigned, randomly, uniformly
and independently to one of the subsets $T_j$. If a vertex
$v$ of $G$ lies in color class
number $j$, and $S_v \cap T_j \neq \emptyset$, then color it by an
arbitrary color in this intersection $S_v \cap T_j$.
The probability that $v$ fails to have such a color is
$$
(1-\frac{1}{k})^{|S_v-R|} \leq (1-\frac{1}{k})^{k \ln m} \leq
\frac{1}{m},
$$
where here we used the fact that $|S_v \cap R| \leq 20k$ for all
$v$.
By linearity of expectation, the expected number of uncolored
vertices at this stage is at most $k$, hence we can fix a
splitting $T_1,\cdots, T_k$ as above so that there are at most
$k$ uncolored vertices. But now we can color these vertices one by
one using the reserve colors. Since for each such vertex $u$,
$|S_u \cap R| \geq k$, each of these vertices has at least
$k$ colors of $R$ in its list and thus we will be able to assign to
it a color that differs from all colors of $R$ assigned to
previous vertices. This completes the proof of the upper bound.
\hfill $\Box$
\subsection{The lower bound}
The proof of the lower bound is essentially the one in
\cite{Al}, with a more careful computation and choice of parameters.
For completeness, we sketch the details.
\begin{prop}
\label{p322}
There exists an $m_0$ so that for all $m > m_0$ and every $k$
$ch(K_{k*m}) > t$ where
$$
t=(k-1-\frac{k}{\ln m}) (\ln m - 4 \ln \ln m)
(1-\frac{\ln m}{m})~ (~=(1+o(1)) k \ln m),
$$
where the $o(1)$-term tends to zero as $m$ and $k$ tend to
infinity.
\end{prop}
\noindent
{\bf Proof:}\,
We consider two possible cases.
\vspace{0.2cm}
\noindent
{\bf Case 1:}\, $k \leq m $.
In this case we prove that $ch(K_{k*m})>s$, where
$$
s= (k-1-\frac{k}{\ln m}) (\ln m - 4 \ln \ln m) ~ (~=(1+o(1)) k \ln m).
$$
Let $S$ be a set of $k (\ln m)^2$ colors, and let
$S_1,S_2, \ldots ,S_m$ be $m$ random subsets of $S$, each chosen
independently and uniformly among all subsets of cardinality
$s$ of $S$, where $s$ is as above. We claim that with positive
probability there is no subset of $S$ of cardinality
at most $|S|/k = (\ln m)^2$ that intersects all
subsets $S_i$. This claim suffices to prove the assertion of the
proposition in this case. Indeed, we simply assign the $m$ vertices
in each color class of $G$ the $m$ lists $S_i$. If there would have
been a proper coloring of $G$ assigning to each vertex a color
from its list, then the set of all colors assigned to vertices
in one of the color classes of $G$ must be of size at most $|S|/k$ and
it must intersect all
lists $S_i$, contradiction. It thus suffices to prove the claim.
Fix a set $T$ of $(\ln m)^2$ colors. The probability that a random
subset of size $s$ of $S$ does not intersect $T$ is
$$
\frac{{{|S|-|T|} \choose s}}{{{|S|} \choose s}}
=\frac{{{(k-1) \ln^2 m} \choose s}}{{{k \ln^2 m } \choose s}}.
$$
This quantity is at least
$$
(\frac{(k-1) \ln^2 m - k \ln m}{k \ln^2 m- k \ln m}) ^s
=(1-\frac{1}{k(1-1/\ln m)})^{[k(1-\frac{1}{\ln m})-1](\ln m -4 \ln
\ln m)}
$$
$$
\geq (\frac{1}{e})^{\ln m-4 \ln \ln m} =\frac{\ln^4 m}{m},
$$
where here we used the fact that for every $q>1$, $(1-1/q)^{q-1}
\geq \frac{1}{e}$.
Therefore, the probability that none of the $m$
random sets $S_i$ misses $T$ is at most
$$
(1-\frac{\ln^{4} m}{m})^m < e^{-\ln^{4}m}.
$$
As the number of choices for $T$ is only
$$
{{k \ln^2 m} \choose {\ln^2 m}} \leq (ek)^{\ln^2 m}
\leq e^{(1+o(1))\ln^3 m},
$$
where here we used the assumption that $k \leq m$, the desired claim
follows, completing the proof of Case 1.
\vspace{0.2cm}
\noindent
{\bf Case 2:}\, $k \geq m$.
In this case, take first the previous construction with $m$ and
$k'=\ln m$. Replace $k$ by the largest integer $k''$ which is
at most $k$ and is divisible by $k'$, that is:
$k''=k'\lfloor k/k' \rfloor$. Note that as $k \geq m$ and $k' = \ln
m$, $k'' \geq k (1-\frac{\ln m}{m})$. Now replace in the
construction for $k'=\ln m$ every color by a group of
$k''/k'$ colors, where all groups are pairwise disjoint,
to get $m$ lists, each of size
$(1+o(1))k'' \ln m=(1+o(1)) k \ln m$, in a set of size
$k'' \ln^2 m$,
so that no
subset of size $\ln^2 m$, that is,
a fraction of $1/k''$ of the colors, intersects
all of them. This shows, as before. that
$ch(K_{m*k''}) > (1+o(1)) k'' \ln m=(1+o(1)) k \ln m$,
and as
$ch(K_{m*k})$ can be only larger (since it contains $K_{m*k''})$ as a
subgraph), this completes the proof.
\hfill $\Box$
\subsection{The existence of the limit}
In this subsection we prove Theorem \ref{t312}. A natural way to try
and prove it is to show that for every
fixed $m$, the function $f(k)=ch(K_{m*k})$ is either sub-additive
or super-additive. In theses cases the existence of the limit would
follow from Fekete's Lemma. Unfortunately this function is not
always super-additive, as shown by the case $m=3$, since
$ch(K_{3*2})=3$ and
$$
ch(K_{3*2k})= \lceil (8k-1)/3 \rceil <3k.
$$
Similarly, the function is not always sub-additive, as shown by the
case of large $m$, where
$ch(K_{m*2})=(1+o(1)) \log_2 m$
and for large $k$,
$$
ch(K_{m*2k})= (1+o(1)) 2k \ln m > (1+o(1)) k \log_2 m.
$$
Still we show that the limit exists by proving that the
above function is nearly sub-additive.
We need the following technical lemma.
\begin{lemma}
\label{l323}
There is a positive integer $s_0$, so that for every integer
$s>s_0$ the following holds. For every real
$c$ satisfying $1/3 \leq c \leq 2/3$ and for every integer $t \geq
2$:
$$
c[(s^{1/3}+3)t^{1/3}-3]^3 -c [(s^{1/3}+3)t^{1/3}-3]^2
\geq
[(s^{1/3}+3)(ct)^{1/3}-3]^3.
$$
\end{lemma}
\noindent
{\bf Proof:}\,
Put
$$
X=c^{1/3}[(s^{1/3}+3)t^{1/3}-3],~~~Y=(s^{1/3}+3)(ct)^{1/3}-3.
$$
Then the above inequality is equivalent to the statement
$$
X^3 -c^{1/3} X^2 \geq Y^3,
$$
that is, to
$$
(X-Y)(X^2+XY+Y^2) \geq c^{1/3} X^2.
$$
Since $1/3 \leq c \leq 2/3$, we have $0.69<c^{1/3}<0.88$.
Thus $X-Y=3-3c^{1/3}>0.36$. For sufficiently large $s$,
$X>Y>0.9X>0$ and thus $XY >0.9 X^2$ and $Y^2>0.8 X^2$.
Therefore
$$
(X-Y)(X^2+XY+Y^2) > 0.36 \cdot 2.7 X^2=0.972 X^2
> 0.88 X^2 > c^{1/3} X^2.
$$
This completes the proof. \hfill $\Box$
We also need the following simple corollary of Chernoff's
Inequality (see, e.g., \cite{AS}, Appendix A.)
\begin{lemma}
\label{l324}
There exists an $s_0>0$ so that for every $s >s_0$, every integer
$t \geq 2$ and every real $c$ satisfying $1/3 \leq c \leq 2/3$,
the probability that the Binomial random variable with parameters
$[(s^{1/3}+3)t^{1/3}-3]^{3}$ and $c$ is at most
$$
c[(s^{1/3}+3)t^{1/3}-3]^3 -c [(s^{1/3}+3)t^{1/3}-3]^2
$$
is smaller than $\frac{1}{(st)^2}$.
\end{lemma}
\noindent
{\bf Proof:},
By Chernoff this probability is smaller than
$$
e^{-\Omega((st)^{1/3})}.
$$
\hfill $\Box$
Using the above, we prove the following.
\begin{prop}
\label{p325}
For every fixed $m$ there exists $k_0=k_0(m)$ so that for all
$k>k_0$ the following holds. If $ch(K_{m*k})=s$ then
for every integer $t \geq 1$
$$
ch(K_{m*kt}) \leq [(s^{1/3}+3)t^{1/3}-3]^3.
$$
\end{prop}
{\bf Proof:}\,
Since trivially $ch(K_{m*k}) \geq ch(K_{1*k})=k$, we can choose
$k_0>m$ so that for $k>k_0$,
$s=ch(K_{m*k})$ is sufficiently large to ensure that
the assertions of Lemma \ref{l323} and Lemma \ref{l324} hold.
Note also that for this $k_0$, $s >m$.
With this $k_0$ we prove the above by induction on $t$.
For $t=1$ there is nothing to prove. Assuming the result
holds for all integers $t'<t$ we prove it for $t$.
Let $G=K_{m*kt}=(V,E)$ have the $kt$ color classes
$U_1,U_2, \cdots U_{kt}$, and suppose we have a list $L_v$
of $[(s^{1/3}+3)t^{1/3}-3]^3$ colors assigned to each vertex
$v \in V$. Put $t_1=\lfloor t/2 \rfloor$,
$t_2=\lceil t/2 \rceil$ and split $V$ into two disjoint
sets $V_1,V_2$, where
$V_1$ consists of all vertices in the first $t_1k$ color classes
$U_j$ and $V_2$ consist of all vertices in the last $t_2$ color
classes $U_j$. Let $G_1$ be the induced subgraph of $G$ on $V_1$
and
$G_2$ the induced subgraph of $G$ on $V_2$. Thus $G_1$ is a copy of
$K_{m*kt_1}$ and $G_2$ is a copy of $K_{m*t_2}$.
Let $S \cup_{v \in V} L_v$ be the set of all colors,
and let $S=S_1 \cup S_2$ be a random partition of it into two
disjoint sets, where each color in $S$ is chosen, randomly and
independently, to lie in $S_1$ with probability $t_1/t$ and to lie
in $S_2$ with probability $t_2/t$.
Our objective is to use only the colors of $S_1$ for the vertices
in $G_1$ and only those of $S_2$ for the vertices in $G_2$.
Note that $1/3 \leq t_1/t \leq t_2/t \leq 2/3$. For each vertex
$v \in V_1$ the set $L_v \cap S_1$ of colors in $S_1$ that belong
to the list of $v$ is of size which is a binomial random variable
with parameters
$[(s^{1/3}+3)t^{1/3}-3]^3$ and $t_1/t$. Therefore, by Lemma
\ref{l324} the probability
that this size is smaller than
$(t_1/t)[(s^{1/3}+3)t^{1/3}-3]^3 -(t_1/t) [(s^{1/3}+3)t^{1/3}-3]^2$
is less than $\frac{1}{(st)^2}$. By the same reasoning the
probability that for a vertex $u \in V_2$ the size of
$L_u \cap V_2$ is smaller than
$(t_2/t)[(s^{1/3}+3)t^{1/3}-3]^3 -(t_2/t) [(s^{1/3}+3)t^{1/3}-3]^2$
is less than $\frac{1}{(st)^2}$. As $s>m, s \geq k$
the total number of vertices is
smaller than $kst<(st)^2$ and hence with positive probability this
does not
happen for any vertex. By Lemma \ref{l323} in this case each vertex
of $G_1$ still has at least
$[(s^{1/3}+3)(t_1)^{1/3}-3]^3$ colors in its list (restricted to
the colors in $S_1$), and a similar statement holds for the vertices
of $G_2$. We can now fix a partition $S=S_1 \cup S_2$ for which this holds
and apply induction to color $G_1$ by the colors from $S_1$ and
$G_2$ by the colors from $S_2$, completing the proof.
\hfill $\Box$
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t312}:}\,
Fix an integer $m \geq 1$.
By the result of \cite{Al} stated in Section 1,
$$
\lim \inf_{k \rightarrow \infty} \frac{ch(K_{m*k})}{k}=q
$$
exists (and is $\Theta(\ln m)$). Fix a small $\epsilon>0$
and let $k>k_0$ be a large integer, where $k_0$ is as in
Proposition \ref{p325}, so that
$$
\frac{ch(K_{m*k})}{k} \leq q +\epsilon.
$$
Put $s=ch(K_{m*k})$. Then $s \leq k(q+\epsilon)$.
By Proposition \ref{p325} for every integer
$t \geq 1$,
$$
ch(K_{m*kt}) \leq [(s^{1/3}+3)t^{1/3}-3]^3
< [s^{1/3}e^{3/s^{1/3}}t^{1/3}]^{3}=ste^{9/s^{1/3}}.
$$
Suppose, further, that $k$ is chosen to be sufficiently large to
ensure that
$$
e^{9/k^{1/3}}<(1+\epsilon).
$$
As $s=ch(K_{m*k}) \geq k$
in this case we have also
$$
e^{9/s^{1/3}}<(1+\epsilon).
$$
Therefore, for every integer $t \geq 1$
$$
ch(K_{m*kt}) \leq
ste^{9/s^{1/3}} <
k(q+\epsilon)t(1+\epsilon).
$$
It follows that for every large integer $p$,
$$
ch(K_{m*p}) \leq
k(q+\epsilon) \lceil p/k \rceil (1+\epsilon)
\leq
k(q+\epsilon) (p+k)/k (1+\epsilon).
$$
Thus
$$
\frac{ch(K_{m*p})}{p}
\leq k(q+\epsilon) (p+k)/(pk) (1+\epsilon)
$$
which for sufficiently large $p$ is at most, say,
$$
(q+\epsilon)(1+\epsilon)^2.
$$
Since, by the result in \cite{Al}, $q =\Theta( \ln m)$
and $\epsilon>0$ can be chosen to be arbitrarily small
this implies that
$$
\lim \sup_{p \rightarrow \infty} \frac{ch(K_{m*p})}{p}
\leq q =
\lim \inf_{p \rightarrow \infty} \frac{ch(K_{m*p})}{p},
$$
completing the proof.
\hfill $\Box$
\section{On vector balancing}
Let $p$ be a prime, let $w_1=e^{2 \pi i/p}$ be the $p$th primitive
root of unity, and define $w_j=w_1^j$ for $0 \leq j \leq p-1$. Let
$n$ be an integer divisible by $p$, and let $B$ be the set of all
$p^n$ vectors of length $n$ in which each coordinate is in the set
$\{1,w_1, \ldots ,w_{p-1} \}$. Let $K(n,p)$ denote the minimum $k$
so that there exists a set $\{v_1, v_2, \ldots ,v_k\}$ of members of
$B$ such that for every $u \in B$ there is some $1 \leq j \leq k$
so that the scalar inner product $v_i \cdot u=0$.
Heged\H{u}s \cite{Heg} proved that for every prime $p$ and $n$
divisible by $p$, $K(n,p) \geq (p-1)n$, extending a result of
\cite{ABCO} where the statement is proved for $p=2$. He also
conjectured that
equality always holds, as is the case for $p=2$, by a simple construction
of Knuth (c.f. \cite{ABCO}).
Our first observation here is that this conjecture
is (very) false for every prime $p \geq 5$ and large $n$.
\begin{prop}
\label{p411}
For every prime $p$ and every $n$ divisible by $p$
\begin{equation}
\label{e411}
K(n,p) \geq \frac{p^n [(n/p)!]^p}{n!}.
\end{equation}
Therefore, for every fixed $p$ and large $n$
\begin{equation}
\label{e412}
K(n,p) \geq (1+o(1)) \frac{(2 \pi)^{(p-1)/2}}{p^{p/2}} \cdot
n^{(p-1)/2}.
\end{equation}
\end{prop}
The proof of Heged\H{u}s is based on Gr\"obner basis methods. In
particular, he established the following result.
\begin{theo}[\cite{Heg}]
\label{t412}
Let $p$ be a prime and let $P(x)=P(x_1,x_2, \ldots ,x_{4p})$ be a
polynomial over $Z_p$ which vanishes over all $\{0,1\}$ vectors of
Hamming weight $2p$ and suppose that there is a $\{0,1\}$-vector
$z$ of Hamming weight $3p$ so that $P(z) \neq 0$ (in $Z_p$).
Then the degree of $P$ is at least $p$.
\end{theo}
An elementary proof of this lemma, due to S. Srinivasan, is
given in \cite{AKV}. Here we describe a variant of this proof
providing a very short derivation
of this lemma from the Combinatorial Nullstellensatz proved in
\cite{Al00}, which is the following.
\begin{theo}
\label{t413}
Let $F$ be an arbitrary field, and let $f=f(x_1, \ldots ,x_n)$
be a polynomial in $F[x_1, \ldots ,x_n]$. Suppose the degree
$deg(f)$ of $f$ is $\sum_{i=1}^n t_i$, where each $t_i$ is a
nonnegative integer, and suppose the coefficient of
$\prod_{i=1}^n x_i^{t_i}$ in $f$ is nonzero. If
$S_1, \ldots ,S_n$ are subsets of $F$ with $|S_i|>t_i$,
then there are $s_1 \in S_1, s_2 \in S_2, \ldots, s_n \in S_n$
so that
$$
f(s_1, \ldots ,s_n) \neq 0.
$$
\end{theo}
\subsection{Proofs}
\noindent
{\bf Proof of Proposition \ref{p411}:}\,
Let $M$ be the collection of all vectors in $B$ in which each
$w_i$ appears in
exactly $n/p$ coordinates and let
$$
m=|M|=\frac{n!}{[(n/p)!]^p}
$$
be its cardinality. We claim that $M$ is the set of all vectors in
$B$ that are orthogonal to the vector ${\bf j}=(1,1, \ldots ,1) \in B$.
Indeed, it is a well known consequence of Eisenstein's
criterion that the minimal polynomial of $w_1$ over the rationals
is the polynomial $1+x+x^2 + \cdots +x^{p-1}$. Therefore, if
$\sum_{i=0}^{p-1} \alpha_i w_i=0$ for some integers $\alpha_i$,
then the polynomial $1+x+x^2 + \cdots +x^{p-1}$ divides
$\sum_{i=0}^{p-1} \alpha_i x^i$, implying that all the coefficients
$\alpha_i$ are equal. This implies the assertion of the claim.
By the claim, the number of vectors in $B$ orthogonal to
${\bf j}$ is exactly $m$, and this is clearly also the
number of vectors in $B$ orthogonal to any other fixed member
of $B$. It follows that if each vector in $B$ is orthogonal to at
least one vector in a subset of cardinality $k=K(n,p)$ of $B$,
then $k \geq p^n/m$, implying (\ref{e411}). The estimate in
(\ref{e412}) follows from (\ref{e411}) by Stirling's Formula.
$\Box$
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t412}:}\,
Without loss of generality assume that $z$ is the vector
starting with $3p$ $1$s followed by $p$ $0$s. Suppose, for
contradiction, that the degree of $P$ is at most $p-1$ and consider
the polynomial
$ f(x_1,x_2 \ldots ,x_{4p}) =f_1-f_2$
where
$$
f_1= P(x)[1-(\sum_{i=1}^{4p} x_i)^{p-1}]
x_1 x_2 \cdots x_{p+1}(1-x_{3p+1})(1-x_{3p+2}) \cdots (1-x_{4p})
$$
and
$$
f_2=P(z) x_1x_2 \cdots x_{3p}(1-x_{3p+1})(1-x_{3p+2}) \cdots
(1-x_{4p}).
$$
The degree of the polynomial $f_1$ is at most $4p-1$, that of
$f_2$ is exactly $4p$, hence the degree of $f$ is $4p$ and the
coefficient of $\prod_{i=1}^{4p} x_i$ in it is $P(z) \neq 0$.
By the
Combinatorial Nullstellensatz (Theorem \ref{t413})
with $F=Z_p$, $n=4p$, $t_i=1$ for all $i$
and $S_i= \{0,1\}$ for all $i$ there is a vector $y=(y_1,y_2, \ldots
,y_{4p}) \in \{0,1\}^{4p}$ so that $f(y_1,y_2, \ldots y_{4p}) \neq
0$. However, the only vector with $\{0,1\}$ coordinates in which
$f_2$ is nonzero is $z$, and as $f_1(z)=f_2(z)=P(z)$, $f(z)=0$.
Thus $y \neq z$ and $f(y)=f_1(y)$. If the Hamming weight of
$y$ is not divisible by $p$ then the term
$[1-(\sum_{i=1}^{4p} y_i)^{p-1}]$ vanishes. If the Hamming weight
of $y$ is $2p$ then the term $P(y)$ vanishes. If it is $0$ or $p$, then the
term $y_1y_2 \cdots y_{p+1}=0$ and if it is $4p$ or $3p$ (and $y \neq z$)
then the term $(1-y_{3p+1})(1-y_{3p+2}) \ldots (1-y_{4p})=0$.
Therefore $f(y)=f_1(y)=0$, contradiction. This completes the proof.
$\Box$
\section{High School Coalitions}
In May 2019 Shay Moran showed me a question posted by a woman
named Ruthi Shaham in a Facebook Group focusing on Mathematics.
She wrote that her son has finished elementary school and was about
to move
to high school. When doing so, each child lists three friends,
and the assignment of children
into classes ensures that each child will have at least one
of these three friends in his class. Ruthi further wrote that
her son heard from five of his schoolmates that they found that
they can make their selections in a way that will ensure that all five will
be scheduled to the same class. She tried to check with a paper and pencil
and couldn't decide whether or not this is possible, but she
suspected it is
impossible.
She thus asked if this is indeed the case, and if so, whether a
larger group
of children can form such a coalition ensuring they will all
necessarily be assigned
to the same class.
In this brief section we show that Ruthi has indeed been right, no
coalition of five children can ensure they will share the same class.
Moreover, no coalition of any size can ensure to share the same class.
This is related to known problems and results
in Graph Theory, as are several variants of the problem
mentioned below.
Here is a more formal formulation of the problem, with general parameters.
Let $N=\{1,2, \ldots ,n\}$ be a finite set of size $n$,
let $k$ and $r$ be integers,
and suppose $n \geq k+1$. For any collection of subsets $S_i$ of $N$,
$(1 \leq i \leq n)$,
with $i \not \in S_i$, and $|S_i|=k$ for all $i$, let
$P(S_1, S_2, \ldots ,S_n)$
be a partition of $N$, so that :
\begin{equation}
\label{e511}
\mbox{For any part~~} N_i \mbox{~~of the partition
and for any~~} j \in N, \mbox{~~if~~} j \in N_i \mbox{~~then~~}
S_j \cap N_i \neq \emptyset.
\end{equation}
Here $N$ denotes the group of children, $S_i$ is the list of friends listed
by child number $i$,
and the partition of $N$ into parts $N_i$ is the partition
of the set of children into classes. The function $P$ represents the
way the children are partitioned into classes $N_i$ given their
choices $S_i$, and the condition (\ref{e511}) is the one ensuring that each
child will have at least one other child from his list in his class.
We say that a subset $R \subset N$
is a successful coalition, if there are choices $S_i, i \in R$ of sets
$S_i$ satisfying $|S_i|=k$ and $i \not \in S_i$ so that for any sets
$S_j \subset N$ with $|S_j|=k$ for all $j \in N-R$, and for any function
$P$ satisfying the conditions above, all elements of $R$ belong to
the same part of the partition $f(S_1, S_2, \ldots ,S_n)$. Note that
by symmetry if a successful coalition of size $r$ is possible
then any set of size $r$ can form such a coalition, and hence we may
always assume that $R=\{1,2, \ldots ,r\}$.
The question of Ruthi is whether or not for $k=3$ there can be a successful
coalition $R$ of size $|R|=5$.
\begin{theo}
\label{t511}
\noindent
\begin{enumerate}
\item
For $k \leq 2$ and every integer $r>1$, every set $R$ of size $r$ can form
a successful coalition.
\item
For any $k \geq 3$ and every $r > 1$ no set of size $r$ can form
a successful
coalition.
\end{enumerate}
\end{theo}
\subsection{Proofs}
Before presenting the general proof, here is a short argument showing
that for $k=3$ no successful coalition of size $5$ is possible. This
proof is a simple application of the probabilistic method.
\vspace{0.2cm}
\noindent
{\bf Claim:}\, Suppose $n \geq 5$, $N=\{1,2, \ldots ,n\}$,
$R=\{1,2, \ldots ,5\}$, and let $S_1, \ldots ,S_5$ be subsets of
$N$, each of size $3$, so that $i \not \in S_i$ for all
$1 \leq i \leq 5$. Then there are subsets $S_j \subset N$,
for $5 \leq j \leq n$ and there is a partition
$P(S_1, \ldots ,S_n)$ of $N$ into two disjoint parts $N_1,N_2$
satisfying (\ref{e511}) such that $R$ intersects both $N_1$ and $N_2$.
\vspace{0.2cm}
\noindent
{\bf Proof:}\, Color the elements of $N$ randomly red and blue, where each
$i \in N$ randomly and independently is red with probability $1/2$ and
blue with probability $1/2$. The probability that all members of $R$
have the same color is $1/16$. For each fixed $i \leq 5$, the probability
that the color of $i$ is different than that of all elements in $S_i$
is $1/8$. Therefore, with probability at least $1-1/16-5/8>0$ none of these
events happens. Hence there is a coloring in which $R$ contains both
red and blue elements, and every $i \in R$ has at least one member
of $S_i$ with the same color as $i$. Fix such a coloring.
Without loss of generality
$1$ is colored red and $2$ is colored blue.
Let $N_1$ be the set of all
elements colored red and let $N_2$ be the set of all elements colored
blue. For each $j \in N_1-R$ let $S_j$ contain $1$ and for each
$j \in N_2-R$ let $S_j$ contain $2$. It is easy to see that the partition
$N=N_1 \cup N_2$ satisfies (\ref{e511}) but $R$ intersects both
$N_1$ and $N_2$, completing the proof. \hfill $\Box$
\vspace{0.2cm}
\noindent
Note that the above proof does not work for $r \geq 8$, thus the proof
of Theorem \ref{t511} requires a different method, which we show next.
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t511}:}\,
The case $k \leq 2$ is very simple. For
$k=1$ simply define $S_i=\{(i+1)(\bmod~~r)\}$ to see that
the coalition $R=\{1,2,\ldots ,r\}$ is successful.
For $k=2$ and $r=2$, $S_1=\{2,3\}$, $S_2=\{1,3\}$ show that
$\{1,2\}$ is successful. For
any larger $r$ add to the above
$S_i=\{1,2\}$ for all $3 \leq i \leq r$.
\vspace{0.2cm}
\noindent
The more interesting part is the proof that for $k\geq 3$ no coalition
of any size $r>1$ can be successful.
The case $r<k$ here is simple.
One possible proof is to repeat the probabilistic
argument described above for the case $k=3,r=5$.
Since for $1<r <k$, $k \geq 3$,
$$
\frac{1}{2^{r-1}} + \frac{r}{2^k} \leq \frac{1}{2}+\frac{k-1}{2^{k}}
\leq \frac{1}{2}+\frac{2}{8} <1
$$
the result follows as before.
(It is also possible to give a direct simple proof
for this case).
For $k \geq 3$, $r \geq k$ consider the digraph whose
set of vertices is $N$, where for each vertex $i$ and each
$j \in S_i$, $ij$ is a directed edge. Thus every outdegree in this digraph
is exactly $k$.
Given the sets
$S_1, \ldots ,S_r$
of outneighbors of the vertices in $R=\{1,2, \ldots ,r\}$
(representing the children attempting
to form a successful coalition), define the sets $S_j$ for $j >r$
in such a way that the induced subgraph on $N-R$ is acyclic.
(For example, we can define $S_j=\{1,2,\ldots ,k\}$ for each
$j>r$, or $S_j=\{j-1, j-2, \ldots, j-k\}$ for each $j > r$. Note that
here we used the fact that $r \geq k$).
The crucial result we use here is a theorem of Thomassen
(\cite{Th}, see also \cite{Al50}
for an extension). This Theorem asserts that any digraph with minimum
outdegree at least $3$ contains two vertex disjoint cycles. Let
$A$ and $B$ be the sets of vertices of these two cycles.
Note that both $A$ and $B$ must contain
a vertex of $R$ (as $N-R$ contains no directed cycles).
Let $A', B'$ be two sets of
vertices satisfying $A \subset A'$, $B \subset B'$ with
$|A'|+|B'|$ maximum subject to the constraint that
every outdegree in $A'$ is at least $1$ and every outdegree in $B'$ is
at least $1$. We claim that $A' \cup B'$ is the set $N$ of
all vertices.
Indeed, otherwise, every $v$ in $C=N-(A' \cup B')$ has no outneighbors in
$A' \cup B'$ (otherwise we could have added it to
either $A'$ or $B'$ contradicting maximality),
so has at least $k \geq 3>1$
outneighbors in $C$ and then we can replace $A'$ by
$A' \cup C$ contradicting
maximality. This proves the claim. The assignment to two groups is now
$N_1=A'$ and $N_2=B'$. Since both $A \subset A'$ and $B \subset B'$ contain
elements of $R$, this shows that $R$ is not a successful coalition,
completing the proof. \hfill $\Box$
\subsection{Variants}
\begin{enumerate}
\item
What if every child is ensured to have at least
two of his choices with him in his class ? In this case, even if
$k$ is arbitrarily large (but $r$ is much larger) we do not know
to prove that a
coalition of $r$ cannot ensure they are all in the same group.
This is identical to one of the open questions in \cite{Al51},
which is the following.
\vspace{0.2cm}
\noindent
{\bf Question:}\, Is there a finite
positive integer $k$ such that every digraph in
which all oudegrees are (at least) $k$ contains
two vertex disjoint subgraphs,
each having minimum outdegree at least $2$ ?
\vspace{0.2cm}
\noindent
On the other hand it is easy to see that this is impossible if
$\frac{1}{2^{r-1}}+ \frac{r (1+k)}{2^{k}}<1$. Indeed, if so
we can split the group of
children randomly into two sets,
red and blue. With positive probability the
specific set of $r$ children trying to form a coalition is not
monochromatic, and also for any child in the coalition there are at
least two of his choices in his group. We can now fix the choices of
all others outside the coalition to ensure they will also be happy with this
partition. It follows that if in this version of the problem
a successful coalition
of size $r$ is possible, then $r$ has to be at least exponential in
$k$.
\item
Suppose we change the rules, and each child lists
$k$ other children that he does {\em not} like, and wishes
not to have many of them
in his class. It can then be shown that for any $k$ there is an example
of choices of the children in which each one lists $k$ others he
prefers to avoid, so that in any partition of the group of children
into $2$ classes, there will always be at least one poor
child sharing the same class with all the $k$ he listed ! This is based
on another result of Thomassen \cite{Th1}: for every $k$ there is a
digraph with minimum outdegree $k$ which contains no even directed cycle.
If $D=(N,E)$
is such a digraph, and $N=V_1 \cup V_2$ is a partition of its
vertex set into two disjoint parts, then, as observed in
\cite{Al51}, there is a vertex in one
of the classes having all its out-neighbors in the same class.
Indeed, otherwise, starting at an arbitrary vertex $v_1$ we can define
an infinite sequence $v_1,v_2, v_3, \ldots$, where each pair
$(v_i,v_{i+1})$ is a directed edge with one end in $V_1$ and one
in $V_2$. As the graph is finite, there is a smallest $j$ such
that there is $i<j$ with $v_i=v_j$, and the cycle
$v_i,v_{i+1},\ldots,v_j=v_i$ is even, contradiction.
On the other hand, by splitting the group of children into
$s \geq 3$ disjoint groups, we can always ensure that each child
will have in his own class at most $2k/s$ of the $k$ children he
wants to avoid. This follows from a result of Keith Ball described
in \cite{Al51}.
\end{enumerate}
\section{$\ell_1$-balls and projections of linear codes}
A remarkable known property of the Binomial distribution $Bin(n,p)$ is
that its median is always either the floor or the ceiling of its
expectation $np$. In particular, if the expectation is an integer
then this is also the median. The following more general result
is proved by
Jogdeo and Samuels in \cite{JS}.
\begin{theo}[\cite{JS}, Theorem 3.2 and Corollary 3.1]
\label{t611}
Let $X=X_1+X_2+ \ldots +X_n$ be a sum of independent indicator random
variables where for each $i$, $Pr(X_i=1)=p_i$ and
$Pr(X_i=0)=1-p_i$. Then the median of $X$ is always the floor or
the ceiling of its expectation $\sum_{i=1}^n p_i$.
\end{theo}
This theorem can be used to derive several interesting results.
Here we describe one quick application and another more complicated
one in which it is convenient (though not absolutely necessary) to
use it, combined with several additional ingredients.
\subsection{ $\ell_1$-balls and Hamming balls in the discrete cube}
If $n$ is even, $d=n/2$ and $x=(1/2,1/2, \ldots ,1/2)$ is the
center of the $n$-dimensional real unit cube $[0,1]^n$, then the
$\ell_1$-ball of radius $d$ centered at $x$ contains all the
$2^n$
points of the discrete cube $\{0,1\}^n$. On the other hand, any
Hamming ball of radius $d$ centered at a vertex $y$ of this
discrete cube
contains only $\sum_{i=0}^d {n \choose i} =(\frac{1}{2}+o(1))
2^n$ points of the cube, where the $o(1)$-term tends to $0$ as $n$
tends to infinity. Madhu Sudan \cite{Su} asked me whether a similar
bound holds for any $\ell_1$-ball of integral radius.
The precise statement of the question is as follows:
\vspace{0.1cm}
\noindent
Is it true that for any positive integer $d$ and for any $\ell_1$-ball
$B$
(centered at any real point in $R^n$) there is a Hamming ball
of the same radius $d$ centered at a point in $\{0,1\}^n$ that
contains at least half the points in $B \cap \{0,1\}^n$ ?
The following stronger result shows that this is indeed the case.
\begin{theo}
\label{t621}
For any real $x=(x_1,x_2, \ldots ,x_n)$ in $R^n$ and
for any subset $A$ of points of
$B(x,d) \cap \{0,1\}^n$, where $B(x,d)$ is the $\ell_1$-ball of
radius $d$
centered
at $x$, and $d$ is an integer, there is $y \in \{0,1\}^n$ so that
$|A \cap B(y,d)| \geq |A|/2.$
\end{theo}
\begin{proof}
Note, first, that we may assume that $x_i \in [0,1]$ for all $i$.
Indeed, otherwise, replace $x_i$ by $1$ if $x_i>1$ and by $0$ if
$x_i<0$. This modification only decreases the $\ell_1$-distance
between $x$ and any point in $\{0,1\}^n$. Therefore $A$ is a subset of
the ball $B(x,d)$ for the modified vector $x$ too. We thus may and will
assume that $x \in [0,1]^n$.
Let $y=(y_1,y_2, \ldots ,y_n)$ be a random binary vector
obtained by choosing, for each i, randomly and
independently, $y_i$ to be $1$ with probability $x_i$ and
$0$ with probability
$(1-x_i)$. For each point $a \in A$, the $\ell_1$-distance
between $y$ and $a$ is a
random variable which is a sum of independent Bernoulli random
variables and its
expectation
is exactly the $\ell_1$ distance between $a$ and $x$, which is at most
$d$.
By Theorem \ref{t611} of Jogedo and Samuels stated above
the probability that this random
variable is at most $d$ is at least a half. It follows by linearity of
expectation that the expected number of
points of $A$ within distance at most $d$ from $y$ is at least
$|A|/2$, and
thus there is a $y$ as needed.
\end{proof}
\subsection{Random projections of linear codes}
Let $F$ be a finite or infinite field, and let $V$ be a linear code
of length $n$, dimension $k$ and minimum relative distance
at least $\delta$
over $F$. Thus $V$ is a subspace of dimension $k$ of $F^n$, and the
number of nonzero coordinates of any nonzero codeword $v \in V$
is at least $\delta n$. Let $m$ be an integer. A
projection of $V$ on $m$ random coordinates is obtained by selecting a
random (multi)set $I$ of $m$ coordinates of $[n]$, chosen with
repetitions. With this random choice of $I$ let $V_m \subset F^m$
be the vector space over $F$ consisting of all
vectors $\{ (v_i)_{i \in I}~: ~ v=(v_1,v_2, \ldots ,v_n) \in V\}$.
One may expect that if $m$ is large, then typically the
vector space $V_m$, considered as a linear code of length $m$ over
$F$, will have dimension $k$ and minimum distance not much smaller
than $\delta m$. This is easy to prove by a standard application of
Chernoff's Inequality and the union bound, provided
$m$ is sufficiently large as a function of $|F|,k$ and
$\delta$. It is, however, not clear at all
that this is the case for $m$ of size
independent of the size of the field $F$ (which may even be
infinite). Such a statement is proved by Saraf and Yekhanin in
\cite{SY}.
\begin{theo}[\cite{SY}, Theorem 3]
\label{t631}
Let $V$ be a linear code of dimension $k$, length $n$ and minimum
relative distance $\delta$ over an arbitrary field $F$.
If $m$ is at least $c(\delta) k$ and $V_m$ is a projection of
$V$ on $m$ random coordinates
then with probability at least
$1-e^{-\Omega(\delta m)}$ the dimension of $V_m$ is $k$ and its
minimum distance is at least $\delta m/8$.
\end{theo}
One can check that the estimate the proof in \cite{SY} provides for
$c(\delta)$ is $b \frac{\log (1/\delta)}{\delta}$ for a
sufficiently large absolute constant $b$. Note, however, that the
minimum relative distance obtained is only $\delta /8$, this loss in the
minimum relative distance is inherent in the approach of \cite{SY}.
Here we show how to apply some of the techniques in the study of
${\varepsilon}$-nets and ${\varepsilon}$-approximations in range spaces with
finite Vapnik-Chervonenkis dimension to get an improved version of
the above theorem in which the relative minimum distance obtained
can be arbitrarily close to $\delta$.
\begin{theo}
\label{t632}
There exists an absolute positive constant $c$ so that the
following holds. Let $B>2$ be an integer, and let
$V$ be a linear code of dimension $k$, length $n$ and minimum
relative distance $\delta$ over an arbitrary field $F$.
If $m$ is at least
$c\frac{B^2 k}{\delta} \log (B/\delta)$
and $V_m$ is a projection of
$V$ on $m$ random coordinates
then with probability at least
$1-e^{-\Omega(\delta m/B^2)}$ the dimension of $V_m$ is $k$ and
its minimum distance
is at least $(\frac{B-1}{B+1})\delta m$.
\end{theo}
Taking $B$ to be a large fixed constant we get that typically the
minimum relative distance of $V_m$ is close to $\delta$, and the
estimates for $m$ and for the failure probability are essentially
as in Theorem \ref{t631}.
We start with a quick reminder of the relevant facts about VC-dimension.
The {\em Vapnik-Chervonenkis dimension} $VC({\cal C})$ of a (finite)
family of binary vectors
${\cal C}$ is the maximum cardinality of a set of coordinates $I$ such
that for every binary vector $(b_i)_{i \in I}$ there is a $C \in
{\cal C}$ so that $C_i=b_i$ for all $i \in I$. (In this case we say
that the set $I$ is {\em shattered} by ${\cal C}$).
Suppose the vectors
in the family are of length $n$. An {\em ${\varepsilon}$-net} for the family is
a subset $I \subset [n]$ such that for every $C \in {\cal C}$ of Hamming
weight at least ${\varepsilon} n$ there is an $i \in I$ so that
$C_i=1$. An ${\varepsilon}$-approximation for the family is
a sub(multi)set $I \in [n]$ so that for every $C \in {\cal C}$
$$
| ~\frac{|\sum_{i=1}^n C_i|}{n}-
\frac{|\sum_{i \in I}^n C_i|}{|I|} ~| < {\varepsilon}.
$$
A basic result proved by Vapnik and Chervonenkis \cite{VC} (with
a logarithmic improvement by Talagrand \cite{Ta}), is that if
$VC({\cal C}) \leq d$ then a random set of $\Theta(\frac{d}{{\varepsilon}^2})$
coordinates is typically an
${\varepsilon}$-approximation.
A similar result, proved by Haussler and
Welzl \cite{HW}, is that for such a ${\cal C}$ a random set of
$\Theta(\frac{d}{{\varepsilon}} \log (1/{\varepsilon}))$ coordinates is typically an
${\varepsilon}$-net.
Another basic combinatorial result is the Sauer-Perles-Shelah
Lemma: if $VC({\cal C}) \leq d$ then the number of distinct projections of
the set of vectors in ${\cal C}$ on any set of $t$ coordinates is at
most $g(d,t)=\sum_{i=0}^d {t \choose i}.$
The relevance of the VC-dimension to projections of linear codes
is the following simple observation.
\begin{claim}
\label{cl633}
Let $F$ be an arbitrary field, and
let $V \subset F^n$ be a linear subspace of dimension $k$ over $F$.
For each vector $v \in C$ let $C=C(v)$ denote the indicator vector
of the support of $v$, that is, $C_i=1$ if $v_i \neq 0$ and
$C_i=0$ is $v_i=0$. Put ${\cal C}=\{C(v): v \in V\}$. Then $VC({\cal C})
= k$.
\end{claim}
\begin{proof}
Since the dimension of $V$ is $k$ it contains a set of $k$ vectors
$v^{(i)}$
such that there is a set $I=\{i_1, i_2, \ldots ,i_k\}$
of $k$ coordinates so that $v^{(i)}_{i_j}$ is $1$ for $i=j$ and
$0$ otherwise. The supports of the set of all linear combinations
with $\{0,1\}$-coefficients
of these vectors shatter the set $I$, implying that
$VC({\cal C}) \geq k$. Conversely, if there is a set of coordinates $J$
shattered by the vectors in ${\cal C}$, then for each $j \in J$ there is
a vector in $V$ with $v_j \neq 0$ and $v_i=0$ for all $i \in J-j$.
These $|J|$ vectors are clearly linearly independent, implying that
$|J| \leq k$ and completing the proof.
\end{proof}
The above claim and the known result stated above about
${\varepsilon}$-approximation for families of vectors with finite
$VC$-dimension suffice to prove a version of Theorem \ref{t632}
with $m=\Theta(\frac{B^2 k}{\delta^2})$. Indeed, we simply consider
a $2\delta/(B+1)$-approximation for the set ${\cal C}$ corresponding
to $V$. Similarly, the result about ${\varepsilon}$-nets shows that
typically the dimension of $V_m$ is $m$.
In order to prove the improved estimate for $m$ stated in
the theorem we show that in the setting
here the bound can be improved to be closer to that in the
theorem about $\delta$-nets. This is proved in the following
result, which applies to general collections of vectors with
a bounded VC-dimension.
\begin{prop}
\label{p634}
There exists an absolute positive constant $c>1$ such that the
following holds.
Let ${\cal C}$ be a family of binary vectors of length $n$, and assume
that $VC({\cal C}) \leq d$. Let $X$ be a random multiset of $m$
coordinates, with $m=c \frac{B^2 d }{{\varepsilon}} \log (B/{\varepsilon})$, where
$B>2$ is an integer. Then
with probability at least $1-e^{-\Omega({\varepsilon} m/B^2)}$, for every
$C \in {\cal C}$ satisfying $\sum_{i=1}^n C_i \geq {\varepsilon} n$
we have $\sum_{i \in X} C_i \geq \frac{B-1}{B+1}{\varepsilon} m.$
\end{prop}
In order to prove the above statement, we need some standard
estimates for large deviations of the hypergeometric distribution.
The estimate we use here was first proved by Hoeffding \cite{Hoe},
see also \cite{JLR}, Theorem 2.10 and Theorem 2.1.
\begin{lemma}[Hoeffding \cite{Hoe}, see also \cite{JLR}]
\label{l635}
Let $H$ be the hypergeometric distribution given by the cardinality
$|R \cap S|$ where $S$ is a random subset of cardinality $m$ in
a set of size $N$ containing a subset $R$ of cardinality
$pN$. Then the probability that $H$ is smaller than
$pm-t$ is at most $e^{-t^2/2pm}$.
\end{lemma}
\vspace{0.1cm}
\noindent
{\bf Proof of Proposition \ref{p634}:}\,
Let $m$ be as in the statement of the proposition and let
$X=(x_1, \ldots ,x_m)$ be a random multiset obtained
by $m$ independent random choices, with repetitions, of elements of $[n]$.
For $C \in {\cal C}$ we let $|C|$ denote $\sum_{i=1}^n C_i$ and let
$|C \cap X|$ denote $|\{i: C_{x_i}=1\}|.$
Let $E_1$ be the
following event:
$$
E_1=\{ \exists C \in {\cal C}: |C| \geq {\varepsilon} n, |C \cap X | <
\frac{B-1}{B+1}{\varepsilon} m \}
$$
To complete the proof we have to show that the probability of $E_1$ is
as small as stated in the proposition.
To do so, we make an additional random choice
and define another event as follows. Independently of the previous
choice, let $T=(y_1, \ldots ,y_{Bm})$ be obtained by $Bm$
independent
random choices of elements of $[n]$.
Let $E_2$ be the event defined by
$$
E_2=\left\{ \exists C \in {\cal C}: |C| \geq {\varepsilon} n,
|C \cap X| < \frac{B-1}{B+1}{\varepsilon} m,
|C \cap T| \geq \lfloor B {\varepsilon} m \rfloor \right\}
$$
\begin{claim}
\label{c636}
$Pr(E_2) \geq \frac{1}{2} Pr{E_1}$.
\end{claim}
\begin{proof}
It suffices to prove that the conditional probability $Pr(E_2 |
E_1)$ is at least $1/2$. Suppose that the event $E_1$ occurs. Then
there is a $C \in {\cal C}$ such that $|C| \geq {\varepsilon} n $ and $|C
\cap X| < \frac{B-1}{B+1} {\varepsilon} m$.
The conditional probability above is clearly
at
least the probability that for this specific $C$, $|C \cap T| \geq
\lfloor B {\varepsilon} m \rfloor $. However $|C \cap T|$ is a binomial random
variable with
expectation at least $ B {\varepsilon} m$, and therefore, by Theorem
\ref{t611} its median is at least the floor of that, implying the
desired result.
\end{proof}
\begin{claim}
\label{c637}
$$
Pr(E_2) \leq g(d,(B+1)m) 2^{-\epsilon m/8(B+1)^2 }
$$
\end{claim}
\begin{proof}
The random choice of $X$ and $T$ can be described in the following
way, which is equivalent to the previous one. First choose $X
\cup T = ( z_1, \ldots ,z_{(B+1)m})$ by making $(B+1)m$ random independent
choices of elements of $[n]$ (with repetitions),
and then choose randomly precisely
$m$ of the elements $z_i$ to be the set $X$, where the remaining elements
$z_j$ form the set $T$. For each member $C \in {\cal C}$
satisfying $|C | \geq {\varepsilon} n $, let $E_C$ be the event that
$$
| C \cap T | \geq \lfloor B {\varepsilon} m \rfloor~~ \mbox{and}~~ |C \cap X| <
\frac{B-1}{B+1}{\varepsilon} m.
$$
A crucial fact is that if $C,C' \in {\cal C}$ are two ranges, $|C |
\geq {\varepsilon} n$
and $|C'| \geq {\varepsilon} n$ and if $C \cap (X \cup T)=
C' \cap (X \cup T )$, then the two events $E_C$ and $E_{C'}$,
when both are conditioned on the choice of $X \cup T$, are
identical. This is because the occurrence of $E_C$ depends only on
the intersection $C \cap (X \cup T) $. Therefore, for any fixed
choice of $X \cup T$, the number of distinct events $E_C$ does not
exceed the number of different sets in the projection
of ${\cal C}$ on the coordinates ${X \cup T}$.
Since the VC-dimension is at most $d$,
this number does not exceed $g(d,(B+1)m)$, by the
Sauer-Perles-Shelah Lemma.
Let us now estimate the probability of a fixed event of the form
$E_C$, given the choice of $X \cup T$.
This probability is at most the probability that a hypergeometric
random variable counting the size of the intersection of
a random set of $m$ elements with a subset $R$ of size at least
$\lfloor B {\varepsilon} m \rfloor$ in a set of size $N=(B+1)m$ is
smaller than $\frac{B-1}{B+1} \epsilon m$. By Lemma \ref{l635},
and using the fact that the choice of $m$ implies that
$\lfloor B {\varepsilon} m \rfloor> (B-1/2) {\varepsilon} m$ this probability is
smaller than $e^{-{\varepsilon} m/8(B+1)^2}$.
\end{proof}
By Claims~\ref{c636} and~\ref{c637}, $Pr(E_1) \leq 2g(d,(B+1)m)
2^{-{\varepsilon} m/8(B+1)^2}$. The assertion of the theorem follows
using the fact that
$$
g(d,(B+1)m)< (\frac{2e(B+1)m}{d})^d.
$$
\hfill $\Box$
\vspace{0.1cm}
\noindent
{\bf Proof of Theorem \ref{t632}:}\,
Let $V$ be a linear code of length $n$, dimension $k$ and minimum
relative distance $\delta$. Let ${\cal C}$ be the set of all indicator
vectors of supports of vectors in $V$. By Claim \ref{cl633} the
VC-dimension of ${\cal C}$ is at most $k$, and by definition the Hamming
weight of each member $C$ of ${\cal C}$ is at least $\delta n$. The
desired result thus follows from Proposition \ref{p634}.
\hfill $\Box$
\section{Connected dominating sets}
The first result in this section was obtained in joint discussions
with Michael Krivelevich \cite{Kr}.
Let $G=(V,E)$ be a connected graph. Let $\gamma(G)$ denote the
minimum size of a dominating set in it, that is, the minimum
cardinality of a set of vertices $X \subset V$ so that each $v \in
V-X$ has at least one neighbor in $X$. Let
$\gamma_c(G)$
denote the minimum size of a connected dominating set of $G$, that
is, the minimum cardinality of a dominating set of vertices $X$ so
that the induced subgraph of $G$ on $X$ is connected.
One of the reasons this parameter has been studied extensively
is the fact that $|V|-\gamma_c(G)$ is exactly the maximum possible
number of leaves in a spanning tree of $G$.
It is well known that if the minimum degree in $G$ is $k$ and its
number of vertices is $n$, then
$\gamma(G) \leq \frac{n (\ln (k+1)+1)}{k+1}$. See \cite{Lo} or
\cite{AS}, Theorem 1.2.2 for a proof. As mentioned in \cite{AS}
this is asymptotically tight for large $k$, see, e.g., \cite{AW}
for a proof that for any ${\varepsilon}>0$ and $k>k_0({\varepsilon})$ a random
$k$-regular graph on $n$ vertices is unlikely to contain a
dominating set of size at most $(1-{\varepsilon}) \frac{n \ln k}{k}$.
Caro, West and Yuster \cite{CWY} proved that for every
connected graph $G$ with $n$ vertices and
minimum degree $k$, $\gamma_c(G)$ is also
not much larger than $\frac{n \ln (k+1)}{k+1}$. The precise
statement of their result is as follows.
\begin{theo}[\cite{CWY}]
\label{t711}
Let $G$ be a connected graph with $n$ vertices and minimum degree
at least $k$. Then
$$
\gamma_c(G) \leq \frac{n(\ln(k+1)+0.5\sqrt{ \ln (k+1)} +145)}{k+1}
$$
\end{theo}
Here we first prove a similar result with a slightly better estimate.
\begin{theo}
\label{t712}
Let $G$ be a connected graph with $n$ vertices and minimum degree
at least $k$. Then
$$
\gamma_c(G) \leq \frac{n(\ln(k+1)+\ln \lceil \ln (k+1) \rceil+4)}{k+1}.
$$
\end{theo}
The main merit here is not the improved estimate, but
the proof, which is much simpler than the one in \cite{CWY}. Like
the proof in \cite{CWY}, it
provides a simple efficient algorithm for finding a connected
dominating set of the required size for a given input graph.
As a byproduct of the proof we get an upper bound for
the difference between $\gamma(G)$ and $\gamma_c(G)$, as stated in
the following theorem.
Define a function $f=f_{n,k}$ mapping $[1,\infty)$ to
$[0,\infty)$
as follows.
For any real $x \geq 1$, let
$x=(y+z) \frac{n}{k+1}$ with $y \geq 0$ an integer
and $z \in [0,1]$ a real:
\begin{enumerate}
\item
If $y=0$ then $ f(x)=\frac{n}{k+1}2z-2.$
\item
If $ y =1$ then
$ f(x)= \frac{n}{k+1}(\frac{z}{y}+ 2)-2.$
\item
If $y \geq 2$ then $ f(x)= \frac{n}{k+1}
( \frac{z}{y}+\frac{1}{y-1}+ \cdots + \frac{1}{1}+2)-2. $
\end{enumerate}
The function $f$ is piecewise linear and monotone increasing. Its
derivative, which exists in all points of $(1,\infty)$ besides the
integral multiples of $\frac{n}{k+1}$, is (weakly) decreasing,
thus $f$ is concave. In addition it satisfies the following.
For every $x = (w+z) \frac{n}{k+1} > \frac{n}{k+1}$ with
$w \geq 1$ an integer and $z \in [0,1]$ a real, and for every $w'$
satisfying $w \leq w' \leq x-1$
\begin{equation}
\label{e7new}
f(x) \geq f(x-w')+1
\end{equation}
Indeed, the derivative of $f(z)$ is at least $\frac{1}{w}$ for every
$z$ in $(x-w',x]$ (besides the integral multiples of
$\frac{n}{k+1}$), and thus $f(x)-f(x-w')$, which is the integral of
this derivative from $x-w'$ to $x$, is at least $w' \cdot
\frac{1}{w} \geq 1$.
\begin{theo}
\label{t713}
Let $G$ be a connected graph with $n$ vertices, minimum degree
at least $k$ and domination number $\gamma=\gamma(G)$.
Then $ \gamma_c(G) \leq \gamma + f_{n,k}(\gamma). $
Therefore
$$
\gamma_c(G) < \gamma+ \frac{n}{k+1} (\ln \lceil \ln (k+1) \rceil +3).
$$
\end{theo}
We also describe an improved argument that provides a better estimate
than the ones in Theorems \ref{t711}, \ref{t712}.
\begin{theo}
\label{t714}
Let $G$ be a connected graph with $n$ vertices and minimum degree
at least $k$. Then
$$
\gamma_c(G) \leq \frac{n}{k+1}(\ln (k+1)+4)-2.
$$
\end{theo}
The proof here too provides an efficient randomized algorithm
for finding a connected dominating set with expected size as in
the theorem. This algorithm can be derandomized and converted into
an efficient deterministic algorithm.
\subsection{Proofs}
In the proofs we use the following simple lemma.
\begin{lemma}
\label{l715}
Let $G=(V,E)$ be a connected graph with $n$ vertices and minimum degree at
least $k$. Let $S \subset V$ be a dominating set of $G$, let $H$
be the induced subgraph of $G$ on $S$, and suppose the number of
its connected components is $x=(y+z)\frac{n}{k+1}$ where
$ y$ is a nonnegative integer and $0 \leq z \leq 1$ is a real.
Then $\gamma_c(G) \leq |S|+f(x)$, where $f=f_{n,k}$ is the function
defined in the previous subsection.
\end{lemma}
\vspace{0.2cm}
\noindent
{\bf Proof:}\,
Starting with the dominating set $S$ we prove, by
induction on $x$, that it is always possible to add to it at most
$f(x)$ additional vertices to get a connected dominating set.
For $x=1$ the given set is already connected, and as $f(1)=0$ the
result in this case is trivial.
If $1<x \leq \frac{n}{k+1}$
we note that as long as there are at least two components, each one
$C$ can be merged to another one by adding at most two vertices. Indeed,
every vertex in the second neighborhood of $C$ is dominated, hence
adding the two vertices of a path from $C$ to any such vertex
merges $C$ to another component. This means that by adding at most
$2 (x-1)=f(x)$ vertices to $S$ we get a connected
dominating set, as needed.
If $x> \frac{n}{k+1}$ pick arbitrarily one vertex $v=v(C)$ in each
of the $x$ connected components of $H$ and let $N(v)$ denote
its closed neighborhood consisting of $v$ and all its neighbors in
$G$. This set is of size at least $k+1$. Therefore there is a
vertex $u$ of $G$ that belongs to at least $\lceil (k+1)x/n \rceil$
of these closed neighborhoods. (This can in fact be slightly improved
as none of the vertices of the dominating set belongs to more than one
such closed neighborhood, but we do not use this improvement
here).
Define $S'=S \cup \{u\}$ and note
that adding $u$ merges at least $\lceil (k+1)x/n \rceil$ components.
Therefore, if $x > w \frac{n}{k+1}$ for an integer $w \geq 1$,
then the number of connected components of the induced subgraph of
$G$ on the dominating set $S'$ is $x-w'$ for some $w' \geq w$.
By induction one can add to $S'$ at most $f(x-w')$ additional
vertices to get a connected dominating set, and the desired result
follows from (\ref{e7new}).
\hfill $\Box$
The proof clearly supplies an efficient deterministic
algorithm for finding a connected dominating set of the required
size, given the initial dominating set $S$.
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t713}:}\,
This is an immediate consequence of Lemma \ref{l715} together with
the obvious fact that if $\gamma(G)=\gamma$ then $G$ contains a
dominating set $S$ of size $\gamma$ with at most $|S|=\gamma$
connected components. The
known fact that $\gamma\leq \frac{n}{k+1}(
\ln (k+1)+1)$ implies that $\gamma \leq \frac{n}{k+1} (y+z)$
with $y=\lceil \ln(k+1) \rceil$ and $z=1$. The definition of the
function $f=f_{n,k}$ thus implies that
$$
f_{n,k}(\gamma) \leq \frac{n}{k+1}(\frac{1}{y}+\frac{1}{y-1}+
\ldots + \frac{1}{1}+2) -2 < \frac{n}{k+1}(\ln y+3),
$$
completing the proof.
\hfill $\Box$
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t712}:}\,
This follows from Theorem \ref{t713} together with the fact that
$\gamma(G) \leq \frac{n}{k+1} (\ln (k+1)+1)$.
\hfill $\Box$
In order to prove Theorem \ref{t714} we need two simple lemmas.
The first one is a known fact, c.f., e.g., \cite{CS}, Formula (3.2).
for completeness we include a short proof.
\begin{lemma}
\label{l716}
For a positive integer $k$ and a real $p \in (0,1)$, let
$B(k,p)$ denote the Binomial random variable with parameters
$k$ and $p$. Then the expectation of $\frac{1}{B(k,p)+1}$ satisfies
$$
E[\frac{1}{B(k,p)+1}]=\frac{1}{(k+1)p}-\frac{(1-p)^{k+1}}{(k+1)p}.
$$
\end{lemma}
\vspace{0.2cm}
\noindent
{\bf Proof:}\, By definition
$$
E[\frac{1}{B(k,p)+1}]
=\sum_{i=0}^k \frac{1}{i+1} {k \choose i} p^i (1-p)^{k-i}
=(1-p)^k \sum_{i=0}^k \frac{1}{i+1}{k \choose i} (\frac{p}{1-p})^i.
$$
By the Binomial formula $(1+x)^k = \sum_{i=0}^k {k \choose i} x^i$.
Integrating we get
$$
\frac{(1+x)^{k+1} -1}{k+1}= \sum_{i=0}^k \frac{1}{i+1} {k \choose
i} x^{i+1}.
$$
Dividing by $x$ and plugging $x=\frac{p}{1-p}$ the desired result
follows.
\hfill $\Box$
\begin{lemma}
\label{l717}
Let $H=(V,E)$ be a graph. For every $v \in V$
let $d_H(v)$ denote the degree of $v$ in $H$. Then the number of
connected components of $H$ is at most $D(H)=\sum_{v \in V}
\frac{1}{d_H(v)+1}$.
\end{lemma}
\vspace{0.2cm}
\noindent
{\bf Proof:}\, The contribution to $D(H)$ from the vertices in any
connected component $C$ of $H$ with $m$ vertices is
$$
\sum_{v \in C} \frac{1}{d(v)+1} \geq \sum_{v \in C}
\frac{1}{m} =1.
$$
\hfill $\Box$
\vspace{0.2cm}
\noindent
{\bf Proof of Theorem \ref{t714}:}\,
Recall that the function $f=f_{n,k}$ defined in the previous
subsection is concave.
Therefore, by Jensen's Inequality, for every positive random
variable $X$, $E[f(X)] \leq f(E[X])$.
Let $G=(V,E)$ be a connected graph with $n$ vertices and
minimum degree at least $k$.
By Lemma \ref{l715} if there is a dominating set $S$ of $G$
and the induced subgraph of $G$ on $S$ has
$x$ connected components, then
\begin{equation}
\label{e711}
\gamma_c(G) \leq |S|+f(x).
\end{equation}
For a dominating set $S$, let $H=H(S)$ be the induced subgraph
of $G$ on $S$, and put $D(H)=\sum_{v \in S} \frac{1}{d_H(v)+1}$
where $d_H(v)$ is the degree of $v$ in $H$.
By Lemma \ref{l717} the number of connected components of
$H$ is at most $D(H)$, and since the function $f=f_{n,k}$ defined
above is monotone increasing this implies, by (\ref{e711}), that
\begin{equation}
\label{e712}
\gamma_c(G) \leq |S| +f(D(H)) =|S|+f (\sum_{v \in S}
\frac{1}{d_H(v)+1}).
\end{equation}
We next describe a random procedure for generating a dominating set
$S$ and complete the proof by upper bounding the expectation of the
right-hand-side of (\ref{e712}). The procedure is the standard one
described in \cite{AS}, Theorem 1.1.2 for generating a dominating
set. Define $p=\frac{\ln(k+1)}{k+1}$ and let $T$ be a random set of
vertices of $G$ obtained by picking, randomly and independently,
each vertex of $G$ to be a member of $T$ with probability $p$.
Let $Y=Y_T$ be the set of all vertices of $G$ that are not
dominated by $T$, that is, all vertices in $V-T$ that have no
neighbors in $T$. The set $S$ defined by $S=T \cup Y_T$ is clearly
dominating. The expected size of $T$ is $np$. The expected size of
$Y_T$ is at most $n(1-p)^{k+1}$, since for any vertex $v$ the
probability it lies in $Y_T$ is exactly $(1-p)^{d_G(v)+1}
\leq (1-p)^{k+1}$, and the bound for the expectation of
$|Y_T|$ follows by linearity
of expectation. We proceed to bound the expectation of
$f(\sum_{v \in S} \frac{1}{d_H(v)+1}).$ By Jensen's Inequality
and the convexity of $f$ mentioned above this
is at most $f(E[\sum_{v \in S} \frac{1}{d_H(v)+1}]).$
Since $f$ is monotone increasing it suffices to bound the
expectation $E[\sum_{v \in S} \frac{1}{d_H(v)+1}]$.
Fix a vertex $v$. The probability it belongs to
$Y_T$ (and hence has degree $0$ in $H$) is
$(1-p)^{d+1}$, where $d$ is its degree in $G$.
The probability it belongs to $T$
and has degree $i$ in $H$ is
$p {d \choose i} p^i (1-p)^{d-i}$.
Therefore, the expectation of $\frac{1}{d_H(v)+1}$
is, by Lemma \ref{l716},
$$
(1-p)^{d+1} + p(\frac{1}{(d+1)p} -\frac{(1-p)^{d+1}}{(d+1)p})
< (1-p)^{k+1}+\frac{1}{k+1}.
$$
Since $(1-p)^{k+1} \leq e^{-p(k+1)}=\frac{1}{k+1}$
this implies, by linearity
of expectation, that
$$
E[\sum_{v \in S} \frac{1}{d_H(v)+1}] \leq \frac{2n}{k+1}.
$$
Using, again, linearity of expectation and the fact that
$f_{n,k}(\frac{2n}{k+1}) =3 \frac{n}{k+1}-2$
we conclude that the expectation of the right-hand-side of
(\ref{e712}) is at most
$$
np+n(1-p)^{k+1}+ 3\frac{n}{k+1}-2 \leq \frac{n}{k+1} (\ln (k+1)+4)-2.
$$
Therefore there is a dominating set $S$ for which this expression
is at most the above quantity,
completing the proof.
\hfill $\Box$
\subsection{Algorithm}
\noindent
The proof of Theorem \ref{t714} clearly supplies a randomized
algorithms generating a connected dominating set of expected size
at most as in the theorem in any given connected input graph
$G=(V,E)$ with
$n$ vertices and minimum degree at least $k$. This algorithm can
be derandomized using the method of conditional expectations,
yielding a polynomial time deterministic algorithm for finding
such a connected dominating set. Here is the
argument. Let $v_1,v_2, \ldots ,v_n$ be an arbitrary numbering of
the vertices of $G$. The algorithm generates a dominating set
$S$ satisfying
$$
|S| +f(D(H)) =|T|+|Y_T| +f (\sum_{v \in S} \frac{1}{d_H(v)+1} )
\leq \frac{n}{k+1} (\ln (k+1)+4)-2,
$$
where $f=f_{n,k}$ is the function defined in the proof of Theorem
\ref{t714}, $H$ is the induced subgraph of $G$ on $S=T \cup Y_T$ and
$D(H)=\sum_{v \in S} \frac{1}{d_H(v)+1}$.
Once such an $S$ is found it is clear that the proof of
the theorem provides an efficient way to construct a connected
dominating set of the required size using it.
The algorithm produces $S$ as above by
going over the vertices $v_i$ in order, where in
step $i$ the algorithm decides whether or not to add $v_i$ to $S$.
Let $S_i$ denote $S \cap \{v_1,v_2, \ldots ,v_i\}$. Thus
$S_0=\emptyset$. For each $i$, $0 \leq i \leq n$, define
a potential function $\psi_i$ in terms of the conditional expectations
of $|S|=|T|+|Y_T|$ given $S_i$, which is denoted by
$E[|S||S_i]$ and the conditional expectation
of $\sum_{v \in S} \frac{1}{d_H(v)+1}$ given $S_i$, denoted by
$E[\sum_{v \in S} \frac{1}{d_H(v)+1}| S_i]$. In this notation
$$
\psi_i=E[|S||S_i]+f(E[D(H) |S_i]= E[|T| | S_i]+E[|Y_T| | S_i]
+f(E[\sum_{v \in S} \frac{1}{d_H(v)+1}| S_i]).
$$
Given the graph $G$ and the set $S_i$, it is not difficult to
compute $\psi_i$ in polynomial time.
Indeed, by linearity of expectation, the conditional expectation
$E[|T||S_i]$ is computed by adding the contribution of each vertex
$v=v_j$ to it. For $j \leq i$ this contribution is $1$ if
$v_j \in T$ and $0$ if $v_j \not \in T$. For $j>i$ the contribution
is $p$. The contribution of $v_j$ to $E[Y_T|S_i]$
is $0$ if
$v_j$ is already dominated by a vertex in $S_i$, and if it is not,
then it is
$(1-p)^s$, where $s$ is the number of neighbors of $v_j$
(including $v_j$ itself if $j>i$)
in the set $V-\{v_1,v_2, \ldots ,v_i\}$.
The conditional expectation $E[\sum_{v \in S} \frac{1}{d_H(v)+1}|
S_i]$ is also computed using linearity of expectation, where the
contribution of each vertex $v_j$ is
$E[\frac{1}{d_H(v_j+1)}| S_i]$. This is also simple to compute
in all cases. We describe here only one representative example.
If $j>i$, $q$ of the neighbors of $v_j$ appear in $S_i$,
and the number of its neighbors in $G$ which lie in
$V-\{v_1, v_2, \ldots ,v_i\}$ is $s$, then
$$
E[\frac{1}{d_H(v_j+1)}| S_i]=p \cdot \sum_{a=0}^{s} {s \choose a}
p^a (1-p)^{s-a}\frac{1}{q+1+a}.
$$
A similar expression exists in every other possible case.
Put $\psi_i=\psi_i^{(T)}+\psi_i^{(Y)}+
\psi_i^{(f)}$, where
$\psi_i{(T)}=E[|T| | S_i]$,
$\psi_i{(Y)}=E[|Y_T| | S_i]$,
and
$\psi_i^{(f)}=f[E(D(H)| S_i]$.
By the definition of conditional expectation
\begin{equation}
\label{e713}
\psi_{i}^{(T)} =p E[|T|~ |~ S_{i+1}=S_i \cup v_{i+1}]
+(1-p) E[|T|~ |~ S_{i+1}=S_i]
\end{equation}
and
\begin{equation}
\label{e7131}
\psi_{i}^{(Y)} =p E[|Y_T|~ |~ S_{i+1}=S_i \cup v_{i+1}]
+(1-p) E[|Y_T|~ |~ S_{i+1}=S_i]
\end{equation}
Similarly, using the fact that the function $f$ is concave
$$
\psi_i^{(f)}=
f(p E[\sum_{v \in H} \frac{1}{d_H(v_j)+1)}| S_{i+1}=S_i \cup v_{i+1}]
+(1-p) E[\sum_{v \in H} \frac{1}{d_H(v_j)+1)}| S_{i+1}=S_i] )
$$
$$
\geq p f(E[\sum_{v \in H} \frac{1}{d_H(v_j)+1)}| S_{i+1}=S_i \cup
v_{i+1}])
+(1-p) f ( E[\sum_{v \in H} \frac{1}{d_H(v_j)+1})| S_{i+1}=S_i] )
$$
$$
\geq \min \{
f(E[\sum_{v \in H} \frac{1}{d_H(v_j)+1)}| S_{i+1}=S_i \cup
v_{i+1}]),
f ( E[\sum_{v \in H} \frac{1}{d_H(v_j)+1)}| S_{i+1}=S_i] ).
$$
Let $\psi_{i+1}^{+}$ denote the value of $\psi_{i+1}$ with
$S_{i+1}=S_i \cup v_{i+1}$ and $\psi_{i+1}^{-}$
denote the value of $\psi_{i+1}$ with $S_{i+1}=S_i$.
By adding the last inequality and (\ref{e713}),(\ref{e7131})
we conclude that
$$
\psi_i \geq \min \{\psi_{i+1}^+, \psi_{i+1}^{-} \}.
$$
Therefore, if the algorithm decides in each step $i+1$ whether
or not to add $v_{i+1}$ to $S_i$ in order to get $S_{i+1}$
by choosing the option that minimizes the value of
$\psi_{i+1}$, then the potential function
$\psi_i$ is a monotone decreasing function of $i$.
Since $\psi_0$ is at most
$\frac{n}{k+1}( \ln (k+1)+4)-2$ by the proof of Theorem
\ref{t714}, so is $\psi_n$. However,
$\psi_n$ is exactly $|S|+f(D(H))$ for the dominating set $S$
constructed by the algorithm. This completes the description
of the algorithm and its correctness.
\subsection{Problem}
We conclude with the following problem.
\vspace{0.1cm}
\noindent
{\bf Problem:}\, Determine or estimate the maximum possible value
of the difference $\gamma_c(G)-\gamma(G)$, where the maximum is
taken over all connected graphs $G$ with $n$ vertices and minimum
degree at least $k$.
\vspace{0.2cm}
\noindent
By Theorem \ref{t713} this maximum is at most $\frac{n}{k+1}(\ln
\lceil \ln (k+1) \rceil +3)$. It is not difficult to show that it
is at least $\lfloor \frac{n}{k+1} \rfloor-1$.
To see this assume, for simplicity, that $k+1$ divides $n$ and
put $m=\frac{n}{k+1}$.
For each $0 \leq i <m$ let $K_i$ be the graph
obtained from a clique on $k+1$ vertices by deleting a single edge
$x_iy_i$. Let $G$ be the $k$-regular graph obtained from the vertex
disjoint union of the $m$ graphs $K_i$ by adding the edges
$y_ix_{i+1}$ for all $0 \leq i <m$, where $x_m=x_0$. For this cycle
of cliques $G$, $\gamma(G)=m=\frac{n}{k+1}$ as shown by a dominating
set consisting of one vertex in each $K_i-\{x_i,y_i\}$ - this is a
minimum dominating set as $G$ is $k$-regular.
On the other hand the induced subgraph on
any connected dominating set must contain at least $m-1$ of the
edges $y_ix_{i+1}$ and their endpoints, and it is not difficult to
check that it must contain at least one additional vertex. Thus
$\gamma_c(G)=2m-1=2 \frac{n}{k+1}-1$. It will be
interesting to close the $\ln \ln (k+1)$ gap between the upper and
lower bounds and decide whether or not the above maximum is
$\Theta(\frac{n}{k+1})$.
\noindent
{\bf Acknowledgment}
I thank Eli Berger, Michael Krivelevich, Shay Moran,
Shubhangi Saraf, Madhu Sudan and Tibor Szab\'o for helpful
discussions.
|
1,116,691,498,418 | arxiv | \section{Introduction}
While the stellarator is a promising magnetic configuration for the realization of steady-state fusion, the geometry of a stellarator must be carefully designed. This is because collisionless charged particle trajectories are not automatically confined in a stellarator, as they are in axisymmetric configurations. Consequently, the quality of confinement depends sensitively on the shape of the confining magnetic field. To optimize a configuration, figures of merit quantifying confinement, along with other physics criteria such as magnetohydrodynamic (MHD) stability, must be considered in numerical optimization of the MHD equilibrium. These figures of merit describing a configuration depend on the shape of the outer plasma boundary or the shape of the electromagnetic coils. It is thus desirable to obtain derivatives with respect to these shapes for optimization of equilibria or identification of sensitivity information. These so-called shape derivatives can be computed by directly perturbing the shape, recomputing the equilibrium, and computing the resulting change to a figure of merit that depends on the equilibrium solution. However, this direct finite-difference approach requires recomputing the equilibrium for each possible perturbation of the shape. For stellarators whose geometry is described by a set of $N_{\Omega}\sim 10^2$ parameters, this requires $N_{\Omega}$ solutions to the MHD equilibrium equations. Despite this computational complexity, gradient-based optimization of stellarators has proceeded with the direct approach (e.g. \cite{Reiman1999,Ku2008,Proll2015}).
The shape gradient quantifies the change in a figure of merit associated with any perturbation to a shape. Thus, if the shape gradient can be obtained, the shape derivative with respect to \textit{any} perturbation is known (more precise definitions of the shape derivative and gradient are given in \S \ref{sec:shape_calculus}). In this work, we provide demonstration of an adjoint approach for computing the shape gradient for functions of MHD equilibria that could be considered within a stellarator configuration optimization. The adjoint approach does not require direct perturbation of a shape, but rather only the solution of one additional force balance equation which depends on the figure of merit of interest. Thus, when $N_{\Omega}$ is very large, as is generally the case for stellarator geometry, the adjoint approach is very advantageous.
In an accompanying work \citep{Antonsen2019}, two adjoint relations are derived: one involving perturbations to the plasma boundary, referred to as the fixed-boundary adjoint relation, and the other involving perturbations to currents in the vacuum region, known as the free-boundary adjoint relation. These can be considered generalizations of the self-adjointness of the force operator that arises in linearized MHD \citep{Bernstein1958}. A summary of these results is presented in \S \ref{sec:adjoint_relation}. These adjoint relations are applied to obtain expressions for the shape gradient of several figures of merit in terms of solutions to an adjoint force balance equation.
Historically, stellarator optimization has been conducted in two stages: in the first, the plasma boundary is varied to optimize an MHD equilibrium for desired physical properties \citep{Nuhrenberg1988}. As a second step, the coils are then optimized to provide the desired plasma boundary. The fixed-boundary adjoint relation provides a means to obtain the shape gradient with respect to the plasma boundary and can be used for the traditional optimization route.
It is also advantageous to consider coupling the coil design with the physics optimization \citep{Strickler2004,Hudson2018,Drevlak2018}, with the aim of obtaining configurations which do not require overly-complex coils. The free-boundary adjoint relation allows the computation of the shape gradient of equilibrium figures of merit with respect to coil geometry, allowing for direct optimization of coils. This approach can also be used to efficiently compute coil tolerances \citep{Landreman2018}.
Although the adjoint relations are based on the equations of linearized MHD, we perform numerical calculations in this work with non-linear MHD solutions with the addition of a small perturbation. In the accompanying paper, numerical calculations of the shape gradient with the adjoint approach were obtained for simple figures of merit that did not require modification of the Variational Moments Equilibrium Code (VMEC) \citep{Hirshman1983}. In this work, we demonstrate that the adjoint approach can be used to compute the shape gradient for other figures of merit that are relevant for the optimization of stellarator equilibria. We obtain expressions for the shape gradients of the vacuum magnetic well (\S \ref{sec:vacuum_well}), magnetic ripple (\S \ref{sec:ripple}), effective ripple in the $1/\nu$ neoclassical regime \citep{Nemov1999} where $\nu$ is the collision frequency (\S \ref{sec:epsilon_eff}), departure from quasisymmetry (\S \ref{sec:quasisymmetry}), and moments of the neoclassical distribution function (\S \ref{sec:neoclassical}) in terms of the solution to an adjoint force balance equation. We present calculations of the shape gradient with the adjoint approach for the vacuum magnetic well, which does not require modification to VMEC. The calculation for the magnetic ripple is computed with a minor modification of the Anisotropic Neumann Inverse Moments Equilibrium Code (ANIMEC) \citep{Cooper19923d}. The adjoint force balance equations needed to compute the shape gradient for the other figures of merit require the addition of a bulk force that will necessitate further modification of an equilibrium or linearized MHD code. Numerical calculations for these figures of merit will, therefore, not be presented in this work.
\section{Shape calculus fundamentals}
\label{sec:shape_calculus}
We now introduce several definitions and relations from the field of shape calculus which will prove useful for calculations in this work. Consider a function, $F(S_P)$, which depends implicitly on the plasma boundary, $S_P$, through the solution to the MHD equilibrium equations with boundary condition $\textbf{B} \cdot \textbf{n}|_{S_P} = 0$ where $\textbf{n}$ is the outward unit normal on $S_P$. We define a functional integrated over the plasma volume, $V_P$,
\begin{gather}
f(S_P) = \int_{V_P} d^3 x \, F(S_P),
\label{eq:f}
\end{gather}
where $S_P$ is the boundary of $V_P$. Consider a vector field describing displacements of the surface, $\delta \textbf{r}$, and a displaced surface $S_{P,\epsilon} = \{ \textbf{r}_0 + \epsilon \delta \textbf{r} : \textbf{r}_0 \in S_P\}$. The shape derivative of $F$ is defined as
\begin{gather}
\delta F(S_P;\delta \textbf{r}) = \lim_{\epsilon \rightarrow 0} \frac{F( S_{P,\epsilon}) - F(S_P)}{\epsilon}.
\end{gather}
The shape derivative of $f$ is defined by the same expression with $F\to f$. Under certain assumptions of smoothness of $\delta F$ with respect to $\delta \textbf{r}$, the shape derivative of the volume-integrated quantity, $f$, can be written in the following way \citep{Delfour2011_4},
\begin{gather}
\delta f(S_P;\delta \textbf{r}) = \int_{V_P} d^3 x \, \delta F(S_P;\delta \textbf{r}) + \int_{S_P} d^2 x \, \delta \textbf{r} \cdot \textbf{n} F.
\label{eq:transport_theorem}
\end{gather}
The first term accounts for the Eulerian perturbation to $F$ while the second accounts for the motion of the boundary. This is referred to as the transport theorem for domain functionals and will be used throughout to compute the shape derivatives of figures of merit of interest.
According to the Hadamard-Zol\'{e}sio structure theorem \citep{Delfour2011}, the shape derivative of a functional of $S_P$ (not restricted to the form of \eqref{eq:f}) can be written in the following form,
\begin{gather}
\delta f(S_P;\delta \textbf{r}) = \int_{S_P} d^2 x \, \delta \textbf{r} \cdot \textbf{n} \mathcal{G},
\label{eq:shape_gradient}
\end{gather}
assuming $\delta f$ exists for all $\delta \textbf{r}$ and is sufficiently smooth.
In the above expression, $\mathcal{G}$ is the shape gradient. This is an instance of the Riesz representation theorem, which states that any linear functional can be expressed as an inner product with an element of the appropriate space \citep{Rudin2006}. As the shape derivative of $f$ is linear in $\delta \textbf{r}$, it can be written in the form of \eqref{eq:shape_gradient}. Intuitively, the shape derivative does not depend on tangential perturbations to the surface. The shape gradient can be computed from derivatives with respect to the set of parameters, $\Omega$, used to discretize $S_P$,
\begin{gather}
\partder{f}{\Omega_i} = \int_{S_P} d^2 x \, \partder{\textbf{r}}{\Omega_i} \cdot \textbf{n} \mathcal{G}.
\label{eq:shape_gradient_system}
\end{gather}
For example, $\Omega = \{ R_{mn}^c, Z_{mn}^s \}$ could be assumed, where these are the Fourier coefficients in a cosine and sine representation of the cylindrical coordinates $(R,Z)$ of $S_P$.
Upon discretization of the right-hand side on a surface, the above takes the form of a linear system that can be solved for $\mathcal{G}$ \citep{Landreman2018}. However, this approach requires performing at least one additional equilibrium calculation for each parameter with a finite-difference approach.
The shape gradient can also be computed with respect to perturbations of currents in the vacuum region. We now consider $f$ to depend on the shape of a set of filamentary coils, $C = \{ C_k \}$, through a free-boundary solution to the MHD equilibrium equations. We consider a vector field of displacements to the coils, $\delta \textbf{r}_{C}$. The shape derivative of $f$ can also be written in shape gradient form,
\begin{gather}
\delta f(C;\delta \textbf{r}_{C}) = \sum_k \int_{C_k} dl \, \textbf{S}_k \cdot \delta \textbf{r}_{C_k},
\label{eq:coil_shape_gradient}
\end{gather}
where $\textbf{S}_k$ is the shape gradient for coil $k$, $C_k$ is the line integral along coil $k$, and the sum is taken over coils. Again, $\textbf{S}_k$ can be computed from derivatives with respect to a set of a parameters describing coil shapes, analogous to \eqref{eq:shape_gradient_system}.
To avoid the cost of direct computation of the shape gradient, we apply an adjoint approach. The shape gradient is thus obtained without perturbing the plasma surface or coil shapes directly, but instead by solving an additional adjoint equation that depends on the figure of merit of interest. We perform the calculation with the direct approach to demonstrate that the same derivative information is computed with either approach.
\section{Adjoint relations for MHD equilibria}
\label{sec:adjoint_relation}
Here we summarize the model for perturbed MHD equilibria and the adjoint relations from the accompanying paper. Throughout we assume the existence of magnetic coordinates such that the magnetic field can be written in the contravariant form as,
\begin{gather}
\textbf{B} = \nabla \psi \times \nabla \theta - \iota(\psi) \nabla \psi \times \nabla \zeta,
\label{eq:magnetic_contravariant}
\end{gather}
where $2\pi\psi$ is the toroidal flux, $\theta$ is a poloidal angle, $\zeta$ is a toroidal angle, and $\iota(\psi)$ is the rotational transform. The equilibrium magnetic field, $\textbf{B}$, is assumed to be in force balance,
\begin{gather}
\frac{\textbf{J} \times \textbf{B}}{c} = \nabla p ,
\label{eq:force_balance}
\end{gather}
where $\textbf{J}$ is the current density, $p(\psi)$ is the plasma pressure, and $c$ is the speed of light. The current density satisfies Ampere's law,
\begin{gather}
\nabla \times \textbf{B} = \frac{4\pi}{c} \textbf{J}.
\label{eq:ampere}
\end{gather}
We will consider a fixed-boundary calculation such that the equilibrium equations \eqref{eq:force_balance}-\eqref{eq:ampere} are solved with a specified value of toroidal flux $2\pi \psi_0$ on a given surface $S_P$, and free-boundary calculations such that they are solved with specified currents in the vacuum region. Two free functions of flux must also be specified, which we take to be $p(\psi)$ and the rotational transform $\iota(\psi)$ or the toroidal current contained within a flux surface,
\begin{gather}
I_T(\psi) = \frac{c}{8\pi^2} \int_0^{2\pi} d \theta \int_0^{2\pi} d \zeta \, \sqrt{g} \textbf{B} \cdot \left( \nabla \zeta \times \nabla \psi \right),
\label{eq:toroidal_current}
\end{gather}
where the Jacobian is $\sqrt{g} = (\nabla \psi \times \nabla \theta \cdot \nabla \zeta)^{-1}$. Fixing the toroidal current is more common in the context of stellarator optimization: for $\beta = 0$, $I_T$ can be taken to vanish while at finite $\beta$ it can be computed to be self-consistent with the neoclassical bootstrap current \citep{Spong2001,Shimizu2018}. We note that specification of an equilibrium state via \eqref{eq:magnetic_contravariant}-\eqref{eq:toroidal_current} is not always possible, as magnetic surfaces may not exist and the necessary periodicity constraints on rational surfaces may not be satisfied in a general 3-dimensional system. At this point we neglect these issues and proceed assuming that \eqref{eq:magnetic_contravariant}-\eqref{eq:toroidal_current} are sufficient to define an equilibrium state.
We now consider a linearization about this equilibrium state resulting from a perturbation to the plasma boundary, $S_P$, the coil shapes, the scalar profiles ($p(\psi)$, $\iota(\psi)$, or $I_T(\psi)$), or the addition of a bulk force. From \eqref{eq:magnetic_contravariant}, the perturbed magnetic field can be expressed in terms of the perturbations to the magnetic coordinates coordinates, ($\delta \psi$, $\delta \theta$, $\delta \zeta$),
\begin{align}
\delta \textbf{B} = \nabla \times [\delta \psi \left(\nabla \theta - \iota(\psi) \nabla \zeta \right) + \nabla \psi \left( \iota(\psi) \delta \zeta - \delta \theta \right) - \delta \Phi(\psi) \nabla \zeta ],
\label{eq:delta_B}
\end{align}
where $\delta \Phi(\psi)$ is the perturbation to the poloidal flux profile such that $\delta \Phi'(\psi) = \delta \iota(\psi)$ is the perturbation to the rotational transform profile. We can express \eqref{eq:delta_B} in terms of the displacement vector,
\begin{align}
\delta \textbf{B} = \nabla \times \left( \bm{\xi} \times \textbf{B} - \delta \Phi(\psi) \nabla \zeta \right),
\end{align}
with
\begin{align}
\bm{\xi} = \frac{\textbf{B}}{B^2} \times \left[\delta \psi (\nabla \theta - \iota(\psi)\nabla \zeta) + \nabla \psi (\iota(\psi)\delta \zeta - \delta \theta ) \right].
\label{eq:displacement}
\end{align}
For perturbations which fix the rotational transform profile, the familiar expression for the perturbed magnetic field from ideal MHD stability theory is recovered.
We define a vector field which defines the displacement of a field line, $\delta \textbf{r}$, such that the perturbation to the field line label $\alpha = \theta - \iota(\psi) \zeta$ and toroidal flux satisfy,
\begin{align}
\delta \psi + \delta \textbf{r} \cdot \nabla \psi &= 0 \\
\delta \alpha + \delta \textbf{r} \cdot \nabla \alpha &= 0,
\end{align}
and $\delta \textbf{r} \cdot \textbf{B} = 0$. Noting that $\delta \alpha = \delta \theta - \iota(\psi) \delta \zeta - \left( \iota'(\psi) \delta \psi + \delta \Phi'(\psi) \right)\zeta $, we find that
\begin{align}
\delta \textbf{r} &= \bm{\xi} + \frac{\textbf{b} \times \nabla \Phi(\psi)}{B} \zeta.
\label{eq:delta_r}
\end{align}
The linearized force balance equation is,
\begin{gather}
\frac{\delta \textbf{J}_{1,2} \times \textbf{B} + \textbf{J} \times \delta \textbf{B}_{1,2}}{c} - \nabla \delta p_{1,2} + \delta \textbf{F}_{1,2} = 0,
\label{eq:perturbed_force_balance}
\end{gather}
where $\delta \textbf{J}_{1,2} = (c/4\pi) \nabla \times \delta \textbf{B}_{1,2}$ is the perturbed current density, $\delta p_{1,2}$ is the perturbed pressure, and $\delta \textbf{F}_{1,2}$ is an additional bulk force. For perturbations that fix the pressure profile, $p(\psi)$, the change to the pressure at fixed position is
\begin{gather}
\delta p_{1,2} = - \bm{\xi}_{1,2} \cdot \nabla p,
\end{gather}
which follows from \eqref{eq:displacement}. Quantities with subscript 1 are called the direct perturbation, and those with subscript 2 are called the adjoint perturbation. Direct perturbations correspond to a specified perturbation to the boundary, $\delta \textbf{r} \cdot \textbf{n} |_{S_P} = \bm{\xi}_1 \cdot \textbf{n} \rvert_{S_P}$, or to the coil shapes, $\delta \textbf{r}_{C}$, with no additional bulk force or perturbation to the profiles. Adjoint perturbations satisfy a modified force balance equation which depends on the figure of merit of interest. We will discuss several examples of adjoint perturbations in the following Sections.
Upon application of the self-adjointness relation for the MHD force operator \citep{Bernstein1958} augmented by the introduction of the perturbed poloidal flux, the following fixed-boundary adjoint relation is obtained,
\begin{multline}
\int_{V_P} d^3 x \, \left(- \bm{\xi}_1 \cdot \delta \textbf{F}_2 + \bm{\xi}_2 \cdot \delta \textbf{F}_1 \right)
- \frac{2\pi}{c} \int_{V_P} d \psi \, \left( \delta I_{T,2}(\psi) \delta \iota_1(\psi) - \delta I_{T,1}(\psi) \delta \iota_2(\psi) \right) \\
- \frac{1}{4\pi} \int_{S_P} d^2 x \, \textbf{n} \cdot \left( \bm{\xi}_2 \delta \textbf{B}_1 \cdot \textbf{B} - \bm{\xi}_1 \delta \textbf{B}_2 \cdot \textbf{B} \right) = 0.
\label{eq:fixed_boundary}
\end{multline}
For perturbed MHD equilibria, the displacement vector $\bm{\xi}$ describes the motion of the boundary ($\bm{\xi} \cdot \textbf{n} |_{S_P} = \delta \textbf{r} \cdot \textbf{n} |_{S_P})$. Thus the shape derivative with respect to the plasma boundary can be expressed with the replacement $\delta \textbf{r} \rightarrow \bm{\xi}$ (Appendix C of \cite{Antonsen2019}). Therefore we see that the boundary term in \eqref{eq:fixed_boundary} is already in the form of a shape gradient \eqref{eq:shape_gradient}. The task thus remains to express the shape derivative of a given figure of merit in terms of the first term in \eqref{eq:fixed_boundary} and convert it to shape gradient form using the fixed-boundary relation.
A similar relation is obtained for perturbation of currents in the vacuum region rather than displacements of the plasma surface,
\begin{multline}
\int_{V_P} d^3 x \, \left(- \bm{\xi}_1 \cdot \delta \textbf{F}_2 + \bm{\xi}_2 \cdot \delta \textbf{F}_1 \right) + \frac{2\pi}{c} \int_{V_P} d \psi \left( \delta \Phi_1(\psi) \der{\delta I_{T,2}(\psi)}{\psi}-\delta \Phi_2(\psi) \der{\delta I_{T,1}(\psi)}{\psi} \right) \\ + \frac{1}{c} \sum_k \left( I_{C_{k}} \int_{C_k} dl \, \left( \delta \textbf{r}_{C_{1,k}}(\textbf{x}) \cdot \textbf{t} \times \delta \textbf{B}_2 -\delta \textbf{r}_{C_{2,k}}(\textbf{x}) \cdot \textbf{t} \times \delta \textbf{B}_1 \right)\right) = 0,
\label{eq:free_boundary}
\end{multline}
where we have made the assumption that the currents are confined to filamentary coils, and the coil shapes are perturbed without perturbations to their currents, $I_{C_k}$. Expressions which do not make these assumptions are provided in \citep{Antonsen2019}. The unit tangent vector along the coil is $\textbf{t}$. We can note that the third term in the above expression is in the form of a coil shape gradient \eqref{eq:coil_shape_gradient}. Thus, this adjoint relation can be applied by expressing the shape derivative of a figure of merit in the form of the first two terms, and the shape gradient is computed from the solution to an adjoint equation.
These relations, \eqref{eq:fixed_boundary} and \eqref{eq:free_boundary}, will now be applied to compute the shape gradients for several figures of merit with the adjoint approach.
\section{Vacuum magnetic well}
\label{sec:vacuum_well}
The averaged radial (i.e. normal to a flux surface) curvature is an important metric for MHD stability \citep{Freidberg2014},
\begin{gather}
\kappa_{\psi} \equiv \left\langle \bm{\kappa} \cdot \left(\partder{\textbf{r}}{\psi} \right)_{\alpha,l}\right\rangle_{\psi} = \left \langle \frac{1}{2B^2} \left(\partder{}{\psi} \left(8\pi p + B^2 \right) \right)_{\alpha,l} \right \rangle_{\psi},
\end{gather}
where the curvature is $\bm{\kappa} = \textbf{b} \cdot \nabla \textbf{b}$, $\textbf{b} = \textbf{B}/B$ is a unit vector in the direction of the magnetic field, $\alpha = \theta - \iota(\psi)\zeta$ is a field line label such that $\textbf{B} = \nabla \psi \times \nabla \alpha$, and $l$ measures length along a field line. Subscripts in the above expression indicate quantities held fixed while computing the derivative. The flux surface average of a quantity $A$ is \begin{gather}
\langle A \rangle_{\psi} = \frac{\int_{-\infty}^{\infty} \frac{dl}{B}\, A}{\int_{-\infty}^{\infty} \frac{dl}{B}} = \frac{\int_0^{2\pi} d \theta \int_0^{2\pi} d \zeta \, \sqrt{g} A}{V'(\psi)}.
\end{gather}
Here $V(\psi)$ is the volume enclosed by the surface labeled by $\psi$. The average radial curvature appears in the ideal MHD potential energy functional for interchange modes, and it provides a stabilizing effect when $p'(\psi) \kappa_{\psi} < 0$. As typically $p'(\psi)<0$, $\kappa_{\psi} >0$ is desirable for MHD stability. In a vacuum field, the expression for the averaged radial curvature reduces to
\begin{gather}
\kappa_{\psi} = - \frac{V''(\psi)}{V'(\psi)}.
\end{gather}
Thus, as volume increases with flux, $V''(\psi)<0$ is advantageous \citep{Helander2014}. The quantity $p'(\psi)V''(\psi)$ also appears in the Mercier criterion for ideal MHD interchange stability \citep{Mercier1974}. Known as the vacuum magnetic well, $V''(\psi)$ has been employed in the optimization of several stellarator configurations (e.g. \cite{Hirshman1999,Henneberg2019}).
We consider the following figure of merit
\begin{gather}
f_W = \int_{V_P} d\psi \, w(\psi)V'(\psi),
\label{eq:f_w_v}
\end{gather}
where $w(\psi)$ is a radial weight function which will be chosen so that \eqref{eq:f_w_v} approximates $V''(\psi)$. This can equivalently be written as
\begin{gather}
f_W = \int_{V_P} d^3 x \, w(\psi).
\end{gather}
\subsection{Fixed-boundary shape gradient}
We consider perturbations about an equilibrium with fixed toroidal current. For the direct perturbation, we have,
\begin{align}
\delta \textbf{F}_1 &= 0 \label{eq:direct_1} \\
\bm{\xi}_1 \cdot \textbf{n} |_{S_P} &= \delta \textbf{r} \cdot \textbf{n} |_{S_P} \label{eq:direct_2} \\
\delta I_{T,1}(\psi) &= 0 \label{eq:direct_3},
\end{align}
for a specified boundary perturbation $\delta \textbf{r} \cdot \textbf{n}$. The shape derivative of $f_W$ is computed upon application of the transport theorem \eqref{eq:transport_theorem}, noting that $\delta \psi = - \bm{\xi}_1 \cdot \nabla \psi$,
\begin{gather}
\delta f_W(S_P;\bm{\xi}_1) = -\int_{V_P} d^3 x \, \bm{\xi}_1 \cdot \nabla w(\psi) + \int_{S_P} d^2 x \, \, \bm{\xi}_1 \cdot \mathbf{n} w(\psi),
\label{eq:df_W}
\end{gather}
where we have assumed $w(\psi)$ to be differentiable. We recast the first term in \eqref{eq:df_W} as a surface integral by applying the fixed-boundary adjoint relation \eqref{eq:fixed_boundary} and prescribing the adjoint perturbation to satisfy the following,
\begin{align}
\delta \textbf{F}_2 &= -\nabla \left(\Delta_P w(\psi) \right) \label{eq:deltaF_vacuum} \\
\bm{\xi}_2 \cdot \textbf{n}|_{S_P} &= 0 \\
\delta I_{T,2}(\psi) &= 0.
\end{align}
Strictly speaking, the adjoint perturbation is the linear response to the bulk force \eqref{eq:deltaF_vacuum}. Rather than solve a linearized force balance equation, we note that the adjoint bulk force takes the form of the gradient of a scalar. This is implemented by perturbing the pressure profile by $\Delta_P w(\psi)$, where $\Delta_P$ is a constant chosen judiciously. Thus a small perturbation is applied to the pressure profile, the non-linear equilibrium is computed, and the change in the fields are recorded. Accordingly $\Delta_P$ must be small enough that non-linear effects are not important, but large enough that round-off error does not dominate.
Upon application of \eqref{eq:fixed_boundary} we obtain the following expression for the shape gradient which depends on the adjoint solution, $\delta \textbf{B}_2$,
\begin{gather}
\mathcal{G}_{W} = \left(w(\psi) + \frac{\delta \textbf{B}_2 \cdot \textbf{B}}{4\pi \Delta_P}\right)_{S_P}.
\label{eq:g_vacuum}
\end{gather}
In Figure \ref{fig:vacuum_well} we present the computation of $\mathcal{G}_{W}$ for the NCSX LI383 equilibrium \citep{Zarnstorff2001} using the the adjoint and direct approaches. We use a weight function
\begin{gather}
w(\psi) = \exp(-(\psi-\psi_{m,1})^2/\psi_w^2)- \exp(-(\psi-\psi_{m,2})^2/\psi_w^2)
\label{eq:weight}
\end{gather}
(see Figure \ref{fig:weight})
such that $f_W$ remains smooth while it approximates $V'(\psi_{m,1})-V'(\psi_{m,2})$ where $\psi_{m,1} = 0.8 \psi_0$, $\psi_{m,2} = 0.1 \psi_0$, and $\psi_{w} = 0.05 \psi_0$. We note that $f_W$ can be interpreted as measuring the change in volume due to the interchange of two flux tubes centered at $\psi_{m,1}$ and $\psi_{m,2}$. If $f_W>0$, this indicates that moving a flux tube radially outward will cause it to expand and lower the potential energy.
All equilibrium calculations are performed with the VMEC code. For the direct approach, derivatives with respect to the Fourier discretization of the boundary ($R_{mn}^c$ and $Z_{mn}^s$) are computed for $m \le 20$ and $|n| \le 10$ using an 8-point centered difference stencil with a polynomial-fitting technique. The direct approach requires 6889 calls to VMEC while the adjoint approach requires two calls. It is clear from Figure \ref{fig:vacuum_well} that the adjoint approach yields the same gradient information as the finite-difference approach, at much lower computational cost. The small difference between Figures \ref{fig:adjoint} and \ref{fig:direct} can be quantified as follows,
\begin{gather}
S_{\text{residual}} = \frac{|S_{\text{adjoint}}-S_{\text{direct}}|}{\sqrt{\int_{S_P} d^2 x \, S_{\text{adjoint}}^2/\int_{S_P} d^2 x }}.
\label{eq:residual}
\end{gather}
The surface-averaged value of $S_{\text{residual}}$ is $3.8\times 10^{-2}$. We note that the number of required equilibrium calculations for the direct shape gradient calculation depends on the Fourier resolution and finite-difference stencil chosen. In this work we present the number of function evaluations required in order for the adjoint and direct shape gradient calculations to agree within a few percent. As the Fourier resolution is increased, the results of the adjoint and direct methods converge to each other.
The residual difference is nonzero due to several sources of error, including discretization error in VMEC. As a result of the assumption of nested magnetic surfaces, MHD force balance \eqref{eq:force_balance} is not satisfied exactly, but a finite force residual is introduced. Error is also introduced by computing $\delta \textbf{B}_2$ with the addition of a small perturbation to a non-linear equilibrium calculation rather than from a linearized MHD solution.
\begin{figure}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=3cm 3cm 3cm 6cm,clip,width=1.0\textwidth]{vprime_surf_adjoint.png}
\caption{Adjoint}
\label{fig:adjoint}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=3cm 3cm 3cm 6cm,clip,width=1.0\textwidth]{vprime_surf_stellopt.png}
\caption{Direct}
\label{fig:direct}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=1cm 6cm 1cm 6cm,clip,width=1.0\textwidth]{weight.pdf}
\caption{Weight function}
\label{fig:weight}
\end{subfigure}
\caption{The shape gradient for $f_W$ \eqref{eq:f_w_v} is computed using the (a) adjoint and (b) direct approaches. (c) The weight function \eqref{eq:weight} used to compute $f_W$. }
\label{fig:vacuum_well}
\end{figure}
\subsection{Coil shape gradient}
The shape derivative of $f_W$ can also be computed with respect to a perturbation of the coil shapes. We consider perturbations about an equilibrium with fixed toroidal current,
\begin{align}
\delta \textbf{F}_1 &= 0 \\
\delta I_{T,1}(\psi) &= 0,
\end{align}
with specified perturbation to the coils shapes, $\delta \textbf{r}_{C_1} \times \textbf{t}$. We prescribe the following adjoint perturbation
\begin{align}
\delta \textbf{F}_2 &=
- \nabla (\Delta_P w(\psi)) \\
\delta I_{T,2}(\psi) &= 0,
\end{align}
with $\delta\textbf{r}_{C_2} \times \textbf{t} = 0$. The same weight function \eqref{eq:weight} is applied, which decreases sufficiently fast that we can approximate $w(\psi_0) = 0$. Upon application of the free boundary adjoint relation \eqref{eq:free_boundary}, we obtain the following coil shape gradient,
\begin{gather}
\textbf{S}_k = \frac{I_{C_k} \textbf{t} \times \delta \textbf{B}_2}{c\Delta_P} \bigg \rvert_{C_k}.
\label{eq:well_coil_shape_gradient}
\end{gather}
The calculation of $\textbf{S}_k$ for each of the 3 unique coil shapes from the NCSX C09R00 coil set\footnote{https://princetonuniversity.github.io/STELLOPT/VMEC\%20Free\%20Boundary\%20Run} \citep{Williamson2005} is shown in Figure \ref{fig:coil_shape_gradient}. The field is computed in the vacuum region for the evaluation of $\delta \textbf{B}_2$ using the DIAGNO code \citep{Gardner1990,Lazerson2012} with a 2-point centered difference stencil. The shape gradient is also computed with a direct approach. The Cartesian components of each coil are Fourier-discretized ($\textbf{X}_m^s$,$\textbf{X}_m^c$), and derivatives are computed with respect to $m \le 40$ with a 4-point centered-difference stencil. The fractional difference between the results obtained with the two approaches is
\begin{gather}
S_{\text{residual},k}^l = \frac{|S_{\text{adjoint},k}^l - S_{\text{direct},k}^l|}{\sqrt{\int_{C_k} dl \, \left(S_{\text{adjoint},k}^l\right)^2/\int_{C_k} dl}}.
\label{eq:residual_coil}
\end{gather}
The line-averaged value of $S_{\text{residual},k}^l$ is $4.1\times 10^{-2}
$. The direct approach required 2917 VMEC calls while the adjoint only required three.
\begin{figure}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=8cm 4cm 4cm 3cm,clip,width=1.0\textwidth]{adjoint_coil_vprime.png}
\caption{Adjoint}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=8cm 4cm 4cm 3cm,clip,width=1.0\textwidth]{stellopt_coil_vprime.png}
\caption{Direct}
\end{subfigure}
\caption{The coil shape gradient for $f_W$ is calculated for each of the 3 unique NCSX coil shapes. The arrows indicate the direction of $\textbf{S}_k$ \eqref{eq:well_coil_shape_gradient}, and their lengths indicate the magnitude scaled according to the legend.}
\label{fig:coil_shape_gradient}
\end{figure}
\section{Ripple on magnetic axis}
\label{sec:ripple}
We now consider a figure of merit which quantifies the ripple near the magnetic axis \citep{Carreras1996,Drevlak2014,Drevlak2018}.
As all physical quantities must be independent of the poloidal angle on the magnetic axis,
this quantifies the departure from quasi-helical or quasi-axisymmetry near the magnetic axis.
We define the magnetic ripple to be,
\begin{align}
f_R &= \int_{V_P} d^3 x \, \widetilde{f}_R
\label{eq:f_R}
\end{align}
with
\begin{subequations}
\begin{align}
\widetilde{f_R} &= \frac{1}{2} w(\psi) \left( B - \overline{B} \right)^2 \\
\overline{B} &= \frac{\int_{V_P} d^3 x \, w(\psi) B}{\int_{V_P} d^3 x \, w(\psi) }
\end{align}
\end{subequations}
and a weight function given by
\begin{align}
w(\psi) = \exp(-\psi^2/\psi_w^2)
\label{eq:weight_ripple}
\end{align}
with $\psi_w = 0.1 \psi_0$.
\subsection{Fixed-boundary shape gradient}
We compute perturbations about an equilibrium with fixed rotational transform
\begin{align}
\delta \textbf{F}_1 &= 0 \\
\bm{\xi}_1 \cdot \textbf{n} \rvert_{S_P} &= \delta \textbf{r} \cdot \textbf{n} \rvert_{S_P} \\
\delta \iota_1(\psi) &= 0.
\end{align}
Noting that the local perturbation to the field strength is
\begin{gather}
\delta B = -\frac{1}{B} \left( B^2 \nabla \cdot \bm{\xi}_1 + \bm{\xi}_1 \cdot \nabla \left(B^2 + 4\pi p \right) + \delta \iota_1(\psi) \textbf{B} \cdot \left(\nabla \psi \times \nabla \zeta \right)\right),
\label{eq:delta_mod_B}
\end{gather}
from \eqref{eq:delta_B},
the shape derivative is computed with the transport theorem \eqref{eq:transport_theorem},
\begin{gather}
\delta f_R(S_P;\bm{\xi}_1) = \int_{S_P} d^2 x \, \bm{\xi}_1 \cdot \textbf{n} \widetilde{f_R} + \int_{V_P} d^3 x \, \left(\partder{\widetilde{f_R}}{B} \delta B + \partder{\widetilde{f_R}}{\psi} \delta \psi \right),
\label{eq:delta_fR}
\end{gather}
where the partial derivative with respect to $B$ is performed at constant $\psi$. We prescribe the following adjoint perturbation,
\begin{align}
\delta \textbf{F}_2 &= - \Delta_P \nabla \cdot \mathbf{P} \label{eq:ripple_F} \\
\bm{\xi}_2 \cdot \textbf{n} |_{S_P} &= 0 \\
\delta \iota_2(\psi) &= 0, \label{eq:ripple_iota}
\end{align}
where $\Delta_P$ is again a constant scale factor. The bulk force perturbation required for the adjoint problem is written as the divergence of an anisotropic pressure tensor, $\textbf{P} = p_{\perp} \textbf{I} + (p_{||}-p_{\perp})\textbf{b}\textbf{b}$ where $\textbf{I}$ is the identity tensor. The parallel and perpendicular pressures are related by the parallel force balance condition,
\begin{gather}
\partder{p_{||}}{B} \bigg \rvert_{\psi} = \frac{p_{||}-p_{\perp}}{B},
\label{eq:par_force_balance}
\end{gather}
which follows from the requirement that $\textbf{b} \cdot \delta \textbf{F}_2 = 0$ \eqref{eq:perturbed_force_balance}. We take the parallel pressure to be
\begin{gather}
p_{||} = \widetilde{f_R}.
\label{eq:p_||}
\end{gather}
Upon application of the fixed-boundary adjoint relation and the expression for the curvature in an equilibrium field, we obtain the following shape gradient,
\begin{gather}
\mathcal{G}_R = \left( p_{\perp} +\frac{\delta \textbf{B}_2 \cdot \textbf{B}}{4\pi \Delta_P} \right)_{S_P}.
\label{eq:well_shape_gradient}
\end{gather}
If instead the toroidal current is held fixed in the direct perturbation as in \eqref{eq:direct_1}-\eqref{eq:direct_3}, then the required adjoint current perturbation is given by
\begin{align}
\delta I_{T,2}(\psi) &= \frac{c \Delta_P}{2\pi} V'(\psi)\left \langle \partder{\widetilde{f}_R}{B} \textbf{b} \cdot \nabla \zeta \times \nabla \psi \right \rangle_{\psi} \label{eq:ripple_I},
\end{align}
with the shape gradient unchanged. See Appendix \ref{app:axis_ripple} for details of the calculation.
To compute the adjoint perturbation \eqref{eq:ripple_F}-\eqref{eq:ripple_I}, we consider the addition of an anisotropic pressure tensor to the non-linear force balance equation,
\begin{gather}
\frac{\textbf{J}' \times \textbf{B}'}{c} = \nabla p' + \Delta_P \nabla \cdot \textbf{P}(\psi',B'),
\label{eq:force_balance_animec}
\end{gather}
where $\textbf{P}(\psi',B') = p_{\perp}(\psi',B') \textbf{I} + \left(p_{||}(\psi',B')-p_{\perp}(\psi',B')\right)\textbf{b}' \textbf{b}'$. Here primes indicate the perturbed quantities (i.e. $B' = B + \delta B$) where unprimed quantities satisfy \eqref{eq:force_balance}. As in \S \ref{sec:vacuum_well}, the perturbation has a scale set by $\Delta_P$ which is chosen to be small enough that the response is linear. Enforcing parallel force balance from \eqref{eq:force_balance_animec} results in the following condition,
\begin{gather}
\partder{p_{||}}{B'} \bigg \rvert_{\psi'} = \frac{p_{||}-p_{\perp}}{B'}.
\label{eq:force_balance_||}
\end{gather}
If we furthermore assume that $\Delta_P \nabla \cdot \textbf{P}$ is small compared with the other terms in \eqref{eq:force_balance_animec}, we can consider it to be a perturbation to the base equilibrium \eqref{eq:force_balance}. In this way, we can apply the perturbed force balance equation \eqref{eq:perturbed_force_balance} with $\delta \textbf{F}_{2} = - \Delta_P \nabla \cdot \textbf{P}(\textbf{B})$, where $\textbf{P}$ is now evaluated with the equilibrium field which satisfies \eqref{eq:force_balance}. Thus the desired pressure tensor \eqref{eq:p_||} can be implemented by evaluating $p_{||}$ at the perturbed field such that \eqref{eq:force_balance_||} is satisfied.
The pressure tensor defined by \eqref{eq:par_force_balance}-\eqref{eq:p_||} has been implemented in the ANIMEC code \citep{Cooper19923d}, which modifies the VMEC variational principle to allow 3D equilibrium solutions with anisotropic pressures to be computed. The ANIMEC code has been used to model equilibria with energetic particle species using pressure tensors based on bi-Maxwellian \citep{Cooper2006} and slowing-down \citep{Cooper2005} distribution functions.
The variational principle assumes that $p_{||}$ only varies on a surface through $B$ and can, therefore, be used to include the required adjoint bulk force.
In Figure \ref{fig:ripple}, we present the computation of $\mathcal{G}_R$ for the NCSX LI383 equilibrium using the adjoint and direct approaches. For the direct approach, derivatives with respect to the Fourier discretization of the boundary are computed for $m \le 11$ and $|n| \le 7$ using an 8-point centered difference stencil. The direct approach required 2761 calls to VMEC while the adjoint approach required two calls. The surface-averaged value of $S_{\text{residual}}$ \eqref{eq:residual} is $3.3\times 10^{-2}$.
\begin{figure}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=3cm 2cm 4cm 6cm,clip,width=1.0\textwidth]{ripple_adjoint.png}
\caption{Adjoint}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=3cm 2cm 4cm 6cm,clip,width=1.0\textwidth]{ripple_stellopt.png}
\caption{Direct}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=1cm 0cm 1cm 1cm,clip,width=1.0\textwidth]{weight_ripple.png}
\caption{Weight function}
\end{subfigure}
\caption{The shape gradient for $f_{R}$ \eqref{eq:f_R} is computed using the (a) adjoint and (b) direct approaches with a weight function \eqref{eq:weight_ripple} shown in (c).}
\label{fig:ripple}
\end{figure}
\iffalse
\subsection{Coil shape gradient}
The shape derivative of $f_R$ is also computed with respect to perturbations of the coil shapes. We consider perturbations about an equilibrium with fixed rotational transform,
\begin{align}
\delta \textbf{F}_1 &= 0 \\
\delta \iota_1 (\psi) &= 0,
\end{align}
with specified perturbations to the coil shapes, $\delta \textbf{r}_{C_1} \times \textbf{t}$. We prescribe the following adjoint perturbation
\begin{align}
\delta \textbf{F}_2 &= - \Delta_P \nabla \cdot \textbf{P} \\
\delta \iota_1(\psi) &= 0,
\end{align}
with $\delta \textbf{r}_{C_2}\times \textbf{t} = 0 $. If instead the toroidal current is held fixed in the direct problem, the prescribed toroidal current for the adjoint problem is given by \eqref{eq:ripple_I}. The same weight function \eqref{eq:weight_ripple} is used, such that we can approximate $w(\psi_0) = 0$. Upon application of the free boundary adjoint relation \eqref{eq:free_boundary}, we obtain the following coil shape gradient,
\begin{align}
\textbf{S}_k &= \frac{I_{C_k} \textbf{t} \times \delta \textbf{B}_2}{c \Delta_P} \bigg \rvert_{C_k}.
\end{align}
The calculation of $\textbf{S}_k$ for each of the 3 unique coil shapes of the NCSX LI383 equilibrium is shown in Figure \ref{fig:coil_shape_gradient_ripple}.
\begin{figure}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[trim=8cm 4cm 1cm 3cm,clip,width=1.0\textwidth]{ripple_coil_adjoint.png}
\caption{Adjoint}
\end{subfigure}
\caption{The coil shape gradient for $f_{R}$ is calculated for each of the 3 unique NCSX coil shapes. The arrows indicate the direction of $\textbf{S}_k$ \eqref{eq:well_coil_shape_gradient}, and their lengths indicate the magnitude scaled according to the legend.}
\label{fig:coil_shape_gradient_ripple}
\end{figure}
\fi
\section{Effective ripple in the $1/\nu$ regime}
\label{sec:epsilon_eff}
The effective ripple in the $1/\nu$ regime \citep{Nemov1999} is a figure of merit which has proven valuable for neoclassical optimization (e.g. \cite{Zarnstorff2001,Ku2008,Henneberg2019}). This quantity characterizes the geometric dependence of the neoclassical particle flux under the assumption of low-collisionality
such that $\epsilon_{\text{eff}}$ is analogous to the helical ripple amplitude, $\epsilon_h$, that appears in the expression of the $1/\nu$ particle flux for a classical stellarator \citep{Galeev1979}. The following expression is obtained for the effective ripple,
\begin{gather}
\epsilon_{\text{eff}}^{3/2}(\psi) = \frac{\pi}{4\sqrt{2}V'(\psi) \epsilon_{\text{ref}}^2} \int_{1/B_{\max}}^{1/B_{\min}} \frac{d \lambda}{\lambda} \, \int_0^{2\pi} d \alpha \, \sum_i \frac{(\partder{}{\alpha}\hat{K}_i(\alpha,\lambda))^2}{\hat{I}_i(\alpha,\lambda)}.
\label{eq:eps_eff}
\end{gather}
Here $\lambda = v_{\perp}^2/(v^2B)$ is the pitch angle, $\alpha = \theta - \iota(\psi) \zeta$ is a field line label, $B_{\min}$ and $B_{\max}$ are the minimum and maximum values of the field strength on a surface labeled by $\psi$, and $\epsilon_{\text{ref}}$ is a reference aspect ratio. We have defined the bounce integrals
\begin{align}
\hat{I}_i(\alpha,\lambda) &= \oint dl \, \frac{v_{||}}{Bv}\label{eq:I_hat} \\
\hat{K}_i(\alpha,\lambda) &= \oint dl \, \frac{v_{||}^3}{Bv^3} \label{eq:K_hat},
\end{align}
where the notation $\oint dl = \sum_{\sigma} \sigma \int_{\zeta_-}^{\zeta_+} d \zeta/\textbf{b} \cdot \nabla \zeta$ indicates integration at constant $\lambda$ and $\alpha$ between successive bounce points where $v_{||}(\zeta_+) = v_{||}(\zeta_-) = 0$ and $\sigma = \text{sign}(v_{||})$. The sum in \eqref{eq:eps_eff} is taken over wells at constant $\lambda$ and $\alpha$ for $\zeta_{-,i} \in [0,2\pi)$.
We consider an integrated figure of merit
\begin{gather}
f_{\epsilon} = \int_{V_P} d^3 x \, w(\psi) \epsilon_{\text{eff}}^{3/2}(\psi),
\label{eq:f_epsilon}
\end{gather}
where $w(\psi)$ is a radial weight function.
We perturb about an equilibrium with fixed toroidal current \eqref{eq:direct_1}-\eqref{eq:direct_3}.
The shape derivative of $f_{\epsilon}$ is computed to be
\begin{gather}
\delta f_{\epsilon}(S_P;\bm{\xi}_1) = \int_{V_P} d^3 x \, \left(\textbf{P}_{\epsilon}: \nabla \bm{\xi}_1 + \delta \iota_1(\psi) \mathcal{I}_{\epsilon}\right)
,
\label{eq:df_epsilon}
\end{gather}
where the double dot (:) indicates contraction between dyadic tensors $\textbf{A}$ and $\textbf{B}$ as $\textbf{A} : \textbf{B} = \sum_{i,j} A_{ij} B_{ji}$, with
\begin{multline}
\mathcal{I}_{\epsilon} = \frac{\pi w(\psi)}{2\sqrt{2} \epsilon_{\text{ref}}^2} \int_{1/B_{\max}}^{1/B} \frac{d \lambda}{\lambda} \, \\ \times \Bigg[\frac{\left(\partder{}{\alpha} \hat{K}(\alpha,\lambda,\zeta) \right)^2}{\hat{I}^2(\alpha,\lambda,\zeta)} \left(-\zeta \textbf{B} \times \nabla \psi \cdot \nabla \left(\frac{|v_{||}|}{vB^2} \right) + \textbf{B} \times \nabla \psi \cdot \nabla \zeta \partder{}{B} \left( \frac{|v_{||}|}{vB} \right) \right)\\ + 2\partder{}{\alpha} \left(\frac{\partder{}{\alpha} \hat{K}(\alpha,\lambda,\zeta) }{\hat{I}(\alpha,\lambda,\zeta)} \right) \left( - \zeta \textbf{B} \times \nabla \psi \cdot \nabla \left(\frac{|v_{||}|^3}{v^3 B^2} \right) +\textbf{B} \times \nabla \psi \cdot \nabla \zeta \partder{}{B} \left(\frac{|v_{||}|^3}{v^3 B} \right) \right)\Bigg]
\end{multline}
and $\textbf{P}_{\epsilon} = p_{||} \textbf{b} \textbf{b} + p_{\perp} (\textbf{I}-\textbf{b}\textbf{b})$ with
\begin{multline}
p_{||} = -\frac{\pi w(\psi)}{2\sqrt{2}\epsilon_{\text{ref}}^2} \int_{1/B_{\max}}^{1/B} \frac{d \lambda}{\lambda} \, \Bigg( \frac{\left(\partder{}{\alpha} \hat{K}(\alpha,\lambda,\zeta) \right)^2}{\hat{I}^2(\alpha,\lambda,\zeta)} \frac{|v_{||}|}{v} +2\partder{}{\alpha} \left(\frac{\partder{}{\alpha} \hat{K}(\alpha,\lambda,\zeta) }{\hat{I}(\alpha,\lambda,\zeta)} \right)\frac{|v_{||}|^3}{v^3 }\Bigg)
\label{eq:p_perp}
\end{multline}
\begin{multline}
p_{\perp} = - \frac{\pi w(\psi)}{2\sqrt{2}\epsilon_{\text{ref}}^2} \int_{1/B_{\max}}^{1/B} \frac{d \lambda}{\lambda} \, \Bigg( \frac{\left(\partder{}{\alpha} \hat{K}(\alpha,\lambda,\zeta) \right)^2}{\hat{I}^2(\alpha,\lambda,\zeta)} \left(\frac{\lambda vB}{2|v_{||}|} + \frac{|v_{||}|}{v} \right) \\ +2\partder{}{\alpha} \left(\frac{\partder{}{\alpha} \hat{K}(\alpha,\lambda,\zeta) }{\hat{I}(\alpha,\lambda,\zeta)} \right)\left( \frac{3\lambda |v_{||}|B}{2v} + \frac{|v_{||}|^3}{v^3} \right) \Bigg).
\label{eq:p_par}
\end{multline}
Derivatives are computed assuming $\epsilon_{\text{ref}}$ is held constant. The bounce integrals are defined with respect to $\zeta$ such that $\hat{I}(\alpha,\lambda,\zeta) = \hat{I}_i$ if $\zeta \in [\zeta_{-,i},\zeta_{+,i}]$ and $\hat{I}(\alpha,\lambda,\zeta) = 0$ if $\lambda B(\alpha,\zeta) > 1$. The same convention is used for $\hat{K}(\alpha,\lambda,\zeta)$.
We prescribe the following adjoint perturbation
\begin{align}
\delta \textbf{F}_2 &= - \Delta_P \nabla \cdot \textbf{P}_{\epsilon} \\
\bm{\xi}_2 \cdot \textbf{n}\rvert_{S_P} &= 0 \\
\delta I_{T,2}(\psi) &= \frac{c }{2 \pi} V'(\psi) \Delta_P \langle \mathcal{I}_{\epsilon}\rangle_{\psi}.
\end{align}
The adjoint bulk force must be consistent with parallel force balance from \eqref{eq:perturbed_force_balance}, which is equivalent to the condition
\begin{gather}
\nabla_{||} p_{||} = \frac{\nabla_{||} B}{B} (p_{||}-p_{\perp}).
\end{gather}
This can be shown to be satisfied by \eqref{eq:p_perp}-\eqref{eq:p_par}, noting that the $\lambda$ integrand vanishes at $1/B$ such that there is no contribution from the parallel gradient acting on the bounds of the integral. There is also no contribution to the parallel gradient from the bounce-integrals, as $|v_{||}|$ vanishes at points of non-zero gradient of $\hat{I}(\alpha,\lambda,\zeta)$ and $\hat{K}(\alpha,\lambda,\zeta)$.
Upon application of the fixed-boundary adjoint relation \eqref{eq:fixed_boundary} and integration by parts, we obtain the following expression for the shape gradient
\begin{gather}
\mathcal{G}_{\epsilon} = \left(p_{\perp} + \frac{\delta \textbf{B} \cdot \textbf{B}}{4\pi \Delta_P}\right)_{S_P}.
\end{gather}
See Appendix \ref{app:1_over_nu} for details of the calculation. The approach demonstrated in this Section could be extended to compute the shape gradients of other figures of merit involving bounce integrals, such as the $\Gamma_c$ metric for energetic particle confinement \citep{Nemov2005} or the variation of the parallel adiabatic invariant on a flux surface \citep{Drevlak2014}.
\section{Departure from quasisymmetry}
\label{sec:quasisymmetry}
Quasisymmetry is desirable as it ensures collisionless confinement of guiding centers. This property follows when the field strength depends on a linear combination of the Boozer angles, $B(\psi,\theta_B,\zeta_B) = B(\psi,M\theta_B-N\zeta_B)$ for fixed integers $M$ and $N$ \citep{Nuhrenberg1988,Boozer1995}. Several stellarator configurations have been optimized to be close to quasisymmetry (e.g. \cite{Reiman1999,Drevlak2013,Henneberg2019,Liu2018}) by minimizing the amplitude of symmetry-breaking Fourier harmonics of the field strength. We will consider a figure of merit that does not require a Boozer coordinate transformation; instead, we use a general set of magnetic coordinates $(\psi,\theta,\zeta)$ to define our figure of merit.
In Boozer coordinates \citep{Boozer1981,Helander2014} ($\psi,\theta_B,\zeta_B$) the covariant form for the magnetic field is
\begin{gather}
\textbf{B} = I(\psi) \nabla \theta_B + G(\psi) \nabla \zeta_B + K(\psi,\theta_B,\zeta_B) \nabla \psi.
\label{eq:boozer_covariant}
\end{gather}
Here $G(\psi) = (2/c) I_P(\psi)$, where $I_P(\psi)$ is the poloidal current outside the $\psi$ surface. The poloidal current can be computed using Ampere's law and expressed as an integral over a surface labeled by $\psi$, $S_P(\psi)$,
\begin{align}
I_P(\psi) &= \frac{c}{4\pi} \int_0^{2\pi} d \zeta \, \textbf{B} \cdot \partder{\textbf{r}}{\zeta} \nonumber \\
&= -\frac{c}{8\pi^2} \int_{S_P(\psi)} d^2 x \, \textbf{B} \cdot \nabla \theta \times \textbf{n}.
\label{eq:poloidal_current}
\end{align}
The quantity $I(\psi) = (2/c) I_T(\psi)$, where $I_T(\psi)$ is the toroidal current inside the $\psi$ surface \eqref{eq:toroidal_current}. We quantify the departure from quasisymmetry in the following way,
\begin{gather}
f_{QS} = \frac{1}{2}\int_{V_P} d^3 x \, w(\psi) \left(\textbf{B} \times \nabla \psi \cdot \nabla B - F(\psi)\textbf{B}\cdot \nabla B\right)^2.
\label{eq:f_QS}
\end{gather}
Here $w(\psi)$ is a radial weight function and
\begin{gather}
F(\psi) = \frac{(M/N)G(\psi) + I(\psi)}{(M/N)\iota(\psi)-1}.
\end{gather}
If $f_{QS} = 0$, then the field is quasisymmetric with mode numbers $M$ and $N$ \citep{Helander2014}, which can be shown using the covariant \eqref{eq:magnetic_contravariant} and contravariant \eqref{eq:boozer_covariant} representations of the magnetic field assuming $B=B(\psi,M\theta_B-N\zeta_B)$ for fixed $M$ and $N$. Note that $f_{QS}$ quantifies the symmetry in Boozer coordinates but can be evaluated in any flux coordinate system.
We consider perturbation about an equilibrium with fixed toroidal current \eqref{eq:direct_1}-\eqref{eq:direct_3}. The perturbations to the Boozer poloidal covariant component is computed using the transport theorem \eqref{eq:transport_theorem},
\begin{align}
\delta G(\psi) &= -\frac{1}{4\pi^2} \int_{S_P(\psi)} d^2 x \, \left(\nabla \cdot \left( \textbf{B} \times \nabla \theta \right) \bm{\xi}_1 \cdot \textbf{n} + \delta \textbf{B} \times \nabla \theta \cdot \textbf{n} \right).
\label{eq:delta_G_1}
\end{align}
In arriving at \eqref{eq:delta_G_1} we have used the fact that spatial derivatives commute with shape derivatives. The first term accounts for the unperturbed current density through the perturbed boundary, and the second accounts for the perturbed current density through the unperturbed boundary. The contribution from the perturbation to the poloidal angle can be shown to vanish. Upon application of \eqref{eq:delta_B} we obtain, noting that $\int_{S_P(\psi)} d^2 x \, A = V'(\psi) \langle A |\nabla \psi | \rangle_{\psi}$ for any quantity $A$,
\begin{multline}
\delta G(\psi) = \\
-\frac{V'(\psi)}{4\pi^2} \left \langle \bm{\xi}_1 \cdot \nabla \psi \nabla \cdot (\textbf{B} \times \nabla \theta ) - \frac{1}{\sqrt{g}} \partder{\textbf{r}}{\zeta} \cdot \nabla \times \left(\bm{\xi}_1 \times \textbf{B} \right) - \frac{\delta \iota_1(\psi)}{ \sqrt{g}^{2}} \partder{\textbf{r}}{\zeta} \cdot \partder{\textbf{r}}{\theta} \right \rangle_{\psi} \label{eq:delta_G},
\end{multline}
Applying the transport theorem \eqref{eq:transport_theorem}, the shape derivative of $f_{QS}$ takes the form,
\begin{multline}
\delta f_{QS}(S_P;\bm{\xi}_1) =\frac{1}{2} \int_{S_P} d^2 x \, \bm{\xi}_1 \cdot \textbf{n} \mathcal{M}^2 w(\psi) + \frac{1}{2} \int_{V_P} d^3 x \, w'(\psi) \delta \psi \mathcal{M}^2 \\
+ \int_{V_P} d^3 x \, w(\psi) \mathcal{M} \left( \delta \textbf{B} \cdot \bm{\mathcal{A}} + \bm{\mathcal{S}} \cdot \nabla \delta B + \textbf{B} \times \nabla \delta \psi \cdot \nabla B - \frac{\delta G(\psi) \textbf{B} \cdot \nabla B}{\iota(\psi)-(N/M)} \right)
\\ + \int_{V_P} d^3 x \, w(\psi) \mathcal{M} \left( \frac{F(\psi)}{\iota(\psi)-(N/M)} \delta \iota_1(\psi) \textbf{B} \cdot \nabla B - \delta \psi F'(\psi)\textbf{B} \cdot \nabla B \right),
\label{eq:df_QS1}
\end{multline}
where $\mathcal{M} = \textbf{B} \times \nabla \psi \cdot \nabla B - F(\psi) \textbf{B} \cdot \nabla B$, $\bm{\mathcal{A}} = \nabla \psi \times \nabla B - F(\psi) \nabla B$, and $\bm{\mathcal{S}} = \textbf{B} \times \nabla \psi - F(\psi) \textbf{B}$. After several steps outlined in Appendix \ref{app:qs}, the shape derivative can be written in the following way,
\begin{gather}
\delta f_{QS}(S_P;\bm{\xi}_1) = \int_{V_P} d^3 x \, \left(\bm{\xi}_1 \cdot \bm{\mathcal{F}}_{QS} + \delta \iota_1(\psi) \mathcal{I}_{QS} \right) + \int_{S_P} d^2 x \, \bm{\xi}_1 \cdot \textbf{n} \mathcal{B}_{QS}
\label{eq:df_QS}
\end{gather}
with
\begin{multline}
\bm{\mathcal{F}}_{QS} =\frac{1}{2} \nabla_{\perp} \left(w(\psi) \mathcal{M}^2 \right)
+ \left((\textbf{b} \times \nabla \psi) \nabla_{||}B + F(\psi) \nabla_{\perp} B \right) w(\psi) \textbf{B} \cdot \nabla \mathcal{M}
\\ + \textbf{B} \times (\nabla \times (\nabla \psi \times \nabla B)) w(\psi) \mathcal{M} -B\nabla_{\perp} \left(w(\psi) \bm{\mathcal{S}} \cdot \nabla \mathcal{M} \right) + \bm{\kappa} Bw(\psi) \bm{\mathcal{S}} \cdot \nabla \mathcal{M} \\ - \nabla \psi \nabla B \cdot \nabla \times \left(w(\psi) \mathcal{M} \textbf{B} \right) +\frac{1}{4\pi^2} \Bigg(- \nabla_{\perp} \left( \frac{w(\psi) V'(\psi) \langle \mathcal{M} \textbf{B} \cdot \nabla B \rangle_{\psi}}{(\iota(\psi)-(N/M))} \right) \left( \textbf{B} \cdot \nabla \psi \times \nabla \theta \right) \\
+\frac{w(\psi) V'(\psi) \langle \mathcal{M} \textbf{B} \cdot \nabla B \rangle_{\psi}}{\iota(\psi)-(N/M)}\left(\nabla \psi\nabla \cdot \left( \textbf{B} \times \nabla \theta \right) - \textbf{B} \times \nabla \times \left(\nabla \psi \times \nabla \theta \right) \right)
\Bigg)
\label{eq:F_QS}
\end{multline}
\begin{multline}
\mathcal{B}_{QS} = -\frac{1}{2}w(\psi) \mathcal{M}^2 + B w(\psi) \bm{\mathcal{S}} \cdot \nabla \mathcal{M} - w(\psi) \mathcal{M} \nabla B \times \textbf{B} \cdot \nabla \psi \\ + \frac{w(\psi)V'(\psi) \langle \mathcal{M}\textbf{B} \cdot \nabla B \rangle_{\psi}}{4\pi^2(\iota(\psi)-(N/M))} \left(\textbf{B} \cdot \nabla \psi \times \nabla \theta \right)
\label{eq:B_QS}
\end{multline}
\begin{multline}
\mathcal{I}_{QS} = - w(\psi) \mathcal{M} \nabla \psi \times \nabla \zeta \cdot \bm{\mathcal{A}} + w(\psi) \left(\bm{\mathcal{S}} \cdot \nabla \mathcal{M}\right) \textbf{b} \cdot \nabla \psi \times \nabla \zeta \\ + \frac{w(\psi) \mathcal{M} \textbf{B} \cdot \nabla B }{\iota(\psi)-(N/M)}\left(F(\psi) -\left \langle \frac{V'(\psi)}{4\pi^2\sqrt{g}^2} \partder{\textbf{r}}{\zeta} \cdot \partder{\textbf{r}}{\theta} \right \rangle_{\psi} \right).
\label{eq:I_QS}
\end{multline}
In \eqref{eq:F_QS}, $\nabla_{||} = \textbf{b} \cdot \nabla$ and $\nabla_{\perp} = \nabla- \textbf{b} \nabla_{||}$ are the parallel and perpendicular gradients. We note that $\bm{\mathcal{F}}_{QS}$ satisfies the parallel force balance condition ($\textbf{b} \cdot \bm{\mathcal{F}}_{QS}=0$) implied by \eqref{eq:perturbed_force_balance}.
We can now prescribe an adjoint perturbation which satisfies,
\begin{align}
\delta \textbf{F}_2 &= \Delta_{QS} \bm{\mathcal{F}}_{QS} \\
\bm{\xi} \cdot \textbf{n} |_{S_P} &= 0 \\
\delta I_{T,2}(\psi) &= \frac{c \Delta_{QS}}{2\pi} V'(\psi) \langle \mathcal{I}_{QS} \rangle_{\psi}.
\end{align}
Upon application of the fixed-boundary adjoint relation we obtain the following shape gradient, \begin{gather}
\mathcal{G}_{QS} = \left(\frac{\delta \textbf{B}_2 \cdot \textbf{B}}{4\pi \Delta_{QS}} + \mathcal{B}_{QS}\right)_{S_P}.
\end{gather}
\section{Neoclassical figures of merit}
\label{sec:neoclassical}
In \S \ref{sec:epsilon_eff}, we considered a figure of merit that quantifies the geometric dependence of the neoclassical particle flux in the $1/\nu$ regime. In applying this model, several assumptions are imposed, such as a small radial electric field, $E_r$, low collisionality, and a simplified pitch-angle scattering collision operator. In this Section, we consider a more general neoclassical figure of merit arising from a moment of the local drift kinetic equation, allowing for optimization at finite collisionality and $E_r$. It is assumed here that the collision time is comparable to the bounce time but shorter than the time needed to complete a magnetic drift orbit. Recently an adjoint method has been demonstrated for obtaining derivatives of neoclassical figures of merit \citep{Paul2019} with respect to local geometric quantities on a flux surface. The adjoint method described in this Section will extend these results, such that shape derivatives with respect to the plasma boundary can be computed.
Consider the following figure of merit,
\begin{gather}
f_{NC} = \int_{V_P} d^3 x \, w(\psi) \mathcal{R}(\psi).
\label{eq:f_NC}
\end{gather}
Here $\mathcal{R}(\psi)$ is a flux surface averaged moment of the neoclassical distribution function, $f_{1}$, which satisfies the local drift kinetic equation (DKE),
\begin{gather}
(v_{||} \textbf{b} + \bm{v}_E)\cdot \nabla f_{1} - C(f_{1}) = - \bm{v}_{\text{m}} \cdot \nabla \psi \partder{f_{0}}{\psi},
\label{eq:DKE}
\end{gather}
where $\bm{v}_E = \textbf{E}\times \textbf{B}/B^2$ is the $\textbf{E} \times \textbf{B}$ drift velocity, $\bm{v}_{\text{m}} \cdot \nabla \psi$ is the radial magnetic drift velocity \eqref{eq:radial_drift}, $f_{0}$ is a Maxwellian \eqref{eq:Maxwellian}, and $C$ is the linearized Fokker-Planck operator. For example, $\mathcal{R}$ can be taken to be the bootstrap current,
\begin{gather}
J_{b} = \sum_s \frac{\langle B \int d^3 v \, f_{1s} v_{||} \rangle_{\psi}}{n_s\langle B^2 \rangle_{\psi}^{1/2}},
\end{gather}
where the sum is taken over species. We note that the geometric dependence that enters the DKE when written in Boozer coordinates only arises through the quantities $\{B, G(\psi), I(\psi), \iota(\psi) \}$. Thus for simplicity, Boozer coordinates will be assumed throughout this Section.
The perturbation to $\mathcal{R}(\psi)$ at fixed toroidal current \eqref{eq:direct_1}-\eqref{eq:direct_3} can be written as,
\begin{gather}
\delta \mathcal{R}(\psi) = \langle S_{\mathcal{R}} \delta B \rangle_{\psi} + \partder{\mathcal{R}(\psi)}{G(\psi)} \delta G(\psi)
+ \partder{\mathcal{R}(\psi)}{\iota (\psi)} \delta \iota_1 (\psi).
\end{gather}
Here $S_{\mathcal{R}}$ is a local sensitivity function which quantifies the change to $\mathcal{R}$ associated with a perturbation of the field strength $\delta B$ defined in the following way. Consider the perturbation to $\mathcal{R}$ resulting from a change in the field strength at fixed $G(\psi)$, $I(\psi)$, and $\iota(\psi)$. The functional derivative of $\mathcal{R}(\psi)$ with respect to $B(\textbf{r})$ can be expressed as,
\begin{gather}
\delta \mathcal{R}(\delta B;B(\textbf{r})) = \left \langle S_{\mathcal{R}} \delta B(\textbf{r}) \right\rangle_{\psi}.
\end{gather}
This is another instance of the Riesz representation theorem: $\delta \mathcal{R}$ is a linear functional of $\delta B$, with the inner product taken to be the flux surface average. Thus $S_{\mathcal{R}}$ can be thought of as analogous to the shape gradient \eqref{eq:shape_gradient}.
The quantities $\{S_{\mathcal{R}},\partial \mathcal{R}(\psi)/\partial G(\psi),\partial \mathcal{R}(\psi)/\partial \iota(\psi) \}$ can be computed with a related adjoint method \citep{Paul2019} with the SFINCS code \citep{Landreman2014}. Here we consider SFINCS to be run on a set of surfaces such that \eqref{eq:f_NC} can be computed numerically. The derivatives computed by SFINCS will appear in the additional bulk force required for the adjoint perturbed equilibrium.
The shape derivative of $f_{NC}$ can be computed on application of the transport theorem \eqref{eq:transport_theorem},
\begin{multline}
\delta f_{NC}(S_P;\bm{\xi}_1) = \int_{S_P} d^2 x \, \bm{\xi}_1 \cdot \textbf{n} w(\psi) \mathcal{R}(\psi) + \int_{V_P} d^3 x \, \delta \psi \partder{}{\psi} \left(w(\psi) \mathcal{R}(\psi)\right)\\
+ \int_{V_P} d^3 x \,
w(\psi) \left(\partder{\mathcal{R}(\psi)}{G(\psi)}\delta G(\psi) + \partder{\mathcal{R}(\psi)}{\iota(\psi)}\delta \iota_1(\psi) + \left\langle S_R \delta B \right\rangle_{\psi} \right).
\label{eq:deltaf_NC}
\end{multline}
After several steps outlined in Appendix \ref{app:nc}, the shape derivative is written in the following form,
\begin{gather}
\delta f_{NC} (S_P;\bm{\xi}_1) = \int_{V_P} d^3 x \, \left(\bm{\xi}_1 \cdot \bm{\mathcal{F}}_{NC} + \delta \iota_1(\psi) \mathcal{I}_{NC} \right) + \int_{S_P} d^3 x \, \bm{\xi}_1 \cdot \textbf{n} \mathcal{B}_{NC}
\label{eq:df_NC}
\end{gather}
with
\begin{align}
\bm{\mathcal{F}}_{NC} &= -\nabla( \mathcal{R}(\psi) w(\psi)) -\nabla \psi (\nabla \times \textbf{B}) \cdot \nabla \theta \partder{\mathcal{R}(\psi)}{G(\psi)} w(\psi) \frac{B^2 \sqrt{g}}{\langle B^2 \rangle_{\psi}} \nonumber \\
&+ \frac{ w(\psi)}{\langle B^2 \rangle_{\psi}} \partder{\mathcal{R}(\psi)}{G(\psi)} \textbf{B} \times \nabla \times \left(\partder{\textbf{r}}{\zeta}B^2 \right) + G(\psi)B^2\nabla \left(\frac{w(\psi)}{\langle B^2 \rangle_{\psi}} \partder{\mathcal{R}(\psi)}{G(\psi)} \right) \nonumber \\
&- \bm{\kappa} w(\psi) S_{\mathcal{R}} B + B \nabla_{\perp} (w(\psi) S_{\mathcal{R}})
\label{eq:F_NC} \\
\mathcal{B}_{NC}
&= w(\psi) \mathcal{R}(\psi) -\frac{w(\psi) B^2}{\langle B^2 \rangle_{\psi}} \partder{\mathcal{R}(\psi)}{G(\psi)} G(\psi)- w(\psi) S_{\mathcal{R}} B \label{eq:B_NC} \\
\mathcal{I}_{NC} &= \partder{\mathcal{R}(\psi)}{G(\psi)} \frac{w(\psi) B^2}{\langle B^2 \rangle_{\psi}\sqrt{g}} \partder{\textbf{r}}{\zeta} \cdot \partder{\textbf{r}}{\theta} + w(\psi) \partder{\mathcal{R}(\psi)}{\iota(\psi)} - w(\psi) S_{\mathcal{R}} \textbf{b} \cdot \nabla \psi \times \nabla \zeta
\label{eq:I_NC}.
\end{align}
The adjoint bulk force $\bm{\mathcal{F}}_{NC}$ is chosen to satisfy parallel force balance required by \eqref{eq:perturbed_force_balance}.
We consider the following adjoint perturbation,
\begin{align}
\delta \textbf{F}_2 &= \Delta_{NC} \bm{\mathcal{F}}_{NC} \\
\bm{\xi}_2 \cdot \textbf{n} \rvert_{S_P} &= 0 \\
\delta I_{T,2}(\psi) &= \frac{c\Delta_{NC}}{2\pi} V'(\psi) \langle \mathcal{I}_{NC} \rangle_{\psi}.
\end{align}
Upon application of the fixed-boundary adjoint relation we obtain the shape gradient,
\begin{gather}
\mathcal{G}_{NC} = \left(\mathcal{B}_{NC} + \frac{\delta \textbf{B}_2 \cdot \textbf{B}}{4\pi \Delta_{NC}}\right)_{S_P}.
\end{gather}
\section{Conclusions}
We have demonstrated that the self-adjointness relations (\S \ref{sec:adjoint_relation}) can be implemented to efficiently compute the shape gradient of figures of merit relevant for stellarator configuration optimization. The shape gradient is obtained by solving an adjoint perturbed force balance equation that depends on the figure of merit of interest. For the vacuum well parameter (\S \ref{sec:vacuum_well}), the additional bulk force required for the adjoint problem is simply the gradient of a function of flux, and so can be implemented by adding a perturbation to the pressure profile. For the magnetic ripple on axis (\S \ref{sec:ripple}), the required bulk force takes the form of the divergence of a pressure tensor that only varies on a surface through the field strength. As this type of pressure tensor is currently treated by the ANIMEC code, this adjoint bulk force is implemented with a minor modification to the code. Computing the shape gradient of $\epsilon_{\text{eff}}^{3/2}$ with the adjoint approach also requires the addition of the divergence of a pressure tensor. However, this pressure tensor varies on a surface through the field line label due to the bounce integrals that appear \eqref{eq:p_par}-\eqref{eq:p_perp}. Thus the variational principle used by the ANIMEC code cannot be easily extended for this application. Similarly, the shape gradients for the quasisymmetry (\S \ref{sec:quasisymmetry}) and neoclassical (\S \ref{sec:neoclassical}) figures of merit require an adjoint bulk force that is not in the form of the divergence of a pressure tensor. This provides an impetus for the development of a flexible perturbed MHD equilibrium code that could enable these calculations. While several 3D ideal MHD stability codes exist \citep{Anderson1990,Schwab1993,Strumberger2016}, only the CAS3D code has been modified in order to perform perturbed equilibrium calculations \citep{Nuhrenberg2003,Boozer2006}. We hope to take advantage of linear MHD calculations for future work through the proper modification of an existing code or development of a new code.
To date, the numerical verification of this adjoint approach for MHD equilibria has been performed with the VMEC and ANIMEC codes, which solve the non-linear force balance equations, \eqref{eq:force_balance} and \eqref{eq:force_balance_animec}. The adjoint perturbed force balance equation \eqref{eq:perturbed_force_balance} is approximated by adding a perturbative bulk force or toroidal current profile, whose characteristic magnitudes are scaled by the $\Delta$ constants (e.g. $\Delta_P$ in \eqref{eq:deltaF_vacuum}). As demonstrated in \cite{Antonsen2019}, these parameters must be small enough that non-linear effects do not become important yet large enough that round-off error does not dominate. This furthermore motivates the development of a perturbed equilibrium code that could eliminate this source of noise.
As demonstrated, this adjoint approach for functions of MHD equilibria is quite flexible and can be applied to many quantities of interest. Because of the demonstrated efficiency in comparison with the direct approach to computing shape gradients, we anticipate many further applications of this method.
\section*{Acknowledgements}
The authors would like to acknowledge productive discussions with Ricardo Nochetto on shape calculus. This work was supported by the US Department of Energy through grants DE-FG02-93ER-54197 and DE-FC02-08ER-54964. The computations presented in this paper have used resources at the National Energy Research Scientific Computing Center (NERSC).
\bibliographystyle{jpp}
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1,116,691,498,419 | arxiv |
\subsection{Structural Lemma}
\label{sec:extendedSecProperties}
In this subsection, we show the following lemma connecting the probabilities $P_i$ to the probabilities $p_j$ from Lemma~\ref{wkeitenc}. The analysis showing the $1/e$-competitiveness of our algorithm
is crucially based on this result. Note that we only use it for $B=2$ but it holds for all $B$.
\begin{lemma}\label{lemmaWkeit}
The probability that $\mathsf{ALG}$ packs element $i \in \mathcal{I}$ is
\begin{numcases}
{P_i=}
p_i & \text{if element $i$ is large,} \label{eq:l1large} \\
i_s^* \cdot p_i + \sum\limits_{x=\mathrm{r}_\mathrm{s}(i) +1}^{B^*} p_{\mathrm{r}'_\mathrm{g}(x)} & \text{if element $i$ is small}, \label{eq:l1small}
\end{numcases}
with $ i_s^* := \min\{\mathrm{r}_\mathrm{s}(i), B\} $ and $ B^* := \min\left\{ B, |\mathcal{I}_S| \right\}$.
\end{lemma}
Observe that \eqref{eq:l1large} follows immediately: Any large element can only be packed when the knapsack is empty, i.e., as the first element. The proof of \eqref{eq:l1small} requires a bit more work.
\begin{definition}
Let $ E_{x,y}^{i,j} $ be the event that the small elements $ i $ and $j$ are packed as the $x$-th and $y$-th items, respectively.
\end{definition}
Note that the event that any item $i \in \mathcal{I}_S$ is packed as $x$-th item, where $x \geq 2$, can be partitioned according to the item packed first. Therefore, for any $i \in \mathcal{I}_S$ and $x \geq 2$,
\begin{equation}\label{HL0}
p_i(x) = \sum\limits_{j \in \mathcal{I}_S} \normalfont \text{Pr} \left[E_{1,x}^{j,i}\right] \,.
\end{equation}
We have the following technical lemmata.
\begin{lemma}\label{HL1}
Let $i \in \mathcal{I}_S$ be any small item and $i_s^* = \min\{\mathrm{r}_\mathrm{s}(i), B\}$.
For $ 2 \leq \ell \leq i_s^* $, it holds that
$ \displaystyle\sum\limits_{j \in \mathcal{I}_S} \normalfont \text{Pr}\left[E_{1,\ell}^{i,j}\right] = p_i $.
\end{lemma}
\begin{proof}
The first step is to show that at least $\ell$ elements are accepted in total, if element $i$ is accepted first. Since element $ i $ has rank $\mathrm{r}_\mathrm{s}(i) $ among the small elements, there are $\mathrm{r}_\mathrm{s}(i) -1$ small elements that are more valuable.
Their position in the input sequence cannot be in the sampling phase, nor before element $ i$ if it is packed first. So there are at least $i_s^*$ small elements that can be packed subsequently.
Therefore, for $ 2 \leq \ell \leq i_s^* $, a small element is packed as $\ell$-th item.
The claim follows by partitioning the event that $i$ is packed first according to the item $j \in \mathcal{I}_S$ packed as $\ell$-th item.
\end{proof}
\begin{lemma}\label{HL2}
For any two small elements $i, j \in \mathcal{I}_S$ and any $x,y \in [B]$, we have
$ \normalfont \text{Pr}\left[E_{x,y}^{i,j}\right] = \normalfont \text{Pr}\left[E_{x,y}^{j,i}\right] $.
\end{lemma}
\begin{proof}
Consider any input sequence of $ E_{x,y}^{i,j} $ and the sequence resulting from swapping the elements $ i $ and $ j $. Since both elements are not part of the sample, the reference element is not changed by the swap. Therefore, no element that was previously accepted will be rejected and none that was previously rejected will be accepted. Only the order of selection changes.
\end{proof}
\begin{lemma}\label{HL3}
For any small items $i,j,m \in \mathcal{I}_S$ with
$ \mathrm{r}_\mathrm{s}(m) > 1$
and $ \mathrm{r}_\mathrm{s}(i) < \mathrm{r}_\mathrm{s}(m)$,
it holds that
$ \normalfont \text{Pr}\left[E_{1,\mathrm{r}_\mathrm{s}(m)}^{m,j}\right] = \normalfont \text{Pr}\left[E_{1,\mathrm{r}_\mathrm{s}(m)}^{i,j}\right] $.
\end{lemma}
\begin{proof}
Consider any input sequence from $E_{1,\mathrm{r}_\mathrm{s}(m)}^{m,j} $. Since $ \mathrm{r}_\mathrm{s}(i) < \mathrm{r}_\mathrm{s}(m) $ applies, element $ i $ lies behind the element with rank $ m $ in the sequence. If both are selected (see $ i_1 $ in Figure \ref{fig:fig3}), this also applies after they have been swapped (see Lemma~\ref{HL2} and $ m_1 $ in Figure \ref{fig:fig3}). If previously only element $ m $ of the two is packed, only element $ i $ (of the two) is selected after their swapping ($ i_2 $ and $ m_2 $ in Figure \ref{fig:fig3}), since in this case, nothing changes in the reference element either. Therefore $ \normalfont \text{Pr}\left[E_{1,\mathrm{r}_\mathrm{s}(m)}^{m,j}\right] \leq \normalfont \text{Pr}\left[E_{1,\mathrm{r}_\mathrm{s}(m)}^{i,j}\right] $ applies.
Now consider any input sequence from $ E_{1,\mathrm{r}_\mathrm{s}(m)}^{i,j} $.
We show that the element $ m $ lies behind the element $ i $ in the sequence since an element is packed as $z$-th item, where $z= \mathrm{r}_\mathrm{s}(m)$.
Assuming this did not apply and $ m $ is in the sample, then there would be at most $ \mathrm{r}_\mathrm{s}(m)-1 $ small elements that can be packed.
In the case that it occurs in the sequence after the sampling phase, but before element $ i $, there must be a more valuable element in the sample (because $ m $ was not packed) and therefore there are again at most $ \mathrm{r}_\mathrm{s}(m) -1 $ small elements that can be selected.
In particular, in both cases, no element is packed as $z$-th item for $ \mathrm{r}_\mathrm{s}(m) $. This is a contradiction to the fact that we consider an input sequence in $ E_{1,\mathrm{r}_\mathrm{s}(m)}^{i,j} $.
Now, using the same argumentation as in the first case, it follows that $ \normalfont \text{Pr}\left[E_{1,\mathrm{r}_\mathrm{s}(m)}^{m,j}\right] \geq \normalfont \text{Pr}\left[E_{1,\mathrm{r}_\mathrm{s}(m)}^{i,j}\right] $, which completes the proof.
\end{proof}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\textwidth]{Ereignis_3_cases_slot.pdf}
\caption{Occurrence of element $ i $ and $ m $ in event $ E_{1,\mathrm{r}_\mathrm{s}(m)}^{m,j} $ and $ E_{1,\mathrm{r}_\mathrm{s}(m)}^{i,j} $}
\label{fig:fig3}
\end{figure}
Using Lemmas~\ref{HL1} to~\ref{HL3}, we are now able to prove Lemma~\ref{lemmaWkeit}.
\begin{proof}[Proof of Lemma~\ref{lemmaWkeit}]
Let $i \in \mathcal{I}$ be any item.
If $i$ is large, it can only be packed as the first item, thus $ P_i = p_i $.
Now, assume that $i$ is small. It holds that
\begin{equation*}
P_i
= \sum\limits_{x=1}^{B^*} p_i(x)
= \underbrace{\sum\limits_{x=1}^{i_s^*} p_i(x)}_{(*)} + \underbrace{ \sum\limits_{x=\mathrm{r}_\mathrm{s}(i) +1}^{B^*} p_i(x) }_{(**)}
\,.
\end{equation*}
We next simplify both starred terms using Lemmas~\ref{HL1} to~\ref{HL3}.
For $ (*) $, it holds that
\begin{align*}
\sum\limits_{x=1}^{i_s^*} p_i(x) &= p_i(1) + \sum\limits_{x=2}^{i_s^*} \sum\limits_{j \in \mathcal{I}_S} \normalfont \text{Pr}\left[E_{1,x}^{j,i}\right] && \text{(Equation~\eqref{HL0})} \\
&= p_i + \sum\limits_{x=2}^{i_s^*} \sum\limits_{j \in \mathcal{I}_S} \normalfont \text{Pr}\left[E_{1,x}^{i,j}\right] && (\text{Lemma \ref{HL2}}) \\
&= p_i + \sum\limits_{x=2}^{i_s^*} p_i(1) && (\text{Lemma \ref{HL1}}) \\
&= i_s^* \cdot p_i. &&
\end{align*}
For $ (**) $, we obtain
\begin{align*}
\sum\limits_{x=\mathrm{r}_\mathrm{s}(i) +1}^{B^*} p_i(x) &= \sum\limits_{x=\mathrm{r}_\mathrm{s}(i) +1}^{B^*} \sum\limits_{j \in \mathcal{I}_S} \normalfont \text{Pr}\left[E_{1,x}^{j,i}\right] && \text{(Equation~\eqref{HL0})} \\
&= \sum\limits_{x=\mathrm{r}_\mathrm{s}(i) +1}^{B^*} \sum\limits_{j \in \mathcal{I}_S} \normalfont \text{Pr}\left[E_{1,x}^{i,j}\right] && (\text{Lemma \ref{HL2}}) \\
&= \sum\limits_{x=\mathrm{r}_\mathrm{s}(i) +1}^{B^*} \sum\limits_{j \in \mathcal{I}_S} \normalfont \text{Pr}\left[E_{1,x}^{\mathrm{r}'_\mathrm{g}(x),j}\right] && (\text{Lemma \ref{HL3}, }\mathrm{r}_\mathrm{s}(i) < x) \\
&= \sum\limits_{x=\mathrm{r}_\mathrm{s}(i) +1}^{B^*} p_{\mathrm{r}'_\mathrm{g}(x),} && (\text{Lemma \ref{HL1}, }2 \leq x = \mathrm{r}_\mathrm{s}(\mathrm{r}'_\mathrm{g}(x)) \leq B)
\end{align*}
which completes the proof.
\end{proof}
The following corollary is an immediate consequence of Lemma~\ref{lemmaWkeit} for $B=2$.
\begin{corollary}\label{Wkeitencorollary}
For $B=2$, the probability that $\mathsf{ALG}$ packs element $ i \in \mathcal{I} $ is
\begin{equation*}
P_i=
\begin{cases}{}
p_i & \text{ if $i$ is large,} \\
p_i +p_{\mathrm{r}'_\mathrm{g}(2)} & \text{ if $i$ is small and $\mathrm{r}_\mathrm{s}(i)=1$,} \\
2 p_i & \text{ if $i$ is small and $\mathrm{r}_\mathrm{s}(i)>1$} \,,
\end{cases}
\end{equation*}
where, if the second most valuable small item does not exist, we set $p_{\mathrm{r}'_\mathrm{g}(2)}=0$.
\end{corollary}
\section{Proof of Propostion~\ref{prop:no-boost}}
\begin{proof}[Proof of Propostion~\ref{prop:no-boost}]
In the first case, the optimum consists of a single element, thus, $ v(\mathsf{OPT}) = v_1 $. Since $\mathsf{ALG}$ chooses this element with probability $ P_1 \geq p_1 $, we have
\begin{equation}\label{eq:firstCase1}
\mathbb{E}[v(\mathsf{ALG})] \geq p_1 \cdot v_1 = p_1 \cdot v(\mathsf{OPT}) \geq (0.35317\pm o(1)) \cdot v(\mathsf{OPT}) \,,
\end{equation}
where for the last inequality we used Lemma~\ref{wkeitenc} with $c = 0.26888$.
Now, assume that the optimal packing contains two elements $x$ and $y$, where we assume $x<y$ w.l.o.g..
Hence, $v(\mathsf{OPT}) = v_x + v_y$. Note that $x$ and $y$ must be the most profitable items among the set of small items, i.e., $ \mathrm{r}_\mathrm{s}(x)=1 \text{ and } \mathrm{r}_\mathrm{s}(y)=2 $.
Next, we bound the expected profit of the packing.
Since $v_i \geq v_x$ for $1 \leq i \leq x$ and $v_i \geq v_y$ for $x+1 \leq i \leq y$,
we obtain
\[
\mathbb{E}[v(\mathsf{ALG})]
\geq \sum\limits_{i=1}^y (P_i \cdot v_i)
= v_x \cdot \sum\limits_{i=1}^{x} P_i + v_y \cdot \sum\limits_{i=x+1}^{y} P_i \,.
\]
%
Define $\lambda_x := \sum_{i=1}^{x} P_i$ and $\lambda_y := \sum_{i=x+1}^{y} P_i$.
If $\lambda_x < \lambda_y $, then $\lambda_y > \lambda_x \geq p_1$
and thus $\mathbb{E}[v(\mathsf{ALG})] \geq v_x \cdot p_1 + v_y \cdot p_1 = p_1 \cdot v(\mathsf{OPT})$,
which gives the same bound as in \eqref{eq:firstCase1}.
%
Therefore, we assume $ \lambda_x \geq \lambda_y $ in the following.
Since $v_x \geq v_y$, applying Chebyshev's sum inequality gives
\begin{equation*}
\mathbb{E}[v(\mathsf{ALG})]
\geq \left( \dfrac{v_x + v_y}{2}\right) \left( \lambda_x + \lambda_y \right)
=\frac{ v(\mathsf{OPT}) }{2}\cdot\sum\limits_{i=1}^{y} P_i
= \frac{v(\mathsf{OPT})}{2}\cdot \left(2p_y+\sum\limits_{i=1}^{y} p_i \right) ,
\end{equation*}
where the last step results from Corollary \ref{Wkeitencorollary} with $ \mathrm{r}_\mathrm{s}(x)=1 \text{ and } \mathrm{r}_\mathrm{s}(y)=2 $.
%
Let $\theta_y = (1/2) \cdot \sum_{i=1}^{y} p_i +p_y$.
By calculating $p_i$ for $1 \leq i \leq 7$ and $c=0.26888$ using Lemma~\ref{wkeitenc}, we obtain the following upper bounds up to additive $o(1)$ terms:
\begin{center}
\begin{tabular}{c|cccccc}
& $y=2$ & $y=3$ & $y=4$ & $y=5$ & $y=6$ & $y=7$ \\
\hline
$\theta_y$ & 0.4115 & 0.3820 & 0.3718 & 0.3678 & 0.3662 & \textbf{0.3656} \\
\end{tabular}
\end{center}
%
Thus, for $ y \in \{2,\ldots,7\} $ we have $\theta_y \geq \theta_7$ for large-enough $n$.
For $ y \in \{8,...,n\} $, we get $\theta_y = (1/2) \cdot \sum_{i=1}^{y} p_i +p_y
\geq (1/2) \cdot \sum_{i=1}^{7} p_i = \theta_7 - p_7$.
Hence, for any $y \geq 2$ it holds that $\theta_y \geq \theta_7 - p_7 \geq 0.35317\pm o(1)$.
Overall, in the second case it holds that
\begin{equation*
\begin{aligned}
\mathbb{E}[v(\mathsf{ALG})] & \geq \frac{v(\mathsf{OPT})}{2}\cdot \left(2p_y+\sum\limits_{i=1}^{y} p_i \right) \geq (\theta_7 - p_7) v(\mathsf{OPT}) \geq (0.35317\pm o(1))\cdot v(\mathsf{OPT}) \,.
\end{aligned}
\end{equation*}
This completes the proof.
\end{proof}
\subsection{Optimal algorithm through $\alpha$-Boosting}
\label{sec:12KSwithBoosting}
The proof of Lemma~\ref{lemma:notOptimalWithoutBoosting} reveals the bottleneck of Algorithm~\ref{alg:extendedSec}: If the optimal solution consists of two elements having a high rank, the probability of selecting those items is small. This problem can be resolved by the concept of \textit{$\alpha$-boosting}.
\begin{definition}[$\alpha$-boosting]
Let $\alpha \geq 1$ be the \textit{boosting factor}.
For any item $i \in \mathcal{I}$, we define its \textit{boosted profit} to be
\[
v'_i = \begin{cases}
\alpha \cdot v_i & \text{if $i$ is small,} \\
v_i & \text{otherwise.} \\
\end{cases}
\]
\end{definition}
In the following, we investigate Algorithm~\ref{alg:extendedSec} enhanced by the concept of $\alpha$-boosting, denoted by $\mathsf{ALG}_\alpha$. This algorithm works exactly as given in the description of Algorithm~\ref{alg:extendedSec}, but works with the boosted profit $v'_i$ instead of the actual profit $v_i$ for any item $i \in \mathcal{I}$.
Note that the unboosted algorithm analyzed in Proposition~\ref{prop:no-boost} is $\mathsf{ALG}_1$.
For the remainder of this subsection, we fix $c = 1/e$. In particular, this implies $p_1 = 1/e\pm o(1)$ and $p_2 = 1/e^2\pm o(1)$ according to Lemma~\ref{wkeitenc}.
So far, we did not specify the boosting factor $\alpha$.
However, the following intuitive reasoning already shows that $\alpha$ should be bounded from above and below: If $\alpha$ is too large, we risk that $\mathsf{ALG_\alpha}$ packs small items with high probability, even when they are not part of the optimal packing. On the other hand, by the result of Proposition~\ref{prop:no-boost} we know that $\mathsf{ALG}_1$ cannot achieve an optimal competitive ratio.
The following theorem provides lower and upper bounds on $\alpha$ such that~$\mathsf{ALG_\alpha}$ is~$(1/e)$-competitive.
\begin{theorem}
\label{theo:optimalAlg12KS}
For $1$-$2$-knapsack, algorithm $\mathsf{ALG}_\alpha$ is $(1/e-o(1))$-competitive if and only if $ 1.400382 \lesssim \alpha \leq e/(e-1) $ and $c=1/e$, assuming $n\rightarrow\infty$.
\end{theorem}
\begin{proof}
For any item $x \in \mathcal{I}$, let $\rho(x)$ denote the global rank of $x$ after boosting.
On a high level, we need to consider two cases.
In the first case, the optimal packing contains a single item $x$.
If $\rho(x) = 1$, we immediately obtain $\mathbb{E} [v(\mathsf{ALG_\alpha})] \geq p_1 v_x = (1/e) \cdot v(\mathsf{OPT})$.
%
Now, suppose $\rho(x) \geq 2$. Let $a$ and $b$ be the items such that $\rho(a)=1$ and $\rho(b)=2$, respectively.
Hence,
\[v'_a > v'_b \geq v'_x \geq v(\mathsf{OPT}) \,.\]
We note that $a$ is small, as otherwise $v_a = v'_a > v(\mathsf{OPT})$.
Moreover, for $\alpha < 2$, item $b$ is large: If $b$ was small, it would follow that $v'_b = \alpha \cdot v_b$ and therefore~$v_a + v_b = v'_a/\alpha + v'_b/\alpha > (2/\alpha) \cdot v(\mathsf{OPT}) > v(\mathsf{OPT})$, contradicting the assumption that the optimal packing contains a single item.
%
Therefore, $a$ is small and $b$ is large, implying
$v_a = v'_a/\alpha > v(\mathsf{OPT})/\alpha$ and $v_b = v'_b \geq v(\mathsf{OPT})$. Hence,
\begin{align}
\mathbb{E}[v(\mathsf{ALG_\alpha})] \geq &\; p_1 \cdot v_a + p_2 \cdot v_b\label{ineq:UB-alpha}\\
= &\; \left(\frac{1}{e}\pm o(1)\right) \cdot \frac{v(\mathsf{OPT})}{\alpha} + \left(\frac{1}{e^2}\pm o(1)\right) \cdot v(\mathsf{OPT})\nonumber\\
\geq &\; \left(\frac{1}{e}\pm o(1)\right) \cdot v(\mathsf{OPT}) \,,\nonumber
\end{align}
where the latter inequality holds for $\alpha \leq e/(e-1)$. Note that, when $v'_a=1$, $v'_b=1-\varepsilon$, and $v'_z=O(\varepsilon)$ for all other items $y$, Inequality~\eqref{ineq:UB-alpha} becomes satisfied with equality as $\varepsilon\to0$. Therefore, $\mathsf{ALG_\alpha}$ is not $(1/e-o(1))$-competitive when $\alpha > e/(e-1)$.
In the remainder of the proof, we consider the case where the optimal packing contains two small items $x$ and $y$, where we assume $v_x > v_y$ without loss of generality.
We set $j := \rho(x)$ and $k := \rho(y)$, where $1 \leq j < k$. Now, let $a_1,\ldots,a_{j-1}$ and $b_{j+1},\ldots,b_{k-1}$ denote the items appearing before $x$ and between $x$ and $y$, respectively, in the ordered sequence of boosted profits:
\[
v'_{a_1} > \ldots > v'_{a_{j-1}} > v'_x > v'_{b_{j+1}} > \ldots > v'_{b_{k-1}} > v'_y \,.
\]
We observe that neither $a$ items nor $b$ items can be small:
Otherwise, the profit of such an item would be strictly larger than $v_y$,
and as any two small items fit together, this item should be in the optimal packing instead of $y$.
Therefore, we have $v_{a_i} = v'_{a_i} > v'_x = \alpha \cdot v_x$ for all $i\in\{1,\dots,j-1\}$ and $v_{b_i} = v'_{b_i} > v'_y = \alpha \cdot v_y$ for all $i\in\{j+1,\dots,k-1\}$.
Now, we can bound the expected profit of $\mathsf{ALG_\alpha}$ as follows:
\begin{align}
\mathbb{E}[v(\mathsf{ALG_\alpha})]
&\geq \left(\sum_{i=1}^{j-1} P_i \cdot \alpha \cdot v_x \right) + P_j \cdot v_x + \left(\sum_{i=j+1}^{k-1} P_i \cdot \alpha \cdot v_y\right) + P_k \cdot v_y \label{eq:boosting1}\\
&= \left(\sum_{i=1}^{j-1} p_i \cdot \alpha \cdot v_x \right) + (p_j + p_k) \cdot v_x + \left(\sum_{i=j+1}^{k-1} p_i \cdot \alpha \cdot v_y\right) + 2p_k \cdot v_y \nonumber\\
&= \underbrace{\left(p_j + p_k + \alpha \cdot \sum_{i=1}^{j-1} p_i \right)}_{\lambda_x} \cdot\; v_x + \underbrace{\left(2p_k + \alpha \cdot \sum_{i=j+1}^{k-1} p_i \right)}_{\lambda_y} \cdot \;v_y\,,\nonumber
\end{align}
where we use Corollary~\ref{Wkeitencorollary} for the first equality.
If $\lambda_x < \lambda_y$ we immediately get $\lambda_x v_x + \lambda_y v_y > \lambda_x (v_x + v_y) \geq p_1 (v_x + v_y) = (1/e) \cdot v(\mathsf{OPT})$.
Therefore, we assume $\lambda_x \geq \lambda_y$ in the following.
By Chebyshev's sum inequality, it holds that
$\lambda_x v_x + \lambda_y v_y \geq (1/2) \cdot (\lambda_x + \lambda_y) \cdot (v_x+v_y)$.
Therefore, the competitive ratio is
\begin{equation}
\label{eq:crBoosting}
\frac{\mathbb{E}[v(\mathsf{ALG_\alpha})]}{v(\mathsf{OPT})} \geq \frac{\lambda_x + \lambda_y}{2}
= \frac{1}{2} \cdot \left( (1-\alpha)\cdot p_j + 3p_k + \alpha \cdot \sum_{i=1}^{k-1} p_i \right) \,.
\end{equation}
%
If $k=2$, it follows that $j=1$ and therefore Equation~\eqref{eq:crBoosting} resolves to
\[
\mathbb{E}[v(\mathsf{ALG_\alpha})] \geq \frac{1}{2} \cdot ( p_1 + 3p_2 ) \cdot v(\mathsf{OPT}) > \frac{1}{e} \cdot v(\mathsf{OPT}) \,,
\]
which holds independently of $\alpha$.
For $k\geq 3$, $\mathsf{ALG_\alpha}$ is $(1/e-o(1))$-competitive by Equation~\eqref{eq:crBoosting} if
\[
\alpha \geq \frac{2/e - p_j -3p_k}{\sum_{i=1}^{k-1} p_i - p_j} =: \theta_{j,k}\,.
\]
It remains to show $\theta_{j,k} \leq 1.400382$ for all $k \geq 3$ and $j$ with $1 \leq j < k$.
For this purpose, we first show
\begin{equation}
\label{eq:alphaBoundKgeq3}
\theta_{j,k} = \frac{2/e - p_j -3p_k}{\sum_{i=1}^{k-1} p_i - p_j}
\leq \frac{2/e - p_1 -3p_k}{\sum_{i=1}^{k-1} p_i - p_1}
= \frac{1/e -3p_k}{\sum_{i=2}^{k-1} p_i}\pm o(1)
~~~~~~~\text{for any $k \geq 3$.}
\end{equation}
Since $p_j$ is decreasing in $j$, the inequality in Equation~\eqref{eq:alphaBoundKgeq3} follows immediately if we can show
$2/e - 3p_k > \sum_{i=1}^{k-1} p_i$ for large-enough $n$.
This inequality is easily verified for $k=3$, as $2/e - 3p_3 > p_1 + p_2$, for large-enough $n$.
For $k \geq 4$, note that $p_k < p_1 - 1/3$, again for large-enough $n$, which is equivalent to $2/e - 3p_k > 1 - p_1$.
Using Observation~\ref{obs:sumOfPi}, we obtain
$\sum_{i=1}^{k-1} p_i < \sum_{i=1}^{n} p_i = 1-c = 1-p_1$.
Combining both inequalities yields Equation~\eqref{eq:alphaBoundKgeq3}.
By computing the last term in Equation~\eqref{eq:alphaBoundKgeq3} for $3 \leq k \leq 10$, we obtain the upper bounds on $\theta_{j,k}$ given in Table~\ref{tab:tabelle3x}, up to additive $o(1)$ terms.
Note that the maximum value is 1.400382.
For $k \geq 11$, we obtain from Equation~\eqref{eq:alphaBoundKgeq3} together with $p_i \geq 0$ for all $i \geq 11$ that
\[
\theta_{j,k} \leq \frac{1/e -3p_k}{\sum_{i=2}^{k-1} p_i}
\leq \frac{1/e}{\sum_{i=2}^{11-1} p_i}
< 1.398875\pm o(1) \,.
\]
\begin{table}[t]
\centering
\caption{Upper bounds on $\theta_{j,k}$ for $ 3 \leq k \leq 10 $ according to Equation~\eqref{eq:alphaBoundKgeq3}.}
\label{tab:tabelle3x}
\begin{tabular}{c|cccccccc}
\toprule
$k$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\midrule
$\frac{1/e -3p_k}{\sum_{i=2}^{k-1} p_i}$ & 1.3475 & 1.3962 & \textbf{1.400382} & 1.3988 & 1.3968 & 1.3952 & 1.3941 & 1.3934 \\
\bottomrule
\end{tabular}
\end{table}
For the lower bound of approximately $1.400382$ on $\alpha$, first note that for $j=1$ and $k=5$, it holds indeed that
\begin{align*}
\theta_{1,5} &=
\dfrac{2/e-p_1-3p_5}{\sum_{i=1}^{5-1} p_i-p_1} = \dfrac{1/e-3p_5}{p_2+p_3+p_4} \pm o(1)\\
&= -\dfrac{51}{16} + \dfrac{9}{4e} + \dfrac{75 - 522 e + 486 e^2}{16 - 96 e + 288 e^2 - 64 e^3}\pm o(1)
\approx 1.400382\pm o(1)\,.
\end{align*}
Next, note that setting $v'_x$,$v'_{b_2}$,$v'_{b_3}$,$v'_{b_4}$, and $v'_y$ all equal to $1+O(\varepsilon)$ and $v'(z)=O(\varepsilon)$ for all other items $z$ makes Inequality~\eqref{eq:boosting1} as well as Inequaltiy~\eqref{eq:crBoosting} tight as $\varepsilon\rightarrow0$. Therefore, the above arguments imply that $\alpha\geq \theta_{1,5}$ if and only if $\mathsf{ALG_\alpha}$ is $(1/e-o(1))$-competitive. This completes the proof.
\end{proof}
\subsection{Related Work}
Kleinberg~\cite{Kleinberg05} first considers $k$-secretary as introduced above, gives an algorithm with competitive ratio $1-\Theta(1/\sqrt{k})$, and shows that this ratio is asymptotically best possible. This result is reproduced by Kesselheim et al.~\cite{KesselheimRTV18} in the more general context of packing LPs. Buchbinder et al.~\cite{BuchbinderJS14} consider the~$(j,k)$-secretary problem in the ordinal setting in which $j$ items can be selected and the goal is to maximize the expected ratio of elements selected from the top $k$ items. They also state the algorithm-design problems as linear programs, which they can only solve for small values of $j$ and $k$, but Chan et al.~\cite{ChanCJ15} can solve them for larger values. Any guarantee for the $(k,k)$-secretary problem carries over to the $k$-secretary problem, but Chan et al.~\cite{ChanCJ15} rule out the other direction. More specifically, Chan et al.'s results include an optimal algorithm for $(2,2)$-secretary with guarantee approximately $0.489$ and a (not necessarily optimal) algorithm for $2$-secretary with guarantee approximately $0.492$. Albers and Ladewig~\cite{AlbersL21} revisit the problem and give simple algorithms with improved (albeit non-optimal) competitive ratios for many fixed values of~$k$.
The knapsack secretary problem is introduced by Babaioff et al.~\cite{BabaioffIKK07} who give a~$1/(10e)$-competitive algorithm, which was subsequently improved by Kesselheim et al.~\cite{KesselheimRTV18} to $1/8.06$ and by Albers, Khan, and Ladewig~\cite{AlbersKL21} to $1/6.65$. Essentially all known $\Omega(1)$-competitive algorithms for the knapsack secretary problem are somewhat wasteful in the competitive ratio, presumably at least partially for the sake of a simpler analysis, in that they randomize between different algorithms that are tailored to respective item sizes. It seems that qualitative progress can only be made by a more fine-grained analysis avoiding such case distinctions.
A variant of the knapsack secretary problem that has recently been considered is the fractional variant in which an item can also be packed fractionally, avoiding situations in which an arriving item cannot be selected at all, even when there is space. The currently best known achievable competitive ratio is $1/4.39$~\cite{GilibertiK21}, also achieved by a blended approach.
It is not difficult to see that no constant competitive ratio can be achieved when the items do not arrive in random but in adversarial order, even in the unit-value case~\cite{Marchetti-SpaccamelaV95}. Starting from this problem, problems in which other assumptions than the order are relaxed are considered as well. For instance, Zhou et al.~\cite{ZhouCL08} consider the version in which each item has a small size; Böckenhauer et al.~\cite{BockenhauerKKR14} and Boyar et al.~\cite{BoyarFL22} introduce advice and untrusted predictions, respectively, to the problem.
Lower bounds for secretary problems in the value setting are rare. For some related problems~\cite{CorreaDFS19,CorreaDFSZ21,abs-2011-01559}, the rich class of strategies can be handled by, for any strategy, identifying an infinite set of values (using Ramsey theory) on which it is much better behaved. It is, however, not clear how such an approach could be applied, e.g., for knapsack secretary since it seems one would need to control how the values in the support are spread out, a property that is irrelevant in the other settings.
\subsection{Our Contribution}
The special case $1$-$2$-knapsack is not only arguably the simplest special case that exhibits features of the knapsack problem distinguishing it from the matroid secretary problem. Since the problem generalizes both the standard secretary problem and $2$-secretary, we believe that settling it in terms of the achievable competitive ratio is also interesting per se.
A good starting point for tackling $1$-$2$-knapsack seems to be the extended secretary algorithm, which is $1/3.08$-competitive in the slightly more general case when all items have size larger than $B/3$~\cite{AlbersKL21}. This algorithm simply ignores the item sizes, samples some prefix of length $cn$ for some optimized constant~$c\in(0,1)$, and afterwards selects all items that surpass the largest value from the sampling phase and that can still be feasibly packed. It is, however, easy to see that this approach cannot achieve $1/e$: Achieving $1/e$ in an instance where the optimal solution consists of a large item requires setting $c=1/e\pm o(1)$. The resulting algorithm will, however, not be $1/e$-competitive in an instance where the optimal solution consists of two small items of equal value, but there are many large items, each slightly more valuable than the individual small items, making sure that the small items are (almost) never selected by the algorithm. In this case, the competitive ratio of the algorithm will be essentially half the probability that the algorithm selects a (large) item, that is, $(1-1/e)/2<1/e$. We denote two instances of the above forms by $\mathcal{I}_1$ and $\mathcal{I}_2$, respectively, in the following. Clearly, it is possible to choose $c$ so as to balance between $\mathcal{I}_1$ and $\mathcal{I}_2$. As a small side result, we show that a ratio of approximately $0.353<1/e$ can be achieved that way.
The key observation leading to our $1/e$-competitive algorithm is that keeping~$c=1/e$ and internally multiplying (\emph{boosting}) values of small items with a suitable constant factor $\alpha>1$ prior to running the extended secretary algorithm may handle both $\mathcal{I}_1$ and $\mathcal{I}_2$: While this is clear for $\mathcal{I}_1$ when the ranking of values does not change through boosting, a small item may overtake the most valuable (large) item. This however means that this small item has relatively large (actual) value. Using that the algorithm also accepts the second-best item with a significant probability ($1/e^2$), we can show that, with the right choice of $\alpha$, we still extract enough value from the small and large items to cover $1/e\cdot v(\mathsf{OPT})$. In $\mathcal{I}_2$, the small items would overtake the large items, significantly improving the expected value achieved by the algorithm; conversely, if they did not overtake, they would not have been harmfully valuable in the first place---again with the right choice of $\alpha$. To sum up, ``$\mathcal{I}_1$ type'' instances impose an upper bound on $\alpha$, and ``$\mathcal{I}_2$ type'' instances impose a lower bound on $\alpha$. We show that the algorithm is~$1/e$-competitive if \emph{and only if} $1.40\lesssim\alpha\leq e/(e-1)\approx 1.58$ where the upper bound comes essentially from the above consideration for $\mathcal{I}_1$. Note that therefore, in particular, our boosting \emph{is} different from ordering the items by their ``bang for the buck'' ratios.
We note that, while $\alpha$-boosting seems reminiscent of $\beta$-filtering~\cite{ChanCJ15} (for $\beta<1$), applying $\beta$-filtering to the extended secretary algorithm will not yield a $1/e$-competitive algorithm. The extended secretary algorithm would be adapted by ignoring items with a value less than $\beta$ times the highest value seen so far. Note that indeed, a ``$\mathcal{I}_1$ type'' instance where all but the most valuable item have a similar small value, one would have to choose $c=1/e\pm o(1)$ again, independently of $\beta$. But such an algorithm would again only be $(1-1/e)/2$-competitive on $\mathcal{I}_2$.
The crux of our analysis is distinguishing all possible cases beyond those covered by $\mathcal{I}_1$ and $\mathcal{I}_2$ in a smart way. To bound the algorithm's value in each of these cases, we precisely characterize the probabilities with which the algorithm selects an item depending on its size and its position in the (boosted) order of values, significantly extending observations made by Albers and Ladewig~\cite{AlbersL21}.
Before tackling the general case and understanding potentially complicated knapsack configurations, we propose considering a clean special case called \mbox{$1$-$B$}-knapsack where items have sizes either $1$ or $B$, and $B$ is large. One may be tempted to think that this special case is difficult in that selecting a small item early on may lead to a blocked knapsack and a horribly inefficient use of capacity, e.g., because all other items are large. On the other hand, when $B$ is large, one can easily avoid such situations by sampling. We do not give a conclusive answer on whether $1/e$ can be matched in this case, but we give some preliminary results.
Unfortunately, a competitive ratio of $1/e$ for $1$-$B$-knapsack cannot be achieved with our boosting approach. The same consideration we made for $\mathcal{I}_1$ earlier (for~\mbox{$1$-$2$}-knapsack) to get an upper bound of $e/(e-1)$ on $\alpha$ still works; in contrast, a generalization of $\mathcal{I}_2$ rules out any constant boosting factor.
We then give another algorithm for $1$-$B$ knapsack which can be viewed as a linear interpolation between the classic secretary algorithm and the algorithm by Kleinberg~\cite{Kleinberg05} for $k$-secretary. We show that it is~$1/(e+1)$-competitive. This algorithm turns out to be \emph{ordinal}, that is, its decisions only depend on the item sizes and the relative order of their values. Remarkably, we are able to show that~$1/(e+1)$ is the best-possible guarantee such algorithms can achieve. We do so by generalizing the factor-revealing linear program due to Buchbinder et al.~\cite{BuchbinderJS14} by adding variables and constraints. Arguing that the LP indeed models our problem becomes more difficult because, in contrast to the setting of Buchbinder et al., at any time, even the size of the next item is random. We do so by showing reductions between our model and an auxiliary batched-arrival model.
\section{Introduction}
\label{sec:intro}
\input{intro}
\section{Preliminaries}
\label{sec:prelims}
\input{prelims}
\section{Matching $1/e$ for 1-2-Knapsack}
\label{sec:1-2}
\input{algorithm}
\input{no-boosting}
\input{boosting}
\section{Ordinal Algorithms for 1-$B$-Knapsack}
\label{sec:ordinal}
\input{impossible}
\section{Conclusion}
\label{sec:conclusion}
\input{conclusion}
\newpage
\bibliographystyle{abbrv}
\subsection{First approach: Without Boosting}
\label{sec:12KSwithoutBoosting}
In this subsection, we study Algorithm~\ref{alg:extendedSec} (as is) for 1-2-knapsack. Unfortunately, there are two instances such that it is impossible to choose the parameter $c$ so that Algorithm~\ref{alg:extendedSec} is $(1/e)$-competitive on both instances.
\begin{lemma}
\label{lemma:notOptimalWithoutBoosting}
For 1-2-knapsack, the competitive ratio of $\mathsf{ALG}$ is at most $0.35767$,
assuming $n \to \infty$.
\end{lemma}
\begin{proof}
Let $ 1 > \epsilon > 0 $ be a constant. We define two instances $\mathcal{I}_1$ and $\mathcal{I}_2$.
In the first instance $\mathcal{I}_1$, all items are large and only one item has substantial profit. Formally, let $v_1 = 1$, $v_i = \epsilon^i$ for $2 \leq i \leq n$, and $s_i = 2$ for all $1 \leq i \leq n$. Then, for instance $\mathcal{I}_1$,
\begin{equation}
\label{eq:extSecAlgUBI1}
\lim_{\epsilon \to 0}~
\mathbb{E}[v(\mathsf{ALG})] = P_1 \cdot v_1 = p_1 \cdot v(\mathsf{OPT}).
\end{equation}
In the second instance $\mathcal{I}_2$, most items are large and essentially of the same profit. However, the optimal packing contains two small items that appear at ranks $n-1$ and $n$. Formally, set $s_i = 2$ for $1 \leq i \leq n-2$, $s_{n-1} = s_{n} = 1$, and~$ v_i = 1 + \epsilon ^i $ for all $ i \in \{1,\ldots,n\} $.
%
As item $n$ never beats any reference item, we have $P_n = 0$. Hence, the algorithm selects only items from $\{1,\ldots,n-1\}$ with positive probability, and always at most one item. For instance $\mathcal{I}_2$, we get
\begin{align}
&\quad\lim_{\epsilon \to 0}~
\mathbb{E}[v(\mathsf{ALG})]
= \lim_{\epsilon \to 0}~ \sum_{i=1}^{n} (P_i \cdot (1+\epsilon^i))
= \sum_{i=1}^{n} p_i\nonumber\\
\overset{\text{Obs.~(\ref{obs:sumOfPi})}}{=}&\quad 1 - c
\leq \frac{1-c}{2} \cdot v(\mathsf{OPT}).\label{eq:extSecAlgUBI2}
\end{align}
Overall, by Equations~\eqref{eq:extSecAlgUBI1} and~\eqref{eq:extSecAlgUBI2}, the competitive ratio as $n\to\infty$ of $\mathsf{ALG}$ is bounded from above by
\begin{equation*}\label{eq:c}
\max\limits_{c \in (0,1) } \min \left\{ p_1,\dfrac{1-c}{2} \right\} = \max\limits_{c \in (0,1) } \min \left\{ c \cdot \ln \dfrac{1}{c},\dfrac{1-c}{2} \right\} \leq 0.35767 \,.
\end{equation*}
This completes the proof.
\end{proof}
As a small side result, we show that this bound is almost tight. The techniques are similar to those used for our main result and presented in the full version of the paper.
\begin{proposition}\label{prop:no-boost}
For $1$-$2$-knapsack, the competitive ratio of $\mathsf{ALG}$ is $ 0.35317-o(1)$,
setting $c=0.26888$ and assuming $n\rightarrow\infty$.
\end{proposition}
|
1,116,691,498,420 | arxiv | \section{Introduction}
Internal symmetries are a familiar feature in quantum field theory with many established properties. For example, places where symmetry realizations change can be associated with the emergence of gapless excitations. Often, the realizations of internal symmetries are constrained by 't Hooft anomaly matching. Additionally, in relativistic QFTs, the Coleman-Mandula theorem \cite{Coleman:1967ad} implies that continuous internal symmetries commute with the Poincare group so that the full symmetry group of the theory, $G$, is a direct product: $G = G_{\rm Poincare} \times G_{\rm internal}$.
All of these features are illustrated in QCD. QCD with $N\geq 3$ colors has a $G_{\rm internal} = [SU(N_F)_V \times SU(N_F)_A \times U(1)_Q]/\left( \mathbb{Z}_{N_F} \times \mathbb{Z}_{N} \right)$ flavor symmetry in the chiral limit where $m_q = 0$, and $G_{\rm QCD} = G_{\rm Poincare} \times G_{\rm internal}$. The $SU(N_F)_A$ part of the internal symmetry group has an 't Hooft anomaly. This can be used to argue that when $m_q=0$ the low-energy effective theory describing fluctuations about the thermodynamic ground state must include some gapless degrees of freedom. For some values of $N_{F}$ and $N$, these gapless degrees of freedom are associated with spontaneous breaking of the $SU(N_F)_A$ symmetry.
Here our focus will be on pure $SU(N)$ Yang-Mills theory
\begin{align}
S = \frac{1}{4g^2} \int d^4x~F^a_{\mu\nu}F^{a\mu\nu} + i \frac{\theta}{16\pi^2} \int d^4x~\epsilon^{\mu\nu\rho\sigma} F^a_{\mu\nu}F^a_{\rho\sigma} \, ,
\end{align}
with $\mu,\nu = 1,\ldots,4$ and $a=1,\ldots,N$. Pure YM theory has no conventional internal symmetries which would act on local operators. It has long been known, however, that it does have a subtler type of internal symmetry, $G_{\rm internal} = \mathbb{Z}_{N}$ center symmetry \cite{Polyakov:1978vu,Susskind:1979up,Gross:1980br,Weiss:1981ev}.
Center symmetry acts non-trivially on certain line operators, but it does not act on local operators.
In the language of \cite{Gaiotto:2014kfa} center symmetry is a ``1-form symmetry", which can be contrasted with e.g. the chiral symmetry of QCD, which is a ``0-form symmetry" whose natural charged objects are local operators.
It turns out that just as with more familiar 0-form symmetries, center symmetry can participate in 't Hooft anomalies \cite{Gaiotto:2017yup}.
In particular there is a mixed 't Hooft anomaly between center symmetry and $CP$ symmetry at $\theta = \pi$ for even $N$, and a closely related notion of ``global inconsistency" for odd N \cite{Gaiotto:2017yup,Tanizaki:2017bam}.
If the conclusions of the Coleman-Mandula theorem were to apply to center symmetry, then center symmetry would commute with $G_{\rm Poincare}$. However, one cannot appeal to this theorem for two reasons. First, the Coleman-Mandula theorem is derived for continuous internal symmetries, while the center symmetry of $SU(N)$ YM theory is discrete. Second, the Coleman-Mandula theorem follows from working out the constraints of symmetries on the S-matrix for relativistic particle scattering, while the charged objects for center symmetry are associated to string-like extended operators.
Indeed, we find that for pure $SU(N)$ YM theory on $\mathbb{R}^{3,1}$, the full symmetry group $G_{\rm YM}$ is generally \emph{not} a direct product:
\begin{align}
G_{\rm YM} \neq G_{\rm Poincare} \times G_{\rm internal}.
\end{align}
In particular, when $N\geq 3$ center symmetry transformations do not commute with a simultaneous transformation of parity and time reversal, $PT$, or with charge conjugation $C$.\footnote{This was noted but not explored in \cite{Aitken:2017ayq}.} $PT$, $C$, and center transformations are symmetries of YM theory for all values of $g$ and $\theta$, so these two symmetries generate a discrete non-Abelian subgroup of $G^{\rm disc}_{\rm YM} \subset G_{\rm YM}$ for $N\geq 3$. However, we will see that the nature of $G^{\rm disc}_{\rm YM}$ depends on $\theta$.
We will show that when $SU(N)$ YM theory is compactified on $\mathbb{R}^3\times S^1$, the discrete 0-form symmetries fit into the group
\begin{align}
G^{\rm discrete}_{\rm YM} = \begin{cases}
D_{2N} \times \mathbb{Z}_2 \times \mathbb{Z}_2
& \theta=0\; \mathrm{mod}\; 2\pi \\
D_{4N} \times \mathbb{Z}_2 \times \mathbb{Z}_2
& \theta=\pi\; \mathrm{mod}\; 2\pi \\
D_{2N} & \text{otherwise}
\end{cases}
\label{eq:ym_theta_sym}
\end{align}
Here, $D_{2N}$ is the dihedral group of symmetries of a regular planar $N$-gon. The dihedral group involves the 0-form part of center symmetry, which acts on Wilson loops which wind around $S^1$, as well as charge conjugation. The $\mathbb{Z}_2 \times \mathbb{Z}_2$ factors are related to parity and time-reversal symmetries. Compactification on a circle simplifies the discussion but is not essential, see Sec. \ref{sec:symmetry} for a discussion concerning the symmetries on $\mathbb{R}^4$.
\begin{figure}[t]
\begin{center}
\includegraphics[width = \textwidth]{symm_fig.pdf}
\caption{[Color Online.] A summary of the symmetries of $SU(N)$ YM theory (right) and of a related $T_N$ toy model from quantum mechanics (left), as a function of $\theta$. }
\label{fig:symm_fig}
\end{center}
\end{figure}
The rest of the paper is concerned with illustrating how these symmetries behave in two different calculable settings. First, we discuss a simple quantum-mechanical toy model in Sec.~\ref{sec:T_N_model} where many of the ideas can be appreciated in the simplest possible context. In Sec.~\ref{sec:YM}, we then explore the symmetries of a calculable deformation of YM theory obtained by a compactifion on a small circle with stabilized center symmetry. This semiclassically calculable regime was uncovered in \cite{Unsal:2008ch}, and intensively explored in related works, see e.g.
\cite{
Unsal:2007vu,Unsal:2007jx,Shifman:2008ja,Shifman:2008cx,Shifman:2009tp,
Cossu:2009sq,Myers:2009df,Simic:2010sv,Unsal:2010qh,Azeyanagi:2010ne,Vairinhos:2011gv,
Thomas:2011ee,Anber:2011gn,Unsal:2012zj,
Poppitz:2012sw,
Poppitz:2012nz,Argyres:2012ka,Argyres:2012vv,
Anber:2013doa,Cossu:2013ora,
Anber:2014lba,Bergner:2014dua,Bhoonah:2014gpa,Li:2014lza,
Anber:2015kea,Anber:2015wha,Misumi:2014raa,
Cherman:2016hcd,Aitken:2017ayq,Anber:2017rch,Anber:2017pak,Anber:2017tug,Anber:2017ezt}. A comparison of the symmetries between our QM toy model and $SU(N)$ YM theory is given in Fig.~\ref{fig:symm_fig}. Our results are summarized in Sec.~\ref{sec:outlook}, and end with some appendices with details on some of our calculations.
In a companion paper \cite{Aitken:2018mbb}, we further explore the vacuum properties of the deformed YM theory as a function of $\theta$.
\section{Non-Abelian global symmetry of YM theory}
\label{sec:symmetry}
In this section we argue that the discrete part of the symmetry group of YM theory $G^{\rm discrete}_{\rm YM}$ includes the dihedral group $D_{2N}$. This involves showing that center symmetry does not commute with charge conjugation $C$. Equivalently, center symmetry does not commute with $PT$ symmetry; our discussion below will only explicitly refer to $C$ for simplicity.
Since center symmetry does not act on any local operators, a non-trivial check of the symmetry group generated by center symmetry and charge conjugation will involve consideration of line operators. For simplicity of exposition, we work in Euclidean space. We will first discuss the symmetries on $\mathbb{R}^3 \times S^1$, and then comment on the generalization to $\mathbb{R}^4$.
First, take spacetime to be $\mathbb{R}^3 \times S^1$, with $S^1$ the $x_4$ direction. The zero-form part of center symmetry acts non-trivially on `Polyakov loops' --- Wilson loops wrapping the circle $\,{\rm tr}\, \Omega= \,{\rm tr}\, P \exp(i \oint dx_4 A_4)$, which are local with respect to $\mathbb{R}^3$.
The action of center symmetry is
\begin{align}
\mathcal{S} \cdot \,{\rm tr}\, \Omega = \omega \,{\rm tr}\, \Omega.
\end{align}
where the exponent of $\omega = e^{2\pi i /N}$ is the charge of $\,{\rm tr}\, \Omega$, and we have denoted the operator implementing center symmetry transformations by $\mathcal{S}$. Of course, $\mathcal{S}^N=1$.
The theory is invariant under charge conjugation symmetry $\mathcal{C}$ at arbitrary $\theta$-angle since topological term
respects $\mathcal{C}$.
Charge conjugation maps ${\cal C}: \Omega \to \Omega^{\dag}=\Omega^{-1}$ so that $\mathcal{C}^2=1$. Let us now work out the group obeyed by $\mathcal{S}$ and $\mathcal{C}$. One can then verify that
\begin{align}
{\cal C} \cdot \mathcal{S} \cdot {\cal C} \cdot \,{\rm tr}\, \Omega = \mathcal{S}^{-1} \cdot \,{\rm tr}\, \Omega.
\end{align}
Thus ${\cal C}$ and $\mathcal{S}$ do not commute. In fact, they obey the defining relations of the dihedral group of symmetries of a regular planar $N$-gon,
\begin{align}
D_{2N}=\{\mathcal{S},\,{\cal C}\; | \; \mathcal{S}^N = 1, \, {\cal C}^2 = 1, \, {\cal C} \, \mathcal{S} \, {\cal C} = \mathcal{S}^{-1} \}.
\label{eq:d2N_algebra}
\end{align}
At $\theta = 0$ YM theory has parity $\mathcal{P}: x_{j} \to -x_{j}, j=1,2,3$ and $x_4$-reflection $\mathcal{R}: x_4 \to L-x_4$ symmetries. There is also an $SO(3)$ Lorentz symmetry associated with the non-compact directions. It is easy to see that center symmetry also does not commute with ${\mathbb R}$ because its behavior when acting on $\,{\rm tr}\, \Omega$ is analogous to charge conjugation, ${\mathbb R} \cdot \,{\rm tr}\, \Omega = \,{\rm tr}\, \Omega^{-1}$. But $[\mathcal{S},\mathcal{P}] =[{\cal C},{\mathbb R}]=[{\cal C},\mathcal{P}] = 0$. However, while ${\mathbb R}$ and $\mathcal{P}$ are manifestly symmetries at $\theta = 0$, they are not symmetries for generic $\theta \neq 0, \pi$.
At $\theta = \pi$ there is either a mixed 't Hooft anomaly or a global inconsistency between center and $CP$ symmetries \cite{Gaiotto:2017yup}, depending on whether $N$ is even or odd. Assuming that center symmetry is not spontaneously broken for all $\theta$, when there is mixed 't Hooft anomaly at $\theta = \pi$ there are two possibilities for the vacuum structure: (1) $CP$ is spontaneously broken or (2) there is a nontrivial topological field theory which matches the anomaly in the IR limit. A global inconsistency condition at $\theta = \pi$ is slightly weaker, and in addition to the two options above, can also be satisfied if there are phase transitions away from $\theta = 0,\pi$ \cite{Gaiotto:2017yup,Tanizaki:2017bam}.
Especially for large $N$, spontaneous breaking of $CP$ at $\theta = \pi$ seems like the most probable way these anomaly/inconsistency conditions would be satisfied, and we assume this is the case in writing expressions in the $\mathbb{R}^4$ limit. On $\mathbb{R}^3 \times S^1$, $CP$ breaking can be shown explicitly. We demonstrate the anomaly/global inconsistency conditions at $\theta = \pi$ imprint themselves on the symmetry group by leading to a central extension. So at $\theta = \pi$, the discrete global symmetry contains a factor of $D_{4N}$, the double-cover of $D_{2N}$.\footnote{The fact that the symmetry group of $SU(2)$ YM theory involves a $D_8$ factor at $\theta = \pi$ was discussed in \cite{Gaiotto:2017yup}.} Taken together, these considerations imply the claim from the introduction in \eqref{eq:ym_theta_sym}.
Note that the dihedral group $D_{4}$ is isomorphic to the Abelian group $\mathbb{Z}_2 \times \mathbb{Z}_2$, but $D_{2N}$ is non-Abelian for all $N>2$. So when $N=2$ the discrete symmetry group of YM theory is Abelian for $\theta \neq \pi$, and becomes non-Abelian only when $\theta =\pi$. However, for all $N > 2$, $G^{\rm discrete}_{\rm YM}$ is non-Abelian for all $\theta$, and the group of zero-form symmetries $\mathcal{S}, \mathcal{P}, {\mathbb R}, {\cal C}$ will be shown to take the form \eqref{eq:ym_theta_sym}.
We now turn our attention to $\mathbb{R}^4$. Here it is helpful to adopt the language of \cite{Gaiotto:2014kfa}, in which center symmetry is viewed as a $p$-form symmetry with $p=1$. The charges of $p$-form symmetries are measured by integrating conserved $d-p-1$ currents on closed $d-p-1$-dimensional manifolds, and are associated to charges of operators supported on manifolds of dimension $p$. The charge of such an operator is non-zero when its worldvolume manifold has non-vanishing linking number with some $d-p-1$-dimensional manifold where one puts the operator generating the symmetry.
In the case of $1$-form center symmetry, the basic charged operators are Wilson lines with appropriate topological properties. In particular, consider an open Wilson line defined on a curve $\gamma$ whose ends go off to infinity in different directions, for instance along $x_4 \to \pm \infty$. One can think of such a line operator $\Omega(\gamma)$ as being associated with a probe fundamental quark-anti-quark pair with separation taken to infinity, with e.g. the quark going to $x_4 \to +\infty$ and the anti-quark going to $x_4 \to -\infty$. (If the $x_4$ direction is compactified to $S^1$, this open Wilson line becomes precisely the Polyakov loop considered earlier.) For our purposes, it will be useful to associate the operator generating center symmetry with the closed two-dimension surface $\Sigma_2$ which spans the $x_1$-$x_2$ plane. In this case, center symmetry acts on $\,{\rm tr}\, \Omega(\gamma)$ as \cite{Gaiotto:2014kfa}
\begin{align}
\mathcal{S} \cdot \,{\rm tr}\,\Omega(\gamma) = \omega^{\ell (\gamma, \Sigma_2) } \,{\rm tr}\,\Omega(\gamma) = \omega^{+1} \,{\rm tr}\,\Omega(\gamma) \,.
\end{align}
where $ \ell (\gamma, \Sigma_2) $ is the linking number of $\gamma$ with $\Sigma_2$ \cite{Gaiotto:2014kfa}, which is
$+1$ in the case above.
Now consider charge conjugation. This symmetry interchanges quarks and anti-quarks, so it acts on $\Omega(\gamma)$ as
\begin{align}
\mathcal{C} \cdot \,{\rm tr}\, \Omega(\gamma)= \,{\rm tr}\,\Omega(\gamma)^{\dag}= \,{\rm tr}\, \Omega(\gamma^{-1})
\end{align}
so $\mathcal{C}$ flips the orientation of $\gamma$. Flipping the orientation of $\gamma$ flips the sign of the linking number of $\gamma$ with $\Sigma_2$, $ \ell (-\gamma, \Sigma_2) = - \ell (\gamma, \Sigma_2) $. The operator group then follows as before,
\begin{align}
{\cal C} \cdot \mathcal{S} \cdot {\cal C} \cdot \,{\rm tr}\,\Omega(\gamma) = \omega^{-1} \,{\rm tr}\,\Omega(\gamma) = \mathcal{S}^{-1} \cdot \,{\rm tr}\,\Omega(\gamma) .
\end{align}
Thus ${\cal C}$ and $\mathcal{S}$ do not commute on $\mathbb{R}^4$. It is also easy to see that $\mathcal{S}$ does not commute with ${\mathbb R}$, the $\theta = 0, \pi$ symmetry operator which now maps $x_4 \to -x_4$.
Rather trivially, symmetries of quantum systems can be associated with groups because, given some state $\ket{\psi}$ in Hilbert space which transforms non-trivially under a symmetry, one can verify that the symmetry action obeys the group axioms. In our case, choosing $\ket{\psi}=\,{\rm tr}\,\Omega \ket{0}$ our remarks above imply that the actions of the ${\cal C}$ and $\mathcal{S}$ transformations obey the group axioms, and combine into the symmetry group $D_{2N}$. Nevertheless, we are dealing with the somewhat unusual situation of considering the combination of a 0-form symmetry and a 1-form symmetry. As this paper was being prepared for submission, Refs.~\cite{Cordova:2018cvg,Benini:2018reh} appeared (see also e.g. Ref.~\cite{Kapustin:2013uxa}), where it is argued that the general algebraic structure appropriate to discuss the mixture of $0$-form and $1$-form symmetries is a ``$2$-group"\cite{BaezLauda}.~\footnote{We are grateful to K.~Jensen and S.~Gukov for discussions on this point.}
\subsection{Physical consequences}
We now comment on some physical consequences of the existence of the dihedral non-Abelian symmetry in $SU(N)$ YM theory. The fact that charge conjugation and center symmetry do not commute means that the associated charge operators cannot be simultaneously diagonalized. This means that if one considers a state that transforms non-trivially under both center and charge conjugation symmetry, one cannot simultaneously specify its center symmetry and charge conjugation quantum numbers. Of course, this means that the existence of the $D_{2N}$ symmetry does \emph{not} imprint itself on the correlation functions of local operators. One must consider correlation functions of appropriate line operators to see the symmetry.
For example, consider $SU(N)$ YM theory with $N>2$ on $\mathbb{R}^3\times S^1$. Finite-energy states transforming under center symmetry can be built out of Wilson loops wrapping $S^1$. Then one can consider scattering amplitudes involving such states, for example at $\theta = 0$. Suppose we choose to specify the center labels of the states. Then the fact that one cannot simultaneously specify the center and charge conjugation quantum numbers --- which is due to the existence of the $D_{2N}$ symmetry --- means that one has to sum over the $C$ quantum numbers for both incoming and outgoing states in computing the scattering amplitudes.
At high temperature, center symmetry is spontaneously broken in pure YM theory. It would be interesting to understand the physical implications of the non-commutativity of center symmetry and e.g. $PT$ symmetry in this setting.
\section{Dihedral symmetries in a quantum mechanical model}
\label{sec:T_N_model}
As a warm up for studying the symmetries and dynamics of $SU(N)$ gauge theory as a function of $\theta$, we will first consider the quantum mechanical system of a particle on a circle, $q(t) = q(t)+2\pi$, in the presence of a potential with $N$ degenerate minima. This class of models are referred to as $T_{N}$ models in \cite{Unsal:2012zj}, where their non-perturbative properties were examined semi-classically.
The Euclidean action of the model is
\begin{align}
S_{T_N}(g,\theta) = \frac{1}{g^{2}}\int dt\,\left[\frac{1}{2}\dot{q}^{2}-\cos\left(Nq\right)\right]-i\frac{\theta}{2\pi}\int dt\,\dot{q}.
\label{eq:tN_definition}
\end{align}
The potential has $N$ degenerate minima at $q_n = \tfrac{2\pi n}{N}, n = 0,1, \cdots N-1$. But the system does not have $N$ degenerate ground states: tunneling/instanton effects typically lift the degeneracies seen in perturbation theory. However, this does not mean that the ground state is always unique. For some values of $\theta$, it turns out to be doubly degenerate. We discuss the ground state structure below from a perspective that we will find useful in YM theory.
Analogies between the 1d $T_N$ model and 4d $SU(N)$ YM theory were previously explored in \cite{Unsal:2012zj}, and a detailed analysis of the symmetries of a very closely related model appeared in \cite{Kikuchi:2017pcp}. The discussion in Sec.~\ref{sec:QM_algebra} thus has overlap with \cite{Kikuchi:2017pcp}, but the subsequent representation-theoretic perspective presented in Sec.~\ref{sec:representation_theory_QM} is new. A discussion of the symmetries of the $T_2$ model as a function of $\theta$ appeared in an appendix of \cite{Gaiotto:2017tne}, but our focus will be on features that appear once $N>2$. Also, a discussion of 't Hooft anomalies from the path integral perspective is given in Appendix~\ref{appendix:path_integral_QM}. This material in this Appendix closely follows the presentation of \cite{Kikuchi:2017pcp}, and we include it here for completeness.
\subsection{Symmetry group as a function of $\theta$}
\label{sec:QM_algebra}
Consider the symmetry group of the $T_N$ theory. Classically, there is a shift symmetry $\mathcal{S}$ as well as `charge conjugation' $\mathcal{C}$ and `time reversal' $\mathcal{T}$ symmetries acting as
\begin{align}
&\mathcal{S}: \; q \to q - 2\pi/N \\
&\mathcal{C}: \; q \to -q\\
&\mathcal{T}: \; t \to - t.
\end{align}
In the quantum theory, the shift symmetry can represented by the operator
\begin{align}
\mathcal{S} = e^{\tfrac{2\pi i}{N} \hat{p}} \, .
\end{align}
where $\hat{p}$ is the momentum operator obeying the canonical commutation relation $[\hat{q}, \hat{p}] = i$. As befits a symmetry operator, $\mathcal{S}$ commutes with the Hamiltonian
\begin{align}
\hat{H} = \frac{1}{2} \left(\hat{p} - \tfrac{\theta}{2\pi}\right)^2 - \cos(N \hat{q}) \, .
\end{align}
Demanding that $\mathcal{T}$ and ${\cal C}$ leave the Hamiltonian invariant in e.g. the coordinate basis, one sees that for $\theta = 0$, the $\mathcal{T}$ and ${\cal C}$ symmetries both act by sending $\hat{p} \to -\hat{p}$, while at $\theta = \pi$, the $\mathcal{T}$ and ${\cal C}$ symmetries both act by sending $\hat{p} \to - \hat{p} +1$. Thus e.g. $\mathcal{T} \hat{p} \mathcal{T}^{-1} = -\hat{p}$ at $\theta = 0$, but $\mathcal{T} \hat{p} \mathcal{T}^{-1} = -\hat{p}+1$ at $\theta = \pi$. One can check that at both $\theta = 0$ and $\theta = \pi$, the ${\cal C}$ and $\mathcal{T}$ operators commute,
\begin{align}
[\mathcal{T},{\cal C}] = 0.
\end{align}
In Minkowski space time-reversal is an anti-unitary operation, so in addition to sending $t \to -t$, $\mathcal{T}$ acts by complex conjugation $\mathcal{T} i \mathcal{T}^{-1}=-i$, in contrast to ${\cal C}$, which is unitary. One can check that this implies that $\left[ \mathcal{T}, \mathcal{S} \right]=0$ in Minkowski space.
${\cal C}$ does not commute with $\mathcal{S}$ at $\theta = 0$. To see this, note the lowest-lying states of the system can be thought of being associated with nodeless wavefunctions $\ket{q_n}$ localized near the $N$ minima. These states are called Wannier states. From $\ket{q_n}$ one can build states with good quantum numbers $\ket{k}$ under $\mathcal{S}$ by a discrete Fourier transform
\begin{align}
\ket{k} = \frac{1}{\sqrt{N}} \sum_{n = 0}^{N-1} \omega^{-n k} \ket{q_n}.
\label{eq:bloch_states}
\end{align}
The states $\ket{k}$ are called Bloch states. Then
\begin{align}\label{eq:tn_S_trans}
\mathcal{S} \ket{k} = \omega^{k} \ket{k}
\end{align}
with $\ket{k}=\ket{k\text{ mod }N}$.
Then one can check that
\begin{align}\label{eq:tn_C_trans}
\mathcal{C} \ket{k} = \begin{cases}
\ket{N-k}
& \theta=0\\
\ket{N-k+1}
& \theta=\pi
\end{cases}.
\end{align}
As a result, at $\theta = 0$ the symmetry operators obey the group
\begin{align}
{\cal C} \mathcal{S} {\cal C}^{-1} =
\mathcal{S}^{-1} \,.
\label{eq:tn_zero_algebra}
\end{align}
Given that $\mathcal{T}^2 = {\cal C}^2 = 1$, the complete symmetry group is isomorphic to
\begin{align}
G^{\theta = 0}_{T_N} = D_{2N} \times \mathbb{Z}_2.
\end{align}
On the other hand, at $\theta =\pi$ we instead obtain
\begin{align}
{\cal C} \mathcal{S} {\cal C}^{-1} =
\omega^{-1}\mathcal{S}^{-1} \,.
\label{eq:tn_pi_algebra}
\end{align}
The appearance of the factor $\omega^{-1}$ on the right-hand side means that the group is not closed in terms of ${\cal C}, \mathcal{T}$ and $\mathcal{S}$. This is a symmetry-group-level indication of a 't Hooft anomaly or global inconsistency between these symmetries. As a result, one of these symmetries must be spontaneously broken at $\theta = \pi$, or there must be a phase transition at some $\theta$ between $0$ and $\pi$.%
\footnote{To decide which of these two symmetries are `actually' broken, it is helpful to note that there is no way to explicitly break $\mathcal{T}$ at $\theta = k = 0$ while preserving $\mathcal{S}$. But if we change the potential
\begin{align}
V = \cos[N q] \to \cos[N (q + \alpha)],
\end{align}
then for any fixed $\alpha \neq 0$, the ${\cal C}$ symmetry $q \to -q$ is explicitly broken, but $\mathcal{S}$ and $\mathcal{T}$ are preserved. One can then verify that this $\mathcal{T}$ and $\mathcal{S}$ remain globally inconsistent at $\theta = \pi$, so that one of them must be broken, and this turns out to be $\mathcal{T}$ \cite{Kikuchi:2017pcp}. Then taking $\alpha$ to 0, we conclude that it is the $\mathcal{T}$ symmetry which is spontaneously broken at $\theta = \pi$ in our variant of the $T_N$ model defined by \eqref{eq:tN_definition}.}
One can try to redefine the operators to get a closed group, for example by
$\tilde{\mathcal{S}}\equiv\omega^{p}\mathcal{S}$. We will refer to $p$ as a Chern-Simons coefficient, since this is how it appears in a path integral description of this system, see \cite{Gaiotto:2017yup,Kikuchi:2017pcp} and Appendix \ref{appendix:path_integral_QM}. This will not spoil the relation $\tilde{\mathcal{S}}^{N}=1$ so long as $p\in\mathbb{\mathbb{Z}}\,\,\text{mod}\,N$. With such a
redefinition \eqref{eq:tn_pi_algebra} becomes
\begin{align}
{\cal C} \tilde{\mathcal{S}} {\cal C}^{-1} =
\omega^{2p-1}\tilde{\mathcal{S}}^{-1} \,.
\label{eq:tn_redefined_algebra}
\end{align}
To keep \eqref{eq:tn_redefined_algebra} isomorphic to \eqref{eq:tn_zero_algebra} requires $2p-1=0\,\text{mod}\,N$.
Now consider the case of odd and even $N$ separately. For even $N$, there is no solution to $2p-1=0\,\,\text{mod\,}N$ for
$p\in\mathbb{\mathbb{Z}}$. Nevertheless, we can get a closed group by taking $p=1/2$. In the path integral description, this gives a Chern-Simons term with an improperly-quantized coefficient. This is associated with a mixed 't Hooft anomaly. In the operator description, the choice $p = 1/2$ gives
\begin{equation}
\mathcal{C}\tilde{\mathcal{S}}\mathcal{C}^{-1}=\tilde{\mathcal{S}}^{-1}.\label{eq:restored alg tnm}
\end{equation}
But now the operator $\tilde{\mathcal{S}}$ satisfies
\begin{equation}
\tilde{\mathcal{S}}^{N}=-1,\qquad\tilde{\mathcal{S}}^{2N}=1 .
\end{equation}
As a result, the symmetry group is now isomorphic to $D_{4N} \times \mathbb{Z}_2$, the central extension of $D_{2N}\times \mathbb{Z}_2$.\footnote{One can think of $D_{4N}$ as the spin group of $D_{2N}$, in the sense that under a $2 \pi$ shift of $q$ (which is a rotation in target space), states goes to minus themselves, and only go back to themselves under a $4 \pi$ shift.} The central extension is the operator-group realization of the anomaly.
For odd $N=2m-1$, $2p-1=0\,\,\text{mod}\,\,N$ is satisfied with the choice $p=\left(N+1\right)/2$. Hence, if we define $\tilde{\mathcal{S}}\equiv\omega^{\left(N+1\right)/2}\mathcal{S}$,
this also reduces to (\ref{eq:restored alg tnm}) since
\begin{align}
\tilde{\mathcal{S}}^{N} & =\omega^{\left(N+1\right)N/2}\mathcal{S}^{N}=\omega^{2mN/2}\left(\omega^{N}\right)^{m}=1.
\end{align}
However, if we insist on preserving the $D_{2N}\times\mathbb{Z}_2$ symmetry at $\theta = 0$, then we must choose the value of Chern-Simons coefficient to be $p=0$ (i.e. the original operator definition). This is the manifestation of the inconsistency condition and results in a centrally extended group $D_{4N} \times \mathbb{Z}_2$ at $\theta = \pi$.
Collecting our results, the symmetry group of the $T_N$ model as a function of $\theta$ is isomorphic to
\begin{equation}
G_{T_N} = \begin{cases}
D_{2N} \times \mathbb{Z}_2
& \theta=0\\
D_{4N} \times \mathbb{Z}_
& \theta=\pi\\
\mathbb{\mathbb{Z}}_{N} & \text{otherwise}.
\end{cases}
\end{equation}
\subsection{Representations of the dihedral group for $\protect\theta=0$ and $\protect\theta=\pi$}
\label{sec:representation_theory_QM}
We now explain how the states of the $T_N$ model fit into the representations of the dihedral group. The value of this discussion is that it relies on the symmetry group structure, rather than the underlying physics, and thus can later be applied almost verbatim to the YM case.
One can construct the $N$-dimensional representation of a dihedral group based on the behavior of the $N$ vacua of the $T_N$ model under the action of charge conjugation (equivalently, time-reversal) and ${\mathbb Z}_N$ shift symmetry. The decomposition of this representation into irreducible representations (irreps) will show us the form of the energy spectrum and provide us another means to see how the degeneracy of the ground state changes between $\theta=0$ and $\pi$. For both $D_{2N}$ and $D_{4N}$, we find results consistent with the operator analysis above.
\begin{table}
\begin{centering}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
& $1\left\{ 1\right\} $ & $2\left\{ r^{\pm1}\right\} $ & $2\left\{ r^{\pm2}\right\} $ & $\cdots$ & $2\left\{ r^{\pm\left(k-1\right)}\right\} $ & $1\left\{r^{k}\right\}$ & $k\left\{ sr^{2b}\right\} $ & $k\left\{ sr^{2b-1}\right\} $\tabularnewline
\hline
${\bf A_{1}}$ & $1$ & $1$ & $1$ & $\cdots$ & $1$ & $1$ & $1$ & $1$\tabularnewline
${\bf A_{2}}$ & $1$ & $1$ & $1$ & $\cdots$ & $1$ & $1$ & $-1$ & $-1$\tabularnewline
${\bf B_{1}}$ & $1$ & $-1$ & $1$ & $\cdots$ & $\left(-1\right)^{k-1}$ & $\left(-1\right)^{k}$ & $1$ & $-1$\tabularnewline
${\bf B_{2}}$ & $1$ & $-1$ & $1$ & $\cdots$ & $\left(-1\right)^{k-1}$ & $\left(-1\right)^{k}$ & $-1$ & $1$\tabularnewline
${\bf E_{1}}$ & $2$ & $2c_{1}$ & $2c_{2}$ & $\cdots$ & $2c_{k-1}$ & $2c_{k}$ & $0$ & $0$\tabularnewline
${\bf E_{2}}$ & $2$ & $2c_{2}$ & $2c_{4}$ & $\cdots$ & $2c_{2k-2}$ & $2c_{2k}$ & $0$ & $0$\tabularnewline
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$\tabularnewline
${\bf E_{k-1}}$ & $2$ & $2c_{k-1}$ & $2c_{2k-2}$ & $\cdots$ & $2c_{\left(k-1\right)^{2}}$ & $2c_{\left(k-1\right)k}$ & $0$ & $0$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{Character table for $D_{2M}=D_{2(2k)}$. Here, $c_{n}=\cos\left(\frac{2\pi n}{M}\right)$. The first row shows the number of elements in the respective conjugacy classes.
\label{tab:D2N_char_tab}}
\end{table}
Let us start by briefly reviewing a few properties of dihedral groups. A more detailed review and discussion of our results is given in Appendix \ref{app:Reps_dihedral}. We will work with a standard presentation of the dihedral group $D_{2M}$, which is given by
\begin{equation}
D_{2M}=\{ r,s|r^{M}=s^{2}=1,srs^{-1}=r^{-1}\}.
\end{equation}
The representations of this group differ for even and odd $M$, so we will consider the two cases separately in what follows.
Below, we will consider the representations which correspond to the low-lying states of the $T_{N}$ model, i.e. the $N$-low lying Bloch states $\ket{k}$, for the cases of $N$ even and odd. Our goal is to understand the representation of the $N$-low lying states. The results are visually summarized in Figs.~\ref{fig:reps_QM_fig} and \ref{fig:reps_QM_fig_2}, which plots the energies of these states as
as function of $\theta$ angle, and are compatible with mixed anomalies/global inconsistencies as well as semi-classics.
\vspace{0.3cm}
\noindent
{\bf Even $N$:}
For $M=N=2k$, the $k+3$ conjugacy classes are\\
\begin{align}
\left\{ 1\right\} ,\left\{ r^{\pm1}\right\} ,\left\{ r^{\pm2}\right\} ,\ldots,\left\{ r^{\pm\left(k-1\right)}\right\} ,\left\{ r^{k}\right\} ,\left\{ sr^{2b}|b=1,\ldots,k\right\} ,\left\{ sr^{2b-1}|b=1,\ldots,k\right\} \label{eq:d_even cc tnm}
\end{align}
where the number of elements in the conjugacy classes are given by
\begin{equation}
\{1,\underset{k-1}{\underbrace{2,2,\ldots,2}},1,k,k\}.\label{eq:d_even cc size tnm}
\end{equation}
A character table for the representations of $D_{2M}$ is given in Table \ref{tab:D2N_char_tab}.
At $\theta=0$, the $N$ low lying states labeled by $\ket{k}$ transform under the action of $D_{2N}$ group elements. The conjugacy classes and number of elements in each class are given by (\ref{eq:d_even cc tnm}) and (\ref{eq:d_even cc size tnm}) with $M=N$. It is straightforward to construct the $N$-dimensional representation associated with $N$-low lying states
under the actions of $\mathcal{S}$ and ${\cal C}$. $\mathcal{S}$ simply introduces a vacuum-dependent phase to each of the states while ${\cal C}$ permutes them. The characters corresponding to the conjugacy classes listed in (\ref{eq:d_even cc size tnm}) are
\begin{equation}
\chi_{\text{even}}^{\theta=0}=\{N,\underset{k-1}{\underbrace{0,0,\ldots,0}},0,2,0\}.
\end{equation}
Character orthogonality then gives the decomposition in terms of irreps\\
\begin{equation} \label{eq:d_even th=0}
R_{\text{even}}^{\theta=0}={\bf A}_{1}\oplus {\bf E_{1}}\oplus {\bf E_{2}}\oplus\ldots\oplus {\bf E_{k-1}}\oplus {\bf B_{1}}
\end{equation}
where ${\bf A_{1}}$ and ${\bf B_{1}}$ are one-dimensional irreps and ${\bf E_{i}}$ is a 2 dimensional irrep (a doublet). ${\bf A_{1}}$ represents the unique ground state of this system which transform trivially under all group operations.
At $\theta=\pi$, per our results of the previous subsection, the symmetry group is now $D_{4N}$. However, we should still construct an $N$-dimensional representations which tells us how the $N$ vacua now transform under this centrally extended group. The characters of this representation are\\
\begin{equation}
\chi_{\text{even}}^{\theta=\pi}=\{N,\underset{N-1}{\underbrace{0,0,\ldots,0}},-N,0,0\}.
\end{equation}
The decomposition in terms of irreps is now given by
\begin{equation} \label{eq:d_even th=pi}
R_{\text{even}}^{\theta=\pi}={\bf \tilde{E}_{1}}\oplus {\bf \tilde{E}_{3}}\oplus\ldots\oplus {\bf \tilde{E}_{2k-1}},
\end{equation}
(with ${\bf \tilde{E}_i}$ now irreps of $D_{4N}$). The fact that the ground state exhibits two-fold degeneracy in this simple quantum mechanics example is a manifestation of the \textquoteright t Hooft anomaly between $\mathbb{\mathbb{Z}}_{N}$ and $\mathbb{\mathbb{Z}}_{2}$ and is tied with the spontaneous breaking of the $\mathbb{\mathbb{Z}}_{2}$ symmetry.
\vspace{0.3cm}
\noindent
{\bf Odd $N$:}
For odd $M=2k+1$, the $k+2$ conjugacy classes are
\begin{equation}
\left\{ 1\right\} ,\left\{ r^{\pm1}\right\} ,\left\{ r^{\pm2}\right\} ,\ldots,\left\{ r^{\pm k}\right\} ,\left\{ sr^{b}|b=1,\ldots,M\right\} \label{eq:d_odd cc tnm}
\end{equation}
where the number of elements in each conjugacy class is now
\begin{equation}
\{1,\underset{k}{\underbrace{2,2,\ldots,2}},N\}.\label{eq:d_odd cc size tnm}
\end{equation}
The corresponding character table is given in Table \ref{tab:D2N+1 char tab tnm}.
\begin{table}
\begin{centering}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& $1\left\{ 1\right\} $ & $2\left\{ r^{\pm1}\right\} $ & $2\left\{ r^{\pm2}\right\} $ & $\cdots$ & $2\left\{ r^{\pm k}\right\} $ & $N\left\{ sr^{2b}\right\} $\tabularnewline
\hline
${\bf A_{1}}$ & $1$ & $1$ & $1$ & $\cdots$ & $1$ & $1$\tabularnewline
${\bf A_{2}}$ & $1$ & $1$ & $1$ & $\cdots$ & $1$ & $-1$\tabularnewline
${\bf E_{1}}$ & $2$ & $2c_{1}$ & $2c_{2}$ & $\cdots$ & $2c_{k}$ & $0$\tabularnewline
${\bf E_{2}}$ & $2$ & $2c_{2}$ & $2c_{4}$ & $\cdots$ & $2c_{2k}$ & $0$\tabularnewline
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$\tabularnewline
${\bf E_{k}}$ & $2$ & $2c_{k}$ & $2c_{2k}$ & $\cdots$ & $2c_{k^{2}}$ & $0$\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\caption{Character table for $D_{2M}=D_{2\left(2k+1\right)}$. Here, $c_{m}=\cos\left(\frac{2\pi m}{M}\right)$.
\label{tab:D2N+1 char tab tnm}}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width = \textwidth]{reps_QM_fig.pdf}
\caption{An illustration of the energy levels of the $T_N$ model for $N=5$ and $N=6$. At $\theta=0$, the ground state is unique, and fits into the one-dimensional ${\bf A_{1}}$ representation of $D_{2N}$, while the excited states fit into either the ${\bf E_{k}}$ representations (which are all two-dimensional) or into the ${\bf B_{1}}$ representation, which is one-dimensional. At $\theta = \pi$, on the other hand, the ground state is always in the two-dimensional $\bf {\widetilde{E}
}_{1}$ representation of $D_{4N}$.}
\label{fig:reps_QM_fig}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width = \textwidth]{rep_plot.pdf}
\caption{A sketch of how the states of the $T_N$ model with $N=5$ and $\theta = 0$ and $\theta =\pi$ fit into the dihedral group $D_{10}$ and $D_{20}$ representations. The Bloch states $\ket{k}$ are defined in \eqref{eq:bloch_states}. }
\label{fig:reps_QM_fig_2}
\end{center}
\end{figure}
At $\theta=0$, the $N$ low lying states transform under the action of $D_{2N}=D_{2\left(2k+1\right)}$ group elements. The characters of the $N$-dimensional representation are given by:\\
\begin{equation}
\chi_{\text{odd}}^{\theta=0}=\{N,\underset{k}{\underbrace{0,0,\ldots,0}},1\}.
\end{equation}
In this case, the decomposition is given by\\
\begin{equation} \label{eq:d_odd th=0}
R_{\text{odd}}^{\theta=0}={\bf A_{1}}\oplus {\bf E_{1}}\oplus {\bf E_{2}}\oplus\ldots\oplus {\bf E_{k}}
\end{equation}
where ${\bf A_{1}}$ is again a one-dimensional irrep and ${\bf E_{i}}$ are 2 dimensional irreps of $D_{2N}$ with $N$ odd. ${\bf A_{1}}$ again represents the unique ground state of this system at $\theta=0$.
For the $T_{N}$ model with odd $N$, there is a global inconsistency condition at $\theta=\pi$ between $\mathcal{S}$ and $\mathcal{T}$ (or ${\cal C}$). As a result, the vacuum cannot remain trivial which indicated either a non-trivial vacuum or a phase transition between $\theta=0$ and $\pi$. We will assume that the inconsistency implies the former such that the $N$ low lying states transform under the action of $D_{4N}=D_{2\left(4k+2\right)}$. This is the group that will give rise to the two-fold degenerate ground state we find as a result of the global inconsistency condition. We also assume the central extension comes about in the same manner as the even $N$ case, where $\tilde{\mathcal{S}}\equiv\omega^{1/2} \mathcal{S}$. Semi-classical instanton analysis \cite{Unsal:2012zj} and numerical diagonalization of \eqref{eq:tN_definition} found in e.g. \cite{Kikuchi:2017pcp} support the resulting degeneracies and $\theta$-dependence from this assumption.
At $\theta=\pi$, the $N$ low lying states transform under the action of $D_{4N}=D_{2\left(4k+2\right)}$ group elements.
The characters of the conjugacy classes in this case are
\begin{equation}
\chi_{\text{odd}}^{\theta=\pi}=\{N,\underset{N-1}{\underbrace{0,0,\ldots,0}},-N,1,-1\}
\end{equation}
and the corresponding decomposition is
\begin{equation} \label{eq:d_odd th=pi}
R_{\text{odd}}^{\theta=\pi}={\bf \tilde{E}_{1}}\oplus {\bf \tilde{E}_{3}}\oplus\ldots\oplus {\bf \tilde{E}_{2k-1}}\oplus {\bf \tilde{B}_{1}}.
\end{equation}
${\bf \tilde{E}_{1}}$ denotes the ground state and exhibits two-fold degeneracy. Other ${\bf \tilde{E}}$-states are excited states, and ${\bf \tilde{B}_{1}}$ is the highest energy state (of the low lying states), which is a singlet.
\section{Dihedral symmetries in Yang-Mills theory on $\mathbb{R}^3\times S^1$}
\label{sec:YM}
We would now like to illustrate Eq.~\eqref{eq:ym_theta_sym} by explicitly looking at symmetry properties of the vacua and excitations of Yang-Mills theory. As is well known, $SU(N)$ YM theory on $\mathbb{R}^4$ is asymptotically free and as such becomes strongly coupled at energy scales small compared to the inverse strong scale, $\Lambda^{-1}$. Hence methods of studying the explicit vacuum structure of the theory are limited. Instead we choose to study YM theory on $\mathbb{R}^3\times S^1$, with a circle size of $L$. In this system the vacuum dynamics are calculable via weak coupling methods, specifically in the limit where $NL\Lambda\ll 1$ and center symmetry is preserved. There has been significant evidence \cite{
Unsal:2007vu,Unsal:2007jx,Shifman:2008ja,Shifman:2008cx,Shifman:2009tp,
Cossu:2009sq,Myers:2009df,Simic:2010sv,Unsal:2010qh,Azeyanagi:2010ne,Vairinhos:2011gv,
Thomas:2011ee,Anber:2011gn,Unsal:2012zj,
Poppitz:2012sw,
Poppitz:2012nz,Argyres:2012ka,Argyres:2012vv,
Anber:2013doa,Cossu:2013ora,
Anber:2014lba,Bergner:2014dua,Bhoonah:2014gpa,Li:2014lza,
Anber:2015kea,Anber:2015wha,Misumi:2014raa,
Cherman:2016hcd,Aitken:2017ayq,Anber:2017rch,Anber:2017pak,Anber:2017tug,Anber:2017ezt} that YM depends smoothly on the parameter $NL\Lambda$, and hence it is conjectured that one can recover results for the theory on $\mathbb{R}^4$ in the large $L$ limit. We will begin by briefly reviewing such a system. Those concerned only with our analysis of the vacuum can skip to Sec. \ref{sec:vacua}.
\subsection{Weak-coupling setup}
\label{sec:YM_setup}
Consider pure $SU(N)$ Yang-Mills theory on $\mathbb{R}^3\times S^{1}$. For small $S^1$, it is known that the $\mathbb{Z}_N$ center symmetry is spontaneously broken \cite{Gross:1980br,Weiss:1981ev}, while for large $S^1$ the symmetry is expected to be restored. The order parameter for the associated phase transition is the expectation value of the trace of powers of
\begin{align}\label{eq:holonomy}
\Omega(x^\mu)=P\exp\left[i\int_{0}^{L}dx_4\,A_{4}\left(x^\mu,x_4\right)\right]
\end{align}
where we have changed conventions slightly and will henceforth use $\mu,\nu=1,2,3$. At large $L$, $\langle \,{\rm tr}\, \Omega^k \rangle = 0$ for $k \neq 0 \textrm{ mod } N$, while at small $L$ $\langle \,{\rm tr}\, \Omega \rangle \neq 0$. However, if one is not interested in interpreting $S^1$ as a Euclidean thermal circle, this phase transition can be avoided by ``center-stabilizing" deformations. One example of such a deformation is the addition of $N_F>1$ massive Majorana adjoint fermions with mass $m_{a} \lesssim 1/(NL)$ \cite{Unsal:2010qh}. Another example is the addition of a double-trace deformation \cite{Unsal:2008ch}. With either deformation, it is believed that center symmetry is then preserved for all $L$, with the benefit that at small $L$ the physics becomes analytically calculable.
We choose to explore the behavior of the symmetries in the center-symmetric phase of the theory that follows from either of deformations referenced above. At small $L$, where quantum fluctuations become small, the holonomy takes the form
\begin{align}
\langle \Omega \rangle = \omega^{-(N-1)/2} \mathrm{diag}(1,\omega, \ldots, \omega^{N-1}),\qquad \omega = e^{2\pi i /N}.
\label{eq:center_sym_holonomy}
\end{align}
up to gauge transformations.
We will analyze the theory at distances large compared to $L$, where the system can be described by a 3D effective field theory. From \eqref{eq:holonomy}, the holonomy eigenvalues above imply that (in a standard gauge-fixed sense) $\left\langle A_4 \right\rangle \ne 0$ , which acts as an adjoint Higgs field in the 3D EFT, and breaks the gauge group down to $U(1)^{N-1}$. The lightest W-bosons have the tree-level mass
\begin{align}
m_W \equiv \frac{2 \pi}{ NL}.
\end{align}
So when $m_W \gg \Lambda$ --- equivalently, when $NL\Lambda \ll 1$ --- the gauge coupling stops running at the scale $m_W$, and the long-distance 3D effective field theory becomes weakly coupled. We focus on this tractable limit for the remainder of this paper.
The lightest fields in the 3D effective field theory are the $U(1)^{N-1}$ gauge bosons, the ``photons". It is useful to note that the associated field strength operators $F_{\mu\nu}^{a}, a =1, \ldots, N-1$, have a gauge-invariant 4D interpolating operator representation given by
\begin{align}
\label{eq:gi_famu}
F^{a}_{\mu \nu}(x_{\mu}) \sim \frac{1}{N} \frac{1}{L} \int d x_4 \sum_{q=1}^{N-1} \omega^{-q a} \,{\rm tr}\, \Omega^{q}(x^{\mu}) F_{\mu \nu}(x_4,x_{\mu}) \, ,
\end{align}
with $F_{\mu \nu}$ the 3D part of the $SU(N)$ non-Abelian field strength. This representation makes clear that the ``color" index can actually be thought of as the discrete Fourier transform of the winding number of a topologically non-trivial state.
In terms of these fields, the tree-level action of the 3D EFT can be written as
\begin{align}
\label{eq:euc_action}
S_{\rm tree} = \frac{L}{4g^2} \int d^3{x} \sum_{a =1}^{N} F^{a}_{\mu \nu} F^{a\mu \nu} \,.
\end{align}
For later notational convenience we have introduced a fictitious $N$th photon in writing this expression. This extra field can be thought of as the diagonal component of a $U(N)$ field strength, and exactly decouples from the physical adjoint fields in our system. Using Eq.~\eqref{eq:gi_famu} one can show that center symmetry acts as
\begin{align}
\mathcal{S}: \; F^{a}_{\mu \nu} \to F^{a+1}_{\mu \nu} \,.
\label{eq:center3d}
\end{align}
In order to analyze the non-perturabtive dynamics of our system, we follow Ref.~\cite{Polyakov:1976fu,Unsal:2008ch} and rewrite \eqref{eq:euc_action} by dualizing the photon, trading $F_{\mu\nu}^{a}$ for a pseudoscalar field $\sigma^{a}$ via the relation
\begin{align}
F^{a}_{\mu \nu}\equiv\frac{\lambda}{4\pi^2}\epsilon_{\mu\nu\rho}\partial^\rho \sigma^a , \qquad\lambda = g^2 N.
\end{align}
This allows us to rewrite \eqref{eq:euc_action} as
\begin{align}
S_{\rm tree, dual} = \lambda m_W \int d^3{x} \sum_{a =1}^{N} (\partial_{\mu}\sigma^a)(\partial^{\mu}\sigma^a) \equiv \lambda m_W \int d^3{x} ~ (\partial_{\mu}\vec{\sigma})^2
\label{eq:perturbative_action}
\end{align}
where we have defined the $N$-component vector of dual photon fields $\vec{\sigma}=(\sigma^1,\ldots,\sigma^N)$.
The dual photons in \eqref{eq:perturbative_action} have no potential to all orders in perturbation theory. So there is no mass gap in perturbation theory. However, the theory has finite-action field configurations that generate a non-perturbative potential for $\vec{\sigma}$. In Appendix \ref{app:Disc_syms}, we review the finite-action solutions of this theory with the smallest action. They come in $N$ distinct types, and are usually called monopole-instantons. They carry topological charge $Q_T = 1/N$, action $S_0=8\pi^2/\lambda$, and carry magnetic charges associated to the simple (co-)roots $\vec{\alpha}_a$ of the affine extension of the $\mathfrak{su}(N)$ Lie algebra. For more details on the non-perturbation solutions and their transformations under the symmetries of the theory, see Appendix \ref{app:Disc_syms}.
As explained in \cite{Unsal:2008ch}, summing over the contributions of the monopole-instanton solutions to the path integral using a dilute-gas approximation (which is well-justified when $NL\Lambda \ll 1$) produces a potential for the dual photons, so that
\begin{align}
S_{\rm \vec{\sigma}} = \int d^3{x}\, \left[ \lambda m_W (\partial_{\mu} \vec{\sigma})^2 + V(\vec{\sigma}) \right]
\label{eq:YM_EFT}
\end{align}
where the non-perturbative potential is given by
\begin{align}
V\left(\vec{\sigma}\right) &= -\frac{A}{\lambda^{2}}m_{W}^{3}e^{-S_0}\sum_{a=1}^{N}\cos\left[\vec{\alpha}_{a}\cdot\vec{\sigma}+\frac{\theta}{N}\right]+ \ldots.
\label{eq:YM_potential}
\end{align}
The ``$\ldots$'' represent higher order contributions which we will neglect here. Here, $A>0$ is an $\mathcal{O}(1)$ scheme-dependent dimensionless constant which will not be important in what follows. The monopole-generated potential depends on the $\theta$ angle because the monopole-instantons have non-vanishing topological charge.
We now show how the YM symmetry group in \eqref{eq:ym_theta_sym} acts in the effective field theory associated to \eqref{eq:YM_EFT}
\subsection{Extrema and symmetries as a function of $\theta$}
\label{sec:vacua}
We now begin our analysis of the vacuum structure of \eqref{eq:YM_EFT}, with the leading order-potential explicitly shown in \eqref{eq:YM_potential}. The dual photon fields live in the weight lattice of $\mathfrak{su}(N)$. The potential has $N$ extrema in the unit cell of the weight lattice at
\begin{equation}
\vec{\sigma}_{k}=\frac{2\pi k}{N}\vec{\rho}, \qquad \mathrm{with}\qquad\vec{\rho}\equiv\sum_{i=1}^{N-1} \vec{\mu}_{i}
\label{eq:YMextrema}
\end{equation}
where $k=0,\ldots,N-1$. Here $\vec{\mu}_{i}$ are the $SU(N)$ fundamental root vectors, and satisfy $\vec{\alpha}_{i}\cdot\vec{\mu}_{j}=\delta_{ij}$, and $\vec{\rho}$ is the Weyl vector satisfying $\vec{\alpha}_{i}\cdot\vec{\rho}=1$ for $i = 1, \ldots, N-1$ and $\vec{\alpha}_{N}\cdot\vec{\rho}= 1 - N$. For example, in a basis where $(\alpha_{a})_b = \delta_{a,b} - \delta_{a+1,b}, \, 1\le a < N$, $\vec{\sigma}_{k}$ takes the form
\begin{align}
\vec{\sigma}_{k} = \frac{2\pi k}{N} (N, N-1, \ldots, 2,1) \, .
\label{eq:weyl_vector}
\end{align}
The non-perturbative 3D energy density evaluated at each of these extrema is
\begin{equation}
V_{k} \equiv V\left(\vec{\sigma} = \vec{\sigma}_k \right)=-N\frac{A}{\lambda^{2}}m_{W}^{3}e^{-S_0}\cos\left(\frac{2\pi k+\theta}{N}\right) + \mathcal{O}(e^{-2S_0}) \,.
\label{eq:pot_vals_tnm}
\end{equation}
For any given $\theta$, the integer $k$ labeling the globally-stable ground state is determined by minimizing \eqref{eq:pot_vals_tnm}. The metastable states of the system will correspond to the subset of extrema with positive curvature in all directions in $\vec{\sigma}$ space.
On any fixed branch, the physics is periodic in $2\pi N$. However, the $k$ that minimizes $V_{k}$ depends on $\theta$. Thus, just from the form of \eqref{eq:pot_vals_tnm}, one can see that as $\theta$ varies in $[0,2\pi)$, the value of $k$ associated with the minimal energy extremum will change in such a way that the physics of the complete system in its ground state has a $\theta$ periodicity of $2\pi$. However, the observables are non-analytic functions of $\theta$, which is associated with jumps in the value of $k$ which minimize the ground state energy density. This is consistent with Witten's conjectured picture \cite{Witten:1978bc,Witten:1998uka} for the $\theta$-dependence of YM theory. Earlier discussions of how $2\pi$ periodicity emerges in the present context were presented in e.g. Refs.~\cite{Unsal:2008ch,Thomas:2011ee,Unsal:2012zj,Poppitz:2012sw,Poppitz:2012nz,Bhoonah:2014gpa,Anber:2017rch}.
Let us now understand how center and coordinate reflection symmetries act on the extrema of \eqref{eq:YMextrema}. To do this, it is useful to work out how these transformations act in compactified YM theory more generally, see Appendix \ref{app:Disc_syms}, and also Ref.~\cite{Aitken:2017ayq}. Here we will focus on reflections of the compactified coordinate $\mathcal{R}$, charge conjugation $\mathcal{C}$, and (0-form) center transformations $\mathcal{S}$. The effective field theory on $\mathbb{R}^3 \times S^1$ is built from the dual photon fields $\sigma_a$, and the action of these transformations
which follows from \eqref{eq:gi_famu} and \eqref{eq:center3d},
is
\begin{align}
\mathcal{S}: \sigma_a &\to \sigma_{a+1}\\
\mathcal{C}: \sigma_a &\to - \sigma_{N-a+1}\\
\mathcal{R}: \sigma_a &\to \begin{cases}
\sigma_{N-a+1}\,, & \theta = 0 \\
\sigma_{N-a+1} -\frac{2\pi(N-a+1)}{N}\,, & \theta = \pi
\end{cases}
\end{align}
Looking at the form of the effective action \eqref{eq:YM_EFT}, it is clear that $\mathcal{S}$ and ${\cal C}$ are symmetries for any $\theta$, as one would expect. The $\mathcal{R}$ coordinate-reflection transformation is a symmetry only if $\theta = 0$ or $\theta = \pi$. Note that when acting on $\vec{\alpha}_{a} \cdot \vec{\sigma}$ at $\theta=\pi$, the reflection symmetry transformation gives
\begin{align}
\mathcal{R} : (\vec{\alpha}_{a} \cdot \vec{\sigma}) \to -\vec{\alpha}_{N-a} \cdot \vec{\sigma} -\frac{2\pi}{N} \,.
\end{align}
The resulting shift in the phase of monopole operators is necessary because a coordinate reflection must be accompanied by a $2\pi$ shift in the $\theta$ angle to be a symmetry of the theory.
One can now easily work out the symmetry group. To do so, consider the action of the symmetry transformations on an operator of the form $e^{i\sigma_a}$. It can be checked that ${\cal C}^{-1} \mathcal{S} {\cal C} = \mathcal{S}^{-1}$, corresponding to a $D_{2N}$ symmetry group, just as one would expect from the general arguments in Sec.~\ref{sec:symmetry}. For the ${\mathbb R}$ and $\mathcal{S}$ symmetries, we obtain
\begin{align}\label{eq:dYM_algebra}
{\mathbb R}^{-1} \mathcal{S} {\mathbb R} = \begin{cases}
\mathcal{S}^{-1} \,, &\theta = 0 \\
\omega \mathcal{S}^{-1} \,, &\theta = \pi \,.
\end{cases}
\end{align}
This corresponds to a $D_{2N}$ group for $\theta = 0$, and a $D_{4N}$ group for $\theta = \pi$. As in in our discussion of the $T_{N}$ model, for even $N$ we interpret the $\theta = \pi$ commutation relations in \eqref{eq:dYM_algebra} to imply the existence of a mixed 't Hooft anomaly between center and time-reversal symmetries, while for odd $N$ we interpret them to imply a global inconsistency between these symmetries.
In total, we find precisely the expected 0-form symmetries of \eqref{eq:ym_theta_sym}, reproduced here for convenience
\begin{align}
G^{\rm discrete}_{\rm YM} = \begin{cases}
D_{2N} \times \mathbb{Z}_2 \times \mathbb{Z}_2
& \theta=0\; \mathrm{mod}\; 2\pi \\
D_{4N} \times \mathbb{Z}_2 \times \mathbb{Z}_2
& \theta=\pi\; \mathrm{mod}\; 2\pi \\
D_{2N} & \text{otherwise}.
\end{cases}
\label{eq:ym_theta_sym_2}
\end{align}
Note that a benefit of our approach we get a simple picture for how the mixed center-$CP$ 't Hooft anomaly of Ref.~\cite{Gaiotto:2017yup} arises (as a central extension of the symmetry group, just like in toy QM examples). Moreover, given that we work in a regime where the dynamics is calculable, we can fully determine the vacuum structure. On the other hand, the general nature of the considerations of Ref.~\cite{Gaiotto:2017yup} have their own benefits. In particular, they are valid regardless of the strength of the coupling in the system. We explore further features of the vacuum structure of \eqref{eq:YM_potential} and higher order corrections in a companion paper \cite{Aitken:2018mbb}.
Turning back to the symmetry transformations of the extrema of the potential, we find that ${\mathbb R}$ acts as
\begin{align}
\mathcal{R} : \vec{\sigma}_{k} \to
\begin{cases}
\vec{\sigma}_{-k} & \theta = 0 \\
\vec{\sigma}_{-k+1} & \theta = \pi.
\end{cases}
\end{align}
while the center transformation rule is
\begin{equation}
\mathcal{S}\::\: \vec{\sigma}_{k} \to \vec{\sigma}_{k} + \frac{2\pi k }{N}\vec c
\label{eq:center_trans_dym}
\end{equation}
where the $N$-vector $\vec{c}$ obeys the relations
\begin{equation}
\vec{\alpha}_{a} \cdot \vec c =\begin{cases}
-N & a = 1\\
0 & 1 < a <N\\
N & a=N
\end{cases}
\end{equation}
For example, in the basis of Eq.~\eqref{eq:weyl_vector}, $\vec c = \left(1,1,\ldots,1,1-N\right)$. The condition that $\mathcal{S}^N\cdot \vec{\sigma}_k = \vec{\sigma}_k$ is related to the periodicity of the $\sigma_{a}$ fields and the quantization of the coefficient of $\vec{c}$ in \eqref{eq:center_trans_dym}.
\section{Summary}
\label{sec:outlook}
We have examined the global symmetries and ground state properties of $SU(N)$ YM theory as a function of the topological $\theta$ angle. The global symmetries were argued to include non-Abelian discrete groups --- specifically, dihedral groups --- for all $\theta$ when $N\ge3$ due to a non-commutativity between center symmetry and charge conjugation. We then examined the vacuum structure of YM theory as a function of $\theta$. First, we warmed up by considering a simple quantum mechanics example whose symmetries also include dihedral groups. We then used the technique of adiabatic circle compactification of YM theory on $\mathbb{R}^3\times S^1$ to illustrate the symmetry structure and some ground state properties in a systematically calculable setting.
\section*{Acknowledgments}
We are grateful to S. Gukov, K.~Jensen, M.~Shifman, T.~Sulejmanpasic, and L.~Yaffe for helpful conversations. We are especially indebted to L. Yaffe for his extensive comments and advice on a draft of the paper. K.~A. is supported by the U.S.~Department of Energy under Grant No.~DE-SC0011637. A.~C. and M.~\"U. thank the KITP for its warm hospitality as part of the program ``Resurgent Asymptotics in Physics and Mathematics'' during the final stages of the research in this paper. Research at KITP is supported by the National Science Foundation under Grant No. NSF PHY11-25915. A.~C. is also supported by the U. S. Department of Energy via grants DE-FG02-00ER-41132, while M.~\"U. is supported U. S. Department of Energy grant DE-FG02-03ER41260.
|
1,116,691,498,421 | arxiv | \section{Introduction and preliminaries}
Let $\sigma=(\sigma_{1},\sigma_{2},\dots,\sigma_{n})$ denote a point on the unit sphere $\mathbb{S}^{n-1}\subset {\mathbb {R}}^n,~n\geq 2,$ and $d\sigma$ be the normalised Lebesgue measure on $\mathbb{S}^{n-1}.$ For continuous functions $f_1, f_2,\dots, f_n$ on ${\mathbb {R}}$ consider the multilinear spherical averages defined by
\begin{eqnarray}\label{def}T(f_{1},f_{2},\dots,f_{n})(x)=\int_{\mathbb{S}^{n-1}}\prod^{n}_{j=1}f_{j}(x-\sigma_j)d\sigma,~x\in {\mathbb {R}}.
\end{eqnarray}
Let $1\leq p_1,p_2,\dots,p_n,r\leq \infty$. We are interested in studying the $L^p$ estimates for the operator $T$ at $(p_1,p_2,\dots,p_n;r)$, i.e.,
\begin{eqnarray}\label{bdd}
\|T(f_{1},f_{2},\dots,f_{n})\|_r\lesssim \prod_{j=1}^n \|f_j\|_{p_j}.
\end{eqnarray}
The notation $A\lesssim B$ (and $A\gtrsim B$) means that there exists an implicit constant $C>0,$ such that $A\leq CB$ (and $A\geq CB$). We will not keep track of the constants and often use the notation as mentioned above. We will also require weaker notion of boundedness of operators between Lorentz spaces as the operator may not always satisfy strong type estimates.
We need to consider a general form of the operator $T$ as it would be required in many of our proofs. Let $\{v_{1},v_{2},\dots ,v_{n}\}$ be linearly independent vectors in ${\mathbb {R}}^n$. Consider the following general form of the operator $T$ given by $$T_v(f_{1},\dots,f_{n}):=\int_{\mathbb{S}^{n-1}}\prod^{n}_{j=1}f_{j}(x-v_{j}\cdot\sigma)d\sigma.$$
Recall that for a given multilinear ($n-$linear) operator one can consider $n$ adjoint operators associated with it. More specificly, we have adjoints of $T_v$ given by
$$\langle T_v^{*j}(f_1,f_2,\dots,f_n),h\rangle :=\langle T_v(f_1,f_2,\dots,f_{j-1},h,f_{j+1},\dots, f_n),f_j\rangle.$$
It is easy to verify that $T_v^{*j}$ is similar to $T_v$ with a different set of linearly independent vectors than that of $T_v$. Using duality arguments boundedness of $T_v$ at $(p_1,p_2,\dots,p_n;r)$ implies the corresponding result for $T_v^{*j}$ at $(p_1,p_2,\dots,p_{j-1},r',p_{j+1},\dots,p_{n};p'_j)$. We will refer to these points as dual points to each other. Here $p'$ denotes the conjugate index to $p$ given by $\frac{1}{p}+\frac{1}{p'}=1$.
In~\cite{oberlin} Oberlin established nesessary and sufficent conditions for the boundedness of $T$ from $\prod_{j=1}^nL^{p}({\mathbb {R}})\rightarrow L^r({\mathbb {R}})$. Later, Bak and Shim~\cite{jbak} extended Oberlin's result improving the range of $p$ and $r$ for the strong type boundedness of $T$. We also refer to ~\cite{oberlin1} for Young's inequality for multilinear convolution operators. In order to describe the known results we require some notation.
Let $R=R(n)$ denote the closed convex hull in ${\mathbb {R}}^2$ of points $O=(0,0), B=(\frac{n-1}{n+1},0), M=(\frac{n+1}{n+3},\frac{2}{n+3}), A=(\frac{n+1}{n+2},1), F=(\frac{1}{n},1),$ see Figure~$1$ for detail. Note that the point $(\frac{1}{p},\frac{1}{r})$ corresponds to $(p,p,\dots,p;r)$. In~\cite{oberlin} Oberlin proved the following result concerning the boundedness of the operator $T$.
\begin{theorem}\cite{oberlin}\label{oberlin} If the operator $T$ is of strong type at $(p,p,\dots,p;r)$ then $(\frac{1}{p},\frac{1}{r})$ lies in the region $R$. Conversely, if $(\frac{1}{p},\frac{1}{r})$ lies in the region $R$ and not on the two closed line segments $AM$ and $MB$, then $T$ is of strong type at $(p,p,\dots,p;r)$. Further, for points $(\frac{1}{p},\frac{1}{r})$ lying on the line segments $AM$ and $MB$, the operator $T$ is of restricted type at $(p,p,\dots,p;r)$, i.e., estimate ~(\ref{bdd}) holds at $(p_1,p_2,\dots,p_n;r)$ for $f_j's$ restricted to characteristic functions.
\end{theorem}
\begin{figure}[H]
\includegraphics[width=.5\textwidth]{Oberlinfig}
\vspace{0mm}
\caption{Region R}
\end{figure}
The question of strong type boundedness of the operator $T$ at points lying on the closed line segments $AM$ and $MB$ remained unresolved for a long time. In 1998, Bak and Shim~\cite{jbak} settled this question and filled in the gap between necessary and sufficient conditions in Theorem~\ref{oberlin} for dimension $n\geq 3.$ More precisely, they proved the following.
\begin{theorem}\cite{jbak}\label{jbak} For $n\geq 3$, the operator $T$ is of strong type at $(p,p,\dots,p;r)$ if, and only if $(\frac{1}{p},\frac{1}{r})$ lies in the region $R$.
\end{theorem}
Bak and Shim~\cite{jbak} also addressed the question in dimension $n=2$. They obtained the following positive and negative results in this case.
\begin{theorem}\cite{jbak}\label{jbak1}
In dimension $n=2$, the following results hold.
\begin{enumerate}
\item $T$ is of strong type at $(\frac{4}{3},\frac{4}{3};1)$.
\item $T$ is bounded from $L^{3,s}({\mathbb {R}})\times L^{3,s}({\mathbb {R}})\rightarrow L^{\infty}({\mathbb {R}})$
if, and only if $0<s\leq 2$. In particular, $T$ fails to be of strong type at $(3,3;\infty)$.
\item Let $H$ denote the point $(\frac{1}{2},\frac{1}{4})$ on the line segment $BM$. If $(\frac{1}{p},\frac{1}{r})$ lies on either of the closed line segments $AM$ and $MH$ then $T$ is of strong type at $(p,p;r)$.
\end{enumerate}
\end{theorem}
Further, we note that in dimension $n=2$ the operator $T$ coincides with the bilinear spherical averaging operator. The bilinear spherical averages and the corresponding bilinear spherical maximal function have been studied by several authors in the recent past.
For functions $f,g \in \mathcal{S}({\mathbb {R}}^d), d\geq 1$, the bilinear spherical average is defined by
\begin{eqnarray*}
\mathcal{A}(f,g)(x):=\int_{\mathbb{S}^{2d-1}}f(x-y)g(x-z)d\sigma(y,z).
\end{eqnarray*}
where $d\sigma(y,z)$ is the normalised Lebesgue measure on the sphere $\mathbb{S}^{2d-1}$.
Observe that the operator $\mathcal A$ for $d=1$ is same as $T$ for $n=2$. The operator $\mathcal A$ and the corrsponding bilinear maximal function was introduced and studied in~\cite{GGIP}. Later, in ~\cite{BGHH, GHH} authors estbalished partial results obtaining $L^{p_1}({\mathbb {R}}^d)\times L^{p_2}({\mathbb {R}}^d)\rightarrow L^p({\mathbb {R}}^d)$ estimates for the bilinear spherical maximal operator for a certain range of $p_1, p_2$ and $p$ with some assumptions on the dimension $d$. Very recently, in~\cite{JL} Jeong and Lee proved $L^{p_1}({\mathbb {R}}^d)\times L^{p_2}({\mathbb {R}}^d)\rightarrow L^p({\mathbb {R}}^d)$ estimates for the maximal operator for the best possible range of exponents $p_1, p_2$ and $p$ for all $d\geq 2.$ They also obtained $L^p$ improving estimates for the bilinear spherical averaging operator $\mathcal A$ for $d\geq 2$. However, the case of dimension $d=1$ has not been addressed so far. We shall fill this gap in this paper. We also refer to the recent papers~\cite{AP,Do, LSK} for further generalisation of the bilinear spherical maximal functions to the multilinear and product type setting.
In this paper our aim is to establish necessary and sufficient conditions on exponents $1\leq p_j, r\leq \infty, j=1,2,\dots,n$ for the $(p_1,p_2,\dots,p_n;r)$ boundedness of the operator $T$.
Note that the results due to Oberlin~\cite{oberlin} and Bak and Shim~\cite{jbak} addressed the same when $p_j=p$ for all $j=1,2,\dots,n.$ We extend their results to the full possible range of exponents and thereby allow the possibility of $p_j$ assuming different values. Moreover, as pointed out earlier the boundedness of $T$ for $n=2$ yields the corresponding $L^p$ improving estimates for the bilinear averaging operator $\mathcal A$ for $d=1$. Our proofs are motivated from the ideas presented in~\cite{jbak,oberlin}. Along with the standard multilinear interpolation theorems the following multilinear interpolation result due to Christ~\cite{Ch} plays an important role.
\begin{lemma}\label{christ}\cite{Ch}
Let $n\geq 2$ and $\Sigma$ be a nontrivial closed $(n-1)$-simplex in the unit cube $[0,1]^n$. Assume that the hyperplane $\Gamma$ generated by $\Sigma$ is not parallel to any of the coordinate axes. If $S$ is a multilinear ($n-$linear) operator such that it is bounded from $L^{p_1,1} \times \dots\times L^{p_n,1}\rightarrow Y$ at all endpoints $(\frac{1}{p_1},\frac{1}{p_2},\dots,\frac{1}{p_n})$ of the simplex $\Sigma$, where $Y$ is a Banach space. Then for $(\frac{1}{p_1},\frac{1}{p_2},\dots,\frac{1}{p_n})$ an interior point of $\Sigma$ and $1\leq q_i\leq \infty$ satisfying $\sum\limits_{i=1}^n \frac{1}{q_i}=1$, the operator $S$ is bounded from $L^{p_1,q_1}\times \dots\times L^{p_n,q_n}\rightarrow Y.$
\end{lemma}
The remaining part of the paper is organized as follow. In Section~\ref{main} we state the main results and describe the necessary region. In Section~\ref{proof:nec} we establish the necessary conditions on exponents for boundedness of the operator. Section~\ref{proof:n=2} is devoted to proving boundedness of the operator for points in the necessary region for $n=2$. Finally, in Section~\ref{proof:all} we complete the proof for $n\geq3.$
\section{Main results}\label{main}
\subsection{Necessary part}
The following result describes necessary conditions on exponents for $L^p$ boundedness of the operator $T$.
\begin{theorem}\label{mainresult:nec}{\bf Necessary conditions.} Let $1\leq p_j,r\leq \infty, j=1,2,\dots,n,$ be given exponents. If the operator $T$ is of strong type at $(p_1,p_2,\dots,p_n;r)$ then the following conditions hold.
\begin{enumerate}[i)]
\item $\frac{1}{r}\leq\sum^{n}_{j=1}\frac{1}{p_{j}}$,
\item $\sum^{n}_{j=1,j\neq k}\frac{1}{p_{j}}+\frac{2}{p_{k}}\leq n-1+\frac{2}{r},$ for $1\leq k\leq n$.
\item $\sum^{n}_{j=1,j\neq k,l}\frac{1}{p_{j}}+\frac{2}{p_{k}}+\frac{2}{p_{l}}\leq n+\frac{1}{r},$ where $1\leq k,l\leq n$ and $k\neq l.$
\end{enumerate}
\end{theorem}
In order to investigate the sufficiency of conditions listed as above for boundedness of the operator $T$ and to state the corresponding results we need to first describe the necessary region.
\subsection*{Necessary region $\mathcal R$ and its endpoints}
Let $\mathcal R=\mathcal R(n)$ denote the closed and bounded region in ${\mathbb {R}}^{n+1}$ enclosed by the hyperplanes determined by the necessary conditions described in Theorem~\ref{mainresult:nec}. We will refer to it as the necessary region $\mathcal R.$ In order to understand the necessary region $\mathcal R$, we need to find its vertices. The vertices of $\mathcal R$ will be referred to as the endpoints. We will see that in dimension $n=2$ it is easy to write down all the endpoints however for large dimensions the number of endpoints is large and it becomes little difficult to describe all of them. Further note that in view of the multilinear interpolation theory, see~\cite{BS,Ch}, it is enough to prove boundedness of the operator $T$ at the endpoints of the convex region $\mathcal R$. Therefore, knowing the endpoints is important to prove the sufficient part to Theorem~\ref{mainresult:nec} which forms the major part of the paper. Another property that will play a crucial role is the fact that dual of an endpoint, in the sense as described in the previous section, remains an endpoint of $\mathcal R$. This can be easily verified and we skip the detail. We will make use of this fact to identify the endpoints. We will see that at some of the endpoints the operator $T$ fails to satisfy strong type estimates. In this scenario, some boundary points become important provided there holds a strong type result at these points. We shall have positive results at some boundary points in our analysis. For an easy reference we keep the notation same as in~\cite{jbak,oberlin} to denote the points already discussed in there. The endpoints of the region $\mathcal R$ are described as follows.
\begin{itemize}
\item Clearly $O=(0,0,\dots,0;0)$ is an endpoint.
\item Point $B=(\frac{n-1}{n+1},\frac{n-1}{n+1},\dots,\frac{n-1}{n+1};0)$ as intersection of $\frac{n+1}{p}=n-1+\frac{2}{r}$ and $\frac{1}{r}=0$. The dual point is $G=(\frac{n-1}{n+1},\dots,\frac{n-1}{n+1},1;\frac{2}{n+1})$. Note that it is intersection of $\sum^{n-1}_{j=1}\frac{1}{p_{j}}+\frac{2}{p_{n}}=n-1+\frac{2}{r}$, $\sum^{n-1}_{j=1,j\neq k}\frac{1}{p_{j}}+\frac{2}{p_{k}}+\frac{2}{p_{n}}=n+\frac{1}{r}$ and $p_{n}=1$. There are $n$ different points of this type.
\item Point $M=(\frac{n+1}{n+3},\frac{n+1}{n+3},\dots,\frac{n+1}{n+3};\frac{2}{n+3})$ as intersection of $\frac{n+1}{p}=n-1+\frac{2}{r}$ and $\frac{n+2}{p}=n+\frac{1}{r}$.
\item Point $E=(\frac{n-1}{n},\frac{n-1}{n},\dots,\frac{n-1}{n},0;0)$ as the intersection of $\sum^{n}_{j=1,j\neq k}\frac{1}{p_{j}}+\frac{2}{p_{k}}= n-1+\frac{2}{r},~k\neq n$, $\frac{1}{p_{n}}=0$ and $\frac{1}{r}=0$. Note that due to symmetry there are $n$ different points of this type. Point $E$ has two different type of dual points. One is of type $P=(\frac{n-1}{n},\frac{n-1}{n},\dots,\frac{n-1}{n},1;1)$ with $n$ different points. The other type of dual is
$K=(\frac{n-1}{n},\frac{n-1}{n},\dots,\frac{n-1}{n},1,0;\frac{1}{n})$, which can be seen as intersection of $\sum^{n}_{j=1,j\neq n-1}\frac{1}{p_{j}}+\frac{2}{p_{n-1}}=n-1+\frac{2}{r}$, $\sum^{n}_{j=1,j\neq k,n-1}\frac{1}{p_{j}}+\frac{2}{p_{k}}+\frac{2}{p_{n-1}}=n+\frac{1}{r} ,(k\neq n)$, $\frac{1}{p_{n-1}}=1$ and $ \frac{1}{p_{n}}=0$. Note that there are $n(n-1)$ many different points of this type.
\item Point $A=(\frac{n+1}{n+2},\frac{n+1}{n+2},\dots,\frac{n+1}{n+2};1)$ as intersection of $\frac{n+2}{p}=n+\frac{1}{r}$ and $\frac{1}{r}=1$. Note that $A$ is an endpoint for $n\geq 3$. When $n=2$, the point $A$ lies on the line segment joining $P$ and $P'$ (see Figure~$2$). The dual of $A$ is given by $A^*=(\frac{n+1}{n+2},\frac{n+1}{n+2},\dots,\frac{n+1}{n+2},0;\frac{1}{n+2})$. There are $n$ different points of this type.
\item It is easy to see the point $C=(0,0,\dots,0,1,0,\dots,0;1)$ is an endpoint and there are $n$ points of this type.
\item For $n\geq 3$, the point $Z=(1,1,\dots,1,1,0;1)$ is the intersection of $\sum^{n}_{j=1,j\neq k,l}\frac{1}{p_{j}}+\frac{2}{p_{k}}+\frac{2}{p_{l}}= n+\frac{1}{r},(k,l\neq n)$, $\frac{1}{r}=1$ and $p_{j}=1$, for $j=1,2,\cdots n-1$. There are $n$ different points of this type. The dual point is given by $Z^*=(1,1,\dots,1,0,0;0)$ which is intersection of $\sum^{n}_{j=1,j\neq k}\frac{1}{p_{j}}+\frac{2}{p_{k}}= n-1+\frac{2}{r}, k\neq n-1,n$; $\frac{1}{r}=0$ and $p_{j}=1, j=1,2,\dots, n-2$. There are $\frac{n(n-1)}{2}$ points of this type. Note that there is no analogue of $Z$ and $Z^*$ for $n=2.$
\item When $n=3$, consider the point $N=(\frac{3}{5},\frac{3}{5},\frac{1}{5};0)$. It has two type of dual points given by $N^{*1}=(1,\frac{3}{5},\frac{1}{5};\frac{2}{5})$ and $N^{*3}=(\frac{3}{5},\frac{3}{5},1;\frac{4}{5})$. Note that $N$ is not an endpoint but lies on the line segment joining $E$ and $B$. We shall see that $T$ is of strong type at $N$.
\end{itemize}
\begin{remark}\label{rem}Note that in the above if we interchange positions of $p_j$ for two different values of $j$ we get another endpoint. This is due to the symmetry of the operator $T$. Points obtained in this fashion will be referred to as similar points. We will state results and demonstrate the proofs only for one point of each type and the corresponding results hold for points that are similar to the ones described.
\end{remark}
The points $O,B,M$ and $A$ have $p_j=p$ for all $j$. These points have already been addressed in~\cite{oberlin}. Even though the point $N$ is not an endpoint, it plays an important role in proving strong type estimates on some part of the boundary. The region $\mathcal R$ is the closed convex hull of all the endpoints listed above (along with their similar points) in ${\mathbb {R}}^{n+1}$.
We list down the endpoints for $n=2$ case separately and it is possible to draw the region $\mathcal R$ in this case, see Figure~$2.$ The endpoints where we have strong type results are marked with boldfaced points. For $n=2$ the endpoints are: $O=(0,0;0), E'=(\frac{1}{2},0;0), E=(0,\frac{1}{2};0), B=(\frac{1}{3},\frac{1}{3};0), K=(0,1;\frac{1}{2}), K'=(1,0;\frac{1}{2}), M=(\frac{3}{5},\frac{3}{5};\frac{2}{5}), C=(0,1;1), C'=(1,0;1), P=(\frac{1}{2},1;1)$,
$P'=(1,\frac{1}{2};1), G'=(1,\frac{1}{3};\frac{2}{3}), G=(\frac{1}{3},1;\frac{2}{3}).$
\begin{figure} \label{fig1}
\includegraphics[width=.6\textwidth]{FIGDM2}
\caption{Region $\mathcal R$ for $n=2$}
\end{figure}
\subsection{Sufficient part : The case of $n=2$}\label{two}
Note that for $n=2$ the operator $T$ takes a simpler form as compared to the higher dimensional analogues. Also, there are fewer endpoints in this case and some of the proofs for $n=2$ provide foundation to deal with the case $n\geq 3$. Therefore, we deal with the case of $n=2$ separately. This treatment also helps us understand the problem better.
As mentioned previously it is enough to prove boundedness of $T$ at the endpoints of $\mathcal R$. Boundedness of $T$ at endpoints for which all the $p_j's$ are equal is already known due to~\cite{jbak,oberlin}. We include these known points in our statement for completion and provide proofs for the remaining points.
\begin{theorem}\label{mainresult:sufn=2}{\bf (Sufficient part for $n=2$)}\label{ $n=2$} In dimension $n=2$ the following estimates hold.
\begin{enumerate}
\item $T$ is of strong type at $O,C$ and $M$.
\item $T$ is of restricted type at $P,G,E,$ and $B$. Moreover, $T$ fails to be of strong type at these points.
\item $T$ is of weak type at the point $K$ and fails to be of strong type at $K$.
\item $T$ does not satisfy strong type estimates at points lying on the open line segment $BE.$
\end{enumerate}
\end{theorem}
\begin{theorem}\label{mainresult:sufall}{\bf (Sufficient part for $n\geq3$)} Let $n\geq 3$. Then,
\begin{enumerate}
\item $T$ is of strong type at $O,B,M,A,A^{*},C,Z,Z^*$ and $G$.
\item $T$ is of restricted type at $E$ and $P$ and it is of restricted weak type at $K$.
\item Moreover, when $n=3$, $T$ is of strong type at the boundary point $N=(\frac{3}{5},\frac{3}{5},\frac{1}{5};0)$ and its dual points $N^{*1}$ and $N^{*3}$.
\end{enumerate}
\end{theorem}
\begin{remark} We have the following remarks concerning the results stated as above.
\begin{enumerate}
\item In Theorems~\ref{mainresult:sufn=2} and~\ref{mainresult:sufall} we have stated the result only for one point of each type. The analogous estimates hold at points which are similar to the ones given in theorems. For example, at point $C'$ (and line segment $BE'$) we have the analogous estimate as that of $C$ (and line segment $BE$).
\item Since the number of endpoints for $n\geq 3$ is large and different type of points have different type of estimates, we do not record results obtained by applying the standard multilinear interpolation arguments to the estimates obtained in Theorems. We have written down certain positive and negative results separately for points lying on the boundary as they require additional arguments along with multilinear interpolation. For example, strong type estimates on the line segment $PP'$ are not recorded in Theorem~\ref{mainresult:sufn=2} as they follow using the multilinear interpolation Lemma~\ref{christ}. However, the failure of strong type estimates on the line segment $BE$ needs to be discussed through examples and hence we have it in Theorem~\ref{mainresult:sufn=2}.
\end{enumerate}
\end{remark}
\section{Proof of Theorem~\ref{mainresult:nec}}\label{proof:nec}
In this section we obtain necessary conditions on exponents for $L^p$ boundedness of the operator $T$. We work with examples considered in \cite{oberlin}. Assume that $T$ is of strong type at $(p_1,p_2,\dots,p_n;r)$ where $1\leq p_1,p_2,\dots,p_n,r\leq \infty$.
For a positive number $L>0$ consider the following setting. Let $f_{j}=\chi_{_{[-L,L]}},~j=1,2,\dots,n$. Observe that for $|x|<L-1$ we have that $T(f_1,f_2,\dots,f_n)(x)\gtrsim 1$ . Therefore, the assumption on $T$ implies that
$$L^{\frac{1}{r}}\lesssim L^{\frac{1}{p_{1}}+\dots+\frac{1}{p_{n}}}$$
for arbitrary large numbers $L$. This yields the first necessary condition in Theorem~\ref{mainresult:nec}, namely $\frac{1}{r}\leq\sum^{n}_{j=1}\frac{1}{p_{j}}.$
Next, consider the functions $f_{j}=\chi_{[-\epsilon,\epsilon]}, j=1,2,\dots,n-1$ and $f_{n}=\chi_{[1-2\epsilon^{2},1+2\epsilon^{2}]}$, where $\epsilon>0$. With this choice of functions for $|x|\leq c\epsilon^{2}$ we have that $T(f_1,f_2,\dots,f_n)(x)\gtrsim \epsilon^{n-1},$ where $c$ is a constant. As earlier we get that $$\epsilon^{n-1+\frac{2}{r}}\lesssim \epsilon^{\frac{1}{p_{1}}+\dots+\frac{1}{p_{n-1}}+\frac{2}{p_{n}}}.$$ Letting $\epsilon\rightarrow 0$, we get the second necessary condition $\frac{1}{p_{1}}+\dots+\frac{1}{p_{n-1}}+\frac{2}{p_{n}}\leq n-1+\frac{2}{r}$ for $k=n.$ Interchanging the roles of functions suitably we get the condition for other values of $k.$
Finally, let $f_{k}=\chi_{[\frac{1}{\sqrt{2}}-\epsilon^{2}, \frac{1}{\sqrt{2}}+\epsilon^{2}]}$, $f_{l}=\chi_{[\frac{-1}{\sqrt{2}}-\epsilon^{2}, \frac{-1}{\sqrt{2}}+\epsilon^{2}]}$ and $f_{j}=\chi_{[-\epsilon,\epsilon]}$ for $j\neq k,l$. Let $J$ denote the box in $\mathbb{R}^{n}$ given by $\chi_J(x_{1},\dots,x_{n})=\prod^{n}_{j=1}f_{j}(x_{j})$. Now observe that the measure of the surface $(J+(t,t,\dots,t))\cap \mathbb{S}^{n-1}$ is of the order of $\epsilon^{n}$ for $|t|\leq \frac{\epsilon}{2n}$. Boundedness of $T$ at the point $(p_{1},p_{2},\dots,p_{n};r)$ implies that $$\epsilon^{n+\frac{1}{r}}\lesssim \epsilon^{\sum^{n}_{j=1,j\neq l,k}\frac{1}{p_{j}}+\frac{2}{p_{k}}+\frac{2}{p_{l}}}.$$
Since $\epsilon$ is arbitrarily small we get the third necessary condition that $\sum^{n}_{j=1,j\neq k,l}\frac{1}{p_{j}}+\frac{2}{p_{k}}+\frac{2}{p_{l}}\leq n+\frac{1}{r}.$
\qed
\section{Proof of Theorem~\ref{mainresult:sufn=2}}\label{proof:n=2}
In this section we prove Theorem~\ref{mainresult:sufn=2}. We demostrate the arguments at each point listed in the theorem separately.
Note that in the case of $n=2$ the operator $T$ is given by
\begin{eqnarray*}\label{def}T(f_{1},f_{2})(x)&=&\int_{\mathbb{S}^{1}}f_{1}(x-\sigma_1)f_{2}(x-\sigma_2)d\sigma\\
&=&\int_{0}^{2\pi}f_{1}(x-\cos t)f_{2}(x-\sin t)dt,~x\in {\mathbb {R}}.
\end{eqnarray*}
Without loss of generality we may assume that $f_1$ and $f_2$ are non-negative functions.
\subsection*{Endpoints $O=(0,0;0), B=(\frac{1}{3},\frac{1}{3};0)$ and $M=(\frac{3}{5},\frac{3}{5};\frac{2}{5})$} These points are already considered in~\cite{jbak,oberlin}. The operator $T$ satisfies strong type estimates at $O=(0,0;0)$ and $M=(\frac{3}{5},\frac{3}{5};\frac{2}{5})$. However, it fails to be of strong type at $B=(\frac{1}{3},\frac{1}{3};0).$ We have weaker estimates at this point, namely $T$ is bounded from $L^{3,s}({\mathbb {R}})\times L^{3,s}({\mathbb {R}})\rightarrow L^{\infty}({\mathbb {R}})$ if $0<s\leq 2$. See, Theorem~\ref{jbak1} and \cite{jbak,oberlin} for detail.
\subsection*{Endpoints $C=(0,1;1)$ and $P=(\frac{1}{2},1;1)$}
First, note that it is easy to verify that $T$ is of strong type at $C$. Whereas from~ \cite{jbak} the estimate at $P$ is known, namely, $T$ maps $L^{2,1}({\mathbb {R}})\times L^{1}({\mathbb {R}})\rightarrow L^{1}({\mathbb {R}}).$ Indeed, one can show that $T$ satisfies strong type estimates at point $(p_1,p_2;r)$ lying on the line segments $CP$, except the point $P$.
For, note that an arbitrary point on the open line segment $CP$ can be written as $(\frac{1}{q},1;1)$ for $2<q<\infty$. Let $h\in L^{\infty}(\mathbb{R})$\\ and make a change of variables to get that
\begin{eqnarray*}
|\langle T(f_1,f_2),h\rangle|&=&|\int_{\mathbb{R}}\left(\int^{1}_{0}f_1(x-\sqrt{1-t^{2}})f_2(x-t)\frac{dt}{\sqrt{1-t^{2}}}\right)h(x)dx|\\
&=&|\int_{\mathbb{R}}f_2(x)\left(\int^{1}_{0}h(x+t)f_1(x+t-\sqrt{1-t^{2}})\frac{dt}{\sqrt{1-t^{2}}}\right)dx|\\
&\leq&\Vert f_2\Vert_{L^{1}}\Vert h\Vert_{L^{\infty}(\mathbb{R})}\Vert f_1\Vert_{L^{q}}\Vert \frac{\chi_{[0,1]}(\cdot)}{\sqrt{(1-\cdot)}}\Vert_{L^{q'}}.
\end{eqnarray*}
Since $1<q'<2$, we have $\Vert \frac{\chi_{[0,1]}(\cdot)}{\sqrt{(1-\cdot)}}\Vert_{L^{q'}}<\infty$. This completes the proof.
Next, we show that $T$ cannot be of strong type at $P$.
For $h\in L^{\infty}(\mathbb{R})$, we can write
\begin{eqnarray*}
\langle T(f_1,f_2),h\rangle
&=&\int_{\mathbb{R}}f_2(x+1)\left(\int^{1}_{-1}h(x+1+t)f_1(x+1+t-\sqrt{1-t^{2}}) \frac{dt}{\sqrt{1-t^{2}}}\right)dx.
\end{eqnarray*}
Therefore, it is enough to show that
$$T_{1}(f_1,h)(x)=\int^{1}_{-1}h(x+1+t)f_{1}(x+1+t-\sqrt{1-t^{2}}) \frac{dt}{\sqrt{1-t^{2}}}$$
is a unbounded function for some choice of $h\in L^{\infty}(\mathbb{R})$ and $f_1\in L^{2}(\mathbb{R})$.
We choose $h=1$ and $f_1(t)=\chi_{[0,\frac{9}{10}]}(|t|)|t|^{-\frac{1}{2}}|\log|t||^{-\frac{2}{3}}.$ Note that $f_1\in L^{2}(\mathbb{R})$. For arbitrarily small $x>0$ we get that,
\begin{eqnarray*}
T_{1}(f_1,h)(x)&\gtrsim&\int^{1-\frac{x^{2}}{2}}_{\frac{3}{4}}|x+1-t-\sqrt{1-t^{2}}|^{-\frac{1}{2}}|\log|x+1-t-\sqrt{1-t^{2}}||^{-\frac{2}{3}}\frac{dt}{\sqrt{1-t^{2}}}\\
&\gtrsim&|\log x|^{-\frac{2}{3}}\int^{1-\frac{x^{2}}{2}}_{\frac{3}{4}}(1-t)^{-1}dt.
\end{eqnarray*}
The second inequality in the above estimate follows by using that
$|x+1-t-\sqrt{1-t^{2}}|\lesssim 1-t$ and $|x+1-t-\sqrt{1-t^{2}}|\gtrsim x^{2}$ for $t\in [\frac{3}{4},1-\frac{x^{2}}{2}]$.
This gives the desired result.
\subsection*{Endpoint $E=(0,\frac{1}{2};0)$ } We shall show that the operator $T$ does not satisfy of strong type at $E=(0,\frac{1}{2};0).$ However, it maps $L^{\infty}({\mathbb {R}})\times L^{2,1}({\mathbb {R}}) $ into $L^{\infty}({\mathbb {R}})$.
Note that after taking out the $\|f_1\|_{\infty}$, it is enough to show that the operator
\begin{eqnarray}\label{Ave}
\tilde{\mathcal{A}}f_{2}(x)=\int_{\mathbb {S}^1}f_{2}(x-\sigma_2)d\sigma,~x\in {\mathbb {R}}
\end{eqnarray}
maps $L^{2,1}({\mathbb {R}})$ into $L^{\infty}({\mathbb {R}})$. This follows using H\"{o}lder's inequality. This point is also used in~\cite{jbak}. Infact, the same argument yields strong type estimate for $T$ on the line segment $OE$ except at point $E$.
Next, we show that the operator $T$ does not verify strong type estimate at $E.$
For, let $f_1=1$ and $f_2(t)=\chi_{[0,\frac{1}{2}]}(|1+t|)|1+t|^{-\frac{1}{2}}|\log|1+t||^{-\frac{2}{3}}$ and note $f_2\in L^2({\mathbb {R}})$. Let $x\in {\mathbb {R}}$ be a small negative number, then we have the following.
\begin{eqnarray*}
T(f_1,f_2)(x)& \gtrsim & \int^{1}_{0}f_2(x-t)\frac{dt}{\sqrt{1-t^{2}}}\\
&\gtrsim & \int^{1-2|x|}_{\frac{1}{2}}\chi_{[0,\frac{1}{2}]}(|1-t+x|)|1-t+x|^{-\frac{1}{2}}|\log|1-t+x||^{-\frac{2}{3}}\frac{dt}{\sqrt{1-t^{2}}}.
\end{eqnarray*}
Note that $|1+x-t|\lesssim |1-t|$ and $|1+x-t|\gtrsim |x|$. Therefore, we get that
\begin{eqnarray*}
T(f_{1},f_{2})(x)&\gtrsim& \int^{1-2|x|}_{\frac{1}{2}}|\log|x||^{-\frac{2}{3}}|1-t|^{-1}dt\\
&=&|\log|x||^{-\frac{2}{3}}\int^{1-2|x|}_{\frac{1}{2}}|1-t|^{-1}dt\\
&\gtrsim &|\log|x||^{-\frac{2}{3}}|\log2|x||.
\end{eqnarray*}
Clearly, the function in the estimate above is not bounded near the origin $x=0$.
\subsection*{Endpoint $K=(0,1;\frac{1}{2})$} The operator $T$ is of weak type at $K$, i.e. it maps $L^{\infty}(\mathbb{R})\times L^{1}(\mathbb{R}) \rightarrow L^{2,\infty}(\mathbb{R})$. Moreover, $T$ fails to be of strong type at $K$.
Note that in view of the standard duality arguments (see \cite{Grafakosclassical}, page $69$), the weak type estimates for the operator $T$ can be deduced by considering the following estimate.
\begin{eqnarray*}
\sup_{\Vert h\Vert_{L^{2,1}}\leq 1}|\int_{\mathbb{R}}T(f_1,f_2)(x)h(x)dx|\lesssim \Vert f_1\Vert_{L^{\infty}} \Vert f_2\Vert_{L^{1}}.
\end{eqnarray*}
Consider
\begin{eqnarray*}
|\int_{\mathbb{R}}T(f_1,f_2)(x)h(x)dx|&=&|\int^{\infty}_{-\infty}\int^{1}_{-1}f_2(x-t)f_1(x-\sqrt{1-t^{2}})\frac{dt}{\sqrt{1-t^{2}}}h(x)dx|\\
&=&|\int^{\infty}_{-\infty}f_2(x)\int^{1}_{-1}h(x+t)f_1(x+t-\sqrt{1-t^{2}})\frac{dt}{\sqrt{1-t^{2}}}dx|\\
&\lesssim&\Vert f_2\Vert_{L^{1}}\Vert f_1\Vert_{L^{\infty}}\Vert h\Vert_{L^{2,1}}.
\end{eqnarray*}
Next, we see that the operator $T$ cannot be bounded from $ L^{\infty}(\mathbb{R})\times L^{1}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$.
Set $f_1=1$ and consider
\begin{eqnarray*}
T(f_1,f_2)(x)&=&\int_{\mathbb{S}^{1}}f_2(x-z)d\sigma(y,z)\\
&=&\int_{\mathbb{R}}\hat{f}_2(\xi)e^{2\pi\iota x\cdot\xi}\left(\int_{\mathbb{S}^{1}}e^{-2\pi\iota(0,\xi)\cdot(y,z)}d\sigma(y,z)\right)d\xi\\
&=&\int_{\mathbb{R}}\hat{f}_2(\xi)\widehat{d\sigma}(0,\xi)e^{2\pi\iota x\cdot\xi}d\xi.
\end{eqnarray*}
Therefore, we get that
\begin{eqnarray}\label{eq101}
\Vert T(f_1,f_2)\Vert^{2}_{L^{2}}=\Vert \widehat{T(f_1,f_2)}\Vert^{2}_{L^{2}}\gtrsim \int_{|\xi|>1}|\hat{f_{2}}(\xi)|^{2}(1+|\xi|)^{-1}d\xi.
\end{eqnarray}
Here we have used decay estimate for the Fourier transform of surface measure, namely,
$$\widehat{d\sigma}(0,\xi)\sim C |\xi|^{\frac{-1}{2}}[e^{2\pi\iota|\xi|}\sum^{\infty}_{j=0}\alpha_{j}|\xi|^{-j}+e^{-2\pi\iota|\xi|}\sum^{\infty}_{j=0}\beta_{j}|\xi|^{-j}],$$
as $|\xi|\rightarrow\infty$, for suitable constants $\alpha_{j},\beta_{j}$ (see \cite{Steinbook}, page $391$).
Choose $f_2\in L^{1}({\mathbb {R}})$ such that $|\hat f_{2}(\xi)|$ decays slower than $(\log|\xi|)^{-\frac{1}{2}}$ for $|\xi|>1.$ This implies that integral in the estimate ~\ref{eq101} diverges and consequently we get the desired result.
\subsection*{Endpoint $G=(\frac{1}{3},1;\frac{2}{3})$}
We show that the operator $T$ maps $ L^{3,\frac{3}{2}}(\mathbb{R})\times L^{1}(\mathbb{R})\rightarrow L^{\frac{3}{2}}(\mathbb{R}).$
Subsequently, we get that $T$ is of restricted type at $G$.
Observe that for the said boundedness result, it is enough to prove that
\begin{eqnarray*}
|\langle T(f_1,f_2),h\rangle|\lesssim \Vert f_1\Vert_{L^{3,\frac{3}{2}}}\Vert f_2\Vert_{L^{1}}\Vert h\Vert_{L^{3}},~~~ h\in L^{3}(\mathbb{R}).
\end{eqnarray*}
Consider
\begin{eqnarray*}
\langle T(f_1,f_2),h\rangle&=&\int_{\mathbb{R}}\left(\int^{2\pi}_{0}f_1(x-\cos\theta)f_2(x-\sin\theta)d\theta\right)h(x)dx\\
&=&\int_{\mathbb{R}}f_2(x)\left(\int^{2\pi}_{0}f_1(x+\sin\theta-\cos\theta)h(x+\sin\theta)d\theta\right)dx\\
&=&\int_{\mathbb{R}}f_2(x)\left(\int_{\mathbb{S}^{1}}f_1(x-\sigma\cdot v_{1})h(x-\sigma\cdot v_{2})d\sigma\right)dx,
\end{eqnarray*}
where $v_{1}=\left( \begin{array}{c} -1 \\ 1 \end{array} \right)$ and $v_{2}=\left( \begin{array}{c} 0 \\ 1 \end{array} \right)$ are linearly independent vectors in $\mathbb{R}^{2}$ and $\sigma=e^{i\theta}\in\mathbb{S}^{1}$.
We consider the following operator
\begin{eqnarray*}
T_v(f_{1},h)(x)=\int_{\mathbb{S}^{1}}f_1(x-\sigma\cdot v_{1})h(x-\sigma\cdot v_{2})d\sigma.
\end{eqnarray*}
Using the same argument as in case of point $E$, we have the following
\begin{eqnarray}\label{T_v}
\Vert T_v(f_{1},h)\Vert_{L^{\infty}}\lesssim \Vert h\Vert_{L^{2,1}} \Vert f_{1}\Vert_{L^{\infty}}~~\text{and}~~\Vert T_v(f_{1},h)\Vert_{L^{\infty}}\lesssim \Vert h\Vert_{L^{\infty}}\Vert f_{1}\Vert_{L^{2,1}}.
\end{eqnarray}
For a small positive number $0<\epsilon<\frac{1}{4}$, we decompose the operator as $T_v(f_{1},h)(x)\lesssim \sum_{j=1,2}U_{j}(f_{1},h)(x)$ for almost every $x\in \mathbb{R}$, where
\begin{eqnarray*}
U_{j}(f_{1},h)(x)=\int_{\{\sigma\in\mathbb{S}^{1}:|\sigma\cdot v_{j}|>\epsilon\}}f_1(x-\sigma\cdot v_{1})h(x-\sigma\cdot v_{2})d\sigma, ~~j=1,2.
\end{eqnarray*}
It is easy to show that (also see Lemma 2 in~\cite{jbak})
\begin{eqnarray}\label{U_j}
\Vert U_{1}(f_{1},h)\Vert_{L^{\infty}}\lesssim \Vert h\Vert_{L^{\infty}}\Vert f_{1}\Vert_{L^{1}}~~\text{and}~~
\Vert U_{2}(h,f_{1})\Vert_{L^{\infty}}\lesssim \Vert h\Vert_{L^{1}}\Vert f_{2}\Vert_{L^{\infty}}
\end{eqnarray}
Observe that $U_j$ also satisfies the estimate~(\ref{T_v}). Therefore, interpolating between these estimates for $U_j$ we get that the desired result holds for $U_j$. Consequently, we get that
$$\Vert T_v(f_{1},h)\Vert_{L^{\infty}}\lesssim \Vert h\Vert_{L^{3}}\Vert f_{1}\Vert_{L^{3,\frac{3}{2}}}.$$
This yields that
\begin{eqnarray*}
|\langle T(f_1,f_2),h\rangle| &\leq& \Vert f_2\Vert_{L^{1}}\Vert T_v(f_1,h)\Vert_{L^{\infty}}\\
&\lesssim &\Vert f_2\Vert_{L^{1}}\Vert f_{1}\Vert_{L^{3,\frac{3}{2}}}\Vert h\Vert_{L^{3}}.
\end{eqnarray*}
Next, we give an example to show that $T$ fails to be of strong type at $G$, i.e., it is unbounded from $L^{3}(\mathbb{R})\times L^{1}(\mathbb{R})$ into $L^{\frac{3}{2}}(\mathbb{R})$.
For, let $f_2\in L^1({\mathbb {R}})$ and $f_1, h\in L^{3}(\mathbb{R})$ and consider
\begin{eqnarray*}
\langle T(f_1,f_2),h\rangle &=& \int_{\mathbb{R}}f_1(x) T_v(f_1,h)(x)dx \end{eqnarray*}
where $T_{{v}}(f_1,h)(x)=\int^{1}_{0}h(x+t)f_{1}(x+t-\sqrt{1-t^{2}})\frac{dt}{\sqrt{1-t^{2}}}$ is the same operator as previously. Since boundedness properties of the operators $T_v$ and $T$ are equivalent, it suffices to prove that $T_{{v}}(f_1,h) \notin L^{\infty}(\mathbb{R})$ for a suitable choice of functions $h$ and $f_1$ in $L^{3}(\mathbb{R})$.
Let $f_1(t)=\chi_{[0,\frac{9}{10}]}(|t|)|t|^{-\frac{1}{3}}|\log|t||^{-\frac{2}{5}}$ and $h(t)=f_{1}(1+t)$. For a small negative real number $x$, one has the following.
\begin{eqnarray*}
T_v(f_1,h)(x)\geq\int^{1-2|x|}_{\frac{1}{2}}\frac{|1+x-t|^{-\frac{1}{3}}}{|\log|1+x+t||^{\frac{2}{5}}}\frac{|x-\sqrt{1-t^{2}}|^{-\frac{1}{3}}}{|\log|x-\sqrt{1-t^{2}}||^{\frac{2}{5}}}\frac{dt}{\sqrt{1-t^{2}}}
\end{eqnarray*}
Observe that for our choice of $x$ and $t$ in the integral above we have $|1+x-t|\lesssim |1-t|$, $|x-\sqrt{1-t^{2}}|\lesssim \sqrt{(1-t)}$, $|\log|1+x-t||\lesssim |\log|x||$ and $|\log|x-\sqrt{1-t^{2}}||\lesssim |\log|x||$. Therefore,
\begin{eqnarray*}
T_v(f_1,h)(x)&\gtrsim& |\log|x||^{-\frac{4}{5}}\int^{1-2|x|}_{\frac{1}{2}}|1-t|^{-1}dt\\
&=&|\log|x||^{-\frac{4}{5}}(-\log2|x|-\log2).
\end{eqnarray*}
This yields the desired result.
\subsection*{Open line segment $BE$} Let $(\frac{1}{p_1},\frac{1}{p_2};0)$ be a point on the open line segment $BE$. Note that one can write $\frac{1}{p_{1}}=\frac{\theta}{3}$ and $\frac{1}{p_{2}}=\frac{\theta}{3}+\frac{1-\theta}{2}$ for $\theta\in (0,1)$. Set $\epsilon=\frac{1-\theta}{6}$ and write $\frac{1}{p_{1}}=\frac{1}{3}-2\epsilon$ and $\frac{1}{p_{2}}=\frac{1}{3}+\epsilon$.
Consider the functions $f_{1}(t)=\chi_{[0,\frac{9}{10}]}(|t|)|t|^{-(\frac{1}{3}-2\epsilon)}|\log|t||^{-\frac{1}{3}}$ and $f_{2}(t)=\chi_{[0,\frac{9}{10}]}(|1+t|)|1+t|^{-(\frac{1}{3}+\epsilon)}|\log|1+t||^{-\frac{1}{2}}$. Then for $x$ near the origin we have
\begin{eqnarray*}
T(f_{1},f_{2})(x)&\geq&\int^{1}_{0}f_{1}(x-\sqrt{1-t^{2}})f_{2}(x-t)\frac{dt}{\sqrt{1-t^{2}}}\\
&\gtrsim& \int^{1-2|x|}_{\frac{3}{4}}f_{1}(x-\sqrt{1-t^{2}})f_{2}(x-t)\frac{dt}{\sqrt{1-t^{2}}}.
\end{eqnarray*}
Observe that for arbitrarily small $x$, $|1+x-t|\lesssim |1-t|$, $|1+x-t|\gtrsim |x|$, $|x-\sqrt{1-t^{2}}|\lesssim \sqrt{1-t^{2}}$ and $|x-\sqrt{1-t^{2}}|\gtrsim |x|$.
Therefore we get,
\begin{eqnarray*}
T(f_{1},f_{2})(x)&\gtrsim& |\log|x||^{-\frac{5}{6}}\int^{1-2|x|}_{\frac{3}{4}}|1-t|^{-1}dt\\
&\gtrsim& -|\log|x||^{-\frac{5}{6}}\log|x|.
\end{eqnarray*}
The above tends to infinity as $x\rightarrow 0$.
This completes the proof of Theorem~\ref{mainresult:sufn=2}.\qed
\begin{remark}
\begin{enumerate}
\item The operator $T$ satisfies strong type estimates at points lying in regions $OEKC$, except at points $E$ and $K$, see Figure~$2.$ Observe that strong type estimates on $OC$ follow by the Riesz-Thorin interpolation.
Next, note that on $OEKC$, we have $\frac{1}{p_{1}}=0$. Therefore, we have
\begin{eqnarray*}
|T(f_{1},f_{2})(x)|\leq \Vert f_{1}\Vert_{L^{\infty}}\tilde{\mathcal{A}}f_{2}(x),
\end{eqnarray*}
where $\tilde{\mathcal A}$ is same as defined earlier in~\ref{Ave} and can be written as
\begin{eqnarray*}
\tilde{\mathcal{A}}f_{2}(x)=\int_{\mathbb{R}}\hat{f_{2}}(\xi)\hat{d\sigma}(\xi,0)e^{2\pi \iota x\cdot\xi}d\xi.
\end{eqnarray*}
Using the estimate $|\hat{d\sigma}(\xi,0)|\lesssim (1+|\xi|)^{-\frac{1}{2}},$ we get the following (see \cite{Hormander})
\begin{eqnarray*}
\tilde{\mathcal{A}}: L^{p_{2}}(\mathbb{R})\rightarrow L^{r}(\mathbb{R}),~~\text{for}~~1<p_{2}\leq2\leq r<\infty~~with~~\frac{1}{p_{2}}-\frac{1}{r}\leq\frac{1}{2}.
\end{eqnarray*}
This implies strong type estimates for $T$ in the region $OEKC$, except on the line segments $OE$ and $KC$. The required estimates are already proved for points $OE,$ whereas for points on $KC$ they can be deduced using the Marcinkiewicz interpolation theorem.
\end{enumerate}
\end{remark}
\section{Proof of Theorem~\ref{mainresult:sufall}}\label{proof:all}
We deal with each point separately. We repeat that we will describe proofs for one endpoint of each type. We use the idea from~\cite{jbak,oberlin}.
\subsection*{Endpoints $O, Z, Z^*, B, M, A, A^*, $ and $ C$}
The boundedness of $T$ at the points $O, Z^*, B, M$ and $A$ is already known due to \cite{jbak,oberlin}.
Moreover, using standard duality arguments one can deduce corresponding estimates at their dual points. In particular, boundedness of $T$ at $Z=(1,1,\dots,1,0;1)$ and $A^*=(\frac{n+1}{n+2},\frac{n+1}{n+2},\dots,\frac{n+1}{n+2},0;\frac{1}{n+2})$ can be deduced from that of $Z^*=(1,1,\dots,1,0,0;0)$ and $A=(\frac{n+1}{n+2},\frac{n+1}{n+2},\dots,\frac{n+1}{n+2};1)$ respectively.
Further, the desired estimates for $T$ at $C$ follows in a straightforward manner.
\subsection*{Endpoint $E=(\frac{n-1}{n},\dots,\frac{n-1}{n},0;0)$}
We show that the operator $T$ is of restricted type at the point $E$.
Let $v_{1},v_{2},\cdots ,v_{n}$ be linearly independent vectors in ${\mathbb {R}}^n$and consider the operator $T_v$ defined as earlier. Further, let $\Lambda:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ be a linear map from ${\mathbb {R}}^n$ to ${\mathbb {R}}^n$ such that $e_{j}\cdot \Lambda x=v_{j}\cdot x,~x\in\mathbb{R}^{n}$. Fix a unit vector $u_{n}$ with $\Lambda u_{n}=c(0,0,\dots,0,1)$ for some $c\in\mathbb{R}\setminus\{0\}$ and let $\{u_{1},u_{2},\dots,u_{n}\}$ be an orthonormal basis of $\mathbb{R}^{n}$.
Let $\eta=(\eta_{1},\dots,\eta_{n-1})$ denote an element of $ \mathbb{S}^{n-2}$ and $d\eta$ be the normalised Lebesgue measure on $\mathbb{S}^{n-2}$. We consider the parametrization of $\mathbb{S}^{n-1}$ given by $$\sigma=\sum^{n-1}_{j=1}r\eta_{j}u_{j}+sgn(r)\sqrt{1-r^{2}}u_{n}.$$ where $-1\leq r\leq 1$.
For convenience we will use the notation $\vec{f}(y)=\prod_{j=1}^nf_j(y_j),$ where $y=(y_1,y_2,\dots,y_n)\in {\mathbb {R}}^n.$ With this we have the following.
\begin{eqnarray*}
&&T(f_{1},f_{2},\dots,f_{n})(x)\\
&&=\int_{\mathbb{S}^{n-1}}\vec{f}\big((x,x,\dots,x)-\Lambda\sigma\big)d\sigma\\
&&=\int^{1}_{-1}\int_{\mathbb{S}^{n-2}}\vec{f}\big((x,x,\dots,x)-sgn(r)\sqrt{1-r^{2}}(0,0,\dots,0,c)-\Lambda(r\sum^{n-1}_{k=1}\eta_{k}u_{k})\big)d\eta |r|^{n-2}\frac{dr}{\sqrt{1-r^{2}}}\\
&&=\int^{1}_{-1}G(r)|r|^{n-2}\frac{dr}{\sqrt{1-r^{2}}},
\end{eqnarray*}
where $G(r)=\int_{\mathbb{S}^{n-2}}\prod^{n-1}_{j=1}f_{j}\big(x- re_{j}\cdot\Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k})\big)f_{n}\big(x- sgn(r)\sqrt{1-r^{2}}c-re_{n}\cdot\Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k})\big)d\eta$. Consider
\begin{eqnarray*}
&&\int^{1}_{-1}G(r)|r|^{n-2}dr\\
&&=\int^{1}_{-1} \int_{\mathbb{S}^{n-2}}\prod^{n-1}_{j=1}f_{j}\big(x- re_{j}\cdot\Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k})\big)f_{n}\big(x- sgn(r)\sqrt{1-r^{2}}c-re_{n}\cdot\Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k})\big)d\eta|r|^{n-2}dr\\
&&\lesssim \Vert f_{n}\Vert_{L^{\infty}}\int^{1}_{0}\int_{\mathbb{S}^{n-2}}\prod^{n-1}_{j=1}f_{j}\big(x- re_{j}\cdot\Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k})\big)d\eta r^{n-2}dr\\
&&\lesssim \Vert f_{n}\Vert_{L^{\infty}} \int_{\mathbb{R}^{n-1}}\prod^{n-1}_{j=1}f_{j}\big(x-e_{j}\cdot \Lambda(\sum^{n-1}_{k=1}x_{k}u_{k})\big)dx_{1}dx_{2}\dots dx_{n-1}\\
&&\lesssim |\Lambda|^{-1}\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1}\Vert f_{j}\Vert_{L^{1}}.
\end{eqnarray*}
On the other hand we have that
\begin{eqnarray}
G(r)&=&\nonumber \int_{\mathbb{S}^{n-2}}\prod^{n-1}_{j=1}f_{j}(x-re_{j}\cdot\Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k}))f_{n}(x-sgn(r)\sqrt{1-r^{2}}c-re_{n}\cdot \Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k}))d\eta\\
&\leq&\nonumber \Vert f_{n}\Vert_{L^{\infty}}\int_{\mathbb{S}^{n-2}}\prod^{n-1}_{j=1}f^{r}_{j}(\frac{x}{r}-e_{j}\cdot\Lambda(\sum^{n-1}_{k=1}\eta_{k}u_{k}))d\eta \\
&\leq& \nonumber r^{-\frac{(n-1)(n-2)}{n}}\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1}\Vert f_{j}\Vert_{\frac{n}{n-2},1} \\
&\simeq & \label{b} r^{-\frac{(n-1)(n-2)}{n}}\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1} |I_{j}|^{\frac{n-2}{n}},
\end{eqnarray}
where $f_{j}=\chi_{I_{j}},~j=1,2,\dots,n-1$.
Note that in the above we have used the boundedness of the operator $T$ at the point $B$ in dimension $n-1$.
Next, we need to consider two cases separately to complete the proof in the following fashion.
\noindent
\textbf{Case 1:} When $\prod^{n-1}_{j=1}|I_{j}|\geq 1$. In this case we use the estimate~\ref{b} as follows.
\begin{eqnarray*}
\int^{1}_{-1}G(r)|r|^{n-2}\frac{dr}{\sqrt{1-r^{2}}}
&\leq &\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1} |I_{j}|^{\frac{n-2}{n}} \int^{1}_{-1}r^{-\frac{(n-1)(n-2)}{n}}|r|^{n-2}\frac{dr}{\sqrt{1-r^{2}}}\\
&\lesssim& \Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1} |I_{j}|^{\frac{n-2}{n}}\\
&\leq & \Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1} |I_{j}|^{\frac{n-1}{n}}.
\end{eqnarray*}
\noindent
\textbf{Case 2:} When $\prod^{n-1}_{j=1}|I_{j}|<1$. We choose $\delta=(\prod^{n-1}_{j=1}|I_{j}|)^{\frac{2}{n}}$ and consider
\begin{eqnarray*}
\int^{1-\delta}_{-1+\delta}G(r)|r|^{n-2}\frac{dr}{\sqrt{1-r^{2}}}
&\leq & \delta^{-\frac{1}{2}}C\int^{1-\delta}_{-1+\delta}G(r)|r|^{n-2}dr\\
&\lesssim & \delta^{-\frac{1}{2}}\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1}|I_{j}|\\
&=&\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1}|I_{j}|^{\frac{n-1}{n}}.
\end{eqnarray*}
For the other part, we have
\begin{eqnarray*}
\int_{\{1-\delta\leq |r|\leq 1\}}G(r)|r|^{n-2}\frac{dr}{\sqrt{1-r^{2}}}
&\lesssim& \Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1}|I_{j}|^{\frac{n-2}{n}}\int_{\{1-\delta\leq |r|
\leq 1\}}|r|^{(n-2)-\frac{(n-1)(n-2)}{n}}\frac{dr}{\sqrt{1-r^{2}}}\\
&\lesssim & \Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1}|I_{j}|^{\frac{n-2}{n}} \delta^{\frac{1}{2}}\\
&=&\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-1}_{j=1}|I_{j}|^{\frac{n-1}{n}}.
\end{eqnarray*}
Here in the estimate above we have used (\ref{b}).
\subsection*{Endpoints $K=(\frac{n-1}{n},\frac{n-1}{n},\frac{n-1}{n},\dots,\frac{n-1}{n},1,0;\frac{1}{n}), P=(\frac{n-1}{n},\frac{n-1}{n},\dots,\frac{n-1}{n},1;1)$ and $G=(\frac{n-1}{n+1},\dots,\frac{n-1}{n+1},1;\frac{2}{n+1})$} First we show that the operator $T_{v}$ (or $T$) is of restricted weak type at $K$ and of restricted type at $P$. This can be proved using the boundedness of $T_{v}$ at the point $E$ along with the duality argument.
In order to prove the boundedness at the point $K$, we need to show that $T_{v}$ maps $L^{\frac{n}{n-1},1}(\mathbb R)\times\dots\times L^{\frac{n}{n-1},1}(\mathbb R)\times L^{1}(\mathbb R)\times L^{\infty}(\mathbb R)\rightarrow L^{n,\infty} (\mathbb R)$.
It suffices to show that $$\sup_{\Vert h\Vert_{L^{\frac{n}{n-1},1}}=1}|\langle T_{v}(f_{1},f_{2},\dots,f_{n}),h\rangle|\lesssim \Vert f_{n-1}\Vert_{L^{1}}\Vert f_{n}\Vert_{L^{\infty}}\prod^{n-2}_{j=1}\Vert f_{j}\Vert_{L^{\frac{n}{n-1},1}}.$$
Now,
\begin{eqnarray*}
|\langle T_{v}(f_{1},f_{2},\dots,f_{n}),h\rangle|
&=&|\langle f_{n-1},T^{*n-1}_{v}(f_{1},f_{2},\dots,f_{n-2},h,f_{n})|\\
&\leq& \Vert f_{n-1}\Vert_{L^{1}}\Vert T^{*n-1}_{v}(f_{1},f_{2},\dots,f_{n-2},h,f_{n})\Vert_{L^{\infty}}.
\end{eqnarray*}
Invoking the boundedness of $T_{v}$ at the point $E$ we get the desired estimate.
\begin{eqnarray*}
|\langle T_{v}(f_{1},f_{2},\dots,f_{n}),h\rangle|\lesssim \Vert f_{n-1}\Vert_{L^{1}} \Vert f_{n}\Vert_{L^{\infty}}\prod^{n-2}_{j=1}\Vert f_{j}\Vert_{L^{\frac{n}{n-1},1}}\Vert h\Vert_{L^{\frac{n}{n-1},1}}.
\end{eqnarray*}
Now, in order to prove restricted type boundedness at the point $P$, we need to show that $T_{v}$ maps $L^{\frac{n}{n-1},1}(\mathbb R)\times \dots\times L^{\frac{n}{n-1},1}(\mathbb R)\times L^{1}(\mathbb R)\rightarrow L^{1} (\mathbb R)$. It suffices to show that
\begin{eqnarray*}
\sup_{\Vert h\Vert_{L^{\infty}}=1}|\langle T_{v}(f_{1},\dots,f_{n}),h\rangle|\lesssim \Vert f_{n}\Vert_{L^{1}}\prod^{n-1}_{j=1}\Vert f_{j}\Vert_{\frac{n}{n-1},1}.
\end{eqnarray*}
This can be proved in a similar manner as the case of boundedness at $K$.
Next, note that the dual of $L^{\frac{n+1}{2}}({\mathbb {R}})$ is $L^{\frac{n+1}{n-1}}({\mathbb {R}})$. Then, using the same reasoning as above, this time with the point $B=(\frac{n-1}{n+1},\dots,\frac{n-1}{n+1};0)$, we get that $T_{v}$ is bounded at $G$.
\subsection*{Strong type estimates for $T$ at $N=(\frac{3}{5},\frac{3}{5},\frac{1}{5};0)$}
When $n=3$ we have the strong typeness at $N=(\frac{3}{5},\frac{3}{5},\frac{1}{5};0)$. This point lies on the segment joining the points $E=(\frac{2}{3},\frac{2}{3},0;0)$ and $B=(\frac{1}{2},\frac{1}{2},\frac{1}{2};0)$. Note that, we do not have strong type estimates at $E$. Therefore, proving strong type estimates for $T$ at $N$ would give us the same on the boudary between $N$ and $B$. Let $\epsilon>0$ be a small number and consider the following operators (see~\cite{jbak} for more details).
\begin{eqnarray*}
U_{j}(f_1,f_2,f_3)(x):=\int_{\{\sigma\in\mathbb{S}^{2}:|v_{j}\cdot \sigma|>\epsilon\}}f_1(x-v_{1}\cdot \sigma)f_{2}(x-v_{2}\cdot \sigma)f_3(x-v_{3}\cdot \sigma)d\sigma, ~~j=1,2,3.
\end{eqnarray*}
See~\cite{jbak} for more detail about $U_{1},U_{2}$ and $U_{3}$.
We know that $U_{1}$ is bounded at the point $(0,1,1;0)$. Also, it is of restricted weak type at $E=(\frac{2}{3},\frac{2}{3},0;0)$ and of strong type at $Z^*=(1,0,0;0)$ using the corresponding estimates for the operator $T$. The interpolation result from~\cite{Ch} yields that $U_1$ maps $L^{\frac{5}{3},2}(\mathbb R)\times L^{\frac{5}{3},2}(\mathbb R)\times L^{5,\infty}(\mathbb R)\rightarrow L^{\infty}(\mathbb R)$. Subsequently, we get that $U_{1}$ is of strong type at $N$. In a simlar way, we can prove the strong typeness of $U_2$ at $N$.
Next, note that $U_{3}$ is of strong type at $(1,1,0;0)$. Further, we know that $T_v$ is strong type bounded at $(0,0,\frac{1}{2};0).$The Riesz-Thorin interpolation theorem for multilinear operators~\cite{BS} yields that $U_3$ is of strong type at $N$. This completes the proof.
The standard duality arguments imply strong type boundedness of $T$ at dual points of $N$.
This completes the proof of Theorem~\ref{mainresult:sufall}. \qed
\section*{Acknowledgement}The first author acknowledges the financial support from the Science and Engineering Research Board (SERB), Government of India, under the grant MATRICS: MTR/2017/000039/Math. The second author is supported by CSIR (NET), file no. 09/1020 (0094)/2016-EMR-I.
|
1,116,691,498,422 | arxiv | \section{Introduction}
Optimal control problems associated to fluid dynamics have been studied by several authors, during the last decades, motivated by the important applications of such type of problems to the industry. In a natural way, most of the first works were devoted to the case of distributed control as this is easier to handle. However, the most challenging problems in applications such as automobile or airplane design, and more recently, in bypass design or boundary reconstruction in medical applications, are modeled by problems where the control is assumed to act on part of the boundary. Actually, boundary control problems are usually harder to deal, specially with respect to optimality conditions, since higher regularity for the solutions is often required. The list of works on the subject is long, and here we only mention a few references \cite{AT90}, \cite{GHS91}, \cite{FS92}, \cite{GM00}, \cite{DK05}, \cite{DT07} and \cite{DY09}.
In this work, and having in mind applications in biomedicine, we will consider the steady Navier-Stokes equations with mixed boundary conditions
\begin{equation}\label{navierstokes}
\left\{ \begin{array}{ll}-\nu\Delta {u}+
{{u}}\cdot \nabla {{u}} +\nabla p= {f}& \qquad
\mbox{in } \Omega,\vspace{2mm} \\
\nabla \cdot{{u}}=0& \qquad \mbox{in } \Omega,\vspace{2mm}\\
\gamma{u}={g}& \qquad \mbox{on } \Gamma_{in},\\
\gamma{{u}}={0}& \qquad \mbox{on } \Gamma_{wall},\\
\nu\partial_n{{u}}-pn={0}& \qquad \mbox{on } \Gamma_{out},
\end{array}\right.\end{equation}
where $\nu$ represents the viscosity of the fluid (possibly divided by its constant density), $f$ the vector force acting on the fluid and $g$ the function imposing the velocity profile on $\Gamma_{in}$. The unknowns are the velocity vector field $u$ and the pressure variable $p$. These equations have been widely used to model and simulate the blood flow in the cardiovascular system (see, for instance, \cite{G08} and the references cited therein). In this type of applications it is often required to represent part of an artery as the computational (bounded) domain $\Omega$. In addition, for the numerical simulations, we impose homogeneous Dirichlet boundary conditions on the surface representing the vessel wall ($\Gamma_{wall}$) and Dirichlet non-homogeneous on the artificial boundary ($\Gamma_{in}$), which is used to truncate the vessel from the upstream region. Besides, on the surface limiting the domain, in the downstream direction ($\Gamma_{out}$), homogeneous Neumann boundary conditions are imposed. In Figure~\ref{figdomain} we can see a longitudinal section of such a domain, where the deformation of $\Gamma_{wall}$ could represent the presence of a plaque of atherosclerosis.
\begin{figure}[h]
\centering
\includegraphics[scale=.45]{Domain}
\centering
\caption{\scriptsize{Representation of the domain $\Omega$}}
\label{figdomain}
\end{figure}
When facing this and other type of pathologies of the cardiovascular system, it is important the evaluation of hemodynamical factors to predict, in a non invasive way, either the evolution of the disease, or the effect of possible therapies. This can be done by relying on the numerical simulations obtained in the domain under analysis. The main difficulty in this strategy lies in the lack of accuracy of the virtual simulations with respect to the real situation. In order to improve the accuracy and make the simulations sound enough, it is possible to use data from measurements of the blood velocity profile, obtained through medical imaging in some smaller parts of the vessel. This can be done through a variational approach, i.e., by setting an optimal control problem with a cost function (or a class of cost functions) of the type
\begin{equation}\label{costfunctional}
J({u},{g})=\beta_1\int_{\Omega_{part}}|{{u}}-{{u}}_d|^2\,dx+\beta_2\int_{\Gamma_{in}}|{ g}|^2\,ds+\beta_3\int_{\Gamma_{in}}|{\nabla_s g}|^2\,ds,
\end{equation}
where ${{u}}_d$ represents the data available only on a part of the domain called $\Omega_{part}$. Note that, while fixing the weights $\beta_1$, $\beta_2$ and $\beta_3$, we determine whether the minimization of $J$ emphasizes more a good approximation of the velocity vector to $u_d$, a ``less expensive'' control $g$ (in terms of the $L^2$-norm), or a smoother control.
An example of $u_d$, measured in $\Omega_{part}$, could be the velocity vectors obtained in several cross sections of the vessel, as represented in Figure~\ref{figdata}.
\begin{figure}[h]
\centering
\includegraphics[scale=1]{stenoData1crop}
\centering
\caption{\scriptsize{Representation of $u_d$ over $\Omega_{part}$}}
\label{figdata}
\end{figure}
Solving the optimal control problem
\begin{equation}\label{controlproblem}
(P)\left\{
\begin{array}{ll}
\text{Minimize}\,\, J({u},{g})\\
\text{subject to}\,\, \eqref{navierstokes}
\end{array}
\right.
\end{equation}
will give us the means of making blood flow simulations more reliable, using known data.
This strategy is not new, and has already been used as a proof of concept in \cite{GTS14} and \cite{TGS14}, where both the Navier-Stokes and the Generalized Navier-Stokes equations were considered to model the blood flow.
Even if it proved to be successful from the numerical point of view, problem $(P)$ has not yet been studied, at least up to the authors knowledge, not even with respect to the existence of solution. In fact, many authors have treated similar problems, considering the same type of cost functionals constrained to the Navier-Stokes equations, but for the case where $\Omega_{part}=\Omega$ and without using mixed boundary conditions. In \cite{DK05} and \cite{DY09} the case with only Dirichlet boundary conditions, and a similar cost functional, was treated. In \cite{GHS91} and \cite{M07} the authors considered $J$ as the cost functional, with $\Omega_{part}=\Omega$, but again they just dealt with Dirichlet boundary conditions. In \cite{FR09} the authors considered a more complex set of mixed boundary condition, but for a different cost functional.
Here we prove the existence of solution for problem $(P)$ regarded in the weak sense. We will make the distinction between different possibilities both for $\Omega_{part}$ and for the parameters $\beta_2$ and $\beta_3$. In order to do that, we will start by setting the existence of a unique weak solution for the state equation (\ref{navierstokes}). The regularity of this solution remains an open problem and will not be treated here. It is important to deal with this issue, before addressing the natural following stages, namely the derivation of optimality conditions for problem $(P)$ and the numerical approximation.
The organization of this paper reads as follows. In Section 2 we give some notation and results needed for this work. The Navier-Stokes equations with mixed boundary conditions are studied in Section 3. Finally, in Section 4, we prove the existence of solution for a class of optimal control problems.
\section{Notation and some useful results}
We consider $\Omega\subset{R}^n$, with $n=2,3$, an open bounded subset with Lipschitz boundary.
The standard Sobolev spaces are denoted by
$$ W^{k,p}(\Omega)=\left\{u\in \Bbb{L}^{p}(\Omega): \,\Vert u\Vert_{W^{k,p}}^p=\sum_{|\alpha|\leq k}\Vert D^{\alpha}u\Vert_{L^p}^p<\infty \right\},$$
where $k\in {I~\hspace{-1.45ex}N} $ and $1<p<\infty$. For $s\in{I~\hspace{-1.45ex}R} $, $W^{s,p}(\Omega)$ is defined by interpolation. The dual space of $ W^{1,p}_0(\Omega)$ is denoted by $ W^{-1,p'}(\Omega)$. We also use $H^s(\Omega)$ to represent the Hilbert spaces $W^{s,2}(\Omega)$. For $\Gamma\subset\partial\Omega$ with positive measure we denote by $H^s(\Gamma)$, $s\geq\frac{1}{2}$, the image of the unique linear continuous trace operator $$\gamma_{\Gamma}:H^{s+\frac{1}{2}}(\Omega)\to H^s (\Gamma),$$
such that $\gamma_{\Gamma}u=u_{|\Gamma}$ for all $u\in H^{s+\frac{1}{2}}(\Omega)\cap C^0(\bar{\Omega})$. In particular, for $s=0$, $H^0(\Gamma)$ is the subspace of $L^2(\Gamma)$ corresponding to the image of the continuous functions in $H^1(\Omega)$. The norm of $H^s(\Gamma)$ is defined similarly to the norm in $H^1(\Omega)$, except that the tangential derivatives on $\Gamma$ should be used (see, for instance, \cite{GHS91}).
Whenever $Y$ is a space of functions $u:\Omega\to R$, we will use the boldface notation $ \mathbf{Y}=Y\times Y\times Y$ for the corresponding space of vector valued functions.
We will also make use of the following Sobolev embedding result:
\begin{lemma}\label{sobolev}
Let $\Omega$ be a bounded set of class $C^1$. Assume that $p<n$ and $p^*=\frac{pn}{n-p}$. Then
\begin{description}
\item[i)] $W^{1,p}(\Omega)\subset L^q$, $\forall q\in [1,p^*[$ with compact embedding.
\item[ii)] $W^{1,p}(\Omega)\subset L^{p*}$, with continuous embedding.
\end{description}
\end{lemma}
\begin{proof}
For the proof see, for instance, \cite{B83}, Corollary IX.14 and Theorem IX.16 - Remark 14ii).
\end{proof}
We consider the spaces of divergence free functions defined by
$${ H}=
\left\{ u\in {H}^1(\Omega)
\mid \nabla\cdot u=0\right\},$$
$${ V}_{wall}=
\left\{\psi\in H_{\Gamma_{wall}}(\Omega)
\mid \nabla\cdot \psi=0\right\}$$
and
$${ V}_D=
\left\{\psi\in H_{\Gamma_D}(\Omega)
\mid \nabla\cdot \psi=0\right\},$$
where $\Gamma_D$ refers to the Dirichlet boundary $\Gamma_{in}\cup\Gamma_{wall}$.
In these definitions, for $\Gamma\in\{\Gamma_{wall},\Gamma_D\}$, we represent by ${H}_{\Gamma}$ the set $${H}_{\Gamma}=\left\{\psi\in H^1(\Omega) \mid \gamma_{\Gamma} \psi=0\right\}.$$
The corresponding norms are defined by
$$\Vert.\Vert_{H}=\Vert.\Vert_{V_{D}}=\Vert.\Vert_{V_{wall}}=\Vert.\Vert_{H^1(\Omega)}.$$
We also define
$${H}_{0}^1(\Gamma)= \left\{v\in L^2(\Gamma)
\mid \nabla_s v \in L^2(\Gamma), \, \gamma_{\partial\Gamma} v=0 \right\}$$ and
$${H}_{00}^{\frac{1}{2}}(\Gamma)= \left\{g\in L^2(\Gamma)
\mid \exists v \in H^{1}(\Omega), \,v_{|_{\partial\Omega}} \in H^{\frac{1}{2}}(\partial\Omega), \, \gamma_{\Gamma} v=g,\, \gamma_{\partial\Omega\setminus\Gamma} v=0 \right\}$$ a closed subspace of $H^{\frac{1}{2}}(\Gamma)$.
Note that we have the continuous embeddings $H_0^1(\Gamma) \subset H_{00}^{\frac{1}{2}}(\Gamma)$ and $ H_{00}^{\frac{1}{2}}(\Gamma)\subset L^2(\Gamma)$ (\cite{DL00}, pp. 397).
Finally, we set
$$\hat{H}^{\frac{1}{2}}(\Gamma_1\cup\Gamma_2)=\left\{(g_1,g_2)\in H_{00}^{\frac{1}{2}}(\Gamma_1)\times H_{00}^{\frac{1}{2}}(\Gamma_2)\mid \int_{\Gamma_1}g_1\cdot n\, ds+\int_{\Gamma_2}g_2\cdot n\, ds=0 \right\}.$$
\section{State Equation}
The well-posedness of system (\ref{navierstokes}) concerning the existence and uniqueness for $g$ within an admissible class is required before studying the existence of solution of the optimal control problem. In \cite{KS98} the authors studied the evolutionary case setting the existence of a solution local in time, for the type of boundary conditions considered here. Concerning the stationary case, in \cite{K98} and \cite{G08} the existence of solution for a similar system was proved. Both authors considered Neumann conditions mixed with Dirichlet homogeneous conditions. In the later it was mentioned that no additional difficulties should be expected with non-homogeneous boundary conditions. In \cite{FR09}, the existence was shown, in the 2D case, for a system with mixed boundary conditions including Dirichlet non-homogeneous. Again the authors mentioned that the 3D case could be proved using the same techniques. For the sake of clearness, we show that system (\ref{navierstokes}) is in fact well-posed in the 3D case, following the ideas of \cite{FR09}.
We first start by considering the Stokes system
\begin{equation}\label{stokes}
\left\{ \begin{array}{ll}-\nu\Delta {u}+\nabla p= {h}& \qquad
\ \mbox{in} \ \Omega,\vspace{2mm} \\
\nabla \cdot{{u}}=0& \qquad \mbox{in} \ \Omega,\vspace{2mm}\\
\gamma{u}={g}& \qquad \mbox{on} \ \Gamma_{in},\\
\gamma{{u}}={0}& \qquad \mbox{on} \ \Gamma_{wall},\\
\nu \partial_n{{u}}-pn={0}& \qquad \mbox{on} \ \Gamma_{out},
\end{array}\right.\end{equation}
\begin{definition}
Let ${g}\in\mathbf{H}_{0}^1(\Gamma_{in}) $, ${h}\in \mathbf{L}^{\frac{3}{2}}(\Omega)$. We call $u\in\mathbf{V}_{{wall}}$ a weak solution of (\ref{stokes}) if $\gamma_{\Gamma_{in}}u=g$ and
\begin{equation} \label{weakstokes}
\quad\nu\int\limits_{\Omega}\nabla u:\nabla v\,dx = \int\limits_{\Omega}{h} v\,dx,
\end{equation}
for all $v\in\mathbf{V}_D$.
\end{definition}
\begin{theorem}.
\begin{description}
\item [i)] There exists a unique solution $ u\in \mathbf{V}_{wall}$ of problem $(\ref{weakstokes})$. For such solution there exists a distribution $p\in\mathbf{L}^{\frac{3}{2}}(\Omega)$ such that $( u,p)\in{V}_{wall}\times{L}^2(\Omega)$ is a solution of (\ref{stokes}) in the sense of distributions. If $u$ and $p$ are smooth enough, then $p$ is unique and the boundary conditions in (\ref{stokes}) are verified point-wise.
\item[ii)] On the other hand, if $( u,p) \in{H}_{\Gamma_{wall}}\times L^{\frac{3}{2}}(\Omega)$ is a solution of problem $(\ref{stokes})$ in the sense of distributions, then $ u$ is a solution of (\ref{weakstokes}).
\end{description}
\end{theorem}
\begin{proof}
\begin{description}
\item[i)] Consider the auxiliar minimization problem
$$\min_A E(u):=\frac{1}{2}\Vert \nabla v\Vert_{\mathbf{L}^2(\Omega)}^2-(h,u)$$
where
$$A=\{u\in \mathbf{H}_{\Gamma_{wall}}, \, \gamma_{\Gamma_{in}}u=g\}.$$
The functional $E:\mathbf{H}^1(\Omega)\to \mathbb{R}$ is continuous and convex on $\mathbf{H}^1(\Omega)$ and thus weakly lower semi-continuous with respect to the $\mathbf{H}^1(\Omega)$ norm. Also, the admissibility set $A$ is sequentially weakly closed.
Finally, since $E$ verifies the coercivity property, the classical theory of the calculus of variations ensures the existence of a unique solution $\bar{u}$ for the minimization problem. Hence, $\bar{u}$ is also the unique solution of the necessary and sufficient optimality condition
$$\nu\int\limits_{\Omega}\nabla u:\nabla v\,dx = \int\limits_{\Omega}{h} v\,dx,\quad \forall v\in\mathbf{H}_{\Gamma_D}$$ and therefore (\ref{weakstokes}) has a unique solution.
If we take $v\in \mathbf{H}_{\Gamma_D}\cup\mathbf{C}_0^{\infty}(\Omega)$ and integrate (\ref{weakstokes}) by parts, we obtain
$$\int_{\Omega}(\nu\Delta \bar{u} +h)\cdot v=0\Leftrightarrow (\Delta \bar{u} +h, v)=0, \quad \forall v\in \mathbf{H}_{\Gamma_D}\cup\mathbf{C}_0^{\infty}(\Omega).$$
Due to the inclusion $L^{\frac{3}{2}}(\Omega)=(L^{{3}}(\Omega))'\subset (W_0^{1,{3}}(\Omega))'=W^{-1,\frac{3}{2}}(\Omega) $, we have $\nu\Delta \bar{u} +h\in \mathbf{W}^{-1,\frac{3}{2}}(\Omega)$. Therefore by De Rham's theorem (\cite{S01} Lemma II.2.2.2) there exits a distribution $p\in L^{\frac{3}{2}}(\Omega)$ such that $\nabla p\in \mathbf{L}^{\frac{3}{2}}(\Omega)$ and $(\nu\Delta \bar{u} +h, v)=(\nabla p,v)$ that is, system (\ref{stokes}) is verified in the sense of distributions.
Let us now assume that $\bar{u}$ and $p$ are smooth and replace $h$ by $-\nu\Delta \bar{u} + \nabla p$ in (\ref{stokes}). Integrating by parts we obtain
$$\int_{\Gamma_{out}}(\nu \partial_n\bar{u}-pn) \cdot v \,ds=0\, ,\quad \forall v\in \mathbf{V}_D.$$
Now consider $w\in\mathbf{C}_0^{\infty}(\Gamma_{out})$ such that $\int_{\Gamma_{out}} w\cdot n\, ds=0$. If we define
\begin{equation}
\bar{w}=\left\{ \begin{array}{ll} w & \qquad \mbox{on } \Gamma_D=\Gamma_{in}\cup\Gamma_{wall}\\
0 & \qquad \mbox{on } \Gamma_{out},
\end{array}\right.\end{equation}
we have
$\bar{w}\in\mathbf{C}_0^{\infty}(\partial\Omega)$ and $\int_{\partial\Omega} \bar{w}\cdot n\, ds=0$. As a result, there exists $v\in\mathbf{V}_D$ such that $\gamma_{\partial\Omega}v=\bar{w}$ and $\gamma_{\Gamma_{out}}v=w$. Consequently,
$$\int_{\Gamma_{out}}(\nu \partial_n\bar{u}-pn) \cdot w \,ds=0,\quad \forall w\in \mathbf{C}_0^{\infty}(\Gamma_{out})\text{ such that } \int_{\Gamma_{out}} w\cdot n\, ds=0.$$
In view of a corollary of the fundamental lemma of the calculus of variations (\cite{D09} Cor.1.25 p.23), we have
$$\nu \partial_n\bar{u}-pn=c_0n \text{ on }\Gamma_{out}, $$ where $c_0$ is a constant.
Let us now take $\bar{p}=p+c$ as another distribution such that (\ref{stokes}) is verified. Then we have
$$0=\int_{\Gamma_{out}}(\nu \partial_n\bar{u}-pn) \cdot v=\int_{\Gamma_{out}}(c-c_0)n \cdot v \,ds\quad \forall v\in \mathbf{V}_D.$$
Choosing $v$ such that $\int_{\Gamma_{out}}n \cdot v \,ds=1$, we conclude that $(\bar{u},\bar{p})$, with $c=c_0$, is the unique solution of (\ref{stokes}).
\item[ii)] If $u \in\mathbf{H}_{\Gamma_{wall}}$ is a solution of (\ref{stokes}) then it is clear that $u\in \mathbf{V}_{wall}$ and, as a result of integration by parts, that (\ref{weakstokes}) is verified.
\end{description}
\end{proof}
Before obtaining an estimate for the Stokes problem, we first recall some related results.
\begin{lemma}\label{generalextension}
Let $g\in\mathbf{H}^{\frac{1}{2}}(\partial\Omega) $ be such that $$\int_{\partial\Omega\setminus\Gamma}g\cdot n\, ds=\int_{\Gamma}g\cdot n\, ds=0.$$ Then there exists $v\in\mathbf{H}$ such that $\gamma v=g$.
\end{lemma}
\begin{proof}See, for instance, \cite{GR86}.
\end{proof}
It is now straightforward to prove the next lemma.
\begin{lemma}\label{inoutextension}
Let $(g_1,g_2)\in \hat{H}^{\frac{1}{2}}(\Gamma_{in}\cup\Gamma_{out})$. Then there is a bounded extension operator $E:\hat{H}^{\frac{1}{2}}(\Gamma_{in}\cup\Gamma_{out})\to\mathbf{V}_{wall}$, $\forall v\in\mathbf{V}_{wall}$, such that for $v=E(g_1,g_2)$ we have $g_1=\gamma_{\Gamma_{in}}v, \, g_2=\gamma_{\Gamma_{out}}v$.
\end{lemma}
As a result, we can obtain the following estimate for the solution.
\begin{lemma}\label{stokesestimate}
Let $\sol:\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})\times\mathbf{L}^{\frac{3}{2}}(\Omega)\to\mathbf{V}_{wall}$ be the solution operator to (\ref{weakstokes}). Then, if $v=\sol(g,h)$, we have
$$\|v \|_{\mathbf{V}_{wall}}^2=\|v \|_{\mathbf{H}^1(\Omega)}^2\leq c \left(\|g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2+\|h\|_{\mathbf{L}^{\frac{3}{2}}(\Omega)}^2\right),$$
where $c>0$ is independent of $(g,h)$.
\end{lemma}
\begin{proof}
Using Lemma~\ref{inoutextension} we see that $v=\E g+\bar{v}$ with $\bar{v}=v-\E g\in \mathbf{V}_D$. Hence
$$\|\nabla v\|_{{\bf L}^2(\Omega)}^2=(\nabla v,\nabla \E g)+(\nabla v,\nabla \bar{v}),$$ which, in view of the definition of weak solution, can be written as
$$\|\nabla v\|_{{\bf L}^2(\Omega)}^2=(\nabla v,\nabla \E g)+\frac{1}{\nu}(h, \bar{v}).$$
We deal with each term of the right-hand side separately. Using Young's inequality, together with the fact that $\E$ is bounded, we have
\begin{align}
|(\nabla v,\nabla \E g)|&\leq c_1\|\nabla v\|_{{\bf L}^2(\Omega)}\|\nabla \E g\|_{{\bf L}^2(\Omega)}\leq c_2\|\nabla v\|_{{\bf L}^2(\Omega)}\| \E g\|_{\mathbf{H}^1(\Omega)}\\
&\leq c_3\|\nabla v\|_{{\bf L}^2(\Omega)}\| g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}\leq \varepsilon \|\nabla v\|_{{\bf L}^2(\Omega)}^2+\frac{c_4}{\varepsilon}\| g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2,
\end{align} for $\varepsilon >0$.
Moreover, using Poincar\'e and Young inequalities and the Sobolev embedding $\mathbf{H}^1(\Omega)\subset \mathbf{L}^3(\Omega)$ (see Lemma~\ref{sobolev}.i), we have
\begin{align}
|( h, \bar{v})|&\leq c_5\|h\|_{{\bf L}^\frac{3}{2}(\Omega)}\|\nabla \bar{v}\|_{{\bf L}^2(\Omega)}\leq \varepsilon \|\nabla \bar{v}\|_{{\bf L}^2(\Omega)}^2+\frac{c_6}{\varepsilon}\| h\|_{{\bf L}^\frac{3}{2}(\Omega)}^2.
\end{align}
And, by similar arguments,
\begin{align}
\|\nabla \bar{v}\|_{{\bf L}^2(\Omega)}^2=\|\nabla v-\nabla \E g\|_{{\bf L}^2(\Omega)}^2&\leq c_7\left(\|\nabla v\|_{{\bf L}^2(\Omega)}^2+\| \E g\|_{\mathbf{H}^1(\Omega)}^2\right)\\
&\leq c_8\left(\|\nabla v\|_{{\bf L}^2(\Omega)}^2+\| g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2\right).
\end{align}
Therefore
$$\|\nabla v\|_{{\bf L}^2(\Omega)}^2\leq \varepsilon (1+c_8) \|\nabla {v}\|_{{\bf L}^2(\Omega)}^2+\frac{c_6}{\varepsilon}\| h\|_{{\bf L}^\frac{3}{2}(\Omega)}^2+(\frac{c_4}{\varepsilon}+c_8\varepsilon)\| g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2$$ and consequently
$$\|v\|_{\mathbf{H}^1(\Omega)}^2\leq c_9\|\nabla v\|_{{\bf L}^2(\Omega)}^2\leq c\left (\| h\|_{{\bf L}^\frac{3}{2}(\Omega)}^2+\| g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2\right)$$ for a certain constant $c>0$.
\end{proof}
We can now prove the existence of a solution for the Navier-Stokes system (\ref{navierstokes}).
\begin{definition}
Let ${g}\in\mathbf{H}_{0}^1(\Gamma_{in}) $, ${f}\in \mathbf{L}^{\frac{3}{2}}(\Omega)$. We say that $u\in\mathbf{V}_{wall}$ is a weak solution of (\ref{navierstokes}) if $\gamma_{\Gamma_{in}}u=g$ and
\begin{equation} \label{weaknavierstokes}
\quad\nu\int\limits_{\Omega}\nabla u:\nabla v\,dx+\int\limits_{\Omega}({u}\cdot\nabla) u v\,dx = \int\limits_{\Omega}{f} v\,dx,
\end{equation}
for all $v\in\mathbf{V}_D$.
\end{definition}
We need the following result.
\begin{lemma}\label{convective}
If $u\in \mathbf{H}^1(\Omega)$, then $u\cdot \nabla u\in \mathbf{L}^{\frac{3}{2}}(\Omega)$ and $\|u\cdot \nabla u\|_{\mathbf{L}^{\frac{3}{2}}(\Omega)}\leq\|u\|_{\mathbf{H}^{1}(\Omega)}^2$.
\end{lemma}
\begin{proof}
Using H\"older's inequality (\cite{B83}, IV.2, Remark 2.) and the Sobolev embedding $H^1(\Omega)\subset L^6(\Omega)$ (see Lemma~\ref{sobolev}.ii)) we have
$$\int_{\Omega}|u\cdot \nabla u|^{\frac{3}{2}}\,dx\leq\|u\|_{\mathbf{L}^6(\Omega)}^{\frac{3}{2}}\|\nabla u\|_{\mathbf{L}^2(\Omega)}^{\frac{3}{2}}\leq c\|u\|_{\mathbf{H}^1(\Omega)}^{\frac{3}{2}} \|\nabla u\|_{\mathbf{L}^2(\Omega)}^{\frac{3}{2}}\leq c\|u\|_{\mathbf{H}^{1}(\Omega)}^3\leq \infty.$$
\end{proof}
\begin{theorem}\label{navierstokesexistence}
Let ${g}\in\mathbf{H}_{0}^1(\Gamma_{in}) $ such that $\|g\|_{\mathbf{H}_{0}^1(\Gamma_{in}) }\leq \rho$, for $\rho>0$ sufficiently small, and ${f}\in \mathbf{L}^{\frac{3}{2}}(\Omega)$. Then, there exists a unique weak solution $u\in\mathbf{V}_{wall}$ of the Navier-Stokes system (\ref{navierstokes}) which verifies
\begin{equation}\label{navierstokesestimate}\|u\|_{\mathbf{H}^1(\Omega)}^2 \leq \alpha \left(\|g\|_{\mathbf{H}_{0}^1(\Gamma_{in}) }^2\right)+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2,
\end{equation}
where $\alpha(s)=c(s^2+s)$.
\end{theorem}
Before proceeding to the proof of the theorem, let us introduce another definition.
\begin{definition}
We define the projection operator $\pro:\mathbf{L}^{\frac{3}{2}}(\Omega)\to \mathbf{\hat{L}}^{\frac{3}{2}}(\Omega)$ as the solution of the equation
$$(\pro h,v)=(h,v),\quad \forall v\in \mathbf{\hat{L}}^3(\Omega),$$ where
$$\mathbf{\hat{L}}^p(\Omega)=\left\{v\in \mathbf{L}^p(\Omega)\mid \nabla \cdot v=0,\, \gamma_{\Gamma_D} (v\cdot n)=0 \right\}.$$
\end{definition}
\begin{proof} [Proof of Theorem~\ref{navierstokesexistence}.]
We look for $h\in\mathbf{\hat{L}}^{\frac{3}{2}}(\Omega)$ such that the corresponding solution to the Stokes system $u=\sol(g,h)$ is also a solution of (\ref{weaknavierstokes}). For this purpose we will use a fixed point argument.
If we replace such $u=\sol(g,h)$ in (\ref{weaknavierstokes}), we get
$$\nu(\nabla \sol ,\nabla v)+(\sol\cdot\nabla \sol,v)=(f,v)\quad \forall v\in \mathbf{V}_D, $$
which, by definition of $\sol$, is equivalent to
\begin{equation}\nonumber
(h, v)+(\sol\cdot\nabla \sol,v)=(f,v)\quad \forall v\in \mathbf{V}_D
\end{equation}
which is also equivalent to
\begin{equation}\label{eqproof}
(h+\sol\cdot\nabla \sol-f,v)=0\quad \forall v\in \mathbf{V}_D.
\end{equation}
Using Lemma~\ref{convective} and the fact that $\mathbf{V}_D$ is dense in $\mathbf{\hat{L}}^3(\Omega)$, we can see that, from equation (\ref{eqproof}), we have
\begin{align}
(\pro (h+\sol\cdot\nabla \sol-f),v)&=0\quad \forall v\in \mathbf{\hat{L}}^3(\Omega)\Leftrightarrow\nonumber\\
(h+\pro (\sol\cdot\nabla \sol-f),v)&=0\quad \forall v\in \mathbf{\hat{L}}^3(\Omega)\Leftrightarrow\nonumber\\
-\pro (\sol\cdot\nabla \sol-f)&=h\, .
\end{align}
We should now prove that the operator $\C:\mathbf{\hat{L}}^{\frac{3}{2}}(\Omega)\to\mathbf{L}^{3}(\Omega)$ defined by
$$\C(h)=-\pro (\sol(g,h)\cdot\nabla \sol(g,h)-f)$$ verifies the contraction property.
Let $h_1,\, h_2\in B_{\delta}$, where $B_{\delta}\subset\mathbf{\hat{L}}^{\frac{3}{2}}(\Omega)$ is a given ball with respect to the $\mathbf{\hat{L}}^{\frac{3}{2}}(\Omega)$ metrics. Then, using H\"older's inequality together with Poincar\'e's inequality, we get
{\small \begin{align}
&\|\C(h_1)-\C(h_2)\|_{\mathbf{\hat{L}}^{\frac{3}{2}}(\Omega)}=\nonumber\\
&\|\pro (\sol(g,h_1)\cdot\nabla \sol(g,h_1)- \sol(g,h_2)\cdot\nabla \sol(g,h_2))\|_{\mathbf{\hat{L}}^{\frac{3}{2}}(\Omega)}=\nonumber\\
&\|\sol(g,h_1)\cdot\nabla \sol(g,h_1)- \sol(g,h_2)\cdot\nabla \sol(g,h_2)\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}\leq\nonumber\\
&\|\sol(g,h_1)\cdot\nabla \sol(g,h_1)-\sol(g,h_2)\cdot\nabla \sol(g,h_1)\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}\nonumber\\
&+\|\sol(g,h_2)\cdot\nabla \sol(g,h_1)- \sol(g,h_2)\cdot\nabla \sol(g,h_2)\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}=\nonumber\\
&\|\sol(0,h_1-h_2)\cdot\nabla \sol(g,h_1)\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}+\|\sol(g,h_2)\cdot\nabla \sol(0,h_1-h_2)\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}\leq\nonumber\\
&\|\sol(0,h_1-h_2)\|_{\mathbf{{L}}^6(\Omega)}\|\nabla \sol(g,h_1)\|_{\mathbf{{L}}^2(\Omega)}+\|\sol(g,h_2)\|_{\mathbf{{L}}^6(\Omega)}\|\nabla \sol(0,h_1-h_2)\|_{\mathbf{{L}}^2(\Omega)}\leq\nonumber\\
&c_1(\|\sol(0,h_1-h_2)\|_{\mathbf{{H}}^1(\Omega)}\|\nabla \sol(g,h_1)\|_{\mathbf{{L}}^2(\Omega)}+\|\sol(g,h_2)\|_{\mathbf{{H}}^1(\Omega)}\|\nabla \sol(0,h_1-h_2)\|_{\mathbf{{L}}^2(\Omega)})\leq\nonumber\\
&c_2\|\nabla\sol(0,h_1-h_2)\|_{\mathbf{{L}}^2(\Omega)}\left(\|\nabla \sol(g,h_1)\|_{\mathbf{{L}}^2(\Omega)}+\|\sol(g,h_2)\|_{\mathbf{{H}}^1(\Omega)}\right).\label{major}
\end{align}}
Using Lemma~\ref{stokesestimate} and the continuous embedding $H_0^1(\Gamma_{in})\subset {H}_{00}^{\frac{1}{2}}(\Gamma_{in})$, we can see that
{\small\begin{align}
(\ref{major})\,&\leq c_3\left(\|h_1-h_2\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2\right)^{\frac{1}{2}}\times\nonumber\\ &\left[\left(\|h_1\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2+\|g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2 \right)^{\frac{1}{2}}+\left(\|h_2\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2+\|g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2 \right)^{\frac{1}{2}} \right] \nonumber\\
&\leq c_4\|h_1-h_2\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}\left[\|h_1\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}+\|h_2\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}+\|g\|_{\mathbf{H}_{0}^{{1}}(\Gamma_{in})} \right] \nonumber\\
&\leq \bar{c}\|h_1-h_2\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)},\nonumbe
\end{align}}
where $\bar{c}$ depends on $\|h_1\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}$, $\|h_2\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}$ and $\|g\|_{\mathbf{H}_{0}^{{1}}(\Gamma_{in})}$. But since $h_1,\, h_2\in B_{\delta}$, we can choose $\delta$ and $\rho$ small enough so that $\bar{c}<1$. Therefore $\sol$ maps $B_{\delta}$ into itself and hence it has a fixed point $\bar{h}$. Since $\bar{c}$ is strictly smaller than $1$, it is easy to see that such fixed point is unique.
As for the estimate (\ref{navierstokesestimate}), let us notice that the fixed point can be obtained as the limit of a sequence $(h_k)$ verifying
$$h_1=\C(0),\,h_2=\C(h_1),\ldots,\, h_k=\C(h_{k-1}),...$$ Since we have $h_k=\sum_{i=1}^k(h_i-h_{i-1})=\sum_{i=1}^k[\C(h_{i-1})-\C(h_{i-2})]$
then, in virtue of Lemma~\ref{convective} and Lemma~\ref{stokesestimate}, we have
\begin{align}
\|\bar{h}\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}=&\|\lim_{k\to\infty}h_k\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}\leq\lim_{k\to\infty}\sum_{i=1}^k\|h_k-h_{k-1}\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}\nonumber\\
\leq&\sum_{i=1}^{\infty}\bar{c}^{\ i-1}\|\C(0)\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}= \frac{\bar{c}}{1-\bar{c}}\|\sol(g,0)\cdot \nabla \sol(g,0)-f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}\nonumber\\
\leq& c_5(\|\sol(g,0)\|_{\mathbf{{H}}^1(\Omega)}^2+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)})\leq c_6(\|g\|_{\mathbf{{H}}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)})\nonumber.\\
\end{align}
Consequently, the solution $u=\sol(\bar{h},g)$ of system (\ref{weaknavierstokes}) is bounded by
\begin{align}
\|u\|_{\mathbf{{H}}^{1}(\Omega)}^2&=\|\sol(g,\bar{h})\|_{\mathbf{{H}}^{1}(\Omega)}^2\leq c_6\left( \|g\|_{\mathbf{{H}}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2+\|\bar{h}\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2\right)\nonumber.\\
&\leq c_7\left( \|g\|_{\mathbf{{H}}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2+\|g\|_{\mathbf{{H}}_{00}^{\frac{1}{2}}(\Gamma_{in})}^4+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2\right)\nonumber.\\
&\leq c_8\left( \|g\|_{\mathbf{{H}}_{0}^{1}(\Gamma_{in})}^2+\|g\|_{\mathbf{{H}}_{0}^{1}(\Gamma_{in})}^4+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2\right)\nonumber.\\
&= \alpha\left( \|g\|_{\mathbf{{H}}_{0}^{1}(\Gamma_{in})}^2\right)+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2\nonumber.
\end{align}
\end{proof}
\begin{remark}\label{remarklessregularity}
In the proof of the previous theorem the fact that $g\in \mathbf{H}_0^1(\Gamma_{in})$ is not essential, and we could alternatively suppose that $g\in\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})$ verifies $\|g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in}) }\leq \rho$. In this case the proof could follow in the same way, but we would get the estimate
\begin{equation}\label{estimatelessregularity}\|u\|_{\mathbf{H}^1(\Omega)}^2 \leq \alpha \left(\|g\|_{\mathbf{H}_{00}^{\frac{1}{2}}(\Gamma_{in})}^2\right)+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2,
\end{equation}
instead of (\ref{navierstokesestimate}).
\end{remark}
\section{Existence results for the optimal control problem}
Consider the admissible control set
$${\cal U}=\left\{g\in H_0^1(\Gamma_{in})
\mid \,\|g\|_{H_0^1(\Gamma)}\leq \rho\right\},$$ where $\rho$ is defined as in Theorem~\ref{navierstokesexistence}.
We can define the weak version of problem $(P)$ as follows: we look for $g\in \cal U$ such that $J(u,g)$ is minimized, where $u$ is the unique weak solution of (\ref{weaknavierstokes}) corresponding to $g$.
\begin{remark}
Note that ${\cal U}$ is just an example of an admissible set, within the abstract set
$${\cal U}_0=\left\{g\in H_0^1(\Gamma_{in}):\text{such that (\ref{weaknavierstokes}) has a unique solution}\right\}.$$
\end{remark}
We can prove the following existence result:
\begin{theorem}\label{existencecontrolfull}
Assume that $\Omega_{part}=\Omega$, $\rho$ is as described above and $\beta_2,\,\beta_3\neq 0$. Then $(P)$ has an optimal solution $(u,g)\in \mathbf{V}_{wall}\times \cal{U}$ in the weak sense.
\end{theorem}
\begin{proof}
First see that for $g=0$ there is a corresponding unique solution $u_0$ to (\ref{weaknavierstokes}) so that $\mathbf{V}_{wall}\times \cal{U}$ is nonempty. This implies that $0\leq J\leq +\infty$.
Let $( u_k,g_k)_k\subset \mathbf{V}_{wall}\times \cal{U} $ be a minimizing sequence, that is, such that
$$J(u_k,g_k)\rightarrow I,\text{ the infimum, when }k\to+\infty.$$ Since ${\cal U}\subset H_0^1(\Gamma_{in})$ is bounded, there exists a subsequence of $(g_k)_k$ which converges weakly to a certain $\bar{g}\in H_0^1(\Gamma_{in})$. Due to (\ref{navierstokesestimate}) we have
$$\|u_k\|_{\mathbf{H}^1(\Omega)}^2 \leq \alpha \left(\|g_k\|_{\mathbf{H}_{0}^1(\Gamma_{in}) }^2\right)+\|f\|_{\mathbf{{L}}^{\frac{3}{2}}(\Omega)}^2,\quad\forall k,$$
and therefore there exists $\bar{u}$ such that $u_k\rightarrow \bar{u}$ weakly in $\mathbf{H}^1(\Omega)$. Indeed, we have $\bar{u}\in \mathbf{V}_{wall}$, as both the divergence operator and the trace operator $\gamma_{\Gamma_{wall}}:H^1(\Omega)\to H^{\frac{1}{2}}(\Gamma_{wall})$ are bounded linear operators. Also, as $\gamma_{\Gamma_{in}}u_k\rightarrow \gamma_{\Gamma_{in}}\bar{u}$, weakly in $\mathbf{H}^{\frac{1}{2}}(\Gamma_{in})$, we have that $\gamma_{\Gamma_{in}}u_k=g_k$ converges weakly in $\mathbf{L}^2(\Gamma_{in})$, both to $\gamma_{\Gamma_{in}}\bar{u }$ and $\bar{g}$. Thus, we must have $\gamma_{\Gamma_{in}}\bar{u}=\bar{g}$. Finally, since the convective term in (\ref{weaknavierstokes}) is weakly continuous in $\mathbf{H}^1(\Omega)$ (see \cite{GR86} p.286) we conclude that $\bar{u}$ is the solution corresponding to $\bar{g}$. Due to the fact that the functional $J$ is both convex and continuous, and therefore strong lower semi-continuous (l.s.c.), it is also l.s.c. with respect to the weak topology (\cite{B83} Remark III.8.6). Consequently,
$$I=\lim_{k} J(u_k,g_k)\geq\liminf_{k}J( u_k, g_k)\geq J(\bar{u},\bar{g})\geq I,$$
and we conclude that $( \bar{u}, \bar{g})$ is a an optimal solution for $(P)$.
\end{proof}
\begin{remark}
The fact that we assume ${\cal U_0}$ bounded in $ H_0^1(\Gamma_{in})$ is a very strong assumption which allows us to prove the result even either if $\beta_2=0$ or $\beta_3=0$. In this latter case, the l.s.c. property of $J$ should be verified with respect to $H^{\frac{1}{2}}(\Gamma_{in})$ rather than $ H_0^1(\Gamma_{in})$.
\end{remark}
\begin{remark} We can also choose an admissible set for the controls that is not necessarily bounded. This is the case when ${\cal U}={\cal U}_0$.
Then, if $\beta_3\neq 0$, from the fact that for a minimizing sequence $(g_k)_k$ we have $$\|g_k\|_{H_0^1(\Gamma_{in})}\leq J(u_k,g_k)\leq +\infty,$$ we can still extract a weakly convergent sequence in $H_0^1(\Gamma_{in})$, so that the proof would follow as above. If $\beta_3=0$, in view of the properties of $ H_0^1(\Gamma_{in})$ (see for instance \cite{GHS91}), we would get $$\|g_k\|_{H_0^1(\Gamma_{in})}\leq\|g_k\|_{L^2(\Gamma_{in})}\leq J(u_k,g_k)\leq +\infty,$$
and the proof could be attained similarly as above.
\end{remark}
We will now consider another choice for $\Omega_{part}$ more connected to the medical applications we have in mind. Let $\Omega$ be a domain representing a blood vessel like in Figure~\ref{figdomain}. Consider $(\Omega_{p_i})_i$ to be a monotone sequence of subsets of $\Omega$, such that
\begin{equation}\label{subsets}\Omega_{p_1}\subset\Omega_{p_2}...\subset\Omega_{p_m}\subset \Omega.
\end{equation}
In addition, assume also that for all $i\in\{1,...,m\}$, we have $$\partial\Omega_{p_i}=\Gamma_{in_i}\cup\Gamma_{wall_i}\cup\Gamma_{out_i}$$
where $\Gamma_{out_i}$, $i\in\{1,...,m\}$, are disjoint surfaces corresponding to cross sections of $\Omega$,
$\Gamma_{in_i}=\Gamma_{in}$,{ and } $\Gamma_{wall_i}=\Gamma_{wall}\cap\overline{\Omega}_{p_i}\neq\emptyset.$
Note that the construction of each $\Omega_{p_i}$ in this way ensures that (\ref{subsets}) is verified, and that each $\Omega_{p_i}$ itself represents a part of the vessel $\Omega$.
Now consider $\Omega_{part}=\cup_{i=1}^ms_i$ where $s_i=\Gamma_{out_i}$, for all $i\in\{1,...,m\}$. An example of such a situation is represented in Figure~\ref{figdata}. We can still establish the existence of solution in this case.
\begin{theorem}\label{existencecontrolsections}
Assume that $\Omega_{part}$ in $J$ is given by $\Omega_{part}=\cup_{i=1}^m{s_i}$, as described above. Then there is an optimal solution to problem $(P)$.
\end{theorem}
\begin{proof}
Let $\gamma_{s_i}:\mathbf{H}^1(\Omega_{p_i})\to\mathbf{H}^{\frac{1}{2}}(s_{i})$ be the family of linear, and bounded, trace operators defining the boundary values, over each surface $s_i$, for functions defined in $\Omega_{p_i}$.
To prove that $J$ is weakly l.s.c, we need to see that it verifies the continuity and convexity properties. Let $u_k\rightarrow u$ in $\mathbf{H}^1(\Omega)$ and consider $\gamma_{s_i}u_d=g_i$ to be the values of the known data over each $s_i$.
In this case $$\left|\int_{\Omega_{part}}(u_k-u_d)^2-(u-u_d)^2\,ds\right|$$ is, in fact,
\begin{align}
&\left|\sum_{i=1}^m\left[\|\gamma_{s_i} u_k-g_i\|_{L^2(s_i)}^2 - \|\gamma_{s_i}u-g_i\|_{L^2(s_i)}^2\right]\right|\leq\nonumber\\
&\left|\sum_{i=1}^m\left[(\|\gamma_{s_i} u_k-\gamma_{s_i}u\|_{L^2(s_i)}+\|\gamma_{s_i}u-g_i\|_{L^2(s_i)}))^2 - \|\gamma_{s_i}u-g_i\|_{L^2(s_i)}^2\right]\right|.\nonumber
\end{align}
Due to the boundness of each $\gamma_{s_i}$ we have that the last term can be bounded from above by
\begin{align}
&\left|\sum_{i=1}^m\left[(c_i\| u_k-u\|_{\mathbf{H}^1(\Omega_{p_i})}+\|\gamma_{s_i}u-g_i\|_{L^2(s_i)}))^2 - \|\gamma_{s_i}u-g_i\|_{L^2(s_i)}^2\right]\right|\leq\nonumber\\
&\left|\sum_{i=1}^m\left[(c_i\| u_k-u\|_{\mathbf{H}^1(\Omega)}+\|\gamma_{s_i}u-g_i\|_{L^2(s_i)}))^2 - \|\gamma_{s_i}u-g_i\|_{L^2(s_i)}^2\right]\right|,\nonumber\\
\end{align}
which goes to zero when $k\rightarrow\infty$.
The convexity follows directly from the fact that
\begin{align}
&\int_{\Omega_{part}}(\frac{u_1+u_2}{2}-u_d)^2\,ds=\sum_{i=1}^m\frac{1}{4}\int_{s_{i}}(\gamma_{s_i}u_1-g_i+\gamma_{s_i}u_2-g_i)^2\,ds\nonumber\\
&\leq\sum_{i=1}^m\frac{1}{4}\int_{s_{i}}2^1[(\gamma_{s_i}u_1-g_i)^2+(\gamma_{s_i}u_2-g_i)^2]\,ds\nonumber\\
&\leq\frac{1}{2}\int_{\Omega_{part}}(u_1-u_d)^2\,ds+\frac{1}{2}\int_{\Omega_{part}}(u_2-u_d)^2\,ds\nonumber.
\end{align}
Therefore $J$ is weakly l.s.c.. The rest of the proof follows as in Theorem~\ref{existencecontrolfull}.
\end{proof}
Lastly, another case that can also be interesting from the applications point of view.
\begin{theorem}\label{existencecontrolpartial}If we consider now $\Omega_{p_i}$ as a family of disjoint subdomains of $\Omega$ and we take $\Omega_{part}=\cup_{i=1}^m\Omega_{p_i}$ in $J$, then problem $(P)$ also has an optimal solution.
\end{theorem}
\begin{proof}
To prove this statement, we will check, once more, that $J$ remains convex and strongly continuous. Concerning the convexity, it follows directly as in Theorem~\ref{existencecontrolsections}. As for the continuity, let $(u_k)_k$ be a convergent sequence to $u$ in $\mathbf{H}^1(\Omega)$, then
\begin{align}
&\left|\int_{\Omega_{part}}(u_k-u_d)^2-(u-u_d)^2\,dx\right|\leq \nonumber\\
&\left|\sum_{i=1}^m\left[(\|u_k-u\|_{L^2(\Omega_{p_i})}+\|u-u_d\|_{L^2(\Omega_{p_i})})^2 - \|u-u_d\|_{L^2(\Omega_{p_i})}^2\right]\right|\leq\nonumber\\
&\left|\sum_{i=1}^m\left[(\|u_k-u\|_{L^2(\Omega)}+\|u-u_d\|_{L^2(\Omega_{p_i})})^2 - \|u-u_d\|_{L^2(\Omega_{p_i})}^2\right]\right|\nonumber
\end{align}
which tends to zero when $k\rightarrow\infty$.
\end{proof}
|
1,116,691,498,423 | arxiv | \part{Motivation and Results}}
\section{Introduction}
\subsection{The Generalized Fermat Equation}
Since Wiles' groundbreaking proof~\cite{Wiles} of Fermat's Last Theorem, attention has shifted towards the study of the \emph{Generalized Fermat Equation (GFE)}, namely, the equation
\begin{equation}
Ax^r+By^q=Cz^p, \qquad p,q,r \in \mathbb{Z}_{\geq 2}, \qquad \frac{1}{r}+\frac{1}{q}+\frac{1}{p}<1,
\label{E:GFE}
\end{equation}
where $A,B,C \in \mathbb{Z}_{\neq 0}$ are pairwise coprime.
We say that a solution $(a,b,c) \in \mathbb{Z}^3$ of \eqref{E:GFE} is {\it
non-trivial} if it satisfies $abc \neq 0$ and we call it {\it primitive} if
$\gcd(a,b,c) = 1$. We call a {\it signature} any triple of exponents~$r,q,p$ as above
and say that~\eqref{E:GFE}
is a {\it Fermat-type equation of signature $(r,q,p)$}. We reserve the letter~$p$ for a prime that is varying while the others exponents are fixed. This way, fixing a prime~$r$, the signatures $(r,r,p)$ and~$(p,p,r)$ correspond to different infinite families of Fermat-type equations.
The GFE is the subject of the following conjecture, which is known to be a consequence of the $ABC$-conjecture (see \cite[Section~5.2]{DG}).
{\bf Conjecture.} Fix $A,B,C \in \mathbb{Z}$ pairwise coprime.
Over all choices of signatures~$(r,q,p)$ the
equation \eqref{E:GFE} admits only finitely many non-trivial primitive solutions; here solutions
where one of $x,y,z$ equals~$1$ are counted
only once, e.g. $2^3 + 1^q = 3^2$ for all~$q$ counts as one solution.
A result of Darmon--Granville~\cite{DG} states that if we fix $A,B,C$ and the signature $(r,q,p)$ then there are only finitely many non-trivial primitive solutions to~\eqref{E:GFE} as predicted.
This is proven by a clever application of the Chevalley-Weil theorem which reduces the result to Faltings' Theorem on the finiteness of rational points on curves of genus $\ge 2$. Apart from this result, almost all progress towards the above conjecture was obtained using extensions of Wiles' proof of FLT that rely on modularity of elliptic curves.
The crucial link between modularity of elliptic curves and Galois representations with Diophantine equations
goes back to the work of Serre~\cite{Serre87} and Darmon~\cite{Darmon44p, DarmonNN2} who studied instances of~\eqref{E:GFE} assuming certain modularity conjectures.
In a nutshell, the core idea is to
map any putative solution of a Diophantine equation to 2-dimensional Galois representations valued in~$\operatorname{GL}_2(\mathbb{F}_p)$ of bounded conductor, and to show these representations arise from modular forms of weight 2 and level equal to their Serre level which should be independent of the solution. A resolution of the equation is obtained, if one can show, by methods for distinguishing residual Galois representations, that the mod~$p$ representations arising from all the modular forms at the Serre level (and weight 2) are not isomorphic to the representations
constructed from
a non-trivial putative solution.
This strategy is known as the \emph{modular method} to solve Diophantine equations.
\subsection{The Darmon program}
Wiles' proof of modularity of semistable elliptic curves over~$\mathbb{Q}$ also gave birth to a new era for modularity lifting theorems. In the last 25 years, this area has seen remarkable progress due to the work of Breuil, Calegari, Diamond, Gee, Geraghty, Kisin, Skinner, Taylor, Thorne,
and others. This progress opened the door for extending the modular method to the setting of totally real fields and Hilbert modularity in~\cite{DF2}.
In practice, when applying the modular method, the association of a residual Galois representation to
a putative solution of a Diophantine equation is provided by elliptic curves. Furthermore, the fact that the representation arises on an elliptic curve allows to study its properties in great detail, as there are many tools available in this setting.
The main steps of this procedure over totally real fields can be summarized as follows.
{\bf Constructing a Frey curve.} Attach an elliptic curve $E/K$ to a putative solution of a Diophantine equation, where $K$ is some
totally real field.
In the case of FLT, following an idea of Frey--Hellegouarch
one considers the curve
\[
\label{E:FreyCurve}
y^2 = x(x-a^p)(x+b^p) \quad \text{ where } \quad
a^p + b^p = c^p, \quad abc \neq 0, \quad a,b,c \in \mathbb{Z}.
\]
Studying different equations require constructing
different curves;
such an elliptic curve is called {\it a Frey elliptic curve} or simply {\it Frey curve} for short.
{\bf Modularity.} Prove modularity of $E/K$.
{\bf Irreducibility.} Prove irreducibility of
$\overline{\rho}_{E,p}$, the mod $p$ Galois representation attached to $E$.
{\bf Level lowering.} Conclude that
$\overline{\rho}_{E,p} \simeq {\overline{\rho}}_{\mathfrak{f},\mathfrak{p}}$ where $\mathfrak{f}$ is a Hilbert
newform over $K$ of parallel weight 2, trivial character and level among finitely many explicit;
here ${\overline{\rho}}_{\mathfrak{f},\mathfrak{p}}$ denotes
the mod~$\mathfrak{p}$ representation attached to~$\mathfrak{f}$ for some $\mathfrak{p} \mid p$.
{\bf Contradiction.}
Compute all the newforms predicted in the previous step; then, for each computed newform~$\mathfrak{f}$ and $\mathfrak{p} \mid p$ in its field of coefficients, show that
$\overline{\rho}_{E,p} \not\simeq \overline{\rho}_{\mathfrak{f},\mathfrak{p}}$. This rules out the isomorphism predicted by level lowering, yielding a contradiction. This final step is also known as {\em the elimination step.}
There are only a few instances of Fermat-type equations having Frey curve attached to them; in fact, those defined over~$\mathbb{Q}$ were already known to Darmon in 1997 (see \cite[p.~14]{DarmonEps} for a list).
Since then, new Frey curves defined over totally real fields were found by Freitas~\cite{F}, which are attached to signature~$(r,r,p)$ where $r \geq 5$ is a prime.
This scarcity prompted Darmon to develop an ambitious program~\cite{DarmonDuke} to tackle Fermat-type
equations with one varying exponent.
A main idea of his program is to replace the use of Frey curves by higher dimensional abelian varieties defined over~$\mathbb{Q}$ that become of $\operatorname{GL}_2$-type over certain totally real fields;
we refer to these varieties as {\it Frey abelian varieties} or simply {\it Frey varieties} for short.
In this way Darmon systematically attaches residual
$2$-dimensional Galois representations with bounded conductor to putative solutions of~\eqref{E:GFE} for all signatures.
However, Darmon's Frey varieties are not always explicit enough to work with and, moreover, applying the rest of his program is challenging because several of the main steps rely on hard open conjectures.
Notably, Conjecture 4.3 of \cite{DarmonDuke} stating that Darmon's Frey varieties have residual Galois representations with large image (which includes irreducibility) except in the CM case, is still wide open.
This conjecture is crucially used to distinguish the mod~$p$ Galois representations attached to a non-trivial solution from those attached to the trivial solutions.
Among the other challenges faced by the program are the difficulty of proving modularity of residual Galois representations over number fields and the lack of modularity lifting theorems in the case of residually reducible image.
Finally, another important difficulty which emerges for signatures with large $r,q$ is that the spaces of Hilbert modular forms become too large to allow explicit computation. This issue was not addressed by Darmon, but it becomes of prime importance when trying to realize his program.
\section{Our contribution to the Darmon program}\label{S:ourApproach}
The main goal of this paper is to understand and expand the current limits of Darmon's approach using higher dimensional Frey varieties.
By exploring the ideas in the Darmon program together
with some of the latest developments surrounding the modular method, we will study in detail an approach to Fermat-type equations of signature $(r,r,p)$ using
a hyperelliptic Frey curve constructed by Kraus~\cite{kraushyper}.
As a Diophantine application of our methods, we will prove Theorem~\ref{T:main} below. Of great importance for this proof is the multi-Frey technique, which we studied in detail in~\cite{BCDF2}. Moreover, for the proof to be efficient we also introduce several theoretical and practical ideas to the contradiction step to reduce the amount of required computations.
This proof gives further evidence that the multi-Frey technique applied with a sufficiently rich set of Frey varieties can be used to `patch together' a complete resolution of a one parameter family of generalized Fermat equations, with optimal bounds on the exponent~$p$. Also, various ideas of Darmon's program
are used successfully for the first time, as the information obtained from the multi-Frey technique
allows to establish the conjectural parts of his program in specific cases of our interest. Moreover,
since for Fermat-type equations of
signature $(r,r,p)$ we have both Frey curves and Frey varieties available, we are able to compare and contrast their use in resolving~\eqref{E:77p}.
For most cases of signatures $(p,p,r)$ or $(q,r,p)$, Frey curves are absent, so the use of Frey varieties becomes essential; the ground work we lay for signature $(r,r,p)$ opens the road to resolving other signatures which afford a Frey variety arising from the Jacobian of a hyperelliptic curve as, for example, Darmon's construction for signature~$(p,p,r)$ \cite{ChenKoutsianas1}.
As mentioned, several aspects of the Darmon program are impractical or conjectural. We now summarize
the main challenges we face and the ways we deal with
them in more detail.
\noindent {\bf The Frey variety.} Darmon's construction of the Frey varieties attached to signature $(r,r,p)$
starts from superelliptic curves and considers certain quotients of them and their Jacobians (see \cite[pp. 422--423]{DarmonDuke}).
Further explicit development of this theory is needed to allow its practical use in Diophantine applications. Instead, we show certain odd Frey representations of signature $(r,r,p)$ (see Section~\ref{S:FreyRep} for a definition) can in fact be given by the Jacobian of a Frey hyperelliptic curve $C_r(a,b)$ due to Kraus~\cite{kraushyper}. To accomplish this, we exhibit a curious relation between Kraus' Frey hyperelliptic curve and Darmon's Frey varieties for equations of signature~$(p,p,r)$ which allows us to prove that its Jacobian $J_r=\Jac(C_r(a,b))$ becomes of $\operatorname{GL}_2$-type over $K=\mathbb{Q}(\zeta_r)^+$, the maximal totally real subfield of~$\mathbb{Q}(\zeta_r)$. Therefore, there are $2$-dimensional Galois representations of $G_K = \Gal({\overline{\Q}} / K)$ attached to~$J_r$.
\noindent {\bf Modularity.}
A remarkable feature of Darmon's Frey varieties is that they share some Galois structures. In particular, in \cite[p. 433]{DarmonDuke} there is a diagram describing the possibility of propagating modularity among them
once suitable modularity lifting theorems become available. We will make this idea work in our setting. Indeed, let $J_r$ be the Jacobian of Kraus' hyperelliptic curve~$C_r(a,b)$; it has dimension~$(r-1)/2$ and becomes of $\operatorname{GL}_2$-type over $K = \mathbb{Q}(\zeta_r)^+$.
From the relation mentioned above between~$J_r$ and Darmon's Frey variety for signature $(p,p,r)$ we first show that the residual representation ${\overline{\rho}}_{J_r,\mathfrak{p}_r} : G_K \to \operatorname{GL}_2(\mathbb{F}_r)$
arises on an elliptic curve and descends to~$G_\mathbb{Q} = \Gal({\overline{\Q}}/\mathbb{Q})$. Here~\(\mathfrak{p}_r\) is the unique prime ideal above~\(r\) in~\(K\). Secondly, we show it is absolutely irreducible when restricted to~\(\Gal({\overline{\Q}}/\mathbb{Q}(\zeta_r))\), allowing us to apply Serre's conjecture, cyclic base change and modularity lifting theorems to conclude modularity
of~$J_r/K$.
\noindent {\bf The conductor.}
A very useful advantage of Kraus' hyperelliptic curves is that we can successfully compute their conductors.
Let $J_r$ be as above. We are interested in the conductor of the $2$-dimensional $p$-adic Galois representations~$\rho_{J_r,\mathfrak{p}}$ attached to~$J_r/K$. This is known to be the cube root of the conductor of~$J_r/K$. The recent remarkable work~\cite{DDMM-local, DDMM-types,best2020users} goes a long way using `cluster pictures' to determine the latter, but it is insufficient for us as it does not apply in even residual characteristic; an additional complication is that we need to determine conductors for varying $a,b,c$. Our approach to determine the conductor at a prime~$\mathfrak{q}$ in $K$ works uniformly independently of the residual characteristic of~$\mathfrak{q}$ as follows. Using explicit calculations with Weierstrass models of hyperelliptic curves we first determine the minimal ramification degree at~$\mathfrak{q}$ of a field $L/K$ where $J_r$ becomes semistable at~$\mathfrak{q}$. Then, from the fact that the local 2-dimension representations~$\rho_{J_r,\mathfrak{p}}|_{D_{\mathfrak{q}}}$ are either principal series, supercuspidal or Steinberg, we pin down which kind is compatible with the field $L$ and then we apply the well known conductor formulas for such representations.
\noindent {\bf Irreducibility.} Let $p$ be prime and~$\mathfrak{p}$ a prime in~$K$ above~$p$.
To apply level lowering results to the residual representations ${\overline{\rho}}_{J_r,\mathfrak{p}}$ attached to $J_r/K$
we first need to show these are irreducible. Our method for computing the conductor gives an exact description of the local representations ${\overline{\rho}}_{J_r,\mathfrak{p}}|_{D_{\mathfrak{q}}}$, where $\mathfrak{q} \mid q \neq p$ is a prime in~$K$.
If such a local representation is supercuspidal then irreducibility holds locally already, otherwise, if we have a principal series we show irreducibility for large enough~$p$. For the particular Diophantine application in Theorem~\ref{T:main},
using the multi-Frey technique, we first show that we can assume that $a$ is even and $b \equiv 1 \pmod{4}$. Under these assumptions, we show that ${\overline{\rho}}_{J_r,\mathfrak{p}}|_{D_{\mathfrak{q}}}$ is a principal series where $\mathfrak{q} \mid 2$. Combining this
with class field theory, we succeed in showing that ${\overline{\rho}}_{J_r,\mathfrak{p}}$ is irreducible for all primes $p \geq 5$.
\noindent {\bf Finiteness of the $p$-torsion representation.}
Another key input for level lowering
is to guarantee that ${\overline{\rho}}_{J,\mathfrak{p}}|_{D_\mathfrak{q}}$ arises on a finite flat group scheme for all~$\mathfrak{q} \mid p$. This implies that
${\overline{\rho}}_{J,\mathfrak{p}}$ arises on a Hilbert newform of parallel weight~$2$, independently of the putative solution.
In the case of Frey curves this follows from standard arguments using the theory of the Tate curve, but for higher dimensional varieties the situation is more delicate. Ellenberg~\cite{Ellenberg} gives
a criterion which is a direct generalization of the usual criterion for elliptic curves, but it is hard to use due to the need to determine a discriminantal set. In~\cite{DarmonDuke}, Darmon
implicitly uses such a theorem for signature $(p,p,r)$, but we are not aware of a complete reference for general signatures. Inspired by Darmon's ideas, we give a criterion
that is easy to apply in practice in many interesting cases; see Theorems~\ref{finiteness} and~\ref{T:finite} in Section~\ref{S:finiteness}.
\noindent {\bf Level lowering and contradiction.}
To obtain a contradiction, we need to
show that ${\overline{\rho}}_{J_r,\mathfrak{p}} \not\simeq {\overline{\rho}}_{g,\mathfrak{P}}$ for all newforms~$g$ in the relevant spaces after level lowering.
Computing the possible newforms~$g$ can be a serious obstruction. Furthermore, aiming to obtain an optimal bound for the exponent can introduce further computational complications; indeed, applying the standard trace comparison method requires, for each newform~$g$, to factor multiple norms from the compositum of~$K$ with the field of coefficients of~$g$, which can be of very large degree. To reduce such computations, we apply level lowering with prescribed inertial types, together with a result that relates the field of coefficient of a newform to its inertial types.
This yields two major benefits: it reduces the number of spaces where we have to eliminate forms and also dramatically cuts down the number of forms we have to consider in the remaining space. We remark this
powerful technique is not available when working with Frey curves (see the last paragraph of Section~\ref{S:Frey7} for further discussion). Despite this improvement, there are still computational challenges left, as illustrated in the proof of Theorem~\ref{T:main} in Section~\ref{S:eliminationJ}. We describe various ways to overcome these issues in Section~\ref{S:enhanements}.
\section{Diophantine applications}
In the works of
Dieulefait--Freitas~\cite{DF1,DF2} and Freitas~\cite{F} several Frey curves were attached to Fermat equations
of signature $(r,r,p)$ for each
prime~$r \geq 5$;
these curves are defined over
totally real subfields of
the $r$-th cyclotomic field~$\mathbb{Q}(\zeta_r)$.
Together with powerful developments of modularity lifting theorems for totally real fields (e.g. \cite{BreuilDiamond}) and consequent modularity of many elliptic curves over these fields, this opened a door to pursue the study of Fermat equations of signature~$(r,r,p)$ of the form
\begin{equation}
x^r + y^r = dz^p, \qquad xyz \ne 0, \qquad \gcd(x,y,z) = 1
\label{E:rrp}
\end{equation}
with $r$ a fixed prime, $d$ a fixed positive integer and $p$ allowed to vary.
Building on these cited works, using a
refined multi-Frey curve approach, we have solved equation \eqref{E:rrp}
with $r=5,13$ and $d=3$ for all exponents $p \geq 2$ (see~\cite{BCDF2} and the joint work with Demb\'el\'e~\cite{BCDDF}).
Moreover, the fourth author~\cite{F} solved it asymptotically for $r=7$ and $d=3$, i.e.,
for all primes $p$ sufficiently large.
As an application of the methods developed in this paper we will optimize this last result. More precisely, we will prove the following.
\begin{theorem}\label{T:main}
For all integers $n \geq 2$, there are no integer solutions $(a,b,c)$ to the equation
\begin{equation}
\label{E:77p}
x^7 + y^7 = 3 z^n
\end{equation}
such that $abc \neq 0$ and $\gcd(a,b,c) = 1$.
\end{theorem}
We remark there is a well known Frey curve~$E_{a,b}/\mathbb{Q}$ attached to~\eqref{E:77p} but it is insufficient to solve the equation (see Section~\ref{S:overQ}).
To highlight the importance of the multi-Frey technique both at the theoretical and computational levels,
we will give four proofs of this theorem
which are divided into two types.
The first type of proof uses the classical modular method with Frey curves over totally real field, which is made possible by a previously unobserved and careful selection of twist of the Frey curve. Indeed, our first proof of this type, given in Section~\ref{S:overCubic}, relies on a Frey curve $F_{a,b}/\mathbb{Q}(\zeta_7)^+$ from~\cite{F} and ideas from~\cite{BCDF2}. Our second proof combines both $E_{a,b}$ with~$F_{a,b}$ to reduce the total computational time considerably, giving a prime example of the advantages of the multi-Frey technique.
The second type of proof adds in the use of Frey abelian varieties. More precisely, we use a multi-Frey approach combining the Frey curves $E_{a,b}$ and~$F_{a,b}$ with Kraus' hyperelliptic curve $C_7(a,b)$ in two different ways. To illustrate the limits of the theory, the first proof of this type uses the curve $C_7(a,b)$ as much as possible, whilst the second proof is designed to minimize the computational time among all proofs we give.
When trying to solve~$\eqref{E:rrp}$ for~$r > 7$ or, more generally, carrying out Darmon's program for other signatures, the spaces of Hilbert newforms involved will quickly become very large. Our work shows that the Frey varieties have additional structures which one can exploit to reduce computations, despite the fact that we have to work with Jacobians of hyperelliptic curves.
The use of Frey varieties seems promising for future efforts to solve GFE asymptotically. For instance, we will prove in this paper the following asymptotic results.
\begin{theorem}
\label{main-asymptotic}
For prime $p$ sufficiently large, there are no integer solutions $(a,b,c)$ to the equation
\begin{equation}
\label{main-equ}
x^{11} + y^{11} = z^p
\end{equation}
such that $abc \not= 0$, $\gcd(a,b,c) = 1$, and $2 \mid a + b$ or $11 \mid a + b$.
\end{theorem}
The trivial solutions $(a,b,c)$ with $ab = 0$ pose a fundamental obstruction for current modular Galois representation theoretic methods to resolve \eqref{main-equ}, which explains the need for local conditions at $2$ or $11$. For small primes $p$, we note the work of \cite{DahmenSiksek}.
The above result is the first time generalized Fermat equations of signature $(11,11,p)$ with unit coefficients have been resolved asymptotically in `unobstructed' congruence classes.
The proof uses a Frey elliptic curve defined over $\mathbb{Q}(\zeta_{11})^+$ and Kraus' Frey hyperelliptic curve $C_{11}(a,b)$, together with a new method we call {\em Legendre descent}. With this new method, the costly computation of Hilbert newforms is mostly avoided, reducing the computation time to a few minutes.
\begin{comment}
As another illustration of the techniques in this paper, we prove the following asymptotic result which covers all unobstructed congruence classes modulo $22$.
\begin{theorem}
\label{main-asymptotic-complete}
For prime $p$ sufficiently large, there are no integer solutions $(a,b,c)$ to the equation
\begin{equation*}
x^{11} + y^{11} = z^p
\end{equation*}
such that $abc \not= 0$, $\gcd(a,b,c) = 1$, and $2 \nmid ab$ or $11 \nmid ab$.
\end{theorem}
The case $2 \nmid a + b, 11 \nmid ab$ (see the proof of Theorem~\ref{main-asymptotic-complete}) which is needed to show the above asymptotic result is not computationally feasible using the Frey curve over $\mathbb{Q}(\zeta_{11})^+$ alone. However, it can be proven using the Frey hyperelliptic curve $C_{11}(a,b)$ and applying the method of Legendre descent together with a new argument invoking strict compatibility in the strong sense.
\end{comment}
\section{Electronic resources}
The {\tt Magma} programs used for the computations needed in this paper are posted in \cite{programs}, where there is included a list of programs, their descriptions, output transcripts, and timings.
\section{Acknowledgments}
We thank Lassina Demb\'el\'e and Angelos Koutsianas for helpful discussions.
We thank also Henri Darmon for conversations about Section~\ref{S:finiteness}.
The fourth author is grateful to Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.
{\large \part{Higher dimensional Frey abelian varieties}\label{Part:higherdimFAV}}
In this part of the paper, we intend to develop theory that allows applications of the modular method to equation~\eqref{E:rrp} using the Jacobian~$J_r$ of the Frey hyperelliptic curve constructed by Kraus~\cite{kraushyper}.
More precisely, we first show in Section~\ref{S:Freyrrp} that there are 2-dimensional Galois representations attached to~$J_r$. Using these representations, we then prove in Section~\ref{S:modularity} that~\(J_r\) is modular, as well as other Frey varieties constructed by Darmon. In Section~\ref{S:irreducibilityrrp}, we give several criteria that allow to prove irreducibility of the residual representations under various assumptions. Finally, assuming irreducibity, we state in Section~\ref{S:levelLowering} level lowering results that are refinements of the classical ones in the context of Frey abelian varieties of higher dimension.
For a field~$k$, fix an algebraic closure~$\overline{k}$ and let $G_k := \Gal(\overline{k}/k)$ be its absolute Galois group.
\section{Connecting Frey representations for signatures $(p,p,r)$ and $(r,r,p)$}
In this section, we first introduce Frey representations which are a key idea in Darmon's program. After recalling some useful background on abelian varieties of~\(\operatorname{GL}_2\)-type and hyperelliptic curves, we give in \S\ref{S:FreyOverKs} an explicit construction of Frey representations of signature~\((r,r,p)\) which differs from Darmon's approach in that it uses hyperelliptic curves instead of superelliptic curves. Our method reveals a surprising relation between Frey representations of signature~\((r,r,p)\) and~\((p,p,r)\) (recall these are different signatures as~$r$ is fixed and $p$ varies). This construction plays a central role in the next section where we relate Kraus' construction to certain specializations of our abelian varieties in the spirit of~\cite[\S1.3]{DarmonDuke}.
\subsection{Frey representations} \label{S:FreyRep}
The following definitions are modifications of Darmon's originals (\cite[Definitions~1.1 and~1.3]{DarmonDuke}) adapted to our needs. Here $K$ is a number field and $\mathbb{F}$ is a finite field of characteristic~$p$ with algebraic closure~${\overline{\F}}$.
Recall that $G_{\overline{K}(t)}$ sits inside $G_{K(t)}$ as normal subgroup.
\begin{definition}\label{D:FreyRep}
A {\it Frey representation} (in characteristic $p$) over $K(t)$ of {\it signature} $(n_i)_{i = 1, \ldots, m}$ with respect to the points
$(t_i)_{i = 1, \ldots, m}$, where $n_i \in \mathbb{N}$ and $t_i \in \mathbb{P}^1(\overline{K})$ is a Galois representation
\begin{equation*}
{\overline{\rho}} : G_{K(t)} \rightarrow \operatorname{GL}_{2}(\mathbb{F})
\end{equation*}
satisfying
\begin{enumerate}
\item the restriction ${\overline{\rho}} \mid_{G_{\overline{K}(t)}}$ has trivial determinant and is irreducible,
\item the projectivization $\mathbb{P} {\overline{\rho}} \mid_{G_{\overline{K}(t)}} : G_{\overline{K}(t)} \rightarrow \text{PSL}_{2}(\mathbb{F})$ of ${\overline{\rho}} \mid_{G_{\overline{K}(t)}}$ is unramified outside the~$t_i$,
\item the projectivization $\mathbb{P} {\overline{\rho}} \mid_{G_{\overline{K}(t)}}$ maps the inertia subgroups at $t_i$ to subgroups of $\operatorname{PSL}_{2}(\mathbb{F})$ generated by elements of order $n_i$, respectively, for $i = 1, \ldots, m$.
\end{enumerate}
\end{definition}
\begin{definition}\label{D:FreyRepEquiv}
Let $\rho_1$, $\rho_2$ be two Frey representations as in Definition~\ref{D:FreyRep} of the same signature with respect to the same points. We say $\rho_1$, $\rho_2$ are equivalent if they are isomorphic over~${\overline{\F}}$ up to a character, that is $\rho_1 \otimes {\overline{\F}}$ is isomorphic to $\rho_2 \otimes \chi$, for a character $\chi : G_{K(t)} \rightarrow {\overline{\F}}^\times$.
\end{definition}
The following lemma is useful for verifying the first condition in Definition~\ref{D:FreyRep}.
\begin{lemma}
\label{frey-irreducible}
Let $\sigma_0, \sigma_1, \sigma_\infty$ be elements of $\operatorname{PSL}_2(\mathbb{F})$ of order $p,p,r$ satisfying $\sigma_0 \sigma_1 \sigma_\infty = 1$, where $p$ and $r$ are prime numbers. Then $\sigma_0, \sigma_1, \sigma_\infty$ generate a subgroup of $\operatorname{PSL}_2(\mathbb{F})$ which acts irreducibly on~$\mathbb{F}^2$.
\end{lemma}
\begin{proof}
By \cite[Lemma 1.11]{DarmonDuke}, the elements $\sigma_0, \sigma_1, \sigma_\infty$ generate all of $\operatorname{PSL}_2(\mathbb{F})$ except in the case $(p,r) = (3,5)$ when it generates a subgroup isomorphic to $A_5$. However, the order of $A_5$ does not divide the order of a Borel subgroup of $\operatorname{PSL}_2(\mathbb{F}_9)$.
\end{proof}
The above definition generalizes the original idea of Frey curve as follows.
In~\cite[\S 1.2]{DarmonDuke} a classification of Frey representations of signature~$(r,r,p)$ and~$(p,p,r)$ with respect to the points $(0, 1, \infty)$ is given and they are shown to arise from a base change of certain abelian varieties over $\mathbb{Q}(t)$. In particular, the varieties obtained by Darmon include the known Frey elliptic curves defined over~$\mathbb{Q}$ (see~\cite[p. 14]{DarmonEps}) in case of prime exponents~$p,r$. On the other hand, Frey elliptic curves genuinely defined over~$K$, such as~$F_{a,b}$ in Section~\ref{S:overCubic} and those in~\cite{F}, are not explained by Darmon's classification for signature~$(r,r,p)$ with respect to the points $(0, 1, \infty)$. Thus, when applying the multi-Frey technique with both kinds of Frey varieties one can expect to exploit independent information arising from each variety. We will see in Section~\ref{S:Frey7} that this combination of Frey varieties plays an essential role in the proof of Theorem~\ref{T:main}.
\begin{remark} Darmon also obtained a similar classification of Frey representations of signature~$(p,q,r)$ for three distinct prime exponents (see \cite[Theorem 1.14]{DarmonDuke}), but these are not considered in this work.
\end{remark}
\subsection{Abelian varieties of~$\operatorname{GL}_2$-type} \label{S:GL2typeAV}
In this paragraph we briefly recall basic facts (see~\cite{RibetGalois}) about abelian varieties of~$\operatorname{GL}_2$-type as they will play an important role in the sequel.
\begin{definition}\label{def:GL2typeAV}
Let $A$ be an abelian variety over a field~$L$ of characteristic~$0$. We say that $A/L$ is of {\it $\operatorname{GL}_2$-type} if there is an embedding $F \hookrightarrow \End_L(A) \otimes \mathbb{Q}$ where $F$ is a number field with $[F:\mathbb{Q}] = \dim A$. We say that $A/L$ of $\operatorname{GL}_2$-type is {\it primitive} if in addition $F \cong \End_L(A) \otimes \mathbb{Q}$.
\end{definition}
\begin{remark}
Often in the literature, $A/L$ being of $\operatorname{GL}_2$-type usually means primitive $\operatorname{GL}_2$-type in our terminology. We prefer to use this terminology because for most arguments in this paper, primitivity is not required; this is also the original terminology of Ribet~\cite{RibetKorea}.
\end{remark}
Let~$A/L$ be an abelian variety
with $F \hookrightarrow \End_L(A) \otimes \mathbb{Q}$ as in Definition~\ref{def:GL2typeAV}. For a prime number~$p$, denote by~$T_p(A)$ the Tate module of~$A$ and write $V_p(A)=T_p(A)\otimes\mathbb{Q}_p$.
According to~\cite[(2.1.1)]{RibetGalois}, we have that~\(V_p(A)\) is a free module of rank~\(2\) over~\(F_p = F\otimes\mathbb{Q}_p\). Moreover, the absolute Galois group~$G_{L}$ of~$L$ acts $F_\mathfrak{p}$-linearly on $V_p(A)$. This gives rise to a strictly compatible system of $2$-dimensional $p$-adic Galois representations (hence justifying the terminology)
\[
\rho_{A,\mathfrak{p}} \; \colon \; G_{L}\xrightarrow[]{\rho_{A,p}}\operatorname{GL}(V_p(A))\simeq\operatorname{GL}_2(F_p)\longtwoheadrightarrow\operatorname{GL}_2(F_\mathfrak{p}),
\]
using the decomposition $F_p \simeq \prod_{\mathfrak{p}\mid p} F_\mathfrak{p}$ where~\(\mathfrak{p}\) runs over the prime ideals above~\(p\) in~\(F\) and $F_\mathfrak{p}$ denotes the completion of~$F$ at~$\mathfrak{p}$.
The representation~$\rho_{A,\mathfrak{p}}$ can be conjugated to take values in~$\operatorname{GL}_2(\mathcal{O}_\mathfrak{p})$ where~$\mathcal{O}_\mathfrak{p}$ stands for the ring of integers in~$F_\mathfrak{p}$. By reduction modulo the maximal ideal, we then get a representation
\[
{\overline{\rho}}_{A,\mathfrak{p}} \; \colon G_{L} \; \longrightarrow\operatorname{GL}_2(\mathbb{F}_\mathfrak{p}),
\]
with values in the residue field~$\mathbb{F}_\mathfrak{p}$ of~$F_\mathfrak{p}$ which is unique up to semi-simplification.
\begin{comment}
\subsection{Hyperelliptic equations}\label{ss:hyperelliptic_equations}
Later we will need to do explicit calculations with models for hyperelliptic curves, particularly to compute the conductor of Frey varieties. In this subsection, we summarize
well known facts that will be relevant for those calculations. The following properties are taken in part from \cite{Lockhart}.
Let $K$ be a local field. Denote by $\mathcal{O}$ the ring of integers of $K$, $k$ the residue field of $\mathcal{O}$, and by $v$ its normalized valuation. Let~$g$ be a positive integer. A (Weierstrass) hyperelliptic equation $E$ over $K$ of genus~$g$ is an equation of the form
\begin{equation}
y^2 + Q(x) y = P(x)
\end{equation}
where $Q, P \in K[x]$, $\deg Q \le g$, $\deg P = 2g + 1$ and $P$ is monic. The discriminant of $E$ is given~by
\begin{equation*}
\Delta_E = 2^{4g} \Delta(P + Q^2/4),
\end{equation*}
where $\Delta(H)$ is the discriminant of $H \in K[x]$.
An algebraic curve $C$ given by a hyperelliptic equation $E$ over $K$ with $\Delta_E \not=0$ is called a hyperelliptic curve over $K$. A hyperelliptic equation $F$ such that $K(F) \cong K(C) = K(E)$ is a called a hyperelliptic model for $C$; here $K(C)$ stands for the function field of~$C$.
Two hyperelliptic models~$E : y^2 + Q(x) y = P(x)$ and~$F : z^2 + T(u) z = S(u)$ for the same hyperelliptic
curve~$C$ over~$K$ are related by a transformation of the shape
\begin{align*}
& x = e^2 u + r, \quad y = e^{2g+1} z + t(u), \quad \text{where} \\
& e \in K^*,\quad r \in K,\quad t \in K[u],\quad \deg(t) \le g.
\end{align*}
The discriminants of the hyperelliptic models are related by
\[
\Delta_F = \Delta_E \, e^{-4g(2g+1)},
\]
hence the valuation of the discriminant modulo $4g(2g+1)$ is an invariant of the isomorphism class of $C$.
A model $\mathcal{C}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ is a $\mathcal{O}$-scheme which is proper and flat over $\mathcal{O}$ such that $\mathcal{C}_K \cong C$ where $\mathcal{C}_K$ denotes the generic fiber of $\mathcal{C}$.
A model $\mathcal{C}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ has good reduction if and only if its reduction mod $v$ is a non-singular curve over the residue field $k$ of~$K$.
A model $\mathcal{C}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ has multiplicative reduction if and only if its reduction mod $v$ is reduced and has only ordinary double points as singularities and at least one singularity (over $\bar k$).
A hyperelliptic model $\mathcal{E}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ is a model over $\mathcal{O}$ arising from hyperelliptic equation $E : y^2 + Q(x) y = P(x)$ over $K$ such that $P, Q \in \mathcal{O}[x]$.
A hyperelliptic curve $C$ over $K$ has good reduction (resp.\ multiplicative reduction) if and only if there is some model $\mathcal{C}$ over $\mathcal{O}$ for $C$ which has good reduction (resp.\ multiplicative reduction).
A hyperelliptic curve $C$ over $K$ has semistable reduction if and only if $C$ has good or multiplicative reduction.
An abelian variety over $K$ has semistable reduction if and only if the linear part of the special fiber of the connected component of its N\'eron model is an algebraic torus.
Let $C$ be a hyperelliptic curve over $K$. It is well known that:
\begin{itemize}
\item $C/K$ has good reduction if and only if $C$ has a hyperelliptic model $\mathcal{E}$ over $\mathcal{O}$ such that $v(\Delta_E) = 0$.
\item Suppose~$K$ has odd residual characteristic.
Then $C/K$ has multiplicative reduction if it has a hyperelliptic model $\mathcal{E}: y^2 = P(x)$ over $\mathcal{O}$ such that the reduction of $P$ mod $v$ has at most double roots and at least one double root over $\bar k$.
\end{itemize}
\end{comment}
\subsection{Hyperelliptic equations}\label{ss:hyperelliptic_equations}
In this subsection, we summarize well known facts (taken in part from \cite{Lockhart}) that will be relevant for later calculations in this section and in~\S\ref{ss:conductor_calculation}.
Let $K$ be a local field. Denote by $\mathcal{O}$ the ring of integers of $K$, $\pi$ a uniformizer in~\(\mathcal{O}\), $k$ the residue field of $\mathcal{O}$, and $v$ its normalized valuation. Let~$g$ be a positive integer.
We refer the reader to the definition of a hyperelliptic equation $E$ over $K$ of genus $g$ and its discriminant $\Delta_E$ in general form described in \cite[Proposition 7.4.24]{Liu-book}, but do not recall it as we will only need the special case in \eqref{odd-degree-case}.
An algebraic curve $C$ given by a hyperelliptic equation $E$ over $K$ with $\Delta_E \not=0$ is called a hyperelliptic curve over $K$. A hyperelliptic equation $F$ such that $K(F) \cong K(C) = K(E)$ is a called a hyperelliptic model for $C$; here $K(C)$ stands for the function field of~$C$.
A model $\mathcal{C}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ is a $\mathcal{O}$-scheme which is proper and flat over $\mathcal{O}$ such that $\mathcal{C}_K \cong C$ where $\mathcal{C}_K$ denotes the generic fiber of $\mathcal{C}$.
A model $\mathcal{C}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ has good reduction if and only if its reduction mod $v$ is a non-singular curve over the residue field $k$ of~$\mathcal{O}$.
A model $\mathcal{C}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ has multiplicative reduction if and only if its reduction mod $v$ is reduced and has only ordinary double points as singularities and at least one singularity (over $\bar k$).
A hyperelliptic model $\mathcal{E}$ over $\mathcal{O}$ for a hyperelliptic curve $C$ over $K$ is a model over $\mathcal{O}$ arising from hyperelliptic equation $E : y^2 + Q(x) y = P(x)$ over $K$ such that $P, Q \in \mathcal{O}[x]$.
A hyperelliptic curve $C$ over $K$ has good reduction (resp.\ multiplicative reduction) if and only if there is some model $\mathcal{C}$ over $\mathcal{O}$ for $C$ which has good reduction (resp.\ multiplicative reduction).
A hyperelliptic curve $C$ over $K$ has semistable reduction if and only if $C$ has good or multiplicative reduction.
An abelian variety over $K$ has semistable reduction if and only if the linear part of the special fiber of the connected component of its N\'eron model is an algebraic torus.
An odd degree hyperelliptic equation $E$ over $K$ of genus~$g$ is an equation of the form
\begin{equation}
\label{odd-degree-case}
y^2 + Q(x) y = P(x)
\end{equation}
where $Q, P \in K[x]$, $\deg Q \le g$, $\deg P = 2g + 1$ and $P$ is monic. The discriminant of $E$ is given~by
\begin{equation*}
\Delta_E = 2^{4g} \Delta(P + Q^2/4),
\end{equation*}
where $\Delta(H)$ is the discriminant of $H \in K[x]$.
Two odd degree hyperelliptic models~$E : y^2 + Q(x) y = P(x)$ and~$F : z^2 + T(u) z = S(u)$ for the same hyperelliptic
curve~$C$ over~$K$ are related by a transformation of the shape
\begin{equation*}
x = e^2 u + r, \quad y = e^{2g+1} z + t(u),
\end{equation*}
where~\(e \in K^*\), \(r \in K\), and \(t \in K[u]\) with \(\deg(t) \le g\). The discriminants of the odd degree hyperelliptic models are related by
\[
\Delta_F = \Delta_E \, e^{-4g(2g+1)},
\]
hence the valuation of the discriminant modulo $4g(2g+1)$ is an invariant of the isomorphism class of $C$.
\begin{proposition}\label{hyperelliptic-good}
Let $C$ be a hyperelliptic curve over $K$ with a $K$-rational Weierstrass point. Then $C$ has good reduction if and only if $C$ has an odd degree hyperelliptic model $\mathcal{E}$ over $\mathcal{O}$ such that $v(\Delta(\mathcal{E})) = 0$.
\end{proposition}
\begin{proof}
If $C$ has an odd degree hyperelliptic model $\mathcal{E}$ over $\mathcal{O}$ such that $v(\Delta(\mathcal{E})) = 0$, then $\mathcal{E}$ is a model with good reduction so $C$ has good reduction.
Suppose $C$ has good reduction, so there exists a model $\mathcal{C}$ of $C$ with good reduction. By \cite[Exercise 8.3.6]{Liu-book}, the hyperelliptic map
\begin{equation*}
\pi: C \rightarrow \mathbb{P}^1_K
\end{equation*}
extends to
\begin{equation*}
\pi : \mathcal{C} \rightarrow \mathbb{P}^1_\mathcal{O}.
\end{equation*}
As a result, $\mathcal{C}_k$ is a non-singular pointed hyperelliptic curve, so using \cite[Proposition 1.2]{Lockhart} we deduce that $C$ has an odd degree hyperelliptic model $\mathcal{E}$.
\end{proof}
Let $C$ be a hyperelliptic curve over $K$.
If $K$ has odd residual characteristic, then $C/K$ has multiplicative reduction if it has a hyperelliptic model $\mathcal{E}: y^2 = P(x)$ over $\mathcal{O}$ such that the reduction of $P$ mod~$v$ has at most double roots and at least one double root over $\bar k$.
Consider now $J/K$ the Jacobian of an hyperelliptic curve defined over~$K$. By Chevalley's theorem, the special fiber of the connected component of the N\'eron model of $J$ is an extension of a linear algebraic group by an abelian variety. We define the toric rank of $J$ as the number of copies of $\mathbb{G}_m$ occurring in this algebraic group. Moreover, $J$ has semistable reduction if this linear algebraic group is an algebraic torus. We say that $J$ has multiplicative reduction if it is semistable and its toric rank is positive.
\begin{remark}
In what follows, we can restrict to odd degree hyperelliptic equations in our applications due to Proposition~\ref{hyperelliptic-good} and the fact that our Frey hyperelliptic curves have a $K$-rational Weierstrass point at $\infty$.
\end{remark}
\subsection{The Frey hyperelliptic curves over $K(s)$ and~$K(t)$}\label{S:FreyOverKs}
Some of the results that follow are based on the arguments in~\cite{ttv}.
We remark that Darmon constructs two hyperelliptic curves over~$\mathbb{Q}(t)$, denoted~$C_r^+(t)$ and~$C_r^-(t)$ in~\cite[\S 1.3]{DarmonDuke}, attached to equations of signature~$(p,p,r)$. Our hyperelliptic curve~$C_r(s)$ for signature~$(r,r,p)$ defined in~\eqref{E:C(s)} below is related to Darmon's~$C_r^-(t)$. This will be important for our argument to work, for example, in the proof of Proposition~\ref{darmonfrey}. Since we have no use for Darmon's curve~$C_r^+(t)$ in this paper, we omit the `-' in our notation~$C_r(s)$.
Let $r \ge 3$ be prime and $\omega_j = \zeta_r^j + \zeta_r^{-j}$ for $j \in \mathbb{Z}$ where we write~$\zeta_r$ for a fixed primitive $r$-th root of unity. Let~$i$ be a fixed primitive fourth root of unity.
Let $K = \mathbb{Q}(\omega_1) = \mathbb{Q}(\zeta_r)^+$ be the maximal totally real subfield of~$\mathbb{Q}(\zeta_r)$.
Let $\mathbb{Q}(s)$ be the field of rational functions over $\mathbb{Q}$ in the variable~$s$ and set
\begin{equation}\label{eq:def_h}
h(x) = \prod_{j=1}^{\frac{r-1}{2}} (x - \omega_j), \qquad f^-(x) = x h(x^2+2) + s.
\end{equation}
Note that $h\in \mathbb{Z}[x]$ is monic of degree~$(r - 1)/2$.
We consider the hyperelliptic curve over $\mathbb{Q}(s)$ given by
\begin{equation} \label{E:C(s)}
C_r(s) \; : \; y^2 = f^-(x) = x h(x^2+2) + s
\end{equation}
and write $J_r(s)$ for its Jacobian.
For studying the relevant properties of $C_r(s)$ and $J_r(s)$ we
will need the following auxiliary lemmas.
Set~$H(x,z) = z^{r - 1}xh\left(\left(\frac{x}{z}\right)^2 + 2\right)\in\mathbb{Z}[x,z^2]$ so that~\(H(x,1) = xh(x^2 + 2)\), and denote by~$T_r$ the $r$-th Chebyshev polynomial of the first kind.
\begin{lemma} \label{L:firstKind}
We have that
\begin{equation*}
H(x,z) = 2 (iz)^r T_r \left( \frac{x}{2iz} \right).
\end{equation*}
In particular, we have
\begin{equation}\label{eq:cheby}
H(x,z) = \sum_{k=0}^{\frac{r - 1}{2}}c_kz^{2k}x^{r - 2k}, \quad\text{where }c_k=\frac{r}{r-k}{r-k \choose k} \in\mathbb{Z},
\end{equation}
and~$c_k$ is divisible by~$r$ for~$k>0$.
\end{lemma}
\begin{proof}
According to~\cite[Section 2.3.2]{Mason}, $T_r$ is explicitly given by the formula
$$
T_r(X)=\sum_{k=0}^{\frac{r - 1}{2}}(-1)^k2^{r-2k-1}c_k X^{r-2k},
$$
and we have the following identities for~$c_k$~:
\[
c_k = 2\binom{r - k}{k} - \binom{r - k - 1}{k} = \frac{1}{2^{r - 2k - 1}}\sum_{j = k}^{\frac{r - 1}{2}}\binom{r}{2j}\binom{j}{k}.
\]
This proves that~$c_k$ is an integer for all~$k\ge0$ and that $c_k\equiv 0\pmod{r}$ for~$k>0$. Note that $H(x,z)$ and $z^rT_r\left(\frac{x}{2iz}\right)$ have degree~$r$ with respect to~$x$. We claim they have the same set of~$r$ distinct complex zeros, hence are scalar multiples of each other, that is there exists a complex number~$\alpha$ such that
\[
H(x,z) = \alpha \cdot z^rT_r \left( \frac{x}{2iz} \right).
\]
Since $H$ is monic and the leading coefficient of
$z^rT_r(\frac{x}{2iz})$ is $1/2i^r$ we conclude
$\alpha = 2 i^r$, as desired.
To complete the proof we will now prove the claim.
Fix $\zeta_r^{1/2}$
a $2r$-th root of unity whose square is $\zeta_r$.
The zeros of $h(x)$
are $\omega_j$ for $j = 1, \ldots, \frac{r-1}{2}$, so that
the zeros of $H(x,z)$ as a polynomial in~$x$ satisfy $x=0$ or
\begin{align*}
\left(\frac{x}{z}\right)^2 = \omega_j - 2 = (\zeta_r^{j/2} - \zeta_r^{-j/2})^2
= \left( 2i \sin \frac{\pi j}{r} \right)^2
\iff x = \pm 2iz \sin \frac{\pi j}{r}
\end{align*}
which are $r-1$ distinct values for $j = 1, \ldots, \frac{r-1}{2}$. We now show that these values of~$x$
also satisfy $T_r \left( \frac{x}{2iz} \right) = 0$.
Indeed, recall the defining identity $T_r(\cos \theta) = \cos r \theta$
and compute
\[
T_r \left( \frac{\pm 2iz \sin \frac{\pi j}{r}}{2iz} \right) =
T_r \left( \cos \left( \frac{\pi}{2} - \frac{\pi (\pm j)}{r} \right) \right) =
\cos \left( \frac{r \pi}{2} - \frac{r \pi (\pm j)}{r} \right) =
\cos \left( \frac{r \pi}{2} - \pi (\pm j) \right) = 0,
\]
where for the first equality we use that $\sin$ is an odd function and for the last that $r$ is odd.
Finally, the equalities above also hold for $j=0$, showing that $x=0$ is also
a root of $z^rT_r \left( \frac{x}{2iz} \right)$.
\end{proof}
\begin{lemma}
\label{L:irreducible}
The polynomial $f^-(x) = x h(x^2+2) + s$ is irreducible in $\mathbb{Q}(s)[x]$.
\end{lemma}
\begin{proof}
Reducing~\(f^-(x)\) modulo $s$ gives the polynomial $x h(x^2+2)$. In the notation of Lemma~\ref{L:firstKind} (applied to~\(z = 1\)), we have
\[
h(x^2 + 2) = \sum_{k=0}^{\frac{r - 1}{2}}c_kx^{r - 2k - 1}\in\mathbb{Z}[x]
\]
with~\(c_0 = 1\), \(c_k\) divisible by~\(r\) for any~\(k\in\{1,\dots, \frac{r - 3}{2}\}\) and~\(c_{\frac{r - 1}{2}} = r\). By Eisenstein's criterion, we get that~\(h(x^2 + 2)\) is irreducible over~\(\mathbb{Q}\). Thus, if $f^-(x)$ is not irreducible in $\mathbb{Q}(s)[x]$, then it must factor into a linear factor and an irreducible factor of degree $r-1$. However, $f^-(x)$ has no roots in~$\mathbb{Q}(s)$.
\end{proof}
\begin{lemma}\label{L:secondKind}
The formal derivative of the polynomial $H(x,z) = z^{r - 1}xh\left(\left(\frac{x}{z}\right)^2 + 2\right)$ with res\-pect to the variable~$x$ is~$z^{r - 1}rh\left(\frac{ix}{z}\right)h\left(-\frac{ix}{z}\right)$.
\end{lemma}
\begin{proof}
Let $U_n(x)$ for $n \geq 1$ be the $n$th Chebyshev polynomial of the second kind.
Using Lemma~\ref{L:firstKind} and the
well known fact $\frac{d}{dx} T_n = n U_{n-1}$
we compute
\[
\frac{d}{dx} H(x,z) = 2 (iz)^r \frac{1}{2iz} \frac{d}{dx} T_r \left( \frac{x}{2iz} \right)
= (iz)^{r-1} r U_{r-1} \left( \frac{x}{2iz} \right).
\]
Next we will show that
\[
U_{r-1} \left( \frac{x}{2i} \right) = i^{r-1}h\left(\frac{ix}{z}\right)h\left(-\frac{ix}{z}\right)
\]
which implies the result.
Indeed,
it is well known that the $r-1$ zeros of $U_{r-1}$ are
$\cos \frac{k\pi}{r}$ for $k=1,\ldots,r-1$.
These are the same as $\pm \cos \frac{2\pi j}{r}$ for $j=1,\ldots,\frac{r-1}{2}$, hence $U_{r-1}\left(\frac{x}{2iz}\right)$ has zeros
$\pm 2iz \cos \frac{2\pi j}{r}$.
On the other hand, the zeros of $h(x)$
are $\omega_j = \zeta_r^j + \zeta_r^{-j} = 2\cos \frac{2\pi j}{r}$
for $j=1,\ldots,\frac{r-1}{2}$, which implies that
\[
U_{r-1}\left( \frac{x}{2iz} \right) = \alpha h\left(\frac{ix}{z}\right)h\left(-\frac{ix}{z}\right)
\]
for some constant~\(\alpha\), as both sides have the same set of $r-1$ distinct zeros. As
the leading coefficient of $U_{r-1}$ is~$2^{r -1}$
and~$h$ is monic, we have $\alpha = i^{r-1}$ as desired.
\end{proof}
\begin{lemma}
\label{evaluation-point}
For any~\(j = 0, \ldots, \frac{r-1}{2}\), we have
\[
H(iz\omega_j,z) = 2(iz)^r\quad\text{and}\quad H(-iz\omega_j,z) = -2(iz)^r.
\]
\end{lemma}
\begin{proof}
For $x = \pm i \omega_jz$ we have $\frac{x}{2iz} = \pm \cos \frac{2 \pi j}{r}$. Using the identity $T_r(\cos \theta) = \cos r \theta$
and the fact that $T_r(x)$ is an odd function (as $r$ is odd),
we compute
\[
T_r\left(\frac{x}{2iz}\right) = T_r \left( \pm \cos \frac{2 \pi j}{r} \right) = \pm \cos 2 \pi j = \pm 1.
\]
The conclusion now follows from Lemma~\ref{L:firstKind}.
\end{proof}
The proof of the following proposition uses the results of the previous lemmas. It will be serve in the study of Kraus' varieties in the next section.
\begin{proposition}\label{P:discriminant}
The discriminant of the polynomial $f^-(x)$ is
\[
\Delta(f^-) = (-1)^\frac{r-1}{2} r^r (s^2+4)^{\frac{r-1}{2}}.
\]
\end{proposition}
\begin{proof}
Let $\alpha \in \overline{\mathbb{Q}(s)}$ be a root of $f^-(x)$, which is irreducible over $\mathbb{Q}(s)$ by Lemma~\ref{L:irreducible}. It is well known that the discriminant of $f^-(x)$ is then given by
\begin{equation}\label{E:disc}
\Delta(f^-) = (-1)^\frac{r(r-1)}{2} N\left(\frac{d}{dx}f^- (\alpha)\right)
\end{equation}
where $N(\cdot)$ is the norm map from~$\mathbb{Q}(s)(\alpha)$ to $\mathbb{Q}(s)$.
Since $r$ is odd, the sets $\{ \omega_j \}$ and
$\{ \omega_{2j} \}$ where~$j$ ranges over
$j = 1, \ldots, \frac{r-1}{2}$ are equal. Moreover, we have
\begin{equation}\label{E:omega2j}
(\pm i \omega_j)^2 + 2 = - (\zeta_r^j + \zeta_r^{-j})^2 + 2 = - (\zeta_r^{2j} + \zeta_r^{-2j}) = - \omega_{2j},
\end{equation}
showing that the zeros of $h(-(x^2+2))$ are $\pm i \omega_j$ for $j = 1, \ldots, \frac{r-1}{2}$. These are also the zeros of
$h(ix) h(-ix)$ and, by comparing leading coefficients,
this gives the identity
\[
h(-(x^2+2)) = (-1)^\frac{r-1}{2} h(ix) h(-ix).
\]
From Lemma~\ref{L:secondKind} applied to~\(z = 1\), equation \eqref{E:disc} and the previous equality we
have that
\begin{align*}
\Delta(f^-) & = (-1)^\frac{r(r-1)}{2} N \left( \frac{d}{dx} f^-(\alpha) \right) =
(-1)^\frac{r(r-1)}{2} N(r h(i \alpha) h(-i\alpha)) \\
& = r^r N((-1)^\frac{(r-1)}{2} h(i \alpha) h(-i\alpha)) = r^r \prod_\beta h(-(\beta^2+2)),
\end{align*}
where $\beta$ ranges through the roots of $f^-(x) = x h(x^2+2) +s$.
To prove the final discriminant formula, we compute as follows
\begin{align*}
\prod_\beta h(-(\beta^2+2)) & = \prod_\beta \prod_{j=1}^{\frac{r-1}{2}} (-\beta^2 -2 - \omega_j) = \prod_{j = 1}^{\frac{r-1}{2}} \prod_\beta (-\beta^2-2 - \omega_{2j}) \quad \text{(as $r$ is odd)} \\
& = \prod_{j = 1}^{\frac{r-1}{2}} \prod_\beta (-\beta^2 - \omega_j^2) = (-1)^\frac{r(r-1)}{2} \prod_{j = 1}^{\frac{r-1}{2}}
f^-(i\omega_j)f^-(-i\omega_j) \quad \text{by } \eqref{E:omega2j} \\
& = (-1)^\frac{r(r-1)}{2} \prod_{j = 1}^{\frac{r-1}{2}} ((i \omega_j) h( (i \omega_j)^2+2) + s) ((-i \omega_j) h( (-i \omega_j)^2+2) + s) \\
& = (-1)^\frac{r(r-1)}{2} (s+2i)^\frac{r-1}{2} (s-2i)^\frac{r-1}{2} \quad \text{by Lemma~\ref{evaluation-point}} \\
& = (-1)^\frac{r-1}{2} (s^2+4)^\frac{r-1}{2} \quad \text{(as $r$ is odd)}.
\end{align*}
\end{proof}
\begin{lemma}
\label{cyclo-relation}
We have the following identity of polynomials:
\begin{equation*}
X^{2r}-1 = X^r (X-1/X) h(X^2+X^{-2}).
\end{equation*}
\end{lemma}
\begin{proof}
Note that
\[ X^r + Y^r =
(X+Y)\prod_{j=1}^\frac{r-1}{2} (X^2 + \omega_j XY + Y^2)
\]
and evaluating at $(X^2,-1)$ leads to
\begin{align*}
X^{2r} - 1 & =
(X^2-1) \prod_{j=1}^\frac{r-1}{2} (X^4 - \omega_j X^2 + 1)
= (X^2-1) X^{r-1}\prod_{j=1}^\frac{r-1}{2} (X^2 - \omega_j + X^{-2}) \\
& = (X^{r+1} - X^{r-1}) h(X^2 + X^{-2}) = X^r (X-1/X) h(X^2+X^{-2}).
\end{align*}
\end{proof}
\begin{proposition}
\label{quotient-tau}
The map~\(\pi : (X,Y)\mapsto \left(X - X^{-1}, Y / X^{\frac{r+1}{2}}\right)\) identifies the curve~\(C_r(s)\) with the quotient of the hyperelliptic curve
\begin{equation*}
D_r(s) \; : \; Y^2 = X (X^{2r}+s X^r - 1)
\end{equation*}
by the involution $\tau: (X,Y) \mapsto (-1/X,(-1)^\frac{r+1}{2} Y/X^{r+1})$.
\end{proposition}
\begin{proof}
Note that~\(\pi(\tau(X,Y)) = \pi(X,Y)\). Suppose that~\((X,Y)\) satisfies~\(Y^2 = X (X^{2r}+s X^r - 1)\). Set~$x = X - X^{-1}$ and $y = Y/X^{\frac{r+1}{2}}$. We have that
\begin{align*}
x h(x^2 + 2) + s & = X^r - X^{-r} + s \\
& = X^{-r} (X^{2r} + s X^{r} - 1) \\
& = X^{-r} (Y^2/X) = (Y/X^{\frac{r+1}{2}})^2 = y^2,
\end{align*}
where the first equality follows from Lemma~\ref{cyclo-relation}. This shows that $\pi$ factors via the
quotient $D_r(s)/\langle \tau \rangle$.
Comparing degrees in the respective function fields, we obtain the result.
\end{proof}
\begin{theorem}
\label{endo-construct}
There is an embedding
\begin{equation*}
K \hookrightarrow \End_{K(s)} (J_r(s)) \otimes \mathbb{Q}.
\end{equation*}
Furthermore, when $s$ is specialized to any element $s_0 \in K$, the above embedding is well-defined.
In particular, $J_r(s_0)/K$ is of $\operatorname{GL}_2$-type with real multiplications by~$K$.
\label{T:GL2type}
\end{theorem}
\begin{proof}
Let $D = D_r(s)$, $C = C_r(s)$, $\tau : D \to D$ and $\pi : D \rightarrow C$ be the curves and maps in Proposition~\ref{quotient-tau}.
There is an injection $\pi^* : H^0(C,\Omega^1) \rightarrow H^0(D,\Omega^1)$ given by pull back, and a projection $\pi_*: H^0(D,\Omega^1) \rightarrow H^0(C,\Omega^1)$ given by the trace map \cite[p.221]{Diamond-Shurman}. The subspace $H^0(C,\Omega^1)$ can thus be described as the $\tau$-invariant subspace of $H^0(D,\Omega^1)$.
Consider the following automorphism of the hyperelliptic curve $D$
\begin{equation}
\label{endomorphism-construction}
[\zeta_r] : (X,Y) \mapsto (\zeta_r X, \zeta_r^\frac{r+1}{2} Y).
\end{equation}
A basis for the $\tau$-invariant regular differentials on$~D$ is
given by
\begin{equation}
\label{basis-differential}
w_j = (X^j + (-1)^{\frac{r+1}{2}-j} X^{r-1-j}) \frac{dX}{Y}
\end{equation}
where $j = 0, \ldots, \frac{r-3}{2}$. We check that:
\begin{equation}
\label{endomorphism}
\left([\zeta_r]^* + [{\zeta_r^{-1}}]^*\right) w_j = \left(\zeta_r^{\frac{r+1}{2}-j-1}+{\bar \zeta_r}^{\frac{r+1}{2}-j-1}\right) w_j.
\end{equation}
Let $J_D$ and $J_C$ be the Jacobians of $D$ and $C$, respectively. The homomorphism
\begin{equation}
\label{picard}
J_C \rightarrow J_D
\end{equation}
induced by $\pi$ using Picard functoriality has finite kernel.
Note that $\psi = [\zeta_r]^* + [\zeta_r^{-1}]^*$ as a map from $J_D$ to $J_D$ is defined over $K(s)$ by \eqref{endomorphism}. We wish to show $\psi$ stabilizes the image of $J_C$ in $J_D$. It suffices to verify this on complex points for every specialization $s \in \mathbb{P}^1(\mathbb{C}) \backslash \left\{ \pm 2i, \infty \right\}$. On complex points, the homomorphism in \eqref{picard} corresponds to the homomorphism
\begin{equation}
H^0(C,\Omega^1)^*/\Lambda_C \rightarrow H^0(D,\Omega^1)^*/\Lambda_D,
\end{equation}
where $\Lambda_C$ and $\Lambda_D$ are the images of $H_1(C,\mathbb{Z})$ and $H_1(D,\mathbb{Z})$ under integration.
By \eqref{endomorphism}, the map induced by $\psi = [\zeta_r]^* + [{\zeta_r^{-1}}]^*$ stabilizes $H^0(C,\Omega^1)^*$ inside $H^0(D,\Omega^1)^*$ and the lattice $\Lambda_D \cap H^0(C,\Omega^1)^*$, which contains $\Lambda_C$ with finite index. It follows that $\End_{K(s)}(J_C) \otimes \mathbb{Q}$ contains $K$ since isogenous abelian varieties have isomorphic endomorphism algebras.
\end{proof}
According to Theorem~\ref{endo-construct} and the construction recalled after Definition~\ref{def:GL2typeAV}, one considers for every prime ideal~$\mathfrak{p}$ of~$K$ above a prime~$p$ the Galois representation
\[
\rho_{J_r(s),\mathfrak{p}} \; \colon \; G_{K(s)}\longrightarrow\operatorname{GL}(V_\mathfrak{p}(J_r(s)))\simeq\operatorname{GL}_2(K_\mathfrak{p})
\]
and its reduction
\[
{\overline{\rho}}_{J_r(s),\mathfrak{p}} \; \colon \; G_{K(s)}\longrightarrow\operatorname{GL}_2(\mathbb{F}_\mathfrak{p})
\]
where~$\mathbb{F}_\mathfrak{p}$ is the residue field of~$K_\mathfrak{p}$.
\begin{proposition}
\label{darmonfrey}
Let $\mathfrak{p}$ be a prime ideal in $K$ above a prime~$p$. Then the representation~${\overline{\rho}}_{J_r(s),\mathfrak{p}}$ is a Frey representation of signature $(p,p,r)$ with respect to the points $(-2i,2i,\infty)$ in the sense of Definition~\ref{D:FreyRep}.
\end{proposition}
\begin{proof}
Let $C_r^-(u)$ be the hyperelliptic curve given by the equation
\begin{equation}\label{E:Cminus}
y^2 = x g(x^2-2) + u,
\end{equation}
where
\begin{equation*}
g(x) = \prod_{j=1}^\frac{r-1}{2} (x+\omega_j).
\end{equation*}
Note that \eqref{E:Cminus} with $u=2-4t$ defines
the curve $C_r^-(t)$ in \cite[p. 420]{DarmonDuke}.
Let $J_r^-(u)$ denote the Jacobian of $C_r^-(u)$. Then ${\overline{\rho}}_{J_r^-(u),\mathfrak{p}}$ is an odd Frey representation of signature $(p,p,r)$ with respect to the points $(-2,2,\infty)$ by the proof of \cite[Theorem 1.10]{DarmonDuke}.
Replacing $x$ by $x/i$, $y$ by $y/i^\frac{1}{2}$ in \eqref{E:Cminus} gives the equation
\begin{equation*}
y^2 = x g(-x^2-2) + i u.
\end{equation*}
Since $g(-X) = (-1)^\frac{r-1}{2} h(X)$, we obtain the equation
\begin{equation*}
y^2 = (-1)^\frac{r-1}{2} x h(x^2+2) + i u.
\end{equation*}
Multiplying by $(-1)^\frac{r-1}{2}$ and setting $s = (-1)^\frac{r-1}{2} i u $ leads to the model
\[
C_r^0 (s) \; : \; (-1)^\frac{r-1}{2} y^2 = x h(x^2+2) + s = f^-(x)
\]
which is the equation for $C_r(s)$ when $\frac{r-1}{2}$ is even and its quadratic twist
by $-1$ when $\frac{r-1}{2}$ is odd. Write $J_r^0(s)$ for the Jacobian of
$C_{r}^0(s)$.
Then,
\begin{equation*}
{\overline{\rho}}_{J_r(s),\mathfrak{p}} \mid_{G_{K(\sqrt{i})(s)}} \cong {\overline{\rho}}_{J_r^0(s),\mathfrak{p}} \mid_{G_{K(\sqrt{i})(s)}} \cong {\overline{\rho}}_{J_r^-(u),\mathfrak{p}} \mid_{G_{K(\sqrt{i})(u)}}.
\end{equation*}
By \eqref{E:C(s)} and Proposition~\ref{P:discriminant}, the curve $C_r(s)$ has good reduction outside $\{\pm 2i, \infty\}$. The conclusion follows by noting that the points $\{\pm 2,\infty\}$ are mapped to $\{\pm 2i, \infty \}$ by $s = (-1)^{\frac{r-1}{2}} i u$, and the order of an inertia subgroup as mapped to $\operatorname{PSL}_2(\mathbb{F}_\mathfrak{p})$ is unaffected by restriction to $G_{K(\sqrt{i})(u)}$, since the extension $K(\sqrt{i})(u)/K(u)$ is unramified. Furthermore, applying Lemma~\ref{frey-irreducible} gives that ${\overline{\rho}}_{J_r(s),\mathfrak{p}} \mid_{G_{\overline{K}(s)}}$ is irreducible.
\end{proof}
Let $t \neq s$ be another variable. Let $K(s,t)$ be the function field where $s$ and $t$ are related by
\begin{equation} \label{E:stRelation}
\frac{1}{s^2+4} = t (1-t).
\end{equation}
We can see both $K(s)$ and $K(t)$ as subfields of $K(s,t)$. Define~$\alpha$ by the following equation
\begin{equation}
\label{eq:sign_for_alpha}
\alpha s = 2t-1
\end{equation}
and note, from the relation \eqref{E:stRelation}, that~$\alpha$ is a square-root of~$t - t^2$. The following lemma uses the notation of Lemma~\ref{L:firstKind}.
\begin{lemma}
\label{model-t}
The quadratic twist by~$\alpha$ of the base change of $C_r(s)$ to $K(s,t)$ has a model
\begin{equation*}
C'_r(t) \; : \; y^2 = x^r + c_1 \alpha^2 x^{r-2} + \ldots + c_{\frac{r-1}{2}} \alpha^{r-1} x + \alpha^{r-1} (2t - 1)
\end{equation*}
defined over $K(t)$. Moreover, the model $C'_r(t)$ has ramification set equal to~$\{0, 1, \infty \}$.
\end{lemma}
\begin{proof}
Because $f^-(x) = H(x,1) + s = x^r + c_1 x^{r-2} + \ldots + c_{\frac{r-1}{2}} x + s$ has only odd powers of $x$ (except for the constant term), we may apply the transformation
\( x \rightarrow \frac{x}{\alpha} \), \( y \rightarrow \frac{y}{\alpha^\frac{r}{2}} \) to obtain, using~\eqref{eq:sign_for_alpha}, the model~\(C'_r(t)\) of the quadratic twist by~\(\alpha\) of~\(C_r(s)\) over $K(t)$ given in the statement. Finally, from~\eqref{E:stRelation} we see that the
ramification set $\{\pm 2i, \infty \}$ of $C_r(s)$
corresponds to the ramification set
$\{ 0, 1, \infty \}$ for $C'_r(t)$.
\end{proof}
Combining Proposition~\ref{P:discriminant} and Lemma~\ref{model-t} with standard formulae for models of hyperelliptic curves (see Section~\ref{ss:hyperelliptic_equations} for a summary)
it follows that~$C_r'(t)$ has discriminant
\begin{equation}\label{eq:disc_C_r_dash}
\Delta(C_r'(t)) = (-1)^{\frac{r - 1}{2}}2^{2(r - 1)}r^r(t(1 - t))^{\frac{(r - 1)^2}{2}}.
\end{equation}
\begin{theorem}
\label{hyperelliptic-darmon}
Let $J'_r(t)$ be the Jacobian of $C'_r(t)$ from Lemma~\ref{model-t}. Then
\begin{enumerate}
\item There is an embedding $K \hookrightarrow \End_{K(t)}(J'_r(t)) \otimes \mathbb{Q}$.
\item For every specialization of $t$ to $t_0 \in \mathbb{P}^1(K) \backslash \left\{ 0, 1, \infty \right\}$, the above embedding is well-defined.
\item For every prime $\mathfrak{p}$ of $K$ above a prime $p$, ${\overline{\rho}}_{J'_r(t),\mathfrak{p}} : G_{K(t)} \rightarrow \operatorname{GL}_2(\mathbb{F}_\mathfrak{p})$ is an odd Frey representation of signature $(r,r,p)$ with respect to the points $(0,1,\infty)$.
\end{enumerate}
\end{theorem}
\begin{proof}
Let $J = J_r(s)$ be the Jacobian of $C = C_r(s)$. Similarly, let~$J'=J'_r(t)$ be the Jacobian of~$C'=C'_r(t)$. By Proposition~\ref{darmonfrey}, the endomorphism algebra $\End_{K(s)}(J) \otimes \mathbb{Q}$ contains $K$ and ${\overline{\rho}}_{J,\mathfrak{p}} : G_{K(s)} \rightarrow \operatorname{GL}_2(\mathbb{F}_\mathfrak{p})$ is a Frey representation of signature
$(p, p, r)$ with respect to the points $(-2i, 2i, \infty)$ for every prime~$\mathfrak{p}$ of $K$ above $p$.
Consider $C$ over $K(s,t)$ which is a degree $2$ extension of $K(t)$ and $K(s)$. Thus, we can identify $\overline{K(s)} = \overline{K(s,t)} = \overline{K(t)}$, and we denote this common field by $\overline{L}$.
Let~$\phi : C' \rightarrow C$ be the isomorphism $(x,y)\mapsto\left(\frac{x}{\alpha},\frac{y}{\alpha^{r/2}}\right)$ given by Lemma~\ref{model-t}. It induces homomorphisms defined over $\overline{L}$
\begin{align*}
\phi^* : J \rightarrow J' \\
\phi_* : J' \rightarrow J
\end{align*}
by Picard and Albanese functoriality respectively. Note that $\phi^*$ and $\phi_*$ are isomorphisms defined over $\overline{L}$ and each is the inverse of the other.
If we have an endomorphism $\psi : J \rightarrow J$, we obtain an endomorphism $\psi' : J' \rightarrow J'$ given by
\begin{equation}
\label{twist-endomorphism}
\psi' = \phi^* \circ \psi \circ \phi_* : J' \rightarrow J',
\end{equation}
using the identification of $J$ and $J'$ under $\phi^*$ and $\phi_*$.
To decide if $\psi'$ is defined over $K(t)$, we may check if the following holds:
\begin{equation}
\label{commute-property}
\psi'^\sigma = \psi',\quad \text{for all $\sigma \in G_{K(t)}$}.
\end{equation}
For $\sigma \in G_{K(t)}$, set $\xi_\sigma = \phi^\sigma \phi^{-1} : C \rightarrow C^\sigma$, which is explicitly given by
\begin{equation}
\label{cocycle}
(x,y) \mapsto \left( x \frac{\alpha}{\alpha^\sigma}, y \frac{\alpha^\frac{r}{2}}{{\alpha^\frac{r}{2}}^\sigma} \right).
\end{equation}
Then, we have
\begin{align}
\notag \psi'^\sigma & = \psi' \\
\notag \iff (\phi^* \circ \psi \circ \phi_*)^\sigma & = \phi^* \circ \psi \circ \phi_* \\
\notag \iff {\phi^*}^{\sigma} \circ \psi^\sigma \circ \phi_*^\sigma & = \phi^* \circ \psi \circ \phi_* \\
\notag \iff \psi^\sigma \circ \phi_*^\sigma \phi_*^{-1} & = ({\phi^*}^{\sigma})^{-1}\phi^* \circ \psi \\
\iff \psi^\sigma \circ {\xi_\sigma}_* & = {\xi_\sigma}_* \circ \psi,
\end{align}
where we have used that $\phi^* = \phi_*^{-1}$ and $({\phi^*}^\sigma)^{-1} = \phi_*^\sigma$.
Let $\psi = [\zeta_r + \zeta_r^{-1}] := [\zeta_r]^* + [\zeta_r^{-1}]^* \in \End_{K(s)}(J) \otimes \mathbb{Q}$ and $\psi'$ the corresponding element in $\End_{K(t)}(J')$ from \eqref{twist-endomorphism}. To verify \eqref{commute-property} holds for all $\sigma \in G_{K(t)}$, it suffices to check that it holds on complex specializations of $t \notin \left\{ 0, 1, \infty \right\}$. Recall that
\begin{equation}
J(\mathbb{C}) \cong H^0(C,\Omega^1)^*/\Lambda_C
\end{equation}
where $\Lambda_C$ is the image of $H_1(C,\mathbb{Z})$ in $H^0(C,\Omega^1)^*$ under the map $\gamma \mapsto \left( w \mapsto \int_\gamma w \right)$. From \eqref{cocycle} and \eqref{endomorphism}, we can verify that the commutation relation \eqref{commute-property} holds when considered as an equality in $\Hom(J(\mathbb{C}),J^\sigma(\mathbb{C}))$ by checking it holds on differentials.
Finally remark that $t = 0, 1$ correspond to $s = \infty$, and $t = \infty$ corresponds to $s = \pm 2i$. Using this correspondence of points, it follows that ${\overline{\rho}}_{J',\mathfrak{p}}$ is a Frey representation of signature $(r,r,p)$ with respect to the points $(0, 1, \infty)$ for every prime $\mathfrak{p}$ of $K$ above $p$. Irreducibility of ${\overline{\rho}}_{J',\mathfrak{p}} \mid_{G_{\bar K(t)}}$ follows from Lemma~\ref{frey-irreducible}.
Furthermore, since the inertia subgroups at $t = 0, 1, \infty$ are generated by an element of order $2r, 2r, 2p$, respectively, it follows that ${\overline{\rho}}_{J',\mathfrak{p}}$ is an odd Frey representation.
\end{proof}
\begin{remark}
In \cite{DarmonDuke}, a superelliptic construction of Frey representations of signature $(r,r,p)$ with respect to the points $(0, 1, \infty)$ is given. Theorem~\ref{hyperelliptic-darmon} shows that there is a hyperelliptic construction (using $C'_r(t)$) of odd Frey representations of signature $(r,r,p)$ with respect to the points $(0, 1, \infty)$.
\end{remark}
\section{Kraus' Frey hyperelliptic curves for signature~$(r,r,p)$}\label{S:Freyrrp}
In this section, we introduce Kraus' Frey varieties
as in~\cite[\S V]{kraushyper} and study their first properties.
In particular, we relate them to the
varieties in \S\ref{S:FreyOverKs}. This allows to attach \(2\)-dimensional Galois representations to them. The study of these associated Galois representations will occupy the rest of this Part~\ref{Part:higherdimFAV} leading in particular to the proof of the modularity results in the next section.
We keep the notation of Section~\ref{S:FreyOverKs}. In particular,
$K= \mathbb{Q}(\zeta_r)^+$ is the maximal totally real subfield of~$\mathbb{Q}(\zeta_r)$ where $\zeta_r$ is a fixed primitive $r$-th root of
unity for~$r\ge3$ a prime.
For $a,b \in \mathbb{Z}$ coprime and such that~$a^r+b^r\not=0$, we
let $C_r(a,b)$ be the hyperelliptic curve of genus~$(r-1)/2$ constructed by Kraus (denoted $C$ in~\cite[\S V]{kraushyper}). Namely, for $ab \neq 0$, we have
\begin{equation}
\label{kraushyper}
C_r(a,b) \; : \; \; y^2 = (ab)^\frac{r-1}{2} x h \left(\frac{x^2}{ab} + 2 \right) + b^r - a^r
\end{equation}
where~$h$ is defined in~\eqref{eq:def_h}.
For $ab =0$ comprimality gives $(a,b)=\pm(0,1)$ or~$\pm(1,0)$ and we set
\begin{equation}\label{E:Jab=0}
C_r(a,b) \; : \; y^2=x^r+b^r-a^r.
\end{equation}
\begin{proposition}
\label{P:Crba}
The curve $C_r(b,a)$ is the quadratic twist by~$-1$ of~$C_r(a,b)$.
\end{proposition}
\begin{proof}This is clear for~\(ab = 0\) and follows from
$C_r(b,a) : y^2 = (ab)^\frac{r-1}{2} x h \left(\frac{x^2}{ab} + 2 \right) - (b^r - a^r)$ when~\(ab\neq 0\) .
\end{proof}
In the next lemma we show that for~\(ab\neq 0\), the hyperelliptic curve~$C_r(a,b)$ is related to a specialization of the model $C'_r(t)$ given
by Lemma~\ref{model-t}. We first we need further notation.
Let~$z_0$ be a fixed square-root of~$ab \neq 0$. We specialize the variables $s$ and~$t$ in \S\ref{S:FreyRep} to be $s = s_0$ and $t = t_0$ where
\begin{equation}
\label{st-identity}
t_0 = \frac{a^r}{a^r+b^r} \quad
\text{ and } \quad
s_0 = \frac{b^r - a^r}{z_0^r}.
\end{equation}
Observe that~\(t_0\in\mathbb{P}^1(\mathbb{Q})\backslash\{0,1,\infty\}\) and that the following relations are satisfied
\begin{equation*}
t_0(1-t_0) = \frac{(a b)^r}{(a^r+b^r)^2}\quad\text{ and }\quad s_0^2 = \frac{(a^r - b^r)^2}{(ab)^r} = \frac{1}{t_0(1-t_0)} - 4.
\end{equation*}
\begin{lemma} \label{L:twistedKraus}
Assume $ab \neq 0$.
The curve~$C'_r(t_0)$ is the quadratic twist of $C_r(a,b)$ by $-\frac{(ab)^\frac{r-1}{2}}{a^r+b^r}$.
\end{lemma}
\begin{proof}
Note that $C_r(a,b)$ can be obtained from $C_r(s_0)$ by replacing $x$ by $\frac{x}{z_0}$ and $y$ by~$\frac{y}{z_0^{r/2}}$, showing that $C_r(a,b)$ is the quadratic twist of $C_r(s_0)$ by~$z_0$. Also, by Lemma~\ref{model-t} and \eqref{st-identity}, $C'_r(t_0)$ is the quadratic twist of $C_r(s_0)$ by $\alpha$ where~$\alpha = \frac{2t_0 - 1}{s_0} = -\frac{z_0^r}{a^r + b^r}$ is given by equation~\eqref{eq:sign_for_alpha}.
Therefore, by composing both twists, the curve~$C'_r(t_0)$ is the quadratic twist of $C_r(a,b)$ by
\[
\frac{\alpha}{z_0} = - \frac{z_0^{r - 1}}{a^r + b^r} = - \frac{(ab)^{\frac{r-1}{2}}}{a^r+b^r}.
\]
\end{proof}
Combining~\eqref{eq:disc_C_r_dash} and Lemma~\ref{L:twistedKraus} with standard formulae for models of hyperelliptic curves (see Section~\ref{ss:hyperelliptic_equations} for a summary) it follows that~$C_r(a,b)$ has discriminant
\begin{equation}\label{E:discriminant}
\Delta(C_r(a,b)) = (-1)^\frac{r-1}{2} 2^{2(r-1)} r^r (a^r+b^r)^{r-1}.
\end{equation}
Note that this latter formula holds in the case~$ab = 0$ as well, with~$C_r(a,b)$ defined as in~\eqref{E:Jab=0}. We write $J_r(a,b)$ for the Jacobian of~$C_r(a,b)$.
\begin{remark}\label{rk:JisCM}
A nice feature of these Frey varieties is that when~\(ab = 0\), then from equation \eqref{E:Jab=0}, we have an endomorphism $(x,y) \mapsto (\zeta_r x, y)$ on~\(C_r(a,b)\) and hence~$J_r(a,b)$ has CM by~$\mathbb{Q}(\zeta_r)$.
\end{remark}
The next result implies that there are $2$-dimensional Galois representations attached to~$J_r$ which is a key property for the detailed study of these abelian varieties done below.
\begin{theorem} \label{T:GL2typeJr}
Assume~\(ab \neq 0\). The base change~$J_r$ of~$J_r(a,b)$ to~$K$ is of $\operatorname{GL}_2$-type. More precisely, there is an embedding
$$K \hookrightarrow \End_K(J_r) \otimes \mathbb{Q}.$$
\end{theorem}
\begin{proof}
This follows from Theorem~\ref{hyperelliptic-darmon} and Lemma~\ref{L:twistedKraus}.
\end{proof}
For $\lambda$ a prime ideal of~$K$ we write~$K_\lambda$ for the corresponding completion of~$K$. Following the construction described after Definition~\ref{def:GL2typeAV}, we denote by
\begin{equation}\label{E:pAdicRepJ}
\rho_{J_r,\lambda} \; \colon \; G_K\longrightarrow\operatorname{GL}_2(K_\lambda)
\end{equation}
the $2$-dimensional $\lambda$-adic Galois representation attached to~$J_r/K$ and by
\[
{\overline{\rho}}_{J_r,\lambda} \; \colon G_K \; \longrightarrow\operatorname{GL}_2(\mathbb{F}_\lambda),
\]
a reduction modulo~$\lambda$ of $\rho_{J_r,\lambda}$, where~$\mathbb{F}_\lambda$ denotes the residue field of~$K_\lambda$ (see~\cite[I.9]{RibetGalois}).
When there exists $c \in \mathbb{Z}$ such that $(a,b,c)$ is a primitive solution to~\eqref{E:rrp}, then~\eqref{E:discriminant} reads
\begin{equation}
\label{Kraus-discriminant}
\Delta(C_r(a,b)) = (-1)^\frac{r-1}{2} 2^{2(r-1)} r^r d^{r-1} c^{p(r-1)}
\end{equation}
and we say that $J_r(a,b)$ is {\em the Frey
variety attached to~$(a,b,c)$}.
We conclude this section with examples of~$C_r(a,b)$ for small values of~$r$:
\begin{align*}
r = 3: & \quad y^2 = x^3 + 3ab x + b^3-a^3, \\
r = 5: & \quad y^2 = x^5 + 5 ab x^3 + 5 a^2 b^2 x + b^5-a^5, \\
r = 7: & \quad y^2 = x^7 + 7 ab x^5 + 14 a^2 b^2 x^3 + 7 a^3 b^3 x + b^7 - a^7, \\
r = 11: & \quad y^2 = x^{11} + 11 ab x^9 + 44 a^2 b^2 x^7 + 77 a^3 b^3 x^5 + 55 a^4 b^4 x^3 + 11 a^5 b^5 x + b^{11} - a^{11}.
\end{align*}
Observe that $r=3$ is the only case where the Frey variety is an elliptic curve; this is a well known curve, which has been used to study the equation $x^3 + y^3 = z^p$ in~\cite{ChenSiksek,Freitas33p,kraus1}.
The curve for $r=7$ will be essential in the proof of Theorem~\ref{T:main} given in Section~\ref{S:Frey7}.
\section{Modularity of Frey varieties}
\label{S:modularity}
As we shall see in \S\ref{S:modularityKraus}, the modularity of~$J_r$ follows by analyzing its mod~$\mathfrak{p}_r$ representation, comparing it to the one arising on the Legendre family and modularity lifting. This comparison to the Legendre family is a key idea in Darmon's program. Indeed, conjecturally the mod~$\mathfrak{p}_r$ representation is modular and plays the role of a `seed' for modularity of all Frey varieties described by Darmon (see diagram in~\cite[p. 433]{DarmonDuke}). The results in this section make this argument unconditional for the Kraus Frey variety~$J_r$ and two of the
varieties introduced by Darmon as well (see \S\ref{S:modularityDarmon}).
\subsection{Modularity of Kraus' varieties $J_r$}\label{S:modularityKraus}
In this paragraph, we keep the notation of Section~\ref{S:Freyrrp}. In particular, $a,b$ are non-zero coprime integers such that~$a^r + b^r \not=0$ and we set~\(t_0 = \frac{a^r}{a^r + b^r}\). Recall that~$J_r$ denotes the base change of~$J_r(a,b)$ to~$K$, where $K=\mathbb{Q}(\zeta_r +\zeta_r^{-1})$. Let $\mathfrak{p}_r$ be the unique prime in~$K$ above~$r$. Since~$r$ is totally ramified in~$K$, its residue field
is $\mathbb{F}_{\mathfrak{p}_r} = \mathbb{F}_r$. In the next sections, the prime~\(\mathfrak{p}_r\) will often be denoted by~\(\mathfrak{q}_r\), but here we use a different notation to emphasize the fact that we think of~\(K\) not as the base field of~\(J_r\) but instead as the field of real multiplication.
As~$\lambda$ varies, the representations $\rho_{J_r,\lambda}$ in~\eqref{E:pAdicRepJ} form a compatible system of Galois representations (\cite[Theorem~(2.1.2)]{RibetGalois}). So $J_r/K$ is modular if and only if one of them is modular if and only if all of them are modular and, in such case, they coincide with the strictly compatible system arising in a Hilbert modular cuspform defined over~$K$ of parallel weight~$2$ and trivial character.
The abelian variety $J_r$ is the base change of~\(J_r(a,b)\) which is defined over~$\mathbb{Q}$. Therefore, for any prime~\(p\), there is a semi-linear action of $G_\mathbb{Q}$ on~$V_p(J_r)$ which extends the $G_K$-action provided by Theorem~\ref{T:GL2typeJr} and satisfies
\begin{equation}
\label{semi-linear-action}
(\alpha v)^\sigma = \alpha^\sigma v^\sigma,
\end{equation}
for $\sigma \in G_\mathbb{Q}$, $v \in V_p(J_r)$, and $\alpha \in K \otimes \mathbb{Q}_p$, where $V_p(J_r)$ is the $p$-adic Tate module of~$J_r$.
We first establish properties of the mod~\(\mathfrak{p}_r\) representation ${\overline{\rho}}_{J_r, \mathfrak{p}_{r}} : G_K \rightarrow \operatorname{GL}_2(\mathbb{F}_r)$ that will be used in the proof of the next two results.
\begin{theorem}\label{T:FreyRep}
The representation ${\overline{\rho}}_{J_r, \mathfrak{p}_{r}}$ is isomorphic to a twist of~${\overline{\rho}}_{L(t_0),r}$, where $L(t)$ is the Legendre family $y^2 = x (x-1) (x-t)$.
Moreover, the representation~${\overline{\rho}}_{J_r, \mathfrak{p}_{r}}$ extends to~$G_\mathbb{Q}$.
\end{theorem}
\begin{proof}
From Theorem~\ref{hyperelliptic-darmon}, we know that
${\overline{\rho}}_{J'_r(t),\mathfrak{p}_r}$ is a Frey representation over~$K$ of signature~$(r,r,r)$ with respect to the points $(0,1,\infty)$. Writing $G_{\mathbb{Q}(t_0)}$ and~\(G_{K(t_0)}\) for $G_\mathbb{Q}$ and~\(G_K\) respectively to make clear that the actions involved are obtained by specialization, it follows from \cite[Theorem 1.5 and~\S1.3]{DarmonDuke} and specialization at~$t=t_0$ that, as representations of $G_{K(t_0)}$, we have
\begin{equation}\label{eq:Legendre_tensor_eps}
{\overline{\rho}}_{J'_r(t_0),\mathfrak{p}_{r}} \cong
{\overline{\rho}}_{L(t_0),r} \otimes \epsilon
\end{equation}
where $\epsilon : G_{K(t_0)}\rightarrow{\overline{\F}}_r^\times$ is a character; both $J'_r(t_0)$ and $L(t_0)$ are non-singular as $t_0 \neq 0, 1, \infty$.
The same conclusion holds for ${\overline{\rho}}_{J_r,\mathfrak{p}_r}$ by Lemma~\ref{L:twistedKraus}, completing the proof of the first assertion.
Because~$r$ is totally ramified in $K$, the action, given (for~\(r = p\)) by~\eqref{semi-linear-action}, of $G_{\mathbb{Q}(t_0)}$ on $V_r(J_r) \otimes_{\mathcal{O}_{\mathfrak{p}_r}} \mathbb{F}_r$, where the tensor product is taken with respect to the reduction map $\mathcal{O}_{\mathfrak{p}_r} \rightarrow \mathbb{F}_r$, is linear and restricts to the action of $G_{K(t_0)}$ given by
${\overline{\rho}}_{J_r,\mathfrak{p}_r}$. This achieves the proof of the second statement.
\end{proof}
\begin{remark}
\label{eps-order-two}
We will show in Lemma~\ref{lem:det} that, in particular, $\det {\overline{\rho}}_{J_r,\mathfrak{p}_r} = \det {\overline{\rho}}_{L(t_0),r} = \chi_r$ is the mod~\(r\) cyclotomic character. Hence, it follows that $\epsilon$ from~\eqref{eq:Legendre_tensor_eps} has order dividing $2$. If ${\overline{\rho}}_{J,\mathfrak{p}_r}$ and ${\overline{\rho}}_{L(t_0),r}$ are unramified at $\mathfrak{q}$, then $\epsilon$ is unramified at $\mathfrak{q}$.
\end{remark}
The proof of the statement below uses the results from Appendix~\ref{app:Filip}.
\begin{theorem}
\label{T:irred7}
Assume~\(r\geq5\). The representation ${\overline{\rho}}_{J_r,\mathfrak{p}_r}$ is absolutely irreducible when restricted to~\(G_{\mathbb{Q}(\zeta_r)}\).
\end{theorem}
\begin{proof}
From Theorem~\ref{T:FreyRep} we know that ${\overline{\rho}}_{J_r,\mathfrak{p}_r}$ satisfies
\begin{equation}\label{E:twistLegendre}
{\overline{\rho}}_{J_r,\mathfrak{p}_r} \simeq {\overline{\rho}}_{L,r} \otimes \chi
\end{equation}
as $G_K$-representations, where $\chi : G_K \to {\overline{\F}}_r^\times$ is a character and $L=L(t_0)$, where $t_0$ is given by~\eqref{st-identity}.
Note that~$L/\mathbb{Q}$ is an elliptic curve (it is non-singular as $ab(a^r+b^r) \not= 0$)
with full $2$-torsion over~$\mathbb{Q}$, as it belongs to the Legendre family.
By~\eqref{E:twistLegendre}, the representation~\({\overline{\rho}}_{J_r,\mathfrak{p}_r}|_{G_{\mathbb{Q}(\zeta_r)}}\) is absolutely irreducible if and only if~\({\overline{\rho}}_{L,r}|_{G_{\mathbb{Q}(\zeta_r)}}\) is absolutely irreducible. We note that we have~\({\overline{\rho}}_{L,r}(G_{\mathbb{Q}(\zeta_r)}) = {\overline{\rho}}_{L,r}(G_\mathbb{Q})\cap\operatorname{SL}_2(\mathbb{F}_r)\).
For~\(L\) non-CM, it follows from Proposition~\ref{prop:B1} and Proposition~\ref{prop:57} for~\(r > 7\) and~\(r \in\{5,7\}\) respectively that~\({\overline{\rho}}_{L,r}(G_{\mathbb{Q}(\zeta_r)}) = \operatorname{SL}_2(\mathbb{F}_r)\). If $\operatorname{SL}_2(\mathbb{F}_r)$ was absolutely reducible it would have to be isomorphic to a subgroup of upper triangular matrices, which is solvable. But $\operatorname{SL}_2(\mathbb{F}_r)$ is not solvable for $r\geq 5$. Hence the result in that case.
Suppose now that $L$ has CM.
Thus $L$ has potentially good reduction everywhere and integral $j$-invariant; more precisely, we obtain
\begin{equation}\label{E:j(L)}
j(L)=2^8\cdot \frac{((a^r+b^r)^2-(ab)^r)^3}{((ab)^r(a^r+b^r))^2} \in \mathbb{Z}.
\end{equation}
Since $a,b$ are coprime we get that $(ab)^{2r}$ divides $2^8$, hence $ab=\pm 1$; from $a^r+b^r \neq 0$, we conclude that~$(a,b)=\pm (1,1)$ and hence~$j(L) = 1728$, and~\(L\) has CM by~\(\mathbb{Q}(i)\). According to~\cite[Proposition~1.14]{zyw}, \({\overline{\rho}}_{L,r}(G_\mathbb{Q})\) is conjugate to the normalizer of a Cartan subgroup~\(C\) of~\(\operatorname{GL}_2(\mathbb{F}_r)\) which is split if~\(r\equiv1\pmod{4}\) and non-split if~\(r\equiv 3\pmod{4}\). In particular, the projective image of~\({\overline{\rho}}_{L,r}(G_\mathbb{Q})\) is a dihedral group~\(D_n\) of order~\(2n\) where~\(n = r - 1\) or~\(r + 1\) according to whether~\(C\) is split or non-split. By the assumption~\(r\geq 5\), we note that we necessarily have~\(n>2\). In particular, \(D_n\) has a unique cyclic subgroup of order~\(n\) given by the projective image of~\(G_{\mathbb{Q}(i)}\). On the other hand, \({\overline{\rho}}_{L,r}(G_{\mathbb{Q}(\zeta_r)}) \) has order~\(2n\) and a projective image~\(H\) of order~\(n\). Assume for a contradiction that~\({\overline{\rho}}_{L,r}|_{G_{\mathbb{Q}(\zeta_r)}}\) is absolutely reducible. Since~\(2n\) is coprime to~\(r\), the representation is decomposable, and thus the group~\(H\) is cyclic. By uniqueness of order $n$ cyclic subgroups of $D_n$, we must have~\(\mathbb{Q}(i) = \mathbb{Q}(\zeta_r)\), which is not the case. This gives the desired contradiction.
\end{proof}
\begin{remark}
For~\(r = 11\) or~\(r\geq 17\), we can give a slightly different proof of the above theorem as follows. The curve~\(L\) (in the notation of the proof) is a quadratic twist (by~\(-(a^r + b^r)\)) of the classical Hellegouarch-Frey curve~\(L' : y^2 = x (x - a^r) (x + b^r)\). Since~\(L'\) has good or bad multiplicative reduction at~\(r\) (and~\(r\geq 7\)), it follows from~\cite[\S\S1.11-1.12 and Proposition~17]{Ser72} that if~\({\overline{\rho}}_{L',r}(G_\mathbb{Q})\neq \operatorname{GL}_2(\mathbb{F}_r)\), then~\({\overline{\rho}}_{L',r}(G_\mathbb{Q})\) is either contained in a Borel subgroup or in the normalizer of a Cartan subgroup~\(C_r'\) of~\(\operatorname{GL}_2(\mathbb{F}_r)\). In the former case, \(L'\) gives rise to a rational point on~\(Y_0(2r)\), contradicting results of Mazur and Kenku (see~\cite[Theorem~1]{Ken82}). In the latter case, using~\cite[Theorem~6.1]{BilPar11} and~\cite[Proposition~2.1]{lemos} when~\(C_r'\) is split or non-split respectively, we get that~\(j(L) = j(L')\in\mathbb{Z}\) and we conclude as in the above proof.
\end{remark}
\begin{theorem}
\label{T:modularity}
The abelian variety $J_r = J_r(a,b)/K$ is modular.
\end{theorem}
\begin{proof}
For~\(r = 3\), we have~\(K = \mathbb{Q}\) and~\(J_r\) is the elliptic curve denoted~\(E(a,b)\) in~\cite{kraus1}. According to the comment at the beginning of \emph{loc. cit.} our modularity result then follows from~\cite{CoDiTa99}. Hence, we can assume~\(r\geq 5\). By Theorem~\ref{T:FreyRep} the representation~${\overline{\rho}}_{J_r,\mathfrak{p}_r} : G_K \rightarrow \operatorname{GL}_2(\mathbb{F}_r)$ extends to an odd representation ${\overline{\rho}}$ of $G_\mathbb{Q}$ which is absolutely irreducible as a consequence of Theorem~\ref{T:irred7}. From Serre's Conjecture \cite{serreconj1,serreconj2}, ${\overline{\rho}}$ is modular, hence ${\overline{\rho}}_{J_r,\mathfrak{p}_r}$ is also modular by cyclic base change \cite{Langlands}. The conclusion that $J_r/K$ is modular now follows from (the full content of) Theorem~\ref{T:irred7} and~\cite[Theorem~1.1]{khareThorne}.
\end{proof}
\begin{remark}
From Remark~\ref{rk:JisCM}, we note that when $ab = 0$, $J_r$ has CM and hence is modular as well.
\end{remark}
We now refine this statement in a way that is going to be used in Parts~\ref{Part:77p} and~\ref{Part:1111p}.
Let~\(\mathfrak{q}\) be a prime ideal in~\(K\) above a rational prime~\(q\). We denote by~\(\Frob_{\mathfrak{q}}\) a Frobenius element at~\(\mathfrak{q}\) in~\(G_K\) and write~\(a_\mathfrak{q}(J_r) = \tr \rho_{J_r,p}(\Frob_\mathfrak{q})\) for a prime~\(p\neq q\) (this definition is independent of the choice of such prime~\(p\)).
\begin{lemma}\label{lem:traces_in_K}
We have
\begin{equation} \label{E:traces}
a_\mathfrak{q}(J_r)^\sigma = a_{\mathfrak{q}^\sigma}(J_r),
\end{equation}
for every~\(\sigma\in G_\mathbb{Q}\). In particular, \(a_\mathfrak{q}(J_r)\) belongs to~\(K\).
\end{lemma}
\begin{proof}
Let~\((v_1,v_2)\) be a basis for~\(V_p(J_r)\) as a free \(K\otimes\mathbb{Q}_p\)-module of rank~\(2\) and denote by~\(A\) the matrix of~\(\rho_{J_r,p}(\Frob_\mathfrak{q})\) with respect to this basis. Let~\(\sigma\) be in~\(G_\mathbb{Q}\). It follows from~\eqref{semi-linear-action} that~\(A^\sigma\) is the matrix of~\(\rho_{J_r,p}(\Frob_{\mathfrak{q}^\sigma})\) with respect to the basis~\((v_1^\sigma,v_2^\sigma)\) and hence
\[
a_{\mathfrak{q}^\sigma}(J_r) = \tr \rho_{J_r,p}(\Frob_{\mathfrak{q}^\sigma}) = \tr A^\sigma = (\tr A)^\sigma = a_\mathfrak{q}(J_r)^\sigma.
\]
\end{proof}
Recall from~\cite[Definition~3.17]{DarmonDuke} that
a Hilbert modular form~$g$ over~$K$ of level~$\mathcal{N}$ is called a $\mathbb{Q}$-form if for all primes
$\mathfrak{q} \nmid \mathcal{N}$ it satisfies
\begin{equation*}
a_{\mathfrak{q}^\sigma}(g) = a_\mathfrak{q}(g)^\sigma,\quad\text{for all $\sigma \in G_\mathbb{Q}$}.
\end{equation*}
The previous discussion and the modularity of~$J_r$ now lead to the following refined statement (see also \cite[Lemma~3.18]{DarmonDuke} for the analogous result for Darmon's Frey varieties $J_r^{\pm}$ assuming modularity, and the next paragraph for a discussion on this assumption).
\begin{theorem}\label{T:Qform}
The variety $J_r$ is associated with a $\mathbb{Q}$-form.
\end{theorem}
\subsection{Modularity of Darmon's varieties $J_r^-$ and $J_{r,r}^-$} \label{S:modularityDarmon}
In~\cite[pp. 420--424]{DarmonDuke} Darmon describes
four Frey varieties for Generalized Fermat Equations of signature~$(r,r,p)$ and~$(p,p,r)$, denoted $J_{r,r}^\pm$ and~$J_r^\pm$, respectively.
More precisely, he defines
abelian varieties $J_{r,r}^\pm(t)$ and~$J_r^\pm(t)$ over~$\mathbb{Q}(t)$ and obtains the Frey varieties by taking an appropriate quadratic twist of the specialization at $t=t_0$, where $t_0$ is given by~\eqref{st-identity} for signature~$(r,r,p)$ and
$t_0 = \frac{a^p}{a^p + b^p}$ for signature~$(p,p,r)$.
Moreover, the base change $J_{r,r}^\pm/K$ and $J_r^\pm/K$ is
of $\operatorname{GL}_2$-type with real multiplications by~$K$.
Our method for proving modularity of~$J_r$ from the previous section also applies to~$J_r^-$ and $J_{r,r}^-$, giving a complete proof of~\cite[Theorem 2.9]{DarmonDuke} for these varieties.
A general inductive strategy on the $\lcm$ of the orders of inertia at $0, 1, \infty$ for proving modularity of fibers in rigid local systems is given in \cite{DarmonRigid}, assuming a generic modularity lifting conjecture.
\begin{theorem} Let $J = J_r^-$ or~$J = J_{r,r}^-$. Then $J/K$ is modular.
\end{theorem}
\begin{proof} This follows similarly to the proof of Theorem~\ref{T:modularity} where Theorem~\ref{T:FreyRep} is replaced by \cite[Theorem 2.6]{DarmonDuke}.
The rest of the proof is the same for~$J = J_{r,r}^-$. When~$J=J_r^-$ the proof is also the same except
that in formula~\eqref{E:j(L)} for $j(L)$ we get $r$ replaced by~$p$ since $t_0 = \frac{a^p}{a^p + b^p}$.
We also make use of the fact that for $ab = 0$, $(a,b) = \pm (1,1), \pm (1,-1)$, these varieties are either singular or have CM.
\end{proof}
\begin{remark}
The mod~$\mathfrak{p}_r$ representations attached to $J_r^+/K$ and~$J_{r,r}^+/K$ descend to~$G_\mathbb{Q}$ and are modular by~\cite[Corollary 2.7]{DarmonDuke}. However,
from~\cite[Theorem 2.6]{DarmonDuke} we see they are reducible, making lifting modularity impossible in general. Sometimes this is possible under some additional local hypothesis (cf. \cite[Theorem~2.9]{DarmonDuke}).
\end{remark}
\begin{remark}
Note that~$J_r^-$ already made an appearance in the proof of Proposition~\ref{darmonfrey}.
\end{remark}
\section{The conductor of $J_r$} \label{S:conductor}
We proved in Sections~\ref{S:Freyrrp} and~\ref{S:modularity} that~$J_r = J_r(a,b)/K$ is a~\(\operatorname{GL}_2\)-type modular abelian variety for $a,b \in \mathbb{Z}$ coprime and non-zero. Therefore, the $\lambda$-adic Galois representations attached to~$J_r$ coincide with the strictly compatible system of representations arising in a Hilbert newform~$g$ defined over~$K$. The strict compatibility of such modular system follows from compatibility with the local Langlands correspondence.\footnote{The fact that regular, algebraic, self-dual cuspidal automorphic forms of~$\operatorname{GL}_n$ over a totally real field give rise to strictly compatible systems is due to the work of many authors; see for example~\cite[Theorem~2.1.1]{BLGGT}. In the case of interest
for our Diophantine application, i.e., Hilbert cuspforms over an odd degree field, this was proved by Carayol~\cite{Carayol86}.}
In particular, the level~$\mathcal{N}$ of~$g$ coincides with the common conductor of $\rho_{J_r,\lambda}$ for all~$\lambda$.
The objective of this section is to determine the ideal~\(\mathcal{N}\) which is given in
Theorem~\ref{T:conductorJI} under certain $2$-adic restrictions on~$a,b$.
The method used is based on exhibiting a local field~$L$ over which $J_r$ has semistable reduction. The classification of 2-dimensional local Weil-Deligne representations guides the choice of~$L$, which we confirm by explicit computations on hyperelliptic equations. The conductor can then be read off by standard formulae once the type of local representation is identified.
The problem of determining the conductors of Jacobians of hyperelliptic curves in general remains an interesting open problem, for which remarkable progress has been obtained recently in \cite{DDMM-local,DDMM-types}. For instance, in \cite{DDMM-local}, one finds a formula which in principle can be used to explicitly determine the conductor exponent of the Jacobian of a hyperelliptic curve over a local field of odd residue characteristic.
The method we use is more specific as it relies on $J_r$ having $\operatorname{GL}_2$-type, as well as on an initial guess for the field of semistable reduction. However, it is in practice efficient and well adapted for computing conductor exponents in parameterized families of hyperellliptic curves, including the case of even residue characteristic.
Before we proceed to the conductor calculation, we recall some standard facts about
local 2-dimensional Galois representations.
\subsection{Inertial types}
Let $\rho : G_{K_\mathfrak{q}} \to \operatorname{GL}_2({\overline{\Q}}_p)$ be a~\(p\)-adic Galois representation where ${K_\mathfrak{q}}$ is a local field with residual characteristic different from~$p$.
When $\rho$ has infinite image of inertia we call it a {\it Steinberg (or Special)} representation. If the image of inertia is finite then we call~$\rho$ a {\it principal series} representation if it is reducible and a {\it supercuspidal} representation if it is irreducible. Moreover, $\rho$ is a principal series if and only if there is an abelian extension $L/K_\mathfrak{q}$ where $\rho|_{G_L}$ is unramified.
Let $K$ be a number field. We let $I_\mathfrak{q} \subset D_\mathfrak{q} \subset G_K$ denote a choice in~$G_K:=\Gal(\overline{K} /K)$ of inertia and decomposition subgroups at a prime~$\mathfrak{q}$, respectively.
Let $J/K$ be an abelian variety of~$\operatorname{GL}_2$-type with~\(F\hookrightarrow\End_K(J)\otimes\mathbb{Q}\) (see~\S\ref{S:GL2typeAV}). The $\lambda$-adic representations $\rho_{J_r,\lambda}$, where~\(\lambda\) runs over the prime ideals in~\(F\), form a strictly compatible system.
In particular, for each prime~$\mathfrak{q}$ of~$K$ and all $\lambda$ in~\(F\) such that $\lambda$ does not divide the residue characteristic of~\(\mathfrak{q}\), the restrictions $\rho_{J,\lambda}|_{D_\mathfrak{q}}$ have the same associated Weil-Deligne representation.
We say that the {\it inertial type} of $J/K$ at a prime~$\mathfrak{q}$ of~\(K\) is Steinberg, principal series or supercuspidal if for some (and hence any) \(\lambda\) not dividing the residue characteristic of~\(\mathfrak{q}\), we have that $\rho_{J,\lambda}|_{D_{\mathfrak{q}}}$ is a
Steinberg, principal series or supercuspidal representation, res\-pectively.
We refer to, for example,~\cite{DPP21} for a summary of definitions and formulae about inertial types.
\subsection{Conductor calculation}\label{ss:conductor_calculation} We assume throughout that~\(r\geq 5\). Let $a,b \in \mathbb{Z}$ be coprime such that $ab(a^r + b^r) \neq 0$. Recall that an equation for~$C_r(a,b)$ is given by~\eqref{kraushyper} which we repeat below for the convenience of the reader:
\[
C_r(a,b) \; : \; \; y^2 = (ab)^\frac{r-1}{2} x h \left(\frac{x^2}{ab} + 2 \right) + b^r - a^r.
\]
Let~$z_0$ be a square-root of~$ab$. Specializing~\eqref{eq:cheby} to~$z = z_0$ yields the following expression for the above model:
\begin{equation}\label{kraushyper_bis}
C_r(a,b) \; : \; \; y^2 = \sum_{k=0}^{\frac{r - 1}{2}}c_k(ab)^{k}x^{r - 2k} + b^r - a^r,\quad\text{where }c_k=\frac{r}{r-k}{r-k \choose k}\in\mathbb{Z}.
\end{equation}
(Note that this latter equation is also compatible with~\eqref{E:Jab=0} when~$ab = 0$, assuming $(ab)^0 = 1$.) This is the model we will consider for~$C_r(a,b)$ in the rest of this section. Recall also the formula~\eqref{E:discriminant} for the discriminant of~\eqref{kraushyper_bis} (also valid for~$ab = 0$, see the discussion after~\eqref{E:discriminant}):
\begin{equation}\label{E:discriminant_bis}
\Delta(C_r(a,b)) = (-1)^\frac{r-1}{2} 2^{2(r-1)} r^r (a^r+b^r)^{r-1}.
\end{equation}
We write $C_r = C_r(a,b)$ and~$J_r = J_r(a,b)$ for simplicity.
We now determine the conductor of~$J_r/K$. For this we use facts from~\S\ref{ss:hyperelliptic_equations}-\S\ref{S:FreyOverKs} and Section~\ref{S:Freyrrp}.
For a prime~$\mathfrak{q}$ of~$K$ we denote by $K_{\mathfrak{q}}$ the completion of $K$ at~$\mathfrak{q}$. For a rational prime~$\ell$ we denote by~$\mathfrak{q}_\ell$ a prime in~$K$ above~$\ell$. Recall that $2$ and~$3$ are unramified in~$K$ and that there is a unique prime ideal~$\mathfrak{q}_r$ in~\(K\) above~$r$.
\begin{proposition}\label{P:typeAt2}
Assume that~$a \equiv 0 \pmod 2$ and $b \equiv 1 \pmod 4$.
Let $L/K_{\mathfrak{q}_2}$ be a finite extension
with ramification index $r$. Then $C_r$ has good reduction over $L$, and moreover, there is no unramified extension of~$K_{\mathfrak{q}_2}$ where $C_r$ attains good reduction.
\end{proposition}
\begin{proof}
Let $L/K_{\mathfrak{q}_2}$ be a finite extension with ramification index $r$, ring of integers~$\mathcal{O}$,
uniformizer~$\pi$ and corresponding valuation~$\upsilon$.
Applying the substitutions $x\rightarrow \pi^2x$, $y \rightarrow \pi^ry + 1$ to~\eqref{kraushyper_bis} and dividing out by $\pi^{2r}$ yields the following model for~$C_r$
\begin{equation*}
y^2 + \frac{2}{\pi^r}y = x^r + \sum_{k=1}^{\frac{r - 1}{2}}c_k\left(\frac{ab}{\pi^4}\right)^{k}x^{r - 2k} + \frac{b^r - a^r - 1}{\pi^{2r}}.
\end{equation*}
The $2$-adic conditions on $a,b$ and the assumption~$r\geq 5$ imply that
\begin{equation*}
\upsilon(ab) \geq \upsilon(2) = r\geq 4\quad\text{and}\quad \upsilon(b^r - a^r - 1) \geq \upsilon(4) = 2r.
\end{equation*}
Therefore the above model of $C_r$ is defined over $\mathcal{O}$. According to~\eqref{E:discriminant_bis} and the formulae in Section~\ref{ss:hyperelliptic_equations}, its discriminant is
$$
(-1)^{\frac{r-1}{2}}\frac{2^{2(r-1)}}{\pi^{2(r-1)r}}\cdot r^r (a^r+b^r)^{r-1}
$$
which is a unit in~$\mathcal{O}$. Hence $C_r$ has good reduction over $L$.
Over an unramified extension of~$K_{\mathfrak{q}_2}$, the valuation of the discriminant of any hyperelliptic model for~$C_r$ is $\equiv 2(r - 1)\pmod{2r(r -1)}$ by~\eqref{E:discriminant_bis} and the results in \S\ref{ss:hyperelliptic_equations}. Hence~$C_r$ cannot have good reduction.
\end{proof}
\begin{corollary} \label{C:inertiaOrder}
Assume $a \equiv 0 \pmod{2}$ and~$b \equiv 1 \pmod{4}$. Let~$\mathfrak{q}_2$ be a prime ideal in~\(K\) above~\(2\).
The inertial type of~$J_r/K$ at~$\mathfrak{q}_2$
is a principal series if $r \mid \# \mathbb{F}_{\mathfrak{q}_2}^*$
and supercuspidal otherwise.
Moreover, for all $\lambda \nmid 2$ in~$K$, the image of inertia~$\rho_{J_r,\lambda}(I_{\mathfrak{q}_2})$ is cyclic of order~$r$.
\end{corollary}
\begin{proof} From Proposition~\ref{P:typeAt2} we know that
$J_r/K_{\mathfrak{q}_2}$ obtains good reduction
over any extension $L/K_{\mathfrak{q}_2}$ with ramification degree~$r$. In particular, this shows that $\rho_{J_r,\lambda}|_{D_{\mathfrak{q}_2}}$ becomes unramified over~$L$, hence the inertial type of $J_r/K$ is not Steinberg at~$\mathfrak{q}_2$. Moreover, the inertial type of~$J_r/K$ at~$\mathfrak{q}_2$ is a principal series if and only if there is an abelian extension $L/K_{\mathfrak{q}_2}$ with ramification degree~$r$ and supercuspidal otherwise. By local class field theory, such an extension exists if and only if $r$ divides $\# \mathbb{F}_{\mathfrak{q}_2}^*$. This proves the first statement.
Also from Proposition~\ref{P:typeAt2}, there is no unramified extension of~$K_{\mathfrak{q}_2}$ over which~$J_r$ has good reduction, hence $\# \rho_{J_r,\lambda}(I_{\mathfrak{q}_2}) \ne 1$ and divides~$r$. The conclusion follows.
\end{proof}
\begin{comment}
\begin{remark}
\label{ab-odd}
By the criterion in \cite{ChenKoutsianas2}, it can be shown for the case $2 \nmid ab$, that $J_r$ has potentially multiplicative reduction at $\mathfrak{q}_2$ and hence its inertial type is a special representation. This precludes the possibility of proving irreducibility of ${\overline{\rho}}_{J_r,\lambda}$ in this case using the local methods (see Proposition~\ref{P:irredSupercuspidal}).
\end{remark}
\end{comment}
Keeping the notation of~\S\ref{S:FreyOverKs}, we write~\(H(x) = \displaystyle{\sum_{k=0}^{\frac{r - 1}{2}}c_k(ab)^{k}x^{r - 2k}\in\mathbb{Z}[x]}\) for the polynomial showing up in the right-hand side of \eqref{kraushyper_bis} (up to the constant term) and recall the standard facto\-rization
\begin{equation}\label{eq:basic_factorization_general}
x^r + y^r = (x + y) \phi_r(x,y)
\end{equation}
where
\[
\phi_r(x,y) = \sum_{k = 0}^{\frac{r - 1}{2}}(-1)^kx^{r - 1 -k}y^k = \prod_{j = 1}^{\frac{r - 1}{2}}\left(x^2 + \omega_j xy + y^2\right).
\]
The following lemma is used in the proofs of Propositions~\ref{P:typeAt7a} and~\ref{P:typeAt7b}.
\begin{lemma}\label{lem:H}
We have~$H(a - b) = a^r - b^r$.
\end{lemma}
\begin{proof}
By Lemma~\ref{L:firstKind} applied to~\(z = z_0\), we have
\begin{align*}
H(a - b) & = (ab)^{\frac{r - 1}{2}}(a - b)h\left(\frac{(a - b)^2}{ab} + 2\right) \\
& = (a - b)\prod_{j = 1}^{\frac{r - 1}{2}}\left(a^2 + b^2 - ab\omega_j\right) \\
& = (a - b)\phi_r(a,-b) \\
& = a^r - b^r,
\end{align*}
as claimed.
\end{proof}
\begin{proposition} \label{P:typeAt7a}
Let $L/K_{\mathfrak{q}_r}$ be a finite extension with ramification index~$4$. If $r \nmid a + b$ then $C_r$ and~$J_r$ have good reduction over~$L$. Moreover, there is no extension of~$K_{\mathfrak{q}_r}$ with ramification index dividing~$2$ where~$C_r$ or~$J_r$ obtain good reduction.
\end{proposition}
\begin{proof}
Let $L / K_{\mathfrak{q}_{r}}$ be finite extension with ramification index~$4$, ring of integers~$\mathcal{O}$, a uniformizer~$\pi$ and~$\upsilon$
the corresponding valuation.
Applying the substitutions $x \rightarrow \pi^2x - (b-a)$, $y\rightarrow\pi^ry$ to~\eqref{kraushyper_bis} and dividing out by $\pi^{2r}$ yields the model~\(y^2 = P(x) + \frac{b^r - a^r}{\pi^{2r}}\) for~$C_r$ where
\begin{align*}
P(x) & = \frac{1}{\pi^{2r}}H(\pi^2x - (b - a)) \\
& = \frac{1}{\pi^{2r}}\sum_{k\ge 0}c_k(ab)^k(\pi^2x - (b - a))^{r - 2k}\quad\text{with } c_k=\frac{r}{r-k}{r-k \choose k} \\
& = \sum_{k\ge 0}c_k\frac{(ab)^k}{\pi^{4k}}\sum_{j\geq0}\binom{r - 2k}{j}\left(-\frac{b - a}{\pi^{2}}\right)^{r - 2k - j}x^j \\
& = \sum_{j\geq0}\sum_{k\ge 0}c_k\frac{(ab)^k}{\pi^{2(r - j)}}\binom{r - 2k}{j}(a - b)^{r - 2k - j}x^j \\
& = \sum_{j\geq0}a_jx^j,
\end{align*}
with~$\binom{n}{m} = 0$ if~$m > n$. We claim that the coefficients~$a_0,\dots,a_r$ of~$P$ satisfy the following properties~:
\begin{enumerate}[(i)]
\item\label{item:p1} $a_0 = \frac{a^r - b^r}{\pi^{2r}}$;
\item\label{item:p2} For~$1\leq j\leq r - 1$, we have~$\upsilon(a_j)\geq0$;
\item\label{item:p3} $a_r = 1$.
\end{enumerate}
The first property follows from Lemma~\ref{lem:H} and the last one from the fact that~\(H\) is monic of degree~\(r\). Let~\(1\leq j\leq r -1\) and let~\(k\geq0\). We have
\[
\upsilon\left(c_k{r - 2k\choose j}\right) \geq \upsilon(r) = 2(r - 1) \geq 2(r - j),
\]
since~\(c_k\) is divisible by~\(r\) for~\(k>0\) by Lemma~\ref{L:firstKind} and~\(r\mid{r \choose j}\). In particular, \(a_j\in\mathcal{O}\) and it follows that the above model of $C_r$ is defined over $\mathcal{O}$. According to~\eqref{E:discriminant_bis} and the formulae in~\S\ref{ss:hyperelliptic_equations}, its discriminant is
$$
(-1)^{\frac{r-1}{2}}\frac{r^r}{\pi^{2 (r-1)r}}\cdot 2^{2(r-1)}(a^r+b^r)^{r-1}
$$
which is a unit in~$\mathcal{O}$ since~\(r\sim \pi^{2(r - 1)}\) and~\(r\nmid a^r + b^r\). Hence $C_r$ has good reduction over $L$.
Over an extension of~$K_{\mathfrak{q}_r}$ with ramification index dividing $2$, the valuation of the discriminant of any hyperelliptic model for~$C_r$ is $\equiv r(r-1)/2, r(r - 1)\pmod{2r(r -1)}$ by~\eqref{E:discriminant_bis} and the results in \S\ref{ss:hyperelliptic_equations}. Hence~$C_r$ cannot have good reduction.
\end{proof}
\begin{corollary} \label{C:inertiaOrderAt7}
Assume $r \nmid a+b$.
The inertial type of~$J_r/K$ at~$\mathfrak{q}_r$
is a principal series if $r \equiv 1 \pmod{4}$
and supercuspidal otherwise.
Moreover, for all $\lambda \nmid r$ in~$K$, the image of inertia~$\rho_{J_r,\lambda}(I_{\mathfrak{q}_r})$ is cyclic of order~$4$.
\end{corollary}
\begin{proof} From Proposition~\ref{P:typeAt7a} we know that
$J_r/K_{\mathfrak{q}_r}$ obtains good reduction
over any extension $L/K_{\mathfrak{q}_r}$ with ramification degree~$4$.
Now the same argument as in the proof of Corollaty~\ref{C:inertiaOrder} shows that $\rho_{J_r,\lambda}|_{D_{\mathfrak{q}_r}}$ is a principal series
when $4$ divides $\# \mathbb{F}_{\mathfrak{q}_r}^* = r-1$ and supercuspidal otherwise. This proves the first statement.
Also from Proposition~\ref{P:typeAt7a}, there is no extension of~$K_{\mathfrak{q}_r}$ with ramification index dividing 2 over which~$J_r$ has good reduction, hence $\# \rho_{J_r,\lambda}(I_{\mathfrak{q}_r}) \ne 1,2$ and divides~$4$. Thus $\# \rho_{J_r,\lambda}(I_{\mathfrak{q}_r}) = 4$.
In this case $\# \rho_{J_r,\lambda}(I_{\mathfrak{q}_r})$ is not a prime number, but the fact that $\rho_{J_r,\lambda}(I_{\mathfrak{q}_r})$ is cyclic follows from the fact that there is a unique tamely ramified extension of order~$4$ of the maximal unramified extension of~$K_{\mathfrak{q}_r}$. (Alternatively, it also follows from the fact that inertia is acting via characters as detailed in the proof of Theorem~\ref{T:conductorJI} below.)
\end{proof}
\begin{proposition} \label{P:typeAt7b}
Let $L/K_{\mathfrak{q}_r}$ be a finite extension with ramification index~$2$. If $r \mid a + b$, then both $C_r$ and~$J_r$ have multiplicative reduction over $L$ and, moreover, there is no unramified extension of~$K_{\mathfrak{q}_r}$ where $C_r$ attains multiplicative reduction.
\end{proposition}
\begin{proof}
Let $L / K_{\mathfrak{q}_{r}}$ be finite extension with ramification index~$2$, ring of integers~$\mathcal{O}$, a uniformizer~$\pi$ and~$\upsilon$
the corresponding valuation. We assume that~$r\mid a + b$. Note that we necessarily have~$ab\neq 0$ since $a$ and~$b$ are coprime.
As in the proof of Proposition~\ref{P:typeAt7a}, write
\[
P(x) = \frac{1}{\pi^{2r}}H(\pi^2x - (b - a)) = \sum_{j\geq0}\sum_{k\ge 0}c_k\frac{(ab)^k}{\pi^{2(r - j)}}\binom{r - 2k}{j}(a - b)^{r - 2k - j}x^j = \sum_{j\geq0}a_jx^j,
\]
where~$\binom{n}{m} = 0$ if~$m > n$.
We claim that the coefficients~$a_0,\dots,a_r$ of~$P$ satisfy the following properties~:
\begin{enumerate}[(i)]
\item\label{item:P1} $a_0 = \frac{a^r - b^r}{\pi^{2r}}$;
\item\label{item:P2} $a_1 = \frac{1}{\pi^{2(r - 1)}}r\phi_r(a,b)\in\mathcal{O}$ and reduces to~$a^{r - 1}$ modulo~$\pi$;
\item\label{item:P3} For~$1<j<r$, $j\not=\frac{r + 1}{2}$, we have~$\upsilon(a_j)>0$;
\item\label{item:P4} $a_{\frac{r + 1}{2}}\in\mathcal{O}$ and reduces to~$2a^{\frac{r + 1}{2}}$ modulo~$\pi$;
\item\label{item:P5} $a_r = 1$.
\end{enumerate}
Assuming these properties, it follows that applying the substitutions $x\rightarrow \pi^2x - (b - a)$, $y \rightarrow \pi^ry$ to~\eqref{kraushyper_bis} and dividing out by $\pi^{2r}$ yields the model \(y^2 = P(x) + \frac{b^r - a^r}{\pi^{2r}}\) for~$C_r$ which is integral and reduces modulo~$\pi$ to
\[
y^2 = x^r + 2a^{\frac{r - 1}{2}}x^{\frac{r + 1}{2}} + a^{r - 1}x = x\left(x^{\frac{r - 1}{2}} + a^{\frac{r - 1}{2}}\right)^2.
\]
Therefore, $C_r$ has multiplicative reduction over $L$.
Let us then prove properties~(\ref{item:P1})-(\ref{item:P5}) above.
\begin{enumerate}[(i)]
\item By Lemma~\ref{lem:H}, we have
\[
a_0 = P(0) = \frac{H(a - b)}{\pi^{2r}} = \frac{a^r - b^r}{\pi^{2r}}.
\]
\item Let~$i$ be a primitive fourth root of unity and recall that~$z_0$ satisfies~$z_0^2 = ab$.
We have $a_1 = \frac{\mathrm d P}{\mathrm d x}(0)$ and hence
\begin{align*}
a_1 & = \frac{1}{\pi^{2r}}\frac{\mathrm d}{\mathrm d x}\left(H(\pi^2x - (b - a))\right)\mid_{x = 0} \\
& = \frac{1}{\pi^{2(r - 1)}}\frac{\mathrm d H}{\mathrm d x}\left(\pi^2x - (b - a)\right)\mid_{x = 0} \\
& = \frac{1}{\pi^{2(r - 1)}}\frac{\mathrm d H}{\mathrm d x}\left(a - b\right) \\
& = \frac{1}{\pi^{2(r - 1)}}(ab)^{\frac{r - 1}{2}}rh\left(i\frac{a - b}{z_0}\right)h\left(-i\frac{a - b}{z_0}\right),\quad\text{by Lemma~\ref{L:secondKind}}. \\
\end{align*}
On the other hand, we compute
\begin{align*}
(ab)^{\frac{r - 1}{2}}h\left(i\frac{a - b}{z_0}\right)h\left(-i\frac{a - b}{z_0}\right) & = (ab)^{\frac{r - 1}{2}}\prod_{m = 1}^{\frac{r - 1}{2}}\left(i\frac{a - b}{z_0} - \omega_m\right)\left(-i\frac{a - b}{z_0} - \omega_m\right) \\
& = (ab)^{\frac{r - 1}{2}}\prod_{m = 1}^{\frac{r - 1}{2}}\left(\omega_m^2 + \frac{(a - b)^2}{ab}\right) \\
& = \prod_{m = 1}^{\frac{r - 1}{2}}\left(a^2 + b^2 + ab\omega_{2m}\right) \\
& = \prod_{m = 1}^{\frac{r - 1}{2}}\left(a^2 + b^2 + ab\omega_{m}\right) \\
& = \phi_r(a,b)\quad\text{by \eqref{eq:basic_factorization_general}.} \\
\end{align*}
Therefore we have proved that $a_1 = \frac{1}{\pi^{2(r - 1)}}r\phi_r(a,b)$. Finally, we have
\begin{align*}
\phi_r(a,b) & = \frac{1}{y}\left((y - a)^r + a^r\right),\quad\text{with~$y = a + b$} \\
& = \frac{1}{y}\left(\sum_{m = 0}^r\binom{r}{m}(-a)^{r - m}y^m + a^r\right) \\
& = \sum_{m = 1}^r\binom{r}{m}(-a)^{r - m}y^{m - 1}\quad\text{(as $r$ is odd)} \\
& \equiv ra^{r - 1}\pmod{r^2}
\end{align*}
since~$y = a + b\equiv 0\pmod{r}$ and~$r\mid\binom{r}{m}$ for~$m = 1,\dots, r - 1$. As~$r\sim \pi^{r - 1}$ in~$L$, we get that~$a_1$ is in~$\mathcal{O}$ and reduces to~$a^{r - 1}$ modulo~$\pi$.
\item For all~$1<j<r$ and all~\(k\geq0\), we have~$\displaystyle{\upsilon\left(c_k\binom{r - 2k}{j}\right) \geq \upsilon(r) = r - 1}$. Therefore, if~$j > \frac{r + 1}{2}$, we have
\[
\upsilon(a_j) \geq \upsilon\left(c_k\binom{r - 2k}{j}\right) - 2(r - j) \geq 2j - r - 1 > 0,
\]
as claimed. Let us now assume~$1<j\leq\frac{r + 1}{2}$. Since $r\mid a + b$, for any integer~\(k\) such that~\(0\leq k\leq \frac{r - j}{2}\), we have
\[
(ab)^k(a - b)^{r - 2k - j} = (-1)^k2^{r - 2k - j}a^{r - j} + ru
\]
for some~$u\in\mathbb{Z}$. Therefore, we have
\[
a_j = \frac{1}{\pi^{2(r - j)}} \sum_{k = 0}^{\frac{r - 1}{2}}\left((-1)^k2^{r - 2k - j}a^{r - j} + ru\right)\binom{r - 2k}{j}c_k = \frac{a^{r - j}}{\pi^{2(r-j)}}\alpha_j + \pi\beta_j
\]
where
\[
\alpha_j = \sum_{k = 0}^{\frac{r - 1}{2}}(-1)^k2^{r - 2k - j}\binom{r - 2k}{j}c_k
\]
and~$\beta_j\in\mathcal{O}$ since~$\binom{r - 2k}{j}c_k\in r\mathbb{Z}$ for all~$k\geq0$ (by Lemma~\ref{L:firstKind}), and~$\upsilon(r^2) = 2(r - 1) > 2(r - j)$ (as $j>1$). Recall from~\cite[Section 2.3.2]{Mason} that
\[
T_r(x) = \sum_{k = 0}^{\frac{r - 1}{2}}(-1)^k2^{r - 2k - 1}c_k x^{r - 2k}
\]
and hence
\[
\alpha_j = \frac{T_r^{(j)}(1)}{j!2^{j - 1}} = \frac{1}{j!2^{j - 1}}\prod_{s = 0}^{j - 1}\frac{r^2 - s^2}{2s + 1}
\]
where the last equality can be deduced
from~\cite[(9.8.35)]{Koekoek}.
Assume now that~\(j\neq \frac{r + 1}{2}\). We have
\[
\alpha_j = r^2 \frac{1}{j!2^{j - 1}}\prod_{s = 1}^{j - 1}\frac{r^2 - s^2}{2s + 1}\in r^2\mathcal{O}
\]
since for~$s \leq j - 1$, we have~$2s + 1 \leq 2j - 1 < r$ and hence~$2s + 1\in\mathcal{O}^\times$. The desired result follows as again~$\upsilon(r^2) = 2(r - 1) > 2(r - j)$.
\item Reasoning as above, and using the same notation, one has
\begin{equation}
\label{middle-term}
\alpha_{\frac{r + 1}{2}} = \frac{T_r^{(\frac{r + 1}{2})}(1)}{\left(\frac{r + 1}{2}\right)!2^{\frac{r - 1}{2}}} = \frac{1}{\left(\frac{r + 1}{2}\right)!2^{\frac{r - 1}{2}}}\prod_{s = 0}^{\frac{r - 1}{2}}\frac{r^2 - s^2}{2s + 1} = \binom{\frac{3r - 1}{2}}{\frac{r + 1}{2}} \equiv 2r \pmod{r^2}
\end{equation}
Since~$r \sim \pi^{r - 1}$ we get the desired result. For the last congruence in \eqref{middle-term}, first note that $\frac{3r-1}{2} = r-1 + \frac{r+1}{2}$. Now
\begin{align*}
\binom{r-1 + \frac{r+1}{2}}{\frac{r+1}{2}} = \binom{r-1 + \frac{r+1}{2}}{r-1} = \frac{(r-1 + \frac{r+1}{2}) \ldots (1 + \frac{r+1}{2})}{(r-1)!} & \equiv 2 r \pmod{r^2} \\
\iff \left( r-1 + \frac{r+1}{2} \right) \ldots \left( 1 + \frac{r+1}{2} \right) & \equiv 2 r (r-1)! \pmod{r^2} \\
\iff \frac{1}{r} \left( r-1 + \frac{r+1}{2} \right) \ldots \left( 1 + \frac{r+1}{2} \right) & \equiv 2 (r-1)! \pmod{r}.
\end{align*}
The last congruence is true as the left hand side taken mod $r$ is the product
\begin{align*}
\left( \frac{r+1}{2} + 1 \right) \cdots (r-1) \cdot \hat r \cdot (1) \cdots \left( \frac{r+1}{2} -1 \right) = \frac{(r-1)!}{\frac{r+1}{2}}.
\end{align*}
\item The polynomial~$H$ is monic of degree~$r$ and hence so is~$P$.
\end{enumerate}
Note that the twist $C_r'/K$ of $C_r/K$ by $\pi^\frac{1}{2}$ has multiplicative reduction. Hence, $C_r$ does not have multiplicative reduction over any unramified extension of~$K_{\mathfrak{q}_r}$.
\end{proof}
\begin{proposition} \label{P:multiplicativeRedJ}
Let $q \not= 2, r$ be a prime and let~\(\mathfrak{q}\) be a prime ideal above~\(q\) in~\(K\). Assume that we have~$q \mid a^r+b^r$. Then $C_r$ and $J_r$ have multiplicative reduction at~$\mathfrak{q}$.
\end{proposition}
\begin{proof}
Let~\(K' = K_{\mathfrak{q}}(i,z_0)\) with~$i$ a primitive fourth root of unity (and~\(z_0\) a square-root of~\(ab\)) and denote by~\(k'\) the residue field of~\(K'\). It is a finite extension of the residue field~\(k\) of~\(K_{\mathfrak{q}}\). We shall prove that the polynomial~\(H(x) + b^r - a^r\) has one single root and~\((r - 1)/2\) double roots in~\(k'\).
By Lemma~\ref{L:firstKind}, we have~$H(x) = H(x,z_0) = 2z_1^rT_r\left(\frac{x}{2z_1}\right)$ where~$z_1 = iz_0$. Hence by Lemma~\ref{evaluation-point}, we have
\[
H(2z_1) = 2z_1^r = H(\omega_jz_1),\quad j= 1,\dots,\frac{r - 1}{2}.
\]
Besides, we have~$z_1^{2r} = (-ab)^r$ and~\((-ab)^r \equiv a^{2r}\pmod{q}\). Up to changing $i$ to~$-i$ if necessary, one may assume that~$z_1^r$ reduces to~$a^r$ in~\(k'\). Hence $2z_1$ and $\omega_jz_1$ are roots of $H(x) + b^r - a^r$ in~$k'$. Moreover, the latter are roots of multiplicity~$>1$ by Lemma~\ref{L:secondKind} applied to~$z = z_0$. This proves the desired result and we conclude that $C_r$ has multiplicative reduction at~$\mathfrak{q}$.
From \cite[Lemma 3.3.5]{Romagny} ($c = 1$ as the special fiber is integral), the toric rank of $J_r$ is positive and hence $J_r$ has multiplicative reduction.
\end{proof}
\begin{proposition}
\label{P:good_reduction}
Let $q \nmid 2r(a^r+b^r)$ be a prime. Then $J_r/\mathbb{Q}_q$ has good reduction.
\end{proposition}
\begin{proof}
If a prime~$q \nmid 2r(a^r + b^r)$ then $q$ does not divide the discriminant given in~\eqref{E:discriminant_bis}; since the model~\eqref{kraushyper_bis} is integral, the curve~$C_r$ has good reduction at~$q$.
\end{proof}
\begin{lemma}\label{lem:det}
Let~\(\lambda\) be a prime ideal in~\(K\) above a rational prime~\(p\). The determinant of the representation~\(\rho_{J_r,\lambda}\) is the \(p\)-adic cyclotomic character.
\end{lemma}
\begin{proof}
We begin by recalling that the determinant of $\rho_{J_r,\lambda}$ is of the form $\varepsilon \cdot \chi_p$, where $\chi_p$ denotes the $p$-adic cyclotomic character and $\varepsilon$ is a character of finite order with values in $K^*$ unramified at the primes of good reduction of $J_r$ (see~\cite[Lemma 3.1]{RibetKorea}).
We also know (cf.~\cite[Lemma 3.2]{RibetKorea}) that the character $\varepsilon \cdot \chi_p$ is totally odd, i.e., it takes the value -1 at any complex conjugation in $G_K$. This is equivalent to saying that the character $\varepsilon$ is totally even, i.e., its fixed field is a totally real extension $F$ of $K$.
We now apply Proposition 3.4 from ~\cite{RibetKorea} and conclude for any prime $w$ of $K$ such that $\rho_{J_r,\lambda}$ is unramified at $w$, denoting by $a_w$ the trace of $\rho_{J_r,\lambda}(\Frob_w)$, we have that:
$$ a_w = \bar{a}_w \cdot \varepsilon(\Frob_w) $$
where $\bar{z}$ denotes the complex conjugated of $z$ for any complex number $z$. Since the traces $a_w$ belong to $K$ by Lemma~\ref{lem:traces_in_K} and $K$ is totally real, this amounts to:
\begin{equation}
\label{prev-equality}
a_w = {a}_w \cdot \varepsilon(\Frob_w)
\end{equation}
As we have already recalled, it follows
from~\cite[Lemma 3.1]{RibetKorea} that the character $\varepsilon$ takes values in $K$ (see also Corollary (3.1) in \cite{Rib77}). Thus, since $K$ is totally real, we know that this character is either trivial or quadratic.
Suppose that $\varepsilon$ is quadratic. Then we apply Cebotarev's density theorem and conclude from \eqref{prev-equality} that $\rho_{J_r,\lambda} = \rho_{J_r,\lambda} \otimes \varepsilon$. In this situation we can deduce (as proved in the second half of Proposition (4.4) in \cite{Rib77}) that if $F/K$ is the quadratic extension fixed by $\varepsilon$, the restriction of $\rho_{J_r,\lambda}$ to $G_F$ has abelian image and $\rho_{J_r,\lambda}$ is induced from a character of $G_F$. In this case, we know from base change that $\rho_{J_r,\lambda}$ is modular and it is attached to a Hilbert modular form $h$ of parallel weight $2$ over $K$ with CM. In particular, the field $F$ has to be a totally imaginary quadratic extension of $K$ and $h$ must correspond to a Gr\"{o}ssencharacter $\psi$ of $F$ (see for example Theorem (4.5) in \cite{Rib77}). But we have already mentioned that the character $\varepsilon$ must be even, thus its fixed field $F$ must be totally real.
This gives a contradiction, thus we conclude that the character $\varepsilon$ has to be trivial.
\end{proof}
The following theorem finally gives the sought after conductor.
\begin{theorem} \label{T:conductorJI}
Assume $a \equiv 0 \pmod{2}$ and $b \equiv 1 \pmod{4}$. The common conductor of the compatible system~\(\{\rho_{J_r,\lambda}\}\) is given by
\[
\mathcal{N} = 2^2 \cdot \mathfrak{q}_r^2 \cdot \mathfrak{n},
\]
where $\mathfrak{n}$ is the squarefree product of all prime ideals dividing~$a^r + b^r$ which are coprime to~$2r$.
In particular, $J_r$ is semistable at all primes not dividing~$2r$.
\end{theorem}
\begin{proof}
By definition of strict compatibility we have that, for each prime~$\mathfrak{q}$ of~$K$ and all $\lambda$ such that $\mathfrak{q} \nmid \Norm(\lambda)$, the
restrictions $\rho_{J_r,\lambda}|_{D_\mathfrak{q}}$
have the same associated Weil-Deligne representation and their conductor
depends only
on $\rho_{J_r,\lambda}|_{I_\mathfrak{q}}$.
Therefore, when calculating the conductor at~$\mathfrak{q}$ in the cases (\ref{item:(a)})--(\ref{item:(d)}) below, without loss of generality, we can assume that $\lambda$ is above a rational prime $p$ such that $\mathfrak{q} \nmid p$.
\begin{enumerate}[(a)]
\item\label{item:(a)} From Corollary~\ref{C:inertiaOrder} we know that the inertial type at~$\mathfrak{q}_2$ of $\rho_{J_r,\lambda}$
(where $\lambda \mid p \neq 2$) is either principal series or supercuspidal and that
$\rho_{J_r,\lambda}(I_{\mathfrak{q}_2})$ is cyclic of order~$r$.
Suppose first the inertial type at~$\mathfrak{q}_2$ of $\rho_{J_r,\lambda}$ is principal series. There is an abelian extension $L/K_{\mathfrak{q}_2}$ with ramification degree~$r$ where $J_r/L$ has good reduction. By local class field theory, such an extension exists if and only if $r \mid \# \mathbb{F}_{\mathfrak{q}_2}^*$ and, in this case, there is a character~$\chi$ of~$G_{K_{\mathfrak{q}_2}}$ fixing~$L$ of conductor~$\mathfrak{q}_2^1$.
Furthermore, since
$\det \rho_{J_r,\lambda}$ is
cyclotomic (hence unramified at~$\mathfrak{q}_2$) by Lemma~\ref{lem:det}
the restriction to inertia $(\rho_{J_r,\lambda} \otimes {\overline{\Q}}_p)|_{I_{\mathfrak{q}_2}}$ is non-trivial and
isomorphic to $(\chi^k \oplus \chi^{-k})|_{I_{\mathfrak{q}_2}}$ for some $k \in \mathbb{Z}$.
Since $\chi^k|_{I_{\mathfrak{q}_2}}$ is either
trivial or of conductor~$\mathfrak{q}_2$ and order~$r$ we conclude that $\rho_{J_r,\lambda}$ is of conductor~$\mathfrak{q}_2^2$ at $\mathfrak{q}_2$.
Suppose now the inertial type at~$\mathfrak{q}_2$ of $\rho_{J_r,\lambda}$
is supercuspidal. A supercuspidal representation is either exceptional or obtained by induction of a character~$\chi$ of a quadratic extension~$M/K_{\mathfrak{q}_2}$
Since $\rho_{J_r,\lambda}(I_{\mathfrak{q}_2})$
is cyclic of odd order~$r$ (hence coprime
to the residual characteristic $2$) it follows that we must be in the case of an induction from $M/K_{\mathfrak{q}_2}$ unramified for if $M/K_{\mathfrak{q}_2}$ is ramified, the induction would be irreducible and not have cyclic image. Moreover, $\rho_{J_r,\lambda}|_{D_{\mathfrak{q}_2}}$ restricted to the subgroup fixing $M/K_{\mathfrak{q}_2}$ is a principal series with corresponding characters $\chi$ and its conjugate~$\chi^s$ where $s$ is the non-trivial element in $\Gal(M/K_{\mathfrak{q}_2})$. These characters have the same conductor.
From the above discussion regarding the principal series case, it follows that $r \mid \# \mathbb{F}^*$ where $\mathbb{F}$ is the residue field of~$M$ and~$\chi$ has conductor exponent~$1$.
The conductor formula together with $M/K_{\mathfrak{q}_2}$ being unramified give that $\rho_{J_r,\lambda}$ is of conductor~$\mathfrak{q}_2^2$ at $\mathfrak{q}_2$.
\item\label{item:(b)} Suppose $r \nmid a+b$.
From Corollary~\ref{C:inertiaOrderAt7} we know that the inertial type of $\rho_{J_r,\lambda}$ at~$\mathfrak{q}_r$ (for~$\lambda \nmid r$)
is either principal series or supercuspidal and
$\rho_{J_r,\lambda}(I_{\mathfrak{q}_r})$ is cyclic of order~$4$.
The rest of the argument follows as in part~(\ref{item:(a)}) (with the extra simplification that there are no exceptional
types in odd characteristic). Thus the conductor of $\rho_{J_r,\lambda}$ at~$\mathfrak{q}_r$ is~$\mathfrak{q}_r^2$.
\item Suppose $r \mid a+b$. From Proposition~\ref{P:typeAt7b} we know that over any finite extension $L/K_{\mathfrak{q}_r}$ with ramification index~$2$ we have that $J_r/L$ has multiplicative reduction and this ramification index is minimal with respect to this property.
From Grothendieck's inertial semistable reduction criterion \cite[Expos\'e 9]{SGA7} we conclude that the action of~$G_L$ on the $p$-adic Tate module~\(V_p(J_r/L)\) of $J_r/L$ is unipotent with infinite inertia image. Since~\(\displaystyle{V_p(J_r/L) = \bigoplus_\mu\rho_{J_r,\mu}|_{G_L}}\) (where~\(\mu\) runs over the prime ideals in~\(K\) above~\(p\)), we see that
$\rho_{J_r,\lambda}|_{G_L}$ has to be unipotent with infinite inertia image, hence
a Steinberg representation. (Note that in fact all the blocks $\rho_{J_r,\mu}|_{G_L}$ are Steinberg due to strict compatibility.)
Thus $\rho_{J_r,\lambda}|_{D_{\mathfrak{q}_r}}$ is a twist of Steinberg by the quadratic character corresponding to $L/K_{\mathfrak{q}_r}$ which has conductor $\mathfrak{q}_r^1$;
therefore $\rho_{J,\lambda}$ is of conductor~$\mathfrak{q}_r^2$ at $\mathfrak{q}_r$.
\item\label{item:(d)} From Proposition~\ref{P:good_reduction}, we know that $C_r$ has good reduction at any prime~$\mathfrak{q} \nmid 2 r(a^r + b^r)$. Moreover, Proposition~\ref{P:multiplicativeRedJ} and Grothendieck's inertial reduction criterion imply that $\rho_{J_r,\lambda}|_{D_{\mathfrak{q}}}$ is a Steinberg representation
at any prime $\mathfrak{q}$ of $K$ above a prime $q \mid a^r+b^r$, $q \not=2, r$. Thus $\rho_{J_r,\lambda}$ has conductor $\mathfrak{q}^1$ at those primes.
\end{enumerate}
The last statement follows directly from Propositions~\ref{P:multiplicativeRedJ} and~\ref{P:good_reduction}.
\end{proof}
\begin{remark}
Parts~(\ref{item:(a)}) and~(\ref{item:(b)}) in the previous proof also follow from the standard fact that if the ramification is tame, i.e., if the order of inertia is coprime to the residual characteristic, then the conductor exponent is $2$ as the dimension of the representation is $2$. We decided to include the above proofs because they can be generalized to the wild case; moreover, this extension determines not only the conductor but in fact it pins down the inertial type. It has been shown at length in the literature that
fully knowing the inertial type can be essential for distinguish Galois representations and successfully applied in the modular method; we refer the reader to ~\cite{BCDF2} for a detailed treatment of this idea using elliptic Frey curves and to~\cite{BCDDF} for the case of an abelian surface.
We now give a hypothetical example of such a generalization;
suppose that we are in the principal series case in part~(\ref{item:(a)}) but instead the degree of $L/K_{\mathfrak{q}_2}$ is~$2r$. Let $\delta$ be the character defining this principal series;
now it follows
by local class field theory that $\delta^2$ has conductor~$\mathfrak{q}_2^1$ and is a power
of the order~$r$ character $\chi$ in the proof of~(\ref{item:(a)}). This yields a concrete bound on the conductor of~$\delta$; moreover, we can compute all the characters with conductor respecting this bound and retain only those satisfying $\delta^2 = \chi^k$. For each character obtained this way we get an explicit principal series representation that may be the inertial type of $J/K$ at~$\mathfrak{q}_2$.
\end{remark}
\section{Irreducibility}
\label{S:irreducibilityrrp}
To apply level lowering results a common hypothesis is that the relevant mod~$p$ representation is absolutely irreducible. When
using the modular method with Frey elliptic curves, there are powerful tools, e.g. Mazur's work on isogenies and Merel's uniform bound on torsion, that often allow to prove sharp lower bounds on~$p$ for which the $p$-torsion representation is absolutely irreducible.
For higher dimensional Frey varieties there are no such general statements;
this is one of the key steps where Darmon's program is conjectural.
The results in this section explore the local information obtained in Section~\ref{S:conductor} to prove irreducibility of the mod~$\mathfrak{p}$ representations attached to~$J_r$ for many values of~$r$ and prime ideals~\(\mathfrak{p}\) with arbitrary (large) residue characteristic~\(p\). Here we use the previous notation. In particular, $r\geq5$ is a prime, $a,b$ are coprime integers such that $ab(a^r + b^r) \neq 0$ and we write~$J_r$ for the base change of~$J_r(a,b)$ to~$K = \mathbb{Q}(\zeta_r)^+$.
We note that for the unique prime ideal~\(\mathfrak{p}_r\) above~\(r\) in~\(K\), we have established the absolute irreducibility of~\({\overline{\rho}}_{J_r,\mathfrak{p}_r}\) in Section~\ref{S:modularity}. Therefore, the results of this section will focus on proving irreducibility for~\({\overline{\rho}}_{J_r,\mathfrak{p}}\) with~\(\mathfrak{p}\) not dividing~\(r\).
\subsection{General irreducibility statements}
\begin{proposition} \label{P:irredSupercuspidal}
Assume that $r \nmid \# \mathbb{F}_{\mathfrak{q}_2}^*$ for the primes~$\mathfrak{q}_2 \mid 2$ in~$K$ and that~\(a,b\) satisfy $a \equiv 0 \pmod{2}$ and~$b \equiv 1 \pmod{4}$. Then, for all primes~$p \neq 2$ and all~$\mathfrak{p} \mid p$ in~$K$, the representation~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is absolutely irreducible.
\end{proposition}
\begin{proof} According to the discussion above, we may assume~\(p\neq r\). Corollary~\ref{C:inertiaOrder} implies that $J_r/K$ has a supercuspidal inertial type at~$\mathfrak{q}_2$. Furthermore,
from part~\eqref{item:(a)} of the proof of Theorem~\ref{T:conductorJI} we know that $\rho_{J_r,\mathfrak{p}}|_{D_{\mathfrak{q}_2}}$ is an irreducible induction from the quadratic unramified extension of~$K_{\mathfrak{q}_2}$ of a character having order~$r$ on inertia. Since $\mathfrak{p} \nmid 2r$ the residual representation ${\overline{\rho}}_{J_r,\mathfrak{p}}|_{D_{\mathfrak{q}_2}}$ is also
an irreducible induction because there is no intersection of the image of inertia with the kernel of reduction. Therefore,
${\overline{\rho}}_{J_r,\mathfrak{p}}$ is absolutely irreducible since $K$ is totally real.
\end{proof}
\begin{proposition} \label{P:irredSupercuspidalAtR}
Assume that $r \not\equiv 1 \pmod{4}$ and that $r \nmid a+b$.
Then, for all primes~$p \neq 2$ and all~$\mathfrak{p} \mid p$ in~$K$, the representation~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is absolutely irreducible.
\end{proposition}
\begin{proof}
According to the discussion before the Proposition~\ref{P:irredSupercuspidal}, we may assume~\(p\neq r\). Corollary~\ref{C:inertiaOrderAt7} implies that $J_r/K$ has a supercuspidal inertial type at~$\mathfrak{q}_r$ with inertia image of order~$4$. The result now follows as in the proof of Proposition~\ref{P:irredSupercuspidal}.
\end{proof}
The previous results give sharp lower bounds for irreducibility of the residual representations attached to~$J_r/K$ under certain contraints on~$r$ and~$a,b \in \mathbb{Z}$. The following result has a weaker conclusion
but holds for all~$r$.
\begin{proposition} \label{P:irredGeneral}
There exists a constant $B_r$, depending only on $r$, such that the following holds.
Assume~\(a,b\) satisfy $ab \neq 0$ and
$2r \nmid a+b$.
Then, for every prime $p>B_r$ and all $\mathfrak{p} \mid p$ in $K$, the representation ${\overline{\rho}}_{J_r,\mathfrak{p}}$ is absolutely irreducible.
\label{prop:big_ired}
\end{proposition}
\begin{proof}
Since $2r\nmid a+b$ we have either $2\nmid a+b$ or $r\nmid a+b$.
From Proposition~\ref{P:typeAt2} and~Proposition~\ref{P:typeAt7a}, we have that~$J_r/K$ has potentially good reduction at the primes~$\mathfrak{q}_2 \mid 2$
or at~$\mathfrak{q}_r$, respectively.
An application of~\cite[Corollary 1]{BCDF1} with $\mathfrak{q} = \mathfrak{q}_2 \mid 2$ or $\mathfrak{q}=\mathfrak{q}_r$ implies that there is a constant $C(K,g,\mathfrak{q})$ such that for every $p>C(K,g,\mathfrak{q})$ the proposition holds. We let $B_r=C(K,g,\mathfrak{q})$ be the maximum of such constants obtained by varying $\mathfrak{q} \mid 2r$. Since $K=\mathbb{Q}(\zeta_r)^+$, $g=\frac{r-1}{2}$ we conclude that $B_r$ depends only on $r$.
We now check that we can apply \cite[Corollary 1]{BCDF1}, that is we have to verify conditions (i)-(iii) of \cite[Theorem 2]{BCDF1}. Indeed, condition~(i) follows from the last statement of Theorem~\ref{T:conductorJI}; condition (ii) follows from Theorem~\ref{T:GL2type}; condition (iii) follows from Proposition~\ref{P:multiplicativeRedJ} because $ab \neq 0$ implies there is a prime $q \neq 2,r$ dividing $a^r + b^r$.
\end{proof}
When~\(r = 7\) (or~\(r = 23\) for instance), Proposition~\ref{P:irredSupercuspidal} does not apply while Proposition~\ref{P:irredSupercuspidalAtR} only gives the desired irreducibility of~\(\rho_{J_7,\mathfrak{p}}\) when~\(7\nmid a + b\) (and~\(p\) odd). We give below another criterion based on results in Class Field Theory that applies in particular to~\(r = 7\).
Write~\(g = \frac{r - 1}{2}\) and let~\(\{\infty_1,\dots,\infty_g\}\) be the set of all infinite places of~\(K\). We define a modulus~\(\mathfrak{m}\) of~\(K\) as a pair~\((\mathfrak{m}_0,\mathfrak{m}_\infty)\) consisting of an integral ideal~\(\mathfrak{m}_0\) of~\(K\) together with a (possibly empty) subset~\(\mathfrak{m}_\infty\) of~\(\{\infty_1,\dots,\infty_g\}\). We also write this formally as~\(\mathfrak{m} = \mathfrak{m}_0\mathfrak{m}_\infty\). Given a modulus~\(\mathfrak{m}\), we write~\(h_\mathfrak{m}\) for the cardinality of the corresponding ray class group (see~\cite[\S3.2]{Coh00}).
We say that a modulus~\(\mathfrak{m} = \mathfrak{m}_0\mathfrak{m}_\infty\) divides a modulus~\(\mathfrak{m}' = \mathfrak{m}_0'\mathfrak{m}_\infty'\), and we write~\(\mathfrak{m}\mid \mathfrak{m}'\), if~\(\mathfrak{m}_0\mid \mathfrak{m}_0'\) (that is~\(\mathfrak{m}_0\supset\mathfrak{m}_0'\)) and~\(\mathfrak{m}_\infty\subset\mathfrak{m}_\infty'\). When~\(\mathfrak{m}\mid\mathfrak{m}'\), we have~\(h_{\mathfrak{m}}\mid h_{\mathfrak{m}'}\) (\emph{loc. cit.}) and if moreover~\(\mathfrak{m}_\infty = \mathfrak{m}_\infty'\), then, by~\cite[Corollary~3.2.4]{Coh00}, \(h_{\mathfrak{m}'}/h_{\mathfrak{m}}\) divides
\begin{equation}\label{E:Cohen}
\phi\left(\mathfrak{m}_0'\mathfrak{m}_0^{-1}\right) = \mathrm{N}_{K/\mathbb{Q}}(\mathfrak{m}_0'\mathfrak{m}_0^{-1})\prod_{\mathfrak{q}\mid \mathfrak{m}_0'\mathfrak{m}_0^{-1}}\left(1 - \frac{1}{\mathrm{N}_{K/\mathbb{Q}}(\mathfrak{q})}\right).
\end{equation}
Write~\(m = (2^{f_2} - 1)(r - 1)\) with~\(f_2\) the residual degree of~\(K\) at~\(2\). Let~\((\epsilon_1,\dots,\epsilon_{g - 1})\) be a basis for the free part of the group of units in~\(K\).
\begin{proposition}\label{P:irred_CFT}
Assume that~\(a,b\) satisfy $a \equiv 0 \pmod{2}$ and~$b \equiv 1 \pmod{4}$, and that we have~:
\begin{enumerate}[(i)]
\item\label{item:Hi} \(g\) is an odd prime number;
\item\label{item:Hii}
the integer~\(h_{2\mathfrak{m}}/h_{\mathfrak{m}}\) is not divisible by~\(r\), where~\(\mathfrak{m} = \mathfrak{q}_r\infty_1\cdots\infty_g\).
\end{enumerate}
If, for some prime ideal~$\mathfrak{p} \mid p$ in~\(K\) with~\(p\nmid 2r\), the representation~\({\overline{\rho}}_{J_r,\mathfrak{p}}\) is not absolutely irreducible, then the following assertions hold:
\begin{enumerate}
\item\label{item:Ccl1} \(p\) is completely split in~\(K\);
\item\label{item:Ccl2} there exists an integer~\(s\) such that \(1\leq s\leq\lfloor\frac{g}{2}\rfloor\) and~\(p\) divides~\(\underset{1\leq i\leq g - 1}{\gcd}\left( \Norm_{K/\mathbb{Q}}(\epsilon_i^{2ms}-1)\right)\);
\item\label{item:Ccl3} there exist~\(s\) distinct prime ideals \(\mathfrak{p}_1,\dots,\mathfrak{p}_s\) above~\(p\) in~\(K\) such that \(r\) divides the integer~\(h_{2\widetilde{\mathfrak{m}}}/h_{\widetilde{\mathfrak{m}}}\) where~\(\widetilde{\mathfrak{m}} = \mathfrak{p}_1\cdots\mathfrak{p}_s\mathfrak{m}\).
\end{enumerate}
\end{proposition}
\begin{remark}
Under assumption~(\ref{item:Hi}), we note that the prime~\(2\) is either totally split or inert in~\(K\). In the former case, we have~\(\#\mathbb{F}_{\mathfrak{q}_2}^* = 1\) for any~\(\mathfrak{q}_2\mid 2\) and Proposition~\ref{P:irredSupercuspidal} already gives an optimal irreducibility result. In the latter case, by formula~\eqref{E:Cohen} above, the integer~\(h_{2\mathfrak{m}}/h_{\mathfrak{m}}\) is a divisor of~\(\phi(2) = 2^{g} - 1 = \#\mathbb{F}_{\mathfrak{q}_2}^*\) where~\(\mathfrak{q}_2\) is the unique prime above~\(2\). In situations where \(2^g - 1\) is divisible by~\(r\) (that is, when~\(r\equiv\pm1\pmod{8}\)), Proposition~\ref{P:irredSupercuspidal} does not apply, but the result above shows that~\({\overline{\rho}}_{J_r,\mathfrak{p}}\) is still absolutely irreducible for all but finitely many values of~\(p\) in an explicit set of primes, assuming that~\(h_{2\mathfrak{m}}/h_{\mathfrak{m}}\) is not divisible by~\(r\).
\end{remark}
\begin{proof}[Proof of Proposition~\ref{P:irred_CFT}]
Assume that the representation~\({\overline{\rho}}_{J_r,\mathfrak{p}}\) is not absolutely irreducible.
Since ${\overline{\rho}}_{J_r,\mathfrak{p}}$ is odd and $K$ is totally real it follows
that ${\overline{\rho}}_{J_r,\mathfrak{p}}: G_K\rightarrow\operatorname{GL}_2(\mathbb{F}_\mathfrak{p})$ is reducible. Here~$\mathbb{F}_{\mathfrak{p}}$ denotes the residual field of $K$ at $\mathfrak{p}$. Recall that $\det {\overline{\rho}}_{J_r,\mathfrak{p}} = \chi_p$ is the mod~$p$ cyclotomic character by Lemma~\ref{lem:det}. Therefore, we have
\[ {\overline{\rho}}_{J_r,\mathfrak{p}} \simeq \begin{pmatrix} \theta & \star\\ 0 & \theta' \end{pmatrix}
\quad \text{with} \quad \theta, \theta' : G_K \rightarrow \mathbb{F}_{\mathfrak{p}}^*
\quad \text{satisfying} \quad \theta \theta' = \chi_p.
\]
For all primes $\mathfrak{q} \nmid p$, we have $\theta'|_{I_\mathfrak{q}} = \theta^{-1}|_{I_\mathfrak{q}}$ (since $\chi_p$ is unramified away from $p$). In particular, the conductor exponent of~$\theta$ and~$\theta'$ is the same at each such prime, and these characters are unramified at primes where the conductor~$N({\overline{\rho}}_{J_r,\mathfrak{p}})$ of~\({\overline{\rho}}_{J_r,\mathfrak{p}}\) has odd valuation. Moreover, we know that $N({\overline{\rho}}_{J_r,\mathfrak{p}})$ divides the conductor~\(\mathcal{N}\) of the compatible system~\(\{\rho_{J_r,\lambda}\}\). According to Theorem~\ref{T:conductorJI}, we have $\mathcal{N} = 2^2 \mathfrak{q}_r^2\mathfrak{n}$ where~\(\mathfrak{n}\) is a squarefree ideal coprime to~\(2r\). It then follows that the conductor of $\theta$ and $\theta'$ away from~$p$ divides $2\mathfrak{q}_r$.
For each prime~$\mathfrak{q} \mid p$,
the restriction to~$I_{\mathfrak{q}}$ of the semisimplification ${\overline{\rho}}_{J_r,\mathfrak{p}}^{ss}$ is isomorphic to either
$\chi_p|_{I_\mathfrak{q}} \oplus 1$ or $\psi \oplus \psi^p$, where~$\psi$ is a fundamental character of level~$2$. Indeed, we know from the results in Section~\ref{S:conductor} that~\(J_r\) has either good or bad multiplicative reduction at~\(\mathfrak{q}\). In the former case, the desired description of~\({\overline{\rho}}_{J_r,\mathfrak{p}}^{ss}|_{I_{\mathfrak{q}}}\) follows from~\cite[Corollaire 3.4.4]{Raynaud}. In the latter case, we rely on the modularity of ~\(J_r\) proved in Section~\ref{S:modularity}: since the reduction of the variety at ~\(\mathfrak{q}\) is multiplicative the corresponding Hilbert modular form is Steinberg locally at ~\(\mathfrak{q}\), and since it is a form of parallel weight $2$ it is well-known (see for instance ~\cite{Geraghty}) that in this case the attached Galois representation $\rho_{J_r,\mathfrak{p}}$ is ordinary locally at ~\(\mathfrak{q}\), thus implying that the restriction to ~$I_{\mathfrak{q}}$ of ${\overline{\rho}}_{J_r,\mathfrak{p}}^{ss}$ is isomorphic to
$\chi_p|_{I_\mathfrak{q}} \oplus 1$.
Since $\theta$, $\theta'$ are valued in
$\mathbb{F}_{\mathfrak{p}}$ which has odd degree over~\(\mathbb{F}_p\) by assumption~(\ref{item:Hi}), the case of fundamental
characters of level 2 is excluded because $\mathbb{F}_{p^2} \not\subset \mathbb{F}_{\mathfrak{p}}$. We conclude that, for each~$\mathfrak{q} \mid p$, we have
\begin{equation}\label{E:inertiaAction}
{\overline{\rho}}_{J_r,\mathfrak{p}}^{ss}|_{I_\mathfrak{q}} \simeq \theta|_{I_\mathfrak{q}} \oplus \theta'|_{I_\mathfrak{q}} \simeq \chi_p|_{I_\mathfrak{q}} \oplus 1
\end{equation}
and exactly one of~\(\theta,\theta'\) is unramified at~\(\mathfrak{q}\), while the other one restricts to~\(I_\mathfrak{q}\) as the cyclotomic character.
\begin{itemize}
\item[(a)] Suppose that one of $\theta$, $\theta'$ is unramified at every prime
dividing $p$. We can assume it is~$\theta$ (after relabeling if needed).
From the above it follows that $\theta$ is a character
of the ray class group of modulus $2\mathfrak{m}$. From Corollary~\ref{C:inertiaOrder} we know that
${\overline{\rho}}_{J_r,\mathfrak{p}}(I_{\mathfrak{q}_2})$ is of order~\(r\) for any~\(\mathfrak{q}_2\) above~\(2\). In particular, \(r\) divides~\(\#\Gal\left(K(2\mathfrak{m})/K(\mathfrak{m})\right) = h_{2\mathfrak{m}}/h_{\mathfrak{m}}\) where~\(K(2\mathfrak{m})\) and~\(K(\mathfrak{m})\) denote the ray class fields of modulus~\(2\mathfrak{m}\) and~\(\mathfrak{m}\) respectively (see~\cite[\S3.2]{Coh00}). This contradicts the second assumption~(\ref{item:Hii}). Therefore this case does not occur.
\item[(b)] If $p$ is inert in $K$, then from \eqref{E:inertiaAction} one of $\theta, \theta'$ is unramified above the prime above $p$ and from (a) we conclude this case does not happen. Hence, $p$ is totally split in $K$ as $K/\mathbb{Q}$ is Galois of prime degree $g$, proving (\ref{item:Ccl1}).
\item[(c)] Assume now that $p$ is totally split in $K$ and that $\theta$, $\theta'$ ramify at some prime above $p$. As $g$ is odd, one of~\(\theta,\theta'\) (say~\(\theta\) after relabeling if necessary) ramifies at strictly less prime ideals than the other. In particular, there exist an integer~\(s\) such that \(1\leq s\leq\lfloor\frac{g}{2}\rfloor\) and~\(s\) distinct prime ideals \(\mathfrak{p}_1,\dots,\mathfrak{p}_s\) above~\(p\) in~\(K\) such that~\(\theta\) ramifies precisely at~\(\mathfrak{p}_1,\dots,\mathfrak{p}_s\) among all prime ideals above~\(p\). Let~\(\mathfrak{q}_2\) be a prime ideal above~\(2\) in~\(K\). From the conductor of~$\theta$, we have
that~$\theta|_{I_{\mathfrak{q}_2}}$ factors (after applying Artin's reciprocity map from class field theory)
via $(\mathcal{O}_K/\mathfrak{q}_2)^* = \mathbb{F}_{2^{f_2}}^*$
and $\theta|_{I_{\mathfrak{q}_r}}$
factors
via $(\mathcal{O}_K/\mathfrak{q}_r)^* = \mathbb{F}_r^*$. It follows that $\theta^{m}$ is unramified away from~\(\mathfrak{p}_1,\dots,\mathfrak{p}_s\) (including infinity) and that~$\theta^{m}|_{I_{\mathfrak{p}_j}} = (\chi_p|_{I_{\mathfrak{p}_j}})^{m}$ for all $1\leq j\leq s$. An application of Lemma~\ref{L:KrausAppendix} below with~\(F = K\), $S=\{ \mathfrak{p}_1,\dots,\mathfrak{p}_s\}$, \(\varphi = \theta^m\),~\(n_\mathfrak{P} = m\) (for any~\(\mathfrak{P}\in S\)) and~\(u = \epsilon_i^2\) gives~$\epsilon_i^{2ms} \equiv 1 \pmod{p}$ for all $1\leq i\leq g - 1$ (note that since $p$ splits completely in~\(K\), the norm maps are the identity map in this setting). In particular, \(p\) divides~\(\underset{1\leq i\leq g - 1}{\gcd}\left( \Norm_{K/\mathbb{Q}}(\epsilon_i^{2ms}-1)\right)\), hence proving~(\ref{item:Ccl2}).
\item[(d)] Following the argument with~$I_{\mathfrak{q}_2}$ as in (a), but where now $\theta$ is a character of the ray class group of modulus $2 \widetilde{\mathfrak{m}}$ and \(\widetilde{\mathfrak{m}} = \mathfrak{p}_1\cdots\mathfrak{p}_s\mathfrak{m}\) gives assertion~(\ref{item:Ccl3}).
\end{itemize}
\end{proof}
To complete the previous proof, it remains to prove the following auxiliary result.
\begin{lemma}\label{L:KrausAppendix}
Let $F$ be a number field with ring of integers~$\mathcal{O}_F$. Let~$p$ be a prime number unramified in~$F$ and~$S_p$ the set of places in~$F$ above~$p$. Let $S \subseteq S_p$ and
$\varphi : G_F \to \mathbb{F}_p^*$ be a character satisfying the following conditions:
\begin{enumerate}
\item $\varphi$ is unramified at all places of~$F$ outside~$S$ (including the places at infinity);
\item\label{item:2inKraus} For all~$\mathfrak{P} \in S$, the restriction $\varphi|_{I_\mathfrak{P}}$ is equal to $(\chi_p|_{I_\mathfrak{P}})^{n_\mathfrak{P}}$ for some positive integer $n_{\mathfrak{P}}$, where $\chi_p$ is the $p$th cyclotomic character.
\end{enumerate}
Then, for all totally positive units~$u \in \mathcal{O}_F^*$, we have
\[
\prod_{\mathfrak{P} \in S} N_\mathfrak{P}(u + \mathfrak{P})^{n_\mathfrak{P}} = 1,
\]
where $N_\mathfrak{P} : (\mathcal{O}_F/\mathfrak{P})^* \to \mathbb{F}_p^*$ is the norm map.
\end{lemma}
\begin{proof} This follows from class field theory. The special case where $n_\mathfrak{P} = 1$ for all~$\mathfrak{P}\in S$ is proved in~\cite[Appendice 1]{Kraus8}. The exact same proof applies for general~$n_\mathfrak{P}$; indeed, see the top of page~24 in {\it loc. cit} for the unique place where hypothesis~(\ref{item:2inKraus}) is used.
\end{proof}
When applied to~\(r = 7\), Proposition~\ref{P:irred_CFT} gives the following result which will be used in Part~\ref{Part:77p}.
\begin{corollary}
For~\(\mathfrak{p} \mid p\), \(p\neq 2\), and~\(a,b\) such that $a \equiv 0 \pmod{2}$ and~$b \equiv 1 \pmod{4}$ the representation ${\overline{\rho}}_{J_7,\mathfrak{p}}$ is absolutely irreducible.
\end{corollary}
\begin{proof}
In the notation of Proposition~\ref{P:irred_CFT}, we have~\(g = 3\) and~\(m = 42\) as the prime~\(2\) is inert in~\(K = \mathbb{Q}(\zeta_7)^+ = \mathbb{Q}(z)\) with~\(z\) defined by $z^3 - z^2 - 2z + 1 = 0$. The ray class group of modulus~\(2\mathfrak{q}_7\infty_1\infty_2\infty_3\) has order~\(2\). Moreover, the group of units in~\(K\) is generated by~\(\{-1,\epsilon_1,\epsilon_2\}\) with~\(\epsilon_1 = z\) and~\(\epsilon_2 = -z + 1\) and the prime numbers~\(p\ge5\), \(p\neq7\) dividing~\(\gcd\left( \Norm_{K/\mathbb{Q}}(\varepsilon_1^{84}-1),\Norm_{K/\mathbb{Q}}(\varepsilon_2^{84}-1)\right)\) are~\(13, 29, 43, 127, 337, 757\) and~\(2017\). They are all completely split in~\(K\). We check that for every prime ideal~\(\mathfrak{p}_1\) over such prime~\(p\), the order of the ray class group of modulus~\(2\mathfrak{q}_7\mathfrak{p}_1\infty_1\infty_2\infty_3\) is either~\(4\) or~\(12\). In particular, it is not divisible by~\(7\) and it then follows from Proposition~\ref{P:irred_CFT} that the representation~\({\overline{\rho}}_{J_7,\mathfrak{p}}$ is absolutely irreducible.
\end{proof}
\subsection{Avoiding the `bad projectively dihedral' case}
The results of the previous paragraph guarantee (under some conditions)
that~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is absolutely irreducible, but for the application of level lowering we will explain in Section~\ref{S:levelLowering} we also need that ~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is not `bad projectively dihedral', that is, ${\overline{\rho}}_{J_r,\mathfrak{p}}|_{G_{K(\zeta_p)}}$ is absolutely irreducible, where $\zeta_p$ is a primitive $p$th root of unity.
For~\(p = r\), this is the content of Theorem~\ref{T:irred7}. For~\(p\nmid 2r\), assuming that~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is absolutely irreducible, this is supplied by the following lemma (generalizing a result of Ribet~\cite{Ribet97BadDihedral} over~$\mathbb{Q}$) when applied to~\(A = J_r\) (see Propositions~\ref{P:multiplicativeRedJ} and~\ref{P:good_reduction} which show that the assumptions of the following lemma are satisfied in this case).
\begin{lemma} \label{L:badDihedral}
Let~$K/\mathbb{Q}$ be a number field.
Let $A/K$ be a~$\operatorname{GL}_2$-type abelian variety with real multiplication by a field $F$.
Let $p >3 $ be a prime unramified in~$K$, and assume that $A$ has good or multiplicative reduction at all primes of~$K$ above~$p$.
Let $\mathfrak{p} \mid p$ be a prime in~$F$ and suppose that ${\overline{\rho}}_{A,\mathfrak{p}} : G_K \to \operatorname{GL}_2(\mathbb{F}_{\mathfrak{p}})$ is absolutely irreducible.
Let $M/K$ be a quadratic extension where some
prime $\mathfrak{p}' \mid p$ in~$K$ ramifies in~$M$.
Then ${\overline{\rho}}_{A,\mathfrak{p}}|_{G_{M}}$ is absolutely irreducible.
\end{lemma}
\begin{proof}
Assume, for a contradiction, that ${\overline{\rho}}_{A,\mathfrak{p}} \otimes {\overline{\F}}_p$ is reducible when restricted to $G_M$. It follows, by~\cite[Theorem~2.12]{Feit}, that the image of
${\overline{\rho}}_{A,\mathfrak{p}}$ lies in the normalizer of a Cartan subgroup of~$\operatorname{GL}_2(\mathbb{F}_{\mathfrak{p}})$
and the projectivization $\mathbb{P}{\overline{\rho}}_{A,\mathfrak{p}}$ has dihedral image. Letting~$H$ be this dihedral image, we have the short exact sequence of groups
$$ 0 \to C \to H \to C_2 \to 0,$$
where $C$ is a cyclic group and~$C_2$ is of order 2.
Morever, the restriction of $\mathbb{P}{\overline{\rho}}_{A,\mathfrak{p}}$ to~$G_{M}$ has image~$C$
and $C_2 \simeq \Gal(M/K)$.
Let~$I_{\mathfrak{p}'} \subset G_K$ be an inertia subgroup at~$\mathfrak{p}'$, where $\mathfrak{p}' \mid p$ is a prime of~$K$ ramified in~$M$.
We claim that
$\mathbb{P}{\overline{\rho}}_{A,\mathfrak{p}}(I_{\mathfrak{p}'}) \subset H$ is cyclic of order $> 2$. Thus, since $H$ is dihedral, we have
$\mathbb{P}{\overline{\rho}}_{A,\mathfrak{p}}(I_{\mathfrak{p}'}) \subset C$ and so
$\Gal(M/K)$ corresponds to an extension where~$\mathfrak{p}'$
is unramified, a contradiction.
Let $\mathfrak{p}'$ be the prime in~$K$ above~$p$ as above. Let $z$ denote the inertial degree of $\mathfrak{p}'$. In this case,
the restriction to~${I_{\mathfrak{p}'}}$
of~${\overline{\rho}}_{A,\mathfrak{p}} \otimes {\overline{\F}}_p$
can be described by two fundamental characters of some level $w$ dividing $z$, or by a pair of conjugated fundamental characters of level $2w$ for some $w$ dividing $z$.
Following \cite[Corollaire 3.4.4]{Raynaud}, we have the following description for the action of ${I_{\mathfrak{p}'}}$:
$$\psi_s^{e_0 + e_1 p + \ldots + e_{s-1} p^{s-1}} \oplus \psi_s^{f_0 + f_1 p + \ldots + f_{s-1} p^{s-1}}$$
where $s= w$ or $s= 2 w$ depending on the case, all the numbers $e_i$ and $f_i$ are $0$ or $1$, and $e_i \neq f_i$ for every $i= 0 , 1, \ldots, s-1$. Observe that here we allow some redundancy, for example if $w>1$, $s=w$ and all $e_i$ are equal to $1$ this agrees with the case of a fundamental character of level $1$, which is a case already covered in the work of Ribet~\cite{Ribet97BadDihedral}.
We want to get a lower bound for the order $k$ of $\mathbb{P}{\overline{\rho}}_{A,\mathfrak{p}}(I_{\mathfrak{p}'})$. This is the smallest positive exponent $k$ such that:
\[
\psi_s^{ k (e_0 + e_1 p + \ldots + e_{s-1} p^{s-1})} = \psi_s^{ k (f_0 + f_1 p + \ldots + f_{s-1} p^{s-1})}.
\]
Thus, since the order of $\psi_s$ is $p^s - 1$, we search for the smallest positive $k$ such that $p^s-1$ divides $k ( (e_0 - f_0) + (e_1 - f_1) p + \ldots + (e_{s-1} - f_{s-1}) p^{s-1})$.
Since $p^s-1 = (p-1) (1+ p + \ldots+ p^{s-1})$, from the inequality
\[
\left| (e_0 - f_0) + (e_1 - f_1) p + \ldots + (e_{s-1} - f_{s-1}) p^{s-1} \right| \leq 1+ p + \ldots+ p^{s-1},
\]
we conclude that $k$ must satisfy $k \geq p-1$, and since we are assuming that $p$ is at least $5$ we conclude that $k \geq 4$.
\end{proof}
\section{Finiteness of specializations of Frey representations}
\label{S:finiteness}
Besides irreducibility, another essential hypothesis for level lowering is that the relevant mod~$p$ representation
is finite at all primes~$\mathfrak{p} \mid p$. In the context of the modular method, this guarantees the representation is of parallel weight 2, hence independent of~$p$ and the putative solution. When working with Frey elliptic curves, the theory of the Tate curve provides a simple criterion, using the valuation of the minimal discriminant, to decide if the $p$-torsion representation is indeed finite. In~\cite{DarmonDuke}, Darmon implicitly uses that an analogous criterion is available for Frey varieties, but we are not aware of a complete reference.
We note also that~\cite{Ellenberg} gives a related criterion which is a direct generalization of the usual criterion for elliptic curves. However, it is hard to use due to the need to determine a discriminantal set.
In this section we give such a finiteness criterion that in particular can be applied to the Frey variety~$J_r$ and suffices
for our Diophantine application below.
More concretely, the objective of this section is to prove Theorem~\ref{T:finite} which is a critical technical result needed for the proof of Proposition~\ref{P:SerreCond}. We shall derive it from the more general Theorem~\ref{finiteness}.
\subsection{Finiteness of residual Galois representations}
Let $\ell$ be a prime. Let $K$ be a finite extension of $\mathbb{Q}_\ell$ with ring of integers $\mathcal{O}$, normalized valuation $v$, and residue field $k$. Let $G$ be the absolute Galois group of $K$, $P \subseteq I$ the wild inertia and inertia subgroup of $G$, respectively.
Let $K^{\operatorname{nr}}$ be the maximal unramified extension of $K$, which is fixed by the inertia subgroup~$I$ of $G$.
Let $K^\text{tr}$ be the maximal tamely unramified extension of $K^{\operatorname{nr}}$, which corresponds to the wild inertia subgroup $P$ of $I$.
We denote also by~$\upsilon$ the extension of~$\upsilon$ to~$K^{\operatorname{nr}}$.
Let ${\overline{\rho}} : G \rightarrow \operatorname{GL}_{2}(\mathbb{F})$ be a representation with $\mathbb{F}$ a finite field of size $p^f$, where $p$ is a prime.
The group ${\overline{\rho}}(I)$ corresponds to the Galois group of a finite totally ramified extension $M/K^{\operatorname{nr}}$ and ${\overline{\rho}}(P)$ is an elementary $p$-group in $\operatorname{GL}_{2}(\mathbb{F})$ which corresponds to a finite tamely ramified subextension $M^t$ of $M/K^{\operatorname{nr}}$.
We have that $M = M^t(x_1^{1/p}, \ldots, x_m^{1/p})$ for some $x_i \in K^{\operatorname{nr}}$ and $p^m = \#{\overline{\rho}}(P)$ by Kummer theory.
We say that ${\overline{\rho}}$ is {\it peu ramifi\'e} if $\upsilon(x_i) \equiv 0 \pmod p$ for all $i$. Note the case that ${\overline{\rho}}$ is tamely ramified (i.e. ${\overline{\rho}}(P) = 1$) is included in the peu-ramifi\'e case.
We say that ${\overline{\rho}}$ is {\it finite} if ${\overline{\rho}}$ arises from a finite flat group
scheme over $\mathcal{O}$.
\begin{theorem}
\label{ppff-equivalence}
${\overline{\rho}}$ is peu ramifi\'e if and only if ${\overline{\rho}}$ is finite.
\end{theorem}
\begin{proof}
This is \cite[Proposition~4.2.1]{Ste19};
see also \cite[Proposition 8.2]{Edixhoven} for another proof in the case $K = \mathbb{Q}_\ell$.
\end{proof}
\begin{theorem}
\label{peu-ramifie-finite}
Suppose the extension cut out by the representation ${\overline{\rho}} : G \rightarrow \operatorname{GL}_{2}(\mathbb{F})$ is contained in an extension of the form $K^\text{nr}(y_1^{1/p}, \ldots, y_n^{1/p})$ in the case $\ell \not= p$, and $K^\text{tr}(y_1^{1/p}, \ldots, y_n^{1/p})$ in the case $\ell = p$.
Assume $y_j \in K^\text{nr}$ and $v(y_j) \equiv 0 \pmod p$ for all $j = 1,\ldots, n$.
Then ${\overline{\rho}}$ is unramified in the case $\ell \not= p$, and finite in the case $\ell = p$.
\end{theorem}
\begin{proof} Recall that $P$ is the largest pro-$\ell$ subgroup of $G$.
If $\ell \not= p$, then the hypotheses imply that ${\overline{\rho}}$ is unramified.
If $\ell = p$, then we have that the extension $M^t(x_1^{1/p}, \ldots, x_m^{1/p})$ corresponding to ${\overline{\rho}}(P)$ is contained in $K^\text{tr}(y_1^{1/p}, \ldots, y_n^{1/p})$. By Kummer theory, the condition $v(y_j) \equiv 0 \pmod p$ for all $j$ implies that $v(x_i) \equiv 0 \pmod p$ for all $i$. Hence, ${\overline{\rho}}$ is peu-ramifi\'e at $\ell = p$. By Theorem~\ref{ppff-equivalence}, ${\overline{\rho}}$ is finite.
\end{proof}
Before proceeding, we introduce some notation that will be of use until the end of this paragraph.
For $z_0 \in \mathbb{P}^1(K)$, let $\pi_0(t) = t - z_0$ if $z_0 \neq \infty$ and $\pi_0(t) = 1/t$ if $z_0 = \infty$. We will write $\pi_0$ for $\pi_0(t)$. Let $R = \mathcal{O}[[\pi_0]]$ and let~$F$ be its field of fractions.
Let $A$ be an abelian variety over the function field $K(t)$ of dimension $d$. Assume~:
\begin{enumerate}[(i)]
\item $A$ has a model over~$R$ with multiplicative reduction at $\pi_0$ with full toric rank;
\item\label{item:abelian_scheme} $A$ extends to an abelian scheme over~$\mathcal{O}((\pi_0))$;
\item\label{item:mult_red} $A$ over~$k[[\pi_0]]$ has multiplicative reduction at $\pi_0$ with full toric rank.
\end{enumerate}
Condition~(\ref{item:abelian_scheme}) ensures that $A(s_0)$ for $s_0 \not= z_0$ in the unit disk centered at $z_0$ is an abelian variety over $K$ and condition~(\ref{item:mult_red}) ensures such $A(s_0)$ has multiplicative reduction over $K$.
Replacing $K$ by a finite unramified extension if necessary, we may also assume that $A$ over~$K[[\pi_0]]$ has split multiplicative reduction at $\pi_0$. By Mumford uniformization \cite{Mumford} (see also~\cite[Theorem~4.5]{Lutkebohmert}), we have that
\begin{equation}
\label{mumford-uniform}
A(\overline{F}) \cong \mathbb{G}_m^d(\overline{F})/Y,
\end{equation}
where $Y$ is a set of periods and $d = \dim A$ (a set of periods $Y$ is a subgroup of $\mathbb{G}_m^d(F)$ isomorphic to $\mathbb{Z}^d$ satisfying an analogue of the Riemann conditions; see~\cite[Definition~1.1]{Mumford}).
Let~$X(T) = \Hom(T,\mathbb{G}_m)$ be the group of (algebraic) characters from $T = \mathbb{G}_m^d$ to~$\mathbb{G}_m$. Denote by~$\delta_1, \ldots, \delta_d$ the component characters for~$X(T)$. Let $y_1, \ldots, y_d$ be a basis for $Y$.
\begin{proposition}
\label{Tate-period}
For all $\chi$ in~$X(T)$ and $y \in Y$, we have that
\(\chi(y) = \pi_0^nu(\pi_0)\) where~$n$ is an integer and~$u(\pi_0)$ is a unit in~$\mathcal{O}[[\pi_0]]$.
\end{proposition}
\begin{proof}
We prove the statement for~$\chi_i(y_j)$ where~$\chi_1, \ldots, \chi_d$ is a basis for~$X(T)$. As $\chi_i(y_j) \in F^\times$, by Weierstrass preparation theorem, it is uniquely of the form
\begin{equation*}
\pi^m \frac{f(\pi_0)}{g(\pi_0)} u(\pi_0),
\end{equation*}
where $\pi$ is a uniformizer for $\mathcal{O}$, $m$ is an integer, $f(\pi_0), g(\pi_0) \in \mathcal{O}[\pi_0]$ are distinguished polynomials in $\pi_0$, and $u(\pi_0)$ is a unit in $\mathcal{O}[[\pi_0]]$. Using functoriality of Mumford uniformization with respect to the base, the periods of $A/k[[\pi_0]]$ are precisely the reductions of the periods of $A/K[[\pi_0]]$, hence $m = 0$.
If $f$ or $g$ is not a power of $\pi_0$, then there is an element $s_0 \not= z_0 \in \mathcal{O}'$ the ring of integers of a finite extension $K'/K$ such that $f(\pi_0)/g(\pi_0)$ and hence $\chi_i(y_j)$ has a zero or pole at $s_0$. Similarly, the periods of $A(s_0)$ are precisely the specializations of the periods of $A/\mathcal{O}[[\pi_0]]$, hence $\chi_i(y_j)$ cannot have a zero or pole at $s_0$.
\end{proof}
\begin{theorem}
\label{finiteness}
Assume further the following two hypothesis~:
\begin{enumerate}
\item\label{item:finiteness1} $A$ is of $\operatorname{GL}_2$-type with real multiplications by $F \hookrightarrow \End_{K(t)}(A) \otimes \mathbb{Q}$;
\item\label{item:finiteness3} there exists~$t_1 \in \mathbb{P}^1(K)$ satisfying $\ord_K(\pi_0(t_1)) > 0$ and $\ord_K(\pi_0(t_1) )\equiv 0 \pmod p$.
\end{enumerate}
Let $\mathfrak{p}$ be any prime of $F$ above a rational prime~$p$. Then $A(t_1)[\mathfrak{p}]$ is unramified (resp.\ finite) if $\ell \not= p$ (resp.\ $\ell = p$).
\end{theorem}
\begin{proof}
Recall that~$\delta_i$ are the component characters for~$X(T)$ and $y_j(\pi_0)$ is a basis for $Y$.
By Proposition~\ref{Tate-period}, we write
\begin{equation*}
\delta_i(y_j(\pi_0)) = \alpha_n \pi_0^n + \alpha_{n+1} \pi_0^{n+1} + \alpha_{n+2} \pi_0^{n+2} + \ldots\in\pi_0^nR
\end{equation*}
where $\alpha_n \in \mathcal{O}^\times$. By assumption~\eqref{item:finiteness3}, we have~$\ord_K(\pi_0(t_1)) > 0$. Therefore, $\delta_i(y_j(\pi_0))$ converges to an element in~$K$ after evaluating at~$t = t_1$. By the second part of assumption~\eqref{item:finiteness3} and the fact that~$\alpha_n$ is a unit, we conclude that~$\ord_K(\delta_i(y_j(\pi_0(t_1)))) \equiv 0 \pmod p$ for all $i,j$. Hence, by assumption~\eqref{item:finiteness1} and~\eqref{mumford-uniform}, we have
\begin{equation*}
K(A(t_1)[\mathfrak{p}]) \subseteq K(A(t_1)[p]) \subseteq K(\zeta_p) \left( \left\{ \delta_i(y_j(\pi_0(t_1)))^{1/p} : i, j = 1, \ldots, d \right\} \right),
\end{equation*}
where $\zeta_p$ is a primitive $p$th root of unity. By Theorem~\ref{peu-ramifie-finite}, $A(t_1)[\mathfrak{p}]$ is unramified (resp.\ finite) if $\ell \not= p$ (resp.\ $\ell = p$).
\end{proof}
\subsection{Specializations coming from solutions}
We now give the criterion to prove finiteness of $\mathfrak{p}$-torsion representations arising on our Frey varieties. We again use the notation of the previous sections. In particular, $r\geq5$ is a prime, $a,b$ are coprime integers such that $a^r + b^r \neq 0$ and we write~$J_r$ for the base change of~$J_r(a,b) = \Jac(C_r(a,b))$ to~$K = \mathbb{Q}(\zeta_r)^+$.
\begin{theorem}\label{T:finite}
Let~\(p\) be a rational prime number.
Let~$\mathfrak{q}$ be a prime in~$K$ not dividing~$2r$ such that~$v_\mathfrak{q}(a^r + b^r)\equiv 0\pmod{p}$. We have the following conclusions~:
\begin{itemize}
\item If $\mathfrak{q}$ does not divide $p$, then~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is unramified at~$\mathfrak{q}$ for all~\(\mathfrak{p}\mid p\) in~\(K\);
\item If $\mathfrak{q}$ divides $p$, then~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is finite at~$\mathfrak{q}$ for all~\(\mathfrak{p}\mid p\) in~\(K\).
\end{itemize}
\end{theorem}
\begin{remark}
When~\(p\nmid r - 1\), we note by~\eqref{E:discriminant} that the condition~\(v_\mathfrak{q}(a^r + b^r)\equiv 0\pmod{p}\) in Theorem~\ref{T:finite} is equivalent to the more usual condition~\(v_\mathfrak{q}(\Delta(C_r)) \equiv 0\pmod{p}\), where~\(\Delta(C_r)\) is the discriminant of~\(C_r = C_r(a,b)\) defined in~\eqref{kraushyper}.
\end{remark}
\begin{proof}[Proof of Theorem~\ref{T:finite}]
Suppose $\mathfrak{q}$ lies above the rational prime $q$ not dividing $2r$.
If $q \nmid a^r+b^r$, then $J_r$ has good reduction at $q$, and we have the desired conclusions.
Let us now assume that~$q\mid a^r + b^r$, and hence~$q\nmid ab$. Recall from Lemma~\ref{model-t} that $C'_r(t)$ is given by
\begin{equation}
C'_r(t) \; : \; y^2 = x^r + c_1 \alpha^2 x^{r-2} + \ldots + c_{\frac{r-1}{2}} \alpha^{r-1} x + \alpha^{r-1} (2t - 1),
\end{equation}
where $\alpha$ satisfies equation~\eqref{eq:sign_for_alpha}. According to equation~\eqref{eq:disc_C_r_dash}, the discriminant of $C'_r(t)$ is given by
\begin{equation*}
\Delta_r'(t) = (-1)^\frac{r-1}{2} 2^{2(r - 1)}r^r \alpha^{(r - 1)^2} = (-1)^{\frac{r - 1}{2}}2^{2(r - 1)}r^r(t(1 - t))^{\frac{(r - 1)^2}{2}}.
\end{equation*}
Letting $u = 1/t$ so $\alpha^2 = (u-1)/u^2$, we obtain
\begin{equation}
C'_r(u) \; : \; y^2 = x^r + c_1 \alpha^2 x^{r-2} + \ldots + c_{\frac{r-1}{2}} \alpha^{r-1} x + \alpha^{r-1} (2 - u)/u,
\end{equation}
whose discriminant is given by
\begin{equation}
\Delta_r'(u) = (-1)^\frac{r-1}{2} 2^{2(r - 1)} r^r \left(\frac{u-1}{u^2}\right)^\frac{(r - 1)^2}{2}.
\end{equation}
Let~$i$ be a fixed primitive fourth root of unity. Replacing $x$ by $x/(iu)$ and $y$ by $y/(iu)^\frac{r}{2}$, yields the hyperelliptic curve
\begin{equation*}
W'_r(u) \; : \; y^2 = x^r - c_1 \alpha^2 u^2 x^{r-2} + \ldots \pm c_{\frac{r-1}{2}} \alpha^{r-1} u^{r-1} x + i^r \alpha^{r-1} u^{r-1} (2 - u) ,
\end{equation*}
with discriminant
\begin{equation}
\label{disc-u}
2^{2(r - 1)} r^r (u-1)^\frac{(r - 1)^2}{2} u^{r-1}.
\end{equation}
Let~$\mathcal{O}'$ be the ring of integers of $K' = K_\mathfrak{q}(i)$, where $K_\mathfrak{q}$ is the completion of $K$ at $\mathfrak{q}$.
Let us write~$t_0 = \frac{a^r}{a^r+b^r}$ and~$J' = \Jac(W_r'(u))$. By Lemma~\ref{L:twistedKraus}, the curve~$C'_r(t_0)$ is a quadratic twist of $C_r(a,b)$ by $-\frac{(ab)^\frac{r-1}{2}}{a^r+b^r}$. Therefore, $W'_r(u_0)$ is the quadratic twist of $C_r(a,b)$ by $-ib^{\frac{r - 1}{2}}/a^{\frac{r + 1}{2}}$. As~$q$ does not divide~$2ab$, the desired result will follow from the same conclusion for~${\overline{\rho}}_{J',\mathfrak{p}}$. Therefore, we wish to apply Theorem~\ref{finiteness} to the abelian variety~$J'$ and~$t_1 = \frac{1}{t_0}$.
From the discriminant formula~\eqref{disc-u}, it follows that~$J'$ extends to an abelian scheme over~$\mathcal{O}'((u))$, since~$r$, $u$ and~$u - 1$ are all invertible in~$\mathcal{O}'((u))$.
Since~$v_q(a^r + b^r)\equiv0\pmod{p}$, we have~$v_\mathfrak{q}(t_1) > 0$ and $v_\mathfrak{q}(t_1) \equiv 0 \pmod p$. In particular, condition~\eqref{item:finiteness3} of Theorem~\ref{finiteness} is satisfied.
To apply Theorem~\ref{finiteness} to~$J'$ we have to show that the above model for $W'_r(u)$ over $\mathcal{O}'[u]$ is such that it has the maximal number of double roots over~$K'[[u]]$ and~$k'[[u]]$ when $u = 0$, where~$k'$ denotes the residue field of~$K'$. We first notice that~$(\alpha u)^2 = u - 1$ reduces to~$-1$ modulo~$u$ and hence $i^r\alpha^{r - 1}u^{r - 1} = i^r((\alpha u)^2)^{\frac{r - 1}{2}}$ reduces to~$i^r(-1)^{\frac{r - 1}{2}} = i^{2r - 1} = i$. Therefore the reduction of~$W'_r(u)$ modulo~$u$ is given by
\[
y^2 = x^r + c_1x^{r - 2} + \ldots + c_{\frac{r - 1}{2}}x + 2i = xh(x^2 + 2) + 2i,
\]
where~$h$ is defined in~\eqref{eq:def_h}. Let us assume~$r\equiv 1\pmod{4}$. From Lemma~\ref{L:firstKind}, we know that $xh(x^2 + 2) = 2iT_r\left(\frac{x}{2i}\right)$ where $T_r(x)$ is the $r$-th Chebyshev polynomial of the first kind. Hence, $-2i$ is a root of~$xh(x^2 + 2) + 2i$ since~$T_r(-1) = -1$; by Lemma~\ref{evaluation-point}, $-i\omega_j$ for~$j = 1,\dots,\frac{r - 1}{2}$ are also roots of~$xh(x^2 + 2) + 2i$. Moreover, according to Lemma~\ref{L:secondKind}, these latter roots have multiplicity~$>1$. Since~$xh(x^2 + 2) + 2i$ has degree~$r$, they are double roots and we have found that this polynomial has the maximal number of double roots.
Finally observe that~$\omega_j-\omega_k = \zeta_r^j(1 - \zeta_r^{k - j})(1 - \zeta_r^{-j - k})$ is a unit~$K'(\zeta_r)$ unless~$j = k$. Therefore, the elements~$-i\omega_j$ in~$\mathcal{O}'$ reduce to different elements in~$k'$. In particular, the above model also has the maximal number of double roots over~$k'[[u]]$, completing the verification of the hypotheses in Theorem~\ref{finiteness} in the case~$r\equiv1\pmod{4}$. We proceed similarly in the case~$r\equiv 3\pmod{4}$.
\end{proof}
\section{Refined level lowering}
\label{S:levelLowering}
This section focuses on the `level lowering' step by combining results from the previous two sections. Our approach replaces the classical level lowering theorems of Fujiwara--Jarvis--Rajaei~\cite{Fuj,Jarv,Raj} by a
result of Breuil--Diamond~\cite[Theorem~3.2.2]{BreuilDiamond}. The latter has more restrictive hypothesis, but it gives modularity of all lifts with prescribed inertial types as long as the types are compatible with the fixed residual representation. As we shall discuss here and again in Section~\ref{S:Frey7}, this approach has various advantages which are fundamental for the elimination step.
We call this approach {\em level lowering with prescribed types} or simply {\em refined level lowering}.
Let~$r, p \geq 5$ be primes, and $d \in \mathbb{Z}_{>0}$
not divisible by any $p$th power. Assume $p \nmid 2rd$.
Suppose $(a,b,c)$ is a non-trivial primitive solution to equation~\eqref{E:rrp}, that is,
\[
a^r + b^r = dc^p, \qquad abc \ne 0, \qquad \gcd(a,b,c) = 1.
\]
Throughout this section, we assume that~\(a,b\) satisfy the following parity condition:
\[
a\equiv0\pmod{2}\quad\text{and}\quad b\equiv1\pmod{4}.
\]
As usual, we let $J_r = J_r(a,b)/K$ be the attached Frey variety, where $K=\mathbb{Q}(\zeta_r + \zeta_r^{-1})$. Recall from Theorem~\ref{T:GL2typeJr} that~$J_r$ is of~$\operatorname{GL}_2$-type with real multiplications by~$K$. Let $\mathfrak{p}$ be a prime ideal in~$K$ dividing~\(p\). We write~\({\overline{\rho}}_{J_r,\mathfrak{p}}\) for the corresponding representation of~\(G_K\) (see Section~\ref{S:Freyrrp}). We denote by~\(\mathfrak{q}_r\) the unique prime ideal above~\(r\) in~\(K\).
\subsection{Finiteness of the representation and Serre level}
For a representation ${\overline{\rho}} : G_K \to \operatorname{GL}_2({\overline{\F}}_p)$ its {\it Serre level~$N({\overline{\rho}})$} is defined as the Artin conductor of~${\overline{\rho}}$ away from~$p$.
The following proposition gives the finite possible values for the Serre level of~\({\overline{\rho}}_{J_r,\mathfrak{p}}\). It uses the results from Theorems~\ref{T:conductorJI} and~\ref{T:finite}.
\begin{proposition} \label{P:SerreCond}
The following assertions hold.
\begin{enumerate}
\item\label{item:Serre_level} The Serre level~$N({\overline{\rho}}_{J_r,\mathfrak{p}})$ of ${\overline{\rho}}_{J_r,\mathfrak{p}}$ divides $2^2 \mathfrak{q}_r^2 \mathfrak{n}_d $
where $\mathfrak{n}_d$ is the squarefree product of the primes $\mathfrak{q} \nmid 2r$ such that $\mathfrak{q} \mid d$.
\item\label{item:finite_rep} For each prime $\mathfrak{q} \mid p$ in~$K$,
the representation ${\overline{\rho}}_{J_r,\mathfrak{p}}|_{D_\mathfrak{q}}$ is finite.
\end{enumerate}
\end{proposition}
\begin{proof}
From Theorem~\ref{T:conductorJI} we know that the conductor of
$\rho_{J,\mathfrak{p}}$ and $\mathcal{N} = 2^2 \mathfrak{q}_r^2 \mathfrak{n}$ (where $\mathfrak{n}$ is the product of all prime ideals dividing~$a^r + b^r$ which are coprime to~$ 2r$) agree away from the places above~\(p\). We have~\(\mathfrak{n}_d\mid\mathfrak{n}\) and we set~\(\mathfrak{n}' = \mathfrak{n}\n_d^{-1}\). Let~\(\mathfrak{q}\) be a prime in~\(K\) dividing~\(\mathfrak{n}'\), but not dividing~\(p\). Then~\(\mathfrak{q}\) is coprime to~\(2rd\) and divides~\(a^r + b^r\). Therefore, $p$ divides the valuation at~$\mathfrak{q}$ of~\(a^r + b^r\) and from the first part of Theorem~\ref{T:finite}, it follows that~\({\overline{\rho}}_{J_r,\mathfrak{p}}\) is unramified at~\(\mathfrak{q}\), hence proving~\eqref{item:Serre_level}.
Let now~\(\mathfrak{q}\) be a prime ideal in~\(K\) above~\(p\). By our assumptions, \(\mathfrak{q}\) is coprime to~\(2rd\), and as before the valuation at~$\mathfrak{q}$ of~\(a^r + b^r\) is divisible by~\(p\). We finally conclude using the second part of Theorem~\ref{T:finite}.
\end{proof}
\subsection{Level lowering}
\begin{theorem}\label{T:levelLowering}
Suppose that ${\overline{\rho}}_{J_r,\mathfrak{p}}$ is absolutely irreducible. Then, there is a Hilbert newform~$g$ over $K$ satisfying the following properties:
\begin{enumerate}[(i)]
\item\label{item:LLitemi} $g$ is of parallel weight 2, trivial character and level $2^2 \mathfrak{q}_r^2 \mathfrak{n}_d$;
\item\label{item:LLitemii} ${\overline{\rho}}_{J_r,\mathfrak{p}} \cong {\overline{\rho}}_{g,\mathfrak{P}}$ for some $\mathfrak{P} \mid p$ in the field of coefficients~$K_g$ of~$g$;
\item\label{item:LLitemiii} for all~$\mathfrak{q}_2 \mid 2$ in~$K$, we have $(\rho_{g,\mathfrak{P}} \otimes {\overline{\Q}}_p)|_{I_{\mathfrak{q}_2}} \simeq \delta \oplus \delta^{-1}$, where~$\delta$ is as character of order~$r$;
\item\label{item:LLitemiv} $K \subset K_g$.
\end{enumerate}
Moreover, if $\mathfrak{n}_d \neq 1$ then~$g$ has no complex multiplication.
\end{theorem}
\begin{proof}
From Theorem~\ref{T:modularity}, $J_r/K$ is modular,
hence~${\overline{\rho}}_{J_r,\mathfrak{p}}$ is modular.
From the proof of part~\eqref{item:(a)} of Theorem~\ref{T:conductorJI}, we know that the inertial type of~$J_r/K$ at any $\mathfrak{q}_2 \mid 2$ is
of the form $(\rho_{J_r,\mathfrak{p}} \otimes {\overline{\Q}}_p)|_{I_{\mathfrak{q}_2}} \simeq \delta \oplus \delta^{-1}$, where~$\delta$ is a character of order~$r$. The same proposition also gives that
$\rho_{J_r,\mathfrak{p}}$ is Steinberg at all~$\mathfrak{q} \nmid 2\mathfrak{q}_r$ dividing~$d$. In particular, since $\rho_{J_r,\mathfrak{p}}$ is a lift of ${\overline{\rho}}_{J_r,\mathfrak{p}}$, these inertial types are compatible with the residual representation~${\overline{\rho}}_{J_r,\mathfrak{p}}$.
By our assumptions and
Lemma~\ref{L:badDihedral} we know that~${\overline{\rho}}_{J_r,\mathfrak{p}}|_{G_{K(\zeta_p)}}$
is absolutely irreducible. Now, for all $p > 5$, the conclusions (\ref{item:LLitemi})--(\ref{item:LLitemiii}) follow from Proposition~\ref{P:SerreCond} and~\cite[Theorem~3.2.2]{BreuilDiamond}.
For $p=5$ (hence $r \geq 7$) the conclusions (\ref{item:LLitemi})--(\ref{item:LLitemiii}) follow by the same results if we additionally show
that~${\overline{\rho}}_{J_r,\mathfrak{p}}(G_{K(\zeta_5)}) \not\cong \operatorname{PSL}_2(\mathbb{F}_5)$. Indeed, this is the case because $\operatorname{PSL}_2(\mathbb{F}_5)$ has order 60,
$[K(\zeta_5): K] = 4$ and
${\overline{\rho}}_{J_r,\mathfrak{p}}(I_{\mathfrak{q}_2})$
has prime order~$r \geq 7$ (because there is no intersection of the image of inertia $\rho_{J_r,\mathfrak{p}}(I_{\mathfrak{q}_2})$ with the kernel of reduction mod~$\mathfrak{p} \mid 5$).
Conclusion~(\ref{item:LLitemiv}) follows from Proposition~\ref{P:coefficientField} below. Finally, if~$\mathfrak{n}_d \neq 1$ then~$g$ has a Steinberg prime which is incompatible with~$g$ having complex multiplication.
\end{proof}
\begin{proposition}\label{P:coefficientField}
Let~$K$ be any totally real field. Let
$g$ a Hilbert modular form over~$K$ with field of coefficients~$K_g$. Let $\lambda$ be a prime in~$K_g$ above the
rational prime~$p$.
Let~$\mathfrak{q} \nmid p$ be a prime in~$K$.
Suppose that $(\rho_{g,\lambda} \otimes {\overline{\Q}}_p)|_{I_\mathfrak{q}} \simeq \delta \oplus \delta^{-1}$, where $\delta$ has order~$n$.
Then $\mathbb{Q}(\zeta_n + \zeta_n^{-1}) \subset K_g$ where $\zeta_n$ is a primitive $n$th root of unity.
\end{proposition}
\begin{proof} We know that the system of Galois representations~$\rho_{g,\lambda} : G_K \to \operatorname{GL}_2(K_{g,\lambda})$ is strictly compatible. Thus,
for all primes $\lambda$ in~$K_g$ such that $\lambda \nmid \Norm(\mathfrak{q})$, the restriction~$\rho_{g,\lambda}|_{I_\mathfrak{q}}$ has the same shape, hence, using the hypothesis, we have
$\zeta_n + \zeta_n^{-1} \in K_{g,\lambda}$ for every such~$\lambda$.
This implies that the extension $K_g(\zeta_n + \zeta_n^{-1})/K_g$ is of degree~$1$ because almost all primes are split in it and hence $\mathbb{Q}(\zeta_n + \zeta_n^{-1}) \subset K_g$.
\end{proof}
\begin{remark}
The previous result relates the field of coefficients of a Hilbert modular form to its inertial types. This allows for
conclusion~(\ref{item:LLitemiv}) in Theorem~\ref{T:levelLowering} which
turns out to be a powerful new elimination technique
only available in the context of Frey abelian varieties which are not elliptic curves; indeed, it is well known that for an elliptic curve~$J$, if the restriction to inertia of~$\rho_{J,p}$ at a prime~$q \neq p$ is of the form $\delta \oplus \delta^{-1}$ then $\delta$ is a character of order $n=2,3,4$ or~$6$. In this case,
Proposition~\ref{P:coefficientField} leads to the trivial conclusion that $\mathbb{Q}(\zeta_n + \zeta_n^{-1}) = \mathbb{Q} \subset K_g$.
\end{remark}
Let $\chi_r$ denote the mod~$r$ cyclotomic character. Note that $\chi_r|_{G_K}$ is a quadratic character.
\begin{corollary}\label{C:levelLowering}
Assume the hypotheses of Theorem~\ref{T:levelLowering}.
Suppose further that $r \mid a+b$.
Then, there is a Hilbert newform~$g$ over $K$ satisfying the following properties:
\begin{enumerate}[(i)]
\item $g$ is of parallel weight 2, trivial character and level $2^2 \mathfrak{q}_r \mathfrak{n}_d$;
\item ${\overline{\rho}}_{J_r,\mathfrak{p}} \otimes \chi_r|_{G_K} \cong {\overline{\rho}}_{g,\mathfrak{P}}$ for some $\mathfrak{P} \mid p$ in the field of coefficients~$K_g$ of~$g$;
\item for all~$\mathfrak{q}_2 \mid 2$ in~$K$, we have $(\rho_{g,\mathfrak{P}} \otimes {\overline{\Q}}_p)|_{I_{\mathfrak{q}_2}} \simeq \delta \oplus \delta^{-1}$, where~$\delta$ is as character of order~$r$;
\item $K \subset K_g$.
\end{enumerate}
Moreover, if $\mathfrak{n}_d \neq 1$ then~$g$ has no complex multiplication.
\end{corollary}
\begin{proof}
Assume all hypotheses of Theorem~\ref{T:levelLowering} plus
$r \mid a+b$.
The quadratic extension $\mathbb{Q}(\zeta_r)/K$ fixed by the quadratic character $\chi_r|_{G_K}$ is unramified away from~$r$ and has ramification index $e=2$ at~$\mathfrak{q}_r$. From Proposition~\ref{P:typeAt7b} and its proof it follows that the Jacobian of the quadratic twist of $C_r/K$ corresponding to~$\mathbb{Q}(\zeta_r)/K$ has conductor~$\mathfrak{q}_r^1$ at~$\mathfrak{q}_r$. Now applying the arguments in the proofs of Proposition~\ref{P:SerreCond} and Theorem~\ref{T:levelLowering} where we replace $\rho_{J_r, \mathfrak{p}}$ by $\rho_{J_r, \mathfrak{p}} \otimes \chi_r|_{G_K}$ yields the corollary.
\end{proof}
{\large \part{A multi-Frey approach
to~$x^7 + y^7 = 3z^p$ using Frey abelian varieties}\label{Part:77p} }
In Section~\ref{S:overCubic}, we prove Theorem~\ref{T:main} using the Frey elliptic curve~$F_{a,b}$ and its twists. We show that using a multi-Frey approach combining~$F_{a,b}$ with the curve~$E_{a,b}$ from Section~\ref{S:overQ} reduces the computational time needed.
In the final sections of this part, we give two more proofs of Theorem~\ref{T:main} using the multi-Frey approach combining the Frey variety~$J_7(a,b)$ from Section~\ref{S:Freyrrp} with
both curves $F_{a,b}$ and~$E_{a,b}$, one of which results in the proof requiring the least computational time of all.
Let~$\zeta_7$ be a primitive $7$th root of unity in~$\mathbb{C}$. Throughout this part, we will use the following factorization and notation:
\begin{equation}\label{eq:basic_factorization}
x^7+y^7=(x+y)\phi_7(x,y)=(x+y)f_1(x,y)f_2(x,y)f_3(x,y)
\end{equation}
where~$\phi_7(x,y)=x^6-x^5y+x^4y^2-x^3y^3+x^2y^4-xy^5+y^6$ and for~$i=1,2,3$,
\[
\omega_i=\zeta_7^i+\zeta_7^{-i},\quad f_i(x,y)=x^2 + \omega_i xy + y^2.
\]
\section{A Frey curve over $\mathbb{Q}$}\label{S:overQ}
Let~$a$ and~$b$ be coprime integers. We consider the elliptic curve~$E_{a,b}$ defined over~$\mathbb{Q}$ by the equation
\begin{equation}\label{eq:EoverQ}
E_{a,b}\ : \ Y^2 = X^3 + a_2X^2 + a_4 X + a_6,
\end{equation}
where
\begin{eqnarray*}
a_2 & = & -(a-b)^2, \\
a_4 & = & -2a^4 + a^3 b - 5a^2 b^2 + ab^3 - 2b^4, \\
a_6 & = & a^6 - 6a^5 b + 8a^4 b^2 - 13a^3 b^3 + 8a^2b^4 - 6ab^5 + b^6.
\end{eqnarray*}
The standard invariants attached to the model~\eqref{eq:EoverQ} are
\begin{eqnarray*}
c_4(E_{a,b}) & = & 2^4\cdot 7(a^4 - a^3b + 3a^2b^2 - ab^3 + b^4), \\
c_6(E_{a,b}) & = & 2^5\cdot 7(a^6 - 15a^5b + 15a^4b^2 - 29a^3b^3 + 15a^2b^4 - 15ab^5 + b^6), \\
\Delta(E_{a,b}) & = & 2^4\cdot7^2\phi_7(a,b)^2.
\end{eqnarray*}
This curve has been considered by Kraus (\cite[\S4.5.1.3]{kraus5}) and by Freitas (up to the change of variables given by~$X'=6^2(X-(a-b)^2/3)$ and~$Y'=6^3Y$, this is the curve denoted~$E_{(a,b)}$ in~\cite[p.~618]{F}; note the sign mistake in the definition of the~$a_4$ coefficient in the published version of \emph{loc. cit.} though).
Let $F/\mathbb{Q}_\ell$ be a finite extension and $F^{\operatorname{unr}}$ its maximal unramified extension. Let $E/F$ and elliptic curve with additive potential good reduction. The extension $F^{\operatorname{unr}}(E[p])$ where $p \geq 3$ is a prime~$\neq \ell$ is independent of~$p$ and it is
the minimal extension of~$F^{\operatorname{unr}}$ where $E$ obtains good reduction.
The degree $e = [F^{\operatorname{unr}}(E[p]) : F^{\operatorname{unr}}]$ is called the semistability defect of~$E$.
\begin{proposition}\label{P:conductorE}
The model~\eqref{eq:EoverQ} of~$E_{a,b}$ is minimal. The conductor~$N_{E_{a,b}}$ of~$E_{a,b}$ is given by $N_{E_{a,b}}=2^{\alpha}7^2\mathrm{rad}_7(\phi_7(a,b))$, where~$\mathrm{rad}_7(\phi_7(a,b))$ is the product of all primes $\neq 7$ dividing~$\phi_7(a,b)$ and
\begin{equation}\label{eq:cond_of_E_at_2}
\alpha = \left\{ \begin{array}{ll}
2 & \quad \text{if $4 \mid ab$},\\
3 & \quad \text{if $2 \parallel ab$ or $4\mid a+b$}, \\
4 & \quad \text{if $2\parallel a+b$}. \\
\end{array} \right.
\end{equation}
Moreover, we have the following properties~:
\begin{itemize}
\item the curve $E_{a,b}$ has additive potentially good reduction at~$7$
with semistability defect $e=3$ or~$6$ if~$7\mid a+b$ or~$7\nmid a+b$, respectively;
\item the curve $E_{a,b}$ has additive potentially good reduction at~$2$ with semistability
defect $e=6$ or~$24$ if~$ab$ is even or~$ab$ is odd, respectively;
\item if~$E_{a,b}$ has bad reduction at a prime~$\ell\not=2,7$, then $\ell\equiv1\pmod{7}$ and~$v_\ell(\Delta(E_{a,b}))=v_\ell(\phi_7(a,b))=v_\ell(a^7+b^7)$.
\end{itemize}
\end{proposition}
\begin{proof}
We first recall that~$\phi_7(a,b)$ and~$a+b$ are coprime away from~$7$. Besides, $7\mid \phi_7(a,b)$ if and only if~$7\mid a+b$, and in that case, we have $v_7(\phi_7(a,b))=1$ (\cite[Lemma~2.1]{DahmenSiksek}). Therefore, the model~\eqref{eq:EoverQ} is minimal at~$7$ with~$(v_7(c_4(E_{a,b})), v_7(\Delta(E_{a,b})))=(2,4)$ or~$(1,2)$ according to whether~$7\mid a+b$ or~$7\nmid a+b$. Hence~$E_{a,b}$ has additive potentially good reduction at~$7$; in particular, $v_7(N_{E_{a,b}})=2$. Its semistability defect~$e$ at~$7$ is equal to the denominator of~$v_7(\Delta(E_{a,b}))/12$ (cf. \cite[p.~312]{Ser72}).
We check that~$(v_2(c_4(E_{a,b})),v_2(c_6(E_{a,b})), v_2(\Delta(E_{a,b})))=(4,5,4)$. Hence the model~\eqref{eq:EoverQ} of~$E_{a,b}$ is minimal at~$2$ and~$E_{a,b}$ has additive potential good reduction at~$2$. According to~\cite[p.~358]{kraus6}, its semistability defect~$e$ at~$2$ is~$6$ or~$24$ if~$ab$ is even or~$ab$ is odd respectively. We now compute the valuation at~$2$ of~$N(E_{a,b})$. By~\cite{papado}, we are in a case~$3$, $4$ or~$5$ in Tate's algorithm and hence $v_2(N(E_{a,b}))$ is~$4$, $3$ or~$2$ respectively. If~$ab$ is even then Proposition~1 in~\emph{loc. cit.} (applied with~$(r,t)=(0,1)$) shows that we are in a case~$\ge4$. Moreover, by Proposition~2 of \emph{loc. cit.} (and in its notation), we have~$b_8\equiv -(ab)^2\pmod{8}$ and hence we are in case~$5$ if~$4\mid ab$ and in case~$4$ if~$2\parallel ab$. Similarly, if~$ab$ is odd, then Propositions~1 and~2 of~\emph{loc. cit.} (applied with~$(r,t)=(1,1)$) show that we are in case~$3$ or~$4$ if~$2\parallel a+b$ or~$4\mid a+b$ respectively.
Let~$\ell\not=2,7$ be a prime of bad reduction. Then, $\ell\mid\phi_7(a,b)$ and, since, $\ell\nmid a+b$, we have~$\ell\equiv 1\pmod{7}$. Moreover, we check that~$c_4(E_{a,b})=-2^4(A_1A_2+A_1A_3+A_2A_3)$ where
\[
A_1=(\omega_3-\omega_2)f_1(a,b),\quad A_2=(\omega_1-\omega_3)f_2(a,b),\quad A_3=(\omega_2-\omega_1)f_3(a,b).
\]
Since $A_1$, $A_2$ and~$A_3$ are coprime away from~$7$ and~$\ell$ divides~$A_1A_2A_3=7\phi_7(a,b)$, we see that~$E_{a,b}$ has bad multiplicative reduction at~$\ell$.
\end{proof}
\begin{lemma}\label{lem:irredE}
Let~$p$ be a prime, $p\ge5$. Then the representation ${\overline{\rho}}_{E_{a,b},p}$ is irreducible.
\end{lemma}
\begin{proof} According to Proposition~\ref{P:conductorE}, the curve~$E_{a,b}$ has additive potential good reduction at~$2$ with semistability defect $e=6$ or~$24$. It then follows from~\cite[cor.~3.4]{billereyIrred} that the representation~${\overline{\rho}}_{E_{a,b},p}$ is irreducible when $p \ge 5$.
\end{proof}
Let us now assume that there exists an integer~$c$ such that~$(a,b,c)$ is a non-trivial primitive solution to equation~\eqref{E:77p}
for an exponent~$n \geq 2$. The case $n = 2$ of Theorem~\ref{T:main} follows from \cite[Theorem 1.1]{BenSki} and the case $n = 3$ from \cite[Theorem 1.5]{BenVatYaz}. The case $n = 7$ is covered by Theorem~2 in \cite[Section~4.3]{Serre87} by noticing that the proof in {\it loc.\ cit.} holds for exponents~$\geq 5$ when $L = 3$. Therefore, we can assume~$n = p$ is prime and~$p\geq 5$, $p\neq 7$.
Write~$E=E_{a,b}$ for simplicity and denote by~$N(\bar{\rho}_{E,p})$ the Serre level
(i.e. Artin conductor away from~$p$)
of~$\bar{\rho}_{E,p}$. If $p \neq 7$, by Proposition~\ref{P:conductorE} and~\cite{kraus7}, then~$N(\bar{\rho}_{E,p})=2^\alpha\cdot7^2$ where~$\alpha$ is defined in~\eqref{eq:cond_of_E_at_2}.
For an integer $M > 0$, let $S_2(M)$ denote the set of cuspforms of weight 2, trivial character and level $M$. By the classical modularity and level-lowering theorems, there exists a newform $f \in S_2\left(2^\alpha\cdot7^2\right)$ and a prime~$\mathfrak{p} \mid p$ in~${\overline{\Q}}$ such that
\begin{equation}
\bar{\rho}_{E,p} \simeq \bar{\rho}_{f,\mathfrak{p}}.
\label{E:iso}
\end{equation}
The following theorem shows this isomorphism cannot hold in many cases.
\begin{theorem}\label{T:overQ}
Let $p = 5$ or $p \geq 11$ be a prime.
Suppose there is a non-trivial primitive
solution~$(a,b,c)$ to equation~\eqref{E:77p} with exponent~$n = p$ and
write $E = E_{a,b}$ for the Frey curve.
Then, we are in one of the following situations:
\begin{enumerate}
\item\label{item1} $4\mid ab$ and~$\bar{\rho}_{E,p}\simeq\bar{\rho}_{E_{1,0},p}$~;
\item\label{item2} $(2 \parallel ab$ or $4\mid a+b)$ and~$\bar{\rho}_{E,p}\simeq\bar{\rho}_{E_{1,-1},p}$.
\end{enumerate}
Moreover, we are in case~\eqref{item1} or in case~\eqref{item2} if $7\nmid a+b$ or~$7\mid a+b$, respectively.
In particular, equation~\eqref{E:77p} does not have non-trivial primitive solutions~$(a,b,c)$ with~$ab$ odd unless~$7\mid a + b$.
\end{theorem}
\begin{proof} Note that when $2 \mid a+b$ then $4 \mid a+b$ due to the shape of \eqref{E:77p}. In particular, from Proposition~\ref{P:conductorE} it follows that the conductor of $E$ at~$2$ is either $2^2$ or $2^3$.
From Proposition~\ref{P:conductorE}, for every prime~$q\not=2,7,p$ and such that~$q\not\equiv 1\pmod{7}$ the curve ~$E$ has good reduction. Then, from
the isomorphisn~\eqref{E:iso}, there exists~$(x,y)\in\{0,\dots,q-1\}^2\backslash\{(0,0)\}$ such that
\[
a_q(f)\equiv a_q(E_{x,y})\pmod{\mathfrak{p}}.
\]
Using these congruences for all such primes~$q\leq40$, we are able to contradict the isomorphism~\eqref{E:iso} for all newforms~$f$, except those corresponding to the elliptic curves~$E_{1,0}$ and~$E_{1,-1}$ in levels~$2^2\cdot7^2$ and~$2^3\cdot7^2$, respectively. These computations only take few seconds.
Now the statement about $7$-adic condition follows directly from the computation of the semistability defect at~$7$ in Proposition~\ref{P:conductorE}.
The last statement follows from cases~(\ref{item1}) and~(\ref{item2}).
\end{proof}
\section{A Frey curve over a totally real cubic field}
\label{S:overCubic}
In this section, we extend the methods of~\cite{F} and prove Theorem~\ref{T:main} using a single Frey curve defined over a totally real cubic field. In Remark~\ref{rk:EandF}, we also discuss how the information from the Frey curve~$E$ obtained in Theorem~\ref{T:overQ} helps us reduce the total running time for the computations needed in this proof.
We keep the notation of Section~\ref{S:overQ}. Put~$z=-\omega_1 = -(\zeta_7+\zeta_7^{-1})$. The number field~$K=\mathbb{Q}(z)$ is defined by $z^3 - z^2 - 2z + 1 = 0$. It has discriminant~$49$ and is the cubic totally real subfield of~$\mathbb{Q}(\zeta_7)$. Let~$\mathcal{O}_K$ denotes its integer ring. We write $\mathfrak{q}_2$, $\mathfrak{q}_3$ and $\mathfrak{q}_7$ for the unique primes in $K$ above 2, 3 and 7, respectively.
Let~$a$ and~$b$ be coprime integers with~$a+b\not=0$. We consider the Frey curve $F_{a,b} := E_{(a,b)}^{(2,1)}$ as defined in \cite[p.~619]{F} given by the model
\begin{equation}\label{eq:FoverK}
F_{a,b}\ : \ Y^2 = X(X-A_{a,b})(X+B_{a,b}),
\end{equation}
where
\begin{eqnarray*}
A_{a,b} & = & (-2 + z + z^2)(a + b)^2 = (\omega_2 - \omega_1)(a + b)^2 \\
B_{a,b} & = & (4 - z^2)(a^2 - zab + b^2) = (2 - \omega_2)(a^2 + \omega_1ab + b^2).
\end{eqnarray*}
The standard invariants of model~\eqref{eq:FoverK} are
\begin{eqnarray*}
c_4(F_{a,b}) & = & 2^4(A_{a,b}^2 + A_{a,b}B_{a,b} + B_{a,b}^2), \\
c_6(F_{a,b}) & = & 2^5(2A_{a,b}^3 + 3A_{a,b}^2B_{a,b} - 3A_{a,b}B_{a,b}^2 - 2B_{a,b}^3), \\
\Delta(F_{a,b}) & = & 2^4\left(A_{a,b}B_{a,b}C_{a,b}\right)^2,
\end{eqnarray*}
where
\[
C_{a,b}=-(A_{a,b} + B_{a,b})=(\omega_1 -2)(a^2 + \omega_2ab + b^2).
\]
Note that $F_{a,b}$ has full $2$-torsion defined over $K$.
Write $F = F_{a,b}$. For~$\delta\in \mathcal{O}_K\backslash\{0\}$, a Weierstrass model for the quadratic twist~$F^{(\delta)}$ of~$F$ by~$\delta$ is given by
\begin{equation}\label{eq:FtwistedoverK}
y^2 = x(x - \delta A_{a,b})(x + \delta B_{a,b}).
\end{equation}
Its standard invariants are given by
\[
c_4\big(F^{(\delta)}\big) = \delta^2 c_4(F),\qquad c_6\big(F^{(\delta)}\big) = \delta^3 c_6(F),\qquad \Delta\big(F^{(\delta)}\big) = \delta^6 \Delta(F).
\]
For a prime ideal~$\mathfrak{q}$ in~$K$ and an ideal~$\mathfrak{a}$, we denote by~$v_\mathfrak{q}(\mathfrak{a})$ the valuation at~$\mathfrak{q}$ of~$\mathfrak{a}$. For an element~$\alpha\in\mathcal{O}_K\backslash\{0\}$, we write~$v_\mathfrak{q}(\alpha)$ for~$v_\mathfrak{q}(\alpha\mathcal{O}_K)$.
\begin{proposition}
Let~$\delta\in\mathcal{O}_K$. Denote by~$N_{F^{(\delta)}}$ the conductor of~$F^{(\delta)}$.
\begin{itemize}
\item If~$\delta$ is coprime to~$\mathfrak{q}_7$, then we have
\begin{equation*}
v_{\mathfrak{q}_7}(N_{F^{(\delta)}}) = \left\{ \begin{array}{ll}
1 & \quad \text{if $7 \mid a + b$},\\
2 & \quad \text{if $7 \nmid a + b$}.\\
\end{array} \right.
\end{equation*}
\item If~$\delta\equiv 1\pmod{\mathfrak{q}_2^2}$, then we have
\begin{equation*}
v_{\mathfrak{q}_2}(N_{F^{(\delta)}}) = \left\{ \begin{array}{ll}
0 & \quad \text{if $4 \parallel a + b$},\\
1 & \quad \text{if $8 \mid a + b$},\\
3 & \quad \text{if $4 \mid ab$}, \\
4 & \quad \text{if $2\parallel a+b$ or~$2\parallel ab$}. \\
\end{array} \right.
\end{equation*}
\item If~$2\parallel ab$ and~$\delta\equiv z^2 - 2 \pmod{\mathfrak{q}_2^2}$, then we have~$v_{\mathfrak{q}_2}(N_{F^{(\delta)}}) = 3$.
\item If~$7 \nmid a + b$ and~$v_{\mathfrak{q}_7}(\delta)$ is odd, then we have~$v_{\mathfrak{q}_7}(N_{F^{(\delta)}}) = 0$.
\end{itemize}
Moreover, a prime ideal~$\mathfrak{q}\nmid \mathfrak{q}_2\cdot\mathfrak{q}_7\cdot\delta$ is a prime of bad reduction for~$F^{(\delta)}$ if and only if it divides~$(a + b)(a^2 + \omega_1ab + b^2)(a^2 + \omega_2ab + b^2)$. In that case, $F^{(\delta)}$ has bad multiplicative reduction at~$\mathfrak{q}$ and we have
\[
v_\mathfrak{q}\big(\Delta(F^{(\delta)})\big) =
\left\{
\begin{array}{ll}
4v_\mathfrak{q}(a^7 + b^7) & \text{if $\mathfrak{q}\mid a + b$}, \\
2v_\mathfrak{q}(a^7 + b^7) & \text{otherwise}. \\
\end{array}
\right.
\]
\label{P:conductorF}
\end{proposition}
\begin{proof} For simplicity, write~$A = A_{a,b}$, $B = B_{a,b}$, $C = C_{a,b}$ and denote by~$(c_4,c_6,\Delta)$ and $\big(c_4^{(\delta)},c_6^{(\delta)},\Delta^{(\delta)}\big)$ the standard invariants of~$F$ and~$F^{(\delta)}$ respectively.
We have that~$(a + b)^2$, $f_1(a,b)$ and~$f_2(a,b)$ are coprime away from~$\mathfrak{q}_7$ and so are~$A$, $B$ and~$C$ since~$\omega_2 - \omega_1$, $2 - \omega_2$ and~$\omega_1 -2$ all generate the prime ideal~$\mathfrak{q}_7$.
Let~$\mathfrak{q}$ be a prime ideal such that $\mathfrak{q}\nmid \mathfrak{q}_2\cdot\mathfrak{q}_7\cdot\delta$. Then~$\mathfrak{q}$ divides~$\Delta^{(\delta)}$ if and only if it divides~$(a + b)(a^2 + \omega_1ab + b^2)(a^2 + \omega_2ab + b^2)$. In that case, $\mathfrak{q}$ does not divide~$c_4^{(\delta)}$. Hence, $F^{(\delta)}$ has bad multiplicative reduction at~$\mathfrak{q}$ and
\begin{align*}
v_\mathfrak{q}\big(\Delta^{(\delta)}\big) = v_\mathfrak{q}(\Delta) & = 2v_{\mathfrak{q}}\left((a+b)^2(a^2 + \omega_1ab + b^2)(a^2 + \omega_2ab + b^2)\right) \\
& =\left\{
\begin{array}{ll}
4v_\mathfrak{q}(a^7 + b^7) & \text{if $\mathfrak{q}\mid a + b$} \\
2v_\mathfrak{q}(a^7 + b^7) & \text{otherwise} \\
\end{array}
\right.
\end{align*}
according to the factorization~\eqref{eq:basic_factorization}.
Let us compute the valuation of the conductor at~$\mathfrak{q}_7$ under the assumptions of the statement.
Assume~$7\mid a+b$ and $\delta$ is coprime to~$\mathfrak{q}_7$. Then, we have~$v_{\mathfrak{q}_7}(A) = 1 + 6v_7(a + b)$ and~$v_{\mathfrak{q}_7}(B) = 1$. Therefore, we have
\[
\big(v_{\mathfrak{q}_7}\big(c_4^{(\delta)}\big),v_{\mathfrak{q}_7}\big(\Delta^{(\delta)}\big)\big) = \big(v_{\mathfrak{q}_7}(c_4),v_{\mathfrak{q}_7}(\Delta)\big) = \big(4,10+12v_7(a+b)\big).
\]
In particular, the model~\eqref{eq:FtwistedoverK} is not minimal at~$\mathfrak{q}_7$, the curve~$F^{(\delta)}$ has bad multiplicative reduction at~$\mathfrak{q}_7$ and we have~$v_{\mathfrak{q}_7}(N_{F^{(\delta)}})=1$.
Assume~$7\nmid a+b$. Then we have~$v_{\mathfrak{q}_7}(A) = v_{\mathfrak{q}_7}(B) = v_{\mathfrak{q}_7}(C) = 1$. In particular, we have~\(v(c_4)\ge2\). If moreover~$\delta$ is coprime to~$\mathfrak{q}_7$, then we have
\[
\big(v_{\mathfrak{q}_7}\big(c_4^{(\delta)}\big), v_{\mathfrak{q}_7}\big(\Delta^{(\delta)}\big)\big) = \big(v_{\mathfrak{q}_7}(c_4), v_{\mathfrak{q}_7}(\Delta)\big) = (\ge 2, 6).
\]
In particular, the model~\eqref{eq:FtwistedoverK} is minimal at~$\mathfrak{q}_7$, the curve~$F^{(\delta)}$ has bad additive reduction at~$\mathfrak{q}_7$ and we have~$v_{\mathfrak{q}_7}(N_{F^{(\delta)}}) = 2$.
If instead we have~$v_{\mathfrak{q}_7}(\delta)$ odd, then we have
\[
\big(v_{\mathfrak{q}_7}\big(c_4^{(\delta)}\big),v_{\mathfrak{q}_7}\big(\Delta^{(\delta)}\big)\big)
= \big(2v_{\mathfrak{q}_7}(\delta) + v_{\mathfrak{q}_7}(c_4), 6v_{\mathfrak{q}_7}(\delta) + v_{\mathfrak{q}_7}(\Delta)\big)
= \big(2v_{\mathfrak{q}_7}(\delta) + v_{\mathfrak{q}_7}(c_4), 6(v_{\mathfrak{q}_7}(\delta) + 1)\big).
\]
Write~\(v_{\mathfrak{q}_7}(\delta) = 2k + 1\). We have~\(\big(v_{\mathfrak{q}_7}\big(c_4^{(\delta)}\big),v_{\mathfrak{q}_7}\big(\Delta^{(\delta)}\big)\big) = \big(v_{\mathfrak{q}_7}(c_4) + 4k + 2, 12(k + 1)\big)\).
Therefore, the model~\eqref{eq:FtwistedoverK} is not minimal at~$\mathfrak{q}_7$ (\cite[Tableau~I]{papado}). Applying the substitution $x \rightarrow x/u^{2(k + 1)}$, $y\rightarrow y/u^{3(k + 1)}$ with~\(u = 2 - z - z^2\) (a generator of~\(\mathfrak{q}_7\)) shows that the curve~$F^{(\delta)}$ has good reduction at~$\mathfrak{q}_7$. Hence we have~$v_{\mathfrak{q}_7}(N_{F^{(\delta)}}) = 0$.
Let us now compute the valuation of the conductor at~$\mathfrak{q}_2$ under the assumptions of the statement.
Assume first that~$2\nmid ab$ and~$\delta\equiv 1\pmod{\mathfrak{q}_2^2}$. Then, we have $2\mid a + b$, $v_{\mathfrak{q}_2}(A) = 2v_2(a + b)$ and~$v_{\mathfrak{q}_2}(B) = v_{\mathfrak{q}_2}(C) = 0$. Moreover~$\delta$ is coprime to~$\mathfrak{q}_2$ and hence we have
$$
\big(v_{\mathfrak{q}_2}\big(c_4^{(\delta)}\big),v_{\mathfrak{q}_2}\big(c_6^{(\delta)}\big),v_{\mathfrak{q}_2}\big(\Delta^{(\delta)}\big)\big) = \big(v_{\mathfrak{q}_2}(c_4),\, v_{\mathfrak{q}_2}(c_6),\, v_{\mathfrak{q}_2}(\Delta)\big) = \big(4,\, 6,\, 4+4v_2(a+b)\big).
$$
If~$v_2(a+b)=1$, then the model~\eqref{eq:FtwistedoverK} corresponds to a case $6$, $7$ or $8$ in Tate's classification with corresponding valuation~$v_{\mathfrak{q}_2}(N_{F^{(\delta)}}) = 4$, $3$ or~$2$ (\cite[Tableau~IV]{papado}). Applying~\cite[Prop.~3]{papado} with, in its notation, $r = 2z$ and $t = 2z(1+z)$ we get that we are in Case~$6$ in Tate's classification. Hence we have~$v_{\mathfrak{q}_2}(N_{F^{(\delta)}}) = 4$ in this case.
Similarly, if~$v_2(a+b)\ge2$, then the model~\eqref{eq:FtwistedoverK} corresponds to either a case $7$ in Tate's classification (with Kodaira type $\mathrm{I}_\nu^*$ where $\nu = 4(v_2(a + b) - 1)\geq4$) or a non-minimal case (\cite[Tableau~IV]{papado}). We apply~\cite[Prop.~4]{papado} with $r = 0$ and~$s = 1 + z$ to conclude that we are in a non-minimal case. Therefore we have good reduction if~$v_2(a+b)=2$ and bad multiplicative reduction otherwise.
Assume now~$2\mid ab$ and~$\delta$ coprime to~$\mathfrak{q}_2$. Then, $A$, $B$ and $C$ reduce to~$\omega_1 + \omega_2$, $\omega_2$ and~$\omega_1$ modulo~$\mathfrak{q}_2$ respectively and we have
\[
\frac{c_4}{2^4}\equiv A^2 + AB + B^2\not\equiv0\pmod{\mathfrak{q}_2}\quad\text{and}\quad\frac{c_6}{2^5}\equiv ABC\not\equiv0\pmod{\mathfrak{q}_2}.
\]
Hence, we have
$$
\big(v_{\mathfrak{q}_2}\big(c_4^{(\delta)}\big),v_{\mathfrak{q}_2}\big(c_6^{(\delta)}\big),v_{\mathfrak{q}_2}\big(\Delta^{(\delta)}\big)\big) = \big(v_{\mathfrak{q}_2}\big(c_4\big),v_{\mathfrak{q}_2}\big(c_6\big),v_{\mathfrak{q}_2}\big(\Delta\big)\big) = \big(4,\, 5,\, 4\big).
$$
Therefore, we are in Case~$3$, $4$ or $5$ of Tate's classification (\cite[Tableau~IV]{papado}) with corres\-ponding valuation~$v_{\mathfrak{q}_2}(N_{F^{(\delta)}}) = 4$, $3$ or~$2$.
If~$\delta\equiv 1\pmod{\mathfrak{q}_2^2}$, then applying~\cite[Prop.~1 and~2]{papado} with~$(r,t) = (1+z+z^2,z)$, we get that we are in Case~$3$ or in Case~$4$ of Tate's classification if~$2\parallel ab$ or $4\mid ab$ respectively.
If~$2\parallel ab$ and~$\delta \equiv z^2 - 2\pmod{\mathfrak{q}_2^2}$, applying~\cite[Prop.~1 and~2]{papado} with~$(r,t) = (z+z^2,1+z+z^2)$, we get that we are in Case~$4$ in Tate's classification. Hence we have~$v_{\mathfrak{q}_2}(N_{F^{(\delta)}}) = 3$.
\end{proof}
\begin{remark}\label{rem:twist}
From Proposition~\ref{P:conductorF} we see that under some $2$-adic or $7$-adic conditions taking a quadratic twist of~$F$ by an appropriate~$\delta$ yields $F_{a,b}^\delta$ with smaller conductor. This conductor reduction is essential to make the modular method using~$F_{a,b}$ computationally feasible
in Section~\ref{S:Frey7first}.
Since this observation is likely to be useful in future applications of the modular method let
us briefly describe how we found
the twist~$\delta = z^2-2$ in Table~\ref{table:delta} which according to Proposition~\ref{P:conductorF} reduces the conductor exponent at~$\mathfrak{q}_2$ from 4 to~$3$ when~$2 \Vert ab$. Indeed, a standard calculation shows the semistability defect of $F_{1,1}$ at~$\mathfrak{q}_2$ is $e=24$,
and so $F_{1,1}$ obtains good reduction
over an extension $L/K_{\mathfrak{q}_2}^{{\operatorname{unr}}}$ such that $\Gal(L/K_{\mathfrak{q}_2}^{{\operatorname{unr}}}) \simeq \operatorname{SL}_2(\mathbb{F}_3)$, and $F_{1,1}$ has an exceptional supercuspidal inertial type at~$\mathfrak{q}_2$. In particular, no quadratic twist will reduce the size of inertia, however these exceptional types tend to appear (up to twist) at odd conductor exponents or large even conductor
exponents (see \cite[Appendix]{dia95} for the case over~$\mathbb{Q}_2$) and since $v_{\mathfrak{q}_2}(N_{F_{1,1}}) = 4$ we might still find a twist having lower odd conductor exponent.
With the help of {\tt Magma} we can compute the $2$-Selmer group of~$K$ unramified outside~$2$ which allows to recover all quadratic extensions of $K$ ramified only at~$\mathfrak{q}_2$. Each such extension corresponds to a possible~$\delta$ and we check the conductor of~$F_{1,1}^\delta$ until it lowers.
\end{remark}
The following result generalizes~\cite[Theorem~11.2]{F}.
\begin{proposition} \label{P:irredOverK}
Let~$p = 5$ or $p \geq 11$ be a prime. Write $F = F_{a,b}$. Then, the representation ${\overline{\rho}}_{F,p}$ is irreducible.
\end{proposition}
\begin{proof} Suppose ${\overline{\rho}}_{F,p}$ is reducible, that is,
\[ {\overline{\rho}}_{F,p} \sim \begin{pmatrix} \theta & \star\\ 0 & \theta' \end{pmatrix}
\quad \text{with} \quad \theta, \theta' : G_K \rightarrow \mathbb{F}_p^*
\quad \text{satisfying} \quad \theta \theta' = \chi_p.\]
The characters $\theta$ and $\theta'$ ramify only at $p$ and additive primes of $F$; the latter can be
$\mathfrak{q}_{7}$ and $\mathfrak{q}_2 = (2)$ according to Proposition~\ref{P:conductorF}. When these primes are
of potentially good reduction, \cite[Theorem~5.1]{Jarv2} implies there is no conductor degeneration mod~$p$. Moreover, in case~$\mathfrak{q}_2$ or~$\mathfrak{q}_7$ are of additive potentially multiplicative reduction, it follows from the theory of the Tate curve that
the conductor of ${\overline{\rho}}_{F,p}$ at such primes is the same as that of~$F$ (even when $\star = 0$ on inertia).
Therefore, at an additive prime $\mathfrak{q}$
both $\theta$, $\theta'$ have conductor
exponent equal to $\upsilon_\mathfrak{q}(N_F)/2$.
We conclude that the possible conductors
away from~$p$ for
$\theta$ and $\theta'$ are divisors of $2^2 \mathfrak{q}_{7}$.
In the notation of \cite[Theorem~1]{FS}
we set $\epsilon_1 = -z$ and $\epsilon_2 = 1+z$,
observe that the unit group of~$K$ is
generated by $\{-1, \epsilon_1, \epsilon_2\}$ and compute $B = 7 \cdot 13^3$.
Thus from the first paragraph of the proof of \cite[Theorem~1]{FS} we
conclude that for $p=11$ and $p \geq 17$ exactly one of $\theta$, $\theta'$
ramifies at $p$. So, replacing $F$ by a $p$-isogenous curve if necessary, we can assume $\theta$ is unramified at $p$.
We conclude that the conductor of $\theta$ divides $2^2 \mathfrak{q}_{7}$.
Let $\infty_1$, $\infty_2$ and $\infty_3$ be the real places of $K$.
The Ray class groups for modulus $2^ 2 \mathfrak{q}_{7} \infty_1 \infty_2 \infty_3$
is isomorphic to
\[
\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}.
\]
Thus $F$ has a quadratic twist with a $p$-torsion point defined over $K$. From the proof of \cite[Theorem 4]{BruNaj}, we see that this is impossible for $p > 13$, concluding the proof
in this case.
Note that for $p = 11$, $F$ gives rise to a $K$-rational point on $X_0(11)$ and for $p = 5$ it gives
a $K$-rational point on $X_0(20)$ (since $F$ has full $2$-torsion over $K$). By the proof of \cite[Lemma 10.4]{AnniSiksek}, the $K$-rational points on $X_0(20)$ are cuspidal and those on $X_0(11)$ correspond to elliptic curves with integral $j$-invariant. Since $F$ has multiplicative
reduction at 3 (by the last statement of Proposition~\ref{P:conductorF}) its $j$-invariant is not integral. We conclude that $F$ cannot correspond to any of these curves, proving the result for $p=5,11$.
Suppose $p = 13$. If one of $\theta$ or $\theta'$ is unramified at all the primes in $K$ above $13$,
replacing $F$ by a $13$-isogenous curve if necessary, we can assume
again the conductor of $\theta$ divides $2^2 \mathfrak{q}_{7}$. Arguing with Ray class groups as above, we conclude that, up to quadratic twist, $F$ has a $13$-torsion point over $K$; since it also
has full $2$-torsion this gives rise to a $K$-rational point on $X_1(26)$.
By the proof of \cite[Lemma 10.4]{AnniSiksek}, this is not possible.
Then, both $\theta$ and $\theta'$ ramify at some prime above $13$.
Let $\mathfrak{p}_1$, $\mathfrak{p}_2$, $\mathfrak{p}_3$ denote the primes of $K$ dividing~$13$.
Since $13$ is unramified in $K$ and $F$ is semistable at the $\mathfrak{p}_i$
it follows that one of $\theta$, $\theta'$ ramifies at one $\mathfrak{p}_i$ and the other
at the other two primes (see \cite[Lemme~1]{Kraus3} and its proof).
So, replacing $F$ by a $13$-isogenous curve if necessary, we may assume $\theta$
is ramified at exactly one prime above 13, say $\mathfrak{p}_j$. Furthermore, since $F$ has either
multiplicative or good ordinary reduction at $\mathfrak{p}_j$, we also know that
$\theta|_{I_{\mathfrak{p}_j}} = \chi_{13}|_{I_{\mathfrak{p}_j}}$. Thus the field cut out by $\theta$ has
modulus dividing $2^2 \mathfrak{q}_7 \mathfrak{p}_{j}$.
The Ray class groups of modulus $2^2 \mathfrak{q}_7 \mathfrak{p}_{1}^{e_1} \mathfrak{p}_{2}^{e_2} \mathfrak{p}_{3}^{e_3} \infty_1 \infty_2 \infty_3$, where $e_1 + e_2 + e_3 = 1$ are all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^5$. Therefore, up to a quadratic twist,
$F$ gives again a $K$-rational point on $X_1(26)$, which is ruled out as above.
\end{proof}
\section{A first proof of Theorem~\ref{T:main} using twists of Frey curves}
\label{S:Frey7first}
Let us now assume that there exists an integer~$c$ such that~ $(a,b,c)$ is a non-trivial primitive solution to equation~\eqref{E:77p} for an exponent~$n \geq 2$. According to the discussion after Lemma~\ref{lem:irredE}, we can assume~$n = p$ is a prime, $p\ge 5$, $p\neq 7$. Write again~$F=F_{a,b}$ for simplicity and define~$\delta\in\mathcal{O}_K\backslash\{0\}$ as in Table~\ref{table:delta}. Note that~$\delta$ is either a unit or $7$ times a unit in~$\mathcal{O}_K$.
\begin{table}[h]
\begin{tabular}{|c|c|c|}
\hline
& $7\nmid a + b$ & $7\mid a + b$ \\
\hline
$2\nmid ab$ & $-7$ & $1$ \\
\hline
$2\parallel ab$ & $-7(z^2 - 2)$ & $z^2 - 2$ \\
\hline
$4\mid ab$ & $-7$ & $1$ \\
\hline
\end{tabular}
\caption{Values of~$\delta$}
\label{table:delta}
\end{table}
Let~$N(\bar{\rho}_{F^{(\delta)},p})$ be the Serre level of the mod~$p$ representation~$\bar{\rho}_{F^{(\delta)},p}$ associated with~$F^{(\delta)}$ (i.e., the prime-to-$p$ part of the Artin conductor). According to Proposition~\ref{P:conductorF} (recall that $3 \mid a+b$ and $p \neq 7$), we have~$N(\bar{\rho}_{F^{(\delta)},p}) = \mathfrak{q}_2^s \mathfrak{q}_3 \mathfrak{q}_7^t$ with
\begin{equation}\label{eq:cond_of_F_delta_at_2_and_7}
s = \left\{ \begin{array}{ll}
1 & \quad \text{if $2 \nmid ab$},\\
3 & \quad \text{if $2 \mid ab$}, \\
\end{array} \right.
\quad\text{and}\quad
t = \left\{ \begin{array}{ll}
0 & \text{if $7 \nmid a + b$}, \\
1& \text{if $7 \mid a + b$}. \\
\end{array} \right.
\end{equation}
Let $S_2(N(\bar{\rho}_{F^{(\delta)},p}))$ denote the space of Hilbert cuspforms of level~$N(\bar{\rho}_{F^{(\delta)},p})$, parallel weight~2 and trivial character. The curve $F$ is modular by~\cite[Corollary 6.4]{F} and hence so is~$F^{(\delta)}$. (Note that we now know that all elliptic curves over totally real cubic fields are modular by the work of Derickx, Najman, and Siksek \cite{DNS20}.) Moreover, by Proposition~\ref{P:irredOverK}, the representation~${\overline{\rho}}_{F,p}$ is irreducible and hence so is~${\overline{\rho}}_{F^{(\delta)},p}$. An application of level lowering theorems for Hilbert modular forms (see~\cite{Fuj,Jarv,Raj}), implies that there exists a Hilbert newform $g \in S_2(N(\bar{\rho}_{F^{(\delta)},p}))$
such that for a prime~$\mathfrak{p} \mid p$ in~${\overline{\Q}}$
we have
\begin{equation}
{\overline{\rho}}_{F^{(\delta)},p} \simeq {\overline{\rho}}_{g,\mathfrak{p}}.
\label{E:iso2}
\end{equation}
In the sequel, we show that this isomorphism is impossible, hence proving Theorem~\ref{T:main}.
Note that when $2 \mid a+b$ then $8 \mid a+b$ due to the shape of~\eqref{E:77p}. Let $q\not=2,3,7,p$ be a rational prime.
From Proposition~\ref{P:conductorF} and~\eqref{E:iso2}, there exists~$(x,y)\in\{0,\dots,q-1\}^2\backslash\{(0,0)\}$ such that
$(a,b) \equiv (x,y) \pmod{q}$
and, for all prime ideals~$\mathfrak{q}$ above~$q$ in~$K$, we have
\begin{equation}\label{eq:congruences}
a_\mathfrak{q}(g)\equiv\left\{
\begin{array}{lll}
a_\mathfrak{q}\big(F_{x,y}^{(\delta)}\big) & \pmod{\mathfrak{p}} & \text{if $F_{x,y}^{(\delta)}$ has good reduction at~$\mathfrak{q}$}, \\
\pm \left(\Norm_{K/\mathbb{Q}}(\mathfrak{q})+1\right) & \pmod{\mathfrak{p}} & \text{otherwise}.
\end{array}
\right.
\end{equation}
If we denote by~$\mathcal{O}$ the integer ring of the coefficient field of~$g$ and by~$b_\mathfrak{q}^{(\delta)}(x,y)$ the right hand side of this congruence, then in particular
\begin{equation}\label{eq:divisibility}
p\mid \prod_{\substack{0\leq x,y\leq q-1 \\ (x,y)\not=(0,0)}}\gcd\left(\Norm_{K_g/\mathbb{Q}}\left((a_\mathfrak{q}(g)-b_\mathfrak{q}^{(\delta)}(x,y))\mathcal{O}~;\ \mathfrak{q}\mid q\right)\right).
\end{equation}
We split the proof of Theorem~\ref{T:main} into two parts according to whether $7$ divides~$a + b$ or not.
\begin{enumerate}[A.]
\item Assume~$7\nmid a + b$. Then, $t = 0$ and hence we have~$N(\bar{\rho}_{F^{(\delta)},p}) = \mathfrak{q}_2\mathfrak{q}_3$ or $\mathfrak{q}_2^3\mathfrak{q}_3$ if $2\nmid ab$ or~$2\mid ab$ respectively. Notice that when~$2\mid ab$, the values of~$\delta$ depends on the valuation at~$2$ of~$ab$ though.
\begin{enumerate}[(i)]
\item\label{item:partAi} When~$2\nmid ab$, we have~$\delta = -7$ and the form~$g$ has level~$\mathfrak{q}_2 \mathfrak{q}_3$. There are two newforms in~$S_2(\mathfrak{q}_2 \mathfrak{q}_3)$. Using the divisibility relation~\eqref{eq:divisibility} with prime ideals above $q = 5$, $13$ and $29$ we eliminate both newforms.
\item When~$2\parallel ab$, we have~$\delta = -7(z^2 - 2)$ and the form~$g$ has level~$\mathfrak{q}_2^3 \mathfrak{q}_3$. There are $47$ newforms in~$S_2(\mathfrak{q}_2^3 \mathfrak{q}_3)$. Using the divisibility relation~\eqref{eq:divisibility} with prime ideals above $q = 5$, $13$, $29$ and~$41$ we eliminate all these newforms, except the exponent $p = 5$ which survives for one form.
We discard it using instead the congruences~\eqref{eq:congruences} themselves (a method we refer to as `refined elimination' in~\cite{BCDF2}) with auxiliary prime $q = 13$.
\item\label{item:partAiii} When~$4\mid ab$, we have~$\delta = -7$ and the form~$g$ has again level~$\mathfrak{q}_2^3 \mathfrak{q}_3$. Using the divisibility relation~\eqref{eq:divisibility} with prime ideals above $q = 5$, $13$, $29$ and~$41$ we eliminate all the $47$ newforms in~$S_2(\mathfrak{q}_2^3 \mathfrak{q}_3)$ for $p \ge 5$, except the exponent $p = 5$ which survives for one form.
We discard it using again refined elimination with auxiliary primes $q = 13$.
\end{enumerate}
Therefore we have proved the result when~$7$ does not divide~$a + b$.
\item\label{item:partB} Let us now assume~$7\mid a + b$.
\begin{enumerate}[(i)]
\item\label{item:partBi} When~$2\nmid ab$, we have~$\delta = 1$ and the form~$g$ has level~$\mathfrak{q}_2 \mathfrak{q}_3\mathfrak{q}_7$. There are five newforms in $S_2(\mathfrak{q}_2 \mathfrak{q}_3 \mathfrak{q}_7)$. Using the divisibility relation~\eqref{eq:divisibility} with prime ideals above $q = 5$, $13$, $29$ and~$41$ we
eliminate all the five newforms for $p \geq 7$ and $p \neq 13$. The exponents $p=5,13$ survive simultaneously for one form $\mathfrak{f}$
with field of coefficients $\mathbb{Q}_{\mathfrak{f}}$ equal to the real cubic subfield of $\mathbb{Q}(\zeta_{13})$.
Using refined elimination with auxiliary prime $q = 29$ we eliminate~$\mathfrak{f}$ for $p=5$. An application of \cite[Theorem~2.1]{Martin} shows there is a
newform $f \in S_2(\mathfrak{q}_2\mathfrak{q}_3\mathfrak{q}_7)$ and a prime $\mathfrak{p}' \mid 13$ in its coefficient
field such that $a_\mathfrak{q}(f) \equiv \Norm(\mathfrak{q}) + 1 \pmod{\mathfrak{p}'}$ for all $\mathfrak{q} \nmid 2\cdot 3 \cdot 7 \cdot 13$.
By comparing the traces for any prime ideal~$\mathfrak{q}$ above~$29$, we conclude that $f=\mathfrak{f}$ and $\mathfrak{p}' = \mathfrak{p}$ is the unique ideal in $\mathbb{Q}_\mathfrak{f}$ above~13.
Therefore, the representation ${\overline{\rho}}_{\mathfrak{f},\mathfrak{p}}$ is reducible and,
since there is no other ideal above~13 in~$\mathbb{Q}_\mathfrak{f}$, we cannot have \eqref{E:iso2}
with $g = \mathfrak{f}$ when $p=13$. (For an argument of this type in a more classical setting, see~\cite[Ex.~2.9]{HalberstadtKraus}.) This completes the proof when~$2\nmid ab$.
\item\label{item:partBii} When~$2\parallel ab$, we have~$\delta = z^2 - 2$ and the form~$g$ is one the $121$ newforms in $S_2(\mathfrak{q}_2^3 \mathfrak{q}_3 \mathfrak{q}_7)$. We perform standard elimination for all the possible forms using the primes above $q = 5$, $13$, and $29$. We discard this way all the newforms for $p \geq 7$, $p\ne 13$. The exponent $p = 5$ survives for three forms
and refined elimination with $q = 29$ (for two of them)
or~$q = 83$ (for the other one) deals with all of them. The exponent $p = 13$ survives for four forms
which we discard using refined elimination with $q = 41$.
\item\label{item:partBiii} When~$4\mid ab$, we have~$\delta = 1$ and the form~$g$ has again level~$\mathfrak{q}_2^3 \mathfrak{q}_3\mathfrak{q}_7$. Standard elimination using the primes above $q = 5$, $13$, and $29$ eliminates all the possible forms except one form for the exponents~$p= 5$ and~$11$,
three forms with exponent~$p = 5$
and four forms with~$p = 13$.
Using refined elimination with $q = 41$, we eliminate the first remaining form (for both exponents~$p = 5$ and~$11$) and the last four forms (for exponent~$p = 13$). We finally discard the exponent~$p = 5$ for the other three forms using refined elimination with~$q = 83$, $41$ or~$29$
\end{enumerate}
Therefore we have proved the result for~$7\mid a + b$ as well.
\end{enumerate}
This finishes the proof of Theorem~\ref{T:main}.
\begin{remark}\label{rk:EandF}
We can make this proof faster by combining information from both curves~$E$ and~$F$. From Theorem~\ref{T:overQ}, which uses Frey curve $E$, to prove Theorem~\ref{T:main} it suffices to deal with items~(\ref{item:partAiii}), (\ref{item:partBi}), and (\ref{item:partBii}) above. This reduces the computation time from $2$ hours to $1$ hour.
\end{remark}
\section{Two multi-Frey approaches to Theorem~\ref{T:main} using Frey abelian varieties}
\label{S:Frey7}
In this section, we will summarize the coming proofs of Theorem~\ref{T:main} and their contributions.
Let $(a,b,c)$ be a non-trivial primitive solution to equation~\eqref{E:77p} for an exponent~$n \geq 2$.
From the discussion after Lemma~\ref{lem:irredE}, we can assume~$n = p$ is a prime, $p\ge 5$ and $p\neq 7$.
Let $C_7 = C_7(a,b)$ be the hyperelliptic Frey curve constructed in Section~\ref{S:Freyrrp} with model
\begin{equation}
\label{E:Kraus7}
y^2 = x^7 + 7 ab x^5 + 14 a^2 b^2 x^3 + 7 a^3 b^3 x + b^7 - a^7 \\
\end{equation}
whose discriminant is
\begin{equation}
\label{Kraus-discriminant0}
\Delta(C_7)= -2^{12} \cdot 7^7 \cdot (a^7+b^7)^{6}.
\end{equation}
To ease notation, we write $C = C_7$ and $J = J_7(a,b) = \Jac(C_7)$ for its Jacobian. The Jacobian~$J$ has dimension~3 and, by Theorem~\ref{T:GL2typeJr}, it becomes of $\operatorname{GL}_2$-type over $K = \mathbb{Q}(\zeta_7)^+$.
The objective of our first proof is to use higher dimensional Frey varieties `as much as possible'.
More precisely, we shall use~$J_7(a,b)$
to prove the following theorem.
\begin{theorem}\label{T:overQvariety}
Let $p = 5$ or $p \geq 11$ be a prime. Then,
the equation~\eqref{E:77p} with exponent~$p$
has no non-trivial primitive solutions $(a,b,c)$
satisfying $a \equiv 0 \pmod{2}$ and $b \equiv 1 \pmod{4}$.
\end{theorem}
The hypothesis of this theorem imply $2 \nmid a+b$ which is the minimal assumption that makes the approach with~$J_7(a,b)$ hopeful. Indeed, when $2 \mid a+b$ we still face the fundamental problem of proving irreducibility of ${\overline{\rho}}_{J_7,\mathfrak{p}}$. Moreover, to make this proof efficient we need to add various techniques to the elimination step which allow for a large reduction of the required computational time; these techniques are detailed in Section~\ref{S:enhanements} and can, in principle, be applied to other instances of~\eqref{E:rrp} or to other Diophantine equations.
By the symmetry of~\eqref{E:77p}, after switching $a$ and $b$, and negating both $a$ and $b$ if necessary, we may assume that the congruences
$a \equiv 0 \pmod{2}$ and $b \equiv 1 \pmod{4}$
are satisfied if and only if $ab$ is even.
Now the multi-Frey technique gives that $ab$ is even: this follows from Theorem~\ref{T:overQ} which regards the curve~$E_{a,b}$ combined with item~\eqref{item:partBi} in Section~\ref{S:overCubic} which uses $F_{a,b}$\footnote{Alternatively, one may also invoke items~\eqref{item:partAi} and~\eqref{item:partBi} in Section~\ref{S:overCubic} which use Frey curve $F$ only.
The running time would be essentially the same.}. Hence Theorem~\ref{T:overQvariety} implies Theorem~\ref{T:main}.
Our second proof of Theorem~\ref{T:main} uses all three Frey varieties and is designed to minimize the total running time, which takes about $1$ minute, compared to $2$ hours using the Frey curve $F_{a,b}$ only (see \S~\ref{T:main}). More precisely, we will use~$J_7(a,b)$ to prove the following.
\begin{theorem}\label{T:overQvariety7adic}
Let $p = 5$ or $p \geq 11$ be a prime. Then,
the equation~\eqref{E:77p} with exponent~$p$
has no non-trivial primitive solutions $(a,b,c)$
satisfying
\[ a \equiv 0 \pmod{2}, \quad b \equiv 1 \pmod{4} \quad \text{ and } \quad 7 \mid a+b.\]
\end{theorem}
Assuming this result, we proceed similarly to the first proof, but appealing to further information from the curve~$F_{a,b}$. Namely, using also part~\eqref{item:partAiii} of Section~\ref{S:overCubic}, one has that~$7$ divides~$a + b$ and $ab$ is even. Now Theorem~\ref{T:overQvariety7adic} implies Theorem~\ref{T:main}, which uses the least computational time of all the proofs.
We remark that both proofs are faster than using only the Frey curves and, moreover, as we shall see below, this improvement is in large part due to properties that are genuine of abelian varieties which are not elliptic curves. Also, the running time of a proof using the modular method is bounded from below by the time required to compute the relevant spaces of newforms.
In this regard, it is usually an advantage to work with levels of smaller norms whenever possible; indeed, this is what makes our final proof the fastest of all and
any proof using the space in part~\eqref{item:partBii} of Section~\ref{S:overCubic} is slow.
However, this is not always the case, as explained in~Section~\ref{S:Avoiding}.
When trying to solve~$\eqref{E:rrp}$ for~$r > 7$ or, more generally, carrying out Darmon's program for other signatures, the spaces of Hilbert newforms involved will quickly become very large. The proofs in this part show the Frey varieties have additional structures which one can exploit to reduce computations, despite the fact that we have to work with Jacobians of hyperelliptic curves.
\section{Challenges in the elimination step}
\label{S:eliminationJ}
When applying the modular method it is often in the elimination step where one faces the biggest challenges,
due to the lack of general methods to distinguish Galois representations. The most simple of such methods is to compare traces of Frobenius as done in the proof of Theorem~\ref{T:main} in Section~\ref{S:overCubic}.
The proof of Theorem~\ref{T:main} below using~$J_7$
also relies heavily on comparing traces. However, difficulties start to arise when we have to compare traces with many newforms or newforms with large coefficients fields.
In this section, using as an example the proof of Theorem~\ref{T:overQvariety}, we first discuss the standard elimination procedure for Frey abelian varieties and then describe two main issues that one faces when applying this procedure.
\subsection{Standard elimination for Frey abelian varieties}
Let $p=5$ or $p \geq 11$ be prime. Suppose that~$(a,b,c)$
satisfies
\[ a^7 + b^7 = 3c^p, \qquad abc \neq 0, \qquad \gcd(a,b,c)=1\]
and $a \equiv 0 \pmod{2}$, $b \equiv 1 \pmod{4}$.
Let $J = J_7(a,b)$ and $\mathfrak{p} \mid p$ in~$K$ (here we see~$K$ as the field of real multiplications of~$J/K$).
From Theorem~\ref{T:levelLowering} properties~(i) and~(ii), we have
\begin{equation}\label{E:iso7}
{\overline{\rho}}_{J,\mathfrak{p}} \cong {\overline{\rho}}_{g,\mathfrak{P}},
\end{equation}
where~$g$ is a Hilbert newform over~$K$ of level $\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7^2$, parallel weight $2$, trivial character and a prime ideal~$\mathfrak{P} \mid p$ in~$K_g$, the coefficient field of~$g$. The corresponding space of cuspforms has dimension 1,350 and it contains 61 Galois conjugacy classes of newforms, which we can calculate with {\tt Magma} in about 8 minutes. To finish the proof, we need to show that, for all such~$g$, the isomorphism~\eqref{E:iso7} is impossible, hence obtaining a contradiciton.
We will now discuss why it is enough to discard~\eqref{E:iso7} for only one newform~$g$ in each Galois conjugacy class, and highlight an important difference to the setting of Frey curves.
Fix an embedding $\iota_K : K \hookrightarrow {\overline{\Q}}$ and note that, since $K/\mathbb{Q}$ is Galois, all such embeddings are of the form $\iota_K \circ \sigma$ where $\sigma \in \Gal(K/\mathbb{Q})$.
Fix also an embedding
$\iota_g : K_g \hookrightarrow {\overline{\Q}}$ and note that, replacing~$\iota_g$ by a different embedding is the same as replacing~$g$ by another form in the same Galois conjugacy class.
The isomorphism~\eqref{E:iso7} means there are primes $\tilde{\mathfrak{p}}$ and~$\tilde{\mathfrak{P}}$ in~${\overline{\Q}}$
such that
$\tilde{\mathfrak{p}} \cap \iota_K(\mathcal{O}_K) = \iota_K(\mathfrak{p})$ and $\tilde{\mathfrak{P}} \cap \iota_g(\mathcal{O}_{K_g}) = \iota_g(\mathfrak{P})$ such that the associated representations
${\overline{\rho}}_{J,\tilde{\mathfrak{p}}} : G_K \to \operatorname{GL}_2({\overline{\F}}_p)$ and ${\overline{\rho}}_{g,\tilde{\mathfrak{P}}} : G_K \to \operatorname{GL}_2({\overline{\F}}_p)$
are isomorphic.
For a prime~$\mathfrak{q} \nmid \mathfrak{q}_2 \mathfrak{q}_3 \mathfrak{q}_r$ of~$K$ and~$\Frob_\mathfrak{q} \in G_K$ a Frobenius element at~$\mathfrak{q}$, we use the standard notation
\[
a_\mathfrak{q}(g) = \tr \rho_{g,\tilde{\mathfrak{P}}}(\Frob_\mathfrak{q}) \quad \text{ and } \quad a_\mathfrak{q}(J) = \tr \rho_{J_7(a,b),\tilde{\mathfrak{p}}}(\Frob_\mathfrak{q}),
\]
where we also assume $J_7(a,b)/K$ has good reduction at~$\mathfrak{q}$.
The smallest subfield of~${\overline{\Q}}$ containing all the traces is the compositum $L = \iota_K(K)\cdot \iota_g(K_g)$. After replacing~$\iota_K$ by $\iota_K \circ \sigma$ if necessary, where $\sigma \in \Gal(K/\mathbb{Q})$,
we can assume there is a prime $\mathfrak{P}_0$ of~$L$ such that
\begin{equation}\label{E:intersection}
\mathfrak{P}_0 \cap \iota_K(\mathcal{O}_K) = \iota_K(\mathfrak{p}) \qquad \text{ and } \qquad \mathfrak{P}_0 \cap \iota_g(\mathcal{O}_{K_g}) = \iota_g(\mathfrak{P}).
\end{equation}
In this case, the isomorphism~\eqref{E:iso7} yields
\begin{equation}\label{E:divisibility}
\iota_K(a_\mathfrak{q}(J)) \equiv \iota_g(a_\mathfrak{q}(g)) \pmod{\mathfrak{P}_0} \implies p \mid \Norm_{L/\mathbb{Q}}(\iota_K(a_\mathfrak{q}(J)) - \iota_g(a_\mathfrak{q}(g))).
\end{equation}
Therefore, if the latter divisibility condition does not hold, then
the pair~$(g,\mathfrak{P})$ does not give rise to~\eqref{E:iso7} for the fixed~$\iota_K$.
Moreover, the previous congruence holds for
all~$\mathfrak{P}_0$ satisfying~\eqref{E:intersection}, thus checking the divisibility condition once eliminates all pairs~$(g,\mathfrak{P})$ for which there is~$\mathfrak{P}_0$ in~$L$ above~$\iota_g(\mathfrak{P})$ satisfying~\eqref{E:intersection}
as the same choice of~$\iota_K$ works.
Finally, checking the divisibility condition \eqref{E:divisibility} for the three possible $\iota_K$ completely eliminates the form~$g$. This means that none of the residual representations attached to~$g$ satisfies~\eqref{E:iso7}.
Since ${\overline{\rho}}_{g,\mathfrak{P}} \simeq {\overline{\rho}}_{g^\tau,\mathfrak{P}^\tau}$ for all embeddings $\tau : K_g \hookrightarrow {\overline{\Q}}$,
we conclude that none of the forms in the Galois conjugacy class of~$g$ can give rise to~\eqref{E:iso7}.
From now on, we will also denote by $K$,~$K_g$ the fields
$\iota_K(K), \iota_g(K_g) \subset {\overline{\Q}}$, respectively.
\subsection{Large fields of coefficients and large auxiliary primes} \label{S:largeFields}
Among the forms~$g$ to be considered the degree of the coefficient fields~$K_g$ can be as large as 54 and the
compositum $K\cdot K_g$ can have degree as large as~$108$.
Moreover, some forms require comparing traces at primes~$\mathfrak{q}$ above rational primes $q > 83$. For each congruence class
$(a,b)$ mod~$q$ we have to take norms of a certain element in $K\cdot K_g$, obtaining a large integer which we need to factor. As soon as $q > 13$ the amount of congruence classes $(a,b)$ mod~$q$ becomes significant, so we have to repeat the described steps multiple times, making the total running time very long.
\subsection{Computing traces of~$J$ at totally split primes}
\label{S:tracesAtSplittingPrimes}
To prove Theorem~\ref{T:overQvariety} we are required to compute the traces $\tr \rho_{J,\mathfrak{p}}(\Frob_\mathfrak{q})$
for primes $\mathfrak{q}$ in~$K$ where $J/K$ has good reduction. Unfortunately, current {\tt Magma} functionality includes only computations of Euler factors for the full Tate module of~$J/\mathbb{Q}$.
To circumvent this issue we proceed as follows.
By Theorem~\ref{T:GL2typeJr},
the Frey variety~$J/K$ is of $\operatorname{GL}_2$-type and has real multiplications by~$K$. Furthermore,
the action of $G_K$ on the $p$-adic Tate module satisfies
\begin{equation}
\label{E:decomposition}
T_p(J/K) \otimes {\overline{\Q}}_p \simeq \rho_{J,\mathfrak{p}} \oplus \rho_{J,\mathfrak{p}_1} \oplus
\rho_{J,\mathfrak{p}_2},
\end{equation}
where the $2$-dimensional
blocks correspond to the different embeddings of~$K$ into~${\overline{\Q}}_p$.
Since $K$ is Galois of prime degree $[K : \mathbb{Q}]=3$, we have 3 independent blocks in~\eqref{E:decomposition} when $p\mathcal{O}_K=\mathfrak{p} \mathfrak{p}_1 \mathfrak{p}_2$ (totally) splits in~$K$, or the blocks are related by an automorphism of the completion~$K_\mathfrak{p}$ (i.e. the coefficient field of $\rho_{J,\mathfrak{p}}$)
when $p$ is inert or (totally) ramified.
Let $\sigma \in G_{\mathbb{Q}}$ be a lift of
a generator of $\Gal(K/\mathbb{Q})$. From the previous discussion, it makes sense to denote the 2-dimensional blocks by
$\rho_{J,\mathfrak{p}}$ $\rho^\sigma_{J,\mathfrak{p}}$ $\rho^{\sigma^2}_{J,\mathfrak{p}}$.
Since $J$ is defined over $\mathbb{Q}$, the relation~\eqref{semi-linear-action} holds,
leading to a connection between inner and outer conjugation, that is,
for all $g \in G_K$, we have
\[
\rho^\sigma_{J,\mathfrak{p}}(g) = \rho_{J,\mathfrak{p}}(\sigma g \sigma^{-1})
\qquad \text{ and } \qquad
\rho^{\sigma^2}_{J,\mathfrak{p}}(g) = \rho_{J,\mathfrak{p}}(\sigma^2 g \sigma^{-2}).
\]
Write $G_{K/\mathbb{Q}}=\Gal(K/\mathbb{Q})$. Let $q$ be a rational prime of
good reduction for $J/\mathbb{Q}$.
Using {\tt Magma} we compute the Euler factor $L_{q}(C,X)$,
take its reverse polynomial and factor it in $K[X]$ as
\begin{align*}
X^6 \cdot L_{q}(C,1/X) = \prod_{\sigma \in G_{K/\mathbb{Q}}} (X^2 - a_{\mathfrak{q}}(J)^\sigma X + N(\mathfrak{q})) = \prod_{\sigma \in G_{K/\mathbb{Q}}} (X^2 - a_{\mathfrak{q}^\sigma}(J) X + N(\mathfrak{q})).
\end{align*}
The previous discussion shows this factorization exists and
the degree 2 factors on the right are the characteristic polynomials
of the action of $\Frob_\mathfrak{q}$ restricted to~$K$ described by each 2-dimensional block. Moreover, since the field of multiplications~$K$ is Galois, totally real of odd degree, and the absolute
values of the roots of each factor are equal to $|\sqrt{N(\mathfrak{q})}|$, it follows that the quadratic factors
above are irreducible over~$K$.
In particular, from~\eqref{E:traces}
we have a set of traces at~$q$
\begin{equation}\label{E:setTraces}
\mathcal{T}_q = \left\{ \tr \rho_{J,\mathfrak{p}}^\sigma (\Frob_\mathfrak{q}) : \sigma \in G_{K/\mathbb{Q}} \right\} =\left\{ a_{\mathfrak{q}}(J)^\sigma : \sigma \in G_{K/\mathbb{Q}} \right\}
=\left\{ a_{\mathfrak{q}^\sigma}(J) : \sigma \in G_{K/\mathbb{Q}} \right\}
\end{equation}
which has exactly one element
in $\mathbb{Z}$ when $q$ is inert in~$K$ and 3 conjugated
elements in $\mathcal{O}_K$ when $q \, \mathcal{O}_K = \mathfrak{q}_1 \mathfrak{q}_2 \mathfrak{q}_3$ splits. In particular, the previous discussion proves the following.
\begin{proposition}\label{P:inertInQ}
Let $a,b \in \mathbb{Z}$ be coprime and such that $a^7 + b^7 \neq 0$.
Write $J = J_7(a,b)$.
Let~$q$ be a prime inert in~$K$. Write $\mathfrak{q} = q\mathcal{O}_K$.
Then, for all primes~$\mathfrak{p} \nmid q$ in~$K$,
the trace of Frobenius of~$\rho_{J,\mathfrak{p}}$ at~$\Frob_\mathfrak{q}$ belongs to~$\mathbb{Z}$.
\end{proposition}
Unfortunately, when $q$ splits and $\#\mathcal{T}_q = 3$ we cannot tell which trace
in~$\mathcal{T}_q$ occurs at each prime~$\mathfrak{q}_i$
for the fixed block~$\rho_{J,\mathfrak{p}}$. This obliges us to
consider all the possibilities when using
splitting primes to compare traces in~\eqref{E:iso7} and
bound the exponent~$p$ using the divisibility condition in~\eqref{E:divisibility}.
The relation \eqref{E:traces} shows that after
fixing the trace at one of the $\mathfrak{q}_i$ the remaining traces
are determined. Therefore, we have to consider three possible cases, one for each element of $\mathcal{T}_q$. Since we have to use `large' auxiliary splitting primes~$q$ (e.g. $q=29$), this multiplies by 3 the issues explained in Section~\ref{S:largeFields}, making the overall computation even slower.
Recall from the discussion surrounding~\eqref{E:divisibility}
that we need to test the divisibility condition $$p \mid \Norm_{L/\mathbb{Q}}(\iota_K(a_{\mathfrak{q}_i}(J)) - \iota_g(a_{\mathfrak{q}_i}(g)))$$
for each embedding $\iota_K \hookrightarrow {\overline{\Q}}$. These embeddings are of the form $\iota_K\circ \sigma$ for $\sigma \in \Gal(K/\mathbb{Q})$, where $\iota_K$ is a fixed embedding. Therefore, we have to find the prime factors
of the norms from $L$ to~$\mathbb{Q}$ of the elements
\begin{equation}
\label{E:difference}
\iota_K(a_{\mathfrak{q}_i}(J)) - \iota_g(a_{\mathfrak{q}_i}(g)),\quad \iota_K(a_{\mathfrak{q}_i}(J)^\sigma) - \iota_g(a_{\mathfrak{q}_i}(g))
\quad
\iota_K(a_{\mathfrak{q}_i}(J)^{\sigma^2}) - \iota_g(a_{\mathfrak{q}_i}(g))
\end{equation}
for a fixed embedding~$\iota_K$. The discussion in the previous paragraph says that we also have to consider the three possible values for $a_{\mathfrak{q}_i}(J) \in \mathcal{T}_q$ given in~\eqref{E:setTraces}. Since the elements of $\mathcal{T}_q$ are conjugate by~$\Gal(K/\mathbb{Q})$ this will permute the elements in~\eqref{E:difference}, hence not adding any additional computations.
Also, for inert primes~$\mathfrak{q}$ we know that $a_\mathfrak{q}(J) \in \mathbb{Z}$ and so $\iota_K(a_\mathfrak{q}(J))$ is the same for all embeddings~$\iota_K$.
\section{Enhancements for the elimination step}
\label{S:enhanements}
Due to the two issues discussed in the previous section, to successfully eliminate all the 61 conjugacy classes of forms by naively applying the standard elimination procedure for Frey abelian varieties takes an inordinate amount of computational time. Although this completes the proof of Theorem~\ref{T:overQvariety}, it takes very long
to reproduce and it suggests that an application to other instances of~\eqref{E:rrp} where the exponent~$r$ or the coefficient~$d$ are larger will be impossible to execute.
In this section, we describe various techniques to improve the elimination step. Since most of it can be applied to the Frey variety $J_r$ attached to~\eqref{E:rrp}, we start by explaining each idea in this general context and then exemplify it with the concrete calculation used to prove Theorem~\ref{T:overQvariety}.
\begin{remark}
When working with Frey elliptic curves the field of real multiplications
~$K = \mathbb{Q}$ and so there is a unique embedding
~$\iota_K : \mathbb{Q} \hookrightarrow {\overline{\Q}}$ to consider. This suggests that total computational time should be reduced in that setting.
However, as we will explain below, there are properties of Frey abelian varieties that we can exploit to reduce the computational
time compared with the proof given in Section~\ref{S:overCubic} where only Frey curves are used.
Furthermore, the asymptotic result in Theorem~\ref{main-asymptotic} for $r=11$ is out of reach for Frey curves but not if we additionally use Frey varieties.
\end{remark}
\subsection{Avoiding unnecessary levels}\label{S:Avoiding}
Note that Proposition~\ref{P:SerreCond} states that $N({\overline{\rho}}_{J,\mathfrak{p}})$
is a divisor of $\mathfrak{q}_2^2 \mathfrak{q}_r^2 \mathfrak{q}_d$ but Theorem~\ref{T:levelLowering} concludes that we only need
to consider forms in the largest level $\mathfrak{q}_2^2 \mathfrak{q}_r^2 \mathfrak{q}_d$. This is a consequence of refined level lowering in the proof of Theorem~\ref{T:levelLowering} which, in particular, forces the presence of the Steinberg primes~$\mathfrak{q} \mid \mathfrak{q}_d$.
For the proof of Theorem~\ref{T:overQvariety}, this means that we only have to consider forms at level $\mathfrak{q}_2^2 \mathfrak{q}_7^2 \mathfrak{q}_3$.
In particular, we avoid dealing with
the level $\mathfrak{q}_2^2 \mathfrak{q}_7^2$.
{\it A priori} this may not look like much of an improvement, as we have to compute at the (unavoidable) larger level~$\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7^2$, but it turns out to cut running time considerably. Indeed, the available algorithms to compute Hilbert newforms of square level are very inefficient, and obtaining the newforms of level $\mathfrak{q}_2^2 \mathfrak{q}_7^2$ takes about 11 hours, whilst those of level~$\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7^2$ are computed under 8 minutes.
\subsection{Restricting the field of coefficients}
\label{S:restrictionKg}
Another consequence of refined level lowering built into Theorem~\ref{T:levelLowering} is conclusion~(iv) of that theorem, namely the constraint $K \subset K_g$.
For the proof of Theorem~\ref{T:overQvariety}, this extra restriction on~$g$ reduces the set of newforms that possibly satisfy~\eqref{E:iso7} from~61 down to 25. The remaining 25 forms are indexed in~\cite{programs} by $i$ in the set
\begin{equation}\label{E:index}
\{ 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 33, 38, 41, 42, 45, 46, 47, 48, 51, 57, 58, 60, 61 \}
\end{equation}
and the degrees of their fields of coefficient
are respectively given by
\begin{equation}\label{E:degrees}
\{ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 9, 9, 12, 12, 15, 18, 18, 21, 21, 36, 36, 54, 54 \}.
\end{equation}
This represents a reduction of the computational time to around 3 hours, where forms 60 and 61 require about 1 hour each, due to the large degree of their coefficient fields.
We remark that this technique is unavailable when using Frey elliptic curves since we always have $\mathbb{Q} \subseteq K_g$.
\subsection{Interchanging $a$ and~$b$}
\label{S:Interchanging}
The Frey curves~$E_{a,b}$ and~$F_{a,b}$ used in Sections~\ref{S:overQ} and~\ref{S:overCubic} are symmetric
in~$a,b$. Thus, when computing all their possible traces of Frobenius at~$\mathfrak{q} \mid q$, it suffices to consider the congruence classes of $(a,b)$ mod~$q$ such that $a \leq b$.
In the case of the Frey hyperelliptic curves~$C_r$ this convenient symmetry does not hold. Instead, from Proposition~\ref{P:Crba} we see that $C_r(a,b)$ and~$C_r(b,a)$ are related by the quadratic twist by~$-1$, which gives the relation of traces
\begin{equation}\label{E:traceTwist-1}
a_\mathfrak{q}(J_r(b,a)) = \chi_{-1}|_{G_K}(\Frob_\mathfrak{q}) a_\mathfrak{q}(J_r(a,b))
\end{equation}
for all~$\mathfrak{q}$ of good reduction for~$J_r(a,b)$, where $\chi_{-1}$ is the character of~$G_\mathbb{Q}$ fixing~$\mathbb{Q}(i)$.
In particular, when $\chi_{-1}|_{G_K}(\Frob_\mathfrak{q}) = 1$
we have the same symmetry as with the Frey elliptic curves;
when $\chi_{-1}|_{G_K}(\Frob_\mathfrak{q}) = -1$ we still only need to consider congruence classes $(a,b)$ mod~$q$ such that $a \leq b$ because we compute $a_\mathfrak{q}(J_r(b,a))$ using~\eqref{E:traceTwist-1}.
\subsection{Identifying quadratic twists}\label{S:twists}
Let $\chi_r$ denote the mod~$r$ cyclotomic character of~$G_\mathbb{Q}$, which has conductor~$r$ and order~$r-1$. In particular, $\chi_r|_{G_K}$ is quadratic and unramified at all primes~$\mathfrak{q} \mid 6$ in~$K$. Therefore, for~$g$ defined over~$K$ of level $\mathfrak{q}_2^2 \mathfrak{q}_d \mathfrak{q}_r^2$, paralell weight~2 and trivial character, the quadratic twist $g\otimes \chi_r|_{G_K}$ is of level $\mathfrak{q}_2^2 \mathfrak{q}_d$, $\mathfrak{q}_2^2 \mathfrak{q}_d \mathfrak{q}_r$ or $\mathfrak{q}_2^2 \mathfrak{q}_d \mathfrak{q}_r^2$.
Moreover, we have the following relation on Fourier coefficients
\begin{equation}\label{E:traceTwist7}
a_\mathfrak{q}(g \otimes \chi_r|_{G_K}) = \chi_r(\Frob_\mathfrak{q}) a_\mathfrak{q}(g) \quad \text{ for all } \mathfrak{q} \nmid 2\cdot \mathfrak{q}_d \cdot \mathfrak{q}_r,
\end{equation}
where $\Frob_\mathfrak{q} \in G_K$ denotes a Frobenius element at~$\mathfrak{q}$.
For the proof of Theorem~\ref{T:overQvariety}, we computed the newforms over $K = \mathbb{Q}(\zeta_7)^+$ of
parallel weight~2 and trivial character and levels
$\mathfrak{q}_2^2 \mathfrak{q}_3$, $\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7$ or $\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7^2$. Comparing fields of coefficients and also a few Fourier coefficients when needed, we identify the
following pairs of twisted newforms
by~$\chi_7$ where at least one newform corresponds
to $i$ in~\eqref{E:index}:
\begin{enumerate}[(i)]
\item the newform $i=38$ arises by twist of a from of level $\mathfrak{q}_2^2 \mathfrak{q}_3$;
\item the newforms $i=16,26,33,41,42,45$ arise by twist of newforms at level $\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7$;
\item the pairs of newforms
indexed by $(i_1,i_2) = (46,47)$,
$(48,51)$, $(57,58)$ and
$(60,61)$ are quadratic twists by~$\chi_7$; this follows from the last statement of Theorem~\ref{T:levelLowering} and the fact that these pairs are determined by the degree of the fields of coefficients, which are 18, 21, 36 and~54, respectively.
\item the pairs of newforms with field of coefficient equal to~$K$ and indexed by $(i_1,i_2) = (12,28)$, $(17,24)$, $(18,19)$, $(20,21)$ and~$(22,23)$ are quadratic twists by~$\chi_7$.
\end{enumerate}
Now the relation~\eqref{E:traceTwist7} gives the following advantages:
\begin{itemize}
\item for primes~$\mathfrak{q}$ in~$K$ such that $\chi_7(\Frob_\mathfrak{q}) = 1$, we only need to compute with one form in each of the pairs listed in (iii) and~(iv);
\item for primes~$\mathfrak{q}$ in~$K$ such that $\chi_7(\Frob_\mathfrak{q}) = -1$, we only need to access the value of the trace of one form in each pair; this still saves significant amounts of time due to the time that~{\tt Magma} needs to access Fourier coefficients at primes of larger norm.
\end{itemize}
\subsection{Coefficients living in strict subfields of~$K_g$}\label{S:inertPrimes}
Let $g$ be a newform defined over~$K$ of level $\mathfrak{q}_2^2 \mathfrak{q}_d \mathfrak{q}_r^2$, parallel weight~2 and trivial character.
Let $\sigma$ be a generator for $\Gal(K/\mathbb{Q})$.
By cyclic base change and Shimura~\cite{shi78}, there exists a newform $g^\sigma$ in the same space such that
\[
K_{g^\sigma} = K_g \quad \text{ and } \quad a_\mathfrak{q}(g^\sigma) = a_{\mathfrak{q}^\sigma}(g) \quad \text{ for all primes $\mathfrak{q}$ in~$K$}.
\]
Thus $\Gal(K/\mathbb{Q})$ acts on the Galois conjugacy classes of newforms.
The orbits of this action of size strictly dividing~$[K : \mathbb{Q}]$ constitute of newforms arising by base change from subfields of~$K$; in particular, those of size~$1$ correspond to forms that are base change from~$\mathbb{Q}$.
Suppose that $g$ and~$g^\sigma$ belong to the same conjugacy class and that~$g$ is not a base change.
Then there exists a subfield $E \subset K_g$ and a non-trivial element
$\tau \in \Gal( K_g/E)$ such that
\begin{equation}\label{E:sigmatau}
a_{\mathfrak{q}}(g^\sigma) = a_{\mathfrak{q}^\sigma}(g) = \tau(a_{\mathfrak{q}}(g))\quad \text{for all primes}\,\,\mathfrak{q} \text{ of } K.
\end{equation}
Since $\Gal(K/\mathbb{Q})$ is cyclic it follows that $\Gal(K_g/E)$ is cyclic of the same order, and the degree of~$E/\mathbb{Q}$
is $[E : \mathbb{Q}] = [K_g : \mathbb{Q}]/[K: \mathbb{Q}]$.
For any prime $\mathfrak{q}$ in~$K$ above a rational prime inert in~$K$, we have $\mathfrak{q}^\sigma = \mathfrak{q}$ for all~$\sigma \in \Gal(K/\mathbb{Q})$ and, therefore, the relation~\eqref{E:sigmatau} implies $a_\mathfrak{q}(g) \in E$. If we also have that $a_\mathfrak{q}(J_r) \in E$ (e.g. this is always the case when $a_\mathfrak{q}(J_r) \in \mathbb{Z}$) then, when comparing traces, we only have to take norms from the smaller extension $E/\mathbb{Q}$.
For the proof of Theorem~\ref{T:overQvariety}, we apply this idea (using the inert primes $q=5,11$) with the forms indexed by $i=60,61$ as follows. We have $[K : \mathbb{Q}] =3$ so the orbits
of the action of $\Gal(K/\mathbb{Q})$ are of size 1 or 3. Since
$i=60,61$ are the only forms whose field
of coefficients has degree 54, their conjugacy
class must be fixed by the action
of $\Gal(K/\mathbb{Q})$. A quick check of the
Fourier coefficients at the primes above~$13$
shows these forms are not a base change. We conclude that their Fourier coefficients at inert primes live in a subfield $E \subset K_g$ of degree~18, which we quickly verify is unique. From Proposition~\ref{P:inertInQ}, the traces of Frobenius of~$J$ at inert primes belong to~$\mathbb{Z} \subset E$; so, when comparing traces at inert primes, we only have to take norms corresponding to the extension $E/\mathbb{Q}$.
We finish this discussion noting that these ideas can actually be applied to all 25 forms we need to eliminate.
Indeed, from the degrees listed in~\eqref{E:degrees}, we see there are at most 2 conjugacy classes of newforms whose field of coefficients has degree~$d_g > 3$, and hence the
same arguments used for the forms $i=60,61$ can be applied; in fact, with a few comparisons of Fourier coefficients, we can also establish~\eqref{E:sigmatau} for the forms with $d_g=3$; however, the time gains are negligible compared to the forms $i=60,61$ which took 2/3 of the overall computational time when only the optimizations from sections~\ref{S:Avoiding} and~\ref{S:restrictionKg} were in place.
\subsection{Optimizing the representation of the field of coefficients}\label{S:optimalRepresentation}
Given the large amount of repetitive calculations involved in the elimination step, having nice polynomials describing fields of coefficients and its subfields can also accelerate the elimination of newforms.
For the proof of Theorem~\ref{T:overQvariety},
this is key for us to deal with the
forms~$g$ indexed by $i=60,61$ whose field of coefficients has degree 54 in an efficient way. The inert primes $q=5,11$ and the arguments explained in section~\ref{S:inertPrimes} are insufficient to eliminate these forms and so we also need to use primes above~$q=13$ which (totally) splits in~$K$.
The corresponding Fourier coefficients have minimal polynomials of degree 54. With {\tt Magma} we can efficiently write
$K_g/\mathbb{Q}$ as a relative extension $K_g/K_6/\mathbb{Q}$
where $K_6$ is of degree~$6$ such that $K \subset K_6$ and the polynomials defining $K_g/K_6$ and $K_6/\mathbb{Q}$ have nicer coefficients. This cuts down the time of using auxiliary prime above~13 considerably.
\begin{comment}
{\color{red} More details on this?}
\end{comment}
\subsection{Inverse $\mathbb{Q}$-forms}\label{S:invQform}
We know from Theorem~\ref{T:Qform} that~$J_r$ is associated with a Hilbert $\mathbb{Q}$-form.
This can also be seen as the special case of the second equality in~\eqref{E:sigmatau}, where we have $K_g = K$ and~$\tau = \sigma$ generates $\Gal(K/\mathbb{Q})$. Therefore, we will call {\it inverse $\mathbb{Q}$-form} to a form~$g$ that satisfies the same relation with $\tau = \sigma^{-1}$.
The following reasoning applies to~$J_r$ but to ease notation we will stick to~$J = J_7$. In this case
we have $\sigma^{-1} = \sigma^2$ and any rational prime~$q \neq 7$ is either inert in~$K=\mathbb{Q}(\zeta_7)^+$ or splits into three prime ideals.
Suppose that~\eqref{E:iso7} holds for~$g$ an inverse $\mathbb{Q}$-form. Since $K_g = K$, after replacing~$g$ by a conjugate if needed, we can assume $\mathfrak{P} \cap \mathcal{O}_K = \mathfrak{p}$.
Let $q$ be a rational prime splitting as~$\mathfrak{q} \mathfrak{q}^\sigma \mathfrak{q}^{\sigma^{-1}}$ in~$K$. Taking traces of Frobenius in~\eqref{E:iso7} gives
\[
a_\mathfrak{q}(g) \equiv a_\mathfrak{q}(J), \quad a_{\mathfrak{q}^\sigma}(g) \equiv a_{\mathfrak{q}^\sigma}(J), \quad a_{\mathfrak{q}^{\sigma^{-1}}}(g) \equiv a_{\mathfrak{q}^{\sigma^{-1}}}(J) \quad \pmod{\mathfrak{p}}
\]
After applying~$\sigma$ to these congruences, using the $\mathbb{Q}$-form property of~$J$ and the inverse $\mathbb{Q}$-form property of~$g$ we obtain (after reordering) that
\[
a_{\mathfrak{q}}(g) \equiv a_{\mathfrak{q}^{\sigma^{-1}}}(J), \quad
a_{\mathfrak{q}^{\sigma}}(g) \equiv a_{\mathfrak{q}}(J), \quad
a_{\mathfrak{q}^{\sigma^{-1}}}(g) \equiv a_{\mathfrak{q}^\sigma}(J) \quad \pmod{\mathfrak{p}^\sigma}
\]
and proceeding similarly with~$\sigma^{-1}$ yields,
\[
a_{\mathfrak{q}}(g) \equiv a_{\mathfrak{q}^{\sigma}}(J), \quad
a_{\mathfrak{q}^{\sigma}}(g) \equiv a_{\mathfrak{q}^{\sigma^{-1}}}(J), \quad
a_{\mathfrak{q}^{\sigma^{-1}}}(g) \equiv a_{\mathfrak{q}}(J) \quad \pmod{\mathfrak{p}^{\sigma^{-1}}}.
\]
Therefore, $p$ divides
\[
\Norm_{K_g/\mathbb{Q}}\big(\gcd\big(a_\mathfrak{q}(g) - a_\mathfrak{q}(J), \;
a_{\mathfrak{q}^\sigma}(g) - a_{\mathfrak{q}^\sigma}(J), \; a_{\mathfrak{q}^{\sigma^{-1}}}(g) - a_{\mathfrak{q}^{\sigma^{-1}}}(J)\big)\big)
\]
if and only if it divides
\[
\Norm_{K_g/\mathbb{Q}}\big(\gcd\big(a_{\mathfrak{q}}(g) -a_{\mathfrak{q}^{\sigma^{-1}}}(J), \;
a_{\mathfrak{q}^{\sigma}}(g) - a_{\mathfrak{q}}(J), \;
a_{\mathfrak{q}^{\sigma^{-1}}}(g) - a_{\mathfrak{q}^\sigma}(J) \big)\big)
\]
if and only if it divides
\[
\Norm_{K_g/\mathbb{Q}}\big(\gcd\big(a_{\mathfrak{q}}(g) - a_{\mathfrak{q}^{\sigma}}(J), \;
a_{\mathfrak{q}^{\sigma}}(g) - a_{\mathfrak{q}^{\sigma^{-1}}}(J), \;
a_{\mathfrak{q}^{\sigma^{-1}}}(g) - a_{\mathfrak{q}}(J) \big)\big).
\]
Observe these three norms differ by a permutation
of~$\{a_{\mathfrak{q}}(J), a_{\mathfrak{q}^\sigma}(J), a_{\mathfrak{q}^{\sigma^{-1}}}(J) \}$ which is the set $\mathcal{T}_q = \{a_{\mathfrak{q}}(J), a_{\mathfrak{q}}(J)^\sigma, a_{\mathfrak{q}}(J)^{\sigma^{-1}}\}$ given in~\eqref{E:traces}. As explained in Section~\ref{S:tracesAtSplittingPrimes}, we are unable to tell which of the traces in $\mathcal{T}_q$ corresponds to each of the primes $\mathfrak{q}$, $\mathfrak{q}^\sigma$ and~$\mathfrak{q}^{\sigma^{-1}}$, but we do know that the value of~$a_\mathfrak{q}(J)$ determines the other two due to the $\mathbb{Q}$-form property, hence there are three possibilities. From the previous discussion, each of these three possibilities leads to one of the three equivalent divisibility conditions with the norms above.
We conclude that, when eliminating an inverse $\mathbb{Q}$-form using an auxiliary prime~$q$ that splits in~$K$, it suffices to consider one of the three possibilities, solving the issue explained in Section~\ref{S:tracesAtSplittingPrimes}.
\section{Proof of Theorems~\ref{T:overQvariety} and~\ref{T:overQvariety7adic}}
Let $p=5$ or $p \geq 11$ be prime. Suppose that~$(a,b,c)$
satisfies
\[ a^7 + b^7 = 3c^p, \qquad abc \neq 0, \qquad \gcd(a,b,c)=1.\]
As above, we let $J = J_7(a,b)/K$ be the attached Frey variety, where $K = \mathbb{Q}(\zeta_7)^+$.
\subsection{Proof of Theorem~\ref{T:overQvariety}}
Suppose $a \equiv 0 \pmod{2}$ and $b \equiv 1 \pmod{4}$.
Let $\mathfrak{p} \mid p$ in~$K$.
From Theorem~\ref{T:levelLowering} we have
\begin{equation}\label{E:iso7II}
{\overline{\rho}}_{J,\mathfrak{p}} \cong {\overline{\rho}}_{g,\mathfrak{P}},
\end{equation}
where~$g$ is a Hilbert newform over~$K$ of level $\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7^2$, parallel weight $2$, trivial character and a prime ideal~$\mathfrak{P} \mid p$ in~$K_g$, the coefficient field of~$g$. Furthermore, $K \subset K_g$. Computing the corresponding space of newform and checking wether $K \subset K_g$ for all of them, implies that $g$ is a form indexed in~\cite{programs} by~$i$ in the set \eqref{E:index}.
For each of these 25 forms
we will obtain a contradiction to the previous isomorphism of representations.
Recall that~$\chi_7$ and $\chi_{-1}$ denote, respectively, the mod~$7$ cyclotomic character and the character of $G_\mathbb{Q}$ fixing $\mathbb{Q}(i)$.
Let~$g$ be the form indexed by~$i$ in the set \eqref{E:index} and assume~\eqref{E:iso7II} holds for~$g$.
Suppose $i=38$. The quadratic twist of~${\overline{\rho}}_{g,\mathfrak{P}}$ by~$\chi_7|_{G_K}$ is unramified at~$\mathfrak{q}_7$ by item~(i) in Section~\ref{S:twists}. Then the same holds for~${\overline{\rho}}_{J,\mathfrak{p}}$ by~\eqref{E:iso7II}, contradicting Propositions~\ref{P:typeAt7a} and~\ref{P:typeAt7b}.
For the remaining 24 forms we will contradict~\eqref{E:iso7II}
by carefully comparing traces. For a prime~$\mathfrak{q} \nmid \mathfrak{q}_2 \mathfrak{q}_3 \mathfrak{q}_r$ of~$K$ and~$\Frob_\mathfrak{q} \in G_K$ a Frobenius element at~$\mathfrak{q}$, we use the standard notation
\[
a_\mathfrak{q}(g) = \tr \rho_{g,\lambda}(\Frob_\mathfrak{q}) \quad \text{ and } \quad a_\mathfrak{q}(J_7(a,b)) = \tr \rho_{J_7(a,b),\lambda}(\Frob_\mathfrak{q}),
\]
where the second equality assumes also that $J_7(a,b)$ has good reduction at~$\mathfrak{q}$.
Suppose now $i \neq 38$ is the set \eqref{E:index} and~\eqref{E:iso7II} holds for the corresponding $g$.
Since $K \subset K_g$, after replacing~$g$ by a conjugate if needed,
we can assume $\mathfrak{P} \cap \mathcal{O}_K = \mathfrak{p}$.
Moreover,
for each rational prime $q \notin \{ 2,3,7 \}$
there exists~$(x,y)\in\{0,\dots,q-1\}^2\backslash\{(0,0)\}$ such that
$(a,b) \equiv (x,y) \pmod{q}$
and, for all primes~$\mathfrak{q} \mid q$ in~$K$, it follows from~\eqref{E:iso7II} that
\begin{equation}\label{E:congJ}
a_\mathfrak{q}(g) \equiv a_\mathfrak{q}\big(J_7(x,y)\big) \pmod{\mathfrak{P}}
\end{equation}
if $J_7(x,y)$ has good reduction at~$\mathfrak{q}$
and
\begin{equation}\label{E:congJll}
a_{\mathfrak{q}}(g)^2 \equiv (\Norm_{K/\mathbb{Q}}(\mathfrak{q})+1)^2 \pmod{\mathfrak{P}}
\end{equation}
otherwise; in the latter case $J_7(x,y)$ has multiplicative reduction at~$\mathfrak{q}$ with level lowering mod~$\mathfrak{p}$ occurring at~$\mathfrak{q}$ (the congruence in this case is often called `the level lowering condition').
For~$q$ a rational prime and $\mathfrak{q} \mid q$ in~$K$,
we set
\[
N_q(g) := \Norm_{K_g/\mathbb{Q}}(a_{\mathfrak{q}}(g)^2 - (\Norm_{K/\mathbb{Q}}(\mathfrak{q})+1)^2).
\]
Since $K$ is Galois, the integer~$N_q(g)$ is independent of the choice
of $\mathfrak{q} \mid q$. From the congruences~\eqref{E:congJ} and~\eqref{E:congJ}, we conclude that $p$ divides the integer
\begin{equation*}
N_q(g)
\prod_{\substack{0\leq x,y\leq q-1 \\ (x,y)\not=(0,0)}}\Norm_{K_g/\mathbb{Q}}\left(\gcd\left(a_\mathfrak{q}(g)-a_\mathfrak{q}\big(J_7(x,y)\big)~;\ \mathfrak{q}\mid q\right)\right).
\end{equation*}
Furthermore, from the discussion in Section~\ref{S:Interchanging}, in particular relation ~\eqref{E:traceTwist-1}, we conclude
that $p$ divides the integer~$B_q(g)$ defined by
\begin{equation}\label{E:Bq1}
N_q(g)
\prod_{\substack{0\leq x \leq y\leq q-1 \\ (x,y)\not=(0,0)}}\Norm_{K_g/\mathbb{Q}}\left(\gcd\left(a_\mathfrak{q}(g) - a_\mathfrak{q}\big(J_7(x,y)\big)~;\ \mathfrak{q}\mid q\right)\right),
\end{equation}
when $\chi_{-1}(\Frob_q) = 1$ and by
\begin{equation}\label{E:Bq2}
N_q(g)
\prod_{\substack{0\leq x \leq y\leq q-1 \\ (x,y)\not=(0,0)}}\Norm_{K_g/\mathbb{Q}}\left(\gcd\left(a_\mathfrak{q}(g)^2 - a_\mathfrak{q}\big(J_7(x,y)\big)^2~;\ \mathfrak{q}\mid q\right)\right),
\end{equation}
when $\chi_{-1}(\Frob_q) = -1$.
Now we deal with the forms
in item~(ii) of Section~\ref{S:twists}, i.e.,
$i=16,26,33,41,42$ or $45$.
Each of these forms can be eliminated for all~$p$ by computing $B_q(g)$ for a combination of the auxiliary primes $q \in \{5, 11, 13\}$,
except for the form $i=16$ and $p=13$;
we deal with this last case using refined elimination with auxiliary prime $q=29$ (see \cite{programs}).
Before dealing with the remaining forms, we discuss how~$B_q(g)$ is related to~$B_q(g \otimes \chi_7)$.
From~\eqref{E:traceTwist7} and the formulas for~$B_q(g)$, we have $N_q(g \otimes \chi_7) = N_q(g)$
and $B_q(g \otimes \chi_7) = B_q(g)$
when $\chi_7(\Frob_\mathfrak{q}) = 1$ or $\chi_{-1}(\Frob_\mathfrak{q}) = -1$; if $\chi_7(\Frob_\mathfrak{q}) = -1$ and $\chi_{-1}(\Frob_\mathfrak{q}) = 1$, then
\begin{equation*}
B_q(g\otimes \chi_7) = N_q(g)
\prod_{\substack{0\leq x \leq y\leq q-1 \\ (x,y)\not=(0,0)}}\Norm\left(\gcd\left(-a_\mathfrak{q}(g) - a_\mathfrak{q}\big(J_7(x,y)\big)~;\ \mathfrak{q}\mid q\right)\right).
\end{equation*}
Moreover, in this latter case, the product $B_q(g)B_q(g\otimes\chi_7)$ divides
\[
\pm N_q(g)^2\prod_{\substack{0\leq x \leq y\leq q-1 \\ (x,y)\not=(0,0)}}\Norm\left(\gcd\left(a_\mathfrak{q}(g)^2 - a_\mathfrak{q}\big(J_7(x,y)\big)^2~;\ \mathfrak{q}\mid q\right)\right).
\]
Now define $A_q(g)$ by~\eqref{E:Bq1}
if $\chi_7(\Frob_\mathfrak{q}) = \chi_{-1}(\Frob_\mathfrak{q}) = 1$ and by~\eqref{E:Bq2} otherwise. The previous discussion implies that if $p \nmid A_q(g)$ then the congruences~\eqref{E:congJ}
and~\eqref{E:congJll} do not hold for both
$g$ and~$g \otimes \chi_7$ for all~$\mathfrak{P} \mid p$ in $K_g = K_{g\otimes \chi_7}$.
Thus we can eliminate a pair of $\chi_7$-twisted forms by computing $A_q(g)$ for only one of them.
Finally we deal with the forms
in items~(iii) and~(iv) of Section~\ref{S:twists}, i.e., the newforms that are quadratic twists by $\chi_7$ of other forms in the same level~$\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7^2$. Indeed,
each of these pairs can be eliminated for all~$p$ by computing $A_q(g)$ for a combination of the auxiliary primes
$q \in \{5, 11, 13,17,29\}$. We remark that for the pair of twisted forms $(60, 61)$ we additionally use the techniques explained in Sections~\ref{S:inertPrimes} and~\ref{S:optimalRepresentation}.
\begin{remark} In the previous proof,
when working with a prime~$q$ splitting in~$K$ we do not have to use all $\mathfrak{q} \mid q$ in~$K$ in the formula for~$B_q(g)$.
In practice, it is usually faster to use only one such~$\mathfrak{q}$ as this avoids computing the $\gcd$; this of course results in more primes dividing $B_q(g)$ but these are easily eliminated by the other auxiliary primes.
There are however particular forms and exponents that are only eliminated if we use two or even three of the primes above~$q$.
\end{remark}
\begin{remark}
In the previous proof, after computing~$B_q(g)$ for a particular~$q$ when using the next auxiliary prime~$q'$ with the same~$g$, we only need to check which prime factors of $B_q(g)$ divide $B_{q'}(g)$. A similar observation applies when computing with~$A_q(g)$.
\end{remark}
\begin{remark} It is interesting to note that the forms $i=60,61$ which were the main catalyzer for the development
of the previous techniques and observations are no longer the forms that take the longest time to be eliminated. The forms which now take the longest are the ones which require the use of the splitting auxiliary prime $q=29$, mainly due to the issue discussed in Section~\ref{S:tracesAtSplittingPrimes}.
\end{remark}
\subsection{Proof of Theorem~\ref{T:overQvariety7adic}}
Suppose $a \equiv 0 \pmod{2}$, $b \equiv 1 \pmod{4}$ and~$7 \mid a+b$.
Let $\mathfrak{p} \mid p$ in~$K$.
From Corollary~\ref{C:levelLowering} we have
\begin{equation}\label{E:iso7III}
{\overline{\rho}}_{J,\mathfrak{p}} \cong {\overline{\rho}}_{g,\mathfrak{P}} \otimes \chi_7|_{G_K},
\end{equation}
where~$g$ is a Hilbert newform over~$K$ of level $\mathfrak{q}_2^2 \mathfrak{q}_3 \mathfrak{q}_7$, parallel weight $2$, trivial character and a prime ideal~$\mathfrak{P} \mid p$ in~$K_g$, the coefficient field of~$g$. Furthermore, $K \subset K_g$. Computing the corresponding space of newform and checking whether $K \subset K_g$ leaves us with 6 newforms to eliminate. Now, similarly to the proof of Theorem~\ref{T:overQvariety}, a comparison of traces of Frobenius using the auxiliary primes $q=5,11,13$ eliminates all 6 forms for all exponents~$p$.
{\large \part{A multi-Frey approach to $x^{11} + y^{11} = z^p$ using Frey abelian varieties} \label{Part:1111p}}
Let $K = \mathbb{Q}(\zeta_{11})^+$ and $\mathcal{O}_K$ denote its ring of integers. Write $\mathfrak{q}_2$ and $\mathfrak{q}_{11}$ for the unique primes in $K$ above 2 and 11, respectively.
In this part, we introduce a Frey elliptic defined over $K$ for the equation~\eqref{main-equ} and tabulate the Diophantine information which can be gleaned from it. Bringing in the Frey hyperelliptic curve $C = C_r$ then allows us to prove Theorem~\ref{main-asymptotic} by exploiting the rich structures that Frey abelian varieties possess.
In particular, the method we call {\em Legendre descent} allows us to perform the elimination step using only classical modular forms to eliminate Hilbert newforms with reducible residual representations.
\bigskip
\section{A Frey curve over a totally real quintic field}
Let $(a,b,c)$ be a primitive solution to \eqref{main-equ} for $p \geq 3$ and such that $a + b \not= 0$.
We consider the Frey curve $F = F_{a,b} := E_{(a,b)}^{(2,1)}$ over $K$ as defined in \cite[p.~619]{F} and given by a model of the form
\begin{equation
F_{a,b}\ : \ Y^2 = X(X-A_{a,b})(X+B_{a,b}),
\end{equation}
where
\begin{align}
A_{a,b} & = (-z^4+5z^2-4) (a+b)^2, \\
B_{a,b} & = (-z^2+4) (a^2 + (z^2-2) a b + b^2),
\end{align}
where
\begin{equation}
C_{a,b} = - A_{a,b} - B_{a,b},
\end{equation}
and $z = \zeta_{11} + \zeta_{11}^{-1}$. The standard invariants of $F_{a,b}$ are given by
\begin{align}
c_4(F_{a,b}) = & 2^4(A_{a,b}^2 + A_{a,b}B_{a,b} + B_{a,b}^2), \\
c_6(F_{a,b}) = & 2^5(2A_{a,b}^3 + 3A_{a,b}^2B_{a,b} - 3A_{a,b}B_{a,b}^2 - 2B_{a,b}^3), \\
\Delta(F_{a,b}) = & 2^4\left(A_{a,b}B_{a,b}C_{a,b}\right)^2.
\end{align}
For~$\delta\in \mathcal{O}_K\backslash\{0\}$, a Weierstrass model for the quadratic twist~$F^{(\delta)}$ of~$F$ by~$\delta$ is given by
\begin{equation
F^{(\delta)}: y^2 = x(x - \delta A_{a,b})(x + \delta B_{a,b}).
\end{equation}
Let
\begin{equation}
\delta = \begin{cases}
-11 & \text{ if } 11 \nmid a + b \\
1 & \text{ if } 11 \mid a + b,
\end{cases}
\end{equation}
Let~$N(\bar{\rho}_{F^{(\delta)},p})$ be the Serre level of the mod~$p$ representation~$\bar{\rho}_{F^{(\delta)},p}$ associated with~$F^{(\delta)}$ (i.e., the prime-to-$p$ part of its Artin conductor).
\begin{proposition}\label{prop:condCurve11}
Suppose $2 \mid a + b$. Then we have that $N(\bar{\rho}_{F^{(\delta)},p}) = \mathfrak{q}_2 \mathfrak{q}_{11}^t$ with
\begin{equation
t = \left\{ \begin{array}{ll}
0 & \text{if $11 \nmid a + b$}, \\
1& \text{if $11 \mid a + b$}. \\
\end{array} \right.
\end{equation}
\end{proposition}
\begin{proof}
If $2 \mid a + b$, then in fact $8 \mid a + b$ and $2 \nmid ab$. Using {\tt Magma}, we can compute the conductor of $F^{(\delta)}$ which is of the form
\begin{equation}
N_{F^{(\delta)}} = \mathfrak{q}_2 \mathfrak{q}_{11}^t \prod_{\mathfrak{q} \mid \Delta(F^{(\delta)}), \mathfrak{q} \nmid \mathfrak{q}_2 \mathfrak{q}_{11}} \mathfrak{q}.
\end{equation}
For this computation, we expand the invariants as series in $a,b$ to determine their valuations exactly and use \cite{papado} to determine the conductor exponents at $\mathfrak{q}_2$ and $\mathfrak{q}_{11}$. We also use the fact that at a prime $\mathfrak{q} \nmid \mathfrak{q}_2 \mathfrak{q}_{11}$, $F^{(\delta)}$ has multiplicative reduction, which follows from \cite{F}.
For $\mathfrak{q} \nmid \mathfrak{q}_2 \mathfrak{q}_{11}$, the valuation $v_\mathfrak{q}(\Delta(F^{(\delta)})) \equiv 0 \pmod p$, so the multiplicative primes $\mathfrak{q} \mid \Delta(F^{(\delta)})$ such that $\mathfrak{q} \nmid \mathfrak{q}_2 \mathfrak{q}_{11}$ do not divide the Serre level $N({\overline{\rho}}_{F^{(\delta)},p})$.
We also have that $v_\mathfrak{q}(N(\bar{\rho}_{F^{(\delta)},p})) = v_\mathfrak{q}(N(\rho_{F^{(\delta)},p})) = v_\mathfrak{q}(N_{F^{(\delta)}}) $ for $\mathfrak{q} = \mathfrak{q}_2$ and $\mathfrak{q} = \mathfrak{q}_{11}$ since the valuation $v_\mathfrak{q}(\Delta(F^{(\delta)})) \not \equiv 0 \pmod p$ for $\mathfrak{q} = \mathfrak{q}_2$ and
for $\mathfrak{q} = \mathfrak{q}_{11}$ when $t=1$.
\end{proof}
Let $S_2(N(\bar{\rho}_{F^{(\delta)},p}))$ denote the space of Hilbert cuspforms of level~$N(\bar{\rho}_{F^{(\delta)},p})$, parallel weight~2 and trivial character. The curve $F$ is modular by~\cite[Corollary 6.4]{F} and hence so is~$F^{(\delta)}$.
\begin{proposition}
The representation~${\overline{\rho}}_{F,p}$ is irreducible for $p$ sufficiently large;
hence the same is true for~${\overline{\rho}}_{F^{(\delta)},p}$.
\end{proposition}
\begin{proof}
See \cite[Theorem 1.4]{F}.
\end{proof}
An application of level lowering theorems for Hilbert modular forms (see~\cite{Fuj,Jarv,Raj}) implies that there is a Hilbert newform $g \in S_2(N(\bar{\rho}_{F^{(\delta)},p}))$
such that for a prime~$\mathfrak{p} \mid p$ in~${\overline{\Q}}$ we have
\begin{equation}
{\overline{\rho}}_{F^{(\delta)},p} \simeq {\overline{\rho}}_{g,\mathfrak{p}}.
\end{equation}
\begin{theorem}
\label{result-from-F}
For prime $p$ sufficiently large, there are no integer solutions $(a,b,c)$ to the equation
\begin{equation}
x^{11} + y^{11} = z^p
\end{equation}
such that $abc \not= 0$, $\gcd(a,b,c) = 1$, and $2 \mid a + b$.
If $q \not= 2,11$ is a fixed prime, we have the same conclusion with the condition $2 \mid a + b$ replaced by $q \mid c$.
\end{theorem}
\begin{proof}
Suppose $2 \mid a + b$. Then $N(\bar{\rho}_{F^{(\delta)},p}) = \mathfrak{q}_2$ or $\mathfrak{q}_2 \mathfrak{q}_{11}$ depending on whether $11 \nmid a + b$ or $11 \mid a + b$ by Proposition~\ref{prop:condCurve11}. There are 1 or 2 newforms at these levels, respectively. These newforms do not have rational coefficients so they are eliminated for $p$ sufficiently large.
Suppose $q \mid c$. Either $q \mid a + b$ and hence every prime $\mathfrak{q}$ of $K$ above $q$ divides $a+b$, or
\begin{equation}
q \mid \frac{a^{11}+b^{11}}{a+b} = a^{10} + \ldots + b^{10}.
\end{equation}
Note that, in the latter case, there exists a prime $\mathfrak{q}$ of $K$ above $q$ which divides $B_{a,b}C_{a,b}$ and so $\mathfrak{q} \mid \Delta(F^{(\delta)})$ for some prime $\mathfrak{q}$ of $K$ in both cases. Finally, in both cases, we obtain a bound on $p$ by the Weil bounds since $q$ is fixed, namely $p$ divides the norm of
\begin{equation}
a_\mathfrak{q}(g) \pm (\Norm_{K/\mathbb{Q}}(\mathfrak{q}) + 1) \not= 0
\end{equation}
from the field of coefficients of $g$ down to $\mathbb{Q}$.
\end{proof}
To obtain Theorem~\ref{main-asymptotic} using only the Frey curve $F$ above requires computation in other levels, in particular, the space of Hilbert newforms of level $\mathfrak{q}_2^3 \mathfrak{q}_{11}$. This space has dimension 12,013 and is not currently feasible to compute.
\section{Frey abelian varieties and the method of Legendre descent}
\label{Legendre-descent}
An obstacle to applying a modular Galois representation theoretic approach to the equations
\begin{equation}
x^r + y^r = z^p
\end{equation}
for larger values of $r$ is the inability to compute the spaces of Hilbert newforms at the required Serre level. In this section, we explain how a certain property enjoyed by the Frey hyperelliptic curves $C = C_r$ can be used to eliminate the Hilbert newforms $f$ at the Serre level, for prime $p$ sufficiently large, using only computations of classical modular forms, and on the assumption that ${\overline{\rho}}_{f,\mathfrak{p}_r}$ is irreducible.
We illustrate this method in the case $r = 11$. Let $(a,b,c)$ be a primitive solution to \eqref{main-equ} such that $2 \nmid a + b$. We may assume from here on that $a \equiv 0 \pmod 2$ and $b \equiv 1 \pmod 4$, at the expense of switching $a$ and $b$, and negating both $a$ and $b$.
Let $C_{11} = C_{11}(a,b)$ be the hyperelliptic Frey curve constructed in Section~\ref{S:Freyrrp}. To ease notation, we write $C = C_{11}$ and $ J_{11}(a,b) = \Jac(C_{11})$ for its Jacobian. We let
\begin{align*}
J = \begin{cases}
J_{11}(a,b) \otimes \chi_{11} & \text{ if } 11 \mid a + b, \\
J_{11}(a,b) & \text{ if } 11 \nmid a + b,
\end{cases}
\end{align*}
where $\chi_{11}$ is as in Corollary~\ref{C:levelLowering}.
The Jacobian~$J$ has dimension~5 and, by Theorem~\ref{T:GL2typeJr}, it becomes of $\operatorname{GL}_2$-type over $K = \mathbb{Q}(\zeta_{11})^+$.
We have shown in \S \ref{S:modularityKraus} that if $ab\not= 0$, then
\begin{enumerate}
\item ${\overline{\rho}}_{J,\mathfrak{p}_{11}} : G_K \rightarrow \operatorname{GL}_2(\overline{\mathbb{Q}}_{11})$ is absolutely irreducible and extends to $G_\mathbb{Q}$,
\item $J/K$ is modular,
\end{enumerate}
where $\mathfrak{p}_{11}$ is the unique prime in $K$ above $11$.
By Theorem~\ref{P:irredSupercuspidal}, ${\overline{\rho}}_{J,\mathfrak{p}}$ is irreducible for $p \not= 2$. From Theorem~\ref{T:levelLowering} and Corollary~\ref{C:levelLowering}, we have that
\begin{equation}
\label{main-modp-isom}
{\overline{\rho}}_{J,\mathfrak{p}} \cong {\overline{\rho}}_{f,\mathfrak{P}}
\end{equation}
for some Hilbert newform $f$ of parallel weight $2$, trivial character, and level $\mathfrak{q}_2^2 \mathfrak{q}_{11}$ or $\mathfrak{q}_2^2 \mathfrak{q}_{11}^2$ accordingly as $11 \mid a + b$ or $11 \nmid a + b$, respectively, and where $\mathfrak{p} \mid p$ is a prime of $K$ and $\mathfrak{P} \mid p$ is a prime of the field of coefficients $K_f$ of $f$.
Enlarging $p$ if needed we can further assume that the field of coefficients~$K_f$ of $f$ is the same as that of~$J$, that is~$K_f = K$. Indeed, the field $K_f$ has been shown to contain $K$. Suppose this containment is strict. By effective Chebotarev, there exists a prime $\mathfrak{q}$ of $K$ such that $a_\mathfrak{q}(f) \notin K$, hence
$a_\mathfrak{q}(f)$ is different from all Galois conjugates of $a_\mathfrak{q}(J) \in K$ as~$K/\mathbb{Q}$ is Galois.
The isomorphism \eqref{main-modp-isom} implies that $p$ divides the norm from $K_f$ to $\mathbb{Q}$ of the non-zero element $\prod_{\sigma \in \Gal(K/\mathbb{Q})} (a_\mathfrak{q}(f) - a_\mathfrak{q}(J)^\sigma)$. This yields a contradiction for $p$ sufficiently large, hence $K_f = K$.
In particular, there is only one representation~${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ attached to~$f$ in residual chacracteristic~$11$. For a representation~${\overline{\rho}}$ we let ${\overline{\rho}}^{ss}$ denotes its semi-simplification.
\begin{theorem}
\label{irred}
For $p$ sufficiently large, a non-trivial primitive solution $(a,b,c)$ to~\eqref{main-equ}
such that $2 \nmid a+b$ gives rise to a newform~$f$ as in~\eqref{main-modp-isom}
with the property that ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ is reducible. Furthermore, we have ${\overline{\rho}}_{f,\mathfrak{p}_{11}}^{ss} \simeq 1 \oplus \chi_{11}$.
\end{theorem}
\begin{proof} Let $f$ be given as above and assume that ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ is irreducible; we will reach a contradiction with large~$p$.
By Clifford theory, ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ is a base change from $\mathbb{Q}$ if and only if ${\overline{\rho}}_{f,\mathfrak{p}_{11}}^\sigma \cong {\overline{\rho}}_{f,\mathfrak{p}_{11}}$ for all elements $\sigma \in \Gal(K/\mathbb{Q})$, where ${\overline{\rho}}_{f,\mathfrak{p}_{11}}^\sigma (\tau):={\overline{\rho}}_{f,\mathfrak{p}_{11}}(\sigma \tau \sigma^{-1})$ for all $\tau \in G_K$.
If ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ is not a base change from $\mathbb{Q}$, then by effective Chebotarev (see for instance \cite{Thorner-Zaman}), there exists a prime $\mathfrak{q}$ of $K$ above a prime $q \nmid 2\cdot 11$, of norm bounded by some fixed constant, and such that $a_{\mathfrak{q}^\sigma}(f) \not\equiv a_{\mathfrak{q}}(f) \pmod {\mathfrak{p}_{11}}$ for some $\sigma \in \Gal(K/\mathbb{Q})$. Observe that the prime~$\mathfrak{q}$ is independent of~$p$.
Thus,
after enlarging~$p$ if needed, we can asssume $q \nmid c$ by Theorem~\ref{result-from-F}.
Since $q \nmid c$ we have that $J$ has good reduction at $\mathfrak{q}$. We can assume further that
\begin{equation}\label{E:mod11}
a_{\mathfrak{q}}(J) \equiv a_{\mathfrak{q}}(f) \pmod {\mathfrak{p}_{11}} \quad \text{ and } \quad a_{\mathfrak{q}^\sigma}(J) \equiv a_{\mathfrak{q}^\sigma}(f) \pmod {\mathfrak{p}_{11}},
\end{equation}
because if one of these does not hold, then either $a_{\mathfrak{q}}(J) \not= a_{\mathfrak{q}}(f)$ or $a_{\mathfrak{q}^\sigma}(J) \not= a_{\mathfrak{q}^\sigma}(f)$, which then yields a bound on $p$ via~\eqref{main-modp-isom}. Since $a_{\mathfrak{q}}(J) \equiv a_{\mathfrak{q}^\sigma}(J) \pmod {\mathfrak{p}_{11}}$ by Theorem~\ref{T:FreyRep}, we thus have that $a_{\mathfrak{q}}(J) \equiv a_{\mathfrak{q}^\sigma}(J) \equiv a_{\mathfrak{q}^\sigma}(f) \not\equiv a_{\mathfrak{q}}(f) \pmod {\mathfrak{p}_{11}}$, contradicting the first part of~\eqref{E:mod11}.
We conclude that ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ is a base change from $\mathbb{Q}$.
To finish the proof, under this base change assumption, we will show there is concrete prime~$\mathfrak{q}$ satisfying
$a_{\mathfrak{q}}(J) \not= a_{\mathfrak{q}}(f)$. Since the possible values for these traces do not depend on~$p$, this yields a contradiction for large~$p$ via~\eqref{main-modp-isom}.
Let $N, k, \chi$ be the Serre level, weight, and character of an extension ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ to $G_\mathbb{Q}$. Since $K/\mathbb{Q}$ is unramified at $2$, we see that $N$ divides $2^2$ (recall that $f$ has level of the form $\mathfrak{q}_2^2 \mathfrak{q}_{11}^i$). For $N = 2$, only $\chi = 1$ is possible. For $N = 2^2$, $\chi = 1$ or $\chi$ is the unique non-trivial character of conductor $2^2$. In the latter case, $\chi(-1) = -1$, but as $K$ is totally real, $\chi$ cannot become trivial after base change to $K$. Thus, we conclude $\chi = 1$ also when $N = 2^2$.
Finally, some twist $\rho_{f,\mathfrak{p}_{11}} \otimes \chi_{11}^i$, $0 \le i \le 9$ will have weight $k \le 12$, where $\chi_{11}$ is the $11$th cyclotomic character. Note twisting by $\chi_{11}$ doesn't affect the Serre level $N$, nor the Serre character $\chi$.
Recall $L(t) : y^2 = x (x-1)( x-t)$ denotes the Legendre family of elliptic curves.
Let $g \in S_k(N)$ be a (classical) newform of weight $k$, trivial character, and level $N$. Let
\begin{align*}
\rho_{g,\mathfrak{p}} : G_\mathbb{Q} \rightarrow \operatorname{GL}_2(\overline{\mathbb{Q}}_p), \\
\rho_{L,p}: G_\mathbb{Q} \rightarrow \operatorname{GL}_2(\mathbb{Q}_p)
\end{align*}
denote the usual $\mathfrak{p}$-adic and $p$-adic Galois representations attached to $g$ and $L$, respectively, where $\mathfrak{p}$ is a prime of $\overline{\mathbb{Q}}_p$ over $p$. Using standard notation, we define
\begin{align*}
a_q(g) & = \tr \rho_{g,\mathfrak{p}}(\Frob_q), \\
a_q(L) & = \tr \rho_{L,p}(\Frob_q), \\
a_{\mathfrak{q}}(g) & = \tr \rho_{g,\mathfrak{p}}(\Frob_\mathfrak{q}), \\
a_{\mathfrak{q}}(L) & = \tr \rho_{L,p}(\Frob_\mathfrak{q}),
\end{align*}
where $\mathfrak{q}$ is a prime of $K$ above the prime $q$, and $\Frob_\mathfrak{q} \in G_K$ (resp.\ $\Frob_q \in G_\mathbb{Q}$) are Frobenius elements at $\mathfrak{q}$ (resp.\ $q$). By the behaviour of Euler factors under base change from $\mathbb{Q}$ to $K$, we have the following relations for $\mathfrak{q}$ a prime of $K$ above $q$,
\begin{align}
& a_{\mathfrak{q}}(g) = BC_{q,k}(a_{q}(g)), \\
& a_{\mathfrak{q}}(L) = BC_{q,2}(a_{q}(L)), \notag
\end{align}
where $BC_{q,k}(x) = x^5 - 5 q^{k-1} x^3 + 5 q^{2 k-2} x$ if $q$ is inert in $K$ and $BC_{q,k}(x) = x$ if $q$ splits completely in $K$.
For each possible choice of $i$, weight and newform $g \in S_k(N)$, we make a choice of small prime $q \not= 2, 11$ such that
\begin{align}
\label{eliminate-cond}
& a_{\mathfrak{q}}(g) \not\equiv \pm a_{\mathfrak{q}}(L) \chi_{11}(\Frob_\mathfrak{q})^i \pmod{\mathfrak{p}_{11}} \\
& a_{\mathfrak{q}}(g) \not\equiv \pm BC_{q,2}(q+1) \chi_{11}(\Frob_\mathfrak{q})^i \pmod{\mathfrak{p}_{11}}, \notag
\end{align}
where $L = L(t)$ is the Legendre curve with $t = a^{11}/(a^{11} + b^{11})$ and under the assumption $L$ has good reduction at $q$. The {\tt Magma} program which is used to find such a prime $q$ checks that \eqref{eliminate-cond}
holds for all $(a_0, b_0) \in \mathbb{F}_q^\times$ satisfying
\begin{equation*}
a_0^{11} b_0^{11} (a_0^{11} + b_0^{11}) \not= 0.
\end{equation*}
Since $L(t_0)$ has good reduction at $q$, where $t_0 = a_0^{11}/(a_0^{11} + b_0^{11})$, we have that $a_\mathfrak{q}(L(t_0)) = a_\mathfrak{q}(L(t))$, where $t_0$ is the reduction of $t$ modulo $q$. Hence, \eqref{eliminate-cond} holds for $q \nmid abc$.
By Theorem~\ref{T:FreyRep} and Remark~\ref{eps-order-two}, we know that $a_{\mathfrak{q}}(J) \equiv \pm a_{\mathfrak{q}}(L) \pmod{\mathfrak{p}_{11}}$ so that
\begin{align*}
& a_{\mathfrak{q}}(J) \chi_{11}(\Frob_\mathfrak{q})^i \equiv \pm a_{\mathfrak{q}}(L) \chi_{11}(\Frob_\mathfrak{q})^i \\
& \not \equiv a_{\mathfrak{q}}(g) \equiv a_{\mathfrak{q}}(f) \chi_{11}(\Frob_\mathfrak{q})^i \pmod {\mathfrak{p}_{11}}.
\end{align*}
Hence $a_{\mathfrak{q}}(J) \not= a_{\mathfrak{q}}(f)$, which then bounds $p$.
Suppose now that $L$ does not have good reduction at $q$ so that $q\mid abc$. By enlarging $p$, Theorem~\ref{result-from-F} shows that $q \nmid c$ and thus $q \mid ab$. Then ${\overline{\rho}}_{L,11}$ is unramified at $q$ and $a_{q}(L) \equiv q+1 \pmod{11}$ (the case $-(q+1)$ is excluded as the reduction type is split multiplicative). But $a_{\mathfrak{q}}(g) \not \equiv \pm BC_{q,2}(q+1) \chi_{11}(\Frob_\mathfrak{q})^i \pmod{\mathfrak{p}_{11}}$ by \eqref{eliminate-cond}. Hence,
\begin{align*}
& a_{\mathfrak{q}}(f) \chi_{11}(\Frob_\mathfrak{q})^i \equiv a_{\mathfrak{q}}(g) \not\equiv \pm BC_{q,2}(q+1) \chi_{11}(\Frob_\mathfrak{q})^i \\
& \equiv a_{\mathfrak{q}}(L) \chi_{11}(\Frob_\mathfrak{q})^i \equiv \pm a_{\mathfrak{q}}(J) \chi_{11}(\Frob_\mathfrak{q})^i \pmod {\mathfrak{p}_{11}},
\end{align*}
so $a_{\mathfrak{q}}(f) \not= a_{\mathfrak{q}}(J)$ and we obtain a bound on $p$.
We list the prime $q$ used for each $N,k$ below (the choice of $q$ can be made to be independent of $i$).
\medskip
\begin{center}
\begin{tabular}{|c|c|}
\hline
$(N,k)$ & $q$ \\
\hline
$(1,12)$ & 17 \\
$(2,8)$ & 13 \\
$(2,10)$ & 3 \\
$(4,6)$ & 3 \\
$(4,10)$ & 7 \\
$(4,12)$ & 19 \\
\hline
\end{tabular}
\end{center}
We conclude that ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ is reducible. We will now prove the last statement. Indeed, since $\det {\overline{\rho}}_{f,\mathfrak{p}_{11}} = \chi_{11}$, we have that
\begin{equation}
{\overline{\rho}}_{f,\mathfrak{p}_{11}} \simeq \begin{pmatrix} \theta_1 & \star\\ 0 & \theta_2 \end{pmatrix}
\quad \text{with} \quad \theta, \theta' : G_K \rightarrow \mathbb{F}_{\mathfrak{p}_{11}}^*
\quad \text{satisfying} \quad \theta \theta' = \chi_{11}.
\end{equation}
Let $\tilde \theta_1, \tilde \theta_2, \tilde \chi_{11}$ be lifts of $\theta_1, \theta_2, \chi_{11}$, respectively, to characters with values in $\overline{\mathbb{Z}} \subseteq \mathbb{C}^*$ and having the same order. The characters $\tilde \theta_1, \tilde \theta_2$ have conductor dividing $2 \mathfrak{q}_{11} \infty_1 \ldots \infty_5$, where $\infty_i$ are the places of $K$ above the infinite place $\infty$ of $\mathbb{Q}$. The conductor exponents of $\tilde \theta_1, \tilde \theta_2$ at $\mathfrak{q}_{11}$ are $\le 1$ because the characters are tamely ramified.
The Ray class group of levels $\mathfrak{q}_{11} \infty_1 \ldots \infty_5$ and $2 \mathfrak{q}_{11} \infty_1 \ldots \infty_5$ both have order $2$. For level $\mathfrak{q}_{11} \infty_1 \ldots \infty_5$, this comes from the character $\tilde \chi_{11}$. Hence, it follows that
\begin{equation}
\label{associated-char}
{\overline{\rho}}_{f,\mathfrak{p}_{11}}^{ss} \simeq 1 \oplus \chi_{11},
\end{equation}
\end{proof}
\begin{comment}
\begin{remark}
If $\rho_{f,\mathfrak{p}}(I_{\mathfrak{q}_2})$ is infinite, then for $p$ sufficiently large, ${\overline{\rho}}_{f,\mathfrak{p}}(I_{\mathfrak{q}_2})$ contains an element of order $p$, but $p$ does not divide the order of ${\overline{\rho}}_{J,\mathfrak{p}}(I_{\mathfrak{q}_2})$, a contradiction. Thus, we may assume $\rho_{f,\mathfrak{p}}(I_{\mathfrak{q}_2})$ is finite. The isomorphism \eqref{main-modp-isom} implies that $\rho_{J,\mathfrak{p}} |_{I_{\mathfrak{q}_2}} \simeq \rho_{f,\mathfrak{p}} |_{I_{\mathfrak{q}_2}}$ for $p$ sufficiently large using Proposition~\ref{residual-inertial}. By Proposition~\ref{P:irredSupercuspidal} and case 4 of \cite[Theorem 1.5]{Jarv2}, the conductor exponent at $\mathfrak{q}_2$ of $\rho_{f,\mathfrak{p}_{11}} \mid_{I_{\mathfrak{q}_2}}$ degenerates upon reduction mod $\mathfrak{p}_{11}$. In fact, it degenerates to conductor exponent $\le 1$, so only the cases $N \le 2$ can occur.
{\color{red} Due to reductions not being well-defined in the reducible case, the Steinberg case can still occur - needs more explanation.}
{\color{red} We can comment this remark out in the release to arxiv while we ponder it and work on getting the rest of the paper ready.}
\end{remark}
\end{comment}
\section{Proof of Theorem~\ref{main-asymptotic}}
The case $2 \mid a + b$ is proven in Theorem~\ref{result-from-F}, so we now assume $2 \nmid a + b$ and the setup at the beginning of Section~\ref{Legendre-descent}, so in particular by \eqref{main-modp-isom}, we have that ${\overline{\rho}}_{J,\mathfrak{p}} \simeq {\overline{\rho}}_{f,\mathfrak{p}}$ where $f$ is a Hilbert newform of level $\mathfrak{q}_2^2 \mathfrak{q}_{11}$ or $\mathfrak{q}_2^2 \mathfrak{q}_{11}^2$, accordingly as $11 \mid a + b$ or $11 \nmid a + b$, respectively.
Let $q = 23$ be an auxiliary prime (which is totally split in $K$).
After enlarging~$p$ if needed, we can asssume $q \nmid c$ by Theorem~\ref{result-from-F}. Thus $a+ b \not\equiv 0 \mod q$ and $J$ has good reduction at any prime $\mathfrak{q}$ of $K$ above $q$. Moreover,
we may assume $a_\mathfrak{q}(f) = a_\mathfrak{q}(J)$ for all~$\mathfrak{q} \mid q$ or we a contradiction after enlarging~$p$.
If $a_\mathfrak{q}(f) = a_\mathfrak{q}(J) \not \equiv \Norm_{K/\mathbb{Q}}(\mathfrak{q})+1 \pmod{\mathfrak{p}_{11}}$,
we get a contradiction with Theorem~\ref{irred}.
Hence, we may now suppose $a_\mathfrak{q}(J) \equiv \Norm_{K/\mathbb{Q}}(\mathfrak{q}) +1 \pmod{\mathfrak{p}_{11}}$ and $a_\mathfrak{q}(J) = a_\mathfrak{q}(f)$ for all primes $\mathfrak{q}$ of $K$ above $q$.
For all $(a_0, b_0) \in \mathbb{F}_q^2$, $(a_0, b_0) \not= (0,0)$, and such that $a_0 + b_0 \not= 0$, we observe from {\tt Magma} computations that the condition $a_\mathfrak{q}(J) \equiv \Norm_{K/\mathbb{Q}}(\mathfrak{q})+1 \pmod{\mathfrak{p}_{11}}$ implies $a_0 b_0 = 0$ and $a_\mathfrak{q}(J)$ takes on one of the following values
\begin{align}
\label{trace-restricted}
4z^3 + 4z^2 - 8z - 8, \\
-4z^3 - 4z^2 + 12z + 4, \notag \\
4z^4 + 4z^3 - 16z^2 - 8z + 8, \notag \\
-4z^4 + 16z^2 - 4z - 12, \notag \\
-4z^3 + 8z - 4, \notag
\end{align}
where $z = \zeta_{11} + \zeta_{11}^{-1}$, and now $J = J(a_0,b_0)$. Since $J$ has good reduction at $\mathfrak{q}$, we have that $a_\mathfrak{q}(J(a,b)) = a_\mathfrak{q}(J(a_0, b_0))$, where $(a_0, b_0)$ is the reduction of $(a,b)$ modulo $q$. Thus, we deduce for $(a,b) \not\equiv (0,0), a+b \not\equiv 0 \pmod q$, we must have that $ab \equiv 0 \pmod q$ and $a_\mathfrak{q}(J)$ takes on one of the values in \eqref{trace-restricted}.
From \eqref{E:traces}, we have that $a_{\mathfrak{q}^\sigma}(J) = a_\mathfrak{q}(J)^\sigma$ for all $\sigma \in G_K$. Thus,
\begin{equation}
\label{Q-form}
\left\{ a_\mathfrak{q}(J)^\sigma : \sigma \in G_K \right\} = \left\{ a_{\mathfrak{q}^\sigma}(J) : \sigma \in G_K \right\}.
\end{equation}
The list in \eqref{trace-restricted} comprises an entire Galois orbit of $a_\mathfrak{q}(J)$ and thus by \eqref{Q-form} we have that the sum of the $a_\mathfrak{q}(J) = a_\mathfrak{q}(f)$ over the primes $\mathfrak{q}$ of $K$ above $q$ is $-12$.
Up to now, we have not invoked the condition $11 \mid a + b$, which we now do to ensure $f$ is of level $\mathfrak{q}_2^2 \mathfrak{q}_{11}$. Without such a condition, the solutions with $ab = 0$ give rise to a newform $f$ at level $\mathfrak{q}_2^2 \mathfrak{q}_{11}^2$ which cannot be eliminated using any current techniques (see paragraph after Theorem~\ref{main-asymptotic}).
To obtain a final contradiction, we need to show for every newform $f$ of level $\mathfrak{q}_2^2 \mathfrak{q}_{11}$, the sum of the $a_\mathfrak{q}(f)$ over the $\mathfrak{q}$ is not $-12$.
Let $\mathfrak{q}_i$ for $i = 1, \ldots, 5$ be the primes of $K$ above $q$. We can compute the Hecke matrices $T_{\mathfrak{q}_i}$ for $i = 1, \ldots, 5$ quickly on the Hilbert newspace of level $\mathfrak{q}_2^2 \mathfrak{q}_{11}$ which has dimension $376$. We check that the rank of $\sum_{i=1}^5 T_{\mathfrak{q}_i} + 12 I$ is $376$, where $I$ is the identity operator. Thus, there can be no newform $f$ at this level with the property that the sum of the $a_{\mathfrak{q}_i}(f)$ over the $\mathfrak{q}_i$ is $-12$.
The computation of the above Hecke matrices can be done in under a minute, and the total time to complete the proof of Theorem~\ref{main-asymptotic} is a few minutes.
\begin{comment}
\begin{remark}
In principle, the Frey curve $F$ can be used to approach the remaining unobstructed congruence class $2 \nmid a + b$, $11 \nmid a + b$, $11 \nmid ab$. It requires the computation of elliptic curves over $K$ of conductor $\mathfrak{q}_2^3$ for an asymptotic result. Although the space of Hilbert newforms at this level only has dimension 1,201, a full computation of newforms or even partial computation of the newforms with rational coefficients is currently out of reach. The reason is that this level requires the indefinite quaternion algebra algorithm \cite{hilbert-computation}, which is slow even for moderate dimensions.
\end{remark}
\end{comment}
\begin{comment}
\section{Proof of Theorem~\ref{main-asymptotic-complete}}
We may assume from Theorem~\ref{main-asymptotic} that $2 \nmid a + b$ and $11 \nmid a + b$. Hence $11 \nmid abc$, where we note that $11 \nmid c$ as $11 \nmid a + b$. By the setup at the beginning of Section~\ref{Legendre-descent}, we have in particular by \eqref{main-modp-isom} that ${\overline{\rho}}_{J,\mathfrak{p}} \simeq {\overline{\rho}}_{f,\mathfrak{p}}$, where $f$ is a Hilbert newform of level $\mathfrak{q}_2^2 \mathfrak{q}_{11}^2$.
\begin{lemma}
Let $G$ be a finite group. Suppose $\mathfrak{p}$ is a prime of $\overline{\mathbb{Q}}$ which does not contain every difference $\chi_\rho(g) - \chi_{\rho'}(g)$, where $\rho, \rho'$ are non-isomorphic representations of $G$ over $\overline{\mathbb{Q}}$ of same dimension, $\chi_\rho, \chi_{\rho'}$ are their characters, and $g \in G$, as well as any prime dividing the order of $G$.
Let $\rho, \rho'$ be two representations of $G$ over $\overline{\mathbb{Q}}$ and denote by ${\overline{\rho}}, {\overline{\rho}}'$ their reductions modulo $\mathfrak{p}$, respectively. Then ${\overline{\rho}} \simeq {\overline{\rho}}'$ implies $\rho \simeq \rho'$.
\end{lemma}
\begin{proof}
Suppose ${\overline{\rho}} \simeq {\overline{\rho}}'$ yet $\rho \not\simeq \rho'$. We therefore have that $\chi_\rho \not= \chi_{\rho'}$ yet the reductions modulo $\mathfrak{p}$ of these two characters are equal. This contradicts the hypothesis on $\mathfrak{p}$. Note also because $\mathfrak{p}$ does not divide the order of $G$, the reductions ${\overline{\rho}}, {\overline{\rho}}'$ are well-defined independently of the choice of lattice for the reduction.
\end{proof}
\begin{corollary}
\label{reduce-isom}
With notation as in the above lemma. There is a constant (depending on $G$ and the dimension of $\rho, \rho'$) such that if ${\overline{\rho}} \simeq {\overline{\rho}}'$ are isomorphic, then $\rho \simeq \rho'$.
\end{corollary}
\begin{proposition}
Let $K$ be a totally real field and fix a prime $\mathfrak{q}$ of $K$ above a prime $q$.
Let $\mathcal{A}$ be a set of modular abelian varieties of $\operatorname{GL}_2$-type of fixed dimension $d$ over $K$ with potentially good reduction at $\mathfrak{q}$.
For $A \in \mathcal{A}$, denote by $\text{WD}_{\mathfrak{q}}(\rho_{A,\lambda})$ the Weil-Deligne representation at $\mathfrak{q}$ corresponding to the strictly compatible system of $\lambda$-adic representations $\rho_{A,\lambda}$ where the $\lambda$ are primes of $\overline{\mathbb{Q}}$ above $\ell \not= q$. Let $\Omega$ be the set of isomorphism classes of the $\text{WD}_{\mathfrak{q}}(\rho_{A,\lambda})$ for $A$ coming from $\mathcal{A}$.
Then $\Omega$ is finite.
\end{proposition}
\begin{proof}
The conductor exponent of the $\rho_{A,\lambda}$ at $\mathfrak{q}$ is bounded as the dimension $d$ is fixed. Hence, there are finitely many isomorphism classes of Weil-Deligne representations $\rho = \text{WD}_\mathfrak{q}(\rho_{A,\lambda})$ when restricted to the inertia group $I_\mathfrak{q}$ of $K$ at $\mathfrak{q}$. Let $\Frob_\mathfrak{q}$ be a Frobenius element in the Weil group $W_\mathfrak{q}$ of $K$ at $\mathfrak{q}$.
After base change to a finite extension of $K$, there are finitely many possibilities for the characteristic polynomial of $\rho(\Frob_\mathfrak{q}^f)$, where $f$ is the inertia degree, using the Weil bounds.
We deduce there are finitely many possibilities for the characteristic polynomial of $\rho(\Frob_\mathfrak{q})$ and hence for the conjugacy classes of $\rho(\Frob_\mathfrak{q})$.
\end{proof}
\begin{proposition}
\label{residual-inertial}
Let $K$ be a totally real field and $\mathfrak{q}$ a prime of $K$. For $p$ sufficiently large we have that
\begin{equation}
{\overline{\rho}}_{f,\mathfrak{p}} \mid_{W_\mathfrak{q}} \simeq {\overline{\rho}}_{f',\mathfrak{p}} \mid_{W_\mathfrak{q}} \implies \rho_{f,\mathfrak{p}} \mid_{W_\mathfrak{q}} \simeq \rho_{f',\mathfrak{p}} \mid_{W_\mathfrak{q}}
\end{equation}
where $f, f'$ are any two Hilbert newforms over $K$ with parallel weight $2$, trivial character, field of coefficients in $K$, and such that $\rho_{f,\mathfrak{p}}(I_{\mathfrak{q}})$ and $\rho_{f',\mathfrak{p}}(I_\mathfrak{q})$ are finite.
\end{proposition}
\begin{proof}
As ${\overline{\rho}}_{f,\mathfrak{p}} \mid_{I_\mathfrak{q}} \simeq {\overline{\rho}}_{f',\mathfrak{p}} \mid_{I_\mathfrak{q}}$, the kernels of ${\overline{\rho}}_{f,\mathfrak{p}} \mid_{I_\mathfrak{q}}$ and ${\overline{\rho}}_{f',\mathfrak{p}} \mid_{I_\mathfrak{q}}$ are the same, which we denote by $N$. Since $p$ is sufficiently large, we may then view $\rho_{f,\mathfrak{p}} \mid_{I_\mathfrak{q}}$ and $\rho_{f',\mathfrak{p}} \mid_{I_\mathfrak{q}}$ as representations of the common finite group $I_\mathfrak{q}/N$. Applying Corollary~\ref{reduce-isom}, we deduce for sufficiently large $p$, that
\begin{equation}
\rho_{f,\mathfrak{p}} \mid_{I_\mathfrak{q}} \simeq \rho_{f',\mathfrak{p}} \mid_{I_\mathfrak{q}}.
\end{equation}
There are finitely many isomorphism classes of Weil-Deligne representations $\rho$ of $W_{\mathfrak{q}}$ which could arise from $\rho_{f,\mathfrak{p}}$ or $\rho_{f',\mathfrak{p}}$ and these have the property that $\rho(I_\mathfrak{q})$ is finite. Fix a model $\rho$ over $\overline{\mathbb{Z}}$ for every such isomorphism class and let $\Omega$ be the set of these models $\rho$. If $\rho \in \Omega$ and $\rho \simeq \rho_{f,\mathfrak{p}} \mid_{W_\mathfrak{q}}$ (resp.\ $\rho_{f',\mathfrak{p}} \mid_{W_\mathfrak{q}}$), then the reduction ${\overline{\rho}}$ of $\rho$ modulo $\mathfrak{p}$ is well-defined independently of the choice of lattice and is isomorphic to ${\overline{\rho}}_{f,\mathfrak{p}} \mid_{W_\mathfrak{q}}$ (resp.\ ${\overline{\rho}}_{f',\mathfrak{p}} \mid_{W_\mathfrak{q}}$).
Let $\text{Frob}_{\mathfrak{q}}$ be a fixed choice of element in $W_\mathfrak{q}$ which maps to an arithmetic Frobenius element. Consider the finite set of elements $\rho(\text{Frob}_{\mathfrak{q}})$ as $\rho$ ranges through the set $\Omega$. For $p$ sufficiently large, two such elements which are not conjugate in $\operatorname{GL}_2(\overline{\mathbb{Z}})$ will remain non-conjugate under reduction modulo any prime $\mathfrak{p}$ of $\overline{\mathbb{Z}}$ above $p$.
We conclude, for $p$ sufficiently large, that
\begin{equation}
\rho_{J,\mathfrak{p}} \mid_{W_\mathfrak{q}} \simeq \rho_{f,\mathfrak{p}} \mid_{W_\mathfrak{q}}.
\end{equation}
\end{proof}
Assume now that ${\overline{\rho}}_{f,\mathfrak{p}_{11}}$ is reducible. It follows from \eqref{associated-char} that
\begin{equation}
{\overline{\rho}}_{f,\mathfrak{p}_{11}}^{ss} \simeq 1 \oplus \chi_{11},
\end{equation}
where $\chi_{11}$ is the mod $11$ cyclotomic character.
We will now work towards a contradiction, on the assumption that $11 \nmid abc$.
For $p$ sufficiently large, the reductions ${\overline{\rho}}_{J,\mathfrak{p}}$ and ${\overline{\rho}}_{f,\mathfrak{p}}$ are irreducible and hence well-defined independently of the choice of lattice. Since we are supposing
\begin{equation}
\label{main-isom}
{\overline{\rho}}_{J,\mathfrak{p}} \simeq {\overline{\rho}}_{f,\mathfrak{p}},
\end{equation}
we have that
\begin{equation}
\label{modp-isom}
{\overline{\rho}}_{J,\mathfrak{p}} \mid_{W_{\mathfrak{q}_{11}}} \simeq {\overline{\rho}}_{f,\mathfrak{p}} \mid_{W_{\mathfrak{q}_{11}}},
\end{equation}
and hence ${\overline{\rho}}_{J,\mathfrak{p}} \mid_{I_{\mathfrak{q}_{11}}} \simeq {\overline{\rho}}_{f,\mathfrak{p}} \mid_I$. We know that $\rho_{J,\mathfrak{p}}(I_{\mathfrak{q}_{11}})$ is finite.
If $\rho_{f,\mathfrak{p}}(I_{\mathfrak{q}_{11}})$ is infinite, then $\rho_{f,\mathfrak{p}}$ is special and some quadratic twist of ${\overline{\rho}}_{f,\mathfrak{p}}$, and hence of ${\overline{\rho}}_{J,\mathfrak{p}}$, has conductor exponent $1$ at $\mathfrak{q}_{11}$. However, by part (b) of the proof of Theorem~\ref{T:conductorJI}, this is not the case. Thus, we may now assume that $\rho_{f,\mathfrak{p}}(I_{\mathfrak{q}_{11}})$ is also finite.
By Proposition~\ref{residual-inertial}, we deduce that ${\overline{\rho}}_{J,\mathfrak{p}} \mid_{W_{\mathfrak{q}_{11}}} \simeq {\overline{\rho}}_{f,\mathfrak{p}} \mid_{W_{\mathfrak{q}_{11}}}$. Using strict compatibility in the strong sense (see \cite[Theorem 5.9]{Boeckle}), we deduce that
\begin{equation}
{\overline{\rho}}_{J,\mathfrak{p}_{11}}^{ss} \mid_{W_{\mathfrak{q}_{11}}} \simeq {\overline{\rho}}_{f,\mathfrak{p}_{11}}^{ss} \mid_{W_{\mathfrak{q}_{11}}}.
\end{equation}
We also know that ${\overline{\rho}}_{J,\mathfrak{p}_{11}} \simeq {\overline{\rho}}_{L,11} \otimes \epsilon$, where $\epsilon$ is a character of order dividing $2$ by Theorem~\ref{T:FreyRep} and Remark~\ref{eps-order-two}. Hence
\begin{equation}
\label{mod11-isom}
{\overline{\rho}}_{L,11}^{ss} \otimes \epsilon \mid_{W_{\mathfrak{q}_{11}}} \simeq {\overline{\rho}}_{f,\mathfrak{p}_{11}}^{ss} \mid_{W_{\mathfrak{q}_{11}}},
\end{equation}
where $L = L(t)$ is the Legendre curve and $t = a^{11}/(a^{11} + b^{11})$.
Since $11 \nmid abc$, $L$ has good reduction at $q$. If $L$ is supersingular at $11$, we obtain a contradiction to \eqref{mod11-isom} as then ${\overline{\rho}}_{L,11} \mid_{W_{\mathfrak{q}_{11}}}$ is irreducible, but ${\overline{\rho}}_{f,\mathfrak{p}_{11}} |_{W_{\mathfrak{q}_{11}}}$ is reducible by Theorem~\ref{irred}.
To see ${\overline{\rho}}_{L,11} |_W$ is irreducible, first note that ${\overline{\rho}}_{L,11}(I_{11})$ is a non-split Cartan subgroup $C'$ which has order $120$ and is cyclic as $C'$ is cyclic. Any subgroup of $C'$ of order dividing $10$ must be a scalar in $C'$. Furthermore any non-scalar element in $C'$ fixes two $\mathbb{F}_{11^2}$-lines but no $\mathbb{F}_{11}$-line. The image ${\overline{\rho}}_{L,11}(I_{\mathfrak{q}_{11}})$ is of index $\le 5$ in ${\overline{\rho}}_{L,11}(I_{11})$ so the order of ${\overline{\rho}}_{L,11}(I_{\mathfrak{q}_{11}})$ has order divisible by $24$. Hence, ${\overline{\rho}}_{L,11}(I_{\mathfrak{q}_{11}})$ contains an element of order $3$ which cannot be scalar. Thus, ${\overline{\rho}}_{L,11}(I_{\mathfrak{q}_{11}})$ fixes no $\mathbb{F}_{11}$-line. It follows that ${\overline{\rho}}_{L,11} \mid_{W_{\mathfrak{q}_{11}}}$ is irreducible.
If $L$ is ordinary at $11$, then $a_{11}(L)$ is a unit in $\mathbb{Z}_{11}$. By \cite[Theorem 6.7]{Br}, we know that
\begin{equation}
\label{ordinary}
{\overline{\rho}}_{L,11}^{ss} \mid_{W_{\mathfrak{q}_{11}}} \simeq
\mu_{a_{11}(L)}^{-1} \oplus \mu_{a_{11}(L)} \chi_{11} \mid_{W_{\mathfrak{q}_{11}}},
\end{equation}
where $\mu_\alpha$ is the unramified character of $W_{11}$ sending a Frobenius element $\text{Frob}_{11}$ to $\alpha$.
By \eqref{associated-char}, \eqref{ordinary}, and the fact that $\epsilon$ has order dividing $2$, we obtain that $a_{\mathfrak{q}_{11}}(L) \equiv \pm 1 \pmod{11}$. Since $L$ has full $2$-torsion, $a_{\mathfrak{q}_{11}}(L)$ is divisible by $4$ which forces $a_{\mathfrak{q}_{11}}(L) \in \left\{ 0, \pm 4 \right\}$, a contradiction.
\end{comment}
\bigskip
{\large \part{Appendices}}
|
1,116,691,498,424 | arxiv | \section{Introduction}
In this paper we want to study the coexistence and implications
between periodic objects of maps on the cylinder
$\Omega = \ensuremath{\mathbb{S}^1}\times \ensuremath{\mathbb{I}},$ of the form:
\[
\map{F}{\begin{pmatrix} \theta \\ x\end{pmatrix}}[
{\begin{pmatrix} R_\omega(\theta)\\f(\theta,x)\end{pmatrix}}
],
\]
where $\ensuremath{\mathbb{S}^1} = \ensuremath{\mathbb{R}} / \ensuremath{\mathbb{Z}}$, $\ensuremath{\mathbb{I}} = [0,1]$,
$R_\omega(\theta) = \theta + \omega \pmod{1}$ with
$\omega \in \ensuremath{\mathbb{R}} \setminus \ensuremath{\mathbb{Q}}$
and $f(\theta,x) = f_{\theta}(x)$ is continuous on both variables.
To study this class of maps, in \cite{FJJK}, were developed clever
techniques that lead to a theorem of the Sharkovski\u{\i} type for
this class of maps and periodic orbits of appropriate objects.
We aim at extending these results and techniques to study the
combinatorial dynamics (\emph{forcing}) and entropy of the
skew-products from the class {\ensuremath{\mathcal{S}(\Omega)}} consisting on all maps of the
above type.
As already remarked in \cite{FJJK}, instead of $\ensuremath{\mathbb{S}^1}$ we could take any
compact metric space~$\Theta$ that admits a minimal homeomorphism
$\map{R}{\Theta}$ such that
$R^{\ell}$ is minimal for every $\ell > 1$.
However, for simplicity and clarity we will remain in the class
$\ensuremath{\mathcal{S}(\Omega)}.$
Before stating the main results of this paper, we will recall the
extension of Sharkovski\u{\i} Theorem to {\ensuremath{\mathcal{S}(\Omega)}} from \cite{FJJK},
together with the necessary notation.
We start by clarifying the notion of a periodic orbit for maps from
{\ensuremath{\mathcal{S}(\Omega)}}. To this end we informally introduce some key notions that
will be defined more precisely in Section~\ref{secDefRes}.
Let $X$ be a compact metric space.
A subset $G \subset X$ is \emph{residual} if it
contains the intersection of a countable family of open dense subsets
in $X.$
In what follows, $\map{\pi}{\Omega}[\ensuremath{\mathbb{S}^1}]$ will denote the
standard projection from $\Omega$ to the circle.
Instead of periodic points we use objects that project over the
whole~$\ensuremath{\mathbb{S}^1},$ called \emph{strips} in \cite[Definition~3.9]{FJJK}.
A \emph{strip in $\Omega$} is a closed set $B \subset \Omega$ such
that $\pi(B) = \ensuremath{\mathbb{S}^1}$ (i.e., $B$ projects on the whole $\ensuremath{\mathbb{S}^1}$) and
$\pi^{-1}(\theta) \cap B$ is a closed interval (perhaps degenerate to a point)
for every $\theta$ in a residual set of $\ensuremath{\mathbb{S}^1}.$
Given two strips $A$ and $B,$ we will write $A < B$ and $A \le B$
(\cite[Definition~3.13]{FJJK}) if there exists a residual set
$G \subset \ensuremath{\mathbb{S}^1},$ such that
for every $(\theta,x) \in A \cap \pi^{-1}(G)$
and $(\theta,y) \in B \cap \pi^{-1}(G)$
it follows that $x < y$ and, respectively, $x \le y$.
We say that the strips $A$ and $B$ are
\emph{ordered}\footnote{This notion will be defined with greater detail
but equivalently in Definition~\ref{orden}.
We are giving here this less technical definition just to simplify this general section.}
(respectively \emph{weakly ordered})
if either $A < B$ or $A > B$
(respectively $A \le B$ or $A \ge B$).
Given $F \in \ensuremath{\mathcal{S}(\Omega)}$ and $n \in \ensuremath{\mathbb{N}}$, a strip $B \subset \Omega$ is
called \emph{$n$-periodic} for $F$ (\cite[Definition~3.15]{FJJK}),
if $F^{n}(B) = B$ and the image sets
$B,\ F(B),\ F^{2}(B),\dots, F^{n-1}(B)$
are pairwise disjoint and pairwise ordered.
To state the main theorem of \cite{FJJK} we need to recall the
\emph{Sharkovski\u{\i} Ordering} (\cite{Shar, Shartrans}).
The \emph{Sharkovski\u{\i} Ordering} is a linear ordering of
$\ensuremath{\mathbb{N}}$ defined as follows:
\begin{align*}
& 3 \gtso{\mbox{\tiny\textup{Sh}}} 5 \gtso{\mbox{\tiny\textup{Sh}}} 7 \gtso{\mbox{\tiny\textup{Sh}}} 9 \gtso{\mbox{\tiny\textup{Sh}}} \dots \gtso{\mbox{\tiny\textup{Sh}}} \\
& 2 \cdot 3 \gtso{\mbox{\tiny\textup{Sh}}} 2 \cdot 5 \gtso{\mbox{\tiny\textup{Sh}}} 2 \cdot 7 \gtso{\mbox{\tiny\textup{Sh}}} 2 \cdot 9 \gtso{\mbox{\tiny\textup{Sh}}} \dots \gtso{\mbox{\tiny\textup{Sh}}} \\
& 4 \cdot 3 \gtso{\mbox{\tiny\textup{Sh}}} 4 \cdot 5 \gtso{\mbox{\tiny\textup{Sh}}} 4 \cdot 7 \gtso{\mbox{\tiny\textup{Sh}}} 4 \cdot 9 \gtso{\mbox{\tiny\textup{Sh}}} \dots \gtso{\mbox{\tiny\textup{Sh}}}\\
& \hspace*{7em} \vdots \\
& 2^n \cdot 3 \gtso{\mbox{\tiny\textup{Sh}}} 2^n \cdot 5 \gtso{\mbox{\tiny\textup{Sh}}} 2^n \cdot 7 \gtso{\mbox{\tiny\textup{Sh}}} 2^n \cdot 9 \gtso{\mbox{\tiny\textup{Sh}}} \dots \gtso{\mbox{\tiny\textup{Sh}}} \\
& \hspace*{7em} \vdots \\
& \cdots \gtso{\mbox{\tiny\textup{Sh}}} 2^n \gtso{\mbox{\tiny\textup{Sh}}} \dots \gtso{\mbox{\tiny\textup{Sh}}}
16 \gtso{\mbox{\tiny\textup{Sh}}} 8 \gtso{\mbox{\tiny\textup{Sh}}} 4 \gtso{\mbox{\tiny\textup{Sh}}} 2 \gtso{\mbox{\tiny\textup{Sh}}} 1.
\end{align*}
In the ordering $\geso{\mbox{\tiny\textup{Sh}}}$ the least element is 1 and the largest
one is 3. The supremum of the set $\{1,2,4,\dots,2^n,\dots\}$ does
not exist.
\begin{ST}[\cite{FJJK}]
Assume that the map $F \in \ensuremath{\mathcal{S}(\Omega)}$ has a $p$-periodic strip.
Then $F$ has a $q$-periodic strip for every $q \ltso{\mbox{\tiny\textup{Sh}}} p.$
\end{ST}
Our first main result (Theorem~\ref{teo-prin-A}) concerns the forcing
relation.
As we will see in detail, the \emph{strips patterns} of
periodic orbits of strips of maps from $\ensuremath{\mathcal{S}(\Omega)}$ can be formalized in a natural
way as cyclic permutations, as in the case of the periodic patterns
for interval maps.
Our first main result states that a cyclic permutation
$\tau$ forces a cyclic permutation $\nu$ as interval patterns if and
only if $\tau$ forces $\nu$ as strips patterns.
Since the Sharkovski\u{\i} Theorem in the interval follows from the
forcing relation, a corollary of Theorem~\ref{teo-prin-A} is
the Sharkovski\u{\i} Theorem for maps from $\ensuremath{\mathcal{S}(\Omega)}$.
Next, an $s$-horseshoe for maps from $\ensuremath{\mathcal{S}(\Omega)}$ can be defined also in a
natural way. Our second main result (Theorem~\ref{teo-prin-B}) states
that if a map $F \in \ensuremath{\mathcal{S}(\Omega)}$ has an $s$-horseshoe then $h(F)$, the
\emph{topological entropy of $F$}, satisfies $h(F) \ge \log(s).$
This is a generalization of the well known result for the interval.
The third main result of the paper (Theorem~\ref{teo-prin-C}) states
that if a map $F \in \ensuremath{\mathcal{S}(\Omega)}$ has a periodic orbit of strips with
strips pattern $\tau,$ then $h(F) \ge h(f_{\tau}),$
where $f_{\tau}$ denotes the \emph{connect-the-dots} interval map
over a periodic orbit with pattern $\tau$.
A corollary of this fact and the lower bounds of the
topological entropy of interval maps from \cite{BGMY} is that,
if the period of $\tau$ is $2^n q$ with $n \ge 0$ and $q \ge 1$ odd,
then $h(F) \ge \tfrac{\log(\lambda_q)}{2^n}$, where $\lambda_1 = 1$
and, for each $q \ge 3,$ $\lambda_q$ is the largest root of the
polynomial $x^q − 2x^{q−2} − 1.$
Moreover,
for every $m=2^n q$ with $n \ge 0$ and $q \ge 1$ odd,
there exists a quasiperiodically forced skew-product on the cylinder $F_m$
with a periodic orbit of strips of period $m$ such that
$h(F_m) = \tfrac{\log(\lambda_q)}{2^n}.$
The paper is organized as follows. In Section~\ref{secDefRes}
we introduce the notation and we state the results in detail and in
Section~\ref{ProofOfteo-prin-A} we prove Theorem~\ref{teo-prin-A}
Finally, in Section~\ref{ProofOfCandD} we prove
Theorems~\ref{teo-prin-B} and \ref{teo-prin-C}.
\section{Definitions and statements of results}\label{secDefRes}
We start by recalling the notion of interval pattern and related
results. Afterwards we will introduce the natural extension to the
class $\ensuremath{\mathcal{S}(\Omega)}$ by defining the cylinder patterns.
In what follows we will denote the class of continuous maps from the
interval $\ensuremath{\mathbb{I}}$ to itself by $\ensuremath{\mathcal{C}^{0}(\I,\I)}.$
\subsection{Interval patterns}\label{IntPat}
Given $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)},$ we say that $p \in \ensuremath{\mathbb{I}}$ is an $n$-periodic point of
$f$ if $f^{n}(p) = p$
and $f^{j}(p) \ne p$ for $j = 1,2,\dots,n-1.$
The set of points $\{p,f(p),f^{2}(p),\ldots,f^{n-1}(p)\}$ will be
called a \emph{periodic orbit}.
A periodic orbit $P = \{p_1,p_2,\ldots,p_n\}$ is said to have the
\emph{spatial labelling} if $p_{1} < p_{2} < \dots < p_{n}.$
In what follows, every periodic orbit will be assumed to have
the spatial labelling unless otherwise stated.
\begin{definition}[Interval pattern]
Let $P = \{p_{1} < p_{2} < \dotsb < p_{n}\}$ be a periodic orbit of a
map $ f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and let $\tau$ be a cyclic permutation over
$\{1,2,\ldots,n\}.$
The periodic orbit $P$ is said to have the
\emph{(periodic) interval pattern} $\tau$ if and only if
$f(p_{i}) = p_{\tau(i)}$ for $i = 1,2,\dots,n.$
The period of $P$, $n,$ will also be called the
\emph{period of $\tau$}.
\end{definition}
\begin{remark}
Every cyclic permutation can occur as interval pattern.
\end{remark}
To study the dynamics of functions from $\ensuremath{\mathcal{C}^{0}(\I,\I)}$ we introduce the
following ordering on the set of interval patterns.
\begin{definition}[Forcing]
Given two interval patterns $\tau$ and $\nu,$ we say that
\emph{$\tau$ forces $\nu,$ as interval patterns}, denoted by
$\tau \Longrightarrow_{\ensuremath{\mathbb{I}}} \nu,$
if and only if \emph{every} $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ that has a periodic
orbit with interval pattern~$\tau$ also has a periodic orbit
with interval pattern~$\nu.$ By \cite[Theorem~2.5]{ALM},
the relation $\Longrightarrow_{\ensuremath{\mathbb{I}}}$ is a partial ordering.
\end{definition}
Next we define a \emph{canonical map} for an interval pattern as
follows.
\begin{definition}[$\tau$-linear map]\label{connect-the-dots}
Let $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and let $P = \{p_{_1},p_{_2},\dots,p_{_n}\}$ be an
$n$-periodic orbit of $f$ with the spatial labelling
($p_{_1} < p_{_2} < \dots < p_{_n}$).
We define the \emph{$P$-linear map $f_P$} as the unique map in $\ensuremath{\mathcal{C}^{0}(\I,\I)}$
such that
$f_P(p_{_i}) = f(p_{_i})$ for $i = 1,2,\ldots,n,$
$f_P$ is affine in each interval of the form $[p_i,p_{i+1}]$ for $i =
1,2,\ldots,n-1,$ and
$f_P$ is constant on each of the two connected components of $\ensuremath{\mathbb{I}}
\setminus [p_1,p_n]$.
The map $f_P$ is also called \emph{$P$-connect-the-dots map}.
Observe that the map $f_P$ is uniquely determined by $P$ and
$f\evalat{P}$.
Let $\tau$ be the pattern of the periodic orbit $P.$
The map $f_P$ will also be called a \emph{$\tau$-linear map} and
denoted by $f_{\tau}.$ Then the maps $f_\tau$ are not unique
but all maps $f_{\tau}\evalat{[\min P, \max P]}$ are topologically
conjugate and, thus, they have the same topological entropy and
periodic orbits.
\end{definition}
The next result is a useful characterization of the forcing relation
of interval patterns in terms of the $\tau$-linear maps.
\begin{theorem}[Characterization of the forcing
relation]\label{carat-forc}
Let $\tau$ and $\nu$ be two interval patterns.
Then, $\tau \Longrightarrow_{\ensuremath{\mathbb{I}}} \nu$ if and only if
$f_{\tau}$ has a periodic orbit with pattern~$\nu$.
\end{theorem}
\subsection{Strips Theory}
In this subsection we introduce a new (more restrictive) kind of strips
with better properties and we study the basic properties that we will need
throughout the paper.
To introduce this new kind of strips we first need to introduce the notion of a
\emph{core} of a set.
Given a compact metric space $(X,d)$ we denote the set of all closed (compact)
subsets of $X$ by $2^X,$ and we endow this space with the Hausdorff metric
\begin{align*}
H_{d}(B,C)
&= \max\{\max_{b\in B}\min_{c\in C} d(c,b),
\max_{c\in C}\min_{b\in B}d (c,b)\}\\
&= \max\{\max_{b\in B} d(b,C), \max_{c\in C} d(c,B)\}.
\end{align*}
It is well known that $(2^X, H_d)$ is compact.
Also, given a set $A$ we will denote the closure of $A$ by $\overline{A}$.
\begin{definition}[\cite{FJJK}]\label{core}
Let $M$ be a subset of $2^{\Omega}.$ We define the \emph{core of $M$}, denoted $M^c$, as
\[
\bigcap_{G\in\mathcal{G}(\ensuremath{\mathbb{S}^1})} \overline{M\cap\pi^{-1}(G)},
\]
where $\mathcal{G}(\ensuremath{\mathbb{S}^1})$ denotes the set of all residual subsets of $\ensuremath{\mathbb{S}^1}.$
Observe that if $M$ is compact, then
$M^c \subset M$ and,
$\pi(M) = \ensuremath{\mathbb{S}^1}$ implies $\pi(M^c) = \ensuremath{\mathbb{S}^1}.$
\end{definition}
This definition of \emph{core} is rather intricate.
Below we settle an equivalent and more useful definition
in the spirit of Lemma~3.2 and Remark~3.3 of \cite{FJJK}.
The role of the residual of continuity in this equivalent definition
is stated without proof in \cite{FJJK} and, hence,
we include the proof for completeness.
Let $M \in 2^{\Omega}$ be such that $\pi(M) = \ensuremath{\mathbb{S}^1}.$
We define the map {\map{\phi_{_M}}{\ensuremath{\mathbb{S}^1}}[2^{\ensuremath{\mathbb{I}}}]} by
$\phi_{_M}(\theta) := M^{\theta},$ and
$\rescont := \set{\theta \in \ensuremath{\mathbb{S}^1}}{\text{$\phi_{_M}$ is continuous at $\theta$}}.$
It can be easily seen that $\phi_{_M}$ is upper semicontinuous (i.e.
for every $\theta \in \ensuremath{\mathbb{S}^1}$ and every open $U \subset \ensuremath{\mathbb{I}}$ such that
$\phi_{_M}(\theta) \subset U,$ $\set{z \in \ensuremath{\mathbb{S}^1}}{\phi_{_M}(z) \subset U}$
is open in $\ensuremath{\mathbb{S}^1}$). Hence, by \cite[Theorem~7.10]{Choq},
the set $\rescont$ is residual.
The set $\rescont$ will be called the \emph{residual of continuity of $M$}.
Given $G \subset \ensuremath{\mathbb{S}^1}$ and a map $\map{\varphi}{G}[2^{\ensuremath{\mathbb{I}}}]$,
$\Graph(\varphi) := \set{(\theta,\varphi(\theta))}{\theta \in G} \subset \ensuremath{\mathbb{S}^1} \times 2^{\ensuremath{\mathbb{I}}}$
denotes the \emph{graph of $\varphi$}.
By abuse of notation we will identify $\Graph(\varphi)$
with the set $\bigcup_{\theta\in G} \{\theta\} \times \varphi(\theta)$.
Hence, we will consider $\Graph(\varphi)$ as a subset of $\Omega$ (or of $G \times \ensuremath{\mathbb{I}}$),
and $\overline{\Graph(\varphi)}$ is a compact subset of $\Omega.$
\begin{lemma}\label{CoreForResidualOfContinuity}
Let $M$ be a compact subset of $\Omega.$ Then,
\[
M^c = \overline{\Graph\left(\phi_{_M}\evalat{G}\right)} = \overline{M \cap \pi^{-1}(G)}
\]
for every residual set $G \subset \rescont.$
Moreover, $M \cap \pi^{-1}(G) = M^c \cap \pi^{-1}(G)$ and $\left(M^c\right)^c = M^c.$
\end{lemma}
\begin{proof}
We start by proving the first statement of the lemma. Notice that if
\begin{equation}\label{CoreForResidualOfContinuity-eq1}
\overline{M \cap \pi^{−1}(\rescont)} \subset \overline{M \cap \pi^{−1}(H)}
\quad\text{for every $H\in \mathcal{G},$}
\end{equation}
then
\[
\overline{M \cap \pi^{−1}(G)} \subset \overline{M \cap\pi^{−1}(\rescont)} \subset
M^c = \bigcap_{H\in\mathcal{G}} \overline{M \cap \pi^{−1}(H)} \subset
\overline{M \cap \pi^{−1} (G)}.
\]
Hence, we only have to prove \eqref{CoreForResidualOfContinuity-eq1}.
Let $H \in \mathcal{G}$ and let $(\theta, x) \in M \cap \pi^{−1}(\rescont)$
(i.e. $\theta \in \rescont$ and $(\theta,x) \in M^{\theta} = \phi_{_M}(\theta)$).
Since $H$ is residual, it is dense in $\ensuremath{\mathbb{S}^1}.$
Therefore, there exists a sequence $\{\theta_n\}_{n=1}^{\infty} \subset H$ converging to $\theta.$
Since $\theta \in \rescont$, $\phi_{_M}$ is continuous in $\theta.$
So, $\lim \phi_{_M}(\theta_n) = \phi_{_M}(\theta)$
and, for every $\varepsilon > 0$ exists $N\in \ensuremath{\mathbb{N}}$ such that
$
d\bigl((\theta,x), \phi_{_M}(\theta_n)\bigr) \le
H_d(\phi_{_M}(\theta), \phi_{_M}(\theta_n)) < \varepsilon
$
for every $n \ge N.$
Since the sets $\phi_{_M}(\theta_n)$ are compact, for every $n\in \ensuremath{\mathbb{N}},$
there exists
$
(\theta_n,x_n) \in \phi_{_M}(\theta_n) \subset M \cap \pi^{−1}(H)
$
such that
$
d\bigl((\theta,x), (\theta_n,x_n)\bigr) = d\bigl((\theta,x), \phi_{_M}(\theta_n)\bigr).
$
Thus, $\lim(\theta_n, x_n) = (\theta,x)$ and, hence,
$(\theta,x) \in \overline{M \cap \pi^{−1}(H)}.$
This implies
$
M \cap \pi^{−1}(\rescont) \subset \overline{M \cap \pi^{−1}(H)}
$
which, in turn, implies \eqref{CoreForResidualOfContinuity-eq1}.
By the first statement,
\[
M \cap \pi^{-1}(G) \subset \overline{M \cap \pi^{-1}(G)} \cap \pi^{-1}(G) = M^c \cap \pi^{-1}(G) \subset M \cap \pi^{-1}(G).
\]
Now, to end the proof of the lemma, take $\widetilde{G} = \rescont \cap \rescont[M^c],$
which is a residual set. By the part of the lemma already proven we have,
\[
\left(M^c\right)^c = \overline{M^c \cap \pi^{-1}\bigl(\widetilde{G}\bigr)} = \overline{M \cap \pi^{-1}\bigl(\widetilde{G}\bigr)} = M^c.
\]
\end{proof}
Now we are ready to define the notion of band.
\begin{definition}[Band]\label{definition-band}
Every strip $A \subset \Omega$ such that $A^c = A$ will be called a \emph{band}.
\end{definition}
\begin{remark}\label{pseudoband}
In view of Lemma~\ref{CoreForResidualOfContinuity}
a band could be equivalently defined as follows:
A \emph{band} is a set of the form $\overline{\Graph(\varphi)},$
where $\varphi$ is a continuous map from a residual set of $\ensuremath{\mathbb{S}^1}$
to the connected elements (intervals) of $2^{\ensuremath{\mathbb{I}}}.$
\end{remark}
Given $F \in \ensuremath{\mathcal{S}(\Omega)},$ a strip $A$ is
\emph{$F-$invariant} if $F(A) \subset A$ and
\emph{$F-$strongly invariant} if $F(A) = A.$
As usual, the intersection of two $F-$invariant strips is
either empty or an $F-$invariant strip.
An invariant strip is called \emph{minimal} if it does not have a strictly contained invariant strip.
\begin{remark}\label{band-properties}
From Corollary~3.5 and Lemmas~3.10 and 3.11 of \cite{FJJK} it follows that the bands in $\Omega$
have the following properties for every map from $\ensuremath{\mathcal{S}(\Omega)}$:
\begin{enumerate}[(1)]
\item The image of a band is a band.
\item Every invariant strip contains an invariant minimal strip.
\item Every invariant minimal strip is a strongly invariant band.
\end{enumerate}
\end{remark}
Moreover, the Sharkovski\u{\i} Theorem for maps from $\ensuremath{\mathcal{S}(\Omega)}$ is indeed,
\begin{ST}[\cite{FJJK}]
Assume that $F \in \ensuremath{\mathcal{S}(\Omega)}$ has a $p$-periodic strip.
Then $F$ has a $q$-periodic band for every $q \ltso{\mbox{\tiny\textup{Sh}}} p.$
\end{ST}
Next we introduce a particular kind of bands that play a key role in this theory
since they allow us to better study and work with the bands.
Given a set $A \subset \Omega$ and $\theta \in \Omega$
we will denote the set $A \cap \pi^{-1}(\theta)$ by $A^{\theta}.$
\begin{definition}
A band $A$ is called \emph{solid} when $A^{\theta}$ is an interval for every $\theta \in \ensuremath{\mathbb{S}^1}$ and
$\delta(A) := \inf\set{\diam(A^{\theta})}{\theta \in \ensuremath{\mathbb{S}^1}} > 0.$
Also, $A$ is called \emph{pinched} if $A^{\theta}$ is a point for
each $\theta$ in a residual subset of $\ensuremath{\mathbb{S}^1}.$
\end{definition}
\begin{remark}\label{solidorpseudocurve}
From \cite[Theorem~4.11]{FJJK} it follows that there are only two kind
of strongly invariant bands: \emph{solid} or \emph{pinched}.
\end{remark}
Despite of the fact that the above notion of pinched band is completely natural,
for several reasons that will become clear later (see also \cite{AMM})
we prefer to view the pinched bands as \emph{pseudo-curves} in the spirit of
Remark~\ref{pseudoband}:
\begin{definition}
Let $G$ be a residual set of $\ensuremath{\mathbb{S}^1}$ and let $\map{\varphi}{G}[\ensuremath{\mathbb{I}}]$
be a continuous map from $G$ to $\ensuremath{\mathbb{I}}.$
The set $\overline{\Graph(\varphi)}$ will be called a \emph{pseudo-curve}.
\end{definition}
The next remark summarizes the basic properties of the pseudo-curves.
\begin{remark}\label{pseudocurves-properties}
The following properties of the pseudo-curves are easy to prove:
\begin{enumerate}[(1)]
\item Every pseudo-curve is a band.
In particular $\pi\left(\overline{\Graph(\varphi)}\right) = \ensuremath{\mathbb{S}^1}.$
\item The image of a pseudo-curve is a pseudo-curve.
Moreover, every invariant pseudo-curve is strongly invariant and minimal.
\end{enumerate}
Now assume that $\overline{\Graph(\varphi)}$ is a pseudo-curve
where $\varphi$ is a map from $G$ to $\ensuremath{\mathbb{I}}.$ Then,
\begin{enumerate}[(1)]\setcounter{enumi}{2}
\item $\rescont[\overline{\Graph(\varphi)}] \supset G$ (see e.g. \cite[Lemma~7.2]{Nadler}).
\item $\overline{\Graph(\varphi)} \cap \pi^{-1}(G) = \Graph(\varphi).$
\end{enumerate}
\end{remark}
Next we want to define a partial ordering in the set of strips.
We recall that a map $g$ from $\ensuremath{\mathbb{S}^1}$ to $\ensuremath{\mathbb{I}}$ is
\emph{lower semicontinuous} (respectively \emph{upper semicontinuous})
at $\theta \in \ensuremath{\mathbb{S}^1}$ if
for every $\lambda < g(\theta)$ (respectively $ \lambda > g(\theta)$)
there exists a neighbourhood $V$ of $\theta$ such that
$\lambda < g(V)$ (respectively $\lambda > g(V)$).
When this condition holds at every point in $\ensuremath{\mathbb{S}^1}$ $g$ is said to be
\emph{lower semicontinuous on $\ensuremath{\mathbb{S}^1}$}
(respectively \emph{upper semicontinuous on $\ensuremath{\mathbb{S}^1}$}).
\begin{definition}[\protect{\cite[Definition~4.1(a)]{FJJK}}]\label{tapas}
Given $A \in 2^{\Omega}$ such that $\pi(A) = \ensuremath{\mathbb{S}^1}$ we define the
functions
\begin{align*}
M_{_A}(\theta) &:= \max\set{x \in \ensuremath{\mathbb{I}}}{(\theta,x) \in A} \\
m_{_A}(\theta) &:= \min\set{x \in \ensuremath{\mathbb{I}}}{(\theta,x) \in A}.
\end{align*}
It can be seen that
$M_{_A}$ is an upper semicontinuous function from $\ensuremath{\mathbb{S}^1}$ to $\ensuremath{\mathbb{I}}$ and
$m_{_A}$ is a lower semicontinuous function from $\ensuremath{\mathbb{S}^1}$ to $\ensuremath{\mathbb{I}}.$
From \cite[Theorem~7.10]{Choq},
each of the functions $m_{_{A}}$ and $M_{_{A}}$ is continuous on a residual
set of $\ensuremath{\mathbb{S}^1}.$
We denote by $\rescont[m_{_{A}}]$ (respectively $\rescont[M_{_{A}}]$) the
residual set of continuity of $m_{_{A}}$ (respectively $M_{_{A}}$).
\end{definition}
\begin{remark}\label{ResConPC}
If $A$ is a pseudo-curve, it follows from \cite[Lemma~7.2]{Nadler} that
$
\rescont[A] = \rescont[M_{_{A}}] = \rescont[m_{_{A}}] = \set{\theta \in \ensuremath{\mathbb{S}^1}}{M_{_A}(\theta) = m_{_A}(\theta)}
$
(that is, $A$ is pinched in $\rescont[A] = \rescont[M_{_{A}}] = \rescont[m_{_{A}}]$) and, hence,
\[
A = \overline{\Graph\left(M_{_{A}}\evalat{\rescont[M_{_{A}}]}\right)}
= \overline{\Graph\left(m_{_{A}}\evalat{\rescont[m_{_{A}}]}\right)}.
\]
\end{remark}
\begin{definition}[\protect{\cite[Definition~3.13]{FJJK}}]\label{orden}
Given two strips $A$ and $B$ we write $A < B$ (respectively $A \le B$)
if there exists a residual set $G$ in $\ensuremath{\mathbb{S}^1}$ such that
$M_{_{A}}(\theta) < m_{_{B}}(\theta)$
(respectively $M_{_{A}}(\theta) \le m_{_{B}}(\theta)$)
for all $\theta \in G.$
We say that two strips are
\emph{ordered} (respectively \emph{weakly ordered})
if either $A < B$ or $A > B$
(respectively $A \le B$ or $A \ge B$).
\end{definition}
\begin{remark}\label{orderedimpliesPDInt}
It follows from the definition that
two (weakly) ordered strips have pairwise disjoint interiors.
\end{remark}
The above ordering can be better formulated in terms of the \emph{covers} of a strip.
\begin{definition}\label{plusminus}
Let $A \subset \Omega$ be a strip.
We define the \emph{top cover of $A$} as the pseudo-curve defined by
$M_{_A}\evalat{\rescont[M_{_{A}}]}$:
\[
A^{+} := \overline{\Graph\left(M_{_A}\evalat{\rescont[M_{_{A}}]}\right)},
\]
and the \emph{bottom cover of $A$} as the pseudo-curve defined by
$m_{_A}\evalat{\rescont[m_{_{A}}]}$:
\[
A^{-} := \overline{\Graph\left(m_{_A}\evalat{\rescont[m_{_{A}}]}\right)}.
\]
\end{definition}
\begin{remark}\label{m-M-func}
The sets $A^{+}$ and $A^{-}$ are bands but in general do not coincide with
$\overline{\Graph\left(M_{_A}\right)}$ and $\overline{\Graph\left(m_{_A}\right)}$
respectively.
Moreover, if $A$ is a pseudo-curve then, from Remark~\ref{ResConPC}, $A^{+} = A^{-} = A.$
\end{remark}
\begin{remark}\label{ordentapas}
Let $A$ and $B$ be two strips.
By Remark~\ref{ResConPC} we have,
$A < B$ if and only if $A^{+} < B^{-}$ and
$A \le B$ if and only if $A^{+} \le B^{-}.$
\end{remark}
To end this subsection we introduce the useful notion of \emph{band between two pseudo-curves}.
Although this definition is inspired in the definition of a \emph{basic strip} from \cite{FJJK}
(see Definition~\ref{Markovsignedstrips}) we follow our approach based in pseudo-curves.
\newcommand{\rescont[M_{_A}] \cap \rescont[m_{_B}]}{\rescont[M_{_A}] \cap \rescont[m_{_B}]}
\begin{definition}
Let $A$ and $B$ be pseudo-curves such that $A < B$. We define the \emph{band between $A$ and $B$} as:
\[
E_{_{AB}} := \overline{\bigcup_{\theta\in \rescont[M_{_A}] \cap \rescont[m_{_B}]} \{\theta\} \times \left(M_{_{A}}(\theta), m_{_{B}}(\theta)\right)}.
\]
\end{definition}
The properties of the set $E_{_{AB}}$ are summarized by:
\begin{lemma}\label{GoodBands}
Let $A$ and $B$ be pseudo-curves such that $A < B$. Then,
\begin{enumerate}[(a)]
\item $E_{_{AB}}^{-} = A$ and $E_{_{AB}}^{+} = B.$
Moreover,
$\left(E_{_{AB}}\right)^{\theta} = \{\theta\} \times \left[M_{_{A}}(\theta), m_{_{B}}(\theta)\right]$
for every $\theta \in \rescont[M_{_A}] \cap \rescont[m_{_B}].$
\item $E_{_{AB}}$ is a band.
\item $E_{_{AB}} := \overline{\Int\left(E_{_{AB}}\right)}.$
In particular, $E_{_{AB}}$ has non-empty interior.
\end{enumerate}
\end{lemma}
\begin{proof}
From the definition of $E_{_{AB}}$ it follows that
\[
\Graph\left(M_{_A}\evalat{\rescont[M_{_A}] \cap \rescont[m_{_B}]}\right)
\subset E_{_{AB}}.
\]
Thus,
\[
A = A^c = \overline{\Graph\left(M_{_A}\evalat{\rescont[M_{_A}] \cap \rescont[m_{_B}]}\right)} \subset E_{_{AB}}
\]
by Remarks~\ref{pseudocurves-properties}(1) and \ref{ResConPC} and Lemma~\ref{CoreForResidualOfContinuity}.
Consequently, $m_{_{E_{_{AB}}}} \le m_{_{A}}.$
Now we will prove that $m_{_{E_{_{AB}}}} \ge m_{_{A}}$ and, hence, $m_{_{E_{_{AB}}}} = m_{_{A}}.$
To see this note that, for every $\theta \in \ensuremath{\mathbb{S}^1},$ there exists a sequence
\[
\{(\theta_n,x_n)\}_{n\in \ensuremath{\mathbb{N}}} \subset
\bigcup_{\theta\in \rescont[M_{_A}] \cap \rescont[m_{_B}]} \{\theta\} \times \left(M_{_{A}}(\theta), m_{_{B}}(\theta)\right)
\]
which converges to $(\theta, m_{_{E_{_{AB}}}}(\theta)).$
Observe that $x_n \ge M_{_{A}}(\theta_n) \ge m_{_{A}}(\theta_n)$ for every $n$.
Therefore, by Remark~\ref{ResConPC}
$
m_{_{E_{_{AB}}}}(\theta) = \lim_{n} x_n \ge \liminf_n m_{_{A}}(\theta_n) \ge m_A(\theta).
$
Since $m_{_{E_{_{AB}}}} = m_{_{A}}$, from Definition~\ref{plusminus} and Remark~\ref{ResConPC}
it follows that $E_{_{AB}}^{-} = A.$
In a similar way we get that $M_{_{E_{_{AB}}}} = M_{_{B}}$ and $E_{_{AB}}^{+} = B.$
Then, by the part already proven and Remark~\ref{ResConPC},
\begin{equation}\label{specialintervalsinEAB}
\left(E_{_{AB}}\right)^{\theta} = \{\theta\} \times \left[M_{_{A}}(\theta), m_{_{B}}(\theta)\right]\quad
\text{for every $\theta \in \rescont[M_{_A}] \cap \rescont[m_{_B}].$}
\end{equation}
This ends the proof of (a).
Now we prove (b).
From the previous statement it follows that $E_{_{AB}}$ is a strip.
Hence, we have to show that $\left(E_{_{AB}}\right)^c = E_{_{AB}}$
which, by Definition~\ref{core}, it reduces to prove that
$E_{_{AB}} \subset \left(E_{_{AB}}\right)^c.$
Moreover, it is enough to show that
\begin{equation}\label{corecontainswhatitshould}
E^{\theta}_{_{AB}} \subset \left(E_{_{AB}}\right)^c\quad
\text{for every $\theta \in \rescont[M_{_A}] \cap \rescont[m_{_B}]$}
\end{equation}
because, by \eqref{specialintervalsinEAB},
\[
E_{_{AB}} \subset \overline{\bigcup_{\theta \in \rescont[M_{_A}] \cap \rescont[m_{_B}]} E^{\theta}_{_{AB}}}
\subset \overline{\left(E_{_{AB}}\right)^c} = \left(E_{_{AB}}\right)^c .
\]
To prove \eqref{corecontainswhatitshould} observe that, since $\rescont[M_{_A}] \cap \rescont[m_{_B}] \cap \rescont[E_{_{AB}}]$
is a residual set (contained in $\rescont[E_{_{AB}}]$), from Lemma~\ref{CoreForResidualOfContinuity} we get
\begin{equation}\label{essentialcoreofEAB}
\left(E_{_{AB}}\right)^c =
\overline{E_{_{AB}} \cap \pi^{-1}\left(\rescont[M_{_A}] \cap \rescont[m_{_B}] \cap \rescont[E_{_{AB}}]\right)} =
\overline{\bigcup_{\theta \in \rescont[M_{_A}] \cap \rescont[m_{_B}] \cap \rescont[E_{_{AB}}]} E^{\theta}_{_{AB}}} .
\end{equation}
In particular,
\[
\bigcup_{\theta \in \rescont[M_{_A}] \cap \rescont[m_{_B}] \cap \rescont[E_{_{AB}}]} E^{\theta}_{_{AB}} \subset \left(E_{_{AB}}\right)^c .
\]
Fix $\theta \in \left(\rescont[M_{_A}] \cap \rescont[m_{_B}]\right) \setminus \rescont[E_{_{AB}}].$
Since $\rescont[M_{_A}] \cap \rescont[m_{_B}] \cap \rescont[E_{_{AB}}]$ is a residual set,
there exists a sequence $\{\theta_n\}_{n=1}^{\infty} \subset \rescont[M_{_A}] \cap \rescont[m_{_B}] \cap \rescont[E_{_{AB}}]$
whose limit is $\theta.$ The continuity of the functions $M_{_{A}}$ and $m_{_{B}}$ in $\rescont[M_{_A}] \cap \rescont[m_{_B}]$
implies that
$\lim M_{_{A}}(\theta_n) = M_{_{A}}(\theta)$
and
$\lim m_{_{B}}(\theta_n) = m_{_{B}}(\theta).$
Therefore, again by \eqref{specialintervalsinEAB},
every point of $E_{_{AB}}^{\theta}$ is limit of points in
$\{E_{_{AB}}^{\theta_n}\}_{n=1}^{\infty}.$
This implies that $E^{\theta}_{_{AB}} \subset \left(E_{_{AB}}\right)^c$ by \eqref{essentialcoreofEAB}.
This ends the proof of (b).
To prove (c) observe that
$\overline{\Int\left(E_{_{AB}}\right)} \subset E_{_{AB}}.$
So, it is enough to show that
\[
\bigcup_{\theta\in \rescont[M_{_A}] \cap \rescont[m_{_B}]} \{\theta\} \times \left(M_{_{A}}(\theta), m_{_{B}}(\theta)\right)
\subset \Int\left(E_{_{AB}}\right).
\]
Take
$(\theta, x) \in \{\theta\} \times \left(M_{_{A}}(\theta), m_{_{B}}(\theta)\right)$
with $\theta \in \rescont[M_{_A}] \cap \rescont[m_{_B}].$
Since $x \ne M_{_{A}}(\theta)$ and $x \ne m_{_{B}}(\theta),$
there exists $\varepsilon > 0$ such that
$x > M_{_{A}}(\theta) + \varepsilon$ and $x < m_{_{B}}(\theta) - \varepsilon.$
On the other hand, the continuity of $M_{_{A}}$ and $m_{_{B}}$ on
$\rescont[M_{_A}] \cap \rescont[m_{_B}]$
implies that there exist $\delta > 0$ such that
$\theta' \in \rescont[M_{_A}] \cap \rescont[m_{_B}]$ and $|\theta - \theta'| < \delta$ implies
$|M_{_{A}}(\theta) - M_{_{A}}(\theta')| < \varepsilon$ and $|m_{_{B}}(\theta) - m_{_{B}}(\theta')| < \varepsilon.$
Now we define
\[
r := \min \left\{\delta, |x-M_{_{A}}(\theta)-\varepsilon|, |x-m_{_{B}}(\theta)+ \varepsilon| \right\} > 0.
\]
Observe that, with this choice of $r,$ $M_{_{A}}(\theta) + \varepsilon \le x -r < x+r \le m_{_{B}}(\theta) - \varepsilon.$
Let
$
U := \set{(\theta', y) \in \Omega}{|\theta - \theta'| < r \text{ and } |x-y| < r}
$
be an open neighbourhood of $(\theta, x).$
We will prove that every $(\theta', y) \in U$ belongs to $E_{_{AB}}.$
If $\theta' \in \rescont[M_{_A}] \cap \rescont[m_{_B}],$
from the choice of $\delta$ and $r,$ it follows that
$
(\theta', y) \in \{\theta'\} \times (x-r,x+r) \subset \{\theta'\} \times \left[M_{_{A}}(\theta'), m_{_{B}}(\theta')\right] \subset E_{_{AB}}.
$
Now assume that $\theta' \notin \rescont[M_{_A}] \cap \rescont[m_{_B}]$
and consider a sequence
$\{\theta_n\}_{n \in \ensuremath{\mathbb{N}}} \subset \rescont[M_{_A}] \cap \rescont[m_{_B}] \cap (\theta-r, \theta+r)$
converging to $\theta'.$
Clearly, $(\theta_n, y) \in U$ for every $n\in \ensuremath{\mathbb{N}}$ and,
by the part already proven, $(\theta_n, y) \in E_{_{AB}}.$
Consequently, since $E_{_{AB}}$ is closed,
$(\theta', y) = \lim (\theta_n, y) \in E_{_{AB}}.$
\end{proof}
\subsection{Strip patterns}
In this subsection we define the notion of strips pattern and forcing
for maps from $\ensuremath{\mathcal{S}(\Omega)}$ along the lines of Subsection~\ref{IntPat}.
\begin{definition}[\protect{\cite[Definition~3.15]{FJJK}}]\label{def-ban}
Let $F \in \ensuremath{\mathcal{S}(\Omega)}.$ We say that a strip $A \subseteq \Omega$
is a \emph{$p$-periodic strip}
if $F^{p}(A) = A$ and the strips $A,F(A),\ldots, F^{p-1}(A)$
are pairwise disjoint and ordered.
The set $\{A,F(A),\ldots, F^{p-1}(A)\}$ is called an
\emph{$n$-periodic orbit of strips}.
By Remarks~\ref{band-properties} and \ref{solidorpseudocurve}, it follows that we can restrict our attention to
two kind of periodic orbit of bands: the solid ones and the pseudo-curves.
\end{definition}
A periodic orbit of strips $\{B_1,B_2,\ldots, B_p\}$
is said to have the \emph{spatial labelling} if
$B_1 < B_2 < \ldots < B_p.$
In what follows we will assume that every periodic orbit of strips
has the spatial labelling.
\begin{definition}[Strip pattern]
Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $\mathcal{B} = \{B_1,B_2,\dots,B_n\}$ be a
periodic orbit of strips.
The \emph{strips pattern of $\mathcal{B}$} is the permutation $\tau$ such that
$F(B_{i}) = B_{\tau(i)}$ for every $i = 1,2,\dotsc,n.$
When a map $F \in \ensuremath{\mathcal{S}(\Omega)}$ has a periodic orbit of strips with
strips pattern $\tau$ we say that \emph{$F$ exhibits the pattern $\tau$.}
\end{definition}
\begin{remark}
Interval and strips patterns are formally the same algebraic objects;
that is cyclic permutations.
\end{remark}
\begin{definition}[Forcing]
Let $\tau$ and $\nu$ be strips patterns.
We say that $\tau$ forces $\nu$ in $\Omega,$
denoted by $\tau \Longrightarrow_{\Omega} \nu,$
if and only if every map $F \in \ensuremath{\mathcal{S}(\Omega)}$
that exhibits the strips pattern~$\tau$ also exhibits the quasiperiodic
pattern~$\nu$.
\end{definition}
The next theorem is the first main result of this paper.
It characterizes the relation $\Longrightarrow_{\Omega}$
by comparison with $\Longrightarrow_{\ensuremath{\mathbb{I}}}.$
\begin{MainTheorem}\label{teo-prin-A}
Let $\tau$ and $\nu$ be patterns (both in $\ensuremath{\mathbb{I}}$ and $\Omega$).
Then,
\[
\tau\Longrightarrow_{\ensuremath{\mathbb{I}}}\nu
\quad\text{if and only if}\quad
\tau\Longrightarrow_{\Omega}\nu.
\]
\end{MainTheorem}
The first important consequence of Theorem~\ref{teo-prin-A} is the
next result which follows from the fact that the Sharkovski\u{\i}
theorem is a corollary of the forcing relation for interval maps.
\begin{corollary}
The Sharkovski\u{\i} Theorem for maps from $\ensuremath{\mathcal{S}(\Omega)}$ holds.
\end{corollary}
\begin{proof}
Assume that $F \in \ensuremath{\mathcal{S}(\Omega)}$ exhibits a $p$-periodic strips pattern
$\tau$ and let $q \in \ensuremath{\mathbb{N}}$ be such that $p \gtso{\mbox{\tiny\textup{Sh}}} q.$
By \cite[Corollary~2.7.4]{ALM}, $\tau \Longrightarrow_{\ensuremath{\mathbb{I}}} \nu$ for
some strips pattern $\nu$ of period $q$. Then, by
Theorem~\ref{teo-prin-A}, $\tau \Longrightarrow_{\Omega} \nu$ and, by
definition, $F$ also has a $q$-periodic orbit of strips (with strips pattern
$\nu$). Then the corollary follows from Remark~\ref{band-properties}(2,3).
\end{proof}
Next we are going to study the relation between the forcing relation
and the topological entropy of maps from $\ensuremath{\mathcal{S}(\Omega)}$. To this end we
introduce the notion of horseshoe in $\ensuremath{\mathcal{S}(\Omega)}.$
Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $A$ and $B$ be bands in $\Omega.$
We say that $A$ \emph{$F$-covers} $B$ if either
$F(A^{-}) \le B^{-}$ and $F(A^{+}) \ge B^{+},$ or
$F(A^{-}) \ge B^{+}$ and $F(A^{+}) \le B^{-}.$
\begin{definition}[Horseshoe]
An \emph{$s$-horseshoe} for a map $F \in \ensuremath{\mathcal{S}(\Omega)}$ is a pair
$(J,\mathcal{D})$ where $J$~is a band and
$\mathcal{D}$ is a set of $s \ge 2$ pairwise weakly ordered bands,
each of them with non-empty interior,
such that $L$ $F-$covers $J$ for every $L \in \mathcal{D}.$
Observe that, by Remark~\ref{orderedimpliesPDInt},
the elements of $\mathcal{D}$ have pairwise disjoint interiors.
\end{definition}
The next theorem is the second main result of the paper. It
relates the topological entropy of maps from $\ensuremath{\mathcal{S}(\Omega)}$ with
horseshoes.
\begin{MainTheorem}\label{teo-prin-B}
Assume that $F \in \ensuremath{\mathcal{S}(\Omega)}$ has an $s$-horseshoe. Then
\[
h(F) \ge \log(s).
\]
\end{MainTheorem}
Next we want to introduce a class of maps that play the role of the
connect-the-dots maps in the interval case and use them to study the
topological entropy in relation with the periodic orbits of strips.
\begin{definition}[Quasiperiodic $\tau$-linear map]
Given a strips pattern $\tau$ we define a \emph{quasiperiodic
$\tau$-linear map} $F_{\tau} \in \ensuremath{\mathcal{S}(\Omega)}$ as:
\[
F_{\tau} (\theta, x) := (R_{\omega}(\theta), f_{\tau}(x))
\]
where $R_\omega$ is the irrational rotation by angle $\omega$ and
$f_{\tau}$ is a $\tau$-linear interval map
(Definition~\ref{connect-the-dots} --- recall that $\tau$ is also an
interval pattern).
\end{definition}
\begin{remark}\label{per-ban}
Since, by definition, $f_{\tau}$ has a periodic orbit with interval
pattern $\tau,$ $F_{\tau}$ has a periodic orbit of bands
(in fact curves which are horizontal circles) with strips pattern
$\tau.$
\end{remark}
The next main result shows that the quasiperiodic
$\tau$-linear maps have minimal entropy among all maps from $\ensuremath{\mathcal{S}(\Omega)}$
which exhibit the strips pattern $\tau$, again as in the interval case.
\begin{MainTheorem}\label{teo-prin-C}
Assume that $F \in \ensuremath{\mathcal{S}(\Omega)}$ exhibits the strips pattern~$\tau$. Then
\[
h(F) \ge h (F_{\tau}) = h (f_{\tau}).
\]
\end{MainTheorem}
Theorem~\ref{teo-prin-C} has an interesting consequence concerning
the entropy of strips patterns that we define as follows.
\begin{definition}[Entropy of strips patterns]
Given a strips pattern $\tau$ we define the \emph{entropy of $\tau$} as
\[
h(\tau) := \inf\set{h(F)}{\text{$F \in \ensuremath{\mathcal{S}(\Omega)}$ and $F$ exhibits
the strips pattern $\tau$}}.
\]
\end{definition}
With this definition, in view of the Remark~\ref{per-ban},
Theorem~\ref{teo-prin-C} can be written as follows:
\setcounter{MainTheorem}{2}
\begin{MainTheorem}
Assume that $F \in \ensuremath{\mathcal{S}(\Omega)}$ exhibits the strips pattern~$\tau$. Then
\[
h(\tau) = h (F_{\tau}) = h (f_{\tau}).
\]
\end{MainTheorem}
By \cite[Corollary~4.4.7]{ALM} and \cite[Lemma~4.4.11]{ALM} we
immediately get the following simple but important corollary of
Theorem~\ref{teo-prin-C} which will allow us to obtain lower bounds of
the topological entropy depending on the set of periods.
\begin{corollary}
Assume that $\tau$ and $\nu$ are strips patterns such that
$\tau \Longrightarrow_{\Omega} \nu.$
Then $h(\tau) \ge h(\nu).$
\end{corollary}
\begin{corollary}
Assume that $F \in \ensuremath{\mathcal{S}(\Omega)}$ has a periodic orbit of strips of
period~$2^{n}q$ with $n \ge 0$ and $q \ge 1$ odd. Then,
\[
h(F) \ge \frac{\log(\lambda_q)}{2^n}
\]
where $\lambda_1 = 1$ and, for each $q \ge 3$ odd, $\lambda_q$ is the
largest root of the polynomial $x^q - 2 x^{q-2} - 1$.
Moreover,
for every $m=2^n q$ with $n \ge 0$ and $q \ge 1$ odd,
there exists a map $F_m \in \ensuremath{\mathcal{S}(\Omega)}$ with a periodic orbit of bands of period $m$
such that $h(F_m) = \tfrac{\log(\lambda_q)}{2^n}.$
\end{corollary}
\begin{proof}
Let $\tau$ denote the strips pattern of a periodic orbit of strips of
$F$ of period~$2^{n}q.$ By Theorem~\ref{teo-prin-C} and \cite{BGMY}
(see also Corollaries~4.4.7 and 4.4.18 of \cite{ALM}) we get that
\[
h(F) \ge h(f_{\tau}) \ge \frac{\log \lambda_q}{2^{n}}.
\]
To prove the second statement we use \cite[Theorem~4.4.17]{ALM}:
for every $m=2^n q$ there exists a primary pattern $\nu_m$ of period $m$ such that
$h(f_{\nu_m}) = \frac{\log \lambda_q}{2^{n}}.$
Then, from Theorem~\ref{teo-prin-C}, we can take $F_m = F_{\nu_m}$.
\end{proof}
\section{Proof of Theorem~\ref{teo-prin-A}}\label{ProofOfteo-prin-A}
To prove Theorem~\ref{teo-prin-A} we need some more notation and
preliminary results.
An important tool in the study of patterns is the Markov graph.
Signed Markov graphs are a specialization of Markov graphs.
Next we define them and clarify the relation with our situation.
A a \emph{combinatorial (directed) signed graph} is defined as a
pair $G = (V, \mathcal{A})$ where
$V$ is a finite set, called the \emph{set of vertices}, and
$\mathcal{A} \subset V \times V \times \{+, -\}$ is called the \emph{set of signed arrows}.
Given a \emph{signed arrow} $\alpha = (I, J, s) \in \mathcal{A},$
$I$ is the \emph{beginning of $\alpha$}, $J$ is the \emph{end of
$\alpha$} and $s$ is the \emph{sign of $\alpha$}.
Such an arrow $\alpha$ is denoted by $I \signedarrow{s} J.$
\subsection{Signed Markov graphs in the interval}
We start by introducing the notion of \emph{signed covering}.
In what follows, $\Bd(A)$ will denote the boundary of $A.$
\begin{definition}\label{sig-arrow}
Let $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and let $I,J \subset \ensuremath{\mathbb{I}}$ be two intervals.
We say that $I$ \emph{positively $F$-covers} $J$, denoted by
$I \signedarrow{+} J$
(or $I \signedarrow[f]{+} J$ if we need to specify the map),
if $f(\min I) \le \min J < \max J \le f(\max I)$ and,
analogously, we say that $I$ \emph{negatively $F$-covers} $J$,
denoted by $I \signedarrow{-} J$ (or $I \signedarrow[f]{-} J$),
if $f(\max I) \le \min J < \max J \le f(\min I).$
Observe that if $I \signedarrow{s_{1}} J_1$ and
$I \signedarrow{s_{2}} J_2$ then $s_{1} = s_{2}.$
We will write $I \signedarrowequal{s_{1}} J$ or $I \signedarrowequal[f]{s_{1}} J$
to denote that $f(I) = J$ and $I \signedarrow[f]{s_{1}} J$
(in particular, $f(\Bd(I)) = \Bd(J)$).
\end{definition}
We associate a signed graph to a periodic orbit of an interval map as
follows.
\begin{definition}\label{Markovsignedinterval}
Let $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and let $P$ be a periodic orbit $f$.
A \emph{$P$-basic interval} is the closure of a connected
component of $[\min P,\max P] \setminus P.$
The \emph{$P$-signed Markov graph of $f$} is the combinatorial signed
graph that has the set of all basic intervals as set of vertices
$V$ and the signed arrows in $\mathcal{A}$ are the ones given by
Definition~\ref{sig-arrow}.
\end{definition}
\begin{remark}\label{graphunique-int}
Observe that the $P$-signed Markov graph of $f$ depends only on
$f\evalat{P}$ or more precisely on the pattern of $P$. It does not
depend on the concrete choice of the points of $P$ and on the graph
of $f$ outside $P$. Consequently, if
$P$ is a periodic orbit of $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and
$Q$ is a periodic orbit of $g \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$
with the same pattern then the $P$-signed Markov graph of
$f$ and the $Q$-signed Markov graph of $g$ coincide.
In particular, the $P$-signed Markov graph of
$f$ and the $P$-signed Markov graph of $f_P$ coincide.
\end{remark}
\subsection{Signed Markov graphs in $\Omega$}
Now we also associate a signed graph to a periodic orbit of strips.
We start by defining the notion of \emph{signed covering} for bands.
It is an improvement of the notion of $F$-covering introduced before.
\begin{definition}[Signed covering \protect{\cite[Definition~4.14]{FJJK}}]\label{signedcovering}
Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $A$ and $B$ be bands in $\Omega.$
We say that $A$ \emph{positively $F$-covers} $B$, denoted by
$A \signedarrow{+} B$
(or $A \signedarrow[F]{+} B$ if we need to specify the map),
if\footnote{%
Although these definitions are formally different from \cite[Definition~4.14]{FJJK},
they are equivalent by \cite[Lemma~4.3(c,d)]{FJJK} and the definitions of the
weak ordering of strips.}\label{footremark}
$F(A^{-}) \le B^{-}$ and $F(A^{+}) \ge B^{+}$
and, analogously, we say that $A$ \emph{negatively $F$-covers} $B$, denoted by
$A \signedarrow{-} B$ (or $A \signedarrow[F]{-} B$),
if $F(A^{-}) \ge B^{+}$ and $F(A^{+}) \le B^{-}.$
Observe that, as in the interval case (see Definition~\ref{sig-arrow}), if
$A \signedarrow{s_1} B_1$ and $A \signedarrow{s_2} B_2,$ then
$s_1 = s_2.$
We will write $A \signedarrowequal{s_{1}} B$ or $A \signedarrowequal[F]{s_{1}} B$
to denote that $F(A) = B$ and $A \signedarrow[F]{s_{1}} B.$
\end{definition}
Next, by using the notion of band between two pseudo-curves,
we will define the analogous of basic interval (\emph{basic band}) and
signed Markov graph for maps from $\ensuremath{\mathcal{S}(\Omega)}$.
\begin{definition}\label{Markovsignedstrips}
Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $\mathcal{B} = \{B_{1},B_{2},\ldots,B_{n}\}$ be
a periodic orbit of strips of $F$ with the spatial labelling
(that is, $B_{1} < B_{2} < \dots < B_{n}$).
For every $i = 1,2,\dots, n-1$ the band
(see Remark~\ref{ordentapas} and Lemma~\ref{GoodBands})
\[
I_{_{B_{i}B_{i+1}}} := E_{_{B_{i}^{+}B_{i+1}^{-}}}
= \overline{\Int\left(E_{_{B_{i}^{+}B_{i+1}^{-}}}\right)}
\]
will be called a \emph{basic band}.
Observe that, from Lemma~\ref{GoodBands}(a),
$I_{_{B_{i}B_{i+1}}}^{-} = B_{i}^{+}$ and $I_{_{B_{i}B_{i+1}}}^{+} = B_{i+1}^{-}.$
The \emph{$\mathcal{B}$-signed Markov graph of $F$} is the combinatorial
signed graph that has the set of all basic bands as set of vertices
$V$ and the signed arrows in $\mathcal{A}$ are the ones given by
Definition~\ref{signedcovering}.
\end{definition}
Clearly, all the basic bands are contained in $E_{_{B_{1}B_{n}}},$
$I_{_{B_{i}B_{i+1}}} \le I_{_{B_{i+1}B_{i+2}}}$ for $i=1,2,\dots, n-2$
and if
$I_{_{B_{i}B_{i+1}}} \cap I_{_{B_{j}B_{j+1}}} \ne \emptyset$ then
$|i-j| = 1.$
\begin{remark}\label{graphunique-strips}
As in the interval case (see Remark~\ref{graphunique-int})
the $P$-signed Markov graph of $F$ is a pattern invariant.
Moreover, if
$P$ is a periodic orbit of $F \in \ensuremath{\mathcal{S}(\Omega)}$ and
$Q$ is a periodic orbit of the interval map $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$
with the same pattern, then the $P$-signed Markov graph of
$F$ and the $Q$-signed Markov graph of $f$ coincide.
In particular, the $P$-signed Markov graph of
$F$ and the $P$-signed Markov graph of $f_P$ coincide.
\end{remark}
The following lemma summarizes the properties of basic bands and
arrows. We will use it in the proof of Theorem~\ref{teo-prin-A}.
\begin{lemma}\label{prop-band}
The following statements hold.
\begin{enumerate}[(a)]
\item Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $A$ and $B$ be bands such that
there is a signed arrow $A \signedarrow{s} B$ from $A$ to $B$ in the
signed Markov graph of $F$. Then,
\begin{enumerate}[({a}.1)]
\item $F(A) \supset B$.
\item $A \signedarrow{s} D$ for every band $D \subset B.$
\item There exists a band $C \subset A$ such that $C \signedarrowequal{s} B.$
Moreover, $F(C^+) \subset B^+$ and $F(C^-) \subset B^-$ if $s = +,$
and $F(C^-) \subset B^+$ and $F(C^+) \subset B^-$ if $s = -.$
\item Assume that $A \signedarrow{s} \widetilde{B}$ with
$B \le \widetilde{B}$ and let $C$ and $\widetilde{C}$ denote the
bands given by (a.3) for $B$ and $\widetilde{B}$ respectively.
Then,
$C \le \widetilde{C}$ if $s = +,$ and
$C \ge \widetilde{C}$ if $s = -.$
\end{enumerate}
\item Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $A$ be a band such that
$A \signedarrow{\pm} A.$
Then there exists a band $A_{\infty} \subset A$ such that
$A_{\infty} \signedarrowequal{\pm} A_{\infty}.$
\end{enumerate}
\end{lemma}
\begin{proof}
Statement (a.1) is \cite[Lemma~4.15]{FJJK} and (a.2)
follows directly from the definitions.
Statements (a.3,4) are \cite[Lemma~4.19]{FJJK} while
statement (b) is \cite[Lemma~4.21]{FJJK}.
\end{proof}
\subsection{Loops of signed Markov graphs}
Given a combinatorial signed Markov graph $G$, a sequence of arrows
$
\alpha = I_0 \signedarrow{s_{0}} I_1 \signedarrow{s_{1}} \cdots \signedarrow{s_{m-1}} I_{m-1}
$
will be called a \emph{path of length $m$}.
The length of $\alpha$ will be denoted by $\abs{\alpha}$.
When a path begins and ends in the same vertex (i.e. $I_{m-1} =
I_{0}$) it will be called a \emph{loop}.
Observe that, then
$
I_1 \signedarrow{s_{1}} I_2 \signedarrow{s_{2}} \cdots \signedarrow{s_{m-2}} I_{m-1} \signedarrow{s_{m}} I_{1}
$
is also a loop in $G.$
This loop is called a \emph{shift} of $\alpha$ and denoted by
$S(\alpha).$
For $n \ge 0,$ we will denote by $S^{n}$ the $n$-th iterate of
the shift. That is,
\[
S^n(\alpha) = I_{j_0} \signedarrow{s_{j_0 }} I_{j_1}
\signedarrow{s_{j_1}} I_{j_2} \signedarrow{s_{j_2}} \cdots
\signedarrow{s_{j_{m-2}}} I_{j_{m-1}},
\]
where $j_r = r + n \pmod{m}.$
Note that $S^{km}(\alpha) = \alpha$ for every $k \ge 0.$
Let
$
\alpha = I_0 \signedarrow{s_{0}} I_1 \signedarrow{s_{1}} \cdots \signedarrow{s_{m-2}} I_{m-1}
$
and
$
\beta = J_0 \signedarrow{r_{0}} J_1 \signedarrow{r_{1}} \cdots \signedarrow{r_{l-2}} J_{l-1}
$
be two paths such that the last vertex of $\alpha$ coincides with
the first vertex of $\beta$ (i.e. $I_{m-1} = J_0$).
The path
$
I_0 \signedarrow{s_{0}} I_1 \signedarrow{s_{1}} \cdots \signedarrow{s_{m-1}} J_0
\signedarrow{r_{0}} J_1 \signedarrow{r_{1}} \cdots \signedarrow{r_{l-1}} J_{l-1}
$
is the \emph{concatenation} of $\alpha$ and $\beta$ and is denoted by
$\alpha\beta.$
In this spirit, for every $n \ge 1,$ $\alpha^n$ will denote the
concatenation of $\alpha$ with himself $n$-times.
the path $\alpha^n$ will be called the \emph{$n$-repetition of
$\alpha$}. Also, $\alpha^\infty$ will denote the infinite path
$\alpha\alpha\alpha\cdots$.
A loop is called \emph{simple} if it is not a repetition of a shorter
loop. Observe that, in that case, the length of the shorter loop
divides the length of the long one.
The next lemma translates the non-repetitiveness of a loop to conditions on its liftings.
Its proof is folk knowledge.
\begin{lemma}\label{simpleisdifferent}
Let $\alpha$ be a signed loop of length $n$ in a combinatorial signed Markov graph $G$.
If $\alpha$ is simple, $S^i(\alpha) \ne S^j(\alpha)$ for every $i \ne j.$
\end{lemma}
Given a path
$
\alpha = I_0 \signedarrow{s_0} I_1 \signedarrow{s_1} \ldots I_{m-1} \signedarrow{s_{m-1}} I_{m}
$
we define the \emph{sign} of $\alpha$, denoted by $\Sign(\alpha),$
as $\prod_{i = 1}^{m} s_i,$
where in this expression we use the obvious multiplication rules:
\begin{align*}
& + \cdot + = − \cdot − = +,\text{ and}\\
& + \cdot − = − \cdot + = −.
\end{align*}
Finally we introduce a (lexicographical) \emph{ordering} in the set
of paths of signed combinatorial graphs.
To this end we start by introducing a linear ordering in the set of
vertices. This ordering is arbitrary but fixed.
In the case of Markov graphs, the spatial labelling of orbits induces
a natural ordering in the set of basic intervals or basic bands,
which is the ordering that we are going to adopt.
More precisely, if $P = \{p_0, p_1,\dots,p_{n-1}\}$ is a periodic
orbit with the spatial labelling, then we endow the set of vertices
(basic intervals) of the associated signed Markov graph with the
following ordering:
\[
[p_0,p_1] < [p_1,p_2] < \dots < [p_{n-2},p_{n-1}].
\]
Analogously, if $\mathcal{B} = \{B_0, B_1,\dots,B_{n-1}\}$ is a periodic orbit
of strips with the spatial labelling, then we endow the set of
vertices (basic intervals) of the associated signed Markov graph with
the following ordering:
\[
I_{_{B_{0}B_{1}}} < I_{_{B_{1}B_{2}}} < \dots < I_{_{B_{n-2}B_{n-1}}}.
\]
Then, the above ordering in the set of vertices naturally induces
a \emph{lexicographical ordering} in the set of paths of the
signed combinatorial graph as follows.
Let
\begin{align*}
\alpha & = I_0 \signedarrow{s_{0}}
I_1 \signedarrow{s_{1}} \cdots
I_{n-1} \signedarrow{s_{n-1}} I_{n}
\ \text{and}\\
\beta &= J_0 \signedarrow{r_{0}}
J_1 \signedarrow{r_{1}} \cdots
J_{m-1} \signedarrow{r_{m-1}} J_{m}
\end{align*}
be paths such that there exists $k \le \min\{n,m\}$
with $I_k \ne J_k$ and $I_i = J_i$ for $i=0,1,\dots,k-1$
(recall that, by Definition~\ref{sig-arrow},
if $I_i = J_i$ then the signs $s_i$ and $r_i$ of the
corresponding arrows coincide).
We write $\alpha < \beta$ if and only if
\[
\begin{cases}
I_k < J_k & \text{when $s = +$, or} \\
I_k > J_k & \text{when $s = -$,}
\end{cases}
\]
where $s = \Sign\left(
I_0 \signedarrow{s_{0}}
I_1 \signedarrow{s_{1}} \cdots
I_{k-1} \signedarrow{s_{k-1}} I_{k}
\right) = s_0s_1\cdots s_{k-1}.$
Next we relate the loops in signed Markov graphs with periodic orbits.
\begin{definition}\label{asoc-int}
Let $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and let $p$ be a periodic point of $f$
and let
\[
\alpha = J_0 \signedarrow{s_{0}} J_1 \signedarrow{s_{1}} \cdots
J_{n-1} \signedarrow{s_{n-1}} J_{0}
\]
be a loop in the $P$-signed Markov graph of $f.$
We say that $\alpha$ and $p$ are \emph{associated}
if $p$ has period $n$ and $f^{i}(p) \in J_{i}$ for every $i = 0,1, \ldots, n-1.$
Observe that in such case $S^{m}(\alpha)$ and $f^m(p)$ are associated for all $m \ge 1.$
\end{definition}
The next lemma relates the ordering of periodic points with the ordering of the associated loops.
Its proof is a simple exercise.
\begin{lemma}\label{lazosintervalasociados}
Let $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and let $f_P$ be a $P$-linear map, where $P$
is a periodic orbit. Let $x$ and $y$ be two distinct periodic points
of $f_P$ associated respectively to two distinct loops $\alpha$ and $\beta$ in
the $P$-signed Markov graph of $f_P.$
Then $x < y$ if and only if $\alpha < \beta.$
Consequently, for every $n \ge 1,$
$f^n(x) < f^n(y)$ if and only if $S^n(\alpha) < S^n(\beta).$
\end{lemma}
The next lemma is folk knowledge but we include the proof
because we are not able to provide an explicit reference for it.
\begin{lemma}\label{lazo-unico}
Let $\tau$ be a pattern and let $f_{\tau} = f_P$ be a $P$-linear
map, where $P$ is a periodic orbit of $f_P$ of pattern $\tau.$
Assume that $\{q_0,q_1,q_2,\ldots,q_m\}$ is a periodic orbit of
$f_{\tau}$ with pattern $\nu \ne \tau.$ Then there exists a unique
loop $\alpha$ in the $P$-signed Markov graph of $f_P$ associated to
$q_0.$
Moreover, $\alpha$ is simple.
\end{lemma}
\begin{proof}
The existence and unicity of the loop $\alpha$ follows from
\cite[Lemma~1.2.12]{ALM}. We have to show that $\alpha$
is simple.
Assume that $\alpha$ is the $k$ repetition of a loop
\[
\beta = J_0 \signedarrow{s_{0}} J_1 \signedarrow{s_{1}} \cdots
J_{\ell-1} \signedarrow{s_{\ell-1}} J_{0}
\]
of length $\ell$ with $k \ge 2$ and $m = k\ell.$
By \cite[Lemma~1.2.6]{ALM}, there exist intervals
$
K_0 \subset J_0, K_1 \subset J_1, \ldots,
K_{\ell-1} \subset J_{\ell-1}
$
such tat $K_i \signedarrowequal{s_i} K_{i+1}$ for
$i=0,1,\dots,\ell-2$ and
$K_{\ell-1} \signedarrowequal {s_{\ell-1}} J_{0}.$
Clearly, since $f_P$ is $P$-linear,
$f_P^\ell\evalat{K_0}$ is an affine map from $K_0$ onto $J_0$.
On the other hand, since $q_0$ is associated to $\alpha = \beta^k$ it
follows that
$
f_P^i(q_0),f_P^{i + \ell}(q_0),\dots,f_P^{i + (k-1)\ell}(q_0) \in J_i
$
for $i=0,1,\dots,\ell-1$ and, consequently,
$q_0,f_P^{\ell}(q_0),\dots,f_P^{(k-1)\ell}(q_0) \in K_0.$
Consequently, since
$f_P^\ell\left(f_P^{(k-1)\ell}(q_0)\right) = f_p^{m}(q_0) = q_0,$
it follows that
$\{q_0,f_P^{\ell}(q_0),\dots,f_P^{(k-1)\ell}(q_0)\}$ is a periodic
orbit $f_P^\ell\evalat{K_0}$ with period $k \ge 2$.
The affinity of $f_P^\ell\evalat{K_0}$ implies that
$f_P^\ell\evalat{K_0}$ is decreasing with slope -1 and $k = 2$.
The fact that $f_P^\ell\evalat{K_0}(K_0) = J_0$ implies that
$K_0 = J_0$ and the endpoints of $J_0$ are also a periodic orbit
of $f_P^\ell\evalat{K_0}$ of period 2. In this situation $P$ and
$\{q_0,q_1,q_2,\ldots,q_m\}$ both have the same period and pattern;
a contradiction.
\end{proof}
Now we want to extend the notion of associated periodic orbit and loop
and Lemma~\ref{lazosintervalasociados} to periodic orbits of strips.
\begin{definition}\label{asoc-band}
Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let and let $\mathcal{B}$ be a periodic orbit of strips of $F$.
We say that a loop
\[
\alpha = J_0 \signedarrow{s_{0}} J_1 \signedarrow{s_{1}} \cdots
J_{n-1} \signedarrow{s_{n-1}} J_{0}
\]
in the $\mathcal{B}$-signed Markov graph of $F$ and a strip $A$ are
\emph{associated} if $A$ is an $n-$periodic strip of $F$ and
$F^{i}(A) \in J_{i}$ for every $i = 0,1, \ldots, n-1.$
Observe that in such case $S^{m}(\alpha)$ and $F^m(A)$ are associated
for all $m \ge 1.$
\end{definition}
The next lemma extends Lemma~1.2.7 and Corollary~1.2.8 of
\cite{ALM} to quasiperiodically forced skew products on the cylinder.
\begin{lemma}\label{lazoasociadobandas}
Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $J_{0},J_{1},\ldots,J_{n-1}$
be basic bands such that
\[
\alpha = J_0 \signedarrow{s_{0}} J_1 \signedarrow{s_{1}} \cdots
J_{n-1} \signedarrow{s_{n-1}} J_{0}
\]
is a simple loop in a signed Markov graph of $F.$
Then there exists a periodic band $C \subset J_0$
associated to $\alpha$ (and hence of period $n$).
Moreover, for every $i,j \in \{0,1,\dots,n-1\},$
$F^i(C) < F^j(C)$ if and only if $S^i(\alpha) < S^j(\alpha).$
\end{lemma}
\begin{proof}
Let $A$ be a basic band and let $B_1 \le B_2 \le \dots \le B_m$ be all
basic bands $F$-covered by $A.$
By Lemma~\ref{prop-band}(a.3,4) there exist bands
$K(A,B_1) \le K(A,B_2) \le \dots \le K(A,B_m)$ contained in $A$
such that $K(A,B_i) \signedarrowequal{s_A} B_i$ for $i=1,2,\dots,m,$
where $s_A$ denotes the sign of all arrows $A \signedarrow{s_A} B_i$
(see Definition~\ref{signedcovering}).
Now we recursively define a family of $2n$ bands in the following way.
We set $K_{2n-1} := K(J_{n-1},J_0) \subset J_{n-1}$ so that
$K_{2n-1} \signedarrowequal{s_{n-1}} J_0.$
Then, assume that $K_{j} \subset J_{j \pmod{n}}$ have already been
defined for $j=i+1,i+2,\dots, 2n-1$ and $i\in \{0,1,\dots, 2n-2\}.$
Since
$J_{\widetilde{\imath}} \signedarrow{s_{\widetilde{\imath}}} J_{i+1 \pmod{n}}$
with $\widetilde{\imath} = i \pmod{n},$ by Lemma~\ref{prop-band}(a.2,3),
there exists a band
$
K_{i} \subset K\left(J_{\widetilde{\imath}},J_{i+1 \pmod{n}}\right) \subset J_{\widetilde{\imath}}
$
such that $K_{i} \signedarrowequal{s_{\widetilde{\imath}}} K_{i+1}.$
Now we claim that for every $i,j \in \{0,1,\dots,n-1\}$
$S^i(\alpha) < S^j(\alpha)$ is equivalent to $K_i \le K_j.$
If $S^i(\alpha) \ne S^j(\alpha)$
there exists $k\in \{0,1,\dots,n-1\}$ such that
\begin{align*}
S^i(\alpha) &= J_{i} \signedarrow{s_i}
J_{i+1} \signedarrow{s_{i+1}} \cdots
J_{k+i-1} \signedarrow{s_{k+i-1}}
J_{k+i} \signedarrow{s_{k+i}} J_{k+i+1} \cdots \text{ and}\\
S^j(\alpha) &= J_{i} \signedarrow{s_i}
J_{i+1} \signedarrow{s_{i+1}} \cdots
J_{k+i-1} \signedarrow{s_{k+i-1}}
J_{k+j} \signedarrow{s_{k+j}} J_{k+j+1} \cdots
\end{align*}
with $J_{k+i \pmod{n}} \ne J_{k+j \pmod{n}}$
(where every sub-index in the above paths must be read modulo $n$).
By construction, $K_{k+i} \subset J_{k+i \pmod{n}}$ and $K_{k+j} \subset J_{k+j \pmod{n}}.$
Hence,
$K_{k+i} \le K_{k+j}$ if and only if $J_{k+i \pmod{n}} < J_{k+j \pmod{n}}.$
By definition
\[
K_{k+i-1} \signedarrowequal{s_{k+i-1 \pmod{n}}} K_{k+i}
\quad\text{and}\quad
K_{k+i-1} \subset K\left(J_{k+i-1 \pmod{n}},J_{k+i \pmod{n}}\right),
\]
and
\[
K_{k+j-1} \signedarrowequal{s_{k+i-1 \pmod{n}}} K_{k+j}
\quad\text{and}\quad
K_{k+j-1} \subset K\left(J_{k+i-1 \pmod{n}},J_{k+j \pmod{n}}\right).
\]
Thus,
$K_{k+i-1} \le K_{k+j-1}$ if and only if
$K_{k+i} \le K_{k+j}$ and $s_{k+i-1 \pmod{n}} = +.$
So, $K_{k+i-1} \le K_{k+j-1}$ if and only if
$J_{k+i \pmod{n}} < J_{k+j \pmod{n}}$ and $s_{k+i-1 \pmod{n}} = +.$
By iterating this argument $k-1$ times backwards we get that
$K_{i} \le K_{j}$ if and only if $J_{k+i \pmod{n}} < J_{k+j \pmod{n}}$
and
\[
\Sign\left(J_{i} \signedarrow{s_i}
J_{i+1} \signedarrow{s_{i+1}} \cdots
J_{k+i-1} \signedarrow{s_{k+i-1}}
J_{k+i}\right) =
s_{i}s_{i+1} \cdots s_{k+i-1} = +
\]
(where every sub-index in the above formula is modulo $n$).
This ends the proof of the claim.
Observe that, since $K_n \subset J_0,$ from the construction
of the sets $K_n$ we get that
$K_{n-1} \subset K_{2n-1},K_{n-2} \subset K_{2n-2},\dots, K_0\subset K_n$
and
$K_{0} \signedarrowequal[F^n]{\Sign(\alpha)} K_{n}.$
Then, by Lemma~\ref{prop-band}(a.2,b) there exists a band
$C \subset K_0 \subset J_0$ such that
$
C \signedarrowequal[F^n]{\text{\scalebox{.7}{$\Sign(\alpha)$}}} C
$
and $F^i(C) \subset K_i \subset J_i$ for $i=0,1,\dots,n-1.$
Since $C$ is a periodic strip, $F^i(C)$ and $F^j(C)$ are either disjoint or equal.
Hence, by the claim,
$F^i(C) < F^j(C)$ if and only if $S^i(\alpha) < S^j(\alpha).$
Now, Lemma~\ref{simpleisdifferent} tells us that
$S^i(\alpha) \ne S^j(\alpha)$ whenever $i \ne j.$
Consequently, $F^i(C) \ne F^j(C)$ whenever $i \ne j$ and $C$ has period $n.$
This ends the proof of the lemma.
\end{proof}
\begin{remark}\label{4horseshoe}
From the construction in the above proof it follows that if
$F \in \ensuremath{\mathcal{S}(\Omega)}$ and
\[
\alpha = J_0 \signedarrow{s_{0}} J_1 \signedarrow{s_{1}} \cdots
J_{n-1} \signedarrow{s_{n-1}} J_{0}
\]
is a loop in the a signed Markov graph of $F$ by basic bands,
then there exist bands
$K_0 = K_0(\alpha) \subset J_0,\ K_1 \subset J_1,\ \ldots,\ K_{n-1} \subset J_{n-1}$
such that $K_{i} \signedarrowequal{s_i} K_{i+1}$ for $i=0,1,\dots, n-2$
and $K_{n-1} \signedarrowequal{s_{n-1}} J_0$.
In particular,
$
K_0 \signedarrowequal[F^n]{\text{\scalebox{.7}{$\Sign(\alpha)$}}} J_0.
$
Moreover, if $\beta$ is another loop such that $\alpha^\infty \ne \beta^\infty,$
then $K_0(\alpha)$ and $K_0(\beta)$ have pairwise disjoint interiors.
\end{remark}
\subsection{Proof of Theorem~\ref{teo-prin-A}}
We start this subsection with a lemma that studies the periodic
orbits of the uncoupled quasiperiodically forced skew-products on the
cylinder (in particular for the maps $F_\tau$).
\begin{lemma}\label{patternequiv}
Let $f \in \ensuremath{\mathcal{C}^{0}(\I,\I)}$ and let $F$ be a map from $\ensuremath{\mathcal{S}(\Omega)}$ such that
$F(\theta,x) = (R_{\omega}(\theta),f(x)).$
Then, the following statements hold.
\begin{enumerate}[(a)]
\item Assume that $P = \{p_{1},p_{2},\ldots,p_{n}\}$ is a periodic
orbit of $f$ with pattern $\tau.$
Then $\ensuremath{\mathbb{S}^1} \times P$ is a periodic orbit of $F$ with pattern $\tau.$
\item If $B$ is a periodic orbit of strips of $F$ with pattern $\tau$ then
there exists a periodic orbit $P$ of $f$ with pattern $\tau$ such
that $\ensuremath{\mathbb{S}^1} \times P$ is a periodic orbit of $F$ with pattern $\tau$ and
$\ensuremath{\mathbb{S}^1} \times P \subset B.$
In particular, every cyclic permutation is a pattern of a function of
$F \in \ensuremath{\mathcal{S}(\Omega)}.$
\end{enumerate}
\end{lemma}
\begin{proof}
The first statement follows directly from the definition of a pattern.
Now we prove (b).
Let $B = \{B_{1},B_{2},\ldots,B_{n}\}$ be periodic orbit of strips of
$F$ with pattern $\tau$ (that is, $F(B_{i}) = B_{\tau(i)}$ for
$i=1,2,\dots,n$).
Since $F = (R_{\omega},f)$ it follows that $F^k = (R^k_{\omega},f^k)$
for every $k\in \ensuremath{\mathbb{N}}$ (so the iterates of $F$ are also uncoupled
quasiperiodically forced skew-products).
So, since $F^n(B_i) = B_i$ for every $i$, it follows that the
strips $B_i$ are horizontal. That is, for every $i$ there exists a
closed interval $J_i \subset \ensuremath{\mathbb{I}}$ such that $B_i = \ensuremath{\mathbb{S}^1} \times J_i.$
Moreover, since the strips are pairwise disjoint, so are the
intervals $J_i$. Clearly, $f(J_{i}) = J_{\tau(i)}$ for every $i$ and,
hence, $f^{n}(J_{1}) = J_{1}.$
So, there exists a point $p_1 \in J_{1}$ such that
$f^{n}(p_1) = p_1$ and
$f^{k}(p_1) \in f^{k}(J_{1}) = J_{\tau^k(1)}$ for $k \ge 0.$
Since the intervals $J_i$ are pairwise disjoint,
the set $P = \{p_1, f(p_1),\dots, f^{n-1}(p_1)\}$ is a periodic orbit
of $f$ of period $n$ such that $\ensuremath{\mathbb{S}^1} \times P \subset B.$
Moreover, if we set $f^{k}(p_1) = p_{\tau^k(1)}$ for
$k=1,2,\dots,n-1$, then $P$ has the spatial labelling and it follows
that the pattern of $P$ is $\tau.$
\end{proof}
\begin{proof}[Proof of Theorem~\ref{teo-prin-A}]
First we prove that $\tau \Longrightarrow_{\Omega} \nu$
implies $\tau \Longrightarrow_{\ensuremath{\mathbb{I}}} \nu.$
The assumption $\tau \Longrightarrow_{\Omega} \nu$
implies that every map $F \in \ensuremath{\mathcal{S}(\Omega)}$ that exhibits the strips
pattern~$\tau$ also exhibits the strips pattern~$\nu$.
In particular, the map $F_{\tau}$ has a periodic orbit of strips with
pattern~$\nu.$
By Lemma~\ref{patternequiv}, $f_{\tau}$ has a periodic orbit with
pattern~$\nu.$
Therefore, $\tau \Longrightarrow_{\ensuremath{\mathbb{I}}}\nu$ by the characterization of
the forcing relation in the interval (Theorem~\ref{carat-forc}).
Now we prove that $\tau \Longrightarrow_{\ensuremath{\mathbb{I}}} \nu$
implies $\tau \Longrightarrow_{\Omega} \nu.$
Clearly, we may assume that $\nu \ne \tau$.
We have to show that every $F \in \ensuremath{\mathcal{S}(\Omega)}$ that has a periodic orbit of strips
$B = \{B_0,B_1,\ldots,B_{n-1}\}$ with strips pattern $\tau$ also
has a periodic orbit of strips with strips pattern $\nu.$
We consider the map $f_{\tau} = f_P$ where $P$ is a periodic orbit
with pattern $\tau.$ By Theorem~\ref{carat-forc}, $f_{\tau}$ has
periodic orbit $Q = \{q_{0}, q_{1}, \dots, q_{n-1}\}$ with
pattern~$\nu.$ Since $Q$ has the spatial labelling, $q_0 = \min Q,$
Since $\nu \ne \tau$, by Lemma~\ref{lazo-unico}, there exists a simple
loop
\[
\alpha = I_{0} \signedarrow{s_0} I_{1}
\signedarrow{s_1} \cdots \signedarrow{s_{n-2}} I_{n-1}
\signedarrow{s_{n-1}} I_0
\]
in the $P$-signed Markov graph of $f_{\tau}$ associated to $q_0.$
Moreover, by Definition~\ref{asoc-int},
\[
\begin{array}{lcl}
q_{0} &\sim&
I_{0} \signedarrow{s_0} I_{1} \signedarrow{s_1} \cdots
\signedarrow{s_{n-2}} I_{n-1} \signedarrow{s_{n-1}} I_0 \\
f_{\tau}(q_{0}) &\sim&
I_{1} \signedarrow{s_1} I_2 \signedarrow{s_2} \cdots
\signedarrow{s_{n-1}} I_{0} \signedarrow{s_0} I_1\\
f_{\tau}^{2}(q_{0}) &\sim&
I_{2} \signedarrow{s_2} I_3 \signedarrow{s_3} \cdots
\signedarrow{s_{n-1}} I_{0} \signedarrow{s_0} I_1
\signedarrow{s_1} I_2\\
\hspace{1em}\vdots && \hspace{8em}\vdots \\
f_{\tau}^{n-1}(q_{0}) &\sim&
I_{n-1} \signedarrow{s_{n-1}} I_0 \signedarrow{s_0} I_1
\signedarrow{s_1} I_{2} \cdots
\signedarrow{s_{n-2}} I_{n-1},
\end{array}
\]
where the symbol $\sim$ means ``associated with''.
By Remark~\ref{graphunique-strips} (see also
Remark~\ref{graphunique-int}), the above loop $\alpha$ also
exists in the $B$-signed Markov graph of $F$ by replacing the
basic intervals~$I_{i} = [q_i, q_{i+1}]$
by the basic bands~$I_{_{B_{i}B_{i+1}}}:$
\[
\alpha = I_{_{B_{0}B_{1}}} \signedarrow{s_0}
I_{_{B_{1}B_{2}}} \signedarrow{s_1} \cdots
\signedarrow{s_{n-2}}
I_{_{B_{n-2}B_{n-1}}} \signedarrow{s_{n-1}}
I_{_{B_{0}B_{1}}}.
\]
By Lemma~\ref{lazoasociadobandas} and Definition~\ref{asoc-band},
$F$ has a periodic band $Q_0$ associated to $\alpha$
(and hence of period $n$), and
\begin{align*}
Q_{0} &\sim&&\hspace*{-0.7em}
I_{_{B_{0}B_{1}}} \signedarrow{s_0}
I_{_{B_{1}B_{2}}} \signedarrow{s_1} \cdots
\signedarrow{s_{n-2}}
I_{_{B_{n-2}B_{n-1}}} \signedarrow{s_{n-1}}
I_{_{B_{0}B_{1}}}\\
F(Q_{0}) &\sim&&\hspace*{-0.7em}
I_{_{B_{1}B_{2}}} \signedarrow{s_1}
I_{_{B_{2}B_{3}}} \signedarrow{s_2} \cdots
\signedarrow{s_{n-2}}
I_{_{B_{n-2}B_{n-1}}} \signedarrow{s_{n-1}}
I_{_{B_{0}B_{1}}}\signedarrow{s_0} I_{_{B_{1}B_{2}}} \\
F^{2}(Q_{0}) &\sim&&\hspace*{-0.7em}
I_{_{B_{2}B_{3}}} \signedarrow{s_2} \cdots
\signedarrow{s_{n-2}}
I_{_{B_{n-2}B_{n-1}}} \signedarrow{s_{n-1}}
I_{_{B_{0}B_{1}}} \signedarrow{s_0}
I_{_{B_{1}B_{2}}} \signedarrow{s_1} I_{_{B_{2}B_{3}}} \\
\hspace{1em}\vdots &&& \hspace{8em}\vdots \\
F^{n-1}(Q_{0}) &\sim&&\hspace*{-0.7em}
I_{_{B_{n-2}B_{n-1}}} \signedarrow{s_{n-1}}
I_{_{B_{0}B_{1}}} \signedarrow{s_0}
I_{_{B_{1}B_{2}}} \signedarrow{s_1} \cdots
\signedarrow{s_{n-2}}
I_{_{B_{n-2}B_{n-1}}}.
\end{align*}
By Lemmas~\ref{lazosintervalasociados} and \ref{simpleisdifferent},
the order of the points $f^{i}_{\tau}(q_{0})$ induces an order on
the shifts of the loop $S^{i}(\alpha),$ with the usual lexicographical
ordering and, by Lemma~\ref{lazoasociadobandas}, the order of the shifts
$S^{i}(\alpha)$ induces the same order on the bands $F^{i}(Q_{0}).$
Thus, for every $i,j \in \{0,1,\dots,n-1\},$ $i \ne j,$
$F^{i}(Q_{0}) < F^{j}(Q_{0})$ if and only if
$f_\tau^{i}(q_{0}) < f_\tau^{j}(q_{0}).$
So, $\{Q_{0},F(Q_0), F^2(Q_0),\dots,F^{n-1}(Q_0)\}$
and $\{q_{0}, q_{1}, \dots, q_{n-1}\}$ have the same pattern $\nu.$
This concludes the proof.
\end{proof}
\section{Proof of Theorems~\ref{teo-prin-B} and
\ref{teo-prin-C}}\label{ProofOfCandD}
We start by proving Theorem~\ref{teo-prin-B}.
The next technical lemma is inspired in \cite[Lemma~4.3.1]{ALM}.
\begin{lemma}\label{hor1}
Let $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let $(J,\mathcal{D})$ be an $s$-horseshoe of
$F.$
Then, there exists $\mathcal{D}_n$,
a set of $s^{n}$ pairwise weakly ordered bands contained in $J,$
each of them with non-empty interior, such that
$(J,\mathcal{D}_n)$ is a $s^{n}$-horseshoe for $F^{n}.$
\end{lemma}
\begin{proof}
We use induction. For $n = 1$ there is nothing to prove.
Suppose that the induction hypothesis holds for some $n$ and
let $D \in \mathcal{D}$ and $C \in \mathcal{D}_{n}.$
Since $C \subset J$ has non-empty interior and $D \signedarrow{\pm} J,$
by Lemma~\ref{prop-band}(a.2,3),
there exists a band $B(D,C) \subset D$ with non-empty interior
such that $B(D,C) \signedarrowequal{\pm} C.$
Moreover, given $C' \in \mathcal{D}_{n}$ with $C' \ne C,$
$B(D,C)$ and $B(D,C')$ can be chosen to be weakly ordered
because $C$ and $C'$ are weakly ordered by assumption.
Since, $C \in \mathcal{D}_{n},$ $B(D,C) \signedarrow[F^{n+1}]{\pm} J.$
Thus, the family
\[
\mathcal{D}_{n+1} =
\set{B(D,C)}{D \in \mathcal{D} \text{ and } C \in \mathcal{D}_n}
\]
consists of $s^{n+1}$ pairwise weakly ordered bands contained in $J,$
each of them with non-empty interior,
such that $B(D,C)$ $F^{n+1}$-covers $J.$
Consequently, $(J,\mathcal{D}_{n+1})$ is an $s^{n+1}$-horseshoe for
$F^{n+1}.$
\end{proof}
\begin{proof}[Proof of Theorem~\ref{teo-prin-B}]
Fix $n > 0.$
By Lemma~\ref{hor1}, $F^{n}$ has a $s^{n}$-horseshoe
$(J,\mathcal{D}).$
Remove the smallest and the biggest band of
$\mathcal{D}$ and call $K$ the smallest band that contains the
remaining elements of $\mathcal{D}.$
Clearly, $K$ is contained in the interior of $J.$
Thus, by Lemma~\ref{prop-band}(a.2,3), each element $D$ of
$\mathcal{D}$ contains in its interior a band $A(D)$
such that $A(D) \signedarrowequal[F^{n}]{\pm} K.$
Then there exists an open cover $\mathcal{B}$ of the strip $J$
(formed by open sets $B$ such $B^{\theta}$
is an open interval for every $\theta \in \ensuremath{\mathbb{S}^1}$),
such that for each $D \in \mathcal{D}\evalat{K},$
the band $A(D)$ intersects only one element $B(D)$ of $\mathcal{B}$
(then it has to be contained in it) and if $D, D' \in
\mathcal{D}\evalat{K}$ with $D \ne D'$ then $B(D)\ne B(D').$
For $D_0,D_1,\ldots,D_{k-1} \in \mathcal{D}\evalat{K}$
the set
$\cap_{i = 0}^{k-1} F^{-n}(A(D_i))$ is non-empty and intersects only
one element of $\mathcal{B}_{F^{n}}^{k},$
namely $\cap_{i = 0}^{k-1} F^{-n}(B(D_i)).$
Therefore the sets $\cap_{i = 0}^{k-1} F^{-n}(A(D_i))$ are different
for different sequences $(D_0,D_1,\ldots,D_{k-1}),$
and thus
\[
\mathcal{N}(\mathcal{B}_{F^{n}}^{k}) \ge (\Card \mathcal{D}-2)^{k},
\]
where $\mathcal{N}(\mathcal{B}_{F^{n}}^{k})$ is defined as in
\cite[Section~4.1]{ALM}.
Hence,
\[
h(F) = \frac{1}{n} h(F^{n}) \ge
\frac{1}{n} h(F^{n}, \mathcal{B}) \ge
\frac{1}{n} \log(\Card(\mathcal{D}) - 2) =
\frac{1}{n} \log(s^{n}-2).
\]
Since $n$ is arbitrary, we obtain $h(F) \ge \log(s).$
\end{proof}
Now we aim at proving Theorem~\ref{teo-prin-C}.
To this end we have to introduce some more notation and preliminary
results concerning the \emph{topological entropy}.
Given a map $f \in \ensuremath{\mathcal{S}(\Omega)}$, $h(F\evalat{I_{\theta}})$ is defined for
every $I_{\theta} := \{\theta\} \times \ensuremath{\mathbb{I}}$
(despite of the fact that it is not $F$-invariant)
by using the Bowen definition of the topological entropy
(c.f. \cite{Bowen, BowenErr}).
Moreover, the Bowen Formula gives
\[
\max \{h(R), h_{\operatorname{fib}}(F) \} \le h(F) \le
h(R) + h_{\operatorname{fib}}(F)
\]
where
\[
h_{\operatorname{fib}}(F) =
\sup\nolimits_{\theta \in \ensuremath{\mathbb{S}^1}} h(F\evalat{I_{\theta}}).
\]
Since $h(R) = 0$, it follows that
$h(F) = h_{\operatorname{fib}}(F).$
In the particular case of the uncoupled maps
$F_{\tau} = (R, f_\tau)$ we easily get the following result:
\begin{lemma}\label{Ft=ft}
Let $\tau$ be a pattern (both in $\ensuremath{\mathbb{I}}$ and $\Omega$). Then
$h(F_{\tau}\evalat{I_{\theta}}) = h (f_{\tau})$
for every $\theta \in \ensuremath{\mathbb{S}^1}.$
Consequently,
\[
h(F_{\tau}) = h_{\operatorname{fib}}(F_{\tau}) = h (f_{\tau}).
\]
\end{lemma}
Given a signed Markov graph $G$ with vertices $I_1, I_2,\dots, I_n$
we associate to it a $n \times n$ \emph{transition matrix}
$T_G = (t_{ij})$ by setting $t_{ij} = 1$ if and only if
there is a signed arrow from the vertex $I_i$ to the
vertex $I_j$ in $G.$ Otherwise, $t_{ij}$ is set to 0.
The spectral radius of a matrix $T,$ denoted by $\rho(T),$ is equal
to the maximum of the absolute values of the eigenvalues of $T.$
\begin{lemma}\label{ent-mat}
Let $P$ be a periodic orbit of strips of $F \in \ensuremath{\mathcal{S}(\Omega)}$ and let
$T$ be the transition matrix of the $P$-signed Markov graph of $F.$
Then
\[
h(F) \ge \max\{0, \log(\rho(T))\}.
\]
\end{lemma}
\begin{proof}
If $\rho(T) \le 1$ then there is nothing to prove.
So, we assume that $\rho(T) > 1.$
Let $J$ be the $i$-th $P$-basic band and let $s$ be the $i$-th
entry of the diagonal of $T^{n}.$
By \cite[Lemma 4.4.1]{ALM} there are $s$ loops of length $n$ in the
$P$-signed Markov graph of $F$ beginning and ending at $J.$
Hence, if $s \ge 2,$ $F^{n}$ has an $s$-horseshoe $(J,\mathcal{D})$
by Remark~\ref{4horseshoe}. By Theorem~\ref{teo-prin-B},
$h(F) = \tfrac{1}{n} h(F^{n}) \ge \tfrac{1}{n} \log(s).$
If there are $k$ basic bands, the trace of $T^{n}$ is not larger
than $k$ times the maximal entry on the diagonal of $T^{n}.$
Hence,
$
h(F) \ge \tfrac{1}{n} \log\left(\tfrac{1}{k} \tr(T^{n}) \right).
$
Therefore, by \cite[Lemma~4.4.2]{ALM},
\[
h(F) \ge \limsup_{n\to\infty} \frac{1}{n}
\log\left(\frac{1}{k}\tr(T^{n})\right) =
\limsup_{n\to\infty} \frac{1}{n} \log(\tr(T^{n}) =
\log(\rho(T)).
\]
\end{proof}
\begin{proof}[Proof of Theorem~\ref{teo-prin-C}]
Let $P$ be a periodic orbit of strips with pattern $\tau$ and let $T$
be the transition matrix of the $P$-signed Markov graph of $F.$
Let $f_\tau = f_Q$ be a $Q$-linear map in $\ensuremath{\mathcal{C}^{0}(\I,\I)},$
where $Q$ is a periodic orbit of $f_Q$ with pattern $\tau.$
In view of Remark~\ref{graphunique-strips} (see also
Remark~\ref{graphunique-int}),
$T$ is also the transition matrix of the $Q$-signed Markov graph of
$f_\tau.$
Consequently, by \cite[Theorem~4.4.5]{ALM},
$
h(f_{\tau}) = \max \{0, \log\left(\rho(T)\right)\}.
$
By Lemmas~\ref{ent-mat} and \ref{Ft=ft},
\[
h(F) \ge \max\{0, \log(\rho(T))\} = h(f_{\tau}) = h(F_{\tau}).
\]
\end{proof}
\bibliographystyle{plain}
|
1,116,691,498,425 | arxiv | \section{Introduction}\label{intro}
\subsection{Motivation}
In his landmark papers \cite{HidaGalois} and \cite{HidaIwasawa}, Hida proved that the $p$-adic Galois representations
attached to ordinary cuspidal Hecke eigenforms by Deligne (\cite{DeligneFormes}, \cite{CarayolReps})
interpolate $p$-adic analytically in the weight variable to a family of $p$-adic representations
whose specializations to integer weights $k\ge 2$ recover
the ``classical" Galois representations attached to weight $k$ cuspidal eigenforms.
Hida's work paved the way for a revolution---
from the pioneering work of Mazur
on Galois deformations to Coleman's construction of $p$-adic families of finite slope overconvergent
modular forms---and began a trajectory of thought whose fruits include some of the most spectacular
achievements in modern number theory.
Hida's proof is constructive and has at its heart the \'etale cohomology of the tower
of modular curves $\{X_1(Np^r)\}_{r}$ over $\Q$. More precisely,
Hida considers the projective limit $H^1_{\et}:=\varprojlim_r H^1_{\et}(X_1(Np^r)_{\Qbar},\Z_p)$
(taken with respect to the trace mappings), which is naturally a module for the
``big" $p$-adic Hecke algebra $\H^*:=\varprojlim_r \H_r^*$, which is itself an algebra
over the completed group ring $\Lambda:=\Z_p[\![1+p\Z_p]\!]\simeq \Z_p[\![T]\!]$
via the diamond operators.
Using the idempotent $e^*\in \H^*$ attached to the (adjoint) Atkin operator $U_p^*$
to project to the part of $H^1_{\et}$ where $U_p^*$ acts invertibly,
Hida proves in \cite[Theorem 3.1]{HidaGalois}
(via the comparison isomorphism between \'etale and topological cohomology and explicit
calculations in group cohomology) that
$e^* H^1_{\et}$ is finite and free as a module over $\Lambda$, and that the resulting Galois representation
\begin{equation*}
\xymatrix{
{\rho: G_{\Q}} \ar[r] & {\Aut_{\Lambda}(e^*H^1_{\et}) \simeq \GL_m(\Z_p[\![T]\!])}
}
\end{equation*}
$p$-adically interpolates the representations attached to ordinary cuspidal eigenforms.
By analyzing the geometry of the tower of modular curves, Mazur and Wiles \cite{MW-Hida}
were able to relate the inertial invariants of the local (at $p$) representation $\rho_p$ to the
\'etale cohomology of the Igusa tower studied in \cite{MW-Analogies}, and in so doing
proved\footnote{Mazur and Wiles treat only the case of tame level $N=1$.}
that the ordinary filtration of the Galois representations attached
to ordinary cuspidal eigenforms interpolates:
both the inertial invariants and covariants
are free of the same finite rank over $\Lambda$ and specialize to the corresponding subquotients
in integral weights $k\ge 2$.
As an application, they provided examples of cuspforms $f$ and primes $p$
for which the specialization of the associated Hida family of Galois representations
to weight $k=1$ is not Hodge-Tate,
and so does not arise from a weight one cuspform via the construction of Deligne-Serre
\cite{DeligneSerre}.
Shortly thereafter, Tilouine \cite{Tilouine} clarified the geometric underpinnings
of \cite{HidaGalois} and \cite{MW-Hida}, and removed most of the restrictions on the $p$-component of the
nebentypus of $f$. Central to both \cite{MW-Hida} and \cite{Tilouine} is a careful study of
the tower of $p$-divisible groups attached to the ``good quotient" modular abelian varieties
introduced in \cite{MW-Iwasawa}.
With the advent of integral $p$-adic Hodge theory,
and in view of the prominent role
it has played in furthering the trajectory initiated by Hida's work,
it is natural to ask
if one can construct Hodge--Tate, de~Rham and crystalline analogues of $e^*H^1_{\et}$,
and if so, to what extent the integral comparison isomorphsms of $p$-adic Hodge theory
can be made to work in $\Lambda$-adic families.
In \cite{OhtaEichler},
Ohta has addressed this question in the case of Hodge cohomology.
Using the invariant differentials on the tower of $p$-divisible groups studied in \cite{MW-Hida} and \cite{Tilouine},
Ohta constructs a $\Lambda \wh{\otimes}_{\Z_p} \Z_p[\mu_{p^{\infty}}]$-module
from which, via an integral version of the Hodge--Tate comparison isomorphism \cite{Tate}
for ordinary $p$-divisible groups, he is able to recover the semisimplification
of the ``semilinear representation"
$\rho_{p}\wh{\otimes} \O_{\mathbf{C}_p}$, where
$\mathbf{C}_p$ is, as usual, the $p$-adic completion of an algebraic closure of $\Q_p$.
Using Hida's results, Ohta proves that his Hodge cohomology analogue of
$e^*H^1_{\et}$ is free of finite rank over $\Lambda\wh{\otimes}_{\Z_p} \Z_p[\mu_{p^{\infty}}]$
and specializes to finite level exactly as one expects. As applications
of his theory, Ohta provides a construction of two-variable $p$-adic $L$-functions
attached to families of ordinary cuspforms differing from that of Kitagawa \cite{Kitagawa},
and, in a subsequent paper \cite{Ohta2},
provides a new and streamlined proof of the theorem of Mazur--Wiles \cite{MW-Iwasawa}
(Iwasawa's Main Conjecture for $\Q$; see also \cite{WilesTotallyReal}).
We remark that Ohta's $\Lambda$-adic Hodge-Tate isomorphism is a crucial ingredient
in the forthcoming proof of Sharifi's conjectures \cite{SharifiConj}, \cite{SharifiEisenstein}
due to Fukaya and Kato \cite{FukayaKato}.
\subsection{Results}\label{resultsintro}
In this paper, we construct the de Rham and crystalline
counterparts to Hida's ordinary $\Lambda$-adic \'etale cohomology and Ohta's
$\Lambda$-adic Hodge cohomology, and we prove appropriate control
and finiteness theorems in each case via a careful study of the geometry of
modular curves and abelian varieties.
We then
prove a suitable $\Lambda$-adic
version of every integral comparison isomorphism one could hope for.
In particular, we are able to recover the entire family
of $p$-adic Galois representations $\rho_{p}$ (and not just its
semisimplification) from our $\Lambda$-adic crystalline cohomology.
As a byproduct of our work, we provide {\em geometric} constructions
of several of the ``cohomologically elusive" semi-linear algebra objects
in $p$-adic Hodge theory, including
the family of \'etale $(\varphi,\Gamma)$-modules attached to $e^*H^1_{\et}$
by Dee \cite{Dee}. As an application of our theory, we give a new
and purely geometric proof of Hida's freeness and control theorems
for $e^*H^1_{\et}$.
In order to survey our main results more precisely, we introduce
some notation. Fix an algebraic closure $\Qbar_p$
of $\Q_p$
as well as a $p$-power compatible sequence
$\{\varepsilon^{(r)}\}_{r\ge 0}$ of primitive $p^r$-th roots of unity in $\Qbar_p$.
We set $K_r:=\Q_p(\mu_{p^r})$ and $K_r':=K_r(\mu_N)$, and we
write $R_r$ and $R_r'$ for the rings of integers
in $K_r$ and $K_r'$, respectively.
Denote by $\scrG_{\Q_p}:=\Gal(\Qbar_p/\Q_p)$ the absolute Galois group
and by $\scrH$
the kernel of the $p$-adic cyclotomic character
$\chi: \scrG_{\Q_p}\rightarrow \Z_p^{\times}$.
We write $\Gamma:=\scrG_{\Q_p}/\scrH \simeq \Gal(K_{\infty}/K_0)$ for the quotient and,
using that $K_0'/\Q_p$ is unramified, we canonically identify $\Gamma$
with $\Gal(K_{\infty}'/K_0')$.
We will denote by $\langle u\rangle$ (respectively $\langle v\rangle_N)$
the diamond operator\footnote{Note that we have $\langle u^{-1}\rangle=\langle u\rangle^*$
and $\langle v^{-1}\rangle_N = \langle v\rangle_N^*$, where $\langle\cdot\rangle^*$
and $\langle \cdot\rangle_N^*$
are the adjoint diamond operators; see \S\ref{tower}.
}
in $\H^*$ attached to $u^{-1}\in \Z_p^{\times}$
(respectively $v^{-1}\in (\Z/N\Z)^{\times}$) and write
$\Delta_r$ for the image of the restriction of $\langle\cdot\rangle :\Z_p^{\times}\hookrightarrow \H^*$
to $1+p^r\Z_p\subseteq \Z_p^{\times}$. For convenience, we put $\Delta:=\Delta_1$,
and for any ring $A$ we write
$\Lambda_{A}:=\varprojlim_r A[\Delta/\Delta_r]$ for the completed group ring
on $\Delta$ over $A$; if $\varphi$ is an endomorphism of $A$, we again write $\varphi$
for the induced endomorphism of $\Lambda_A$ that acts as the identity on $\Delta$.
Finally, we denote by $X_r:=X_1(Np^r)$ the
usual modular curve over $\Q$ classifying (generalized) elliptic curves
with a $[\mu_{Np^r}]$-structure,
and by $J_r:=J_1(Np^r)$ its Jacobian.
Our first task is to construct a de Rham analogue of Hida's $e^*H^1_{\et}$.
A na\"ive idea would be to mimic Hida's construction, using the
(relative) de Rham cohomology of $\Z_p$-integral models of the modular curves $X_r$
in place of $p$-adic \'etale cohomology. However, this approach fails due
to the fact that $X_r$ has bad reduction
at $p$, so the relative de Rham
cohomology of integral models does not provide good
$\Z_p$-lattices in the de Rham cohomology of $X_r$ over $\Q_p$.
To address this problem, we use the canoninical integral structures in de Rham cohomology
studied in \cite{CaisDualizing} and the canonical integral model $\X_r$ of $X_r$
over $R_r$ associated to the moduli problem
$([\bal\ \Gamma_1(p^r)]^{\varepsilon^{(r)}\text{-}\mathrm{can}};\ [\mu_N])$
\cite{KM} to construct well-behaved integral ``de Rham cohomology" for the tower of modular
curves. For each $r$, we obtain a short exact sequence of free $R_r$-modules
with semilinear $\Gamma$-action and comuting $\H_r^*$-action
\begin{equation}
\xymatrix{
0\ar[r] & {H^0(\X_r, \omega_{\X_r/R_r})} \ar[r] & {H^1(\X_r/R_r)} \ar[r] & {H^1(\X_r,\O_{\X_r})}
\ar[r] & 0
}\label{finiteleveldRseq}
\end{equation}
which is co(ntra)variantly functorial in finite $K_r$-morphisms of the generic fiber $X_r$,
and whose scalar extension to $K_r$ recovers the Hodge filtration of $H^1_{\dR}(X_r/K_r)$.
Extending scalars to $R_{\infty}$ and taking projective limits,
we obtain a short exact sequence of $\Lambda_{R_{\infty}}$-modules with semilinear $\Gamma$-action
and commuting linear $\H^*$-action
\begin{equation}
\xymatrix{
0\ar[r] & {H^0(\omega)} \ar[r] & {H^1_{\dR}} \ar[r] & {H^1(\O)}
}.\label{dRseq}
\end{equation}
Our first main result (see Theorem \ref{main}) is that the ordinary part of (\ref{dRseq})
is the correct de Rham analogue of Hida's ordinary $\Lambda$-adic \'etale cohomology:
\begin{theorem}
There is a canonical short exact sequence of finite free $\Lambda_{R_{\infty}}$-modules
with semilinear $\Gamma$-action and commuting linear $\H^*$-action
\begin{equation}
\xymatrix{
0\ar[r] & {e^*H^0(\omega)} \ar[r] & {e^*H^1_{\dR}} \ar[r] & {e^*H^1(\O)} \ar[r] & 0
}.\label{orddRseq}
\end{equation}
As a $\Lambda_{R_{\infty}}$-module, $e^*H^1_{\dR}$ is free of rank $2d$, while each of the flanking terms
in $(\ref{orddRseq})$ is free of rank $d$, for $d=\sum_{k=3}^{p+1}\dim_{\mathbf{F}_p} S_k(\Gamma_1(N);\mathbf{F}_p)^{\ord}$.
Applying $\otimes_{\Lambda_{R_{\infty}}} R_{\infty}[\Delta/\Delta_r]$ to $(\ref{orddRseq})$
recovers the ordinary part of the scalar extension of $(\ref{finiteleveldRseq})$ to $R_{\infty}$.
\end{theorem}
We then show that the $\Lambda_{R_{\infty}}$-adic Hodge filtration (\ref{orddRseq}) is very nearly ``auto dual".
To state our duality result more succintly,
for any ring homomorphism $A\rightarrow B$,
we will write $(\cdot)_B:=(\cdot)\otimes_A B$ and $(\cdot)_B^{\vee}:=\Hom_B((\cdot)\otimes_A B , B)$ for these
functors from $A$-modules to $B$-modules. If $G$ is any group of automorphisms of $A$ and $M$
is an $A$-module with a semilinear action of $G$, for any ``crossed"
homomorphism\footnote{That is, $\psi(\sigma\tau) = \psi(\sigma)\cdot\sigma\psi(\tau)$ for all $\sigma,\tau\in\Gamma$,}
$\psi:G\rightarrow A^{\times}$ we will write $M(\psi)$ for the
$A$-module $M$ with ``twisted" semilinear $G$-action given by $g\cdot m:=\psi(g)g m$.
Our duality theorem is (see Proposition \ref{dRDuality}):
\begin{theorem}\label{dRdualityThm}
The natural cup-product auto-duality of
$(\ref{finiteleveldRseq})$ over $R_r':=R_r[\mu_N]$ induces a canonical
$\Lambda_{R_{\infty}'}$-linear and $\H^*$-equivariant isomorphism
of exact sequences
\begin{equation*}
\xymatrix{
0\ar[r] & {e^*H^0(\omega)(\langle\chi\rangle\langle a\rangle_N)_{\Lambda_{R_{\infty}'}}}
\ar[r]\ar[d]^-{\simeq} &
{e^*H^1_{\dR}(\langle\chi\rangle\langle a\rangle_N)_{\Lambda_{R_{\infty}'}}}
\ar[r]\ar[d]^-{\simeq} &
{e^*H^1(\O)(\langle\chi\rangle\langle a\rangle_N)_{\Lambda_{R_{\infty}'}}}
\ar[r]\ar[d]^-{\simeq} & 0\\
0\ar[r] & {(e^*H^1(\O))^{\vee}_{\Lambda_{R_{\infty}'}}} \ar[r] &
{({e^*H^1_{\dR}})^{\vee}_{\Lambda_{R_{\infty}'}}} \ar[r] &
{(e^*H^0(\omega))^{\vee}_{\Lambda_{R_{\infty}'}}} \ar[r] & 0
}
\end{equation*}
that is compatible with the natural action of $\Gamma \times \Gal(K_0'/K_0)\simeq \Gal(K_{\infty}'/K_0)$
on the bottom row and the twist of the natural action on the top row by the $\H^*$-valued character
$\langle \chi\rangle \langle a\rangle_N$,
where
$a(\gamma) \in (\Z/N\Z)^{\times}$ is
determined for $\gamma\in \Gal(K_0'/K_0)$ by
$\zeta^{a(\gamma)}=\gamma\zeta$ for every $N$-th root of unity $\zeta$.
\end{theorem}
We moreover prove that, as one would expect, the $\Lambda_{R_{\infty}}$-module $e^*H^0(\omega)$
is canonically isomorphic to the module $eS(N,\Lambda_{R_{\infty}})$ of ordinary $\Lambda_{R_{\infty}}$-adic
cusp forms of tame level $N$; see Corollary \ref{LambdaFormsRelation}.
To go further, we study the tower of $p$-divisible groups attached to the ``good quotient" modular abelian
varieties introduced by Mazur-Wiles \cite{MW-Iwasawa}. To avoid technical complications with logarithmic
$p$-divisible groups, following \cite{MW-Hida} and \cite{OhtaEichler}, we will henceforth remove the trivial tame character by working with the sub-idempotent ${e^*}'$ of $e^*$ corresponding to projection to the part
where $\mu_{p-1}\subseteq \Z_p^{\times}\simeq \Delta$ acts {\em non}-trivially.
As is well-known (e.g. \cite[\S9]{HidaGalois} and \cite[Chapter 3, \S2]{MW-Iwasawa}), the $p$-divisible group $G_r:={e^*}'J_r[p^{\infty}]$
over $\Q$ extends to a $p$-divisible group $\mathcal{G}_r$ over $R_r$, and we write
$\o{\mathcal{G}}_r:=\mathcal{G}_r\times_{R_r} \mathbf{F}_p$ for its special fiber. Denoting by $\ensuremath{\mathbf{D}}(\cdot)$
the contravariant Dieudonn\'e module functor on $p$-divisible groups over $\mathbf{F}_p$,
we form the projective limits
\begin{equation}
\ensuremath{\mathbf{D}}_{\infty}^{\star}:=\varprojlim_r \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\star})\quad\text{for}\quad \star\in \{\et,\mult,\Null\},
\label{DlimitsDef}
\end{equation}
taken along the mappings induced by
$\o{\mathcal{G}}_r\rightarrow \o{\mathcal{G}}_{r+1}$. Each of these is naturally a $\Lambda$-module
equipped with linear (!) Frobenius $F$ and Verscheibung $V$ morphisms satisfying $FV=VF=p$,
as well as a linear action of $\H^*$ and a ``geometric inertia" action of $\Gamma$, which reflects
the fact that the generic fiber of $\mathcal{G}_r$ descends to $\Q_p$.
The $\Lambda$-modules (\ref{DlimitsDef}) have the expected structure (see Theorem \ref{MainDieudonne}):
\begin{theorem}\label{DieudonneMainThm}
There is a canonical split short exact sequence of finite and free $\Lambda$-modules
\begin{equation}
\xymatrix{
0 \ar[r] & {\ensuremath{\mathbf{D}}^{\et}_{\infty}} \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}} \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\mult}} \ar[r] & 0.
}.\label{Dieudonneseq}
\end{equation}
with linear $\H^*$ and $\Gamma$-actions.
As a $\Lambda$-module, $\ensuremath{\mathbf{D}}_{\infty}$ is free of rank $2d'$, while $\ensuremath{\mathbf{D}}_{\infty}^{\et}$
and $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$ are free of rank $d'$, where $d':=\sum_{k=3}^p \dim_{\mathbf{F}_p} S_k(\Gamma_1(N);\mathbf{F}_p)^{\ord}$.
For $\star\in \{\mult,\et,\Null\}$, there are canonical isomorphisms
\begin{equation*}
\ensuremath{\mathbf{D}}_{\infty}^{\star}\mathop{\otimes}\limits_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\star})
\end{equation*}
which are compatible with the extra structures.
Via the canonical splitting of $(\ref{Dieudonneseq})$, $\ensuremath{\mathbf{D}}_{\infty}^{\star}$ for $\star=\et$
$($respetively $\star=\mult$$)$ is
identified with the maximal subpace of $\ensuremath{\mathbf{D}}_{\infty}$ on which $F$ $($respectively $V$$)$ acts
invertibly .
The Hecke operator $U_p^*\in \H^*$ acts as $F$ on $\ensuremath{\mathbf{D}}_{\infty}^{\et}$ and as $\langle p\rangle_N V$ on
$\ensuremath{\mathbf{D}}_{\infty}^{\mult}$,
while $\Gamma$ acts trivially on $\ensuremath{\mathbf{D}}_{\infty}^{\et}$ and via $\langle \chi(\cdot)\rangle^{-1}$
on $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$.
\end{theorem}
We likewise have the appropriate ``Dieudonn\'e"
analogue of Theorem \ref{dRdualityThm} (see Proposition \ref{DieudonneDuality}):
\begin{theorem}\label{DDuality}
There is a canonical $\H^*$-equivariant isomorphism of exact sequences of $\Lambda_{R_0'}$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\et}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}} \ar[r]\ar[d]^-{\simeq} &
{\ensuremath{\mathbf{D}}_{\infty}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} &
{\ensuremath{\mathbf{D}}_{\infty}^{\mult}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} & 0 \\
0\ar[r] & {(\ensuremath{\mathbf{D}}_{\infty}^{\mult})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\ensuremath{\mathbf{D}}_{\infty})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\ensuremath{\mathbf{D}}_{\infty}^{\et})^{\vee}_{\Lambda_{R_0'}}}\ar[r] & 0
}
\end{equation*}
that is $\Gamma\times \Gal(K_0'/K_0)$-equivariant,
and intertwines $F$
$($respectively $V$$)$ on the top row with $V^{\vee}$
$($respectively $F^{\vee}$$)$ on the bottom.\footnote{Here, $F^{\vee}$ (respectively $V^{\vee}$)
is the map taking a linear functional $f$ to $\varphi^{-1}\circ f\circ F$
(respectively $\varphi\circ f\circ V$), where $\varphi$
is the Frobenius automorphism of $R_0'=\Z_p[\mu_N]$.}
\end{theorem}
Just as Mazur-Wiles are able to relate the ordinary-filtration of ${e^*}'H^1_{\et}$
to the \'etale cohomology of the Igusa tower, we can interpret the slope filtraton (\ref{Dieudonneseq})
in terms of the crystalline cohomology of the Igusa tower as follows.
For each $r$, let $I_r^{\infty}$ and $I_r^0$ be the two ``good" irreducible components of
$\X_r\times_{R_r}\mathbf{F}_r$ (see Remark \ref{MWGood}), each of which is isomorphic to the Igusa curve $\Ig(p^r)$
of tame level $N$ and $p$-level $p^r$. For $\star\in \{0,\infty\}$ we form the projective limit
\begin{equation*}
H^1_{\cris}(I^{\star}):=\varprojlim_{r} H^1_{\cris}(I_r^{\star}/\Z_p);
\end{equation*}
with respect to the trace mappings on crystalline cohmology induced by the canonical
degeneracy maps on Igusa curves.
Then $H^1_{\cris}(I^{\star})$ is naturally a $\Lambda$-module with linear Frobenius $F$ and Verscheibung $V$ endomorphisms.
Letting $f'$ be the idempotent of $\Lambda$ corresponding to projection to the part
where $\mu_{p-1}\subseteq \Delta\hookrightarrow \Lambda$ acts nontrivially, we prove
(see Theorem \ref{DieudonneCrystalIgusa}):
\begin{theorem}
There is a canonical isomorphism of $\Lambda$-modules, compatible with $F$ and $V,$
\begin{equation}
\ensuremath{\mathbf{D}}_{\infty} =\ensuremath{\mathbf{D}}_{\infty}^{\mult}\oplus \ensuremath{\mathbf{D}}_{\infty}^{\et}\simeq
f'H^1_{\cris}(I^{0})^{V_{\ord}} \oplus
f'H^1_{\cris}(I^{\infty})^{F_{\ord}}.\label{crisIgusa}
\end{equation}
which preserves the direct sum decompositions of source and target.
This isomorphism is Hecke and $\Gamma$-equivariant, with $U_p^*$ and $\Gamma$
acting as $\langle p\rangle_N V\oplus F$ and
$ \langle \chi(\cdot)\rangle^{-1}\oplus \id$, respectively,
on each direct sum.
\end{theorem}
We note that our ``Dieudonn\'e module" analogue (\ref{crisIgusa}) is a significant
sharpening of its \'etale counterpart \cite[\S4]{MW-Hida}, which is formulated
only up to isogeny (i.e. after inverting $p$). From $\ensuremath{\mathbf{D}}_{\infty}$,
we can recover the $\Lambda$-adic Hodge filtration (\ref{orddRseq}), so
the latter is canonically split (see Theorem \ref{dRtoDieudonneInfty}):
\begin{theorem}\label{dRtoDieudonne}
There is a canonical $\Gamma$ and $\H^*$-equivariant isomorphism of
exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {{e^*}'H^0(\omega)} \ar[r]\ar[d]^-{\simeq} &
{{e^*}'H^1_{\dR}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O)} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\ensuremath{\mathbf{D}}_{\infty}\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\ensuremath{\mathbf{D}}_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] & 0
}\label{dRcriscomparison}
\end{gathered}
\end{equation}
where the mappings on bottom row are the canonical inclusion and projection morphisms
corresponding to the direct sum decomposition $\ensuremath{\mathbf{D}}_{\infty}=\ensuremath{\mathbf{D}}_{\infty}^{\mult}\oplus \ensuremath{\mathbf{D}}_{\infty}^{\et}$.
In particular, the Hodge filtration exact sequence $(\ref{orddRseq})$ is canonically
split, and admits a canonical descent to $\Lambda$.
\end{theorem}
We remark that under the identification (\ref{dRcriscomparison}),
the Hodge filtration (\ref{orddRseq}) and slope filtration (\ref{Dieudonneseq})
correspond, but in the opposite directions. As a consequence of Theorem
\ref{dRtoDieudonne}, we deduce (see Corollary \ref{MFIgusaDieudonne} and
Remark \ref{MFIgusaCrystal}):
\begin{corollary}
\label{OhtaCor}
There is a canonical isomorphism of finite free $\Lambda$ $($respectively $\Lambda_{R_0'}$$)$-modules
\begin{equation*}
{e}'S(N,\Lambda) \simeq \ensuremath{\mathbf{D}}_{\infty}^{\mult}
\qquad\text{respectively}\qquad
e'\H\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_0'} \simeq \ensuremath{\mathbf{D}}_{\infty}^{\et}(\langle a\rangle_N)\mathop{\otimes}\limits_{\Lambda}{\Lambda_{R_0'}}
\end{equation*}
that intertwines $T\in \H:=\varprojlim \H_r$ with $T^*\in \H^*$, where we let
$U_p^*$ act as $\langle p\rangle_N V$ on $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$ and as $F$ on $\ensuremath{\mathbf{D}}_{\infty}^{\et}$.
The second of these isomorphisms is in addition $\Gal(K_0'/K_0)$-equivariant.
\end{corollary}
We are also able to recover the semisimplification of ${e^*}'H^1_{\et}$ from $\ensuremath{\mathbf{D}}_{\infty}$.
Writing $\I\subseteq \scrG_{\Q_p}$
for the inertia subgroup at $p$, for any $\Z_p[\scrG_{\Q_p}]$-module $M$, we denote by $M^{\I}$ (respectively $M_{\I}:=M/M^{\I}$)
the sub (respectively quotient) module of invariants (respectively covariants) under $\I$.
\begin{theorem}\label{FiltrationRecover}
There are canonical isomorphisms of $\Lambda_{W(\o{\mathbf{F}}_p)}$-modules
with linear $\H^*$-action and semilinear actions of $F$, $V$, and $\scrG_{\Q_p}$
\begin{subequations}
\begin{equation}
\ensuremath{\mathbf{D}}_{\infty}^{\et} \mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)}
\simeq ({e^*}'H^1_{\et})^{\I}\mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)}
\label{inertialinvariants}
\end{equation}
and
\begin{equation}
\ensuremath{\mathbf{D}}_{\infty}^{\mult}(-1) \mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)}
\simeq ({e^*}'H^1_{\et})_{\I}\mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)}.
\label{inertialcovariants}
\end{equation}
\end{subequations}
Writing $\sigma$ for the
Frobenius automorphism of $W(\o{\mathbf{F}}_p)$,
the isomorphism $(\ref{inertialinvariants})$ intertwines $F\otimes \sigma$ with $\id\otimes\sigma$
and $\id\otimes g$ with $g\otimes g$ for $g\in \scrG_{\Q_p}$, whereas $(\ref{inertialcovariants})$
intertwines $V\otimes \sigma^{-1}$ with $\id\otimes\sigma^{-1}$ and $g\otimes g$ with $g\otimes g$,
where $g\in\scrG_{\Q_p}$ acts on the Tate twist
$\ensuremath{\mathbf{D}}_{\infty}^{\mult}(-1):=\ensuremath{\mathbf{D}}_{\infty}\otimes_{\Z_p}\Z_p(-1)$ as
$\langle \chi(g)^{-1}\rangle \otimes \chi(g)^{-1}$.
\end{theorem}
Theorem \ref{FiltrationRecover} gives the following ``explicit" description
of the semisimplification of ${e^*}'H^1_{\et}$:
\begin{corollary}
For any $T\in (\H^{*\ord})^{\times}$, let
$\lambda(T):\scrG_{\Q_p}\rightarrow \H^{*\ord}$ be the
unique continuous $($for the $p$-adic topology on $\H^{*\ord}$$)$
unramified character whose value on $($any lift of$)$
$\mathrm{Frob}_p$ is $T$.
Then $\scrG_{\Q_p}$ acts on $({e^*}'H^1_{\et})^{\I}$ through the
character $\lambda({U_p^*}^{-1})$ and on $({e^*}'H^1_{\et})_{\I}$
through $\chi^{-1} \cdot \langle \chi^{-1}\rangle \lambda(\langle p\rangle_N^{-1}U_p^*)$.
\end{corollary}
We remark that Corollary \ref{OhtaCor} and Theorem \ref{FiltrationRecover}
combined give a refinement of the main result of \cite{OhtaEichler}.
We are furthermore able to recover the main theorem of \cite{MW-Hida}
(that the ordinary filtration of ${e^*}'H^1_{\et}$ interpolates
$p$-adic analytically):
\begin{corollary}\label{MWmainThmCor}
Let $d'$ be the integer of Theorem $\ref{DieudonneMainThm}$. Then each of
$({e^*}'H^1_{\et})^{\I}$ and $({e^*}'H^1_{\et})_{\I}$ is a free
$\Lambda$-module of rank $d'$, and for each $r\ge 1$ there are canonical
$\H^*$ and $\scrG_{\Q_p}$-equivariant isomorphisms of $\Z_p[\Delta/\Delta_r]$-modules
\begin{subequations}
\begin{equation}
({e^*}'H^1_{\et})^{\I} \mathop{\otimes}\limits_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq
{e^*}'H^1_{\et}({X_r}_{\Qbar_p},\Z_p)^{\I}\label{HidaResultSub}
\end{equation}
\begin{equation}
({e^*}'H^1_{\et})_{\I} \mathop{\otimes}\limits_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq
{e^*}'H^1_{\et}({X_r}_{\Qbar_p},\Z_p)_{\I}
\label{HidaResultQuo}
\end{equation}
\end{subequations}
\end{corollary}
To recover the full $\Lambda$-adic local Galois representation ${e^*}'H^1_{\et}$,
rather than just its semisimplification,
it is necessary to work with the full Dieudonn\'e {\em crystal} of $\mathcal{G}_r$ over $R_r$.
Following Faltings \cite{Faltings} and Breuil (e.g. \cite{Breuil}), this is equivalent
to studying the evaluation of the Dieudonn\'e crystal of $\mathcal{G}_r\times_{R_r} R_r/pR_r$
on the ``universal" divided power thickening $S_r\twoheadrightarrow R_r/pR_r$,
where $S_r$ is the $p$-adically completed PD-hull
of the surjection $\Z_p[\![u_r]\!]\twoheadrightarrow R_r$
sending $u_r$ to $\varepsilon^{(r)}-1$. As the rings $S_r$ are too unwieldly
to directly construct a good crystalline analogue of Hida's
ordinary \'etale cohomology, we must functorially descend
the ``filtered $S_r$-module" attached to $\mathcal{G}_r$ to the much simpler ring $\mathfrak{S}_r:=\Z_p[\![u_r]\!]$.
While such a descent is provided
(in rather different ways) by the work of Breuil--Kisin and Berger--Wach, neither of these
frameworks is suitable for our application: it is essential for us
that the formation of this descent to $\mathfrak{S}_r$ commute with base change as one moves up
the cyclotomic tower, and it is not at all clear that this holds for Breuil--Kisin modules
or for the Wach modules of Berger. Instead, we use the theory of \cite{CaisLau}, which works with frames and
windows \`a la Lau and Zink to provide the desired functorial descent to a ``$(\varphi,\Gamma)$-module" $\m_r(\mathcal{G}_r)$
over $\mathfrak{S}_r$.
We view $\mathfrak{S}_r$
as a $\Z_p$-subalgebra of $\mathfrak{S}_{r+1}$ via the map sending $u_r$ to $\varphi(u_{r+1}):=(1+u_{r+1})^p -1$,
and we write $\mathfrak{S}_{\infty}:=\varinjlim \mathfrak{S}_r$ for
the rising union\footnote{As explained in Remark \ref{Slimits},
the $p$-adic completion of $\mathfrak{S}_{\infty}$ is actually a very nice ring:
it is canonically and Frobenius equivariantly isomorphic to $W(\mathbf{F}_p[\![u_0]\!]^{\perf})$,
for $\mathbf{F}_p[\![u_0]\!]^{\perf}$ the perfection of the $\mathbf{F}_p$-algebra $\mathbf{F}_p[\![u_0]\!]$.
}
of the $\mathfrak{S}_r$, equiped with its Frobenius {\em automorphism} $\varphi$ and commuting action of
$\Gamma$ determined by $\gamma u_r:=(1+u_r)^{\chi(\gamma)} - 1$.
We then form the projective limits
\begin{equation*}
\m_{\infty}^{\star}:=\varprojlim (\m_r(\mathcal{G}_r^{\star})\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty})\quad\text{for}\quad \star\in\{\et,\mult,\Null\}
\end{equation*}
taken along the mappings induced by $\mathcal{G}_{r}\times_{R_r} R_{r+1}\rightarrow \mathcal{G}_{r+1}$
via the functoriality of $\m_r(\cdot)$
and its compatibility with base change.
These are $\Lambda_{\mathfrak{S}_{\infty}}$-modules equipped with a semilinear action of $\Gamma$,
a linear and commuting action of $\H^*$, and a commuting $\varphi$ (respectively $\varphi^{-1}$) semilinear endomorphism $F$ (respectively $V$)
satisfying $FV=VF = \omega$, for $\omega:=\varphi(u_1)/u_1 = u_0/\varphi^{-1}(u_0)\in \mathfrak{S}_{\infty}$,
and they provide
our crystalline analogue of Hida's ordinary \'etale cohomology (see Theorem \ref{MainThmCrystal}):
\begin{theorem}
There is a canonical short exact sequence of finite free $\Lambda_{\mathfrak{S}_{\infty}}$-modules
with linear $\H^*$-action, semilinear $\Gamma$-action, and commuting semilinear
endomorphisms $F,$ $V$ satisfying $FV=VF=\omega$
\begin{equation}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}} \ar[r] & {\m_{\infty}} \ar[r] & {\m_{\infty}^{\mult}} \ar[r] & 0
}.\label{CrystallineAnalogue}
\end{equation}
Each of $\m_{\infty}^{\star}$ for $\star\in \{\et,\mult\}$ is free of rank $d'$ over
$\Lambda_{\mathfrak{S}_{\infty}}$, while $\m_{\infty}$ is free of rank $2d'$, where
$d'$ is as in Theorem $\ref{DieudonneMainThm}$. Extending scalars on $(\ref{CrystallineAnalogue})$
along the canonical surjection
$\Lambda_{\mathfrak{S}_{\infty}}\twoheadrightarrow \mathfrak{S}_{\infty}[\Delta/\Delta_r]$ yields the short exact
sequence
\begin{equation*}
\xymatrix{
0 \ar[r] & {\m_r(\mathcal{G}_r^{\et})\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}} \ar[r] &
{\m_r(\mathcal{G}_r)\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}} \ar[r] &
{\m_r(\mathcal{G}_r^{\mult})\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}} \ar[r] &
0
}
\end{equation*}
compatibly with $\H^*$, $\Gamma$, $F$ and $V$.
\end{theorem}
Again, in the spirit of Theorems \ref{dRdualityThm} and \ref{DDuality},
there is a corresponding ``autoduality" result for $\m_{\infty}$
(see Theorem \ref{CrystalDuality}). To state it, we must work over the ring
$\mathfrak{S}_{\infty}':=\varinjlim_r \Z_p[\mu_N][\![u_r]\!]$, with the inductive limit taken along the
$\Z_p$-algebra maps sending $u_r$ to $\varphi(u_{r+1})$.
\begin{theorem}
Let $\mu:\Gamma\rightarrow \Lambda_{\mathfrak{S}_{\infty}}^{\times}$ be the crossed homomorphism
given by $\mu(\gamma):=\frac{u_1}{\gamma u_1}\chi(\gamma) \langle \chi(\gamma)\rangle$.
There is a canonical $\H^*$ and $\Gal(K_{\infty}'/K_0)$-compatible isomorphism of
exact sequences
\begin{equation*}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_{\infty}^{\et}(\mu \langle a\rangle_N)_{\Lambda_{\mathfrak{S}_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}(\mu \langle a\rangle_N)_{\Lambda_{\mathfrak{S}_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}^{\mult}(\mu \langle a\rangle_N)_{\Lambda_{\mathfrak{S}_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0\ar[r] & {(\m_{\infty}^{\mult})_{\Lambda_{\mathfrak{S}_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty})_{\Lambda_{\mathfrak{S}_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty}^{\et})_{\Lambda_{\mathfrak{S}_{\infty}'}}^{\vee}} \ar[r] & 0
}
\end{gathered}
\end{equation*}
that intertwines $F$
$($respectively $V$$)$ on the top row with
$V^{\vee}$ $($respectively $F^{\vee}$$)$ on the bottom.
\end{theorem}
The $\Lambda_{\mathfrak{S}_{\infty}}$-modules $\m_{\infty}^{\et}$ and $\m_{\infty}^{\mult}$
have a particularly simple structure (see Theorem \ref{etmultdescent}):
\begin{theorem}
There are canonical $\H^*$, $\Gamma$, $F$ and $V$-equivariant isomorphisms
of $\Lambda_{\mathfrak{S}_{\infty}}$-modules
\begin{subequations}
\begin{equation}
\m_{\infty}^{\et} \simeq \ensuremath{\mathbf{D}}_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda} \Lambda_{\mathfrak{S}_{\infty}},
\end{equation}
intertwining $F$ $($respetcively $V$$)$ with
$F\otimes \varphi$ $($respectively $F^{-1}\otimes \omega\cdot \varphi^{-1}$$)$ and $\gamma\in \Gamma$
with $\gamma\otimes\gamma$, and
\begin{equation}
\m_{\infty}^{\mult}\simeq \ensuremath{\mathbf{D}}_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda} \Lambda_{\mathfrak{S}_{\infty}},
\end{equation}
intertwing $F$ $($respectively $V$$)$ with $V^{-1} \otimes \omega \cdot\varphi$
$($respectively $V\otimes\varphi^{-1}$$)$
and $\gamma$ with $\gamma\otimes \chi(\gamma)^{-1} \gamma u_1/u_1)$.
In particular, $F$ $($respectively $V$)
acts invertibly on $\m_{\infty}^{\et}$ $($respectively $\m_{\infty}^{\mult}$$)$.
\end{subequations}
\end{theorem}
From $\m_{\infty}$, we can recover $\ensuremath{\mathbf{D}}_{\infty}$ and ${e^*}'H^1_{\dR}$,
with their additional structures (see Theorem \ref{SRecovery}):
\begin{theorem}\label{MinftySpecialize}
Viewing $\Lambda$ as a $\Lambda_{\mathfrak{S}_{\infty}}$-algebra via the map induced by $u_r\mapsto 0$,
there is a canonical isomorphism of short exact sequences of finite free $\Lambda$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}}} \Lambda}\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}}} \Lambda}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}}} \Lambda} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\et}} \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}} \ar[r] &
{\ensuremath{\mathbf{D}}_{\infty}^{\mult}} \ar[r] & 0
}
\end{equation*}
which is $\Gamma$ and $\H^*$-equivariant and carries $F\otimes 1$ to $F$
and $V\otimes 1$ to $V$.
Viewing $\Lambda_{R_{\infty}}$ as a $\Lambda_{\mathfrak{S}_{\infty}}$-algebra via the map
$u_r\mapsto (\varepsilon^{(r)})^p - 1$, there is a canonical
isomorphism of short exact sequences of
$\Lambda_{R_{\infty}}$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}}} \Lambda_{R_{\infty}}}
\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}}} \Lambda_{R_{\infty}}}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}}} \Lambda_{R_{\infty}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {{e^*}'H^1(\O)} \ar[r]_{i} &
{{e^*}'H^1_{\dR}} \ar[r]_-{j} & {{e^*}'H^0(\omega)} \ar[r] & 0
}
\end{equation*}
with $i$ and $j$ the canonical sections
given by the splitting in Theorem $\ref{dRtoDieudonne}$.
\end{theorem}
To recover Hida's ordinary \'etale cohomology from $\m_{\infty}$,
we introduce the ``period" ring of Fontaine\footnote{Though we use the notation introduced by Berger and Colmez.}
$\wt{\ensuremath{\mathbf{E}}}^+:=\varprojlim \O_{\mathbf{C}_p}/(p)$, with the projective limit
taken along the $p$-power mapping; this is a perfect valuation ring of characteristic $p$
equipped with a canonical action of $\scrG_{\Q_p}$ via ``coordinates". We write $\wt{\ensuremath{\mathbf{E}}}$
for the fraction field of $\wt{\ensuremath{\mathbf{E}}}^+$ and
$\wt{\a}:=W(\wt{\ensuremath{\mathbf{E}}})$ for its ring of Witt vectors, equipped
with its canonical Frobenius automorphism $\varphi$ and $\scrG_{\Q_p}$-action induced by Witt functoriality.
Our fixed choice of $p$-power compatible sequence $\{\varepsilon^{(r)}\}$
determines an element $\u{\varepsilon}:=(\varepsilon^{(r)}\bmod p)_{r\ge 0}$
of $\wt{\ensuremath{\mathbf{E}}}^+$, and we $\Z_p$-linearly embed $\mathfrak{S}_{\infty}$ in $\wt{\a}$ via
$u_r\mapsto \varphi^{-r}([\u{\varepsilon}]-1)$ where $[\cdot]$ is the Teichm\"uller
section. This embedding is $\varphi$ and $\scrG_{\Q_p}$-compatible, with $\scrG_{\Q_p}$
acting on $\mathfrak{S}_{\infty}$ through the quotient $\scrG_{\Q_p}\twoheadrightarrow \Gamma$.
\begin{theorem}
\label{RecoverEtale}
Twisting the structure map $\mathfrak{S}_{\infty}\rightarrow \wt{\a}$ by the Frobenius automorphism $\varphi$,
there is a canonical isomorphism of short exact sequences of $\Lambda_{\wt{\a}}$-modules
with $\H^*$-action
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\varphi} \Lambda_{\wt{\a}}}
\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\varphi} \Lambda_{\wt{\a}}}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\varphi} \Lambda_{\wt{\a}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {({e^*}'H^1_{\et})^{\I}\mathop{\otimes}\limits_{\Lambda} \Lambda_{\wt{\a}}} \ar[r] &
{{e^*}'H^1_{\et}\mathop{\otimes}\limits_{\Lambda} \Lambda_{\wt{\a}}} \ar[r] &
({e^*}'H^1_{\et})_{\I}\mathop{\otimes}\limits_{\Lambda} \Lambda_{\wt{\a}}\ar[r] & 0
}\label{FinalComparisonIsom}
\end{gathered}
\end{equation}
that is $\scrG_{\Q_p}$-equivariant for the ``diagonal" action of $\scrG_{\Q_p}$
$($with $\scrG_{\Q_p}$ acting on $\m_{\infty}$ through $\Gamma$$)$
and intertwines
$F\otimes \varphi$ with $\id\otimes\varphi$ and $V\otimes\varphi^{-1}$ with $\id\otimes \varphi^{-1}$.
In particular, there is a canonical isomorphism of $\Lambda$-modules, compatible
with the actions of $\H^*$ and $\scrG_{\Q_p}$,
\begin{equation}
{e^*}'H^1_{\et} \simeq \left( \m_{\infty}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\varphi}
\Lambda_{\wt{\a}}\right )^{F\otimes\varphi = 1}.\label{RecoverEtaleIsom}
\end{equation}
\end{theorem}
Theorem \ref{RecoverEtale} allows us to give a new proof of Hida's finiteness and control theorems
for ${e^*}'H^1_{\et}$:
\begin{corollary}[Hida]\label{HidasThm}
Let $d'$ be as in Theorem $\ref{DieudonneMainThm}$. Then
${e^*}'H^1_{\et}$ is free $\Lambda$-module of rank $2d'$.
For each $r\ge 1$ there is a canonical isomorphism of $\Z_p[\Delta/\Delta_r]$-modules
with linear $\H^*$ and $\scrG_{\Q_p}$-actions
\begin{equation*}
{e^*}'H^1_{\et} \mathop{\otimes}\limits_{\Lambda} \Z_p[\Delta/\Delta_r] \simeq {e^*}'H^1_{\et}({X_r}_{\Qbar_p},\Z_p)
\end{equation*}
which is moreover compatible with the isomorphisms $(\ref{HidaResultSub})$ and $(\ref{HidaResultQuo})$
in the evident manner.
\end{corollary}
We also deduce a new proof of the following duality result
\cite[Theorem 4.3.1]{OhtaEichler} ({\em cf.} \cite[\S6]{MW-Hida}):
\begin{corollary}[Ohta]\label{OhtaDuality}
Let $\nu:\scrG_{\Q_p}\rightarrow \H^*$ be the
character $\nu:=\chi\langle \chi\rangle \lambda(\langle p\rangle_N)$.
There is a canonical $\H^*$ and $\scrG_{\Q_p}$-equivariant isomorphism
of short exact sequences of $\Lambda$-modules
\begin{equation*}
\xymatrix{
0 \ar[r] & {({e^*}'H^1_{\et})^{\I}(\nu)}
\ar[d]^-{\simeq} \ar[r] &
{{e^*}'H^1_{\et}(\nu)}\ar[d]^-{\simeq} \ar[r] &
{({e^*}'H^1_{\et})_{\I}(\nu)}
\ar[d]^-{\simeq}\ar[r] & 0 \\
0 \ar[r] & {\Hom_{\Lambda}(({e^*}'H^1_{\et})_{\I},\Lambda)} \ar[r] &
{\Hom_{\Lambda}({e^*}'H^1_{\et},\Lambda)} \ar[r] &
{\Hom_{\Lambda}(({e^*}'H^1_{\et})^{\I},\Lambda)}\ar[r] & 0
}
\end{equation*}
\end{corollary}
The $\Lambda$-adic splitting
of the ordinary filtration of $e^*H^1_{\et}$ was considered by
Ghate and Vatsal \cite{GhateVatsal}, who prove (under certain
technical hypotheses of ``deformation-theoretic nature")
that if the $\Lambda$-adic family $\mathscr{F}$ associated to
a cuspidal eigenform $f$
is primitive and $p$-distinguished, then the associated
$\Lambda$-adic local Galois representation $\rho_{\mathscr{F},p}$
is split split if and only if
some arithmetic specialization of $\mathscr{F}$
has CM \cite[Theorem 13]{GhateVatsal}.
We interpret the $\Lambda$-adic splitting of the ordinary filtration
as follows:
\begin{theorem}\label{SplittingCriterion}
The short exact sequence $(\ref{CrystallineAnalogue})$ admits
a $\Lambda_{\mathfrak{S}_{\infty}}$-linear splitting which is compatible with $F$, $V$,
and $\Gamma$ if and only if the ordinary filtration of ${e^*}'H^1_{\et}$
admits a $\Lambda$-linear spitting which is compatible with the action of $\scrG_{\Q_p}$.
\end{theorem}
\subsection{Overview of the article}\label{Overview}
Section \ref{Prelim} is preliminary: we review the integral $p$-adic cohomology theories
of \cite{CaisDualizing} and \cite{CaisNeron}, and summarize the relavant facts concerning
integral models of modular curves
from \cite{KM} that we will need. Of particular importance is a description of the
$U_p$-correspondence in characteristic $p$, due to Ulmer \cite{Ulmer}, and recorded in Proposition \ref{UlmerProp}.
In \S\ref{DiffCharp}, we study the de Rham and crystalline cohomolgy of the Igusa tower, and prove
the key ``freeness and control" theorems that form the technical characteristic $p$ backbone of this paper.
Via an almost combinatorial argument using the description of $U_p$ in characteristic $p$,
we then relate the cohomology of the Igusa tower to the mod $p$ reduction of the ordinary part
of the (integral $p$-adic) cohomology of the modular tower.
Section \ref{PhiGammaCrystals} is a summary of the theory developed in \cite{CaisLau},
which uses Dieudonn\'e crystals of $p$-divisible groups to
provide a ``cohomological" construction of the $(\varphi,\Gamma)$-modules attached
to potentially Barsotti--Tate representations. It is precisely this theory
which allows us to construct our crystalline analogue of Hida's ordinary $\Lambda$-adic
\'etale cohomology.
Section \ref{results} constitutes the main body of this paper, and the reader who
is content to refer back to \S\ref{Prelim}--\ref{PhiGammaCrystals} as needed should skip directly there.
In \S\ref{TowerFormalism}, we develop a
commutative algebra formalism for working with projective limits of ``towers" of cohomology
that we use frequently in the sequel. Using the canonical lattices in de Rham
cohomology studied in \cite{CaisDualizing} (and reviewed in \S\ref{GD}), we construct
our $\Lambda$-adic de Rham analogue of Hida's ordinary $\Lambda$-adic \'etale cohomology in \S\ref{ordfamdR},
and we show that the expected freeness and control results follow
by reduction to characteristic $p$ from the structure
theorems for the de Rham cohomology of the Igusa tower established in \S\ref{DiffCharp}.
Using work of Ohta \cite{OhtaEichler}, in \S\ref{ordforms} we relate the Hodge filtration of our $\Lambda$-adic de Rham
cohomology to the module of $\Lambda$-adic cuspforms.
In section \ref{BTfamily}, we study the tower of $p$-divisible groups whose cohomology
allows us to construct our $\Lambda$-adic Dieudonn\'e and crystalline analogues of
Hida's \'etale cohomlogy in \S\ref{OrdDieuSection} and \S\ref{OrdSigmaSection}, respectively.
We establish $\Lambda$-adic comparison isomorphisms between each of these cohomologies
using the integral comparison isomorphisms of \cite{CaisNeron} and \cite{CaisLau}, recalled in \S\ref{Universal}
and \S\ref{pDivPhiGamma}, respectively. This enables us to give a new proof of Hida's freeness and
control theorems and of Ohta's duality theorem in \S\ref{OrdSigmaSection}.
As remarked in \S\ref{resultsintro}, and following \cite{OhtaEichler} and \cite{MW-Hida}, our construction
of the $\Lambda$-adic Dieudonn\'e and crystalline counterparts to Hida's \'etale cohomology
excludes the trivial eigenspace for the action of $\mu_{p-1}\subseteq \Z_p^{\times}$
so as to avoid technical complications with logarithmic $p$-divisible groups.
In \cite{Ohta2}, Ohta uses the ``fixed part" (in the sense of Grothendieck \cite[2.2.3]{GroModeles})
of N\'eron models with semiabelian reduction to extend his results on
$\Lambda$-adic Hodge cohomology to allow trivial tame nebentype character.
We are confident that by using Kato's logarithmic Dieudonn\'e theory \cite{KatoDegen}
one can appropriately generalize our results in \S\ref{OrdDieuSection} and \S\ref{OrdSigmaSection} to include
the missing eigenspace for the action of $\mu_{p-1}$.
\subsection{Notation}\label{Notation}
If $\varphi:A\rightarrow B$ is any map of rings, we will often write $M_B:=M\otimes_{A} B$
for the $B$-module induced from an $A$-module $M$ by extension of scalars.
When we wish to specify $\varphi$, we will write $M\otimes_{A,\varphi} B$.
Likewise, if $\varphi:T'\rightarrow T$ is any morphism of schemes, for any $T$-scheme $X$
we denote by $X_{T'}$ the base change of $X$ along $\varphi$.
If $f:X\rightarrow Y$ is any morphism of $T$-schemes,
we will write $f_{T'}: X_{T'}\rightarrow Y_{T'}$
for the morphism of $T'$-schemes obtained from $f$ by base change along $\varphi$.
When $T=\Spec(R)$ and $T'=\Spec(R')$ are affine, we abuse notation and write
$X_{R'}$ or $X\times_{R} R'$ for $X_{T'}$.
We will frequently work with schemes over a discrete valuation ring $R$.
We will often write $\X,\mathcal{Y},\ldots$ for schemes over $\Spec(R)$,
and will generally use $X,Y,\ldots$ (respectively $\o{\X},\o{\mathcal{Y}},\ldots$)
for their generic (respectively special) fibers.
\subsection{Acknowledgements}
It is a pleasure to thank Laurent Berger, Brian Conrad, Adrian Iovita, Joseph Lipman, Tong Liu, and Romyar Sharifi
for enlightening conversations and correspondence. I am especially grateful to Haruzo Hida, and Jacques
Tilouine for their willingness to answer many questions concerning their work.
This paper owes a great deal to the work of Masami Ohta, and I heartily thank him
for graciously hosting me during a visit to Tokai University in August, 2009.
\tableofcontents
\section{Preliminaries}\label{Prelim}
This somewhat long section is devoted to recalling the geometric
background we will need in our constructions. Much (though not all)
of this material is contained in \cite{CaisDualizing}, \cite{CaisNeron}
and \cite{KM}.
\subsection{Dualizing sheaves and de {R}ham cohomology}\label{GD}
We begin by describing a certain modification of the usual de Rham complex for non-smooth curves.
The hypercohomology of this (two-term) complex is in general much better behaved than algebraic de Rham
cohomology and will enable us to construct our $\Lambda$-adic de Rham cohomology.
We largely refer to \cite{CaisDualizing}, but remark that our
treatment here is different in some places and better suited to our purposes.
\begin{definition}\label{curvedef}
A {\em curve} over a scheme $S$ is a morphism $f:X\rightarrow S$
of finite presentation which is a flat local complete
intersection\footnote{That is, a {\em syntomic morphism} in the sense of
Mazur \cite[II, 1.1]{FontaineMessing}. Here, we use the definition of l.c.i. given
in \cite[Exp. \Rmnum{8}, 1.1]{SGA6}.}
of pure relative dimension 1 with geometrically reduced fibers.
We will often say that $X$ is a curve over $S$ or that $X$ is
a relative $S$-curve when $f$ is clear from context.
\end{definition}
\begin{proposition}\label{curveproperties}
Let $f:X\rightarrow S$ be a flat morphism of finite type. The following are equivalent:
\begin{enumerate}
\item The morphism $f:X\rightarrow S$ is a curve.\label{fiscrv}
\item For every $s\in S$, the fiber $f_s:X_s\rightarrow \Spec k(s)$ is a curve.\label{fiberscrv}
\item For every $x\in X$ with $s=f(x)$, the local ring
$\O_{X_s,x}$ is a complete intersection\footnote{That is, the quotient of a regular local ring by
a regular sequence.} and $f$ has geometrically reduced fibers of pure dimension 1.\label{localringcrv}
\end{enumerate}
Moreover, any base change of a curve is again a curve.
\end{proposition}
\begin{proof}
Since $f$ is flat and of finite presentation, the definition of local complete
intersection that we are using ({\em i.e.} \cite[Exp. \Rmnum{8}, 1.1]{SGA6}) is equivalent
to the definition given in \cite[$\mathrm{\Rmnum{4}}_4$, 19.3.6]{EGA} by \cite[Exp. \Rmnum{8}, 1.4]{SGA6};
the equivalence of (\ref{fiscrv})--(\ref{localringcrv}) follows immediately.
The final statement of the proposition is an easy consequence of \cite[$\mathrm{\Rmnum{4}}_4$, 19.3.9]{EGA}.
\end{proof}
\begin{corollary}\label{curvecorollary}
Let $f:X\rightarrow S$ be a finite type morphism of pure relative dimension $1$.
\begin{enumerate}
\item If $f$ is smooth, then it is a curve.\label{smoothcrv}
\item If $X$ and $S$ are regular and $f$ has geometrically reduced fibers
then $f$ is a curve.\label{regcrv}
\item If $f$ is a curve then it is Gorenstein and hence also Cohen Macaulay.\label{crvCM}
\end{enumerate}
\end{corollary}
\begin{proof}
The assertion (\ref{smoothcrv}) is obvious, and (\ref{regcrv}) follows from the
fact that a closed subscheme of a regular scheme
is regular if and only if it is defined (locally) by a regular sequence; {\em cf.} \cite[6.3.18]{LiuBook}.
Finally, (\ref{crvCM})
follows from Proposition \ref{curveproperties} (\ref{localringcrv}) and
the fact that every local ring
that is a complete intersection is Gorenstein and hence Cohen Macaulay
(see, e.g., Theorems 18.1 and 21.3 of \cite{matsumura}).
\end{proof}
Fix a relative curve $f:X\rightarrow S$.
We wish to apply Grothendieck duality theory to $f$, so we
henceforth assume that $S$ is a noetherian scheme of finite Krull dimension\footnote{Nagata gives
an example \cite[A1, Example 1]{nagata} of an affine and regular noetherian scheme of infinite Krull dimension,
so this hypotheses is not redundant.} that is Gorenstein and excellent, so that
that $\O_S$ is a dualizing complex for $S$ \cite[V,\S10]{RD}.
Since $f$ is CM by Corollary \ref{curvecorollary} (\ref{crvCM}),
by \cite[Theorem 3.5.1]{GDBC}) the relative dualizing complex $f^!\O_S$ has a
unique nonzero cohomology sheaf, which is in degree $-1$,
and we define the {\em relative dualizing sheaf} for $X$ over $S$ (or for $f$) to be:
\begin{equation*}
\omega_f=\omega_{X/S} := H^{-1}(f^!\O_S).
\end{equation*}
Since the fibers of $f$ are Gorenstein, $\omega_{X/S}$ is an invertible $\O_X$-module by \cite[V, Proposition 9.3, Theorem 9.1]{RD}. The formation of $\omega_{X/S}$ is compatible
with arbitrary base change on $S$ and \'etale localization on $X$ \cite[Theorem 3.6.1]{GDBC}.
\begin{remark}\label{abstractdualizing}
Since $S$ is Gorenstein and of finite Krull dimension and $f^!$ carries dualizing complexes for $S$ to dualizing complexes for $X$
(see \cite[\Rmnum{5}, \S8]{RD}),
the sheaf $\omega_{X/S}$ (thought of as a complex concentrated in some degree) is a dualizing complex for the abstract scheme $X$.
\end{remark}
\begin{proposition}\label{canmap}
Let $X\rightarrow S$ be a relative curve. There is a canonical
map of $\O_X$-modules
\begin{equation}
\xymatrix{
{c_{X/S}: \Omega^1_{X/S}} \ar[r] & {\omega_{X/S}}
}\label{cmap}
\end{equation}
whose formation commutes with any base change $S'\rightarrow S$, where
$S'$ is noetherian of finite Krull dimension, Gorenstein, and excellent.
Moreover, the restriction of $c_{X/S}$ to any $S$-smooth subscheme
of $X$ is an isomorphism.
\end{proposition}
\begin{proof}
See \cite{elzeinapp}, especially Th\'eor\`eme \Rmnum{3}.1, and {\em cf.} \cite[6.4.13]{LiuBook}.
\end{proof}
\begin{definition}\label{complexregdiff}
We define the two-term $\O_S$-linear complex (of $\O_S$-flat coherent $\O_X$-modules) concentrated in degrees 0 and 1
\begin{equation}
\xymatrix{
{\omega_f^{\bullet}=\omega_{X/S}^{\bullet}:=\O_X} \ar[r]^-{d_S} & {\omega_{X/S}}
}
\end{equation}
where $d_S$ is the composite of the map (\ref{cmap}) and the universal
$\O_S$-derivation $\O_X\rightarrow \Omega^1_{X/S}$. We view $\omega_{X/S}^{\bullet}$
as a filtered complex via ``{\em la filtration b\^ete}" \cite{DeligneHodge2},
which provides an exact triangle
\begin{equation}
\xymatrix{
{\omega_{X/S}[-1]} \ar[r] & {\omega^{\bullet}_{X/S}} \ar[r] & {\O_X}
}\label{HodgeFilComplex}
\end{equation}
in the derived category that we call the {\em Hodge Filtration} of $\omega^{\bullet}_{X/S}$.
\end{definition}
Since $c_{X/S}$ is an isomorphism over the $S$-smooth locus $X^{\sm}$ of $f$ in $X$, the complex $\omega^{\bullet}_{X/S}$ coincides with
the usual de Rham complex over $X^{\sm}$. Moreover, it follows immediately from
Proposition \ref{canmap} that the formation of $\omega_{X/S}^{\bullet}$ is compatible with
any base change $S'\rightarrow S$ to a noetherian scheme $S'$ of finite Krull dimension that is
Gorenstein and excellent.
\begin{definition}
Let $f:X\rightarrow S$ relative curve over $S$. For each nonnegative integer $i$, we define
\begin{equation*}
\mathscr{H}^i(X/S):=\R^i f_*\omega_{X/S}^{\bullet}.
\end{equation*}
When $S=\Spec R$ is affine, we will write $H^i(X/R)$ for the global sections of the $\O_S$-module
$\mathscr{H}^i(X/S)$.
\end{definition}
The complex $\omega_{X/S}^{\bullet}$ and its filtration (\ref{HodgeFilComplex})
behave extremely well with respect to duality:
\begin{proposition}\label{GDuality}
Let $f:X\rightarrow S$ be a proper curve over $S$. There is a canonical
quasi-isomorphism
\begin{equation}
\omega_{X/S}^{\bullet} \simeq \R\scrHom_X^{\bullet}(\omega_{X/S}^{\bullet},\omega_{X/S}[-1])
\label{DualityIsom}
\end{equation}
which is compatible with the filtrations on both sides induced by $(\ref{HodgeFilComplex})$.
In particular:
\begin{enumerate}
\item There is a natural quasi-isomorphism
\begin{equation*}
\R f_*\omega^{\bullet}_{X/S}\simeq \R\scrHom_X^{\bullet}(\R f_*\omega^{\bullet}_{X/S},\O_S)[-2]
\end{equation*}
which is compatible with the filtrations induced by $(\ref{HodgeFilComplex})$.\label{DualityOnS}
\item If $\rho:Y\rightarrow X$ is any finite morphism of proper curves over $S$,
then there is a canonical quasi-isomorphism
\begin{equation*}
\R\rho_*\omega_{Y/S}^{\bullet} \simeq \R\scrHom_X^{\bullet}
(\R\rho_*\omega_{Y/S}^{\bullet},\omega_{X/S}[-1]).
\end{equation*}
that is compatible with filtrations.\label{DualityRho}
\end{enumerate}
\end{proposition}
\begin{proof}
For the first claim, see the proofs of Lemmas 4.3 and 5.4 in \cite{CaisDualizing}, noting that although
$S$ is assumed to be the spectrum of a discrete valuation ring and the definition of curve in that paper
differs somewhat from the definition here, the arguments themselves apply {\em verbatim} in our
context. The assertion (\ref{DualityOnS}) (respectvely (\ref{DualityRho})) follows from this by
applying $\R f_*$ (respectively $\R\rho_*$) to both sides of (\ref{DualityIsom}) and appealing
to Grothendieck duality
\cite[Theorem 3.4.4]{GDBC} for the proper map $f$ (respectively $\rho$); see the proofs
of Lemma 5.4 and Proposition 5.8 in \cite{CaisDualizing} for details.
\end{proof}
In our applications, we need to understand the cohomology $H^i(X/S)$ for a proper
curve $X\rightarrow S$ when $S$ is either the spectrum of a discrete valuation
ring $R$ of mixed characteristic $(0,p)$ or the spectrum of a perfect field.
We now examine each of these situations in more detail.
First suppose that $S:=\Spec(R)$ is the spectrum of a discrete valuation ring $R$ having field of fractions $K$ of characteristic zero and perfect residue field $k$ of characteristic $p>0$, and fix a normal curve $f:X\rightarrow S$ that is proper over $S$ with smooth and geometrically connected
generic fiber $X_K$. This situation is studied extensively
in \cite{CaisDualizing}, and we content ourselves with a summary
of the results we will need.
To begin, we recall the following ``concrete" description of the relative dualizing sheaf:
\begin{lemma}\label{ConcreteDualizingDescription}
Let $i:U\hookrightarrow X$ be any Zariski open subscheme of $X$
whose complement consists of finitely many points of codimension $2$
$($necessarily in the closed fiber of $X$$)$. Then the canonical map
\begin{equation*}
\xymatrix{
{\omega_{X/S}} \ar[r] & {i_*i^*\omega_{X/S} \simeq i_*\omega_{U/S}}
}
\end{equation*}
is an isomorphism. In particular, $\omega_{X/S}\simeq i_*\Omega^1_{U/S}$
for any Zariski open subscheme $i:U\hookrightarrow X^{\sm}$ whose
complement consists of finitely many points of codimension two.
\end{lemma}
\begin{proof}
The first assertion is \cite[Lemma 3.2]{CaisNeron}. The second
follows from this, since $X^{\sm}$ contains the generic fiber
and the generic points of the closed fiber by our definition of curve.
\end{proof}
\begin{proposition}\label{ComplexFunctoriality}
Let $\rho:Y\rightarrow X$ be a finite morphism of normal and proper $S$-curves.
\begin{enumerate}
\item Attached to $\rho$ are natural
pullback and trace morphisms of complexes
\begin{equation*}
\xymatrix{
{\rho^*: \omega^{\bullet}_{X/S}} \ar[r] & {\rho_*\omega^{\bullet}_{Y/S}}
}
\quad\text{and}\quad
\xymatrix{
{\rho_*: \rho_*\omega^{\bullet}_{Y/S}} \ar[r] & {\omega^{\bullet}_{X/S}}
}
\end{equation*}
which are of formation compatible with \'etale localization on $X$ and
flat base change on $S$ and
are dual via the duality of Proposition $\ref{GDuality}$ $(\ref{DualityRho})$.
\label{FunctorialityProps1}
\item For any $S$-smooth point $y\in {Y}^{\sm}$ with image $x:=\rho(y)$
that lies in $X^{\sm}$, the induced mappings of complexes of $\O_{X,x}$-modules
$\omega^{\bullet}_{X/S,x}\rightarrow \omega^{\bullet}_{Y/S,y}$
and
$\omega^{\bullet}_{Y/S,y}\rightarrow \omega^{\bullet}_{X/S,x}$
coincide with the usual pullback and trace mappings on de Rham complexes
attached to the finite flat morphism of smooth schemes
$\Spec(\O_{Y,y})\rightarrow \Spec(\O_{X,x})$.\label{FunctorialityProps2}
\end{enumerate}
\end{proposition}
\begin{proof}
The assertions of (\ref{FunctorialityProps1}) follow from the proofs of Propositions 4.5 and 5.5
of \cite{CaisDualizing}, while (\ref{FunctorialityProps2})
is a straightforward consequence of the very construction of $\rho_*$ and $\rho^*$
as given in \cite[\S4]{CaisDualizing}.
\end{proof}
Since the generic fiber of $X$ is a smooth and proper curve over $K$,
the Hodge to de Rham spectral sequence degenerates \cite{DeligneIllusie}, and there
is a functorial short exact sequence of $K$-vector spaces
\begin{equation}
\xymatrix{
0\ar[r] & {H^0(X_K,\Omega^1_{X_K/K})} \ar[r] & {H^1_{\dR}(X_K/K)} \ar[r] & {H^1(X_K,\O_{X_K})} \ar[r] & 0
}\label{HodgeFilCrv}
\end{equation}
which we call the {\em Hodge filtration} of $H^1_{\dR}(X_K/K)$.
\begin{proposition}\label{HodgeIntEx}
Let $f:X\rightarrow S$ be a normal curve that is proper over $S=\Spec(R)$.
\begin{enumerate}
\item There are natural isomorphisms of free $R$-modules of rank $1$
\begin{equation*}
H^0(X/R)\simeq H^0(X,\O_X)\quad\text{and}\quad H^2(X/R)\simeq H^1(X,\omega_{X/S}),
\end{equation*}
which are canonically $R$-linearly dual to each other.
\item There is a canonical short exact sequence of finite free $R$-modules,
which we denote $H(X/R)$,
\begin{equation*}
\xymatrix{
0\ar[r] & {H^0(X,\omega_{X/S})} \ar[r] & {H^1(X/R)} \ar[r] & {H^1(X,\O_X)} \ar[r] & 0
}
\end{equation*}
that recovers the Hodge filtration $(\ref{HodgeFilCrv})$ of $H^1_{\dR}(X_K/K)$ after
extending scalars to $K$. \label{CohomologyIntegral}
\item Via the canonical cup-product auto-duality of $(\ref{HodgeFilCrv})$,
the exact sequence $H(X/R)$ is naturally isomorphic to
its $R$-linear dual.\label{CohomologyDuality}
\item The exact sequence $H(X/R)$ is contravariantly
$($respectively covariantly$)$ functorial in finite morphisms $\rho:Y\rightarrow X$
of normal and proper $S$-curves via pullback $\rho^*$ $($respectively trace $\rho_*$$)$;
these morphisms recover the usual pullback and trace mappings on Hodge filtrations after extending scalars
to $K$ and are adjoint with respect to the canonical cup-product autoduality of $H(X/R)$
in $(\ref{CohomologyDuality})$. \label{CohomologyFunctoriality}
\end{enumerate}
\end{proposition}
\begin{proof}
By Raynaud's ``{\em crit\`ere de platitude cohomologique}" \cite[Th\'eor\`me 7.2.1]{Raynaud}
(see also \cite[Proposition 2.7]{CaisDualizing}), our requirement that curves have geometrically
reduced fibers implies that $f:X\rightarrow S$ is cohomologically flat.\footnote{In other words, the $\O_S$-module $f_*\O_X$ commutes with
arbitrary base change.}
The proposition now follows from Propositions 5.7--5.8 of \cite{CaisDualizing}.
\end{proof}
We now turn to the case that $S=\Spec(k)$ for a perfect field $k$ and
$f:X\rightarrow S$ is a proper and geometrically connected curve over $k$.
Recall that $X$ is required to be geometrically reduced, so that the
$k$-smooth locus $U:=X^{\sm}$ is the complement of finitely many closed
points in $X$.
\begin{proposition}\label{HodgeFilCrvk}
Let $X$ be a proper and geometrically connected curve over $k$.
\begin{enumerate}
\item There are natural isomorphisms of 1-dimensional $k$-vector spaces
\begin{equation*}
H^0(X/k)\simeq H^0(X,\O_X)\quad\text{and}\quad H^2(X/k)\simeq H^1(X,\omega_{X/k}),
\end{equation*}
which are canonically $k$-linearly dual to each other.\label{H0H2overk}
\item There is a natural short exact sequence, which we denote $H(X/k)$
\begin{equation*}
\xymatrix{
0 \ar[r] & {H^0(X,\omega_{X/k})} \ar[r] & {H^1(X/k)} \ar[r] &
{H^1(X,\O_{X})} \ar[r] & 0
}\label{HodgeDegenerationField}
\end{equation*}
which is canonically isomorphic to its own $k$-linear dual.\label{HodgeExSeqk}
\end{enumerate}
\end{proposition}
\begin{proof}
Consider the long exact cohomology sequence arising from the exact
triangle (\ref{HodgeFilComplex}).
Since $X$ is proper over $k$, geometrically connected and reduced,
the canonical map $k\rightarrow H^0(X,\O_X)$ is an isomorphism, and it follows that
the map $d:H^0(X,\O_X)\rightarrow H^0(X,\omega_{X/k})$ is zero,
whence the map $H^0(X/k)\rightarrow H^0(X,\O_X)$ is an isomorphism.
Thanks to Proposition \ref{GDuality} (\ref{DualityOnS}), we have a canonical
quasi-isomorphism
\begin{equation}
\R\Gamma(X,\omega_{X/k}^{\bullet})\simeq
\R\Hom_k^{\bullet}(\R\Gamma(X,\omega_{X/k}^{\bullet}),k)[-2]\label{GDFieldExplicit}
\end{equation}
that is compatible with the filtrations induced by (\ref{HodgeFilComplex}).
Using the spectral sequence
\begin{equation*}
E_2^{m,n}:=\Ext_k(\mathbf{H}^{-n}(X,\omega_{X/k}^{\bullet}))
\implies H^{m+n}(\R\Hom_k^{\bullet}(\R\Gamma(X,\omega_{X/k}^{\bullet}),k))
\end{equation*}
and the vanishing of $\Ext_k^m(\cdot,k)$ for $m>0$,
we deduce that $H^2(X/k)\simeq H^0(X/k)^{\vee}$ is 1-dimensional over $k$.
Since Grothendieck's trace map $H^1(X,\omega_{X/k})\rightarrow k$ is an isomorphism,
we conclude that the {\em surjective} map of 1-dimensional $k$-vector spaces
$H^1(X,\omega_{X/k})\rightarrow H^2(X/k)$ must be an isomorphism. It follows that
the map $d:H^1(X,\O_X)\rightarrow H^1(X,\omega_{X/k})$ is zero as well, as desired.
The fact that that the resulting short exact sequence in (\ref{HodgeDegenerationField}) is
canonically isomorphic to its $k$-linear dual, and the fact that
the isomorphisms in (\ref{H0H2overk}) are $k$-linearly dual are
now easy consequences of the isomorphism (\ref{GDFieldExplicit}).
\end{proof}
We now suppose that $k$ is algebraically closed, and
following \cite[\S5.2]{GDBC}, we recall Rosenlicht's
explicit description \cite{Rosenlicht} of the relative dualizing sheaf $\omega_{X/k}$
and of Grothendieck duality.
Denote by $k(X)$ the ``function field" of $X$, {\em i.e.} $k(X):=\prod_i k(\xi_i)$
is the product of the residue fields at the finitely many generic points of
$X$, and write $j:\Spec(k(X))\rightarrow X$ for the canonical map.
By definition, the {\em sheaf of meromorphic differentials on $X$}
is the pushforward $\u{\Omega}^1_{k(X)/k}:=j_*\Omega^1_{k(X)/k}$.
Our hypothesis that $X$ is reduced implies that it is smooth
at its generic points, so $j$ factors through the open immersion
$i:U:=X^{\sm}\hookrightarrow X$. By \cite[Lemma 5.2.1]{GDBC}, the canonical map
of $\O_X$-modules
\begin{equation}
\xymatrix{
{\omega_{X/k}} \ar[r] & {i_*i^*\omega_{X/k}\simeq i_*\Omega^1_{U/k}}
}\label{OmegaSubSheaf}
\end{equation}
is injective, and it follows that $\omega_{X/k}$ is a subsheaf of
$\u{\Omega}^1_{k(X)/k}$. Rosenlicht's theory gives a concrete description of this subsheaf,
as we now explain.
Let $\pi:\nor{X}\rightarrow X$ be the normalization of $X$.
We have a natural identification
of ``function fields" $k(\nor{X})=k(X)$ and hence a canonical isomorphism
$\pi_* \u{\Omega}^1_{k(\nor{X})/k}\simeq \u{\Omega}^1_{k(X)/k}$
of sheaves on $X$.
\begin{definition}\label{OmegaReg}
Let $\omega_{X/k}^{\reg}$ be the sheaf of $\O_{X}$-modules
whose sections over any open $V\subseteq X$ are those
meromorphic differentials $\eta$ on $\pi^{-1}(V)\subseteq \nor{X}$
which satisfy
\begin{equation}
\sum_{y\in \pi^{-1}(x)} \res_y(s\eta)=0
\end{equation}
for all $x\in V(k)$ and all $s\in \O_{X,x}$, where $\res_{y}$
is the classical residue map on meromorphic differentials on
the smooth (possibly disconnected) curve $\nor{X}$ over the algebraically closed field $k$.
\end{definition}
\begin{remark}\label{OmegaRegMero}
Let $\Irr(X)$ be the set of irreducible components of $X$.
Since $\pi$ is an isomorphism over $U$ and $X$ is smooth at its generic
points, $\nor{X}$ is the disjoint union of the smooth, proper, and irreducible
$k$-curves $\nor{I}$ for $I\in \Irr(X)$.
Therefore, a meromorphic differential $\eta$ on $\nor{X}$
may be viewed as a tuple
$\eta = \left(\eta_{\nor{I}}\right)_{I\in \Irr(X)}$,
with $\eta_{\nor{I}}$ a meromorphic differential on the smooth and irreducible
curve $\nor{I}$.
The condition for a meromorphic differential $\eta$ on $\pi^{-1}(V)$
to be a section of $\omega_{X/k}^{\reg}$ over $V$ is then
\begin{equation*}
\sum_{y\in \pi^{-1}(x)} \res_y(s_y\eta_{\nor{I}_y}) = 0
\end{equation*}
for all $x\in V(k)$ and all $s\in \O_{X,x}$, where $\nor{I}_y$ is the
unique connected component of $\nor{X}$ on which $y$ lies and $s_y$
is the image of $s$ under the canonical map $\O_{X,x}\rightarrow \O_{\nor{I}_y,y}$.
\end{remark}
As any holomorphic differential on $\nor{X}$ has zero residue at every closed point,
the pushforward $\pi_*\Omega^1_{\nor{X}/k}$ is naturally a subsheaf of $\omega_{X/k}^{\reg}$,
and this inclusion is an equality at every $x\in U(k)$ since
$\pi$ is an isomorphism over $U$. It likewise follows from the definition
that any section of $\omega^{\reg}_{X/k}$
must be holomorphic at every smooth point of $X$, so there is a natural inclusion
\begin{equation}
\xymatrix{
{\omega^{\reg}_{X/k}} \ar@{^{(}->}[r] & {i_*\Omega^1_{U/k}}
}\label{OmegaRegIncl}
\end{equation}
which is an isomorphism over $U$. Moreover, by \cite[Lemma 5.2.2]{GDBC},
any section of $\omega^{\reg}_{X/k}$ has poles at the
finitely many non-smooth points of $X$ with order bounded by a constant depending
only on $X$, and it follows that $\omega^{\reg}_{X/k}$ is a coherent sheaf on $X$.
Since (\ref{OmegaRegIncl}) is an isomorphism at the generic points of $X$, we have
a quasi-coherent flasque resolution
\begin{equation*}
\xymatrix{
0\ar[r] & {\omega_{X/k}^{\reg}} \ar[r] & {\u{\Omega}^1_{k(X)/k}} \ar[r] &
{\displaystyle\bigoplus_{x\in X^0} {i_x}_*\left(\u{\Omega}^1_{k(X)/k,x}/\omega_{X/k,x}^{\reg}\right)}
\ar[r] & 0
},
\end{equation*}
where $X^0$ is the set of closed points of $X$ and $i_x:\Spec(\O_{X,x})\rightarrow X$
is the canonical map. The associated long exact cohomology
sequence yields an exact sequence of $k$-vector spaces
\begin{equation}
\xymatrix{
{\Omega^1_{k(X)/k}} \ar[r] &
{\displaystyle\bigoplus_{x\in X^0} \left(\u{\Omega}^1_{k(X)/k,x}/\omega_{X/k,x}^{\reg}\right)}
\ar[r] & {H^1(X,\omega^{\reg}_{X/k})} \ar[r] & 0
}.\label{TrRegCon}
\end{equation}
For $x\in X^0$, the $k$-linear ``residue" map
\begin{equation*}
\xymatrix{
{\res_x:\Omega^1_{k(X)/k,x}} \ar[r] & k
}
\quad\text{defined by}\quad
\res_x(\eta):=\sum_{y\in \pi^{-1}(x)} \res_y(\eta)
\end{equation*}
kills $\omega_{X/k,x}^{\reg}$, and the induced composite map
\begin{equation*}
\xymatrix{
{\Omega^1_{k(X)/k}} \ar[r] &
{\displaystyle\bigoplus_{x\in X^0}
\left(\u{\Omega}^1_{k(X)/k,x}/\omega_{X/k,x}^{\reg}\right)}
\ar[r]^-{\sum \res_x} & k
}
\end{equation*}
is zero by the residue theorem on the (smooth) connected components of
$\nor{X}$. Thus, from (\ref{TrRegCon}) we obtain a $k$-linear ``trace map"
\begin{equation}
\xymatrix{
{\res_X:H^1(X,\omega^{\reg}_{X/k})} \ar[r] & k
}\label{ResidueMap}
\end{equation}
which coincides with the usual residue map when $X$ is smooth.
Rosenlicht's explicit description of the relative dualizing sheaf and of Grothendieck
duality for $X/k$ is:
\begin{proposition}[Rosenlicht]\label{Rosenlicht}
Let $X$ be a proper and geometrically connected curve over $k$ with $k$-smooth
locus $U$. Viewing $\omega_{X/k}$ and $\omega^{\reg}_{X/k}$
as subsheaves of $i_*\Omega^1_{U/k}$ via $(\ref{OmegaSubSheaf})$ and
$(\ref{OmegaRegIncl})$, respectively, we have an equality
\begin{equation*}
\omega_{X/k}=\omega_{X/k}^{\reg}\quad\text{inside}\quad i_*\Omega^1_{U/k}.
\end{equation*}
Under this identification,
Grothendieck's trace map $H^1(X,\omega_X)\rightarrow k$
coincides with $-\res_X$.
\end{proposition}
\begin{proof}
See \cite[Theorem 5.2.3]{GDBC}.
\end{proof}
We now return to the situation that $S=\Spec(R)$ for a discrete valuation ring $R$ with fraction field $K$
of characteristic zero and perfect residue field $k$ of characteristic $p>0$.
\begin{lemma}\label{ReductionCompatibilities}
Let $X$ be a normal and proper curve over $S=\Spec(R)$ with smooth and geometrically
connected generic fiber, and denote by $\o{X}:=X_k$
the special fiber of $X$; it is a proper and geometrically connected curve over $k$
by Proposition $\ref{curveproperties}$ $(\ref{fiberscrv})$.
\begin{enumerate}
\item The canonical base change map
\begin{equation*}
\xymatrix{
0 \ar[r] & {H^0(X,\omega_{X/S})\mathop{\otimes}\limits_R k} \ar[r]\ar[d]^-{\simeq} &
{H^1(X/R) \mathop{\otimes}\limits_R k} \ar[r]\ar[d]^-{\simeq} &
{H^1(X,\O_X)\mathop{\otimes}\limits_R k} \ar[r]\ar[d]^-{\simeq} & 0\\
0\ar[r] & {H^0(\o{X},\omega_{\o{X}/k})} \ar[r] & {H^1(\o{X}/k)} \ar[r] &
{H^1(\o{X},\O_{\o{X}})} \ar[r] & 0
}
\end{equation*}
is an isomorphism. \label{BaseChngDiagram}
\item Let $\rho:Y\rightarrow X$ be a finite morphism of
normal and proper curves over $S$ with smooth and geometrically connected
generic fibers.
The canonical diagrams $($one for $\rho^*$ and one for $\rho_*$$)$
\begin{equation*}
\xymatrix{
{H^0(Y,\omega_{Y/S})\mathop{\otimes}\limits_R k} \ar@<0.5ex>[r]^-{\rho_*\otimes 1}\ar[d]_-{\simeq} &
\ar@<0.5ex>[l]^-{\rho^*\otimes 1} {H^0(X,\omega_{X/S})\mathop{\otimes}\limits_R k} \ar[d]^-{\simeq}\\
{H^0(\o{Y},\omega_{\o{Y}/k})}\ar@{^{(}->}[d]_-{(\ref{OmegaSubSheaf})} &
{H^0(\o{Y},\omega_{\o{Y}/k})}\ar@{^{(}->}[d]^-{(\ref{OmegaSubSheaf})}\\
{H^0(\nor{\o{Y}}, \Omega^1_{k(\nor{\o{Y}})/k})} \ar@<0.5ex>[r]^-{\nor{\o{\rho}}_*} &
\ar@<0.5ex>[l]^-{{\nor{\o{\rho}}}^*} {H^0(\nor{\o{X}}, \Omega^1_{k(\nor{\o{X}})/k})}
}
\end{equation*}
commute, where ${\nor{\o{\rho}}}^*$ and $\nor{\o{\rho}}_*$ are the usual pullback
and trace morphisms on meromorphic differential forms associated to the finite
flat map $\nor{\o{\rho}}:\nor{\o{Y}}\rightarrow \nor{\o{X}}$
of smooth curves over $k$.\label{PTBCCompat}
\end{enumerate}
\end{lemma}
\begin{proof}
Since $X$ is of relative dimension 1 over $S$, the cohomologies $H^1(X,\O_X)$ and $H^1(X,\omega_{X/S})$
both commute with base change, and they are both free over $R$ by Proposition \ref{HodgeIntEx}.
We conclude that $H^i(X,\O_X)$ and $H^i(X,\omega_{X/S})$ commute with base change for all $i$ and
hence that the left and right vertical maps in the base change diagram (\ref{BaseChngDiagram}) (whose
rows are exact by Propositions \ref{HodgeIntEx} and \ref{HodgeFilCrvk})
are isomorphisms. It follows that the middle vertical map in
(\ref{BaseChngDiagram}) is an isomorphism as well.
The compatibility of pullback and trace under base change to the special fibers,
as asserted by the diagram in (\ref{PTBCCompat}),
is a straightforward consequence of Proposition \ref{ComplexFunctoriality}
(\ref{FunctorialityProps2}), using the facts that $X$ and $Y$ are
smooth at generic points of closed fibers and that $\o{\rho}:\o{Y}\rightarrow\o{X}$
takes generic points to generic points as noted in the proof of Lemma \ref{ConcreteDualizingDescription}.
\end{proof}
\subsection{Universal vectorial extensions and Dieudonn\'e crystals}\label{Universal}
There is an alternate description of the short exact sequence $H(X/R)$ of Proposition
\ref{HodgeIntEx} (\ref{CohomologyIntegral}) in terms of
Lie algebras and N\'eron models of Jacobians that will allow us to relate
this cohomology to Dieudonn\'e modules. To explain this description and its connection
with crystals, we first recall some facts from \cite{MM} and \cite{CaisNeron}.
Fix a base scheme $T$, and let $G$ be an fppf sheaf of abelian groups over $T$.
A {\em vectorial extension} of $G$ is a short exact sequence (of fppf sheaves of abelian
groups)
\begin{equation}
\xymatrix{
0 \ar[r] & {V} \ar[r] & {E} \ar[r] & {G} \ar[r] & 0.
}\label{extension}
\end{equation}
with $V$ a vector group (i.e. an fppf abelian sheaf which is locally represented by a product of $\Ga$'s).
Assuming that $\Hom(G,V)=0$ for all vector groups $V$, we say that a vectorial extension
(\ref{extension}) is {\em universal} if, for any vector group $V'$ over $T$,
the pushout map $\Hom_T(V,V')\rightarrow \Ext^1_T(G,V')$
is an isomorphism. When a universal vectorial extension of $G$ exists, it is
unique up to canonical isomorphism and covariantly functorial in morphisms $G'\rightarrow G$
with $G'$ admitting a universal extension.
\begin{theorem}\label{UniExtCompat}
Let $T$ be an arbitrary base scheme.
\begin{enumerate}
\item If $A$ is an abelian scheme over $T$, then a universal vectorial
extension $\E(A)$ of $A$ exists, with $V=\omega_{\Dual{A}}$,
and is compatible with arbitrary base change on $T$.
\label{UniExtCompat1}
\item If $p$ is locally nilpotent on $T$ and $G$ is a $p$-divisible group over
$T$, then a universal vectorial extension $\E(G)$ of $G$ extsis, with $V=\omega_{\Dual{G}}$,
and is compatible with arbitrary base change on $T$.
\label{UniExtCompat2}
\item If $p$ is locally nilpotent on $T$ and $A$ is an abelian scheme over $T$ with
associated $p$-divisible group $G:=A[p^{\infty}]$, then the canonical map of fppf sheaves
$G\rightarrow A$ extends to a natural map
\begin{equation*}
\xymatrix{
0 \ar[r] & {\omega_{\Dual{G}}} \ar[r]\ar[d] & {\E(G)} \ar[r]\ar[d] & {G}\ar[d] \ar[r] & 0\\
0 \ar[r] & {\omega_{\Dual{A}}} \ar[r] & {\E(A)} \ar[r] & {A} \ar[r] & 0
}
\end{equation*}
which induces an isomorphism of the corresponding short exact sequences of Lie algebras.
\label{UniExtCompat3}
\end{enumerate}
\end{theorem}
\begin{proof}
For the proofs of (\ref{UniExtCompat1}) and (\ref{UniExtCompat2}), see
\cite[\Rmnum{1}, \S1.8 and \S1.9]{MM}. To prove (\ref{UniExtCompat3}), note that
pulling back the universal vectorial extension of $A$ along $G\rightarrow A$
gives a vectorial extension $\E'$ of $G$ by $\omega_{\Dual{A}}$. By universality, there then exists
a unique map $\psi:\omega_{\Dual{G}}\rightarrow \omega_{\Dual{A}}$ with the property
that the pushout of $\E(G)$ along $\psi$ is $\E'$, and this gives the map on universal extensions.
That the induced map on Lie algebras is an isomorphism follows from \cite[\Rmnum{2}, \S 13]{MM}.
\end{proof}
For our applications, we will need a generalization of the universal extension
of an abelian scheme to the setting of N\'eron models; in order to describe this
generalization, we first recall the explicit description of the universal
extension of an abelian scheme in terms of rigidified extensions.
For any commutative $T$-group scheme $F$,
a {\em rigidified extension of $F$ by $\Gm$ over $T$} is a pair $(E,\sigma)$
consisting of an extension (of fppf abelian sheaves)
\begin{equation}
\xymatrix{
0 \ar[r] & {\Gm} \ar[r] & {E} \ar[r] & {F} \ar[r] & 0
}\label{ExtRigDef}
\end{equation}
and a splitting $\sigma: \Inf^1(F)\rightarrow E$ of the pullback of (\ref{ExtRigDef})
along the canonical closed immersion $\Inf^1(F)\rightarrow F$. Two rigidified
extensions $(E,\sigma)$ and $(E',\sigma')$ are equivalent if there
is a group homomorphism $E\rightarrow E'$ carrying $\sigma$ to $\sigma'$
and inducing the identity on $\Gm$ and on $F$.
The set $\Extrig_T(F,\Gm)$ of equivalence classes of rigidified extensions over $T$ is naturally a group
via Baer sum of rigidified extensions\cite[\Rmnum{1}, \S2.1]{MM}, so the functor on $T$-schemes
$T'\rightsquigarrow \Extrig_{T'}(F_{T'},\Gm)$ is naturally a group functor that is
contravariant in $F$ via pullback (fibered product).
We write $\scrExtrig_T(F,\Gm)$ for the fppf sheaf of abelian groups associated to this functor.
\begin{proposition}[Mazur-Messing]\label{MMrep}
Let $A$ be an abelian scheme over an arbitrary base scheme $T$.
The fppf sheaf $\scrExtrig_T(A,\Gm)$ is represented by a smooth and separated $T$-group scheme,
and there is a canonical short exact sequence of smooth group schemes over $T$
\begin{equation}
\xymatrix{
0\ar[r] & {\omega_A} \ar[r] & {\scrExtrig_T(A,\Gm)} \ar[r] & {\Dual{A}} \ar[r] & 0
}.\label{univextabelian}
\end{equation}
Furthermore, $(\ref{univextabelian})$ is naturally isomorphic to the universal extension of $\Dual{A}$ by a vector
group.
\end{proposition}
\begin{proof}
See \cite{MM}, $\Rmnum{1}, \S2.6$ and Proposition 2.6.7.
\end{proof}
In the case that $T=\Spec R$ for $R$ a discrete valuation ring of mixed
characteristic $(0,p)$ with fraction field $K$,
we have the following genaralization of Proposition \ref{MMrep}:
\begin{proposition}
Let $A$ be an abelian variety over $K$, with dual abelian variety $\Dual{A}$, and
write $\ensuremath{\mathcal{A}}$ and $\Dual{\ensuremath{\mathcal{A}}}$ for the N\'eron models of $A$ and $\Dual{A}$ over $T=\Spec(R)$.
Then the fppf abelian sheaf $\scrExtrig_T(\ensuremath{\mathcal{A}},\Gm)$ on the category of smooth $T$-schemes
is represented by a smooth and separated $T$-group scheme. Moreover, there
is a canonical short exact sequence of smooth group schemes over $T$
\begin{equation}
\xymatrix{
0\ar[r] & {\omega_{\ensuremath{\mathcal{A}}}} \ar[r] & {\scrExtrig_T(\ensuremath{\mathcal{A}},\Gm)} \ar[r] & {\Dual{\ensuremath{\mathcal{A}}}^0} \ar[r] & 0
}\label{NeronCanExt}
\end{equation}
which is contravariantly functorial in $A$ via homomorphisms of abelian varieties over $K$.
The formation of $(\ref{NeronCanExt})$ is compatible with smooth base change on $T$; in particular,
the generic fiber of $(\ref{NeronCanExt})$ is the universal extension of $\Dual{A}$ by a vector group.
\end{proposition}
\begin{proof}
Since $R$ is of mixed characteristic $(0,p)$ with perfect residue field,
this follows from Proposition 2.6 and the discussion following Remark 2.9 in \cite{CaisNeron}.
\end{proof}
In the particular case that $A$ is the Jacobian of a smooth, proper and geometrically
connected curve $X$ over $K$ which is the generic fiber of a normal proper curve $\X$
over $R$, we can relate the exact sequence of Lie algebras attached to (\ref{NeronCanExt})
to the exact sequence $H(X/R)$ or Proposition \ref{HodgeIntEx} (\ref{CohomologyIntegral}):
\begin{proposition} \label{intcompare}
Let $\X$ be a proper relative curve over $T=\Spec(R)$ with smooth generic fiber $X$ over $K$.
Write $J:=\Pic^0_{X/K}$ for the Jacobian of $X$ and $\Dual{J}$ for its dual,
and let $\mathcal{J}$, $\Dual{\mathcal{J}}$ be the corresponding N\'eron models over $R$.
There is a canonical homomorphism of exact sequences of finite free $R$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\Lie\omega_{\mathcal{J}}} \ar[r]\ar[d] & {\Lie\scrExtrig_T(\mathcal{J},\Gm)} \ar[r]\ar[d]
& {\Lie \Dual{\mathcal{J}}^0} \ar[r]\ar[d] & 0\\
0 \ar[r] & {H^0(\X,\omega_{\X/T})} \ar[r] & {H^1(\X/R)} \ar[r] & {H^1(\X,\O_{\X})}
\ar[r] & 0
}
\end{gathered}\label{IntegralComparisonMap}
\end{equation}
that is an isomorphism when $\X$ has rational singularities.\footnote{Recall that $\X$ is
said to have {\em rational singularities} if it admits a resolution of singularities
$\rho:\X'\rightarrow \X$ with the natural map $R^1\rho_*\O_{{\X'}}=0$. Trivially, any
regular $\X$ has rational singularities.}
For any finite morphism $\rho:\mathcal{Y} \rightarrow \X$ of $S$-curves satisfying the above hypotheses,
the map $(\ref{IntegralComparisonMap})$ intertwines $\rho_*$
$($respectively $\rho^*$$)$ on the bottom row with $\Pic(\rho)^*$
$($respectively $\Alb(\rho)^*$$)$ on the top.
\end{proposition}
\begin{proof}
See Theorem 1.2 and (the proof of) Corollary 5.6 in \cite{CaisNeron}.
\end{proof}
\begin{remark}\label{canonicalproperty}
Let $X$ be a smooth and geometrically connected curve over $K$
admitting a normal proper model $\X$ over $R$ that is a curve
having rational singularities.
It follows from Proposition \ref{intcompare}
and the N\'eron mapping property
that $H(\X/R)$ is a {\em canonical integral structure}
on the Hodge filtration (\ref{HodgeFilCrv}): it is
independent of the choice of proper model $\X$ that is normal with rational singularities,
and
is functorial in finite morphisms $\rho:Y\rightarrow X$
of proper smooth curves over $K$ which admit models over $R$ satisfying these hypotheses.
These facts can be proved in greater generality by appealing to resolution of singularities
for excellent surfaces and the flattening techniques of Raynaud--Gruson \cite{RayGrus};
see \cite[Theorem 5.11]{CaisDualizing} for details.
\end{remark}
We will need to relate universal extensions of $p$-divisible to their Dieudonn\'e crystals.
In order to explain how this goes, we begin by recalling some basic facts
from crystalline Dieudonn\'e theory, as discussed in \cite{BBM}.
Fix a perfect field $k$ and set $\Sigma:=\Spec(W(k))$, considered as a PD-scheme via the canonical divided powers on the ideal $pW(k)$. Let $T$ be
a $\Sigma$-scheme on which $p$ is locally nilpotent (so $T$ is naturally a PD-scheme over $\Sigma$), and
denote by $\Cris(T/\Sigma)$ the big crystalline site of $T$ over $\Sigma$,
endowed with the {\em fppf} topology (see \cite[\S 2.2]{BBM1}).
If $\mathscr{F}$ is a sheaf on $\Cris(T/\Sigma)$ and $T'$ is any PD-thickening of $T$,
we write $\mathscr{F}_{T'}$ for the associated {\em fppf} sheaf on $T'$.
As usual, we denote by $i_{T/\Sigma}:T_{fppf}\rightarrow (T/\Sigma)_{\Cris}$
the canonical morphism of topoi, and
we abbreviate $\underline{G}:={i_{T/\Sigma}}_{*}G$ for any fppf sheaf $G$ on $T$.
Let $G$ be a $p$-divisible group over $T$, considered as an fppf abelian sheaf on $T$.
As in \cite{BBM}, we define the (contravariant) {\em Dieudonn\'e crystal of $G$ over $T$} to be
\begin{equation}
\ensuremath{\mathbf{D}}(G) := \scrExt^1_{T/\Sigma}(\underline{G},\O_{T/\Sigma}).\label{DieudonneDef}
\end{equation}
It is a locally free crystal in $\O_{T/\Sigma}$-modules, which is contravariantly functorial
in $G$ and of formation compatible with base change along PD-morphisms $T'\rightarrow T$ of $\Sigma$-schemes
thanks to 2.3.6.2 and Proposition 2.4.5 $(\rmnum{2})$ of \cite{BBM}.
If $T'=\Spec(A)$ is affine, we will simply write $\ensuremath{\mathbf{D}}(G)_A$ for the finite locally free $A$-module
associated to $\ensuremath{\mathbf{D}}(G)_{T'}$.
The structure sheaf $\O_{T/\Sigma}$ is canonically an extension of $\u{\mathbf{G}}_a$
by the PD-ideal $\mathcal{J}_{T/\Sigma}\subseteq \O_{T/\Sigma}$, and by applying $\scrHom_{T/\Sigma}(\underline{G},\cdot)$
to this extension one obtains (see Propositions 3.3.2 and 3.3.4 as well as
Corollaire 3.3.5 of \cite{BBM})
a short exact sequence (the {\em Hodge filtration})
\begin{equation}
\xymatrix{
0\ar[r] & {\scrExt^1_{T/\Sigma}(\underline{G},\mathcal{J}_{T/\Sigma})}\ar[r] &
{\ensuremath{\mathbf{D}}(G)}\ar[r] &
{\scrExt^1_{T/\Sigma}(\underline{G},\u{\mathbf{G}}_a)}\ar[r] & 0
}\label{HodgeFilCrys}
\end{equation}
that is contravariantly functorial
in $G$ and of formation compatible with base change along PD-morphisms
$T'\rightarrow T$ of $\Sigma$-schemes.
The following ``geometric" description of the value of (\ref{HodgeFilCrys}) on a PD-thickening of the
base will be essential for our purposes:
\begin{proposition}\label{BTgroupUnivExt}
Let $G$ be a fixed $p$-divisible group over $T$ and let $T'$ be any
$\Sigma$-PD thickening of $T$. If $G'$ is any lifting of $G$ to a $p$-divisible
group on $T'$, then there is a natural isomorphism
\begin{equation*}
\xymatrix{
0 \ar[r] & {\omega_{G'}} \ar[r]\ar[d]^-{\simeq} & {\scrLie(\E(\Dual{G'}))} \ar[r]\ar[d]^-{\simeq} &
{\scrLie (\Dual{G'})}\ar[r]\ar[d]^-{\simeq} & 0\\
0\ar[r] & {\scrExt^1_{T/\Sigma}(\underline{G},\mathcal{J}_{T/\Sigma})_{T'}}\ar[r] & {\ensuremath{\mathbf{D}}(G)_{T'}}\ar[r] &
{\scrExt^1_{T/\Sigma}(\underline{G},\underline{\mathbf{G}}_a)_{T'}}\ar[r] & 0
}
\end{equation*}
that is moreover compatible with base change in the evident manner.
\end{proposition}
\begin{proof}
See \cite[Corollaire 3.3.5]{BBM} and \cite[\Rmnum{2}, Corollary 7.13]{MM}.
\end{proof}
\begin{remark}\label{MessingRem}
In his thesis \cite{Messing}, Messing showed that the Lie algebra of the universal extension
of $\Dual{G}$ is ``crystalline in nature" and used this as the {\em definition}\footnote{Noting
that it suffices to define the crystal $\ensuremath{\mathbf{D}}(G)$ on $\Sigma$-PD thickenings $T'$
of $T$ to which $G$ admits a lift.} of $\ensuremath{\mathbf{D}}(G)$.
(See chapter $\Rmnum{4}$, \S2.5 of \cite{Messing} and especially 2.5.2). Although we
prefer the more intrinsic description (\ref{DieudonneDef}) of
\cite{MM} and \cite{BBM}, it is ultimately Messing's original
definition that will be important for us.
\end{remark}
\subsection{Integral models of modular curves}\label{tower}
We record some basic facts about integral models of modular curves that will be needed in what follows.
We assume that the reader is familiar with \cite{KM}, and will freely use the notation and terminology
therein. Throughout, we fix a prime $p$ and a positive integer $N$ not divisible by $p$.
\begin{definition}
Let $r$ be a nonnegative integer and $R$ a ring containing a fixed choice $\zeta$ of
primitive $p^r$-th root of unity
in which $N$ is invertible.
The moduli problem
$\scrP_{r}^{\zeta}:=([\bal\ \Gamma_1(p^r)]^{\zeta\text{-}\mathrm{can}}; [\mu_N]])$
on $(\Ell/R)$
assigns to $E/S$ the set of quadruples $(\phi:E\rightarrow E',P,Q ; \alpha) $
where:
\begin{enumerate}
\item $\phi:E\rightarrow E'$ is a $p^r$-isogeny.
\item $P\in \ker\phi(S)$ and $Q\in \ker\phi^t(S)$ are generators of
$\ker\phi$ and $\ker\phi^t$, respectively,
which pair to $\zeta$ under the canonical pairing
$\langle\cdot,\cdot\rangle_{\phi}: \ker\phi\times\ker\phi^t\rightarrow \mu_{\deg\phi}$
\cite[\S2.8]{KM}.
\item $\alpha:\mu_N\hookrightarrow E[N]$ is a closed immersion of $S$-group schemes.
\end{enumerate}
\end{definition}
\begin{proposition}\label{XrRepresentability}
If $N \ge 4$, then the moduli problem $\scrP_{r}^{\zeta}$ is represented
by a regular scheme $\ensuremath{\mathbf{M}}(\scrP_r^{\varepsilon})$ that is flat of pure relative dimension
$1$ over $\Spec(R)$. The moduli scheme $\ensuremath{\mathbf{M}}(\scrP_r^{\zeta})$ admits a canonical
compactification $\o{\ensuremath{\mathbf{M}}}(\scrP_r^{\zeta})$, which is regular and proper flat of pure
relative dimension $1$ over $\Spec(R)$.
\end{proposition}
\begin{proof}
Using that $N$ is a unit in $R$, one first shows that for $N\ge 4$, the moduli problem
$[\mu_N]$ on $(\Ell/R)$ is representable over $\Spec(R)$ and finite \'etale;
this follows from 2.7.4, 3.6.0, 4.7.1 and 5.1.1 of \cite{KM}, as $[\mu_N]$
is isomorphic to $[\Gamma_1(N)]$ over any $R$-scheme containing a fixed choice
of primitive $N$-th root of unity (see also \cite[8.4.11]{KM}). By \cite[4.3.4]{KM},
to prove the first assertion it is then enough to show that
$[\bal \Gamma_1(p^r)]^{\zeta\text{-}\mathrm{can}}$ on $(\Ell/R)$ is relatively representable and
regular, which (via \cite[9.1.7]{KM}) is a consequence of \cite[7.6.1 (2)]{KM}.
For the second assertion, see \cite[\S8]{KM}.
\end{proof}
Recall that we have fixed a compatible sequence $\{\varepsilon^{(r)}\}_{r\ge1}$
of primitive $p^r$-th roots of unity in $\o{\Q}_p$.
\begin{definition}\label{XrDef}
We set $\X_r:=\o{\ensuremath{\mathbf{M}}}(\scrP_r^{\varepsilon^{(r)}})$, viewed as a scheme over $T_r:=\Spec(R_r)$.
\end{definition}
There is a canonical action of $\Z_p^{\times}\times (\Z/N\Z)^{\times}$
by $R_r$-automorphisms of $\X_r$,
defined at the level of the underlying moduli problem by
\begin{equation}
{(u,v)\cdot (\phi:E\rightarrow E',P,Q; \alpha)} :={(\phi:E\rightarrow E',uP, u^{-1}Q; \alpha\circ v)}
\label{balcanaction}
\end{equation}
as one checks by means of the computation
$\langle uP,u^{-1}Q\rangle_{\phi} = \langle P,Q\rangle^{uu^{-1}}_{\phi} = \langle P,Q \rangle_{\phi}$.
Here, we again write $v:\mu_N\rightarrow\mu_N$ for the automorphism of $\mu_N$
functorially defined by $\zeta \mapsto \zeta^v$ for any $N$-th root of unity $\zeta$.
We refer to this action of $\Z_p^{\times}\times (\Z/N\Z)^{\times}$ as the {\em diamond operator}
action, and will denote by $\langle u \rangle$ (respectively $\langle v \rangle_N$) the
automorphism induced by $u\in \Z_p^{\times}$ (respectively $v\in (\Z/N\Z)^{\times}$).
There is also an $R_r$-semilinear ``geometric inertia" action of $\Gamma:=\Gal(K_{\infty}/K_0)$
on $\X_r$, which allows us to descend the generic fiber of $\X_r$ to $K_0$.
To explain this action,
for $\gamma \in \Gamma$ and any $T_r$-scheme $T'$, let us write
$T'_{\gamma}$ for the base change of $T'$ along the morphism $T_r\rightarrow T_r$
induced by $\gamma\in \Aut(R_r)$.
There is a canonical functor $(\Ell/(T_r)_{\gamma})\rightarrow (\Ell/T_r)$ obtained by viewing an elliptic curve
over a $(T_r)_{\gamma}$-scheme $T'$ as
the same elliptic curve over the same base $T'$,
viewed as a $T_r$-scheme via the projection $(T_r)_{\gamma}\rightarrow T_r$.
For a moduli problem $\scrP$ on $(\Ell/T_r)$, we denote by $\gamma^*\scrP$
the moduli problem on $(\Ell/(T_r)_{\gamma})$ obtained by composing $\scrP$ with this functor;
see \cite[4.1.3]{KM}.
Each $\gamma\in \Gamma$ gives rise to
a morphism of moduli problems $\gamma: \scrP_r^{\varepsilon^{(r)}}\rightarrow \gamma^*\scrP_r^{\varepsilon^{(r)}}$
via
\begin{equation}
{\gamma(\phi:E\rightarrow E',P,Q; \alpha)} :=
{(\phi_{\gamma}:E_{\gamma}\rightarrow E'_{\gamma},\chi(\gamma)^{-1}P_{\gamma},Q_{\gamma}; \alpha_{\gamma})}
\label{gammamapsModuli}
\end{equation}
where the subscript of $\gamma$ means ``base change along $\gamma$" (see \S\ref{Notation}).
Since
\begin{equation*}
\langle \chi(\gamma)^{-1}P_{\gamma}, Q_{\gamma}\rangle_{\phi_{\gamma}} =
\gamma\langle P,Q\rangle_{\phi}^{\chi(\gamma)^{-1}} = \langle P,Q \rangle_{\phi}
\end{equation*}
this really is a morphism of moduli problems on $(\Ell/T_r)$. We thus obtain
a morphism of $T_r$-schemes
\begin{equation}
\xymatrix{
{\gamma:\X_r} \ar[r] & {(\X_r)_{\gamma}}
}\label{gammamaps}
\end{equation}
for each $\gamma\in \Gamma$, compatibly with change in $\gamma$.
The induced semilinear action of $\Gamma$ on the generic fiber of ${\X_r}$
provides a descent datum with respect to the canonical map $\Spec(K_r)\rightarrow \Spec(K_0)$,
which is necessarily effective as this map is \'etale. Thus, there is a unique
scheme $X_r$ over $K_0=\Q_p$ with $(X_r)_{K_r}\simeq (\X_r)_{K_r}$;
as the diamond operators visibly commute with the action of $\Gamma$, they
act on $X_r$ by $\Q_p$-automorphisms in a manner that is compatible with this identification.
\begin{remark}\label{genfiberrem}
We may identify $X_r$ with the base change to $\Q_p$ of the
modular curve $X_1(Np^r)$ over $\Q$ classifying pairs $(E,\alpha)$
of a generalized elliptic curve $E/S$ together with an embedding of $S$-group schemes
$\alpha:\mu_{Np^r}\hookrightarrow E^{\sm}$ whose image meets each irreducible
component in every geometric fiber.
If instead we were to use the geometric inertia action on $\X_r$ induced by
\begin{equation*}
{\gamma(\phi:E\rightarrow E',P,Q; \alpha)} :=
{(\phi_{\gamma}:E_{\gamma}\rightarrow E'_{\gamma},P_{\gamma},\chi(\gamma)^{-1}Q_{\gamma}; \alpha_{\gamma})},
\end{equation*}
then the resulting descent $X_r'$ of the generic fiber of $\X_r$ to $\Q_p$ would be canonically
isomorphic to the base change to $\Q_p$ of the modular curve $X_1(Np^r)'$ over $\Q$
classifying generalized elliptic curves $E/S$ with an embedding of $S$-group schemes
$\Z/Np^r\Z\hookrightarrow E^{\sm}[Np^r]$ whose image meets each irreducible
component in every geometric fiber. Of course, $X_1(Np^r)$ (respectively $X_1(Np^r)'$)
is the canonical model of the upper half-plane quotient $\Gamma_1(Np^r)\backslash \h^*$
with $\Q$-rational cusp cusp $i\infty$ (respectively $0$).
\end{remark}
Recall (\cite[\S6.7]{KM}) that over any base scheme $S$, a cyclic $p^{r+1}$-isogeny of elliptic curves
$\phi:E\rightarrow E'$ admits a ``standard factorization" (in the sense of \cite[6.7.7]{KM})
\begin{equation}
\xymatrix{
E=:E_0 \ar[r]^-{\phi_{0,1}} & E_1 \ar[r] \cdots & E_{r} \ar[r]^-{\phi_{r,r+1}} & E_{r+1}:=E'
}.\label{standardFac}
\end{equation}
For each pair of nonnegative integers $a<b\le r+1$ we will write
$\phi_{a,b}$ for the composite $\phi_{a,a+1}\circ\cdots\circ\phi_{b-1,b}$
and $\phi_{b,a}:=\phi_{a,b}^t$ for the dual isogeny. Using this notion, we define
``degeneracy maps" $\rho,\sigma:\X_{r+1} \rightrightarrows \X_r$ (over the map
$T_{r+1}\rightarrow T_r$) at the level of underlying moduli problems
as follows ({\em cf.}: \cite[11.3.3]{KM}):
\begin{equation}
\begin{aligned}
&{\rho(\phi:E_0\rightarrow E_{r+1},P,Q; \alpha)}:=
{(\phi_{0,r}:E_0\rightarrow E_{r},pP,\phi_{r+1,r}(Q); \alpha)}\\
&{\sigma(\phi: E_0\rightarrow E_{r+1},P,Q; \alpha)}:=
{(\phi_{1,r+1}:E_1\rightarrow E_{r+1},\phi_{0,1}(P),pQ; \phi_{0,1}\circ \alpha)}
\end{aligned}\label{XrDegen}
\end{equation}
By the universal property of fiber products, we obtain morphisms
$T_{r+1}$-schemes
\begin{equation}
\xymatrix{
{\X_{r+1}} \ar@<0.5ex>[r]^-{\rho}\ar@<-0.5ex>[r]_-{\sigma} & {\X_r\times_{T_r} T_{r+1}}
}.\label{rdegen}
\end{equation}
that are compatible with the diamond operators and
the geometric inertia action of $\Gamma$.
\begin{remark}
On generic fibers, the morphisms (\ref{rdegen}) uniquely descend to degeneracy mappings
$\rho,\sigma:X_{r+1}\rightrightarrows X_r$ of smooth curves over $\Q_p$. Under the identification
$X_r\simeq X_1(Np^r)_{\Q_p}$ of Remark \ref{genfiberrem}, the map $\rho$ corresponds
to the ``standard" projection, induced by ``$\tau\mapsto \tau$" on the complex upper half-plane,
whereas $\sigma$ corresponds to the morphism induced by ``$\tau\mapsto p\tau$."
\end{remark}
Recall that we have fixed a choice of primitive $N$-th root of unity $\zeta_N$ in $\o{\Q}_p$.
The Atkin Lehner ``involution" $w_{\zeta_N}$ on $\X_r\times_{R_r} R_r'$ is defined as
in \cite[\S8]{pAdicShimura}. Following \cite[11.3.2]{KM}, we define the Atkin Lehner
automorphism $w_{\varepsilon^{(r)}}$ of $\X_r$ over $R_r$ on the
underlying moduli problem $\scrP_r^{\varepsilon^{(r)}}$ as
\begin{equation*}
{w_{\varepsilon^{(r)}} (\phi:E\rightarrow E',P,Q; \alpha)} :=
{(\phi^t:E'\rightarrow E,-Q,P;\ \phi\circ\alpha )}
\label{AtkinLehnerInv}
\end{equation*}
We then define $w_r:=w_{\varepsilon^{(r)}} \circ w_{\zeta_N}=w_{\zeta_N}\circ w_{\varepsilon^{(r)}}$; it is an automorphism of $\X_r \times_{R_r} R_r'$
over $R_r':=R_r[\mu_N]$.
\begin{proposition}\label{ALinv}
For all $(u,v)\in \Z_p^{\times}\times(\Z/N\Z)^{\times}$ and all
$\gamma\in \Gal(K_{\infty}'/K_0)$, the identities
\begin{align*}
w_r \langle u\rangle \langle v\rangle_N &= \langle v\rangle_N^{-1}\langle u^{-1}\rangle w_r \\
(\gamma^*w_r)\gamma & = \gamma w_r\langle \chi(\gamma)\rangle^{-1}\langle a(\gamma)\rangle_N^{-1}\\
w_r^2 &= \langle -p^r\rangle_N\langle -N\rangle \\
\rho w_{r+1} &= w_{r}\sigma\\
\sigma w_{r+1} &= \langle p\rangle_N w_r\rho
\end{align*}
hold, with $a:\Gal(K_{\infty}'/K_0)\rightarrow (\Z/N\Z)^{\times}$ the character determined
by $\gamma\zeta=\zeta^{a(\gamma)}$ for all $\zeta\in \mu_N(\Qbar_p)$.
\end{proposition}
\begin{proof}
This is an easy consequence of definitions.
\end{proof}
In order to describe the special fiber of $\X_r$, we must first introduce Igusa curves:
\begin{definition}
Let $r$ be a nonnegative integer. The moduli problem $\I_r:=([\Ig(p^r)]; [\mu_N])$ on
$(\Ell/\mathbf{F}_p)$ assigns to $(E/S)$ the set of triples $(E,P;\alpha)$ where $E/S$ is an elliptic curve
and
\begin{enumerate}
\item $P\in E^{(p^r)}(S)$ is a point that generates the $r$-fold iterate of
Verscheibung $V^{(r)}:E^{(p^r)}\rightarrow E$.
\item $\alpha:\mu_N\hookrightarrow E[N]$ is a closed immersion of $S$-group schemes.
\end{enumerate}
\end{definition}
\begin{proposition}
If $N\ge 4$, then the moduli problem $\I_r$ on $(\Ell/\mathbf{F}_p)$ is represented by a smooth
affine curve $\ensuremath{\mathbf{M}}(\I_r)$ over $\mathbf{F}_p$ which admits a canonical smooth
compactification $\o{\ensuremath{\mathbf{M}}}(\I_r)$.
\end{proposition}
\begin{proof}
One argues as in the proof of Proposition \ref{XrRepresentability}, using \cite[12.6.1]{KM}
to know that $[\Ig(p^r)]$ is relatively representable on $(\Ell/\mathbf{F}_p)$,
regular 1-dimensional and finite flat over $(\Ell/\mathbf{F}_p)$.
\end{proof}
\begin{definition}\label{IgusaDef}
Set $\Ig_r:=\o{\ensuremath{\mathbf{M}}}(\I_r)$; it is a smooth, proper, and geometrically connected $\mathbf{F}_p$-curve.
\end{definition}
There is a canonical action of the diamond operators $\Z_p^{\times}\times (\Z/N\Z)^{\times}$ on the moduli problem
$\I_r$ via $(u,v)\cdot (E,P; \alpha):= (E,uP; v\circ\alpha)$; this induces a corresponding action on $\Ig_r$
by $\mathbf{F}_p$-automorphisms. We again write $\langle u\rangle$ (respectively $\langle v\rangle_N$) for the
action of $u\in \Z_p^{\times}$ (respectively $v\in (\Z/N\Z)^{\times}$).
Thanks to the ``backing up theorem" \cite[6.7.11]{KM}, one also has natural degeneracy maps
\begin{equation}
\xymatrix{
{\rho:\Ig_{r+1}} \ar[r] & {\Ig_r}
}\qquad\text{induced by}\qquad
\rho(E,P;\alpha):= (E,VP,\alpha)
\label{Vmapsch}
\end{equation}
on underlying moduli problems. This map is visibly
equivariant for the diamond operator action on source and target.
Let $\SS_r$ be the (reduced) closed subscheme of $\Ig_r$
that is the support of the coherent ideal sheaf of relative differentials $\Omega^1_{\Ig_r/\Ig_0}$;
over the unique degree 2 extension of $\mathbf{F}_p$, this scheme breaks up as a disjoint union of rational
points---the supersingular points. The map (\ref{Vmapsch})
is finite of degree $p$, generically \'etale and totally (wildly) ramified over
each supersingular point.
We can now describe the special fiber of $\X_r$:
\begin{proposition} \label{redXr}
The scheme $\o{\X}_r:=\X_r\times_{T_r} \Spec(\mathbf{F}_p)$ is the disjoint union, with crossings at the supersingular points, of the following proper, smooth $\mathbf{F}_p$-curves:
for each pair $a,b$ of nonnegative integers with $a+b=r$, and for each $u\in (\Z/p^{\min(a,b)}\Z)^{\times}$,
one copy of
$\Ig_{\max(a,b)}$.
\end{proposition}
We refer to \cite[13.1.5]{KM} for the definition of ``disjoint union with crossings at the supersingular points".
Note that the special fiber of $\X_r$ is (geometrically) reduced; this will be crucial in our later work. We often write $I_{(a,b,u)}$ for the irreducible component of $\o{\X}_r$ indexed by the triple $(a,b,u)$ and will refer to it as the {\em $(a,b,u)$-component} (for fixed $(a,b)$ we have $I_{(a,b,u)}=\Ig_{\max(a,b)}$ for all $u$).
For the proof of Proposition \ref{redXr}, we refer the reader to \cite[13.11.2--13.11.4]{KM},
and content ourselves with recalling
the correspondence between (non-cuspidal) points of the $(a,b,u)$-component and
$[\bal \Gamma_1(p^r)]^{1\text{-}\mathrm{can}}$-structures on elliptic curves.\footnote{Note that under the canonical ring homomorphism
$R_r\twoheadrightarrow \mathbf{F}_p$, our fixed choice $\varepsilon^{(r)}$ of primitive $p^r$-th root of unity
maps to $1\in \mathbf{F}_p$, which {\em is} a primitive $p^r$-th root of unity by definition \cite[9.1.1]{KM},
as it is a root of the $p^r$-th cyclotomic polynomial over $\mathbf{F}_p$!}
Let $S$ be any $\mathbf{F}_p$ scheme, fix an ordinary elliptic curve $E_0$ over $S$, and
let $(\phi:E_0\rightarrow E_r,P,Q; \alpha)$ be an element of $\scrP_{r}^1(E_0/S)$.
By \cite[13.11.2]{KM}, there exist unique nonnegative integers $a,b$
with the property that the cyclic $p^r$-isogeny $\phi$ factors as a purely inseparable cyclic $p^a$-isogeny
followed by an \'etale $p^b$-isogeny (this {\em is} the standard factorization of $\phi$).
Furthermore, there exists a unique elliptic curve $E$ over $S$ and $S$-isomorphisms
$E_0\simeq E^{(p^b)}$ and $E_r\simeq E^{(p^a)}$ such that the cyclic $p^r$ isogeny $\phi$ is:
\begin{equation*}
\xymatrix{
{E_0\simeq E^{(p^b)}} \ar[r]^-{F^a} & {E^{(p^r)}} \ar[r]^-{V^b} & {E^{(p^a)} \simeq E_r}
}
\end{equation*}
and $P\in E^{(p^b)}(S)$ (respectively $Q\in E^{(p^a)}$) is an Igusa structure of level $p^b$ (respectively $p^a$) on $E$ over $S$.
When $a\ge b$ there is a unique unit $u\in (\Z/p^{b}\Z)^{\times}$ such that $V^{a-b}(Q)=uP$ in $E^{(p^b)}(S)$ and when
$b\ge a$ there is a unique unit $u\in (\Z/p^a\Z)^{\times}$ such that $uV^{b-a}(P)=Q$ in $E^{(p^a)}(S)$.
Thus, for $a\ge b$ (respectively $b\ge a$) and fixed $u$, the data $(E,Q;p^{-b}V^b\circ\alpha)$
(respectively $(E,P;p^{-b}V^b\circ\alpha)$)
gives an $S$-point of the $(a,b,u)$-component
$\Ig_{\max(a,b)}$.
Conversely, suppose given $(a,b,u)$ and an $S$-valued point of $\Ig_{\max(a,b)}$ which is neither a cusp nor a supersingular point (in the sense that it corresponds to an ordinary elliptic curve with extra structure).
If $a\ge b$ and $(E,Q;\alpha)$ is the given $S$-point of $\Ig_{a}$ then we set
$P:=u^{-1}V^{a-b}(Q)$, while if $b\ge a$ and $(E,P;\alpha)$ is the given $S$-point of $\Ig_b$ then we set $Q:=uV^{b-a}P$. Due to \cite[13.11.3]{KM}, the data
\begin{equation*}
(\xymatrix{
{E^{(p^b)}} \ar[r]^-{F^a} & {E^{(p^r)}} \ar[r]^-{V^b} & {E^{(p^a)}, P,Q; F^b\circ\alpha}
})
\end{equation*}
gives an $S$-point of $\ensuremath{\mathbf{M}}(\scrP_r^{1})$. These constructions are visibly inverse to each other.
\begin{remark}\label{rEvenChoice}
When $r$ is even and $a=b=r/2$, there is a choice to be made as to how one identifies the
$(r/2,r/2,u)$-component of $\o{\X}_r$ with $\Ig_{r/2}$: if
$(\phi:E_0\rightarrow E_r,P,Q;\alpha)$ is an element of $\scrP_r^1(E_0/S)$
which corresponds to a point on the $(r/2,r/2,u)$-component, then for $E$ with
$E_0\simeq E^{(p^{r/2})}\simeq E_r$,
{\em both} $(E,P; p^{-r/2}V^{r/2}\circ\alpha)$ and $(E,Q;p^{-r/2}V^{r/2}\circ\alpha)$ are
$S$-points of $\Ig_{p^{r/2}}$.
Since $uP=Q$,
the corresponding closed immersions
$\Ig_{r/2}\hookrightarrow \o{\X}_r$
are twists of each other by the automorphism $\langle u\rangle$ of the source.
We will {\em consistently choose} $(E,Q;p^{-r/2}V^{r/2}\circ\alpha)$ to identify the $(r/2,r/2,u)$-component
of $\o{\X}_r$ with $\Ig_{r/2}$.
\end{remark}
\begin{remark}\label{MWGood}
As in \cite[pg.~236]{MW-Hida}, we will refer to $I_r^{\infty}:=I_{(r,0,1)}$ and $I_r^0:=I_{(0,r,1)}$
as the two ``good" components of $\o{\X}_r$.
The $\Q_p$-rational cusp $\infty$ of $X_r$ induces a section
of $\X_r\rightarrow T_r$ which meets $I_r^{\infty}$, while
the section
induced by the $K_r'$-rational cusp $0$ meets $I_r^0$.
It is precisely these irreducible components of
$\o{\X}_r$ which contribute to the ``ordinary" part of cohomology.
We note that $I_r^{\infty}$ corresponds to the image of $\Ig_r$ under the map $i_1$
of \cite[pg. 236]{MW-Hida}, and corresponds to the component of $\o{\X}_r$ denoted by $C_{\infty}$
in \cite[pg. 343]{Tilouine}, by $C_r^{\infty}$ in \cite[pg. 231]{Saby}
and, for $r=1$, by $I$ in \cite[\S 7]{tameness}.
\end{remark}
By base change, the degeneracy mappings (\ref{rdegen}) induces morphisms
$\o{\rho},\o{\sigma}:\o{\X}_{r+1}\rightrightarrows \o{\X}_r$
of curves over $\mathbf{F}_p$ which admit the following descriptions on irreducible components:
\begin{proposition}\label{pr1desc}
Let $a,b$ be nonnegative integers with $a+b=r+1$ and $u\in (\Z/p^{\min(a,b)}\Z)^{\times}$.
The restriction of the map $\o{\sigma}: \o{\X}_{r+1}\rightarrow \o{\X}_r$ to the $(a,b,u)$-component
of $\o{\X}_{r+1}$ is:
\begin{equation*}
\begin{cases}
\xymatrix{
{\Ig_{a}=I_{(a,b,u)}} \ar[r]^-{ F\circ \rho} & {I_{(a-1,b,u)}=\Ig_{a-1}}
}
&:\quad b < a \le r+1 \\
\xymatrix{
{\Ig_{b}=I_{(a,b,u)}} \ar[r]^-{\langle u\rangle^{-1}F} &
{I_{(a-1,b,u\bmod p^{a-1})}=\Ig_{b}}
}
&:\quad a = b = r/2 \\
\xymatrix{
{\Ig_{b}=I_{(a,b,u)}} \ar[r]^-{F} & {I_{(a-1,b,u\bmod p^{a-1})}=\Ig_{b}}
}
&:\quad a < b < r+1 \\
\xymatrix{
{\Ig_{r+1}=I_{(0,r+1,1)}} \ar[r]^-{\langle p\rangle_N\rho} & {I_{(0,r,1)}=\Ig_{r}}
}
&:\quad (a,b,u)=(0,r+1,1)
\end{cases}
\end{equation*}
and the restriction of the map $\overline{\rho}: \o{\X}_{r+1}\rightarrow \o{\X}_r$ to the $(a,b,u)$-component
of $\o{\X}_{r+1}$ is:
\begin{equation*}
\begin{cases}
\xymatrix{
{\Ig_{r+1}=I_{(r+1,0,1)}} \ar[r]^-{\rho} & {I_{(r,0,1)}=\Ig_{r}}
}
&:\quad (a,b,u)=(r+1,0,1)\\
\xymatrix{
{\Ig_{a}=I_{(a,b,u)}} \ar[r]^-{F} & {I_{(a,b-1,u\bmod p^{b-1})}=\Ig_{a}}
}
&:\quad b < a+1 \le r+1 \\
\xymatrix{
{\Ig_{b}=I_{(a,b,u)}} \ar[r]^-{\langle u\rangle F\circ \rho} & {I_{(a,b-1,u)}=\Ig_{b-1}}
}
&:\quad a+1 = b = r/2+1\\
\xymatrix{
{\Ig_{b}=I_{(a,b,u)}} \ar[r]^-{F\circ \rho} & {I_{(a,b-1,u)}=\Ig_{b-1}}
}
&:\quad a+1 < b \le r+1 \\
\end{cases}
\end{equation*}
Here, for any $\mathbf{F}_p$-scheme $I$, the map $F:I\rightarrow I$ is the absolute Frobenius morphism.
\end{proposition}
\begin{proof}
Using the moduli-theoretic definitions (\ref{XrDegen}) of the degeneracy maps,
the proof is a pleasant exercise in tracing through
the functorial correspondence between the points of $\o{\X}_r$
and points of $\Ig_{(a,b,u)}$. We leave the details to the reader.
\end{proof}
We likewise have a description of the automorphism
of $\o{\X}_r$ induced via base change by
the geometric inertia action\footnote{
Since $\Gamma$ acts trivially on $\mathbf{F}_p$,
for each $\gamma\in \Gamma$ the base change of the $R_r$-morphism $\gamma: \X_r\rightarrow (\X_r)_{\gamma}$
along the map induced by the canonical surjection $R_r\twoheadrightarrow \mathbf{F}_p$
is an $\mathbf{F}_p$-morphism $\o{\gamma}:\o{\X}_r\rightarrow (\o{\X}_r)_{\gamma}\simeq \o{\X}_r$.
}
(\ref{gammamapsModuli}) of $\Gamma$:
\begin{proposition}\label{AtkinInertiaCharp}
Let $a,b$ be nonnegative integers with $a+b=r$ and $u\in (\Z/p^{\min(a,b)}\Z)^{\times}$.
For $\gamma\in \Gamma$, the restriction of $\o{\gamma}:\o{\X}_{r}\rightarrow \o{\X}_r$ to
the $(a,b,u)$-component of $\o{\X}_{r}$ is:
\begin{equation*}
\begin{cases}
\xymatrix{
{\Ig_{a}=I_{(a,b,u)}} \ar[r]^-{\id} &
{I_{(a,b,\chi(\gamma)u)}=\Ig_{a}}
}
&:\quad b\le a \le r\\
\xymatrix{
{\Ig_{b}=I_{(a,b,u)}} \ar[r]^-{\langle \chi(\gamma)\rangle^{-1}} & {I_{(a,b,\chi(\gamma)u)}=\Ig_{b}}
}
&:\quad a < b \le r \\
\end{cases}
\end{equation*}
\label{InertiaCharp}
\end{proposition}
Following \cite[\S7--8]{Ulmer}, we now define a correspondence $\pi_1,\pi_2:\mathcal{Y}_r\rightrightarrows\X_r$ on $\X_r$ over $R_r$ which naturally
extends the correspondence on $X_r$ giving the Hecke operator $U_p$ (see below for a brief discussion
of correspondences).
\begin{definition}
Let $r$ be a nonnegative integer and $R$ a ring containing a fixed choice $\zeta$ of
primitive $p^r$-th root of unity
in which $N$ is invertible.
The moduli problem
$\scrQ_{r}^{\zeta}:=([\Gamma_0(p^{r+1}); r,r]^{\zeta\text{-}\mathrm{can}}; [\mu_N])$ on $(\Ell/R)$
assigns to $E/S$ the set of quadruples $(\phi:E\rightarrow E',P,Q;\alpha)$
where:
\begin{enumerate}
\item $\phi$ is a cyclic $p^{r+1}$-isogeny
with standard factorization
\begin{equation*}
\xymatrix{
E=:E_0 \ar[r]^-{\phi_{0,1}} & E_1 \ar[r] \cdots & E_{r} \ar[r]^-{\phi_{r,r+1}} & E_{r+1}:=E'
}
\end{equation*}
\item $P\in E_1(S)$ and $Q\in E_r(S)$ are generators of
$\ker\phi_{1,r+1}$ and $\ker \phi_{r,0}$, respectively, and satisfy
\begin{equation*}
\langle P, \phi_{r,r+1}(Q) \rangle_{\phi_{1,r+1}}=\langle \phi_{1,0}(P),Q \rangle_{\phi_{0,r}}=\zeta.
\end{equation*}
\item $\alpha:\mu_N\hookrightarrow E[N]$ is a closed immersion of $S$-group schemes.
\end{enumerate}
\end{definition}
\begin{proposition}\label{YrRepresentability}
If $N \ge 4$, then the moduli problem $\scrQ_{r}^{\zeta}$ is represented
by a regular scheme $\ensuremath{\mathbf{M}}(\scrQ_r^{\zeta})$ that is flat of pure relative dimension
$1$ over $\Spec(R)$. This scheme admits a canonical
compactification $\o{\ensuremath{\mathbf{M}}}(\scrP_r^{\zeta})$, which is regular and proper flat of pure
relative dimension $1$ over $\Spec(R)$.
\end{proposition}
\begin{proof}
As in the proof of Proposition \ref{XrRepresentability}, it suffices to prove that
$[\Gamma_0(p^{r+1}); r,r]^{\zeta\text{-}\mathrm{can}}$
is relatively representable and regular, which follows from \cite[7.6.1]{KM};
see also \S7.9 of {\em op.~cit.}
\end{proof}
\begin{definition}\label{YrDef}
We set $\mathcal{Y}_r:=\o{\ensuremath{\mathbf{M}}}(\scrQ_r^{\varepsilon^{(r)}})$, viewed as a scheme over $T_r=\Spec(R_r)$.
\end{definition}
The scheme $\mathcal{Y}_r$ is equipped with an action of the diamond operators $\Z_p^{\times}\times (\Z/N\Z)^{\times}$,
as well as a ``geometric inertia" action of $\Gamma$ given moduli-theoretically
exactly as in (\ref{balcanaction}) and (\ref{gammamapsModuli}).
The ``semilinear'' action of $\Gamma$ is equivalent to a descent datum---necessarily effective---on the
generic fiber of $\mathcal{Y}_r$, and we denote by $Y_r$ the resulting unique $\Q_p$-descent
of $(\mathcal{Y}_r)_{K_r}$.
\begin{remark}\label{Yrgeniden}
We may identify $Y_r$ with the base change to $\Q_p$ of the modular curve $X_1(Np^r;Np^{r-1})$
over $\Q$ classifying triples $(E_1,\alpha,C)$ where $E_1$ is a generalized elliptic curve,
$\alpha:\mu_{Np^r}\hookrightarrow E^{\sm}_1[Np^r]$ is an embedding of group schemes
whose image meets each irreducible component in every geometric fiber,
and $C$ is a locally free subgroup scheme of rank $p$ in $E^{\sm}_1[p]$ with the property that
$C\cap\im\alpha = 0$.
Note that $X_1(Np^r; Np^{r-1})$ is the canonical model
over $\Q$ with rational cusp $i\infty$ of the modular curve
$\Gamma_{r+1}^r\backslash \h^*$, for
$\Gamma_{r+1}^r:=\Gamma_1(p^r)\cap \Gamma_0(p^{r+1})$.
\end{remark}
There is a canonical morphism of curves $\pi:\X_{r+1}\rightarrow \mathcal{Y}_r$ over $T_{r+1}\rightarrow T_r$
induced by the morphism
\begin{equation}
\begin{gathered}
\scrP_{r+1}^{\varepsilon^{(r)}}\rightarrow \scrQ_r^{\varepsilon^{(r)}}\quad\text{given by}
\quad{\pi(\phi:E\rightarrow E',P,Q;\alpha)} := {(\phi:E\rightarrow E',\phi_{0,1}(P),\phi_{r+1,r}(Q); \alpha)}.
\end{gathered}\label{XtoY}
\end{equation}
One checks that $\pi$ is equivariant with respect to the action of the diamond operators
and of $\Gamma$, and so descends to a map $\pi:Y_r\rightarrow X_r$ of smooth curves over $\Q_p$.
It is likewise straightforward to check that the two projection maps $\sigma,\rho: \X_{r+1}\rightrightarrows \X_r$
of (\ref{XrDegen}) factor through $\pi$ via unique maps of $T_r$-schemes
$\pi_1,\pi_2: \mathcal{Y}_{r}\rightrightarrows \X_r$, given as morphisms of underlying
moduli problems on $(\Ell/R_r)$
\begin{equation}
\begin{aligned}
&{\pi_1(\phi:E_0\rightarrow E_{r+1},P,Q;\alpha)}:=
{(E_1\xrightarrow{\phi_{1,r+1}} E_{r+1},P,\phi_{r,r+1}(Q);\ \phi_{0,1}\circ\alpha)}\\
&{\pi_2(\phi:E_0\rightarrow E_{r+1},P,Q;\alpha)}:=
{(E_0\xrightarrow{\phi_{0,r}} E_{r},\phi_{1,0}(P),Q; \alpha)}
\end{aligned}\label{Upcorr}
\end{equation}
That these morphisms are well defined and
that one has $\rho=\pi\circ \pi_2$ and $\sigma=\pi\circ \pi_1$
is easily verified (see \cite[\S7]{Ulmer} and compare to
\cite[\S11.3.3]{KM}). They are moreover
finite of generic degree $p$, equivariant for the diamond operators,
and $\Gamma$-compatible; in particular, $\pi_1,\pi_2$ descend to finite
maps $\pi_1,\pi_2:Y_r\rightrightarrows X_r$
over $\Q_p$.
Via our identifications in Remarks \ref{genfiberrem} and \ref{Yrgeniden}, the map $\pi_1$
corresponds to the usual ``forget $C$" map, while $\pi_2$ corresponds to ``quotient by $C$".
We stress that the ``standard" degeneracy map $\rho:X_{r+1}\rightarrow X_r$ factors
through $\pi_2$ (and not $\pi_1$).
\begin{proposition}\label{redYr}
The scheme $\o{\mathcal{Y}}_r:=\mathcal{Y}_r\times_{T_r} \Spec(\mathbf{F}_p)$ is the disjoint union, with crossings
at the supersingular points, of the following proper, smooth $\mathbf{F}_p$-curves:
for each pair of nonnegative integers $a,b$ with $a+b=r+1$ and for each $u\in (\Z/p^{\min(a,b)\Z})^{\times}$,
one copy of
\begin{equation*}
\begin{cases}
\Ig_{\max(a,b)} &\text{if}\ ab\neq 0\\
\Ig_{r} &\text{if}\ (a,b)=(r+1,0)\ \text{or}\ (0,r+1)
\end{cases}
\end{equation*}
\end{proposition}
We will write $J_{(a,b,u)}$ for the irreducible component of $\o{\mathcal{Y}}_r$
indexed by $(a,b,u)$, and refer to it as the $(a,b,u)$-component; again,
$J_{(a,b,u)}$ is independent of $u$.
The proof of Proposition \ref{redYr} is a straightforward adaptation of the arguments of
\cite[13.11.2--13.11.4]{KM} (see also \cite[Proposition 8.2]{Ulmer}).
We recall the correspondence between non-cuspidal points of the $(a,b,u)$-component and
$[\Gamma_0(p^{r+1}); r,r]^{1\text{-}\mathrm{can}}$-structures on elliptic curves.
Fix an ordinary elliptic curve $E_0$ over an $\mathbf{F}_p$-scheme $S$,
and let $(\phi:E_0\rightarrow E_{r+1},P,Q; \alpha)$ be an element of
$\scrQ_r^{1}(E_0/S)$. As before, there exist unique
nonnegative integers $a,b$ with $a+b=r+1$
and a unique elliptic curve $E/S$
with the property that the cyclic $p^{r+1}$-isogeny $\phi$ factors as
\begin{equation*}
\xymatrix{
{E_0\simeq E^{(p^b)}} \ar[r]^-{F^a} & {E^{(p^{r+1})}} \ar[r]^-{V^b} & {E^{(p^a)} \simeq E_{r+1}}
}.
\end{equation*}
First suppose that $ab\neq 0$. Then the point $P\in E^{(p^{b+1})}(S)$ (respectively $Q\in E^{(p^{a+1})}(S)$)
is an $[\Ig(p^b)]$ (respectively $[\Ig(p^a)]$) structure on $E^{(p)}$ over $S$.
If $a\ge b$, there is a unit $u\in (\Z/p^b\Z)^{\times}$ such that $V^{a-b}(Q)=uP$ in $E^{(p^{b+1})}(S)$,
while if $a\le b$ then there is a unique $u\in (\Z/p^a\Z)^{\times}$ with $uV^{b-a}(P)=Q$ in $E^{(p^{a+1})}(S)$.
For $a\ge b$ (respectively $a < b$), and fixed $u$, the data $(E^{(p)}, Q ; p^{1-b}V^{b-1}\circ\alpha)$
(respectively $(E^{(p)}, P; p^{1-b}V^{b-1}\circ \alpha)$) gives an $S$-point of the $(a,b,u)$-component
$\Ig_{\max(a,b)}$. If $b=0$ (respectively $a=0$), then $Q\in E^{(p^r)}(S)$ (respectively $P\in E^{(p^r)}(S)$)
is an $[\Ig(p^r)]$-structure on $E=E_0$ (respectively $E = E_{r+1}$).
In these extremal cases, the data $(E,Q;\alpha)$
(respectively $(E,P; p^{-r-1}V^{r+1}\circ\alpha)$)
gives an $S$-point of the $(r+1,0,1)$-component (respectively $(0,r+1,1)$-component)
$Ig_r$.
Conversely, suppose given $(a,b,u)$ and an $S$-point of $\Ig_{\max(a,b)}$ which is neither
cuspidal nor supersingular. If $r+1 > a\ge b$ and $(E,Q;\alpha)$ is the given point of $\Ig_a$,
then we set $P:=u^{-1}V^{a-b}(Q)\in E^{(p^b)}(S)$, while if $r+1 > b\ge a$ and $(E,P;\alpha)$ is the given point of
$\Ig_b$, we set $Q:=uV^{b-a}P\in E^{(p^{a})}(S)$. Then
\begin{equation*}
(\xymatrix{
{E^{(p^{b-1})}} \ar[r]^-{F} & {E^{(p^b)}} \ar[r]^-{F^{a-1}} & {E^{(p^r)}} \ar[r]^-{V^{b-1}} &
{E^{(p^a)}} \ar[r]^-{V} & {E^{(p^{a-1})}}, P,Q; F^{b-1}\circ\alpha
})
\end{equation*}
is an $S$-point of $\ensuremath{\mathbf{M}}(\scrQ_r^{1})$.
If $b=0$ and $(E,Q,\alpha)$ is an $S$-point of $\Ig_r$, then we let $P\in E^{(p)}(S)$ be the identity
section and we obtain an $S$-point $(F^{r+1}:E\rightarrow E^{(p^{r+1})},P,Q;\alpha)$
of $\ensuremath{\mathbf{M}}(\scrQ_r^1)$.
If $a=0$ and $(E,P,\alpha)$ is an $S$-point of $\Ig_r$, then we let $Q\in E^{(p)}(S)$
be the identity section and we obtain an $S$-point $(V^{r+1}:E^{(p^{r+1})}\rightarrow E,P,Q;F^{r+1}\circ\alpha)$
of $\ensuremath{\mathbf{M}}(\scrQ_r^1)$.
Using the descriptions of $\o{\X}_r$ and $\o{\mathcal{Y}}_r$ furnished by
Propositions \ref{redXr} and \ref{redYr}, we can calculate the restrictions of
the degenercy maps $\o{\pi}_1,\o{\pi}_2:\o{\mathcal{Y}}_r\rightrightarrows \o{\X}_r$
to each irreducible component
of the special fiber of $\mathcal{Y}_r$. The following is due to Ulmer\footnote{We warn the reader, however,
that Ulmer omits the effect of the degeneracy maps on $[\mu_N]$-structures, so his formulae
are slightly different from ours.}
\cite[Proposition 8.3]{Ulmer}:
\begin{proposition}\label{UlmerProp}
Let $a,b$ be nonnegative integers with $a+b=r+1$ and $u\in (\Z/p^{\min(a,b)}\Z)^{\times}$.
The restriction of the map $\o{\pi}_1: \o{\mathcal{Y}}_r\rightarrow \o{\X}_r$ to the $(a,b,u)$-component
of $\o{\mathcal{Y}}_r$ is:
\begin{equation*}
\begin{cases}
\xymatrix{
{\Ig_{r}=J_{(r+1,0,1)}} \ar[r]^-{F} & {I_{(r,0,1)}=\Ig_{r}}
}
&:\quad (a,b,u)=(r+1,0,1)\\
\xymatrix{
{\Ig_{a}=J_{(a,b,u)}} \ar[r]^-{\rho} & {I_{(a-1,b,u)}=\Ig_{a-1}}
}
&:\quad b < a < r+1 \\
\xymatrix{
{\Ig_{b}=J_{(a,b,u)}} \ar[r]^-{\langle u^{-1}\rangle} & {I_{(a-1,b,u\bmod p^{a-1})}=\Ig_{b}}
}
&:\quad a=b=(r+1)/2\\
\xymatrix{
{\Ig_{b}=J_{(a,b,u)}} \ar[r]^-{\id} & {I_{(a-1,b,u\bmod p^{a-1})}=\Ig_{b}}
}
&:\quad a < b < r+1 \\
\xymatrix{
{\Ig_{r}=J_{(0,r+1,1)}} \ar[r]^-{\langle p\rangle_N} & {I_{(0,r,1)}=\Ig_{r}}
}
&:\quad (a,b,u)=(0,r+1,1)
\end{cases}
\end{equation*}
and the restriction of the map $\overline{\pi}_2: \o{\mathcal{Y}}_r\rightarrow \o{\X}_r$ to the $(a,b,u)$-component
of $\o{\mathcal{Y}}_r$ is:
\begin{equation*}
\begin{cases}
\xymatrix{
{\Ig_{r}=J_{(r+1,0,1)}} \ar[r]^-{\id} & {I_{(r,0,1)}=\Ig_{r}}
}
&:\quad (a,b,u)=(r+1,0,1)\\
\xymatrix{
{\Ig_{a}=J_{(a,b,u)}} \ar[r]^-{\id} & {I_{(a,b-1,u\bmod p^{b-1})}=\Ig_{a}}
}
&:\quad b < a+1 \le r+1 \\
\xymatrix{
{\Ig_{b}=J_{(a,b,u)}} \ar[r]^-{\langle u \rangle \rho} & {I_{(a,b-1,u)}=\Ig_{b-1}}
}
&:\quad a +1 = b =r/2 + 1 \\
\xymatrix{
{\Ig_{b}=J_{(a,b,u)}} \ar[r]^-{\rho} & {I_{(a,b-1,u)}=\Ig_{b-1}}
}
&:\quad a+1 < b < r+1 \\
\xymatrix{
{\Ig_{r}=J_{(0,r+1,1)}} \ar[r]^-{F} & {I_{(0,r,1)}=\Ig_{r}}
}
&:\quad (a,b,u)=(0,r+1,1)
\end{cases}
\end{equation*}
\end{proposition}
\begin{proof}
The proof is similar to the proof of Proposition \ref{pr1desc}, using the correspondence between
irreducible components of $\mathcal{Y}_r$, $\X_r$ and Igusa curves that we have explained, together with the moduli-theoretic
definitions (\ref{Upcorr}) of the degeneracy mappings. We leave the details to the reader.
\end{proof}
We end this section with a brief discussion
of correspondences on curves and their induced action on cohomology and Jacobians,
which we then apply to the specific case of modular curves.
Fix a ring $R$ and a proper normal curve $X$ over $S=\Spec R$. Throughtout this discussion, we assume either that
$R$ is a discrete valuation ring of mixed characteristic $(0,p)$ with perfect residue field,
or that $R$ is a perfect field (and hence the normal $X$ is smooth).
\begin{definition}
A {\em correspondence $T:=(\pi_1,\pi_2)$ on $X$} is an ordered pair $\pi_1,\pi_2:Y\rightrightarrows X$
of finite $S$-morphisms of normal and $S$-proper curves. The {\em transpose}
of a correspondence $T:=(\pi_1,\pi_2)$ on $X$ is the correspondence
on $X$ given by the ordered pair $T^*:=(\pi_2,\pi_1)$.
\end{definition}
Thanks to Proposition \ref{HodgeIntEx} (\ref{CohomologyFunctoriality}), any correspondence
$T=(\pi_1,\pi_2)$ on $X$ induces an $R$-linear endomorphism of the short exact sequence
$H(X/R)$ via ${\pi_1}_*\pi_2^*$. By a slight abuse of notation, we denote this endomorphism
by $T$; as endomorphisms of $H(X/R)$ we then have
\begin{equation}
T= {\pi_1}_*\pi_2^*\qquad\text{and}\qquad T^* = {\pi_2}_* \pi_1^*.
\label{HeckeDef}
\end{equation}
Given a finite map $\pi:X\rightarrow X$, we will consistently view $\pi$ as a correspondence on $X$
via the association $\pi\rightsquigarrow (\id,\pi)$.
In this way, we may think of correspondences
on $X$ as ``generalized endomorphisms." This perspective can be made more compelling as follows.
First suppose that $R$ is a field, and fix a correspondence $T$ given by an ordered pair
$\pi_1,\pi_2:Y\rightrightarrows X$ of finite morphisms of smooth and proper curves.
Then $T$ and its transpose $T^*$ induce endomorphisms of the Jacobian $J_X:=\Pic^0_{X/R}$ of $X$, which we again
denote by the same symbols, via
\begin{equation}
T:=\Alb(\pi_2)\circ \Pic^0(\pi_1)\qquad\text{and}\qquad T^*:= \Alb(\pi_1) \circ \Pic^0(\pi_2)
\label{JacHecke}
\end{equation}
Note that when $T=(\id,\pi)$ for a morphism $\pi:X\rightarrow X$, the induced endomorphisms
(\ref{JacHecke}) of $J_X$ are given by $T=\Alb(\pi)$ and $T^*:=\Pic^0(\pi)$.\footnote{Because of this fact,
for a general correspondence $T=(\pi_1,\pi_2)$
the literature often refers to the induced endomorphism $T$ (respectively $T^*$) of $J_X$
as the {\em Albanese} (respectively {\em Picard}) or
{\em covariant} (respectively {\em contravariant}) action of the correspondence
$(\pi_1,\pi_2)$. Since the definitions (\ref{JacHecke}) of $T$ and $T^*$ {\em both}
literally involve Albanese and Picard functoriality, we find this old terminology
confusing, and eschew it in favor of the consistent notation we have introduced.
}
Abusing notation, we will simply write $\pi$ for the endomorphism $\Alb(\pi)$ of $J_X$
induced by the correspondence $(1,\pi)$, and $\pi^*$ for the endomorphism $\Pic^0(\pi)$
induced by $(\pi,1) = (1,\pi)^*$. When $\pi:X\rightarrow X$ is an automorphism, an easy argument
shows that $\pi^* = \pi^{-1}$ as automorphisms of $J_X$.
With these definitions, the canonical filtration compatible isomorphism
$H^1_{\dR}(X/R) \simeq H^1_{\dR}(J_X/R)$ is $T$ (respectively $T^*$)-equivariant with respect to
the action (\ref{HeckeDef}) on $H^1_{\dR}(X/R)$ and the action on $H^1_{\dR}(J_X/R)$
induced by pullback along the endomorphisms (\ref{JacHecke}); see \cite[Proposition 5.4]{CaisNeron}.
Now suppose that $R$ is a discrete valuation ring with fraction field $K$
and fix a correspondence $T$ on $X$ given by a pair of finite morphisms
of normal curves $\pi_1,\pi_2:Y\rightrightarrows X$. Let us write $T_K$
for the induced correspondence on the (smooth) generic fiber $X_K$ of $X$.
Via (\ref{JacHecke}) and the N\'eron mapping property, $T_K$ and $T_K^*$
induces endomorphisms of the N\'eron model $J_X$ of the Jacobian of $X_K$,
which we simply denote by $T$ and $T^*$, respectively.
Thanks to Proposition \ref{intcompare}, the filtration compatible morphism
(\ref{IntegralComparisonMap}) is $T$- and $T^*$-equivariant for the given action (\ref{HeckeDef}) on $H^1(X/R)$
and the action on $\Lie\scrExtrig_R(J_X,\Gm)$ induced by (\ref{JacHecke}) and the
(contravariant) functoriality of $\scrExtrig_R(\cdot,\Gm)$.
\begin{remark}
As in Remark \ref{canonicalproperty}, if $X$ is a normal proper curve over $R$
with rational singularities, then any correspondence on $X_K$
induces a filtration compatible endomorphism of $H^1(X/R)$
via its action on $J_{X_K}$, the N\'eron mapping property, and
the isomorphism (\ref{IntegralComparisonMap}) of Proposition \ref{intcompare}.
\end{remark}
We now specialize this discussion to the case of the modular curve $X_1(Np^r)$ over $\Q$.
For any prime $\ell$, one defines the Hecke correspondences
$T_{\ell}$ for $\ell\nmid Np$ and $U_{\ell}$ for $\ell | Np$ on $X_1(Np^r)$
as in \cite[\S8]{pAdicShimura}
({\em cf.} also \cite[\S3]{tameness} and \cite[Chapter 2, \S5.1--5.8]{MW-Iwasawa},
though be aware that the latter works instead with the modular curves $X_1(Np^r)'$
of Remark \ref{genfiberrem}).
If $\ell\neq p$, we have similarly defined correspondences
$T_{\ell}$ and $U_{\ell}$ on $\Ig_r$ over $\mathbf{F}_p$ (see \cite[Chapter 2, \S5.4--5.5]{MW-Iwasawa}).
For $\ell\neq p$, the Hecke correspondences extend to correspondences on $\X_r$ over $R_r$,
essentially by the same definition, while for $\ell=p$ the correspondence $U_p:=(\pi_1,\pi_2)$
on $\X_r$ is defined using the maps (\ref{Upcorr}). We use the same symbols to
denote the induced endomorphisms (\ref{JacHecke}) of the Jacobian $J_1(Np^r)$.
\begin{definition}
We write $\H_r(\Z)$ (respectively $\H_r^*(\Z))$
for the $\Z$-subalgebra of $\End_{\Q}(J_1(Np^r))$ generated by the
Hecke operators $T_{\ell}$ (respectively $T_{\ell}^*$)
for $\ell\nmid Np$ and $U_{\ell}$ (respectively $U_{\ell}^*$) for $\ell | Np$,
and the diamond operators $\langle u\rangle$ (respectively $\langle u\rangle^*$)
for $u\in \Z_p^{\times}$ and
$\langle v\rangle_N$ (respectively $\langle v\rangle_N^*$) for $v\in (\Z/NZ)^{\times}$.
For any commutative ring $A$, we set $\H_r(A):=\H_r(\Z)\otimes_{\Z} A$
and $\H_r^*(A):=\H_r^*(\Z)\otimes_{\Z} A$, and for ease of notation
we set $\H_r:=\H_r(\Z_p)$ and $\H_r^*:=\H_r^*(\Z_p)$.
\end{definition}
The relation between the Hecke algebras $\H_r$ and $\H_r^*$ is explained by the following:
\begin{proposition}\label{AtkinInterchange}
Denote by $w_r$ the automorphism of $({J_r})_{K_r'}$ induced by the correspondence $(1,w_r)$
on $({X_r})_{K_r'}$ over $K_r'$. Viewing $\H_r$ and $\H_r^*$ as $\Z_p$-subalgebras
of $\End_{K_r'}(({J_r})_{K_r'})\otimes_{\Z}\Z_p$, conjugation by $w_r$ carries
$\H_r$ isomorphically onto $\H_r^*$: that is, $w_rT=T^*w_r$ for all Hecke operators $T$.
\end{proposition}
\begin{proof}
This is standard; see, e.g., \cite[pg. 336]{Tilouine}, \cite[2.1.8]{OhtaEichler},
or \cite[Chapter 2, \S5.6 (c)]{MW-Iwasawa}.
\end{proof}
\section{Differentials on modular curves in characteristic \texorpdfstring{$p$}{p}}\label{DiffCharp}
We now analyze the ``modified de Rham cohomology"
(\S\ref{GD}) of the special fibers of the modular
curves $\X_r/R_r$, and we relate its ordinary part to the de Rham cohomology of the ``Igusa Tower."
\subsection{The Cartier operator}
Fix a perfect field $k$ of characteristic $p > 0$ and write $\varphi:k\rightarrow k$ for the $p$-power Frobenius map.
In this section, we recall the basic theory of the Cartier operator
for a smooth and proper curve over $k$. As we will only need the theory
in this limited setting, we will content ourselves with a somewhat {\em ad hoc} formulation of it.
Our exposition follows \cite[\S10]{SerreTopology}, but
the reader may consult \cite[\S5.5]{Oda} or \cite{CartierNouvelle} for a more general treatment.
Let $X$ be a smooth and proper curve over $k$ and write $F:X\rightarrow X$
for the absolute Frobenius map;
it is finite and flat and is a morphism over the endomorphism of $\Spec(k)$ induced by $\varphi$.
Let $D$ be an effective Cartier (=Weil) divisor on $X$ over $k$, and write $\O_X(-D)$ for
the coherent (invertible) ideal sheaf determined by $D$.
The pullback map $F^*:\O_{X}\rightarrow {F}_*\O_{X}$ carries the ideal sheaf
$\O_X(-nD) \subseteq \O_{X}$ into ${F}_*\O_X(-npD)$, so we obtain a canonical $\varphi$-semilinear pullback map on cohomology
\begin{equation}
\xymatrix{
{F^*:H^1(X,\O_X(-nD))} \ar[r] & H^1(X,\O_X(-npD)).
}\label{Fpullback}
\end{equation}
By Grothendieck--Serre duality, (\ref{Fpullback}) gives a $\varphi^{-1}$-semilinear ``trace"
map\footnote{This map coincides with Grothendieck's trace morphism on dualizing sheaves
attached to the finite map $F$.} of $k$-vector spaces
\begin{equation}
\xymatrix{
{V:={F}_*:H^0(X,\Omega^1_{X/k}(npD))} \ar[r] & {H^0(X,\Omega^1_{X/k}(nD))}
}\label{cartier}
\end{equation}
\begin{proposition}\label{CartierOp}
Let $X/k$ be a smooth and proper curve, $D$ an effective Cartier divisor on $X$,
and $n$ a nonnegative integer.
\begin{enumerate}
\item There is a unique $\varphi^{-1}$-linear endomorphism $V:={F}_*$
of $H^0(X,\Omega^1_{X/k}(nD))$ which is dual, via Grothendieck-Serre
duality, to pullback by absolute Frobenius on $H^1(X,\O_X(-nD))$.
\label{CartierExists}
\item The map $V$ ``improves poles" in the sense that it factors through the canonical inclusion
\begin{equation*}
\xymatrix{
{H^0(X,\Omega^1_{X/k}(\lceil\frac{n}{p}\rceil D)} \ar@{^{(}->}[r] &
{H^0(X,\Omega^1_{X/k}(nD))}
}.
\end{equation*}
\label{CartierImproves}
\item If $\rho:Y\rightarrow X$ is any finite morphism of smooth proper curves over $k$,
and $\rho^*D$ is the pullback of $D$ to $Y$, then
the induced pullback and trace maps
\begin{equation*}
\xymatrix{
H^0(Y,\Omega^1_{Y/k}(n\rho^*D)) \ar@<0.5ex>[r]^-{\rho^*} &
\ar@<0.5ex>[l]^-{\rho_*} H^0(X,\Omega^1_{X/k}(nD))
}
\end{equation*}
commute with $V$.
\label{CartierCommutes}
\item Assume that $k$ is algebraically closed. Then
for any meromorphic differential $\eta$ on $X$ and any closed point $x$ of $X$, the formula
\begin{equation*}
\res_x(V\eta)^p = \res_x(\eta)
\end{equation*}
holds, where $\res_x$ is the canonical ``residue at $x$ map" on meromorphic
differentials.
\label{CartierResidue}
\end{enumerate}
\end{proposition}
\begin{proof}
Both (\ref{CartierExists}) and (\ref{CartierImproves}) follow from our discussion, while
(\ref{CartierCommutes}) follows (via duality) from the fact that the $p$-power
map commutes with any ring homomorphism. Finally, (\ref{CartierResidue}) follows from the fact
that the canonical isomorphism $H^1(X,\Omega^1_{X/k})\rightarrow k$ induced by the residue map coincides
with the negative of Grothendieck's trace isomorphism ({\em cf.}
Proposition \ref{Rosenlicht}), together with the fact that Grothendieck's
trace morphism is compatible with compositions; see Appendix B and Corollary 3.6.6 of \cite{GDBC}.
\end{proof}
\begin{remark}\label{poletrace}
Quite generally, if $\rho:Y\rightarrow X$ is any finite morphism of
smooth curves over $k$
and $y$ is any $k$-point of $Y$ with $x=\rho(y)\in X(k)$,
then for any meromorphic differential $\eta$ on $Y$ we have
\begin{equation}
\ord_x(\rho_*\eta) \le \left\lceil \frac{\ord_y(\eta)}{e}\right\rceil\label{poleimprovement}
\end{equation}
where $e$ is the ramification index of the extension of discrete valuation rings
$\O_{X,x}\rightarrow \O_{Y,y}$.
Indeed, if $\I_x$ and $\I_y$ denote the ideal sheaves of the reduced closed subschemes $x$ and $y$,
then the pullback map $\O_X\rightarrow \rho_*\O_Y$ carries $\I_x^n$ into $\rho_*\I_y^{ne}$.
Passing to the map on $H^1$'s and using Grothendieck duality,
we deduce that $\rho_*$ carries $H^0(Y,\Omega^1_{Y/k}\otimes\I_y^{-ne})$ into $
H^0(X,\Omega^1_X\otimes\I_x^{-n})$, whence the estimate (\ref{poleimprovement}).
If moreover $k$ is algebraically closed, then we have ({\em cf.} \cite[Theorem 4]{TateResidues})
\begin{equation}
\res_x(\rho_*\eta) = \res_y(\eta).
\end{equation}\label{TateFormula}
\end{remark}
We recall the following (generalization of a) well-known lemma of Fitting:
\begin{lemma}\label{HW}
Let $A$ be a commutative ring, $\varphi$ an automorphism of $A$,
and $M$ be an $A$-module equipped with a $\varphi$-semilinear
endomorphism $F:M\rightarrow M.$
Assume that one of the following holds:
\begin{enumerate}
\item $M$ is a finite length $A$-module.\label{finlen}
\item $A$ is a complete noetherian adic ring,\label{top}
with ideal of definition $I\subsetneq A$, and $M$ is a finite $A$-module.
\end{enumerate}
Then there is a unique direct sum decomposition
\begin{equation}
M = M^{F_{\ord}} \oplus M^{F_{\nil}},\label{FittingDecomp}
\end{equation}
where $M^{\varphi_{\ord}}$
is the maximal $\varphi$-stable
submodule of $M$ on which $F$ is bijective, and $M^{F_{\nil}}$
is the maximal $F$-stable submodule of $M$ on which $F$
is $($topologically$)$ nilpotent. The assignment $M\rightsquigarrow M^{F_{\star}}$
for $\star=\ord, \nil$ is an exact functor on the category of $($left$)$
$A[F]$-modules verifying $(\ref{finlen})$ or $(\ref{top})$.
\end{lemma}
\begin{proof}
For the proof in case (\ref{finlen}), we refer to \cite[\Rmnum{6}, 5.7]{LazardGroups},
and just note that one has:
\begin{equation*}
M^{F_{\ord}}:=\bigcap_{n\ge 0} \im(F^n)\quad\text{and}\quad
M^{F_{\nil}}:=\bigcup_{n\ge 0} \ker(F^n),
\end{equation*}
where one uses that $\varphi$ is an automorphism to know that the image and kernel
of $F^n$ are $A$-submodules of $M$.
It follows immediately from this that the association $M\rightsquigarrow M^{F_{\star}}$
is a functor from the category of left $A[F]$-modules of finite $A$-length
to itself. It is an exact functor because the canonical inclusion $M^{F_{\star}}\rightarrow M$
is an $A[F]$-direct summand.
In case (\ref{top}), our hypotheses ensure that $M/I^nM$ is a noetherian and Artinian
$A$-module, and hence of finite length, for all $n$. Our assertions in this situation
then follow immediately from (\ref{finlen}), via the uniqueness of (\ref{FittingDecomp}),
together with fact that $M$ is finite as an $A$-module, and hence $I$-adically complete (as $A$ is).
\end{proof}
We apply \ref{HW} to the $k$-vector space $M:=H^0(X,\Omega^1_{X/k})$
equipped with the $\varphi^{-1}$ semilinear map $V$:
\begin{definition}\label{ordnil}
The $k[V]$-module $H^0(X,\Omega^1_{X/k})^{V_{\ord}}$ is called the
{\em $V$-ordinary subspace} of holomorphic differentials on $X$. It is
the maximal $k$-subspace of $H^0(X,\Omega^1_{X/k})$ on which $V$ is bijective.
The nonnegative integer $\gamma_X:=\dim_k H^0(X,\Omega^1_{X/k})^{V_{\ord}}$
is called the {\em Hasse-Witt invariant} of $X$.
\end{definition}
\begin{remark}\label{DualityOfFVOrd}
Let $D$ be any effective Cartier divisor.
Since $V:={F}_*$ and $F:=F^*$ are adjoint under the canonical perfect
$k$-pairing between $H^0(X,\Omega^1_{X/k}(D))$ and $H^1(X,\O_X(-D))$, this pairing
restricts to a perfect duality pairing
\begin{equation}
\xymatrix{
{H^0(X,\Omega^1_{X/k}(D))^{V_{\ord}} \times H^1(X,\O_X(-D))^{F_{\ord}}} \ar[r] & {k}
}.\label{DualityOfFVOrdMap}
\end{equation}
In particular (taking $D=0$) we also have $\gamma_X=\dim_k H^1(X,\O_X)^{F_{\ord}}$.
\end{remark}
The following ``control lemma" is a manifestation of the fact that the Cartier
operator improves poles (Proposition \ref{CartierOp}, (\ref{CartierImproves})):
\begin{lemma}\label{sspoles}
Let $X$ be a smooth and proper curve over $k$ and $D$ an effective Cartier divisor
on $X$. Considering $D$ as a closed subscheme of $X$, we write $D_{\red}$ for associated
reduced closed subscheme.
\begin{enumerate}
\item For all positive integers $n$, the canonical morphism
\begin{equation*}
H^0(X,\Omega^1_{X/k}(D_{\red})) \rightarrow H^0(X,\Omega^1_{X/k}(nD))
\end{equation*}
induces a natural isomorphism on $V$-ordinary subspaces.\label{VControl}
\item For each positive integer $n$, the canonical map
\begin{equation*}
H^1(X, \O_X(-nD)) \rightarrow H^1(X,\O_X(-D_{\red}))
\end{equation*}
induces a natural isomorphism on $F$-ordinary subspaces.\label{FControl}
\item The identifications in $(\ref{VControl})$ and $(\ref{FControl})$ are canonically
$k$-linearly dual, via Remark $\ref{DualityOfFVOrd}$.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows immediately from Proposition \ref{CartierOp}, (\ref{CartierImproves})
and Remark \ref{DualityOfFVOrd}.
\end{proof}
Now let $\pi:Y\rightarrow X$ be a finite branched covering of smooth, proper and geometrically connected curves over $k$ with group $G$ that is a $p$-group.
Let $D_X$ be any effective Cartier divisor on $X$ over $k$ with support containing the ramification locus of $\pi$,
and put $D_Y=\pi^*D_X$. As in Lemma \ref{sspoles}, denote by $D_{X,\red}$ and $D_{Y,\red}$ the underlying reduced closed subschemes;
as $D_{Y,\red}$ is $G$-stable, the $k$-vector spaces
$H^0(Y,\Omega^1_{Y/k}(nD_{Y,\red}))$ and $H^1(Y,\O_Y(-nD_{Y,\red})$
are canonically $k[G]$-modules for any $n\ge 1$. The following theorem of Nakajima
is the key to the proofs of our structure theorems for $\Lambda$-modules:
\begin{proposition}[Nakajima]\label{Nakajima}
Assume that $\pi$ is ramified,
let $\gamma_X$ be the Hasse-Witt invariant of $X$ and set $d:=\gamma_X-1+\deg (D_{X,\red})$.
Then for each positive integer $n$:
\begin{enumerate}
\item The $k[G]$-module
$H^0(Y,\Omega^1_{Y/k}(nD_{Y,\red}))^{V_{\ord}}$ is free of rank $d$ and independent
of $n$.\label{NakajimaOne}
\item The $k[G]$-module $H^1(Y,\O_Y(-nD_{Y,\red}))^{F_{\ord}}$ is naturally
isomorphic to the contragredient of $H^0(Y,\Omega^1_{Y/k}(nD_{Y,\red}))^{V_{\ord}}$;
as such, it is $k[G]$-free of rank $d$ and independent of $n$.
\label{NakajimaTwo}
\end{enumerate}
\end{proposition}
\begin{proof}
The independence of $n$ is simply Lemma \ref{sspoles};
using this, the first assertion is then equivalent to Theorem 1 of \cite{Nakajima}.
The second assertion is immediate from Remark \ref{DualityOfFVOrd},
once one notes that for $g\in G$ one has the identity $g_*=(g^{-1})^*$
on cohomology (since $g_*g^*=\deg g = \id$), so $g^*$ and $(g^{-1})^*$
are adjoint under the duality pairing (\ref{DualityOfFVOrdMap}).
\end{proof}
We end this section with
a brief explanation of the relation between the
de Rham cohomology of $X$ over $k$ and the Dieudonn\'e module
of the $p$-divisible group of the Jacobian of $X$. This will allow
us to give an alternate description of the $V$-ordinary (respectively $F$-ordinary)
subspace of $H^0(X,\Omega^1_{X/k})$ (respectively $H^1(X,\O_X)$) which
will be instrumental in our applications.
Pullback by the absolute Frobenius gives a semilinear endomorphism
of the Hodge filtration $H(X/k)$ of $H^1_{\dR}(X/k)$
which we again denote by $F=F^*$.
Under the canonical autoduality of $H(X/k)$ provided by Proposition \ref{HodgeFilCrvk}
(\ref{HodgeDegenerationField}) ,
we obtain $\varphi^{-1}$-semilinear endomorphism
\begin{equation}
\xymatrix{
{V:={F}_*: H^1_{\dR}(X/k)} \ar[r] & {H^1_{\dR}(X/k)}
}\label{CartierOndR}
\end{equation}
whose restriction to $H^0(X,\Omega^1_{X/k})$ coincides with (\ref{cartier}).
Let $A$ be the ``Dieudonn\'e ring", {\em i.e.}~the (noncommutative if $k\neq \mathbf{F}_p$)
ring $A:=W(k)[F,V]$, where
$F$, $V$ satisfy $FV=VF=p$,
$F\alpha=\varphi(\alpha)F$, and $V\alpha=\varphi^{-1}(\alpha)V$ for all $\alpha\in W(k)$.
We view $H^1_{\dR}(X/k)$ as a left $A$-module
in the obvious way.
\begin{proposition}[Oda]\label{OdaDieudonne}
Let $J:=\Pic^0_{X/k}$ be the Jacobian of $X$ over $k$ and $G:=J[p^{\infty}]$
its $p$-divisible group. Denote by $\ensuremath{\mathbf{D}}(G)$ the contravariant
Dieudonn\'e crystal of $G$, so the Dieudonn\'e module $\ensuremath{\mathbf{D}}(G)_W$ is naturally a left $A$-module, finite and
free over $W:=W(k)$.
\begin{enumerate}
\item There are canonical isomorphisms of left $A$-modules
\begin{equation*}
H^1_{\dR}(X/k)\simeq \ensuremath{\mathbf{D}}(J)_{k}\simeq \ensuremath{\mathbf{D}}(G)_k.
\end{equation*}\label{OdaIsom}
\item For any finite morphism $\rho:Y\rightarrow X$ of smooth and proper curves
over $k$, the identification of $(\ref{OdaIsom})$ intertwines $\rho_*$
with $\ensuremath{\mathbf{D}}(\Pic^0(\rho))$ and $\rho^*$ with $\ensuremath{\mathbf{D}}(\Alb(\rho))$.\label{OdaIsomFunctoriality}
\item Let $G=G^{\et}\times G^{\mult}\times G^{\loc}$
be the canonical direct product decomposition of $G$ into its maximal \'etale,
multiplicative, and local-local subgroups.
Via the identification of $(\ref{OdaIsom})$, the canonical mappings
in the exact sequence $H(X/k)$ induce natural isomorphisms of left $A$-modules
\begin{equation*}
H^0(X,\Omega^1_{X/k})^{V_{\ord}} \simeq \ensuremath{\mathbf{D}}(G^{\mult})_k
\quad\text{and}\quad
H^1(X,\O_X)^{F_{\ord}} \simeq \ensuremath{\mathbf{D}}(G^{\et})_k
\end{equation*}\label{GetaleGmult}
\item The isomorphisms of $(\ref{GetaleGmult})$ are dual to each other, using the duality pairing
of Remark $\ref{DualityOfFVOrd}$
together with the canonical isomorphism $\ensuremath{\mathbf{D}}(G)_k^t\simeq \ensuremath{\mathbf{D}}(\Dual{G})_k$
and the autoduality of $G$ resulting from the autoduality of $J$.\label{BBMDuality}
\end{enumerate}
\end{proposition}
\begin{proof}
Using the characterizing properties of the Cartier operator
defined by Oda \cite[Definition 5.5]{Oda} and the explicit
description of the autoduality of $H^1_{\dR}(X/k)$ in terms
of cup-product and residues, one checks that the endomorphism
of $H^1_{\dR}(X/k)$ in \cite[Definition 5.6]{Oda} is adjoint
to $F^*$, and therefore coincides with the endomorphism
$V:={F}_*$ in (\ref{CartierOndR}); {\em cf.}
the proof of \cite[Proposition 9]{SerreTopology}.
We recall that one has a canonical isomorphism
\begin{equation}
H^1_{\dR}(X/k)\simeq H^1_{\dR}(J/k)\label{dRIdenJac}
\end{equation}
which is compatible with Hodge filtrations and duality (using the canonical
principal polarization to identify $J$ with its dual) and which, for
any finite morphism of smooth curves $\rho:Y\rightarrow X$ over $k$,
intertwines $\rho_*$ with $\Pic^0(\rho)^*$ and $\rho^*$ with $\Alb(\rho)^*$; see
\cite[Proposition 5.4]{CaisNeron}, noting that the proof given there works over any field $k$,
and {\em cf.}~Proposition \ref{intcompare}. It follows from these compatibilities
and the fact that the Cartier operator as defined in \cite[Definition 5.5]{Oda} is functorial
that the identification (\ref{dRIdenJac}) is moreover an isomorphism of left $A$-modules,
with the $A$-structure on $H^1_{\dR}(J/k)$ defined as in \cite[Definition 5.8]{Oda}.
Now by \cite[Corollary 5.11]{Oda} and \cite[Theorem 4.2.14]{BBM},
for any abelian variety $B$ over $k$,
there is a canonical isomorphism of left $A$-modules
\begin{equation}
H^1_{\dR}(B/k)\simeq \ensuremath{\mathbf{D}}(B)_k\label{AbVarDieuMod}
\end{equation}
Using the definition of this isomorphism in Proposition 4.2 and Theorem 5.10 of \cite{Oda},
it is straightforward (albeit tedious\footnote{Alternately,
one could appeal to
\cite{MM}, specifically to Chapter I, 4.1.7, 4.2.1, 3.2.3, 2.6.7
and to Chapter II, \S13 and \S15 (see especially Chapter II, 13.4 and 1.6).
See also \S2.5 and \S4 of \cite{BBM}.
})
to check that for any homomorphism $h:B'\rightarrow B$
of abelian varieties over $k$, the identification (\ref{AbVarDieuMod}) intertwines $h^*$ and $\ensuremath{\mathbf{D}}(h)$.
Combining (\ref{dRIdenJac}) and (\ref{AbVarDieuMod})
yields (\ref{OdaIsom}) and (\ref{OdaIsomFunctoriality}).
Now since $V={F}_*$ (respectively $F=F^*$) is the zero endomorphism of $H^1(X,\O_X)$
(respectively $H^0(X,\O_X)$), the canonical mapping
\begin{equation*}
\xymatrix{
{H^0(X,\Omega^1_{X/k})} \ar@{^{(}->}[r] & {H^1_{\dR}(X/k)\simeq \ensuremath{\mathbf{D}}(G)_k}
}
\quad\text{respectively}\quad
\xymatrix{
{\ensuremath{\mathbf{D}}(G)_k\simeq H^1_{\dR}(X/k)} \ar@{->>}[r] & {H^1(X,\O_X)}
}
\end{equation*}
induces an isomorphism on $V$-ordinary (respectively $F$-ordinary) subspaces.
On the other hand, by Dieudonn\'e theory one knows that
for {\em any} $p$-divisible group $H$, the semilinear endomorphism $V$
(respectively $F$) of $\ensuremath{\mathbf{D}}(H)_W$
is bijective if and only if $H$ is of multiplicative type (respectively \'etale).
The (functorial) decomposition $G=G^{\et}\times G^{\mult}\times G^{\loc}$
yields a natural isomorphism of left $A$-modules
\begin{equation*}
\ensuremath{\mathbf{D}}(G)_W\simeq \ensuremath{\mathbf{D}}(G^{\et})_W\oplus \ensuremath{\mathbf{D}}(G^{\mult})_W\oplus \ensuremath{\mathbf{D}}(G^{\loc})_W,
\end{equation*}
and it follows that the natural maps $\ensuremath{\mathbf{D}}(G^{\mult})_W\rightarrow \ensuremath{\mathbf{D}}(G)_W$,
$\ensuremath{\mathbf{D}}(G)_W\rightarrow \ensuremath{\mathbf{D}}(G^{\et})_W$ induce isomorphisms
\begin{equation}
\ensuremath{\mathbf{D}}(G^{\mult})_W \simeq \ensuremath{\mathbf{D}}(G)^{V_{\ord}}_W\quad\text{and}\quad
\ensuremath{\mathbf{D}}(G)^{F_{\ord}}_W\simeq \ensuremath{\mathbf{D}}(G^{\et})_W,\label{VordMultFordEt}
\end{equation}
respectively, which gives (\ref{GetaleGmult}). Finally, (\ref{BBMDuality})
follows from Proposition 5.3.13 and the proof of Theorem 5.1.8 in \cite{BBM},
using Proposition 2.5.8 of {\em op.~cit.}~and the compatibility of the isomorphism
(\ref{dRIdenJac}) with duality (for which see
\cite[Theorem 5.1]{colemanduality} and {\em cf.} \cite[Lemma 5.5]{CaisNeron}).
\end{proof}
\subsection{The Igusa tower}\label{IgusaTower}
We apply Proposition \ref{Nakajima} to the Igusa tower
(Definition \ref{IgusaDef}).
The canonical degeneracy map
$\rho:I_{r}\rightarrow I_1$ defined by (\ref{Vmapsch}) is
finite \'etale outside\footnote{We will frequently
write simply $\SS$ for the divisor $\SS_r$ on $I_r$
when $r$ is clear from context.
}
$\SS:=\SS_r$ and totally (wildly) ramified
over $\SS_1$, and so makes $I_r$ in to a branched cover
of $I_1$ with group $\Delta/\Delta_r$.
The cohomology groups $H^0(I_r,\Omega^1_{I_r/\mathbf{F}_p}(\SS))$
and $H^1(I_r,\O_{I_r}(-\SS))$ are therefore naturally
$\mathbf{F}_p[\Delta/\Delta_r]$-modules.
\begin{proposition}\label{IgusaStructure}
Let $r$ be a positive integer, write $\gamma$ for the $p$-rank of
$J_1(N)_{\mathbf{F}_p}$, and set $\delta:=\deg\SS$.
\begin{enumerate}
\item The $\mathbf{F}_p[\Delta/\Delta_r]$-modules $H^0(I_r,\Omega^1_{I_r/\mathbf{F}_p}(\SS))^{V_{\ord}}$
and $H^1(I_r,\O_{I_r}(-\SS))^{F_{\ord}}$ are both free of rank $d:=\gamma+\delta-1$.
Each is canonically isomorphic to the contragredient of the other.
\label{IgusaFreeness}
\item For any positive integer $s\le r$, the canonical trace mapping
associated to $\rho:I_r\rightarrow I_s$ induces natural isomorphisms of $\mathbf{F}_p[\Delta/\Delta_s]$-modules
\begin{subequations}
\begin{equation*}
\xymatrix{
{\rho_*:H^0(I_r, \Omega^1_{I_r/\mathbf{F}_p}(\SS))^{V_{\ord}}
\mathop{\otimes}\limits_{\mathbf{F}_p[\Delta/\Delta_r]} \mathbf{F}_p[\Delta/\Delta_s]} \ar[r]^-{\simeq} &
{H^0(I_s, \Omega^1_{I_s/\mathbf{F}_p}(\SS))^{V_{\ord}}}
}
\end{equation*}
\begin{equation*}
\xymatrix{
{\rho_*:H^1(I_r, \O_{I_r}(-\SS))^{F_{\ord}}
\mathop{\otimes}\limits_{\mathbf{F}_p[\Delta/\Delta_r]} \mathbf{F}_p[\Delta/\Delta_s]} \ar[r]^-{\simeq} &
{H^1(I_s, \O_{I_s}(-\SS))^{F_{\ord}}}
}
\end{equation*}
\end{subequations}
\label{IgusaControl}
\end{enumerate}
\end{proposition}
\begin{remark}
Using the moduli interpretation of $I_r$
and calculations on formal groups of universal elliptic curves,
one can show \cite[Lemma 12.9.3]{KM} that pullback induces a canonical identification
\begin{equation*}
\rho^*\Omega^1_{I_s/k}=\Omega^1_{I_r/k}(-p^{r-1}(p^r-p^s)\cdot\SS).
\end{equation*}
If $n$ is any positive integer, it follows easily from this that $\rho^*$ identifies
$H^0(I_s,\Omega^1_{I_s/k}(n\cdot\SS))$ with the $\Delta_s/\Delta_r$-invariant subspace of
$H^0(I_r, \Omega^1_{I_r/k}(-N_{r,s}(n)\cdot\SS))$, for
$N_{r,s}(n)=p^{r-1}(p^r-p^s)-p^{r-s}n$.
In particular, via pullback, $H^0(I_1,\Omega^1_{I_1/k}(p^r-p))$
is canonically identified with the $\Delta/\Delta_r$-invariant subspace
of $H^0(I_r,\Omega^1_{I_r/k})$, so the $k$-dimension of this subspace
grows {\em exponentially} with $r$. In this light, it is
remarkable that the $V$-ordinary subspace has controlled growth.
We will not use these facts in what follows, though see Remark \ref{FullIgusaStructure}.
\end{remark}
In order to prove Proposition \ref{IgusaStructure}, we require the following Lemma
({\em cf.} \cite[p. 511]{MW-Analogies}):
\begin{lemma}\label{MW}
Let $\pi: Y\rightarrow X$ be a finite flat and generically \'etale morphism of
smooth and geometrically irreducible curves over a field $k$.
If there is a geometric point of $X$ over which $\pi$ is totally ramified then the induced map
of $k$-group schemes $\Pic(\pi):\Pic_{X/k}\rightarrow \Pic_{Y/k}$ has trivial scheme-theoretic kernel.
\end{lemma}
\begin{proof}
The hypotheses and the conclusion are preserved under extension of $k$, so we may
assume that $k$ is algebraically closed. We fix a $k$-point $x\in X(k)$ over which
$\pi$ is totally ramified, and let $y\in Y(k)$ be the unique $k$-point of $Y$ over $x$.
To prove that $\Pic_{X/k}\rightarrow \Pic_{Y/k}$
has trivial kernel, it suffices to prove that the map of groups
$\pi_R^*:\Pic(X_R)\rightarrow \Pic(Y_R)$
is injective for every Artin local $k$-algebra $R$.
We fix such a $k$-algebra, and denote by $x_R\in X_R(R)$ and $y_R\in Y_R(R)$
the points obtained from $x$ and $y$ by base change. Let $\L$ be a line bundle
on $X_R$ whose pullback to $Y_R$ is trivial; our claim is that we may choose
a trivialization $\pi^*\L\xrightarrow{\simeq} \O_{Y_R}$ of $\pi^*\L$
over $Y_R$ which descends to $X_R$. In other words, by descent theory,
we assert that we may choose a trivialization of $\pi^*\L$ with the property that the
two pullback trivializations under the canonical projection maps
\begin{equation}
\xymatrix{
{Y_R \times_{X_R} Y_R} \ar@<-0.5ex>[r]_-{\rho_2}\ar@<0.5ex>[r]^-{\rho_1} & {Y_R}
}\label{TwoPullback}
\end{equation}
coincide.
We first claim that the $k$-scheme $Z:=Y\times_X Y$ is connected and generically reduced.
Since $\pi$ is totally ramified over $x$,
there is a unique geometric point $(y,y)$ of $Z$ mapping to $x$ under the canonical map
$Z\rightarrow X$. Since this map is moreover finite flat (because $\pi:Y\rightarrow X$
is finite flat due to smoothness of $X$ and $Y$), every connected component of $Z$ is finite flat
onto $X$ and so passes through $(y,y)$. Thus, $Z$ is connected.
On the other hand, $\pi:Y\rightarrow X$ is generically
\'etale by hypothesis, so there exists a dense open subscheme $U\subseteq X$
over which $\pi$ is \'etale. Then $Z\times_X U$ is \'etale---and hence smooth---over $U$
and the open immersion $Z\times_X U\rightarrow Z$ is schematically dense
as $U\rightarrow X$ is schematically dense and $\pi$ is finite
and flat. As $Z$ thus contains a $k$-smooth and dense subscheme,
it is generically reduced.
Fix a choice $e$ of $R$-basis of the fiber $\L(x_R)$ of $\L$ at $x_R$.
As any two trivializations of $\pi^*\L$ over $Y_R$ differ by an element of
$R^{\times}$, we may uniquely choose a trivialization which on $x_R$-fibers
\begin{equation}
\xymatrix{
{\L(x_R)\simeq \pi^*\L(y_R)}\ar[r]^-{\simeq} & {\O_{Y_R}(y_R)\simeq R}
}\label{TrivOnYunit}
\end{equation}
carries $e$ to $1$. The obstruction to the two pullback trivializations
under (\ref{TwoPullback}) being equal is a global unit on $Y_R\times_{X_R} Y_R$.
But since $Y_R\times_{X_R} Y_R = (Y\times_X Y)_R$, we have by flat base change
\begin{equation*}
H^0(Y_R\times_{X_R} Y_R, \O_{Y_R\times_{X_R} Y_R}) = H^0(Y\times_X Y,\O_{Y\times_X Y})\otimes_k R=R
\end{equation*}
where the last equality rests on the fact that $Y\times_X Y$ is connected, generically reduced,
and proper over $k$.
Thus, the obstruction to the two pullback trivializations being equal is an element of $R^{\times}$,
whose value may be calculated at any point of $Y_R\times_{X_R} Y_R$. By our choice (\ref{TrivOnYunit})
of trivialization of $\pi^*\L$, the value of this obstruction at the point $(y_R,y_R)$ is 1,
and hence the two pullback trivializations coincide as desired.
\end{proof}
\begin{proof}[Proof of Proposition $\ref{IgusaStructure}$]
Since $\rho:I_r\rightarrow I_s$ is a finite branched cover with group $\Delta_s/\Delta_r$
and totally wildly ramified over $\SS_s$,
we may apply Proposition \ref{Nakajima}, which gives (\ref{IgusaFreeness}).
To prove (\ref{IgusaControl}), we work over $k:=\o{\mathbf{F}}_p$ and
argue as follows. Since $\rho:I_r\rightarrow I_{s}$
is of degree $p^{r-s}$ and totally ramified over $\SS_{s}$, we have $\rho^*\SS_{s}=p^{r-s}\cdot\SS$;
it follows that pullback induces a map
\begin{equation}
\xymatrix{
{H^1(I_{s},\O_{I_{s}}(-\SS_{s}))} \ar[r]^-{\rho^*} & {H^1(I_r,\O_{I_r}(-\SS))}
}\label{pbH1poles}
\end{equation}
which we claim is {\em injective}.
To see this,
we observe that the long exact cohomology sequence attched to
the short exact sequence of sheaves on $I_r$
\begin{equation*}
\xymatrix{
0\ar[r] & {\O_{I_r}(-\SS)} \ar[r] & {\O_{I_r}} \ar[r] & {\O_{\SS}} \ar[r] & 0
}
\end{equation*}
(with $\O_{\SS}$ a skyscraper sheaf supported on $\SS$)
yields a commutative diagram with exact rows
\begin{equation}
\begin{gathered}
\xymatrix@C=15pt{
0\ar[r] & {H^0(I_{s},\O_{I_{s}})} \ar[r]\ar[d] &
{H^0(I_{s},\O_{\SS_{s}})} \ar[r]\ar[d] &
{H^1(I_{s},\O_{I_{s}}(-\SS_{s}))} \ar[r]\ar[d] &
{H^1(I_{s},\O_{I_{s}})} \ar[r]\ar[d] & 0\\
0\ar[r] & {H^0(I_r,\O_{I_r})} \ar[r] &
{H^0(I_r,\O_{\SS})} \ar[r] & {H^1(I_r,\O_{I_r}(-\SS))} \ar[r] &
{H^1(I_r,\O_{I_r})} \ar[r] & 0
}
\end{gathered}
\label{pbH1inj}
\end{equation}
The left-most vertical arrow are is an isomorphism because $I_r$ is geometrically connected for all $r$.
Since $\SS$ is reduced, we have
$H^0(I_r,\O_{\SS})=k^{\deg\SS}$
for all $r$, so since $\rho:I_r\rightarrow I_{s}$ totally ramifies over every
supersingular point, the second left-most vertical arrow in
(\ref{pbH1inj}) is also an isomorphism.
Now the rightmost vertical map in (\ref{pbH1inj}) is
identified with the map on Lie algebras
${\Lie \Pic^0_{I_{s}/k}} \rightarrow {\Lie\Pic^0_{I_r/k}}$
induced by $\Pic^0(\rho)$,
which is injective thanks to Lemma \ref{MW} and the left-exacness of the functor $\Lie$.
An easy diagram chase
using (\ref{pbH1inj}) then shows that (\ref{pbH1poles}) is injective, as claimed.
Using again the equality $\rho^*(\SS_s)=p^{r-s}\cdot\SS_r$,
pullback of meromorphic differentials yields a mapping
\begin{equation}
\xymatrix{
{H^0(I_s,\Omega^1_{I_s/k}(\SS))} \ar[r] & {H^0(I_r, \Omega^1_{I_r/k}(p^{r-s}\cdot\SS))}
}\label{diffPBinj}
\end{equation}
which is injective since $\rho:I_r\rightarrow I_s$ is separable.
Dualizing the injective maps (\ref{pbH1poles}) and (\ref{diffPBinj}), we see that the canonical
trace mappings
\begin{subequations}
\begin{equation}
\xymatrix{
{H^0(I_r,\Omega^1_{I_r/k}(\SS))} \ar[r]^-{\rho_*} &
{H^0(I_{s},\Omega^1_{I_{s}/k}(\SS))}
}
\end{equation}
\begin{equation}
\xymatrix{
{H^1(I_r,\O_{I_r}(-p^{r-s}\cdot \SS))} \ar[r]^-{\rho_*} & {H^1(I_{s},\O_{I_{s}}(-\SS))}
}
\end{equation}
\end{subequations}
are surjective for all $r\ge s\ge 1$.
Passing to $V$- (respectively $F$-) ordinary parts and using
Lemma \ref{sspoles} (\ref{VControl}), we conclude that the canonical trace
mappings attached to $I_r\rightarrow I_s$ induce {\em surjective} maps
as in Proposition \ref{IgusaStructure} (\ref{IgusaControl}). By (\ref{IgusaFreeness}),
these mappings are then surjective
mappings of free $\mathbf{F}_p[\Delta/\Delta_s]$-modules of the same rank, and are hence
isomorphisms.
\end{proof}
\begin{remark}\label{FullIgusaStructure}
If $G$ is any cyclic group of $p$-power order, then the
representation theory of $G$ is rather easy, even over a field $k$ of characteristic $p$.
Denoting by $\gamma$
any fixed generator of $G$, for each integer $d$ with $1\le d\le \#G$, there is a unique
indecomposable representation of $G$ of dimension $d$, given explicitly by the
$k[G]$-module $V_d:=k[G]/(\gamma-1)^d$.
By using Artin-Schreier theory for a $G$-cover of proper smooth curves $Y\rightarrow X$ over $k$,
for any $G$-stable Cartier divisor $D$ on $Y$ it is possible to determine
the multiplicity of $V_d$ in the $k[G]$-module $H^0(Y,\Omega^1_{Y/k}(D))$ purely in terms
of the ramification data of $Y\rightarrow X$. This is carried out for $D=\emptyset$
in \cite{ValentiniMadan}. For the $G:=\Delta/\Delta_r$-cover $I_{r}\rightarrow I_1$, one finds
\begin{equation*}
H^0(I_r,\Omega^1_{I_r/k})\simeq k[G]^{g(I_1)} \oplus \left(k[G]/(\gamma-1)^{p^{r-1}-1}\right)^{p(\deg\SS_1)-1}
\oplus\bigoplus_{d=1}^{p^{r-1}-2} \left(k[G]/(\gamma-1)^{d}\right)^{p(\deg\SS_1)}
\end{equation*}
as $k[G]$-modules, where $g(I_1)$ is the genus of $I_1$.
\end{remark}
The space of meromorphic differentials $H^0(I_1,\Omega^1_{I_1/\mathbf{F}_p}(\SS))$
has a natural action of $\mathbf{F}_p^{\times}$ via the diamond operators $\langle \cdot\rangle$,
and the eigenspaces for this action are intimitely connected with mod $p$ cusp forms:
\begin{proposition}\label{MFmodp}
Let $S_k(N;\mathbf{F}_p)$ be the space of weight $k$ cuspforms for $\Gamma_1(N)$ over $\mathbf{F}_p$,
and denote by $H^0(I_r,\Omega^1_{I_1/\mathbf{F}_p}(\SS))(k-2)$ the subspace
of $H^0(I_r,\Omega^1_{I_1/\mathbf{F}_p}(\SS))$ on which $\mathbf{F}_p^{\times}$ acts through the
character $\langle u \rangle\mapsto u^{k-2}$.
For each $k$ with $ 2 < k < p+1$, there are canonical isomorphisms
of $\mathbf{F}_p$-vector spaces
\begin{equation}
\addtocounter{equation}{1}
S_k(N;\mathbf{F}_p) \simeq H^0(I_1,\Omega^1_{I_1/\mathbf{F}_p})(k-2) \simeq H^0(I_1,\Omega^1_{I_1/\mathbf{F}_p}(\SS))(k-2)
\tag{$\arabic{section}.\arabic{subsection}.\arabic{equation}_k$}
\label{WtkIsom}
\end{equation}
which are equivariant for the Hecke operators, with $U_p$ acting as usual on modular forms
and as the Cartier operator $V$
on differential forms. For $k=2,$ $p+1$, we have the following commutative diagram:
\begin{equation*}
\xymatrix{
{S_2(N;\mathbf{F}_p)} \ar[r]^-{\simeq} \ar@{^{(}->}[d]_-{\cdot A} &
{H^0(I_1,\Omega^1_{I_1/\mathbf{F}_p})(0)} \ar@{^{(}->}[d] \\
{S_{p+1}(N;\mathbf{F}_p)} \ar[r]^-{\simeq} & {H^0(I_1,\Omega^1_{I_1/\mathbf{F}_p}(\SS))(0)} \\
}
\end{equation*}
where $A$ is the Hasse invariant.
\end{proposition}
\begin{proof}
This follows from Propositions 5.7--5.10 of \cite{tameness}, using Lemma \ref{CharacterSpaces};
we note that our forward reference to Lemma \ref{CharacterSpaces} does not result in circular reasoning.
\end{proof}
\begin{remark}\label{dMFmeaning}
For each $k$ with $2\le k\le p+1$, let us write $d_k:=\dim_{\mathbf{F}_p} S_k(N;\mathbf{F}_p)^{\ord}$
for the $\mathbf{F}_p$-dimension of the subspace of weight $k$ level $N$ cuspforms over $\mathbf{F}_p$
on which $U_p$ acts invertibly.
As in Proposition \ref{IgusaStructure} (\ref{IgusaFreeness}),
let $\gamma$ be the $p$-rank of the Jacobian of
$X_1(N)_{\mathbf{F}_p}$ and $\delta:=\deg\SS$. It follows immediately from Proposition
\ref{MFmodp} that we have the equality
\begin{equation}
d :=\gamma+\delta-1 = \sum_{k=3}^{p+1} d_k.
\end{equation}
\end{remark}
\subsection{Structure of the ordinary part of \texorpdfstring{$H^0(\o{\X}_r,\omega_{\o{\X}_r/\mathbf{F}_p})$}{
H0(Xr,omega)}
}\label{OrdStruct}
Keep the notation of \S\ref{IgusaTower} and let $\X_r/R_r$ be as in Definition \ref{XrDef}.
As before, we denote by $\o{\X}_r:=\X_r\times_{R_r} \mathbf{F}_p$ the special fiber of $\X_r$; it is a
curve over $\mathbf{F}_p$ in the sense of Definition \ref{curvedef}.
In this section, using Rosenlicht's theory of the dualizing sheaf as explained in \S\ref{GD}
and the explicit description of $\o{\X}_r$ given by Proposition \ref{redXr},
we will compute the {\em ordinary part} of the cohomology $H(\o{\X}_r/\mathbf{F}_p)$
in terms of the de Rham cohomology of the Igusa tower.
For notational ease, as in Remark \ref{MWGood} we write $I_r^{\infty}:=I_{(r,0,1)}$ and
$I_r^0:=I_{(0,r,1)}$ for the two ``good" components of $\o{\X}_r$.
Each of these components is abstractly isomorphic to the Igusa curve $\Ig(p^r)$ of level $p^r$ over
$X_1(N)_{\mathbf{F}_p}$, and we will henceforth make this identification; for $s\le r$, we will
write simply $\rho:I_r^{\star}\rightarrow I_s^{\star}$ for the
the canonical degeneracy map induced by (\ref{Vmapsch}).
Using Proposition \ref{UlmerProp}, one checks that
the $\H_r$-correspondences on $\X_r$ restrict to the $\H_r$-correspondences on $I_r^{\infty}$,
(the point is that the degeneracy maps defining $U_p$ on $\X_r$ restrict to a correspondence
on $I_r^{\infty}$), while the $\H_r^*$-correspondences
on $\X_r$ restrict to the $\H_r^*$-correspondences on $I_r^{0}$.
In particular, $U_p=(F,\langle p\rangle_N)$ on $I_r^{\infty}$ and $U_p^*=(F,\id)$
on $I_r^0$.
For $\star=0,\infty$, we denote by $i_r^{\star}: I_r^{\infty}\hookrightarrow \o{\X}_r$ the canonical closed immersion.
\begin{proposition}\label{charpord}
For each positive integer $r$, pullback of differentials along
$i_r^{0}$ $($respectively $i_r^{\infty}$$)$ induces a natural and $\H_r^*$ $($resp.
$\H_r$$)$-equivariant isomorphism of $\mathbf{F}_p[\Delta/\Delta_r]$-modules
\begin{equation}
\xymatrix{
{e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r})} \ar[r]^-{\simeq}_-{(i_r^{0})^*} &
{H^0(I_r^{0},\Omega^1_{I_r^{0}}(\SS))^{V_{\ord}}}
},
\ \text{resp.}\
\xymatrix{
{e_rH^0(\o{\X}_r,\omega_{\o{\X}_r})} \ar[r]^-{\simeq}_-{(i_r^{\infty})^*} &
{H^0(I_r^{\infty},\Omega^1_{I_r^{\infty}}(\SS))^{V_{\ord}}}
}.
\label{I'compIsom}
\end{equation}
which is $\Gamma$-equivariant for the ``geometric inertia action" $(\ref{gammamaps})$
on $\o{\X}_r$ and the action $\gamma\mapsto \langle \chi(\gamma)\rangle^{-1}$ on $I_r^0$
$($respectively the trivial action on $I_r^{\infty}$$)$.
The isomorphisms $(\ref{I'compIsom})$ induce identifications that are
compatible with change in $r$:
the four diagrams formed
by taking
the interior or the exterior arrows
\begin{equation}
\begin{gathered}
\xymatrix@R=30pt{
{e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r})}
\ar@<-0.5ex>[r]_-{\langle p\rangle_N^{r} (i_r^0)^*}
\ar@<0.5ex>[r]^-{F_*^r (i_r^0)^*} \ar@<-0.5ex>[d]_-{\rho_*} &
{H^0(I_r^{0},\Omega^1_{I_r^{0}}(\SS))^{V_{\ord}}} \ar@<0.5ex>[d]^-{\rho_*} \\
{e_{s}^*H^0(\o{\X}_{s},\omega_{\o{\X}_{s}})}
\ar@<-0.5ex>[r]_-{F_*^s (i_s^0)^*}
\ar@<0.5ex>[r]^-{\langle p\rangle_N^{s} (i_s^0)^*}
\ar@<-0.5ex>[u]_-{\sigma^*} &
{H^0(I_s^{0},\Omega^1_{I_s^{0}}(\SS))^{V_{\ord}}} \ar@<0.5ex>[u]^-{\rho^*}
}
\quad\raisebox{-24pt}{and}\quad
\xymatrix@R=30pt{
{e_rH^0(\o{\X}_r,\omega_{\o{\X}_r})} \ar@<-0.5ex>[r]_-{(i_r^{\infty})^*}
\ar@<0.5ex>[r]^-{F_*^r (i_r^{\infty})^*}
\ar@<-0.5ex>[d]_-{\sigma_*} &
{H^0(I_r^{\infty},\Omega^1_{I_r^{\infty}}(\SS))^{V_{\ord}}} \ar@<0.5ex>[d]^-{\rho_*} \\
{e_{s}H^0(\o{\X}_{s},\omega_{\o{\X}_{s}})} \ar@<-0.5ex>[r]_-{F_*^s (i_s^{\infty})^*}
\ar@<0.5ex>[r]^-{(i_s^{\infty})^*}
\ar@<-0.5ex>[u]_-{\rho^*} &
{H^0(I_s^{\infty},\Omega^1_{I_s^{\infty}}(\SS))^{V_{\ord}}} \ar@<0.5ex>[u]^-{\rho^*}
}
\end{gathered}\label{rCompatDiagrams}
\end{equation}
are all commutative for $s\le r$.
Via the automorphism $\o{w}_{r}$ of $\o{\X}_r$ and the identification
$I_{r}^0\simeq \Ig(p^r)\simeq I_{r}^{\infty}$, the first diagram of $(\ref{rCompatDiagrams})$
is carried isomorphically and compatibly on to the second.
The same assertions hold true if we replace $\o{\X}_r$ with $\nor{\o{\X}}_r$ and $\Omega^1_{I_r^{\star}}(\SS)$
with $\Omega^1_{I_r^{\star}}$ throughout.
\end{proposition}
\begin{proof}
We may and do work over $k:=\o{\mathbf{F}}_p$, and we abuse notation slightly by
writing $\o{\X}_r$ for the {\em geometric} special fiber of $\X_r$.
If $X$ is an $\mathbf{F}_p$-scheme, we likewiseagain write $X$ it's base change to $k$,
and we write $F:X\rightarrow X$ for the base change of the absolute Frobenius of
$X$ over $\mathbf{F}_p$ to $k$.
Let $\nor{\o{\X}}_r\rightarrow \o{\X}_r$ be the
normalization map; by Proposition \ref{redXr}, we know that
$\nor{\o{\X}}_r$ is the disjoint union of proper smooth and irreducible
Igusa curves $I_{(a,b,u)}$ indexed by triples $(a,b,u)$ with
with $a,b$ nonnegative integers satisfying $a+b=r$ and
$u\in(\Z/p^{\min(a,b)}\Z)^{\times}$. Via Proposition \ref{Rosenlicht},
we identify $\omega_{\o{\X}_r/k}$ with Rosenlicht's sheaf $\omega_{\o{\X}_r/k}^{\reg}$ of
regular differentials, and we simply write $\omega_{\o{\X}_r}$ for this sheaf.
By Definition \ref{OmegaReg} and Remark \ref{OmegaRegMero}, we have a functorial
injection of $k$-vector spaces
\begin{equation}
\xymatrix{
{H^0(\o{\X}_r,\omega_{\o{\X}_r})}\ar@{^{(}->}[r] &
{H^0(\nor{\o{\X}}_r,\underline{\Omega}^1_{k(\nor{\o{\X}}_r)})\simeq
\prod\limits_{(a,b,u)}\Omega^1_{k(I_{(a,b,u)})}}
}\label{dualizing2prod}
\end{equation}
with image precisely those elements $(\eta_{(a,b,u)})$ of the product that satisfy
$\sum \res_{x_{(a,b,u)}}(s\eta_{(a,b,u)}) =0$
for each supersingular point $x\in \o{\X}_r(k)$ and all $s\in \O_{\o{\X}_r,x}$, where
$x_{(a,b,u)}$ is the unique point of $I_{(a,b,u)}$ lying
over $x$ and the sum is over all triples $(a,b,u)$ as above.
We henceforth identify $\eta\in H^0(\o{\X}_r,\omega_{\o{\X}_r})$ with its image
under (\ref{dualizing2prod}), and we denote by
$\eta_{(a,b,u)}$ the $(a,b,u)$-component of $\eta$.
Recall from (\ref{HeckeDef})
that the correspondence $U_p:=(\pi_1,\pi_2)$
on $\X_r$ given by the
degeneracy maps $\pi_1,\pi_2:\mathcal{Y}_r\rightrightarrows \X_r$ of (\ref{Upcorr})
yields endomorphisms $U_p:=(\pi_1)_*\circ\pi_2^*$ and $U_p^*:=(\pi_2)_*\circ\pi_1^*$
of $H^0(\X_r,\omega_{\X_r/R_r})$;
we will again denote by $U_p$ and $U_p^*$ the induced endomorphisms $U_p\otimes 1$ and $U_p^*\otimes 1$ of
\begin{equation*}
H^0(\o{\X}_r,\omega_{\o{\X}_r}) \simeq H^0(\X_r,\omega_{\X_r/R_r})\otimes_{R_r} k,
\end{equation*}
where the isomorphism is
the canonical one of Lemma \ref{ReductionCompatibilities} (\ref{BaseChngDiagram}).
By the functoriality of normalization,
we have an induced correspondence $U_p:=(\nor{\o{\pi}}_1,\nor{\o{\pi}}_2)$
on $\nor{\o{\X}}_r$,
and we write $U_p$ and $U_p^*$ for the resulting endomorphisms (\ref{HeckeDef})
of $H^0(\nor{\o{\X}}_r,\underline{\Omega}^1_{k(\nor{\o{\X}}_r)})$.
By Lemma \ref{ReductionCompatibilities} (\ref{PTBCCompat}),
the map (\ref{dualizing2prod}) is then $U_p$ and $U_p^*$-equivariant.
The Hecke correspondences away from $p$
and the diamond operators act on the source of (\ref{dualizing2prod})
via ``reduction modulo $p$" and on the target via the induced
correspondences in the usual way (\ref{HeckeDef}), and
the map (\ref{dualizing2prod}) compatible with these
actions
thanks to Lemma
\ref{ReductionCompatibilities} (\ref{PTBCCompat}). Similarly,
the semilinear ``geometric inertia" action of $\Gamma:=\Gal(K_{\infty}/K_0)$
on $\X_r$ induces a linear action on $\nor{\o{\X}}_r$ as in Proposition
\ref{AtkinInertiaCharp} (\ref{InertiaCharp}), and the map (\ref{dualizing2prod}) is equivariant
with respect to these actions.
We claim that for {\em any} meromorphic differential $\eta = (\eta_{(a,b,u)})$ on $\nor{\o{\X}}_r$, we
have
\begin{subequations}
\begin{equation}
\left({U_p}\eta \right)_{(a,b,u)} = \begin{cases}
F_*\eta_{(r,0,1)} &:\quad (a,b,u)=(r,0,1)\\
\rho_*\eta_{(a+1,b-1,u)} &:\quad 0 < b \le a \\
\sum\limits_{\substack{u'\in (\Z/p^{a+1}\Z)^{\times} \\ u'\equiv u\bmod p^{a}}}
\langle u'\rangle \eta_{(a+1,b-1,u)}
&:\quad r\ \text{odd},\ a=b-1\\
\sum\limits_{\substack{u'\in (\Z/p^{a+1}\Z)^{\times} \\ u'\equiv u\bmod p^{a}}}
\rho^*\langle u'\rangle \eta_{(a+1,b-1,u')}
&:\quad r\ \text{even},\ a=b-2\\
\sum\limits_{\substack{u'\in (\Z/p^{a+1}\Z)^{\times}\\ u'\equiv u\bmod p^a}}
\rho^*\eta_{(a+1,b-1,u')} &:\quad 0 \le a < b-2\\
\end{cases}\label{Up1}
\end{equation}
The proof of this claim is an easy exercise using the definition of $U_p$, the explicit description of
the maps $\nor{\o{\pi}}_1$ and $\nor{\o{\pi}}_2$
given in Proposition \ref{UlmerProp}, and the fact that $F^*$ kills any global
meromorphic differential form on a scheme of characteristic $p$.
In a similar manner, one derives the explicit description
\begin{equation}
\left(U_p^*\eta \right)_{(a,b,u)} = \begin{cases}
\langle p\rangle_N^{-1}F_*\eta_{(0,r,1)} &:\quad (a,b,u)=(0,r,1)\\
\rho_*\eta_{(a-1,b+1,u)} &:\quad 0 < a < b \\
\langle u\rangle^{-1}\rho_*\eta_{(a-1,b+1,u)} &:\quad r\ \text{even},\ b=a\\
\sum\limits_{\substack{u'\in (\Z/p^{b+1}\Z)^{\times} \\ u'\equiv u\bmod p^{b}}}
\langle u'\rangle^{-1} \eta_{(a-1,b+1,u')}
&:\quad r\ \text{odd},\ b=a-1\\
\sum\limits_{\substack{u'\in (\Z/p^{b+1}\Z)^{\times}\\ u'\equiv u\bmod p^b}}
\rho^*\eta_{(a-1,b+1,u')} &:\quad 0 \le b < a-1\\
\end{cases}\label{Up2}
\end{equation}
\end{subequations}
The crucial observation for our purposes is that for $0 < b \le r$, the $(a,b,u)$-component of
${U_p}\eta$ depends only on the $(a+1,b-1,u')$-components of $\eta$ for varying $u'$, and similarly
for $0<a\le r$ the $(a,b,u)$-component of $U_p^*\eta$ depends only on the $(a-1,b+1,u')$-components
of $\eta$.
By induction, we deduce
\begin{subequations}
\begin{equation}
\left(U_p^n\eta \right)_{(a,b,u)} = \begin{cases}
\rho_*^b F_*^{n-b}\eta_{(r,0,1)} &:\quad b \le a \\
\sum\limits_{\substack{u'\in (\Z/p^{b}\Z)^{\times}\\ u'\equiv u\bmod p^a}}
\langle u'\rangle \rho_*^a F_*^{n-b}\eta_{(r,0,1)} &:\quad a < b
\end{cases}\label{Upn1}
\end{equation}
and
\begin{equation}
\left({U_p^*}^n\eta \right)_{(a,b,u)} = \begin{cases}
\rho_*^a\langle p\rangle_N^{a-n}F_*^{n-a}\eta_{(0,r,1)} &:\quad a < b \\
\sum\limits_{\substack{u'\in (\Z/p^{a}\Z)^{\times}\\ u'\equiv u\bmod p^b}}
\langle u'\rangle^{-1} \rho_*^b\langle p\rangle_N^{a-n} F_*^{n-a}\eta_{(0,r,1)} &:\quad b \le a
\end{cases}\label{Upn2}
\end{equation}
\end{subequations}
for any $n\ge r\ge 1$.
For any $r>0$ and for $\star=\infty, 0$ we define maps
\begin{align*}
\xymatrix{
{\gamma_r^{\star}: H^0(I_r^{\star},\Omega^1_{I_r^{\star}}(\SS))^{V_{\ord}}} \ar[r] &
{H^0(\nor{\o{\X}}_r,\underline{\Omega}^1_{k(\nor{\o{\X}}_r)})}
}
\end{align*}
by
\begin{subequations}
\begin{equation}
(\gamma_{r}^{\infty}(\eta))_{(a,b,u)}: = \begin{cases}
\rho_*^bF_*^{-b}\eta &:\quad b\le a\\
\sum\limits_{\substack{u'\in (\Z/p^{b}\Z)^{\times}\\ u'\equiv u\bmod p^a}}
\langle u'\rangle \rho_*^a F_*^{-b}\eta &:\quad a < b \\
\end{cases}\label{betadef}
\end{equation}
and
\begin{equation}
(\gamma_{r}^{0}(\eta))_{(a,b,u)}: = \begin{cases}
\rho_*^a\langle p\rangle_N^{a}F_*^{-a}\eta &:\quad a < b\\
\sum\limits_{\substack{u'\in (\Z/p^{a}\Z)^{\times}\\ u'\equiv u\bmod p^b}}
\langle u'\rangle^{-1} \rho_*^b \langle p\rangle_N^{a} F_*^{-a}\eta_{(0,r,1)} &:\quad b \le a
\end{cases}\label{gammadef}
\end{equation}
\end{subequations}
These maps are well-defined because $F_*=V$ is invertible on the $V$-ordinary subspace,
and they are immediately seen to be injective by looking at $(r,0,1)$-components.
Note moreover that the $(a,b,u)$-component of $\gamma_r^{\star}(\eta)$ is independent of $u$.
We claim that
the maps $\gamma_r^{\star}$ have image in
$H^0(\nor{\o{\X}}_r,\omega_{\nor{\o{\X}}_r})$ (i.e. that they factor through (\ref{dualizing2prod})).
To see this, we proceed as follows.
Suppose that $x$ is any supersingular point on $\o{\X}_r$ and $s\in \O_{\o{\X}_r,x}$ is arbitrary.
By Proposition \ref{Rosenlicht} and Definition \ref{OmegaReg},
we must check that the sum of the residues of $s\gamma^{\infty}(\eta)$
at all $k$-points of $\nor{\o{\X}}_r$ lying over $x$ is zero.
Using (\ref{betadef}), we calculate that this sum is equal to
\begin{align}
\sum_{b \le a}
\sum_{u\in (\Z/p^b\Z)^{\times}}\res_{x_{(a,b,u)}}(s\rho_*^bF_*^{-b}\eta)
+\sum_{a < b} \sum_{u\in (\Z/p^b\Z)^{\times}}
\res_{x_{(a,b,u)}}(s\langle u\rangle \rho_*^aF_*^{-b}\eta)
\label{residuecalc1}
\end{align}
where $x_{(a,b,u)}$ denotes the unique point of the $(a,b,u)$-component of $\nor{\o{\X}}_r$ over $x$,
and the outer sums range over all nonnegative integers $a,b$ with $a+b=r$.
We claim that for any meromorphic differential $\omega$ on $I_{(a,b,u)}$ and any
supersingular point $y$ of $I_{(a,b,u)}$ over $x$, we have
\begin{subequations}
\begin{equation}
\res_y(\omega) = \res_y(\langle u\rangle\omega)\label{DiamondFix}
\end{equation}
for all $u\in \Z_p^{\times}$, and, if in addition $\omega$ is $V$-ordinary,
\begin{equation}
\res_y(s\omega) = s(x)\res_y(\omega)\label{FunctionPullOut}
\end{equation}
\end{subequations}
Indeed, (\ref{DiamondFix}) is a consequence of (\ref{TateFormula}),
using the fact that the automorphism
$\langle u\rangle$ of $I_{(a,b,u)}$ fixes every supersingular point,
while (\ref{FunctionPullOut}) is deduced by thinking about formal expansions of differentials
at $y$ and using the fact that a $V$-ordinary meromorphic differential has at worst simple poles
thanks to Lemma \ref{sspoles}. Via (\ref{DiamondFix})--(\ref{FunctionPullOut}), we reduce the
sum (\ref{residuecalc1}) to
\begin{align}
\sum_{a + b = r}
\sum_{u\in (\Z/p^b\Z)^{\times}} s(x)\res_{x_{(a,b,u)}}(\rho_*^{\min(a,b)}F_*^{-b}\eta)
&= \sum_{a + b = r}
\varphi(p^b)s(x)\res_{x_{(a,b,1)}}(\rho_*^{\min(a,b)}F_*^{-b}\eta)\nonumber\\
&=s(x)\res_{x_{(r,0,1)}}(\eta) - s(x)\res_{x_{(r,0,1)}}(F_*^{-1}\eta)
\label{twoterm}
\end{align}
where the first equality above follows from the fact
that for {\em fixed} $a,b$, the points $x_{(a,b,u)}$ for varying $u\in (\Z/p^{\min(a,b)}\Z)^{\times}$
are all identified with the {\em same} point on $\Ig(p^{\max(a,b)})$, and the second
equality is a consequence of (\ref{TateFormula}), since $\rho(x_{(r,0,1)})=x_{(r-1,1,1)}$.
As $\eta$ is $V$-ordinary, there exists a $V$-ordinary meromorphic differential $\xi$
on $I_r^0$ with $\eta=F_*\xi$; substituting this expression for $\eta$ in to (\ref{twoterm})
and applying (\ref{TateFormula}) once more, we conclude that (\ref{twoterm}) is zero,
as desired.
That $\gamma_r^{0}$ has image in $H^0(\o{\X}_r,\omega_{\o{\X}_r/k})$
follows from a nearly identical calculation, and we omit the details.
It follows immediately from our calculations (\ref{Up1})--(\ref{Up2}) and the definitions
(\ref{betadef})--(\ref{gammadef}) that
the relations $U_p\circ \gamma_r^{\infty}=\gamma_r^{\infty}\circ F_*$
and ${U_p^*}\circ\gamma^0_r=\gamma_r^0\circ \langle p\rangle_N^{-1}F_*$ hold.
Since $F_*$ is invertible on the source of $\gamma_r^{\star}$,
it follows immediately
that $\gamma_r^0$ has image
contained in $e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r})$ and that
$\gamma_r^{\infty}$ has image contained in
$e_rH^0(\o{\X}_r,\omega_{\o{\X}_r})$.
To see that these containments an equalities, we proceed as follows.
Suppose that $\xi\in e_rH^0(\o{\X}_r,\omega_{\o{\X}_r})$ is arbitrary. We claim that
the meromorphic differential $\xi_{(r,0,1)}$ on $I_{r}^{\infty}$ has at worst {\em simple}
poles along $\SS$ (and is holomorphic outside $\SS$). Indeed, for each $n>0$ we may find
$\xi^{(n)}\in {e_r}H^0(\o{\X}_r,\omega_{\o{\X}_r})$ with $\xi=U_p^n\xi^{(n)}$.
As discussed in \S\ref{GD}, when viewed as a meromorphic differential on $\nor{\o{\X}}_r$
any section of $\omega_{\o{\X}_r}$
has poles of order bounded
by a constant depending only on $r$ (see \cite[Lemma 5.2.2]{GDBC}).
Since $F:I_r^{\infty}\rightarrow I_r^{\infty}$ is inseparable of degree $p$ (so
totally ramified over every
supersingular point), it follows from Remark \ref{poletrace} that there
exists $n > r$ such that
the meromorphic differential
$F_*^{n}\xi^{(n)}_{(r,0,1)}$ has at worst simple poles along $\SS$;
by the formula (\ref{Upn1}) for $U_p^n$, we conclude that the same is true of
$$\xi_{(r,0,1)} = (U_p^n\xi^{(n)})_{(r,0,1)}=F_*^n\xi^{(n)}_{(r,0,1)}.$$
Applying this with $\xi^{(r)}$ in the role of $\xi$, and using
(\ref{Upn1}) and (\ref{betadef}) we calculate
\begin{equation}
\xi=U_p^{r}\xi^{(r)} = \gamma_r^{\infty}(F_*^r\xi^{(r)}_{(r,0,1)}),
\end{equation}
so $\gamma_r^{\infty}$ surjects onto $e_rH^0(\o{\X}_r,\omega_{\o{\X}_r})$ and is
hence an isomorphism onto this image. A nearly identical argument shows that
$\gamma_r^{0}$ is an isomorphism onto $e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r})$.
Since pullback of meromorphic differentials along
$i_r^{\infty}:I_r^{\infty}\hookrightarrow \nor{\o{\X}}_r$ is given by projection
\begin{equation}
\xymatrix@C=35pt{
{H^0(\nor{\o{\X}}_r,\underline{\Omega}^1_{k(\nor{\o{\X}}_r)})\simeq
\prod\limits_{(a,b,u)} H^0(I_{(a,b,u)},\u{\Omega}^1_{k(I_{(a,b,u)})})} \ar[r]^-{\proj_{(r,0,1)}} &
{H^0(I_r^{\infty},\u{\Omega}^1_{k(I_r^{\infty})})}
}\label{PBisProj}
\end{equation}
onto the $(r,0,1)$-component,
the composition of $\gamma_r^{\infty}$ and (the restriction of)
$(i_r^{\infty})^*$ in either
order is the identity map. Since $i_r^{\infty}$ is compatible with the $\H_r$-correspondences,
the resulting isomorphism (\ref{I'compIsom})
is $\H_r$-equivariant (with $U_p$ acting on the target via $F_*$). Similarly,
since the ``geometric inertia" action (\ref{gammamaps}) of
$\Gamma$ on $\X_r$ is compatible via $i_r^{\infty}$ with the trivial action
on $I_r^{\infty}$ by Proposition \ref{AtkinInertiaCharp},
the isomorphism (\ref{I'compIsom}) is equivariant for these actions of $\Gamma$.
A nearly identical analysis shows that $(i_r^{0})^*$ is $\H_r^*$-compatible
(with $U_p^*$ acting on the target as $\langle p\rangle_N^{-1} F_*$) and $\Gamma$-equivariant for the
action of $\Gamma$ on $I_r^{0}$ via $\langle \chi(\cdot)\rangle^{-1}$
The commutativity of
the four diagrams in (\ref{rCompatDiagrams}) is an immediate consequence
of the descriptions of the degeneracy mappings $\o{\rho},\o{\sigma}$ on $\nor{\o{\X}}_r$ furnished by Proposition
\ref{pr1desc} and the explication (\ref{PBisProj}) of pullback by $i_r^{\star}$ in terms
of projection. That $\o{w}_r$ interchanges the two diagrams
in (\ref{rCompatDiagrams}) is an immediate consequence of Proposition \ref{ALinv}.
Finally, that the assertions of Proposition \ref{charpord} all hold if $\o{\X}_r$ and
$\Omega^1_{I_r^{\star}}(\SS)$
are replaced by $\nor{\o{\X}}_r$ and $\Omega^1_{I_r^{\star}}$, respectively, follows from a
a similar---but much simpler---argument. The point is that the maps $\gamma_r^{\star}$
of (\ref{betadef})--(\ref{gammadef})
visibly carry $H^0(I_r^{\star},\Omega^1_{I_r^{\star}})^{V_{\ord}}$ into
$H^0(\nor{\o{\X}}_r,\Omega_{\nor{\o{\X}}_r}^1)$, from which it follows
via our argument that they induce the claimed isomorphisms.
\end{proof}
Since $\o{\X}_r$ is a proper and geometrically connected curve over $\mathbf{F}_p$, Proposition \ref{HodgeFilCrvk}
(\ref{HodgeDegenerationField}) provides short exact sequences of
$\mathbf{F}_p[\Delta/\Delta_r]$-modules with linear $\Gamma$ and $\H_r^*$ (respectively $\H_r$)-action
\begin{subequations}
\begin{equation}
\xymatrix{
0\ar[r] & {e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r/\mathbf{F}_p})} \ar[r] & {e_r^*H^1(\o{\X}_r/\mathbf{F}_p)} \ar[r] &
{e_r^*H^1(\o{\X}_r,\O_{\o{\X}_r})} \ar[r] & 0
}\label{sesincharp1}
\end{equation}
respectively
\begin{equation}
\xymatrix{
0\ar[r] & {e_rH^0(\o{\X}_r,\omega_{\o{\X}_r/\mathbf{F}_p})} \ar[r] & {e_rH^1(\o{\X}_r/\mathbf{F}_p)} \ar[r] &
{e_rH^1(\o{\X}_r,\O_{\o{\X}_r})} \ar[r] & 0
}\label{sesincharp2}
\end{equation}
\end{subequations}
which are canonically $\mathbf{F}_p$-linearly dual to each other. We likewise have such exact sequences
in the case of $\nor{\o{\X}}_r$; note that since $\nor{\o{\X}}_r$ is smooth,
the short exact sequence $H(\nor{\o{\X}}_r/\mathbf{F}_p)$ is simply the Hodge filtration
of $H^1_{\dR}(\nor{\o{\X}}_r/\mathbf{F}_p)$.
\begin{corollary}\label{SplitIgusa}
The absolute Frobenius morphism of $\o{\X}_r$ over $\mathbf{F}_p$ induces a natural
$\mathbf{F}_p[\Delta/\Delta_r]$-linear, $\Gamma$-compatible, and $\H_r^*$
$($respectively $\H_r$$)$ equivariant splitting of $(\ref{sesincharp1})$
$($respectively $(\ref{sesincharp2})$$)$.
Furthermore, for each $r$ we have natural isomorphisms of split short exact sequences
\begin{subequations}
\begin{equation}
\xymatrix@C=15pt{
0\ar[r] & {e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r/\mathbf{F}_p})} \ar[r]\ar[d]_-{F_*^r (i_r^0)^*}^-{\simeq} &
{e_r^*H^1(\o{\X}_r/\mathbf{F}_p)} \ar[r]\ar[d]^-{\simeq} &
{e_r^*H^1(\o{\X}_r,\O_{\o{\X}_r})} \ar[r] & 0\\
0 \ar[r] & {H^0(I_r^{0},\Omega^1(\SS))^{V_{\ord}}} \ar[r] &
{H^0(I_r^{0},\Omega^1(\SS))^{V_{\ord}}\oplus
H^1(I_r^{\infty},\O(-\SS))^{F_{\ord}}} \ar[r] &
{H^1(I_r^{\infty},\O(-\SS))^{F_{\ord}}} \ar[r]\ar[u]_-{\VDual{(i_r^{\infty})^*}}^-{\simeq} & 0
}\label{LowerIsom1}
\end{equation}
\begin{equation}
\xymatrix@C=15pt{
0\ar[r] & {e_rH^0(\o{\X}_r,\omega_{\o{\X}_r/\mathbf{F}_p})}
\ar[r]\ar[d]_-{F_*^r (i_r^{\infty})^*}^-{\simeq} &
{e_rH^1(\o{\X}_r/\mathbf{F}_p)} \ar[r]\ar[d]^-{\simeq} &
{e_rH^1(\o{\X}_r,\O_{\o{\X}_r})} \ar[r] & 0\\
0 \ar[r] & {H^0(I_r^{\infty},\Omega^1(\SS))^{V_{\ord}}} \ar[r] &
{H^0(I_r^{\infty},\Omega^1(\SS))^{V_{\ord}}\oplus
H^1(I_r^{0},\O(-\SS))^{F_{\ord}}} \ar[r] &
{H^1(I_r^{0},\O(-\SS))^{F_{\ord}}} \ar[r]
\ar[u]_-{\VDual{(i_r^{0})^*}\langle p\rangle_N^{-r}}^-{\simeq} & 0
}\label{UpperIsom1}
\end{equation}
\end{subequations}
which are compatible with the extra structures.
The identification
$(\ref{LowerIsom1})$ $($respectively $(\ref{UpperIsom1})$$)$ is moreover
compatible with change in $r$ using the trace mappings attached
to $\rho: I_r^{\star}\rightarrow I_{r-1}^{\star}$ and to
$\o{\rho}:\o{\X}_r\rightarrow \o{\X}_{r-1}$ $($respectively
$\o{\sigma}:\o{\X}_r\rightarrow \o{\X}_{r-1}$$)$.
The same statements hold true if we replace $\o{\X}_r$, $\Omega^1_{I_r^{\star}}(\SS)$,
and $\O_{I_r^{\star}}(-\SS)$ with $\nor{\o{\X}}_r$, $\Omega^1_{I_r^{\star}}$,
and $\O_{I_r^{\star}}$, respectively.
\end{corollary}
\begin{proof}
Pullback by the absolute Frobenius endomorphism of $\o{\X}_r$
induces an endomorphism of (\ref{sesincharp1}) which
kills $H^0(\o{\X}_r,\omega_{\o{\X}_r/\mathbf{F}_p})$
and so yields a morphism of $\mathbf{F}_p[\Delta/\Delta_r]$-modules
\begin{equation}
\xymatrix{
{e_r^*H^1(\o{\X}_r,\O_{\o{\X}_r})}\ar[r] & {{e_r^*}H^1(\o{\X}_r/\mathbf{F}_p)}
}\label{H1Splitting}
\end{equation}
that
is $\Gamma$ and $\H_r^*$-compatible
and projects to the endomorphism $F^*$ of $e_r^*H^1(\o{\X}_r,\O_{\o{\X}_r})$.
On the other hand, Proposition \ref{charpord} gives a natural $\Gamma$ and $\H_r^*$-equivariant
isomorphism of $\mathbf{F}_p[\Delta/\Delta_r]$-modules
\begin{equation}
\xymatrix{
{H^1(I_r^{\infty},\O_{I_r^{\infty}}(-\SS))^{F_{\ord}}}\ar[r]^-{\VDual{(i_r^{\infty})^*}} &
{e_r^*H^1(\o{\X}_r,\O_{\o{\X}_r})}
}.\label{IsomOnH1}
\end{equation}
As this isomorphism intertwines $F^*$ on source and target,
we deduce that $F^*$ acts invertibly on
${e_r^*H^1(\o{\X}_r,\O_{\o{\X}_r})}$. We may therefore pre-compose (\ref{H1Splitting}) with $(F^*)^{-1}$
to obtain a canonical splitting of (\ref{sesincharp1}), which by construction is
$\mathbf{F}_p[\Delta/\Delta_r]$-linear and compatible with $\Gamma$ and $\H_r^*$.
The existence of (\ref{LowerIsom1}) as well
as its compatibility with $\Gamma$, $\H_r^*$ and with change in $r$ now follows
immediately from Proposition \ref{charpord} and duality (see Remark \ref{DualityOfFVOrd}).
The corresponding assertions for the exact sequence (\ref{sesincharp2}) and the
diagram (\ref{UpperIsom1}) are proved similarly, and we leave the details to the reader.
A nearly identical argument shows that the same assertions hold true when $\o{\X}_r$,
$\Omega^1_{I_r^{\star}}(\SS)$,
and $\O_{I_r^{\star}}(-\SS)$ are replaced by $\nor{\o{\X}}_r$,
$\Omega^1_{I_r^{\star}}$, and $\O_{I_r^{\star}}$, respectively.
\end{proof}
\begin{corollary}\label{FreenessInCharp}
The exact sequences
$(\ref{sesincharp1})$ and $(\ref{sesincharp2})$ are split short exact sequences of
free $\mathbf{F}_p[\Delta/\Delta_r]$-modules
whose terms have $\mathbf{F}_p[\Delta/\Delta_r]$-ranks
$d$, $2d$, and $d$, respectively, for $d$ as in
Remark $\ref{dMFmeaning}$.
For $s\le r$, the degeneracy maps $\rho,\sigma:\X_r\rightrightarrows \X_s$
induce natural isomorphisms of
exact sequences
\begin{align*}
&\xymatrix{
{\rho_*:e_r^*H(\o{\X}_r/\mathbf{F}_p) \mathop{\otimes}\limits_{\mathbf{F}_p[\Delta/\Delta_s]} \mathbf{F}_p[\Delta/\Delta_r]} \ar[r]^-{\simeq} &
{e_s^*H(\o{\X}_s/\mathbf{F}_p)}
}\\
&\xymatrix{
{\sigma_*:e_rH(\o{\X}_r/\mathbf{F}_p) \mathop{\otimes}\limits_{\mathbf{F}_p[\Delta/\Delta_s]} \mathbf{F}_p[\Delta/\Delta_r]} \ar[r]^-{\simeq} &
{e_sH(\o{\X}_s/\mathbf{F}_p)}
}
\end{align*}
that are $\Gamma$ and $\H_r^*$ $($respectively $\H_r$$)$ equivariant.
\end{corollary}
\begin{proof}
This follows immediately from Proposition \ref{IgusaStructure} and Corollary \ref{SplitIgusa}.
\end{proof}
\begin{remark}
We warn the reader that the na\"ive analogue of Corollary \ref{FreenessInCharp}
in the case of $\nor{\o{\X}}_r$ is false: while
$H^0(I_r,\Omega^1(\SS))^{V_{\ord}}$ is a free $\mathbf{F}_p[\Delta/\Delta_r]$-module, the submodule
of holomorphic differentials need {\em not} be.
Over $k=\o{\mathbf{F}}_p$, the residue map gives a short exact sequence of $k[\Delta/\Delta_r]$-modules
\begin{equation*}
\xymatrix{
0\ar[r] & {H^0(I_r,\Omega^1_{I_r/k})^{V_{\ord}}} \ar[r] & {H^0(I_r,\Omega^1_{I_r/k}(\SS))^{V_{\ord}}}
\ar[r] & {\ker\left(k^{\delta} \xrightarrow{\sum} k\right)} \ar[r] & 0
}
\end{equation*}
with middle term that is free over $k[\Delta/\Delta_r]$;
see Theorem 2 of \cite{Nakajima}. The splitting of this exact sequence is then equivalent
to the projectivity---hence freeness---of $H^0(I_r,\Omega^1_{I_r/k})^{V_{\ord}}$
over $k[\Delta/\Delta_r]$.
\end{remark}
In order to formulate the correct analogue of Corollary \ref{FreenessInCharp}
in the case of $\nor{\o{\X}}_r$, we proceed as follows.
Denote by $\tau:\mathbf{F}_p^{\times} \rightarrow \Z_p^{\times}$ the Teichm\"uller character,
and for any $\Z_p$-module $M$ with a linear action of $\mathbf{F}_p^{\times}$ and any
$j\in \Z/(p-1)\Z$, let
\begin{equation*}
M(j):=\{m\in M\ :\ d\cdot m = \tau(d)^jm\ \text{for all}\ d\in \mathbf{F}_p^{\times}\}
\end{equation*}
be the subspace of $M$ on which $\mathbf{F}_p^{\times}$ acts via $\tau^j$.
As $\#\mathbf{F}_p^{\times}=p-1$ is a unit in $\Z_p^{\times}$, the submodule $M(j)$
is a direct summand of $M$.
Explicitly, the idenitity of $\Z_p[\mathbf{F}_p^{\times}]$ admits the decomposition
\begin{equation}
1 = \sum_{j\in \Z/(p-1)\Z} f_j\quad\text{with}\quad
f_j:=\frac{1}{p-1}\sum_{g\in \mathbf{F}_p^{\times}} \tau^{-j}(g)\cdot g
\label{GpRngIdem}
\end{equation}
into mutually orthogonal idempotents $f_j$, and we have $M(j)=f_jM$.
In applications, we will consistently need to remove the trivial eigenspace
$M(0)$ from $M$, as this eigenspace in the $p$-adic Galois representations
we consider is not potentially crystalline at $p$. We will write
\begin{equation}
f':=\sum_{\substack{j\in \Z/(p-1)\Z \\ j\neq 0}} f_j\label{TeichmullerIdempotent}
\end{equation}
for the idempotent of $\Z_p[\mathbf{F}_p^{\times}]$ corresponding to projection away
from the 0-eigenspace for $\mathbf{F}_p^{\times}$.
Applying these considerations
to the identifications of split exact sequences in Corollary \ref{SplitIgusa}, which
are compatible with the canonical diamond operator action of $\Z_p^{\times}\simeq\mathbf{F}_p^{\times}\times \Delta$
on both rows, we obtain a corresponding identifiction of split exact sequences
of $\tau^j$-eigenspaces, for each $j\bmod p-1$.
The following is a generalization of \cite[Proposition 8.10 (2)]{tameness}:
\begin{lemma}\label{CharacterSpaces}
Let $j$ be an integer with $j\not\equiv 0\bmod p-1$. For each $r$, there are canonical isomorphisms
\begin{equation}
\xymatrix{
{H^0(I_r,\Omega^1_{I_r})(j)}\ar[r]^-{\simeq} & {H^0(I_r,\Omega^1_{I_r}(\SS))(j)}
}\qquad\text{and}\qquad
\xymatrix{
{H^1(I_r,\O(-\SS))(j)}\ar[r]^-{\simeq} & {H^1(I_r,\O)(j)}
}\label{IgusaEigen}
\end{equation}
The normalization map
$\nu:\nor{\o{\X}}_r\rightarrow \o{\X}_r$ induces a natural isomorphism of split
exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {{e_r^*}H^0(\o{\X}_r,\Omega^1_{\nor{\o{\X}}_r})(j)} \ar[r]\ar[d]_-{\nu_*}^-{\simeq} &
{{e_r^*}H^1_{\dR}(\nor{\o{\X}}_r/\mathbf{F}_p)(j)} \ar[r]\ar[d]^-{\simeq} &
{{e_r^*}H^1(\nor{\o{\X}}_r,\O_{\nor{\o{\X}}_r})(j)} \ar[r] & 0 \\
0\ar[r] & {e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r/\mathbf{F}_p})(j)} \ar[r] &
{e_r^*H^1(\o{\X}_r/\mathbf{F}_p)(j)} \ar[r] &
{e_r^*H^1(\o{\X}_r,\O_{\o{\X}_r})(j)} \ar[r]\ar[u]_-{\nu^*}^-{\simeq} & 0
}\label{splitholo}
\end{gathered}
\end{equation}
where the central vertical arrow is deduced from the outer two vertical arrows via the
splitting of both rows by the Frobenius endomorphism.
The same assertions hold if we replace $e_r^*$ with $e_r$.
\end{lemma}
\begin{proof}
The first map in (\ref{IgusaEigen}) is injective, as it is simply the canonical inclusion.
To see that it is an isomorphism,
we may work over $k:=\o{\mathbf{F}}_p$. If $\eta$ is {\em any} meromorphic differential on
$I_r$ on which $\mathbf{F}_p^{\times}$
acts via the character $\tau^j$,
then since the diamond operators fix every supersingular point on $I_r$ we have
\begin{equation*}
\res_x(\eta) = \res_x(\langle u\rangle\eta) = \tau^j(u)\res_x(\eta)
\end{equation*}
for any $x\in \SS(k)$ and all $u\in \mathbf{F}_p^{\times}$. As
$j\not\equiv 0\bmod p-1$, so $\tau^j$ is nontrivial, we must therefore have $\res_x(\eta)=0$ for all
supersingular points $x$. If in addition $\eta$ is holomorphic
outside $\SS$ with at worst simple poles along $\SS$, then $\eta$ must be holomorphic everywhere,
so the first map in (\ref{IgusaEigen}) is
surjective, as desired. The second mapping in (\ref{IgusaEigen}) is dual to the first, and
hence an isomorphism as well.
Now for each $j\not\equiv 0\bmod p-1$,
we have a commutative diagram
\begin{equation}
\begin{gathered}
\xymatrix{
{e_r^*H^0(\nor{\o{\X}}_r,\Omega^1_{\nor{\o{\X}}_r})(j)}
\ar@{^{(}->}[r]^-{\nu_*}\ar[d]_-{(i_r^0)^*}^-{\simeq} &
{e_r^*H^0(\o{\X}_r,\omega_{\o{\X}_r})(j)} \ar[d]^-{(i_r^0)^*}_-{\simeq} \\
{H^0(I_r^0,\Omega^1_{I_r^0})(j)^{V_{\ord}}} \ar@{^{(}->}[r]^-{\simeq} &
{H^0(I_r^0,\Omega^1_{I_r^0}(\SS))(j)^{V_{\ord}}}
}
\label{linkingDiagram}
\end{gathered}
\end{equation}
of $\mathbf{F}_p[\Delta/\Delta_r]$-modules with $\Gamma$ and $\H_r^*$-action in which
the two vertical arrows are isomorphisms by Proposition \ref{charpord} and the bottom horizontal mapping is
an isomorphism as we have just seen. We conclude that the top horizontal
arrow of (\ref{linkingDiagram}) is an isomorphism as well.
Thus, the left vertical map in (\ref{splitholo}) is an isomorphism, so the same is true
of the right vertical map by duality. The diagram (\ref{splitholo}) then follows at once
from the fact the both rows are canonically split by the Frobenius endomorphism, thanks to
Corollary \ref{SplitIgusa}.
A nearly identical argument shows that the same assertions hold if we replace $e_r^*$
with $e_r$ throughout.
\end{proof}
If $A$ is any $\Z_p[\mathbf{F}_p^{\times}]$-algebra and $a\in A$, we will
write $a':=f'a$ for the product of $a$ with the idempotent $f'$
of (\ref{TeichmullerIdempotent}), or equivalently the
projection of $a$ to the complement
of the trivial eigenspace for $\mathbf{F}_p^{\times}$. We will apply this
to $A=\H_r,\,\H_r^*$, viewed as $\Z_p[\mathbf{F}_p^{\times}]$-algebras
in the usual manner, via the diamond operators and the Teichm\"uller
section $\tau:\mathbf{F}_p^{\times}\hookrightarrow \Z_p^{\times}$.
\begin{proposition}\label{NormalizationCoh}
For each $r$ there are natural isomorphisms of split short exact sequences
\begin{subequations}
\begin{equation}
\begin{gathered}
\xymatrix@C=15pt{
0\ar[r] & {{e_r^*}'H^0(\o{\X}_r,\Omega^1_{\nor{\o{\X}}_r})}
\ar[r]\ar[d]_-{F_*^r (i_r^0)^*}^-{\simeq} &
{{e_r^*}'H^1_{\dR}(\nor{\o{\X}}_r/\mathbf{F}_p)} \ar[r]\ar[d]^-{\simeq} &
{{e_r^*}'H^1(\nor{\o{\X}}_r,\O_{\nor{\o{\X}}_r})} \ar[r] & 0\\
0 \ar[r] & {f'H^0(I_r^{0},\Omega^1)^{V_{\ord}}} \ar[r] &
{f'H^0(I_r^{0},\Omega^1)^{V_{\ord}}\oplus
f'H^1(I_r^{\infty},\O)^{F_{\ord}}} \ar[r] &
{f'H^1(I_r^{\infty},\O)^{F_{\ord}}} \ar[r]\ar[u]_-{\VDual{(i_r^{\infty})^*}}^-{\simeq} & 0
}\label{LowerIsom2}
\end{gathered}
\end{equation}
\begin{equation}
\begin{gathered}
\xymatrix@C=15pt{
0\ar[r] & {e_r'H^0(\nor{\o{\X}}_r,\Omega^1_{\nor{\o{\X}}_r})}
\ar[r]\ar[d]_-{F_*^r (i_r^{\infty})^*}^-{\simeq} &
{e_r'H^1_{\dR}(\nor{\o{\X}}_r/\mathbf{F}_p)} \ar[r]\ar[d]^-{\simeq} &
{e_r'H^1(\nor{\o{\X}}_r,\O_{\nor{\o{\X}}_r})} \ar[r] & 0\\
0 \ar[r] & {f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}} \ar[r] &
{f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}\oplus
f'H^1(I_r^{0},\O)^{F_{\ord}}} \ar[r] &
{f'H^1(I_r^{0},\O)^{F_{\ord}}} \ar[r]
\ar[u]_-{\VDual{(i_r^{0})^*}\langle p\rangle_N^{-r}}^-{\simeq} & 0
}\label{UpperIsom2}
\end{gathered}
\end{equation}
\end{subequations}
Setting $d':=\sum_{k=3}^p d_k$ where $d_k:=\dim_{\mathbf{F}_p} S_k(N;\mathbf{F}_p)^{\ord}$
as in Remark $\ref{dMFmeaning}$,
the terms in the top rows of $(\ref{LowerIsom2})$ and $(\ref{UpperIsom2})$
are free $\mathbf{F}_p[\Delta/\Delta_r]$-modules of ranks $d'$, $2d'$, and $d'$.
The identification
$(\ref{LowerIsom2})$ $($respectively $(\ref{UpperIsom2})$$)$ is
$\Gamma$ and $\H_r^*$ $($respectively $\H_r$$)$-equivariant, and
compatible with change in $r$ using the trace mappings attached
to $\rho: I_r^{\star}\rightarrow I_s^{\star}$ and to
$\o{\rho}:\o{\X}_r\rightarrow \o{\X}_{s}$ $($respectively
$\o{\sigma}:\o{\X}_r\rightarrow \o{\X}_{s}$$)$.
\end{proposition}
\begin{proof}
This follows immediately from Corollaries \ref{SplitIgusa}--\ref{FreenessInCharp}
and Lemma \ref{CharacterSpaces}, using the fact that the group ring $\mathbf{F}_p[\Delta/\Delta_r]$
is local, so any projective $\mathbf{F}_p[\Delta/\Delta_r]$-module
is free.
\end{proof}
As usual, we write $\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}[p^{\infty}]$ for the $p$-divisible group
of the Jacobian of
$\nor{\o{\X}}_r$ over $\mathbf{F}_p$; it is equipped
with canonical actions of
$\H_r$ and $\H_r^*$, as well as a ``geometric inertia"
action of $\Gamma$ over $\mathbf{F}_p$.
\begin{definition}\label{pDivGpSpecial}
We define $\Sigma_r:={e_r^*}'\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}[p^{\infty}]$,
equipped with the induced actions of $\H_r^*$ and $\Gamma$.
\end{definition}
We will employ Proposition \ref{NormalizationCoh} and Oda's description (Proposition \ref{OdaDieudonne}) of
Dieudonn\'e modules in terms of de Rham cohomology to analyze the structure of
$\Sigma_r$.
\begin{proposition}\label{GisOrdinary}
For each $r$,
there is a natural isomorphism of $A:=\Z_p[F,V]$-modules
\begin{equation}
\ensuremath{\mathbf{D}}(\Sigma_r)_{\mathbf{F}_p} \simeq {e_r^*}'H^1_{\dR}(\nor{\o{\X}}_r/\mathbf{F}_p)\simeq
f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}\oplus f'H^1(I_r^0,\O)^{F_{\ord}}.\label{DieudonneDesc}
\end{equation}
which is compatible with $\H_r^*$, $\Gamma$, and change in $r$ and which
carries $\ensuremath{\mathbf{D}}(\Sigma_r^{\mult})_{\mathbf{F}_p}$ $($respectively $\ensuremath{\mathbf{D}}(\Sigma_r^{\et})_{\mathbf{F}_p}$$)$ isomorphically
onto $f'H^0(I_r^{0},\Omega^1)^{V_{\ord}}$ $($respectively
$f'H^1(I_r^{\infty},\O)^{F_{\ord}}$$)$. In particular,
$\Sigma_r$ is ordinary.
\end{proposition}
\begin{proof}
First note that since the identifications (\ref{LowerIsom2}) and (\ref{UpperIsom2})
are induced by the canonical closed immersions $i_r^{\star}:I_r^{\star}\hookrightarrow \nor{\o{\X}}_r$,
they are compatible with the natural actions of Frobenius and the Cartier operator.
The isomorphism (\ref{DieudonneDesc}) is therefore an immediate consequence of
Propositions \ref{OdaDieudonne} and \ref{NormalizationCoh}. Since this isomorphism is
compaible with $F$ and $V$, we have
\begin{subequations}
\begin{equation}
\ensuremath{\mathbf{D}}(\Sigma_r^{\mult})_{\mathbf{F}_p}
\simeq \ensuremath{\mathbf{D}}(\Sigma_r)_{\mathbf{F}_p}^{V_{\ord}}
\simeq f'H^0(I_r^{0},\Omega^1)^{V_{\ord}}
\end{equation}
and
\begin{equation}
\ensuremath{\mathbf{D}}(\Sigma_r^{\et})\otimes_{\Z_p}\mathbf{F}_p
\simeq \ensuremath{\mathbf{D}}(\Sigma_r)_{\mathbf{F}_p}^{F_{\ord}}
\simeq f'H^1(I_r^{\infty},\O)^{F_{\ord}}
\end{equation}
\end{subequations}
and we conclude that the canonical inclusion
$\ensuremath{\mathbf{D}}(\Sigma_r^{\mult})_{\Z_p}\oplus\ensuremath{\mathbf{D}}(\Sigma_r^{\et})_{\Z_p}\hookrightarrow \ensuremath{\mathbf{D}}(\Sigma_r)_{\Z_p}$
is surjective, whence $\Sigma_r$ is ordinary by Dieudonn\'e theory.
\end{proof}
We now analyze the ordinary $p$-divisible group $\Sigma_r$ in more detail.
Since $\nor{\o{\X}}_r$
is the disjoint union of proper smooth and irreducible Igusa curves $I_{(a,b,u)}$
(see Proposition \ref{redXr}) with $I_r^0:=I_{(0,r,1)}$ and $I_r^{\infty}=I_{(r,0,1)}$,
we have
a canonical identification
\begin{equation}
\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p} = \prod_{(a,b,u)} \Pic^0_{I_{(a,b,u)}/\mathbf{F}_p}.
\label{Pic0Iden}
\end{equation}
For $\star=0,\infty$ let us write $j_r^{\star}:=\Pic^0_{I_r^{\star}/\mathbf{F}_p}$
for the Jacobian of $I_r^{\star}$ over $\mathbf{F}_p$.
The canonical closed immersions
$i_r^{\star}:I_r^{\star}\hookrightarrow \nor{\o{\X}}_r$ yield (by Picard and Albanese functoriality)
homomorphisms of abelian varieties over $\mathbf{F}_p$
\begin{equation}
\xymatrix{
{\Alb(i_r^{\star}):j_r^{\star}} \ar[r] & {\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}}
}
\quad\text{and}\quad
\xymatrix{
{\Pic^0(i_r^{\star}):\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}} \ar[r] & {j_r^{\star}}
}.\label{AlbPicIncl}
\end{equation}
Via the identification (\ref{Pic0Iden}), we know that $j_r^{\star}$ is a direct factor
of $\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}$; in these terms $\Alb(i_r^{\star})$ is the unique mapping
which projects to the identity on $j_r^{\star}$ and to the zero map on all other factors,
while $\Pic^0(i_r^{\star})$ is simply projection onto the factor $j_r^{\star}$.
As $\Sigma_r$ is a direct factor of ${f' \Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}[p^{\infty}]}$,
these mappings induce homomorphisms of $p$-divisible groups over $\mathbf{F}_p$
\begin{subequations}
\begin{equation}
\xymatrix@C=35pt{
{f'j_r^{0}[p^{\infty}]^{\mult}} \ar[r]^-{\Alb(i_r^{0})} &
{f' \Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}[p^{\infty}]^{\mult}} \ar[r]^-{\proj} &
{\Sigma_r^{\mult}}
}\label{Alb0}
\end{equation}
\begin{equation}
\xymatrix@C=35pt{
{\Sigma_r^{\et}} \ar[r]^-{\incl} &
{f' \Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}[p^{\infty}]^{\et}} \ar[r]^-{\Pic^0(i_r^{\infty})} &
{f'j_r^{\infty}[p^{\infty}]^{\et}}
}\label{Picinfty}
\end{equation}
\end{subequations}
which we (somewhat abusively) again denote by $\Alb(i_r^{0})$ and $\Pic^0(i_r^{\infty})$, respectively.
The following is a sharpening of \cite[Chapter 3, \S3, Proposition 3]{MW-Iwasawa}
(see also \cite[Proposition 3.2]{Tilouine}):
\begin{proposition}\label{MWSharpening}
The mappings $(\ref{Alb0})$ and $(\ref{Picinfty})$ are isomorphisms. They induce
a canonical split short exact sequences of $p$-divisible groups over $\mathbf{F}_p$
\begin{equation}
\xymatrix@C=45pt{
0 \ar[r] & {f'j^0_r[p^{\infty}]^{\mult}} \ar[r]^-{\Alb(i_r^0)\circ V^r} &
{\Sigma_r} \ar[r]^-{\Pic^0(i_r^{\infty})} & {f'j^{\infty}_r[p^{\infty}]^{\et}} \ar[r] & 0
}\label{pDivUpic}
\end{equation}
which is:
\begin{enumerate}
\item $\Gamma$-equivariant for the geometric inertia action on $\Sigma_r$, the trivial
action on $f'j_r^{\infty}[p^{\infty}]^{\et}$, and the action via $\langle \chi(\cdot) \rangle^{-1}$
on $f'j_r^{0}[p^{\infty}]^{\mult}$. \label{GammaCompatProp}
\item $\H_r^*$-equivariant with $U_p^*$ acting on $f'j_r^{\infty}[p^{\infty}]^{\et}$
as $F$ and on $f'j_r^{0}[p^{\infty}]^{\mult}$ as $\langle p\rangle_N V$.
\item Compatible with change in $r$ via the mappings $\Pic^0(\rho)$ on $j_r^{\star}$ and $\Sigma_r$.
\label{ChangerProp}
\end{enumerate}
\end{proposition}
\begin{proof}
It is clearly enough to prove that the sequence (\ref{pDivUpic}) induced by $(\ref{Alb0})$ and $(\ref{Picinfty})$
is exact.
Since the contravariant Dieudonn\'e module functor from the category of $p$-divisible groups
over $\mathbf{F}_p$ to the category of $A$-modules which are $\Z_p$ finite and free is an exact
anti-equivalence, it suffices to prove such exactness
after applying $\ensuremath{\mathbf{D}}(\cdot)_{\Z_p}$.
As the resulting sequence consist of finite free $\Z_p$-modules,
exactness may be checked modulo $p$ where it follows immediately from
Propositions \ref{NormalizationCoh} and \ref{GisOrdinary}.
The claimed compatibility with $\Gamma$, $\H_r^*$, and change in $r$ is deduced from
Propositions \ref{AtkinInertiaCharp},
\ref{UlmerProp}, and \ref{pr1desc}, respectively.
\end{proof}
\begin{remark}
It is possible to give a short proof of Proposition \ref{MWSharpening}
along the lines of \cite{MW-Iwasawa} or \cite{Tilouine} by using Proposition \ref{UlmerProp} directly.
We stress, however, that our approach via Dieudonn\'e modules gives more refined information,
most notably that the Dieudonn\'e module of $\Sigma_r[p]$ is free as an $\mathbf{F}_p[\Delta/\Delta_r]$-module.
This fact will be crucial in our later arguments.
\end{remark}
\section{Dieudonn\'e crystals and \texorpdfstring{$(\varphi,\Gamma)$}{(Phi,Gamma)}-modules}\label{PhiGammaCrystals}
In this section, we summarize the main results of \cite{CaisLau},
which provides a classification of $p$-divisible groups
over $R_r$ by certain semi-linear algebra structures.
These structures---which arise naturally via the Dieudonn\'e crystal functor---
are cyclotomic analogues of Breuil and Kisin modules, and are closely
related to Wach modules.\footnote{See \cite{CaisLau} for the precise relationship.}
\subsection{\texorpdfstring{$(\varphi,\Gamma)$}{(Phi,Gamma)}-modules attached to \texorpdfstring{$p$}{p}-divisible groups}\label{pDivPhiGamma}
Fix a perfect field $k$ of characteristic $p$.
Write $W:=W(k)$ for the Witt vectors of $k$ and $K$ for its fraction field,
and denote by $\varphi$ the unique automorphism of $W(k)$ lifting the $p$-power map
on $k$. Fix an algebraic closure $\overline{K}$ of $K$, as well
as a compatible sequence $\{\varepsilon^{(r)}\}_{r\ge 1}$ of primitive $p$-power roots of unity in $\o{K}$,
and set $\scrG_K:=\Gal(\o{K}/K)$.
For $r\ge 0$, we put $K_r:=K(\mu_{p^r})$ and $R_r:=W[\mu_{p^r}]$,
and we set $\Gamma_r:=\Gal(K_{\infty}/K_r)$, and $\Gamma:=\Gamma_0$.
Let $\mathfrak{S}_r:=W[\![u_r]\!]$ be the power series ring in one variable $u_r$ over $W$,
viewed as a topological ring via the $(p,u_r)$-adic topology.
We equip $\mathfrak{S}_r$ with the unique continuous action of $\Gamma$ and extension of $\varphi$
determined by
\begin{align}
&\gamma u_r := (1+u_r)^{\chi(\gamma)} -1\quad \text{for $\gamma\in \Gamma$} && \text{and} &&
\varphi(u_r) := (1+u_r)^p -1.\label{gamphiact}
\end{align}
We denote by $\O_{\E_r}:=\widehat{\mathfrak{S}_r[\frac{1}{u_r}]}$ the $p$-adic completion of the localization
${\mathfrak{S}_r}_{(p)}$, which is a complete discrete valuation ring with uniformizer $p$ and
residue field $k(\!(u_r)\!)$. One checks that the actions of $\varphi$ and $\Gamma$
on $\mathfrak{S}_r$ uniquely extend to $\O_{\E_r}$.
For $r>0$, we write $\theta: \mathfrak{S}_r\twoheadrightarrow R_r$ for the continuous and $\Gamma$-equivariant $W$-algebra
surjection
sending $u_r$ to $\varepsilon^{(r)}-1$, whose kernel is the principal ideal generated by
the Eisenstein polynomial $E_r:=\varphi^r(u_r)/\varphi^{r-1}(u_r)$,
and we denote by $\tau:\mathfrak{S}_r\twoheadrightarrow W$ the continuous and $\varphi$-equivariant surjection of $W$-algebras
determined by $\tau(u_r)=0$.
We lift the canonical inclusion $R_r\hookrightarrow R_{r+1}$
to a $\Gamma$- and $\varphi$-equivariant $W$-algebra injection
${\mathfrak{S}_r} \hookrightarrow {\mathfrak{S}_{r+1}}$
determined by $u_r\mapsto \varphi(u_{r+1})$;
this map uniquely extends to a continuous injection
$\O_{\E_r}\hookrightarrow \O_{\E_{r+1}}$, compatibly with $\varphi$ and $\Gamma$.
We will frequently identify $\mathfrak{S}_r$ (respectively $\O_{\E_r}$) with its image in $\mathfrak{S}_{r+1}$
(respectively $\O_{\E_{r+1}}$),
which coincides with the image of $\varphi$ on $\mathfrak{S}_{r+1}$ (respectively $\O_{\E_{r+1}})$.
Under this convention, we have
$E_{r}(u_r) = E_1(u_1) = u_0/u_1$ for all $r>0$, so we will simply write $\omega:=E_r(u_r)$
for this common element of $\mathfrak{S}_r$ for $r>0$.
\begin{definition}
We write $\textswab{BT}_{\mathfrak{S}_r}^{\varphi}$ for the category of {\em Barsotti-Tate modules over $\mathfrak{S}_r$},
{\em i.e.} the category whose objects are pairs $(\m,\varphi_{\m})$ where
\begin{itemize}
\item $\m$ is a free $\mathfrak{S}_r$-module of finite rank.
\item $\varphi_{\m}:\m\rightarrow \m$ is a $\varphi$-semilinear
map whose linearization has cokernel killed by $\omega$,
\end{itemize}
and whose morphisms are $\varphi$-equivariant $\mathfrak{S}_r$-module homomorphisms.
We write $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$ for the subcategory of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi}$
consisting of objects $(\m,\varphi_{\m})$ which admit a semilinear $\Gamma$-action
(in the category $\textswab{BT}_{\mathfrak{S}_r}^{\varphi}$) with the property that $\Gamma_r$ acts trivially on
$\m/u_r\m$. Morphisms in $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$ are $\varphi$ and $\Gamma$-equivariant
morphisms of $\mathfrak{S}_r$-modules.
We often abuse notation by writing $\m$ for the pair
$(\m,\varphi_{\m})$ and $\varphi$ for $\varphi_{\m}$.
\end{definition}
If $(\m,\varphi_{\m})$ is any object of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$, then
$1\otimes\varphi_{\m}:\varphi^*\m\rightarrow \m$
is injective with cokernel killed by $\omega$, so there is a unique
$\mathfrak{S}_r$-linear homomorphism $\psi_{\m}:\m\rightarrow \varphi^*\m$
with the property that the composition of $1\otimes\varphi_{\m}$ and $\psi_{\m}$ (in either order)
is multiplication by $\omega$. Clearly, $\varphi_{\m}$ and $\psi_{\m}$ determine eachother.
\begin{definition}\label{DualBTDef}
Let $\m$ be an object of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$. The {\em dual
of $\m$} is the object $(\m^{t},\varphi_{\m^{t}})$ of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$
whose underlying $\mathfrak{S}_r$-module is $\m^{t}:=\Hom_{\mathfrak{S}_r}(\m,\mathfrak{S}_r)$, equipped with
the $\varphi$-semilinear endomorphism
\begin{equation*}
\xymatrix@C=32pt{
{\varphi_{\m^{t}}: \m^{t}} \ar[r]^-{1\otimes \id_{\m^{t}}} & {\varphi^*\m^{t} \simeq (\varphi^*\m)^{t}}
\ar[r]^-{\psi_{\m}^{t}} & {\m^{t}}
}
\end{equation*}
and the commuting action of $\Gamma$ given for $\gamma\in \Gamma$ by
\begin{equation*}
(\gamma f)(m) := \chi(\gamma)^{-1}\varphi^{r-1}(\gamma u_r/u_r)\cdot\gamma (f(\gamma^{-1} m )).
\end{equation*}
\end{definition}
There is a natural notion of base change for Barsotti--Tate modules.
Let $k'/k$ be an algebraic extension (so $k'$ is automatically perfect), and write $W':=W(k')$,
$R_r':=W'[\mu_{p^r}]$, $\mathfrak{S}_r':=W'[\![u_r]\!]$, and so on.
The canonical
inclusion $W\hookrightarrow W'$ extends to a $\varphi$ and $\Gamma$-compatible
$W$-algebra injection $\iota_r:\mathfrak{S}_r\hookrightarrow \mathfrak{S}_{r+1}'$, and extension
of scalars along $\iota_r$ yields a canonical
canonical base change functor ${\iota_r}_*: \textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}\rightarrow \textswab{BT}_{\mathfrak{S}_{r+1}}^{\varphi,\Gamma}$
which one checks is compatible with duality.
Let us write $\pdiv_{R_r}^{\Gamma}$ for the subcategory of $p$-divisible groups over $R_r$
consisting of those objects and morphisms which descend (necessarily uniquely) to $K=K_0$ on generic fibers.
By Tate's Theorem, this is of course equivalent to the full subcategory of $p$-divisible
groups over $K_0$ which have good reduction over $K_r$. Note that for any algebraic extension $k'/k$,
base change along the inclusion $\iota_r:R_r\hookrightarrow R_{r+1}'$ gives a covariant functor
${\iota_r}_*:\pdiv_{R_r}^{\Gamma}\rightarrow \pdiv_{R_{r+1}'}^{\Gamma}$.
The main result of \cite{CaisLau} is the following:
\begin{theorem}\label{CaisLauMain}
For each $r>0$, there is a contravariant functor
$\m_r:\pdiv_{R_r}^{\Gamma}\rightarrow \textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$ such that:
\begin{enumerate}
\item The functor $\m_r$ is an exact equivalence of categories, compatible with duality.
\label{exequiv}
\item The functor $\m_r$ is of formation compatible with base change:
for any algebraic extension $k'/k$, there is a natural isomorphism
of composite functors ${\iota_r}_*\circ \m_r \simeq \m_{r+1}\circ {\iota_{r}}_*$ on $\pdiv_{R_r}^{\Gamma}$.
\label{BaseChangeIsom}
\item For $G\in \pdiv_{R_r}^{\Gamma}$, put $\o{G}:=G\times_{R_r} k$ and $G_0:=G\times_{R_r} R_r/pR_r$.
\begin{enumerate}
\item There is a functorial and $\Gamma$-equivariant isomorphism of $W$-modules
\begin{equation*}
\m_r(G)\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi\circ \tau} W \simeq \ensuremath{\mathbf{D}}(\o{G})_W,
\end{equation*}
carrying
$\varphi_{\m}\otimes \varphi$ to $F:\ensuremath{\mathbf{D}}(\o{G})_W\rightarrow \ensuremath{\mathbf{D}}(\o{G})_W$
and $\psi_{\m}\otimes 1$ to $V\otimes 1: \ensuremath{\mathbf{D}}(\o{G})_W \rightarrow \varphi^*\ensuremath{\mathbf{D}}(\o{G})_W$.
\label{EvaluationONW}
\item There is a functorial and $\Gamma$-equivariant isomorphism of $R_r$-modules
\begin{equation*}
\m_r(G)\mathop{\otimes}\limits_{\mathfrak{S}_r,\theta\circ\varphi} R_{r} \simeq \ensuremath{\mathbf{D}}(G_0)_{R_r}.
\end{equation*}\label{EvaluationONR}
\end{enumerate}
\end{enumerate}
\end{theorem}
We wish to explain how to functorially recover the $\scrG_K$-representation afforded
by the $p$-adic Tate module $T_pG_K$ from $\m_r(G)$. In order to do so, we must first recall
the necessary period rings; for a more detailed synopsis of these rings and their properties,
we refer the reader to \cite[\S6--\S8]{Colmez}.
As usual, we put\footnote{Here we use the notation introduced by Berger and Colmez; in Fontaine's
original notation, this ring is denoted $\R$.}
$$\wt{\ensuremath{\mathbf{E}}}^+:=\varprojlim_{x\mapsto x^p} \O_{\c_K}/(p),$$
equipped with its canonical $\scrG_K$-action via ``coordinates"
and $p$-power Frobenius map $\varphi$.
This is a perfect ({\em i.e.} $\varphi$ is an automorphism) valuation ring
of charteristic $p$ with residue field $\overline{k}$
and fraction field $\wt{\ensuremath{\mathbf{E}}}:=\Frac(\wt{\ensuremath{\mathbf{E}}}^+)$ that is algebraically closed.
We view $\wt{\ensuremath{\mathbf{E}}}$ as a topological field via its valuation topology, with respect to which
it is complete.
Our fixed choice of $p$-power compatible
sequence $\{\varepsilon^{(r)}\}_{r\ge 0}$
induces an element $\u{\varepsilon}:=(\varepsilon^{(r)}\bmod p)_{r\ge 0}$ of $\wt{\ensuremath{\mathbf{E}}}^+$
and we set $\ensuremath{\mathbf{E}}_{K}:=k(\!(\u{\varepsilon} - 1)\!)$, viewed as a
topological\footnote{The valuation $v_{\ensuremath{\mathbf{E}}}$ on $\wt{\ensuremath{\mathbf{E}}}$ induces the usual discrete valuation
on $\ensuremath{\mathbf{E}}_{K,r}$, with the unusual
normalization $1/p^{r-1}(p-1)$.} subring of $\wt{\ensuremath{\mathbf{E}}}$; note that this is
a $\varphi$- and $\scrG_K$-stable subfield of $\wt{\ensuremath{\mathbf{E}}}$ that is independent
of our choice of $\u{\varepsilon}$. We write $\ensuremath{\mathbf{E}}:=\ensuremath{\mathbf{E}}_K^{\sep}$
for the separable closure of $\ensuremath{\mathbf{E}}_K$ in the algebraically closed field $\wt{\ensuremath{\mathbf{E}}}$.
The natural $\scrG_K$-action on $\wt{\ensuremath{\mathbf{E}}}$ induces a canonical identification
$\Gal(\ensuremath{\mathbf{E}}/\ensuremath{\mathbf{E}}_{K}) = \scrH:=\ker(\chi)\subseteq \scrG_K$, so
$\ensuremath{\mathbf{E}}^{\scrH}=\ensuremath{\mathbf{E}}_{K}$.
If $E$ is any subring of $\wt{\ensuremath{\mathbf{E}}}$, we write $E^+:=E\cap \wt{\ensuremath{\mathbf{E}}}^+$
for the intersection (taken inside $\wt{\ensuremath{\mathbf{E}}}$).
We now construct Cohen rings for each of the above subrings of $\wt{\ensuremath{\mathbf{E}}}$.
To begin with, we put
\begin{equation*}
\wt{\a}^+:=W(\wt{\ensuremath{\mathbf{E}}}^+),\qquad\text{and}\qquad \wt{\a}:=W(\wt{\ensuremath{\mathbf{E}}});
\end{equation*}
each of these rings is equipped with a canonical Frobenius automorphism $\varphi$
and action of $\scrG_K$ via Witt functoriality.
Set-theoretically identifying $W(\wt{\ensuremath{\mathbf{E}}})$ with $\prod_{m=0}^{\infty} \wt{\ensuremath{\mathbf{E}}}$ in the usual way, we
endow each factor with its valuation topology and give $\wt{\a}$ the product topology.\footnote{This
is what is called the {\em weak topology} on $\wt{\a}$.
If each factor of $\wt{\ensuremath{\mathbf{E}}}$ is instead given the discrete topology, then the
product topology on $\wt{\a}=W(\wt{\ensuremath{\mathbf{E}}})$ is the familiar $p$-adic
topology, called the {\em strong} topology.}
The $\scrG_K$ action on $\wt{\a}$ is then continuous
and the canonical $\scrG_K$-equivariant $W$-algebra surjection
$\theta:\wt{\a}^+\rightarrow \O_{\c_K}$ is continuous when
$\O_{\c_K}$ is given its usual $p$-adic topology.
For each $r\ge 0$, there is a unique continuous $W$-algebra map $j_r:\O_{\E_r}\hookrightarrow \wt{\a}$
determined by $j_r(u_r):=\varphi^{-r}([\u{\varepsilon}] - 1)$. These maps are
moreover $\varphi$ and $\scrG_K$-equivariant, with $\scrG_K$ acting on $\O_{\E_r}$ through the quotient
$\scrG_K\twoheadrightarrow \Gamma$, and compatible with change in $r$.
We define $\a_{K,r}:=\im(j_r:\O_{\E_r}\rightarrow \wt{\a}),$
which is
naturally a $\varphi$ and $\scrG_K$-stable subring of $\wt{\a}$ that is independent of our choice
of $\u{\varepsilon}$.
We again omit the subscript when $r=0$.
Note that $\a_{K,r}=\varphi^{-r}(\a_K)$ inside $\wt{\a}$, and that $\a_{K,r}$
is a discrete valuation ring with uniformizer $p$ and residue field $\varphi^{-r}(\ensuremath{\mathbf{E}}_K)$
that is purely inseparable over $\ensuremath{\mathbf{E}}_K$.
We define $\a_{K,\infty}:=\bigcup_{r\ge 0} \a_{K,r}$
and write $\wt{\a}_K$ (respectively $\wh{\a}_K$)
for the closure of $\a_{K,\infty}$ in $\wt{\a}$ with respect to the weak
(respectively strong) topology.
Let $\a_{K,r}^{\sh}$ be the strict Henselization of $\a_{K,r}$
with respect to the separable closure of its residue field inside $\wt{\ensuremath{\mathbf{E}}}$.
Since $\wt{\a}$ is strictly Henselian, there is a unique local morphism
$\a_{K,r}^{\sh}\rightarrow \wt{\a}$ recovering the given inclusion on residue fields,
and we henceforth view $\a_{K,r}^{\sh}$ as a subring of $\wt{\a}$.
We denote by $\a_r$ the topological closure
of $\a_{K,r}^{\sh}$ inside $\wt{\a}$ with respect to the strong topology,
which
is a $\varphi$ and $\scrG_K$-stable subring of $\wt{\a}$,
and we note that $\a_r = \varphi^{-r}(\a)$ and $\a_r^{\scrH}= \a_{K,r}$
inside $\wt{\a}$.
We note also that the canonical map $\Z_p\hookrightarrow \wt{\a}^{\varphi=1}$
is an isomorphism, from which it immediately follows that the same is true
if we replace $\wt{\a}$ by any of its subrings constructed above.
If $A$ is any subring of $\wt{\a}$, we define $A^+:=A\cap \wt{\a}^+$,
with the intersection taken inside $\wt{\a}$.
\begin{remark}\label{Slimits}
We will identify $\mathfrak{S}_r$ and $\O_{\E_r}$ with their respective images
$\a_{K,r}^+$ and $\a_{K,r}$ in $\wt{\a}$ under $j_r$.
Writing $\mathfrak{S}_{\infty}:=\varinjlim \mathfrak{S}_r$
and $\O_{\E_{\infty}}:=\varinjlim \mathfrak{S}_r$, we likewise
identify $\mathfrak{S}_{\infty}$ with $\a_{K,\infty}^+$ and $\O_{\E_{\infty}}$
with $\a_{K,\infty}$.
Denoting by $\wh{\mathfrak{S}}_{\infty}$ (respectively $\wt{\mathfrak{S}}_{\infty}$) the $p$-adic (respectively
$(p,u_0)$-adic) completions, one has
\begin{equation*}
\wh{\mathfrak{S}}_{\infty} = \wh{\a}_K^+ = W(\ensuremath{\mathbf{E}}_K^{\rad,+})\quad\text{and}\quad
\wt{\mathfrak{S}}_{\infty} = \wt{\a}_K^+ = W(\wt{\ensuremath{\mathbf{E}}}_K^{+}),
\end{equation*}
for $\ensuremath{\mathbf{E}}_K^{\rad}:=\cup_{r\ge 0} \varphi^{-r}(\ensuremath{\mathbf{E}}_K)$ the radiciel ($=$perfect) closure of
$\ensuremath{\mathbf{E}}_K$ in $\wt{\ensuremath{\mathbf{E}}}$ and $\wt{\ensuremath{\mathbf{E}}}_K$ its topological completion.
Via these identifications, $\omega :=u_0/u_1\in \a_{K,1}^+$ is
a principal generator
of $\ker(\theta:\wt{\a}^+\twoheadrightarrow \O_{\mathbf{C}_K})$.
\end{remark}
We can now explain the functorial relation between $\m_r(G)$ and $T_pG_K$:
\begin{theorem}\label{comparison}
Let $G\in \pdiv_{R_r}^{\Gamma}$, and write $H^1_{\et}(G_K):=(T_pG_K)^{\vee}$
for the $\Z_p$-linear dual of $T_pG_K$.
There is a canonical mapping
of finite free $\a_r^+$-modules with semilinear Frobenius and $\scrG_K$-actions
\begin{equation}
\xymatrix{
{\m_r(G)\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} \a_r^+} \ar[r] & {H^1_{\et}(G_K)\otimes_{\Z_p} \a_r^+}
}
\end{equation}
that is injective with cokernel killed by $u_1$.
Here, $\varphi$ acts as $\varphi_{\m_r(G)}\otimes \varphi$ on source
and as $1\otimes\varphi$ on target, while $\scrG_K$ acts diagonally
on source and target through the quotient $\scrG_K\twoheadrightarrow \Gamma$
on $\m_r(G)$.
In particular, there is a natural $\varphi$ and $\scrG_K$-equivariant
isomorphism
\begin{equation}
{\m_r(G)\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} \a_r} \simeq {H^1_{\et}(G_K)\otimes_{\Z_p} \a_r}.
\label{comparisonb}
\end{equation}
These mappings are compatible with duality and with change in $r$ in the obvious manner.
\end{theorem}
\begin{corollary}\label{GaloisComparison}
For $G\in \pdiv_{R_r}^{\Gamma}$, there are functorial isomorphisms of $\Z_p[\scrG_K]$-modules
\begin{subequations}
\begin{align}
T_pG_K &\simeq \Hom_{\mathfrak{S}_r,\varphi}(\m_r(G),\a_r^+)\\
H^1_{\et}(G_K) &\simeq (\m_r(G) \mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} \a_r)^{\varphi_{\m_r(G)}\otimes \varphi=1}.
\label{FontaineModule}
\end{align}
\end{subequations}
which are compatible with duality and change in $r$. In the first isomorphism,
we view $\a_r^+$ as a $\mathfrak{S}_r$-algebra via the composite of the usual structure map with
$\varphi$.
\end{corollary}
\begin{remark}
By definition, the map $\varphi^r$ on $\O_{\E_r}$ is injective with image $\O_{\E}:=\O_{\E_0}$,
and so induces a $\varphi$-semilinear isomorphism of $W$-algebras
$\xymatrix@C=15pt{{\varphi^{r}:\O_{\E_r}} \ar[r]^-{\simeq}&{\O_{\E}} }$.
It follows from (\ref{FontaineModule}) of Corollary \ref{GaloisComparison} and Fontaine's theory of
$(\varphi,\Gamma)$-modules
over $\O_{\E}$ that $\m_r(G)\otimes_{\mathfrak{S}_r,\varphi^r} \O_{\E}$ {\em is} the $(\varphi,\Gamma)$-module
functorially associated to the $\Z_p[\scrG_K]$-module $H^1_{\et}(G_K)$.
\end{remark}
For the remainder of this section, we recall the construction of the functor $\m_r$, both because we shall need
to reference it in what follows, and because we feel it is enlightening.
For details, including the proofs of Theorems \ref{CaisLauMain}--\ref{comparison}
and Corollary \ref{GaloisComparison}, we refer the reader to \cite{CaisLau}.
Fix $G\in \pdiv_{R_r}^{\Gamma}$ and set $G_0:=G\times_{R_r}{R_r/pR_r}.$
The $\mathfrak{S}_r$-module $\m_r(G)$ is a functorial descent of the evaluation of
the Dieudonn\'e crystal $\ensuremath{\mathbf{D}}(G_0)$ on a certain ``universal" PD-thickening of $R_r/pR_r$,
which we now describe.
Let $S_r$ be the $p$-adic completion of the PD-envelope of $\mathfrak{S}_r$ with respect to the
ideal $\ker\theta$, viewed as a (separated and complete) topological ring via the $p$-adic topology.
We give $S_r$ its PD-filtration: for $q\in \Z$ the ideal $\Fil^q S_r$ is the
topological closure of the ideal generated by $\{\alpha^{[n]}\,:\,\alpha\in \ker\theta,\,n\ge q\}$.
By construction, the map $\theta:\mathfrak{S}_r\twoheadrightarrow R_r$
uniquely extends to a continuous surjection of $\mathfrak{S}_r$-algebras $S_r\twoheadrightarrow R_r$
(which we again denote by $\theta$) whose kernel $\Fil^1 S_r$ is equipped with topologically
PD-nilpotent\footnote{Here we use our assumption that $p>2$.} divided powers.
One shows that there is a unique continuous endomorphism $\varphi$ of $S_r$ extending $\varphi$ on $\mathfrak{S}_r$,
and that $\varphi(\Fil^1 S_r)\subseteq pS_r$; in particular, we may define
$\varphi_1: \Fil^1 S_r\rightarrow S_r$ by $\varphi_1:=\varphi/p$,
which is a $\varphi$-semilinear homomorphism of $S_r$-modules. Note that
$\varphi_1(E_r)$ is a unit of $S_r$, so the image of $\varphi_1$ generates
$S_r$ as an $S_r$-module.
Since the action of $\Gamma$ on $\mathfrak{S}_r$ preserves $\ker\theta$, it follows from the universal mapping property of
divided power envelopes and $p$-adic continuity considerations that this action uniquely extends to
a continuous and $\varphi$-equivariant action of $\Gamma$ on $S_r$ which
is compatible with the PD-structure and the filtration.
Similarly, the transition map $\mathfrak{S}_r\hookrightarrow \mathfrak{S}_{r+1}$ uniquely extends to a
continuous $\mathfrak{S}_r$-algebra homomorphism $:S_r\rightarrow S_{r+1}$ which is moreover compatible with filtrations
(because $E_r(u_r)=E_{r+1}(u_{r+1})$ under our identifications),
and for nonnegative integers $s < r$ we view $S_r$ as an $S_s$-algebra
via these maps.
\begin{definition}
Let $\textswab{BT}_{S_r}^{\varphi}$ be the category of triples $(\mathscr{M},\Fil^1\mathscr{M}, \varphi_{\mathscr{M},1})$ where
\begin{itemize}
\item $\mathscr{M}$ is a finite free $S_r$-module and $\Fil^1\mathscr{M}\subseteq \mathscr{M}$ is an $S_r$-submodule.
\item $\Fil^1\mathscr{M}$ contains $(\Fil^1 S_r)\mathscr{M}$ and the quotient $\mathscr{M}/\Fil^1\mathscr{M}$ is a free
$S_r/\Fil^1S_r=R_r$-module.
\item $\varphi_{\mathscr{M},1}:\Fil^1\mathscr{M}_r\rightarrow \mathscr{M}$ is a $\varphi$-semilinear map whose image
generates $\mathscr{M}$ as an $S_r$-module.
\end{itemize}
Morphisms in $\textswab{BT}_{S_r}^{\varphi}$ are $S_r$-module homomorphisms which are compatible with the
extra structures. As per our convention, we will often write $\mathscr{M}$ for a triple
$(\mathscr{M},\Fil^1\mathscr{M},\varphi_{\mathscr{M},1})$, and $\varphi_1$ for $\varphi_{\mathscr{M},1}$ when it can cause
no confusion. We denote by $\textswab{BT}_{S_r}^{\varphi,\Gamma}$ the subcategory of $\textswab{BT}_{S_r}^{\varphi}$
consisting of objects $\mathscr{M}$ that are equipped
with a semilinear action of $\Gamma$ which preserves $\Fil^1\mathscr{M}$, commutes with $\varphi_{\mathscr{M},1}$,
and whose restriction to $\Gamma_r$ is trivial on $\mathscr{M}/u_r\mathscr{M}$; morphisms in $\textswab{BT}_{S_r}^{\varphi,\Gamma}$
are $\Gamma$-equivariant morphisms in $\textswab{BT}_{S_r}^{\varphi}$.
\end{definition}
The kernel of the surjection $S_r/p^nS_r\twoheadrightarrow R_r/pR_r$ is the image of the ideal
$\Fil^1 S_r + pS_r$, which by construction is equipped topologically PD-nilpotent divided powers.
We may therefore define
\begin{equation}
\mathscr{M}_r(G)=\ensuremath{\mathbf{D}}(G_0)_S:=\varprojlim_n \ensuremath{\mathbf{D}}(G_0)_{S/p^nS},
\end{equation}
which is a finite free $S_r$-module that depends contravariantly functorially on $G_0$.
We promote $\mathscr{M}_r(G)$ to an object of $\textswab{BT}_{S_r}^{\varphi,\Gamma}$ as follows.
As the quotient map $S_r\twoheadrightarrow R_r$ induces a PD-morphism of PD-theckenings
of $R_r/pR_r$, there is a natural isomorphism of free $R_r$-modules
\begin{equation}
\mathscr{M}_r(G)\otimes_{S_r} R_r \simeq \ensuremath{\mathbf{D}}(G_0)_{R_r}.\label{surjR}
\end{equation}
By Proposition \ref{BTgroupUnivExt}, there is a canonical ``Hodge" filtration $\omega_G \subseteq \ensuremath{\mathbf{D}}(G_0)_{R_r}$,
which reflects the fact that $G$ is a $p$-divisible group over $R_r$ lifting $G_0$,
and we define $\Fil^1\mathscr{M}_r(G)$ to be the preimage of $\omega_G$ under the
composite of the isomorphism (\ref{surjR}) with the natural surjection
$\mathscr{M}_r(G)\twoheadrightarrow \mathscr{M}_r(G)\otimes_{S_r} R_r$; note that this depends on $G$ and not just on
$G_0$. The Dieudonn\'e crystal is compatible with arbitrary base change, so the
relative Frobenius $F_{G_0}:G_0\rightarrow G_0^{(p)}$ induces an canonical morphism of $S_r$-modules
\begin{equation*}
\xymatrix{
{\varphi^*\ensuremath{\mathbf{D}}(G_0)_{S_r} \simeq \ensuremath{\mathbf{D}}(G_0^{(p)})_{S_r}} \ar[r]^-{\ensuremath{\mathbf{D}}(F_{G_0})} & {\ensuremath{\mathbf{D}}(G_0)_{S_r}}
},
\end{equation*}
which we may view as a $\varphi$-semilinear map $\varphi_{\mathscr{M}_r(G)}:\mathscr{M}_r(G)\rightarrow \mathscr{M}_r(G)$.
As the relative Frobenius map $\omega_{G_0^{(p)}}\rightarrow \omega_{G_0}$ is zero,
it follows that the restriction of $\varphi_{\mathscr{M}_r(G)}$ to $\Fil^1 \mathscr{M}_r(G)$ has image contained in
$p\mathscr{M}_r(G)$, so we may define $\varphi_{\mathscr{M}_r(G),1}:=\varphi_{\mathscr{M}_r(G)}/p$, and one proves as in
\cite[Lemma A.2]{KisinFCrystal}
that the image of $\varphi_{\mathscr{M}_r(G),1}$ generates $\mathscr{M}_r(G)$ as an $S_r$-module.
It remains to equip $\mathscr{M}_r(G)$ with a canonical semilinear action of $\Gamma$.
Let us write $G_{K_r}$ for the generic fiber of $G$ and $G_{K}$ for its unique
descent to $K=K_0$. The existence of this descent is reflected by the
existence of a commutative diagram with cartesian square
\begin{equation}
\begin{gathered}
\xymatrix{
{G_{K}\mathop{\times}\limits_K K_r} \ar@/^/[rrd]^-{1\times \gamma} \ar@/_/[ddr] \ar@{.>}[dr]|-{\gamma} & & \\
&{\big(G_{K}\mathop{\times}\limits_K K_r\big)_{\gamma}} \ar[r]_-{\rho_1} \ar[d]^-{\rho_2}\ar@{} [dr] |{\square} &
{G_{K}\mathop{\times}\limits_K K_r} \ar[d]\\
&{\Spec(K_r)} \ar[r]_-{\gamma} &{\Spec(K_r)}
}
\end{gathered}
\label{GammaAction}
\end{equation}
for each $\gamma\in \Gamma$, compatibly with change in $\gamma$; here, the subscript of $\gamma$ denotes base change
along the map of schemes induced by $\gamma$.
Since $G$ has generic fiber $G_{K_r}=G_K\times_K K_r$, Tate's Theorem ensures that the
dotted arrow above uniquely extends to an isomorphism
of $p$-divisible groups over $R_r$
\begin{equation}
\xymatrix{
{G}\ar[r]^-{\gamma} & {G_{\gamma}}
},\label{TateExt}
\end{equation}
compatibly with change in $\gamma$.
By assumption, the action of $\Gamma$ on $S_r$ commutes with the divided powers
on $\Fil^1 S_r$ and induces the given action on the quotient $S_r\twoheadrightarrow R_r$;
in other words, $\Gamma$ acts by automorphisms on the object
$(\Spec(R_r/pR_r)\hookrightarrow \Spec(S_r/p^nS_r))$ of $\Cris((R_r/pR_r)/W)$.
Since $\ensuremath{\mathbf{D}}(G_0)$ is a crystal, each $\gamma\in \Gamma$ therefore gives an $S_r$-linear map
\begin{equation*}
\xymatrix{
{\gamma^*\ensuremath{\mathbf{D}}(G_0)_{S_r} \simeq \ensuremath{\mathbf{D}}((G_0)_{\gamma})_{S_r}} \ar[r] & {D(G_0)_{S_r}}
}
\end{equation*}
and hence an $S_r$-semilinear (over $\gamma$) endomorphism $\gamma$ of $\mathscr{M}_r(G)$.
One easily checks that the resulting action of $\Gamma$ on $\mathscr{M}_r(G)$
commutes with $\varphi_{\mathscr{M},1}$ and preserves $\Fil^1\mathscr{M}_r(G)$.
By the compatibility of $\ensuremath{\mathbf{D}}(G_0)$ with base change and the obvious fact that
the $W$-algebra surjection $S_r\twoheadrightarrow W$ sending $u_r$ to $0$
is a PD-morphism over the canonical surjection $R_r/pR_r\twoheadrightarrow k$,
there is a natural isomorphism
\begin{equation}
\mathscr{M}_r(G)\otimes_{S_r} W \simeq \ensuremath{\mathbf{D}}(\o{G})_W.
\end{equation}
It follows easily from this and the diagram (\ref{GammaAction})
that the action of $\Gamma_r$ on $\mathscr{M}_r(G)/u_r\mathscr{M}_r(G)$
is trivial.
To define $\m_r(G)$, we functorially descend the $S_r$-module $\mathscr{M}_r(G)$
along the structure morphism $\alpha_r:\mathfrak{S}_r\rightarrow S_r$. More precisely,
for $\m\in \textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$, we define
${\alpha_r}_*(\m):=(M,\Fil^1M,\Phi_1)\in \textswab{BT}_{S_r}^{\varphi,\Gamma}$ via:
\begin{equation}
\begin{gathered}
M:=\m\mathop{\otimes}\limits_{\mathfrak{S}_r,\alpha_r\circ\varphi} S_r\qquad\text{with diagonal $\Gamma$-action}\\
\Fil^1 M :=\left\{ m\in M\ :\ \varphi_{\m}\otimes\id (m) \in \m\otimes_{\mathfrak{S}_r} \Fil^1 S_r
\subseteq \m\otimes_{\mathfrak{S}_r} S_r \right\} \\
\xymatrix{
{\Phi_1: \Fil^1 M} \ar[r]^-{\varphi_{\m}\otimes\id} & { \m\mathop{\otimes}\limits_{\mathfrak{S}_r} \Fil^1 S_r}
\ar[r]^-{\id\otimes\varphi_1} & {\m\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} S_r = M}
}.
\end{gathered}
\label{BreuilSrDef}
\end{equation}
The following is the key technical point of \cite{CaisLau}, and is proved using
the theory of windows:
\begin{theorem}\label{Lau}
For each $r$, the functor ${\alpha_r}_*:\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}\rightarrow \textswab{BT}_{S_r}^{\varphi,\Gamma}$
is an equivalence of categories, compatible with change in $r$.
\end{theorem}
\begin{definition}
For $G\in \pdiv_{R_r}^{\Gamma}$, we write $\m_r(G)$
for the functorial descent of $\mathscr{M}_r(G)$ to an object of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$
as guaranteed by Theorem \ref{Lau}.
By construction, we have a natural isomorphism
of functors ${\alpha_r}_*\circ \m_r\simeq \mathscr{M}_r$ on $\pdiv_{R_r}^{\Gamma}$.
\end{definition}
\begin{example}\label{GmQpZpExamples}
Using Messing's description of the Dieudonn\'e crystal of a $p$-divisible group
in terms of the Lie algebra of its universal extension (cf. remark \ref{MessingRem}),
one calculates that for $r\ge 1$
\begin{subequations}
\begin{equation}
\m_r(\Q_p/\Z_p) = \mathfrak{S}_r,\qquad \varphi_{\m_r(\Q_p/\Z_p)}:= \varphi,\qquad \gamma:=\gamma
\label{MrQpZp}
\end{equation}
\begin{equation}
\m_r(\mu_{p^{\infty}}) = \mathfrak{S}_r,\qquad \varphi_{\m_r(\mu_{p^{\infty}})}:= \omega\cdot\varphi,
\qquad \gamma:=\chi(\gamma)^{-1}\varphi^{r-1}(\gamma u_r/u_r)\cdot \gamma
\label{MrMu}
\end{equation}
\end{subequations}
with $\gamma\in \Gamma$ acting as indicated.
Note that both $\m_r(\Q_p/\Z_p)$ and $\m_r(\Gm[p^{\infty}])$
arise by base change from their incarnations when $r=1$,
as follows from the fact that $\omega = \varphi(u_1)/u_1$ and
$\varphi^{r-1}(\gamma u_r/u_r)=\gamma u_1/u_1$ via our identifications.
\end{example}
\subsection{The case of ordinary \texorpdfstring{$p$}{p}-divisible groups}
When $G\in \pdiv_{R_r}^{\Gamma}$ is ordinary,
one can say significantly more about the structure of the $\mathfrak{S}_r$-module $\m_r(G)$.
To begin with, we observe that for arbitrary $G\in \pdiv_{R_r}^{\Gamma}$,
the formation of the maximal \'etale quotient of $G$ and of the maximal
connected and multiplicative-type sub $p$-divisible groups of $G$ are functorial in $G$,
so each of $G^{\et}$, $G^0$, and $G^{\mult}$ is naturally an object of $\pdiv_{R_r}^{\Gamma}$
as well. We thus (functorially) obtain objects
$\m_r(G^{\star})$ of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi, \Gamma}$ which admit particularly simple
descriptions when $\star=\et$ or $\mult$, as we now explain.
As usual, we write $\o{G}^{\star}$ for the special fiber of $G^{\star}$ and $\ensuremath{\mathbf{D}}(\o{G}^{\star})_W$
for its Dieudonn\'e module.
Twisting the $W$-algebra structure on $\mathfrak{S}_r$ by the automorphism $\varphi^{r-1}$
of $W$,
we define objects of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$
\begin{subequations}
\begin{equation}
\m_r^{\et}(G) : = \ensuremath{\mathbf{D}}(\o{G}^{\et})_W\mathop{\otimes}\limits_{W,\varphi^{r-1}} \mathfrak{S}_r,
\qquad \varphi_{\m_r^{\et}}:= F\otimes \varphi,
\qquad \gamma:=\gamma \otimes \gamma
\label{MrEtDef}
\end{equation}
\begin{equation}
\m_r^{\mult}(G) : = \ensuremath{\mathbf{D}}(\o{G}^{\mult})_W\mathop{\otimes}\limits_{W,\varphi^{r-1}} \mathfrak{S}_r,
\qquad \varphi_{\m_r^{\mult}}:= V^{-1}\otimes E_r\cdot\varphi,
\qquad \gamma:=\gamma \otimes \chi(\gamma)^{-1}\varphi^{r-1}(\gamma u_r/u_r)\cdot \gamma
\label{MrMultDef}
\end{equation}
\end{subequations}
with $\gamma\in \Gamma$ acting as indicated.
Note that these formulae make sense and do indeed give objects of $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$
as $V$ is
invertible\footnote{A $\varphi^{-1}$-semilinear map of $W$-modules $V:D\rightarrow D$
is {\em invertible} if there exists a $\varphi$-semilinear endomorphism $V^{-1}$ whose composition
with $V$ in either order is the identity. This is easily seen to be equivalent to
the invertibility of the linear map $V\otimes 1: D\rightarrow \varphi^* D$, with $V^{-1}$
the composite of $(V\otimes 1)^{-1}$ and the $\varphi$-semilinear map $\id\otimes 1:D\rightarrow \varphi^*D$.
}
on $\ensuremath{\mathbf{D}}(\o{G}^{\mult})_W$ and $\gamma u_r/u_r \in \mathfrak{S}_r^{\times}$.
It follows easily from these definitions that $\varphi_{\m_r^{\star}}$
linearizes to an isomorphism when $\star=\et$ and has image
contained in $\omega\cdot \m_r^{\mult}(G)$ when $\star=\mult$
Of course, $\m_r^{\star}(G)$
is contravariantly functorial in---and depends only on---the closed fiber $\o{G}^{\star}$ of $G^{\star}$.
\begin{proposition}\label{EtaleMultDescription}
Let $G$ be an object of $\pdiv_{R_r}^{\Gamma}$ and let $\m_r^{\et}(G)$ and $\m_r^{\mult}(G)$
be as above. The map $F^r:G_0 \rightarrow G_0^{(p^r)}$
$($respectively $V^r:G_0^{(p^r)}\rightarrow G_0$$)$ induces a natural isomorphism
in $\textswab{BT}_{\mathfrak{S}_r}^{\Gamma}$
\begin{equation}
\m_r(G^{\et}) \simeq \m_r^{\et}(G)\qquad\text{respectively}\qquad
\m_r(G^{\mult}) \simeq \m_r^{\mult}(G).\label{EtMultSpecialIsoms}
\end{equation}
These identifications are compatible with change in $r$
in the sense that for $\star=\et$ $($respectively $\star=\mult$$)$ there is a canonical
commutative diagram in $\textswab{BT}_{\mathfrak{S}_{r+1}}^{\Gamma}$
\begin{equation}
\begin{gathered}
\xymatrix{
{\m_{r+1}(G^{\star}\times_{R_r} R_{r+1})}
\ar[r]_-{\simeq}^-{(\ref{EtMultSpecialIsoms})}\ar[d]_-{\simeq} &
{\m_{r+1}^{\star}(G\times_{R_r} R_{r+1})} \ar@{=}[r] &
{\ensuremath{\mathbf{D}}(\o{G}^{\star})_W\mathop{\otimes}\limits_{W,\varphi^r} \mathfrak{S}_{r+1}}
\ar[d]^-{F\otimes\id\ (\text{respectively}\ V^{-1}\otimes\id)}_-{\simeq} \\
{\m_r(G^{\star})\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{r+1}} \ar[r]^-{\simeq}_-{(\ref{EtMultSpecialIsoms})} &
{\m_r^{\star}(G)\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{r+1}} \ar@{=}[r] &
{\ensuremath{\mathbf{D}}(\o{G}^{\star})_W\mathop{\otimes}\limits_{W,\varphi^{r-1}} \mathfrak{S}_{r+1}}
}
\end{gathered}
\label{EtMultSpecialIsomsBC}
\end{equation}
where the left vertical isomorphism is deduced from Theorem $\ref{CaisLauMain}$ $(\ref{BaseChangeIsom}).$
\end{proposition}
\begin{proof}
For ease of notation, we will write $\m_r^{\star}$ and
and $\ensuremath{\mathbf{D}}^{\star}$ for $\m_r^{\star}(G)$ and $\ensuremath{\mathbf{D}}(\o{G}^{\star})_W$, respectively.
Using (\ref{BreuilSrDef}), one finds that $\mathscr{M}_r^{\et}:={\alpha_r}_*(\m_r^{\et})\in \textswab{BT}_{S_r}^{\varphi,\Gamma}$
is given by the triple
\begin{subequations}
\begin{equation}
\mathscr{M}_r^{\et}:=(\ensuremath{\mathbf{D}}^{\et}\otimes_{W,\varphi^r} S_r,\ \ensuremath{\mathbf{D}}^{\et}\otimes_{W,\varphi^r} \Fil^1 S_r,\
F\otimes \varphi_1)
\end{equation}
with $\Gamma$ acting diagonally on the tensor product. Similarly,
${\alpha_r}_*(\m_r^{\mult})$ is given by the triple
\begin{equation}
(\ensuremath{\mathbf{D}}^{\mult}\otimes_{W,\varphi^r} S_r,\ \ensuremath{\mathbf{D}}^{\mult}\otimes_{W,\varphi^r} S_r,\
V^{-1} \otimes v_r\cdot\varphi)\label{WindowMultCase}
\end{equation}
\end{subequations}
where $v_r=\varphi(E_r)/p$ and $\gamma\in \Gamma$ acts on $\ensuremath{\mathbf{D}}^{\mult}\otimes_{W,\varphi^r} S_r$
as $\gamma \otimes \chi(\gamma)^{-1} \varphi^r(\gamma u_r/u_r)\cdot \gamma$.
Put $\lambda := \log(1+u_0)/{u_0},$
where $\log(1+X):\Fil^1 S_r\rightarrow S_r$ is the usual (convergent for the $p$-adic topology) power series
and $u_0$ is viewed as an element of $S_r$ via the structure map $S_0\rightarrow S_r$
(concretely, $u_0= \varphi^r(u_r)$).
For each $r\ge 0$, one checks that $\lambda$ admits the convergent product expansion
$\lambda=\prod_{i\ge 0} \varphi^i(v_r)$, so $\lambda\in S_r^{\times}$ and
\begin{equation}
\frac{\lambda}{\varphi(\lambda)} = \varphi(E_r)/p= v_r\qquad\text{and}\qquad
\frac{\lambda}{\gamma\lambda} = \chi(\gamma)^{-1}\varphi^r(\gamma u_r/u_r) \quad\text{for}\
\gamma\in \Gamma.\label{lambdaTransformation}
\end{equation}
It follows from (\ref{lambdaTransformation})
that the $S_r$-module automorphism of $\ensuremath{\mathbf{D}}^{\mult}\otimes_{W,\varphi^r} S_r$
given by multiplication by $\lambda$
carries (\ref{WindowMultCase}) isomorphically onto the object of $\textswab{BT}_{S_r}^{\varphi,\Gamma}$
given by the triple
\begin{equation}
\mathscr{M}_r^{\mult}:=(\ensuremath{\mathbf{D}}^{\mult}\otimes_{W,\varphi^r} S_r,\ \ensuremath{\mathbf{D}}^{\mult}\otimes_{W,\varphi^r} S_r,\
V^{-1}\otimes\varphi)
\end{equation}
with $\Gamma$ acting {\em diagonally} on the tensor product.
On the other hand, since $G_0^{\et}$ (respectively $G_0^{\mult}$) is \'etale
(respectively of multiplicative type) over $R_r/pR_r$, the relative Frobenius
(respectively Verscheibung) morphism of $G_0$ induces isomorphisms
\begin{subequations}
\begin{equation}
\xymatrix{
{G_0^{\et}} \ar[r]_-{\simeq}^-{F^r} & {(G_0^{\et})^{(p^r)} \simeq
{\varphi^r}^*\o{G}^{\et} \times_k R_r/pR_r}
}\label{Ftrick}
\end{equation}
respectively
\begin{equation}
\xymatrix{
{G_0^{\mult}} & \ar[l]^-{\simeq}_-{V^r} {(G_0^{\mult})^{(p^r)}
\simeq {\varphi^r}^*\o{G}^{\mult} \times_k R_r/pR_r}
}\label{Vtrick}
\end{equation}
\end{subequations}
of $p$-divisible groups over $R_r/pR_r$, where we have used the fact that the map $x\mapsto x^{p^r}$
of $R_r/pR_r$ factors as $R_r/pR_r \twoheadrightarrow k \hookrightarrow R_r/pR_r$
in the final isomorphisms of both lines above. Since the Dieudonn\'e crystal is compatible
with base change and the canonical map $W\rightarrow S_r$ extends to a PD-morphism
$(W,p)\rightarrow (S_r, pS_r+\Fil^1 S_r)$ over $k\rightarrow R_r/pR_r$,
applying $\ensuremath{\mathbf{D}}(\cdot)_{S_r}$ to (\ref{Ftrick})--(\ref{Vtrick}) yields natural isomorphisms
$\ensuremath{\mathbf{D}}(G_0^{\star})_{S_r} \simeq \ensuremath{\mathbf{D}}^{\star}\otimes_{W,\varphi^r} S_r$ for $\star=\et,\mult$
which carry $F$ to $F\otimes \varphi$. It is a straightforward exercise using the construction
of $\mathscr{M}_r(G^{\star})$ given in \S\ref{pDivPhiGamma} to check
that these isomorphisms extend to give isomorphisms $\mathscr{M}_r(G^{\et}) \simeq \mathscr{M}_r^{\et}$
and $\mathscr{M}_r(G^{\mult}) \simeq \mathscr{M}_r^{\mult}$ in $\textswab{BT}_{S_r}^{\varphi,\Gamma}$.
By Theorem \ref{Lau},
we conclude that we have natural isomorphisms in $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$
as in (\ref{EtMultSpecialIsoms}). The commutativity of (\ref{EtMultSpecialIsomsBC})
is straightforward, using the definitions of the base change isomorphisms.
\end{proof}
Now suppose that $G$ is ordinary.
As $\m_r$ is exact by Theorem \ref{CaisLauMain} (\ref{exequiv}),
applying $\m_r$ to the connected-\'etale sequence of $G$ gives
a short exact sequence in $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$
\begin{equation}
\xymatrix{
0\ar[r] & {\m_r(G^{\et})} \ar[r] & {\m_r(G)} \ar[r] & {\m_r(G^{\mult})} \ar[r] & 0
}\label{ConEtOrdinary}
\end{equation}
which is contravariantly functorial and exact in $G$.
Since $\varphi_{\m_r}$ linearizes to an isomorphism on $\m_r(G^{\et})$
and is topologically nilpotent on $\m_r(G^{\mult})$, we think of
(\ref{ConEtOrdinary}) as the ``slope flitration" for Frobenius acting on $\m_r(G)$.
On the other hand, Proposition \ref{BTgroupUnivExt} and Theorem \ref{CaisLauMain} (\ref{EvaluationONR})
provide
a canonical ``Hodge filtration" of $\m_r(G)\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} R_r\simeq \ensuremath{\mathbf{D}}(G_0)_{R_r}$:
\begin{equation}
\xymatrix{
0\ar[r] & {\omega_{G}} \ar[r] & {\ensuremath{\mathbf{D}}(G_0)_{R_r}} \ar[r] & {\Lie(G^t)} \ar[r] & 0
}\label{HodgeFilOrd}
\end{equation}
which is contravariant and exact in $G$.
Our assumption that $G$ is ordinary yields
({\em cf.} \cite{KatzSerreTate}):
\begin{lemma}\label{HodgeFilOrdProps}
With notation as above, there are natural and $\Gamma$-equivariant
isomorphisms
\begin{equation}
\Lie(G^t)\simeq \ensuremath{\mathbf{D}}(G_0^{\et})_{R_r} \qquad\text{and} \qquad \ensuremath{\mathbf{D}}(G_0^{\mult})_{R_r}\simeq \omega_G.
\label{FlankingIdens}
\end{equation}
Composing these isomorphisms with the canonical maps obtained by applying $\ensuremath{\mathbf{D}}(\cdot)_{R_r}$
to the connected-\'etale sequence of $G_0$
yield functorial $R_r$-linear splittings of the Hodge filtration $(\ref{HodgeFilOrd})$.
Furthermore, there is a canonical and $\Gamma$-equivariant isomorphism of split exact
sequences of $R_r$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\omega_{G}} \ar[r]\ar[d]^-{\simeq} & {\ensuremath{\mathbf{D}}(G_0)_{R_r}} \ar[r]\ar[d]^-{\simeq} &
{\Lie(G^t)} \ar[r]\ar[d]^-{\simeq} & 0\\
0 \ar[r] & {\ensuremath{\mathbf{D}}(\o{G}^{\mult})_W\mathop{\otimes}\limits_{W,\varphi^r} R_r} \ar[r]_-{i} &
{\ensuremath{\mathbf{D}}(\o{G})_W\mathop{\otimes}\limits_{W,\varphi^r} R_r} \ar[r]_-{j} &
{\ensuremath{\mathbf{D}}(\o{G}^{\et})_W\mathop{\otimes}\limits_{W,\varphi^r} R_r}\ar[r] & 0
}
\end{gathered}
\label{DescentToWIsom}
\end{equation}
with $i,j$ the inclusion and projection mappings
obtained from the canonical direct sum decomposition
$\ensuremath{\mathbf{D}}(\o{G})_W\simeq \ensuremath{\mathbf{D}}(\o{G}^{\mult})_W\oplus \ensuremath{\mathbf{D}}(\o{G}^{\et})_W$.
\end{lemma}
\begin{proof}
Applying $\ensuremath{\mathbf{D}}(\cdot)_{R_r}$ to the connected-\'etale sequence
of $G_0$ and using Proposition \ref{BTgroupUnivExt} yields a commutative diagram
with exact columns and rows
\begin{equation}
\begin{gathered}
\xymatrix{
& & 0\ar[d] & 0 \ar[d] & \\
& 0 \ar[r]\ar[d] & {\omega_{G}}\ar[r]\ar[d] &
{\omega_{G^{\mult}}} \ar[r]\ar[d] & 0\\
0\ar[r] & {\ensuremath{\mathbf{D}}(G_0^{\et})_{R_r}} \ar[r]\ar[d] & {\ensuremath{\mathbf{D}}(G_0)_{R_r}}\ar[r]\ar[d] &
{\ensuremath{\mathbf{D}}(G_0^{\mult})_{R_r}} \ar[r]\ar[d] & 0\\
0 \ar[r] & {\Lie({G^{\et}}^t)} \ar[r]\ar[d] & {\Lie(G^t)}\ar[r]\ar[d] &
0 & \\
& 0 & 0 & &
}
\end{gathered}
\label{OrdinaryDiagram}
\end{equation}
where we have used the fact that that the invariant differentials
and Lie algebra of an \'etale $p$-divisible group
(such as $G^{\et}$ and ${G^{\mult}}^t\simeq {G^t}^{\et}$)
are both zero. The isomorphisms (\ref{FlankingIdens})
follow at once. We likewise immediately see that the short exact sequence
in the center column of (\ref{OrdinaryDiagram}) is functorially and $R_r$-linearly
split. Thus, to prove the claimed identification in (\ref{DescentToWIsom}),
it suffices to exhibit natural isomorphisms of free $R_r$-modules with $\Gamma$-action
\begin{equation}
\ensuremath{\mathbf{D}}(G_0^{\et})_{R_r} \simeq \ensuremath{\mathbf{D}}(\o{G}^{\et})_W\mathop{\otimes}\limits_{W,\varphi^r} R_r
\qquad\text{and}\qquad
\ensuremath{\mathbf{D}}(G_0^{\mult})_{R_r} \simeq \ensuremath{\mathbf{D}}(\o{G}^{\mult})_W\mathop{\otimes}\limits_{W,\varphi^r} R_r,
\label{TwistyDieuIsoms}
\end{equation}
both of which follow easily by applying $\ensuremath{\mathbf{D}}(\cdot)_{R_r}$ to
(\ref{Ftrick}) and (\ref{Vtrick}) and using the compatibility
of the Dieudonn\'e crystal with base change as in the proof of Proposition (\ref{EtaleMultDescription}).
\end{proof}
From the slope filtration (\ref{ConEtOrdinary}) of $\m_r(G)$
we can recover both the (split) slope filtration of $\ensuremath{\mathbf{D}}(\o{G})_W$
and the (split) Hodge filtration (\ref{HodgeFilOrd}) of $\ensuremath{\mathbf{D}}(G_0)_{R_r}$:
\begin{proposition}\label{MrToHodge}
There are canonical and $\Gamma$-equivariant isomorphisms of short exact sequences
\begin{subequations}
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_r(G^{\et})\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi\circ\tau} W} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G)\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi\circ\tau} W} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G^{\mult})\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi\circ\tau} W} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\ensuremath{\mathbf{D}}(\o{G}^{\et})_W} \ar[r] & {\ensuremath{\mathbf{D}}(\o{G})_W} \ar[r] & {\ensuremath{\mathbf{D}}(\o{G}^{\mult})_W}
\ar[r] & 0
}\label{MrToDieudonneMap}
\end{gathered}
\end{equation}
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_r(G^{\et})\mathop{\otimes}\limits_{\mathfrak{S}_r,\theta\circ\varphi} R_r} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G)\mathop{\otimes}\limits_{\mathfrak{S}_r,\theta\circ\varphi} R_r} \ar[r]\ar[d]^-{\simeq} &
{\m_r(G^{\mult})\mathop{\otimes}\limits_{\mathfrak{S}_r,\theta\circ\varphi} R_r} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\Lie(G^t)} \ar[r]_-{i} & {\ensuremath{\mathbf{D}}(G_0)_{R_r}} \ar[r]_-{j} & {\omega_{G}} \ar[r] & 0\\
}
\end{gathered}\label{MrToHodgeMap}
\end{equation}
\end{subequations}
Here, $i:\Lie(G^t)\hookrightarrow \ensuremath{\mathbf{D}}(G_0)_{R_r}$
and $j:\ensuremath{\mathbf{D}}(G_0)_{R_r}\twoheadrightarrow \omega_{G}$
are the canonical splittings of Lemma $\ref{HodgeFilOrdProps}$,
the top row of $(\ref{MrToHodgeMap})$ is obtained from $(\ref{ConEtOrdinary})$ by extension of scalars,
and the isomorphism $(\ref{MrToDieudonneMap})$ intertwines $\varphi_{\m_r(\cdot)}\otimes \varphi$ with $F\otimes \varphi$
and $\psi\otimes 1$ with $V\otimes 1$.
\end{proposition}
\begin{proof}
This follows immediately from Theorem \ref{CaisLauMain} (\ref{EvaluationONW}) and Lemma \ref{HodgeFilOrdProps}.
\end{proof}
\section{Results and Main Theorems}\label{results}
In this section, we will state and prove our main results as described in \S\ref{resultsintro}.
Throughout, we will keep the notation of \S\ref{resultsintro} and of
\S\ref{pDivPhiGamma} with $k:=\mathbf{F}_p$.
\subsection{The formalism of towers}\label{TowerFormalism}
In this preliminary section, we set up a general commutative algebra framework
for dealing with the various projective limits of cohomology modules
that we will encounter.
\begin{definition}
A {\em tower of rings} is an inductive system $\mathscr{A}:=\{A_r\}_{r\ge 1}$ of local rings with local
transition maps. A {\em morphism of towers} $\mathscr{A}\rightarrow \mathscr{A}'$
is a collection of local ring homomorphisms $A_r\rightarrow A_r'$ which are compatible
with change in $r$.
A {\em tower of $\mathscr{A}$-modules} $\mathscr{M}$
consists of the following data:
\begin{enumerate}
\item For each integer $r\ge 1$, an $A_r$-module $M_r$.
\item A collection of $A_r$-module homomorphisms
$\varphi_{r,s}:M_r\rightarrow M_{s}\otimes_{A_{s}} A_r $
for each pair of integers $r\ge s\ge 1$, which are compatible
in the obvious way under composition.
\end{enumerate}
A {\em morphism of towers of $\mathscr{A}$-modules} $\mathscr{M}\rightarrow \mathscr{M}'$
is a collection of $A_r$-module homomorphisms $M_r\rightarrow M_r'$ which
are compatible with change in $r$ in the evident manner. For a tower of
rings $\mathscr{A}=\{A_r\}$, we will write $A_{\infty}$ for the inductive limit,
and for
a tower of $\mathscr{A}$-modules $\mathscr{M}=\{M_r\}$, we set
\begin{equation*}
M_B := \varprojlim_r \left( M_r\otimes_{A_r} B\right)\quad\text{and write simply}\quad
M_{\infty}:=M_{A_{\infty}},
\end{equation*}
for any $A_{\infty}$-algebra $B$, with the projective limit taken with respect to the induced transition maps.
\end{definition}
\begin{lemma}\label{Technical}
Let $\mathscr{A}=\{A_r\}_{r\ge 0}$ be a tower of rings
and suppose that $I_r\subseteq A_r$ is a sequence of proper principal ideals
such that $A_r$ is $I_r$-separated and
the image of $I_{r}$ in $A_{r+1}$ is contained in $I_{r+1}$ for all $r$.
Write $I_{\infty}:=\varinjlim I_r$
for the inductive limit, and set $\o{A}_r:=A_r/I_r$ for all $r$.
Let $\mathscr{M}=\{M_r,\rho_{r,s}\}$ be a tower of $\mathscr{A}$-modules
equipped with an action\footnote{That is, a homomorphism of
groups $\Delta\rightarrow \Aut_{\mathscr{A}}(\mathscr{M})$,
or equivalently, an $A_r$-linear action of $\Delta$ on $M_r$
for each $r$ that is compatible with change in $r$.
}
of $\Delta$ by $\mathscr{A}$-automorphisms. Suppose that
$M_r$ is free of finite rank over $A_r$ for all $r$, and that
$\Delta_r$ acts trivially on $M_r$.
Let $B$ be an $A_{\infty}$-algebra, and observe that $M_B$
is canonically a module over the completed group ring $\Lambda_B$.
Assume that $B$ is either flat over $A_{\infty}$ or
that $B$ is a flat $\o{A}_{\infty}$-algebra,
and that the following two conditions hold for all $r>0$
\begin{enumerate}
\setcounter{equation}{1}
\renewcommand{\theenumi}{\theequation{\rm\alph{enumi}}}
{\setlength\itemindent{10pt}
\item $\o{M}_r:=M_r/I_rM_r$ is a free $\o{A}_r[\Delta/\Delta_r]$-module of rank
$d$ that is independent of $r$.\label{freehyp}}
{\setlength\itemindent{10pt}
\item For all $s\le r$ the induced maps
$\xymatrix@1{
{\overline{\rho}_{r,s}: \o{M}_r}\ar[r] &
{\o{M}_{s}\otimes_{\o{A}_{s}} \o{A}_{r}}
}$
are surjective.\label{surjhyp}}
\end{enumerate}
Then:
\begin{enumerate}
\item $M_r$ is a free $A_r[\Delta/\Delta_r]$-module of rank $d$ for all $r$.\label{red2pfree}
\item The induced maps of $A_r[\Delta/\Delta_{s}]$-modules
\begin{equation*}
\xymatrix{
{M_r \otimes_{A_r[\Delta/\Delta_r]} A_r[\Delta/\Delta_{s}]} \ar[r] & {M_{s}\otimes_{A_{s}} A_r}
}
\end{equation*}
are isomorphisms for all $r\ge s$.\label{red2psurj}
\item $M_B$ is a finite free $\Lambda_{B}$-module of rank $d$. \label{MBfree}
\item For each $r$, the canonical map
\begin{equation*}
\xymatrix{
{M_B\otimes_{\Lambda_B} B[\Delta/\Delta_r]} \ar[r] & {M_r\otimes_{A_r} B}
}
\end{equation*}
is an isomorphism of $B[\Delta/\Delta_r]$-modules.
\label{ControlLimit}
\item If $B'$ is any
$B$-algebra which is flat over
$A_{\infty}$ or $\o{A}_{\infty}$, then the canonical map
\begin{equation*}
\xymatrix{
{M_B\otimes_{\Lambda_B} \Lambda_{B'}} \ar[r] & {M_{B'}}
}
\end{equation*}
is an isomorphism of finite free $\Lambda_{B'}$-modules.\label{CompletedBaseChange}
\end{enumerate}
\end{lemma}
\begin{proof}
For notational ease, let us put $\Lambda_{A_r,s}:=A_r[\Delta/\Delta_{s}]$ for all pairs of nonnegative integers
$r,s$. Note that $\Lambda_{A_r,s}$ is a local $A_r$-algebra, so the principal
ideal $\widetilde{I}_r:=I_r\Lambda_{A_r,s}$ is
contained in the radical of $\Lambda_{A_r,s}$.
Let us fix $r$ and choose a principal generator $f_r\in A_r$ of
$I_r$ (hence also of $\widetilde{I}_r$). The module $M_r$ is obviously finite
over $\Lambda_{A_r,r}$ (as it is even finite over $A_r$), so by hypothesis (\ref{freehyp})
we may choose $m_1,\ldots,m_{d}\in M_r$
with the property that the images of the $m_i$ in $\o{M}_r=M_r/\widetilde{I}_rM_r$
freely generate $\o{M}_r$ as an
$\o{A}_r[\Delta/\Delta_r]=\Lambda_{A_r,r}/\widetilde{I}_r$-module.
By Nakayama's Lemma \cite[Corollary to Theorem 2.2]{matsumura}, we conclude
that $m_1,\ldots,m_{d}$ generate $M_r$ as a $\Lambda_{A_r,r}$-module.
If
\begin{equation}
\sum_{i=1}^{d} x_i m_i =0\label{genreln}
\end{equation}
is any relation on the $m_i$ with $x_i\in \Lambda_{A_r,r}$, then necessarily
$x_i\in \widetilde{I}_r\Lambda_{A_r,r}$,
and we claim that $x_i\in \widetilde{I}_r^j$ for all $j\ge 0$. To see this, we proceed by induction and suppose
that our claim holds for $j\le N$. Since $\widetilde{I}_r$ is principal,
for each $i$ there exists $x_i'\in \Lambda_{A_r,r}$
with $x_i = f_r^N x_i'$, and the relation (\ref{genreln}) reads $f_r^Nm=0$ with $m\in M_r$ given by
$m:=\sum_{i=1}^{d} x_i'm_i.$
Since $M_r$ is free as an $A_r$-module, it is in particular torsion free, so we conclude that
$m=0$. Since the images of the $m_i$ {\em freely} generate $M_r/\wt{I}_rM_r$,
it follows that $x_i'\in \widetilde{I}_r$ and hence that $x_i\in \widetilde{I}_r^{N+1}$,
which completes the induction.
By our assumption that $A_r$ is $I_r$-adically separated, we must have $x_i=0$ for all $i$
and the relation (\ref{genreln})
is trivial. We conclude that $m_1,\ldots,m_{d}$ freely generate $M_r$ over $\Lambda_{A_r,r}$, giving
(\ref{red2pfree}).
To prove (\ref{red2psurj}), note that our assumption (\ref{surjhyp}) that the maps $\overline{\rho}_{r,s}$ are
surjective for all $r\ge s$ implies that the same is true of the maps $\rho_{r,s}$
(again by Nakayama's Lemma) and hence that the induced map of
$\Lambda_{A_r,s}$-modules in (\ref{red2psurj}) is surjective.
As this map is then a surjective map of free $\Lambda_{A_r,s}$-modules of the same
rank $d$, it must be an isomorphism.
Since the kernel of the canonical surjection $\Lambda_{A_r,r}\twoheadrightarrow \Lambda_{A_r,s}$
lies in the radical of $\Lambda_{A_r,r}$, we deduce by Nakayama's Lemma that
any lift to $M_r$ of a $\Lambda_{A_r,s}$-basis of $M_{s}\otimes_{A_{s}} A_r$
is a $\Lambda_{A_r,r}$-basis of $M_r$.
It follows easily from this that the projective limit $M_B$ is a free $\Lambda_{B}$-module of rank
$d$ for any flat $A_{\infty}$-algebra $B$. The corresponding assertions for any flat
$\o{A}_{\infty}$-algebra $B$ follow similarly, using the hypotheses (\ref{freehyp})
and (\ref{surjhyp}) directly, and this gives (\ref{MBfree}).
Observe that the mapping of (\ref{ControlLimit}) is obtained from the canonical
surjection $M_B\twoheadrightarrow M_r\otimes_{A_r} B$ by extension of scalars,
keeping in mind the natural identification
$M_r\otimes_{A_r} B \otimes_{\Lambda_B} B[\Delta/\Delta_r] \simeq M_r \otimes_{A_r} B.$
It follows at once that this mapping is surjective. By (\ref{red2pfree}) and (\ref{MBfree}), we conclude that
the mapping in (\ref{ControlLimit}) is a surjection of free $B[\Delta/\Delta_r]$-modules
of the same rank and is hence an isomorphism as claimed.
It remains to
prove (\ref{CompletedBaseChange}).
Extending scalars,
the canonical maps
$M_B\twoheadrightarrow M_r\otimes_{A_r} B$ induce surjections
\begin{equation*}
\xymatrix{
{M_{B}\otimes_{\Lambda_B} \Lambda_{B'}} \ar@{->>}[r] &
{(M_r\otimes_{A_r} B)\otimes_{\Lambda_B} \Lambda_{B'} \simeq M_r\otimes_{A_r} B'}
}
\end{equation*}
that are compatible in the evident manner with change in $r$. Passing to inverse
limits gives the mapping $M_{B}\otimes_{\Lambda_{B}}\Lambda_{B'}\rightarrow M_{B'}$
of (\ref{CompletedBaseChange}). Due to (\ref{MBfree}),
this is then a map of finite free $\Lambda_{B'}$-modules of the same rank,
so to check that it is an isomorphism it suffices by Nakayama's Lemma
to do so after applying $\otimes_{\Lambda_{B'}} B'[\Delta/\Delta_r]$, which is
an immediate consequence of (\ref{ControlLimit}).
\end{proof}
We record the following elementary commutative algebra fact,
which will be extremely useful to us:
\begin{lemma}\label{fflatfreedescent}
Let $A\rightarrow B$ be a local homomorphism of local rings which makes $B$ into
a flat $A$-algebra, and let $M$ be an arbitrary $A$-module.
Then $M$ is a free $A$-module of finite rank if and only if $M\otimes_A B$
is a free $B$-module of finite rank.
\end{lemma}
\begin{proof}
First observe that since $A\rightarrow B$ is local and flat, it is faithfully flat.
We write $M=\varinjlim M_{\alpha}$ as the direct limit of its finite $A$-submodules,
whence $M\otimes_A B = \varinjlim (M_{\alpha}\otimes_A B)$ with each of $M_{\alpha}\otimes_A B$
naturally a finitely generated $B$-submodule of $M\otimes_A B$. Assume that $M\otimes_A B$
is finitely generated as a $B$-module. Then there exists $\alpha$ with
$M_{\alpha}\otimes_A B\rightarrow M\otimes_A B$ surjective, and as $B$ is faithfully flat over $A$,
this implies that $M_{\alpha}\rightarrow M$ is surjective, whence $M$ is finitely generated over $A$.
Suppose in addition that $M\otimes_A B$ is free as a $B$-module. In particular, $M\otimes_A B$
is $B$-flat, which implies by faithful flatness of $B$ over $A$ that $M$ is $A$-flat
(see, e.g. \cite[Exercise 7.1]{matsumura}). Then $M$ is a finite flat module over the local ring $A$,
whence it is free as an $A$-module by \cite[Theorem 7.10]{matsumura}.
\end{proof}
Finally, we analyze duality for towers with $\Delta$-action.
\begin{lemma}\label{LambdaDuality}
With the notation of Lemma $\ref{Technical}$,
let
$\mathscr{M}:=\{M_r,\rho_{r,s}\}$ and $\mathscr{M}':=\{M_r',\rho_{r,s}'\}$ be two
towers of $\mathscr{A}$-modules with $\Delta$-action satisfying $(\ref{freehyp})$ and $(\ref{surjhyp})$.
Suppose that for each $r$ there exist $A_r$-linear perfect duality pairings
\begin{equation}
\xymatrix{
{\langle \cdot,\cdot \rangle_{r}:M_r\times M_r'} \ar[r] & A_r
}\label{pairinghyp}
\end{equation}
with respect to which $\delta$ is self-adjoint for all $\delta\in \Delta$,
and which satisfy the compatibility condition\footnote{By abuse of notation,
for any map of rings $A\rightarrow B$ and any $A$-bilinear pairing of $A$-modules
$\langle\cdot,\cdot\rangle:M\times M'\rightarrow A$, we again write
$\langle\cdot,\cdot\rangle: M_B\times M_B'\rightarrow B$ for the $B$-bilinear pairing induced
by extension of scalars.}
\begin{equation}
\langle\rho_{r,s}m, \rho_{r,s}'m'\rangle_{s} =
\sum_{\delta\in \Delta_{s}/\Delta_{r}} \langle m,\delta^{-1} m'\rangle_{r}
\label{pairingchangeinr}
\end{equation}
for all $r\ge s$. Then
for each $r$, the pairings
$
\xymatrix@1{
{(\cdot,\cdot)_{r}: M_r \times M_r'} \ar[r] & \Lambda_{A_r,r} }
$
defined by
\begin{equation*}
(m,m')_{r} := \sum_{\delta\in \Delta/\Delta_r} \langle m , \delta^{-1} m'\rangle_r \cdot \delta
\end{equation*}
are $\Lambda_{A_r,r}$-bilinear and perfect, and compile to give a $\Lambda_B$-linear
perfect pairing
\begin{equation*}
\xymatrix{
{(\cdot,\cdot)_{\Lambda_B}: M_B \times M_B'} \ar[r] & {\Lambda_B}
}.
\end{equation*}
In particular, $M_B'$ and $M_B$ are canonically $\Lambda_B$-linearly dual to eachother.
\end{lemma}
\begin{proof}
An easy reindexing argument shows that $(\cdot,\cdot)_{r}$ is $\Lambda_{A_r,r}$-linear
in the right factor, from which it follows that it is also $\Lambda_{A_r,r}$-linear
in the left due to our assumption that $\delta\in \Delta$ is self-adjoint with respect to
$\langle\cdot, \cdot\rangle_{r}$. To prove that
$(\cdot,\cdot)_{r}$ is a perfect duality pairing, we analyze the $\Lambda_{A_r,r}$-linear map
\begin{equation}
\xymatrix@C=45pt{
{M_r} \ar[r]^-{m\mapsto (m,\cdot)_r} & {\Hom_{\Lambda_{A_r,r}}(M_r',\Lambda_{A_r,r})}
}.\label{GroupRingDuality}
\end{equation}
Due to Lemma \ref{Technical}, both $M_r$ and $M_r'$ are free $\Lambda_{A_r,r}$-modules,
necessarily of the same rank by the existence of the perfect $A_r$-duality pairing (\ref{pairinghyp}).
It follows that (\ref{GroupRingDuality}) is a homomorphism of free $\Lambda_{A_r,r}$-modules
of the same rank. To show that it is an isomorphism it therefore suffices to prove it is surjective,
which may be checked after extension of scalars along the augmentation map
$\Lambda_{A_r,r}\twoheadrightarrow A_r$ by Nakayama's Lemma.
Consider the diagram
\begin{equation}
\begin{gathered}
\xymatrix@C=38pt{
{M_r \mathop{\otimes}\limits_{\Lambda_{A_r,r}} A_r} \ar[r]^-{(\ref{GroupRingDuality})\otimes 1}
\ar[d]_-{\rho_{r,1}\otimes 1}^-{\simeq} &
{\Hom_{\Lambda_{A_r,r}}(M_r',\Lambda_{A_r,r})\mathop{\otimes}\limits_{\Lambda_{A_r,r}} A_r} \ar[r]^-{\xi}_-{\simeq} &
{\Hom_{A_r}(M_r'\mathop{\otimes}\limits_{\Lambda_{A_r,r}} A_r, A_r)} \\
{M_1\mathop{\otimes}\limits_{A_1} A_r} \ar[rr]^-{\simeq} & &{\Hom_{A_r}(M_1'\mathop{\otimes}\limits_{A_1} A_r, A_r)}
\ar[u]_-{(\rho_{r,1}'\otimes 1)^{\vee}}^-{\simeq}
}
\end{gathered}
\label{XiDiagram}
\end{equation}
where $\xi$ is the canonical map sending $f\otimes \alpha$ to $\alpha(f\otimes 1)$,
and the bottom horizontal arrow is obtained by $A_r$-linearly extending the canonical
duality map $m\mapsto \langle m,\cdot\rangle_1$. On the one hand, the vertical
maps in (\ref{XiDiagram}) are isomorphisms thanks to Lemma \ref{Technical} (\ref{red2psurj}),
while the map $\xi$ and the bottom horizontal arrow are isomorphisms
because arbitrary extension of scalars commutes with linear duality
of {\em free} modules.\footnote{Quite generally, for any ring $R$, any $R$-modules
$M$, $N$, and any $R$-algebra $S$, the canonical map
\begin{equation*}
\xymatrix{
{\xi_M:\Hom_R(M,N)\otimes_R S} \ar[r] & {\Hom_S(M\otimes_R S, N\otimes_R S)}
}
\end{equation*}
sending $f\otimes s$ to $s(f\otimes \id_S)$ is an isomorphism if $M$ is finite and free over $R$.
Indeed, the map $\xi_R$ is visibly an isomorphism, and one checks that $\xi_{M_1\oplus M_2}$
is naturally identified with $\xi_{M_1}\oplus \xi_{M_2}$.
}
On the other hand, this diagram commutes because (\ref{pairingchangeinr}) guarantees the relation
\begin{equation*}
\xymatrix@C=15pt{
{\langle \rho_{r,1}m , \rho_{r,1}'m' \rangle_1} \ar@^{=}[r]^-{(\ref{pairingchangeinr})} &
{\displaystyle\sum_{\delta\in \Delta/\Delta_r} \langle m,\delta^{-1}m'\rangle_r
\equiv (m,m')_r \bmod{I_{\Delta}}}
}
\end{equation*}
where $I_{\Delta}=\ker(\Lambda_{A_r,r}\twoheadrightarrow A_r)$ is the augmentation ideal
We conclude that (\ref{GroupRingDuality}) is an isomorphism, as desired.
The argument that the corresponding map with the roles of $M_r$ and $M_r'$ interchanged
is an isomorphism proceeds {\em mutatis mutandis}.
Using the definition
of $(\cdot,\cdot)_r$ and (\ref{pairingchangeinr}), one has more generally that
\begin{equation*}
(\rho_{r,s}m,\rho_{r,s}'m')_{s} \equiv (m,m')_r \bmod
\ker(\Lambda_{A_r,r}\twoheadrightarrow \Lambda_{A_r,s})
\end{equation*}
for all $r\ge s$. In particular, the pairings $(\cdot,\cdot)_r$ induce, by extension
of scalars, a $\Lambda_B$-bilinear pairing
\begin{equation*}
\xymatrix{
{(\cdot,\cdot)_{\Lambda_B}: M_B\times M_{B}'} \ar[r] & {\Lambda_B}
}
\end{equation*}
which satisfies the specialization property
\begin{equation}
(\cdot,\cdot)_{\Lambda_B} \equiv (\cdot, \cdot)_{r} \bmod \ker(\Lambda_B \twoheadrightarrow \Lambda_{B,r}).
\label{pairingspecialize}
\end{equation}
From $(\cdot,\cdot)_{\Lambda_B}$ we obtain in the usual way duality morphisms
\begin{equation}
\xymatrix@C=45pt{
{M_B} \ar[r]^-{m\mapsto (m,\cdot)_{\Lambda_B}} & \Hom_{\Lambda_B}(M_{B}',\Lambda_B)
}\quad\text{and}\quad
\xymatrix@C=45pt{
{M_B'} \ar[r]^-{m'\mapsto (\cdot,m')_{\Lambda_B}} & \Hom_{\Lambda_B}(M_{B},\Lambda_B)
}\label{LambdaDualityMaps}
\end{equation}
which we wish to show are isomorphisms. Due to Lemma \ref{Technical} (\ref{MBfree}),
each of (\ref{LambdaDualityMaps}) is a map of finite free $\Lambda_B$-modules of the same rank,
so we need only show that these mappings are surjective. As the kernel of
$\Lambda_B\twoheadrightarrow \Lambda_{B,r}$ is contained in the radical of $\Lambda_B$,
we may by Nakayama's Lemma check
such surjectivity after extension of scalars along $\Lambda_B\twoheadrightarrow \Lambda_{B,r}$
for any $r$, where it follows from (\ref{pairingspecialize}) and the fact that $M_r$ and $M_{s}$
are free $\Lambda_{A_r,r}$-modules, so that the extension of scalars of the perfect
duality pairing $(\cdot,\cdot)_r$ along the canonical map
$\Lambda_{A_r,r}\rightarrow \Lambda_{B,r}$ is again perfect.
\end{proof}
\subsection{Ordinary families of de Rham cohomology}\label{ordfamdR}
Let $\{\X_r/T_r\}_{r\ge 0}$ be the tower of modular curves
introduced in \S\ref{tower}.
As $\X_r$ is regular and proper flat over $T_r=\Spec(R_r)$ with geometrically reduced fibers, it is a curve
in the sense of Definition \ref{curvedef} (thanks to Corollary \ref{curvecorollary})
which moreover satisfies the hypotheses of Proposition \ref{HodgeIntEx}.
Abbreviating
\begin{align}
& H^0(\omega_{r}):=H^0(\X_{r},\omega_{\X_{r}/S_{r}}), & & H^1_{\dR,r}:= H^1(\X_{r}/R_{r}), & & H^1(\O_{r}):=H^1(\X_{r},\O_{\X_{r}}),\label{shfcoh}
\end{align}
Proposition \ref{HodgeIntEx} (\ref{CohomologyIntegral}) provides a canonical short exact sequence
$H(\X_r/R_r)$ of finite
free $R_r$-modules
\begin{equation}
\xymatrix{
0\ar[r] & {H^0(\omega_r)} \ar[r] & {H^1_{\dR,r}} \ar[r] & {H^1(\O_r)} \ar[r] & 0
}\label{HodgeFilIntAbbrev}
\end{equation}
which recovers the Hodge filtration of $H^1_{\dR}(X_r/K_r)$ after inverting $p$.
The Hecke correspondences on $\X_r$ induce, via Proposition \ref{HodgeIntEx} (\ref{CohomologyFunctoriality})
(or by Proposition \ref{intcompare} and Remark \ref{canonicalproperty}),
canonical actions of $\H_r$ and $\H_r^*$ on $H(\X_r/R_r)$ via $R_r$-linear endomorphisms.
In particular, $H(\X_r/R_r)$ is canonically a short exact sequence of $\Z_p[(\Z/Np^r\Z)^{\times}]$-modules
via the diamond operators.
Similarly, pullback along (\ref{gammamaps}) yields $R_r$-linear morphisms
$H((\X_r)_{\gamma}/R_r)\rightarrow H(\X_r/R_r)$ for each $\gamma\in \Gamma$;
using the fact that hypercohomology commutes with flat base change
(by \v{C}ech theory), we obtain an action of $\Gamma$ on $H(\X_r/R_r)$
which is $R_r$-semilinear over the canonical action of $\Gamma$ on $R_r$
and which commutes with the actions of $\H_r$ and $\H_r^*$ as the Hecke operators are defined over
$K_0=\Q_p$.
For $r\ge s$, we will need to work with the base change $\X_s\times_{T_s} T_r$,
which is a curve over $T_r$ thanks to Proposition \ref{curveproperties}.
Although $\X_s\times_{T_s} T_r$ need no longer be regular as $T_{r}\rightarrow T_{s}$ is not smooth when $r> s$,
we claim that it is necessarily {\em normal}.
Indeed, this follows from the more general assertion:
\begin{lemma}\label{normalcrit}
Let $V$ be a discrete valuation ring and $A$ a finite type Cohen-Macaulay $V$-algebra with smooth generic fiber and geometrically reduced
special fiber. Then $A$ is normal.
\end{lemma}
\begin{proof}
We claim that $A$ satisfies Serre's ``$R_1+S_2$"-criterion for normality \cite[Theorem 23.8]{matsumura}. As $A$ is assumed
to be CM, by definition of Cohen-Macaulay $A$ verifies $S_i$ for all $i\ge 0$, so we need only show that each localization of $A$ at
a prime ideals of codimension 1 is regular. Since $A$ has geometrically reduced special fiber, this special fiber is in particular
smooth at its generic points. As $A$ is flat over $V$ (again by definition of CM), we deduce that the (open) $V$-smooth locus
in $\Spec A$ contains the generic points of the special fiber and hence contains all codimension-1 points (as the generic
fiber of $\Spec A$ is assumed to be smooth). Thus $A$ is $R_1$, as desired.
\end{proof}
We conclude that $\X_{s}\times_{T_s} T_r$ is a normal curve, and
we obtain from Proposition \ref{HodgeIntEx}
a canonical short exact sequence of finite free $R_r$-modules $H(\X_{s}\times_{T_s}T_r/R_r)$
which recovers the Hodge filtration of $H^1_{\dR}(X_{s}/K_r)$ after inverting $p$.
As hypercohomology commutes with flat base change and the formation of the
relative dualizing sheaf and the structure sheaf are
compatible with arbitrary base change, we have a natural isomorphism of short exact sequences of free $R_r$-modules
\begin{equation}
H(\X_{s}\times_{T_s} T_r/R_r) \simeq H(\X_{s}/R_{s})\otimes_{R_{s}} R_r. \label{bccompat}
\end{equation}
In particular, we have $R_r$-linear actions of $\H_{s}^*$, $\H_r$ and an $R_r$-semilinear action of
$\Gamma$ on $H(\X_s\times_{T_s} T_r/R_r)$. These actions moreover commute with one another.
Consider now the canonical degeneracy map $\rho: \X_r\rightarrow \X_{s}\times_{T_s} T_r$ of curves over $T_r$
induced by (\ref{rdegen}).
As $\X_r$ and $\X_{s}\times_{T_s} T_r$ are normal and proper curves over $T_r$,
we obtain from Proposition \ref{HodgeIntEx} (\ref{CohomologyFunctoriality})
canonical trace mappings of short exact sequences
\begin{equation}
\xymatrix{
{\rho_* : H(\X_{r}/R_r)} \ar[r] & {H(\X_{s}\times_{T_s}T_r/R_r)
\simeq H(\X_{s}/R_{s})\otimes_{R_{s}}R_r}
}\label{trmap}
\end{equation}
which recover the usual trace mappings on de Rham cohomology after inverting $p$;
as such, these mappings are Hecke and $\Gamma$-equivariant, and
compatible with change in $r,s$ in the obvious way.
Tensoring these maps (\ref{trmap}) over $R_r$ with $R_{\infty}$,
we obtain projective systems of free $R_{\infty}$
with semilinear $\Gamma$-action and commuting, linear
$\H^*:=\varprojlim_r \H_r^*$ action:
\begin{definition}\label{limitmods}
We write
\begin{align*}
&H^0(\omega):=\varprojlim_r \left(H^0(\omega_r) \mathop{\otimes}\limits_{R_r} {R}_{\infty}\right),
&& H^1_{\dR}:=\varprojlim_r \left(H^1_{\dR,r}\mathop{\otimes}\limits_{R_r} {R}_{\infty} \right),
&& H^1(\O) := \varprojlim_r \left(H^1(\O_r)\mathop{\otimes}\limits_{R_r} {R}_{\infty}\right)
\end{align*}
for the projective limit with respect to the maps induced by $\rho_*$,
each of which is naturally a module for
$\Lambda_{R_{\infty}}={R}_{\infty}[\![\Delta]\!]$,
and is equipped with a semilinear $\Gamma$-action
and a linear $\H^*$-action.
\end{definition}
Although we have a left exact sequence of $\Lambda_{R_\infty}$-modules
with semilinear $\Gamma$-action and $\H^*$-action
\begin{equation*}
\xymatrix{
0\ar[r] & {H^0(\omega)} \ar[r] & {H^1_{\dR}} \ar[r] & {H^1(\O)}
},
\end{equation*}
this sequence is almost certainly not right exact. It is moreover unlikely that
any of the $\Lambda_{R_{\infty}}$-modules in
Definition \ref{limitmods} are finitely generated. The situation
is much better if we pass to {\em ordinary parts}:
\begin{theorem}\label{main}
Let $e^*$ be the idempotent of $\H^*$ associated to $U_p^*$ and let $d$ be
the positive integer defined as in Proposition $\ref{IgusaStructure}$ $(\ref{IgusaFreeness})$.
Then $e^*H^0(\omega)$, $e^*H^1_{\dR}$ and $e^*H^1(\O)$ are free $\Lambda_{R_{\infty}}$-modules
of ranks $d$, $2d$, and $d$ respectively, and there is a canonical short exact sequence
of free $\Lambda_{R_{\infty}}$-modules with linear $\H^*$-action and $R_{\infty}$-semilinear
$\Gamma$-action
\begin{equation}
\xymatrix{
0\ar[r] & {e^*H^0(\omega)} \ar[r] & {e^*H^1_{\dR}} \ar[r] & {e^*H^1(\O)} \ar[r] & 0
}.\label{mainthmexact}
\end{equation}
For each positive integer $r$, applying $\otimes_{\Lambda_{R_{\infty}}} R_{\infty}[\Delta/\Delta_r]$
to $(\ref{mainthmexact})$ yields the short exact sequence
\begin{equation}
\xymatrix{
0\ar[r] & {e^*H^0(\omega_r)\mathop{\otimes}\limits_{R_r} R_{\infty}} \ar[r] &
{e^*H^1_{\dR}\mathop{\otimes}\limits_{R_r} R_{\infty}} \ar[r] &
{e^*H^1(\O)\mathop{\otimes}\limits_{R_r} R_{\infty}} \ar[r] & 0
},\label{mainthmexact2}
\end{equation}
compatibly with the actions of $\H^*$ and $\Gamma$.
\end{theorem}
\begin{proof}
Applying $e^*$ to the short exact sequence $H(\X_r/R_r)$ yields a short exact sequence
\begin{equation}
\xymatrix{
0\ar[r] & {e^*H^0(\omega_r)}\ar[r] & {e^*H^1_{\dR,r}} \ar[r] & {e^*H^1(\O_r)} \ar[r] & 0
}\label{hitwithidem}
\end{equation}
of $R_r[\Delta/\Delta_r]$-modules with linear $\H_r^*$-action and $R_r$-semilinear $\Gamma$-action in which each term is
free as an $R_r$-module.\footnote{Indeed, $e^*M$ is a direct summand of $M$ for any $\H_r^*$-module $M$,
and hence $R_r$-projective ($=R_r$-freee) if $M$ is.}
Similarly, for each pair of nonnegative integers $r\ge s$,
the trace mappings (\ref{trmap}) induce a commutative diagram with exact rows
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {e^*H^0(\omega_r)} \ar[r]\ar[d]_-{\rho_*} &
{e^*H^1_{\dR,r}} \ar[r]\ar[d]_-{\rho_*} & {e^*H^1(\O_r)}
\ar[r]\ar[d]^-{\rho_*} & 0\\
0\ar[r] & {e^*H^0(\omega_{s})\otimes_{R_{s}}R_r} \ar[r] &
{e^*H^1_{\dR,{s}}\otimes_{R_{s}}R_r} \ar[r] & {e^*H^1(\O_{s})\otimes_{R_{s}}R_r}
\ar[r] & 0
}
\end{gathered}
\label{piecetogether}
\end{equation}
We will apply Lemma \ref{Technical} with $A_r=R_r$, $I_r=(\pi_r)$, $B=R_{\infty}$
nd with $M_r$ each one of the terms in (\ref{hitwithidem}).
In order to do this, we must check that the hypotheses (\ref{freehyp}) and
(\ref{surjhyp}) are satisfied.
Applying $\otimes_{R_r} \mathbf{F}_p$ to the short exact sequence (\ref{hitwithidem})
and using the fact that the idempotent $e^*$ commutes with tensor products,
we obtain, thanks to Lemma \ref{ReductionCompatibilities} (\ref{BaseChngDiagram}),
the short exact sequence of $\mathbf{F}_p$-vector spaces (\ref{sesincharp1}).
By Corollary \ref{FreenessInCharp}, the three terms of (\ref{sesincharp1}) are free
$\mathbf{F}_p[\Delta/\Delta_r]$-modules
of ranks $d$, $2d$, and $d$ respecvitely, so (\ref{freehyp})
is satisfied for each of these terms. Similarly, applying $\otimes_{R_r} \mathbf{F}_p$
to the diagram (\ref{piecetogether})
yields a diagram which by Corollary \ref{SplitIgusa} is naturally isomorphic to
the diagram of $\mathbf{F}_p[\Delta/\Delta_r]$-modules with split-exact rows
\begin{equation*}
\xymatrix@C=15pt{
0 \ar[r] & {H^0(I_r^{\infty},\Omega^1(\SS))^{V_{\ord}}} \ar[r]\ar[d]_-{\rho_*} &
{H^0(I_r^{\infty},\Omega^1(\SS))^{V_{\ord}}\oplus
H^1(I_r^0,\O(-\SS))^{F_{\ord}}} \ar[r]\ar[d]|-{\rho_*\oplus \rho_*} &
{H^1(I_r^0,\O(-\SS))^{F_{\ord}}} \ar[r]\ar[d]^-{\rho_*} & 0 \\
0 \ar[r] & {H^0(I_{s}^{\infty},\Omega^1(\SS))^{V_{\ord}}} \ar[r] &
{H^0(I_{s}^{\infty},\Omega^1(\SS))^{V_{\ord}}\oplus
H^1(I_{s}^0,\O(-\SS))^{F_{\ord}}} \ar[r] &
{H^1(I_{s}^0,\O(-\SS))^{F_{\ord}}} \ar[r] & 0
}
\end{equation*}
Each of the vertical maps in this diagram is surjective due to Proposition \ref{IgusaStructure}
(\ref{IgusaControl}), and we conclude that the hypothesis (\ref{surjhyp}) is satisfied
as well. Furthermore, the vertical maps in (\ref{piecetogether}) are then surjective by Nakayama's
Lemma, so applying $\otimes_{R_r} R_{\infty}$
yields an inverse system of short exact sequences in which the first term satisfies the Mittag-Leffler
condition. Passing to inverse limits is therefore (right) exact, and we obtain the short
exact sequence (\ref{mainthmexact}).
\end{proof}
Due to Proposition \ref{HodgeIntEx} (\ref{CohomologyDuality}), the short exact sequence
(\ref{HodgeFilIntAbbrev}) is auto-dual with respect to the canonical cup-product pairing $(\cdot,\cdot)_r$
on $H^1_{\dR,r}$. We extend scalars along $R_r\rightarrow R_r':=R_r[\mu_N]$, so that the Atkin-Lehner
``invoultion" $w_r$ is defined, and consider the ``twisted" pairing on ordinary parts
\begin{equation}
\xymatrix{
{\langle \cdot,\cdot\rangle _r : ({e^*}H^1_{\dR,r})_{R_r'} \times ({e^*}H^1_{\dR,r})_{R_r'}} \ar[r] & {R_r'}
}\qquad\text{given by}\qquad \langle x, y\rangle_r := (x, w_r {U_p^*}^r y).
\label{TwistdRpairing}
\end{equation}
It is again perfect and satisfies $\langle T^* x,y\rangle =\langle x, T^* y \rangle$
for all $x,y\in (e^*H^1_{\dR,r})_{R_r'}$ and $T^*\in \H_r^*$.
\begin{proposition}\label{dRDuality}
The pairings $(\ref{TwistdRpairing})$ compile to give a perfect $\Lambda_{R_{\infty}'}$-linear
duality pairing
\begin{equation*}
\xymatrix{
{\langle\cdot,\cdot\rangle_{\Lambda_{R_{\infty}'}}:
({e^*}H^1_{\dR})_{\Lambda_{R_{\infty}'}} \hspace{-1ex}\times ({e^*}H^1_{\dR})_{\Lambda_{R_{\infty}'}}}
\ar[r] & {\Lambda_{R_{\infty}'}}
}\ \text{given by}\
\langle x , y \rangle_{\Lambda_{R_{\infty}'}} :=
\varprojlim_r\sum_{\delta\in \Delta/\Delta_r} \langle x_r, \langle \delta^{-1}\rangle^* y_r\rangle_r\cdot\delta
\end{equation*}
for $x=\{x_r\}_r$ and $y=\{y_r\}_r$ in $(e^*H^1_{\dR})_{\Lambda_{R_{\infty}'}}$.
The pairing $\langle \cdot,\cdot \rangle_{\Lambda_{R_{\infty}}'}$ induces
a canonical isomorphism
\begin{equation*}
\xymatrix{
0\ar[r] & {e^*H^0(\omega)(\langle\chi\rangle\langle a\rangle_N)_{\Lambda_{R_{\infty}'}}}
\ar[r]\ar[d]^-{\simeq} &
{e^*H^1_{\dR}(\langle\chi\rangle\langle a\rangle_N)_{\Lambda_{R_{\infty}'}}}
\ar[r]\ar[d]^-{\simeq} &
{e^*H^1(\O)(\langle\chi\rangle\langle a\rangle_N)_{\Lambda_{R_{\infty}'}}}
\ar[r]\ar[d]^-{\simeq} & 0\\
0\ar[r] & {(e^*H^1(\O))^{\vee}_{\Lambda_{R_{\infty}'}}} \ar[r] &
{({e^*H^1_{\dR}})^{\vee}_{\Lambda_{R_{\infty}'}}} \ar[r] &
{(e^*H^0(\omega))^{\vee}_{\Lambda_{R_{\infty}'}}} \ar[r] & 0
}
\label{mainthmdualityisom}
\end{equation*}
that is $\H^*$-equivariant and compatible with the natural action
of $\Gamma \times \Gal(K_0'/K_0)\simeq \Gal(K_{\infty}'/K_0)$
on the bottom row and the twist
$\gamma\cdot m := \langle \chi(\gamma)\rangle\langle a(\gamma)\rangle_N \gamma m$
of the natural action on the top,
where
$a(\gamma) \in (\Z/N\Z)^{\times}$ is
determined by
$\zeta^{a(\gamma)}=\gamma\zeta$ for every $\zeta\in \mu_N(\Qbar_p)$.
\end{proposition}
\begin{proof}[Proof of Proposition $\ref{dRDuality}$]
That $\langle\cdot,\cdot \rangle_{\Lambda_{R_{\infty}'}}$ is a perfect duality pairing
follows easily from Lemma \ref{LambdaDuality}, using Theorem \ref{main} and the formalism of
\S\ref{TowerFormalism}, once we check that the twisted pairings (\ref{TwistdRpairing})
satisfy the hypothesis (\ref{pairinghyp}). By the definition (\ref{TwistdRpairing})
of $\langle\cdot, \cdot\rangle_r$, this amounts to the computation
\begin{align*}
({\rho_1}_* x, w_r {U_p^*}^r{\rho_1}_* y)_r = (x, \rho_1^* w_r {U_p^*}^r{\rho_1}_*y)_{r+1}
&=(x, w_{r+1} {U_p^*}^r \rho_2^*{\rho_1}_*y)_{r+1} \\
&= \sum_{\delta\in \Delta_r/\Delta_{r+1}}(x, w_{r+1} {U_p^*}^{r+1} \langle \delta^{-1}\rangle^* y)_{r+1}
\end{align*}
where we have used Proposition \ref{ALinv} and the identity
$\rho_2^*{\rho_1}_* = U_p^*\sum_{\delta\in \Delta_r/\Delta_{r+1}} \langle\delta^{-1}\rangle^*$
on $H^1_{\dR,r+1}$, which follows from\footnote{The reader will check that our forward reference
to \S\ref{BTfamily} does not involve any circular reasoning.}
Lemma \ref{MFtraceLem} by using Lemma \ref{LieFactorization} and
Proposition \ref{intcompare}. We obtain an isomorphism of short exact sequences of
$\Lambda_{R_{\infty}'}$-modules as in (\ref{mainthmdualityisom}), which it remains to show is
$\Gamma\times\Gal(K_0'/K_0)$-equivariant
for the specified actions. For this, we compute that for $\gamma\in\Gal(K_{\infty}'/K_0)$,
\begin{equation*}
\langle \gamma x,\gamma y\rangle_{r} =
(\gamma x, w_r {U_p^*}^r \gamma y)_r =
(\gamma x, \gamma w_r {U_p^*}^r \langle \chi(\gamma)^{-1} \rangle\langle a(\gamma)^{-1}\rangle_N y)_r
= \gamma \langle x,\langle \chi(\gamma)^{-1} \rangle\langle a(\gamma)^{-1}\rangle_N y\rangle_r,
\end{equation*}
where we have used Proposition \ref{ALinv} and the fact that the cup product is Galois-equivarant.
It now follows easily from definitions that
\begin{equation*}
\langle \gamma x,\gamma y\rangle_{\Lambda_{R_{\infty}'}} =
\langle \chi(\gamma)^{-1} \rangle \gamma \langle x, \langle a(\gamma)^{-1}\rangle_N y
\rangle_{\Lambda_{R_{\infty}'}},
\end{equation*}
and the claimed $\Gamma\times\Gal(K_0'/K_0)$-equivariance of (\ref{mainthmdualityisom}) is equivalent to this.
\end{proof}
\begin{remark}
For an open subgroup $H$ of $\scrG_K$ and any $H$-stable subfield
$F$ of $\c_{K}$, denote by $\Rep_F(H)$ the category of finite-dimensional
$F$-vector spaces that are equipped with a continuous semilinear
action of $H$. Recall \cite{DSen} that classical Sen theory provides a functor
$\ensuremath{\mathbf{D}}_{\Sen}:\Rep_{\c_K}(\scrG_K)\rightarrow \Rep_{K_{\infty}}(\Gamma)$
which is quasi-inverse to $(\cdot)\otimes_{K_{\infty}} \c_K$. Furthermore,
for any $W\in \Rep_{\c_K}(\scrG_K)$, there is a unique $K_{\infty}$-linear
operator $\Theta_{D}$ on $D:=\ensuremath{\mathbf{D}}_{\Sen}(W)$ with the property that
$\gamma x = \exp(\log \chi(\gamma)\cdot \Theta_D)(x)$ for all $x\in D$
and all $\gamma$ in a small enough open neighborhood of $1\in \Gamma$.
We expect that for $W$ any specialization of $e^*H^1_{\et}$ along a continuous homomorphism
$\Lambda\rightarrow K_{\infty}$, there is a canonical isomorphism between $D:=\ensuremath{\mathbf{D}}_{\Sen}(W\otimes {\c_K})$
and the corresponding specialization of $e^*H^1_{\dR}$,
with the Sen operator $\Theta_D$
induced by the Gauss-Manin connections on $H^1_{\dR,r}$. In this way, we might think
of $e^*H^1_{\dR}$
as a $\Lambda$-adic avatar of ``$\ensuremath{\mathbf{D}}_{\Sen}(e^*H^1_{\et}\otimes_{\Lambda} \Lambda_{\O_{\c_K}})$."
We hope to pursue these connections in future work.
\end{remark}
\subsection{Ordinary \texorpdfstring{$\Lambda$}{Lambda}-adic modular forms}\label{ordforms}
In this section, we discuss the relation between $e^*H^0(\omega)$
and ordinary $\Lambda_{R_{\infty}}$-adic cuspforms as defined by Ohta \cite[Definition 2.1.1]{OhtaEichler}.
We begin with some preliminaries on modular forms.
For a ring $A$, a congruence subgroup $\Gamma$, and a nonnegative integer $k$, we will write $S_k(\Gamma;A)$ for the space of weight $k$ cuspforms for $\Gamma$ over $A$; we put $S_k(\Gamma):=S_k(\Gamma;\Qbar)$.
If $\Gamma',$ $\Gamma$ are congruence subgroups and $\gamma\in \GL_2(\Q)$ satisfies
$\gamma^{-1}\Gamma'\gamma\subseteq \Gamma$, then there is a
canonical injective ``pullback" map
on modular forms
$\xymatrix@1{{\iota_{\gamma}:S_k(\Gamma)} \ar@{^{(}->}[r] & {S_k(\Gamma')}}$
given by $\iota_{\gamma}(f):=f\big|_{\gamma^{-1}}$.
When $\Gamma'\subseteq \Gamma$, {\em unless specified to the contrary}, we will
always view $S_k(\Gamma)$ as a subspace of $S_k(\Gamma')$ via $\iota_{\id}$.
As $\gamma\Gamma'\gamma^{-1}$ is necessarily of finite index in $\Gamma$,
one also has a canonical ``trace" mapping
\begin{equation}
\xymatrix{
{\tr_{\gamma}:S_k(\Gamma')} \ar[r] & {S_k(\Gamma)}
}
\qquad\text{given by}\qquad
\tr_{\gamma}(f):=\sum_{\delta\in \gamma^{-1}\Gamma'\gamma\backslash\Gamma} (f\big|_{\gamma})\big|_{\delta}
\label{MFtrace}
\end{equation}
with the property that $\tr_{\gamma}\circ\iota_{\gamma}$ is multiplication by $[\Gamma: \gamma^{-1}\Gamma'\gamma]$
on $S_k(\Gamma)$.
We define
\begin{equation*}
S_2^{\infty}(\Gamma_r;R_r):=S_2(\Gamma_r;R_r)\qquad\text{and}\qquad
S_2^{0}(\Gamma_r;R_r):=
\{f\in S_2(\Gamma_r; \Qbar_p)\ :\ f\big|_{w_r} \in S_2^{\infty}(\Gamma_r;R_r) \},
\end{equation*}
By definition, $S_2^{\star}(\Gamma_r;R_r)$ for $\star=0,\infty$ are $R_r$-submodules
of $S_2(\Gamma_r;K_r')$ that are carried isomorphically onto eachother by the automorphism
$w_r$ of $S_2(\Gamma_r;K_r')$. Note that $S_2^{\star}(\Gamma_r;R_r)$ is precisely the
$R_r$-submodule consisting of cuspforms whose formal expansion at the cusp $\star$
has coefficients in $R_r$.
As the Hecke algebra $\H_r$ stabilizes
$S_2^{\infty}(\Gamma_r; R_r)$, it follows immediately from Proposition \ref{AtkinInterchange} that
$S_2^0(\Gamma_R;R_r)$ is stable under the action of $\H_r^*$ on $S_2(\Gamma_r; K_r)$.
Furthermore, $\Gal(K_r'/K_0)$ acts on $S_2(\Gamma_r;K_r')\simeq S_2(\Gamma_r;\Q_p)\otimes_{\Q_p} K_r'$
through the second tensor factor, and this action leaves stable the $R_r$-submodule $S_2^{\infty}(\Gamma_r; R_r)$.
The second equality of Proposition \ref{ALinv} then implies that $S_2^0(\Gamma_r;R_r)$
is also a $\Gal(K_r'/K_0)$-stable $R_r$-submodule of $S_2(\Gamma_r;K_r')$.
A straightforward computation shows that
the direct factor $\Gal(K_0'/K_0)$ of $\Gal(K_r'/K_0)$ acts trivially on $S_2^{\infty}(\Gamma_r;R_r)$
and through $\langle a\rangle_N^{-1}$ on $S_2^0(\Gamma_r;R_r)$.
We can interpret $S_2^{\star}(\Gamma_r;R_r)$ geometrically as follows.
As in Remark \ref{MWGood}, for $\star= \infty, 0$
let $I_r^{\star}$ be the irreducible component
of $\o{\X}_r$ passing through the cusp $\star$,
and denote by $\X_{r}^{\star}$ the complement
in $\X_r$ of all irreducible components of $\o{\X}_r$ distinct from $I_{r}^{\star}$.
By construction, $\X_r$ and $\X_r^{\star}$ have the same generic fiber $X_r\times_{\Q_p} K_r$.
Using Proposition \ref{redXr}, it is not hard to show that the diamond operators
induce automorphisms of $\X_r^{\star}$, and one checks via Proposition \ref{AtkinInertiaCharp}
that the ``semilinear" action (\ref{gammamaps}) of $\gamma\in \Gamma$ on $\X_r$
carries $\X_r^{\star}$ to $(\X_r^{\star})_{\gamma}$ for all $\gamma$.
\begin{lemma}\label{Edixhoven}
Formal expansion at the $R_r$-point $\infty$ $($respectively $R_r'$-point $0$$)$ of $\X_r^{\star}$
induces an isomorphism of $R_r$-modules
\begin{equation}
H^0(\X_r^{\infty},\Omega^1_{\X_r^{\infty}/R_r}) \simeq S_2^{\infty}(\Gamma_r;R_r)
\quad\text{respectively}\quad
H^0(\X_r^{0},\Omega^1_{\X_r^{0}/R_r})(\langle a\rangle_N^{-1}) \simeq S_2^{0}(\Gamma_r;R_r)
\end{equation}
which is equivariant for the natural action of $\Gamma$ and $\H_r$ $($respectively $\H_r^*$$)$ on source
and target and, in the case of the second isomorphism, intertwines the action of $\Gal(K_0'/K_0)$
via $\langle a\rangle_N^{-1}$ on source with the natural action on the target.
\end{lemma}
\begin{proof}
The proof is a straightforward
adaptation of the proof of \cite[Proposition 2.5]{EdixhovenComparison}.
\end{proof}
Now $\X_r\rightarrow S_r$ is smooth outside the supersingular points, so there is a canonical
closed immersion $\iota_r^{\star}:\X_r^{\star}\hookrightarrow \X_r^{\sm}$.
Using Lemmas \ref{ConcreteDualizingDescription} and \ref{Edixhoven},
pullback of differentials along $\iota_r^{\star}$ gives a natural map
\begin{equation}
\xymatrix{
{H^0(\X_r,\omega_{\X_r/T_r})\simeq H^0(\X_r^{\sm},\Omega^1_{\X_r^{\sm}/T_r})} \ar[r]^-{(\iota_r^{\star})^*} &
{H^0(\X_r^{\star},\Omega^1_{\X_r^{\star}/T_r}) \simeq S_2^{\star}(\Gamma_r;R_r)}
}\label{OmegasComparison}
\end{equation}
which is an isomorphism after inverting $p$.
In particular,
the map (\ref{OmegasComparison}) is injective, $\Gamma$-equivariant, and compatible with
the natural action of $\H_r$ (respectively $\H_r^*$) on source and target for $\star=\infty$ (respectively
$\star=0$), and in the case of $\star=0$ intertwines the action of $\Gal(K_0'/K_0)$ via the character
$\langle a\rangle_N^{-1}$ on source with the natural action on the target.
\begin{remark}
The image of $(\ref{OmegasComparison})$ for $\star=\infty$
is naturally identified
with the space of weight $2$ cuspforms for $\Gamma_r$ whose formal expansion
at {\em every} cusp has $R_r$-coefficients.
\end{remark}
Applying the idempotent $e$ (respectively $e^*$) to (\ref{OmegasComparison}) with $\star=\infty$
(respectively $\star=0$) gives an injective homomorphism
\begin{subequations}
\begin{equation}
\xymatrix{
{eH^0(\X_r,\omega_{\X_r/T_r})} \ar@{^{(}->}[r] & {eS_2^{\infty}(Np^r;R_r)}
}\label{OmegasComparisonOrd}
\end{equation}
respectively
\begin{equation}
\xymatrix{
{e^*H^0(\X_r,\omega_{\X_r/T_r})(\langle a\rangle_N^{-1})} \ar@{^{(}->}[r] & {e^*S_2^0(Np^r;R_r)}
}\label{OmegasComparisonOrd0}
\end{equation}
\end{subequations}
which is compatible with the canonical actions of $\Gamma$ and of $\H_r$ (respectively $\H_r^*$) on
source and target and in the case of (\ref{OmegasComparisonOrd}) is $\Gal(K_0'/K_0)$-equivariant.
\begin{proposition}\label{MFGeometryIsom}
The mappings $(\ref{OmegasComparisonOrd})$ and $(\ref{OmegasComparisonOrd0})$ are isomorphisms.
\end{proposition}
\begin{proof}
We treat the case of $(\ref{OmegasComparisonOrd})$; the proof that (\ref{OmegasComparisonOrd})
is an isomorphism goes through {\em mutatis mutandis}.
We must show that (\ref{OmegasComparisonOrd}) is surjective.
To do this, let $\nu\in e_rS_2^{\infty}(Np^r;R_r)$ be arbitrary. Since (\ref{OmegasComparisonOrd})
is an isomorphism after inverting $\pi_r$, there exists a least nonnegative integer $d$
such that $\pi_r^d\nu$ is in the image of (\ref{OmegasComparisonOrd}).
Assume that $d\ge 1$, and let $\eta\in eH^0(\X_r,\omega_{\X_r/R_r})$
be any element mapping to $\pi_r^d \nu$. For an irreducible component $I$ of $\o{\X}_r$,
write $I^h$ for the complement of the super-singular points in $I$, and denote by
$i_r^{\infty}:I_r^{\infty,h}\hookrightarrow \X_r^{\infty}$ the canonical immersion.
We then have a commutative diagram
\begin{equation}
\begin{gathered}
\xymatrix@C=40pt{
{H^0(\o{\X}_r,\omega_{\o{\X}_r/R_r})} \ar[r]^-{(\ref{OmegasComparison})\bmod \pi_r}\ar@{^{(}->}[d] &
{H^0(\X_r^{\infty},\Omega^1_{\X_r^{\infty}/R_r})\mathop{\otimes}\limits_{R_r} \mathbf{F}_p}\ar[d]^-{(i_r^{\infty})^*} \\
{\displaystyle\prod\limits_{I\in \Irr(\o{\X}_r)} H^0(I^h,\Omega^1_{I^h/\mathbf{F}_p})}\ar[r]_-{\proj_{\infty}} &
{H^0(I_r^{\infty,h},\Omega^1_{I_r^{\infty,h}/\mathbf{F}_p})}
}\label{etamaps20}
\end{gathered}
\end{equation}
where the left vertical mapping follows from Definition \ref{OmegaReg} and Remark \ref{OmegaRegMero}
({\em cf}. the proof of Proposition \ref{charpord}), while the bottom map is simply projection.
Our assumption that $d\ge 1$ implies that the image of $\o{\eta}:=\eta\bmod \pi_r$
under the composite of the right vertical and top horizontal maps in (\ref{etamaps20})
is zero and hence, viewing
$\o{\eta}=(\eta_{(a,b,u)})$ as a meromorphic differential on the normalization of $\o{\X}_r$,
we have $\eta_{(r,0,1)}=\proj_{\infty}(\o{\eta})=0$.
Using the formula (\ref{Upn1}), we deduce that $U_p^n\o{\eta}=0$ for $n$ sufficiently large.
But $U_p$ acts invertibly on $\eta$ (and hence on $\o{\eta}$) so we necessarily have that
$\o{\eta}=0$ or what is the same thing that $\eta\bmod \pi_r=0$. We conclude that
$\pi^{d-1}\nu$ is in the image of (\ref{OmegasComparisonOrd}), contradicting the minimality
of $d$. Thus $d=0$ and (\ref{OmegasComparisonOrd}) is surjective.
\end{proof}
For $s \le r$, Ohta shows \cite[2.3.4]{OhtaEichler} that the trace mapping
$\tr_{\id}:S_k(\Gamma_r; K_r)\rightarrow S_k(\Gamma_s; K_s)\otimes_{K_s} K_r$
attached to the inclusion $\Gamma_r\subseteq \Gamma_s$
carries $S_k^0(\Gamma_r;R_r)$ into $S_k^0(\Gamma_s;R_s)\otimes_{R_s} R_r$, so that the projective limit
\begin{equation*}
\mathfrak{S}_k^*(N,R_{\infty}) : = \varprojlim_{\tr_{\id}} S_k^0(\Gamma_r; R_r)\otimes_{R_r} R_{\infty}
\end{equation*}
makes sense. It is canonically a $\Lambda_{R_{\infty}}$-module, equipped with an action of $\H^*$,
a semilinear action of $\Gamma$, and a natural action of $\Gal(K_0'/K_0)$.
On the other hand,
let $eS(N;\Lambda_{R_{\infty}})\subseteq \Lambda_{R_{\infty}}[\![q]\!]$
be the space of ordinary $\Lambda_{R_{\infty}}$-adic
cuspforms of level $N$, as defined in \cite[2.5.5]{OhtaEichler}.
This space is equipped with an action of $\H$ via the usual formulae on formal
$q$-expansions (see, for example \cite[\S1.2]{WilesLambda}), as well as an action
of $\Gamma$ via its $q$-coefficient-wise action on $\Lambda_{R_{\infty}}[\![q]\!]$.
\begin{theorem}[Ohta]\label{OhtaThm}
Then there is a canonical isomorphism of $\Lambda_{R_{\infty}}$-modules
\begin{equation}
\xymatrix{
{eS(N;\Lambda_{R_{\infty}})} \ar[r]^-{\simeq} & {e^*\mathfrak{S}_2^*(N,R_{\infty})}
}\label{LambdaForms}
\end{equation}
that intertwines the action of $T\in \H$ on the source with that of $T^*\in \H^*$
on the target, for all $T\in \H$. This isomorphism is $\Gal(K_{\infty}'/K_0)$-equivariant
for the natural action of $\Gal(K_{\infty}'/K_0)$ on ${e^*\mathfrak{S}_2^*(N,R_{\infty})}$
and the twisted action $\gamma\cdot \mathscr{F} := \langle \chi(\gamma)\rangle^{-1}\langle a(\gamma)\rangle_N^{-1}
\gamma\mathscr{F}$ on $eS(N;\Lambda_{R_{\infty}})$.
\end{theorem}
\begin{proof}
For the definition of the canonical map (\ref{LambdaForms}), as well as the proof that it is an isomorphism,
see Theorem 2.3.6 and its proof in \cite{OhtaEichler}. With the conventions of \cite{OhtaEichler},
the claimed compatibility of (\ref{LambdaForms}) with Hecke operators is a consequence of \cite[2.5.1]{OhtaEichler},
while the $\Gal(K_{\infty}'/K_0)$-equivariance of (\ref{LambdaForms}) follows from \cite[Proposition 3.5.6]{OhtaEichler}.
\end{proof}
\begin{corollary}\label{LambdaFormsRelation}
There is a canonical isomorphism of $\Lambda_{R_{\infty}}$-modules
\begin{equation}
eS(N;\Lambda_{R_{\infty}})(\langle \chi\rangle^{-1})\simeq e^* H^0(\omega)
\end{equation}
that intertwines the action of $T\in \H$ on the source with $T^*\in \H^*$
on the target and is $\Gamma$-equivariant for the canonical action of $\Gamma$
on $e^*H^0(\omega)$ and the twisted action $\gamma\cdot \mathscr{F}:=\langle \chi(\gamma)\rangle^{-1} \gamma\mathscr{F}$
on $eS(N;\Lambda_{R_{\infty}})$.
\end{corollary}
\begin{proof}
This follows immediately from Proposition \ref{MFGeometryIsom} and Theorem \ref{OhtaThm}.
\end{proof}
\subsection{\texorpdfstring{$\Lambda$}{Lambda}-adic Barsotti-Tate groups}\label{BTfamily}
In order to define a crystalline analogue of Hida's ordinary $\Lambda$-adic \'etale cohomology,
we will apply the theory of \S\ref{PhiGammaCrystals} to a certain ``tower"
$\{\mathcal{G}_r/R_r\}_{r\ge 1}$ of $p$-divisible groups (a $\Lambda$-adic Barsotti Tate group
in the sense of Hida \cite{HidaNotes}, \cite{HidaNotes2}) whose construction involves
artfully cutting out certain $p$-divisible
subgroups of $J_r[p^{\infty}]$ over $\Q$ and the ``good reduction'' theorems of Langlands-Carayol-Saito.
The construction of $\{\mathcal{G}_r/R_r\}_{r\ge 1}$ is certainly well-known (e.g. \cite[\S1]{MW-Hida},
\cite[Chapter 3, \S1]{MW-Iwasawa}, \cite[Definition 1.2]{Tilouine}
and \cite[\S 3.2]{OhtaEichler}), but as we shall need substantially finer information about the $\mathcal{G}_r$
than is available in the literature, we devote this section to recalling their construction and properties.
For nonnegative integers $i\le r$, write
$\Gamma_r^i:=\Gamma_1(Np^i)\cap \Gamma_0(p^r)$ for the intersection $($taken inside $\SL_2(\Z)$$)$,
so $\Gamma_r=\Gamma_r^r$.
We will need the following fact
({\em cf.} \cite[pg. 339]{Tilouine}, \cite[2.3.3]{OhtaEichler})
concerning the trace mapping $(\ref{MFtrace})$ attached to the canonical inclusion
$\Gamma_{r}\subseteq \Gamma_i$ for $r\ge i$; for notational clarity,
we will write $\tr_{r,i}:S_k(\Gamma_r)\rightarrow S_k(\Gamma_i)$ for this map.
\begin{lemma}\label{MFtraceLem}
Fix integers $i\le r$ and let $\tr_{r,i}:S_k(\Gamma_r)\rightarrow S_k(\Gamma_i)$
be the trace mapping $(\ref{MFtrace})$ attached to the inclusion
$\Gamma_r\subseteq \Gamma_i$. For $\alpha:=\left(\begin{smallmatrix} 1 & 0 \\ 0 & p\end{smallmatrix}\right)$,
we have an equality
of $\o{\Q}$-endomorphisms of $S_k(\Gamma_{r})$
\begin{equation}
\iota_{\alpha^{r-i}}\circ \tr_{r,i} = (U_p^*)^{r-i}
\sum_{\delta\in \Delta_i/\Delta_{r}} \langle \delta \rangle.
\label{DualityIdentity}
\end{equation}
\end{lemma}
\begin{proof}
We have
index $p^{r-i}$ inclusions of groups $\Gamma_{r} \subseteq \Gamma_{r}^i \subseteq \Gamma_i$
with $\Gamma_{r}$ normal in $\Gamma_{r}^i$, as it is the kernel of the canonical
surjection $\Gamma_{r}^i\twoheadrightarrow \Delta_i/\Delta_{r}$.
For each $\delta\in \Delta_i/\Delta_{r}$, we
fix a choice of $\sigma_{\delta}\in \Gamma_{r}^i$ mapping to $\delta$
and calculate that
\begin{equation}
\Gamma_i = \coprod_{\delta\in \Delta_i/\Delta_{r}} \coprod_{j=0}^{p^{r-i}-1}
\Gamma_{r}\sigma_{\delta} \varrho_j
\qquad\text{where}\qquad
\varrho_j:=\begin{pmatrix} 1 & 0 \\ jNp^i & 1\end{pmatrix}.\label{CosetDecomp}
\end{equation}
On the other hand, for each $0\le j < p^{r-i}$ one has the equality of matrices in $\GL_2(\Q)$
\begin{equation}
p^{r-i}\varrho_j \alpha^{-(r-i)} = \tau_{r} \begin{pmatrix} 1 & -j \\ 0 & p^{r-i} \end{pmatrix} \tau_{r}^{-1}
\qquad\text{for}\qquad \tau_{r} := \begin{pmatrix} 0 & -1 \\ Np^{r} & 0 \end{pmatrix}.
\label{EasyMatCalc}
\end{equation}
The claimed equality (\ref{DualityIdentity}) follows easily from (\ref{CosetDecomp}) and (\ref{EasyMatCalc}), using
the equalities of operators $(\cdot)\big|_{\sigma_{\delta}}=\langle \delta\rangle $
and $U_p^* = w_{r} U_p w_{r}^{-1}$ on $S_k(\Gamma_{r})$ (see Proposition \ref{AtkinInterchange}).
\end{proof}
Perhaps the most essential ``classical" fact for our purposes is that
the Hecke operator $U_p$ acting on spaces of modular forms ``contracts" the $p$-level, as is made precise by the following:
\begin{lemma}\label{UpContract}
If $f\in S_k(\Gamma_r^i)$ then $U_p^{d}f$ is in the image of the canonical map
$\iota_{\id}:S_k(\Gamma_{r-d}^i)\hookrightarrow S_k(\Gamma_r^i)$ for each integer $d\le r-i$. In particular,
$U_p^{r-i}f$ is in the image of $S_k(\Gamma_i)\hookrightarrow S_k(\Gamma_r^i)$.
\end{lemma}
Certainly Lemma \ref{UpContract} is well-known
(e.g. \cite{Tilouine}, \cite{HidaNotes}, \cite{Ohta1});
because of its importance in our subsequent applications, we sketch a proof (following
the proof of \cite[Lemma 1.2.10]{Ohta1}; see also \cite[\S 2]{HidaNotes}).
We note that $\Gamma_r\subseteq \Gamma_r^i$ for all $i\le r$, and the resulting inclusion
$S_k(\Gamma_r^i)\hookrightarrow S_k(\Gamma_r)$ has image
consisting of forms on $\Gamma_r$ which are eigenvectors for the diamond operators
and whose associated character has conductor with $p$-part dividing $p^{i}$.
\begin{proof}[Proof of Lemma $\ref{UpContract}$]
Fix $d$ with $0\le d\le r-i$ and let $\alpha:=\left(\begin{smallmatrix} 1 & 0 \\ 0 & p\end{smallmatrix}\right)$
be as in Lemma \ref{MFtraceLem};
then $\alpha^d$ is an element of the commeasurator of $\Gamma_{r}^i$ in $\SL_2(\Q)$. Consider the following
subgroups of $\Gamma_{r-d}^i$:
\begin{align*}
H&:= \Gamma_{r-d}^i \cap \alpha^{-d}\Gamma_{r}^i\alpha^d\\
H'&:= \Gamma_{r-d}^i \cap \alpha^{-d}\Gamma_{r-d}^i\alpha^d,
\end{align*}
with each intersection taken inside of $\SL_2(\Q)$. We claim that $H=H'$ inside $\Gamma_{r-d}^i$.
Indeed, as $\Gamma_{r}^i\subseteq \Gamma_{r-d}^i$, the inclusion $H\subseteq H'$ is clear.
For the reverse inclusion, if $\gamma:=\left(\begin{smallmatrix} * & * \\ x & *\end{smallmatrix}\right)\in \Gamma_{r-d}^i$,
then we have $\alpha^{-d}\gamma\alpha^d = \left(\begin{smallmatrix} * & * \\ p^{-d}x & *\end{smallmatrix}\right)$,
so if this lies in $\Gamma_{r-d}^i$ we must have $x\equiv 0\bmod p^r$ and hence $\gamma\in \Gamma_r^i$.
We conclude that the coset spaces $H\backslash\Gamma_{r-d}^i$ and $H'\backslash\Gamma_{r-d}^i$
are equal. On the other hand, for {\em any} commeasurable subgroups $\Gamma,\Gamma'$ of a group $G$
and any $g$ in the commeasurator of $\Gamma$ in $G$,
an elementary computation shows that we have a bijection of coset spaces
\begin{align*}
(\Gamma'\cap g^{-1}\Gamma g )\backslash \Gamma' \simeq \Gamma\backslash\Gamma g\Gamma'
\end{align*}
via $(\Gamma'\cap g^{-1}\Gamma g)\gamma\mapsto \Gamma g\gamma$.
Applying this with $g=\alpha^d$ in our situation and using the
decomposition
\begin{equation*}
\Gamma_{r-d}^i \alpha^d \Gamma_{r-d}^i = \coprod_{j=0}^{p^{d}-1} \Gamma_{r-d}^i\begin{pmatrix} 1 & j \\ 0 & p^{d}\end{pmatrix}
\end{equation*}
(see, e.g. \cite[proposition 3.36]{Shimura}), we deduce that we also have
\begin{equation}\label{disjointHecke}
\Gamma_r^i \alpha^d \Gamma_{r-d}^i = \coprod_{j=0}^{p^{d}-1} \Gamma_r^i\begin{pmatrix} 1 & j \\ 0 & p^{d}\end{pmatrix}.
\end{equation}
Writing $U:S_k(\Gamma_r^i)\rightarrow S_k(\Gamma_{r-d}^i)$ for the ``Hecke operator" given by
(e.g. \cite[\S3.4]{Ohta1})
$\Gamma_r^i \alpha^d \Gamma_{r-d}^i$, an easy computation using \ref{disjointHecke} shows that the composite
\begin{equation*}
\xymatrix{
S_k(\Gamma_r^i) \ar[r]^-{U} & S_{k}(\Gamma_{r-d}^i) \ar@{^{(}->}[r] & S_k(\Gamma_r^i)
}
\end{equation*}
coincides with $U_p^d$ on $q$-expansions. By the $q$-expansion principle, we deduce that $U_p^d$ on $S_k(\Gamma_r^i)$
indeed factors through the subspace $S_k(\Gamma_{r-d}^i)$, as desired.
\end{proof}
For each integer $i$ and any character $\varepsilon:(\Z/Np^i\Z)^{\times}\rightarrow \Qbar^{\times}$, we denote by
$S_2(\Gamma_i,\varepsilon)$
the $\H_i$-stable subspace of weight 2 cusp forms for $\Gamma_i$ over $\Qbar$
on which the diamond operators act through $\varepsilon(\cdot)$.
Define
\begin{equation}
\o{V}_r := \bigoplus_{i=1}^r\bigoplus_{\varepsilon } S_2(\Gamma_i,\varepsilon)
\label{VrDef}
\end{equation}
where the inner sum is over all Dirichlet characters defined modulo $Np^i$ whose $p$-parts are {\em primitive}
({\em i.e.} whose conductor has $p$-part exactly $p^i$).
We view $\o{V}_r$ as a $\Qbar$-subspace of $S_2(\Gamma_r)$ in the usual way
({\em i.e.} via the embeddings $\iota_{\id}$).
We define $\o{V}_r^*$ as the direct sum (\ref{VrDef}), but viewed
as a subspace of $S_2(\Gamma_r)$ via the ``nonstandard" embeddings
$\iota_{\alpha^{r-i}}:S_2(\Gamma_i)\rightarrow S_2(\Gamma_r)$.
As in (\ref{TeichmullerIdempotent}), we write $f'$ for the idempotent of $\Z_p[\mu_{p-1}]$
corresponding to ``projection away from the trivial $\mu_{p-1}$-eigenspace."
From the formulae (\ref{GpRngIdem})
we see that $h':=(p-1)f'$ lies in the subring $\Z[\mu_{p-1}]$ of $\Z_p[\mu_{p-1}]$
and satisfies $h'^2 = (p-1)h'$.
We define endomorphisms of $S_2(\Gamma_r)$:
\begin{equation}
U_r^*:=h'\circ (U_p^*)^{r+1} = (U_p^*)^{r+1}\circ h'\quad\text{and}\quad
U_r:=h'\circ (U_p)^{r+1} = (U_p)^{r+1}\circ h'.
\label{UrDefinition}
\end{equation}
\begin{corollary}\label{UpProjection}
As subspaces of $S_2(\Gamma_r)$ we have $w_r(\o{V}_r^*)=\o{V}_r$.
The space $\o{V}_r$ $($respectively $\o{V}_r^*$$)$ is naturally an $\H_r$ $($resp. $\H_r^*$$)$-stable
subspace of $S_2(\Gamma_r)$, and admits a canonical descent to $\Q$.
Furthermore, the endomorphisms
$U_r$ and $U_r^*$ of $S_2(\Gamma_r)$
factor through $\o{V}_r$ and $\o{V}_r^*$, respectively.
\end{corollary}
\begin{proof}
The first assertion follows from the relation $w_r\circ \iota_{\alpha^{r-i}}=\iota_{\id}\circ w_i$
as maps $S_2(\Gamma_i)\rightarrow S_2(\Gamma_r)$, together with the fact that $w_i$ on $S_2(\Gamma_i)$
carries $S_2(\Gamma_i,\varepsilon)$ isomorphically onto $S_2(\Gamma_i,\varepsilon^{-1})$.
The $\H_r$-stability of $\o{V}_r$ is clear as each of $S_2(\Gamma_i,\varepsilon)$ is an $\H_r$-stable subspace
of $S_2(\Gamma_r)$; that $\o{V}_r^*$ is $\H_r^*$-stable follows from this and the comutation
relation $T^* w_r = w_r T$ of Proposition \ref{AtkinInterchange}. That $\o{V}_r$ and $\o{V}_r^*$
admit canonical descents to $\Q$ is clear, as $\scrG_{\Q}$-conjugate Dirichlet characters have equal
conductors. The final assertion concerning the endomorphisms $U_r$ and $U_r^*$
follows easily from Lemma \ref{UpContract},
using the fact that $h':S_2(\Gamma_r)\rightarrow S_2(\Gamma_r)$
has image contained in $\bigoplus_{i=1}^r S_k(\Gamma_r^i)$.
\end{proof}
\begin{definition}
We denote by $V_r$ and $V_r^*$ the canonical descents to $\Q$ of $\o{V}_{r}$
and $\o{V}_r^*$, respectively.
\end{definition}
Following \cite[Chapter \Rmnum{3}, \S1]{MW-Iwasawa} and \cite[\S2]{Tilouine},
we recall the construction of certain ``good" quotient abelian varieties of $J_r$ whose
cotangent spaces are naturally identified with $V_r$ and $V_r^*$. In what follows,
we will make frequent use of the following elementary result:
\begin{lemma}\label{LieFactorization}
Let $f:A\rightarrow B$ be a homomorphism of commutative group varieties
over a field $K$ of characteristic $0$. Then:
\begin{enumerate}
\item The formation of $\Lie$ and $\Cot$ commutes with the formation of kernels and images: the kernel $($respectively image$)$ of $\Lie(f)$ is canonically
isomorphic to the Lie algebra of the kernel $($respectively image$)$ of $f$, and
similarly for cotangent spaces at the identity.
In particular, if $A$ is connected and $\Lie(f)=0$ $($respectively $\Cot(f)=0$$)$
then $f=0$.\label{ExactnessOfLie}
\item Let $i:B'\hookrightarrow B $ be a closed immersion of commutative
group varieties over $K$ with $B'$ connected. If $\Lie(f)$ factors through $\Lie(i)$
then $f$ factors $($necessarily uniquely$)$ through $i$.
\label{InclOnLie}
\item Let $j:A\twoheadrightarrow A''$ be a surjection of commutative
group varieties over $K$ with connected kernel. If $\Cot(f)$
factors through $\Cot(j)$ then $f$ factors $($necessarily uniquely$)$ through $j$.
\label{LieFactorizationSurj}
\end{enumerate}
\end{lemma}
\begin{proof}
The key point is that because $K$ has characteristic zero, the functors $\Lie(\cdot)$
and $\Cot(\cdot)$ on the category of commutative group schemes are {\em exact}.
Indeed, since $\Lie(\cdot)$ is always left
exact, the exactness of $\Lie(\cdot)$ follows easily from the fact that any
quotient mapping
$G\twoheadrightarrow H$
of group varieties in characteristic zero is smooth (as the kernel is
a group variety over a field of characteristic zero and hence automatically smooth),
so the induced map on Lie algebras is a surjection.
By similar reasoning
one shows that the right exact $\Cot(\cdot)$ is likewise exact,
and the first part of (\ref{ExactnessOfLie}) follows easily. In particular,
if $\Lie(f)$ is the zero map then $\Lie(\im(f))=0$, so $\im(f)$ is zero-dimensional.
Since it is also smooth, it must be \'etale. Thus, if $A$ is connected,
then $\im(f)$ is both connected and \'etale, whence it is a single point; by
evaluation of $f$ at the identity of $A$ we conclude that $f=0$.
The assertions (\ref{InclOnLie}) and (\ref{LieFactorizationSurj})
now follow immediately by using universal mapping properties.
\end{proof}
To proceed with the construction of good quotients of $J_r$, we now consider the diagrams of ``degeneracy mappings" of curves over $\Q$ for $i=1,2$
\begin{equation}
\addtocounter{equation}{1}
\xymatrix{
{X_r} \ar[r]^-{\pi} & {Y_r} \ar[r]^-{\pi_i} & {X_{r-1}}
}
\tag{$\arabic{section}.\arabic{subsection}.\arabic{equation}_i$}
\label{DegeneracyDiag}
\end{equation}
where $\pi$ and $\pi_i$ are the maps induced by (\ref{XtoY}) and (\ref{Upcorr}),
respectively. These mappings covariantly (respectively contravariantly) induce
mappings on the associated Jacobians via Albanese (respectively Picard) functoriality.
Writing $JY_r:=\Pic^0_{Y_r/\Q}$ and setting $K_1^i:=JY_1$ for $i=1,2$
we inductively define abelian subvarieties $\iota_r^i:K_r^i\hookrightarrow JY_r$ and abelian variety quotients
$\alpha_r^i:J_r\twoheadrightarrow B_r^i$ as follows:
\begin{equation}
\addtocounter{equation}{1}
B_{r-1}^i:= J_{r-1}/\Pic^0(\pi)(K_{r-1}^i)
\qquad\text{and}\qquad
K_{r}^i:=\ker(JY_r \xrightarrow{\alpha_{r-1}^i\circ\Alb(\pi_i)} B_{r-1}^i)^0
\tag{$\arabic{section}.\arabic{subsection}.\arabic{equation}_i$}
\label{BrDef}
\end{equation}
for $r\ge 2$, $i=1,2$, with $\alpha_{r-1}^i$ and $\iota_r^i$ the obvious mappings;
here, $(\cdot)^0$ denotes the connected component of the identity of $(\cdot)$.
As in \cite[\S 3.2]{OhtaEichler}, we have modified Tilouine's construction
\cite[\S2]{Tilouine} so that kernel of $\alpha_r$ is connected; {\em i.e.} is an abelian
subvariety of $J_r$ ({\em cf.} Remark \ref{TilouineReln}).
Note that we have a commutative diagram of abelian varieties over $\Q$
for $i=1,2$
\begin{equation}
\addtocounter{equation}{1}
\begin{gathered}
\xymatrix@C=50pt@R=26pt{
& {J_{r-1}}\ar@{->>}[r]^-{\alpha_{r-1}^i} &
{B_{r-1}^i} \ar@{=}[d] \\
{K_r^i} \ar@{^{(}->}[r]^-{\iota_r^i}\ar@{=}[d] &
{JY_r} \ar[r]^-{\alpha_{r-1}^i\circ \Alb(\pi_i)} \ar[d]|-{\Pic^0(\pi)}\ar[u]|-{\Alb(\pi_i)} &
B_{r-1}^i \\
K_r^i \ar[r]_-{\Pic^0(\pi)\circ \iota_r} & {J_r} \ar@{->>}[r]_-{\alpha_r^i} & {B_r^i}
}
\end{gathered}
\tag{$\arabic{section}.\arabic{subsection}.\arabic{equation}_i$}
\label{BrDefiningDiag}
\end{equation}
with bottom two horizontal rows that are complexes.
\begin{warning}\label{GoodQuoWarning}
While the bottom row of (\ref{BrDefiningDiag}) is exact in the middle by definition of $\alpha_r^i$,
the central row is {\em not} exact in the middle: it follows from
the fact that $\Alb(\pi_i)\circ\Pic^0(\pi_i)$ is multiplication by $p$
on $J_{r-1}$ that the component group of the kernel of
$\alpha_{r-1}^i\circ\Alb(\pi_i):JY_r\rightarrow B_{r-1}^i$ is nontrivial
with order divisible by $p$.
Moreover, there is no mapping $B_{r-1}^i\rightarrow B_r^i$
which makes the diagram (\ref{BrDefiningDiag}) commute.
\end{warning}
In order to be consistent with the literature, we adopt the following convention:
\begin{definition}\label{BalphDef}
We set $B_r:=B_r^2$ and $B_r^*:=B_r^1$, with $B_r^i$ defined inductively by (\ref{BrDef}).
We likewise set $\alpha_r:=\alpha_r^2$ and $\alpha_r^*:=\alpha_r^1$.
\end{definition}
\begin{remark}\label{TilouineReln}
We briefly comment on the relation between our quotient $B_r$
and the ``good" quotients of $J_r$ considered by Ohta \cite{OhtaEichler},
by Mazur-Wiles \cite{MW-Iwasawa}, and by Tilouine \cite{Tilouine}.
Recall \cite[\S2]{Tilouine} that Tilouine constructs\footnote{The notation Tilouine
uses for his quotient is the same as the notation we have used for our (slightly modified)
quotient. To avoid conflict, we have therefore chosen to
alter his notation.} an abelian variety quotient
$\alpha_r':J_r\twoheadrightarrow B_r'$ via an inductive
procedure nearly identical to the one used to define $B_r=B_r^1$:
one sets $K_1':=JY_1$, and for $r\ge 2$ defines
\begin{equation*}
B_{r-1}':= J_{r-1}/\Pic^0(\pi)(K_{r-1}')
\qquad\text{and}\qquad
K_{r}':=\ker(JY_r \xrightarrow{\alpha_{r-1}'\circ\Alb(\pi_2)} B_{r-1}').
\end{equation*}
Using the fact that the
formation of images and identity components commutes,
one shows via a straightforward induction argument that
$\alpha_r:J_r\twoheadrightarrow B_r$
identifies $B_r$ with $J_r/(\ker\alpha_r')^0$; in particular,
our $B_r$ is the same as Ohta's \cite[\S3.2]{OhtaEichler}
and Tilouine's quotient $\alpha_r':J_r\rightarrow B_r'$ uniquely
factors through $\alpha_r$ via an isogeny $B_r\twoheadrightarrow B_r'$
which has degree divisible by $p$ by Warning \ref{GoodQuoWarning}.
Due to this fact, it is {\em essential} for our purposes to work with $B_r$ rather than $B_r'$.
Of course, following \cite[3.2.1]{OhtaEichler}, we could have simply
{\em defined} $B_r$ as $J_r/(\ker\alpha_r')^0$,
but we feel that the construction we have given is more natural.
On the other hand, we remark that $B_r$
is naturally a quotient of the ``good" quotient $J_r\twoheadrightarrow A_r$ constructed
by Mazur-Wiles in \cite[Chapter \Rmnum{3}, \S1]{MW-Iwasawa},
and the kernel of the corresponding surjective homomorphism
$A_r\twoheadrightarrow B_r$
is isogenous to $J_0\times J_0$.
\end{remark}
\begin{proposition}\label{BrCotIden}
Over $F:=\Q(\mu_{Np^r})$, the automorphism $w_r$ of ${J_r}_F$ induces
an isomorphism of quotients ${B_r}_{F}\simeq {B_r^*}_F$.
The abelian variety $B_r$ $($respectively $B_r^*$$)$ is the unique quotient of $J_r$ by a $\Q$-rational
abelian subvariety with the property that
the induced map on cotangent spaces
\begin{equation*}
\xymatrix{
{\Cot(B_r)} \ar@{^{(}->}[r]_-{\Cot(\alpha_r)} &
{\Cot(J_r)\simeq S_2(\Gamma_r;\Q)}
}
\quad\text{respectively}\quad
\xymatrix{
{\Cot(B_r^*)} \ar@{^{(}->}[r]_-{\Cot(\alpha_r^*)} &
{\Cot(J_r)\simeq S_2(\Gamma_r;\Q)}
}
\end{equation*}
has image precisely $V_r$ $($respectively $V_r^*$$)$.
In particular, there are canonical actions
of the Hecke algebras\footnote{We must warn the reader that
Tilouine \cite{Tilouine} writes $\H_r(\Z)$ for the
$\Z$-subalgebra of $\End(J_r)$ generated by the Hecke operators
acting via the $(\cdot)^*$-action ({\em i.e.} by ``Picard" functoriality)
whereas our $\H_r(\Z)$ is defined using the $(\cdot)_*$-action.
This discrepancy is due primarily to the fact that Tilouine
identifies {\em tangent} spaces of modular abelian varieties
with spaces of modular forms, rather than cotangent spaces as is our convention.
Our notation for regarding Hecke algebras
as sub-algebras of $\End(J_r)$ agrees with that of Mazur-Wiles
\cite[Chapter \Rmnum{2}, \S5]{MW-Iwasawa}, \cite[\S7]{MW-Hida} and
Ohta \cite[3.1.5]{OhtaEichler}.
}
$\H_r(\Z)$ on $B_r$ and $\H_r^*(\Z)$ on $B_r^*$ for which $\alpha_r$ and $\alpha_r^*$
are equivariant.
\end{proposition}
\begin{proof}
By the construction of $B_r^i$ and Proposition \ref{ALinv},
the automorphism $w_{r}$ of $J_{r,F}$ carries $\ker(\alpha_r)$
to $\ker(\alpha_r^*)$ and induces an isomorphsm $B_{r,F} \simeq B_{r,F}^*$ over $F$
that intertwines the action of $\H_r$ on $B_r$ with $\H_r^*$ on $B_r^*$.
The isogeny $B_r\twoheadrightarrow B_r'$ of Remark \ref{TilouineReln} induces
an isomorphism on cotangent spaces, compatibly with the inclusions into
$\Cot(J_r)$. Thus, the claimed identification of the image of
$\Cot(B_r)$ with $V_r$ follows from \cite[Proposition 2.1]{Tilouine}
(using \cite[Definition 2.1]{Tilouine}). The claimed uniqueness
of $J_r\twoheadrightarrow B_r$ follows easily from Lemma \ref{LieFactorization}
(\ref{LieFactorizationSurj}). Similarly, since the subspace $V_r$
of $S_2(\Gamma_r)$ is stable under $\H_r$, we conclude from
Lemma \ref{LieFactorization} (\ref{LieFactorizationSurj}) that for any
$T\in \H_r(\Z)$, the induced morphism $J_r\xrightarrow{T} J_r\twoheadrightarrow B_r$
factors through $\alpha_r$, and hence that $\H_r(\Z)$ acts on $B_r$
compatibly (via $\alpha_r$) with its action on $J_r$.
\end{proof}
\begin{lemma}\label{Btower}
There exist unique morphisms $B_r^*\leftrightarrows B_{r-1}^*$
of abelian varieties over $\Q$ making
\begin{equation*}
\xymatrix{
{J_{r}} \ar[r]^-{\alpha_r^*} \ar[d]_-{\Alb(\sigma)} &{B_r^*} \ar[d] \\
{J_{r-1}} \ar[r]_-{\alpha_{r-1}^*} & {B_{r-1}^*}
}\qquad\raisebox{-18pt}{and}\qquad
\xymatrix{
{J_{r}} \ar[r]^-{\alpha_r^*} &{B_r^*} \\
{J_{r-1}} \ar[u]^-{\Pic^0(\rho)} \ar[r]_-{\alpha_{r-1}^*} & {B_{r-1}^*}\ar[u]
}
\end{equation*}
commute; these maps are moreover $\H_r^*(\Z)$-equivariant.
By a slight abuse of notation, we will simply write $\Alb(\sigma)$ and $\Pic^0(\rho)$
for the induced maps on $B_r^*$ and $B_{r-1}^*$, respectively.
\end{lemma}
\begin{proof}
Under the canonical identification of $\Cot(J_r)\otimes_{\Q}\o{\Q}$ with $S_2(\Gamma_r)$,
the mapping on cotangent spaces induced by $\Alb(\sigma)$ (respectively
$\Pic^0(\rho)$) coincides with $\iota_{\alpha}:S_2(\Gamma_{r-1})\rightarrow S_2(\Gamma_r)$
(respectively $\tr_{r,r-1}:S_2(\Gamma_r)\rightarrow S_2(\Gamma_{r-1})$).
As the kernel of $\alpha_r^*:J_r\twoheadrightarrow B_r^*$ is connected by definition,
thanks to Lemma \ref{LieFactorization} (\ref{LieFactorizationSurj}) it suffices
to prove that $\iota_{\alpha}$ (respectively $\tr_{r,r-1}$)
carries $V_{r-1}^*$ to $V_{r}^*$ (respectively $V_{r}^*$ to $V_{r-1}^*$).
On one hand, the composite
$\iota_{\alpha}\circ \iota_{\alpha^{r-1-i}}:S_2(\Gamma_i,\varepsilon)\rightarrow S_2(\Gamma_r)$
coincides with the embedding $\iota_{\alpha^{r-i}}$, and it follows immediately from the
definition of $V_r^*$ that $\iota_{\alpha}$ carries $V_{r-1}^*$ into $V_r^*$.
On the other hand, an easy calculation using (\ref{DualityIdentity}) shows that
one has equalities of maps $S_2(\Gamma_i,\varepsilon)\rightarrow S_2(\Gamma_r)$
\begin{equation*}
\iota_{\alpha}\circ \tr_{r,r-1}\circ \iota_{\alpha^{(r-i)}} = \begin{cases}
\iota_{\alpha^{(r-i)}}pU_p^* & \text{if}\ i< r \\
0 & \text{if}\ i=r
\end{cases}.
\end{equation*}
Thus,
the image of $\iota_{\alpha}\circ\tr_{r,r-1}:V_r^*\rightarrow S_2(\Gamma_r)$
is contained in the image of $\iota_{\alpha}:V_{r-1}^*\rightarrow S_2(\Gamma_r)$;
since $\iota_{\alpha}$ is injective, we conclude that
the image of $\tr_{r,r-1}:V_r^*\rightarrow S_2(\Gamma_{r-1})$ is contained in $V_{r-1}^*$
as desired.
\end{proof}
\begin{proposition}\label{GoodRedn}
The abelian varieties $B_r$ and $B_r^*$ acquire good reduction over $\Q_p(\mu_{p^r})$.
\end{proposition}
\begin{proof}
See \cite[Chap \Rmnum{3}, \S2, Proposition 2]{MW-Iwasawa}
and {\em cf.} \cite[\S9, Lemma 9]{HidaGalois}.
\end{proof}
As in \S\ref{OrdStruct}, we denote by ${e^*}':=f'e^*\in \H^*$ and $e':=f'e\in \H$
the sub-idempotents of $e^*$ and $e$, respectively,
corresponding to projection away from the trivial eigenspace of $\mu_{p-1}$.
\begin{proposition}\label{GoodRednProp}
The maps $\alpha_r$ and $\alpha_r^*$
induce isomorphisms of $p$-divisible groups over $\Q$
\begin{equation}
{e^*}'J_r[p^{\infty}] \simeq {e^*}'B_r^*[p^{\infty}]\quad\text{and}\quad
{e}'J_r[p^{\infty}] \simeq {e}'B_r[p^{\infty}],
\label{OrdBTisoms}
\end{equation}
respectively, that are $\H^*$ $($respectively $\H$$)$ equivariant
and compatible with change in $r$ via $\Alb(\sigma)$ and $\Pic^0(\rho)$
$($respectively $\Alb(\rho)$ and $\Pic^0(\sigma)$$)$.
\end{proposition}
We view the maps (\ref{UrDefinition})
as endomorphisms of $J_r$ in the obvious way, and again write
$U_r^*$ and $U_r$ for the induced endomorphism of
$B_r^*$ and $B_r$, respectively. To prove Proposition \ref{GoodRednProp},
we need the following geometric incarnation of
Corollary \ref{UpProjection}:
\begin{lemma}\label{UFactorDiagLem}
There exists a unique $\H_r^*(\Z)$ $($respectively $\H_r(\Z)$$)$-equivariant
map $W_r^*:B_r^*\rightarrow J_r$ $($respectively $W_r:B_r\rightarrow J_r$$)$
of abelian varieties over $\Q$ such that the diagram
\begin{equation}
\begin{gathered}
\xymatrix@C=30pt@R=35pt{
{J_r}\ar[d]_-{U_r^*}
\ar@{->>}[r]^-{\alpha_r^*} & {B_r^*} \ar[dl]|-{W_r^*} \ar[d]^-{U_r^*} \\
{J_r}\ar@{->>}[r]_-{\alpha_r^*} & {B_r^*}
}
\quad\raisebox{-24pt}{respectively}\quad
\xymatrix@C=30pt@R=35pt{
{J_r}\ar[d]_-{U_r}
\ar@{->>}[r]^-{\alpha_r} & {B_r} \ar[dl]|-{W_r} \ar[d]^-{U_r} \\
{J_r}\ar@{->>}[r]_-{\alpha_r} & {B_r}
}\label{UFactorDiag}
\end{gathered}
\end{equation}
commutes.
\end{lemma}
\begin{proof}
Consider the endomorphism of $J_r$ given by $U_r$.
Due to Corollary \ref{UpProjection},
the induced mapping on cotangent spaces factors through the inclusion
$\Cot(B_r)\hookrightarrow \Cot(J_r)$. Since the kernel of the quotient
mapping $\alpha_r:J_r\twoheadrightarrow B_r$ giving rise to this inclusion is connected,
we conclude from Lemma \ref{LieFactorization} (\ref{LieFactorizationSurj})
that $U_r$ factors uniquely through $\alpha_r$
via an $\H_r$-equivariant morphism $W_r:B_r\rightarrow J_r$.
The corresponding statements for $B_r^*$ are proved similarly.
\end{proof}
\begin{proof}[Proof of Proposition $\ref{GoodRednProp}$]
From (\ref{UFactorDiag}) we get commutative diagrams of $p$-divisible groups
over $\Q$
\begin{equation}
\begin{gathered}
\xymatrix{
{e^*}'{J_r}[p^{\infty}]\ar[d]_-{U_r^*}^-{\simeq} \ar[r]^-{\alpha_r^*} &
{e^*}'{B_r^*}[p^{\infty}] \ar[dl]|-{W_r^*} \ar[d]^-{U_r^*}_-{\simeq} \\
{e^*}'{J_r}[p^{\infty}]\ar[r]_-{\alpha_r^*} &
{e^*}'{B_r^*}[p^{\infty}]
}
\quad\raisebox{-24pt}{and}\quad
\xymatrix{
e'{J_r}[p^{\infty}]\ar[d]_-{U_r}^-{\simeq} \ar[r]^-{\alpha_r} &
e'{B_r}[p^{\infty}] \ar[dl]|-{W_r} \ar[d]^-{U_r}_-{\simeq} \\
e'{J_r}[p^{\infty}]\ar[r]_-{\alpha_r} &
e'{B_r}[p^{\infty}]
}
\label{UFactorDiagpDiv}
\end{gathered}
\end{equation}
in which all vertical arrows are isomorphisms due to the very definition of the
idempotents ${e^*}'$ and $e'$. An easy diagram chase then shows that {\em all}
arrows must be isomorphisms.
\end{proof}
We will write $\mathcal{B}_r$, $\mathcal{B}^*_r$, and $\mathcal{J}_r$, respectively, for the N\'eron models of
the base changes $(B_r)_{K_r}$, $(B_r^*)_{K_r}$ and $(J_r)_{K_r}$
over $T_r:=\Spec(R_r)$; due to Proposition \ref{GoodRednProp}, both $\mathcal{B}_r$ and $\mathcal{B}_r^*$ are abelian
schemes over $T_r$.
By the N\'eron mapping property, there are canonical actions of $\H_r(\Z)$ on $\mathcal{B}_r$, $\mathcal{J}_r$
and of $\H_r^*(\Z)$ on $\mathcal{B}_r^*$, $\mathcal{J}_r$ over $R_r$ extending the actions on generic fibers
as well as ``semilinear" actions of $\Gamma$
over the $\Gamma$-action on $R_r$ ({\em cf.} (\ref{GammaAction})).
For each $r$, the N\'eron mapping property further provides diagrams
\begin{equation}
\begin{gathered}
\xymatrix{
{\mathcal{J}_r \times_{T_r} T_{r+1}}\ar@<-1ex>[d]_{\Pic^0(\rho)} \ar[r]^-{\alpha_r^*}
& {\mathcal{B}_r^* \times_{T_r} T_{r+1}}\ar@<1ex>[d]^{\Pic^0(\rho)} \\
{\mathcal{J}_{r+1}} \ar[r]_-{\alpha_{r+1}^*} \ar@<-1ex>[u]_-{\Alb(\sigma)} & \ar@<1ex>[u]^-{\Alb(\sigma)} {\mathcal{B}_{r+1}^*}
}
\quad\raisebox{-24pt}{respectively}\quad
\xymatrix{
{\mathcal{J}_r \times_{T_r} T_{r+1}}\ar@<-1ex>[d]_{\Pic^0(\sigma)} \ar[r]^-{\alpha_r}
& {\mathcal{B}_r \times_{T_r} T_{r+1}}\ar@<1ex>[d]^{\Pic^0(\sigma)} \\
{\mathcal{J}_{r+1}} \ar[r]_-{\alpha_{r+1}}\ar@<-1ex>[u]_-{\Alb(\rho)} & \ar@<1ex>[u]^-{\Alb(\rho)} {\mathcal{B}_{r+1}}
}
\label{Nermaps}
\end{gathered}
\end{equation}
of smooth commutative group schemes over $T_{r+1}$ in which the inner and outer rectangles commute, and
all maps are $\H_{r+1}^*(\Z)$ (respectively $\H_{r+1}(\Z)$) and $\Gamma$ equivariant.
\begin{definition}\label{ordpdivdefn}
We define $\mathcal{G}_r:={e^*}'\left(\mathcal{B}_r^*[p^{\infty}]\right)$ and
we write $\mathcal{G}_r':=\mathcal{G}_r^{\vee}$ for its Cartier dual,
each of which is canonically an object of $\pdiv_{R_r}^{\Gamma}$.
For each $r\ge s$, noting that $U_p^*$ is an automorphism of $\mathcal{G}_r$,
we obtain from (\ref{Nermaps}) canonical morphisms
\begin{equation}
\xymatrix@C=45pt{
{\rho_{r,s}:\mathcal{G}_{s}\times_{T_{s}} T_{r}} \ar[r]^-{\Pic^0(\rho)^{r-s}} & {\mathcal{G}_{r}}
}
\qquad\text{and}\qquad
\xymatrix@C=70pt{
{\rho_{r,s}' : \mathcal{G}_{s}'\times_{T_{s}} T_{r}} \ar[r]^-{{({U_p^*}^{-1}\Alb(\sigma))^{\vee}}^{r-s}} & {\mathcal{G}_{r}'}
}\label{pdivTowers}
\end{equation}
in $\pdiv_{R_r}^{\Gamma}$, where $(\cdot)^{i}$ denotes the $i$-fold composition, formed in the obvious manner.
In this way, we get towers of $p$-divisible groups
$\{\mathcal{G}_r,\rho_{r,s}\}$ and $\{\mathcal{G}_r',\rho_{r,s}'\}$;
we will write $G_r$ and $G_r'$ for the unique descents of the generic fibers of $\mathcal{G}_r$
and $\mathcal{G}_r'$ to $\Q_p$, respectively.\footnote{Of course, $G_r'=G_r^{\vee}$.
Our non-standard notation $\mathcal{G}_r'$ for the Cartier
dual of $\mathcal{G}_r$ is preferrable, due to the fact that $\rho_{r,s}'$ is
{\em not} simply the dual of $\rho_{r,s}$; indeed, these two mappings go in opposite
directions!}
We let $T^*\in \H_r^*$ act on $\mathcal{G}_r$
through the action of $\H_r^*(\Z)$ on $\mathcal{B}_r^*$,
and on $\mathcal{G}_r'=\mathcal{G}_r^{\vee}$ by duality ({\em i.e.} as $(T^*)^{\vee}$).
The maps (\ref{pdivTowers}) are then $\H_r^*$-equivariant.
\end{definition}
By Proposition \ref{GoodRednProp}, $G_r$
is canonically isomorphic to ${e^*}'J_r[p^{\infty}]$, compatibly with the action of $\H_r^*$.
Since $J_r$ is a Jacobian---hence principally polarized---one might expect that $\mathcal{G}_r$
is isomorphic to its dual in $\pdiv_{R_r}^{\Gamma}$. However, this is {\em not quite} the case
as the canonical isomorphism $J_r\simeq J_r^{\vee}$ intertwines the actions of $\H_r$
and $\H_r^*$, thus interchanging the idempotents ${e^*}'$ and $e'$. To describe
the precise relationship between $\mathcal{G}_r^{\vee}$ and $\mathcal{G}_r$, we proceed as follows.
For each $\gamma\in \Gal(K_r'/K_0)\simeq \Gamma\times \Gal(K_0'/K_0)$,
let us write $\phi_{\gamma}: {G_r}_{K_r'}\xrightarrow{\simeq} \gamma^*({G_r}_{K_r'})$
for the descent data isomorphisms encoding the unique $\Q_p=K_0$-descent of ${G_r}_{K_r'}$ furnished by $G_r$.
We ``twist" this descent data by the $\Aut_{\Q_p}(G_r)$-valued character $\langle \chi\rangle\langle a\rangle_N$
of $\Gal(K_{\infty}'/K_0)$:
explicitly, for $\gamma\in \Gal(K_{r}'/K_0)$ we set
$\psi_{\gamma}:= \phi_{\gamma}\circ \langle \chi(\gamma)\rangle\langle a(\gamma)\rangle_N$
and note that since $\langle \chi(\gamma)\rangle\langle a(\gamma)\rangle_N$
is defined over $\Q_p$, the map $\gamma\rightsquigarrow \psi_{\gamma}$
really does satisfy the cocycle condition.
We denote by $G_r(\langle \chi\rangle\langle a\rangle_N)$ the unique $p$-divisible group over $\Q_p$
corresponding to this twisted descent datum.
Since the diamond operators commute with the Hecke operators, there is a canonical
induced action of $\H_r^*$ on $G_r(\langle \chi\rangle\langle a\rangle_N)$.
By construction, there is a canonical
$K_r'$-isomorphism
$G_r(\langle \chi\rangle\langle a\rangle_N)_{K_r'}\simeq {G_r}_{K_r'}$. Since
$G_r$ acquires good reduction over $K_r$ and the $\scrG_{K_r}$-representation
afforded by the Tate module of $G_r(\langle \chi\rangle\langle a\rangle_N)$
is the twist of $T_pG_r$ by the {\em unramified} character $\langle a\rangle_N$,
we conclude that $G_r(\langle \chi\rangle\langle a\rangle_N)$ also acquires
good reduction over $K_r$, and we denote the resulting object of $\pdiv_{R_r}^{\Gamma}$
by $\mathcal{G}_r(\langle \chi\rangle\langle a\rangle_N)$.
\begin{proposition}\label{GdualTwist}
There is a natural $\H_r^*$-equivariant isomorphism of $p$-divisible groups over $\Q_p$
\begin{equation}
G_r' \simeq G_r(\langle \chi\rangle \langle a\rangle_N)
\label{GrprimeGr}
\end{equation}
which uniquely extends to an isomorphism of the corresponding objects in $\pdiv_{R_r}^{\Gamma}$
and is compatible with change in $r$ using $\rho_{r,s}'$ on $G_r'$ and $\rho_{r,s}$ on $G_r$.
\end{proposition}
\begin{proof}
Let $\varphi_r: J_r\rightarrow J_r^{\vee}$ be the canonical principal polarization over $\Q_p$;
one then has the relation $\varphi_r\circ T = (T^*)^{\vee}\circ \varphi_r$
for each $T\in \H_r(\Z)$. On the other hand, the $K_r'$-automorphism
$w_r: {J_r}_{K_r'}\rightarrow {J_r}_{K_r'}$ intertwines $T\in \H_r(\Z)$ with $T^*\in \H_r^*(\Z)$.
Thus, the $K_r'$-morphism
\begin{equation*}
\xymatrix{
{\psi_r:{J_r}_{K_r'}^{\vee}} \ar[r]^-{({U_p^*}^{r})^{\vee}} &
{{J_r}_{K_r'}^{\vee}} \ar[r]^-{\varphi_r^{-1}}_{\simeq} & {J_r}_{K_r'}
\ar[r]^-{w_r}_{\simeq} & {{J_r}_{K_r'}}
}
\end{equation*}
is $\H_r^*(\Z)$-equivariant.
Passing to the induced map on $p$-divisible groups and applying ${e^*}'$, we
obtain from Proposition \ref{GoodRednProp} an $\H_r^*$-equivariant isomorphism
of $p$-divisible groups $\psi_r: {G_r'}_{K_r'} \simeq {G_r}_{K_r'}$. As
\begin{equation*}
\xymatrix@C=35pt{
{{J_r}_{K_r'}} \ar[r]^-{\langle \chi(\gamma)\rangle \langle a\rangle_N w_r}\ar[d]_-{1\times \gamma} &
{{J_r}_{K_r'}}\ar[d]^-{1\times \gamma} \\
{({J_r}_{K_r'})_{\gamma}} \ar[r]_-{\gamma^*(w_r)} & {({J_r}_{K_r'})_{\gamma}}
}
\end{equation*}
commutes for all $\gamma\in \Gal(K_r'/K_0)$ by Proposition \ref{ALinv},
the $K_r'$-isomorphism $\psi_r$ uniquely descends to an
$\H_r^*$-equivariant isomorphism (\ref{GrprimeGr})
of $p$-divisible groups over $\Q_p$.
By Tate's Theorem, this identification
uniquely extends to an isomorphism of the corresponding objects in $\pdiv_{R_r}^{\Gamma}$.
The asserted compatibility with change in $r$ boils down to the commutativity of the diagrams
\begin{equation*}
\begin{gathered}
\xymatrix{
{{e^*}'J_s[p^{\infty}]^{\vee}} \ar[r]^-{({U_p^*}^{s})^{\vee}} \ar[d]_-{{({U_p^*}^{-1}\Alb(\sigma))^{\vee}}^{r-s}}&
{{e^*}'J_s[p^{\infty}]^{\vee}} \ar[d]^-{{\Alb(\sigma)^{\vee}}^{r-s}} \\
{{e^*}'J_r[p^{\infty}]^{\vee}} \ar[r]_-{({U_p^*}^{r})^{\vee}} & {{e^*}'J_r[p^{\infty}]^{\vee}} \\
}
\quad\raisebox{-22pt}{and}\quad
\xymatrix{
{{J_s}_{K_r'}^{\vee}} \ar[r]^-{\varphi_s^{-1}} \ar[d]_-{{\Alb(\sigma)^{\vee}}^{r-s}}
& {J_s}_{K_r'} \ar[r]^-{w_s} \ar[d]|-{\Pic^0(\sigma)^{r-s}} & {{J_s}_{K_r'}}\ar[d]^-{\Pic^0(\rho)^{r-s}}\\
{{J_r}_{K_r'}^{\vee}} \ar[r]_-{\varphi_r^{-1}}
& {J_r}_{K_r'} \ar[r]_-{w_r} & {{J_r}_{K_r'}}
}
\end{gathered}
\end{equation*}
for all $s\le r$. The commutativity of the first diagram is clear, while that of the second follows
from Proposition \ref{ALinv} and the fact that
for {\em any} finite morphism $f:Y\rightarrow X$ of smooth curves over a field $K$,
one has $\varphi_Y\circ \Pic^0(f)=\Alb(f)^{\vee}\circ \varphi_X$, where
$\varphi_{\star}:J_{\star}\rightarrow J_{\star}^{\vee}$ is the canonical
principal polarization on Jacobians for $\star=X,Y$ (see, for example, the proof of Lemma 5.5 in \cite{CaisNeron}).
\end{proof}
We now wish to relate the special fiber of $\mathcal{G}_r$ to the $p$-divisible
group $\Sigma_r:={e^*}'\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}[p^{\infty}]$
of Definition \ref{pDivGpSpecial}. In order to do this, we proceed as follows.
Since $\X_r$ is regular, and proper flat over $R_r$ with (geometrically) reduced special fiber,
$\Pic^0_{\X_r/R_r}$ is a smooth $R_r$-scheme by \S8.4 Proposition 2 and \S9.4 Theorem 2 of \cite{BLR}.
By the N\'eron mapping property, we thus have a natural mapping $\Pic^0_{\X_r/R_r}\rightarrow \mathcal{J}_r^0$
that recovers the canonical identification on generic fibers, and is in fact an isomorphism
by \cite[\S9.7, Theorem 1]{BLR}. Composing with the map $\alpha_r^*:\mathcal{J}_r\rightarrow \mathcal{B}_r^*$
and passing to special fibers
yields a homomorphism of smooth commutative algebraic groups over $\mathbf{F}_p$
\begin{equation}
\xymatrix{
{\Pic^0_{\o{\X}_r/\mathbf{F}_p}} \ar[r]^-{\simeq} & {\o{\mathcal{J}}_r^0} \ar[r] & {\o{\mathcal{B}}^*_r}
}\label{PicToB}
\end{equation}
Due to \cite[\S9.3, Corollary 11]{BLR},
the normalization map $\nor{\o{\X}}_r\rightarrow \o{\X}$ induces a surjective homomorphism
$\Pic^0_{\o{\X}_r/\mathbf{F}_p}\rightarrow {\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}}$
with kernel that is a smooth, connected {\em linear} algebraic group over $\mathbf{F}_p$.
As any homomorphism from an affine group variety to an abelian variety is zero,
we conclude that (\ref{PicToB}) uniquely factors through this quotient, and we obtain
a natural map of abelian varieties:
\begin{equation}
\xymatrix{
{\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}} \ar[r] & {\o{\mathcal{B}}_r^*}
}\label{AbVarMaps}
\end{equation}
that is necessarily equivariant for the actions of $\H_r^*(\Z)$ and $\Gamma$.
As in \ref{AlbPicIncl},
we write
$j_r^{\star}:=\Pic^0_{I_r^{\star}/\mathbf{F}_p}$ the Jacobian of $I_r^{\star}$ for $\star=0,\infty$.
The following Proposition relates the special fiber of $\mathcal{G}_r$ to the
$p$-divisible group $\Sigma_r$ of Definition \ref{pDivGpSpecial}, and thus
enables an explicit description of the special fiber
of $\mathcal{G}_r$ in terms of the $p$-divisible groups of $j_r^{\star}$
({\em cf}. \S3 and \S4, Proposition 1 of \cite{MW-Hida} and pgs. 267--274 of \cite{MW-Iwasawa}).
\begin{proposition}\label{SpecialFiberOrdinary}
The mapping $(\ref{AbVarMaps})$ induces an isomorphism
of $p$-divisible groups over $\mathbf{F}_p$
\begin{equation}
\o{\mathcal{G}}_r := {{e^*}'\o{\mathcal{B}}_r^*[p^{\infty}]} \simeq
{{e^*}'\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}[p^{\infty}]}=:\Sigma_r
\label{OnTheNose}
\end{equation}
that is $\H_r^*$ and $\Gamma$-equivariant and compatible with change in $r$
via the maps $\rho_{r,s}$ on $\o{\mathcal{G}}_r$ and the maps $\Pic^0(\rho)^{r-s}$ on $\Sigma_r$.
In particular,
$\mathcal{G}_r/R_r$ is an ordinary $p$-divisible group, and for each $r$ there is a canonical exact sequence,
compatible with change in $r$ via $\rho_{r,s}$ on $\o{\mathcal{G}}_r$ and $\Pic^0(\rho)^{r-s}$ on $j_r^{\star}[p^{\infty}]$
\begin{equation}
\xymatrix@C=45pt{
0 \ar[r] & {f'j_r^{0}[p^{\infty}]^{\mult}} \ar[r]^-{\Alb(i_r^{0})\circ V^r} &
{\o{\mathcal{G}}_r} \ar[r]^-{\Pic^0(i_r^{\infty})} & {f'j_r^{\infty}[p^{\infty}]^{\et}} \ar[r] & 0
}\label{GrSpecialExact}
\end{equation}
where $i_r^{\star}:I_r^{\star}\hookrightarrow \nor{\o{\X}}_r$ are the canonical closed immersions
for $\star=0,\infty$. Moreover, $(\ref{GrSpecialExact})$ is compatible with the actions of
$\H^*$ and $\Gamma$, with $U_p^*$ $($respectively $\gamma\in \Gamma$$)$
acting on $f'j_r^{0}[p^{\infty}]^{\mult}$ as $\langle p\rangle_N V$
$($respectively $\langle \chi(\gamma)\rangle^{-1}$$)$
and on $f'j_r^{\infty}[p^{\infty}]^{\et}$ as $F$ $($respectively $\id$$)$.
\end{proposition}
\begin{proof}
The diagram (\ref{UFactorDiag}) induces a corresponding diagram of
N\'eron models over $R_r$ and hence of special fibers over $\mathbf{F}_p$.
Arguing as above, we obtain a commutative diagram of abelian
varieties
\begin{equation}
\begin{gathered}
\xymatrix@C=30pt@R=35pt{
{\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}}\ar[d]_-{U_r^*}
\ar[r]^-{\o{\alpha}^*_r} & {\o{\mathcal{B}}_r^*} \ar[dl]^-{W_r^*} \ar[d]^-{U_r^*} \\
{\Pic^0_{\nor{\o{\X}}_r/\mathbf{F}_p}}\ar[r]_-{\o{\alpha}^*_r} & {\o{\mathcal{B}}_r^*}
}\label{UFactorDiagmodp}
\end{gathered}
\end{equation}
over $\mathbf{F}_p$. The proof of \ref{GoodRednProp} now goes through {\em mutatis mutandis} to give the claimed
isomorphism (\ref{OnTheNose}). The rest follows immediately from Proposition \ref{MWSharpening}.
\end{proof}
\subsection{Ordinary families of Dieudonn\'e modules}\label{OrdDieuSection}
Let $\{\mathcal{G}_r/R_r\}_{r\ge 1}$ be the tower of $p$-divisible groups given by Definition \ref{ordpdivdefn}.
From the canonical morphisms $\rho_{r,s}: \mathcal{G}_{s}\times_{T_{s}} T_r\rightarrow \mathcal{G}_{r}$ we obtain
a map on special fibers $\o{\mathcal{G}}_{s}\rightarrow \o{\mathcal{G}}_r$ over $\mathbf{F}_p$
for each $r\ge s$; applying the contravariant Dieudonn\'e module functor
$\ensuremath{\mathbf{D}}(\cdot):=\ensuremath{\mathbf{D}}(\cdot)_{\Z_p}$ yields a projective
system of finite free $\Z_p$-modules $\{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\}_r$ with compatible linear endomorphisms $F,V$
satisfying $FV=VF=p$.
\begin{definition}\label{DinftyDef}
We write $\ensuremath{\mathbf{D}}_{\infty}:=\varprojlim_r \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)$ for the projective limit
of the system $\{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\}_r$. For $\star\in \{\et,\mult\}$
we write $\ensuremath{\mathbf{D}}_{\infty}^{\star}:=\varprojlim_r \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\star})$
for the corresponding projective limit.
\end{definition}
Since $\H_r^*$ acts by endomorphisms on $\o{\mathcal{G}}_r$, compatibly with change in $r$,
we obtain an action of $\H^*$ on $\ensuremath{\mathbf{D}}_{\infty}$ and on $\ensuremath{\mathbf{D}}_{\infty}^{\star}$.
Likewise, the ``geometric inertia action" of $\Gamma$ on $\o{\mathcal{G}}_r$ by automorphisms
of $p$-divisible groups over $\mathbf{F}_p$ gives an
action of $\Gamma$ on $\ensuremath{\mathbf{D}}_{\infty}$ and $\ensuremath{\mathbf{D}}_{\infty}^{\star}$.
As $\o{\mathcal{G}}_r$ is ordinary by Proposition \ref{SpecialFiberOrdinary}, applying $\ensuremath{\mathbf{D}}(\cdot)$
to the (split) connected-\'etale squence of $\o{\mathcal{G}}_r$ gives, for each $r$,
a functorially split exact sequence
\begin{equation}
\xymatrix{
0 \ar[r] & {\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\et})} \ar[r] & {\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)} \ar[r] &
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})} \ar[r] & 0
}\label{DieudonneFiniteExact}
\end{equation}
with $\Z_p$-linear actions of $\Gamma$, $F$, $V$, and $\H_r^*$.
Since projective limits commute with finite direct sums, we obtain
a split short {\em exact} sequence of $\Lambda$-modules with linear $\H^*$ and $\Gamma$-actions
and commuting linear endomorphisms $F,V$ satisfying $FV=VF=p$:
\begin{equation}
\xymatrix{
0 \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\et}} \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}} \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\mult}} \ar[r] & 0
}.\label{DieudonneInfiniteExact}
\end{equation}
\begin{theorem}\label{MainDieudonne}
As in Proposition $\ref{NormalizationCoh}$, set $d':=\sum_{k=3}^p \dim_{\mathbf{F}_p} S_k(N;\mathbf{F}_p)^{\ord}$.
Then:
\begin{enumerate}
\item $\ensuremath{\mathbf{D}}_{\infty}$ is a free $\Lambda$-module of rank $2d'$, and $\ensuremath{\mathbf{D}}_{\infty}^{\star}$
is free of rank $d'$ over $\Lambda$ for $\star\in \{\et,\mult\}$.
\label{MainDieudonne1}
\item For each $r\ge 1$, applying $\otimes_{\Lambda} \Z_p[\Delta/\Delta_r]$ to
$(\ref{DieudonneInfiniteExact})$ yields the short exact sequence $(\ref{DieudonneFiniteExact})$,
compatibly with $\H^*$, $\Gamma$, $F$ and $V$.
\label{MainDieudonne2}
\item Under the canonical splitting of $(\ref{DieudonneInfiniteExact})$, $\ensuremath{\mathbf{D}}_{\infty}^{\et}$
is the maximal subspace of $\ensuremath{\mathbf{D}}_{\infty}$ on which $F$ acts invertibly, while
$\ensuremath{\mathbf{D}}_{\infty}^{\mult}$ corresponds to the maximal subspace of $\ensuremath{\mathbf{D}}_{\infty}$ on which $V$ acts
invertibly.
\label{MainDieudonne3}
\item The Hecke operator $U_p^*$ acts as $F$ on $\ensuremath{\mathbf{D}}_{\infty}^{\et}$ and as $\langle p\rangle_NV$ on $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$.
\label{MainDieudonne4}
\item $\Gamma$ acts trivially on $\ensuremath{\mathbf{D}}_{\infty}^{\et}$ and via $\langle \chi\rangle^{-1}$
on $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$.
\label{MainDieudonne5}
\end{enumerate}
\end{theorem}
\begin{proof}
We apply Lemma \ref{Technical} with $A_r=\Z_p$, $I_r=(p)$,
and with $M_r$ each one of the terms in (\ref{DieudonneFiniteExact}).
Due to Proposition \ref{GisOrdinary}, there is a natural isomorphism of split short exact sequences
\begin{equation*}
\xymatrix{
0 \ar[r] & {\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\et})_{\mathbf{F}_p}} \ar[r]\ar[d]^-{\simeq} &
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)_{\mathbf{F}_p}} \ar[r] \ar[d]^-{\simeq}&
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})_{\mathbf{F}_p}} \ar[r] \ar[d]^-{\simeq}& 0 \\
0 \ar[r] & {f'H^1(I_r^0,\O)^{F_{\ord}}}\ar[r] &
{f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}\oplus f'H^1(I_r^0,\O)^{F_{\ord}}} \ar[r] &
{f'H^0(I_r^{\infty},\Omega^1)^{V_{\ord}}} \ar[r] & 0
}
\end{equation*}
that is compatible with change in $r$ using the trace mappings attached to
$\rho:I_r^{\star}\rightarrow I_{s}$ and the maps on Dieudonn\'e modules
induced by $\o{\rho}_{r,s}:\o{\mathcal{G}}_{s} \rightarrow \o{\mathcal{G}}_r$.
The hypotheses (\ref{freehyp}) and (\ref{surjhyp})
of Lemma \ref{Technical} are thus satisfied with $d'$ as in the statement of the theorem,
thanks to Proposition \ref{IgusaStructure} (\ref{IgusaFreeness})--(\ref{IgusaControl})
and Lemma \ref{CharacterSpaces}. We conclude from Lemma \ref{Technical}
that (\ref{MainDieudonne1}) and (\ref{MainDieudonne2}) hold.
As $F$ (respectively $V$) acts invertibly on $\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\et})$ (respectively
$\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})$) for all $r$, assertion (\ref{MainDieudonne3}) is clear, while
(\ref{MainDieudonne4}) and (\ref{MainDieudonne5}) follow immediately from
Proposition \ref{SpecialFiberOrdinary}.
\end{proof}
As in Proposition \ref{dRDuality}, the short exact sequence (\ref{DieudonneInfiniteExact})
is very nearly ``auto dual":
\begin{proposition}\label{DieudonneDuality}
There is a canonical isomorphism of short exact sequences of $\Lambda_{R_0'}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\et}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}} \ar[r]\ar[d]^-{\simeq} &
{\ensuremath{\mathbf{D}}_{\infty}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} &
{\ensuremath{\mathbf{D}}_{\infty}^{\mult}(\langle \chi \rangle\langle a\rangle_N)_{\Lambda_{R_0'}}}\ar[r]\ar[d]^-{\simeq} & 0 \\
0\ar[r] & {(\ensuremath{\mathbf{D}}_{\infty}^{\mult})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\ensuremath{\mathbf{D}}_{\infty})^{\vee}_{\Lambda_{R_0'}}} \ar[r] &
{(\ensuremath{\mathbf{D}}_{\infty}^{\et})^{\vee}_{\Lambda_{R_0'}}}\ar[r] & 0
}
\end{gathered}
\label{DmoduleDuality}
\end{equation}
that is $\H^*$ and $\Gamma\times \Gal(K_0'/K_0)$-equivariant,
and intertwines $F$
$($respectively $V$$)$ on the top row with $V^{\vee}$
$($respectively $F^{\vee}$$)$ on the bottom.
\end{proposition}
\begin{proof}
We apply the duality formalism
of Lemma \ref{LambdaDuality}.
Let us write
$\rho_{r,s}':\o{\mathcal{G}}_r'\rightarrow \o{\mathcal{G}}_s'$ for the maps on special fibers
induced by (\ref{pdivTowers}).
Thanks to Proposition \ref{GdualTwist}, the definition \ref{ordpdivdefn} of $\o{\mathcal{G}}_r':=\o{\mathcal{G}}_r^{\vee}$,
the natural isomorphism $\mathcal{G}_r\times_{R_r} R_r' \simeq \mathcal{G}_r(\langle \chi\rangle\langle a\rangle_N)\times_{R_r} R_r'$,
and the compatibility of the Dieudonn\'e module functor with duality, there are natural
isomorphisms of $R_0'$-modules
\begin{equation}
\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)(\langle \chi\rangle\langle a\rangle_N) \mathop{\otimes}\limits_{\Z_p} R_0' \simeq
\ensuremath{\mathbf{D}}(\o{\mathcal{G}_r(\langle \chi\rangle\langle a\rangle_N)})\mathop{\otimes}\limits_{\Z_p} R_0'
\simeq \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r')\mathop{\otimes}\limits_{\Z_p} R_0' = \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\vee})\mathop{\otimes}\limits_{\Z_p} R_0'\simeq
(\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r))_{R_0'}^{\vee}
\label{evpairingDieudonne}
\end{equation}
that are $\H^*_r$-equivariant, $\Gal(K_r'/K_0)$-compatible
for the standard action $\sigma\cdot f (m):=\sigma f(\sigma^{-1}m)$
on the $R_0'$-linear dual of $\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\otimes_{\Z_p} R_0'$,
and compatible with change in $r$ using $\rho_{r,s}$
on $\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)$ and $\rho_{r,s}'$ on $\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r')$. We claim that the resulting
perfect ``evaluation" pairings
\begin{equation}
\xymatrix{
{\langle\cdot,\cdot\rangle_r : \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)(\langle \chi\rangle\langle a\rangle_N)\mathop{\otimes}\limits_{\Z_p}{R_0'}
\times \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\mathop{\otimes}\limits_{\Z_p}{R_0'}} \ar[r] & {R_0'}
}\label{pdivSpecialTwistPai}
\end{equation}
satisfy the compatibility hypothesis (\ref{pairingchangeinr}) of Lemma \ref{LambdaDuality}.
Indeed, the stated compatibility of (\ref{evpairingDieudonne}) with change in $r$
and the very definition (\ref{pdivTowers}) of the transition maps $\rho_{r,s}'$
implies that for $r\ge s$
\begin{equation*}
\langle \ensuremath{\mathbf{D}}(\Pic^0(\rho)^{r-s} x), y\rangle_s = \langle x , \ensuremath{\mathbf{D}}({U_p^*}^{s-r}\Alb(\sigma)^{r-s}) y\rangle_r,
\end{equation*}
so our claim follows from the equality in $\End_{\Q_p}(J_{r+1})$
\begin{equation}
\Pic(\rho)\circ \Alb(\sigma) = U_p^* \sum_{\delta\in \Delta_r/\Delta_{r+1}} \langle \delta^{-1}\rangle,
\label{PicAlbRelation}
\end{equation}
which, as in the proof of Proposition \ref{dRDuality}, follows from Lemma \ref{MFtraceLem} via Lemma
\ref{LieFactorization}.
Again, by the $\H_r^*$-compatibility of (\ref{evpairingDieudonne}), the action of $\H_r^*$
is self-adjoint with resect to (\ref{pdivSpecialTwistPai}), so
Lemma \ref{LambdaDuality} gives a perfect $\Gal(K_{\infty}'/K_0)$-compatible duality pairing
$\langle\cdot,\cdot \rangle: \ensuremath{\mathbf{D}}_{\infty}(\langle \chi\rangle\langle a\rangle_N)
\otimes_{\Lambda} \Lambda_{R_0'} \times \ensuremath{\mathbf{D}}_{\infty}\otimes_{\Lambda} \Lambda_{R_0'} \rightarrow \Lambda_{R_0'}$
with respect to which $T^*$ is self-adjoint for all $T^*\in \H^*$.
That the resulting isomorphism (\ref{DmoduleDuality})
intertwines
$F$ with $V^{\vee}$
and $ V $ with $F^{\vee}$ is an immediate consequence of the compatibility of the Dieudonn\'e module
functor with duality.
\end{proof}
We can interpret $\ensuremath{\mathbf{D}}_{\infty}^{\star}$ in terms of the crystalline cohomology of the
Igusa tower as follows. Let $I_r^0$ and $I_r^{\infty}$ be the two ``good"
components of $\o{\X}_r$ as in Remark \ref{MWGood}, and form the projective limits
\begin{equation*}
H^1_{\cris}(I^{\star}) := \varprojlim_{r} H^1_{\cris}(I_r^{\star})
\end{equation*}
for $\star\in \{\infty,0\}$, taken with respect to the trace maps on crystalline
cohomology (see \cite[\Rmnum{7}, \S2.2]{crystal2}) induced by the canonical degeneracy mappings
$\rho:I_{r}^{\star}\rightarrow I_{s}^{\star}$.
Then $H^1_{\cris}(I^{\star})$
is naturally a $\Lambda$-module (via the diamond operators), equipped with
a commuting action of $F$ (Frobenius) and $V$ (Verscheibung) satisfying $FV=VF=p$.
Letting $U_p^*$ act as $F$ (respectively $\langle p\rangle_N V$) on $H^1_{\cris}(I^{\star})$ for $\star=\infty$
(respectively $\star=0$) and the Hecke operators outside $p$ (viewed as correspondences
on the Igusa curves) act via pullback and trace at each level $r$, we obtain
an action of $\H^*$ on $H^1_{\cris}(I^{\star})$. Finally, we
let $\Gamma$ act trivially on $H^1_{\cris}(I^{\star})$ for $\star=\infty$
and via $\langle\chi^{-1}\rangle$ for $\star=0$.
\begin{theorem}\label{DieudonneCrystalIgusa}
There is a canonical $\H^*$ and $\Gamma$-equivariant
isomorphism of $\Lambda$-modules
\begin{equation*}
\ensuremath{\mathbf{D}}_{\infty} = \ensuremath{\mathbf{D}}_{\infty}^{\mult}\oplus \ensuremath{\mathbf{D}}_{\infty}^{\et} \simeq
f'H^1_{\cris}(I^{0})^{V_{\ord}} \oplus
f'H^1_{\cris}(I^{\infty})^{F_{\ord}}
\end{equation*}
which respects the given direct sum decompositions and is compatible with $F$ and $V$.
\end{theorem}
\begin{proof}
From the exact sequence (\ref{GrSpecialExact}), we obtain for each $r$ isomorphisms
\begin{equation}
\xymatrix@C=55pt{
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})} \ar[r]^-{\simeq}_-{V^r \circ \ensuremath{\mathbf{D}}(\Alb(i_r^{0}))} &
{f'\ensuremath{\mathbf{D}}(j_r^{0}[p^{\infty}])^{V_{\ord}}}
}\qquad\text{and}\qquad
\xymatrix@C=55pt{
{f'\ensuremath{\mathbf{D}}(j_r^{\infty}[p^{\infty}])^{F_{\ord}}} \ar[r]^-{\simeq}_-{\ensuremath{\mathbf{D}}(\Pic^0(i_r^{\infty}))} &
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\et})}
}\label{IgusaInterpretation}
\end{equation}
that are $\H^*$ and $\Gamma$-equivariant (with respect to the actions
specified in Proposition \ref{SpecialFiberOrdinary}), and compatible with change in $r$
via the mappings $\ensuremath{\mathbf{D}}(\rho_{r,s})$ on $\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\star})$ and $\ensuremath{\mathbf{D}}(\rho)$
on $\ensuremath{\mathbf{D}}(j_r^{\star}[p^{\infty}])$. On the other hand, for {\em any} smooth and proper curve
$X$ over a perfect field $k$ of characteristic $p$,
thanks to \cite{MM} and \cite[\Rmnum{2}, \S3 C Remarque 3.11.2]{IllusiedR}
there are natural isomorphisms
of $W(k)[F,V]$-modules
\begin{equation}
\ensuremath{\mathbf{D}}(J_X[p^{\infty}]) \simeq H^1_{\cris}(J_X/W(k)) \simeq H^1_{\cris}(X/W(k))\label{MMIllusie}
\end{equation}
that for any finite map of smooth proper curves $f:Y\rightarrow X$ over $k$
intertwine $\ensuremath{\mathbf{D}}(\Pic(f))$ and $\ensuremath{\mathbf{D}}(\Alb(f))$ with trace and pullback by $f$ on crystalline cohomology,
respectively. Applying this to $X=I_r^{\star}$ for $\star=0,\infty$, appealing to
the identifications (\ref{IgusaInterpretation}), and passing to inverse limits completes the proof.
\end{proof}
Applying the idempotent $f'$ of (\ref{TeichmullerIdempotent}) to the Hodge filtration (\ref{mainthmexact})
yields a short exact sequence of free $\Lambda_{R_{\infty}}$-modules with
semilinear $\Gamma$-action and linear commuting action of $\H^*$:
\begin{equation}
\xymatrix{
0 \ar[r] & {{e^*}'H^0(\omega)} \ar[r] & {{e^*}'H^1_{\dR}} \ar[r] & {{e^*}'H^1(\O)} \ar[r] & 0
}.\label{LambdaHodgeFilnomup}
\end{equation}
The key to relating (\ref{LambdaHodgeFilnomup}) to the slope filtration (\ref{DieudonneInfiniteExact})
is the following comparison isomorphism:
\begin{proposition}\label{KeyComparison}
For each positive integer $r$, there is a natural isomorphism of short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\omega_{\mathcal{G}_r}} \ar[r]\ar[d]_-{\simeq} & {\ensuremath{\mathbf{D}}(\mathcal{G}_{r,0})_{R_r}} \ar[r]\ar[d]^-{\simeq} &
{\Lie(\Dual{\mathcal{G}}_r)} \ar[r]\ar[d]^-{\simeq} & 0 \\
0\ar[r] & {{e^*}'H^0(\omega_r)} \ar[r] & {{e^*}'H^1_{\dR,r}} \ar[r] & {{e^*}'H^1(\O_r)} \ar[r] & 0
}
\end{gathered}\label{CollectedComparisonIsom}
\end{equation}
that is compatible with $\H_r^*$, $\Gamma$, and change in $r$ using
the mappings $(\ref{pdivTowers})$ on the top row and the maps $\rho_*$ on the bottom.
Here, the bottom row is obtained from $(\ref{HodgeFilIntAbbrev})$ by applying ${e^*}'$
and the top row is the Hodge filtration of $\ensuremath{\mathbf{D}}(\mathcal{G}_{r,0})_{R_r}$ given by
Proposition $\ref{BTgroupUnivExt}$.
\end{proposition}
\begin{proof}
Let $\alpha_r^*: J_r\twoheadrightarrow B_r^*$ be the map of Definition \ref{BalphDef}.
We claim that $\alpha_r^*$ induces a canonical isomorphism of short exact sequences of free
$R_r$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\omega_{\mathcal{G}_r}}\ar[d]_-{\simeq} \ar[r] & {\ensuremath{\mathbf{D}}(\mathcal{G}_{r,0})_{R_r}} \ar[d]_-{\simeq}\ar[r] &
{\Lie(\mathcal{G}_r^t)} \ar[d]^-{\simeq}\ar[r] & 0 \\
0 \ar[r] & {{e^*}'\omega_{\mathcal{J}_r}} \ar[r] & {{e^*}'\Lie\scrExtrig(\mathcal{J}_r,\Gm)} \ar[r] &
{{e^*}'\Lie({\mathcal{J}_r^t}^0)} \ar[r] & 0
}
\end{gathered}\label{HodgeToExtrigMap}
\end{equation}
that is $\H_r^*$ and $\Gamma$-equivariant and compatible with change in $r$ using
the map on N\'eron models induced by $\Pic^0(\rho)$ and the maps (\ref{pdivTowers})
on $\mathcal{G}_r$.
Granting this claim, the proposition then follows immediately from Proposition \ref{intcompare}.
To prove our claim, we introduce the following notation:
set $V:=\Spec(R_r)$, and for $n\ge 1$ put $V_n:=\Spec(R_r/p^nR_r)$. For any scheme
(or $p$-divisible group) $X$ over $V$, we put $X_n:=X\times_V V_n$.
If $\ensuremath{\mathcal{A}}$ is a N\'eron model over $V$,
we will write $H(\ensuremath{\mathcal{A}})$ for the short exact sequence of free $R_r$-modules obtained by
applying $\Lie$ to the canonical extension (\ref{NeronCanExt}) of $\Dual{\ensuremath{\mathcal{A}}}^0$.
If $G$ is a $p$-divisible group over $V$, we similalry
write $H(G_n)$ for the short exact sequence of Lie algebras associated to the universal extension
of $G_n^t$ by a vector group over $V_n$ (see Theorem \ref{UniExtCompat}, (\ref{UniExtCompat2})).
If $\ensuremath{\mathcal{A}}$ is an abelian scheme over $V$
then we have natural and compatible (with change in $n$) isomorphisms
\begin{equation}
H(\ensuremath{\mathcal{A}}_n[p^{\infty}])\simeq H(\ensuremath{\mathcal{A}}_n)\simeq H(\ensuremath{\mathcal{A}})/p^n,\label{AbSchpDiv}
\end{equation}
thanks to Theorem \ref{UniExtCompat}, (\ref{UniExtCompat3}) and (\ref{UniExtCompat1}); in particular, this
justifies our slight abuse of notation.
Applying the contravariant functor ${e^*}'H(\cdot)$ to the diagram of N\'eron models over $V$
induced by (\ref{UFactorDiag}) yields a commutative diagram of short exact sequences of
free $R_r$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
{{e^*}'H(\mathcal{J}_r)} & {{e^*}'H(\mathcal{B}_r^*)}\ar[l] \\
{{e^*}'H(\mathcal{J}_r)} \ar[u]^-{U_r^*}\ar[ur] & {{e^*}'H(\mathcal{B}_r)}\ar[u]_-{U_r^*}\ar[l]
}
\end{gathered}
\end{equation}
in which both vertical arrows are isomorphisms by definition of ${e^*}'$. As in the proofs of Propositions
\ref{GoodRednProp} and \ref{SpecialFiberOrdinary}, it follows that
the horizontal maps must be isomorphisms as well:
\begin{equation}
{e^*}'H(\mathcal{J}_r)\simeq {e^*}'H(\mathcal{B}_r^*)
\label{alphaIdenOrd}
\end{equation}
Since these isomorphisms are induced via the N\'eron mapping property and the functoriality
of $H(\cdot)$ by the $\H_r^*(\Z)$-equivariant map $\alpha_r^*:J_r\twoheadrightarrow B_r^*$,
they are themselves $\H_r^*$-equivariant. Similarly, since $\alpha_r^*$ is defined
over $\Q$ and compatible with change in $r$ as in Lemma \ref{Btower}, the isomorphism
(\ref{alphaIdenOrd}) is compatible with the given actions of $\Gamma$ (arising via the N\'eron
mapping property from the semilinear action of $\Gamma$ over $K_r$ giving the descent
data of ${J_r}_{K_r}$ and ${B_r}_{K_r}$ to $\Q_p$) and change in $r$.
Reducing (\ref{alphaIdenOrd}) modulo $p^n$ and using the canonical isomorphism (\ref{AbSchpDiv}) yields
the identifications
\begin{equation}
{e^*}'H(\mathcal{J}_r)/p^n\simeq {e^*}'H(\mathcal{B}_r^*)/p^n \simeq {e^*}'H(\mathcal{B}_{r,n}^*[p^{\infty}])
\simeq H({e^*}'\mathcal{B}_{r,n}^*[p^{\infty}]) =: H(\mathcal{G}_{r,n})\label{ModPowersIsom}
\end{equation}
which are clearly compatible with change in $n$, and which are easily checked
(using the naturality of (\ref{AbSchpDiv}) and our remarks above) to be
$\H_r^*$ and $\Gamma$-equivariant, and compatible with change in $r$.
Since the surjection $R_r\twoheadrightarrow R_r/pR_r$ is a PD-thickening,
passing to inverse limits (with respect to $n$) on (\ref{ModPowersIsom}) and using
Proposition \ref{BTgroupUnivExt} now completes the proof.
\end{proof}
\begin{corollary}\label{RelationToHodgeCor}
Let $r$ be a positive integer. Then the short exact sequence of free $R_r$-modules
\begin{equation}
\xymatrix{
0\ar[r] & {{e^*}'H^0(\omega_r)} \ar[r] & {{e^*}'H^1_{\dR,r}} \ar[r] & {{e^*}'H^1(\O_r)} \ar[r] & 0
}\label{TrivialEigenHodge}
\end{equation}
is functorially split; in particular,
it is split compatibly with the actions of $\Gamma$ and $\H_r^*$.
Moreover, $(\ref{TrivialEigenHodge})$ admits a functorial descent to $\Z_p$:
there is a natural isomorphism of split short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {{e^*}'H^0(\omega_r)} \ar[r]\ar[d]_-{\simeq} &
{{e^*}'H^1_{\dR,r}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O_r)} \ar[r]\ar[d]^-{\simeq} & 0\\
0 \ar[r] & {\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})\mathop{\otimes}\limits_{\Z_p} R_r} \ar[r] &
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\mathop{\otimes}\limits_{\Z_p} R_r}\ar[r] &
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{et})\mathop{\otimes}\limits_{\Z_p} R_r} \ar[r] & 0
}
\end{gathered}\label{DescentZp}
\end{equation}
that is $\H^*$ and $\Gamma$ equivariant, with
$\Gamma$ acting trivially on $\o{\mathcal{G}}_r^{\et}$ and through $\langle \chi\rangle^{-1}$ on $\o{\mathcal{G}}_r^{\mult}$.
The identification $\ref{DescentZp}$ is compatible with change in $r$ using the maps $\rho_*$ on the top
row and the maps induced by
\begin{equation*}
\xymatrix@C=35pt{
{\o{\mathcal{G}}_r=\o{\mathcal{G}}_r^{\mult} \times \o{\mathcal{G}}_r^{\et}} \ar[r]^{V^{-1}\times F} &
{\o{\mathcal{G}}_r^{\mult} \times \o{\mathcal{G}}_r^{\et}=\o{\mathcal{G}}_r} \ar[r]^-{\o{\rho}} &
{\o{\mathcal{G}}_{r+1}}
}
\end{equation*}
on the bottom row.
\end{corollary}
\begin{proof}
Consider the isomorphism (\ref{CollectedComparisonIsom}) of Proposition \ref{KeyComparison}.
As $\mathcal{G}_r$ is an ordinary $p$-divisible group by Proposition \ref{SpecialFiberOrdinary},
the top row of (\ref{CollectedComparisonIsom}) is functorially split
by Lemma \ref{HodgeFilOrdProps}, and this gives our first assertion.
Composing the inverse of (\ref{CollectedComparisonIsom})
with the isomorphism (\ref{DescentToWIsom}) of Lemma \ref{HodgeFilOrdProps} gives
the claimed identification (\ref{DescentZp}).
That this isomorphism is compatible with change in $r$ via the specified maps
follows easily from the construction of (\ref{DescentToWIsom}) via
(\ref{TwistyDieuIsoms}).
\end{proof}
We can now prove Theorem \ref{dRtoDieudonne}. Let us recall the statement:
\begin{theorem}\label{dRtoDieudonneInfty}
There is a canonical isomorphism of
short exact sequences of finite free $\Lambda_{R_{\infty}}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {{e^*}'H^0(\omega)} \ar[r]\ar[d]^-{\simeq} &
{{e^*}'H^1_{\dR}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O)} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\ensuremath{\mathbf{D}}_{\infty}\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] &
{\ensuremath{\mathbf{D}}_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_{\infty}}} \ar[r] & 0
}
\end{gathered}
\end{equation}
that is $\Gamma$ and $\H^*$-equivariant.
Here, the mappings on bottom row are the canonical inclusion and projection morphisms
corresponding to the direct sum decomposition $\ensuremath{\mathbf{D}}_{\infty}=\ensuremath{\mathbf{D}}_{\infty}^{\mult}\oplus \ensuremath{\mathbf{D}}_{\infty}^{\et}$.
In particular, the Hodge filtration exact sequence $(\ref{LambdaHodgeFilnomup})$ is canonically
split, and admits a canonical descent to $\Lambda$.
\end{theorem}
\begin{proof}
Applying $\otimes_{R_r} R_{\infty}$ to $(\ref{DescentZp})$ and passing to projective limits
yields an isomorphism of split exact sequences
\begin{equation*}
\xymatrix{
0\ar[r] & {{e^*}'H^0(\omega)} \ar[r]\ar[d]_-{\simeq} &
{{e^*}'H^1_{\dR}} \ar[r]\ar[d]^-{\simeq} & {{e^*}'H^1(\O)} \ar[r]\ar[d]^-{\simeq} & 0\\
0 \ar[r] & {\varprojlim\limits_{\o{\rho}\circ V^{-1}} \left(\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})\mathop{\otimes}\limits_{\Z_p} R_{\infty}\right)} \ar[r] &
{\varprojlim\limits_{\o{\rho}\circ (V^{-1}\times F)}\left(\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\mathop{\otimes}\limits_{\Z_p} R_{\infty}\right)}\ar[r] &
{\varprojlim\limits_{\o{\rho}\circ F}\left(\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{et})\mathop{\otimes}\limits_{\Z_p} R_{\infty}\right)} \ar[r] & 0
}
\end{equation*}
On the other hand, the isomorphisms
$ \xymatrix@1@C=37pt{
{\o{\mathcal{G}}_r = \o{\mathcal{G}}_r^{\mult}\times \o{\mathcal{G}}_r^{\et} } \ar[r]^-{V^{-r}\times F^{r}} &
{\o{\mathcal{G}}_r^{\mult}\times \o{\mathcal{G}}_r^{\et} =\o{\mathcal{G}}_r}
}
$
induce an isomorphism of projective limits
\begin{equation*}
\xymatrix{
{\varprojlim\limits_{\o{\rho}}\left(\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\mathop{\otimes}\limits_{\Z_p} R_{\infty}\right)} \ar[r]^-{\simeq} &
{\varprojlim\limits_{\o{\rho}\circ (V^{-1}\times F)}\left(\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\mathop{\otimes}\limits_{\Z_p} R_{\infty}\right)}
}
\end{equation*}
which is visibly compatible with the the canonical splittings of source and target.
The result now follows from
Lemma \ref{Technical} (\ref{CompletedBaseChange}) and the proof of Theorem \ref{MainDieudonne},
which guarantee that the canonical mapping
$\ensuremath{\mathbf{D}}_{\infty}\otimes_{\Lambda}\Lambda_{R_{\infty}}\rightarrow
\varprojlim_{\o{\rho}} (\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r)\otimes_{\Z_p}R_{\infty})$
is an isomorphism respecting the natural splittings.
\end{proof}
As in \S\ref{ordforms}, for any subfield $K$ of $\c_p$ with ring of integers $R$, we denote by
$eS(N;\Lambda_R)$ the module of ordinary $\Lambda_R$-adic cuspforms of level $N$
in the sense of \cite[2.5.5]{OhtaEichler}. Following our convention of \S\ref{OrdStruct},
we write $e'S(N;\Lambda_R)$ for the direct summand of $eS(N;\Lambda_R)$
on which $\mu_{p-1}\hookrightarrow \Z_p^{\times}\subseteq \H$ acts
nontrivially.
\begin{corollary}\label{MFIgusaDieudonne}
There is a canonical isomorphism of finite free $\Lambda$-modules
\begin{equation}
{e}'S(N;\Lambda) \simeq \ensuremath{\mathbf{D}}_{\infty}^{\mult}
\label{LambdaFormsCrystalline}
\end{equation}
that intertwines $T\in \H$ on $e'S(N;\Lambda)$ with $T^*\in \H^*$
on $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$,
where $U_p^*$ acts on $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$as $\langle p\rangle_N V$.
\end{corollary}
\begin{proof}
We claim that there are natural isomorphisms of finite free $\Lambda_{R_{\infty}}$-modules
\begin{equation}
\ensuremath{\mathbf{D}}_{\infty}^{\mult} \otimes_{\Lambda} \Lambda_{R_{\infty}} \simeq
{e^*}'H^0(\omega) \simeq {e}'S(N,\Lambda_{R_{\infty}}) \simeq
e'S(N,\Lambda)\otimes_{\Lambda} \Lambda_{R_{\infty}}\label{TakeGammaInvariants}
\end{equation}
and that the resulting composite isomorphism
intertwines $T^*\in \H^*$ on $\ensuremath{\mathbf{D}}_{\infty}^{\mult}$ with $T\in \H$ on
$e'S(N,\Lambda)$ and is $\Gamma$-equivariant, with $\gamma\in\Gamma$
acting as $\langle \chi(\gamma)\rangle^{-1}\otimes \gamma$ on each tensor product.
Indeed, the first and second isomorphisms are
due to Theorem \ref{dRtoDieudonneInfty} and Corollary \ref{LambdaFormsRelation}, respectively,
while the final isomorphism is a consequence of the definition
of $e'S(N;\Lambda_R)$ and the facts that this $\Lambda_R$-module is free of finite rank
\cite[Corollary 2.5.4]{OhtaEichler} and specializes as in \cite[2.6.1]{OhtaEichler}.
Twisting the
$\Gamma$-action on the source and target of the composite (\ref{TakeGammaInvariants}) by
$\langle \chi \rangle$ therefore gives a $\Gamma$-equivariant isomorphism
\begin{equation}
\ensuremath{\mathbf{D}}_{\infty}^{\mult} \otimes_{\Lambda} \Lambda_{R_{\infty}} \simeq
S(N,\Lambda)\otimes_{\Lambda} \Lambda_{R_{\infty}}\label{TwistedGammaIsom}
\end{equation}
with $\gamma\in \Gamma$ acting as $1\otimes \gamma$ on source and target. Passing to $\Gamma$-invariants on
(\ref{TwistedGammaIsom}) yields (\ref{LambdaFormsCrystalline}).
\end{proof}
\begin{remark}\label{MFIgusaCrystal}
Via Proposition \ref{DieudonneDuality} and the natural $\Lambda$-adic
duality between $e\H$ and $eS(N;\Lambda)$ \cite[Theorem 2.5.3]{OhtaEichler}, we obtain
a canonical $\Gal(K_0'/K_0)$-equivariant isomorphism of $\Lambda_{R_0'}$-modules
\begin{equation*}
e'\H\mathop{\otimes}\limits_{\Lambda} \Lambda_{R_0'} \simeq \ensuremath{\mathbf{D}}_{\infty}^{\et}(\langle a\rangle_N)\mathop{\otimes}\limits_{\Lambda}{\Lambda_{R_0'}}
\end{equation*}
that intertwines $T\otimes 1$ for $T\in \H$ acting on $e'\H$ by multiplication
with $T^*\otimes 1$, with $U_p^*$ acting on $\ensuremath{\mathbf{D}}_{\infty}^{\et}(\langle a\rangle_N)$ as $F$.
From Theorem \ref{DieudonneCrystalIgusa} and Corollary \ref{MFIgusaDieudonne}
we then obtain canonical
isomorphisms
\begin{equation*}
e'S(N;\Lambda)\simeq f'H^1_{\cris}(I^0)^{V_{\ord}}\qquad\text{respectively}\qquad
e'\H\mathop{\otimes}\limits_{\Lambda}\Lambda_{R_0'} \simeq
f'H^1_{\cris}(I^{\infty})^{F_{\ord}}(\langle a\rangle_{N})\mathop{\otimes}\limits_{\Lambda}\Lambda_{R_0'}
\end{equation*}
intertwing $T$ (respectively $T\otimes 1$) with $T^*$ (respectively $T^*\otimes 1$)
where $U_p^*$ acts on crystalline cohomology as $\langle p\rangle_N V$
(respectively $F\otimes 1$). The second of these isomorphisms is moreover $\Gal(K_0'/K_0)$-equivariant.
\end{remark}
In order to relate the slope filtration (\ref{DieudonneInfiniteExact}) of $\ensuremath{\mathbf{D}}_{\infty}$
to the ordinary filtration of ${e^*}'H^1_{\et}$, we
require:
\begin{lemma}
Let $r$ be a positive integer let $G_r={e^*}'J_r[p^{\infty}]$
be the unique $\Q_p$-descent of the generic fiber of $\mathcal{G}_r$, as in Definition $\ref{ordpdivdefn}$.
There are canonical isomophisms of free $W(\o{\mathbf{F}}_p)$-modules
\begin{subequations}
\begin{equation}
\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\et})\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p) \simeq \Hom_{\Z_p}(T_pG_r^{\et},\Z_p)\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p)
\label{etalecase}
\end{equation}
\begin{equation}
\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})(-1)\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p) \simeq \Hom_{\Z_p}(T_pG_r^{\mult},\Z_p)
\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p).
\label{multcase}
\end{equation}
that are $\H_r^*$-equivariant and $\scrG_{\Q_p}$-compatible for the diagonal action on source and target,
with $\scrG_{\Q_p}$ acting trivially on $\ensuremath{\mathbf{D}}(\o{\mathcal{G}}^{\et}_r)$ and via $\chi^{-1}\cdot \langle \chi^{-1}\rangle$
on $\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})(-1):=\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})\otimes_{\Z_p} \Z_p(-1)$. The isomorphism
$(\ref{etalecase})$ intertwines $F\otimes\sigma$ with $1\otimes \sigma$
while $(\ref{multcase})$ intertwines $V\otimes\sigma^{-1}$ with $1\otimes\sigma^{-1}$.
\end{subequations}
\end{lemma}
\begin{proof}
Let $\mathcal{G}$ be any object of $\pdiv_{R_r}^{\Gamma}$ and write $G$ for the unique
descent of the generic fiber $\mathcal{G}_{K_r}$ to $\Q_p$. We recall that the semilinear $\Gamma$-action
on $\mathcal{G}$ gives the $\Z_p[\scrG_{K_r}]$-module
$T_p\mathcal{G}:=\Hom_{\O_{\mathbf{C}_p}}(\Q_p/\Z_p,\mathcal{G}_{\O_{\mathbf{C}_p}})$
the natural structure of $\Z_p[\scrG_{\Q_p}]$-module via $g\cdot f:= g^{-1}\circ g^*f\circ g$.
It is straightforward to check that the natural map $T_p\mathcal{G}\rightarrow T_pG$, which is an isomorphism
of $\Z_p[\scrG_{K_r}]$-modules by Tate's theorem, is an isomorphism of $\Z_p[\scrG_{\Q_p}]$-modules
as well.
For {\em any} \'etale $p$-divisible group $H$ over a perfect field $k$, one has a canonical
isomorphism of $W(\o{k})$-modules with semilinear $\scrG_k$-action
\begin{equation*}
\ensuremath{\mathbf{D}}(H)\mathop{\otimes}\limits_{W(k)} W(\o{k}) \simeq \Hom_{\Z_p}(T_pH,\Z_p)\mathop{\otimes}\limits_{\Z_p} W(\o{k})
\end{equation*}
that intertwines $F\otimes\sigma$ with $1\otimes\sigma$ and $1\otimes g$
with $g\otimes g$ for $g\in \scrG_k$; for example, this can be deduced
by applying \cite[\S4.1 a)]{BBM1} to $H_{\o{k}}$
and using the fact that
the Dieudonn\'e crystal is compatible with base change.
In our case, the \'etale $p$-divisible group $\mathcal{G}_r^{\et}$ lifts $\o{\mathcal{G}}_r^{\et}$
over $R_r$, and we obtain a natural isomorphism of $W(\o{\mathbf{F}}_p)$-modules
with semilinear $\scrG_{K_r}$-action
\begin{equation*}
\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\et})\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p) \simeq \Hom_{\Z_p}(T_p\mathcal{G}_r^{\et},\Z_p)\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p).
\end{equation*}
By naturality in $\mathcal{G}_r$, this identification respects the semilinear $\Gamma$-actions
on both sides (which are trivial, as $\Gamma$ acts trivially on $\mathcal{G}_r^{\et}$); as explained
in our initial remarks, it is precisely this action which allows us to view $T_p\mathcal{G}_r^{\et}$ as a
$\Z_p[\scrG_{\Q_p}]$-module, and we deduce (\ref{etalecase}). The proof of (\ref{multcase})
is similar, using the natural isomorphism (proved as above) for any multiplicative $p$-divisible group $H/k$
\begin{equation*}
\ensuremath{\mathbf{D}}(H)\mathop{\otimes}\limits_{W(k)} W(\o{k}) \simeq T_p\Dual{H}\mathop{\otimes}\limits_{\Z_p} W(\o{k}),
\end{equation*}
which intertwines $V\otimes\sigma^{-1}$ with $1\otimes\sigma^{-1}$
and $1\otimes g$ with $g\otimes g$, for $g\in \scrG_k$.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{FiltrationRecover}$ and Corollary $\ref{MWmainThmCor}$]
For a $p$-divisible group $H$ over a field $K$, we will write $H^1_{\et}(H):=\Hom_{\Z_p}(T_pH,\Z_p)$;
our notation is justified by the standard fact that, for $J_X$ the Jacobian
of a curve $X$ over $K$, there is a natural isomorphisms of $\Z_p[\scrG_K]$-modules
\begin{equation}
H^1_{\et}(J_X[p^{\infty}]) \simeq H^1_{\et}(X_{\Kbar},\Z_p).\label{etalecohcrvjac}
\end{equation}
It follows from (\ref{etalecase})--(\ref{multcase}) and
Theorem \ref{MainDieudonne} (\ref{MainDieudonne1})--(\ref{MainDieudonne2})
that $H^1_{\et}(G_r^{\star})\otimes_{\Z_p} W(\o{\mathbf{F}}_p)$ is a free
$W(\o{\mathbf{F}}_p)[\Delta/\Delta_r]$-module of rank $d'$ for $\star\in \{\et,\mult\}$,
and hence that $H^1_{\et}(G_r^{\star})$ is a free $\Z_p[\Delta/\Delta_r]$-module
of rank $d'$ by Lemma \ref{fflatfreedescent}. In a similar manner, using
the faithful flatness of $W(\o{\mathbf{F}}_p)[\Delta/\Delta_r]$ over $\Z_p[\Delta/\Delta_r]$,
we deduce that the canonical trace mappings
\begin{equation}
\xymatrix{
{H^1_{\et}(G_r^{\star})} \ar[r] & {H^1_{\et}(G_{r'}^{\star})}
}\label{cantraceetale}
\end{equation}
are surjective for all $r\ge r'$.
By Lemma \ref{Technical}, we conclude that
$H^1_{\et}(G_{\infty}^{\star}):=\varprojlim_r H^1_{\et}(G_r^{\star})$
is a free $\Lambda$-module of rank $d'$ and that there are canonical
isomorphisms of $\Lambda_{W(\o{\mathbf{F}}_p)}$-modules
\begin{equation*}
H^1_{\et}(G_{\infty}^{\star})\mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)} \simeq \varprojlim_r \left(H^1_{\et}(G_r^{\star})\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p)\right)
\end{equation*}
for $\star\in \{\et,\mult\}$. Since we likewise have canonical identifications
\begin{equation*}
\ensuremath{\mathbf{D}}_{\infty}^{\star}\mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)} \simeq \varprojlim_r \left(\ensuremath{\mathbf{D}}(G_r^{\star})\mathop{\otimes}\limits_{\Z_p} W(\o{\mathbf{F}}_p)\right)
\end{equation*}
thanks to Lemma \ref{Technical} and (the proof of) Theorem \ref{MainDieudonne},
passing to inverse limits on (\ref{etalecase})--(\ref{multcase}) gives a canonical
isomorphism of $\Lambda_{W(\o{\mathbf{F}}_p)}$-modules
\begin{equation}
\ensuremath{\mathbf{D}}_{\infty}^{\star}\mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)} \simeq
H^1_{\et}(G_{\infty}^{\star})\mathop{\otimes}\limits_{\Lambda} \Lambda_{W(\o{\mathbf{F}}_p)}\label{dieudonneordfillimit}
\end{equation}
for $\star\in \{\et,\mult\}$.
Applying the functor $H^1_{\et}(\cdot)$ to the connected-\'etale sequence of $G_r$
yields a short exact sequence of $\Z_p[\scrG_{\Q_p}]$-modules
\begin{equation*}
\xymatrix{
0\ar[r] & {H^1_{\et}(G_r^{\et})} \ar[r] & {H^1_{\et}(G_r)} \ar[r] & {H^1_{\et}(G_r^{\mult})}\ar[r] & 0
}
\end{equation*}
which naturally identifies ${H^1_{\et}(G_r^{\star})}$ with the invariants (respectively covariants)
of $H^1_{\et}(G_r)$ under the inertia subgroup $\I\subseteq \scrG_{\Q_p}$ for $\star=\et$ (respectively
$\star=\mult$).
As $G_r={e^*}'J_r[p^{\infty}]$ by definition, we deduce from this and (\ref{etalecohcrvjac})
a natural isomorphism of short exact sequences of $\Z_p[\scrG_{\Q_p}]$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {H^1_{\et}(G_r^{\et})} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_r)} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_r^{\mult})} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {({e^*}'H^1_{\et,r})^{\I}} \ar[r] &
{{e^*}'H^1_{\et,r}} \ar[r] &
{({e^*}'H^1_{\et,r})_{\I}} \ar[r] & 0
}
\end{gathered}\label{inertialinvariantsseq}
\end{equation}
where for notational ease abbreviate $H^1_{\et,r}:=H^1_{\et}({X_r}_{\Qbar_p},\Z_p)$.
As the trace maps (\ref{cantraceetale}) are surjective, passing to inverse limits
on (\ref{inertialinvariantsseq}) yields an isomorphism of short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {H^1_{\et}(G_{\infty}^{\et})} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_{\infty})} \ar[r]\ar[d]^-{\simeq} &
{H^1_{\et}(G_{\infty}^{\mult})} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {\varprojlim_r ({e^*}'H^1_{\et,r})^{\I}} \ar[r] &
{\varprojlim_r {e^*}'H^1_{\et,r}} \ar[r] &
{\varprojlim_r ({e^*}'H^1_{\et,r})_{\I}} \ar[r] & 0
}
\end{gathered}\label{limitetaleseq}
\end{equation}
Since inverse limits commute with group invariants, the bottom row of (\ref{limitetaleseq})
is canonically isomorphic to the ordinary filtration of Hida's ${e^*}'H^1_{\et}$,
and Theorem \ref{FiltrationRecover} follows immediately from (\ref{dieudonneordfillimit}).
Corollary \ref{MWmainThmCor} is then an easy consequence of Theorem \ref{FiltrationRecover} and
Lemma \ref{fflatfreedescent};
alternately one can prove Corollary \ref{MWmainThmCor} directly from Lemma \ref{Technical},
using what we have seen above.
\end{proof}
\subsection{Ordinary families of \texorpdfstring{$\mathfrak{S}$}{S}-modules}\label{OrdSigmaSection}
We now study the family of Dieudonn\'e crystals attached to
the tower of $p$-divisible groups $\{\mathcal{G}_{r}/R_r\}_{r\ge 1}$.
For each pair of positive integers $r\ge s$,
we have a morphism $\rho_{r,s}: \mathcal{G}_{s}\times_{T_{s}} T_r\rightarrow \mathcal{G}_{r}$
in $\pdiv_{R_{r}}^{\Gamma}$;
applying the contravariant functor $\m_r:\pdiv_{R_r}^{\Gamma}\rightarrow \textswab{BT}_{\mathfrak{S}_r}^{\Gamma}$
studied in \S\ref{pDivPhiGamma} to the map on connected-\'etale sequences induced by $\rho_{r,s}$
and using the exactness of $\m_r$ and its compatibility with base change
(Theorem \ref{CaisLauMain}), we obtain
maps of exact sequences in
$\textswab{BT}_{\mathfrak{S}_{r}}^{\Gamma}$
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_r(\mathcal{G}_r^{\et})} \ar[r]\ar[d]_-{\m_r(\rho_{r,s})} &
{\m_r(\mathcal{G}_r)} \ar[r]\ar[d]^-{\m_r(\rho_{r,s})} & {\m_r(\mathcal{G}_r^{\mult})}
\ar[r]\ar[d]^-{\m_r(\rho_{r,s})} & 0\\
0\ar[r ] & {\m_{s}(\mathcal{G}_{s}^{\et})\mathop{\otimes}\limits_{\mathfrak{S}_{s}} \mathfrak{S}_r} \ar[r] &
{\m_r(\mathcal{G}_{s})\mathop{\otimes}\limits_{\mathfrak{S}_{s}} \mathfrak{S}_r} \ar[r] & {\m_r(\mathcal{G}_{s}^{\mult})\mathop{\otimes}\limits_{\mathfrak{S}_{s}} \mathfrak{S}_r} \ar[r] & 0
}\label{BTindLim}
\end{gathered}
\end{equation}
\begin{definition}\label{DieudonneLimitDef}
Let $\star=\et$ or $\star=\mult$ and define
\begin{align}
\m_{\infty}&:=\varprojlim_r \left(\m_r(\mathcal{G}_r) \mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}\right)
&
\m_{\infty}^{\star}&:=\varprojlim_r \left(\m_r(\mathcal{G}_r^{\star}) \mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}\right),
\label{UncompletedDieudonneLimit}
\end{align}
with the projective limits taken with respect to the mappings induced by (\ref{BTindLim}).
\end{definition}
Each of (\ref{UncompletedDieudonneLimit}) is naturally a module over
the completed group ring $\Lambda_{\mathfrak{S}_{\infty}}$
and is equipped with a semilinear action of $\Gamma$ and a $\varphi$-semilinear
Frobenius morphism defined by $F:=\varprojlim (\varphi_{\m_r}\otimes \varphi)$.
Since $\varphi$ is bijective on $\mathfrak{S}_{\infty}$, we also have a $\varphi^{-1}$-semilinear
Verscheibung morphism defined as follows. For notational ease, we provisionally
set $M_r:=\m_r(\mathcal{G}_r)\otimes_{\mathfrak{S}_r} \mathfrak{S}_{\infty}$ and we define
\begin{equation}
\xymatrix@C=50pt{
{V_r: M_r} \ar[r]^{m\mapsto 1\otimes m} &
{{\varphi^{-1}}^*M_r }
\ar[r]^-{{\varphi^{-1}}^*(\psi_{\m_r}\otimes 1)} &
{{\varphi^{-1}}^*\varphi^*M_r\simeq M_r}
}
\end{equation}
with $\psi_{\m_r}$ as above Definition \ref{DualBTDef}. It is easy to see
that the $V_r$ are compatible with $r$, and we
put $V:=\varprojlim V_r$ on $\m_{\infty}$. We define Verscheibung
morphisms on $\m_{\infty}^{\star}$ for $\star=\et,\mult$
similarly. As the composite of $\psi_{\m_r}$ and $1\otimes\varphi_{\m_r}$
in either order is multiplication by $E_r(u_r) = u_0/u_1=:\omega$, we have
\begin{equation*}
FV = VF = \omega.
\end{equation*}
Due to the functoriality of $\m_r$, we moreover have a
$\Lambda_{\mathfrak{S}_{\infty}}$-linear action of
$\H^*$
on each of (\ref{UncompletedDieudonneLimit})
which commutes with $F$, $V$, and $\Gamma$.
\begin{theorem}\label{MainThmCrystal}
As in Proposition $\ref{NormalizationCoh}$, set $d':=\sum_{k=3}^p \dim_{\mathbf{F}_p} S_k(N;\mathbf{F}_p)^{\ord}$.
Then $\m_{\infty}$ $($respectively $\m_{\infty}^{\star}$ for $\star=\et,\mult$$)$
is a free $\Lambda_{\mathfrak{S}_{\infty}}$-module of rank $2d'$ $($respectively $d'$$)$
and there is a canonical short exact sequence of $\Lambda_{\mathfrak{S}_{\infty}}$-modules
with linear $\H^*$-action and semi linear actions of $\Gamma$, $F$ and $V$
\begin{equation}
\xymatrix{
0\ar[r] & {\m_{\infty}^{\et}} \ar[r] & {\m_{\infty}} \ar[r] & {\m_{\infty}^{\mult}} \ar[r] & 0
}.\label{DieudonneLimitFil}
\end{equation}
Extension of scalars of $(\ref{DieudonneLimitFil})$
along the quotient
$\Lambda_{\mathfrak{S}_{\infty}}\twoheadrightarrow \mathfrak{S}_{\infty}[\Delta/\Delta_r]$
recovers the exact sequence
\begin{equation}
\xymatrix{
0\ar[r] & {\m_r(\mathcal{G}_r^{\et})\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}} \ar[r] &
{\m_r(\mathcal{G}_r)\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}} \ar[r] &
{\m_r(\mathcal{G}_r^{\mult})\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}} \ar[r] & 0
}.
\end{equation}
for each integer $r>0$, compatibly with $\H^*$, $\Gamma$, $F$, and $V$.
\end{theorem}
\begin{proof}
Since $\varphi$ is an automorphism
of $\mathfrak{S}_{\infty}$, pullback by $\varphi$ commutes with
projective limits of $\mathfrak{S}_{\infty}$-modules.
As the canonical $\mathfrak{S}_{\infty}$-linear map $\varphi^*\Lambda_{\mathfrak{S}_{\infty}}\rightarrow \Lambda_{\mathfrak{S}_{\infty}}$
is an isomorphism of rings (even of $\mathfrak{S}_{\infty}$-algebras), it therefore suffices
to prove the assertions of Theorem \ref{MainThmCrystal} after pullback by $\varphi$,
which will be more convenient due to the relation between
$\varphi^*\m_r(\mathcal{G}_r)$ and the Dieudonn\'e crystal of $\mathcal{G}_r$.
Pulling back (\ref{BTindLim}) by $\varphi$ gives a commutative diagram
with exact rows
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\varphi^*\m_r(\mathcal{G}_r^{\et})} \ar[r]\ar[d] &
{\varphi^*\m_r(\mathcal{G}_r)} \ar[r]\ar[d] & {\varphi^*\m_r(\mathcal{G}_r^{\mult})}
\ar[r]\ar[d] & 0\\
0\ar[r] & {\varphi^*\m_{s}(\mathcal{G}_{s}^{\et})\mathop{\otimes}\limits_{\mathfrak{S}_{s}} \mathfrak{S}_r} \ar[r] &
{\varphi^*\m_r(\mathcal{G}_{s})\mathop{\otimes}\limits_{\mathfrak{S}_{s}} \mathfrak{S}_r} \ar[r] &
{\varphi^*\m_r(\mathcal{G}_{s}^{\mult})\mathop{\otimes}\limits_{\mathfrak{S}_{s}} \mathfrak{S}_r} \ar[r] & 0
}
\end{gathered}\label{BTindLimPB}
\end{equation}
and we apply Lemma \ref{Technical} with $A_r:=\mathfrak{S}_r$, $I_r:=(u_r)$,
$B=\mathfrak{S}_{\infty}$, and with $M_r$ each one of the
terms in the top row of (\ref{BTindLimPB}).
The isomorphism (\ref{MrToDieudonneMap})
of Proposition \ref{MrToHodge}
ensures, via Theorem \ref{MainDieudonne} (\ref{MainDieudonne1}), that the hypothesis (\ref{freehyp})
is satisfied.
Due to the functoriality of (\ref{MrToDieudonneMap}),
for any $r\ge s$, the mapping obtained from (\ref{BTindLimPB}) by reducing
modulo $I_r$ is identified with the mapping on (\ref{DieudonneFiniteExact}) induced (via functoriality
of $\ensuremath{\mathbf{D}}(\cdot)$) by $\o{\rho}_{r,s}$.
As was shown in the proof of Theorem (\ref{MainDieudonne}), these mappings are surjective
for all $r\ge s$, and we conclude that hypothesis (\ref{surjhyp}) holds as well.
Moreover,
the vertical mappings of (\ref{BTindLimPB}) are then surjective by Nakayama's Lemma,
so as in the proof of Theorems \ref{main} and \ref{MainDieudonne} (and keeping in mind
that pullback by $\varphi$ commutes with projective limits of $\mathfrak{S}_{\infty}$-modules),
we obtain, by applying $\otimes_{\mathfrak{S}_r} \mathfrak{S}_{\infty}$ to (\ref{BTindLimPB}), passing
to projective limits, and pulling back by $(\varphi^{-1})^*$,
the short exact sequence (\ref{DieudonneLimitFil}).
\end{proof}
\begin{remark}
In the proof of Theorem \ref{MainThmCrystal}, we could have alternately applied Lemma \ref{Technical}
with $A_r=\mathfrak{S}_r$ and $I_r:=(E_r)$, appealing to the identifications (\ref{MrToHodgeMap})
of Proposition \ref{MrToHodge} and (\ref{CollectedComparisonIsom}) of Proposition \ref{KeyComparison},
and to Theorem \ref{main}.
\end{remark}
The short exact sequence (\ref{DieudonneLimitFil}) is closely related to its $\Lambda_{\mathfrak{S}_{\infty}}$-linear
dual. In what follows, we write $\mathfrak{S}_{\infty}':=\varinjlim_r \Z_p[\mu_N][\![ u_r]\!]$,
taken along the mappings $u_r\mapsto \varphi(u_{r+1})$; it is naturally a $\mathfrak{S}_{\infty}$-algebra.
\begin{theorem}\label{CrystalDuality}
Let $\mu:\Gamma\rightarrow \Lambda_{\mathfrak{S}_{\infty}}^{\times}$ be the crossed homomorphism
given by $\mu(\gamma):=\frac{u_1}{\gamma u_1}\chi(\gamma) \langle \chi(\gamma)\rangle$.
There is a canonical $\H^*$ and $\Gal(K_{\infty}'/K_0)$-equivariant isomorphism of
exact sequences of $\Lambda_{\mathfrak{S}_{\infty}'}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0\ar[r] & {\m_{\infty}^{\et}(\mu \langle a\rangle_N)_{\Lambda_{\mathfrak{S}_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}(\mu \langle a\rangle_N)_{\Lambda_{\mathfrak{S}_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} &
{\m_{\infty}^{\mult}(\mu \langle a\rangle_N)_{\Lambda_{\mathfrak{S}_{\infty}'}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0\ar[r] & {(\m_{\infty}^{\mult})_{\Lambda_{\mathfrak{S}_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty})_{\Lambda_{\mathfrak{S}_{\infty}'}}^{\vee}} \ar[r] &
{(\m_{\infty}^{\et})_{\Lambda_{\mathfrak{S}_{\infty}'}}^{\vee}} \ar[r] & 0
}
\end{gathered}
\label{MinftyDuality}
\end{equation}
that intertwines $F$
$($respectively $V$$)$ on the top row with
$V^{\vee}$ $($respectively $F^{\vee}$$)$ on the bottom.
\end{theorem}
\begin{proof}
We first claim that there is a natural isomorphism of $\mathfrak{S}_{\infty}'[\Delta/\Delta_r]$-modules
\begin{equation}
\m_r(\mathcal{G}_r)(\mu\langle a\rangle_N)\otimes_{\mathfrak{S}_r} \mathfrak{S}_{\infty}' \simeq
\Hom_{\mathfrak{S}_{\infty}'}(\m_r(\mathcal{G}_r)\otimes_{\mathfrak{S}_r}\mathfrak{S}_{\infty}', \mathfrak{S}_{\infty}')
\label{twistyisom}
\end{equation}
that is $\H^*$-equivariant and $\Gal(K_{\infty}'/K_0)$-compatible for the standard action
$\gamma\cdot f(m):=\gamma f(\gamma^{-1}m)$ on the right side, and that intertwines
$F$ and $V$ with $V^{\vee}$ and $F^{\vee}$, respectively.
Indeed, this follows immediately from the identifications
\begin{equation}
{\m_r(\mathcal{G}_r)(\langle \chi \rangle\langle a\rangle_N)\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}'} \simeq
{\m_r(\mathcal{G}_r')\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}'=:\m_r(\mathcal{G}_r^{\vee})\mathop{\otimes}\limits_{\mathfrak{S}_r}\mathfrak{S}_{\infty}'}
\simeq {\m_r(\mathcal{G}_r)^{\vee}_{\mathfrak{S}_{\infty}'}}
\label{GrTwist}
\end{equation}
and the definition (Definition \ref{DualBTDef}) of duality in $\textswab{BT}_{\mathfrak{S}_r}^{\varphi,\Gamma}$; here, the
first isomorphism in (\ref{GrTwist}) results from Proposition \ref{GdualTwist}
and Theorem \ref{CaisLauMain} (\ref{BaseChangeIsom}), while the final
identification is due to Theorem \ref{CaisLauMain} (\ref{exequiv}).
The identification (\ref{twistyisom}) carries $F$ (respectively $V)$
on its source to $V^{\vee}$ (respectively $F^{\vee}$) on its target due to the compatibility
of the functor $\m_r(\cdot)$ with duality (Theorem \ref{CaisLauMain} (\ref{exequiv})).
From (\ref{twistyisom})
we obtain a natural $\Gal(K_r'/K_0)$-compatible evaluation pairing of $\mathfrak{S}_{\infty}'$-modules
\begin{equation}
\xymatrix{
{\langle\cdot,\cdot\rangle_r: \m_r(\mathcal{G}_r)(\mu\langle a\rangle_N) \mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}'
\times \m_r(\mathcal{G}_r)\mathop{\otimes}\limits_{\mathfrak{S}_r} \mathfrak{S}_{\infty}'} \ar[r] & {\mathfrak{S}_{\infty}'}
}\label{crystalpairingdefs}
\end{equation}
with respect to which the natural action of $\H^*$ is self-adjoint, due to the
fact that (\ref{GrTwist}) is $\H^*$-equivariant by Proposition \ref{GdualTwist}.
Due to the compatibility with change in $r$ of the identification (\ref{GrprimeGr}) of Proposition \ref{GdualTwist}
together with the definitions (\ref{pdivTowers}) of $\rho_{r,s}$ and
$\rho_{r,s}'$,
the identification (\ref{GrTwist}) intertwines the map induced by $\Pic^0(\rho)$ on its source
with the map induced by ${U_p^*}^{-1}\Alb(\sigma)$ on its target. For $r\ge s$, we therefore have
\begin{equation*}
\langle \m_r(\rho_{r,s})x , \m_r(\rho_{r,s})y \rangle_s =
\langle x, \m_r({U_p^*}^{s-r}\Pic^0(\rho)^{r-s}\Alb(\sigma)^{r-s})y\rangle_r =
\sum_{\delta\in \Delta_s/\Delta_r} \langle x, \delta^{-1} y \rangle_r,
\end{equation*}
where the final equality follows from (\ref{PicAlbRelation}).
Thus, the perfect pairings (\ref{crystalpairingdefs}) satisfy the compatibility condition
(\ref{pairingchangeinr}) of Lemma \ref{LambdaDuality} which, together with
Theorem \ref{MainThmCrystal}, completes the proof.
\end{proof}
The $\Lambda_{\mathfrak{S}_{\infty}}$-modules $\m_{\infty}^{\et}$ and $\m_{\infty}^{\mult}$ admit
canonical descents to $\Lambda$:
\begin{theorem}\label{etmultdescent}
There are canonical $\H^*$, $\Gamma$, $F$ and $V$-equivariant isomorphisms
of $\Lambda_{\mathfrak{S}_{\infty}}$-modules
\begin{subequations}
\begin{equation}
\m_{\infty}^{\et} \simeq \ensuremath{\mathbf{D}}_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda} \Lambda_{\mathfrak{S}_{\infty}},
\end{equation}
intertwining $F$ $($respetcively $V$$)$ with
$F\otimes \varphi$ $($respectively $F^{-1}\otimes \omega\cdot \varphi^{-1}$$)$ and $\gamma\in \Gamma$
with $\gamma\otimes\gamma$, and
\begin{equation}
\m_{\infty}^{\mult}\simeq \ensuremath{\mathbf{D}}_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda} \Lambda_{\mathfrak{S}_{\infty}},
\end{equation}
\end{subequations}
intertwing $F$ $($respectively $V$$)$ with $V^{-1} \otimes \omega \cdot\varphi$
$($respectively $V\otimes\varphi^{-1}$$)$
and $\gamma$ with $\gamma\otimes \chi(\gamma)^{-1} \gamma u_1/u_1$.
In particular, $F$ $($respectively $V$)
acts invertibly on $\m_{\infty}^{\et}$ $($respectively $\m_{\infty}^{\mult}$$)$.
\end{theorem}
\begin{proof}
We twist the identifications (\ref{EtMultSpecialIsoms}) of Proposition
\ref{EtaleMultDescription} to obtain natural isomorphisms
\begin{equation*}
\xymatrix@C=40pt{
{\m_r(\mathcal{G}_r^{\et})} \ar[r]^-{\simeq}_-{F^r \circ (\ref{EtMultSpecialIsoms})} &
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\et})_{\Z_p}\otimes_{\Z_p} \mathfrak{S}_r}
}\qquad\text{and}\qquad
\xymatrix@C=40pt{
{\m_r(\mathcal{G}_r^{\mult})} \ar[r]^-{\simeq}_-{V^{-r} \circ (\ref{EtMultSpecialIsoms})} &
{\ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\mult})_{\Z_p}\otimes_{\Z_p} \mathfrak{S}_r}
}
\end{equation*}
that are $\H_r^*$-equivariant and, Thanks to \ref{EtMultSpecialIsomsBC},
compatible with change in $r$ using the maps on source and target
induced by $\rho_{r,s}$. Passing to inverse limits and appealing to Lemma \ref{Technical}
and (the proof of) Theorem \ref{MainDieudonne}, we deduce for $\star=\et,\mult$
natural isomorphisms of $\Lambda_{\mathfrak{S}_{\infty}}$-modules
\begin{equation*}
\m_{\infty}^{\star} \simeq \varprojlim_r \left( \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\star})_{\Z_p}\otimes_{\Z_p} \mathfrak{S}_{\infty}\right)
\simeq \ensuremath{\mathbf{D}}_{\infty}^{\star}\otimes_{\Lambda} \Lambda_{\mathfrak{S}_{\infty}}
\end{equation*}
that are $\H^*$-equivariant and satisfy the asserted compatibility with respect to
Frobenius, Verscheibung, and the action of $\Gamma$ due to
Proposition \ref{EtaleMultDescription} and the definitions (\ref{MrEtDef})--(\ref{MrMultDef}).
\end{proof}
We can now prove Theorem \ref{MinftySpecialize}, which asserts that
the slope filtration (\ref{MinftySpecialize}) of $\m_{\infty}$
specializes, on the one hand, to the slope filtration (\ref{DieudonneInfiniteExact})
of $\ensuremath{\mathbf{D}}_{\infty}$, and on the other hand to the Hodge filtration (\ref{LambdaHodgeFilnomup})
(in the opposite direction!) of ${e^*}'H^1_{\dR}$. We recall the precise statement:
\begin{theorem}\label{SRecovery}
Let $\tau:\Lambda_{\mathfrak{S}_{\infty}}\twoheadrightarrow \Lambda$ be the $\Lambda$-algebra
surjection induced by $u_r\mapsto 0$.
There is a canonical $\Gamma$ and $\H^*$-equivariant
isomorphism of split exact sequences of finite free $\Lambda$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\tau} \Lambda}\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\tau} \Lambda}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\tau} \Lambda} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}^{\et}} \ar[r] & {\ensuremath{\mathbf{D}}_{\infty}} \ar[r] &
{\ensuremath{\mathbf{D}}_{\infty}^{\mult}} \ar[r] & 0
}
\end{gathered}\label{OrdFilSpecialize}
\end{equation}
which carries $F\otimes 1$ to $F$ and $V\otimes 1$ to $V$.
Let $\theta\circ\varphi:\Lambda_{\mathfrak{S}_{\infty}}\rightarrow \Lambda_{R_{\infty}}$
be the $\Lambda$-algebra surjection induced by $u_r\mapsto (\varepsilon^{(r)})^p-1$.
There is a canonical $\Gamma$ and $\H^*$-equivariant
isomorphism of split exact sequences of finite free $\Lambda_{R_{\infty}}$-modules
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] & {\m_{\infty}^{\et}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\theta\varphi} \Lambda_{R_{\infty}}}
\ar[d]_-{\simeq} \ar[r] &
{\m_{\infty}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\theta\varphi} \Lambda_{R_{\infty}}}\ar[r] \ar[d]_-{\simeq}&
{\m_{\infty}^{\mult}\mathop{\otimes}\limits_{\Lambda_{\mathfrak{S}_{\infty}},\theta\varphi} \Lambda_{R_{\infty}}} \ar[r]\ar[d]_-{\simeq} & 0\\
0 \ar[r] & {{e^*}'H^1(\O)} \ar[r]_{i} &
{{e^*}'H^1_{\dR}} \ar[r]_-{j} & {{e^*}'H^0(\omega)} \ar[r] & 0
}
\end{gathered}
\end{equation}
where $i$ and $j$ are the canonical sections
given by the splitting in Theorem $\ref{dRtoDieudonne}$.
\end{theorem}
\begin{proof}
To prove the first assertion, we apply Lemma \ref{Technical}
with $A_r=\mathfrak{S}_r,$ $I_r=(u_r)$, $B=\mathfrak{S}_{\infty}$, $B'=\Z_p$ (viewed as
a $B$-algebra via $\tau$) and $M_r=\m_r^{\star}$ for $\star\in \{\et,\mult,\Null\}$.
Thanks to (\ref{MrToDieudonneMap}) in the case $G=\mathcal{G}_r$,
we have a canonical identification $\o{M}_r:=M_r/I_rM_r \simeq \ensuremath{\mathbf{D}}(\o{\mathcal{G}}_r^{\star})_{\Z_p}$
that is compatible with change in $r$ in the sense that the induced projective
system $\{\o{M}_r\}_{r}$ is identified with that of Definition \ref{DinftyDef}.
It follows from this and
Theorem \ref{MainDieudonne} (\ref{MainDieudonne1})--(\ref{MainDieudonne2}) that
the hypotheses (\ref{freehyp})--(\ref{surjhyp}) are satisfied,
and (\ref{OrdFilSpecialize}) is an isomorphism by Lemma \ref{Technical} (\ref{CompletedBaseChange}).
In exactly the same manner,
the second assertion follows by appealing to
Lemma \ref{Technical} with $A_r=\mathfrak{S}_r$, $I_r=(E_r)$, $B=\mathfrak{S}_{\infty}$, $B'=R_{\infty}$
(viewed as a $B$-algebra via $\theta\circ\varphi$)
and $M_r=\m_r^{\star}$, using (\ref{MrToHodgeMap}) and
Theorem \ref{main} to verify the hypotheses (\ref{freehyp})--(\ref{surjhyp}).
\end{proof}
\begin{proof}[Proof of Theorem $\ref{RecoverEtale}$ and Corollary $\ref{HidasThm}$]
Applying Theorem \ref{comparison} to (the connected-\'etale sequence of) $\mathcal{G}_r$
gives a natural isomorphism of short exact sequences
\begin{equation}
\begin{gathered}
\xymatrix{
0 \ar[r] &{\m_r(\mathcal{G}_r^{\et})\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} \a_r } \ar[r]\ar[d]^-{\simeq} &
{\m_r(\mathcal{G}_r)\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} \a_r} \ar[r]\ar[d]^-{\simeq} &
{\m_r(\mathcal{G}_r^{\mult})\mathop{\otimes}\limits_{\mathfrak{S}_r,\varphi} \a_r} \ar[r]\ar[d]^-{\simeq} & 0 \\
0 \ar[r] & {H^1_{\et}(\mathcal{G}_r^{\et})\mathop{\otimes}\limits_{\Z_p} \a_r} \ar[r] &
{H^1_{\et}(\mathcal{G}_r)\mathop{\otimes}\limits_{\Z_p} \a_r} \ar[r] &
{H^1_{\et}(\mathcal{G}_r^{\mult})\mathop{\otimes}\limits_{\Z_p}\a_r}\ar[r] & 0
}
\end{gathered}
\label{etalecompdiag}
\end{equation}
Due to Theorem \ref{MainThmCrystal}, the terms in the top row of \ref{etalecompdiag}
are free of ranks $d'$, $2d'$, and $d'$ over $\wt{\a}_r[\Delta/\Delta_r]$, respectively,
so we conclude from Lemma \ref{fflatfreedescent} (with $A=\Z_p[\Delta/\Delta_r]$
and $B=\a_r[\Delta/\Delta_r]$) that $H^1_{\et}(\mathcal{G}_r^{\star})$ is a free $\Z_p[\Delta/\Delta_r]$-module
of rank $d'$ for $\star=\{\et,\mult\}$ and that $H^1_{\et}(\mathcal{G}_r)$ is free of rank $2d'$
over $\Z_p[\Delta/\Delta_r]$. Using the fact that
$\Z_p\rightarrow \a_r$ is faithfully flat, it then follows
from the surjectivity of the vertical maps in (\ref{BTindLimPB})
(which was noted in the proof of Theorem \ref{MainThmCrystal})
that the canonical trace mappings $H^1_{\et}(\mathcal{G}_r^{\star})\rightarrow H^1_{\et}(\mathcal{G}_{r'}^{\star})$
for $\star\in \{\et,\mult,\Null\}$ are surjective for all $r\ge r'$.
Applying Lemma \ref{Technical} with $A_r=\Z_p$, $M_r:=H^1_{\et}(\mathcal{G}_r^{\star})$,
$I_r=(0)$, $B=\Z_p$ and $B'=\wt{\a}$, we conclude that $H^1_{\et}(\mathcal{G}_{\infty}^{\star})$
is free of rank $d'$ (respectively $2d'$) over $\Lambda$ for $\star=\et,$ $\mult$ (respectively $\star=\Null$),
that the specialization mappings
\begin{equation*}
\xymatrix{
{H^1_{\et}(\mathcal{G}_{\infty}^{\star})\mathop{\otimes}\limits_{\Lambda} \Z_p[\Delta/\Delta_r]} \ar[r] &
{H^1_{\et}(\mathcal{G}_r^{\star})}
}
\end{equation*}
are isomorphisms, and that the canonical mappings for $\star\in \{\et,\mult,\Null\}$
\begin{equation}
\xymatrix{
{H^1_{\et}(\mathcal{G}_{\infty}^{\star})\mathop{\otimes}\limits_{\Lambda} \Lambda_{\wt{\a}}} \ar[r] &
{\varprojlim_r \left(H^1_{\et}(\mathcal{G}_r^{\star})\mathop{\otimes}\limits_{\Z_p} \wt{\a}\right)}
}\label{etaleswitcheroo}
\end{equation}
are isomorphisms. Invoking the isomorphism (\ref{limitetaleseq})
gives Corollary \ref{HidasThm}. By Lemma \ref{Technical} with $A_r=\mathfrak{S}_r$, $M_r=\m_r(\mathcal{G}_r^{\star})$,
$I_r=(0)$, $B=\mathfrak{S}_{\infty}$ and $B'=\wt{\a}$, we similarly conclude from (the proof of) Theorem
\ref{MainThmCrystal} that the canonical mappings for $\star\in \{\et,\mult,\Null\}$
\begin{equation}
\xymatrix{
{\m_{\infty}^{\star}\mathop{\otimes}\limits_{\mathfrak{S}_{\infty},\varphi} \Lambda_{\wt{\a}}} \ar[r] &
{\varprojlim_r \left(\m_r(\mathcal{G}_r^{\star})\mathop{\otimes}\limits_{\mathfrak{S}_r} \wt{\a}\right)}
}\label{crystalswitcheroo}
\end{equation}
are isomorphisms.
Applying $\otimes_{\a_r} \wt{\a}$ to the diagram (\ref{etalecompdiag}),
passing to inverse limits, and using the isomorphisms
(\ref{etaleswitcheroo}) and (\ref{crystalswitcheroo}) gives (again invoking (\ref{limitetaleseq}))
the isomorphism (\ref{FinalComparisonIsom}).
Using the fact that the inclusion $\Z_p\hookrightarrow \wt{\a}^{\varphi=1}$
is an equality, the isomorphism (\ref{RecoverEtaleIsom}) follows immediately from
(\ref{FinalComparisonIsom}) by taking $F\otimes\varphi$-invariants.
\end{proof}
Using Theorems \ref{RecoverEtale} and \ref{CrystalDuality} we can give a new proof of
Ohta's duality theorem \cite[Theorem 4.3.1]{OhtaEichler} for the $\Lambda$-adic ordinary
filtration of ${e^*}'H^1_{\et}$ (see Corollary \ref{OhtaDuality}):
\begin{theorem}\label{OhtaDualityText}
There is a canonical $\Lambda$-bilinear and perfect duality pairing
\begin{equation}
\langle \cdot,\cdot\rangle_{\Lambda}: {e^*}'H^1_{\et}\times {e^*}'H^1_{\et}\rightarrow \Lambda
\quad\text{determined by}\quad
\langle x,y\rangle_{\Lambda} \equiv \sum_{\delta\in \Delta/\Delta_r}
(x , w_r {U_p^*}^r\langle\delta^{-1}\rangle^*y)_r \delta \bmod I_r
\label{EtaleDualityPairing}
\end{equation}
with respect to which the action of $\H^*$ is self-adjoint; here,
$(\cdot,\cdot)_r$ is the usual cup-product pairing on $H^1_{\et,r}$ and
$I_r:=\ker(\Lambda\twoheadrightarrow \Z_p[\Delta/\Delta_r])$.
Writing $\nu:\scrG_{\Q_p}\rightarrow \H^*$ for the character
$\nu:=\chi\langle\chi\rangle \lambda(\langle p\rangle_N)$, the
pairing $(\ref{EtaleDualityPairing})$ induces a canonical
$\scrG_{\Q_p}$ and $\H^*$-equivariant isomorphism of exact sequences
\begin{equation*}
\xymatrix{
0 \ar[r] & {({e^*}'H^1_{\et})^{\I}(\nu)}
\ar[d]^-{\simeq} \ar[r] &
{{e^*}'H^1_{\et}(\nu)}\ar[d]^-{\simeq} \ar[r] &
{({e^*}'H^1_{\et})_{\I}(\nu)}
\ar[d]^-{\simeq}\ar[r] & 0 \\
0 \ar[r] & {\Hom_{\Lambda}(({e^*}'H^1_{\et})_{\I},\Lambda)} \ar[r] &
{\Hom_{\Lambda}({e^*}'H^1_{\et},\Lambda)} \ar[r] &
{\Hom_{\Lambda}(({e^*}'H^1_{\et})^{\I},\Lambda)}\ar[r] & 0
}
\end{equation*}
\end{theorem}
\begin{proof}
The proof is similar to that of Proposition \ref{dRDuality}, using
Corollary \ref{HidasThm} and applying Lemma \ref{LambdaDuality} ({\em cf.}
the proof of \cite[Theorem 4.3.1]{OhtaEichler} and of \cite[Proposition 4.4]{SharifiConj}).
Alternatively, one can prove Theorem \ref{OhtaDualityText} by appealing to Theorem \ref{CrystalDuality}
and isomorphism (\ref{RecoverEtaleIsom}) of Theorem \ref{RecoverEtale}.
\end{proof}
\begin{proof}[Proof of Theorem $\ref{SplittingCriterion}$]
Suppose first that (\ref{DieudonneLimitFil}) admits a $\Lambda_{\mathfrak{S}_{\infty}}$-linear
splitting $\m_{\infty}^{\mult}\rightarrow \m_{\infty}$
which is compatible with $F$, $V$, and $\Gamma$.
Extending scalars along $\Lambda \rightarrow \Lambda_{\wt{\a}}\xrightarrow{\varphi}\Lambda_{\wt{\a}}$
and taking $F\otimes\varphi$-invariants
yields, by Theorem \ref{RecoverEtale}, a $\Lambda$-linear and $\scrG_{\Q_p}$-equivariant map
$({e^*}'H^1_{\et})_{\I}\rightarrow {e^*}'H^1_{\et}$
whose composition with the canonical projection
${e^*}'H^1_{\et}\twoheadrightarrow ({e^*}'H^1_{\et})_{\I}$
is necessarily the identity.
Conversely, suppose that the ordinary filtration of ${e^*}'H^1_{\et}$ is $\Lambda$-linearly
and $\scrG_{\Q_p}$-equivariantly split. Applying $\otimes_{\Lambda} \Z_p[\Delta/\Delta_r]$
to this splitting gives, thanks to Corollary \ref{HidasThm} and the isomorphism
(\ref{inertialinvariantsseq}), a $\Z_p[\scrG_{\Q_p}]$-linear splitting of
\begin{equation*}
\xymatrix{
0 \ar[r] & {T_pG_r^{\mult}} \ar[r] & {T_pG_r} \ar[r] & {T_pG_r^{\et}}\ar[r] & 0
}
\end{equation*}
which is compatible with change in $r$ by construction.
By $\Gamma$-descent and Tate's theorem, there is a natural isomorphism
\begin{equation*}
{\Hom_{\pdiv_{R_r}^{\Gamma}}(\mathcal{G}_r^{\et},\mathcal{G}_r)}\simeq {\Hom_{\Z_p[\scrG_{\Q_p}]}(T_pG_r^{\et},T_pG_r)}
\end{equation*}
and we conclude that the connected-\'etale sequence of $\mathcal{G}_r$ is split (in the category
$\pdiv_{R_r}^{\Gamma}$), compatibly with change in $r$. Due to the functoriality
of $\m_r(\cdot)$, this in turn implies that
the top row of (\ref{BTindLim}) is split in $\textswab{BT}_{\mathfrak{S}_r}^{\Gamma}$,
compatibly with change in $r$, which is easily seen to imply the splitting of (\ref{DieudonneLimitFil}).
\end{proof}
\bibliographystyle{amsalpha_noMR}
|
1,116,691,498,426 | arxiv | \section{Introduction}
The TRAPPIST-1 planetary system was discovered by \citet{Gillon_2016} and \citet{Gillon_2017}, using the Transiting Planets and PlanetIsimals Small Telescope \citep{Gillon_2011, Gillon_2013}. TRAPPIST-1 h is the most outer planets detected in this system, its detection was first suggested in \citet{Gillon_2017}, but later confirmed in \citet{Luger_2017b}. Further observations using Spitzer and K2 photometry followed the discovery to better constrain planetary parameters \citep{Delrez_2018, Ducrot_2018, Burdanov_2019, Ducrot_2020}. Since then, important scientific efforts have been carried out to observe, characterise, and model the seven planets orbiting this M8-type star. This is motivated by the fact that the TRAPPIST-1 system offers the most favourable conditions to study rocky planets in the habitable zone, that is to say planets that could harbour liquid water on their surface as defined in \citet{Kasting_1993}.
TRAPPIST-1 is close (39.14 light years), cool (2559 K), and small (0.117 R$_\odot$), making it favourable for observations \citep{Gillon_2017}. On the other hand, the star is also the limiting factor in studying the atmosphere of TRAPPIST-1 planets. M-type stars stay for millions of years in the pre-main sequence (PMS) phase, during which planets are exposed to strong non-thermal extreme UV (EUV) and far-UV irradiation, which is expected to lead to atmospheric hydrodynamical escape \citep{Vidal-Madjar_2003, Bourrier_2017a} and a runaway greenhouse effect \citep{Ramirez_2014}. TRAPPIST-1 is a very cold M-dwarf, but it is supposedly very active with strong flaring events \citep{Vida_2017} and EUV flux \citep{Wheatley_2017}. Atmospheric erosion might have stripped all planets in the TRAPPIST-1 system of their atmospheres \citep{lammer_2003, Bolmont_2017a}. Whether or not an atmosphere was sustained depends on the initial amount of accreted volatiles during the planetary formation phase, and the intensity of the atmospheric escape due to the star activity.
\begin{table}
\caption{Stellar and planetary parameters used in this work}
\centering
\begin{tabular}{l | c} \hline \hline
Parameter & Value \\ \hline
Spectral type & M8-V \\
R$_{\rm s}$ (R$_\odot$) & 0.1170 $\pm$ 0.0036 \\
M$_{\rm s}$ (M$_\odot$) & 0.0802 $\pm$ 0.0073 \\
T$_{\rm s}$ (K) & 2559 $\pm$ 50 \\
$\log$(g) & 5.21 \\
Fe/H & 0.040 $\pm$ 0.080 \\ \hline
R$_{\rm p}$ (R$_{\oplus}$) & 0.752 $\pm$ 0.032\\
M$_{\rm p}$ (M$_{\oplus}$) & 0.331 $^{+0.056}_{-0.049}$ \\
a (AU) & 0.059 $\pm$ 0.004\\
T$_{\rm eff}$ (K) & 173 $\pm$ 4\\
S (S$_\oplus$) & 0.165 $\pm$ 0.025 \\
a/R$_{\rm s}$ & 109 $\pm$ 4 \\
i (deg) & 89.76$^{+0.05}_{-0.03}$\\
e\tablefootmark{a} & 0 \\
b & 0.45 \\
P$_{\rm orb}$ (days) & 18.767 $^{+0.004}_{-0.003}$\\
T$_{\rm mid}$ (BJD$_{\rm TDB}$) & 2\,458\,751.06983 $\pm$ 0.00021\tablefootmark{b} \\ \hline
\end{tabular}
\tablefoot{Values are from \citet{Gillon_2017} and \citet{Luger_2017a}.
\tablefoottext{a}{Fixed to zero}
\tablefoottext{b}{Obtained in this work}}
\label{table1:parameter}
\end{table}
The TRAPPIST-1 planetary system is very compact, all the planets are within 0.06 AU and they are co-planar \citep{Luger_2017a, Luger_2017b, Delrez_2018}. In addition to this, they all have a circularised orbit with eccentricities below 0.01 \citep{Gillon_2017, Luger_2017b} and present gravitational interactions forming a resonant chain, thus suggesting that the system had a relatively peaceful history. TRAPPIST-1 h is the furthest and the smallest known planet of this planetary system. It has a radius of 0.752$\pm$0.032 R$\oplus$ and a mass of 0.331$^{+0.056}_{-0049}$ M$\oplus$ \citep{Luger_2017b, Gillon_2017}, which suggests a density similar to that of Mars ($\sim$ 4000 kg/m$^3$). The planetary parameters are detailed in Table \ref{table1:parameter} along with stellar and orbital parameters of the system.
Two possible formation scenarios have been proposed for the TRAPPIST-1 system and in particular for TRAPPIST-1 h. The first one suggests that all the planets that formed beyond the water frost line migrated inwards, causing the resonance, and they are now located between planets g and h. This possibility was proposed in the discovery papers \citet{Gillon_2017} and \citet{Luger_2017b}, but also detailed in \citet{Ormel_2017}, \citet{Tamayo_2017}, and \citet{Coleman_2019}. If TRAPPIST-1 h formed far from the host star, it could be volatiles-rich because the atmospheric escape would only remove between 1 and 10\% of the total planet mass \citep{Tian_Ida_2015, Bolmont_2017a, Bourrier_2017a, Turbet_2020b}. TRAPPIST-1 h could also have formed in situ, and a short migration or an eccentricity damping could have caused the resonant chain \citep{MacDonald_2018}. In this case, the planet is probably dry \citep{Turbet_2020b} because of the strong atmospheric erosion.
On the other hand, TRAPPIST-1 h, being the furthest planet of the system, might have had a more important quantity of initial gas than inner planets. It could have formed with TRAPPIST-1 f and g in a different part of the proto-planetary disk leading to a different bulk composition \citep{Papaloizou_2018, Turbet_2020b}. Volatiles could also have been brought after by cometary impacts or degassing \citep{Kral_2018, Dencs_2019, Turbet_2020b, Kimura_2020}, and this is favoured for outer planets because volatiles' impacts dominate over the impact erosion mechanism \citep{Kral_2018}.
For close-in planetary systems, the effects of gravitational tides by the star on the planets are important and shape the orbital dynamics, that is to say they slow down the rotation rate, reduce the obliquity, and circularise the orbit. As shown in \citet{Turbet_2018}, the evolution timescales for TRAPPIST-1 h is 7 million years for the rotation and 80 million years for the obliquity. Knowing the age of the TRAPPIST-1 system, which is 8 billion years \citep{Burgasser_2017}, it is likely that TRAPPIST-1 h is in a synchronous rotation state. However, tidal heating is unlikely to be the dominant interior heating process for outer planets \citep{Turbet_2018, Makarov_2018, Dobos_2019} as compared with direct atmospheric warming. The received stellar flux is indeed two orders of magnitude higher than the tidal heating for TRAPPIST-1 h \citep{Turbet_2020b}. It is then unlikely that TRAPPIST-1 h tidal heating caused the melting of the mantle leading to the out-gassing of volcanic gases \citep{Turbet_2020b}.
As of today, TRAPPIST-1 h is the only planet of the system for which the near-infrared (NIR) spectrum (1.1-1.7$\mu$m) from the Hubble Space Telescope (HST) Wide Field Camera 3 Grism 141 (WFC3/G141) has not been published. The other planets' spectra have already been studied with different pipelines and stellar contamination models in \citet{de_Wit_2016} and \citet{de_Wit_2018}, \citet{Zhang_2018}, and \citet{Wakeford_2019}. From these analyses, we learned that the TRAPPIST-1 planets probably do not have an H$_2$, He extended atmosphere. However, it was impossible to rule out this hypothesis using only HST/WFC3 \citep{de_Wit_2018,Moran_2018}. All spectra are consistent with flat spectra and could be fitted with different models including a high altitude cloud cover and/or a high metallicity hydrogen-rich atmosphere. A featureless spectrum could also be the result of the absence of an atmosphere around these planets. However, \citet{Bourrier_2017a} and \citet{Bourrier_2017b} analysed Lyman-$\alpha$ HST/STIS transits of TRAPPIST-1 b and c and detected a decrease in the flux, which might hint at the presence of an extended hydrogen exosphere.
We present the first attempt to characterise the atmosphere of the seventh planet of the system, TRAPPIST-1 h. In Sec. \ref{sec:1.1}, we analyse the HST/WFC3 G141 raw data using the python package Iraclis \citep{Tsiaras_2016b,Tsiaras_2016a, Tsiaras_2018} and detail the stellar contamination models used to correct our spectrum in Sec. \ref{sec:1.2}. Section \ref{sec:1.3} presents the atmospheric characterisation of TRAPPIST-1 h. First, different atmospheric scenarios are discussed based on the recent review by \citet{Turbet_2020b}. Then, we detail the atmospheric retrieval set-ups we performed using the Bayesian radiative transfer code TauREx3 \citep{Alrefaie2019}\footnote{\url{https://github.com/ucl-exoplanets/TauREx3_public}}. Finally, we discuss our findings in Sec. \ref{sec:3}.
\section {Data analysis}\label{sec:1}
\subsection{Hubble WFC3 data reduction and extraction}\label{sec:1.1}
We used the raw spatially scanned spectroscopic images obtained from Proposal 15\,304 (PI: Julien de Wit) in the Mikulski Archive for Space Telescope\footnote{\url{https://archive.stsc"i.e"du/hst/}}. Three transit observations of TRAPPIST-1 h were acquired using the Grism 256 aperture and 256 x 256 sub-array with an exposure time of 112.08 s. We refer to the data taken in July 2017, September 2019, and July 2020 as Observations 1, 2, and 3, respectively. Each visit is made up of four HST orbits, with 60 exposures in Observation 2 and 50 exposures for Observations 1 and 3, each being made in the forward spatial scan mode.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{visit3_wlc.pdf}}
\resizebox{\hsize}{!}{\includegraphics{visit2_wlc.pdf}}
\resizebox{\hsize}{!}{\includegraphics{visit4_wlc.pdf}}
\caption{White light curve fits for the three visits on TRAPPIST-1h (top: July 2017, middle: September 2019, and bottom: July 2020). For every observation, we show the de-trended flux (colour points) and the best fit model (dotted lines) along with the residuals from the best fit model.}
\label{fig:wlc_fitting}
\end{figure}
\begin{figure*}[htpb]
\centering
\includegraphics[width=17cm]{visit2_splc.pdf}
\caption{Spectral light curve fits of the September 2019 visit (Observation 2) for the transmission spectra of TRAPPIST-1 h. An artificial offset in the y-axis was applied for clarity. For each light curve, the left panel shows the de-trended spectral light curves with the best fit model in dotted lines with the centred wavelength and the right panel shows the residuals and values for the standard deviation ($\sigma$) in ppm, the reduced Chi-squared ($\tilde{\chi}^2$), and the auto-correlation (R$^2$).}
\label{fig:spclc_obs2}
\end{figure*}
\begin{table*}[htpb!]
\caption{Combined transit depth, associated uncertainties and limb-darkening coefficients.}
\centering
\begin{tabular}{cccccccc} \hline \hline
Wavelength ($\mu$m) & Bandwidth ($\mu$m) & Transit depth (ppm) & Error (ppm) & \multicolumn{4}{c}{Limb-darkening coefficients}\\
& & & & a1 & a2 & a3 & a4 \\
\hline
1.1262 & 0.0308 & 3128.22 & 129.30 & 2.0139 &-1.6261 & 0.8709 &-2.0398\\
1.1563 & 0.0293 & 2981.61 & 132.61 & 2.1956 & -2.0725 & 1.2403 & -0.3147 \\
1.1849 & 0.0279 & 3224.87 & 121.09 & 2.1292 & -1.9036 & 1.0964 &-0.2708\\
1.2123 & 0.0269 & 3275.06 & 112.69 & 1.9514 & -1.5303 & 0.8020 & -0.1849\\
1.2390 & 0.0265 & 3476.20 & 95.97 & 1.9236 & -1.5957 & 0.8838 & -0.2137\\
1.2657 & 0.0269 & 3264.82 & 95.65 & 2.0255 & -1.8405 & 1.0765 & -0.2698\\
1.2925 & 0.0267 & 3589.24 & 115.10 & 2.1105 & -2.1495 & 1.3561 & -0.3578 \\
1.3190 & 0.0263 & 3686.39 & 110.98 & 2.1650 & -2.2486 & 1.4262 &-0.3772\\
1.3454 & 0.0265 & 3368.20 & 126.87 & 1.2204 & -0.1088 & -0.1857 & 0.0789\\
1.3723 & 0.0274 & 2900.07 & 108.34 & 1.0023 & 0.4493 & -0.6644 & 0.2195\\
1.4000 & 0.0280 & 3271.69 & 109.38 & 0.9553 & 0.4582 &-0.6187 & 0.1988\\
1.4283 & 0.0285 & 3321.59 & 103.01 & 0.7774 & 0.7086 &-0.7252 & 0.2128\\
1.4572 & 0.0294 & 3111.09 & 113.41 & 0.9247 & 0.4694 &-0.6071 & 0.1921\\
1.4873 & 0.0308 & 3070.67 & 113.98 & 1.0279 & 0.2998 &-0.530181 & 0.18063\\
1.5186 & 0.0318 & 3037.95 & 112.45 & 1.2541 & -0.1103 & -0.2727 & 0.1188\\
1.5514 & 0.0337 & 3125.30 & 102.78 & 1.5025 & -0.6408 & 0.1247 & 0.0082\\
1.5862 & 0.0360 & 3472.00 & 114.20 & 1.7942 & -1.3368 & 0.6809 &-0.1553\\
1.6237 & 0.0390 & 3045.52 & 95.82 & 1.9296 & -1.7566 & 1.0358 & -0.2629 \\ \hline
1.3750 & 0.5500 & 3268.70 & 51.38 & 2.009 & -1.7704 & 1.0225 &-0.2546
\\\hline
\end{tabular}
\tablefoot{ The final transmission spectrum was computed in ppm using the three HST/WFC3 G141 transit observations from July 2017, September 2019, and July 2020 on TRAPPIST-1 h.}
\label{table2:spectrum}
\end{table*}
To reduce and analyse the data, we used Iraclis \footnote{\url{https://github.com/ucl-exoplanets/Iraclis}} \citep{Tsiaras_2016b,Tsiaras_2016a, Tsiaras_2018}, a publicly available pipeline, dedicated to the analysis of the scanned spectroscopic observations obtained with the near-infrared grisms (G102, G141) of Hubble's Wide Field Camera 3. The reduction of the raw observations follows these steps: zero-read subtraction, reference pixels correction, non-linearity correction, dark current subtraction, gain conversion, sky background subtraction, flat-field correction, and corrections for bad pixels and cosmic rays. For all three observations, we used the reduced spatially scanned spectroscopic images to extract the white and spectral light curves. We used the default 'low' resolution from Iraclis for the spectral light curves bins, which correspond to a resolving power of around 50 at 1.4${\rm \mu}$m.
Using the extracted light curves and the time of the observations, we first looked for contamination from others TRAPPIST-1 planets transits using the python package PyLightcurve \citep{Tsiaras_2016a} \footnote{\url{https://github.com/ucl-exoplanets/pylightcurve}}. The planets and transit parameters were set to those of \citet{Gillon_2017}. TRAPPIST-1 c was also transiting during the second orbit of the first observation (July 2017) and we then suppressed this orbit from the rest of the analysis. We plot in the appendix \ref{appendix:predicted_transits} the extracted raw flux and the corresponding predicted transits of TRAPPIST-1 planets for the three visits.
The first orbit always presents a stronger wavelength-dependent ramp than the other orbits and is usually suppressed from the analysis. However, we decided to keep the first HST orbit in every transit observation in order to conserve an out-of-transit baseline and correctly fit the transit parameters. Indeed, every attempt was made to keep as many exposures as possible. For Observations 1 and 2, we removed the first two exposures of these first orbits, but kept all exposures of every subsequent orbit. However, for Observation 3, an adequate fit could only be obtained by removing the first exposure of every orbit, a practice which is normal as these exposures present significantly lower counts than the following exposures \citep[e.g.][]{Deming_2013, Tsiaras_2016b, Edwards_2021}.
We fitted the white light curves and the spectral light curves using the transit model from PyLightcurve \citep{Tsiaras_2016a} and the Markov chain Monte Carlo (MCMC) method implemented in emcee \citep{Foreman_Mackey_2013}. For the white light curve fitting of all the observations, the only free planetary parameters are the mid-transit time and the planet-to-star radius ratio. The other planetary parameters were fixed to the values from \citet{Luger_2017a} (a/R$_{\rm s}$=109$\pm$4 and i=89.76$^{\circ}$) and stellar parameters are from \citet{Gillon_2017} (T$_{\rm s}$=2559$\pm$50 K, log(g)=5.21, Fe/H=0.04). We also fitted for the coefficients ra, r$b_1$, and r$b_2$. We adopted the parameterisation of \citet{Claret_2012} and \citet{Claret_2013} with four parameters to describe the limb-darkening coefficient. We used the PHOENIX database \citep{Claret_2018} and ExoTETHyS package \citep{Morello_2020} to obtain the limb-darkening coefficients for the white light curve analysis but also in every wavelength bin for the spectral curves fitting (see Table \ref{table2:spectrum}).
We accounted for the ramp time-dependent systematic effect in the white light curve fitting using the following formula with t being the time, t$_0$ the beginning time of each HST orbit, T$_0$ the mid-transit time, n$_{\rm scan}$ a normalisation factor, ra the slope of a linear systematic trend, and (rb$_1$ , rb$_2$ ) the coefficients of the exponential systematic trend along each HST orbit:
\begin{equation}
R_w(t)=n_w^{scan}(t)(1-r_a(t-T_0))(1-r_{b_1}e^{r_{b_2}(t-t_0))}
\label{eq1}
.\end{equation}
We then fitted for the planet-to-star radius ratio in every wavelength band. We used the white light curve divide method \citep{Kreidberg_2014a} along with a spectral-dependent visit-long slope \citep{Tsiaras_2018} model to account for the systematic effects as follows, with $\chi_\lambda$ being the slope for the wavelength-dependent systematic effects along each orbit, LC$_{\rm w}$ the white light curve signal, and M$_{\rm w}$ the white light curve best fit model:
\begin{equation}
R_\lambda(t)=n_\lambda^{scan}(t)(1-\chi_\lambda(t-T_0))\frac{LC_w}{M_w}
\label{eq2}
.\end{equation}
The white light curve fits for the three different observations are shown in Fig. \ref{fig:wlc_fitting}. The planet-to-star radius ratio are found to be compatible with 0.0575$\pm$0.0006 for Observation 1, 0.0565$\pm$0.0009 for Observation 2, and 0.0575$\pm$0.0012 for Observation 3. We found the following mid-transit times in BJD$_{\rm TDB}$: 2\,458\,319.4282 $\pm$ 0.00020 for Observation 1, 2\,458\,751.06983 $\pm$ 0.00021 for Observation 2, and 2\,459\,051.3428 $\pm$ 0.00021 for Observation 3. The spectral light curve fit for the second observation is presented in Fig. \ref{fig:spclc_obs2}, while the two other spectral light curve fits are in appendix \ref{appendix:spcl_obs2_obs3}. We computed the final transmission spectrum by combining the three spectral fits using a weighted mean of the transmission spectra. After the initial white light curve fit, the errors on each exposure were scaled to match the root mean square of the residuals. The white fitting was then performed a second time with these scaled errors. A similar scaling was also applied to the spectral light curves. This method ensures that the recovered uncertainties on the transit depth are not underestimated \citep{Tsiaras_2016b}. The transmission spectra and the recovered final transit depth are overplotted in Fig. \ref{fig:all_transits}, along with the corresponding residuals. We note a rise in the transit depth around 1.3$\mu$m. All three observations exhibit similar features over these regions, suggesting this is of astrophysical origin and part of the transit spectrum and not a contamination, or poor fitting, of a single visit. We also present in appendix \ref{appendix:wlc_spcl_all} the three white light curve fits in the same plot using a planet-to-star radius ratio weighted by the mean of the three white light curve best fits for the transit model. The combined extracted spectrum and the uncertainties are presented in Table \ref{table2:spectrum}.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{T1H_spectra_low.pdf}}
\resizebox{\hsize}{!}{\includegraphics{T1H_residuals_low.pdf}}
\caption{Recovered transit depths for the three observations and combined transmission spectrum with 1 and 2$\sigma$ uncertainty ranges (top). First, we suppressed the white light curve values from each visit raw flux, then, we computed the weighted mean, and finally we added the mean white light curve value to obtain the transit depth. Residuals are from the spectral light curves analysis and the combined spectrum (bottom).}
\label{fig:all_transits}
\end{figure}
\subsection{Modelling the stellar contamination}\label{sec:1.2}
TRAPPIST-1 is known for presenting a heterogeneous photosphere that can lead to a misinterpretation of the transmission spectra. The goal of this section is to use different existing models to correct our spectrum for a stellar contribution. The star presents a $\sim$ 1$\%$ photometric variability in the I+z bandpass interpreted as active regions rotating in and out of view \citep{Gillon_2016}.
\citet{Rackham_2018} show that it would cause a non-negligible effect (transit light source effect, TLSE) on the transmission spectrum if the variability is consistent with rotational modulations. Several previous studies have examined the stellar surface models using a variety of methods, but their results are not consistent. In the present study, three stellar models from three studies, \citet{Zhang_2018}, \citet{Morris_2018a}, and \citet{Wakeford_2019}, were introduced and examined.
Table \ref{tab:stellar_model} shows the temperature and the covering fraction of each component for each model. We note that $T_i$ is the temperature, $f_i$ is the covering fraction at the photosphere, and $f'_i$ is the covering fraction at the transit chord. The M18 model is the best fit model from \citet{Morris_2018a}. Z18 is the best fit contamination model taken from Table 16 in \citet{Zhang_2018}. W19 is the $3T_{c+m}$ model from \citet{Wakeford_2019}. We note that what we call the W19 model here is not the best fit model in their analysis, as they conclude that TLSE is not significant in their data, but they did not exclude $3T_{c+m}$.
\begin{table}[htpb!]
\caption{Summary of the adopted TRAPPIST-1 stellar models.}
\centering
\begin{tabular}{c|ccc}\hline \hline
Model & Z18 & M18 & W19 \\ \hline
$T_1$(K) & $2000$ & $2500$ & $2400$ \\
$T_2$(K) & $2400$ & $5300$ & $3000$ \\
$T_3$(K) & $3000$ & $-$ & $5800$ \\ \hline
$f_1$& $0.38$ & $0.999952$ & $0.64$ \\
$f_2$ &$0.14$ & $4.8\times10^{-5}$ & $0.35$\\\hline
$f'_1$ & $0.10$ & $1.0$ & $0.646$\\
$f'_2$ & $0.45$ & $0.0$ & $0.354$\\\hline
\end{tabular}
\label{tab:stellar_model}
\end{table}
We define the wavelength-dependent contamination factor $\varepsilon_\lambda$ as
\begin{equation}
\delta_\lambda = \varepsilon_\lambda \times \delta_{real,\lambda}
,\end{equation}
where $\delta_\lambda$ is the measured transit depth and $\delta_{real,\lambda}$ is the actual transit depth. For each stellar surface model, $\varepsilon_\lambda$ was calculated as
\begin{eqnarray}
\varepsilon_\lambda &= \frac{f'_1 S_{1,\lambda} + f'_2 S_{2,\lambda} + f'_3 S_{3,\lambda}}{f_1 S_{1,\lambda} + f_2 S_{2,\lambda} + f_3 S_{3,\lambda}}\\
f_3 &= 1-f_1-f_2 \\
f'_3 &= 1-f'_1-f'_2
,\end{eqnarray}
where $S_{i,\lambda}$ is the stellar flux of each temperature component. We used the BT-Settl model for each temperature, with the metallicity
[Fe/H]= 0 dex and the stellar surface gravity log g at 5.2, from the SVO theoretical spectra web server (\footnote{\url{http://svo2.cab.inta-csic.es/theory/newov2/}}).
\subsection{Atmospheric modelling}\label{sec:1.3}
\subsubsection{Possible atmospheric scenarios}\label{sec:1.3.1}
\citet{Turbet_2020b} reviewed the different atmospheric scenarios for TRAPPIST-1 planets. We discuss the different possibilities mentioned for TRAPPIST-1 h, such as a H$_2$/He rich atmosphere, a H$_2$O envelope, as well as a O$_2$, a CO$_2$, a CH$_4$/NH$_3$, or a N$_2$ dominated atmosphere. First, numerical modelling using mass and radius measurements have shown that a H$_2$/He envelope is unlikely for all TRAPPIST-1 planets. \citet{Turbet_2020b} constructed a mass-radius relation using the \citet{Grimm_2018} atmospheric climate calculation and estimated that for a 'cold' scenario assuming 100 x solar metallicity and based on TRAPPIST-1 h irradiation, the maximum hydrogen to core mass fraction is $4\times10^{-4}$ for a clear atmosphere. Using the estimation of \citet{Wheatley_2017} for the EUV flux received by the planet ($10^{2}$ erg.s$^{-1}$.cm$^{-2}$) and the results from \citet{Bolmont_2017a}, \citet{Bourrier_2017a} and \citet{Bourrier_2017b}, they computed the equivalent mass loss over the age of the system (8 billion years) and found $10^{22}$ kg (i.e. $5\times10^{-3}$ mass fraction). A hydrogen-rich envelope could be ripped out in $\sim$100 million years for TRAPPIST-1 h \citep{Turbet_2020b}, meaning that this type of atmosphere is not completely impossible but unstable and unlikely to be sustained around this low mass planet. The recent publication by \citet{Hori_2020} has also shown that the total mass loss over the planet lifetime is supposedly higher than the initial amount of accreted gas.
Regarding a water-rich atmosphere scenario, \citet{Turbet_2019a}, \citet{Turbet_2020a} and \citet{Turbet_2020b} estimated the water content in TRAPPIST-1 planets by taking the runaway greenhouse limit into account, while \citet{Bourrier_2017a} investigated the hydrodynamic water loss. Combining those two pieces of information leads to the conclusion that TRAPPIST-1 h could have lost less than three Earth oceans and could have retained water in its atmosphere or surface. \citet{Lincowski_2019} show that O$_2$ atmospheres would be the best candidate for TRAPPIST-1 planets as a remnant of H$_2$O erosion and \citet{Wordsworth_2018} determine that O$_2$ build-up is limited to one bar for TRAPPIST-1h.
We note that NH$_3$ and CH$_4$ are highly sensitive to photo-dissociation \citep{Turbet_2018} and for TRAPPIST-1h to sustain a CH$_4$ or a NH$_3$ rich atmosphere would require an important source of those species. Assuming an Earth-like methane production rate, the planet could have a concentration up to 0.3\% \citep{Rugheimer_2015}. However, methane or ammonia photolysis rates could decrease via the formation of high altitude clouds or hazes \citep{Sagan_1997, Wolf_2010, Arney_2016}.
An Earth-like atmosphere, that is one bar and a N$_2$ rich atmosphere, might be stable against stellar wind for TRAPPIST-1 h if CO$_2$ is abundant \citep{Dong_2018, Dong_2019}; CO$_2$ could accumulate in TRAPPIST-1 planets \citep{Lincowski_2019} because it is less sensitive to atmospheric escape \citep{Dong_2017,Dong_2018, Dong_2019}. However, \citet{Turbet_2018} and \citet{Turbet_2020b} show that TRAPPIST-1 h would probably experience a CO$_2$ collapse. The planet is far from the star and probably tidally locked, favouring CO$_2$ surface condensation. Furthermore, CO and O$_2$ could also be found in the case of a CO$_2$ rich atmosphere due to the photo-dissociation of CO$_2$ and the low recombination of CO and O$_2$ \citep{Gao_2015, Hu_2020}.
Finally, a water ocean at the surface of TRAPPIST-1 h, implying a potential habitability, is unlikely. As the planet is beyond the CO$_2$ collapse region, the atmosphere does not warm the surface \citep{Turbet_2020b}. To counterbalance the CO$_2$ condensation, the planet would require a very thick CO$_2$ atmosphere with volcanic gases such as H$_2$ and CH$_4$, but, as explained above, neither H$_2$ nor CH$_4$ are expected to be stable in the TRAPPIST-1 h atmosphere \citep{Pierrehumbert_2011, Wordsworth_2017, Ramirez_2017, Lincowski_2018, Turbet_2018,Turbet_2019b, Turbet_2020a}. Very few observational constraints have been brought on TRAPPIST-1 planets, leaving a wide range of atmospheric possibilities. The goal of the following section is to analyse the TRAPPIST-1 h IR spectrum with regards to the predictions mentioned above in order to bring new constraints and prepare further observations.
\subsubsection{Retrieval analysis set-up}\label{sec:1.3.2}
We used TauREx3 \citep{Alrefaie2019,Alrefaie2021} and the nested sampling algorithm Multinest \citep{Feroz_2009} with an evidence tolerance of 0.5 and 1500 live points to perform the atmospheric retrieval analysis. TauREx3 is a fully Bayesian code that maps the atmospheric parameters space to find the best fit model for the transmission spectrum. It includes the molecular line lists from the ExoMol project \citep{Tennyson_2016,Chubb_2021}, HITEMP \citep{Tennyson_2018}, and HITRAN \citep{Rothman_1987, Rothman_2013}. We simulated the atmosphere assuming a constant temperature-pressure profile and every layer of the simulated atmosphere is uniformly distributed in log spaced, with a total of 100 ranging from $10^{-2}$ to $10^5$ Pa. We included the collision-induced absorption (CIA) of H2-H2 \citep{abel_2011, fletcher_2018}, H$_2$-He \citep{abel_2012}, and Rayleigh scattering. We used a wide range of temperatures (50-1000K) to adjust the temperature of the planet, using the effective temperature ($\sim$173 K) as the initial value. The planetary radius was also fitted as a free parameter in the model and its value ranges from $\pm 50\%$ of the published value reported in Table \ref{table1:parameter}. The planetary radius fitted corresponds to the bottom of the atmosphere, that is the radius of the planet assumed to be at one bar here. Clouds were included using a simple grey opacity model and the top clouds pressure varies from $10^{-2}$ to $10^5$ Pa. We considered the following opacity sources: H$_2$O \citep{Polyansky_2018}, CO$_2$ \citep{Rothman_2010}, NH$_3$ \citep{Yurchenko_2011}, and CO \citep{Yurchenko_2014}.
We performed two different atmospheric retrievals by forcing a primary and then a secondary atmosphere. We modelled the TRAPPIST-1 h atmosphere using H$_2$, He, and N$_2$ as fill gas and H$_2$O, CO, CO$_2$, NH$_3$, and CH$_4$ as trace gases. We note that H$_2$, He, and N$_2$ do not display features in the spectrum; they contribute to the continuum and shape the mean molecular weight. The ratio between H$_2$ and He abundances was fixed to the solar value of 0.17, while the ratio between the abundance of N$_2$ over the abundance of H$_2$ varied between $10^{-12}$ and $10^{-2}$ for the primary model and between $10^{-12}$ and $10^{4}$ for the secondary scenario. The mean molecular weight can then evolve towards higher values and we were able to test a Hydrogen rich and then a Nitrogen rich atmosphere. The abundance of the other molecular absorption sources were included in the fit as a volume mixing ratio, allowing us to vary between $10^{-12}$ and $10^{-2}$.
A flat-line model, only including a cloud deck, was performed to assess the significance of the different scenarios compared to a baseline. A baseline is representative of the lack of an atmosphere (e.g. an atmosphere with no spectral features) or a flat spectrum that can only be fitted by a high altitude cloud deck. The significance was computed using a Bayes factor, that is the difference of logarithm evidence between the best fit model and the baseline model. The Bayesian evidence was computed using Bayes' theorem for a set of $\theta$ parameters in a model H for the data D \citep{Feroz_2009}
\begin{equation}
P(\theta|D, H)=\frac{P(D|\theta, H)P(\theta|H)}{P(D|H)}
,\end{equation}
where P($\theta$|D, H) $\equiv$P($\theta$) is the posterior probability distribution, P(D|$\theta$, H) $\equiv$L($\theta$) is the likelihood, P($\theta$|H)$\equiv$ $\pi$($\theta$) is the prior, and P(D|H)$\equiv$ E is the Bayesian evidence. The nested sampling method estimates the Bayesian evidence of a given likelihood volume and the evidence can be expressed as follows:
\begin{equation}
E=\int L(\theta) \pi(\theta) \, \mathrm{d}\theta
.\end{equation}
To compare the two H$_0$ and H$_1$ models, in our case the flat-line model and the primary or secondary scenario, we can compute the respective posterior probabilities, given the observed data set D,
\begin{equation}
\frac{P(H_1|D)}{P(H_0|D)}= \frac{P(D|H_1)P(H_1)}{P(D|H_0)P(H_0)}=\frac{E_1 P(H_1)}{E_0 P(H_0)}
,\end{equation}
where P(H$_1$)/P(H$_0$) is the a priori probability ratio for the two models, which can often be set to unity \citep{Feroz_2009}. We used the logarithm version of the model selection to compute the Bayes factor, $\Delta$log(E) between the flat-line and the tested model. This factor is also called the atmospheric detectability index (ADI) in \citet{Tsiaras_2018} and defined as a positive value. The significance ($\sigma$) represents the strength of a detection and it was estimated using a \citet{Kass_1995}, \citet{Trotta_2008}, and \citet{Benneke_2013} formalism. We used Table 2 in \citet{Trotta_2008} and Table 2 in \citet{Benneke_2013} to find the equivalence between the Bayes factor and the significance $\sigma$ and evaluate the strength of a detection. A Bayes factor of 1 corresponds to a 2.1$\sigma$ detection and is considered weak, a Bayes factor greater than 3 (3$\sigma$) is considered significant, and one superior to 11 (5$\sigma$) is considered as a strong detection.
For the rest of the paper, we define $\Delta$log(E)=log($\textrm{E}_\textrm{Atmospheric Model}$)-log($\textrm{E}_\textrm {Flat line}$). The atmospheric model can be considered a better fit compared to the flat line if the $\Delta$log(E) is superior to 3.
\section{Results}\label{sec:2}
\subsection{Atmospheric retrieval results}\label{sec:2.1}
There is no evidence of molecular absorption in the recovered spectrum of TRAPPIST-1h from the two retrieval results. Both primary and secondary retrieval analyses have logarithm evidence (109.92 and 110.18, respectively) comparable to the one of the flat-line model, that is 110.55.
This result favours the scenario of a planet with no atmosphere, that is the presence of a high cloud layer in a primary atmosphere or a secondary envelope. It is consistent with previous work on other TRAPPIST-1 planets \citep{de_Wit_2018,Wakeford_2019,Zhang_2018}.
\begin{figure}[htpb]
\centering
\resizebox{\hsize}{!}{\includegraphics{best_fit_taurex_T1H_wide.pdf}}
\resizebox{\hsize}{!}{\includegraphics{stellar_models_WFC3_G141_wide.pdf}}
\caption{Best-fit models to TRAPPIST-1 h HST WFC3 G141 data from atmosphere retrievals (top) and stellar contamination models based on \citet{Zhang_2018}, \citet{Wakeford_2019}, and \citet{Morris_2018a} (bottom).}
\label{fig:best_fit}
\end{figure}
Figure \ref{fig:best_fit} (top) shows the extracted spectrum with the best-fit atmospheric results: flat-line (red), primary (blue), and secondary model (purple). The flat-line and the secondary best fit models are similarly flat with a transit depth around 3220 ppm, while the primary models are found around 3274 ppm. This difference is due to the different radius and temperature estimations depending on the scale height and the weight of the atmosphere. We present the correlations among parameters for the primary and the secondary model in Fig. \ref{fig:posteriors_taurex}. We overplotted the two posterior distributions for a direct comparison, but the values are displayed for the secondary best-fit model. The primary model posterior distribution alone is presented in Appendix \ref{appendix:posteriors_primary}.
The secondary atmospheric retrieval analysis estimates the radius to be 0.69$^{+0.03}_{-0.07}$R$_{\oplus}$ and the temperature reaches 345$^{+326}_{-196}$K. The mean molecular weight distribution is bi-modal, and the code is able to retrieve two solutions: a light atmosphere with a 2.3 g/mol mean molecular weight and a heavier solution with a mean molecular weight reaching 25.35$^{+2.46}_{-23.02}$ g/mol corresponding to a 16 km scale height. This is correlated to the abundance of N$_2$ retrieved as the ratio of inactive gases, that is log(N$_2$/H$_2$). When we allowed this ratio to increase beyond one, the best fit value was constrained to 1.01$^{+1.18}_{-6.13}$. Yet, we note the presence of a second solution, around seven, which corresponds to the primary analysis retrieved value and creates the bi-modal distribution in the mean molecular weight. Nitrogen is the only parameter that impacts the value of the mean molecular weight as no constraints can be put on H$_2$O, CH$_4$, CO, CO$_2$, and NH$_3$.
\begin{table*}[htpb!]
\centering
\caption{Statistical results of the atmospheric retrieval analysis and the stellar contamination modelling on TRAPPIST-1h HST WFC3 G141 data.}
\label{table:stats}
\begin{tabular}{ l|c c c c } \hline\hline
Model & $\chi^2$ &$\tilde{\chi}^2$ & log(E) & $\Delta$log(E)\\ \hline
Flat-line & 64.95 & 3.61 & 110.55 & N/A\\
Atmosphere primary &65.20 & 3.62 & 109.92 & -0.63 \\
Atmosphere secondary &65.25 & 3.62 & 110.18 & -0.37\\
Stellar \citet{Zhang_2018} & 54.02 & 3.00 &N/A & N/A \\
Stellar \citet{Wakeford_2019} & 60.70 & 3.37 &N/A & N/A\\
Stellar \citet{Morris_2018a} &63.91 & 3.55 & N/A & N/A \\
\hline
Corrected by \citet{Zhang_2018} \\
Flat-line &54.94 & 3.05 & 115.48 & N/A\\
Atmosphere primary &50.54 & 2.81 & 115.02 & -0.46\\
Atmosphere secondary & 54.77 & 3.04 & 115.20 & -0.28\\
\hline
Corrected by \citet{Wakeford_2019} \\
Flat-line & 61.41 & 3.41 & 112.13 & N/A\\
Atmosphere primary & 61.91 & 3.44 &111.51 &-0.62\\
Atmosphere secondary & 61.80 &3.43 & 112.01 &-0.12\\
\hline
Corrected by \citet{Morris_2018a} \\
Flat-line & 65.11 & 3.62 &110.39& N/A \\
Atmosphere primary & 65.28 &3.63 &109.79 &-0.60\\
Atmosphere secondary & 65.23 &3.62 & 110.12 &-0.27\\
\hline
\end{tabular}
\tablefoot{Chi-squared ($\chi^2$) and reduced chi-squared ($\tilde{\chi}^2$2) were computed using the result of the retrieval best-fit model and the stellar contamination models. Bayesian logarithm evidence (log(E)) and the Bayes factor ($\Delta$log(E)) were computed when applicable, i.e. only for the atmospheric retrieval analysis.}
\end{table*}
From both posteriors distributions, we found the anti-correlation between the radius, temperature, and layer for top clouds, that is the radius decreases with increasing temperature and decreasing layer for top clouds. The latter is found really high in the atmosphere, log(P$_{\rm clouds}$)=1.02$^{+1.90}_{-1.72}$, which corresponds to a cloud layer at approximately 10$^{-4}$ bar. Considering the pressure of this layer, it is likely that these clouds may not be condensation clouds, but rather photochemical mists or hazes with particles big enough not to have a spectral slope.
From those two retrievals analyses, we show that the atmosphere must be either secondary (probably dominated by nitrogen) or primary with a very high photochemical haze layer. The two retrieval analyses have similar statistical results so we cannot favour one solution. We cannot rule out the hypothesis of a lack of an atmosphere either as the log(E) of the flat-line model remains the highest. We can reject a primary clear atmosphere as expected for this planet as the primary model does indeed require a layer of clouds to correctly fit the spectrum (see Sec. \ref{sec:3.1}).
\begin{figure*}[htpb!]
\centering
\includegraphics[width=17cm]{posteriors_combined.pdf}
\caption{Posterior distributions for the primary (blue) and the secondary retrieval (purple) on the extracted TRAPPIST-1h spectrum. Only the values from the secondary best-fit analysis are displayed. }
\label{fig:posteriors_taurex}
\end{figure*}
\subsection{Including the stellar contamination}\label{sec:2.2}
We present in Fig. \ref{fig:best_fit} (bottom) the stellar contamination models and in Table \ref{table:stats} the statistical results on both the atmosphere and stellar models. We computed the chi-squared ($\chi2$) and the reduced chi-squared ($\tilde{\chi}^2$2) for all models and indicate the logarithm evidence (log(E)) from the retrieval analysis.
The stellar contamination model of \citet{Zhang_2018} is favoured, according to the chi-squared computation but none of the models we tested here are significant and can explain variations in the TRAPPIST-1h spectrum. In particular, the rise in the transit depth around 1.3$\mu$m is not reproduced. To account for stellar contamination, we corrected our HST/WFC3 extracted spectrum by subtracting the stellar contributions using the \citet{Zhang_2018}, \citet{Wakeford_2019}, and \citet{Morris_2018a} formalism.
\begin{figure}[htpb!]
\centering
\resizebox{\hsize}{!}{\includegraphics{best_fit_taurex_T1H_zhang_wide.pdf}}
\resizebox{\hsize}{!}{\includegraphics{best_fit_taurex_T1H_wakeford_wide.pdf}}
\resizebox{\hsize}{!}{\includegraphics{best_fit_taurex_T1H_morris_wide.pdf}}
\caption{Best fit models to TRAPPIST-1 h HST WFC3 G141 data after subtraction of stellar contamination contributions according to the \citet{Zhang_2018} (top), \citet{Wakeford_2019} (middle), and \citet{Morris_2018a} (bottom) formalism. }
\label{fig:best_fit_corrected}
\end{figure}
We present in Table \ref{table4bis:spectra_corrected} the transit depth after subtraction of the stellar contamination for the three models and conduct the same retrieval analysis as in Sec. \ref{sec:2.1} on those corrected spectra. We overplotted all the different spectra as a comparison in appendix \ref{appendix:corrected_spectra}. Statistical results on the retrieved corrected spectra are detailed in Table \ref{table:stats}. For all the corrected spectra, the flat line is the favoured model, but the correction by \citet{Zhang_2018} leads to the highest log(E). We present in Fig. \ref{fig:best_fit_corrected} best-fit atmospheric retrieval results on the three spectra, while posterior distributions are in Appendix \ref{appendix:posteriors_taurex_Zhang}, \ref{appendix:posteriors_taurex_Wakeford}, and \ref{appendix:posteriors_taurex_Morris}. We note transit depth variations at 1.2 $\mu$m, 1.45$\mu$m, and 1.6$\mu$m on the primary best fit model on the spectrum corrected by \citet{Zhang_2018}. This is due to the contribution of CO$_2$ to the best-fit solution, but the amount of CO$_2$ is not constrained (see posterior distributions in Appendix \ref{appendix:posteriors_taurex_Zhang}) and the model is not statistically significant. We also observe variations in the transit depth around 1.5$\mu$m on the primary best fit model on the spectrum corrected by \citet{Morris_2018a}. This is due to the absorption of ammonia. Once again, this absorption is not constrained in terms of abundance and the log(E) remains below the one of the flat line. As an indication, we put the best-fit opacity contributions from those two models in appendix \ref{appendix:contributions}. The correction made here to the spectra does improve the retrieval statistical results in the case of the \citet{Zhang_2018} correction, but it does not lead to molecular detection and does not allow us to provide further constraints on the atmosphere of TRAPPIST-1 h.
To better constrain the stellar contamination, we also tried to add the optical value found in \citet{Luger_2017a} using K2 photometry. As seen in the plot of Appendix \ref{appendix:stellar_contamination_K2} and in the $\chi^2$ computation results in Appendix \ref{appendix:stats_K2}, the existing stellar models discussed here fit the spectrum poorly. First, we cannot ensure inter-instrument calibration at this accuracy and combining a different transit depth could lead to misinterpretations of the spectrum \citep{Yip_2020}. In addition, it is possible that the stellar spot distribution has changed in the intervening time between the observations, but this is unlikely as they are not that far apart. K2 light curves were taken between 15 December 2016 and 4 March 2017, while the HST data were taken in July 2017, September 2019, and July 2020. A more likely explanation is that for both K2 and HST data, multiple transits were stacked, regardless of the phase of the star's rotation. If the stellar rotational phase and activity were different from time to time, the effect in the transmission spectrum would be suppressed when they are stacked. Adding this point does not further constrain the stellar contamination models in the case of TRAPPIST-1h.
\begin{table}[htpb!]
\caption{Corrected transit depth in ppm using stellar contamination models.}
\centering
\begin{tabular}{p{2.5cm} p{1.5cm} p{1.5cm} p{1.5cm}} \hline \hline
Wavelength ($\mu$m) & \multicolumn{3}{c}{Transit depth (ppm)} \\
\hline
1.1262 & 3072.99 & 3193.93 & 3129.83\\
1.1563 & 2934.85 & 3032.81 & 2983.10 \\
1.1849 & 3171.13 & 3259.82 & 3226.35\\
1.2123 & 3202.19 & 3283.50 & 3276.38\\
1.2390 & 3406.64 & 3477.46 & 3478.10\\
1.2657 & 3207.19 & 3257.41 & 3266.07\\
1.2925 & 3520.20 & 3564.16 & 3590.50 \\
1.3190 & 3643.19 & 3673.80 & 3687.80\\
1.3454 & 3388.59 & 3396.28 & 3369.86\\
1.3723 & 2948.59 & 2939.83 & 2901.69\\
1.4000 & 3341.90 & 3318.92 & 3273.55\\
1.4283 & 3414.44 & 3373.99 & 3323.56\\
1.4572 & 3193.82 & 3149.00 & 3112.83\\
1.4873 & 3141.10 & 3090.15 & 3072.24\\
1.5186 & 3093.99 & 3037.20 & 3039.34\\
1.5514 & 3165.61 & 3101.84 & 3126.55\\
1.5862 & 3499.04 & 3418.90 & 3473.19\\
1.6237 & 3057.41 & 2980.84 & 3046.45
\\\hline
References & 1 & 2 & 3\\ \hline
\end{tabular}
\label{table4bis:spectra_corrected}
\tablebib{ (1) \citet{Zhang_2018}; (2)\citet{Wakeford_2019}; (3) \citet{Morris_2018a}}
\end{table}
\section{Discussion} \label{sec:3}
\begin{figure*}[htpb!]
\centering
\includegraphics[width=\columnwidth]{best_fit_taurex_H2O_0.001_T1B.pdf}
\includegraphics[width=\columnwidth]{best_fit_taurex_H2O_0.001_T1C.pdf}
\includegraphics[width=\columnwidth]{best_fit_taurex_H2O_0.001_T1D.pdf}
\includegraphics[width=\columnwidth]{best_fit_taurex_H2O_0.001_T1E.pdf}
\includegraphics[width=\columnwidth]{best_fit_taurex_H2O_0.001_T1F.pdf}
\includegraphics[width=\columnwidth]{best_fit_taurex_H2O_0.001_T1G.pdf}
\includegraphics[width=\columnwidth]{best_fit_taurex_H2O_0.001_T1H.pdf}
\caption{Comparison of the best-fit atmospheric results for TRAPPIST-1 planetary spectra in the case of a forced primary clear atmosphere with a volume mixing ratio of water fixed to 10$^{-3}$ in a H-dominated atmosphere. We used spectra from \citet{Zhang_2018} for the TRAPPIST-1 b to g retrievals and present the results in units of scale height. }
\label{fig:primary_T1system}
\end{figure*}
\subsection{Primary clear atmosphere}\label{sec:3.1}
We show in Sec. \ref{sec:2.1} that the HST/WFC3 extracted spectrum was compatible with either a secondary or a primary cloudy and hazy atmosphere if we retain the hypothesis of a presence of an atmosphere. In this section, we explore the case of a primary clear atmosphere by fixing the molecular absorption of the different species to 10$^{-3}$, which forces spectral features. The temperature was allowed to vary between $\pm$20$\%$ of the equilibrium one (173K) and the radius was fitted between $\pm$50$\%$ of the published value. We tested six different opacity sources, H$_2$O, CO$_2$, CO, CH$_4$, NH$_3,$ and N$_2,$ separately by running a retrieval for each sources. We included collision-induced absorption and Rayleigh scattering and fixed the He/H$_2$ ratio to 0.17. The atmosphere was simulated as in Sec. \ref{sec:2.1}, with 100 layers ranging between 10$^{-2}$ and 10$^{5}$ Pa.
We measured the size of a clear atmosphere in the case of TRAPPIST-1 h in those six configurations and show that a primary clear atmosphere is rejected in each case. We present in Table \ref{table:primary_clear} best-fit results from the six tested scenarios. We indicate the radius, the temperature, and the mean molecular weight, and we estimate the corresponding scale height. For comparison, we also indicate the results from the flat-line model of Sec. \ref{sec:2.1}. Statistical results, that is to say the logarithm evidence, from primary clear models are below the one of the flat line with an absolute difference of 3 or more, while including H$_2$O, CH$_4$, NH$_3$, or N$_2$. This result indicates that a primary clear atmosphere is rejected with high confidence (i.e. 3$\sigma$). Primary clear atmospheric scenarios with traces of CO or CO$_2$ have higher $\Delta$log(E), remaining below the one of the flat-line model, but they cannot be rejected as firmly as the others (see also Fig. \ref{fig:delta_logE_mmw} in Sec. \ref{sec:3.2}).
\begin{table*}[htpb]
\centering
\caption{Best-fit atmospheric results and derived parameters for a primary clear retrieval analysis. }
\label{table:primary_clear}
\begin{tabular}{ l| c c c c c c c c} \hline\hline
Model & R$_{\rm P}$(R$_{\rm \oplus}$) & T(K) & $\mu$(g/mol) & H(km) & $\chi^2$ &$\tilde{\chi}^2$ & log(E) & $\Delta$log(E) \\
\hline
Flat-line & 0.61$\pm${0.110} & 296$\pm 225$ & 2.30 & 71.28 &64.95& 3.61 &110.55 & N/A \\
H$_2$O & 0.70$\pm${0.003} & 140$\pm 2$ & 2.32 & 75.27 & 128.68 &7.15 & 74.78 & -35.77 \\
CO$_2$ & 0.71$\pm${0.003} & 157$\pm 26$ & 2.35 & 86.84 & 68.37&3.80 & 108.99 & -1.56\\
CO & 0.72$\pm${0.003} & 158$\pm 25$ & 2.33 & 88.84 &70.53 &3.92 & 107.93 & -2.62 \\
CH$_4$ & 0.69$\pm${0.003} & 139$\pm 2$ & 2.32 & 72.20 &146.35 &8.13 & 65.48 & -45.07\\
NH$_3$ & 0.68$\pm${0.003} & 140$\pm 3$ & 2.32 & 70.56 &107.58 & 5.98 & 85.38 & -25.17\\
N$_2$ & 0.72$\pm${0.003} & 156$\pm 26$ & 2.33 & 87.85 &69.71 & 3.87& 106.82 & -3.73\\ \hline
\end{tabular}
\tablefoot{The primary clear atmospheric scenario was simulated including the molecular absorption with a volume mixing ratio fixed to 10$^{-3}$ in a H-dominated atmosphere.}
\end{table*}
A primary clear atmosphere scenario was previously rejected for TRAPPIST-1 planets using HST/WFC3 G141 spectra \citep{de_Wit_2018, Zhang_2018, Wakeford_2019}. Performing the same exercise with a fixed 10$^{-3}$ water abundance for the others six TRAPPIST spectra from \citet{Zhang_2018}, we also confirm that a primary clear atmospheric model does not fit their spectra. We note that we simulated planet atmospheres with the same 100 layers between 10$^{-2}$ and 10$^{-5}$ even though they have a different size, radius, and mass. We present in Fig. \ref{fig:primary_T1system} the best-fit atmospheric retrieval results in the case of a Hydrogen-dominated atmosphere with water as a trace gas (the volume mixing ratio was fixed to 10$^{-3}$) for the seven planets of the TRAPPIST-1 system. The results are presented in number of scale height and we can see that TRAPPIST-1 planets are unlikely to possess a clear atmosphere dominated by hydrogen with water in a low quantity. The comparison of logarithm evidence between a flat-line model and a primary clear atmosphere for all seven planets is detailed in appendix \ref{appendix:primary_clear_T1}. This is in agreement with theoretical modelling as detailed in Sec. \ref{sec:1.3.1} and in \citet{Turbet_2020a} and \citet{Hori_2020}.
\subsection{Steam atmosphere}\label{sec:3.2}
\begin{figure*}[htpb]
\centering
\includegraphics[width=\columnwidth]{best_fit_taurex_CO2_0.2_T1H_wide.pdf}
~
\includegraphics[width=\columnwidth]{best_fit_taurex_CO_0.2_T1H_wide.pdf}
\includegraphics[width=\columnwidth]{best_fit_taurex_N2_0.8_T1H_wide.pdf}
~
\includegraphics[width=\columnwidth]{best_fit_taurex_NH3_0.8_T1H_wide.pdf}
\caption{Comparison of the best-fit atmospheric results for the TRAPPIST-1h spectrum in the case of four forced secondary clear atmospheres with a volume mixing ratio of CO$_2$ fixed to 0.2 (upper left),CO to 0.2 (upper right), N$_2$ to 0.8 (bottom left), and ammonia to 0.8 (bottom right). }
\label{fig:steam_taurex}
\end{figure*}
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{logE_mmw_figure.pdf}}
\caption{Comparison of the log evidence for a flat line to that of single molecule retrievals where the abundance of the molecule is fixed and no clouds were included. We represent the Delta log(E) with respect to the mean molecular weight of the tested atmospheres. The region between dashes represents the set of Bayes factor values for which it is not possible to conclude compared to a flat line, that is with absolute $\Delta$log(E) below 3. Models below the large dashed lines are strongly disfavoured compared to the flat line. }
\label{fig:delta_logE_mmw}
\end{figure}
From the review of the possible atmospheric scenario \citep{Turbet_2020a}, TRAPPIST-1h could have a water-, methane-, ammonia-, nitrogen-, or even a carbon-dioxide-rich atmosphere depending on the evolution of the planet, on the species collapses, and on the photo-chemistry. A steam atmosphere is unlikely for TRAPPIST-1h as it would require the planet to have retained its atmosphere and to be stable against stellar wind (see Sec. \ref{sec:1.3.1}). However, we tested those different hypothesis by using a similar approach as in Sec. \ref{sec:3.1}, but allowing for heavier atmospheres by increasing the volume mixing ratio of the tested molecular absorber from 0.01 to 0.8 progressively. Best-fit atmospheric results along with derived parameters and statistical criteria are detailed in Appendix \ref{table:secondary_clear}. We note that some forced secondary steam atmospheric models have log(E) equal to or slightly above the one of the flat-line model. The difference in log(E) is above one for one case, with the CO-rich atmosphere having a volume mixing ratio fixed to 0.2. This model has a $\Delta$log(E) of 1.01 corresponding to 2.1$\sigma$ confidence, hence a 'weak detection' in \citet{Benneke_2013} classification. The best-fit spectrum of the models presenting an elevated log(E) are plotted in Fig. \ref{fig:steam_taurex}. They correspond to the model of 20$\%$ CO$_2$ and CO and 80$\%$ of N$_2$ and NH$_3$. Carbon dioxide and carbon monoxide have similar absorption features in the HST/WFC3 wavelength range, which leads to similar best-fit results. Moreover, as the features are very small, we obtained the same volume mixing ratios for those species. We note that N$_2$ acts as a fill gas in the atmosphere as it does not have features; the best-fit spectrum is then similar to that of the flat line. We can add a CO-rich atmosphere to the possible atmospheric scenario for TRAPPIST-1h.
We note that $\Delta$log(E) remains below one for most of the other tested models, meaning that they are not statistically significant. We present in Fig. \ref{fig:delta_logE_mmw} the comparison of the log evidence for a flat line to that of single molecule retrievals from the primary clear analysis of Sec. \ref{sec:3.1} and the secondary models of Sec. \ref{sec:3.2} following the formalism of Fig.6 in \citet{Mugnai_2021}. We decided to represent the $\Delta$log(E) with respect to the mean molecular weight of the modelled atmospheres to compare the different scenarios as similar molecular abundances could lead to different weights and metallicities. Primary atmospheric models with a metallicity below 50 times solar (i.e. mmw=2.70 g/mol) are rejected with more than 5$\sigma$ confidence (i.e. the Bayes factor is inferior to -11), except for CO and CO$_2$. In addition, if the atmosphere was primary, it would be unlikely that it does not contain any water. The equivalence between abundances, the mean molecular weight, and solar metallicity is presented in Appendix Table \ref{table:secondary_clear} and a figure of all $\Delta$log(E) with respect to the abundances is presented in Appendix \ref{appendix:delta_logE}. The area between dashes represents the set of Bayes factor values for which it is not possible to conclude compared to a flat line, that is with absolute $\Delta$log(E) below 3. Models with a $\Delta$log(E) between -3 and -11 can be significantly rejected compared to a flat line, while the ones below -11 are strongly disfavoured.
\subsection{Impact of changing the spectral resolution}
\begin{figure}[htpb!]
\centering
\resizebox{\hsize}{!}{\includegraphics{best_fit_taurex_T1H_vlow_wide.pdf}}
\resizebox{\hsize}{!}{\includegraphics{best_fit_taurex_T1H_high_wide.pdf}}
\caption{Best fit models to TRAPPIST-1 h HST WFC3 G141 data using a very low (top) and a high (bottom) resolution with a resolving power of 25 and 70 around 1.4$\mu$m. }
\label{fig:best_fit_resolution}
\end{figure}
Neither stellar contamination nor atmospheric absorption can explain the rise in the transit depth around 1.3 $\mu$m. This is probably due to scattering noise remaining after the extraction and the spectral light curve fitting. By changing the resolution of the data extraction, we investigated if the scattering at 1.3$\mu$m remains significant and if a single narrow band of the spectrum caused this 'feature'. We performed the same data analysis as in Sec. \ref{sec:2.1} using two other binning resolutions with a resolving power of 25 and 70 around 1.4$\mu$m, respectively. We also performed the same retrieval analysis using the primary, the secondary, and the flat-line set-ups on the two spectra. We obtained similar results; the flat-line model is the best fit according to the Bayes factor.
We present in Fig.\ref{fig:best_fit_resolution} the best-fit atmospheric results on the two spectra of TRAPPIST-1 h. The log(E) of the flat line is 47.52 and 163.62 for the very low and the high resolution spectra whereas the log(E) of the primary model reaches 47.04 and 163.15, respectively. Log(E) of the secondary model are also below the one of the flat-line model, that is 47.42 and 163.42. Changing the resolution of the spectrum does not flatten or increase the rise of the transit depth at 1.3$\mu$m and it was recovered in each case. Results from Sec. \ref{sec:2.2} are confirmed while using different resolutions. A flat-line model remains the best-fit to the TRAPPIST-1 h spectrum.
\section{Conclusion}
Terrestrial planets with a secondary envelope are challenging to characterise especially given the low resolution and narrow wavelength coverage of HST. Here, we have presented a transmission spectrum of a 0.7 R${\rm \oplus}$ planet, TRAPPIST-1h, the seventh planet of the highly studied TRAPPIST-1 system. This planet is the furthest and the smallest planet of the system, yet we were able to obtain a spectrum by combining three different HST observations. We cannot make a strong claim from the analysis of the spectrum as it is better fitted using a flat line. However, we were able to rule out with more than 3$\sigma$ confidence a primary clear atmosphere, as for the other TRAPPIST planets. Given these observations, we are not yet able to distinguish between a featureless cloudy H-dominated atmosphere and a clear or cloudy secondary envelope with smaller spectral features. The two models have similar statistical significance and cannot be distinguished from retrieval analysis. We cannot completely rule out the possibility that TRAPPIST-1h has lost its atmosphere over its lifetime either as the evidence for a flat-line model is favoured. We tested secondary clear atmospheric scenarios and found that a CO-rich atmosphere with a volume mixing ratio of 0.2 in an hydrogen atmosphere obtained the best statistical result with a Bayes factor of 1.01 (i.e. a 2.1$\sigma$ detection). Yet this result is not significant enough and is mostly driven by the last points of the spectrum. This could be due to stellar activity even though all the stellar contamination models tested here were not able to reproduce those points and the rise of the transit depth around 1.3$\mu$m. Other absorbing species, such as H$_2$S or H$_2$CO, could also create features around 1.3$\mu$m, but they are unlikely to be produced in the TRAPPIST-1 h atmosphere with such a high level of absorption. The feature is likely caused by either stellar contamination, or by the planet. However, as previously stated in this paper, we cannot find an explanation for it. We note that the while these scatter data points will cause the atmospheric model to be poorly fit, the same is true of the flat and cloudy models. Therefore, as each will feature these points equally poorly, the evidence between the two will be independent of this and so not overly affected. Future observations with the James Webb Space Telescope (JWST) will hopefully remove the ambiguity; however, as shown in Fig. \ref{fig:delta_logE_mmw}, we can rule out clear H/He atmospheres with high confidence. It is then necessary to obtain more data on this planet and on the other six planets of the system to prove the presence of an atmosphere and better constrain the nature of this intriguing planetary system.
\begin{acknowledgements}
This study makes use of observations with the NASA/ESA Hubble Space Telescope, obtained at the
Space Telescope Science Institute (STScI) operated by the Association of Universities for Research in Astronomy. The publicly available HST observations presented here were taken as part of proposal 15304, led by Julien de Wit. These were obtained from the Hubble Archive which is part of the Mikulski Archive for Space Telescopes.
\end{acknowledgements}
\bibliographystyle{aa}
|
1,116,691,498,427 | arxiv | \section{Introduction}
Social segregation is a primary problem for our well-being, and for
the policy-making of our governments. The most basic questions
regarding social segregation concern its quantification, and the
prediction and prevention of its onset and its outcomes. Attempts to
approach the problem from a quantitative viewpoint date back to the
late 1960s, with a model proposed by the economist Thomas
C.~Schelling~\cite{Schelling1971,Schelling1969}. In this model,
individuals are embedded in a two-dimensional lattice, and are
characterized by a threshold ``tolerance'' to other individual
opinions. This model naturally attracted the attention of statistical
physics because of its analogy with Blume-Emery-Griffiths and Potts
models, and more in general with binary mixtures and interfacial
dynamics.
It shows a complex phase diagram, including threshold phenomena (phase
transitions) where opinions separate spatially and may form
patterns~\cite{DallAsta2008,Gauvin2010,Rogers2012,Gauvin2009}.
Schelling's model demonstrates that even mild preferences for a set of
agents for defining themselves as a local minority can produce strong
spatial segregation patterns, challenging the common view that
discrimination is a necessary condition for segregation.
While spatial ``steric'' interactions and dimensionality are very
important in Schelling's model, human interactions can in most cases
be described as
network-like~\cite{Newman2002,Watts1998,Amaral2000,Barthelemy2003a,Barthelemy2011}.
In a situation with (nearly) immutable convictions and limited
tolerance to other opinions, individuals sharing the same conviction
might find themselves severed from society even if their potential for
social interaction is not limited by spatial constraints. Such a
situation is very dangerous for society, for the danger of triggering
self-propelled distortions of reality shared between many individuals.
For example, this is particularly relevant in the on-line world of
social networks. The diffusion of on-line non-intermediated
unverified and polarized contents and the spread of misinformation is
becoming a pressing problem for our society. One of the most relevant
driving forces has been recognised as the echo-chamber
effect~\cite{Sirbu2013,Zollo2015,DelVicario2016a}. It consists in the
formation of segregated clusters of users who share some strong common
opinions, increasingly reinforcing these ideas and thus becoming
impenetrable to news diverging from their point of view.
Thus, another possible approach (relatively less explored) may attempt
to describe segregation using opinion-based network models, such as
the voter model~\cite{Castellano2009,Sood2005,Suweis2012}. The complex
networks literature provides many examples of segregation in the
structure of relationships (from school friendship to value- and
belief-oriented partitioning) empirical
data~\cite{Girvan2002,Newman2004}. However, the literature on complex
networks models focuses mostly on how opinion dynamics is shaped by
network-like human interactions, i.e., on how individuals change their
mind based the opinions of others~\cite{Ben1996,Sood2005,Suweis2012}.
Such a framework is not well-suited to describe segregation, where
precisely the opposite occurs, i.e., human interactions change
following stable ``opinions'', or other more general
individual-specific factors (as it happens in Schelling's model).
Indeed, some of these factors may be very strongly rooted in
individuals, such as convictions, religious and cultural factors, and
even immutable physical or racial features.
A comparativelly smaller thread of
studies~\cite{Holme2006,Castellano2009,Durrett2012,Min2017} has
considered the coevolution of network connections and opinions. In
such models, individuals can both change their mind and change their
connections, and segregated states can emerge, depending on the
intrinsic time scales of these
processes~\cite{Holme2006,Durrett2012}. However, the conditions for
reaching segregated states are not the main focus of these
investigations, which are typically focused on the conditions for
reaching consensus. In order to understand the factors leading to
segregated states, it is important to address the case where node
attributes (convictions) are persistent.
There is very little work in the literature addressing such situation
on networks. A fairly recent study~\cite{Henry2011}, considered the
emergence of segregation in a social network by a model with
continuous opinions and an individual ``aversion bias'' favoring the
severing of connections with increasing difference of opinions, in
favor of random rewiring. They proved the existence of attractor
steady states with given segregation levels that are independent of
initial conditions, and characterized the time scales of convergence
to these states.
However, this study did not address the possibility and existence of
the threshold phenomena that are ubiquitious in Schelling's
model. Such phenomena are important to address, as argued in the
previous paragraphs.
Here, we define an alternative model of segregation on networks based
on \emph{discrete} convictions, and we study it through analytical
calculations and direct simulation. In our model, individuals may
choose to follow other individuals based on sharing the same
conviction, or based on their popularity (regardless of
conviction). The trade-off between these two moves defines a
transition between a well-mixed and a segregated state. A threshold
parameter, analogous (but not equivalent) to the ``tolerance''
parameter in Schelling's model, weighs the two different possible
choices. We analyze this model in the case of binary states of the
agents (two possible convictions, such as Democrats and Republicans),
and we are able to fully characterize the conditions for the emergence
of phase transitions the relaxation time scales of the system in the
segregated and non-segregated phases. Importantly, in order for
transitions to exist, the conviction move has to occur on the same
time scale of the popularity move, regardless of the size of the
community being segregated. Finally, we show that minority
convictions segregate more easily, and we characterize this phenomenon
quantitatively.
\begin{figure}
\centering
\includegraphics[width=0.38\textwidth]{figure1}
\caption{\label{figure:moves} Illustration of the action of the model
basic moves. Nodes represent agents and colors represent
convictions. Edges represent directed social connections (A follows
B if an edge is sent from A to B). The selected edge to be removed
is in both cases $e_{1\to 2}$. In a conviction move, the new target
can be chosen only among the blue nodes (in the sketch this move
creates the edge $e_{1\to 0}$), while in a popularity move the new
target can be chosen regardless of its opinion, so that every node
with an in-degree greater than 0 is a potential candidate (in the
sketch this moves creates the edge $e_{1\to 4}$).}
\end{figure}
\section{Definition of the model}
Our model describes a social network as a directed graph where
individuals (nodes) follow other individual's opinions by sending
directed edges to their corresponding nodes. The initial condition is
a random directed graph $G_0(N,m,h)$ made of $N\in\mathbb{N}$
nodes. Each node has fixed outdegree $m\in\mathbb{N}$. A fraction
$h\in[0;1]$ of individuals hold a certain conviction, which we
identify with the color \emph{red} (as opposed to the probability
$1-h$ of holding the opposite conviction, i.e. being colored in
\emph{blue}). The total number of edges $M=N\cdot m$ defines the size
of our system. The graph is constructed through the associated
adjacency matrix by filling randomly with $m$ ones the matrix rows of
a zero matrix (we exclude the matrix diagonal elements which would
indicate self-edges). As a consequence of this construction procedure,
the in-degrees follow a Poisson distribution with average value $m$ (as
in an Erd\~os-R\'enyi random graph \cite{Erdos1960}).
The network evolves at \emph{fixed} conviction, by choosing at each
step one of two possible rewiring moves (Fig.~\ref{figure:moves})
accordingly to the choice parameter $\varphi\in[0;1]$
:
\begin{itemize}
\item with probability $\varphi$ a \emph{conviction move} chooses
randomly one among all the edges $e_{i\to j}$ between two nodes
holding different convictions (which we will call ``heterogeneous'' edges),
deletes, chooses uniformly a new target node $k$ holding the same
conviction as $i$ and creates a new ``homogeneous'' edge $e_{i\to k}$;
\item alternatively, with probability $1-\varphi$, a \emph{popularity
move} which chooses randomly one edge $e_{i\to j}$ among all the
edges of the network, deletes it, and creates a new edge
$e_{i\to k}$ with a target $k$ chosen among all the nodes with a
preferential attachment criterion, i.e. with a probability equal to
the in-degree of the target node normalized by the total number of
edges $M$.
\end{itemize}
It is important to underline the fact that the opinion move selects
the edge to be removed in the basket of the heterogeneous edges. As it
will be more clear in the following, this choice is essential in order
to obtain a threshold phenomenon for segregation.
We quantify the segregation using as order parameter the total number
of homogeneous edges connecting nodes with the same conviction. In the
initial condition ($t=0$), and for $M$ sufficiently large, the
densities of the four different kinds of edges (red to red, blue to
blue, red to blue and blue to red) are:
\begin{align}
e_0(rr) &=h^2 \nonumber\\
e_0(bb) &=(1-h)^2 \nonumber\\
e_0(rb) &=e_0(br)=h(1-h) \ \ .
\end{align}
More in general, for every step $t>0$, the link densities are
functions of this parameter order parameter. Indeed, since
$\Omega_t:=M(e_t(rr)+e_t(bb))$, one has
\begin{align}
e_t(rr) &=\frac{h^2}{h^2+(1-h)^2} \frac{\Omega_t}{M} \nonumber\\
e_t(bb) &=\frac{(1-h)^2}{h^2+(1-h)^2} \frac{\Omega_t}{M} \nonumber\\
e_t(rb) &=e_0(br)=\frac{M-\Omega_t}{2M} \ .
\end{align}
We define a segregated phase as a state where, for large networks,
typically all the heterogeneous edges disappear, leaving the network
with only edges between like-minded nodes, characterized by a
saturation of the order parameter to the maximum value $\Omega_t=M$.
\section{Results}
\begin{figure*}
\centering
\includegraphics[width=0.6\textwidth]{figure2}
\caption{\label{figure:ph_transition} A threshold phenomenon to a
segregated state appears for a critical value of the choice
parameter $\varphi_c$. \textbf{a)} Evolution of the fraction of
homogeneous links. The plot shows the order parameter normalized by
the total number of edges $M$ plotted against sweeps. The curves are
obtained by simulating the evolution of the same initial random
graph $G_0(N=500,m=5,h=1/2)$ for different values of $\varphi$. For
low $\varphi$, the long-time value of $\Omega_{\infty}(\varphi)$
relaxes to a steady state where the edges connecting nodes with
different colors fluctuate around a finite value, while as $\varphi$
grows, it reaches one (a segragated state) in a finite time. The
right-hand panel shows some illustrative simulation snapshots, where
the network is visualized with a spring model based on shared links.
\textbf{b)} Plot of the mean order parameter at steady state versus
the choice parameter $\varphi$ comparing the analytical results
(solid line) of Eq.~\ref{eq:Omega_SS_phi} with numerical simulations
for different sizes of the network $M$ (symbols). This analysis
supports a segregation transition for $\varphi_c=1/3$ (for $h=1/2$).
\textbf{c)} Fluctuations scale linearly with the size of the
system. Plot of the dispersion of the order parameter from the
simulations in panel b (symbols). As the size of the network grows,
the variability across realizations peaks around the critical value
$\varphi_c=1/3$ reflecting the prediction of
Eq.~\ref{eq:Omega2_SS_phi} (solid line).}
\end{figure*}
\subsection{A transition to a segregated state emerges at a critical
point}
By construction of the model dynamics, conviction moves favor the
transition to a segregated phase, while popularity moves try to
reestablish the disorder and will also affect the in-degree
distribution. Moreover, we expect networks characterized by asymmetric
densities of opinions ($h\neq 1/2$) to reach a segregated phase more
easily.
Starting by the same initial random graph $G_0$, we evolved the
network for different values of $\varphi$ and at each step we recorded
the order parameter $\Omega_t(\varphi)$, starting from initial
conditions with $\Omega_0=1/2$ for $h=1/2$
(Fig.~\ref{figure:ph_transition}a), representing the fraction of
homogeneous edges (connecting individuals with equal convictions). For
low values of $\varphi$, the system does not segregate, but they reach
a balance between popularity- and conviction-based moves. As the value
of $\varphi$ increases, conviction-based moves become increasingly
dominant, and the steady-state value of the order parameter increases
until it reaches the maximum possible value $M$, indicating that
typically the number of heterogeneous edges is negligible compared to
the total number of edges, and the system reaches a segregated
phase. This behavior suggests the existence of a critical value
$\varphi_c$ of the choice parameter, above which the steady state of
the network is always in a segregated phase.
In order to find the critical value of the choice parameter
analytically, we used a mean-field approach, based on an estimate of
the average variation $\Delta\Omega_t$ at every step. Conviction moves
increase $\Omega_t$ by 1, while popularity moves might act differently
depending on the probability of picking an edge of a certain kind, and
also on the kind of the new edge created. The resulting mean-field equation is
\begin{equation}
\Delta\left<\Omega_{t}(\varphi,h)\right> =
\underbrace{\vartheta\varphi}_{\text{conv. move}} +
\underbrace{(1-\varphi)\left[\vartheta
p_{t}^{+}(h)-p_{t}^{-}(h)\right]}_{\text{pop. move}} \ ,
\label{eq:midfield}
\end{equation}
where the Heaviside step function $\vartheta:=\theta(M-\Omega_t)$
excludes forbidden moves once the segregation state is
reached, while $p_t^{\pm}(h)$ are the probabilities of respectively
increasing and decreasing the order parameter with a popularity move.
In the continuum time limit, and for $h=1/2$ (for a more general
derivation for every $h\in[0;1]$ see section \ref{subsec:meanfield})
Eq.~\eqref{eq:midfield} gives the following differential equation for the
average value of the order parameter
\begin{equation}
\partial_{t}\left< \Omega_{t} (\varphi) \right>
=\vartheta\frac{1+\varphi}{2}-(1-\varphi)\frac{1+\vartheta}{2}\frac{\left<
\Omega_{t}(\varphi)\right>}{M} \ .
\label{eq:dt_Omega}
\end{equation}
This equation can be explicitly integrated (for $\varphi\neq1$),
yielding the time dependence for the average value of the
order parameter,
\begin{align}\label{eq:Omega_time}
\frac{\left< \Omega_{t} (\varphi) \right>}{M}
&=
\left[ \left( 1-\frac{1}{2}\vartheta\right) -
\frac{\vartheta}{1+\vartheta}\frac{1+\varphi}{1-\varphi}\right]
e^{-(1-\varphi)\frac{1+\vartheta}{2M}t} +
\nonumber\\
&+\frac{\vartheta}{1+\vartheta}\frac{1+\varphi}{1-\varphi}\ \ .
\end{align}
In the pre-segregation regime (where $\Omega_t<M$ and therefore
$\vartheta=1$) the relaxation is then exponential with
characteristic time
\begin{equation}
\tau_{\Omega}=\frac{M}{1-\varphi}.
\label{eq:tau_Omega}
\end{equation}
Hence, the asymptotic value
\begin{equation}
\frac{\left< \Omega_{\infty} (\varphi)
\right>}{M}=\min_{\varphi\in[0;1)} \left\{
1,\frac{1+\varphi}{2(1-\varphi)}\right\} \label{eq:Omega_SS_phi}
\end{equation}
will be reached for times
$t\gg\tau_{\Omega}$. Fig.~\ref{figure:ph_transition}b compares this
prediction with direct simulations. The model behaves as expected
already for relatively small-sized networks ($M=100$) and gradually
moves towards the predicted curve as the size of the system grows. By
setting $\left< \Omega_{\infty} (\varphi) \right>=1$ in
Eq.~\ref{eq:Omega_SS_phi} and solving for $\varphi$ one finds the
critical value of the choice parameter at which the transition occurs,
which for $h=1/2$ is $\varphi_c=1/3$. This transition has a clear
similarity with second order phase transitions~\cite{Landau1980} ,
because of a discontinuity in the first derivative of $\Omega_t$ with
respect to $\varphi$. The analogy identifies the order parameter
$\Omega$ with the magnetization, while the role of the temperature is
played here by the choice parameter $\varphi$.
The fluctuations of the order parameter also characterize the
transition. These can be estimated by the second cumulant moment
$\text{Var}[\Omega_{\infty}(\varphi)]$. A peak in amplitude of the
fluctuations at the critical value $\varphi_c$ should signal the
transition. In the social segregation interpretation, this means that
the transition to a segregated state is also marked by sudden growth
and shrinkage of its connections to the rest of the world. In order to
access the fluctuations analytically, we explicitly considered the
master equation~\cite{Gardiner1985}). Calling $P_t(\Omega)$ the
probability of having $\Omega$ homogeneous edges at time $t$ the master
equation is defined as
\begin{equation}
\partial_t P_t (\Omega)=\sum_{\Omega'\neq\Omega}
W(\Omega|\Omega')P_t(\Omega')-
W(\Omega'|\Omega)P_t(\Omega) \ ,
\label{eq:ME}
\end{equation}
where $W(\Omega|\Omega')$ are the transition rates of moving from a
network with $\Omega'$ homogeneous edges to a network of $\Omega$ edges, which
for our system (always in the case of $h=1/2$) is
\begin{align}\label{eq:ME_rates}
W(\Omega|\Omega')
&=\delta_{\Omega',\Omega-1}\left[
\varphi+(1-\varphi)\frac{M-\Omega'}{2M} \right] +
\nonumber\\
&+\delta_{\Omega',\Omega+1}(1-\varphi)\frac{\Omega'}{2M}
+\delta_{\Omega',\Omega}\frac{1-\varphi}{2} \ .
\end{align}
In the above equation, the first row describes the contribution of
both the opinion and popularity moves to an increase in $\Omega$,
while the second row describes the contributions of the popularity
move to respectively decrease and keep unaltered the order
parameter. Then we define the factorial moment generating function
\begin{equation}
G(s,t)=\sum_{\Omega=0}^{M}s^{\Omega}P_t(\Omega) \ ,
\label{eq:FGMF}
\end{equation}
where $s\in\mathbb{R}$ is the dual parameter of $\Omega$. Combining
Eqs.~\eqref{eq:ME} and~\eqref{eq:FGMF} (see
Appendix~\ref{subsec:ME_FMGF}) yields the following partial
differential equation,
\begin{equation}\label{eq:FGMF_dyn}
\partial_t G(s,t)=G(s,t)\frac{1+\varphi}{2}\left( s-1 \right)+
\partial_s G(s,t)\frac{1-\varphi}{2M}\left( 1-s^2 \right) \ .
\end{equation}
By evaluating $\partial_s^n[\partial_t G(s,t)|_{s=1}]$ for every
$n\in\mathbb{N}$ we obtain a closed system of time-only differential
equations giving the exact dynamics (including the transient phase) of
all the factorial moments. The first factorial moment coincides with
the average, so we find again Eq.~\ref{eq:dt_Omega}, whereas the
second factorial moment gives $\left < \Omega_t^2 \right > $ and hence
the variance. Taking the long-time limit we obtain an analytical
expression for the fluctuations
\begin{equation}
\frac{\text{Var} [\Omega_{\infty}
(\varphi)]}{M}=
\begin{cases}
\frac{1+\varphi}{4(1-\varphi)} & \text{for } \varphi\leq1/3\\
0 & \text{for } \varphi>1/3
\end{cases}
\label{eq:Omega2_SS_phi}
\end{equation}
Fig.~\ref{figure:ph_transition}c shows that as the size of $M$ (number
of edges) of the network grows, the simulations tend to agree with
this large-$M$ prediction, showing a behavior that resembles that of
the susceptibility in second-order phase transitions, with
fluctuations amplitude scaling linearly in $M$.
By means of the generating function formalism, we can go further and
calculate exactly the stationary solution of the Master Equation
(\ref{eq:ME}) with transition rates given by
Eq. (\ref{eq:ME_rates}). The resulting stationary probability function
$P_{\text{stat}}$ is (see Appendix~\ref{subsec:ME_STATSOL} for
detailed calculations):
\begin{equation}\label{Pstaz}
P_{\text{stat}}(\Omega )=
\frac{2^{-\frac{M (\varphi +1)}{1-\varphi}}
\left(\frac{M (\varphi+1)}{1-\varphi }\right)^{(\Omega)}}
{\Omega !},
\end{equation}
where $x^{(\Omega )}$ is the factorial power of $x$ and it is given by
$\frac{\Gamma (x+1)}{\Gamma (-\Omega +x+1)}$. From Eq.~(\ref{Pstaz})
we can then define the entropy of the system
$S(\varphi)=-\sum_{\Omega=0}^{M\rightarrow\infty}
P_{\text{stat}}(\Omega) \log[P_{\text{stat}}(\Omega )]$ and its
derivative with respect to the choice parameter $\varphi$. As Figure
\ref{figure:entropy-phase-transition} shows, by plotting $S(\varphi)$
and $\partial_{\varphi}S(\varphi)$ we can effectively see that the
system undergoes a genuine phase transition.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figure-samir3.pdf}
\caption{\label{figure:entropy-phase-transition} Entropy is
characterized by a discontinuity in correspondence with the critical
value of the choice parameter $\varphi_c$. A) The entropy of the
system as a function of the order parameter $\varphi$ for different
system size $M$. B) Its derivative with respect to $\varphi_c$. The
dashed gray line represent the predicted critical threshold
$\varphi_c=1/3$.}
\end{figure}
\subsection{Overlap of time scales is necessary for a segregation
transition to exist}
We now discuss more in detail an essential ingredient for a
segregation sharp transition to exist, the fact that the conviction
move occurs on the same time scale of the popularity move, regardless
of the size of heteorogeneous edges in the system. In other words, the
conviction move is realized at each step with probability $\varphi$
drawing directly from the basket of heterogeneous edges in order to
observe the transition.
We can understand this result by considering a similar model in which
the opinion move is, for instance, defined as follows. Select an edge
randomly among all the $M$ edges of the network (rather then from the
basket of the heterogeneous ones) and if the edge is heterogeneous
execute the conviction move, otherwise leave the network unaltered and
move on by executing a new step. In this model the mean-field
equation, Eq.~\ref{eq:midfield} will take an additional term
representing the heterogeneous edge density multiplying the conviction
move term,
\begin{equation}
\Delta\left<\Omega_{t}\right>=\underbrace{\vartheta\varphi\frac{M-\left<\Omega_t\right>}{M}}_{\text{op. move
variant}}+(1-\varphi)\left[\vartheta
p_{t}^{+}(h)-p_{t}^{-}(h)\right] \ .
\label{eq:midfield_variant}
\end{equation}
The critical value $\varphi_c$ is found setting
$\Delta\left<\Omega_{t}\right>$ to zero and the average value of the
order parameter saturates to its maximum value $M$. Substituting these
quantities one immediately finds that the contribution of the opinion
move disappears, leaving us with the equation
$(1-\varphi_c)\left[\vartheta p_{t}^{+}(h)-p_{t}^{-}(h)\right]=0$
which has the only trivial solution $\varphi_c=1$ (that represents a
model in which only opinion based move are executed). In other words,
a segregated phase is found only in the trivial case where the agents
only choose their connections by conviction.
This analysis also gives a general condition for the existence of a
transition, which is that the conviction move has to be such that the
multiplicative factor introduced in the opinion move term in
Eq.~\eqref{eq:midfield_variant} translates into a function
$f(\Omega_t)$ characterized by the condition $f(M)\neq0$).
A possible justification for this forcing in the opinion move can be
found by considering some realistic situations characterized by a
segregation phenomenon driven by strong convictions (ethnicity,
political orientation, religious beliefs, etc.). If an agent is left
only with opposite minded neighbors, it is likely going to be the
first one to decide to sever a connection and rewire with someone with
the same conviction. For this reason, we believe that direct targeting
of heterogeneous connection in an environment of strong convictions
might be a realistic assumption.
\subsection{The popularity move broadens the in-degree distribution in
the unsegregated phase, but does not affect the transition point. }
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{figure4}
\caption{\label{figure:in-degrees} Preferential attachment from the
popularity move broadens hte in-degree distribution. \textbf{a)}
Empirical survival distribution function (ESDF) of the in-degree
distributions of networks evolved for different values of
$\varphi$. The plot was obtained by evolving an initial random graph
$G_0(N=100,m=5,h=1/2)$ for $t=10^6$ steps (the in-degrees are
normalized with respect to the total number of edges $M=500$). The
broadening of the distribution indicates the increasing presence of
bigger attractors in the evolved networks. \textbf{b)} Two
different trends for the Fano factor of the in-degrees are observed
in the regions below and above the segregation transition. The plot
reports the Fano factor of the in-degrees distributions shown in
panel a versus the choice parameter $\varphi$. In the region above
the critical value of the choice parameter $\varphi_c=1/3$ the
deviation from a Poisson distribution ($F(k_{in})=1$) is
small, while the unsegregated region shows a
super-exponential departure (the vertical axis is in log-scale)
towards larger dispersions as $\varphi$ decreases.}
\end{figure}
We proceed by considering the role of the popularity move in setting
the in-degree distribution and in the segregation transition. The
initial random graph $G_0(N,m,h)$ has by definition
Poisson-distributed in-degrees $k_{in}$ for large $N$,
with a mean equal do the fixed outdegree of every node of the network
$m$. As the network evolves, the distribution of the in-degrees
changes at each popularity move, because the most popular nodes are
more likely to be chosen as a target for the newly created edges. This
determines a departure from the initial distribution towards
heavier-tailed distributions, in analogy with the ``rich gets richer''
principle that usually characterizes social
networks~\cite{Castellano2009}.
In order to properly characterize this behavior evaluated the
empirical survival distribution function (ESDF) of the in-degree
distributions of evolved graphs $G_t$ for different values of the
choice parameter. The ESDF indicates the probability of observing a
node $i$ with in-degree $k_{in}(i)$ greater then a certain value
$k_{in}$, and is defined as
\begin{equation}
\text{ESDF}(k_{in})=\frac{1}{M}\sum_{i=0}^{M}\theta(k_{in}-k_{in}(i))
\ ,
\end{equation}
Fig.~\ref{figure:in-degrees}a shows that when $\varphi=1$ the initial
distribution is unaltered (the dashed line represents the distribution
for the initial random graph $G_0$), but as $\varphi$ decreases the
in-degree distributions take increasingly heavier tails.
The same phenomenon can be quantified by a single broadness parameter
such as the Fano factor of the in-degrees $F(k_{in})$, defined as
\begin{equation}
F(k_{in})=\frac{\text{Var}[k_{in}]}{\left< k_{in} \right>} \ .
\end{equation}
This parameter is 1 for a Poisson distribution, whereas greater values
indicate larger dispersion. Fig.~\ref{figure:in-degrees} shows this
parameter plotted as a function of the choice parameter $\varphi$. The
Fano Factor increases as popularity-based moves become more probable
(as $\varphi$ goes to zero). Moreover two different trends appear to
characterize the region below and above the critical value
$\varphi_c=1/3$.
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{pop_random5}
\caption{\label{figure:in-degrees_no_pop}. Skewed node popularity does
not affect segregation. \textbf{ab)} Same plots as
Fig.~\ref{figure:in-degrees}, for a model in which the popularity
move is changed with a rewiring on a uniformly chosen random
node. This model shows the same phase transition as the original one
(and in particular the plots in Fig.~\ref{figure:ph_transition}bc
are identical), but the transition is not accompanied by changes in
node degree.}
\end{figure}
Finally, although we found that popularity-based rewiring increases
the dispersion of social connections in the unsegregated regime, this
preferential attachment ingredient does not affect the segregation
transition in any way, as we have verified by substituting
popularity-based rewiring with random rewiring in our simulations
(Fig.~\ref{figure:in-degrees_no_pop}).
Although one may expect that the presence of popular individuals may
help avoiding the emergence of segregation due to their capacity of
attracting new nodes regardless of their opinion, this does not happen
in this model. The reason is easily understood from
Eq.~\eqref{eq:midfield} and \eqref{eq:midfield_variant}, which govern
the dynamics of the order parameter, where it is clear that the
in-degree distribution never comes into play.
\subsection{Minority convictions segregate more easily}
\label{sec:min_conv}
\begin{figure}
\centering
\includegraphics[]{figure6}
\caption{\label{figure:critical_h} Minority convictions tend to
segregate more easily. \textbf{a)} Average value of the order
parameter $\left< \Omega_t \right>$ in networks evolved from initial
networks $G_0(100,5,h)$ for different values of $h\leq1/2$ (the
results for $h>1/2$ are the same due to the symmetry $h\to1-h$). As
the the density of nodes holding a certain conviction decreases, the
networks will reach a segregated phase for lower values of
$\varphi$. \textbf{b)} Simulations confirm the analytical
prediction for the critical points of the model. The critical points
(symbols) are ectracted from the curves in panel a, for different
values of $h$, and compared with the prediction described by
Eq.~\ref{eq:critical_h} (solid line).}
\end{figure}
The results presented up to this point were obtained under the
hypothesis of equally represented convictions condition ($h=1/2$). A
more generic case describes minority versus majority convictions,
characterized by different values of $h$. The differences from the
symmetric case concern both the characteristic time $\tau_{\Omega}$
needed to reach the steady state and the critical value $\varphi_c$ at
which the transition to a segregated phase occurs.
In order to study this asymmetric situation we write a mean-field
equation valid for every value of $h\in[0,1]$. Starting from
Eq.~\ref{eq:midfield}, we just need to specify how the terms
$p^{\pm}_t(h)$ depend on $h$ (see section \ref{subsec:meanfield}),
\begin{align}
p_t^+(h) &=\frac{M-\left<\Omega_t\right>}{2M} \nonumber\\
p_t^-(h) &=\frac{h(1-h)}{h^2+(1-h)^2}\frac{\left<\Omega_t\right>}{M} \ .
\end{align}
The resulting mean-field equation can be integrated in the continuum
limit as in the symmetric case $h=1/2$, yielding the dynamics of the
average value of the order parameter. The critical value $\varphi_c$
on the asymmetry $h$ is obtained again by imposing the segregation
regime conditions $\Delta\left<\Omega_t\right>=0$ and
$\left<\Omega_t\right>=M$. Solving for $\varphi$ gives
\begin{equation}
\varphi_c(h)=\frac{h(1-h)}{1-h(1-h)}
\label{eq:critical_h}
\end{equation}
for the critical value. This relation satisfies the red-blue symmetry
$\varphi_c(h)=\varphi_c(1-h)$ with maximum value $\varphi_c(1/2)=1/3$
(as in Eq.~\ref{eq:Omega_SS_phi}) for the symmetric
case. Fig.~\ref{figure:critical_h}b compares the predicted critical
point from Eq.~\ref{eq:critical_h} to simulations of evolved networks
for different values of $h$ Fig.~\ref{figure:critical_h}a. This
analysis shows that a situation characterized by a minority conviction
favors segregation for lower values of the choice parameter,
indicating that the symmetric situation is the one in which
segregation can be more easily avoided (the situation is analogous to
the miscibility gap for phase segregation in a binary mixture).
The characteristic duration of the transient before a steady state is
reached is also affected by the presence of a minority conviction. The
solution of the mean-field equation gives
\begin{equation}
\tau_{\Omega}(h)=\frac{2M\left[ h^2+(1-h)^2 \right]}{1-\varphi} \ ,
\end{equation}
i.e., the characteristic relaxation time will increase for asymmetric
convictions. This time scale is important in cases where the
segregation dynamics competes with the spreading of
consensus~\cite{Holme2006,Durrett2012}.
\subsection{Scale-invariance close to the transition}
The limit of large system size, $M\rightarrow \infty$, is better
analyzed in terms of a finite-size scaling ansatz, typical of critical
phenomena \cite{Fisher1967,hahne2006critical}. We define the
normalized choice parameter
\begin{equation}
t = \frac{\varphi - \varphi_c}{\varphi_c} \ .
\end{equation}
and the intensive order parameter
\begin{equation}
m = \frac{M-\Omega_{\infty}}{M}
\end{equation}
so that
\begin{equation}
\langle m \rangle= 1-\frac{\langle \Omega_{\infty} \rangle}{M}=
\langle \frac{M-\Omega_{\infty}}{M} \rangle
\end{equation}
and we assume that $\langle m \rangle$, which in principle depends on
both $M$ and $t$ separately, is an homogeneous function of $t$ and a
suitable power of $M$, that is
\begin{equation}
\langle m \rangle =
|t|^{\beta} \tilde{f}_1(M^{y} t)
\label{eq:scaling_omega}
\end{equation}
in the large (small) $M$ ($t$) limit with $M^{y} t$ fixed. $y$ and
$\beta$ are exponents that are expected to be independent of the
microscopic details of the dynamical model, characterizing the
transition point, while $f$ is a scaling function, which might depend
on the model specificities. Since we expect that $m$ is non-zero
(zero) for $t<0$ ($t>0$) the scaling function $f$ should behave
asymptotically as
\begin{equation}
\lim_{x\rightarrow +\infty} \tilde{f}_1(x) = 0, \qquad
\lim_{x\rightarrow -\infty}\tilde{f}_1(x) = \textrm{constant} > 0
\end{equation}
In order to estimate the two scaling exponents $\beta$ and $y$, we plot
$m |t|^{-\beta}$ versus $M^{y} t$ and determine the exponents so that
the best collapse of the different curves is obtained. Indeed one
should obtain a different curve for each value of $M$ as $t$ varies
and this is what we observe for generic pair $\beta$ and $y$. However
for $\beta = 1$ and $y = 1/2$ the various curves collapse in a range
of $x \equiv M^{y} t$ that increases as $M$ becomes larger and larger
as Fig.\ref{figure:collapse}, panel (a), shows.
The same analysis leads to the following scaling ansatz for the
variance of $m$ (corresponding to $\text{Var}[\Omega_{\infty}]/M^2)$
in terms of the original extensive order parameter):
\begin{equation}
\text{Var}[m] = t^{2} \tilde{f}_2(M^{1/2} t)
\label{eq:scaling_varomega}
\end{equation}
and the corresponding collapse is shown in Fig.\ref{figure:collapse},
panel (b). Both scaling Eqs.(\ref{eq:scaling_omega}) and
(\ref{eq:scaling_varomega}) are captured by the more general scaling
ansatz of the distribution function of $m$
\begin{equation}
P(m, t, M) = |t|^{-1} \tilde{P}(mt^{-1},\ M^{1/2} t)
\label{eq:pdf}
\end{equation}
\begin{figure}
\centering \includegraphics[width=0.48\textwidth]{collapses7}
\caption{\label{figure:collapse} The fraction of homogeneous edges
and its variance obey scaling. \textbf{a)} Scaling collapse for
the fraction of homogenous edges. \textbf{b)} Scaling collapse
for the variance. The $x$ and $y$ axes of both plots compare the
functions predicted by Eqs. \ref{eq:scaling_omega} and
\ref{eq:scaling_varomega}. The symbols correspond to data points
from simulations at different network size above and below the
segragation transition point.}
\end{figure}
\subsection{A model with pure intra-specific aversion leads to an
equivalent segregation threshold behavior.}
Motivated by the literature on segregation models based on aversion
between unlike individuals~\cite{Schelling1971,Henry2011}, we asked
whether the same threshold phenomenon observed in our model could be
present in case of conviction moves that were based purely on aversion
bias.
To this end, we defined a model variant where the conviction move
(with probability $\varphi$) chooses randomly one heterogeneous edge,
between two nodes holding different convictions and rewires it to a
random node. In this variant, the popularity move (with probability
$1-\varphi$ at each step) remains the same. Under this variant,
Eq.~\eqref{eq:midfield} becomes
\begin{equation}
\Delta\left<\Omega_{t}(\varphi,h)\right> =
\underbrace{\vartheta\frac{\varphi}{2}}_{\text{conv. move}} +
\underbrace{(1-\varphi)\left[\vartheta
p_{t}^{+}(h)-p_{t}^{-}(h)\right]}_{\text{pop. move}} \ ,
\label{eq:aversion}
\end{equation}
immediately leading to the expression,
\begin{equation}
\frac{\left< \Omega_{\infty} (\varphi)
\right>}{M}=\min_{\varphi\in[0;1)} \left\{
1,\frac{1}{2(1-\varphi)}\right\} \label{eq:Omega_SS_phi2}
\end{equation}
for the mean fraction of heterogeneous edges.
By setting $\left< \Omega_{\infty} (\varphi) \right>=1$ in
Eq.~\ref{eq:Omega_SS_phi2} and solving for $\varphi$ one finds again
the critical value, which for $h=1/2$ is $\varphi_c=1/2$.
An analogous reasoning
can be followed for solving for the higher moments of the distribution
of $\Omega$.
Fig.~\ref{figure:aversion} shows that direct simulations of the
aversion bias model are fully in line with these theoretical
predictions. Thus, we conclude that aversion alone is sufficient to
produce a sudden segregation threshold.
\begin{figure}
\centering \includegraphics[width=0.48\textwidth]{aversion8}
\caption{\label{figure:aversion} The sudden transition
to a segregated state remains in a model with aversion bias
only. \textbf{a)} Mean order parameter at steady state versus
the choice parameter $\varphi$ comparing theory (solid line) with
numerical simulations for different sizes of the network $M$
(symbols). This analysis supports a segregation transition for
$\varphi_c=1/2$ (for $h=1/2$). \textbf{b)} The dispersion of the
order parameter (symbols) shows the same behavior as the standard
model (compare with Fig~\ref{figure:ph_transition}).}
\end{figure}
\section{Discussion and Conclusions}
Social segregation is ubiquitous in our society, and manifests itself
as fragmentation of social networks at all scales, in countries,
cities, schools, firms, governmental agencies, etc. Its consequences
may lead to a wide range of nefastous phenomena ranging from
inefficient planning to war.
It is driven by diverse and enormously complex sociological, cultural,
environmental and economic dilemmas, which are unlikely to be solved
in the near future.
However, since the pioneering work of
Schelling~\cite{Schelling1969,DallAsta2008,Gauvin2010,Henry2011}
there is increasing agreement that there may be common quantitative
traits in the ``macroscopic'' dynamics of segregation that emerge from
this complexity.
A quantitative understanding of the consequences of such simple
features on the dynamics of a social network may be important to
develop efficient estimators to be used in real-life examples to
detect and prevent segregation phenomena.
The framework developed here shows that complete segregation in a
network setting without any spatial aspects can emerge as a threshold
phenomenon that corresponds to a genuine phase transition. Close to
such transition point, small perturbations of the system can cause
very large rearrangements in the state.
Importantly, we have shown that such transition point is scale
invariant, hence ``universal'' in the statistical physics sense. This
supports the hypothesis that close to this critical point more
detailed descriptions of social interactions are not necessary, since
a wide class of models may behave similarly.
We can also parallel this model with available physical models for the
separation of phases and mixtures. For example, binary mixtures can
be described in a coarse-grained way as a set of particles of two
kinds filling a cubic lattice, with an energy cost for particles of
one kind sitting next to particles of the other kind. This system
(equivalent to an Ising model) shows a spatial phase separation when
temperature is lowered. Contrary to this case, in our model set on a
network a concept of distance is missing, since all individuals can
potentially interact with any other agent in each move.
However, we can parallel our results to a variant of the above model
where instead of the usual ``local'' fraction of lattice sites
occupied by each kind of particle, we write the free energy in terms
of the parameter used here, i.e., the fraction of homogeneous edges
$e_h = - \Omega/M$. The energetic term is simply $ -\chi e_h $. In
order to write the entropy, we consider the network as a gas of edges
formed by connecting nodes. We compute the number of ways to assign
$\Omega$ edges out of $M$, considering that each edge is spurious if
two colors of the same kind are selected. The resulting free energy
is $ \beta F = e_h \log(e_h)+ (1-e_h) \log(1-e_h) - e_h \chi
$. Minimizing this free energy and comparing with the equations
governing our model shows that they are different, and our model
cannot be reconducted to this simple case. The question remains open
on whether there is a simple equilibrium model recapitulating the
phase-separation behavior shown by our segregation model.
Segregation in social networks may be driven by both homophyly (the
choice of social interactions with like individuals) and aversion.
These ingredients are mixed in different proportion in the existing
literature. Our basic model contains both, since in the
conviction-based rewirings interactions between dissimilar partners are
rewired in favor of homogeneous ones.
Schelling's model~\cite{Schelling1971} shows that aversion from
dissimilar network partners alone, coupled with a random selection of
new partners, may be sufficient to induce segregation. Our analysis of
a model variant where the conviction-based rewiring is based on pure
aversion supports this conclusion. Indeed, this variant shows the same
type of threshold phenomenon, in full quantitative agreement with the
main model. The (expected) quantitative change is that in the case of
pure aversion the transition point is shifted to higher values of the
choice parameter $\varphi$, compared to the case where both aversion and
homophyly are in place.
Overall, our analysis supports the conclusion that whether
conviction-based rewiring is based on aversion or homophyly is not a
key ingredient for the existence of a segregation threshold.
Instead, the important feature to determine a threshold phenomenon for
segregation is that the the conviction-based rewiring of the network
(based on aversion or homophyly, or both) occurs on the same time
scale of the popularity-based rewirings (i.e. the establishment of
social interactions that are non-discriminant). In the alternative
scenario in which, e.g., each kind of rewiring occurs proportionally
to the number of extant interactions, segregation occurs smoothly. In
such situation, at all levels of the bias in establishing interactions
(quantified by the choice parameter $\varphi$) the network maintains a
finite fraction of interactions between dissimilar individuals.
\begin{acknowledgments}
The authors would like to thank Mirta Galesic for useful feedback,
and Alessandro Civeriati, Andrea Possenti and Sara Cerioli for
preliminary work on this project.
\end{acknowledgments}
|
1,116,691,498,428 | arxiv | \section{Introduction}
\label{introduction}
\subsection{Motivations and overview}
Theoretical studies on black holes in asymptotically anti-de Sitter
spacetimes have attracted substantial attention since the advent of the
anti-de Sitter/conformal field theory (AdS/CFT) correspondence
\cite{maldacena1,gubser1,witten1}. In particular, the quasinormal-mode (QNM)
spectra of various types of asymptotically AdS black holes have been analyzed
since then (see Refs. \cite{wang1,govi1,wang2,zhu,wang3,konoplya1, konoplya2,
starinets1,aros,musiri1,musiri2,crisostomo,fernando,konoplya3,siopsis1,wang4,
giammatteo1,jing1,maeda,siopsis2,zhang,zhang2,zhidenko,rao,kout1,amado,aliev}
for a sample).
According to the AdS/CFT correspondence, an asymptotically AdS black hole is,
in the CFT side, associated to a system in thermal equilibrium whose
temperature is the Hawking temperature of the black hole. In such a context,
blackhole perturbations correspond to small deviations from equilibrium of
the CFT thermal system, and the characteristic damping time of perturbations,
which is given by the inverse of the imaginary part of the fundamental QNM
frequency, is a measure of the dynamical timescale of approach to thermal
equilibrium of the corresponding conformal field theory \cite{horowitz1}.
The literature on QNM of AdS black holes includes studies taking into account
a variety of different aspects such as the topology of the event horizon, the
number of dimensions of the spacetime, the particular type of perturbation
fields considered, and also the special parameters which characterize each
different black hole itself. Each one of these variant properties reflects on
the dual CFT. For instance, assuming the $(3+1)$-dimensional AdS spacetime
contains a plane-symmetric black hole, then the holographic field theory is
defined over the $(2+1)$-dimensional Minkowski spacetime, which is the
conformal boundary of the bulk AdS spacetime. Moreover, different blackhole
parameters characterize different dual plasmas in the CFT side, and different
equilibrium states of such systems at the boundary.
An important issue in the study of the vibrational modes of black holes is
the choice of appropriate boundary conditions. In the case of asymptotically
flat spacetimes, the solutions to the wave equations governing linear
perturbations are, near the boundaries, given by plane wave functions. QNM
are then defined as solutions which satisfy physically well motivated
boundary conditions, namely, purely ingoing waves at the horizon and purely
outgoing waves at infinity (see Refs. \cite{kokkotas, nollert} for reviews).
For anti-de Sitter black holes, on the other hand, the condition at the
future horizon is the same as for asymptotically flat spacetimes, but now
there are no natural conditions to be imposed on the perturbation variables
at the AdS infinity. These can be Dirichlet, Neumann, or Robin boundary
conditions, depending on whether it is required that the field perturbations,
their derivatives or a combination of both vanish at the AdS boundary,
respectively. In the study of the evolution of a massless scalar field in
$(3+1)$-, $(4+1)$-, and $(6+1)$-dimensional Schwarzschild-AdS spacetimes,
Horowitz and Hubeny \cite{horowitz1} computed the corresponding
quasinormal-mode
spectra by imposing Dirichlet boundary conditions on such a field at
infinity. This option was well justified in that context, since by writing
the radial part of the Klein-Gordon equation in a Schr\"odinger-like form,
the resulting effective potential diverges at that boundary. The same
boundary condition was used to study massless scalar and electromagnetic
perturbations of $(2+1)$-dimensional Ba\~{n}ados-Teitelboim-Zanelli (BTZ)
black holes \cite{banados}. For BTZ black holes, an analytical closed form
for the quasinormal frequencies was derived \cite{cardoso1}, and it was
verified that the quasinormal frequencies correspond exactly to the poles of
retarded correlation functions in the dual $(1+1)$-dimensional CFT
\cite{birm1}. It was also suggested in Ref. \cite{son1} that the relation
between quasinormal modes and singularities of correlation functions should
also hold for scalar fields in higher-dimensions, as far as the frequencies
are computed by imposing Dirichlet boundary conditions on such fields at AdS
infinity.
In the meantime, two fundamental difficulties arise when considering
gravitational and/or electromagnetic perturbations of AdS black holes,
particularly in higher dimensional spacetimes. The first problem is related
to the arbitrariness in the choice of gauge-invariant perturbation fields. In
fact, there is an infinity of gauge-invariant combinations of metric (or
vector potential) fluctuations that can be used as fundamental variables
governing the gravitational (or electromagnetic) perturbations. The second
problem is related to the ambiguity in defining appropriate boundary
conditions for the quasinormal modes. A traditional way to face such
arbitrariness is opting for master variables that lead to equations
generalizing those for perturbations in asymptotically flat spacetimes. That
is to say, variables are chosen in such a way to put the radial part of the
fundamental equations into a Schr\"odinger-like form. From now on, the
corresponding master variables shall be called the Regge-Wheeler-Zerilli
(RWZ) variables.\footnote{In the first study of gravitational QNM in AdS
spacetimes, Cardoso and Lemos \cite{cardoso2} used the same kind of variables
as the early works in asymptotically flat spacetimes by Regge and Wheeler
\cite{reggewheeler}, and by Zerilli\cite{zerilli}.}
With such a choice of variables, it was investigated gravitational and/or
electromagnetic perturbations of the Schwarzschild-AdS
\cite{cardoso2,cardoso3,konoplya4,cardoso4,lopez,musiri3,friess1},
Reissner-Nordstr\"om-AdS \cite{berti1,natario,shu}, and Kerr-AdS
\cite{giammatteo2} black holes, as well as the perturbations of black holes
with non-spherical topologies \cite{birm2,kout2}, including the
plane-symmetric ones \cite{cardoso5,miranda1,miranda2}. Analogously to the
massless scalar field case, in all of these works the quasinormal modes were
computed by imposing Dirichlet boundary conditions on the master fields at
infinity. Alternative boundary conditions for the same
Regge-Wheeler-Zerilli variables have been discussed in Refs.
\cite{moss,micha}.
A different route was taken by N\'u\~{n}ez and Starinets \cite{nunez}, who
defined the quasinormal frequencies of a perturbation in an asymptotically
AdS spacetime as ``the locations in the complex frequency plane of the poles
of the retarded correlator of the operators dual to that perturbation''. To
compute the real-time correlation functions, they suggested using the
Lorentzian AdS/CFT prescription of Refs. \cite{son1, herzog1}.
The quasinormal-mode definition supplied by N\'u\~{n}ez and Starinets was
explored in Ref. \cite{kovtun1}, where a new set of
fundamental variables was introduced to study electromagnetic and
gravitational perturbations of $(4+1)$-dimensional plane-symmetric black
holes
(or black branes, for short). It was shown there that the imposition of
Dirichlet boundary conditions on such a new set of gauge-invariant variables
at infinity leads exactly to the quasinormal frequencies
associated to the corresponding black branes. In the present work these
kind of fundamental variables shall be called the Kovtun-Starinets (KS)
variables.
An important consequence of the N\'u\~{n}ez-Starinets approach \cite{nunez}
is that the resulting quasinormal-mode spectra present a set of dispersion
relations, here called hydrodynamic QNM, that behave like diffusion, shear,
and sound wave modes in the long-wavelength, low-frequency limit
\cite{kovtun1}. These results are totally consistent with what is expected
from the CFT point of view, and they provide a non-trivial test of the
AdS/CFT correspondence. It is also worth noticing that neither the
electromagnetic diffusion mode nor the gravitational sound wave mode are
obtained by imposing Dirichlet boundary conditions on the RWZ
master variables. For Schwarzschild-AdS and topological-AdS
$(3+1)$-dimensional black holes, it was only possible to obtain sound wave
modes in the gravitational quasinormal spectra by requiring that a specific
combination of the master field and its derivative vanishes at infinity
\cite{micha,siopsis3,alsup}.
\subsection{The present work}
\subsubsection{General procedure}
In this work the definition of QNM given by N\'u\~nez and Starinets
\cite{nunez} is applied to compute the quasinormal frequencies associated to
electromagnetic and gravitational perturbations of $(3+1)$-dimensional
plane-symmetric AdS black holes. The overall procedure is similar to that of
Ref. \cite{kovtun1} and consists of the following steps:
\begin{itemize}
\item[(1)] Initially the translation invariance of the static plane-symmetric
AdS spacetimes is used to Fourier transform the fluctuation fields with
respect to time and to the two Cartesian coordinates $(x,y)$ of the plane.
\item[(2)] With the spatial wave vector chosen to be in the $y$-direction,
both the electromagnetic and the gravitational perturbation fields are
separated into two sets according to their behavior under the transformation
$x\rightarrow -x$: odd (axial, or transverse), and even (polar, or
longitudinal) perturbations.
\item[(3)] Each sector of perturbation fields is governed by a set of
linearized differential equations. In all of the cases studied here, the
complete set of perturbation equations can be decoupled in order to obtain a
unique second-order differential equation, which is the fundamental equation
of that perturbation sector. The fundamental equations are written in
terms of gauge-invariant combinations of the perturbation fields, extending
the original definitions of Kovtun-Starinets variables \cite{kovtun1} to
$(3+1)$-dimensional spacetimes.
\item[(4)] Then, the standard AdS/CFT prescription of Ref. \cite{son1} is
applied to express the real-time $R$-symmetry current and stress-energy
tensor correlators in terms of quantities which represent the asymptotic
behavior of perturbations near the AdS-space boundary. Such a procedure
shows that the imposition of Dirichlet boundary conditions on
Kovtun-Starinets variables at infinity leads to the poles of the CFT
correlation functions, and therefore, according to the N\'u\~{n}ez-Starinets
definition of QNM, to the quasinormal spectra of the plane-symmetric AdS
black holes.
\item[(5)] With well defined boundary conditions and a set of decoupled
fundamental equations, the hydrodynamical limit of the QNM spectra is then
analyzed. This limit is reached for perturbation modes in which the frequency
and the wavenumber are much smaller than the Hawking temperature of
the black hole.
\item[(6)] The last step is numerically compute the electromagnetic
and gravitational quasinormal dispersion relations for different
blackhole parameters. For such a purpose, the Horowitz-Hubeny
method \cite{horowitz1}, which reduces the problem of finding
QNM frequencies to that of obtaining the roots of infinite polynomial
equations, is used.
\end{itemize}
\subsubsection{Main results}
Among the new results found in the present work, it is worth
mentioning the following ones.
\begin{itemize}
\item First, the derivation of the electromagnetic diffusion mode and the
gravitational sound wave mode is performed by means of a traditional QNM
calculation. These modes were earlier obtained by Herzog \cite{herzog2,
herzog3}, who utilized the AdS/CFT prescription \cite{son1} to directly
compute the hydrodynamic limit of the CFT $R$-symmetry current and
stress-energy tensor correlators.
\item Second, it is found that the procedure of imposing Dirichlet boundary
conditions on the gauge-invariant KS variables breaks the
isospectrality between the axial and polar electromagnetic QNM, that follows
from RWZ variables. As a result, the polar electromagnetic
quasinormal modes are totally new, since the KS variable with
Dirichlet boundary condition yields a different spectrum when compared to the
RWZ variable with the same kind of boundary conditions. Regarding
to the electromagnetic axial perturbations, the dispersion relations found
here enlarge previous results of Ref. \cite{cardoso5}.
\item Third, it is found in addition that the complete spectra of
electromagnetic QNM present a tower of purely damped modes which tend to
Matsubara frequencies characteristic to bosonic systems in the
long-wavelength regime. However, these quasinormal modes do not exist for all
wavenumbers. In fact, there is a saturation value for the wavenumber above
which the electromagnetic purely damped modes disappear.
\item Fourth, another result to be mentioned is the difference between the
spectrum of the gravitational polar perturbations, computed by using the
KS variable, and that obtained using the RWZ master
variable \cite{miranda1}. The differences are specially significant when the
fluctuation wavenumber is of the same order of the magnitude of the
blackhole temperature.
\item And last but not least, the dispersion relations calculated here
complete the previous results for axial gravitational QNM of
$(3+1)$-dimensional plane-symmetric AdS black holes \cite{cardoso5,miranda1}.
\end{itemize}
\subsubsection{Structure of the paper}
The layout of the present article is as follows. Sect. \ref{m2branes}
contains a brief summary of the relation between the plane-symmetric
$\mbox{AdS}_{4}$ black holes and the eleven-dimensional supergravity solution
associated with a stack of $N$ M2-branes. In the sequence a detailed study of
the electromagnetic quasinormal modes is performed (Sect. \ref{qnm-eletro}):
The basic equations are obtained in Sect. \ref{flut-eletro} and the
connection between the blackhole perturbations and the CFT $R$-symmetry
currents is explored in Sect. \ref{green-electro}; the hydrodynamic modes of
the electromagnetic perturbations are studied in Sect.
\ref{hydro-eletro}, and the general dispersion relations of
electromagnetic QNM are reported in Sect. \ref{dispersion-eletro}. The
gravitational quasinormal modes are studied in Sect. \ref{grav-qnm}:
The basic equations are obtained in Sect. \ref{flut-grav};
\ref{green-gravit} is devoted to investigate the relation between the
gravitational QNM and the stress-energy tensor correlators in the
holographic CFT; the hydrodynamic modes of the gravitational
perturbations are studied in Sect. \ref{hydro-grav}, and the numerical
results for the dispersion relations of the remaining gravitational QNM
are presented in Sect. \ref{dispersion-gravit}. The article is completed, in
Sect. \ref{consid-final}, with the analysis and interpretation of the main
results.
\subsection{Notation and conventions}
Natural units are going to be used throughout this paper, i.e., the speed of
light $c$, Boltzmann constant $k_B$, and Planck constant $\hbar$ are all set
to unity, $c=k_{B}=\hbar=1$. Regarding to notation, capital Latin indices
$M,\,N,\,...$ vary over the coordinates of the whole AdS spacetime, while
Greek indices $\mu,\,\nu,\,...$ label different coordinates at the boundary,
and small Latin indices $i,\,j,\,...$ vary only over the spacelike
coordinates at the boundary. The convention for the metric signature and for
all the definitions of curvature tensors follow Ref. \cite{misner}.
\section{M2-branes and the plane-symmetric black holes}
\label{m2branes}
\subsection{The background spacetime}
Since the QNM definition of N\'u\~{n}ez-Starinets \cite{nunez}, that is
adopted in this work, makes heavy use of the relation between AdS black holes
and conformal field theories at finite temperature, it becomes important to
review here how the plane-symmetric AdS$_4$ black holes arise in the context
of the AdS/CFT conjecture.\footnote{The brief summary presented in this
section is based on material found in Refs. \cite{herzog2, aharo, herzog4}.}
A fundamental role in the AdS/CFT correspondence\footnote{The interest here
is the AdS$_4$/CFT$_3$ correspondence.} is played by
extended two-dimensional objects known as M2-branes \cite{townsend}.
The world-volume theory
of $N$ M2-branes is a $(2+1)$-dimensional non-Abelian Yang-Mills theory
which presents $\mathcal{N}=8$ supersymmetries in addition to a
$SU(N)$ gauge group. The coupling constant of the theory
flows to strong coupling in the infrared limit, and it is believed that
the flow is to an infrared-stable fixed point that describes
a superconformal field theory \cite{seiberg}.
This CFT also has an emerging $R$-charge symmetry which
is expanded to $SO(8)$.
From the supergravity point of view, a stack of $N$ M2-branes
is described by a nonextremal solution to the
supergravity equations of motion,
characterized by the metric \cite{herzog2, horowitz2, itzhaki}
\begin{equation}
ds^2=H^{-2/3}(\widetilde{r})\left[-\mathfrak{h}(\widetilde{r})dt^2+dx^{2}+
dy^{2}\right]+H^{1/3}(\widetilde{r})\left[\mathfrak{h}^{-1}(\widetilde{r})
d\widetilde{r\,}^{2}+\widetilde{r\,}^2 d\Omega_{7}^{2}\right],
\label{metM2brana}
\end{equation}
where
\begin{equation}
H(\widetilde{r})=1+\left(\frac{R}{\widetilde{r}}\right)^{6}
\qquad\mbox{and}\qquad\mathfrak{h}(\widetilde{r})=1-
\left(\frac{\widetilde{r}_{0}}{\widetilde{r}}\right)^{6},
\label{hdef1}
\end{equation}
and by a four-form field whose dual Hodge is given by
\begin{equation}
\star F_{4}=F_{7}=6R^{6}
\mbox{Vol}(S^{7}){\boldsymbol{\varepsilon}},
\end{equation}
where $\boldsymbol{\varepsilon}$ stands for the Levi-Civita
tensor on $S^{7}$.
According to the AdS/CFT correspondence \cite{maldacena1,gubser1,witten1},
the $(2+1)$-dimensional $\mathcal{N}=8$ CFT is dual to M-theory on the
background spacetime \eqref{metM2brana}. Furthermore, the quantization
condition on the ${F}_{4}$ flux connects the parameter $R$ to the number of
branes $N$ \cite{klebanov1}:
\begin{equation}
R^{9}\pi^{5}=N^{3/2}\kappa_{11}^{2}\sqrt{2},
\label{relparam}
\end{equation}
where $\kappa_{11}$ is the gravitational coupling strength in
$(10+1)$-dimensional supergravity.
In the large $N$ limit ($N\gg 1$), one can consider only the near-horizon
region ($\widetilde{r}\ll R$) of the spacetime \eqref{metM2brana}.
Function $H(\widetilde{r})$ then reduces to
$H(\widetilde{r})=R^{6}/\widetilde{r\,}^{6}$. Moreover,
defining a new radial coordinate by $r=\widetilde{r\,}^{2}/2R$,
metric \eqref{metM2brana} becomes
\begin{equation}
ds^2=\frac{4r^2}{R^2}\left[-\mathfrak{h}(r)dt^2+
dx^{2}+dy^{2}\right]+\frac{R^2}{4r^2}
\frac{dr^2}{\mathfrak{h}(r)}
+R^{2}d\Omega_{7}^{2}.
\label{nearhorizon}
\end{equation}
The AdS part of the metric \eqref{nearhorizon},
associated to the coordinates $\{t,x,y,r\}$, is identical to the
solution of Einstein equations with negative cosmological term
corresponding to a $(3+1)$-dimensional plane-symmetric AdS black hole
\cite{lemos0,lemos1, huang, cai}:
\begin{equation}
ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+
\frac{r^{2}}{L^{2}}(dx^{2}+dy^{2}),
\label{background}
\end{equation}
where the horizon function $f(r)$ is given by
\begin{equation}
f(r)=\left(\frac{r}{L}\right)^{2}\,\mathfrak{h}(r)=
\left(\frac{r}{L}\right)^{2}\left(1-\frac{r_{0}^{3}}{r^{3}}\right),
\label{fdef1}
\end{equation}
and the seven-sphere radius $R$ has been rewritten as $R=2L$, with $L$ now
representing the AdS radius of the spacetime \eqref{background}. Parameters
$r_0$ and $L$ are related to the blackhole Hawking temperature $T$ by
\begin{equation}
T=\frac{3}{4\pi}\frac{r_{0}}{L^{2}}.
\label{hawk-temperature}
\end{equation}
\subsection{Normalization of the field action}
The full theory is the eleven-dimensional supergravity on
$\mbox{AdS}_{4}\times S^{7}$, and the existence of a compact seven-sphere
enables one to consistently reduce the theory to Einstein-Maxwell theory on
$\mbox{AdS}_{4}$ \cite{herzog4, duff1, berenstein1}. The main objective in
summarizing such a procedure here is to make explicit the dependence of the
action for the fields in the AdS$_{4}$ spacetime on the number of colours
$N$, which is one of the parameters characterizing the holographic CFT.
Upon Kaluza-Klein dimensional reduction, the Maxwell gauge field
$A_{\scriptscriptstyle{M}}$ arises from a combination of metric and ${F}_{4}$ form
perturbations in the eleven-dimensional supergravity. This field corresponds
to a $U(1)$ subgroup of the $SO(8)$ symmetry group of the complete spacetime
\eqref{metM2brana}. The mechanism of dimensional reduction also furnishes
the $(3+1)$-dimensional Einstein-Maxwell action with a negative cosmological
constant $\Lambda=-3/L^{2}$:
\begin{equation}
S=\frac{1}{2\kappa_{4}^{2}}\int d^{4}x\sqrt{-g}\left(\mathcal{R}
+\frac{6}{L^{2}}-L^{2}F_{\scriptscriptstyle{MN}}F^{\scriptscriptstyle{MN}}\right),
\label{acaocompleta}
\end{equation}
where $\mathcal{R}$ denotes the Ricci scalar and $F_{\scriptscriptstyle{MN}}$ is the
electromagnetic strength tensor, and for the purposes of the present
analysis the electromagnetic Lagrangian $\mathcal{L}_{em} \sim
F_{\scriptscriptstyle{MN}}F^{\scriptscriptstyle{MN}}$ is considered as a perturbation on the gravitational
Lagrangian $\mathcal{L}_{gr} \sim \mathcal{R} +{6}/{L^{2}} $. It is
assumed that the gravitational coupling constants in four and eleven
dimensions are related by means of the seven-sphere volume \cite{herzog4},
\begin{equation}
\frac{1}{2\kappa_{4}^{2}}=\frac{R^{7}
\mbox{Vol}(S^{7})}{2\kappa_{11}^2}.
\end{equation}
Then, considering that the volume of a unitary seven-sphere
is $\mbox{Vol}(S^{7})=\pi^{4}/3$, and using the standard
normalization \eqref{relparam} for $\kappa_{11}$,
it is found
\begin{equation}
\frac{1}{2\kappa_{4}^{2}}=\frac{\sqrt{2}N^{3/2}}{24\pi L^{2}}.
\label{gravity-const}
\end{equation}
Action \eqref{acaocompleta} with the gravitational constant $\kappa_{4}$
given in terms of the number of colours $N$ and of the anti-de Sitter
radius $L$ is the desired result, which is needed for the development of
the present work.
\section{Electromagnetic quasinormal modes}
\label{qnm-eletro}
\subsection{Perturbation equations}
\label{flut-eletro}
In the AdS/CFT context, the electromagnetic field in the AdS bulk couples to
the CFT $R$-symmetry currents at the spacetime boundary. Hence, in order to
construct the current-current two-point correlation functions in the CFT, it
is necessary to consider fluctuations of the gauge field $A_{\scriptscriptstyle{M}}$. Such a
field is implicitly defined by
\begin{equation}
F_{\scriptscriptstyle{MN}}=\partial_{\scriptscriptstyle{M}}A_{\scriptscriptstyle{N}}-
\partial_{\scriptscriptstyle{N}}A_{\scriptscriptstyle{M}},
\label{strength}
\end{equation}
with $F_{\scriptscriptstyle{MN}}$ satisfying equations of motion derived from the action
\eqref{acaocompleta}. Therefore, considering the electromagnetic field as a
perturbation on the background spacetime of metric \eqref{background}, the
resulting equations of motion for $A_{M}$ are the usual Maxwell equations
\begin{equation}
\partial_{\scriptscriptstyle{M}}\left(\sqrt{-g}g^{\scriptscriptstyle{MA}}
g^{\scriptscriptstyle{NB}}F_{\scriptscriptstyle{AB}}\right)=0,
\label{maxwell}
\end{equation}
where $g_{\scriptscriptstyle{MN}}$ stands for the metric components given by
\eqref{background}.
When looking for solutions to Eqs. \eqref{maxwell}, by taking into account
the isometries of the background metric \eqref{background}, it is convenient
to decompose the gauge field in terms of Fourier transforms as follows
\begin{equation}
A_{\scriptscriptstyle{M}}(t,x,y,r)=\frac{1}{(2\pi)^{3}}\int{\! d\omega\, dk_{x}\, dk_{y}\,}
e^{-i\omega t+ik_{x}x+ik_{y}y}\widetilde{A}_{\scriptscriptstyle{M}}(\omega,k_{x},k_{y},r).
\label{EMfourier}
\end{equation}
Furthermore, without loss of generality, in the plane-symmetric background
spacetime \eqref{background} one may choose the wave three-vector $k$ in
the form $k_\mu=(k_0,k_x,k_y)=(-\omega,0,q)$. This is carried out
through an appropriate rotation in the $x-y$ plane, in such a way that the
Fourier modes of the gauge field propagate along the $y$ direction only. With
such a choice, the electromagnetic perturbations $ A_{\scriptscriptstyle{M}}$ can be split
into two independent sets according to their behavior under parity operation,
$x\rightarrow -x$:
\begin{itemize}
\item Axial (odd, or transverse) perturbations:
$A_{x}$;
\item Polar (even, or longitudinal) perturbations:
$A_{t}$, $A_{y}$, $A_{r}$.
\end{itemize}
Since these two sets of perturbations are orthogonal sets, they can
be studied separately, as it is done in the following.
\subsubsection{Equations for axial perturbations}
Axial electromagnetic perturbations are governed by the transverse component
of Maxwell equations \eqref{maxwell}, which gives
\begin{equation}
f\frac{d^2 A_{x}}{dr^2}+\frac{df}{dr}\frac{d A_{x} }{dr}+
\left(\frac{\omega^2 r^2-q^2 L^{2}f}{fr^2}\right) A_{x}=0,
\label{axeletro1}
\end{equation}
where, to simplify notation, the tilde was dropped, $\widetilde A_x
\rightarrow A_x$. Moreover, it follows from Eqs. \eqref{strength} and
\eqref{EMfourier}, together with $k_{\mu}=(-\omega,0,q)$, that $A_{x}$ is
proportional to the transverse component of the electric field: $
E_{x}=i\omega A_{x}$. Therefore, being a gauge-invariant quantity, $A_{x}$ is
also a good candidate as master variable for axial perturbations. In fact, it
is possible to cast Eq. \eqref{axeletro1} into a Schr\"odinger-like form
\cite{cardoso5}
\begin{equation}
\left(\frac{d^{\,2}}{dr_{\ast}^{2}}+\omega^{2}\right)
\Psi^{\scriptscriptstyle{(-)}}=f\left(\frac{qL}{r}\right)^{2}\,
\Psi^{\scriptscriptstyle{(-)}},
\label{ondaeletro1}
\end{equation}
where $\Psi^{\scriptscriptstyle{(-)}}(r)=A_{x}(r)$, and the tortoise coordinate $r_{\ast}$
is defined in terms of the radial coordinate $r$ by
\begin{equation}
\frac{dr}{dr_{\ast}} = {f(r)} \label{tortoise}.
\end{equation}
For the present purposes it is convenient to change coordinates to the
inverse radius $u=r_{0}/r$, and then, by writing Eq. \eqref{ondaeletro1} in
terms of $E_{x}$ it results
\begin{equation}
E_{x}^{''}+\frac{\mathfrak{h}^{'}}{\mathfrak{h}}E_{x}^{'}+
\frac{\mathfrak{w}^2-\mathfrak{q}^2\mathfrak{h}}{\mathfrak{h}^2}E_{x}=0,
\label{fieletro}
\end{equation}
where the primes denote derivation with respect to the variable
$u$, and $\mathfrak{w}$ and $\mathfrak{q}$ are respectively the
normalized frequency and wavenumber, defined by
\begin{equation}
\mathfrak{w}=\frac{3\omega}{4\pi T}\quad\qquad\mbox{and}
\qquad\quad\mathfrak{q}=\frac{3q}{4\pi T},
\label{wq-normalizados}
\end{equation}
with $T$ being the Hawking temperature of the black hole, given by Eq.
\eqref{hawk-temperature}. Function $\mathfrak{h}$ is obtained from Eqs.
\eqref{hdef1}, or from Eq. \eqref{fdef1}, and in terms of the variable
$u=r_0/r$ reads
\begin{equation}
\mathfrak{h}\equiv \mathfrak{h}(u)=1-u^3\,.
\label{hdef2}
\end{equation}
Quantity $E_x$ shall be the fundamental gauge-invariant
variable to be used in the present analysis of the QNM modes for axial
electromagnetic perturbations.
\subsubsection{Equations for polar perturbations}
Differently from the axial electromagnetic perturbations, the components of
the gauge field $A_{\scriptscriptstyle{M}}$ corresponding to the set of polar perturbations
are not gauge invariant. The gauge freedom can then be used in order to
simplify the relevant equations of motion. In fact, the invariance of Maxwell
equations under the gauge transformation $A_{\scriptscriptstyle{M}}\rightarrow
A_{\scriptscriptstyle{M}}+\partial_{\scriptscriptstyle{M}}\lambda$ allows choosing $\lambda$ in such a way
that one of the components $A_t,\,A_y,$ or $A_r$ vanishes. For instance, it
is possible to work in the so-called radial gauge, in which $A_r=0$
\cite{herzog2}. In this gauge, Maxwell equations \eqref{maxwell}
corresponding to the polar perturbations are
\begin{equation}
\omega r^{2}\frac{d}{dr}A_{t}+qL^{2}f\frac{d}{dr}A_{y}=0,
\label{eqeletro1}
\end{equation}
\begin{equation}
r^{2}\frac{d^2}{dr^2}A_{t}+2r\frac{d}{dr}A_{t}-\frac{L^{2}}{f}
\left(q\omega A_{y}+q^{2}A_{t}\right)=0,
\label{eqeletro2}
\end{equation}
\begin{equation}
f\frac{d^2}{dr^2}A_{y}+\frac{df}{dr}\frac{d}{dr}A_{y}
+\frac{1}{f}\left(\omega qA_{t}+\omega^{2}A_{y}\right)=0.
\label{eqeletro3}
\end{equation}
Notice that this set of equations does not constitute a linearly independent
system, and any subset composed by two of such equations determines the two
remaining unknown components of the gauge field, $A_t$ and $A_y$.
From now on, $A_t$ or $A_y$ could be adopted as a primary variable
and equations \eqref{eqeletro1}-\eqref{eqeletro3} could be
decoupled in order to find a unique differential equation for one of
these functions. However, to avoid the inconvenient of dealing with
gauge dependent quantities, it is interesting to use
the electric field components, which are gauge-invariant quantities.
Even though this choice eliminates gauge-dependent potential fields,
a residual ambiguity is left: from the electric field components
$E_{r}=dA_{t}/dr$ and $E_{y}=i(q A_{t}+\omega A_{y})$, what is the best choice?
The answer to the last question is not simple and both of the possible
answers have been tried in the literature. For instance, inspired by
preceding works studying blackhole perturbations in asymptotically flat
spacetimes \cite{ruffini}, Cardoso and Lemos opted for the radial component
$E_r$ in studying QNM of Schwarzschild-AdS black holes \cite{cardoso2},
and plane-symmetric AdS black holes \cite{cardoso5}. Introducing a new
variable
$\Psi^{\scriptscriptstyle{(+)}}(r)=r^{2}E_{r}(r)$ and using Eqs. \eqref{eqeletro1} and
\eqref{eqeletro2}, they were able to reduce the system of equations into a
unique ordinary differential equation of Schr\"odinger type
\begin{equation}
\left(\frac{d^{\,2}}{dr_{\ast}^{2}}+\omega^{2}\right)
\Psi^{\scriptscriptstyle{(+)}}=f\left(\frac{qL}{r}\right)^{2}
\Psi^{\scriptscriptstyle{(+)}},
\label{ondaeletro2}
\end{equation}
which has the same form as the fundamental equation for axial perturbations,
Eq. \eqref{ondaeletro1}. An interesting consequence of this fact is that the
QNM spectra for both the axial and polar perturbations, with the same
boundary conditions, are identical \cite{cardoso2,cardoso5}. As a matter of
fact, an open question left behind in the early works computing
electromagnetic QNM of AdS black holes is the lack of a convincing physical
justification for the choice of Dirichlet boundary conditions at infinity for
both the polar and the axial perturbations. Such an issue will be considered
in the sequence of this work.
As far as one is interested in computing the polar electromagnetic
quasinormal frequencies, it will be shown in Sect. \ref{green-electro} that
$E_y$ is more appropriate as a fundamental variable than $E_r$. This was
the choice made, for instance, by Kovtun and Starinets in studying QNM of
black branes in $(4+1)$-dimensional spacetimes \cite{kovtun1}. Following
these
authors, in the present work $E_y$ is adopted as the fundamental variable
to be used to determine the QNM spectrum of polar electromagnetic
perturbations. Having made this choice, one then writes equations in terms of
the independent variable $u=r_0/r$. With this, Eqs. \eqref{eqeletro1} and
\eqref{eqeletro2} written for $E_{y}$ lead to
\begin{equation}
E_{y}^{''}+\frac{\mathfrak{w}^{2}\mathfrak{h}^{'}}{\mathfrak{h}\left(
\mathfrak{w}^2-\mathfrak{q}^{2}\mathfrak{h}\right)}E_{y}^{'}+
\frac{\left(\mathfrak{w}^2-\mathfrak{q}^{2}\mathfrak{h}
\right)}{\mathfrak{h}^2}E_{y}=0,
\label{yeletro}
\end{equation}
where $\mathfrak{w}$ and $\mathfrak{q}$ are respectively the normalized
frequency and wavenumber, defined by Eqs. \eqref{wq-normalizados},
and $\mathfrak{h}$ is given by Eq. \eqref{hdef2}.
According to the N\'u\~{n}ez-Starinets QNM definition \cite{nunez}, once one
has found the fundamental perturbation equations for a field on the AdS
spacetime, the next step is establishing explicit relations between the
perturbation variables and the corresponding retarded Green functions in the
holographic CFT. It is exactly from these relations that will emerge the
boundary conditions to be imposed on the perturbation fields at infinity,
viz, the conditions that lead to the singularities of the two-point
correlation functions in the boundary field theory, and consequently to the
quasinormal frequencies of the fluctuation modes. Such a task is performed in
what follows.
\subsection{$R$-current correlation functions}
\label{green-electro}
In the case of electromagnetic perturbations, the AdS/CFT correspondence
\cite{maldacena1,gubser1,witten1} tells that, in the strong coupling, large
$N$ limit, the information on the thermal correlation functions of the
$R$-symmetry currents are encoded into the electric field components $E_{j}$
($j=x,y$), which are solutions to the differential equations \eqref{fieletro}
and \eqref{yeletro}, respectively. It can be shown that, close to the horizon
($u\approx 1$), such functions are given approximately by
$E_{j}=\mathfrak{h}^{\pm i\mathfrak{w}/3}$, where the negative (positive)
exponent corresponds to ingoing (outgoing) waves. Moreover, depending on the
sign of the exponent, the boundary values of the perturbation functions act
as sources of retarded or advanced Green functions in the dual CFT. To
compute the retarded two-point functions, one has to opt for the negative
exponent. It is also necessary to know the asymptotic form of the
perturbation functions close to the infinite boundary ($u\approx 0$). A
simple analysis shows that the solutions of equations \eqref{fieletro} and
\eqref{yeletro} which satisfy incoming-wave condition at horizon
present the
following behavior around $u=0$:
\begin{gather}
E_{x}=\mathcal{A}_{(x)}(\mathfrak{w},\mathfrak{q})
+...\;+\mathcal{B}_{(x)}(\mathfrak{w},\mathfrak{q})u+ ... \,,
\label{assintelet1}\\
E_{y}=\mathcal{A}_{(y)}(\mathfrak{w},\mathfrak{q})+...\,+
\mathcal{B}_{(y)}(\mathfrak{w},\mathfrak{q})u+...\, ,
\label{assintelet2}
\end{gather}
where ellipses denote higher powers of $u$ for each one of the independent
solutions. Symbols $\mathcal{A}_{(j)}(\mathfrak{w},
\mathfrak{q})$ and $\mathcal{B}_{(j)}(\mathfrak{w},
\mathfrak{q})$, introduced in the above equations, stand for the connection
coefficients associated to the corresponding differential equations for
$E_x$ and $E_y$.
To proceed further and calculate the correlation functions, the
electromagnetic action at the boundary needs to be determined.
It is usual to split the action as $S_{\scriptscriptstyle{EM}} =
S_{\scriptscriptstyle{horizon}}+S_{\scriptscriptstyle{boundary}}$. Using the equations of motion and the
preceding definitions it follows
\begin{equation}
S_{\scriptscriptstyle{boundary}}=\frac{\chi}{2}\;\underset{u\rightarrow 0}{\mbox{lim}}
\,\int\frac{d\mathfrak{w} d\mathfrak{q}}{(2\pi)^2}\left[\frac{\mathfrak{h}}{
\mathfrak{w}^{2}-\mathfrak{q}^{2}\mathfrak{h}}E_{y}^{'}(u,k)E_{y}(u,
-k)+\frac{\mathfrak{h}}{\mathfrak{w}^{2}}
E_{x}^{'}(u,k)E_{x}(u,-k)\right],
\label{acaoeletro}
\end{equation}
where
\begin{equation}
\chi=\frac{8\pi TL^{2}}{3\kappa_{4}^{2}}
=\frac{(2N)^{3/2}T}{9}
\end{equation}
is the electric susceptibility of the dual system \cite{herzog2,herzog4}.
In order to apply the Lorentzian AdS/CFT prescription of Ref. \cite{son1},
the asymptotic solutions \eqref{assintelet1} and \eqref{assintelet2} are used
to write the derivatives of the electric field in terms of the boundary
values of the three-vector potential $A_{\mu}^{0}(k)=A_{\mu}(u\rightarrow 0,
k)$. The $R$-current correlation functions $C_{\mu\nu}$ are proportional
to the coefficients of the terms containing the product
$A_{\mu}^{0}(k)A_{\nu}^{0}(-k)$ that appears in the action
\eqref{acaoeletro}. It is then found:
\begin{equation}
\begin{aligned}
&C_{tt}=\chi\frac{\mathfrak{q}^{2}}{\left(\mathfrak{w}^{2}
-\mathfrak{q}^{2}\right)}\frac{\mathcal{B}_{(y)}(\mathfrak{w},\mathfrak{q})}
{\mathcal{A}_{(y)}(\mathfrak{w},\mathfrak{q})},
&\qquad& C_{yy}=\chi\frac{\mathfrak{w}^{2}}{\left(\mathfrak{w}^{2}
-\mathfrak{q}^{2}\right)}\frac{\mathcal{B}_{(y)}(\mathfrak{w},\mathfrak{q})}
{\mathcal{A}_{(y)}(\mathfrak{w},\mathfrak{q})},\\
&C_{ty}=-\chi\frac{\mathfrak{w}\mathfrak{q}}
{\left(\mathfrak{w}^{2}-\mathfrak {q}^{2}\right)}\frac{\mathcal{B}_{(y)}
(\mathfrak{w},\mathfrak{q})}{\mathcal{A}_{(y)}(\mathfrak{w},\mathfrak{q})},
&\qquad& C_{xx}= \chi\frac{\mathcal{B}_{(x)}(\mathfrak{w},\mathfrak{q})}
{\mathcal{A}_{(x)}(\mathfrak{w},\mathfrak{q})}.\label{celetro}
\end{aligned}
\end{equation}
Moreover, for a $(2+1)$-dimensional CFT at finite temperature, the
current-current correlation functions can be written in terms of the
transverse and longitudinal self-energies
$\Pi^{T}(\mathfrak{w},\mathfrak{q})$ and
$\Pi^{L}(\mathfrak{w},\mathfrak{q})$, respectively (See Appendix
\ref{apen-correlations} for a summary of such relations). Hence, comparing
Eqs. \eqref{celetro} to Eqs. \eqref{tself-en}--\eqref{lself-en3} of Appendix
\ref{apen-correlations}, one finds
\begin{equation}
\Pi^{T}(\mathfrak{w},\mathfrak{q})=\chi\frac{\mathcal{B}_{(x)}(\mathfrak{w},
\mathfrak{q})}{\mathcal{A}_{(x)}(\mathfrak{w},\mathfrak{q})},
\qquad\qquad\Pi^{L}(\mathfrak{w},\mathfrak{q})=\chi
\frac{\mathcal{B}_{(y)}(\mathfrak{w},\mathfrak{q})}{
\mathcal{A}_{(y)}\mathfrak{w},\mathfrak{q})}.
\end{equation}
These results show that the retarded two-point correlation functions are
fully determined by the ratio between the connection coefficients of
equations \eqref{fieletro} and \eqref{yeletro}. Furthermore, the poles of
the thermal correlation functions are given by the zeros of the coefficients
$\mathcal{A}_{(x)}(\mathfrak{w},\mathfrak{q})$ and
$\mathcal{A}_{(y)}(\mathfrak{w}, \mathfrak{q})$. According to Ref.
\cite{nunez}, the poles of $C_{\mu\nu}$ define the electromagnetic QNM
frequencies of the black hole localized in the AdS spacetime. Such
frequencies are then obtained by imposing Dirichlet boundary conditions
on the electric field components $E_{x}$ and $E_{y}$ at $u=0$, with $E_{x}$
and $E_{y}$ being functions that satisfy also an incoming-wave condition at
the horizon.
\subsection{QNM and the gauge-invariant variables}
At this stage one could ask whether imposing Dirichlet conditions at the
boundary ($u=0$) on the Regge-Wheeler-Zerilli (RWZ) variables
$\Psi^{\scriptscriptstyle{(\pm)}}$ would produce the same QNM spectra as the spectra
obtained by imposing the same boundary conditions onto the Kovtun-Starinets
(KS) variables $E_{x,y}$. For the transverse electromagnetic sector, the
answer to this question is quite easy to find. In fact, variables
$\Psi^{\scriptscriptstyle{(-)}}$ and $E_{x}$ are proportional to each other, so that both of
the obtained QNM spectra, either using the KS variable or using the RWZ
variable, are identical. In the case of polar electromagnetic perturbations,
equations for the KS variable $E_{y}$ and for the RWZ variable
$\Psi^{\scriptscriptstyle{(+)}}$ do not have the same form and, in addition, $E_{y}$ and
$\Psi^{\scriptscriptstyle{(+)}}$ are independent variables, so that the answer is not
immediate. In fact, as it is shown below (see Sects. \ref{hydro-eletro} and
\ref{dispersion-eletro}), the QNM spectrum obtained from $E_{y}$ is
different from the QNM spectrum obtained from $\Psi^{\scriptscriptstyle{(+)}}$.
\subsection{Dispersion relations for the hydrodynamic QNM}
\label{hydro-eletro}
The hydrodynamic limit of perturbations corresponds to the small frequency
($\mathfrak{w}\ll 1$) and small wavenumber ($\mathfrak{q}\ll 1$) region of
the spectrum of the respective Fourier modes. In general, the quasinormal
modes can be classified according the behavior of the dispersion relations in
the hydrodynamic limit, and in this respect there are two classes. There is a
set of QNM for which the frequency $\mathfrak{w}(\mathfrak{q})$ vanishes when
$\mathfrak{q}\rightarrow 0$. Such modes are named here hydrodynamic
quasinormal modes. But there is another kind of QNM for which the
corresponding frequency in the long-wavelength limit is nonzero. To
distinguish these two kind of modes from each other, the modes belonging to
the later kind are denominated non-hydrodynamic quasinormal modes. In this
section, the dispersion relations of the electromagnetic hydrodynamic QNM are
studied by means of analytical and numerical methods. The electromagnetic
non-hydrodynamic QNM shall be object of study in the next section.
From the CFT point of view, it is expected that at least one of the
electromagnetic QNM should show the typical behavior of a diffusion mode
in the hydrodynamic limit. Trying to find such a mode, one then
looks for solutions to Eqs. \eqref{fieletro} and \eqref{yeletro} in
the form of power series in $\mathfrak{w}$ and $\mathfrak{q}$,
under the assumption $\mathfrak{w}\sim\mathfrak{q}$. Written
in terms of the variables $F_{j}=\mathfrak{h}^{i\mathfrak{w}/3}E_{j}$
(for $j=x,y$), which are more appropriate for the present analysis,
perturbation equations \eqref{fieletro} and \eqref{yeletro} may be cast as
\begin{equation}
F_{j}^{''}+\frac{u^{2}}{\mathfrak{h}}\left(2i\mathfrak{w}-3a_{j}
\right)F_{j}^{'}+\frac{1}{\mathfrak{h}^{2}}\left[i\mathfrak{w}(2u+u^{4}
-3a_{j}u^{4})+\mathfrak{w}^{2}(1-u^{4})-\mathfrak{q}^{2}\mathfrak{h}
\right]F_{j}=0,
\label{perturbativa1}
\end{equation}
where $a_{x}=1$ and $a_{y}=\mathfrak{w}^{2}/(\mathfrak{w}^{2}-
\mathfrak{q}^{2}\mathfrak{h})$.
After relabelling parameters as
$\mathfrak{w}\rightarrow\lambda\mathfrak{w}$ and $\mathfrak{q}
\rightarrow\lambda\mathfrak{q}$ with $\lambda\ll 1$,
it is assumed that solutions of Eqs. \eqref{perturbativa1}
can be expanded in the form
\begin{equation}
F_{j}(u)=F_{j}^{0}(u)+\lambda F_{j}^{1}(u)+\lambda^{2}
F_{j}^{2}(u)+...,\;
\label{expansaoeletro}
\end{equation}
where the coefficients $F_{j}^{\alpha}(u)$, with $\alpha = 0,1,2, ...,$
represent arbitrary functions of variable $u$, and which are also homogeneous
functions of degree $\alpha$ on $\mathfrak{w}$ and $\mathfrak{q}$.
The boundary condition of being ingoing waves at the horizon imposed on
$E_j$, when translated to the new functions $F_j$, implies their dominant
terms in expansion \eqref{expansaoeletro} must assume constant values close
the horizon ($u\approx 1$). Then, in terms of the expansion
\eqref{expansaoeletro} one has the following conditions
\begin{equation}
F_{j}^{0}(1)=\mbox{constant},\qquad F_{j}^{1}(1)=F_{j}^{2}(1)=...=0.
\label{contorno1}
\end{equation}
It is now possible to solve Eqs. \eqref{perturbativa1} order by order and,
after imposing the boundary conditions given in Eqs. \eqref{contorno1}, the
following expansions are found:
\begin{equation}
E_{x}=C_{x}\mathfrak{h}^{-i\mathfrak{w}/3}\bigg[1-
i\mathfrak{w}\frac{\sqrt{3}}{3}\left(\frac{\pi}{3}
-\arctan{\frac{1+2u}{\sqrt{3}}}\right)+\frac{i\mathfrak{w}}{2}
\ln{\frac{1+u+u^2}{3}}+\mathcal{O}(\mathfrak{w}^{2})\bigg],
\label{fiserie}
\end{equation}
\begin{equation}
E_{y}=C_{y}\mathfrak{h}^{-i\mathfrak{w}/3}\bigg[1+
\frac{i\mathfrak{q}^{2}}{\mathfrak{w}}(1-u)-i\mathfrak{w}
\frac{\sqrt{3}}{3}\left(\frac{\pi}{3}-\arctan{\frac{1+2u}{
\sqrt{3}}}\right)+\frac{i\mathfrak{w}}{2}\ln{\frac{1+u+u^2}{3}}
+\mathcal{O}(\mathfrak{w}^{2})\bigg],\label{yserie}
\end{equation}\\
where $C_{x}$ and $C_{y}$ are arbitrary normalization constants.
One finds from Eq. \eqref{fiserie} no solution satisfying the Dirichlet
condition at the AdS spacetime boundary, $E_x(0) =0$, and which is at the
same time compatible with the hydrodynamic approximation
$\mathfrak{w},\mathfrak{q}\ll 1$. This means there is no axial
electromagnetic hydrodynamic QNM, and no $R$-charge diffusion in the
transverse direction to the spatial wave vector, as expected from the CFT
point of view. On the other hand, the condition $E_{y}(0)=0$ and Eq.
\eqref{yserie} lead to\footnote{This result was also found through direct
calculation of the hydrodynamic limit of correlation functions by Herzog
\cite{herzog2,herzog3}. The comparison to that result is in fact a test for
the analysis performed in Sect. \ref{green-electro}.}
\begin{equation}
\mathfrak{w}=-i\mathfrak{q}^{2}
\qquad\Longrightarrow\qquad
\omega=-\frac{3i}{4\pi T}q^2,
\end{equation}
from where one can read the diffusion coefficient $D=3/4\pi T$.
It is worth noticing that this diffusion mode is not found if one uses the
RWZ master variable $\Psi^{\scriptscriptstyle{(+)}}$ instead of $E_{y}$.
\FIGURE{
\centering\epsfig{file=cpuroeletro3b2.eps, height=9.41cm,
width=6.0cm, angle=270}
\caption{The dispersion relation for the only electromagnetic hydrodynamic
QNM (solid line), which is purely damped, $\mathfrak{w}=-i\mathfrak{w}_{I}$,
and corresponds to a polar perturbation. The dotted line is the diffusion
mode
$\mathfrak{w}_{I}=\mathfrak{q}^{2}$, which approaches the
quasinormal frequency in the hydrodynamic limit
$\mathfrak{w},\mathfrak{q}\ll 1$.}
\label{hidroeletro}}
As seen above, the hydrodynamic limit of perturbation equations comprises a
very special interval in the space of parameters $\mathfrak{w}$ and
$\mathfrak{q}$. Besides the physical relevance of this regime, it
corresponds to the very rare situations where analytical expressions
can be found for the quasinormal frequencies. In the great majority of
cases, numerical methods have to be employed in order to find
the complete dispersion relations
$\mathfrak{w}\times \mathfrak{q}$. In the sequence, the Horowitz-Hubeny
method \cite{horowitz1} is used to compute the dispersion relation
$\mathfrak{w}=-i\mathfrak{w}_{I}(\mathfrak{q})$ for the
electromagnetic hydrodynamic QNM, which in the limit of small wavenumbers
corresponds to the diffusion mode found above. The result is shown in Fig.
\ref{hidroeletro}. One sees the deviation of the exact dispersion relation
curve (solid line) from the hydrodynamic limit curve $\mathfrak{w}_{I} =
\mathfrak{q}^{2}$ (dotted line). Another interesting fact is that the
quasinormal frequency $\mathfrak{w}(\mathfrak{q})$ disappears
for $\mathfrak{q}$ larger than approximately $0.557$. This is
characteristic to all the electromagnetic purely damped modes,
as analyzed in the next section (see Table \ref{valoreslimites}).
\subsection{Dispersion relations for the non-hydrodynamic QNM}
\label{dispersion-eletro}
As mentioned earlier, one of the goals of the present analysis is to
obtain the electromagnetic quasinormal modes of plane-symmetric
AdS$_4$ spacetimes and to compare the present results with the
results of Ref. \cite{cardoso5}. As verified in the hydrodynamic limit
(Sect. \ref{hydro-eletro}), the QNM spectra calculated here may be quite
different from the spectra obtained in that work because
of the use of different fundamental variables, and therefore a more careful
search study on the dispersion relations of these modes is justified.
\subsubsection{Purely damped modes}
\label{purelydamped}
For small values of $\mathfrak{q}$, electromagnetic perturbations
of AdS black holes present a special set of
modes which are purely damped. These are not usual QNM since the real part
of the frequencies vanishes eliminating the oscillatory behavior of the
perturbations which is characteristic of QNM. Furthermore, the
frequencies $\mathfrak{w}(\mathfrak{q})$ of such modes cannot be, in
general, associated to hydrodynamic poles since most of the purely
damped modes have nonvanishing frequencies in limit as the wave\-number
$\mathfrak{q}$ goes to zero. To distinguish from the regular QNM, the purely
imaginary frequencies of both the axial and the polar perturbations shall be
labelled by a special quantum number, $n_s$, that assumes just integer
values, starting by $n_s=0$ for the hydrodynamic diffusion mode. As it was
shown above, the axial sector of electromagnetic perturbations does not
present quasinormal modes in the hydrodynamic limit, so that the set
of axial purely damped modes starts at $n_s=1$.
\FIGURE{
\centering\epsfig{file=puroeletro3.eps, height=9.41cm,
width=6.0cm, angle=270}
\caption{The dispersion relations for polar (solid lines) and
axial (dashed lines) purely damped electromagnetic QNM. The dotted (lowest)
line is the diffusion mode $\mathfrak{w}_{s}=\mathfrak{q}^{2}$, which
approaches the $n_s =0$ quasinormal frequency in the hydrodynamic limit
$\mathfrak{w},\mathfrak{q}\ll 1$. The insert shows the
behavior of the dispersion relations for higher quasinormal
frequencies.}
\label{eletropuro}}
An interesting property of electromagnetic QNM of AdS black holes in
$(3+1)$-di\-men\-sion\-al spacetimes has been recently discovered by
Herzog and collaborators \cite{herzog4}: The current-current
correlators are analytical functions at $\mathfrak{q}=0$, meaning
that there are no quasinormal frequencies for null wavenumber.
It can be shown that such a property is a consequence of
the well known duality relation between electric and magnetic
fields in vacuum. In fact, using the invariance of Maxwell equations under
the duality operation, electric field $\leftrightarrow$ magnetic field, and
the invariance of the correlation functions under rotations in the case of
null wavenumber (zero momentum), it was shown that the transverse and
longitudinal self-energies, $\Pi^{T}(\mathfrak{w},0)$ and
$\Pi^{L}(\mathfrak{w},0)$, are well behaved functions of the frequency for
all values of $\mathfrak{w}$.
On the other hand, as verified through the numerical results for purely
damped modes, there are quasinormal frequencies even for wavenumbers very
close to zero. In fact, as shown in Fig. \ref{eletropuro}, the small
wavenumber limit ($\mathfrak{q}=\epsilon$, with $\epsilon$ very small but
non-zero\footnote{In this specific case, the time of computation spent by
the numerical code based on the Horowitz-Hubeny method \cite{horowitz1}
written to find the quasinormal frequencies becomes very large as
$\epsilon$ approaches zero.}) of the corresponding purely imaginary
quasinormal frequencies, $\mathfrak{w}=-i\mathfrak{w}_{s}$, is given
approximately by
\begin{equation}
\mathfrak{w}_{s}=\frac{3}{2}n_{s}\qquad\Longrightarrow\qquad
\omega_{s}=\omega_{n_{s}}\equiv 2\pi T n_{s},
\end{equation}
where $\omega_{n_{s}}$ are the Matsubara frequencies of a generic quantum bosonic
system. Moreover, as it is also seen from Fig. \ref{eletropuro}, the
hydrodynamic pole $n_{s}=0$ is similar to other purely damped modes. As a
matter of fact, the only property that distinguishes a particular purely
damped mode from another is the behavior of these modes around
$\mathfrak{q}=0$: The only QNM satisfying the condition $\lim_{\mathfrak{q}
\rightarrow 0} \mathfrak{w}(\mathfrak{q}) =0$ is the hydrodynamic mode.
\TABLE{
\begin{tabular}{cccccc}
\hline\hline
\multicolumn{3}{c}{Polar} &
\multicolumn{3}{c}{Axial}\\
\cline{1-6}
$n_{s}$ & $\mathfrak{q}_{\mbox{\scriptsize{lim}}}\times 10^{3}$
& $\mathfrak{w}_{\mbox{\scriptsize{lim}}}$ (interval) &
$\quad n_{s}$ & $\mathfrak{q}_{\mbox{\scriptsize{lim}}}\times
10^{3}$ & $\mathfrak{w}_{\mbox{\scriptsize{lim}}}$ (interval)\\
\hline
(0,1) & $557.319\;$ & $[0.648111$, $0.648429]$ &
$\quad$(1,2) & $339.330$ & $[2.04771$, $2.04811]$ \\
(2,3) & $162.034\;$ & $[3.52507$, $3.52562]$ &
$\quad$(3,4) & $71.8726$ & $[5.01701$, $5.01788]$ \\
(4,5) & $31.0102\;$ & $[6.51286$, $6.51384]$ &
$\quad$(5,6) & $13.1892$ & $[8.00994$, $8.01169]$ \\
(6,7) & $5.55917\;$ & $[9.50785$, $9.51033]$ &
$\quad$(7,8) & $2.32839$ & $[11.0057$, $11.0100]$ \\
(8,9) & $0.970660\;$ & $[12.5041$, $12.5097]$ &
$\quad$(9,10) & $0.403180$ & $[14.0009$, $14.0114]$ \\
\hline\hline
\end{tabular}
\centering
\caption{Approximate limiting values of frequencies
$\mathfrak{w}=-i\mathfrak{w}_{\mbox{\scriptsize{lim}}}$
and wavenumbers $\mathfrak{q}_{\mbox{\scriptsize{lim}}}$
for purely damped electromagnetic modes. The brackets
indicate that the actual limiting values lie between
the two indicated endpoints in each case.}
\label{valoreslimites}}
At the opposite side of the quasinormal spectrum, i.e., for larger values of
$\mathfrak{q}$, there are saturation points at a maximum wavenumber value,
$\mathfrak{q}_{\mbox{\scriptsize{lim}}}$, beyond which the specific mode
disappears (See, however, Figs. \ref{eletroaxial} and \ref{eletropolar}). This
seems to happen for everyone of the modes, with
$\mathfrak{q}_{\mbox{\scriptsize{lim}}}$ decreasing for higher overtones (cf.
Fig. \ref{eletropuro}). The curves representing dispersion relations
associated to two different but contiguous modes meet each other exactly at
the saturation point, i.e., the two dispersion relation curves coincide at
that point. The axial modes group in pairs according to the relation
$n_{s}=\{(1,2),(3,4),(5,6),...\}$, while the polar modes are paired as
$n_{s}=\{(0,1),(2,3),(4,5),...\}$. The approximate limiting values
$\mathfrak{w}_{\mbox{\scriptsize{lim}}}$ and
$\mathfrak{q}_{\mbox{\scriptsize{lim}}}$ for $n_{s}=0,1,2,...,10$ are shown
in Table \ref{valoreslimites}.
The existence of a meeting point between two contiguous dispersion relation
curves suggests that for the special wavenumber values
$\mathfrak{q}=\mathfrak{q}_{\mbox{\scriptsize{lim}}}$ the corresponding
quasinormal frequencies
$\mathfrak{w}=-i\mathfrak{w}_{\mbox{\scriptsize{lim}}}$ represent double poles
of the CFT current-current correlation functions. A strong support to such a
conclusion comes from the behavior of the connection coefficients
$\mathcal{A}_{(j)}(\mathfrak{w}, \mathfrak{q})$ as a function of
$\mathfrak{w}=\mathfrak{w}_{R}-i\mathfrak{w}_{I}$ for small values of
$\mathfrak{q}$ and $\mathfrak{w}_{R}=0$. In Fig. \ref{funcaocompleta} it is
shown the profile of $\mathcal{A}_{(x)}(\mathfrak{w},\mathfrak{q})$ for
$\mathfrak{q}=\mathfrak{q}_{\mbox{\scriptsize{lim}}} \simeq 0.0132$ and
$0<\mathfrak{w}_{I}<9$. For this wavenumber, there are six purely imaginary
quasinormal frequencies encompassing the $n_{s}=1$ to $n_{s}=6$ axial QNM,
which by definition are the points where
$\mathcal{A}_{(x)}(\mathfrak{w},\mathfrak{q})=0$. In particular, for $n_{s}=5$
and $n_{s}=6$, the zeros of $\mathcal{A}_{(x)} (\mathfrak{w},\mathfrak{q})$
coincide, indicating that the corresponding quasinormal frequencies are
identical (see also Fig. \ref{eletropuro}).
The multiplicity of specific quasinormal frequencies as poles of
the correlation functions may also be verified through the derivatives of
$\mathcal{A}_{(j)}(\mathfrak{w},\mathfrak{q})$ with respect to
$\mathfrak{w}$. This was done numerically, by taking fixed values of
$\mathfrak{q}$ and letting $\mathfrak{q} \rightarrow
\mathfrak{q}_{\mbox{\scriptsize{lim}}}$. It was found that the first
derivative of $\mathcal{A}_{(j)}(\mathfrak{w},\mathfrak{q})$ in relation
to $\mathfrak{w}$ vanishes when $\mathfrak{w}=
-i\mathfrak{w}_{\mbox{\scriptsize{lim}}}$ and
$\mathfrak{q}=\mathfrak{q}_{\mbox{\scriptsize{lim}}}$, but it
is not zero for other values of the wavenumber.
This result proves that $\mathfrak{w}=-i\mathfrak{w}_{\mbox{\scriptsize{lim}}}$
corresponds to, at least, a double zero of $\mathcal{A}_{(j)}(\mathfrak{w},
\mathfrak{q})$, and consequently, to a double pole of the
corresponding current-current correlation functions.
\FIGURE{
\centering\epsfig{file=funcaonova2.eps, height=9.91cm,
width=7.0cm, angle=270} \caption{The connection coefficient
$\mathcal{A}_{(x)}(\mathfrak{w}, \mathfrak{q})$ for $\mathfrak{w}_{R}=0$,
$\mathfrak{w}_{I}=(0,9)$ and
$\mathfrak{q}=\mathfrak{q}_{\mbox{\scriptsize{lim}}}\simeq 0.0132$. The points
represent the $n_{s}=1,2,..,6$ purely imaginary quasinormal frequencies
associated to the axial electromagnetic perturbations. Note the coincidence of
the points corresponding to $n_{s}=5$ and $n_{s}=6$, indicating a possible
doubleness of the related quasinormal frequency.}
\label{funcaocompleta}}
\subsubsection{Ordinary quasinormal modes}
The electromagnetic perturbations of AdS black holes present also a family of
regular (ordinary) quasinormal modes whose frequencies have nonzero real and
imaginary parts. The numerical results for the quasinormal frequencies of the
first five regular modes are shown respectively in Figs.
\ref{eletroaxial} and \ref{eletropolar} for axial and polar fluctuations.
The form of the dispersion relations $\mathfrak{w}_{R}(\mathfrak{q})$
and $\mathfrak{w}_{I}(\mathfrak{q})$ indicates a connection between
the electromagnetic ordinary QNM and the family of purely damped modes
discussed in the last section. As shown in Figs. \ref{eletroaxial} and
\ref{eletropolar}, each regular quasinormal frequency only appears for
$\mathfrak{q}$ larger than a minimum wavenumber value, which (to a good
approximation) coincides with the limiting value of the corresponding pair
of purely damped modes, $\mathfrak{q}_{\mbox{\scriptsize{lim}}}$.
That is to say, all the dispersion relations, $\mathfrak{w} (\mathfrak{q})$,
for the ordinary QNM begin at the points $(\mathfrak{w}_{\mbox{\scriptsize{lim}}},
\mathfrak{q}_{\mbox{\scriptsize{lim}}})$, with the real parts starting from
zero value, $\displaystyle{ \mathfrak{w}_{R}(\mathfrak{q}\rightarrow
\mathfrak{q}^+_{\mbox{\scriptsize{lim}}})=0}$, while the imaginary parts
start at $\mathfrak{w}_{I}(\mathfrak{q}\rightarrow\mathfrak{q}^+
_{\mbox{\scriptsize{lim}}})=\mathfrak{w}_{\mbox{\scriptsize{lim}}}$.
For higher wavenumber values, the ordinary electromagnetic modes
show a sequence of quasinormal frequencies whose imaginary parts grow with
the principal quantum number $n$. In this respect, Figs. \ref{eletroaxial}
and \ref{eletropolar} show a simiral behavior for the real parts of the
quasinormal electromagnetic frequencies.
\FIGURE{
\centering\epsfig{file=realeletroaxial_new.eps, height=7.137cm,
width=4.968cm, angle=270}
\centering\epsfig{file=imageletroaxial_new.eps, height=7.137cm,
width=4.968cm, angle=270}
\caption{The real (left) and imaginary (right) parts of the frequencies
$\mathfrak{w}=3\omega/4\pi T$
for the first five ordinary axial electromagnetic modes
as a function of the normalized wavenumber $\mathfrak{q}=3q/4\pi T$.
The quantum number $n$ arranges the regular polar QNM in increasing
order of values of $\mathfrak{w}_{I}$. In the right it is also shown
the frequencies $\mathfrak{w}_{s}=3\omega_{s}/4\pi T$ associated to
the axial electromagnetic purely damped QNM.}
\label{eletroaxial}}
\FIGURE{
\centering\epsfig{file=realeletropolar_new.eps, height=7.137cm,
width=4.968cm, angle=270}
\centering\epsfig{file=imageletropolar_new.eps, height=7.137cm,
width=4.968cm, angle=270}
\caption{The real (left) and imaginary (right) parts of the frequencies
$\mathfrak{w}=3\omega/4\pi T$ for the first five ordinary polar
electromagnetic modes as a function of the normalized wavenumber
$\mathfrak{q}=3q/4\pi T$. As for the axial modes, the quantum number $n$
arranges the regular QNM from lower to higher values of $\mathfrak{w}_{I}$.
In the right it is also shown the frequencies $\mathfrak{w}_{s}=
3\omega_{s}/4\pi T$ associated to the polar electromagnetic purely
damped QNM.}
\label{eletropolar}}
\FIGURE{
\centering\epsfig{file=eletroexpandido2.eps, height=7.337cm,
width=5.668cm, angle=270}
\centering\epsfig{file=imagexpandido2.eps, height=7.337cm,
width=5.668cm, angle=270}
\caption{Graphs of the dispersion relations of the first five regular polar
electromagnetic QNM for large values of $\mathfrak{q}$. The figure on the
left hand side shows the real part of the frequency, $\mathfrak{w}_{R}$,
while the graph on the right hand side is for the imaginary part,
$\mathfrak{w}_{I}$.}
\label{eletroexpandido}}
The quasinormal frequencies found here show that AdS black holes are not
good oscillators. As it is well known, an interesting way of measuring the
quality of an oscillator is by means of its quality factor
$Q=\mathfrak{w}_{R}/2\mathfrak{w}_{I}$. In general, in the region of small
wavenumbers, the electromagnetic QNM have very small quality factors, $Q\ll
1$. For instance, taking $\mathfrak{q}=0.557319$ and considering the
fundamental polar mode one obtains $Q=7.24\times 10^{-5}$, a quality factor
typical to highly damped oscillatory systems. On the other hand, quality
factors of the order of unity are found for large wavenumber values, such as
$Q=1$ for $\mathfrak{q}$ around $1.11$, and $Q= 85.9$ for
$\mathfrak{q}=40$.
From the holographic field theory point of view, the real part of the
frequencies may be interpreted as quasiparticle excitation energies in the
dual plasma defined at the conformal boundary of the AdS spacetime. However,
such an interpretation only
makes sense for excitations, or quasinormal modes, with
large quality factors. In fact, according to Heisenberg uncertainty
principle, the uncertainty in the energy of a quasiparticle is of the
order of $\omega_{I}$ (in units of $\hbar$). Hence, quality factors smaller
than unity imply in energy uncertainties larger than the energies of
the quasiparticles themselves, making the interpretation of energy
excitations as quasiparticles meaningless.
In the $Q\gg 1$ regime, where the quasiparticle interpretation is feasible,
Fig. \ref{eletroexpandido} shows that the dispersion relations of the
ordinary electromagnetic QNM frequencies have real parts approaching
straight lines of the form $\mathfrak{w}_{R}=\mathfrak{q}+\mathfrak{b}_{n}$,
where $\mathfrak{b}_n$ depends only on the specific mode $n$. This means the
functions $\omega_{R}(q)$ approach the usual energy-momentum relation associated
to a zero rest mass particle, $\omega_{R}=q$, as $T\rightarrow 0$. Furthermore,
the characteristic damping time ($\tau=1/\omega_{I}$) of the
electromagnetic fluctuations diverges in the limit
$\mathfrak{q}\rightarrow\infty$, i.e., the functions $\omega_I(q)$ tend to
zero for large wavenumbers. All of the above results are consistent with
the expected properties of poles of correlation functions in quantum field
theories at zero temperature \cite{herzog4}.
\section{Gravitational quasinormal modes}
\label{grav-qnm}
Even though gravitational perturbations of plane-symmetric black holes in
$(3+1)$-dimensional AdS spacetimes have been analyzed in some extent
\cite{herzog2, herzog3, cardoso5, miranda1, miranda2}, the
quasi\-nor\-mal-mode
dispersion relations, with the use of the KS variables, were not found yet,
and hence the comparison with the spectra obtained by using the RWZ
gauge-invariant variables was not performed. This is done next.
\subsection{Fundamental equations for gravitational fluctuations}
\label{flut-grav}
As usual, gravitational perturbations are described here in terms of linear
metric fluctuations, which means the metric for the perturbed spacetime is
written as $g_{\scriptscriptstyle{MN}}=g^{\scriptscriptstyle{0}}_{\scriptscriptstyle{MN}} + h_{\scriptscriptstyle{MN}}$, where
$h_{\scriptscriptstyle{MN}}$ is considered as a perturbation in the background metric
$g^{\scriptscriptstyle{0}}_{\scriptscriptstyle{MN}}$, given by Eq. \eqref{background}. Among the variety
of possible gauge choices in studying metric perturbations, an interesting
choice is the so-called radial gauge, in which the coordinate system is
chosen in such a way that $h_{rt}=h_{rx}=h_{ry}=h_{rr}=0$. Since one of
the aims here is to investigate the relation among the perturbations of
plane-symmetric AdS$_4$ black holes and the correlation functions in the dual
CFT, it is convenient to use the radial gauge formalism to study the metric
fluctuations. A brief description of this formalism is given in the
following.
As it was done in the previous section when studying the electromagnetic
perturbations, the isometries of spacetime \eqref{background} allow Fourier
transforming coordinates $t$, $x$ and $y$, and writing metric fluctuations
$h_{\scriptscriptstyle{MN}}$ as
\begin{equation}
h_{\scriptscriptstyle{MN}}(t,x,y,r)= \frac{1}{(2\pi)^{3}} \int \!{d\omega\, dk_{x}\,
dk_{y}\,} e^{-i\omega t+ik_{x}x+ik_{y}y}
\widetilde{h}_{\scriptscriptstyle{MN}}(\omega,k_{x},k_{y},r).
\label{gravfourier}
\end{equation}
Again the wave three-vector may be chosen as $k_\mu=(-\omega,0,q)$,
and hence metric perturbations can be split into two disjoint sets. Namely,
the axial (transverse) sector of gravitational perturbations is
characterized by the quantities $h_{tx}$ and $h_{yx}$, and the polar
(longitudinal) sector of perturbations is composed by $h_{tt}$,
$h_{xx}$, $h_{yy}$, and $h_{ty}$.
\subsubsection{Axial perturbations in the radial gauge}
Now one needs to find evolution equations for the axial (transverse)
gravitational perturbations in the radial gauge, that are composed by the
metric fluctuations $h_{tx}$ and $h_{yx}$. The task is carried out following
the usual procedure of the theory of linear perturbations. The linearized
Einstein equations corresponding to the axial sector of gravitational
perturbations yield a set of three coupled differential equations for
$h_{tx}$ and $h_{yx}$ \cite{herzog2}. Of course, such equations are not
independent from each other, and one of them may be eliminated as a
combination of the other two. A resulting system of linearly independent
equations (after Fourier transforming them) which is interesting for the
present study is the following:
\begin{gather}
H_{tx}^{'}+\frac{\mathfrak{q}\mathfrak{h}}{\mathfrak{w}}
H_{yx}^{'}=0,\label{axial1}\\
H_{tx}^{''}-\frac{2}{u}H_{tx}^{'}-\frac{\mathfrak{q}}{\mathfrak{h}}
\left(\mathfrak{w}H_{yx}+\mathfrak{q} H_{tx}\right)=0,
\label{axial2}
\end{gather}
where $H_{tx}$ and $H_{yx}$ are defined by
\begin{equation}
H_{tx}=\frac{L^{2}}{r^{2}}\,h_{tx},\qquad\qquad
H_{yx}=\frac{L^{2}}{r^{2}}\,h_{yx}.
\label{axialh-ij}
\end{equation}
As in the case of electromagnetic perturbations, a mandatory condition that a
candidate for fundamental variable must satisfy is being gauge invariant,
which for metric perturbations means the candidate has to be invariant under
infinitesimal coordinate transformations. Inspired once again in the work by
Kovtun and Starinets \cite{kovtun1}, among the different combinations of
axial functions $H_{tx}$ and $H_{yx}$ which furnish gauge-invariant
quantities, one takes
\begin{equation}
Z_{1}=i\left(qH_{tx}+\omega H_{yx}\right)
\label{mestreaxial}
\end{equation}
as the fundamental gauge-invariant function of
the axial gravitational perturbations.
Decoupling the system of differential equations \eqref{axial1} and
\eqref{axial2} in terms of the fundamental variable $Z_{1}$,
it results the solely second-order differential equation
\begin{equation}
Z_{1}^{''}-\frac{2(\mathfrak{w}^{2}-\mathfrak{q}^{2}
\mathfrak{h})\mathfrak{h}-u\mathfrak{h}^{'}
\mathfrak{w}^{2}}{u\mathfrak{h}(\mathfrak{w}^{2}-\mathfrak{q}^{2}
\mathfrak{h})}Z_{1}^{'}
+\frac{\mathfrak{w}^{2}-\mathfrak{q}^{2}
\mathfrak{h}}{\mathfrak{h}^{2}}Z_{1}=0.
\label{yeqaxial}
\end{equation}
Solutions to this equation satisfying the QNM boundary conditions are
studied below.
\subsubsection{Polar perturbations in the radial gauge}
The polar (longitudinal) sector of gravitational perturbations in the radial
gauge is described by metric fluctuations $h_{tt}$, $h_{xx}$, $h_{yy}$ and
$h_{ty}$. These components of the metric perturbation tensor are used to
define new quantities
\begin{equation}
H_{tt}=\frac{1}{f}\,h_{tt},\qquad
H_{xx}=\frac{L^{2}}{r^{2}}\,h_{xx},\qquad
H_{yy}=\frac{L^{2}}{r^{2}}\,h_{yy},\qquad
H_{ty}=\frac{L^{2}}{r^{2}}\,h_{ty},
\label{polarh-ij}
\end{equation}
which are more appropriate to deal with during calculations to obtain
perturbation equations. Hence, the polar components of linearized Einstein
equations furnish a set of seven coupled equations for the variables defined
in Eqs. \eqref{polarh-ij}. Only four of such a set are independent equations
and, among the possible choices, the more interesting for the present work
are
\begin{align}
H_{ty}^{'}=&\frac{2u\mathfrak{w}\mathfrak{q}}{b(u)} \left(H_{xx}
-H_{tt}\right)+ \frac{ua(u)}{2\mathfrak{q}\mathfrak{h}b(u)}
\left(\mathfrak{w}H_{xx} +\mathfrak{w}H_{yy}+ 2\mathfrak{q}H_{ty}\right)
-\frac{4\mathfrak{w}\mathfrak{h}}{\mathfrak{q} b(u)}H_{tt}^{'},
\label{polar1}\\ H_{xx}^{'}=&\frac{\mathfrak{w}ua(u)}{2\mathfrak{q}^{2}
\mathfrak{h}^{2}b(u)} \left(\mathfrak{w}H_{xx} +\mathfrak{w}H_{yy}
+2\mathfrak{q}H_{ty}\right) -\frac{c(u)}{\mathfrak{q}^{2}b(u)}H_{tt}^{'}+
\frac{2u\mathfrak{w}^{2}} {\mathfrak{h}b(u)} H_{xx}+\frac{ua(u)}{2
\mathfrak{h}b(u)}H_{tt}, \label{polar2}\\ \notag\\[-0.6cm]
H_{yy}'=&\frac{2u}{\mathfrak{h}b(u)}\left[\mathfrak{w}^{2} H_{xx}
+\mathfrak{w}^{2}H_{yy} +\mathfrak{q}^{2}\mathfrak{h} \left(H_{tt}
-H_{xx}\right) +2\mathfrak{q}\mathfrak{w} H_{ty}\right]
+\left(\frac{4\mathfrak{h}}{b(u)} +\frac{c(u)}{\mathfrak{q}^{2}
b(u)}\right)H_{tt}^{'} \nonumber\\ -&\frac{u}{2\mathfrak{h}b(u)}
\left[4\mathfrak{w}^{2} H_{xx} +c(u)H_{tt}\right]
-\frac{\mathfrak{w}uc(u)}{2\mathfrak{q}^{2} \mathfrak{h}^{2}b(u)}
\left(\mathfrak{w}H_{xx} +\mathfrak{w}H_{yy} +2\mathfrak{q}H_{ty}\right),
\label{polar3}\\ \notag\\[-0.6cm]
\!\!\!H_{tt}^{''}=&\frac{2\mathfrak{w}^{2}}{\mathfrak{h}b(u)} \left(H_{xx}
+H_{yy}\right) +\frac{2\mathfrak{q}}{\mathfrak{h}b(u)}
\left(2\mathfrak{w}H_{ty} +\mathfrak{q}\mathfrak{h} H_{tt}\right)
+\frac{(2+u^{3})}{2u\mathfrak{h}b(u)} \left[\mathfrak{q}^{2}H_{xx}
+(8+u^3)H_{tt}^{'}\right], \label{polar4}
\end{align}
where, as above, the primes denote derivatives with respect to the variable
$u=r_0/r$, and coefficients $a(u)= 3u^{4}-12u-4\mathfrak{w}^{2}$,
$b(u)=\mathfrak{h}+3$, and $c(u)=4\mathfrak{w}^{2}-\mathfrak{q}^{2}b(u)$
were introduced to simplify notation.
Finally, a gauge-invariant function $Z_2$ is built as a particular
combination of the metric perturbations
\begin{equation}
Z_2=4\omega qH_{ty}+2\omega^{2}H_{yy} +\left[q^{2}(3-\mathfrak{h})
-2\omega^{2}\right] H_{xx}+2q^{2}\mathfrak{h}H_{tt},
\label{mestrepolar}
\end{equation}
for which, uncoupling the equations of motion \eqref{polar1}-\eqref{polar4},
it is found the following second-order differential equation
\begin{equation}
Z_{2}^{''}-\frac{4\mathfrak{w}^{2}(2+u^{3})+\mathfrak{q}^{2}
d(u)}{u\mathfrak{h}c(u)}Z_{2}^{'} +\frac{4\mathfrak{w}^{4}
+\mathfrak{q}^{4}\mathfrak{h}b(u) -\mathfrak{q}^{2}e(u)}
{\mathfrak{h}^{2}c(u)}Z_{2}= 0,
\label{yeqpolar}
\end{equation}
where $d(u)=4u^{3}-5u^{6}-8$ and
$e(u)=9u^{4}\mathfrak{h}+\mathfrak{w}^{2}(8-5u^{3})$.
Equations \eqref{yeqaxial} and \eqref{yeqpolar} are the fundamental
equations that are going to be used in the next sections to compute
the QNM spectra associated to gravitational perturbations
of plane AdS$_4$ black holes.
\subsection{Stress-energy tensor correlation functions}
\label{green-gravit}
For the gravitational perturbations, the AdS/CFT correspondence
establishes a relation among the solutions of Eqs. \eqref{yeqaxial} and
\eqref{yeqpolar} and the stress-energy tensor of the dual CFT. From this
relation, the stress-energy tensor correlators can be determined, and in
order to do that the explicit form of the fields in the bulk AdS spacetime
has to be known. More precisely, in order to impose the ingoing-wave
condition at the horizon, and to map AdS to CFT quantities at the boundary
of the spacetime, the asymptotic form of the metric perturbation
functions close to the horizon and at the boundary are necessary.
In the horizon neighborhood ($u\approx 1$), the gravitational
gauge-invariant variables $Z_1$ and $Z_2$ have a similar behavior as the
electric field components (see Sect. \ref{green-electro}), viz,
$Z_{1,2}\sim\mathfrak{h}^{\pm i\mathfrak{w}/3}$. As in the electromagnetic
case, to compute the retarded Green functions, one has to choose
the solutions corresponding to the negative imaginary power,
$Z_{1,2}\sim\mathfrak{h}^{-i\mathfrak{w}/3}$. On the other hand,
at the conformal boundary of the AdS spacetime, the metric fluctuations
are such that
\begin{gather}
Z_{1}=\mathcal{A}_{(1)}(\mathfrak{w},\mathfrak{q})
+...\;+\mathcal{B}_{(1)}(\mathfrak{w},\mathfrak{q})u^{3}+...\, ,
\label{assintgrav1}\\
Z_{2}=\mathcal{A}_{(2)}(\mathfrak{w},\mathfrak{q})+...\;+
\mathcal{B}_{(2)}(\mathfrak{w},\mathfrak{q})u^{3}+...\, ,
\label{assintgrav2}
\end{gather}
where ellipses denote higher powers of $u$, and quantities
$\mathcal{A}_{(1)}(\mathfrak{w},\mathfrak{q})$ and
$\mathcal{B}_{(1)}(\mathfrak{w},\mathfrak{q})$, and
$\mathcal{A}_{(2)}(\mathfrak{w}, \mathfrak{q})$ and
$\mathcal{B}_{(2)}(\mathfrak{w},\mathfrak{q})$
are the connection coefficients related to the differential equations
\eqref{yeqaxial} and \eqref{yeqpolar}, respectively.
For the remaining of this section, as usual, the gravitational perturbations
are split into axial and polar sectors and the analysis of the corresponding
actions, coming from Eq. \eqref{acaocompleta}, are performed separately for
both of the perturbation types.
\subsubsection{Axial sector}
It is well known that in the calculation of two-point correlation functions
from the gravitational action only quadratic terms in metric perturbations
need to be considered. Moreover, according to the Lorentzian AdS/CFT
prescription \cite{son1} (see also Ref. \cite{policastro1}), in order to
obtain the CFT retarded correlators the relevant terms are the quadratic
terms in the derivatives of $H_{\mu\nu}$. Hence, collecting all of the
contributions coming from the gravitational part of action
\eqref{acaocompleta}, one gets
\begin{equation}
S^{\scriptscriptstyle{(2)}}=\frac{P}{2}\int du\,d^{3}x\frac{1}{u^2}
\left[H_{tx}^{'2}-\mathfrak{h}
H_{yx}^{'2}\right]+...\,,
\label{acaoaxial1}
\end{equation}
where
\begin{equation}
P=\left(\frac{4\pi T}{3}\right)^{3}\frac{L^{2}}{2\kappa_{4}^2}=
\frac{8\sqrt{2}}{81}\pi^{2}N^{3/2}T^{3}
\label{pressao}
\end{equation}
is interpreted as the pressure of the dual plasma \cite{herzog3}.
Now expressing functions $H_{tx}^{'}$ and $H_{yx}^{'}$ in terms of the axial
fundamental variable $Z_{1}$ through Eqs. \eqref{axial1}-\eqref{mestreaxial},
substituting the resulting relations into Eq. \eqref{acaoaxial1}, and making
use of the fundamental equation \eqref{yeqaxial}, it is found the following
action (at the boundary)
\begin{equation}
S_{\scriptscriptstyle{boundary}}^{\scriptscriptstyle{(2)}}=\frac{P}{2}\;\underset{u\rightarrow 0}{
\mbox{lim}}\,\int\frac{d\mathfrak{w}\, d\mathfrak{q}}{(2\pi)^{2}}\,
\frac{\mathfrak{h}}{u^{2}(\mathfrak{w}^{2}-\mathfrak{q}^{2}\mathfrak{h})}
Z_{1}^{'}(u,k)Z_{1}(u,-k)+ S_{\scriptscriptstyle{CT}}^{\scriptscriptstyle{(2)}},
\label{axial-acao2}
\end{equation}
where the contact terms represented by $S_{\scriptscriptstyle{CT}}^{\scriptscriptstyle{(2)}}$ do not carry
derivatives of the metric perturbation functions. In the calculation of the
correlation functions, after Fourier transformation, the contact terms give
rise to derivatives of the Dirac delta function. Their removal can be done
through the holographic renormalization, with the inclusion of appropriate
counter terms in the supergravity action \cite{bianchi}.
Besides using Eq. \eqref{mestreaxial}, the asymptotic expansion given by Eq.
\eqref{assintgrav1} is used to write the derivative of the gauge-invariant
quantity $Z_{1}$ in terms of the boundary values of the perturbation fields
$H_{\mu\nu}^{0}(k) =H_{\mu\nu}(u\rightarrow 0,k)$, and then the AdS/CFT
prescription \cite{son1} can be applied to the present case in order to
calculate the retarded correlation functions of the holographic stress-energy
tensor $T^{\mu\nu}$. The result is
\begin{align}
&G_{tx,tx}=-3P\frac{\mathfrak{q}^{2}}{
(\mathfrak{w}^{2}-\mathfrak{q}^{2})}\frac{\mathcal{B}_{(1)}
(\mathfrak{w},\mathfrak{q})}{\mathcal{A}_{(1)}
(\mathfrak{w},\mathfrak{q})},
\label{correlacao-tftf}\\
&G_{tx,yx}=3P\frac{\mathfrak{w}\mathfrak{q}}{
(\mathfrak{w}^{2}-\mathfrak{q}^{2})}\frac{\mathcal{B}_{(1)}
(\mathfrak{w},\mathfrak{q})}{\mathcal{A}_{(1)}
(\mathfrak{w},\mathfrak{q})},
\label{correlacao-tfyf}\\
&G_{yx,yx}=-3P\frac{\mathfrak{w}^{2}}{
(\mathfrak{w}^{2}-\mathfrak{q}^{2})}\frac{\mathcal{B}_{(1)}
(\mathfrak{w},\mathfrak{q})}{\mathcal{A}_{(1)}
(\mathfrak{w},\mathfrak{q})}.
\label{correlacao-yfyf}
\end{align}
As it happens for the current-current correlations functions, one can
find general expressions for the two-point thermal functions associated
to the stress-energy tensor which hold for any scale invariant
$(2+1)$-dimensional field theory (see Appendix \ref{apen-correlations}).
For fluctuations of the transverse momentum density in the CFT
one has the correlators
\begin{align}
&G_{tx,tx}= \frac{\mathfrak{q}^{2}} {2\left(\mathfrak{w}^2
-\mathfrak{q}^{2}\right)} \,G_{1}(\mathfrak{w},\mathfrak{q}),
\label{emtcorrfun1}\\
&G_{tx,yx}= -\frac{\mathfrak{w}\mathfrak{q}}{2\left(\mathfrak{w}^2
-\mathfrak{q}^{2}\right)}\,G_{1}(\mathfrak{w},\mathfrak{q}),
\label{emtcorrfun2}\\
&G_{yx,yx}=\frac{\mathfrak{w}^{2}}{2\left(\mathfrak{w}^2
-\mathfrak{q}^{2}\right)} \,G_{1}(\mathfrak{w},\mathfrak{q}).
\label{emtcorrfun3}
\end{align}
Therefore, by comparing the general expressions of Eqs.
\eqref{emtcorrfun1}-\eqref{emtcorrfun3} to the results given in Eqs.
\eqref{correlacao-tftf}-\eqref{correlacao-yfyf}, the following scalar
function is found
\begin{equation}
G_{1}(\mathfrak{w},\mathfrak{q})=-6P\,\frac{\mathcal{B}_{(1)}
(\mathfrak{w},\mathfrak{q})}{\mathcal{A}_{(1)}(\mathfrak{w},
\mathfrak{q})}.\label{G_1}
\end{equation}
It is then seen that Dirichlet condition imposed on the fundamental variable
$Z_{1}$ at the boundary, $Z_{1}(0)=\mathcal{A}_{(1)}(\mathfrak{w},
\mathfrak{q})=0,$ leads straightforwardly to the poles of the correlation
functions $G_{tx,tx}$, $G_{tx,yx}$ and $G_{yx,yx}$. As a consequence of this
result, such a requirement also yields the quasinormal spectrum associated to
the axial gravitational perturbation modes of plane-symmetric black holes.
\subsubsection{Polar sector}
The procedure to be applied to the polar sector of gravitational
perturbations is the same as for the axial sector. The starting point here is
the part of the boundary gravitational action built with the quadratic terms
in the polar metric perturbations, which is given by \cite{herzog3}
\begin{equation}
\begin{split}
S_{\scriptscriptstyle{boundary}}^{\scriptscriptstyle{(2)}}=\frac{P}{2}\;
\underset{u\rightarrow 0}{\mbox{lim}}\,\int d^{3}x\bigg[&
\frac{1}{4}\left(2H_{tt}^{2}-8H_{ty}^{2}+H_{tt}H_{xx}+
H_{tt}H_{yy}\right)-\frac{1}{4}\left(H_{xx}-H_{yy}\right)^{2}\\
&-\frac{\mathfrak{h}}{2u^{2}}\left(H_{ty}^{2}+
H_{xx}H_{yy}-H_{tt}H_{xx}
-H_{tt}H_{yy}\right)^{'}\bigg].
\end{split}
\label{polar-acao1}
\end{equation}
By using the relation among polar metric fluctuations and the gauge-invariant
variable $Z_{2}$, Eq. \eqref{mestrepolar}, and the equations of motion
\eqref{polar1}-\eqref{polar4}, the boundary action \eqref{polar-acao1} can be
cast into the form
\begin{equation}
S_{\scriptscriptstyle{boundary}}^{\scriptscriptstyle{(2)}}=\frac{P}{2}\;\underset{u\rightarrow
0}{\mbox{lim}}
\,\int\frac{d\mathfrak{w}d\mathfrak{q}}{(2\pi)^{2}}\,
\frac{\mathfrak{h}}{u^{2}\left[4\mathfrak{w}^{2}-
\mathfrak{q}^{2}(4-u^{3})\right]^{2}}\,Z_{2}^{'}(u,k)
Z_{2}(u,-k)+S_{\scriptscriptstyle{CT}}^{\scriptscriptstyle{(2)}},
\label{polar-acao2}
\end{equation}
where the contact terms $S_{\scriptscriptstyle{CT}}^{\scriptscriptstyle{(2)}}$ do not contain derivatives
of metric perturbations. One now uses the asymptotic expansion
\eqref{assintgrav2} to write the derivative of the gauge-invariant variable
$Z_2$ in terms of boundary values of the polar metric perturbations
$H_{\mu\nu}^{0}(k)$. After substituting the resulting expression into the
action \eqref{polar-acao2}, the appropriate functional
derivatives\footnote{Functional derivatives in the sense defined by the
Lorentzian AdS/CFT prescription of Ref. \cite{son1}.} of the action with
respect to
the independent fields $H_{tt}^{0}(k)$, $H_{ty}^{0}(k)$, $H_{yy}^{0}(k)$ and
$H_{xx}^{0}(k)$ are performed. Notice, however, that in the case of polar
gravitational perturbations, the use of the AdS/CFT prescription is not
direct. In fact, it is first necessary to identify explicitly how the metric
perturbations couple to the stress-energy tensor at the boundary. As
discussed in Refs. \cite{liu, policastro2}, such a coupling
is given by
\begin{equation}
-\frac{1}{2}\int dt\, d^{2}x\,h_{\;\,\mu}^{\nu}T_{\;\;\nu}^{\mu}=
-\frac{1}{2}\int\,dt\,d^{2}x\left[H_{tt}^0T^{tt}
+H_{xx}^{0}T^{xx} +H_{yy}^{0}T^{yy}+2H_{ty}^{0}T^{ty}\right].
\end{equation}
Taking this coupling into account, the covariant components of the polar
correlation functions are found
\begin{equation}
G_{\mu\nu,\alpha\beta}=Q_{\mu\nu,\alpha\beta}
G_{2}(\mathfrak{w},\mathfrak{q}),
\end{equation}
where the scalar function $G_{2}(\mathfrak{w},\mathfrak{q})$
is given by
\begin{equation}
G_{2}(\mathfrak{w},\mathfrak{q})=
-6P\frac{\mathcal{B}_{(2)}(\mathfrak{w},\mathfrak{q})}{
\mathcal{A}_{(2)}(\mathfrak{w},\mathfrak{q})}+\mbox{contact terms},
\end{equation}
and tensor $Q_{\mu\nu,\alpha\beta}$ is given in Appendix
\ref{apen-correlations} (see also \cite{kovtun1}). This result shows
definitely that Dirichlet boundary condition imposed on the fundamental
variable $Z_{2}$ at infinity leads to the poles of the function
$G_{2}(\mathfrak{w},\mathfrak{q})$, and, by definition, to the quasinormal
frequencies of polar gravitational vibration modes.
\subsection{QNM and the gauge-invariant variables}
As in the study of electromagnetic quasinormal modes, a comparison between
results found using RWZ variables with the ones obtained using
KS gauge-invariant variables to describe gravitational
perturbations deserves to be made. As in that case, while the axial
quasinormal spectrum is independent of the choice of the fundamental
variable, polar quasinormal spectrum strongly depends on it.
A strong evidence that the QNM spectra obtained using either RWZ or KS
variables to describe axial gravitational perturbations are identical
comes from the study of the hydrodynamic limit of such perturbations as
performed in Ref. \cite{miranda1} and in Sect. \ref{hydro-grav} (see below).
As a matter of fact, even though different methods have been employed in
each case, both of the quasinormal spectra present a typical hydrodynamic
shear mode with diffusion coefficient $D=1/4\pi T$, independently if one
uses variable $Z^{\scriptscriptstyle{(-)}}$, or variable $Z_{1}$. Furthermore, it can be
shown that the explicit relation between RWZ and KS variables is
\begin{equation}
Z_{1}=\frac{f}{r^{2}}\partial_{r} \left[rZ^{\scriptscriptstyle{(-)}}\right],
\end{equation}
and at the AdS spacetime boundary, axial fundamental variables $Z^{\scriptscriptstyle{(-)}}$
and $Z_{1}$ are proportional to each other,
\begin{equation}
Z_{1}(u)\big|_{u=0}=\frac{1}{L^{2}}Z^{\scriptscriptstyle{(-)}}(u)\big|_{u=0}\, ,
\end{equation}
what proves that the two spectra obtained from $Z^{\scriptscriptstyle{(-)}}$ and $Z_{1}$
are indeed identical to each other.
The situation in the polar sector is quite diverse from what happens in the
axial sector. As it is shown in the sequence of the present work (see Sects.
\ref{hydro-grav} and \ref{dispersion-gravit}), RWZ and KS variables with the
same boundary conditions generate different quasinormal frequencies. In
particular, it is shown in Sect. \ref{hydro-grav} that the hydrodynamic limit
of $Z_{2}$ contains a sound wave mode which is not seen in the quasinormal
spectrum obtained from $Z^{\scriptscriptstyle{(+)}}$.
\subsection{Dispersion relations for the hydrodynamic QNM}
\label{hydro-grav}
For the fluctuations of the stress-energy tensor in the dual field theory,
hydrodynamics predicts a shear mode in the transverse (axial) sector and a
sound wave mode in the longitudinal (polar) sector. As it is going to be
shown below, these modes also appear in the gravitational QNM spectra of the
plane black holes as long as one investigates the regime of small
frequencies and wavenumbers, $\mathfrak{w}\rightarrow 0$ and
$\mathfrak{q}\rightarrow 0$. Proceeding analogously to the case of
electromagnetic perturbations studied in Sect. \ref{hydro-eletro}, the
following change of variables is done
\begin{equation}\label{Z312}
H_{j}(u)=\mathfrak{h}^{i\mathfrak{w}/3}Z_{j}(u),\quad\qquad j=1,2.
\end{equation}
Functions $H_{1}$ and $H_{2}$ are then expanded in power series of
$\mathfrak{w}$ and $\mathfrak{q}$. Besides being approximate solutions to
Eqs. \eqref{yeqaxial} and \eqref{yeqpolar} in the hydrodynamic limit, such
series must represent ingoing waves at the horizon, namely,
$H_{j}(u)\big|_{u=1}= {\rm constant}$. These conditions are fulfilled by the
following expansions:
\begin{equation}
Z_{1}=C_{1}\mathfrak{h}^{-i\mathfrak{w}/3}
\left[1+i\frac{\mathfrak{q}^{2}\mathfrak{h}}{3\mathfrak{w}}
+{\mathcal{O}}(\mathfrak{w}^{2})\right],\\
\end{equation}
\begin{equation}
Z_{2}=C_{2}\mathfrak{h}^{-i\mathfrak{w}/3}\left[2-u^{6}
-4\frac{\mathfrak{w}^{2}}{\mathfrak{q}^{2}}-
\frac{4i\mathfrak{w}\mathfrak{h}}{3}+{\mathcal{O}}(\mathfrak{w}^{2})
\right],\\
\end{equation}
where $C_{1}$ and $C_{2}$ are arbitrary normalization constants. Dirichlet
conditions at the spacetime boundary, $u=0$, are now imposed on both
the axial and the polar fundamental variables, $Z_{1}(0)=0$ and $Z_{2}(0)=0$.
The first variable leads to an axial quasinormal mode identified to the shear
mode with
\begin{equation}
\omega=-iDq^{2},
\label{shear}
\end{equation}
while the second variable furnishes a dispersion relation
characteristic to a sound wave mode
\begin{equation}
\omega=\pm\frac{1}{\sqrt{2}}q-\frac{iDq^{2}}{2}\, ,
\label{soundwave}
\end{equation}
where $D$ is the diffusion coefficient, given by
\begin{equation}
D=\frac{\eta}{\varepsilon+P}=\frac{1}{4\pi T}, \label{Dcoeff}
\end{equation}
with $\eta$ and $\varepsilon$ being respectively the shear coefficient and the
energy density of the dual system.
\FIGURE{
\centering \epsfig{file=shearmode.eps, height=9.41cm, width=6.0cm, angle=270}
\caption{The dispersion relation (imaginary part) for the axial
gravitational
hydrodynamic mode (solid line), in comparison to the shear mode,
$\mathfrak{q}^{2}/3$ (dotted line), and to the algebraically special
mode, $\mathfrak{q}^{4}/6$ (dashed line).}
\label{hidrograv_axial}}
Among other interesting results that can be found, combining Eq.
\eqref{Dcoeff} to the thermodynamic Euler relation $P=-\varepsilon+Ts\Rightarrow
s=(\varepsilon+P)/T$, one gets the ratio between the shear coefficient and the
entropy density of the dual plasma,
\begin{equation}
\frac{\eta}{s}=\frac{\eta\,T}{\varepsilon+P}=\frac{1}{4\pi},
\end{equation}
or, in conventional units, $\eta/s=\hbar/4\pi k_{\scriptscriptstyle{B}}$. This ratio is the
same for all finite temperature field theories with a dual gravitational
description in the AdS spacetime \cite{kovtun2}. It is also speculated that
$\eta/s=1/4\pi$ represents a lower bound --the KSS bound-- of such a ratio
for all fluids in nature.
The complete dispersion relations for the gravitational hydrodynamic
QNM are obtained by means of the Horowitz-Hubeny method
\cite{horowitz1}, and, for the sake of simplicity, the numerical results for
the axial and polar sectors are analyzed separately.\\
\noindent
{\it (i) Axial modes}: The extension of the dispersion relation of the axial
hydrodynamic QNM for large values of $\mathfrak{q}$ was already done in Ref.
\cite{miranda1}, but for completeness it is also shown in Fig.
\ref{hidrograv_axial}. As the normalized wavenumber $\mathfrak{q}$ reaches
values beyond of the hydrodynamic regime, the magnitude of the QNM frequency
$\mathfrak{w}=-i\mathfrak{w}_{I}$ increases faster than the magnitude of the
shear mode frequency $\mathfrak{w}=-i\mathfrak{q}^{2}/3$, and for very large
wavenumber values, $\mathfrak{q}\gg 1$, the hydrodynamic QNM frequency
approaches the algebraically special frequency
$\mathfrak{w}=-i\mathfrak{q}^{4}/6$.\\
\FIGURE{
\centering\epsfig{file=hidrorealpolar_new.eps, height=7.137cm,
width=4.968cm, angle=270}
\centering\epsfig{file=hidroimagpolar_new.eps, height=7.137cm,
width=4.968cm, angle=270}
\caption{Real and imaginary parts, respectively, of the dispersion
relation $\mathfrak{w}(\mathfrak{q})$ for the hydrodynamic longitudinal
gravitational mode (solid lines). The dashed lines in both of the figures
correspond to the sound wave mode, given by Eq. \eqref{soundwave}, and the
dotted curve in the figure on the left hand side is the relation
$\mathfrak{w}_{R}=\mathfrak{q}$.}
\label{hidrogravpolar}}
\FIGURE{
\centering\epsfig{file=hidroderivada.eps, height=7.137cm,
width=4.968cm, angle=270}
\centering\epsfig{file=hidroderimag.eps, height=7.137cm,
width=4.968cm, angle=270}
\caption{The group velocity $c_{s}=d\mathfrak{w}_{R}/d\mathfrak{q}$ and the
derivative $d\mathfrak{w}_{I}/d\mathfrak{q}$ as a function of the normalized
wavenumber $\mathfrak{q}$ for the polar hydrodynamic gravitational mode
(solid lines).
The dashed line in the figure on the left is the sound velocity for a
(2+1)-dimensional CFT, $c_{s}=1/\sqrt{2}$, and the
dotted line in the same figure represents the speed of light, $c=1$.}
\label{hidroderivada}}
\noindent
{\it (ii) Polar modes}: In this sector of perturbations, RWZ and KS variables
submitted to the same boundary conditions generate different quasinormal
frequencies. As seen above, the hydrodynamic limit of $Z_{2}$ presents a
sound wave mode which is not present in the quasinormal spectrum obtained
from $Z^{\scriptscriptstyle{(+)}}$. The extended dispersion relations for the longitudinal
hydrodynamic QNM are shown in Fig. \ref{hidrogravpolar}. The real part of the
frequency clearly shows the transition from the hydrodynamic regime
$\mathfrak{w}_{R}= \mathfrak{q}/\sqrt{2}$, at low wavenumbers, to a regime
characterized by collisionless dual plasma in which the dispersion relation
$\mathfrak{w}_{R}(\mathfrak{q})$ approaches the ultra-relativistic relation
$\mathfrak{w}_{R}=\mathfrak{q}$. Between these two extreme regimes, the group
velocity, defined by $c_{s}=d\mathfrak{w}_{R}/d\mathfrak{q}$, assumes values
that are higher than the speed of light. The graphs in Fig.
\ref{hidroderivada} show that $c_{s}>1$ for all wavenumbers larger than
$\mathfrak{q}\simeq 1.336$, and that the minimum decaying time (the maximum
of $\mathfrak{w}_{I}$) corresponds to $\mathfrak{q}\simeq 3.213$. Notice,
however, that $d\mathfrak{w}_{R}/d\mathfrak{q}$ surpasses the speed of light
at wavenumber values lying outside the hydrodynamic regime and therefore that
superluminal group velocity cannot be interpreted as the sound velocity in
the corresponding media.
\subsection{Dispersion relations for the non-hydrodynamic QNM}
\label{dispersion-gravit}
Conventionally, a non-hydrodynamic QNM is every mode for which the dispersion
relation presents a gap in the limit $\mathfrak{q}\rightarrow 0 $. That is to
say, the quasinormal frequency $\mathfrak{w}(\mathfrak{q})$ tends to a
nonzero value in the limit where the wavenumber goes to zero. These kind of
gravitational perturbation modes are studied in this section.
As done in the case of electromagnetic perturbations, the QNM obtained by
using RWZ gravitational variables $Z^{\scriptscriptstyle{(\pm)}}$, obeying
Schr\"odinger-like equations, are compared to the QNM obtained by using the
KS gauge-invariant quantities $Z_{1,2}$, which lead to the
poles of stress-energy tensor correlators in the $\mathcal{N}=8$
super-Yang-Mills field theory. In particular, the results obtained here from
$Z_{1,2}$ are compared to the results of Refs. \cite{cardoso5,miranda1}.
\subsubsection{Purely damped modes}
The spectra of gravitational perturbations of plane-symmetric $AdS_{4}$
black holes do not present non-hydrodynamic quasinormal frequencies
with vanishing real part. In fact, the only purely damped mode
of the gravitational perturbations is the axial hydrodynamic QNM
which was already investigated in the last section.
\subsubsection{Ordinary quasinormal modes}
\label{nonhydro-gravit}
As usual, the study of the dispersion relations of regular non-hydrodynamic
gravitational QNM is more conveniently performed by considering axial and
polar sectors of such perturbations separately.\\
\noindent
{\it (i) Axial modes}: As shown above, in the case of gravitational axial
modes both of the fundamental variables $Z_{1}$ and $Z^{\scriptscriptstyle{(-)}}$ yield the
same QNM spectrum. Even though a detailed study of the fundamental
non-hydrodynamic QNM (based on the RWZ master variable) was performed in Ref.
\cite{miranda1}, higher overtones of axial gravitational QNM were not fully
investigated. Hence, the aim here is to complete the analysis by including
such higher overtones. Fig. \ref{gravaxial} shows the numerical results for
the dispersion relations of the first five axial quasinormal modes:
$n=1$,..., $5$. The general forms of the curves
are approximately the same for all values of $n$:
At intermediate values of $\mathfrak{q}$, there is a local minimum in the
real part of the frequency $\mathfrak{w}_{R}(\mathfrak{q})$, and for large
values of the wavenumber, every dispersion relation
$\mathfrak{w}_{R}(\mathfrak{q})$ tends to some straight line parallel to the
ultra-relativistic energy-momentum relation,
$\mathfrak{w}_{R}=\mathfrak{q}$.\\
\FIGURE{
\centering\epsfig{file=regravaxial.eps, height=7.137cm,
width=4.968cm, angle=270}
\centering\epsfig{file=imgravaxial.eps, height=7.137cm,
width=4.968cm, angle=270}
\caption{The first five quasinormal frequencies of non-hydrodynamic axial
gravitational modes, $\mathfrak{w}=(3\omega_{R}/4\pi T)-i(3\omega_{I}/4\pi
T)$, as a function of the normalized wavenumber $\mathfrak{q}=3q/4\pi T$.
The quantum number $n$ arranges the modes in growing order according to
the strength of the imaginary parts of the frequencies.}
\label{gravaxial}}
\FIGURE{
\centering\epsfig{file=regravpolar.eps, height=7.137cm,
width=4.968cm, angle=270}
\centering\epsfig{file=imgravpolar.eps, height=7.137cm,
width=4.968cm, angle=270}
\caption{The first five quasinormal frequencies of non-hydrodynamic
polar gravitational modes, $\mathfrak{w}=(3\omega_{R}/4\pi T)
-i(3\omega_{I}/4\pi T)$, as a function of the normalized
wavenumber $\mathfrak{q}=3q/4\pi T$, ordered as in
the case of axial gravitational modes.}
\label{gravpolar} }
\noindent
{\it (ii) Polar modes}:
Contrary to the axial perturbations, the longitudinal gravitational
fluctuations present different spectra as one takes $Z_{2}$ or $Z^{\scriptscriptstyle{(+)}}$
as fundamental variable. The numerical results for these modes, based on the
gauge-invariant variable $Z_{2}$, are shown in Fig. \ref{gravpolar}. Notice
that now the real part of the quasinormal frequency $\mathfrak{w}_{R}$ is a
monotonic increasing function of $\mathfrak{q}$, showing neither local maxima
nor local minima which, on contrary, appear in the quasinormal spectrum for
the RWZ master variable $Z^{\scriptscriptstyle{(+)}}$ used in Refs.
\cite{cardoso5,miranda1}. A comparison between the QNM spectrum found from
the RWZ variable (cf. Ref. \cite{miranda1}) and that found from the
KS variable used here can also be done through the data
presented in Table \ref{tabelapolar}. It is clearly seen the similarity
between the two spectra in the two asymptotic regions of wavenumber values.
Both of the spectra are approximately the same for small values and also for
large values of $\mathfrak{q}$. The main differences happen in the regime
where the normalized wavenumber $\mathfrak{q}$ is of the order of unity,
which means that $q$ is of the order of the blackhole Hawking temperature.
Moreover, the real parts of the quasinormal frequencies in both of the
spectra are relatively closer to each other when compared to the
corresponding imaginary parts. This implies that, even though the QNM
oscillation frequencies are essentially the same for both choices of
variables, the decaying timescales ($\tau=1/\omega_{I}$) are significantly
smaller for the KS choice.
\TABLE{
\begin{tabular}{lccccc}
\hline\hline
& \multicolumn{2}{c}{Kovtun-Starinets} & \multicolumn{2}{c}
{Regge-Wheeler-Zerilli}\\
\cline{2-5}
$\quad\mathfrak{q}\quad $ & $\qquad\mathfrak{w}_{R}\qquad $ &
$\qquad\mathfrak{w}_{I}\qquad $
& $\qquad\mathfrak{w}_{R}\qquad $ &
$\qquad\mathfrak{w}_{I}\qquad $\\
\hline
0.004 & 1.84942 & 2.66385 & 1.84945 & 2.66384\\
0.04 & 1.84964 & 2.66379 & 1.85027 & 2.66248\\
0.4 & 1.87207 & 2.65770 & 1.92488 & 2.52658\\
1 & 2.00603 & 2.62917 & 2.03016 & 1.92213\\
2 & 2.60256 & 2.56803 & 2.30526 & 1.55218\\
5 & 5.57791 & 2.25854 & 5.25618 & 1.27974\\
10 & 10.5703 & 1.94304 & 10.2839 & 1.07342\\
\hline\hline
\end{tabular}
\centering
\caption{Numerical results for the first non-hydrodynamic quasinormal mode
associated to the polar gravitational perturbations. The second and third
columns show respectively the values of real and imaginary parts of the
frequencies obtained by using the KS variable $Z_{2}$, and the last two
columns present the results obtained by using the RWZ variable
$Z^{\scriptscriptstyle{(+)}}$, which are also shown in Ref. \cite{miranda1}.}
\label{tabelapolar}}
\section{Final comments and conclusion}
\label{consid-final}
One of the important issues dealt with in the present work is related to the
choice of appropriate variables in order to determine the QNM spectra of AdS
black holes. It is argued that Kovtun-Starinets gauge-invariant quantities,
together with incoming-wave condition at horizon and Dirichlet boundary
condition at infinity, should be used in order to find the correct
quasinormal dispersion relations. The resulting spectrum for a given
perturbation is in general different from what is obtained using other
kind of variables with the same boundary conditions.
In the case of electromagnetic perturbations, contrary to what is obtained
using the so-called Regge-Wheeler-Zerilli quantities $\Psi^{\scriptscriptstyle{(\pm)}}$ where
the spectra for both the axial and the polar modes are the same,
Kovtun-Starinets variables present a different spectrum for each mode type.
Also, both these sectors of electromagnetic perturbations present purely
damped modes whose dispersion relations, in the limit of small wavenumbers,
approach the bosonic Matsubara frequencies $\omega=-2i\pi Tn_{s}$. This
result can be compared to the ``quasi-Matsubara'' frequencies, $\omega=2\pi
Tn_{s}(1-i)$, that have been found for zero wavenumber fluctuations of
$(4+1)$-dimensional black branes \cite{nunez}. The studies show the emergence
of infinite sequences of bosonic Matsubara frequencies for both $(3+1)$- and
$(4+1)$-dimensional black branes, but the particular behavior of the QNM
dispersion relations at zero wavenumber is quite different in each case. In
$(3+1)$ dimensions the real part of the frequencies is zero for very small
wavenumber values, while in $(4+1)$ dimensions real and imaginary parts of the
frequencies are both finite at zero wavenumber. Moreover, as pointed out in
Sect. \ref{dispersion-eletro} and discussed in detail in Ref. \cite{herzog4},
the invariance of Maxwell equations under the electric
field $\leftrightarrow$ magnetic field duality operation in
$(3+1)$-dimensional spacetimes implies that there are no electromagnetic QNM
at zero wavenumber. Such a duality invariance does not hold in higher
dimensional spacetimes, what justifies the different behavior of zero
wavenumber electromagnetic QNM found here when compared to the results of Ref.
\cite{nunez}.
Other special property of electromagnetic fluctuations deserving to be
mentioned here is the cutoff in the dispersion relations of
purely damped QNM at a particular value of the wavenumber,
$\mathfrak{q}=\mathfrak{q}_{\mbox{\scriptsize{lim}}}$. This cutoff implies in
an abrupt change in the behavior of the fundamental quasinormal mode, but
not in the thermalization time $\tau$. For wavenumbers in the interval
$0<\mathfrak{q}<\mathfrak{q}_{\mbox{\scriptsize{lim}}}$, parameter $\tau$ is
given by the first purely damped mode, $\tau=1/\omega_{s}$, while for
wavenumber values above $\mathfrak{q}_{\mbox{\scriptsize{lim}}}$, the
characteristic decaying time are governed by the fundamental ordinary QNM,
$\tau=1/\omega_{I}$. Since the imaginary parts of the frequencies of these
modes are equal for $\mathfrak{q}=\mathfrak{q}_{\mbox{\scriptsize{lim}}}$, the
thermalization time changes continuously for wavenumbers close to
$\mathfrak{q}_{\mbox{\scriptsize{lim}}}$.
The numerical results for the gravitational QNM show that the thermalization
time of axial modes for wavenumbers in the interval $0<\mathfrak{q}<1.935$ is
determined by the hydrodynamic mode. This means that, at least in the limit
$\mathfrak{q}\ll 1$, where the shear mode is a good approximation for the
QNM, the thermalization time $\tau$ is a linear function of Hawking
temperature, $\tau\simeq 4\pi T/q^{2}$. As in the case of
electromagnetic fluctuations, the transition to the regime where the
thermalization time is determined by the first regular axial QNM is
continuous. Such a transition happens for the values $\mathfrak{q}\simeq
1.935$ and $\mathfrak{w}_{I}\simeq 2.296$. Also, at this point the
thermalization time reaches its minimum value, $\tau\simeq 0.104/T$. On the
other hand, for polar gravitational perturbations the decaying time $\tau$
is always determined by the hydrodynamic QNM which reduces to the sound wave
mode in the small wavenumber limit $\mathfrak{q}\ll 1$.
Finally, the behavior of the group velocity $c_{s}$ shown in Fig.
\ref{hidroderivada}, which is greater than unity for all
$\mathfrak{q}>1.336$, deserves further analysis. First note that apparent
superluminal propagation of this type, which at first sight seems to violate
causality, has been found in other relativistic quantum systems. For
instance, Scharnhorst \cite{schar} has shown in the case of Casimir effect
that when vacuum fluctuations obey periodic boundary conditions, the two-loop
corrections to the polarization tensor lead to superluminal photon
propagation. Also, it is argued that the physically meaningful propagation
velocity and, consequently, the one that defines the light cones in
spacetime, is the front wave speed $v_{\mbox{\scriptsize{wf}}}$, which is
given by the limit of the phase speed $v_{\mbox{\scriptsize{ph}}}=\omega/q$
when $\omega\rightarrow\infty$ (see, e.g., Ref. \cite{shore} for a review).
The graph on the left in Fig. \ref{hidroderivada} shows that
$v_{\mbox{\scriptsize{ph}}}$ approaches $c=1$ at high frequencies, and,
therefore, the oscillations in the super-Yang-Mills plasma do not violate
causality.
\section*{Acknowledgments}
We thank conversations with Vitor Cardoso, Marc Casals, and Jos\'e
P. S. Lemos. ASM thanks Funda\-\c{c}\~{a}o Universidade Federal do ABC for
hospitality. VTZ thanks Funda\c c\~ao de Amparo \`a
Pesquisa do Estado de S\~ao Paulo (FAPESP) for financial help. VTZ is
supported by a fellowship from Conselho Nacional de Desenvolvimento Cient\'\i
fico e Tecnol\'ogico of Brazil (CNPq).
|
1,116,691,498,429 | arxiv | \chapter{Introduction} \label{sec::Intro}
Since the advent of quantum mechanics, the quantum nature of the interaction between light and atoms has been an active area of study. With the emergence of quantum information science came novel applications for large atomic ensembles interacting with light, so-called atom-light interfaces. For quantum information processing and quantum communication, these systems can form networks with light acting as a carrier of information and the ensemble acting as a quantum memory or repeater \cite{Fleischhauer2002,Polzik2004,Kimble2008,DLCZ,Kuzmich2004}. Collective degrees of freedom of the atomic ensemble are potential platforms for continuous variable quantum computing, in which light is utilized to perform gates and operations \cite{ContinuousVar}. Light can also mediate entangling interactions between the atoms in the ensemble, creating non-classical spin squeezed states for use in metrology \cite{KuzBig00,appel09,Takano2009,VulSqueezingClock,Koschorreck2010,TakTak05}.
The controllability of the atomic ensemble is vital for all of these applications. Quantum memories and repeaters require long coherence times that permit the storage of information in the spin degrees of freedom without degradation. The ability to create states that are non-Gaussian in the collective spin of the atomic ensemble is essential for continuous variable quantum computing. Spin squeezed states require the generation of either interatomic entanglement or entanglement within the internal spin degrees of freedom of the atoms. In addition to having applications in quantum information processing, control of large atomic ensembles is of interest in fundamental physics. Creating non-classical spin states in mesoscopic systems can shed light on poorly understood phenomena like the quantum-classical boundary and many-body entanglement.
The ability to produce spin squeezed states is a critical benchmark in the effort to control the collective spin of a large atomic ensemble. Spin squeezed states contain quantum correlations that reduce the variance of a spin component below the standard quantum limit (SQL). These correlations, which can be produced by entanglement between the atoms in the ensemble, serve as a measure for the strength of the collective control achievable in the atom-light interface. Beyond quantum control, spin squeezed states are of practical interest in metrology, where their quantum correlations can be harnessed to improve the precision of atomic clocks \cite{Wineland94, VulSqueezingClock} and magnetometers \cite{KosMitSq, BudkerSq}.
\section{Context of Dissertation}
This dissertation studies the generation of spin squeezed states in large ensembles of alkali atomic spins. While ensembles of alkali atoms figure prominently in experimental demonstrations and theoretical investigations of spin squeezing \cite{KuzBig00,appel09,Takano2009,VulSqueezingClock,Koschorreck2010,TakTak05}, this dissertation focuses on a property of alkali atoms seldom studied in the context of spin squeezing, control over the internal hyperfine ground spin. Ensembles of alkalis are unique platforms in that the spin of the hyperfine ground state is fully controllable \cite{ASmith13, MerkelControlPRA}. We seek to integrate control over the internal spins of the atoms that compose the ensemble with control over the collective spin. Squeezing of the collective spin and interatomic entanglement are generated by the Faraday interaction, which couples the collective spin to the polarization of the light. As we demonstrate in this dissertation, control over the internal spins of the alkalis has a surprising impact on both the strength of the Faraday interaction and the generation of the interatomic entanglement that contributes to spin squeezing.
Specifically, we find that by preparing the internal spins in a state with large projection noise variance, we can maximize the resolution of the collective spin in a measurement mediated by the light. This increases the measurement backaction on the ensemble, a signature of the enhanced entanglement generation between the atoms. Through the application of subsequent control on the internal spins, this entanglement can be converted into metrologically useful spin squeezing. Although spin squeezing is a phenomenon involving the collective spin of the atomic ensemble, it is enhanced substantially by control over the internal spin.
Much of the work on spin squeezing in both alkali ensembles and other platforms has focused on the case in which the ensemble is composed of two-level systems or ``qubits". In an ensemble of alkalis, this corresponds to a hyperfine spin quantum number of $f=1/2$. This dissertation places particular emphasis on ensembles where the alkali atomic spins are ``qudits", i.e. $f\geq 1/2$. The higher spin case is especially interesting from the perspective of internal spin control because of the greater number of internal degrees of freedom. Whereas for $f=1/2$ spin squeezing is achieved solely by the generation of interatomic entanglement, when $f>1/2$ entanglement between the internal degrees of freedom creates squeezing of the internal spin. Internal spin squeezing can be combined with interatomic entanglement when $f>1/2$ for a substantial enhancement in the overall squeezing of the ensemble.
Optical pumping due to spontaneous emission is the most significant source of decoherence in the atomic ensemble. In most previous work on spin squeezing, optical pumping is treated phenomenologically. We study the effects of optical pumping in the ensemble using a master equation derived from first principles in Ref. \cite{DeuJes09}. This treatment of optical pumping reveals a variety of interesting effects. Most relevant to our study is the substantial role played by the spin size, $f$, in the decay of spin squeezing due to optical pumping. Many of the optical pumping processes that damage spin squeezing are suppressed for larger $f$. Furthermore, ``transfers of coherence" that occur only when $f>1/2$ increase the robustness of interatomic entanglement. Even without internal spin squeezing, we find that alkali atoms with $f>1/2$ generate more squeezing than those with $f=1/2$ because they are less susceptible to decoherence.
Theoretical models of spin squeezing in large atomic ensembles often treat the light that couples to the atoms as a plane wave. In experimental implementations of squeezing, employing plane-like waves is undesirable because they are poorly mode matched to the radiation pattern of the ensemble. The coupling between the light and the ensemble is enhanced by utilizing paraxial beams that match the paraxial radiation emitted by a spatially extended atomic ensemble. In this dissertation, we analyze the spin squeezing produced by a quantum nondemolition (QND) measurement of the ensemble's collective spin mediated by a paraxial beam, rather than a plane wave. Because the intensity of the paraxial beam is spatially varying, mode matching, spin squeezing and optical pumping are influenced by the geometries of the probe and the atomic ensemble. We the find optimal geometries of the ensemble and probe for both mode matching and spin squeezing in the presence of optical pumping. To our knowledge, this is the first treatment of optical pumping in a three-dimensional, inhomogeneous atom-light interface.
\section{Summary of Dissertation}
The remainder of this dissertation is organized as follows. Chapter \ref{sec::Squeezing} introduces spin squeezing and defines several important concepts, such as the collective spin and the standard quantum limit (SQL). We analyze the entanglement and quantum correlations that produce spin squeezing. Also discussed is the use of spin squeezed states in metrological applications, such as magnetometry. Special emphasis is placed on the role of the ensemble's mean spin in metrological applications, where it increases the resolvability of measurement outcomes. We examine several methods of quantifying spin squeezing and introduce the Wineland squeezing parameter as a measure of spin squeezing and metrological usefulness.
Chapter \ref{sec::ALinterface} examines the interaction between the light and the atomic ensemble in detail. From the AC-Stark Hamiltonian that couples the atomic spins and the polarization of the light, we derive the Faraday interaction. We show that the Faraday interaction creates entanglement between the light and ensemble that is enhanced when the ensemble is prepared in a state with large projection noise variance. The Holstein-Primakoff approximation on the light and the multilevel Holstein-Primakoff approximation on the atoms enable us to treat the combined system of the light and ensemble as a multimode Gaussian state on two effective modes. We outline properties of Gaussian states that are useful for modeling the dynamics of the light and ensemble, including the covariance matrix update formalism.
In Chapter \ref{Sec:Protocols}, we introduce protocols that utilize the Faraday interaction to create spin squeezing in the atomic ensemble. In all of these protocols, the light mediates a nonlinear interaction in the ensemble's collective spin that creates interatomic entanglement. The strength of this interaction ultimately depends upon the initial entanglement between the light and atoms. The evolution of the light and ensemble under each protocol is modeled using the covariance matrix update formalism. Through this formalism, we show that the protocols create squeezing of a quadrature in a phase plane defined by the multilevel Holstein-Primakoff approximation. While this quadrature squeezing does not necessarily imply spin squeezing, it scales monotonically with interatomic entanglement. Through control over the internal spins of the atoms, we show how this entanglement is converted to metrologically relevant spin squeezing. Additional control can squeeze the internal spins of the atoms, further enhancing spin squeezing.
Chapter \ref{sec::OpticalPumping} introduces the primary source of decoherence in the system, optical pumping due to the spontaneous scattering of photons. Optical pumping, as we demonstrate, damages spin squeezing by destroying beneficial interatomic entanglement, increasing the collective spin variance and causing the mean spin to decay. Some of these damaging effects can be offset by internal spin control. The extent to which optical pumping damages spin squeezing depends substantially on the initial state preparation of the ensemble. We outline specific properties of state preparations that determine the ensemble's susceptibility to decoherence. For particular state preparations and for $f>1/2$, ``transfers of coherence" can preserve interatomic entanglement in the presence of optical pumping.
Chapter \ref{sec::ModHPCovar} explores the dynamics of the ensemble and light as the system undergoes both squeezing interactions and decoherence due to optical pumping. Whereas we previously made the multilevel Holstein-Primakoff approximation on the ensemble by treating each atom as a qubit embedded in the higher dimensional hyperfine spin, preserving transfers of coherence requires that we treat the atoms as embedded qutrits.
We modify the multilevel Holstein-Primakoff approximation to accommodate the ensemble of embedded qutrits, which enables us to treat the ensemble as a Gaussian state on two modes. The effects of optical pumping can be expressed as an update on the covariance matrix of this Gaussian state, as can the coherent squeezing dynamics. Using the covariance matrix update formalism, we conduct a variety of numerical simulations to explore the influence of the spin size $f$ and the state preparation of the ensemble upon the achievable squeezing.
The state preparation of the ensemble has substantial influence on the coherent generation of squeezing and upon decoherence of the ensemble due to optical pumping. In Chapter \ref{sec::Beyond}, we use optimal control techniques to determine the state preparations of the ensemble that maximize squeezing for multiple values of $f$. We specialize to the case of squeezing by quantum nondemolition (QND) measurement, which can be expressed in differential form. Each state preparation determines a unique set of coupled differential equations that give the evolution of the ensemble under both QND measurement and optical pumping. Utilizing a numerical search, we find the state preparations that maximize the spin squeezing determined by the solution to these differential equations.
In Chapter \ref{paraxial}, we extend our previous treatment of spin squeezing to a three-dimensional atom-light interface. We model the light as a paraxial Gaussian beam, which is more closely mode matched to the radiation pattern of the ensemble. The inhomogenous nature of the light and ensemble necessitates the introduction of ``spin waves", which are collective spin operators that take into account the non-uniform coupling between the light and ensemble. QND measurement of a spin wave, in addition to decoherence of the ensemble from optical pumping and scattering of the light into transverse spatial modes outside of the probe, can be modeled through a system of coupled differential equations. We solve for the geometries of the probe and ensemble that maximize both spin squeezing and mode matching for $f=1/2$ and $f=4$.
Chapter \ref{conclusion} concludes the dissertation and discusses possible extensions of this work.
\begin{table}[ht]
\centering
\begin{tabular}{c l }
\hline
Chapters & Publication \\ [0.5ex]
\hline
&\\
4, 5, 6& L. M. Norris, C. M. Trail, P.S. Jessen and I. H. Deutsch, \\&\emph{Enhanced Squeezing of a Collective Spin via Control }\\&\emph{of Its Qudit Subsystems}, Phys. Rev. Lett. \textbf{109}, 173603 (2012).\\&\\
4,5,6,7&L. M. Norris, C. M. Trail, P.S. Jessen and I. H. Deutsch, \\&\emph{Spin Squeezing Ensembles of Qudits} (In preparation).\\&\\
8&B. Q. Baragiola, L. M. Norris, E. Monta$\tilde{\text{n}}$o, P. G. Mickelson, \\&P. S. Jessen and I. H. Deutsch, \emph{Three-dimensional light-matter} \\&\emph{interface for collective spin squeezing in atomic ensembles}, \\&Phys. Rev. A \textbf{89}, 033850 (2014).\\\\
\hline
\end{tabular}
\label{table:nonlin}
\caption{List of publications and the corresponding chapters of this dissertation.}
\end{table}
\chapter{Spin Squeezing}\label{sec::Squeezing}
Spin squeezed states contain quantum correlations that reduce the variance in a component of the collective spin below the standard quantum limit. In the pages that follow, we discuss precisely what is meant by the collective spin and the standard quantum limit. Additionally, we examine the interatomic entanglement and the types of quantum correlations that lead to spin squeezing. Because producing spin squeezed states is our focus, we introduce the Wineland squeezing parameter as a means of quantifying the metrologically useful spin squeezing that we create. The Wineland squeezing parameter is motivated by the application of spin squeezed states in magnetometry. Through the squeezing parameter, we outline some of the preliminary differences between generating spin squeezing in ensembles of ``qudits" versus ensembles of ``qubits".
\section{Collective Spin}
We study spin squeezing in a large ensemble of $N_A$ identical spins with quantum number $f$. Each spin $i$ in the ensemble has an angular momentum operator $\hat{\mathbf{f}}^{(i)}$ associated with its internal spin. Throughout this text, lowercase letters denote operators on the internal spins of the constituent spins that form the ensemble. We seek to squeeze the \emph{collective} spin of the ensemble, the components of which are the summation of the components of the internal spins,
\begin{align}
\hat{F}_{n}=\sum_{i=1}^{N_A}\hat{f}_n^{(i)},
\end{align}
where $n\in\{x,y,z\}$.
A state of the ensemble with particular significance in metrology is the spin coherent state (SCS), the state in which all of the internal spins are polarized along the same spatial direction. The SCS along $x$, for instance, is the eigenstate of $\hat{F}_x$ with maximal spin eigenvalue,
\begin{align}
\ket{SCS(x)}=\ket{f,\,m_x=f}^{\otimes N_A}=\ket{F=N_Af,\,M_x=N_Af}.
\end{align}
While $\ket{SCS(x)}$ has no variance in $\hat{F}_x$, the variance of $\ket{SCS(x)}$ in the collective spin components orthogonal to $x$ defines the standard quantum limit (SQL) of the ensemble. For $N_A>>1$, the spin coherent state is approximately Gaussian distributed in the collective spin components orthogonal to $x$. Consider the projection of $\ket{SCS(x)}$ onto the eigenstates of $\hat{F}_z$,
\begin{align}\label{eq::SCSBinomial}
\ket{SCS(x)}&=\frac{1}{2^{N_Af}}\sum_{M_z}\sqrt{\binom{2N_Af}{N_Af+M_z}}\ket{F=N_Af,\,M_z}\\\label{eq::SCSGauss}
&\approx (\pi N_Af)^{-1/4}\sum_{M_z}\text{exp}\left(-\frac{M_z^2}{2N_Af}\right)\ket{F=N_Af,\,M_z}.
\end{align}
The approximate equality in this expression arises from applying the central limit theorem to the binomial distribution in \erf{eq::SCSBinomial}. The probability of measuring the ensemble in the $\hat{F}_z$ eigenstate $\ket{F=N_Af,M_z}$ is given by the Gaussian probability density function
\begin{align}
P(M_z)=\frac{1}{\sqrt{\pi N_Af}}\text{exp}\left(-\frac{M_z^2}{N_Af}\right).
\end{align}
The variance of $ P(M_z)$, $\Delta F_z^2=N_Af/2$, is the SQL, the smallest nonzero variance possible without quantum correlations.
\section{Spin Squeezing}\label{sec::SpinSqDef}
Spin squeezed states of the ensemble have quantum correlations either between the constituent spins or between the internal degrees of freedom of the constituent spins, permitting the variance of a collective spin component to fall below the SQL \cite{Wineland94, KitagawaUeda93}. For an ensemble symmetric under interchange of the constituent spins, this can be seen through a decomposition of the collective spin variance in terms of the internal spin components,
\begin{align}\label{variance}
\Delta F_z^2=N_A(N_A-1)\langle\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}\rangle_{i\neq j}+N_A\Delta f_z^2.
\end{align}
The first term in \erf{variance} is proportional to the covariance between any two spins in the ensemble, while the second term depends upon the variance of the internal spin. For a spin coherent state of the ensemble along $x$, $\langle\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}\rangle_{i\neq j}=0$ and $\Delta f_z^2=f/2$. Entanglement between the constituent spins can produce states with $\langle\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}\rangle_{i\neq j}<0$, however, reducing the collective variance. Alternatively, $\Delta F_z^2$ can fall below the SQL if the variance of each internal spin is less than $f/2$ \cite{Chaudhury07, Fernholz08}. Either or both of these mechanisms produce states that are ``squeezed" in $\hat{F}_z$, having variance $\Delta F_z^2\leq N_Af/2$. For an ensemble state with $\expect{\hat{F}_x}=N_Af$, the Heisenberg uncertainty principle mandates that the orthogonal collective spin components satisfy $\Delta F_y^2\Delta F_z^2\geq (N_Af/2)^2$. Therefore, a state squeezed in $\hat{F}_z$ must be ``anti-squeezed" in $\hat{F}_y$, implying $\Delta F_y^2\geq N_Af/2$.
\section{Quantifying Spin Squeezing}\label{sec::QuantSqueezing}
Squeezed states were first proposed in the context of
light \cite{Stoler71,Lu72,Hollenhorst79}. Consider a mode of an optical field where the phase space position and momentum quadratures are defined as
\begin{align}
&\hat{X}=\frac{1}{\sqrt{2}}\left(\hat{a}^\dag+\hat{a}\right)\;\;\;\;\text{and}\\
&\hat{P}=\frac{i}{\sqrt{2}}\left(\hat{a}^\dag-\hat{a}\right).
\end{align}
The standard quantum limit is set by the phase space variance of a spin coherent state, given by $\Delta X^2=\Delta P^2=1/2$. A state is, thus, squeezed in the quadrature $\hat{X}$ if $\Delta X^2<1/2$.
This squeezing is quantified by the ``quadrature squeezing parameter",
\begin{align}
\zeta_q=2\Delta X^2.
\end{align}
A state is squeezed when $\zeta_q<1$ with a smaller $\zeta_q$ indicating more squeezing.
We could easily define an analogous parameter for the ensemble, scaling linearly with the variance of a collective spin component. For example, consider
\begin{align}
\zeta=\frac{2}{N_Af}\Delta F_z^2.
\end{align}
While $\zeta<1$ does indicate spin squeezing, this parameter has no dependence on the three dimensional structure of the collective spin. The three dimensional structure of the collective spin, as we shall see, is critical in metrological applications for spin squeezed states.
\begin{figure}[H]
\centering
\includegraphics[scale=.38]{MeanSpin.pdf}
\caption{(a) An atomic ensemble is prepared with the mean spin along $x$. (b) A magnetic field along $y$ of strength $B_y$ causes the spin of the ensemble to rotate about $y$ by an angle $\phi$. By measuring $\hat{F}_z$, the rotation angle, $\phi$, can be determined. The fundamental resolution is determined by the quantum uncertainty of the spin projection (projection noise), shown here as an uncertainty patch.}\label{fig::MeanSpin}
\end{figure}
Consider an example involving magnetometry. Suppose the ensemble is prepared in an initial state with the mean collective spin aligned along $x$, i.e. $\hat{\mathbf{F}}=\expect{\hat{F}_x}\mathbf{e}_x$, as depicted in Fig. \ref{fig::MeanSpin} (a). If the ensemble is exposed to a magnetic field of unknown strength in the $y$-direction, given by $\mathbf{B}=B_y\mathbf{e}_y$, the collective spin will precess about $\hat{F}_y$. The precession angle, $\phi=\gamma_g B_y \Delta t$, is proportional to the field strength along $y$, the gyromagnetic ratio $\gamma_g$ and the interaction time $\Delta t$. Our objective is to deduce the field strength, $B_y$, by measuring $\phi$. If $\phi$ is small, it can be determined by measuring $\hat{F}_z$, as
\begin{align}
\hat{F}_z=\expect{\hat{F}_x}\text{sin} \phi.
\end{align}
The variance of $\phi$ in the measurement of $\hat{F}_z$ is given by
\begin{align} \label{AngRes}
\Delta\phi^2=\frac{\Delta F_z^2}{\expect{\hat{F}_x}^2}.
\end{align}
It is natural that the angular resolution should depend on the variance of the measured spin component, $\Delta F_z^2$. The angular resolution also depends on the ``mean spin" $\expect{\hat{F}_x}$, which explains why the three dimensional structure of the collective spin is essential for metrology. The mean spin, which is always a component orthogonal to the squeezed component, acts like a lever arm in the magnetometer, as shown in Fig. \ref{fig::MeanSpin}. A larger mean spin makes the measured displacement in $\hat{F}_z$ larger and more resolvable, leading to a more precise estimate of $\phi$. A squeezing parameter that takes into account the metrological usefulness of a squeezed state must depend on the mean spin.
The previous example involving magnetometry is equivalent to Ramsey interferometry for atomic clocks when $f=1/2$. In the context of Ramsey interferometry, Wineland proposed a parameter that quantifies both squeezing and metrological usefulness \cite{Wineland94}. The Wineland squeezing parameter is defined as
\begin{align}\label{eq::SqParameter}
\zeta_m=\frac{(\Delta\phi^2)}{(\Delta\phi^2)_{SCS}}=\frac{2N_Af\Delta F_z^2}{\langle\hat{F}_x\rangle^2}.
\end{align}
Here, $(\Delta\phi^2)_{SCS}$ is the angular resolution of a magnetometer that uses a spin coherent state and
$(\Delta\phi^2)$ is the angular resolution of a magnetometer that uses the state we wish to quantify. A state for which $\zeta_m<1$ improves the resolution of a magnetometer over a an SCS. Because the maximal value of $\expect{\hat{F}_x}$ is $N_Af$, a state with $\zeta_m<1$ also has $\Delta F_z^2<SQL$. A smaller variance combined with a still sizable mean spin make a spin squeezed state metrologically useful. Fig. \ref{fig::magnetometry} compares a magnetometer using a SCS and one using a squeezed state. From this point onward in the text, we consider a squeezed state to be a state for which $\zeta_m<1$.
\begin{figure}[H]
\centering
\includegraphics[scale=.6]{magnetometry2.pdf}
\caption{Using an ensemble of spins to measure a magnetic field. (a) The ensemble is prepared in a spin coherent state along the $x$-axis with with transverse variance $\Delta F_{y,z}^2=N_Af/2$. (b) After the ensemble is subjected to a magnetic field along $y$, its mean spin rotates by an angle $\phi$ about $y$. The angle $\phi$ is deduced by measuring $\hat{F}_z$. The precision with which we can determine $\phi$ is, therefore, proportional to $\Delta F_{z}^2=N_Af/2$. (c) The ensemble is prepared in a state squeezed in $\hat{F}_z$, i.e. with $\zeta_m<1$, and a mean spin along the $x$-axis. (d) After being subjected to a magnetic field along $y$, the mean spin of the ensemble rotates by an angle $\phi$ about $y$. Our ability to resolve the angle $\phi$ is enhanced over (b) because $\Delta F_z^2<N_Af/2$. }\label{fig::magnetometry}
\end{figure}
\section{Collective Spin Squeezing and $f$}
The Wineland squeezing parameter sheds light on the relationship between collective spin squeezing and the internal spin size, $f$. By substituting \erf{variance} and $\expect{\hat{F}_x}=N_A\expect{\hat{f}_x}$ into \erf{eq::SqParameter}, the squeezing parameter becomes
\begin{align}
\zeta_m=\frac{2N_A\Delta F_z^2}{\expect{\hat{F}_x}^2}=\frac{2N_Af\expect{\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}}_{i\neq j}}{\expect{\hat{f}_x^{(i)}}^2}+\zeta_m^{(i)}
\end{align}
for $N_A>>1$. As before, $\expect{\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}}_{i\neq j}$ is the covariance between any two different spins in the ensemble. The last term on the right-hand side of this expression is the value of the Wineland squeezing parameter for any single spin in the ensemble, given by
\begin{align}
\zeta_m^{(i)}=\frac{2f\Delta f_z^{(i)\,2}}{\expect{f_x^{(i)}}^2}.
\end{align}
Like the variance $\Delta F_z^2$, $\zeta_m$ can be reduced in two different ways. Entanglement between the atoms creates negative correlations for which $\expect{\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}}_{i\neq j}<0$. Alternatively, the internal spins can be squeezed, causing $\zeta_m^{(i)}<1$. This latter option exists only when $f>1/2$, however, which we can demonstrate by calculating the squeezing parameter of an arbitrary qubit.
Consider a $f=1/2$ qubit in the state $\ket{\psi}=\text{cos}\theta\ket{\uparrow}+e^{i\phi}\text{sin}\theta\ket{\downarrow}$, where \\$\ket{\uparrow}$ denotes $\ket{f=1/2,m_z=1/2}$ and $\ket{\downarrow}$ denotes $\ket{f=1/2,m_z=-1/2}$. The variance and mean spin of this state are $\Delta f_z^2=\text{cos}^2\theta\text{sin}^2\theta$ and $\expect{\hat{f}_x}=\text{cos}\theta\text{sin}\theta\text{cos}\phi$, respectively. The squeezing parameter of a single qubit is, therefore,
\begin{align}
\zeta_m^{f=1/2}=\frac{1}{\text{cos}^2\phi}.
\end{align}
Because the minimum value of $\zeta_m^{f=1/2}$ is 1, a qubit cannot be squeezed. Thus, entanglement is solely responsible for spin squeezing in an ensemble of spins with $f=1/2$. When $f\geq 1/2$, on the other hand, both entanglement and squeezing of the internal spin generate spin squeezing.
\chapter{Atom-Light Interface}\label{sec::ALinterface}
Large atomic ensembles interacting with light or ``atom-light interfaces" are platforms for many spin squeezing protocols and experimental demonstrations of spin squeezing \cite{KuzBig00,appel09,Takano2009,VulSqueezingClock,Koschorreck2010,TakTak05}. Typically, the correlations between the atomic spins that create spin squeezing are generated through the atoms' mutual coupling to the light. In this chapter, we analyze the interaction between the light and atoms in detail. In particular, we concentrate on the entanglement generated between the light and ensemble, which is essential for creating spin squeezing. We also develop the mathematical formalism necessary to model the effects of spin squeezing protocols on the light and ensemble. By applying variations of the Holstein-Primakoff approximation, we greatly simplify the representation of the light and ensemble states \cite{HP,MultilevelHP}. This enables us to utilize the properties of Gaussian states to describe the joint evolution of the light and ensemble.
\section{The Faraday Interaction}\label{sec::FaradayH}
We study spin squeezing in an ensemble composed of $N_A>>1$ alkali atoms. The atoms, which are identically prepared in one of the ground hyperfine manifolds with spin $f$, interact with a probe laser far detuned from the D1 or D2 line, as depicted in Fig. \ref{fig::AlkaliLines}. For a probe with sufficiently large detuning, $\Delta$, and weak intensity, the excited hyperfine manifolds of the atoms can be adiabatically eliminated. The resulting interaction couples the magnetic sublevels of the spin-$f$ ground hyperfine manifold to the polarization modes of the optical field. This is described by the ac-Stark Hamiltonian, which can be decomposed into irreducible tensor components as follows \cite{DeuJes09},
\begin{align}\label{eq::ACstark}
&\hat{H}=\frac{\hbar\chi_0}{\Delta t}\sum_{i=1}^{N_A}[C^{(0)}\hat{S}_0+C^{(1)}\hat{f}_z^{(i)}\hat{S}_3\\
&+C^{(2)}((\hat{f}_x^{(i)2}-\hat{f}_y^{(i)2})\hat{S}_1/2-(3\hat{f}_z^{(i)\;2}-\hat{\mathbf{f}}^{(i)\;2})\hat{S}_0/6+(\hat{f}_x^{(i)}\hat{f}_y^{(i)}+\hat{f}_y^{(i)}\hat{f}_x^{(i)})\hat{S}_2/2)]\notag,
\end{align}
where $\chi_0=(\sigma_0\Gamma)/(2A\Delta)$, $\sigma_0$ is the resonant cross section for unit oscillator strength, $\Gamma$ is the excited state linewidth, $A$ is the cross sectional area of the probe and $\Delta t$ is the interaction time. Here, the light's direction of propagation is along $z$ and $\hat{\textbf{S}}$ is the quantized Stokes' vector of the light with components
\begin{align}\label{Stokes}
&\hat{S}_0=\frac{1}{2}(\hat{a}_x^\dag\hat{a}_x+\hat{a}_{y}^\dag\hat{a}_{y})\;\;\;\;\;
\hat{S}_1=\frac{1}{2}(\hat{a}_x^\dag\hat{a}_{x}-\hat{a}_{y}^\dag\hat{a}_y)\\\notag
&\hat{S}_2=\frac{1}{2}(\hat{a}_{x}^\dag\hat{a}_y+\hat{a}_y^\dag\hat{a}_{x})\;\;\;\;\;
\hat{S}_3=\frac{1}{2i}(\hat{a}_{x}^\dag\hat{a}_{y}-\hat{a}_y^\dag\hat{a}_x),
\end{align}
where $x$ and $y$ denote orthogonal linearly polarized modes. The Stokes' components 1 through 3 are analogous to angular momentum operators, satisfying the $su(2)$ commutation relations, $[\hat{S}_i,\hat{S}_j]=i\epsilon_{ijk}\hat{S}_k$.
\begin{figure}
\centering
\includegraphics[scale=.55]{AlkaliLines.pdf}
\caption{Energy levels of an alkali atom with nuclear spin $i$ and ground hyperfine spins $f_+$ and $f_-$, both greater than $1/2$. Atoms in the ensemble are prepared in one of the ground hyperfine manifolds with spin $f$ equal to $f_+$ or $f_-$. The probe laser is detuned from the excited state hyperfine splitting of the D1 or D2 line. The spin quantum numbers of the excited hyperfine states are denoted $f'$.}
\label{fig::AlkaliLines}
\end{figure}
The Hamiltonian in \erf{eq::ACstark} is decomposed into scalar, vector and tensor terms with coefficients $C^{(0)}$, $C^{(1)}$ and $C^{(2)}$, respectively. The scalar term shifts the atomic energy levels by an amount proportional to the probe intensity. Because this shift is independent of the atomic state, the scalar term has no influence on the dynamics of the atom-light interface and can be discarded. The vector component, commonly known as the Faraday interaction, couples the circular polarization of the probe to the collective spin component in the direction of the light's propagation. This interaction generates atom-light entanglement, the crucial first step in many spin squeezing protocols. The tensor component, on the other hand, induces an undesirable birefringence on the light as well as nonlinear dynamics on the internal state of the atomic spins \cite{Smith03}. The magnitudes of the constants $C^{(K)}$ determine the contribution of each term to the dynamics. While the full form of the $C^{(K)}$ is given in \cite{DeuJes09}, for our purposes it suffices to know that each constant depends upon the detuning with $C^{(1)}\propto 1/\Delta$ and $C^{(2)}\propto 1/\Delta^2$ to leading order in $1/\Delta$.
Although it is smaller than the vector component by an order of $1/\Delta$, the tensor term is not negligible over the time scale during which spin squeezing occurs. The damaging birefringence effects of the tensor term are removed in the presence of a large bias magnetic field along the direction of the light's propagation \cite{Baragiola14}. This averages to zero the coupling between the light and the atomic spin components transverse to $z$. This effect is evident when we transform into a frame rotating at the Larmor frequency of the bias, $\Omega_B$, via the unitary $\hat{U}_B(t)=\text{exp}(-i\Omega_Bt\hat{F}_z)$. The terms in the tensor component that depend upon the transverse internal spin components become
\begin{align}
&\expect{\hat{U}_B^\dag(t)\hat{f}_x^{(i)2}\hat{U}_B(t)}_t=(\hat{f}_x^{(i)2}+\hat{f}_y^{(i)2})/2\\
&\expect{\hat{U}_B^\dag(t)\hat{f}_y^{(i)2}\hat{U}_B(t)}_t=(\hat{f}_x^{(i)2}+\hat{f}_y^{(i)2})/2
\end{align}
and
\begin{align}
&\expect{\hat{U}_B^\dag(t)(\hat{f}_x^{(i)}\hat{f}_y^{(i)}+\hat{f}_y^{(i)}\hat{f}_x^{(i)})\hat{U}_B(t)}_t=0,
\end{align}
where $\expect{\cdot}_t$ denotes a time average. Transformed into the rotating frame and time averaged, the AC Stark Hamiltonian is
\begin{align}\label{eq::ACstarkRot}
&\hat{H}=\frac{\hbar\chi_0}{\Delta t}\sum_{i=1}^{N_A}[C^{(1)}\hat{f}_z^{(i)}\hat{S}_3
-C^{(2)}(3\hat{f}_z^{(i)\;2}-\hat{\mathbf{f}}^{(i)\;2})\hat{S}_0/6].
\end{align}
The residual tensor component of the Hamiltonian nonlinearly couples the internal spins of the atoms to the intensity of the probe. This coupling can, in principle, be eliminated with the internal spin control techniques covered in Sec. \ref{sec::IntControl}. In practice, one can also apply two probes detuned from the D1 and D2 lines to cancel the residual tensor component without affecting the vector component \cite{Baragiola14}.
After removing the tensor term, the Faraday interaction is the dominant contribution to the Hamiltonian. We rewrite the Faraday Hamiltonian as
\begin{align}\label{eq::FaradayDef}
\hat{H}=\frac{\hbar\chi}{\Delta t}\hat{S}_3\hat{F}_z,
\end{align}
where $\chi=g_f(\sigma_0/A)(\Gamma/6\Delta)$ is the Faraday rotation angle, $g_f$ is the Land\'{e} g-factor, $\Gamma$ is the linewidth of the transition, $A$ is the beam area and $\sigma_0=3\lambda^2/2\pi$ is the resonant cross section for a unit oscillator strength. The Faraday interaction is an effective spin-spin interaction that couples the polarization of the light to the collective spin of the ensemble. The linear polarization of the light rotates by an amount proportional to the $z$-component of the ensemble's collective spin. Similarly, the collective spin of the ensemble rotates about $z$ by an amount proportional to the light's circular polarization, quantified by $\hat{S}_3$. Pictorially, these effects can be represented as rotations in the Poincar\'{e} and Bloch spheres as shown in Fig. \ref{fig::FaradaySpheres}. Note that the Faraday rotation angle, $\chi$, is proportional to the Land\'{e} g-factor, $g_f$. In the higher hyperfine ground manifold, the Land\'{e} g-factor is given by $g_f=f^{-1}$, while $g_f=(f+1)^{-1}$ in the lower manifold. As a consequence, the strength of the Faraday interaction decreases with increasing $f$ in both manifolds.
\begin{figure}
\centering
\includegraphics[scale=.4]{FaradaySpheres.pdf}
\caption{The Poincar\'{e} sphere of the light (left) and the Bloch sphere of the atomic ensemble (right). The $x$, $y$ and $z$ axes of the Poincar\'{e} sphere represent $x/y$ linear polarization, diagonal/anti-diagonal linear polarization and right/left circular polarization, respectively. The Faraday interaction, given by the Hamiltonian $\hat{H}=\hbar\chi\hat{S}_3\hat{F}_z/\Delta t$, creates a rotation of the light's linear polarization in the Poincar\'{e} sphere about $\hat{S}_3$ by an angle $\theta_L$, which is proportional to the collective spin $\hat{F}_z$ of the ensemble. The Faraday interaction generates a similar rotation of the ensemble's collective spin about $\hat{F}_z$ by an angle $\theta_A$ proportional to the light's circular polarization, quantified by $\hat{S}_3$. }\label{fig::FaradaySpheres}
\end{figure}
\section{Entanglement and the Faraday Interaction}\label{sec::EntangleFaraday}
The most important aspect of the Faraday interaction in relation to spin squeezing is the entanglement it generates between the light and the ensemble. Consider an initial state of the ensemble that can be decomposed in terms of the collective spin eigenstates of $\hat{F}_z$ as $\ket{\Phi_A}=\sum_{M_z=-N_Af}^{N_Af}C(M_z)\ket{N_Af,M_z}$. For an initial state of the light $\ket{\Phi_L}$, the Faraday interaction produces the entangled state
\begin{align}\label{eq::FaradayState}
\ket{\Phi_{AL}}\!=e^{-i\chi\hat{S}_3\hat{F}_z}\ket{\Phi_L}\ket{\Phi_A}=\!\!\!\!\sum_{M_z=-N_Af}^{N_Af}\!\!\!\!C(M_z)\left(e^{-i\chi\hat{S}_3M_z}\ket{\Phi_L}\right)\ket{N_Af,M_z}.
\end{align}
Each $\hat{F}_z$ eigenstate of the ensemble with eigenvalue $M_z$ is coupled to a state of the light that has been rotated about $\hat{S}_3$ by an angle proportional to $M_z$. Measuring the rotation angle of the light provides information about the associated $\hat{F}_z$ eigenstate of the ensemble and vice versa, an indicator of entanglement.
More quantitatively, the strength of the entanglement between the light and atoms can be determined by calculating the purity of the reduced density operator of the light after tracing out the ensemble. If the initial state of the ensemble consists of $N_A>>1$ separable identically prepared atomic spins, it is approximately Gaussian by the central limit theorem. The $C(M_z)$ coefficients in $\ket{\Phi_A}$ can, thus, be written as $C(M_z)=(2\pi\Delta F_z^2)^{-1/4}\text{exp}(-\frac{M_z^2}{4\Delta F_z^2})$. Consider an initial state of the light $\ket{N_L}_x$, where all photons are linearly polarized along $x$ and $N_L$ is the number of photons in a pulse of time $\Delta t$. We can take advantage of the $su(2)$ algebra of the Stokes' components to write the initial state as an effective spin eigenstate of $\hat{S}_1$, $\ket{\Phi_L}=\ket{N_L/2,M_1=N_L/2}$. Here, the effective total angular momentum is $N_L/2$ and $M_i$ denotes an eigenvalue of $\hat{S}_i$. In the basis of eigenstates of $\hat{S}_3$, $\ket{\Phi_L}=(\pi N_L/2)^{-1/4}\sum_{M_3=-N_L/2}^{N_L/2}\text{exp}\left(-\frac{M_3^{\;2}}{N_L}\right)\ket{N_L/2,M_3}$, analogous to a spin coherent state. In the limit of continuous $M_z$, the reduced density operator of the light is
\begin{align}
\hat{\rho}_L&=\text{Tr}_A(\ket{\Phi_{AL}}\bra{\Phi_{AL}})\\\notag
&=\sum_{M_3,\,M_3'=-N_L/2}^{N_L/2}\sqrt{\frac{2}{\pi N_L}}e^{-\frac{M_3^{2}+M_3'^{2}}{N_L}}
e^{-\frac{1}{2}(M_3-M_3')^2\Delta F_z^2\chi^2}\ket{N_L/2,M_3}\bra{N_L/2,M_3'}.
\end{align}
In the limit of continuous $M_3$ and $M_3'$, the purity is
\begin{align}\label{eq::purityXi}
\text{Tr}(\rho_L^2)=\frac{1}{\sqrt{1+\chi^2N_L\Delta F_z^2}}.
\end{align}
The purity decreases with the quantity
\begin{align}\label{eq::1stXi}
\xi=\chi^2N_L\Delta F_z^2,
\end{align}
which we call the ``collective spin coupling constant". A larger collective spin coupling constant, therefore, signifies greater entanglement between the light and ensemble.
While it seems natural that the entanglement between the light and ensemble should increase with $N_L$ and $\chi$, the presence of $\Delta F_z^2$ in the collective spin coupling constant is counterintuitive. The relationship between the ``projection noise" $\Delta F_z^2$ and entanglement is explained by considering a measurement of the ensemble's collective spin mediated by the Faraday interaction. In this form of measurement, the polarization of the light serves as a meter for the ensemble's collective spin. Recall that the Faraday interaction rotates the initial state of the light about $\hat{S}_3$ by an amount proportional to the projections of the ensemble's collective spin in $\hat{F}_z$, as shown in \erf{eq::FaradayState}. If the ensemble is initially linearly polarized along $x$, $\expect{\hat{S}_2}=0$. By measuring the displacement of the light in $\hat{S}_2$, the rotation angle and corresponding $\hat{F}_z$ eigenstate of the ensemble can be deduced. Measuring the $\hat{S}_2$ Stokes component of the light is, thus, an indirect measurement of the ensemble's collective spin component $\hat{F}_z$.
To understand the role of the projection noise, we examine the measurement of $\hat{S}_2$ in greater detail. Consider an apparatus used to measure $\hat{S}_2$ in absence of the atomic ensemble, depicted in Fig. \ref{fig::QNDwoutAtoms} (a) and (c). The light travels along $z$ until it encounters a polarizing beam splitter, whereupon half of the light is diverted to a photodetector that counts photons with $+45^\circ$ polarization and the other half is diverted to a photodetector that counts photons with $-45^\circ$ polarization. The subtracted intensities measured by the two photodetectors yields a measurement of $\hat{S}_2=\frac{1}{2}(\hat{a}_{+45^\circ}^\dag\hat{a}_{+45^\circ}-\hat{a}_{-45^\circ}^\dag\hat{a}_{-45^\circ})$. The variance in this measurement, $\Delta S_2^2$, is due to ``shot noise" arising from the vacuum fluctuations in the y-polarized mode. For an initial state of the light that is horizontally polarized, shown in Fig. \ref{fig::QNDwoutAtoms} (b), the shot noise variance is $\Delta S_2^2=N_L/4$. Figure \ref{fig::QNDwAtoms} (a) and (d) shows the same measurement apparatus, except that the light passes through the atomic ensemble en route to the photodetectors. If initial state of the light is $\ket{N_L}_x$, as depicted in Fig. \ref{fig::QNDwAtoms} (b), the state of the light after the Faraday interaction is shown schematically in Fig. \ref{fig::QNDwAtoms} (c). This state consists of a superposition of the states $\ket{N_L}_x$, all of which have been rotated by an angle $\chi M_z$ about $\hat{S}_3$, corresponding to projections of the ensemble spin state in the $\hat{F}_z$ eigenbasis. For small rotation angles, $\chi M_z$, the displacement of each superposition in $\hat{S}_2$ is
\begin{align}\label{eq::S2displacement}
\expect{\hat{S}_2}_{M_z}=(N_L/2)\text{sin}(\chi M_z)\approx N_L\chi M_z/2.
\end{align}
As shown in Fig. \ref{fig::QNDwAtoms} (d), the mean of the measurement signal from the photodetectors is equal to the displacement of one of the superpositions of $\ket{N_L}_x$. The variance of the signal over the time interval of the measurement is the shot noise variance of $\ket{N_L}_x$, which is once again $\Delta S_2^2=N_L/4$.
\begin{figure}[H]
\centering
\includegraphics[scale=.45]{QNDwoutAtoms.pdf}
\caption{The intrinsic shot noise of the light. (a) The light passes through the experimental apparatus, which contains no atoms. (b) The state of the light, $x$ polarized, is an eigenstate of $\hat{S}_1$. The light has variance in the Stokes' components $\hat{S}_2$ and $\hat{S}_3$, corresponding to the diagonal/anti-diagonal polarization and circular polarization of the light. This variance, given by $\Delta S_2^2=\Delta S_3^2=N_L/4$, is known as the shot noise. (c) Measuring the diagonal/anti-diagonal polarization gives a signal with mean zero, since $\expect{\hat{S}_2}=0$. The variance in the signal is the shot noise, $\Delta S_2^2$. }\label{fig::QNDwoutAtoms}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.42]{QNDwAtoms.pdf}
\caption{The effect of atomic projection noise on atom-light entanglement. (a) The light passes through the experimental apparatus, where it interacts with the atoms. (b) The light, prior to interacting with the atoms, is in an eigenstate of $\hat{S}_1$. The variance in the transverse components, $\Delta S_2^2=\Delta S_3^2=N_L/4$, is the shot noise. (c) If the ensemble is prepared in an initial state $\ket{\Phi_A}=\sum_{M_z=-N_Af}^{N_Af}C(M_z)\ket{N_Af,M_z}$ with $\expect{\hat{F}_z}=0$, the Faraday interaction causes no net rotation. Instead, the Faraday interaction couples each $\hat{F}_z$ eigenstate with a state of the light that has been rotated about $\hat{S}_3$ by an angle proportional to the eigenvalue $M_z$, $\sum_{M_z=-N_Af}^{N_Af}C(M_z)e^{-i\chi M_z\hat{S}_3}\ket{N_L}_x\ket{N_Af,M_z}$. The spread in rotation angles about $\hat{S}_3$ increases with $\Delta F_z^2$ or the ``projection noise" of the ensemble. (d) When $\hat{S}_2$ is measured by the polarimeter, the mean of the signal corresponds to the rotation angle about $\hat{S}_3$, while the variance of the signal is the shot noise. The greater spread in rotation angles created by increased atomic projection noise makes the rotation angle more resolvable beneath the shot noise. Because the rotation angle corresponds to an $\hat{F}_z$ eigenstate of the ensemble, information about the ensemble's collective spin is also more resolvable with increased projection noise. A measurement of $\hat{S}_2$, thus, creates more measurement backaction in the ensemble, indicating greater atom-light entanglement.
}\label{fig::QNDwAtoms}
\end{figure}
Being able to deduce the value of $\hat{F}_z$ by measuring $\hat{S}_2$ requires that the value of the displacement in \erf{eq::S2displacement} is resolvable beneath the shot noise of the signal. The displacement is resolvable as long as $\expect{\hat{S}_2}_{M_z}>\Delta S_2$. The smallest separation between values of $\hat{F}_z$ that is resolvable beneath the shot noise of the light is, therefore,
\begin{align}
(\Delta F_z)_{SN}=\frac{1}{\chi\sqrt{N_L}}.
\end{align}
We refer to $(\Delta F_z^2)_{SN}$ as the shot noise resolution of a measurement of $\hat{F}_z$. Note that a large spread of measurement outcomes is more resolvable than a smaller one. It follows that different measurement outcomes of $\hat{F}_z$ are more resolvable in a measurement of the light when the ensemble has a large projection noise, $\Delta F_z^2$. As illustrated in Fig. \ref{fig::StatePreps2}, such an ensemble has a greater spread of $\hat{F}_z$ eigenstates, producing a greater spread of rotation angles in the state of the light after the Faraday interaction. This leads to larger relative displacements of $\hat{S}_2$ and greater resolution of $\hat{F}_z$. The ability to probe one system and obtain information about another is a signature of entanglement. Indeed, the increased resolution of $\hat{F}_z$ that results from a larger projection noise indicates greater entanglement between the light and atoms. For this reason, the collective spin coupling constant increases with the projection noise. The collective spin coupling constant from \erf{eq::1stXi} can alternatively be expressed as
\begin{align}\label{eq::2ndXI}
\xi=\frac{(\Delta F_z^2)_{PN}}{(\Delta F_z^2)_{SN}}=\chi^2N_L\Delta F_z^2,
\end{align}
where $(\Delta F_z^2)_{PN}$ denotes the projection noise. As demonstrated in \erf{eq::purityXi}, this is the key quantity in determining the entanglement between the light and ensemble. The resolvability of $\hat{F}_z$, and the entanglement generated between the light and ensemble, increases with the ratio of the projection noise to the shot noise resolution.
\begin{figure}[H]
\centering
\includegraphics[scale=.47]{StatePreps3.pdf}
\caption{The Faraday interaction increases the variance of the light's polarization by an amount proportional to the projection noise of the ensemble's collective spin along $\hat{F}_z$. (a) Variance induced on the light's polarization by an ensemble prepared in a spin coherent state, $\ket{f,\, m_x=f}^{\otimes N_A}$. The projection noise of the spin coherent state is $(\Delta F_z^2)_{PN}=N_Af/2$, the standard quantum limit. (b) Variance induced upon the light's polarization by an ensemble prepared in $\ket{f,\, m_x=0}^{\otimes N_A}$, a state with a larger projection noise than the spin coherent state. This enhanced projection noise, $(\Delta F_z^2)_{PN}=N_Af(f+1)/2$, induces more variance upon the light's polarization, increasing the resolvability of translations corresponding to eigenvalues of $\hat{F}_z$.}\label{fig::StatePreps2}
\end{figure}
\section{Holstein-Primakoff Approximations }\label{sec::HP}
Both the state of the light and the state of the atomic ensemble belong to Hilbert spaces with extremely large dimensions. The state of the light is specified by its two polarization modes, $x$ and $y$, and fixed excitation number $N_L$. In the Schwinger representation, this corresponds to a spin state with total angular momentum number $N_L/2$ and Hilbert space dimension $N_L+1$, where $N_L$ is on the order of $10^8$ for realistic parameters. In the case of $f=1/2$, the collective spin state of the ensemble has total angular momentum quantum number $N_A/2$ and Hilbert space dimension $N_A+1$. Although this dimensionality is large, since $N_A \sim 10^6$, the situation is appreciably worse when $f>1/2$. Consider an ensemble state $\ket{\Psi}=\ket{\psi}^{\otimes N_A}$, where $\ket{\psi}$ is a state of the $2f+1$ dimensional internal spin. This state can be decomposed in terms of collective spin states with different total angular momentum quantum numbers, $F$,
\begin{align}
\ket{\Psi}=\sum_F\sum_{M=-F}^FC_M^F\ket{F,M}.
\end{align}
In the case of $f=1/2$, all $C_M^F=0$ except when $F=N_A/2$. This symmetry enables the dimension of the Hilbert space to be reduced from $2^{N_A}$ to $N_A+1$. When $f>1/2$, $\ket{\Psi}$ has projections onto collective spin states with many different total angular momentum quantum numbers $F$. Reducing the dimensionality of the relevant Hilbert space below $(2f+1)^{N_A}$ requires determining the $C_M^F$, which is an open problem. Without analytic methods, we must instead rely upon an approximation. By applying variations of the Holstein-Primakoff (HP) approximation to both the light and the atomic ensemble, we can treat each system as a state on a single bosonic mode \cite{HP}.
\subsection{Holstein-Primakoff Transformation}
The Holstein-Primakoff transformation is a map between angular momentum operators, or any generating set of $su(2)$, and bosonic creation and annihilation operators. Consider the angular momentum operators $\hat{J}_+$, $\hat{J}_-$ and $\hat{J}_z$. In terms of the creation and annihilation operators $\hat{a}^\dag$ and $\hat{a}$, these angular momentum operators can be expressed as
\begin{align}\label{eq::HPJplus}
\hat{J}_+=\hat{J}_y+i\hat{J}_z=\sqrt{2J}\sqrt{1-\frac{\hat{a}^\dag\hat{a}}{2J}}\hat{a}\\\label{eq::HPJminus}
\hat{J}_-=\hat{J}_y-i\hat{J}_z=\sqrt{2J}\hat{a}^\dag\sqrt{1-\frac{\hat{a}^\dag\hat{a}}{2J}}
\end{align}
and
\begin{align}\label{eq::HPJx}
\hat{J}_x=J-\hat{a}^\dag\hat{a},
\end{align}
where $J$ is the total angular momentum quantum number.
The Holstein-Primakoff transformation is an exact correspondence. If we desire to reduce the dimensionality of the relevant Hilbert space associated with the angular momentum $\hat{\mathbf{J}}$, we must make an approximation. In cases where $J$ is very large, the expressions in Eqs. (\ref{eq::HPJplus}), (\ref{eq::HPJminus}) and (\ref{eq::HPJx}) become
\begin{align}
\hat{J}_+=\hat{J}_y+i\hat{J}_z\approx\sqrt{2J}\hat{a}\\
\hat{J}_-=\hat{J}_y-i\hat{J}_z\approx\sqrt{2J}\hat{a}^\dag
\end{align}
and
\begin{align}
\hat{J}_x\approx J.
\end{align}
This mapping is referred to as the Holstein-Primakoff approximation. The operator $\hat{J}_x$ is treated as a classical quantity with no variance, while the operators $\hat{J}_+$ and $\hat{J}_-$ remain quantum. By making the Holstein-Primakoff approximation, we have restricted the Hilbert space to states where $\expect{\hat{J}_x}\approx J$.
\subsection{Holstein-Primakoff Approximation on the Light}
Because the Stokes' components satisfy the $su(2)$ commutation relations, we can write these operators in terms of creation and annihilation operators through the Holstein-Primakoff transformation. For the squeezing protocols we will later consider, the initial state of the light is linearly polarized along $x$, which is equivalent to the effective angular momentum eigenstate of $\hat{S}_1$ with maximal spin projection, $\ket{N_L/2, M_1=N_L/2}$. Since the effective total angular momentum of this state is $N_L/2>>1$, we can apply the Holstein-Primakoff approximation to the Stokes' components. The Stokes' component $\hat{S}_1$ becomes a classical quantity with
\begin{align}\label{eq::S1NL}
\hat{S}_1=\frac{1}{2}(\hat{a}_x^\dag\hat{a}_x-\hat{a}_y^\dag\hat{a}_y)\approx N_L/2.
\end{align}
The number of excitations in the $x$ mode, likewise, becomes effectively classical with $\hat{a}_x^\dag\hat{a}_x\approx N_L$. The creation and annihilation operators on the $y$ mode, on the other hand, are non-classical, containing all of the quantum uncertainty associated with the light. The operators on the light that remain quantum depend upon $\hat{a}_y^\dag$ and $\hat{a}_y$ with
\begin{align}
\hat{S}_+=\hat{S}_2+i\hat{S}_3\approx\sqrt{N_L}\hat{a}_y
\end{align}
and
\begin{align}
\hat{S}_-=\hat{S}_2-i\hat{S}_3\approx\sqrt{N_L}\hat{a}_y^\dag.
\end{align}
The Stokes' components $\hat{S}_2$ and $\hat{S}_3$ become
\begin{align}\label{eq::S2Xy}
\hat{S}_2=\sqrt{\frac{N_L}{2}}\hat{X}_y
\end{align}
and
\begin{align}\label{eq::S3Py}
\hat{S}_3=\sqrt{\frac{N_L}{2}}\hat{P}_y,
\end{align}
where $\hat{X}_y=(\hat{a}_y^\dag+\hat{a}_y)/\sqrt{2}$ and $\hat{P}_y=i(\hat{a}_y^\dag-\hat{a}_y)/\sqrt{2}$ are the position and momentum quadratures on the mode $y$.
Interestingly, we can arrive at the same approximation directly from the definition of the Stokes' components. Again, we take the initial state of the light to be linearly polarized along $x$ so that $\langle\hat{S}_1\rangle=N_L/2$. Because $\expect{\hat{S}_1}$ is large relative to the uncertainties of the orthogonal Stokes' components, given by $\Delta S_2=\Delta S_3=\sqrt{N_L}/2$, the state of the light is confined to a small locally flat region of the Poincar\'{e} sphere as shown in Fig. \ref{fig::HPlight}. In this region, the number of $x$-polarized photons remains approximately constant at $N_L$, implying $\hat{a}_x^\dag\approx\hat{a}_x\approx\sqrt{N_L}$. Under this approximation, the Stokes' vector components in \erf{Stokes} become
\begin{align}
\hat{S}_1&\approx N_L/2\\
\hat{S}_2&\approx\frac{\sqrt{N_L}}{2}(\hat{a}_y+\hat{a}_y^\dag)=\sqrt{\frac{N_L}{2}}\hat{X}_y
\end{align}
and
\begin{align}
\hat{S}_3&\approx\frac{\sqrt{N_L}}{2i}(\hat{a}_y-\hat{a}_y^\dag)=\sqrt{\frac{N_L}{2}}\hat{P}_y,
\end{align}
equivalent to Eqs. (\ref{eq::S1NL}), (\ref{eq::S2Xy}) and (\ref{eq::S3Py}).
\begin{figure}[H]
\centering
\includegraphics[scale=.77]{HPlight.pdf}
\caption{The Holstein-Primakoff approximation on the light. The light, completely polarized along $x$, is in the initial state $\ket{N_L}_x$ for which $\hat{S}_1\approx N_L/2$. In this regime, the uncertainties of the transverse Stokes' components $\Delta S_2=\Delta S_3=\sqrt{N_L}/2$ are much smaller than $\hat{S}_1$. The state of the light, thus, occupies a small locally flat region of the Poincar\'{e} sphere. In this region, the Stokes' components $\hat{S}_2$ and $\hat{S}_3$ are well approximated by the position and momentum quadratures $\hat{X}_y$ and $\hat{P}_y$.}\label{fig::HPlight}
\end{figure}
Regardless of how we obtain the Holstein-Primakoff approximation, the result is that the state of the light is specified by a single bosonic mode $y$. The initial state of the light, completely polarized along $x$, corresponds to the vacuum state, $\ket{0}_y$. So long as the light undergoes weak interactions that do not lead to large displacements on the Poincar\'{e} sphere, implying that the light remains in a region where $\expect{\hat{S}_1}\approx N_L/2$, its state remains well approximated by a single mode. The Faraday interaction fits this criteria, as $\chi<<1$ for realistic parameters.
\subsection{Holstein-Primakoff Approximation on the Atomic \\Ensemble}\label{sec::HPEnsemble}
In a manner similar to the light, we can greatly simplify the state of the atomic ensemble through the Holstein-Primakoff approximation. We first consider a spin coherent state of the ensemble along $x$ for $f=1/2$, which is given by \\$\ket{N_A/2,M_x=N_A/2}=\ket{1/2,m_x=1/2}^{\otimes N_A}$. This is an exact analogy to the case in which the light is linearly polarized along $x$ with the angular momentum operators $\hat{F}_x$, $\hat{F}_y$ and $\hat{F}_z$ taking the place of the Stokes' components $\hat{S}_1$, $\hat{S}_3$ and $\hat{S}_2$, respectively. This parallel is made especially clear when we express the collective angular momentum operators of the ensemble in the Schwinger representation. The Schwinger representation is another mapping from angular momentum operators to bosonic creation and annihilation operators. Unlike the Holstein-Primakoff transformation, however, the Schwinger representation expresses angular momentum operators in terms of two bosonic modes. Consider the collective spin operators acting on the $f=1/2$ atomic ensemble written in the basis of $\hat{f}_x$ eigenstates \\$\ket{\uparrow}=\ket{f=1/2,m_x=1/2}$ and $\ket{\downarrow}=i\ket{f=1/2,m_x=-1/2}$,
\begin{align}\label{eq::FxHalf}
\hat{F}_x=\frac{1}{2}\sum_{i=1}^{N_A}\left(\ket{\uparrow}\bra{\uparrow}_i-\ket{\downarrow}\bra{\downarrow}_i\right)\\\label{eq::FyHalf}
\hat{F}_y=\frac{i}{2}\sum_{i=1}^{N_A}\left(\ket{\downarrow}\bra{\uparrow}_i-\ket{\uparrow}\bra{\downarrow}_i\right)
\end{align}
and
\begin{align}\label{eq::FzHalf}
\hat{F}_z=\frac{1}{2}\sum_{i=1}^{N_A}\left(\ket{\downarrow}\bra{\uparrow}_i+\ket{\uparrow}\bra{\downarrow}_i\right).
\end{align}
The phase of the state $\ket{\downarrow}$ has been selected so that the collective spin operator $\hat{F}_z$ corresponds with the derivation in Sec. \ref{sec::MultiHPEnsemble}, which employs the multilevel Holstein-Primakoff approximation. In the expressions above, the term $\sum_{i=1}^{N_A}\ket{\downarrow}\bra{\uparrow}_i$ that occurs in Eqs. (\ref{eq::FxHalf}) and (\ref{eq::FyHalf}) can be viewed as annihilating an atom in the state $\ket{\uparrow}$ and creating an atom in the state $\ket{\downarrow}$. Likewise, the term $\sum_{i=1}^{N_A}\ket{\uparrow}\bra{\downarrow}_i$ creates an atom in the state $\ket{\uparrow}$ and annihilates an atom in the state $\ket{\downarrow}$. The terms $\sum_{i=1}^{N_A}\ket{\uparrow}\bra{\uparrow}_i$ and $\sum_{i=1}^{N_A}\ket{\downarrow}\bra{\downarrow}_i$ in \erf{eq::FzHalf} are number operators, quantifying the number of atoms in states $\ket{\uparrow}$ and $\ket{\downarrow}$, respectively. The Schwinger representation recasts the collective spin operators in terms of creation and annihilation operators on the ``modes" $\uparrow$ and $\downarrow$,
\begin{align}\label{eq::FxHalf2}
\hat{F}_x=\frac{1}{2}\left(\hat{a}_{\uparrow}^\dag\hat{a}_{\uparrow}-\hat{a}_{\downarrow}^\dag\hat{a}_{\downarrow}\right)\\\label{eq::FyHalf2}
\hat{F}_y=\frac{i}{2}\left(\hat{a}_{\downarrow}^\dag\hat{a}_{\uparrow}-\hat{a}_{\uparrow}^\dag\hat{a}_{\downarrow}\right)
\end{align}
and
\begin{align}\label{eq::FzHalf2}
\hat{F}_z=\frac{1}{2}\left(\hat{a}_{\downarrow}^\dag\hat{a}_{\uparrow}+\hat{a}_{\uparrow}^\dag\hat{a}_{\downarrow}\right).
\end{align}
Note that these are identical to the Stokes' components with the modes $x$ and $y$ being replaced by the modes $\uparrow$ and $\downarrow$.
Because the ensemble prepared in a spin coherent state for $f=1/2$ has total angular momentum $F=N_A/2>>1$, the Holstein-Primakoff approximation holds on the collective spin operators. The collective spins take a form identical to the Stokes' components under the Holstein-Primakoff approximation,
\begin{align}
\hat{F}_x&\approx N_A/2\\
\hat{F}_y&\approx
\frac{\sqrt{N_A}}{2i}\big(\hat{a}_{\downarrow}-\hat{a}_{\downarrow}^\dag\big)=\sqrt{\frac{N_A}{2}}\hat{P}_{\downarrow}
\end{align}
and
\begin{align}
\hat{F}_z&\approx\frac{\sqrt{N_A}}{2}\big(\hat{a}_{\downarrow}+\hat{a}_{\downarrow}^\dag\big)=\sqrt{\frac{N_A}{2}}\hat{X}_{\downarrow}.
\end{align}
The collective spin component $\hat{F}_x$ is treated classically, analogous to $\hat{S}_1$. The $\uparrow$ mode is, likewise, the treated classically with $\hat{a}_{\uparrow}^\dag=\hat{a}_{\uparrow}=\sqrt{N_A}$, while the $\downarrow$ mode is quantum. The state of the ensemble is specified solely by the mode $\downarrow$. The initial spin coherent state with each atom in the state $\ket{\uparrow}$, is equivalent to the vacuum, $\ket{0}_{\downarrow}$. As shown in Fig. \ref{fig::HPatoms}, the Holstein-Primakoff approximation can also be understood pictorially on the Bloch sphere similarly to the Poincar\'{e} sphere in the case of the light. When $\expect{\hat{F}_x}=N_A/2>>1$, it is much larger than the uncertainties of the transverse collective spin components, $\Delta F_y=\Delta F_z=\sqrt{N_A}/2$. The state of the ensemble is, thus, confined to a small region of the Bloch sphere that can be treated as a locally flat plane. In this region, the transverse collective spin components are well approximated by quadratures on a single bosonic mode.
\begin{figure}[H]
\centering
\includegraphics[scale=.42]{HPatoms.pdf}
\caption{The Holstein-Primakoff approximation on the ensemble prepared in a spin coherent state. The HP approximation is applied in the exact same manner as it was on the light in Fig. \ref{fig::HPlight}. Because the atoms are completely spin polarized along $x$, $\hat{F}_x\approx N_Af$. Because the transverse collective spin uncertainties $\Delta F_y$ and $\Delta F_z$ are so much smaller than $\hat{F}_x$, the state of the ensemble occupies a small, locally flat region of the Bloch sphere. The collective spin operators $\hat{F}_y$ and $\hat{F}_z$ are approximated as the position and momentum quadratures $\hat{P}_\downarrow$ and $\hat{X}_\downarrow$.}\label{fig::HPatoms}
\end{figure}
\subsection{Multilevel Holstein-Primakoff Approximation on the Ensemble}\label{sec::MultiHPEnsemble}
For an ensemble with $f=1/2$ prepared in a spin coherent state, the Holstein-Primakoff approximation allows us to treat the state of the ensemble and all associated operators as being on a single bosonic mode. We would like to do this for a more general ensemble state where $f\geq1/2$, however. This can be accomplished through the multilevel Holstein-Primakoff approximation of Kurucz and M\o lmer \cite{MultilevelHP}. Consider a separable initial state of the ensemble of the form $\ket{\Psi}=\ket{\uparrow}^{\otimes N_A}$, where each atom is identically prepared in some arbitrary state $\ket{\uparrow}$ in the spin-$f$ ground manifold. We should stress that, unlike in the previous section, $\ket{\uparrow}$ is not the state of a qubit, but the state of a $2f+1$ dimensional \textit{qudit}. To first order in $\chi$ when $\bra{\uparrow}\hat{f}_z\ket{\uparrow}$=\,0 and $\ket{\uparrow}$ is not an eigenstate of $\hat{f}_z$, the Faraday interaction maps each atom to an orthogonal state $\ket{\downarrow}$,
\begin{eqnarray}
(\mathbb{I}^{(j)}-i\chi\hat{S}_3\hat{f}_z^{(j)})\ket{\uparrow}^{(j)}=\ket{\uparrow}^{(j)}-i\chi\hat{S}_3\sqrt{(\Delta f_z^2)_\uparrow}\ket{\downarrow}^{(j)},
\end{eqnarray}
where $(\Delta f_z^2)_\uparrow$ is shorthand for $\bra{\uparrow}(\Delta\hat{f}_z)^2\ket{\uparrow}$. For small time scales, therefore, we can approximate each atomic spin as a qubit embedded in the larger $2f+1$ dimensional hyperfine manifold. The states defining our embedded qubit, $\ket{\uparrow}$ and $\ket{\downarrow}$, we refer to as the ``fiducial state" and the ``coupled state''. In terms of the fiducial state, the coupled state is given by
\begin{align}\label{eq::coupledDef}
\ket{\downarrow}=\frac{\hat{f}_z\ket{\uparrow}}{\sqrt{(\Delta\hat{f}_z^2)_\uparrow}},
\end{align}
Here, we have assumed for simplicity that $\expect{\hat{f}_z}_\uparrow=\expect{\hat{f}_z}_\downarrow=0$, where $\expect{\hat{f}_z}_\psi$ is shorthand for $\bra{\psi}\hat{f}_z\ket{\psi}$. We treat the most general case, where the means of the fiducial and coupled states are not restricted to being zero, in Chapter \ref{sec::Beyond}. Also note that no coupled state exists when the fiducial state is an eigenstate of $\hat{f}_z$.
Restricting each atomic spin to the fiducial and coupled states, the collective spin $\hat{F}_z$ becomes
\begin{align}\label{FzExpand}
\hat{F}_z=\sum_{i=1}^{N_A}\sqrt{(\Delta f_z^2)_\uparrow}(\ket{\uparrow}\bra{\downarrow}_i+\ket{\downarrow}\bra{\uparrow}_i).
\end{align}
Each atom is now an ``embedded qubit" in the larger $2f+1$ dimensional hyperfine manifold. We can use the same techniques that we employed in the case of the light and the $f=1/2$ ensemble.
The Schwinger representation enables us to express $\hat{F}_z$ in terms of creation and annihilation operators on a pair of oscillator modes $\uparrow$ and $\downarrow$,
\begin{align}\label{eq::Schwinger}
\hat{F}_z&\approx\sqrt{(\Delta f_z^2)_\uparrow}(\hat{a}_\uparrow^\dag\hat{a}_\downarrow+\hat{a}_\downarrow^\dag\hat{a}_\uparrow).
\end{align}
The operators $\hat{a}_{\uparrow(\downarrow)}^\dag$ and $\hat{a}_{\uparrow(\downarrow)}$ create and annihilate an atom in the state $\ket{\uparrow\!\!(\downarrow)}$. Because the number of atoms in the fiducial state remains approximately equal to $N_A>>1$ when $\chi<<1$, we can treat the $\uparrow$ mode as classical by taking $\hat{a}_{\uparrow}^\dag\approx\hat{a}_{\uparrow}\approx\sqrt{N_A}$. Upon making this approximation,
\begin{eqnarray}\label{eq::MeanZeroPosition}
\hat{F}_z\approx\sqrt{2N_A(\Delta f_z^2)_\uparrow}\hat{X}_{\downarrow}.
\end{eqnarray}
where $\hat{X}_{\downarrow}$ is the position quadrature on the $\downarrow$ mode.
In terms of operators on the embedded qubit ensemble, the position quadrature is defined as
\begin{align}
\hat{X}_\downarrow&=\frac{1}{\sqrt{2}}(\hat{a}_\downarrow^\dag+\hat{a}_\downarrow)\notag\\
&\label{eq::position}
\approx\frac{1}{\sqrt{2N_A}}\sum_{i=1}^{N_A}(\ket{\uparrow}\bra{\downarrow}_i+\ket{\downarrow}\bra{\uparrow}_i).
\end{align}
While it is not related to a collective spin component in the same manner as $\hat{X}_{\downarrow}$, a momentum quadrature is similarly defined as an operator on the embedded qubit ensemble,
\begin{align}
\hat{P}_{\downarrow}&=\frac{i}{\sqrt{2}}(\hat{a}_\downarrow^\dag-\hat{a}_\downarrow)\notag\\
&\label{eq::momentum}
\approx\frac{i}{\sqrt{2N_A}}\sum_{i=1}^{N_A}(\ket{\downarrow}\bra{\uparrow}_i-\ket{\uparrow}\bra{\downarrow}_i).
\end{align}
In the limit where nearly all of the $N_A$ atoms remain in the fiducial state, the ensemble quadratures obey the canonical commutation relations,
\begin{align}
[\hat{X}_\downarrow,\hat{P}_\downarrow]=\frac{i}{N_A}\sum_{i=1}^{N_A}(\ket{\uparrow}\bra{\uparrow}_i-\ket{\downarrow}\bra{\downarrow}_i)\approx i.
\end{align}
After making the multilevel Holstein-Primakoff approximation, the state of the ensemble can be succinctly expressed in terms of a single oscillator mode. Similar to the light, the initial state of the ensemble with each atom prepared in $\ket{\uparrow}$ corresponds to the vacuum state, $\ket{0}_\downarrow$. The ensemble effectively resides in a phase plane defined by the position and momentum quadratures, $\hat{X}_\downarrow$ and $\hat{P}_\downarrow$. This approximation holds for a weak Faraday interaction that does not transfer atoms outside the embedded qubit appreciably.
\section{Phase Plane Faraday Interaction}\label{Sec::PhasePlaneFaraday}
After making the Holstein-Primakoff approximation on the light and the multilevel Holstein-Primakoff approximation on the ensemble, the Faraday interaction acts on the effective modes $y$ and $\downarrow$. By combining Eqs. (\ref{eq::FaradayDef}), (\ref{eq::S3Py}) and (\ref{eq::MeanZeroPosition}), we can write the Faraday Hamiltonian in terms of the quadratures on modes $y$ and $\downarrow$ as
\begin{align}\label{eq::HPfaraday}
\hat{H}=\frac{\hbar\chi}{\Delta t}\sqrt{N_LN_A(\Delta f_z^2)_\uparrow}\hat{P}_y\hat{X}_\downarrow.
\end{align}
This interaction generates entanglement between the light and atoms by coupling the modes $y$ and $\downarrow$. In the phase plane picture, the Faraday interaction generates a translation of the light in $\hat{X}_y$ by an amount proportional to the position of the atoms, $\hat{X}_\downarrow$. Likewise, the Faraday interaction translates the state of the ensemble along $\hat{P}_\downarrow$ by an amount depending on the the momentum of the light, $\hat{P}_y$. Note that the strength of the coupling between the light and ensemble increases with the projection noise of the ensemble, given by $(\Delta F_z^2)=N_A(\Delta f_z^2)_\uparrow$. This is another analytic demonstration that greater ensemble projection noise leads to increased entanglement between the light and atoms.
\section{Internal Spin Control}\label{sec::IntControl}
The previous sections have demonstrated that the strength of the Faraday interaction and the resulting atom-light entanglement increase with the initial projection noise of the ensemble. Because the initial projection noise is proportional to the variance of the fiducial state, it follows that we can enhance the Faraday interaction by preparing each atom in a fiducial state, $\ket{\uparrow}$, with larger internal spin variance, $(\Delta\hat{f}_z^2)_\uparrow$. Through a combination of radio-frequency and microwave magnetic fields, the hyperfine ground state of an alkali atom is completely controllable \cite{MerkelControlPRA}. This control has been experimentally demonstrated in the $f=3$ and $f=4$ ground manifolds of cesium \cite{ASmith13}. Controllability of the ground hyperfine spin enables us to apply any unitary transformation $\hat{u}$ to the space spanned by the magnetic sublevels of $f$. For the ensemble, control over the internal hyperfine spins of the atoms permits local unitary transformations of the form $\hat{u}^{\otimes N_A}$. Because of this restriction, the ensemble can be prepared in any state of the form $\ket{\uparrow}^{\otimes N_A}$. Internal spin control also enables us to apply arbitrary rotations and a variety of other useful transformations, which will be detailed in later sections.
\subsection{State Preparations}\label{sec::StatePreps}
To more concretely demonstrate how state preparation influences entanglement generation, we analyze the performance of squeezing protocols using three specific fiducial states. The results we present can be generalized to any fiducial state, however. The the fiducial states we consider have varying amounts of projection noise, leading to different degrees of atom-light entanglement and spin squeezing. In later chapters, we will also use these fiducial states to explore the impact of state preparation upon the decoherence of the ensemble. These studies demonstrate that state preparation greatly influences achievable spin squeezing.
The first state preparation we consider is the familiar spin coherent state ($SCS$) in which the fiducial state is the maximal spin projection along $x$,
\begin{align}
\ket{\uparrow_{SCS}}&=\ket{f,m_x=f}.
\end{align}
The coupled state in the multilevel HP approximation is
\begin{align}
\ket{\downarrow_{SCS}}=i\ket{f,m_x=f-1},
\end{align}
since $\Delta\hat{f}_z\ket{f,m_x=f} =i\sqrt{f/2}\:\ket{f,m_x=f-1}$, where $\left(\Delta f_z^2\right)_{\uparrow_{SCS}}=f/2$. The SCS has the smallest initial projection noise of any state preparation we consider. As a consequence, the collective spin coupling constant, which is given by $\xi(\uparrow_{SCS})=\\\gamma\Delta t OD/(18f)$, decays the most markedly with $f$.
The ``cat" preparation, in which each atom is prepared in the fiducial state
\begin{align}
\ket{\uparrow_{\text{cat}}}&=\frac{1}{\sqrt{2}}\left(\ket{f,m_z=f}+\ket{f,m_z=-f}\right),
\end{align}
has the largest projection noise of any initially separable ensemble state with
\\$(\Delta f_z^2)_{\uparrow_{\text{cat}}}=f^2$. From Eq. (\ref{eq::coupledDef}), the coupled state for the cat preparation
\begin{align}
\ket{\downarrow_{\text{cat}}}=\frac{1}{\sqrt{2}}\left(\ket{f,m_z=f}-\ket{f,m_z=-f}\right).
\end{align}
Due to its sizable initial projection noise, the cat preparation exhibits the largest collective spin coupling constant, $\xi(\uparrow_{\text{cat}})=\gamma\Delta t OD/9$. Interestingly, $\xi(\uparrow_{\text{cat}})$ is also independent of $f$.
Although the cat preparation generates the largest coherent interaction strength, it is extremely susceptible to decoherence, as our analysis will later show. Because of this, we consider an additional state preparation, ``$m_x=0$", with an intermediate projection noise between the $SCS$ and the cat. In the this preparation, each atom is prepared in the magnetic sublevel with zero spin projection along $x$,
\begin{align}
\ket{\uparrow_0}=\ket{f,m_x=0}.
\end{align}
The variance of the fiducial state, $(\Delta f_z^2)_{\uparrow_0}=f(f+1)/2$, scales quadratically with $f$ like the cat state. By again using Eq. (\ref{eq::coupledDef}), we determine coupled state to be
\begin{align}\label{coupledMx0}
\ket{\downarrow_0}=\frac{i}{\sqrt{2}}\left(\ket{f,m_z=-1}-\ket{f,m_z=1}\right).
\end{align}
While the collective spin coupling constant $\xi(\uparrow_0)=\gamma\Delta t OD(f+1)/(18f)$ still decreases with $f$, it does so at much reduced rate compared to the $SCS$.
\section{Gaussian States}\label{sec::GaussianStates}
Making the HP approximation on the light and the multilevel HP approximation on the atoms enables us to treat both systems on equal footing. The states of the light and the atoms become states of bosonic modes, each initially prepared in the vacuum state. This fact is particularly significant, as the vacuum state is a Gaussian state in phase space. Moreover, the interaction between the light and atomic ensemble preserves this Gaussianity to good approximation. Gaussian states, which have positive Gaussian-distributed Wigner functions, possess many useful properties that greatly simplify the description of the atom-light interface. Below we highlight several of the properties we will later utilize to characterize light-mediated squeezing and decoherence of the atomic ensemble. For more comprehensive reviews of Gaussian states, see Refs. \cite{Giedke02, PlenioEisert, Wang07, Weedbrook12, Adesso14}. Throughout this section, we consider the most general Gaussian state on a set of bosonic modes $1,...,n$. Specializing to the case of the atom-light system requires that we consider only two such modes, one associated with the ensemble, labeled $\downarrow$, and one associated with the light, labeled $y$.
A Gaussian state $\hat{\rho}\,$ on modes $1,...,n$ is fully specified by the first and second order moments of the phase space quadratures $\hat{X}_1,\hat{P}_1,...,\hat{X}_n,\hat{P}_n$. Because we are primarily concerned with entanglement generation, we need only consider the variances and covariances of the quadratures, which contain the atom-light and interatomic correlations. Information regarding these correlations is stored in the $2n\times 2n$ covariance matrix, $\Sigma$, with elements
\begin{align}
\Sigma_{ij}=\frac{\langle\Delta\hat{\textbf{d}}_i\Delta\hat{\textbf{d}}_j+\Delta\hat{\textbf{d}}_j\Delta\hat{\textbf{d}}_i\rangle}{2},
\end{align}
where $\hat{\textbf{d}}=\{\hat{X}_1,\hat{P}_1,...,\hat{X}_n,\hat{P}_n\}^T$ and $\Delta\hat{\textbf{d}}_i=\hat{\textbf{d}}_i-\langle\hat{\textbf{d}}_i\rangle$. All covariance matrices corresponding to physical states satisfy
\begin{align}\label{continuousUncert}
\Sigma +\frac{i}{2}\sigma\geq 0,
\end{align}
where the matrix $\sigma$, known as the symplectic matrix, is defined in terms of the canonical commutation relations
\begin{align}
\sigma_{jk}=-i[\hat{d}_j,\hat{d}_k]
\end{align}
or
\begin{eqnarray}\label{eq::sympMatrix}
\sigma=\bigoplus_{i=1}^{n}\left(\begin{matrix} 0 & 1 \\ -1 & 0\end{matrix}\right).
\end{eqnarray}
As a consequence of \erf{continuousUncert}, the Heisenberg uncertainty relations are fulfilled on pairs of conjugate quadratures \cite{SymplecticGeo}.
Up to a translation in phase space, the Wigner function of a Gaussian state depends purely upon the covariance matrix,
\begin{align}
W(\textbf{d})=\frac{1}{(2\pi)^n\sqrt{\text{det}(\Sigma)}}e^{-\frac{1}{2}\textbf{d}^T\Sigma^{-1}\textbf{d}},
\end{align}
where $\textbf{d}=\{X_1,P_1,...,X_n,P_n\}^T$. A vacuum state in all modes, such as the initial state of the atom-light system, has the covariance matrix
\begin{align}\label{eq::vacuumCov}
\Sigma_0=\bigoplus_{i=1}^{n}\left(\begin{matrix} 1/2 & 0 \\ 0 & 1/2\end{matrix}\right).
\end{align}
The covariance matrix $\Sigma_0$ generates a symmetric, Gaussian-distributed Wigner function
\begin{align}
W_0(\textbf{d})=\frac{1}{\pi^n}e^{-(X_1^2+P_1^2+...+X_n^2+P_n^2)}
\end{align}
with zero mean and a variance of 1/2 in all quadratures. As the light and ensemble evolve via a Faraday based squeezing protocol, the Gaussian character of the combined state is preserved. Because it stores all information regarding entanglement, we need only track the evolution of the covariance matrix. Below, we describe how the covariance matrix evolves as the corresponding Gaussian state undergoes unitary and dissipative dynamics.
The evolution of $\hat{\rho}$ under any operation that preserves Gaussianity can be represented in the Heisenberg picture as a transformation of the covariance matrix. We first consider unitary maps that preserve Gaussianity, which are generated by Hamiltonians of the form
\begin{align}\label{eq::GaussH}
\hat{H}=\sum_{i,j}h^{(i,j)}(\hat{\textbf{d}}_i\hat{\textbf{d}}_j+\hat{\textbf{d}}_j\hat{\textbf{d}}_i)
\end{align}
for real $h^{(i,j)}$ \cite{PlenioEisert}. Associated with each Gaussian-preserving $\hat{H}$ is a \emph{symplectic map} $S_H$, which dictates the evolution of both the first order moments and the covariance matrix. Under a unitary map generated by $\hat{H}$, the central first order moments and the covariance matrix of $\hat{\rho}$ transform as
\begin{align}
\Delta\hat{\textbf{d}}'= S_H\Delta\hat{\textbf{d}}
\end{align}
and
\begin {align}
\Sigma'= S_H\Sigma S_H^{T}.
\end{align}
All symplectic maps satisfy
\begin {align}\label{symplectic1}
S_H\sigma S_H^{T}=\sigma,
\end{align}
which ensures that \erf{continuousUncert} is preserved on the output covariance matrix $\Sigma'$.
Unitary maps that preserve Gaussianity are a subset of the more general class of completely positive maps that preserve Gaussianity, so-called Gaussian channels. Associated with each Gaussian channel $\mathcal{E}$ acting on $\hat{\rho}\,$ is a matrix $M_\mathcal{E}$ and a symmetric, positive semidefinite matrix $N_\mathcal{E}$. Under $\mathcal{E}$, the evolution of the central first order moments and covariance matrix are given by
\begin{align}
\Delta\hat{\textbf{d}}'= M_\mathcal{E}\Delta\hat{\textbf{d}}
\end{align}
and
\begin{align}
\Sigma'= M_\mathcal{E}\Sigma M_\mathcal{E}^{T}+N_\mathcal{E}.
\end{align}
Preserving \erf{continuousUncert} on $\Sigma'$ requires that $M_\mathcal{E}$ and $N_\mathcal{E}$ satisfy \cite{PlenioEisert}
\begin{align}\label{ChannelCond}
N_\mathcal{E}+\frac{i}{2}\sigma-\frac{i}{2}M_\mathcal{E}\sigma M_\mathcal{E}^T\geq 0.
\end{align}
The matrix $N_\mathcal{E}$, referred to as the noise component, increases the variance of the quadratures. Consequently, $\mathcal{E}$ is called a \emph{noise channel} in the case that $M_\mathcal{E}=\mathbb{I}$. When $N_\mathcal{E}=0$, condition \erf{ChannelCond} is equivalent to \erf{symplectic1} and $M_\mathcal{E}$ is a symplectic map. In the case where $N_\mathcal{E}\neq0$, $\mathcal{E}$ is a dissipative transformation. Gaussian channels of this sort frequently arise when $\hat{\rho}\,$ becomes entangled with another system that is subsequently traced out, such as an environment.
In addition to Gaussian channels, operations that preserve Gaussianity include several forms of measurement, one of the most important being homodyne detection. Suppose that we wish to perform homodyne detection on the $n$th mode of the $n$-mode Gaussian state $\hat{\rho}$. The initial covariance matrix of $\hat{\rho}$ can be written in terms of the submatrices $A$, $B$ and $C$ as
\begin{align}
\Sigma=\left(\begin{matrix} A & C \\ C^T & B\end{matrix}\right),
\end{align}
where $A$ is the covariance matrix of modes 1 through $n-1$, $B$ is the covariance matrix of mode $n$, and the matrix $C$ contains the covariances between modes 1 through $n-1$ and $n$. Consider a homodyne measurement of $\hat{X}_n$, which we will denote by $h[\hat{X}_n]$. In a homodyne measurement with perfect efficiency, the state of the $n$th mode is projected onto an eigenstate of $\hat{X}_n$ with no variance. Although the $n$th mode is no longer Guassian, the state remains Gaussian on modes $1,...,\,n-1$ and has the covariance matrix
\begin{eqnarray}\label{HCovariance}
\Sigma'=h[{\hat{X}_n}](\Sigma)=A-C(\mathbb{P} B\mathbb{P})^{-1}C^T,
\end{eqnarray}
where $\mathbb{P}=\text{diag}(1,0)$ and the inverse symbol denotes the Moore-Penrose pseudoinverse \cite{PlenioEisert}. As evidenced in \erf{HCovariance}, the Gaussian state on modes 1 through $n$ evolves deterministically, independent of the measurement outcome of $\hat{X}_n$.
\chapter{Squeezing Protocols} \label{Sec:Protocols}
As discussed in Sec. \ref{sec::EntangleFaraday}, the Faraday interaction creates entanglement between the light and atoms with strength given by the collective spin coupling constant, $\xi$. It is entanglement between the atoms, however, that creates spin squeezing. In squeezing protocols that utilize the Faraday interaction, a mode of the light acts as a quantum data bus, inducing entanglement through its mutual coupling to all atoms. Atom-light entanglement can be converted into interatomic entanglement through measurement of the light, which subjects the ensemble to measurement backaction \cite{KuzMan98,Koschorreck2010,Takano2009}. In other protocols, the light mediates an effective nonlinear interaction on the ensemble \cite{TakTak05,TraDeu10}. The commonality in all protocols is that spin squeezing ultimately depends upon the strength of the initial entanglement between the light and atoms.
Using the covariance matrix update rules of Gaussian states, we can describe these squeezing protocols in a simple, compact manner. We represent the ensemble and light collectively as a multimode Guassian state with $\hat{\textbf{d}}=(\hat{X}_\downarrow,\hat{P}_\downarrow,\hat{X}_y,\hat{P}_y)^T$, where the initial vacuum state of the system has the covariance matrix given in \erf{eq::vacuumCov}. Because the protocols produce an ensemble state that is squeezed in the $\hat{X}_\downarrow$-$\hat{P}_\downarrow$ phase plane, we employ the quadrature squeezing parameter, $\zeta_q$, to quantify squeezing. The relationship between the Wineland spin squeezing parameter, $\zeta_m$, and the quadrature squeezing parameter, $\zeta_q$, will be explored in Sec. \ref{Sec::SqParameters}.
\section{Quantum Nondemolition (QND) Measurement}\label{sec::QNDmeas}
Quantum nondemolition measurement was one of the earliest techniques used to create spin squeezing in atomic ensembles \cite{KuzMan98,Koschorreck2010,Takano2009}. A schematic of this protocol is shown in Fig. \ref{fig::QND}. Squeezing by QND measurement employs the procedure described in Sec. \ref{sec::EntangleFaraday}, in which the collective spin component $\hat{F}_z$ of the ensemble is determined through a measurement of the Stokes' component $\hat{S}_2$ of the light. Because we have made the Holstein-Primakoff approximation, phase space quadratures take the place of the collective spin component $\hat{F}_z$ and the Stokes' component $\hat{S}_2$. In the phase plane picture, the position quadrature $\hat{X}_\downarrow$ of the ensemble is inferred through a measurement of the position quadrature $\hat{X}_y$ of the light.
\begin{figure}[H]
\centering
\includegraphics[scale=.35]{QNDsetup.pdf}
\caption{Schematic of the experimental setup for the QND measurement protocol. Horizontally polarized probe light passes through the ensemble, causing a rotation of its linear polarization. The magnitude of this rotation, given by the value of the quadrature $\hat{X}_y$, is proportional to the collective spin of the ensemble, $\hat{F}_z$, or the quadrature, $\hat{X}_\downarrow$, in the generalized Holstein-Primakoff approximation. A balanced polarimeter at $\pm45^\circ$ measures the quadrature $\hat{X}_y$, an effective measurement of $\hat{F}_z$. This causes squeezing of the ensemble along $\hat{F}_z$ by measurement backaction. In the generalized Holstein-Primakoff picture, this is equivalent to phase plane squeezing of the quadrature $\hat{X}_\downarrow$. }
\label{fig::QND}
\end{figure}
The light and atoms are first entangled by the Faraday interaction, the action of which over a small time $\Delta t$ is described by the symplectic matrix
\begin{align}\label{eq::FaradayS}
S_F(\Delta t)=\left(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 &1& 0 & -\sqrt{\xi}\\ \sqrt{\xi} & 0&1& 0\\0 & 0 & 0 & 1\end{matrix}\right).
\end{align}
Equation (\ref{eq::FaradayS}) follows from the evolution of the quadratures in the Heisenberg picture. The atom-light entanglement is evidenced in the transformed quadratures,
\begin{align}\label{firstpass}
\left(\begin{matrix}\hat{X}_{\downarrow}(\Delta t)\\ \hat{P}_{\downarrow}(\Delta t)\\\hat{X}_{y}(\Delta t)\\\hat{P}_{y}(\Delta t)\end{matrix}\right)=
\left(\begin{matrix}\hat{X}_{\downarrow}(0)\\ \hat{P}_{\downarrow}(0)-\sqrt{\xi}\hat{P}_{y}(0)\\\hat{X}_{y}(0)+\sqrt{\xi}\hat{X}_{\downarrow}(0)\\\hat{P}_{y}(0)\end{matrix}\right).
\end{align}
Note that the position quadrature of the light, $\hat{X}_y$, contains information about the position quadrature of the ensemble, $\hat{X}_\downarrow$. A measurement of the light by the balanced polarimeter in Fig. \ref{fig::QND} behaves like a homodyne measurement with the probe light acting as a local oscillator. Performing homodyne detection on $\hat{X}_y$ is an effective measurement of the ensemble's $\hat{X}_\downarrow$, inducing backaction on the ensemble that reduces the variance of $\hat{X}_\downarrow$. For an arbitrary covariance matrix at time $t$, this protocol is described by the update
\begin{align}\label{eq::QNDupdate}
\Sigma_{\downarrow}(t+\Delta t)=h[\hat{X}_y]\left(S_F(\Delta t)\Sigma(t) S_F^{T}(\Delta t)\right).
\end{align}
For the initial vacuum state of the atom-light system, the covariance matrix of the ensemble resulting from the QND measurement protocol is
\begin{align}\label{eq::QNDcov}
\Sigma_{\downarrow}(\Delta t)
=\frac{1}{2}\left(\begin{matrix}(1+\xi)^{-1}& 0\\ 0 &1+\xi\end{matrix}\right).
\end{align}
The squeezing of the position quadrature $\hat{X}_\downarrow$, which follows from \erf{eq::QNDcov}, is
\begin{align}
\zeta_q=\frac{1}{1+\xi}.
\end{align}
Note that $\zeta_q$ decreases with increasing $\xi$. This signifies that squeezing, like the atom-light entanglement, improves with a larger collective spin coupling constant.
\section{Double Pass Protocols}\label{sec::DP}
Generating spin squeezing through a double pass geometry, in which the light and atoms interact twice via the Faraday interaction, was first proposed in \cite{TakTak05}. Over the two passes, the light mediates an effective atom-atom interaction that generates interatomic entanglement and squeezing. This process is depicted in Fig. \ref{fig::DPsetup}.
\begin{figure}
\centering
\includegraphics[scale=.5]{DPsetupNew.pdf}
\caption{Schematic of the experimental setup for the double pass protocols. Upon first entering the apparatus, light passes through the ensemble and undergoes Faraday rotation. The light is then reflected back towards the ensemble, passing twice through a $\lambda/8$ wave plate. The light then passes through the ensemble a second time, once again undergoing Faraday rotation. In the quantum eraser protocol, the light's polarization is measured upon exiting the ensemble. A magnetic field is applied to the ensemble along $z$ conditioned upon the measurement value, rotating the collective spin. }\label{fig::DPsetup}
\end{figure}
The light and ensemble are initially entangled by the Faraday interaction, transforming the quadratures as in Eq. (\ref{firstpass}). Recall that after the first pass, the position quadrature of the light $\hat{X}_y$ contains information about the position quadrature of the ensemble $\hat{X}_\downarrow$. After the first pass, the light proceeds twice through a $\lambda/8$ wave plate, rotating the $\hat{X}_y$ and $\hat{P}_y$ quadratures by $\pi/2$,
\begin{align}
\left(\begin{matrix}\hat{X}_{\downarrow}(\Delta t)\\ \hat{P}_{\downarrow}(\Delta t)\\\hat{X}_{y}(\Delta t)\\\hat{P}_{y}(\Delta t)\end{matrix}\right)=
\left(\begin{matrix}\hat{X}_{\downarrow}(0)\\ \hat{P}_{\downarrow}(0)-\sqrt{\xi}\hat{P}_{y}(0)\\-\hat{P}_{y}(0)\\\hat{X}_{y}(0)+\sqrt{\xi}\hat{X}_{\downarrow}(0)\end{matrix}\right).
\end{align}
The light passes through the ensemble a second time with the $\hat{P}_y $ quadrature containing information about $\hat{X}_\downarrow$. The Faraday interaction then couples $\hat{P}_y $ to $\hat{X}_\downarrow$, creating an effective $\hat{X}_\downarrow^2$ interaction. The nonlinear nature of this interaction is alternatively illustrated by decomposing the protocol into unitary operators,
\begin{align}\label{Udp}
\hat{U}_{DP}=\hat{U}_{\text{Faraday}}\hat{U}_{\frac{\lambda}{8}}\hat{U}_{\frac{\lambda}{8}}\hat{U}_{\text{Faraday}}=e^{i\sqrt{\xi}(\hat{X}_y-\hat{P}_y)\hat{X}_\downarrow}e^{i\frac{\xi}{2}\hat{X}_\downarrow^2}.
\end{align}
The first exponential on the right hand side of \erf{Udp} does not create squeezing. Rather, it indicates the residual entanglement between the light and atoms after the second pass. The second exponential, however, is a well studied spin squeezing interaction known as one-axis twisting \cite{KitagawaUeda93}. One axis-twisting is a shearing interaction that creates squeezing in the $\hat{X}_\downarrow$-$\hat{P}_{\downarrow}$ phase plane as depicted in Fig. \ref{fig::Twisting}.
\begin{figure}
\centering
\includegraphics[scale=.4]{twisting3.pdf}
\caption{The effect of the one-axis twisting interaction, $e^{i\xi\hat{X}_\downarrow^2/2}$, in the phase plane of the ensemble. One-axis twisting imparts a translation of the ensemble state along $\hat{P}_\downarrow$ by an amount proportional to the state's displacement in $\hat{X}_\downarrow$. This shearing creates squeezing in the $\hat{X}_\downarrow$-$\hat{P}_\downarrow$ phase plane.}\label{fig::Twisting}
\end{figure}
For an arbitrary covariance matrix at time $t$, the double pass protocol is described by the update
\begin{align}
\Sigma_{\downarrow}(t+2\Delta t)=\text{Tr}_y\left(S_F(\Delta t)R_{\frac{\lambda}{4}}S_F(\Delta t)\Sigma(t)S_F^T(\Delta t)R_{\frac{\lambda}{4}}^TS_F^T(\Delta t)\right).
\end{align}
Here, $\text{Tr}_y$ denotes taking the partial trace over the light, which is performed by discarding all entries of the covariance matrix involving operators on the $y$ mode. After tracing out the light, we are left with the ``reduced" covariance matrix of the ensemble, $\Sigma_{\downarrow}$. The eigenvalues of $\Sigma_{\downarrow}$ are the variances of the squeezed and anti-squeezed quadratures in phase space. The reduced density matrix can be diagonalized by performing a phase space rotation via a symplectic rotation matrix
\begin{align}
R(\theta)=\left(\begin{matrix}\text{cos}\theta&-\text{sin}\theta\\
\text{sin}\theta&\text{cos}\theta\end{matrix}\right).
\end{align}
Applying the double pass protocol to the initial vacuum state and rotating the resulting reduced covariance matrix by an angle
\begin{align}
\phi=-\text{tan}^{-1}\left(\frac{2\xi}{2\xi+\xi^2+\xi\sqrt{\xi^2+4\xi+8}}\right)
\end{align}
yields
\begin{align}
&R(\phi)\Sigma_{\downarrow}R(\phi)^T=\\&\frac{1}{4}\left(\begin{matrix}2+2\xi+\xi^2-\xi\sqrt{\xi^2+4\xi+8}&0\\
0&2+2\xi+\xi^2+\xi\sqrt{\xi^2+4\xi+8}\end{matrix}\right).
\end{align}
This indicates that the double pass protocol produces squeezing in the phase plane along the angle $\pi-\phi$. For $\xi>>1$, the squeezing parameter becomes
\begin{align}
\zeta_q\approx\frac{2}{\xi}.
\end{align}
Although the double pass performs slightly worse than the QND measurement protocol, it does not require a high quantum efficiency measurement of the light.
\subsection{Quantum Eraser}
Although the ensemble is squeezed after the double pass protocol, the squeezing is limited because the light and atoms are still entangled after the second pass. This entanglement is evidenced in \erf{Udp} and in the quadrature evolution after the second pass,
\begin{align}\label{eq::doublepassQuads}
\left(\begin{matrix}\hat{X}_{\downarrow}(2\Delta t)\\ \hat{P}_{\downarrow}(2\Delta t)\\\hat{X}_{y}(2\Delta t)\\\hat{P}_{y}(2\Delta t)\end{matrix}\right)=
\left(\begin{matrix}\hat{X}_{\downarrow}(0)\\ \hat{P}_{\downarrow}(0)-\sqrt{\xi}\hat{P}_{y}(0)-\sqrt{\xi}\hat{X}_{y}(0)-\xi\hat{X}_{\downarrow}(0)\\ -\hat{P}_{y}(0)+\sqrt{\xi}\hat{X}_{\downarrow}(0)\\ \hat{X}_{y}(0)+\sqrt{\xi}\hat{X}_{\downarrow}(0)\end{matrix}\right).
\end{align}
Taking the partial trace over the light, in effect discarding the light while it is still entangled to the atoms, causes the ensemble to decohere and limits squeezing. This decoherence can be remedied with a quantum eraser, a measurement that disentangles two systems \cite{Scully91,TraDeu10}. Homodyne detection on the $y$ mode projects the light into a pure state, separable from the ensemble. Note from \erf{eq::doublepassQuads} that the quadrature $\hat{X}'_{y}=(\hat{X}_{y}-\hat{P}_{y})/\sqrt{2}$ contains no atomic information. A measurement of $\hat{X}'_{y}$, thus, disentangles the atoms from the light without causing measurement backaction on the ensemble \cite{TraDeu10}. The Kraus operator that evolves the ensemble conditioned on a measurement value $X_y'$ of $\hat{X}'_{y}$ is given by
\begin{align}
\bra{X_y'}\hat{U}_{DP}\ket{0}_y=e^{i\sqrt{2\xi}X_y'\hat{X}_\downarrow}e^{i\frac{\xi}{2}\hat{X}_\downarrow^2}\bra{X_y'}0\rangle_y.
\end{align}
We see that the net effect of the measurement is a translation of the ensemble along $\hat{P}_\downarrow$. This translation can be cancelled by applying a magnetic field along the $z$ axis, resulting in a pure one-axis twisting interaction. In the covariance matrix update picture, the Takeuchi protocol with the quantum eraser is given by
\begin{align}\label{eq::QE}
\Sigma_{\downarrow}(t+2\Delta t)=h[{\hat{X}'_y}]\left(S_F(\Delta t)R_{\frac{\lambda}{4}}S_F(\Delta t)\Sigma(t)S_F^T(\Delta t)R_{\frac{\lambda}{4}}^TS_F^T(\Delta t)\right).
\end{align}
By applying \erf{eq::QE} to the covariance matrix of the initial vacuum state, we find the resultant squeezing. For $\xi>>1$,
\begin{align}
\zeta_q\approx\frac{1}{\xi^2}.
\end{align}
This quadratic scaling with $\xi$ is a substantial improvement over the double pass alone.
\subsection{Phase-matching}
In eliminating an important source of decoherence on the ensemble, the quantum eraser generates a pure one-axis twisting interaction. The spin squeezing generated by the double pass protocol can be enhanced even further, however. Decomposing the one-axis twisting interaction in terms of creation and annihilation operators on the $\downarrow$ mode yields
\begin{align}\label{eq::twisting}
e^{i\frac{\xi}{2}\hat{X}_\downarrow^2}=\text{exp}\left(i\frac{\xi}{4}(\hat{a}_\downarrow^{\dag 2}+\hat{a}_\downarrow^{2})+i\frac{\xi}{2}\hat{a}_\downarrow^{\dag}\hat{a}_\downarrow+i\frac{\xi}{4}\right).
\end{align}
The first term on the right hand side is a Bogoliubov transformation, a pure squeezing interaction. The second term is a rotation in the $\hat{X}_\downarrow$-$\hat{P}_\downarrow$ phase plane, while the last term is a phase that is irrelevant to the dynamics. Due to the rotation term, the pure squeezing generated by the Bogoliubov transformation takes place along a variable axis. By eliminating this rotation or ``phase-matching", we can ensure that squeezing occurs along a consistent axis \cite{TraDeu10}. To achieve this, we apply the double pass with the quantum eraser over a very small interaction time $2\Delta t/n$ followed by a rotation in the phase plane by an angle $\xi/2n$, which can be generated with internal spin control. Through a Trotter expansion, we alternatingly apply the double pass plus quantum eraser and the rotation over infinitesimal time steps. This procedure generates a Bogoliubov transformation without extraneous rotation,
\begin{align}\label{bogoliubov}
\hat{U}_{PM}&=\text{lim}_{n\rightarrow\infty}\left(e^{-i\frac{\xi}{2n}\hat{a}_\downarrow^\dag\hat{a}_\downarrow}e^{i\frac{\xi}{2n}\hat{X}_\downarrow^2}\right)^n=e^{i\frac{\xi}{4}(\hat{a}_\downarrow^{\dag2}+\hat{a}_\downarrow^2)}.
\end{align}
The phase matching procedure can also be implemented as a symplectic transformation on the covariance matrix. The one-axis twisting unitary in \erf{eq::twisting} applied over a small interaction time $2\Delta t/n$ becomes the unitary transformation
\begin{align}\label{eq::SmallTwisting}
\hat{U}_n=e^{i\frac{\xi}{2n}\hat{X}_\downarrow^2}.
\end{align}
Because this unitary is generated by a Hamiltonian of the form in \erf{eq::GaussH}, it preserves Gaussianity and acts upon the covariance matrix via a symplectic map, $S_n$.
To first order in $\xi/n$, the singular value decomposition of $S_n$ is
\begin{align}
S_n=R(\theta_-) D R(-\theta_+)
\end{align}
where
\begin{align}
D=\left(\begin{matrix}e^{-\frac{\xi}{2n}}& 0\\ 0& e^{\frac{\xi}{2n}}\end{matrix}\right)
\end{align}
and the angles $\theta_{\pm}\approx\pi/4\pm\frac{\xi}{4n}$. Alternating the one-axis twisting interaction and rotations of $\theta_+-\theta_-\approx\xi/(2n)$, creates pure squeezing along a consistent axis,
\begin{align}\label{eq::phasematchingTheta}
...R(\theta_+-\theta_-)S_{1}^{(n)}R(\theta_+-\theta_-)S_{1}^{(n)}=R(\theta_+) D^n R(-\theta_-).
\end{align}
Up to a rotation in phase space, the resulting symplectic map is
\begin{align}\label{exponentialSq}
D^n=\left(\begin{matrix}e^{-\frac{\xi}{2}}& 0\\ 0& e^{\frac{\xi}{2}}\end{matrix}\right).
\end{align}
Applying this symplectic map to the vacuum covariance matrix produces squeezing that scales exponentially with $\xi$,
\begin{align}
\zeta_q=e^{-\xi}.
\end{align}
Through a combination of feedback and internal spin controls, the scaling of spin squeezing with the collective spin coupling constant is substantially enhanced.
\section{Quadrature Squeezing vs. Spin Squeezing}\label{Sec::SqParameters}
The spin squeezing protocols discussed in Sections \ref{sec::QNDmeas} and \ref{sec::DP} create squeezing in the phase plane defined by the ensemble quadratures $\hat{X}_\downarrow$ and $\hat{P}_\downarrow$. When the fiducial state is a spin coherent state, this is equivalent to squeezing a component of the collective spin. For more general fiducial states, however, the relationship between squeezing in the phase plane and squeezing of a collective spin component is not immediately clear. To shed light on this, we revisit equations (\ref{eq::position}) and (\ref{eq::momentum}), which show that the quadratures have an alternative interpretation as operators on the ensemble of embedded qubits. Recall that each embedded qubit $j$ is defined on a basis consisting of the fiducial and coupled states, $\ket{\uparrow}_j$ and $\ket{\downarrow}_j$, within the $2f+1$ dimensional hyperfine spin of each atom $j$. Consider the collective spin components of the embedded qubit ensemble given by
\begin{align}
&\hat{\Sigma}_x=\frac{1}{2}\Sigma_{j=1}^N\hat{\sigma}_x^{(j)} \;\;\;\text{and}
\\ &\hat{\Sigma}_y=\frac{1}{2}\Sigma_{j=1}^N\hat{\sigma}_y^{(j)},
\end{align}
where
\begin{align}
&\hat{\sigma}_x^{(j)}=\ket{\downarrow}\bra{\uparrow}_j+\ket{\uparrow}\bra{\downarrow}_j\;\;\;\; \text{and}\\
&\hat{\sigma}_y^{(j)}=i(\ket{\downarrow}\bra{\uparrow}_j-\ket{\uparrow}\bra{\downarrow}_j)
\end{align}
are the Pauli spin operators on a qubit $j$ with basis states $\ket{\uparrow}_j$ and $\ket{\downarrow}_j$. From equations (\ref{eq::position}) and (\ref{eq::momentum}),
\begin{align}
&\hat{X}_\downarrow\approx\sqrt{2}\hat{\Sigma}_x \;\;\;\text{and}
\\ &\hat{P}_\downarrow\approx\sqrt{2}\hat{\Sigma}_y.
\end{align}
Squeezing in the phase plane is, thus, equivalent to squeezing the collective spin of the embedded qubit ensemble in a plane defined by $\hat{\Sigma}_x$ and $\hat{\Sigma}_y$.
Squeezing of the embedded qubit ensemble does not necessarily imply squeezing of the atomic ensemble composed of spin-$f$ qudits, however. While $\hat{\Sigma}_x$ is proportional to $\hat{F}_z$ under the multilevel HP approximation, $\hat{\Sigma}_y$ does not in general relate to a collective spin component of the qudit ensemble when $f\geq1/2$. Squeezing in the plane defined by $\hat{\Sigma}_x$ and $\hat{\Sigma}_y$, nonetheless, reveals valuable information about the state of the qudit ensemble. Recall that because a qubit cannot be internally squeezed, spin squeezing in an ensemble of qubits is the product of entanglement alone. Squeezing in the $\hat{\Sigma}_x$-$\hat{\Sigma}_y$ plane, which is synonymous with squeezing in the $\hat{X}_\downarrow$-$\hat{P}_\downarrow$ phase plane, implies entanglement between the atoms in the ensemble. We can, furthermore, view the squeezing parameter $\zeta_q$, as a measure of this interatomic entanglement. While the squeezing protocols outlined in Sections \ref{sec::QNDmeas} and \ref{sec::DP} do not necessarily squeeze a component of the collective spin $\hat{F}_n$, they generate entanglement between the atoms in the ensemble.
While $\zeta_q<1$ indicates the presence of interatomic entanglement, the question remains as to how exactly $\zeta_q$ relates to the metrological spin squeezing parameter, $\zeta_m$. In terms of the phase plane squeezing parameter, we can express the metrological spin squeezing parameter as
\begin{align}\label{parameters}
\zeta_m=\zeta_m^\uparrow\zeta_q,
\end{align}
where $\zeta_m^\uparrow$ is the metrological squeezing parameter of a single atomic spin prepared in the fiducial state,
\begin{align}
\zeta_m^\uparrow=\frac{2f\left(\Delta f_z^2\right)_\uparrow}{\langle\hat{f}_x\rangle^2}.
\end{align}
From \erf{parameters}, it is clear that when the fiducial state is anti-squeezed, i.e. $\zeta_m^\uparrow>1$, squeezing of the collective spin cannot occur unless there is a sufficiently high degree of squeezing in the phase plane or, equivalently, interatomic entanglement. To achieve metrologically relevant spin squeezing $\zeta_q$ must be very small, specifically $\zeta_q<(\zeta_m^\uparrow)^{-1}$.
\section{Post-Processing}\label{sec::postprocessing}
Our protocol for enhancing the Faraday interaction relies on preparing the atoms in fiducial states, such as $\ket{\uparrow_\text{cat}}$ and $\ket{\uparrow_0}$, which increases projection noise variance of the ensemble thereby maximizing the resolvability of a collective spin projection in a measurement of the light. Increasing the projection noise seems antithetical to the goal of squeezing. Indeed, equation (\ref{parameters}) appears to suggest that we strengthen the Faraday interaction, and the resulting interatomic entanglement, at the expense of metrologically relevant squeezing. Through internal spin control, however, we can convert this enhanced interatomic entanglement into metrologically relevant squeezing.
Note that when the ensemble is prepared in a spin coherent state, \\$\bra{\uparrow_{SCS}}\hat{f}_x\ket{\uparrow_{SCS}}=f$ and $\bra{\uparrow_{SCS}}(\Delta\hat{f}_z)^2\ket{\uparrow_{SCS}}=f/2$, implying $\zeta_m^{\uparrow_{SCS}}=1$. By \erf{parameters}, the phase plane squeezing and metrologically relevant spin squeezing are equivalent, i.e. $\zeta_m=\zeta_q$. We can take advantage of this relationship by mapping the squeezing created in the $\hat{X}_{\downarrow}$-$\hat{P}_{\downarrow}$ phase plane into the $\hat{X}_{\downarrow_{SCS}}$-$\hat{P}_{\downarrow_{SCS}}$ phase plane. Using internal spin control, we can generate the partial isometry
\begin{align}\label{Uscs}
\hat{U}_{SCS}=\bigotimes_{i=1}^{N_A}\left(\ket{\uparrow_{SCS}}\bra{\uparrow}_i+\ket{\downarrow_{SCS}}\bra{\downarrow}_i\right).
\end{align}
On each atom $i$, $\hat{U}_{SCS}$ maps an arbitrary fiducial state to $\ket{\uparrow_{SCS}}_i$ and an arbitrary coupled state to $\ket{\downarrow_{SCS}}_i$. The phase plane quadratures are transformed as
\begin{align}
&\hat{U}_{SCS}^\dag\hat{X}_\downarrow\hat{U}_{SCS}=\hat{X}_{\downarrow_{SCS}}\;\;\;\;\text{and}\\
&\hat{U}_{SCS}^\dag\hat{P}_\downarrow\hat{U}_{SCS}=\hat{P}_{\downarrow_{SCS}}.
\end{align}
An ensemble state $\ket{\zeta}$ that is squeezed in the $\hat{X}_{\downarrow}$-$\hat{P}_{\downarrow}$ phase plane will be mapped to a state $\ket{\zeta'}=\hat{U}_{SCS}\ket{\zeta}$ that is squeezed in the $\hat{X}_{\downarrow_{SCS}}$-$\hat{P}_{\downarrow_{SCS}}$ phase plane. We can see this more clearly by looking at the quadrature variances,
\begin{align}
&\bra{\zeta}(\Delta\hat{X}_\downarrow)^2\ket{\zeta}=\bra{\zeta'}(\Delta\hat{X}_{\downarrow_{SCS}})^2\ket{\zeta'}\;\;\;\;\text{and}\label{eq::UscsVarX}\\
&\bra{\zeta}(\Delta\hat{P}_\downarrow)^2\ket{\zeta}=\bra{\zeta'}(\Delta\hat{P}_{\downarrow_{SCS}})^2\ket{\zeta'}.\label{eq::UscsVarP}
\end{align}
As a result of Eqs. (\ref{eq::UscsVarX}) and (\ref{eq::UscsVarP}), the phase plane squeezing parameter $\zeta_q$ is preserved by $\hat{U}_{SCS}$. The metrological squeezing parameter of the fiducial state, however, is not preserved,
\begin{align}
\zeta_m^\uparrow\rightarrow\frac{2f\bra{\uparrow}\hat{U}_{SCS}^\dag(\Delta\hat{f}_z)^2\hat{U}_{SCS}\ket{\uparrow}}{\bra{\uparrow}\hat{U}_{SCS}^\dag\hat{f}_x\hat{U}_{SCS}\ket{\uparrow}^2}=\zeta_m^{\uparrow_{SCS}}=1.
\end{align}
The metrologically relevant spin squeezing parameter, consequently, transforms as:
\begin{align}
\zeta_m=\zeta_m^\uparrow\zeta_q\rightarrow\zeta_m^{\uparrow_{SCS}}\zeta_q=\zeta_q.
\end{align}
The partial isometry $\hat{U}_{SCS}$ converts squeezing in the phase plane into metrologically relevant spin squeezing.
By combining state preparation, a Faraday-based squeezing protocol and ``post-processing" in the form of a partial isometry, we can create metrologically relevant spin squeezing that increases with the initial projection noise variance of the ensemble \cite{NorDeu12}. The steps of this procedure are shown in Fig. \ref{fig::SqScheme}.
The enhanced entanglement created by preparing the ensemble in a state with larger projection noise does not come at the expense of metrologically relevant squeezing. Indeed, through this series of controls, squeezing produced by the Faraday interaction is increased substantially. For QND measurement, in particular, one can obtain more squeezing than what would seem possible given the shot noise resolution.
\begin{figure}[H]
\centering
\includegraphics[scale=.8]{SqScheme.pdf}
\caption{Using internal spin control to enhance spin squeezing. (a) Internal spin control is first used to prepare each atom in the fiducial state, $\ket{\uparrow}$, creating the ensemble state $\ket{\uparrow}^{\otimes N_A}$. The fiducial state is chosen so that the ensemble has a larger value of projection noise, $\Delta F_z^2=N_A (\Delta f_z^2)_\uparrow$. (b) A Faraday spin squeezing protocol is applied to the ensemble. Interatomic entanglement is created through the coupling of the light and ensemble. The coupling between the light and ensemble and the interatomic entanglement that results increases with the initial projection noise. (c) The interatomic entanglement generated by the squeezing protocol creates squeezing in the phase plane of the HP quadratures $\hat{X}_\downarrow$ and $\hat{P}_\downarrow$. The greater the squeezing, the greater the interatomic entanglement. (d) By applying a partial isometry, the squeezing in the phase plane is converted into squeezing of the collective spin. (e) The end result is a spin squeezed state of the ensemble with $\zeta_m<1$.
}\label{fig::SqScheme}
\end{figure}
\subsection{Internal Spin Squeezing}\label{sec::IntSpinSqueeze}
While the partial isometry in \erf{Uscs} enables us to convert interatomic entanglement into metrologically relevant spin squeezing, it is not the optimal map. A partial isometry that maps a fiducial state to an internally spin squeezed state $\ket{\uparrow_S}$, for which $\zeta_m^{\uparrow_S}<1$, generates an even larger amount of metrologically relevant squeezing.
As the squeezed state $\ket{\uparrow_S}$, we choose the Yurke state \cite{Combes05,YurkeState}, which is defined for integer $f$ as
\begin{equation}\label{eq::defYur}
\ket{\uparrow_{\text{y}}} \equiv \frac{\sin\alpha}{\sqrt{2}} \ket{f, m_z=1} +\cos\alpha \ket{f, m_z=0} +\frac{\sin\alpha}{\sqrt{2}} \ket{f, m_z=-1}.
\end{equation}
The Yuke state is maximally squeezed as $\alpha\rightarrow 0$, meaning that $\zeta_m^{\uparrow_\text{y}}$ reaches the Heisenberg limit of $1/f$. For arbitrary $\alpha$, the squeezing parameter of the Yurke state is
\begin{align}
\zeta_m^{\uparrow_\text{y}}= \frac{1}{(f+1)\text{cos}^2\alpha}.
\end{align}
The state that couples to the Yurke state in the multilevel HP approximation is
\begin{equation}
\ket{\downarrow_{\text{y}}}=\frac{1}{\sqrt{2}}(\ket{f, m_z=1}-\ket{f, m_z=-1}),
\end{equation}
since $\Delta\hat{f}_z\ket{\uparrow_{\text{y}}}=\text{sin}\alpha\ket{\downarrow_{\text{y}}}$ and $(\Delta\hat{f}_z^2)_{\uparrow_\text{y}}=\text{sin}^2\alpha$.
While the original Yurke state given in \erf{eq::defYur} is only defined for integer $f$, we introduce a ``half-integer Yurke state" for half-integer $f$,
\begin{align}
\ket{\uparrow_\text{hy}}\!=\!\frac{\text{sin}\alpha}{\sqrt{2}}\ket{f,m_z=3/2}\!+\!\text{cos}\alpha\ket{f,m_z=1/2}
\!+\!\frac{\text{sin}\alpha}{\sqrt{2}}\ket{f,m_z=-1/2}.
\end{align}
The squeezing parameter of the half-integer Yurke state is
\begin{align}
\zeta_m^{\uparrow_\text{hy}}=\frac{4f}{\text{cos}^2\alpha(\sqrt{(f+3/2)(f-1/2)}+f+1/2)^2}.
\end{align}
Although the squeezing parameter of the half-integer Yurke state has a more complicated dependence on $f$ than the squeezing parameter of the Yurke state, it is also maximally squeezed, scaling roughly with $1/f$ as $\alpha\rightarrow 0$. The state coupled to the half-integer Yurke state in the multilevel HP approximation is
\begin{align}
\ket{\downarrow_{\text{hy}}}=\frac{1}{\sqrt{2}}(\ket{f, m_z=3/2}-\ket{f, m_z=-1/2}),
\end{align}
as $\Delta\hat{f}_z\ket{\uparrow_{\text{hy}}}=\text{sin}\alpha\ket{\downarrow_{\text{hy}}}$ and $(\Delta\hat{f}_z^2)_{\uparrow_\text{hy}}=\text{sin}^2\alpha$.
To enhance metrologically relevant squeezing through squeezing of the internal spin, we take an approach similar to post-processing with the spin coherent state. Using internal spin control, we generate the partial isometry
\begin{align}
\hat{U}_{\text{y}}=\bigotimes_{i=1}^{N_A}\Bigg\{\begin{matrix}\left(\ket{\uparrow_\text{y}}\bra{\uparrow}_i+\ket{\downarrow_{\text{y}}}\bra{\downarrow}_i\right)&\text{integer $f$}\\
\left(\ket{\uparrow_\text{hy}}\bra{\uparrow}_i+\ket{\downarrow_{\text{hy}}}\bra{\downarrow}_i\right)&\text{half-integer $f$}\end{matrix}\;\;\;.
\end{align}
This partial isometry maps squeezing from the $\hat{X}_\downarrow$- $\hat{P}_\downarrow$ phase plane to the $\hat{X}_{\downarrow_\text{y}}$- $\hat{P}_{\downarrow_\text{y}}$ phase plane for integer $f$ and to the $\hat{X}_{\downarrow_\text{hy}}$- $\hat{P}_{\downarrow_\text{hy}}$ phase plane for half-integer $f$.
Like $\hat{U}_{SCS}$, the partial isometry $\hat{U}_{\text{y}}$ preserves both the quadrature variances and the phase plane squeezing parameter, $\zeta_q$. The metrological squeezing parameter is transformed, however,
\begin{align}
\zeta_m=\zeta_m^\uparrow\zeta_q\rightarrow\Bigg\{\begin{matrix}\zeta_m^{\uparrow_\text{y}}\zeta_q=\frac{\zeta_q}{\text{cos}^2\alpha(f+1)}&\text{integer $f$}\\
\zeta_m^{\uparrow_\text{hy}}\zeta_q=\frac{4f\zeta_q}{\text{cos}^2\alpha(\sqrt{(f+3/2)(f-1/2)}+f+1/2)^2}&\text{half-integer $f$}\end{matrix}\;\;\;.
\end{align}
When $\alpha= 0$, the internal spin squeezing produces a multiplicative enhancement on the order of $1/f$ compared to the partial isometry $\hat{U}_{\text{SCS}}$. Also note that the gains produced by internal spin squeezing increase with $f$.
\chapter{Decoherence Due to Optical Pumping}\label{sec::OpticalPumping}
In the previous chapter, we demonstrated that internal spin control in the form of state preparation and post-processing can enhance the performance of Faraday-based squeezing protocols. While the gains are significant, achievable spin squeezing is ultimately limited by the various sources of decoherence in the system. Under realistic experimental conditions, optical pumping of the atoms due to spontaneous scattering of photons is the most significant source of decoherence. Because the rate of photon scattering increases with the entangling strength of the Faraday interaction, any attempt to squeeze the ensemble inevitably produces decoherence. Preparation of the ensemble with internal spin control adds another layer of complexity, as the effects of optical pumping are state dependent. Achievable spin squeezing is limited by tradeoffs between the increased entanglement generation and the increased susceptibility to decoherence that come with the choice of a fiducial state.
Although this chapter concerns the effect of optical pumping upon the atoms in the ensemble, it should be noted that optical pumping arising from spontaneous photon scattering produces decoherence of the light as well. Absorption followed by spontaneous emission causes the light to be diffusely scattered outside the spatial mode of the probe, meaning that photons are lost. This notwithstanding, the number of photons per pulse is much greater than the number of atoms. Additionally, the atomic ensemble continually interacts with fresh pulses of light. While the effects of spontaneous photon scattering accumulate on the ensemble, this is not true of the light. For these reasons, loss of photons due to diffuse scattering has minimal impact and can be ignored.
\section{Fundamentals of Optical Pumping}
\begin{figure}
\centering
\includegraphics[scale=.35]{OPqubit.pdf}
\caption{Optical pumping in the case of $f=1/2$ for an atom being driven on a $f=1/2$ to $f'=1/2$ transition. The probe is linearly polarized along $x$. Solid lines represent absorption of a photon from the probe and dashed lines represent spontaneous emission of a photon. In (a), the quantization axis of the atom is parallel to the polarization of the probe, meaning that the probe is $\pi$ polarized. In (b), the quantization axis of the atom is orthogonal to the polarization of the probe, meaning that the probe is a superposition of $\sigma_+$ and $\sigma_-$ light. }
\label{fig::OPqubit}
\end{figure}
To develop an intuition for the effects of optical pumping, we first examine a simple case for $f=1/2$. Depicted in Fig. \ref{fig::OPqubit} is an atom with ground state angular momentum $f=1/2$ being driven on a transition to an excited state with angular momentum $f'=1/2$. Initially, the atom is in the state $\ket{f=1/2,m}$, where the spin projection number is $m=\pm1/2$. The probe photons driving the atom carry angular momentum with spin $s=1$. When the atom absorbs a probe photon, it is excited to the $f'=1/2$ manifold and its spin projection will change by an amount $\Delta m$ depending upon the polarization of the probe relative to the quantization axis of the atomic spin. If the probe is right circularly polarized with respect to the atomic quantization axis, which is called $\sigma_+$ light, $\Delta m=1$. Conversely, if the probe is left circularly polarized with respect to the atomic quantization axis, which is referred to as $\sigma_-$ light, $\Delta m=-1$. When the probe is linearly polarized along the quantization axis, which is called $\pi$ light, the spin projection of the atom does not change, i.e. $\Delta m=0$.
Figure \ref{fig::OPqubit} depicts two possible orientations of the atomic quantization axis with respect to the probe polarization. In Fig. \ref{fig::OPqubit} (a), the atom is quantized along the $x$ axis, which is parallel to the linear polarization of the probe. This $\pi$ light induces the transitions, $\ket{f=1/2,m=\pm1/2}\rightarrow\ket{f'=1/2,m'=\pm1/2}$. In Fig. \ref{fig::OPqubit} (b), the atom is quantized along $z$, which is perpendicular to the linear polarization of the probe. With respect to the quantization axis, the light's linear polarization along $x$ can be expressed in the basis of right and left circular polarization as $\mathbf{e}_x=(\mathbf{e}_L-\mathbf{e}_R)/\sqrt{2}$. This makes the probe a superposition of $\sigma_+$ and $\sigma_-$ light, which causes the transitions $\ket{f=1/2,m=\pm1/2}\rightarrow\ket{f'=1/2,m'=\mp1/2}$.
After absorbing a probe photon and being excited up to the $f'$ manifold, the atom returns to the ground state by spontaneously emitting a photon. Due to conservation of angular momentum, spontaneous emission can change the spin projection of the atom. Like absorption, this change depends on the polarization of the emitted photon with respect to the atomic quantization axis. Spontaneous emission of a $\sigma_+$ photon changes the spin projection by $\Delta m=-1$, which induces the transition \\$\ket{f'=1/2,1/2}\rightarrow\ket{f=1/2,-1/2}$ in Figs. \ref{fig::OPqubit} (a) and (b). When the atom in Figs. \ref{fig::OPqubit} (a) and (b) spontaneously emits a $\sigma_-$ photon, changing the spin projection by $\Delta m=1$, it undergoes the transition $\ket{f'=1/2,-1/2}\!\rightarrow\!\ket{f=1/2,1/2}$. Spontaneous emission of a $\pi$ photon does not change the spin projection number. In Figs. \ref{fig::OPqubit} (a) and (b), spontaneous emission of a $\pi$ photon induces the transitions $\ket{f'=1/2,\pm1/2}\rightarrow\ket{f=1/2,\pm1/2}$. From Figs. \ref{fig::OPqubit} (a) and (b), it is evident that the total change in the spin projection of the atom from one cycle of absorption and emission depends on the polarizations of both the absorbed and emitted photons.
\section{Optical Pumping in the Higher Spin Alkali}
Optical pumping affects higher spin alkali atoms in a similar fashion. As in the $f=1/2$ case, absorption of a $\sigma_\pm$ or $\pi$ photon changes the spin projection of the atom by $\Delta m=\pm 1$ or $\Delta m=0$. Likewise, emission of a $\sigma_\pm$ or $\pi$ photon alters the spin projection by $\Delta m=\mp 1$ or $\Delta m=0$. There are several important differences, however. One of the most significant differences is due to the presence of the second ground hyperfine manifold when $f>1/2$. Recall that when $f>1/2$, an alkali atom has two ground hyperfine manifolds with spins $f_\pm=i\pm1/2$, where $i$ is the nuclear spin. The ground hyperfine manifold with spin $f$, in which we prepare the atoms, can be either $f_+$ or $f_-$. For reference, the energy levels of an alkali are depicted in Fig. \ref{fig::AlkaliLines}. Like $f=1/2$, a higher spin alkali atom initially in the $f$ manifold transitions to a higher energy level when it absorbs a probe photon. Depending on the color of the photon it spontaneously emits, however, the atom might be optically pumped either back into the spin-$f$ manifold or into the other ground hyperfine manifold. When the detuning of the probe is sufficiently small compared to the ground state hyperfine splitting, atoms pumped into the other ground manifold are far off resonance, meaning that they are effectively lost from the system. For higher spin alkalis, optical pumping reduces the number of atoms in the ensemble in addition to changing the states of the atoms. Examples of optical pumping when the atom is prepared in a ground hyperfine manifold with $f=4$ are shown in Figs. \ref{fig::f4pi} and \ref{fig::f4sigma}.
\begin{figure}[H]
\centering
\includegraphics[scale=.38]{f4pi.pdf}
\caption{An alkali atom is prepared in the ground hyperfine manifold with $f=4$. The quantization axis of the atom is parallel to the linear polarization of the probe. The solid line represents absorption of a photon from the probe and the dashed lines represent spontaneous emission of a photon. The atom absorbs a $\pi$ photon and transitions to the excited hyperfine multiplet. By spontaneously emitting a $\sigma_+$, $\sigma_-$ or $\pi$ photon, the atom transitions back to one of the ground hyperfine manifolds. Whether the atom is optically pumped into $f=4$ or $f_-=3$ depends on the color of the emitted photon.}
\label{fig::f4pi}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.38]{f4sigma.pdf}
\caption{An alkali atom is prepared in the ground hyperfine manifold with $f=4$. The quantization axis of the atom is perpendicular to the linear polarization of the probe, making the light a superposition of $\sigma_+$ and $\sigma_-$. Solid lines represent absorption of a photon from the probe and dashed lines represent spontaneous emission of a photon. The grey dashed lines denote transitions with negligibly small probability due to destructive interference. The atom absorbs the superposition of $\sigma_+$ and $\sigma_-$ and transitions to the excited hyperfine multiplet. By spontaneously emitting a $\sigma_+$, $\sigma_-$ or $\pi$ photon, the atom transitions back to one of the ground hyperfine manifolds. The atom can be optically pumped into $f=4$ or $f_-=3$, depending on the color of the emitted photon. }
\label{fig::f4sigma}
\end{figure}
\subsection{Optical Pumping Master Equation}
For an alkali atom prepared in the spin-$f$ ground hyperfine manifold interacting with a probe linearly polarized along $x$ and detuned from the D1 or D2 line, optical pumping is approximately described by the master equation \cite{DeuJes09}
\begin{align}\label{eq::Masteri}
\frac{d\hat{\rho}^{(i)}}{dt}\Big|_{\text{op}}=-\frac{2\gamma_s}{9}\hat{\rho}^{(i)}+\frac{g_f^2\gamma_s}{9}\left(\hat{f}_y^{(i)}\hat{\rho}^{(i)}\hat{f}_y^{(i)}+\hat{f}_z^{(i)}\hat{\rho}^{(i)}\hat{f}_z^{(i)}\right).
\end{align}
Here, the photon scattering rate is $\gamma_s=(N_L/\Delta t)\,(\sigma_0/A)\,(\Gamma^2/4\Delta^2)$. This equation neglects the tensor effects in optical pumping when the detuning is large compared to the excited state hyperfine splitting, but small compared to the fine structure splitting. The internal spin operators $\hat{f}_y^{(i)}$ and $\hat{f}_z^{(i)}$ act on the spin-$f$ manifold. The effect of loss into the other ground manifold is taken into account by the fact that this master equation is not trace preserving when $f>1/2$. Although this master equation acts only on the $f$ manifold and, consequently, does not track the state of the atom in the other ground manifold, the probability of finding the atom in the $f$ manifold decreases with time due to loss. This is sufficient to determine the evolution of the ensemble observables that are relevant to spin squeezing since they are composed of sums of internal spin operators on the $f$ manifold.
To better understand how the master equation describes optical pumping, we decompose it into the form
\begin{align}\label{MostGenMasteri}
\frac{d\hat{\rho}^{(i)}}{dt}\Big|_{\text{op}}=-\Gamma_{\text{op}}\hat{\rho}^{(i)}+\sum_q\hat{W}_q^{(i)}\hat{\rho}^{(i)}\hat{W}_q^{(i)\dag}.
\end{align}
Here, $\Gamma_{\text{op}}=2\gamma_s/3$ is the total rate of optical pumping events and the $\hat{W}_q^{(i)}$ are jump operators. The jump operators update the state of atom in the $f$ manifold conditioned upon the polarization of the emitted photon. The index $q$ can take the value $+$, $0$ or $-$, representing the emission of a $\sigma_+$, $\pi$ or $\sigma_-$ photon, respectively. Assuming the detuning is large compared to the excited state hyperfine splitting, the exact form of the jump operators depends upon the quantization of the atomic spin relative to the probe polarization. For example, consider an atom quantized parallel to the linear polarization of the probe along $x$. The jump operators in this instance are approximately \cite{DeuJes09}
\begin{align}\label{piPlus}
&\hat{W}_{+}^{(i)}=\frac{g_f}{3}\sqrt{\frac{\gamma_s}{2}}\hat{f}_{-}^{(i)},\\\label{pi0}
&\hat{W}_{0}=\frac{2\sqrt{\gamma_{s}}}{3}\mathbb{I}^{(i)}
\end{align}
and
\begin{align}\label{piMinus}
&\hat{W}_{-}^{(i)}=\frac{g_f}{3}\sqrt{\frac{\gamma_s}{2}}\hat{f}_{+}^{(i)},
\end{align}
which correspond to absorption of a $\pi$ photon and emission of a $\sigma_+$, $\pi$ and $\sigma_-$ photon, respectively. By referring to Fig. \ref{fig::f4pi}, we see that each of these jump operators updates the state of the atom how one would expect. The jump operators $\hat{W}_\pm^{(i)}$ change the spin projection by $\Delta m=\mp 1$ and the jump operator $\hat{W}_0^{(i)}$ leaves the spin projection unchanged. Alternatively, we can consider an atom quantized along $z$, perpendicular to the linear polarization of the probe along $x$. For a quantization axis along $z$ in the limit of large detuning, the jump operators are approximately
\begin{align}\label{sigmaPlus}
&\hat{W}_{+}^{(i)}=\sqrt{\frac{\gamma_s}{2}}\left(\frac{2}{3}\hat{\mathbb{I}}^{(i)}-\frac{g_f}{3}\hat{f}_z^{(i)}\right), \\\label{sigma0}
&\hat{W}_0=\frac{g_f\sqrt{\gamma_{s}}}{3}\hat{f}_y^{(i)}
\end{align}
and
\begin{align}\label{sigmaMinus}
&\hat{W}_{-}=\sqrt{\frac{\gamma_s}{2}}\left(\frac{2}{3}\hat{\mathbb{I}}^{(i)}+\frac{g_f}{3}\hat{f}_z^{(i)}\right),
\end{align}
corresponding to the absorption of a photon in a superposition of $\sigma_+$ and $\sigma_-$ followed by the emission of a $\sigma_+$, $\pi$ and $\sigma_-$ photon, respectively. By referring to Fig. \ref{fig::f4sigma}, we can verify the action of these jump operators. When the detuning is significantly larger than the excited state hyperfine splitting, interference between transitions to the different spins $f'$ in the excited hyperfine multiplet eliminates changes in the spin projection of $\Delta m \pm 2 $. The jump operators $\hat{W}_{\pm}^{(i)}$, therefore, effectively describe absorption of a $\sigma_\pm$ photon followed by emission of of a $\sigma_\pm$ photon. These transitions leave the atomic state unchanged, as indicated by the jump operators. The jump operator $\hat{W}_0$ leaves the atom in a superposition of states with $\Delta m \pm 1$, as it should from Fig. \ref{fig::f4sigma}.
Ultimately, the polarization of the probe should dictate the effects of optical pumping and not the quantization axis we choose for the atomic spin. If we take the jump operators defined in Eqs. (\ref{piPlus})-(\ref{piMinus}) for a parallel quantization axis and substitute them into the general master equation in \erf{MostGenMasteri}, we will obtain the master equation in \erf{eq::Masteri}. Likewise, if we take the jump operators defined in Eqs. (\ref{sigmaPlus})-(\ref{sigmaMinus}) for a perpendicular quantization axis and substitute them into the general master equation in \erf{MostGenMasteri}, we will again obtain the master equation in \erf{eq::Masteri}. The quantization axis of the atom does not have any effect on optical pumping. We are, thus, free to choose the quantization axis that is more convenient for a particular atomic state or problem at hand.
\subsection{Ensemble Master Equation}
The master equation describing the evolution of a single atom under optical pumping in \erf{eq::Masteri} is generalized to describe the ensemble of $N_A$ atoms as follows,
\begin{align}\label{eq::MasterEnsem}
\frac{d\hat{\rho}}{dt}\Big|_{\text{op}}=-\frac{2\gamma_s}{9}\sum_{i=1}^{N_A}\mathbb{I}^{(i)}\hat{\rho}+\frac{g_f^2\gamma_s}{9}\sum_{i=1}^{N_A}\left(\hat{f}_y^{(i)}\hat{\rho}\hat{f}_y^{(i)}+\hat{f}_z^{(i)}\hat{\rho}\hat{f}_z^{(i)}\right).
\end{align}
This equation gives the evolution of the ensemble state, $\hat{\rho}$, as each atom, $i$, undergoes optical pumping independently. For an ensemble identical under interchange of atoms, taking the index $i$ in \erf{eq::MasterEnsem} to $k$ instead of $N_A$ gives the evolution of any group of $k$ atoms in the ensemble,
\begin{align}\label{eq::Masterk}
\frac{d\hat{\rho}^{(1, ..., k)}}{dt}\Big|_{\text{op}}=&-\frac{2\gamma_s}{9}\sum_{i=1}^{k}\mathbb{I}^{(i)}\hat{\rho}^{(1, ..., k)}\\\notag&+\frac{g_f^2\gamma_s}{9}\sum_{i=1}^{k}\left(\hat{f}_y^{(i)}\hat{\rho}^{(1, ..., k)}\hat{f}_y^{(i)}+\hat{f}_z^{(i)}\hat{\rho}^{(1, ..., k)}\hat{f}_z^{(i)}\right).
\end{align}
Because the master equation is not trace preserving, care must be taken in deriving equations of motion for ensemble observables from \erf{eq::MasterEnsem}. The equation of motion for a $k$th order correlation function $\expect{\hat{o}^{(1)}...\hat{o}^{(k)}}$, for example, follows directly from $\hat{\rho}^{(1, ..., k)}$ in \erf{eq::Masterk},
\begin{align}\label{eq::kCorrelation}
\frac{d}{dt}\expect{\hat{o}^{(1)}&...\hat{o}^{(k)}}=\text{Tr}\Big(\hat{\rho}^{(1, ..., k)}\hat{o}^{(1)}...\hat{o}^{(k)}\Big) \\\notag
=&-\frac{2k\gamma_s}{9}\expect{\hat{o}^{(1)}...\hat{o}^{(k)}}
+\frac{g_f^2\gamma_s}{9}\sum_{i=1}^{k}\Big\langle\left(\hat{f}_y^{(i)}\hat{o}^{(i)}\hat{f}_y^{(i)}+\hat{f}_z^{(i)}\hat{o}^{(i)}\hat{f}_z^{(i)}\right)
\Pi_{j\neq i}\hat{o}^{(j)}\Big\rangle.
\end{align}
\section{Effects of Optical Pumping}\label{Sec::OPEvents}
Given a basic understanding of optical pumping, we now examine how it affects the ensemble state and the observables relevant to spin squeezing. Optical pumping has several harmful effects, which include a decay of the negative correlations that create spin squeezing, a decay of the mean spin and an increase in the collective spin variance that we are trying to squeeze. We explore optical pumping on an ensemble initially prepared in $\ket{\uparrow}^{\otimes N_A}$ with fiducial state $\ket{\uparrow}$ and coupled state $\ket{\downarrow}$. To avoid specifying a quantization axis for the atomic spins, we employ a master equation of the form
\begin{align}\label{MostGenMasterk}
\frac{d\hat{\rho}^{(1, ..., k)}}{dt}\Big|_{\text{op}}=-\Gamma_{\text{op}}\sum_{i=1}^{k}\mathbb{I}^{(i)}\hat{\rho}^{(1, ..., k)}+\sum_{i=1}^{k}\sum_q\hat{W}_q^{(i)}\hat{\rho}^{(1, ..., k)}\hat{W}_q^{(i)\dag},
\end{align}
where the $\hat{W}_q^{(i)}$ are arbitrary jump operators and $\hat{\rho}^{(1, ..., k)}$ is the density operator of any $k$ atoms.
\subsection{Growth of the Collective Spin Projection Variance}\label{sec::VarianceOP}
To understand the impact of optical pumping on the collective spin variance, we first study the interatomic entanglement that causes this variance to become ``squeezed". If the Bogoliubov transformation in \erf{bogoliubov} is rotated by $\pi/4$, it creates pure squeezing of the quadrature $\hat{X}_\downarrow$, which is proportional to $\hat{F}_z$ under the multilevel HP approximation. To lowest order in $\xi/N_A<<1$, the spin squeezed state created by the rotated Bogoliubov transformation is
\begin{align}
\ket{\psi}_\text{sq}\approx\ket{\uparrow}^{\otimes N_A}-\frac{\xi}{4N_A}\sum_{i\neq j}\ket{\downarrow_i\downarrow_j}\ket{\uparrow}_{\neq i, j}^{\otimes\left(N_A-2\right)}.
\label{SqueezedState}
\end{align}
This is an entangled state where every two atoms $i$ and $j$ are pairwise correlated. The density operator corresponding to this squeezed state is
\begin{align}\label{densityOp}
\hat{\rho}_{\text{sq}}\approx&\left(\ket{\uparrow}\bra{\uparrow}\right)^{\otimes N_A}\\\notag&-
\frac{\xi}{4N_A}\sum_{i\neq j}(\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\downarrow_j}+\ket{\downarrow_i\downarrow_j}\bra{\uparrow_i\uparrow_j})\left(\ket{\uparrow}\bra{\uparrow}\right)_{\neq i,j}^{\otimes N_A-2}.
\end{align}
Note that $\hat{\rho}_{\text{sq}}$ contains the term
\begin{align}\label{eq::pairwiseCoherence}
\hat{c}(i,j)=\left(\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\downarrow_j}+\ket{\downarrow_i\downarrow_j}\bra{\uparrow_i\uparrow_j}\right),
\end{align}
which is the coherence between two pairwise entangled atoms. To understand how pairwise entanglement influences spin squeezing, we can calculate the correlation term in \erf{variance} from $\hat{c}(i,j)$,
\begin{align}
\langle\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}\rangle_{i\neq j}\approx-\frac{\xi}{2N_A}\text{Tr}(\hat{c}(i,j)\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)})
=-\xi(\Delta f_z^2)_\uparrow/N_A.
\label{negCorr}
\end{align}
Recall that collective spin squeezing requires this correlation term to be negative in absence of internal spin squeezing. Equation (\ref{negCorr}) demonstrates that negative correlations are fed by coherences of entangled, pairwise correlated states. Note also that the magnitude of the correlation term is proportional to the variance of the fiducial state.
Pairwise correlations decay at a rate dependent upon the choice of fiducial state, making some state preparations more robust than others. To see this, we calculate the evolution of the correlation term for the initial squeezed state in \erf{densityOp} over a small time step $\Delta t$ using the master equation in \erf{MostGenMasterk},
\begin{align}\label{corrDecay}
\langle\Delta\hat{f}_z^{(i)}\Delta&\hat{f}_z^{(j)}(\Delta t)\rangle_{i\neq j}\approx\left(1-2\Gamma_\text{op}\Delta t\right)\langle\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}(0)\rangle_{i\neq j}\\\notag&-\frac{\xi}{N_A}\sqrt{(\Delta f_z^2)_\uparrow}\Delta t
\sum_q(\bra{\uparrow}\hat{W}_q^{\dag}\Delta\hat{f}_z\hat{W}_q\ket{\downarrow}+\bra{\downarrow}\hat{W}_q^{\dag}\Delta\hat{f}_z\hat{W}_q\ket{\uparrow}).
\end{align}
The first term in this equation represents a decay of the initial correlation term due to all optical pumping events. The second term, which updates the correlation term based upon the state of the atoms optically pumped back into the $f$ manifold, is proportional to the quantity
\begin{align}\label{update}
C(\uparrow)&=\sqrt{(\Delta f_z^2)_\uparrow}\sum_q\left(\bra{\uparrow}\hat{W}_q^{\dag}\Delta\hat{f}_z\hat{W}_q\ket{\downarrow}+\bra{\downarrow}\hat{W}_q^{\dag}\Delta\hat{f}_z\hat{W}_q\ket{\uparrow}\right)\\
&=2\sqrt{(\Delta f_z^2)_\uparrow}\sum_q\text{Re}\left[\bra{\widetilde{q}_\uparrow}\Delta\hat{f}_z\ket{\widetilde{q}_\downarrow}\right].
\end{align}
Here, $\ket{\widetilde{q}_\uparrow}=\hat{W}_q\ket{\uparrow}$ and $\ket{\widetilde{q}_\downarrow}=\hat{W}_q\ket{\downarrow}$ are the unnormalized states to which the fiducial and coupled states are mapped after an optical pumping event. If the fiducial and coupled states are mapped to states such that $C(\uparrow)>0$, negative correlations are maintained by optical pumping. If $C(\uparrow)<0$, on the other hand, optical pumping produces positive correlations. Consequently, squeezing decays more rapidly for fiducial states where $C(\uparrow)<0$. Pairwise correlations also decay due to loss into the other ground hyperfine manifold. Even if optical pumping back into the $f$ manifold preserves negative correlations, the total rate of optical pumping events is larger than the rate of replenishment into the $f$ manifold, signifying a net loss of atoms and a decay of negative correlations.
As the ensemble undergoes optical pumping, the collective variance $\Delta F_z^2$ is affected both by the decay of negative correlations and by the variances of the states in the $f$ manifold to which the atoms are optically pumped. The variance of the initial squeezed state in \erf{densityOp} after a small time step $\Delta t$ is
\begin{align}\label{SmallTvar}
\Delta F_z^2(\Delta t)\approx&\left(1-\Gamma_\text{op}\Delta t\right)\Delta F_z^2(0)+
\Gamma_\text{op} N_A\xi\Delta t(\Delta f_z^2)_\uparrow\notag\\&-
N_A\xi\Delta tC(\uparrow)+
N_A\Delta t\sum_q\bra{\widetilde{q}_\uparrow}(\Delta\hat{f}_z)^2\ket{\widetilde{q}_\uparrow}.
\end{align}
The first and second terms in \erf{SmallTvar} are the decay of the collective variance and the negative correlations, respectively, due to all optical pumping events. The third term is the familiar update on the correlation term, while the fourth term is a ``noise injection" that results from optical pumping back into the $f$ manifold. To a first order approximation, the noise term only depends upon the variance of the state $\ket{\widetilde{q}_\uparrow}$ to which the fiducial state is mapped. This occurs because the population of atoms in the fiducial state remains much larger than the population of atoms in the coupled state. As in the case of the correlation term, loss into the other ground hyperfine manifold causes the collective variance to decay. The behavior of the collective variance under optical pumping ultimately depends upon competition between the noise injection, the decay of the correlation term and loss.
\subsection{Decay of Mean Spin}
In addition to increasing the variance of the collective spin, optical pumping causes the mean spin to decay. Recall that the spin squeezing parameter in \erf{eq::SqParameter} is inversely proportional to the square of the mean spin, $\expect{\hat{F}_x}$. A smaller mean spin, thus, results in less metrologically useful spin squeezing. To good approximation when $N_A>>1$ and $\chi<<1$, the mean spin is given by
\begin{align}\label{eq::FxPop}
\expect{\hat{F}_x}=\expect{\hat{f}_x}_\uparrow N_\uparrow
+\expect{\hat{f}_x}_\downarrow N_\downarrow.
\end{align}
Here, $N_\psi=\sum_{i=1}^{N_A}\ket{\psi}\bra{\psi}_i$ is a ``population" quantifying the number of atoms in state $\ket{\psi}$. Because the ensemble begins in an eigenstate of the populations, $N_\uparrow$ and $N_\downarrow$ have no initial variance. While $N_\uparrow$ and $N_\downarrow$ accumulate variance through optical pumping, this variance influences the covariances of the ensemble observables as a second order effect in the scattering rate, $\gamma_s$. We, thus, approximate the populations as c-numbers.
Equation (\ref{eq::FxPop}) enables us to track the decay of the mean spin through the populations. The evolution of the populations under optical pumping is governed primarily by two processes, loss and ``spin flips". Loss into the other ground hyperfine manifold is a familiar concept. Spin flips, on the other hand, occur when an atom is optically pumped from the fiducial state to the coupled state. While an atom could also be flipped from the coupled state to the fiducial state in principle, this process does not contribute appreciably to the dynamics since the ensemble is prepared with every atom in the fiducial state. The rate at which an atom in state $\ket{\psi}$, where $\psi\in\{\uparrow,\downarrow\}$, is lost into the other ground hyperfine manifold is
\begin{align}\label{lossPsi}
\Gamma_{\text{loss},\psi}=\Gamma_{\text{op}}-\sum_q|\bra{\psi}\hat{W}_q\ket{\psi}|^2.
\end{align}
The rate of spin flips from $\ket{\uparrow}$ to $\ket{\downarrow}$ is given by
\begin{align}\label{eq::FlipRate}
\Gamma_{\text{flip}}=\sum_q|\bra{\downarrow}\hat{W}_q\ket{\uparrow}|^2.
\end{align}
In terms of the loss and spin flip rates, the evolution of the mean spin over a small time step $\Delta t$ is given by
\begin{align}\label{eq::meanSpinEvol}
\expect{\hat{F}_x(\Delta t)}=&\expect{\hat{f}_x}_\uparrow\hat{N}_\uparrow(\Delta t)
+\expect{\hat{f}_x}_\downarrow\hat{N}_\downarrow(\Delta t)\\\notag
\approx&\expect{\hat{f}_x}_\uparrow(1-\Gamma_{\text{loss},\uparrow}\Delta t)\hat{N}_\uparrow(0)+
\\&\expect{\hat{f}_x}_\downarrow\big[(1-\Gamma_{\text{loss},\downarrow}\Delta t)\hat{N}_\downarrow(0)
+\Gamma_\text{flip}\Delta t\hat{N}_\uparrow(0)\big].\notag
\end{align}
From \erf{eq::meanSpinEvol}, it is evident that loss causes the mean spin to decay. Because loss also causes the collective variance to decay, however, damage to spin squeezing is much reduced. In general, spin flips are more damaging to spin squeezing. Spin flips destroy negative correlations at a rate depending upon $C(\uparrow)$.
Because an atom returns to the $f$ manifold after a spin flip, there is no reduction in the collective variance. The extent to which spin flips cause the mean spin to decay depends upon the values of $\expect{\hat{f}_x}_{\uparrow'}$ and $\expect{\hat{f}_x}_{\downarrow'}$, where $\ket{\uparrow'}$ and $\ket{\downarrow'}$ are the states to which the fiducial and coupled states are mapped after post-processing. The partial isometries discussed in Sec. \ref{sec::postprocessing} map the atoms to fiducial and coupled states for which $\expect{\hat{f}_x}_{\uparrow'}>\expect{\hat{f}_x}_{\downarrow'}$, ensuring that every spin flip event reduces the mean spin.
\section{Internal Spin Control and Optical Pumping}\label{sec::ControlOP}
To make the multilevel Holstein-Primakoff approximation, we modeled the atoms as an ensemble of embedded qubits composed of the fiducial and coupled states. As we have seen, optical pumping back into the $f$ manifold takes the fiducial and coupled states to the states $\ket{\widetilde{q}_\uparrow}$ and $\ket{\widetilde{q}_\downarrow}$, respectively, which may not be contained in the subspace spanned by the embedded qubit. Through internal spin control, however, we have the option of eliminating atoms that have been pumped outside the embedded qubit subspace. Using microwave pulses, which induce transitions between the ground hyperfine manifolds of alkali atoms, we can generate a map that takes all states in the $f$ manifold to the other ground hyperfine manifold, except for the fiducial and coupled states. For example, suppose that our ensemble has been prepared in a spin coherent state, where the fiducial and coupled states are $\ket{\uparrow_{SCS}}=\ket{f,m_x=f}$ and $\ket{\downarrow_{SCS}}=i\ket{f,m_x=f-1}$. Also suppose that the $f$ manifold is the ground hyperfine manifold with the larger spin quantum number, the spin quantum number of the other ground hyperfine manifold being $f-1$. Through internal spin control, we can generate the unitary map
\begin{align}
\hat{U}_-=&\bigotimes_{i=1}^{N_A}\!\Big(\ket{f,m_x=f}\bra{f,m_x=f}_i\!+\!\ket{f,m_x=f-1}\bra{f,m_x=f-1}_i\\
&+\sum_{m=-f}^{f-2}\ket{f-1,m_x=m+1}\bra{f,m_x=m}_i\Big).
\end{align}
The unitary $\hat{U}_-$ has no effect upon the embedded qubit, but maps all other states in the $f$ manifold to the ground manifold with spin $f-1$. After applying $\hat{U}_-$ or a similar unitary operation on the internal spin, all optical pumping events outside the embedded qubit are equivalent to loss.
In addition to ensuring the validity of the embedded qubit approximation, this control has a substantial impact on the evolution of the projection noise. Recall that the noise injection in \erf{SmallTvar} increases with the variance of $\ket{\widetilde{q}_\uparrow}$. If a component of $\ket{\widetilde{q}_\uparrow}$ is contained outside the subspace spanned by the embedded qubit, mapping all states outside the embedded qubit subspace to the other ground manifold will reduce the noise injection. If $\ket{\widetilde{q}_\uparrow}$ lies entirely outside of the embedded qubit subspace, its contribution to the noise injection will be completely eliminated. Consequently, the only states contributing to the noise injection are the fiducial and coupled state. The amount that the noise injection contributes to the overall collective spin variance ultimately depends upon the magnitudes of $\bra{\uparrow'}(\Delta\hat{f}_z)^2\ket{\uparrow'}$ and $\bra{\downarrow'}(\Delta\hat{f}_z)^2\ket{\downarrow'}$, where $\ket{\uparrow'}$ and $\ket{\downarrow'}$ are the states to which the fiducial and coupled states are mapped after post-processing.
\section{A Detailed Look at The SCS}
To apply the ideas of the previous sections, we now examine the effects of optical pumping on the SCS preparation in detail. For conceptual purposes, we neglect the bias magnetic field along the $z$-axis. Optical pumping in the presence of the bias magnetic field will be treated in Chapter \ref{sec::ModHPCovar}. Because the fiducial state of the SCS preparation is $\ket{\uparrow_{SCS}}=\ket{f,m_x=f}$, it is natural to take the quantization axis of the atomic spin to be along $x$, parallel to the probe polarization. The optical pumping processes permitted in this configuration are depicted in Fig. \ref{fig::SCSpi} (a) and (b). Note that when an atom absorbs a $\pi$ photon and then returns to the $f$ manifold by emitting a $\pi$ photon, its state is unchanged. Therefore, we concentrate on the two other optical pumping processes within the $f$ manifold: (1) the atom absorbs a $\pi$ polarized photon from the probe and emits a $\sigma_+$ photon and (2) the atom absorbs a $\pi$ polarized photon from the probe and emits a $\sigma_-$ photon. These optical pumping processes are described by the jump operators $\hat{W}_\pm$ in Eqs. (\ref{piPlus}) and (\ref{piMinus}). Because the expected number of atoms in the fiducial state remains large and an atom in the magnetic sublevel $m_x=f$ cannot emit a $\sigma_-$ photon, process (2) does not contribute appreciably to the decoherence of the atomic state. Process (1), thus,
is the dominant effect for the SCS.
\begin{figure}[H]
\centering
\includegraphics[scale=.6]{SCSpi.pdf}
\caption{Optical pumping processes permitted in the $SCS$ preparation. In (a) and (b), the polarization of the probe is parallel to the quantization axis of the atomic spin, meaning that it is $\pi$ polarized. Fig. (a) shows the absorption of a $\pi$ polarized photon followed by the emission of a $\pi$ polarized photon. This process does not change the spin state of the atom. Fig. (b) shows the absorption of a $\pi$ polarized photon followed by the emission of a $\sigma_+$ polarized photon, causing the projection of the atom's angular momentum along $x$ to decrease by 1. Not pictured is the absorption of a $\pi$ polarized photon and the emission of a $\sigma_-$ photon. This process is prohibited when the atom remains in the fiducial state, the state with maximal spin projection along $x$. Although the atom can emit a $\sigma_-$ photon after it has been optically pumped into the coupled state, this is a second order effect. Consequently, the absorption of a $\pi$ polarized photon and the emission of a $\sigma_-$ photon does not contribute appreciably to the dynamics when the ensemble is prepared in the $SCS$ preparation.}
\label{fig::SCSpi}
\end{figure}
Process (1) is a spin flip, sending $\ket{\uparrow_{SCS}}=\ket{f,m_x=f}$ to $\ket{\downarrow_{SCS}}=\\i\ket{f,m_x=f-1}$ up to a global phase, as depicted in Fig. \ref{fig::SCSpi} (b). While the fiducial state ``flips" to the coupled state, the coupled state is sent to another state, $\ket{f,m_x=f-2}$, up to a global phase. Using internal spin control, we could eliminate this state by sending it to the other ground manifold. However, an examination of the pairwise entanglement that creates spin squeezing reveals that this is not the optimal thing to do. Consider how the squeezed state in \erf{densityOp} for an initial fiducial state $\ket{\uparrow_{SCS}}$ transforms when one of the atoms undergoes process (1),
\begin{align}\label{eq::NegCorTransferCoherence}
\hat{W}_+^{(1)}&\hat{\rho}_\text{sq}\hat{W}_+^{(1) \dag}\propto2f\ket{\downarrow_{SCS}}\bra{\downarrow_{SCS}}_1\left(\ket{\uparrow_{SCS}}\bra{\uparrow_{SCS}}\right)^{\otimes (N_A-1)}_{\neq 1}\\\notag&-\!
\frac{\xi\sqrt{f(2f-1)}}{N_A}\!\!\sum_{j\neq 1}\!(\ket{\downarrow_{SCS}}_1\ket{\uparrow_{SCS}}_j\bra{f-2}_1\bra{\downarrow_{SCS}}_j\!+\!\text{h.c.})\\\notag&\;\;\;\times\left(\ket{\uparrow_{SCS}}\bra{\uparrow_{SCS}}\right)_{\neq 1,j}^{\otimes N_A-2}
\\\notag&-\!
\frac{\xi f}{2N_A}\ket{\downarrow_{SCS}}\bra{\downarrow_{SCS}}_1\!\!\sum_{i\neq j,\;i,j\neq 1}\!(\ket{\uparrow_{SCS}}_i\ket{\uparrow_{SCS}}_j\bra{\downarrow_{SCS}}_i\bra{\downarrow_{SCS}}_j\!+\!\text{h.c.})\\\notag&\;\;\;\times\left(\ket{\uparrow_{SCS}}\bra{\uparrow_{SCS}}\right)_{\neq 1,i,j}^{\otimes N_A-3}.
\end{align}
Here, $\ket{f-2}$ is shorthand for $\ket{f,m_x=f-2}$. In the state above, atom 1 has been ``flipped". From this state, we can calculate the contribution of atom 1 to the negative correlations in \erf{variance},
\begin{align}\notag
\langle\Delta\hat{f}_z^{(j)}\Delta\hat{f}_z^{(1)}\rangle_{j\neq 1}=&-\frac{\xi\sqrt{f(2f-1)}}{N_A}\bra{f-2}_1\bra{\downarrow_{SCS}}_j\Delta\hat{f}_z^{(j)}\Delta\hat{f}_z^{(1)}\ket{\downarrow_{SCS}}_1\ket{\uparrow_{SCS}}_j\\&\;\;\;\;+\text{c.c.}\\
=&-\frac{\xi f(2f-1)}{N_A}\label{transferCorr2}.
\end{align}
After the optical pumping event, pairwise coherences between $\ket{\downarrow_{SCS}}$ and \\$\ket{f-2}$ take the place of pairwise coherences between $\ket{\uparrow_{SCS}}$ and $\ket{\downarrow_{SCS}}$, feeding the negative correlations that generate spin squeezing. By mapping $\ket{f-2}$ to the other ground hyperfine manifold, we destroy negative correlations.
To understand this another way, we can write $\hat{f}_z$ in the basis of eigenstates of $\hat{f}_x$ as
\begin{align}\label{fzSCStransfer}
\hat{f}_z=&\sqrt{\frac{f}{2}}\Big(i\ket{f-1}\bra{f}-i\ket{f}\bra{f-1}\Big)\\\notag
&+\sqrt{\frac{2f-1}{2}}\Big(i\ket{f-2}\bra{f-1}-i\ket{f-1}\bra{f-2}\Big).
\end{align}
The collective spin $\hat{F}_z$ becomes
\begin{align}
\hat{F}_z=\sqrt{\frac{f}{2}}\hat{\Sigma}_{f-1}+\sqrt{\frac{2f-1}{2}}\hat{\Sigma}_{f-2},
\end{align}
where $\hat{\Sigma}_{f-1}=\sum_{j=1}^{N_A}(i\ket{f-1}\bra{f}_j-i\ket{f}\bra{f-1}_j)$ and $\hat{\Sigma}_{f-2}=\\\sum_{j=1}^{N_A}(i\ket{f-2}\bra{f-1}_j-i\ket{f-1}\bra{f-2}_j)$. Note that $\hat{\Sigma}_{f-1}$ is related to the position quadrature $\hat{X}_\downarrow$ since $\ket{\uparrow_{SCS}}=\ket{f}$ and $\ket{\downarrow_{SCS}}=i\ket{f-1}$. In terms of the operators $\hat{\Sigma}_{f-1}$ and $\hat{\Sigma}_{f-2}$, the collective spin variance is given by
\begin{align}
\Delta F_z^2=&\frac{f}{2}\Delta \Sigma_{f-1}^2+\frac{\sqrt{f(2f-1)}}{2}\expect{\Delta\hat{\Sigma}_{f-1}\Delta\hat{\Sigma}_{f-2}+\Delta\hat{\Sigma}_{f-2}\Delta\hat{\Sigma}_{f-1}}\\\notag&+\frac{2f-1}{2}\Delta \Sigma_{f-2}^2.
\end{align}
The second term in this expression, which is the covariance between the operators $\hat{\Sigma}_{f-1}$ and $\hat{\Sigma}_{f-2}$, contains negative correlations that contribute to spin squeezing. These correlations are fed by pairwise coherences in of the form in \erf{eq::NegCorTransferCoherence} between atoms 1 and $j$. If we eliminate the state $\ket{f-2}$, the second term disappears and the variance becomes
\begin{align}
\Delta F_z^2=&\frac{f}{2}\Delta \Sigma_{f-1}^2+\frac{2f-1}{2}N_{\downarrow_{SCS}}.
\end{align}
When $\ket{f-2}$ is absent, negative correlations that reduce the variance are lost. Though it may seem counterintuitive, preserving $\ket{f-2}$ in addition to the fiducial and coupled states produces more spin squeezing.
\section{Transfers of Coherence}\label{sec::TransferState}
While internal spin control can reduce the noise injection and allow us to continue modeling our atoms as embedded qubits, eliminating all states besides the fiducial and coupled states is not always advantageous, as shown in the previous section. In the case of the SCS preparation, we saw that preserving the state $\ket{f,m_x=f-2}$ maintains negative correlations that create spin squeezing. We call a state such as $\ket{f,m_x=f-2}$, which contributes to negative correlations in the presence of optical pumping, a ``transfer state". In this section, we explore the existence of transfer states for arbitrary state preparations.
We first follow a procedure similar to Sec. \ref{sec::MultiHPEnsemble} and expand $\hat{{f}}_z$ in terms of the fiducial state, the coupled state and a third state, $\ket{\wr}$, orthogonal to both the fiducial and the coupled states,
\begin{align}\notag
&\hat{f}_z\approx(\ket{\uparrow}\bra{\uparrow}+\ket{\downarrow}\bra{\downarrow}+\ket{\wr}\bra{\wr})\hat{f}_z(\ket{\uparrow}\bra{\uparrow}+\ket{\downarrow}\bra{\downarrow}+\ket{\wr}\bra{\wr})
\\\label{newFz}=&\sqrt{(\Delta f_z^2)_\uparrow}\left(\ket{\uparrow}\bra{\downarrow}\!+\!\ket{\downarrow}\bra{\uparrow}\right)
+\sqrt{(\Delta f_z^2)_\downarrow\!-\!(\Delta f_z^2)_\uparrow}\left(\ket{\downarrow}\bra{\wr}\!+\!\ket{\wr}\bra{\downarrow}\right).
\end{align}
Note that by the definition of the coupled state, $\bra{\wr}\hat{f}_z\ket{\uparrow}=\sqrt{(\Delta f_z^2)_\uparrow}\bra{\wr}\downarrow\rangle=0$, which explains the absence of terms coupling $\ket{\uparrow}$ and $\ket{\wr}$ in \erf{newFz}. For simplicity, we have also assumed that $\expect{\hat{f}_z}_\uparrow=\expect{\hat{f}_z}_\downarrow=\expect{\hat{f}_z}_\wr=0$. States that satisfy this criteria, such as the SCS, cat and $m_x=0$ state preparations, are natural to consider for spin squeezing. This assumption will be relaxed in Chapter \ref{sec::Beyond}.
The third orthogonal state in \erf{newFz}, which we denoted $\ket{\wr}$, is the transfer state. Although we will demonstrate this more rigorously below, compare \erf{newFz} to the expansion of $\hat{f}_z$ for the SCS preparation in \erf{fzSCStransfer}. The state $-\ket{f,m_x=f-2}$ in \erf{fzSCStransfer}, which is the transfer state for the SCS preparation, takes the place of $\ket{\wr}$. From \erf{newFz}, we can determine a general expression for the transfer state in terms of the coupled and fiducial states. When $\sqrt{(\Delta f_z^2)_\downarrow\!-\!(\Delta f_z^2)_\uparrow}\neq 0$, the transfer state is given by
\begin{align}\label{CoherenceState}
\ket{\wr}=\frac{1}{\sqrt{(\Delta f_z^2)_\downarrow\!-\!(\Delta f_z^2)_\uparrow}}\left(\hat{f}_z\ket{\downarrow}-\sqrt{(\Delta f_z^2)_\uparrow}\ket{\uparrow}\right).
\end{align}
As discussed in Sec. \ref{sec::VarianceOP}, the behavior of negative correlations under optical pumping is governed by the quantity $C(\uparrow)$, given in \erf{update}. Recall that when $C(\uparrow)>0$, negative correlations are preserved. Substituting \erf{newFz} into \erf{update} yields an expression for $C(\uparrow)$ in terms of the fiducial, coupled and transfer states,
\begin{align}\label{expandedC}
C(\uparrow)=&(\Delta f_z^2)_\uparrow\sum_q\text{Re}[\langle\widetilde{q}_\uparrow|\uparrow\rangle\langle\downarrow|\widetilde{q}_\downarrow\rangle+\langle\widetilde{q}_\uparrow|\downarrow\rangle\langle\uparrow|\widetilde{q}_\downarrow\rangle]
\\\notag&+\sqrt{(\Delta f_z^2)_\uparrow((\Delta f_z^2)_\downarrow-(\Delta f_z^2)_\uparrow)}\sum_q\text{Re}[\langle\widetilde{q}_\uparrow|\downarrow\rangle\langle\wr|\widetilde{q}_\downarrow\rangle+\langle\widetilde{q}_\uparrow|\wr\rangle\langle\downarrow|\widetilde{q}_\downarrow\rangle].
\end{align}
The first term in this expression depends upon coherences between $\ket{\uparrow}$ and $\ket{\downarrow}$. The second term, which depends upon coherences between $\ket{\downarrow}$ and $\ket{\wr}$, determines the influence of the transfer state on negative correlations.
To gain insight, we analyze the second term in \erf{expandedC} in greater detail. The components of the second term for each $q$ are proportional to
\begin{align}\label{transferTerm}
C_{\wr}(\uparrow,q)=\text{Re}\left[\langle q_\uparrow|\downarrow\rangle\langle\wr|q_\downarrow\rangle+\langle q_\uparrow|\wr\rangle\langle\downarrow| q_\downarrow\rangle\right],
\end{align}
where $\ket{q_\uparrow}$ and $\ket{q_\downarrow}$ are the normalized versions of
$\ket{\widetilde{q}_\uparrow}=\hat{W}_q\ket{\uparrow}$ and \\$\ket{\widetilde{q}_\downarrow}=\hat{W}_q\ket{\downarrow}$. We first consider two cases in which $C_{\wr}(\uparrow,q)$ is maximal,
\begin{align}
&(1) \;\;\;\langle q_\uparrow|\downarrow\rangle\langle\wr|q_\downarrow\rangle=1 \;\;\;\text{and}\;\;\; \langle q_\uparrow|\wr\rangle\langle\downarrow| q_\downarrow\rangle=0
\end{align}
and
\begin{align}
&(2) \;\;\;\langle q_\uparrow|\downarrow\rangle\langle\wr|q_\downarrow\rangle=0 \;\;\;\text{and}\;\;\; \langle q_\uparrow|\wr\rangle\langle\downarrow| q_\downarrow\rangle=1.
\end{align}
Recall that the noise injection in \erf{SmallTvar} is proportional to $\bra{q_\uparrow}(\Delta\hat{f}_z)^2\ket{q_\uparrow}$, the variance of the state to which the fiducial state is optically pumped. In case (2), the fiducial state is mapped to the transfer state. As a consequence, the transfer state contributes to the noise injection. Although preserving $\ket{\wr}$ maximizes $C(\uparrow)$ and the magnitude of the correlation term, this can be offset by a large injection of noise. Whether or not it is beneficial to retain the transfer state depends on the strength of its contribution to the negative correlations relative to the noise injection. If the contribution to the noise injection is larger, it is optimal to eliminate the noise injection by mapping the transfer state to the other hyperfine manifold. The situation is different for case (1), in which the fiducial state is mapped to the coupled state and the coupled state to the transfer state. In this case, the noise injection is proportional to $\bra{\downarrow}(\Delta\hat{f}_z)^2\ket{\downarrow}$. Even if the noise injection is larger than the magnitude of the correlation update term, eliminating the coupled state through microwave control would destroy all beneficial pairwise correlations between $\ket{\uparrow}$ and $\ket{\downarrow}$. It is instead advantageous to maximize $C(\uparrow)$ by retaining $\ket{\wr}$. This is the case in the SCS preparation.
From Eqs. (\ref{SmallTvar}) and (\ref{expandedC}), we can determine the exact contribution of the transfer state to the negative correlations and to the noise injection. However, \erf{SmallTvar} presupposes that the ensemble was initially in a pure, highly squeezed state. In practice, we apply our squeezing protocols to states which are not squeezed at the initial time. As a consequence, the magnitude of the correlation update term is nearly always smaller than the noise injection. In this case, any contribution that the transfer state makes to the noise injection should be eliminated by mapping the transfer state to the other ground manifold. Preserving the transfer state is beneficial when the following conditions hold:
\begin{align}\label{eq::TofUp}
&(1)\; T(\uparrow)=\sum_q\text{Re}\left[\langle q_\uparrow|\downarrow\rangle\langle\wr| q_\downarrow\rangle\right]>0
\end{align}
and
\begin{align}
\label{eq::up2squiggle}
&(2)\; N(\uparrow)=\sum_q|\langle q_\uparrow|\wr\rangle|^2=0.
\end{align}
The first condition ensures that the transfer state contributes positively to $C(\uparrow)$, while the second condition guarantees that the transfer state does not contribute to the noise injection. There is a deeper physical meaning to condition (1), however. Recall from \erf{negCorr} that negative correlations are fed by pairwise coherences between the fiducial and coupled states of the form $\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\downarrow_j}+\text{h.c.}$. Condition (1) being satisfied indicates that an optical pumping process transforms these coherences from $\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\downarrow_j}+\text{h.c.}$ to $\ket{\uparrow_i\downarrow_j}\bra{\downarrow_i\wr_j}+\text{h.c.}$. When $ T(\uparrow)>0$, these transformed coherences feed negative correlations and generate spin squeezing. For this reason, we call this optical pumping process a ``transfer of coherence" \cite{CohenTannoudji75}. We can take advantage of a transfer of coherence, if one exists, by preserving the transfer state $\ket{\wr}$ and mapping all other states outside the embedded qubit to the other ground manifold.
\chapter{Optical Pumping and the Covariance Matrix Update Formalism}\label{sec::ModHPCovar}
Having closely examined optical pumping, the principle source of decoherence in our system, we now seek to explore its effect on the achievable spin squeezing. As discussed in the previous chapter, retaining the transfer state in addition to the fiducial and coupled states can preserve negative correlations that would otherwise be lost to optical pumping. In cases where preserving the transfer state is beneficial, we model each atom as an embedded qutrit consisting of the fiducial, coupled and transfer states. Whereas the relevant ensemble observables in the embedded qubit case were the quadratures $\hat{X}_\downarrow$ and $\hat{P}_\downarrow$, an expanded number of ensemble observables is required when we treat the atoms as embedded qutrits. Working in the Heisenberg picture, we determine the evolution of the means and covariances of these observables under both coherent squeezing dynamics and decoherence from optical pumping. For particular fiducial states, the multilevel Holstein-Primakoff approximation can be revised to accommodate the transfer state. When this approximation holds, the ensemble is a Gaussian state on two oscillator modes. This enables us to utilize the Gaussian formalism of Sec. \ref{sec::GaussianStates} and the squeezing protocols of Chapter \ref{Sec:Protocols} to track the evolution of the ensemble and light through the covariance matrix.
\section{New Ensemble Observables}\label{sec::NewObs}
Preserving the transfer state in addition to the coupled and fiducial states produces an ensemble where each atomic spin is modeled as an embedded qutrit with basis states $\ket{\uparrow}$, $\ket{\downarrow}$ and $\ket{\wr}$. Operators on the qutrit state of a single atomic spin can be decomposed in terms of an operator basis
\begin{align}\label{eq::qutritOp1}
\hat{x}_{\downarrow\uparrow}&=\frac{1}{\sqrt{2}}\left(\ket{\downarrow}\bra{\uparrow}+\ket{\uparrow}\bra{\downarrow}\right)\\
\hat{y}_{\downarrow\uparrow}&=\frac{i}{\sqrt{2}}\left(\ket{\downarrow}\bra{\uparrow}-\ket{\uparrow}\bra{\downarrow}\right)\\
\hat{x}_{\wr\downarrow}&=\frac{1}{\sqrt{2}}\left(\ket{\wr}\bra{\downarrow}+\ket{\downarrow}\bra{\wr}\right)\\
\hat{y}_{\wr\downarrow}&=\frac{i}{\sqrt{2}}\left(\ket{\wr}\bra{\downarrow}-\ket{\downarrow}\bra{\wr}\right)\\
\hat{x}_{\uparrow\wr}&=\frac{1}{\sqrt{2}}\left(\ket{\uparrow}\bra{\wr}+\ket{\wr}\bra{\uparrow}\right)\\
\hat{y}_{\uparrow\wr}&=\frac{i}{\sqrt{2}}\left(\ket{\uparrow}\bra{\wr}-\ket{\wr}\bra{\uparrow}\right)\\
\hat{n}_{\uparrow}&=\ket{\uparrow}\bra{\uparrow}\\
\hat{n}_\downarrow&=\ket{\downarrow}\bra{\downarrow}
\end{align}
and
\begin{align}\label{eq::qutritOp9}
\hat{n}_\wr&=\ket{\wr}\bra{\wr}.
\end{align}
Note that the operators $\hat{x}_{ij}$ and $\hat{y}_{ij}$ are Pauli spin operators expressed in the basis $\{\ket{i},\ket{j}\}$, but with a different normalization convention. By summing these operators over all atoms, we obtain a basis for collective operators on the ensemble of embedded qutrits,
\begin{align}
\hat{X}_{\downarrow\uparrow}&=\sum_{i=1}^{N_A}\hat{x}_{\downarrow\uparrow}^{(i)}=\frac{1}{\sqrt{2}}\sum_i\left(\ket{\downarrow}\bra{\uparrow}_i+\ket{\uparrow}\bra{\downarrow}_i\right)\label{eq::Xdownup}\\
\hat{Y}_{\downarrow\uparrow}&=\sum_{i=1}^{N_A}\hat{p}_{\downarrow\uparrow}^{(i)}=\frac{i}{\sqrt{2}}\sum_{i=1}^{N_A}\left(\ket{\downarrow}\bra{\uparrow}_i-\ket{\uparrow}\bra{\downarrow}_i\right)\\
\hat{X}_{\wr\downarrow}&=\sum_{i=1}^{N_A}\hat{x}_{\wr\downarrow}^{(i)}=\frac{1}{\sqrt{2}}\sum_{i=1}^{N_A}\left(\ket{\wr}\bra{\downarrow}_i+\ket{\downarrow}\bra{\wr}_i\right)\\
\hat{Y}_{\wr\downarrow}&=\sum_{i=1}^{N_A}\hat{p}_{\wr\downarrow}^{(i)}=\frac{i}{\sqrt{2}}\sum_{i=1}^{N_A}\left(\ket{\wr}\bra{\downarrow}_i-\ket{\downarrow}\bra{\wr}_i\right)\label{eq::Psquiggledown}\\
\hat{X}_{\uparrow\wr}&=\sum_{i=1}^{N_A}\hat{x}_{\uparrow\wr}^{(i)}=\frac{1}{\sqrt{2}}\sum_{i=1}^{N_A}\left(\ket{\uparrow}\bra{\wr}_i+\ket{\wr}\bra{\uparrow}_i\right)\\
\hat{Y}_{\uparrow\wr}&=\sum_{i=1}^{N_A}\hat{p}_{\uparrow\wr}^{(i)}=\sum_{i=1}^{N_A}\frac{i}{\sqrt{2}}\left(\ket{\uparrow}\bra{\wr}_i-\ket{\wr}\bra{\uparrow}_i\right)\label{eq::Sigmayupwr}\\
N_\uparrow&=\sum_{i=1}^{N_A}\hat{n}_{\uparrow}^{(i)}=\sum_{i=1}^{N_A}\ket{\uparrow}\bra{\uparrow}_i\label{eq::PopUp}\\
N_\downarrow&=\sum_{i=1}^{N_A}\hat{n}_\downarrow^{(i)}=\sum_{i=1}^{N_A}\ket{\downarrow}\bra{\downarrow}_i
\end{align}
and
\begin{align}
N_\wr&=\sum_{i=1}^{N_A}\hat{n}_\wr^{(i)}=\sum_{i=1}^{N_A}\ket{\wr}\bra{\wr}_i.\label{eq::PopWr}
\end{align}
The first six ensemble observables in Eqs. (\ref{eq::Xdownup})-(\ref{eq::Sigmayupwr}) are collective pseudo spin operators on the ensemble of embedded qutrits. Note that the first two pseudo spins, $\hat{X}_{\downarrow\uparrow}$ and $\hat{Y}_{\downarrow\uparrow}$, are related to the ensemble quadratures, $\hat{X}_\downarrow$ and $\hat{P}_\downarrow$. By writing $\hat{X}_{\downarrow\uparrow}$ and $\hat{Y}_{\downarrow\uparrow}$ in the Schwinger representation and linearizing the operators in the $\uparrow$ mode, we obtained $\hat{X}_\downarrow$ and $\hat{P}_\downarrow$ in Sec. \ref{sec::HPEnsemble}. The final three ensemble observables in Eqs. (\ref{eq::PopUp})-(\ref{eq::PopWr}) are the familiar populations, which quantify the number of atoms in the fiducial, coupled and transfer states.
The central focus of this chapter is solving for the squeezing parameter as a function of time while the ensemble undergoes both spin squeezing and optical pumping. Recall that the metrological squeezing parameter in \erf{eq::SqParameter} depends upon the collective spin variance, the mean spin and the total number of atoms in the $f$ manifold. All of these quantities can be expressed in terms of the means and covariances of the unnormalized quadratures and the populations. In absence of post-processing via internal spin control, the collective spin variance, the mean spin and the total atom number are given by
\begin{align}
\Delta F_z^2&=v(\uparrow)^2\Delta\hat{X}_{\downarrow\uparrow}^{2}+2v(\uparrow)w(\uparrow)\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S
+w(\uparrow)^2\Delta\hat{X}_{\wr\downarrow}^{2},\label{eq::FzVarNewObs}\\
\expect{\hat{F}_x}&=\expect{\hat{f}_x}_\uparrow N_\uparrow+\expect{\hat{f}_x}_\downarrow N_\downarrow+\expect{\hat{f}_x}_\wr N_\wr
\label{eq::FxNewObs}
\end{align}
and
\begin{align}
N=N_\uparrow+ N_\downarrow+N_\wr.\label{eq::NNewObs}
\end{align}
Here, the functions $v(\uparrow)$ and $w(\uparrow)$ depend on the variances of the fiducial and coupled states,
\begin{align}\label{eq::vDef}
v(\uparrow)=\sqrt{2(\Delta f_z^2)_\uparrow}
\end{align}
and
\begin{align}\label{eq::wDef}
w(\uparrow)=\sqrt{2(\Delta f_z^2)_\downarrow-2(\Delta f_z^2)_\uparrow}.
\end{align}
By tracking the collective pseudo spins and populations as a function of time, we can calculate the squeezing parameter.
\section{Equations of Motion Under Optical Pumping}
In order to determine the evolution of the correlation functions, we must describe their dynamics under optical pumping. We utilize the master equation description of optical pumping introduced in Chapter \ref{sec::OpticalPumping}. In a realistic implementation of a squeezing protocol based on the Faraday interaction, one introduces a bias magnetic field in the direction of the light's propagation along $z$ to fix the quantization axis of the atoms. The bias field necessitates that we transform into a frame rotating at the Larmor frequency about $z$, as described in Sec. \ref{sec::FaradayH}. In the rotating frame, the the master equation in \erf{eq::MasterEnsem} becomes
\begin{align}\label{eq::MasterRotating}
\frac{d\hat{\rho}}{dt}\Big|_{\text{op}}=&-\frac{2\gamma_s}{9}\hat{\rho}\sum_{i=1}\mathbb{I}^{(i)}+
\frac{g_f^2\gamma_s}{9}\sum_{i=1}\left(\hat{f}_z^{(i)}\hat{\rho}\hat{f}_z^{(i)}
+\frac{1}{2}\hat{f}_y^{(i)}\hat{\rho}\hat{f}_y^{(i)}+\frac{1}{2}\hat{f}_x^{(i)}\hat{\rho}\hat{f}_x^{(i)}\right)\\\label{eq::MasterD}
=&\gamma_s\sum_i\mathcal{D}^{(i)}(\hat{\rho}).
\end{align}
This master equation describes optical pumping of the ensemble resulting from the absorption of photons with equal probability of being polarized along $x$ or $y$.
\subsection{Dynamics of First Order Moments}
To begin, we determine the equation of motion for first order collective operators under optical pumping. By summing \erf{eq::MasterRotating} over a single $i$, we obtain the equation of motion for the density matrix of an individual atom,
\begin{align}\label{eq::SingleMaster}
\frac{d\hat{\rho}^{(i)}}{dt}\Big|_{\text{op}}=\gamma_s\mathcal{D}^{(i)}(\hat{\rho}^{(i)}).
\end{align}
Evolution of a first order collective operator $\hat{O}=\sum_{i=1}^{N_A}\hat{o}^{(i)}$ follows from \erf{eq::SingleMaster},
\begin{align}\label{eq::FirstOrderEvol}
\frac{d}{dt}\hat{O}\Big|_{\text{op}}=\gamma_s\sum_{i=1}^{N_A}\mathcal{D}^{(i)}(\hat{o}^{(i)}).
\end{align}
This expression for $d\hat{O}/dt|_{\text{op}}$ can be decomposed in terms of the collective pseudo spins and populations as
\begin{align}\label{eq::FirstOrderEvolBasis1}
\frac{d}{dt}\hat{O}\Big|_{\text{op}}=&\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{x}_{\downarrow\uparrow})\hat{X}_{\downarrow\uparrow}+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{p}_{\downarrow\uparrow})\hat{P}_{\downarrow\uparrow}+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{x}_{\wr\downarrow})\hat{X}_{\wr\downarrow}\\\notag&+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{p}_{\wr\downarrow})\hat{P}_{\wr\downarrow}+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{x}_{\uparrow\wr})\hat{X}_{\uparrow\wr}+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{p}_{\uparrow\wr})\hat{P}_{\uparrow\wr}\\\notag
&+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{n}_\uparrow)N_\uparrow+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{n}_\downarrow)N_\downarrow
+\gamma_s\text{Tr}(\mathcal{D}(\hat{o})\hat{n}_\wr)N_\wr.
\end{align}
Calculating the mean spin and total atom number requires knowledge of the populations in the fiducial, coupled and transfer states. From \erf{eq::FirstOrderEvolBasis1}, the evolution of a population $N_\phi$ is given by
\begin{align}\label{eq::FirstOrderPopulations}
\frac{dN_\phi}{dt}\Big|_{\text{op}}=&\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{x}_{\downarrow\uparrow})\expect{\hat{X}_{\downarrow\uparrow}}+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{p}_{\downarrow\uparrow})\expect{\hat{P}_{\downarrow\uparrow}}\\\notag&+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{x}_{\wr\downarrow})\expect{\hat{X}_{\wr\downarrow}}+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{p}_{\wr\downarrow})\expect{\hat{P}_{\wr\downarrow}}\\\notag&+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{x}_{\uparrow\wr})\expect{\hat{X}_{\uparrow\wr}}+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{p}_{\uparrow\wr})\expect{\hat{P}_{\uparrow\wr}}\\\notag
&+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{n}_\uparrow)N_\uparrow+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{n}_\downarrow)N_\downarrow
+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{n}_\wr)N_\wr,
\end{align}
for $\phi\in\{\uparrow,\,\downarrow,\,\wr\}$. For an ensemble prepared in the state $\ket{\uparrow}^{\otimes N_A}$, the means of the collective pseudo spins are initially zero. The population of atoms in the fiducial state, on the other hand, is extremely large with $N_\uparrow>>1$. Although the means of the pseudo spins can become nonzero if they are coupled to $N_\uparrow$ by their equations of motion, they remain much smaller than $N_\uparrow$. Because their influence upon the populations is second order in the scattering rate, the means of the pseudo spins can be eliminated from the equation of motion above, leaving
\begin{align}\label{eq::FirstOrderPopulations2}
\frac{dN_\phi}{dt}\Big|_{\text{op}}=\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{n}_\uparrow)N_\uparrow+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{n}_\downarrow)N_\downarrow
+\gamma_s\text{Tr}(\mathcal{D}(\hat{n}_\phi)\hat{n}_\wr)N_\wr.
\end{align}
The result is a set of closed differential equations that couple the populations to one another.
\subsection{Dynamics of Second Order Moments}\label{sec::2ndOrderOP}
We next turn our attention to calculating the equation of motion for the second order collective operators. Specifically, we focus on the covariances between collective operators, which are essential for determining the collective spin variance. Deriving these equations of motion requires the master equation describing the evolution of any two atoms in the ensemble, $i$ and $j$,
\begin{align}\label{eq::rhoij}
\frac{d\hat{\rho}^{(i,j)}}{dt}\Big|_{\text{op}}=\gamma_s\mathcal{D}^{(i)}(\hat{\rho}^{(i,j)})+\gamma_s\mathcal{D}^{(j)}(\hat{\rho}^{(i,j)}).
\end{align}
This master equation follows from taking the sum in \erf{eq::MasterRotating} over the two indices $i$ and $j$. From $d\hat{\rho}^{(i,j)}/dt$ we obtain the evolution of a second-order correlation function,
\begin{align}\label{eq::2ndCorFun}
\frac{d}{dt}\expect{\Delta\hat{o}^{(i)}\!\Delta\hat{a}^{(j)}}_{S,\,i\neq j}\Big|_{\text{op}}=&\gamma_s\expect{\Delta\mathcal{D}^{(i)}(\hat{o}^{(i)})\Delta\hat{a}^{(j)}}_{S,\,i\neq j}\\\notag&+\gamma_s\expect{\Delta\hat{o}^{(i)}\Delta\mathcal{D}^{(j)}(\hat{a}^{(j)})}_{S,\,i\neq j}.
\end{align}
Here and throughout, the notation $\expect{\Delta\hat{x}\Delta\hat{y}}_S$ on two operators $\hat{x}$ and $\hat{y}$ is shorthand for the covariance $\expect{\Delta\hat{x}\Delta\hat{y}+\Delta\hat{y}\Delta\hat{x}}/2$.
For two collective operators $\hat{O}=\sum_i\hat{o}^{(i)}$ and $\hat{A}=\sum_i\hat{a}^{(i)}$, the covariance depends on both first and second order correlation functions, i.e. both single-atom and two-atom expectation values,
\begin{align}
\expect{\Delta\hat{O}\Delta\hat{A}}_S=\sum_{i\neq j}\expect{\Delta\hat{o}^{(i)}\Delta\hat{a}^{(j)}}_S
+\sum_i\expect{\Delta\hat{o}^{(i)}\Delta\hat{a}^{(i)}}_S.
\end{align}
To obtain the evolution of this covariance under optical pumping, we employ the equations of motion for first and second order correlation functions given in Eqs. (\ref{eq::FirstOrderEvol}) and (\ref{eq::2ndCorFun}), which yield
\begin{align}\label{eq::CovarEvol}
\frac{d}{dt}\expect{\Delta\hat{O}\Delta\hat{A}}_S\Big|_{\text{op}}
=&\gamma_s\sum_{i=1}^{N_A}\expect{\Delta\mathcal{D}^{(i)}(\hat{o}^{(i)})\Delta\hat{A}}_S\\\notag
&+\gamma_s\sum_{j=1}^{N_A}\expect{\Delta\hat{O}\Delta\mathcal{D}^{(j)}(\hat{a}^{(j)})}_S\\\notag
&+\frac{\gamma_s}{2}\sum_{i=1}^{N_A}\expect{\mathcal{D}^{(i)}(\{\hat{o}^{(i)},\hat{a}^{(i)}\})}
-\frac{\gamma_s}{2}\sum_{i=1}^{N_A}\expect{\{\mathcal{D}^{(i)}(\hat{o}^{(i)}),\hat{a}^{(i)}\}}\\\notag
&-\frac{\gamma_s}{2}\sum_{i=1}^{N_A}\expect{\{\hat{o}^{(i)},\mathcal{D}^{(i)}(\hat{a}^{(i)})\}}.
\end{align}
We refer to the final three terms, collectively as the ``noise term". Note that the noise term is a first order collective operator, while the remaining terms of \erf{eq::CovarEvol} are second order. The contribution to the noise term from each atom is proportional to the superoperator
\begin{align}\label{eq::noiseSuperOp}
\mathcal{N}(\hat{o},\hat{a})=\frac{1}{2}\mathcal{D}(\{\hat{o},\hat{a}\})
-\frac{1}{2}\{\mathcal{D}(\hat{o}),\hat{a}\}
-\frac{1}{2}\{\hat{o},\mathcal{D}(\hat{a})\},
\end{align}
which acts upon the two internal spin operators $\hat{o}$ and $\hat{a}$.
Utilizing the collective operator basis of the pseudo spins and populations, we can decompose the equation of motion for $\expect{\Delta\hat{O}\Delta\hat{A}}_S$ in a manner analogous to the equation of motion for the first order operator, $\hat{O}$. The evolution of $\expect{\Delta\hat{O}\Delta\hat{A}}_S$ becomes
\begin{align}\label{eq::CovarEvol2}
\frac{d}{dt}\expect{\Delta\hat{O}\Delta\hat{A}}_S\Big|_{\text{op}}
=&\gamma_s\sum_{\hat{X}\in\mathcal{S}}\text{Tr}(\mathcal{D}(\hat{o})\hat{x})\expect{\Delta\hat{X}\Delta\hat{A}}_S\\\notag
&+\gamma_s\sum_{\hat{X}\in\mathcal{S}}\text{Tr}(\mathcal{D}(\hat{a})\hat{x})\expect{\Delta\hat{O}\Delta\hat{X}}_S\\\notag
&+\gamma_s\text{Tr}(\mathcal{N}(\hat{o},\hat{a})\hat{n}_\uparrow)N_\uparrow
+\gamma_s\text{Tr}(\mathcal{N}(\hat{o},\hat{a})\hat{n}_\downarrow)N_\downarrow\\\notag
&+\gamma_s\text{Tr}(\mathcal{N}(\hat{o},\hat{a})\hat{n}_\wr)N_\wr
\end{align}
where $\hat{X}=\sum_{i=1}^{N_A}\hat{x}^{(i)}$ and $\mathcal{S}=\{\hat{X}_{\downarrow\uparrow},\hat{Y}_{\downarrow\uparrow},
\hat{X}_{\wr\downarrow},\hat{Y}_{\wr\downarrow},\hat{X}_{\uparrow\wr},\hat{Y}_{\uparrow\wr}\}$.
Because we treat the populations as c-numbers, $\Delta N_\uparrow\approx\Delta N_\downarrow\approx\Delta N_\wr\approx 0$. The covariances in the equation of motion for $\expect{\Delta\hat{O}\Delta\hat{A}}_S$ are, therefore, between the collective pseudo spins alone. As in the case of the equation of motion for the populations, the means of the collective pseudo spins remain small, enabling us to decompose the noise term as a sum of populations.
\section{Revising the Multilevel Holstein Primakoff \\Approximation}\label{sec::NewQuads}
In order to implement the squeezing protocols of Chapter \ref{Sec:Protocols}, we represented the ensemble as a Gaussian state on a single bosonic mode. This was accomplished through the multilevel Holstein Primakoff approximation, which relied upon describing each atom as an embedded qubit consisting of the fiducial and coupled states. As we have argued, retaining the transfer state in addition to the fiducial and coupled states can increase the robustness of a state preparation to optical pumping. In this section, we modify the multilevel Holstein-Primakoff approximation in order to preserve the transfer state. When the master equation describing optical pumping and the fiducial state satisfy certain properties, the ensemble becomes a Gaussian state on two effective collective spin modes. We adapt the squeezing protocols presented in Chapter \ref{Sec:Protocols} to accommodate the state of the light and atomic ensemble, which becomes a three mode Gaussian. Optical pumping can be expressed as a Gaussian channel upon the system covariance matrix, enabling us to combine coherent squeezing dynamics with dissipation.
\subsection{Effective Collective Spin Modes}\label{sec::EffModes}
In Sec. \ref{sec::NewObs}, we introduced the collective pseudo spin operators, which together with the populations, form a basis for operators on the ensemble of embedded qutrits. Of the collective pseudo spins defined in Eqs. (\ref{eq::Xdownup})-(\ref{eq::Sigmayupwr}), the operators $\hat{X}_{\downarrow\uparrow}$, $\hat{Y}_{\downarrow\uparrow}$, $\hat{X}_{\wr\downarrow}$ and $\hat{Y}_{\wr\downarrow}$ contain the coherences that generate spin squeezing. Recall that pairwise coherences between the fiducial and coupled states and between the coupled and transfer states create negative correlations that reduce $\Delta F_z^2$. Consider the commutator between a collective pseudo spin with subscript $\downarrow\uparrow$ and another with subscript $\wr\!\downarrow$. For $\hat{d}\in\{\hat{X},\hat{Y}\}$, this commutator is of the form
\begin{align}\label{eq::unnormDiffModes}
[\hat{d}_{\downarrow\uparrow},\hat{d}_{\wr\downarrow}]=\sum_{i=1}^{N_A}\left(C_{\uparrow\wr}\ket{\uparrow}\bra{\wr}_i+C_{\wr\uparrow}\ket{\wr}\bra{\uparrow}_i\right),
\end{align}
where $C_{\uparrow\wr}$ and $C_{\wr\uparrow}$ are complex constants. In cases where coherences between the fiducial and transfer states are negligible, the quadratures with subscripts $\downarrow\uparrow$ and $\wr\!\!\downarrow$ approximately commute like two bosonic modes.
In general, coherences between the fiducial and transfer states are not negligible. As discussed in Sec. \ref{Sec::OPEvents}, squeezing protocols develop pairwise coherences between the fiducial coupled states of the form $\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\downarrow_j}+\text{h.c.}$. For certain fiducial states, optical pumping can transform these pairwise coherences into coherences of the form $\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\wr_j}+\text{h.c.}$. While $\expect{\ket{\wr}\bra{\uparrow}}=\expect{\ket{\uparrow}\bra{\wr}}=0$ when these pairwise coherences are present, second order moments involving coherences between the fiducial and transfer states are not necessarily zero. Consider, for example, the covariance $\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\uparrow\wr}}_S$. The pairwise coherences $\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\wr_j}+\text{h.c.}$ contribute positively to this second order moment, since
\begin{align}
\bra{\downarrow_i\wr_j}(\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\uparrow\wr})_S\ket{\uparrow_i\uparrow_j}=\frac{1}{2}.
\end{align}
Consequently, we cannot neglect the value of the commutator in \erf{eq::unnormDiffModes} if the pairwise coherences $\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\wr_j}+\text{h.c.}$ are present.
In order for operators on the ``modes" $\downarrow\uparrow$ and $\wr\!\downarrow$ to commute, the fiducial state and the master equation describing optical pumping must satisfy certain properties. If the condition in \erf{eq::up2squiggle} holds, which guarantees that preserving the transfer state is beneficial for spin squeezing, optical pumping creates the pairwise coherences $\ket{\uparrow_i\uparrow_j}\bra{\downarrow_i\wr_j}+\text{h.c.}$ only when there exists a jump operator $\hat{W}_q$ such that both \\$\bra{\uparrow}\hat{W}_q\ket{\uparrow}\neq 0$ and $\bra{\downarrow}\hat{W}_q\ket{\wr}\neq 0$. Pairwise coherences between the fiducial and transfer states do not develop as long as
\begin{align}\label{eq::ModeCondition}
\sum_q\text{Re}[\bra{\uparrow}\hat{W}_q\ket{\uparrow}\bra{\downarrow}\hat{W}_q^{\dag}\ket{\wr}]=0.
\end{align}
For the remainder of this section, we consider only the case in which this condition holds. In Chapter \ref{sec::Beyond}, we treat the most general case in which pairwise coherences between the fiducial and transfer state are permitted to develop.
The condition in \erf{eq::ModeCondition} being satisfied implies that
\begin{align}
[\hat{d}_{\downarrow\uparrow},\hat{d}_{\wr\downarrow}]\approx 0.
\end{align}
The ensemble becomes a state on two effective collective spin modes, $\downarrow\uparrow$ and $\wr\!\downarrow$. The collective pseudo spins $\hat{X}_{\downarrow\uparrow}$, $\hat{Y}_{\downarrow\uparrow}$, $\hat{X}_{\wr\downarrow}$ and $\hat{Y}_{\wr\downarrow}$ are conjugate observables on these effective oscillator modes. There are several key differences between the collective pseudo spins and the quadratures $\hat{X}_\downarrow$ and $\hat{P}_\downarrow$ derived in Sec. \ref{sec::HPEnsemble}. First, the collective pseudo spins do not obey the canonical commutation relations. Instead,
\begin{align}\label{eq::commutator1}
[\hat{X}_{\downarrow\uparrow},\hat{Y}_{\downarrow\uparrow}]=i(N_\uparrow-N_\downarrow)
\end{align}
and
\begin{align}\label{eq::commutator2}
[\hat{X}_{\wr\downarrow},\hat{Y}_{\wr\downarrow}]=i(N_\downarrow-N_\wr).
\end{align}
Furthermore, because the populations $N_\uparrow$, $N_\downarrow$ and $N_\wr$ are not constant under optical pumping, the commutation relations are time-varying. From the commutators in equations (\ref{eq::commutator1}) and (\ref{eq::commutator2}), we can deduce the uncertainty relations on conjugate collective pseudo spins,
\begin{align}\label{eq::uncert1}
\Delta X_{\downarrow\uparrow}^{2}\Delta Y_{\downarrow\uparrow}^{2}\geq\frac{(N_\uparrow-N_\downarrow)^2}{4},
\end{align}
and
\begin{align}\label{eq::uncert2}
\Delta X_{\wr\downarrow}^{2}\Delta Y_{\wr\downarrow}^{2}\geq\frac{(N_\downarrow-N_\wr)^2}{4}.
\end{align}
Note that the lower bounds in the uncertainty relations are, likewise, time-varying.
To place the atoms and the light on similar footing, we define ``collective pseudospins" on the $y$ mode of the light,
\begin{align}
\hat{X}_y=\sqrt{\frac{N_L}{2}}(\hat{a}_y^\dag+\hat{a}_y)
\end{align}
and
\begin{align}
\hat{Y}_y=i\sqrt{\frac{N_L}{2}}(\hat{a}_y^\dag-\hat{a}_y).
\end{align}
These operators are equivalent to the HP quadratures of the light with a different normalization convention. The ``collective pseudospins" of the light satisfy the commutation relation
\begin{align}\label{eq::Commute3}
[\hat{X}_y,\hat{Y}_y]=iN_L
\end{align}
and the uncertainty relation
\begin{align}\label{eq::uncert3}
\Delta X_{y}^{2}\Delta Y_{y}^{2}\geq\frac{N_L^2}{4}.
\end{align}
Because the number of photons is approximately unchanged by spontaneous emission, the commutation and uncertainty relations are constant in time unlike those of the ensemble collective pseudospins.
The commutation relations of all observables on the light and ensemble can be expressed concisely as
\begin{align}
[\hat{\textbf{d}}_j,\hat{\textbf{d}}_k]=i(n\sigma)_{jk}
\end{align}
where $\hat{\textbf{d}}=\{\hat{X}_{\downarrow\uparrow},\hat{Y}_{\downarrow\uparrow},\hat{X}_{\wr\downarrow},\hat{Y}_{\wr\downarrow},\hat{X}_y,\hat{Y}_y\}^T$. Here, $\sigma$ is the symplectic matrix defined in \erf{eq::sympMatrix} and the matrix $n$ is given by
\begin{eqnarray}
n=\left(\begin{matrix} N_\uparrow-N_\downarrow& 0& 0& 0& 0& 0 \\ 0 & N_\uparrow-N_\downarrow&0&0& 0& 0
\\0&0&N_\downarrow-N_\wr&0& 0& 0\\0&0&0&N_\downarrow-N_\wr&0& 0\\0&0&0&0&N_L&0\\0&0&0&0&0&N_L
\end{matrix}\right).
\end{eqnarray}
For the collective pseudo spins, the matrix $n\sigma$ takes the place of the symplectic matrix, $\sigma$.
\subsection{Gaussianity}\label{sec::Gauss}
In absence of optical pumping, the ensemble is a Gaussian state on a single mode, $\downarrow$. The previous section demonstrated that when optical pumping is taken into account, the ensemble can be approximated as a state on two collective spin modes, labeled by $\downarrow\uparrow$ and $\wr\!\downarrow$. In this section, we show that the ensemble is Gaussian on these modes. The light and ensemble, thus, form a multimode Gaussian state that can be evolved via the covariance matrix update formalism.
In our new definition of the ensemble modes, the mode $\downarrow\uparrow$ takes the place of the mode $\downarrow$. Like the mode $\downarrow$, an excitation in $\downarrow\uparrow$ represents an atom taken from the fiducial state to the coupled state. Likewise, the initial state of the ensemble with each atom prepared in the fiducial state is equivalent to $\ket{0}_{\downarrow\uparrow}$, the vacuum state in the mode $\downarrow\uparrow$. Similarly, excitations in the mode $\wr\!\downarrow$ correspond to an atom taken from the coupled state to the transfer state. The vacuum state in the mode $\wr\!\downarrow$ corresponds to an ensemble state with a comparatively large number of atoms in the coupled state and none in the transfer state. Due to the condition in \erf{eq::up2squiggle}, which prohibits optical pumping directly from the fiducial state to the transfer state, any population in the transfer state is the result of a second order optical pumping event. Because population in the coupled state accumulates due to spin flips, which are first order optical pumping events, the population of atoms in the coupled state is always substantially larger than the population of atoms in the transfer state. As soon as a spin flip event transfers population to the coupled state, the ensemble becomes a state on two modes. A spin flip event also transfers the beneficial coherences between the fiducial and coupled states into coherences between the coupled and transfer states. By evolving the density matrix in \erf{densityOp} under a spin flip, it can be seen that the coherences between the coupled and transfer states remain smaller than any population in the coupled state by an order of $\xi/N_A$. This is also true of the coherences between the fiducial and coupled states, which are smaller than the population in the fiducial state by an order of $\xi/N_A$.
In terms of the collective pseudo spins, the Wigner function of the initial vacuum state, $\ket{0}_{\downarrow\uparrow}\ket{0}_y$, of the light and the ensemble is
\begin{align}
W(X_{\downarrow\uparrow},Y_{\downarrow\uparrow},X_{y},Y_{y})=
\frac{1}{\pi^2N_AN_L}e^{-\frac{X_{\downarrow\uparrow}^{2}+Y_{\downarrow\uparrow}^{2}}{N_A}}
e^{-\frac{X_{y}^{2}+Y_{y}^{2}}{N_L}}.
\end{align}
From \erf{MostGenMasterk}, the evolution of the ensemble density matrix, $\hat{\rho}_A$, under optical pumping over a small time step $\Delta t$ is approximately
\begin{align}\label{eq::OPrho}
\hat{\rho}_A(\Delta t)\approx&\left(1-\Gamma_{\text{op}}\Delta tN_A+\Delta t N_A\sum_q|\bra{\uparrow}\hat{W}_q\ket{\uparrow}|^2\;\right)\ket{0}\bra{0}_{\downarrow\uparrow}\ket{0}\bra{0}_{\wr\downarrow}\\\notag
&+\Delta t\sqrt{N_A}\left(\sum_q\bra{\uparrow}\hat{W}_q\ket{\uparrow}\bra{\uparrow}\hat{W}_q^\dag\ket{\downarrow}\;\right)\ket{0}\bra{1}_{\downarrow\uparrow}\ket{0}\bra{0}_{\wr\downarrow}\\\notag
&+\Delta t\sqrt{N_A}\left(\sum_q\bra{\downarrow}\hat{W}_q\ket{\uparrow}\bra{\uparrow}\hat{W}_q^\dag\ket{\uparrow}\;\right)\ket{1}\bra{0}_{\downarrow\uparrow}\ket{0}\bra{0}_{\wr\downarrow}\\\notag
&+\Delta t\left(\sum_q|\bra{\downarrow}\hat{W}_q\ket{\uparrow}|^2\right)\left(\sum_{i=1}^{N_A}\ket{\downarrow}\bra{\downarrow}_i\ket{\uparrow}\bra{\uparrow}_{\neq i}^{\otimes N_A-1}\right)\ket{0}\bra{0}_{\wr\downarrow}.
\end{align}
In the expression above, $\ket{1}_{\downarrow\uparrow}=\sum_{i=1}^{N_A}\ket{\downarrow}_i\ket{\uparrow}_{\neq i}^{\otimes N_A-1}/\sqrt{N_A}$, which signifies the presence of an atom in the coupled state. After the first optical pumping event, whereupon population is transferred into the coupled state, the combined system of the ensemble and light becomes a state on three modes. Defining creation and annihilation operators on the ensemble modes as $\hat{a}_{mn}^\dag=\sum_{i=1}^{N_A}\ket{m}\bra{n}_i/\sqrt{N_A}$ and $\hat{a}_{mn}=\sum_{i=1}^{N_A}\ket{n}\bra{m}_i/\sqrt{N_A}$ for $m,\,n\in \{\uparrow,\downarrow,\wr\}$, we can calculate the Wigner function of the system from $\hat{\rho}_A(\Delta t)$. The Wigner function of the system after undergoing optical pumping for a time $\Delta t$ is approximately
\begin{align}
W_A(\mathbf{d})\approx W(X_{\downarrow\uparrow},Y_{\downarrow\uparrow})
\frac{1}{\pi N_\downarrow}e^{-\frac{X_{\wr\downarrow}^{2}+Y_{\wr\downarrow}^{2}}{N_\downarrow}}\frac{1}{\pi N_L}e^{-\frac{X_{y}^{2}+Y_{y}^{2}}{N_L}},
\end{align}
where $W(X_{\downarrow\uparrow},Y_{\downarrow\uparrow})$ is the Wigner function of the ensemble on the mode $\downarrow\uparrow$, given by
\begin{align}\label{eq::OPwigner}
W(X_{\downarrow\uparrow},Y_{\downarrow\uparrow})=&\frac{1}{\pi N_A}\Big(1-\Delta t N_A \Gamma_{\text{op}}+\Delta t N_A\sum_q|\bra{\uparrow}\hat{W}_q\ket{\uparrow}|^2\\\notag&\;\;\;\;\;\;\;\;\;\;+\Delta t N_A\sum_q|\bra{\downarrow}\hat{W}_q\ket{\uparrow}|^2\;\Big)e^{-\frac{X_{\downarrow\uparrow}^2+Y_{\downarrow\uparrow}^2}{N_A}}\\\notag
&+\frac{\Delta t 2^{3/2}}{\pi N_A}\text{Re}\Big(\bra{\uparrow}\hat{W}_q\ket{\uparrow}\bra{\uparrow}\hat{W}_q^\dag\ket{\downarrow}(X_{\downarrow\uparrow}+iY_{\downarrow\uparrow})\Big)e^{-\frac{X_{\downarrow\uparrow}^2+Y_{\downarrow\uparrow}^2}{N_A}}.
\end{align}
While the first term of $W(X_{\downarrow\uparrow},Y_{\downarrow\uparrow})$ is Gaussian, the second is non-Gaussian. Note, however, that the second term is smaller than the first by an order of $N_A$.
Thus, to good approximation, the state of the ensemble is Gaussian on mode $\downarrow\uparrow$ after the first optical pumping event. The combined state of the light and ensemble becomes a multimode Gaussian on three modes.
\subsection{Covariance Matrix Update Formalism for \\Variable Atom Number}\label{sec::CovMarixOPUpdate}
Because the combined state of the ensemble and the light is a multimode Gaussian, we can adapt the formalism outlined in Sec. \ref{sec::GaussianStates}, which enables us to completely specify the state of the system by its covariance matrix. In terms of the collective pseudo spins, the covariance matrix is defined analogously to the covariance matrix in Sec. \ref{sec::GaussianStates}, with elements given by
\begin{align}\label{eq::newCovMatrix}
\widetilde{\Sigma}_{ij}=\left\langle\frac{\Delta\hat{\textbf{d}}_i\Delta\hat{\textbf{d}}_j+\Delta\hat{\textbf{d}}_j\Delta\hat{\textbf{d}}_i}{2}\right\rangle,
\end{align}
for $\hat{\textbf{d}}=\{\hat{X}_{\downarrow\uparrow},\hat{Y}_{\downarrow\uparrow},\hat{X}_{\wr\downarrow},\hat{Y}_{\wr\downarrow},\hat{X}_y,\hat{Y}_y\}^T$ and $\Delta\hat{\textbf{d}}_i=\hat{\textbf{d}}_i-\langle\hat{\textbf{d}}_i\rangle$.
The covariance matrix corresponds to a physical state provided that
\begin{align}\label{covcondition2}
\widetilde{\Sigma} +\frac{i}{2}n\sigma \geq 0.
\end{align}
As a consequence of \erf{covcondition2}, the uncertainty relations in equations (\ref{eq::uncert1}), (\ref{eq::uncert2}) and (\ref{eq::uncert3}) are satisfied. From \erf{eq::newCovMatrix}, the covariance matrix of the initial vacuum state is
\begin{align}
\widetilde{\Sigma}_0=\frac{1}{2}\left(\begin{matrix}N_A&0&0&0&0&0\\
0&N_A&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&0&0\\
0&0&0&0&N_L&0\\
0&0&0&0&0&N_L\end{matrix}\right).
\end{align}
Note that this covariance matrix satisfies \erf{covcondition2}, since the populations $N_\downarrow$ and $N_\wr$ are zero at the initial time.
Both unitary and dissipative evolution on the system can be expressed as update maps upon the covariance matrix. Unitary transformations act upon the covariance matrix via a map $S$, similar to a symplectic transformation,
\begin{align}
\widetilde{\Sigma}'=S\,\widetilde{\Sigma}\, S^{T}.
\end{align}
Rather than fulfilling the symplectic condition of \erf{symplectic1}, however, $S$ satisfies
\begin {eqnarray}
S\, n\sigma\, S^{T}=n\sigma,
\label{symplectic}
\end{eqnarray}
which ensures that \erf{covcondition2} is preserved on the covariance matrix. Dissipative evolution of the system can be expressed as a Gaussian channel, transforming the covariance matrix as
\begin {eqnarray}\label{eq::UnormGaussChannel}
\widetilde{\Sigma}'= M\, \widetilde{\Sigma} \,M^{T}+N,
\end{eqnarray}
where $N$ is a positive semi-definite matrix.
Equation (\ref{covcondition2}) holds on $\widetilde{\Sigma}'$ provided that
\begin {eqnarray}
N+\frac{i}{2}n\sigma-\frac{i}{2}M\, n\sigma\, M^T\geq 0.
\end{eqnarray}
The squeezing protocols of Sec. \ref{Sec:Protocols} are easily modified to act on the covariance matrix $\widetilde{\Sigma}$ with its alternative normalization and additional mode. Some of the symplectic matrices employed in the squeezing protocols must be adapted to the new normalization convention and their dimensionality increased to accommodate the ensemble mode $\wr\!\downarrow$. For instance, the symplectic rotation matrix $R(\theta)$, which acts upon the ensemble covariance matrix in the double pass squeezing protocols, must be generalized to rotate both modes $\downarrow\uparrow$ and $\wr\!\downarrow$ by an angle $\theta$. The rotation matrix becomes
\begin{align}
R(\theta)=\left(\begin{matrix}\text{cos}\theta&-\text{sin}\theta&0&0\\
\text{sin}\theta&\text{cos}\theta&0&0\\
0&0&\text{cos}\theta&-\text{sin}\theta\\
0&0&\text{sin}\theta&\text{cos}\theta\end{matrix}\right).
\end{align}
The symplectic matrix corresponding to the Faraday interaction on modes $\downarrow\uparrow$, $\wr\!\downarrow$ and $y$ is also altered by the new normalization convention, taking the form \begin{align}
\widetilde{S}_{F}=\left(\begin{matrix} 1 & 0 & 0 & 0 & 0 & 0\\ 0 &1& 0 & 0 & 0 & -\sqrt{\xi_\uparrow}(N_\uparrow-N_\downarrow)\\ 0 & 0 & 1 & 0 & 0 & 0
\\ 0 & 0 & 0 &1& 0 & -\sqrt{\xi_\downarrow}(N_\downarrow-N_\wr)\\
\sqrt{\xi_\uparrow}N_L & 0&\sqrt{\xi_\downarrow}N_L& 0 & 1 &0\\0 & 0 & 0 & 0 & 0 & 1\end{matrix}\right),
\end{align}
where $\xi_\uparrow=\chi^2(\Delta f_z^2)_\uparrow$ and $\xi_\downarrow=\chi^2((\Delta f_z^2)_\downarrow-(\Delta f_z^2)_\uparrow)$. The procedure for homodyne measurement of a quadrature, outlined in \erf{HCovariance}, is independent of the covariance matrix normalization. Similarly unchanged is the method for taking the partial trace over a mode of the system.
\section{Covariance Matrix Update for Optical \\Pumping}\label{sec::covUpdate}
The master equation in \erf{eq::MasterRotating} can be described as a Gaussian channel acting upon the covariance matrix of the light and ensemble, taking the form of \erf{eq::UnormGaussChannel}. Because the effects of optical pumping due to spontaneous emission are negligible upon the light, the mean update matrix and noise matrix have the structure
\begin{align}\label{eq::entireM}
M_\gamma=M_{A\gamma}\oplus\mathbb{I}_y
\end{align}
and
\begin{align}\label{eq::entireN}
N_\gamma=N_{A\gamma}\oplus\left(\begin{matrix}0&0\\0&0\end{matrix}\right)_y,
\end{align}
where $M_{A\gamma}$ and $N_{A\gamma}$ are matrices acting on the ensemble modes and the $y$ subscript denotes a matrix acting on the light.
We first determine the matrix $M_{A\gamma}$ through the equations of motion for the collective pseudo spins, which follow from the evolution of a first order moment under optical pumping given in \erf{eq::FirstOrderEvolBasis1}. Recall that first and second order moments involving coherences between the fiducial and transfer states can be neglected as long as $\sum_q\text{Re}[\bra{\uparrow}\hat{W}_q\ket{\uparrow}\bra{\downarrow}\hat{W}_q^{\dag}\ket{\wr}]=0$. For the master equation in \erf{eq::MasterRotating}, this condition is equivalent to $\bra{\uparrow}\mathcal{D}(\ket{\uparrow}\bra{\downarrow})\ket{\wr}= 0$.
When this condition is satisfied, we can discard the terms in \erf{eq::FirstOrderEvolBasis1} involving the quadratures $\hat{X}_{\uparrow\wr}$ and $\hat{Y}_{\uparrow\wr}$.
After dispensing with $\hat{X}_{\uparrow\wr}$ and $\hat{Y}_{\uparrow\wr}$, we utilize \erf{eq::FirstOrderEvolBasis1} to calculate the evolution of the first order central moment, $\Delta\hat{O}=\hat{O}-\expect{\hat{O}}$.
The equation of motion for $\Delta\hat{O}$ is
\begin{align}
\frac{d}{dt}\Delta\hat{O}|_{\text{op}}=&\;\gamma_s\text{Tr}\left(\mathcal{D}(\hat{o})\hat{x}_{\downarrow\uparrow}\right)\Delta\hat{X}_{\downarrow\uparrow}+
\gamma_s\text{Tr}\left(\mathcal{D}(\hat{o})\hat{p}_{\downarrow\uparrow}\right)\Delta\hat{Y}_{\downarrow\uparrow}\\\notag&+
\gamma_s\text{Tr}\left(\mathcal{D}(\hat{o})\hat{x}_{\wr\downarrow}\right)\Delta\hat{X}_{\wr\downarrow}
+\gamma_s\text{Tr}\left(\mathcal{D}(\hat{o})\hat{p}_{\wr\downarrow}\right)\Delta\hat{Y}_{\wr\downarrow}\\\notag&+
\gamma_s\text{Tr}\left(\mathcal{D}(\hat{o})\hat{n}_\uparrow\right)\Delta N_\uparrow
+\gamma_s\text{Tr}\left(\mathcal{D}(\hat{o})\hat{n}_\downarrow\right)\Delta N_\downarrow\\\notag&
+\gamma_s\text{Tr}\left(\mathcal{D}(\hat{o})\hat{n}_\wr\right)\Delta N_\wr.
\end{align}
Because we treat the populations as c-numbers, $\Delta N_\uparrow\approx \Delta N_\downarrow\approx \Delta N_\wr\approx0$. The resulting equation of motion for $\Delta\hat{O}$ is independent of the populations.
By integrating the equation of motion over a small time $\Delta t$, we obtain
\begin{align}\label{eq::CentralFirstIntegrate}
\Delta\hat{O}(\Delta t)\approx&\,\Delta\hat{O}(0)\\\notag&+\gamma_s\Delta t\,\text{Tr}\left(\mathcal{D}(\hat{o})\hat{x}_{\downarrow\uparrow}\right)\Delta\hat{X}_{\downarrow\uparrow}(0)+
\gamma_s\Delta t\, \text{Tr}\left(\mathcal{D}(\hat{o})\hat{y}_{\downarrow\uparrow}\right)\Delta\hat{Y}_{\downarrow\uparrow}(0)\\\notag&+
\gamma_s\Delta t\, \text{Tr}\left(\mathcal{D}(\hat{o})\hat{x}_{\wr\downarrow}\right)\Delta\hat{X}_{\wr\downarrow}(0)
+\gamma_s\Delta t\, \text{Tr}\left(\mathcal{D}(\hat{o})\hat{y}_{\wr\downarrow}\right)\Delta\hat{Y}_{\wr\downarrow}(0).
\end{align}
Taking $\hat{O}\in\{\hat{X}_{\downarrow\uparrow},\hat{Y}_{\downarrow\uparrow},\hat{X}_{\wr\downarrow},\hat{Y}_{\wr\downarrow}\}$ in \erf{eq::CentralFirstIntegrate} gives the us the update matrix for the central moments of the quadratures,
\begin{align}\label{eq::Mgamma}
&M_{A\gamma}(\Delta t)=\mathbb{I}+\\\notag&\gamma_s\Delta t\left(\begin{matrix}
\text{Tr}\left(\mathcal{D}(\hat{x}_{\downarrow\uparrow})\hat{x}_{\downarrow\uparrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{x}_{\downarrow\uparrow})\hat{y}_{\downarrow\uparrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{x}_{\downarrow\uparrow})\hat{x}_{\wr\downarrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{x}_{\downarrow\uparrow})\hat{y}_{\wr\downarrow}\right)\\
\text{Tr}\left(\mathcal{D}(\hat{y}_{\downarrow\uparrow})\hat{x}_{\downarrow\uparrow})\right)&\text{Tr}\left(\mathcal{D}(\hat{y}_{\downarrow\uparrow})\hat{y}_{\downarrow\uparrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{y}_{\downarrow\uparrow})\hat{x}_{\wr\downarrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{y}_{\downarrow\uparrow})\hat{y}_{\wr\downarrow}\right)\\
\text{Tr}\left(\mathcal{D}(\hat{x}_{\wr\downarrow})\hat{x}_{\downarrow\uparrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{x}_{\wr\downarrow})\hat{y}_{\downarrow\uparrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{x}_{\wr\downarrow})\hat{x}_{\wr\downarrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{x}_{\wr\downarrow})\hat{y}_{\wr\downarrow}\right)\\
\text{Tr}\left(\mathcal{D}(\hat{y}_{\wr\downarrow})\hat{x}_{\downarrow\uparrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{y}_{\wr\downarrow})\hat{y}_{\downarrow\uparrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{y}_{\wr\downarrow})\hat{x}_{\wr\downarrow}\right)&\text{Tr}\left(\mathcal{D}(\hat{y}_{\wr\downarrow})\hat{y}_{\wr\downarrow}\right)
\end{matrix}\right).
\end{align}
Update matrices for the different state preparations are given in Appendix \ref{sec::MgammaUps}.
We now turn our attention to the noise matrix $N_{A \gamma}$, which we derive from the equations of motion for the second order moments. Integrating \erf{eq::CovarEvol2} over a small time $\Delta t$ yields
\begin{align}\label{eq::CovarEvolInt}
\expect{\Delta\hat{O}\Delta\hat{A}}_S(\Delta t)
=&\expect{\Delta\hat{O}\Delta\hat{A}}_S(0)
+\gamma_s\Delta t\sum_{\hat{X}\in\mathcal{S}}\text{Tr}(\mathcal{D}(\hat{o})\hat{x})\expect{\Delta\hat{X}\Delta\hat{A}}_S\\\notag
&+\gamma_s\Delta t\sum_{\hat{X}\in\mathcal{S}}\text{Tr}(\mathcal{D}(\hat{a})\hat{x})\expect{\Delta\hat{O}\Delta\hat{X}}_S\\\notag
&+\gamma_s\Delta t\text{Tr}(\mathcal{N}(\hat{o},\hat{a})\hat{n}_\uparrow)N_\uparrow
+\gamma_s\Delta t\text{Tr}(\mathcal{N}(\hat{o},\hat{a})\hat{n}_\downarrow)N_\downarrow\\\notag
&+\gamma_s\Delta t\text{Tr}(\mathcal{N}(\hat{o},\hat{a})\hat{n}_\wr)N_\wr.
\end{align}
When $\hat{O}$ and $\hat{A}$ are collective pseudo spins, the first three terms of this equation arise from the action of $M_{A\gamma}$. Note that these terms are moments of second order collective operators. The noise matrix contains the final three terms, which are moments of first order collective operators. Because a second order scattering event is required to populate the transfer state, the population $N_\wr$ is negligible. The noise matrix becomes \pagebreak
\begin{align}
N_{A\gamma}(\Delta t)=\gamma_s\Delta t \sum_{\psi\in\{\uparrow,\downarrow\}}N_\psi\times
\end{align}
\vspace{-8 mm}
\begingroup\makeatletter\def\f@size{10}\check@mathfonts
\def\maketag@@@#1{\hbox{\m@th\large\normalfont#1}}
\begin{align}
\notag&
\left(\begin{matrix}
\text{Tr}\left(\mathcal{N}(\hat{x}_{\downarrow\uparrow},\hat{x}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{x}_{\downarrow\uparrow},\hat{p}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{x}_{\downarrow\uparrow},\hat{x}_{\wr\downarrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{x}_{\downarrow\uparrow},\hat{p}_{\wr\downarrow})\hat{n}_\psi\right)\\
\text{Tr}\left(\mathcal{N}(\hat{p}_{\downarrow\uparrow},\hat{x}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{p}_{\downarrow\uparrow},\hat{p}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{p}_{\downarrow\uparrow},\hat{x}_{\wr\downarrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{p}_{\downarrow\uparrow},\hat{p}_{\wr\downarrow})\hat{n}_\psi\right)\\
\text{Tr}\left(\mathcal{N}(\hat{x}_{\wr\downarrow},\hat{x}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{x}_{\wr\downarrow},\hat{p}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{x}_{\wr\downarrow},\hat{x}_{\wr\downarrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{x}_{\wr\downarrow},\hat{p}_{\wr\downarrow})\hat{n}_\psi\right)\\
\text{Tr}\left(\mathcal{N}(\hat{p}_{\wr\downarrow},\hat{x}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{p}_{\wr\downarrow},\hat{p}_{\downarrow\uparrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{p}_{\wr\downarrow},\hat{x}_{\wr\downarrow})\hat{n}_\psi\right)&
\text{Tr}\left(\mathcal{N}(\hat{p}_{\wr\downarrow},\hat{p}_{\wr\downarrow})\hat{n}_\psi\right)
\end{matrix}\right)
\end{align}
\endgroup
Noise matrices for the various state preparations are given in Appendix \ref{sec::MgammaUps}.
Because the noise matrix, the mean spin and the total atom number all depend upon $N_\uparrow$ and $N_\downarrow$, we must derive a similar update matrix to evolve the populations in time. We focus only upon the populations of atoms in the fiducial and coupled states, since the population in the transfer state is negligible. Neglecting $N_\wr$ in \erf{eq::FirstOrderPopulations2} and integrating the equation of motion over a small time step $\Delta t$ yields
\begin{align}\label{eq::NumEvol3}
N_\psi(\Delta t)\approx N_\psi(0)+\gamma_s\Delta t\,\text{Tr}(\mathcal{D}(\hat{n}_\psi)\hat{n}_\uparrow)N_\uparrow(0)
+\gamma_s\Delta t\, \text{Tr}(\mathcal{D}(\hat{n}_\psi)\hat{n}_\downarrow)N_\downarrow(0).
\end{align}
Taking $\psi\in\{\uparrow,\downarrow\}$, we obtain the update matrix
\begin{align}\label{eq::popUpdate}
J_\gamma(\Delta t)=\mathbb{I}+\gamma_s\Delta t\left(\begin{matrix}
\text{Tr}(\mathcal{D}(\hat{n}_\uparrow)\hat{n}_\uparrow)&\text{Tr}(\mathcal{D}(\hat{n}_\uparrow)\hat{n}_\downarrow)\\
\text{Tr}(\mathcal{D}(\hat{n}_\downarrow)\hat{n}_\uparrow)&\text{Tr}(\mathcal{D}(\hat{n}_\downarrow)\hat{n}_\downarrow)
\end{matrix}\right).
\end{align}
The matrix $J_\gamma(\Delta t)$ evolves a vector of the populations forward in time by $\Delta t$,
\begin{align}
\left(\begin{matrix}N_\uparrow(t+\Delta t)\\N_\downarrow(t+\Delta t)\end{matrix}\right)
=J_\gamma(\Delta t)
\left(\begin{matrix}N_\uparrow(t)\\N_\downarrow(t)\end{matrix}\right).
\end{align}
Because the QND and double pass squeezing protocols have no appreciable effect upon the populations, $J_\gamma(\Delta t)$ is the only update matrix required to evolve $N_\uparrow$ and $N_\downarrow$.
\section{Simulating Gaussian Dynamics}\label{sec::GaussSim}
After deriving the covariance matrix and population updates for optical pumping, we can revisit the squeezing protocols of Chapter \ref{Sec:Protocols} to numerically evaluate their performance when decoherence is taken into account. To simulate both the coherent and dissipative evolution of the light and the atomic ensemble, we follow a procedure similar to Ref. \cite{MadMol}. By alternating over very small time increments the covariance matrix updates corresponding to a squeezing interaction and to optical pumping, we approximate the evolution of the system as it undergoes both processes simultaneously.
We first outline our approach for simulating the QND measurement protocol in the presence of optical pumping. Optical pumping acts on the covariance matrix by a Gaussian channel $\mathcal{P}$, which depends upon the mean update matrix and the noise matrix,
\begin{align}
\widetilde{\Sigma}(t+\Delta t)&=\mathcal{P}[\widetilde{\Sigma}(t)]\\\notag
&=M_\gamma(\Delta t)\widetilde{\Sigma}(t)M_\gamma^T(\Delta t)+N_\gamma(\Delta t).
\end{align}
Here, $M_\gamma$ and $N_\gamma$ are the update matrices defined in Eqs. (\ref{eq::entireM}) and (\ref{eq::entireN}), acting on the covariance matrix of both the light and ensemble. The QND measurement protocol acts upon the covariance matrix by a map $\mathcal{Q}$,
\begin{align}
\widetilde{\Sigma}(t+\Delta t)&=\mathcal{Q}[\widetilde{\Sigma}(t)]
\\\notag&=h[\hat{X}_y](S_F(\Delta t)\widetilde{\Sigma}(t)S_F^T(\Delta t))
\oplus\widetilde{\Sigma}_{0y}.
\end{align}
This map differs from the update described in \erf{eq::QNDupdate} in that after the $\hat{X}_y$ quadrature is measured via homodyne detection, a vacuum state covariance matrix of the light is appended to the covariance matrix of the ensemble. This represents a fresh pulse of light entering the experimental apparatus. In the next iteration of the protocol, this light will interact with the ensemble and subsequently undergo homodyne detection. The covariance matrix is propagated forward in time by alternating the maps $\mathcal{P}$ and $\mathcal{Q}$ for some number of iterations $n$,
\begin{align}
\widetilde{\Sigma}(n\Delta t)=(\mathcal{Q}\cdot\mathcal{P})^n[\widetilde{\Sigma}(0)].
\end{align}
Through this procedure, we obtain the covariance matrix of the light and ensemble after the system undergoes QND measurement for a time $n\Delta t$.
We next consider the phase-matching protocol. The first step in the phase-matching protocol is implementing the one-axis twisting interaction, which can be combined with optical pumping to form a Gaussian channel $\mathcal {T}$,
\begin{align}
\widetilde{\Sigma}(t+2\Delta t)&=\mathcal{T}[\widetilde{\Sigma}(t)]
\\\notag&=h[\hat{X}'_y]\left(S_F(\Delta t)\mathcal{P}[R_{\frac{\lambda}{4}}S_F(\Delta t) \mathcal{P}[\widetilde{\Sigma}(t)]S_F^T(\Delta t)R_{\frac{\lambda}{4}}^T]S_F^T(\Delta t)\right)\oplus\!\widetilde{\Sigma}_{0y}
\end{align}
As in the map $\mathcal{Q}$, a vacuum state covariance matrix of the light is appended to the covariance matrix of the ensemble after the quantum eraser, representing a fresh pulse of light. Similar to the coherent phase matching process depicted in \erf{eq::phasematchingTheta}, the one-axis twisting interaction is followed by a rotation of the ensemble given by the map $\mathcal{R}$,
\begin{align}
\mathcal{R}[\widetilde{\Sigma}(t)]=R(\theta_\text{opt})\widetilde{\Sigma}(t)R(\theta_\text{opt})^T.
\end{align}
For the coherent version of the phase matching protocol in \erf{eq::phasematchingTheta}, the ensemble is rotated by an angle $\xi/(2n)$ at each iteration. Here, the presence of noise from optical pumping changes the optimal angle for generating spin squeezing. A numerical optimization is used to determine the angle, $\theta_\text{opt}$, that maximizes squeezing at each iteration of the protocol. Like the case of QND measurement, we evolve the covariance matrix by alternating the maps $\mathcal{T}$ and $\mathcal{R}$ for some number of iterations $n$. This produces the covariance matrix at time $2n\Delta t$,
\begin{align}
\widetilde{\Sigma}(2n\Delta t)=(\mathcal{R}\cdot\mathcal{T})^n[\widetilde{\Sigma}(0)].
\end{align}
Note that, unlike the case of QND measurement, each iteration of the phase matching protocol takes a time $2\Delta t$ because of the double pass.
In addition to the covariance matrix, we must also simulate the evolution of the populations. Because they evolve only by optical pumping, the populations at time $t=n\Delta t$ are given by
\begin{align}
\left(\begin{matrix}N_\uparrow(n\Delta t)\\N_\downarrow(n\Delta t)\end{matrix}\right)
=J_\gamma(\Delta t)^n
\left(\begin{matrix}N_\uparrow(0)\\N_\downarrow(0)\end{matrix}\right)
\end{align}
for both the QND measurement and phase matching protocols. Here, $J_\gamma(\Delta t)$ is the update matrix given in \erf{eq::popUpdate}.
\section{Post-Processing Internal Spin Control}\label{sec::postprocessing2}
The final step in creating spin squeezing is post-processing via a partial isometry that acts identically on the internal spin state of each atom. In this section, we modify the partial isometries of Sections \ref{sec::postprocessing} and \ref{sec::IntSpinSqueeze} to accommodate the transfer state. This enables us to derive an expression for the squeezing parameter following the application of a post-processing partial isometry. When combined with the numerical techniques of the previous section, this expression determines the amount of squeezing achievable in the atomic ensemble.
Consider an initial state preparation with fiducial, coupled and transfer states $\ket{\uparrow},\;\ket{\downarrow}$ and $\ket{\wr}$. We seek to map the ensemble to a final state preparation with fiducial, coupled and transfer states $\ket{\uparrow'},\;\ket{\downarrow'}$ and $\ket{\wr'}$. To preserve the mean spin and relevant correlations, a fiducial state must always be mapped to a fiducial state, a coupled state to a coupled state and a transfer state to a transfer state. The partial isometry that implements this map is given by
\begin{align}\label{UArb}
\hat{U}_{\uparrow'}=\bigotimes_{i=1}^{N_A}\left(\ket{\uparrow'}\bra{\uparrow}_i+\ket{\downarrow'}\bra{\downarrow}_i+\ket{\wr'}\bra{\wr}_i\right).
\end{align}
This partial isometry leaves the total atom number invariant, but transforms the collective spin variance in \erf{eq::FzVarNewObs} as
\begin{align}
\Delta F_z^2&=v(\uparrow')^2\Delta\hat{X}_{\downarrow\uparrow}^{2}+2v(\uparrow')w(\uparrow')\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S
+w(\uparrow')^2\Delta\hat{X}_{\wr\downarrow}^{2},
\end{align}
where $w(\uparrow)$ and $v(\uparrow)$ are defined in Eqs. (\ref{eq::vDef}) and (\ref{eq::wDef}). In cases where it is not beneficial to preserve the transfer state, $\Delta\hat{X}_{\wr\downarrow}^{2}=N_\downarrow/2$ and $\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S=0$. The collective variance after the partial isometry is then
\begin{align}
\Delta F_z^2&=v(\uparrow')^2\Delta\hat{X}_{\downarrow\uparrow}^{2}+w(\uparrow')^2N_\downarrow/2.
\end{align}
The partial isometry also transforms the mean spin in \erf{eq::FxNewObs}, which is given by
\begin{align}
\expect{\hat{F}_x}&=\expect{\hat{f}_x}_{\uparrow'} N_\uparrow+\expect{\hat{f}_x}_{\downarrow'} N_\downarrow+\expect{\hat{f}_x}_{\wr'} N_\wr.
\end{align}
The final term in this expression is absent in cases where it is not beneficial to preserve the transfer state. With the transformed collective variance and mean spin, the metrological squeezing parameter becomes
\begin{align}\label{eq::unnormSqParam2}
\zeta_m=&2f(N_\uparrow+N_\downarrow+N_\wr)\times\\\notag
&\frac{v(\uparrow')^2\Delta\hat{X}_{\downarrow\uparrow}^{2}+2v(\uparrow')w(\uparrow')\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S
+w(\uparrow')^2\Delta\hat{X}_{\wr\downarrow}^{2}}{\left(\expect{\hat{f}_x}_{\uparrow'} N_\uparrow+\expect{\hat{f}_x}_{\downarrow'} N_\downarrow
+\expect{\hat{f}_x}_{\wr'} N_\wr\right)^2}
\end{align}
when the transfer state is preserved and
\begin{align}\label{eq::noTransfer}
\zeta_m=&2f(N_\uparrow+N_\downarrow)
\frac{v(\uparrow')^2\Delta\hat{X}_{\downarrow\uparrow}^{2}
+w(\uparrow')^2N_\downarrow/2}{\left(\expect{\hat{f}_x}_{\uparrow'} N_\uparrow+\expect{\hat{f}_x}_{\downarrow'} N_\downarrow\right)^2}
\end{align}
when the transfer state is eliminated.
The effect of post-processing upon the squeezing parameter is not as evident as it was in Sec. \ref{sec::postprocessing} in absence of decoherence. The squeezing parameter of \erf{parameters} can be recovered, however, by eliminating all except the leading order terms in Eqs. (\ref{eq::unnormSqParam2}) and (\ref{eq::noTransfer}). By preserving $\Delta\hat{X}_{\downarrow\uparrow}^{2}$ and $N_\uparrow$, we obtain
\begin{align}\label{eq::unnormSqParam3}
\zeta_m \approx \frac{2fN_\uparrow v(\uparrow')^2\Delta\hat{X}_{\downarrow\uparrow}^{2}}{\left(\expect{\hat{f}_x}_{\uparrow'}N_\uparrow\right)^2}= \frac{4fN_\uparrow^2 (\Delta\hat{f}_z)_{\uparrow'}\Delta\hat{X}_\downarrow^2}{\left(\expect{\hat{f}_x}_{\uparrow'}N_\uparrow\right)^2}=\zeta_m^{\uparrow'}\zeta_q.
\end{align}
While $\zeta_m$ and this approximate expression differ in the presence of optical pumping, we see that post-processing improves squeezing by mapping the majority of atoms, which remain in the fiducial state $\ket{\uparrow}$, to a new fiducial state, $\ket{\uparrow'}$, with more internal spin squeezing.
In sections \ref{sec::postprocessing} and \ref{sec::IntSpinSqueeze}, we examined partial isometries that map from an arbitrary state preparation to the $SCS$, Yurke and half-integer Yurke state preparations. From \erf{eq::unnormSqParam2}, we can determine the squeezing parameters that result
when these partial isometries are modified to accommodate the transfer state. In cases where the transfer state is eminimated, we obtain the squeezing parameter by taking $\Delta\hat{X}_{\wr\downarrow}^{2}\rightarrow N_\downarrow/2$, $\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S\rightarrow0$ and $N_\wr\rightarrow 0$. When the transfer state is preserved, mapping to the SCS preparation requires that $\ket{\wr}$ is mapped to $\ket{\wr_{SCS}}=-\ket{f,m_x=f-2}$, the transfer state in the $SCS$ preparation. Implementing this partial isometry transforms the squeezing parameter as
\begin{align}\label{eq::SCSSqParamOP}
\zeta_m=&2f(N_\uparrow+N_\downarrow+N_\wr)\times\\\notag
&\frac{f\Delta\hat{X}_{\downarrow\uparrow}^{2}+2\sqrt{f(2f-1)}\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S+(2f-1)\Delta\hat{X}_{\wr\downarrow}^{2}}{(fN_\uparrow+(f-1)N_\downarrow+(f-2)N_\wr)^2}.
\end{align}
Mapping to the Yurke preparation similarly requires that $\ket{\wr}$ be mapped to
\begin{align}
\ket{\wr_\text{y}}=\frac{\text{cos}\alpha}{\sqrt{2}}\ket{f,m_z=1}-\text{sin}\alpha\ket{f,m_z=0}+\frac{\text{cos}\alpha}{\sqrt{2}}\ket{f,m_z=-1},
\end{align}
which is the transfer state in the Yurke preparation. The squeezing parameter that results from this partial isometry is
\begin{align}\label{eq::YurkeSqParamOP}
\zeta_m=&2f(N_\uparrow+N_\downarrow+N_\wr)\times\\\notag
&\frac{\text{sin}^2\alpha\Delta\hat{X}_{\downarrow\uparrow}^{2}+2\text{cos}\alpha\text{sin}\alpha\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S+\text{cos}^2\alpha\Delta\hat{X}_{\wr\downarrow}^{2}}{f(f+1)\text{cos}^2\alpha\;\text{sin}^2\alpha \;N_\uparrow^2}.
\end{align}
The transfer state in the half-integer Yurke preparation takes a similar form,
\begin{align}
\ket{\wr_\text{hy}}\!=\!\frac{\text{cos}\alpha}{\sqrt{2}}\ket{f,m_z=3/2}\!-\!\text{sin}\alpha\ket{f,m_z=1/2}\!+\!\frac{\text{cos}\alpha}{\sqrt{2}}\ket{f,m_z=-1/2}.
\end{align}
The partial isometry to the half-integer Yurke preparation produces the squeezing parameter
\begin{align}\label{eq::YurkeLikeSqParamOP}
\zeta_m=&8f(N_\uparrow+N_\downarrow+N_\wr)\times\\\notag
&\frac{\text{sin}^2\alpha\Delta\hat{X}_{\downarrow\uparrow}^{2}+2\text{cos}\alpha\text{sin}\alpha\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S+\text{cos}^2\alpha\Delta\hat{X}_{\wr\downarrow}^{2}}
{(\sqrt{(f+3/2)(f-1/2)}+f+1/2)^2\;\text{sin}^2\alpha\;\text{cos}^2\alpha \;N_\uparrow^2}.
\end{align}
Recall that the squeezing of the Yurke and the half-integer Yurke states is maximal as $\alpha\rightarrow0$. Because the terms $\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S$ and $\Delta\hat{X}_{\wr\downarrow}^{\,2}$ are nonzero in the presence of optical pumping, however, the squeezing parameters in \erf{eq::YurkeSqParamOP} and \erf{eq::YurkeLikeSqParamOP} become infinite as $\alpha\rightarrow 0$. This occurs because atoms are optically pumped into the coupled states, which are infinitely anti-squeezed. In practice, when employing a partial isometry to the Yurke or half-integer Yurke state, we use a numerical search to determine the optimal nonzero $\alpha$. The optimal $\alpha$ maximizes the contribution of the squeezed fiducial state, while minimizing the contribution of the anti-squeezed coupled state.
\section{Results}\label{sec::HPResults}
The modified multilevel Holstein-Primakoff approximation along with the numerical methods introduced in this chapter enable us to determine the amount of squeezing achievable in the presence of optical pumping. The master equation describing optical pumping in the rotating frame and the SCS, cat, and $m_x=0\,$ state preparations satisfy the condition $\bra{\uparrow}\mathcal{D}(\ket{\uparrow}\bra{\downarrow})\ket{\wr}= 0$, ensuring that the modified multilevel Holstein-Primakoff approximation is valid. This allows us to quantitatively examine the influence of the fiducial state upon both coherent squeezing and decoherence of the ensemble. Using numerical techniques, we can also explore the impact of spin size and post-processing upon spin squeezing. In this section, we present a variety of numerical simulations showing achievable squeezing for various state preparations, spin sizes, post-processing partial isometries and squeezing protocols. In an effort to demonstrate what might be feasible in a laboratory setting, we have used experimentally realistic values of parameters relating to the light and atomic ensemble with $OD=300$, $N_A=10^6$, $N_L=3\times 10^8$, $\sigma_0/A=3\times 10^{-4}$ and $\Gamma/\Delta=10^{-3}$ for all simulations.
\subsection{Testing the modified multilevel HP}
Before presenting numerical results relating to achievable squeezing, we test the validity of the modified multilevel Holstein-Primakoff approximation, which enabled us to treat the ensemble as a Gaussian state on two modes. For the case of $f=1$, the exact differential equations describing the evolution of the mean spin, the collective variance and the populations under QND measurement and optical pumping form a closed set that can be easily solved through standard numerical methods.
From Ref. \cite{MadMol}, the collective variance under continuous QND measurement of $\hat{F}_z$ obeys the nonlinear differential equation
\begin{align}
\frac{d}{dt}\Delta F_z^2\big|_\text{QND}=-\kappa(\Delta F_z^2)^2.
\end{align}
Here, $\kappa=\chi^2N_L/\Delta t$ is the ``measurement strength". Further details about the differential formulation of continuous QND measurement will be provided in Chapter \ref{sec::Beyond}. The equation of motion for the variance under both continuous QND measurement and optical pumping follows from the evolution of second order moments under optical pumping given in \erf{eq::CovarEvol},
\begin{align}\label{eq::NumVar}
\frac{d}{dt}\Delta F_z^2=-\kappa(\Delta F_z^2)^2-\frac{2\gamma_s}{9}\Delta F_z^2+\frac{\gamma_s}{9}\expect{\hat{N}_1+\hat{N}_0+\hat{N}_{-1}}.
\end{align}
Here, we have decomposed the noise term as a sum of the population operators $\hat{N}_1$, $\hat{N}_0$ and $\hat{N}_{-1}$, defined as $\hat{N}_m=\sum_{i=1}^{N_A}\ket{f,m_x=m}\bra{f,m_x=m}_i$. Unlike the multilevel HP approximation, we treat these populations as operators, rather than c-numbers. Because continuous QND measurement negligibly effects the populations and mean spin, their equations of motion depend soley on optical pumping. From the equation of motion for first order moments under optical pumping given in \erf{eq::FirstOrderEvol}, the populations and mean spin satisfy
\begin{align}
\frac{d}{dt}\expect{\hat{N}_1}=-\frac{\gamma_s}{9}\expect{\hat{N}_1}+\frac{\gamma_s}{18}\expect{\hat{N}_0},
\end{align}
\begin{align}
\frac{d}{dt}\expect{\hat{N}_0}=-\frac{2\gamma_s}{9}\expect{\hat{N}_0}+\frac{\gamma_s}{18}\expect{\hat{N}_1}+\frac{\gamma_s}{18}\expect{\hat{N}_{-1}},
\end{align}
\begin{align}
\frac{d}{dt}\expect{\hat{N}_{-1}}=-\frac{\gamma_s}{9}\expect{\hat{N}_{-1}}+\frac{\gamma_s}{18}\expect{\hat{N}_0}
\end{align}
and
\begin{align}\label{eq::NumMeanSpin}
\frac{d}{dt}\expect{\hat{F}_x}=-\frac{\gamma_s}{6}\expect{\hat{F}_x}.
\end{align}
The variance and populations form a closed set of differential equations that can be numerically solved, while the mean spin is an exponential that can be solved analytically.
In Fig. \ref{fig::HPvsExact}, we compare the squeezing generated by the solution of Eqs. (\ref{eq::NumVar}) through (\ref{eq::NumMeanSpin}) with the squeezing predicted by a simulation of the QND measurement protocol utilizing the modified multilevel Holstein-Primakoff approximation. We consider an ensemble of atoms with $f=1$, initially prepared in a spin coherent state. The solutions deviate very little in the time vicinity of peak squeezing with the peak squeezing predicted by both models differing only by .07 dB. From this plot, it is evident that neglecting the coherences between the fiducial and transfer states, treating $\downarrow\uparrow$ and $\wr\!\downarrow$ as two commuting modes and approximating the populations as c-numbers has little effect upon the predicted squeezing.
The case of $f=1$ is unique in that each atomic spin is an actual qutrit, rather than being a qutrit embedded in a higher dimensional qudit. For atoms with larger spin, the microwave internal spin control introduced in Sec. \ref{sec::ControlOP} can be utilized to map all states in the $f$-manifold to the other ground hyperfine manifold, with the exception of the fiducial, coupled and transfer states. When this control is applied, higher spin atoms are in effect qutrits. The results in Fig. \ref{fig::HPvsExact} are, thus, indicative of the deviations that we expect between the two models for higher spins as well as for $f=1$.
\begin{figure}
\centering
\includegraphics[scale=.4]{HPvsExact.pdf}
\caption{Comparison of the modified multilevel Holstein-Primakoff approximation (red) with the numerical solution of the exact differential equations (blue) for the case of an ensemble with $f=1$ prepared in the $SCS$. The inverse of the metrological squeezing parameter is plotted against time in units of the scattering rate, $\gamma_s$. The maximal squeezing predicted by these models differs by .07dB. At the end of the plotted time interval, the models differ by .44 dB. }\label{fig::HPvsExact}
\end{figure}
\subsection{Performance of Squeezing Protocols for Different State Preparations}
The initial fiducial state of the ensemble has an enormous impact on the coherent generation of squeezing and decoherence due to optical pumping. The interatomic entanglement generated by the squeezing protocols increases with the variance of the fiducial state. Other properties of the fiducial state, such as the spin flip rate and atom loss rate, govern the robustness of the ensemble to optical pumping. The modified multilevel HP approximation and the numerical methods developed in this chapter allow us to assess the performance of the SCS, cat, and $m_x=0$ preparations in the presence of optical pumping.
\begin{figure}
\centering
\includegraphics[scale=.35]{StateStats.pdf}
\caption{The variance of the fiducial state, the collective spin coupling constant, the spin flip rate of the fiducial state and the total loss rate from the $f$ manifold for the SCS, $m_x=0$ and cat state preparations. Note that the spin flip and loss rates for the cat state are for $f\geq 1$. When $f=1/2$, the spin flip and loss rates for the cat are identical to those of the SCS. The spin flip and loss rates are calculated with the master equation in the rotating frame given in \erf{eq::MasterRotating}.}\label{fig::StateStats}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=.39]{StateStatPlots.pdf}
\caption{Dependence of relevant quantities for the SCS and $m_x=0$ state preparations upon $f$. Plotted are the quantities $9\xi(OD\,\gamma_s\tau)^{-1}$ (purple), $\Gamma_\text{flip}\gamma_s^{-1}$ (light blue) and $\Gamma_\text{loss}\gamma_s^{-1}$ (red).}\label{fig::StateStatPlots}
\end{figure}
To guide our intuition, the table in Fig. \ref{fig::StateStats} lists quantities relevant to the performance of each state preparation. These quantities - the variance of the fiducial state, the collective spin coupling constant, the spin flip rate of the fiducial state and the total loss rate of atoms from the $f$ manifold - are essential to interpreting the numerical results. Recall that the squeezing generated by the Faraday interaction increases with the collective spin coupling constant, $\xi$, which is proportional to the projection noise fluctuations of the fiducial state, $(\Delta f_z^2)_\downarrow$. Spin flips, the most damaging optical pumping processes, cause the mean spin to decay and inject noise back into the system, thereby increasing the variance of the collective spin. While loss can result in an even greater decay of the mean spin, it is less damaging to spin squeezing because it also causes the collective variance to decay. Each of these quantities depend not only on the state preparation, but on the size of the hyperfine spin, $f$. Figure \ref{fig::StateStatPlots} provides a visual depiction of how these quantities vary with $f$.
We first examine the performance of the state preparations for $f=4$, corresponding to the larger ground hyperfine manifold of cesium. Figures \ref{fig::4Plots} (a) and (b) show the squeezing and the reduction in the collective variance generated by the double-pass phase matching protocol combined with a partial isometry to the SCS preparation. The same quantities for the QND measurement protocol combined with a partial isometry to the SCS preparation are plotted in (c) and (d). Also shown in Fig. \ref{fig::4Plots} (e) is the decay of the mean spin, which is governed by optical pumping alone. Unsurprisingly, phase matching outperforms the QND measurement protocol. Our primary interest is the performance of the state preparations, however. The cat state has the largest fiducial state projection noise, but also the largest rate of spin flips. In addition, for this preparation, there is no transfer of coherence to mitigate the effect of spin flips. Consequently, the protocols rapidly generate squeezing on the cat preparation, but this squeezing also decays at a rapid rate. The increase in $\Delta F_z^2$ for the cat preparation in plots (b) and (d) at longer times demonstrates the deleterious effect of spin flips. In contrast, the collective variances of the SCS and $m_x=0$ preparations appear to asymptote; the excess noise injection is balanced by squeezing and loss. Interestingly, the cat preparation's large rate of spin flips relative to loss make its mean spin the most robust to optical pumping. While a ``lost" atom contributes nothing to the mean spin, an atom that undergoes a spin flip from the fiducial to the coupled state contributes $\expect{\hat{f}_x}_{\downarrow'}$, where $\ket{\downarrow'}$ is the state to which the coupled state is mapped by the post-processing partial isometry. For a partial isometry to the SCS preparation, $\expect{\hat{f}_x}_{\downarrow'}=f-2$, meaning that a spin flip event reduces the mean spin by 2 as opposed to $f$ in the case of loss. Although the $m_x=0$ preparation has a smaller enhancement of the fiducial state projection noise compared to the cat, its reduced rate of spin flips make it more robust to optical pumping. Accordingly, the $m_x=0$ preparation outperforms both the cat and the SCS preparation, which has the smallest initial projection noise variance of the three state preparations. For the phase matching and QND protocols, the $m_x=0$ preparation achieves a peak squeezing of 13.4 dB and 9.3 dB, respectively. Curiously, the $m_x=0$ preparation has a large rate of loss, which causes its mean spin to decay the fastest due to optical pumping. The enhanced squeezing of the collective variance for the $m_x=0$ preparation compensates for this, however. The SCS preparation has the smallest fiducial state projection noise and spin flip rate. As a consequence, the SCS preparation is much more robust to optical pumping, but its peak squeezing is significantly smaller than that of the cat or $m_x=0$ because the entangling effect of the Faraday interaction is weaker.
To illustrate the role of the spin size in determining squeezing, Fig. \ref{fig::2Plots} replicates the same plots in Fig. \ref{fig::4Plots} with $f=2$ instead of $f=4$. As in the $f=4$ case, the $m_x=0$ preparation outperforms both the SCS and the cat preparations, attaining a peak squeezing of 13.7 dB for the double-pass phase matching protocol and 9.5 dB for the QND protocol. Unlike the $f=4$ case, the performances of the SCS and the cat preparations are comparable for $f=2$. For the phase matching protocol, the SCS and cat preparations achieve nearly identical peak squeezing values of 11.0 dB and 11.1dB, respectively, though the peak value is reached much faster for the cat state. For the QND measurement protocol, the SCS and cat preparations reach 7.8 dB and 8.1dB, respectively. The improved performance of the SCS preparation relative to the cat can be explained by the behavior of the mean spin. Note that the mean spin of the SCS preparation in Fig. \ref{fig::2Plots} (e) decays the least, followed by the cat and the $m_x=0$ state. The order is different in the $f=4$ case, in which the mean spin of the cat decays the least, followed by the SCS and the $m_x=0$ state. Whereas the loss rate of the SCS preparation was substantially higher than the cat state preparation for $f=4$, their loss rates are nearly equal when $f=2$. The spin flip rate for the cat at $f=2$ is much larger, however. This causes the mean spin of the cat state to decay faster. Consequently, the squeezing produced by the SCS preparation is much improved when $f=2$. Despite this, the SCS preparation is still the lowest performer. Also similar to the $f=4$ case, the cat state preparation is the least robust to optical pumping with its squeezing decaying the quickest. The SCS is the most robust, followed by the $m_x=0$ state. For both $f=4$ and $f=2$, the $m_x=0$ state preparation strikes the most optimal balance between enhanced squeezing due to the variance of the fiducial state and robustness to optical pumping.
\begin{figure}[H]
\centering
\includegraphics[scale=.6]{Spin4fPlots062014.pdf}
\caption{Performance of the state preparations for $f=4$, SCS (green), cat (black) and $m_x=0$ (blue). For post-processing, a partial isometry mapping each state preparation to the SCS preparation was applied to the ensemble. Plot (a) shows the squeezing generated by each state preparation for the phase matching protocol. Plot (b) shows the corresponding variance, normalized by the variance of the spin coherent state. Plot (c) shows the squeezing generated by each state preparation for the QND measurement protocol with the corresponding variance, normalized by the variance of the spin coherent state shown in plot (d). For both squeezing protocols, the decay of the mean spin normalized by the mean spin of the spin coherent state is shown in plot (e). }\label{fig::4Plots}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[scale=.6]{Spin2fPlots062014.pdf}
\caption{Performance of the state preparations for $f=2$, SCS (green), cat (black) and $m_x=0$ (blue). For post-processing, a partial isometry mapping each state preparation to the SCS preparation was applied to the ensemble. Plot (a) shows the squeezing generated by each state preparation for the phase matching protocol. The corresponding variance, normalized by the variance of the spin coherent state, is plotted in (b). Plot (c) shows the squeezing generated by each state preparation for the QND measurement protocol with the corresponding variance, normalized by the variance of the spin coherent state shown in plot (d). For both squeezing protocols, the decay of the mean spin normalized by the mean spin of the spin coherent state is shown in plot (e).}\label{fig::2Plots}
\end{figure}
\subsection{Effect of Partial Isometries}
To convert the interatomic entanglement generated by the Faraday interaction into metrologically relevant squeezing, we have focused on internal spin control via two partial isometries. The first partial isometry, which maps an arbitrary state preparation to the SCS preparation, creates squeezing that depends upon interatomic entanglement alone, since $\zeta_m^{\uparrow_{SCS}}=1$. The second partial isometry maps an arbitrary state preparation to the Yurke or half-integer Yurke preparation, for which $\zeta_m^{\uparrow_{\text{y}}}<1$ and $\zeta_m^{\uparrow_{\text{hy}}}<1$. Because the Yurke states are squeezed, the spin squeezing generated by these partial isometries depends on internal spin squeezing as well as interatomic entanglement.
Fig. \ref{fig::PICompare} examines the effect of internal spin squeezing on the collective spin squeezing achieved by the different state preparations. For $f=4$, Fig. \ref{fig::PICompare} shows the squeezing generated by QND measurement combined with either a partial isometry to the SCS preparation or a partial isometry to the Yurke preparation. This plot is generated using the formulas in Eqs. (\ref{eq::SCSSqParamOP}) and (\ref{eq::YurkeSqParamOP}), which give the squeezing that results from a Faraday-effect squeezing protocol combined with a partial isometry in the presence of optical pumping. The parameter $\alpha$ in \erf{eq::YurkeSqParamOP}, which determines the squeezing of the Yurke state, is chosen by a numerical optimization. In absence of decoherence, the partial isometry to the Yurke preparation produces a multiplicative enhancement of the squeezing parameter by $(f+1)^{-1}$ as compared to the SCS preparation. When the inverse of the squeezing parameter is plotted in dB for $f=4$, as in Fig. \ref{fig::PICompare}, this enhancement appears as an increase in the amount of squeezing by 7.0 dB.
The dashed lines, which correspond to the Yurke partial isometry, are translated upward by nearly 7.0 dB at small times. The upward translation decreases, however, as time progresses and the state decoheres due to optical pumping. Nonetheless, as Fig. \ref{fig::PICompare} attests, collective spin squeezing is enhanced substantially by the internal spin squeezing resulting from the partial isometry to the Yurke preparation.
\begin{figure}
\centering
\includegraphics[scale=.4]{PICompare.pdf}
\caption{The performance of the state preparations SCS (green), cat (black) and $m_x=0$ (blue) for $f=4$ under the QND measurement protocol with optical pumping. Squeezing is plotted versus time in units of scattering rate. Here, two different partial isometries are used to convert interatomic entanglement into spin squeezing, a map to the SCS preparation (solid), a map to the Yurke preparation (dashed). }\label{fig::PICompare}
\end{figure}
\subsection{Scaling of Squeezing with $f$}
Figure \ref{fig::fPlots} further examines the dependence of squeezing on the spin size, $f$. Plotted in Fig. \ref{fig::fPlots} is the peak squeezing generated by the QND and phase matching protocols for partial isometries to both the SCS and Yurke preparations. We focus first on Fig. \ref{fig::fPlots} (a) and (c), which show the peak squeezing of the QND and phase matching protocols on different state preparations with a partial isometry to the SCS preparation. Plots (a) and (c) exhibit similar behavior, except at small values of $f$. Perhaps most significantly, both plots attain their maximum value of squeezing at $f=2$, not $f=1/2$, which one might naively expect given that the collective spin coupling constant decreases monotonically with increasing $f$. The $m_x=0$ preparation performs the best for both the QND and phase matching protocols, peaking at $f=2$ and then gradually declining due to the decreased collective spin coupling constant. The $m_x=0$ preparation likely peaks at $f=2$ because it is the smallest spin for which there is a transfer of coherence, giving the ensemble additional robustness to decoherence. A smaller spin is favorable since the collective spin coupling constant decreases with increasing spin. For both protocols, the performance of the $SCS$ preparation falls off the most rapidly, beginning at $f=1$. This occurs because the collective spin coupling constant of the SCS preparation, which is proportional to $1/f$, has the greatest decline with increasing $f$. Although the spin flip rate of the $SCS$ preparation also decreases with $1/f$, this does not compensate for loss in the coherent interaction strength. For the cat preparation, the collective spin coupling constant, the rate of loss, and the rate of spin flips are all constant with $f$. This is reflected by the relative stability of the cat preparation after $f=1$.
Fig. \ref{fig::fPlots} (b) and (d) depict the dependence of the peak squeezing generated by the QND measurement and phase matching protocols on different state preparations combined with internal spin squeezing. The internal spin squeezing is generated by a partial isometry to the Yurke preparations. The comparative performances of the state preparations in plots (b) and (d) is the same as for (a) and (c). A significant difference between these plots, however, is that spin squeezing largely improves as $f$ increases in plots (b) and (d). Although the collective spin coupling constant decreases with increasing $f$, internal spin squeezing generates a greater amount of collective spin squeezing for atoms with larger $f$. Internal spin squeezing more than compensates for the loss in coherent interaction strength as $f$ increases.
\begin{figure}[H]
\centering
\includegraphics[scale=.55]{fPlots.pdf}
\caption{Peak squeezing vs. spin size for SCS (green), cat (black) and $m_x=0$ (blue). These plots show the performance of the QND and phase matching squeezing protocols, as quantified by the inverse of the metrological squeezing parameter, versus the size of the hyperfine spin $f$. The topmost plots show the performance of the QND squeezing protocol with (a) a partial isometry to the SCS preparation and (b) a partial isometry to the Yurke preparation for integer $f$ and a partial isometry to the half-integer Yurke preparation for half-integer $f$. The bottom plots show the performance of the phase matching protocol with (c) a partial isometry to the SCS preparation and (d) a partial isometry to the Yurke preparation for integer $f$ and a partial isometry to the half-integer Yurke preparation for half-integer $f$.}\label{fig::fPlots}
\end{figure}
\subsection{Effect of the Transfer State}
\begin{figure}
\centering
\includegraphics[scale=.48]{TransferEffect.pdf}
\caption{Performance of state preparations with and without the transfer state for the QND measurement protocol and $f=4$. A partial isometry mapping to the SCS preparation is applied to the $m_x=0$ preparation. Plot (a) shows the squeezing generated by the SCS state preparation for $f=4$ when the transfer state is preserved (green solid) and when the transfer state is eliminated via internal spin control (light green dashed). Preserving the transfer state improves spin squeezing by 1.4 dB. Plot (b) shows the squeezing generated by the $m_x=0$ state preparation for $f=4$ when the transfer state is preserved (blue solid) and when the transfer state is eliminated (light blue dashed). For the $m_x=0$ state preparation, spin squeezing is improved by 1.9 dB when the transfer state is preserved.}\label{fig::transfer}
\end{figure}
Sec. \ref{sec::TransferState} introduced the counterintuitive idea that preserving the transfer state, rather than removing it by mapping it to the other ground hyperfine manifold with internal spin control, can improve spin squeezing in the presence of optical pumping. This idea is explored in Fig. \ref{fig::transfer}, which compares the squeezing generated by two state preparations under QND measurement and optical pumping when the transfer state is preserved and when it is eliminated. For $f=4$, the plots in Fig. \ref{fig::transfer} feature the SCS preparation and the $m_x=0$ preparation with a final partial isometry to the SCS preparation. From Fig. \ref{fig::transfer}, it is evident that preserving the transfer state has a significant effect upon the achievable spin squeezing. In the case of the SCS preparation, shown in Fig. \ref{fig::transfer} (a), preserving the transfer state improves peak squeezing by 1.4 dB. For the $m_x=0$ preparation in Fig. \ref{fig::transfer} (b), preserving the transfer state results in a 1.9 dB improvement of spin squeezing. In addition to increasing the magnitude of the peak squeezing, the presence of the transfer state also reduces the rate at which spin squeezing decays. Overall, retaining the transfer state greatly improves the performance of both state preparations in the presence of optical pumping.
\chapter{Optimal Spin States for QND Squeezing}\label{sec::Beyond}
Thus far, the techniques we have used to determine the achievable spin squeezing in the atomic ensemble have been suited only to particular fiducial states. In Chapters \ref{sec::ALinterface} and \ref{sec::OpticalPumping}, we derived the coupled and transfer states from the fiducial state by making the assumption that $\expect{\hat{f}_z}_\uparrow=\expect{\hat{f}_z}_\downarrow=\expect{\hat{f}_z}_\wr=0$.
The modified version of the multilevel Holstein-Primakoff approximation developed in Chapter \ref{sec::ModHPCovar} was applicable to ensembles prepared in fiducial states that satisfied the condition in \erf{eq::ModeCondition}, ensuring that pairwise coherences involving the fiducial and transfer states did not develop. While the SCS, cat, and $m_x=0$ state preparations satisfy both $\expect{\hat{f}_z}_\uparrow=\expect{\hat{f}_z}_\downarrow=\expect{\hat{f}_z}_\wr=0$ and \erf{eq::ModeCondition}, this is not necessarily true of an arbitrary fiducial state. In this chapter we develop techniques to model the effects of squeezing and optical pumping for any fiducial state.
The techniques of this chapter rely on expressing a squeezing protocol in differential form. The double pass squeezing protocols, such as phase matching, are not readily converted to differential form. Because the only way to model these protocols is as a series of updates upon the covariance matrix of a Gaussian state, we must rely upon the modified multilevel Holstein-Primakoff approximation of the previous chapter. A relatively simple differential form exists for the QND measurement protocol, however. The effects of QND measurement and optical pumping can, thus, be modeled more easily on any fiducial state. We utilize this method to perform a numerical search over the space of all fiducial states in order to find the state preparation that maximizes squeezing in the presence of optical pumping.
\section{Coupled and Transfer States for Arbitrary\\ Fiducial States}
Determining squeezing when the ensemble is prepared in an arbitrary fiducial state requires that we generalize the definitions of the coupled and transfer states to the case where $\expect{\hat{f}_z}_\uparrow$, $\expect{\hat{f}_z}_\downarrow$ and $\expect{\hat{f}_z}_\wr$ are not necessarily zero. In Sec. \ref{sec::MultiHPEnsemble}, we derived the coupled state by considering the effect of the Faraday interaction on a single atom prepared in the fiducial state. Here, we do the same thing, but relax the assumption that $\expect{\hat{f}_z}_\uparrow=0$. For $\chi<<1$,
\begin{align}\label{eq::CoupledNo0Mean}
e^{-i\chi\hat{S}_3\hat{f}_z}\ket{\uparrow}&\approx(\mathbb{I}-i\chi\hat{S}_3\hat{f}_z)\ket{\uparrow}\\\notag
&=\ket{\uparrow}-i\chi\hat{S}_3\expect{\hat{f}_z}_\uparrow\ket{\uparrow}
-i\chi\hat{S}_3(\hat{f}_z-\expect{\hat{f}_z}_\uparrow)\ket{\uparrow}.
\end{align}
The Faraday interaction maps the fiducial state back to itself and to an orthogonal state, proportional to $(\hat{f}_z-\expect{\hat{f}_z}_\uparrow)\ket{\uparrow}=\Delta\hat{f}_z\ket{\uparrow}$. This orthogonal state is, by definition, the coupled state. From \erf{eq::CoupledNo0Mean}, the coupled state is given by
\begin{align}\label{eq::CoupledDefNo0}
\ket{\downarrow}=&\frac{\Delta\hat{f}_z\ket{\uparrow}}{\sqrt{(\Delta f_z^2)_\uparrow}}\\\notag
=&\frac{\Delta\hat{f}_z\ket{\uparrow}}{v(\uparrow)/\sqrt{2}}.
\end{align}
The transfer state arises by considering the effect of the Faraday interaction on the coupled state, which depends on
\begin{align}\label{eq::TransferNo0Mean}
\hat{f}_z\ket{\downarrow}=\expect{\hat{f}_z}_\downarrow\ket{\downarrow}+\sqrt{(\Delta f_z^2)_\uparrow}\ket{\uparrow}+
\Big((\hat{f}_z-\expect{\hat{f}}_\downarrow)\ket{\downarrow}-\sqrt{(\Delta f_z^2)_\uparrow}\ket{\uparrow}\Big).
\end{align}
The Faraday interaction maps the coupled state to a superposition of itself, the fiducial state and another state orthogonal to both the coupled and the fiducial states, proportional to $(\hat{f}_z-\expect{\hat{f}}_\downarrow)\ket{\downarrow}-\sqrt{(\Delta f_z^2)_\uparrow}\ket{\uparrow}$. This orthogonal state is, by definition, the transfer state. From \erf{eq::TransferNo0Mean}, the transfer state is given by
\begin{align}\label{TransferStateNo0}
\ket{\wr}&=\frac{1}{\sqrt{(\Delta f_z^2)_\downarrow-(\Delta f_z^2)_\uparrow}}\left(\Delta\hat{f}_z\ket{\downarrow}-\sqrt{(\Delta f_z^2)_\uparrow}\ket{\uparrow}\right)\\\notag
&=\frac{1}{w(\uparrow)}\left(\sqrt{2}\Delta\hat{f}_z\ket{\downarrow}-v(\uparrow)\ket{\uparrow}\right).
\end{align}
From Eqs. (\ref{eq::CoupledDefNo0}) and (\ref{TransferStateNo0}), we see that the definitions of the coupled and transfer states are readily generalized to the case of nonzero $\expect{\hat{f}_z}_\uparrow$ and $\expect{\hat{f}_z}_\downarrow$.
Using the generalized definitions of the coupled and transfer states, the internal spin component $\hat{f}_z$ can be expressed in the embedded qutrit basis as
\begin{align}\label{eq::fzno0}
\hat{f}_z=&\sqrt{(\Delta f_z^2)_\uparrow}\left(\ket{\uparrow}\bra{\downarrow}+\ket{\downarrow}\bra{\uparrow}\right)+\expect{\hat{f}_z}_\uparrow\ket{\uparrow}\bra{\uparrow}+\expect{\hat{f}_z}_\downarrow\ket{\downarrow}\bra{\downarrow}
\\\notag&+\sqrt{(\Delta f_z^2)_\downarrow-(\Delta f_z^2)_\uparrow}\left(\ket{\downarrow}\bra{\wr}+\ket{\wr}\bra{\downarrow}\right)+\expect{\hat{f}_z}_\wr\ket{\wr}\bra{\wr}.
\end{align}
Note that the second term contains coherences between the fiducial and the coupled state and the fourth term contains coherences between the coupled and transfer state, both of which are responsible for generating negative correlations in spin squeezed states. Treating the expectation values $\expect{\hat{f}_z}_\uparrow$, $\expect{\hat{f}_z}_\downarrow$ and $\expect{\hat{f}_z}_\wr$ as nonzero contributes additional terms to $\hat{f}_z$, but preserves the essential coherences.
\subsection{Collective Spin Variance}
In Chapter \ref{sec::ModHPCovar}, we determined the collective spin variance $\Delta F_z^2$ by tracking the evolution of covariances between the collective pseudospin operators. Here, we show that the exact same thing is possible with the more general definitions of the coupled and transfer states. Due to the presence of the nonzero means in this expression, the collective spin depends upon both the collective pseudospins and the populations. From \erf{eq::fzno0},
\begin{align}
\hat{F}_z=v(\uparrow)\hat{X}_{\downarrow\uparrow}+w(\uparrow)\hat{X}_{\wr\downarrow}
+\expect{\hat{f}_z}_\uparrow N_\uparrow+\expect{\hat{f}_z}_\downarrow N_\downarrow+\expect{\hat{f}_z}_\wr N_\wr.
\end{align}
Because the populations are treated as c-numbers, however, they cancel from the uncertainty of the collective spin,
\begin{align}
\Delta\hat{F}_z=\hat{F}_z-\expect{\hat{F}_z}\approx v(\uparrow)\Delta\hat{X}_{\downarrow\uparrow}+w(\uparrow)\Delta\hat{X}_{\wr\downarrow}.
\end{align}
The collective spin variance,
\begin{align}
\Delta F_z^2=v(\uparrow)^2\Delta X_{\downarrow\uparrow}^2+
2v(\uparrow)w(\uparrow)\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S+
w(\uparrow)^2\Delta X_{\wr\downarrow}^2,
\end{align}
takes the exact same form as \erf{eq::FzVarNewObs}.
\subsection{Faraday Interaction}
Because the Faraday interaction depends upon the collective spin $\hat{F}_z$, it takes a different form when the means $\expect{\hat{f}_z}_\uparrow$, $\expect{\hat{f}_z}_\downarrow$, and $\expect{\hat{f}_z}_\wr$ are nonzero. In terms of the collective pseudospins and populations of the ensemble,
\begin{align}\label{eq::HPfaraday}
\hat{H}=&\frac{\hbar\chi}{\Delta t}\hat{S}_3\Big(v(\uparrow)\hat{X}_{\downarrow\uparrow}+w(\uparrow)\hat{X}_{\wr\downarrow}\Big)\\\notag&+\frac{\hbar\chi}{\Delta t}\hat{S}_3\Big(\expect{\hat{f}_z}_\uparrow N_\uparrow+\expect{\hat{f}_z}_\downarrow N_\downarrow+\expect{\hat{f}_z}_\wr N_\wr\Big).
\end{align}
The first term in the expression is the familiar analogue of the Faraday interaction in the phase plane of the ensemble. The final term in this expression is absent when we assume that $\expect{\hat{f}_z}_\uparrow=\expect{\hat{f}_z}_\downarrow=\expect{\hat{f}_z}_\wr=0$. Even when these expectations are nonzero, this term has little effect on the state of the ensemble or the entanglement between the light and atoms. Because the populations are treated as c-numbers, the final term commutes with all ensemble observables and, thus, has no influence on the ensemble dynamics. For the light, this term generates a rotation of the light's polarization in the Poincar\'{e} sphere about $\hat{S}_3$. Because there is no projection noise contribution from the populations, the fluctuations in polarization induced by the ensemble on the light are unchanged by the presence of the final term. This term, therefore, has no role in creating entanglement between the light and ensemble. The final term also has no influence over preexisting entanglement. In the phase plane picture, the final term generates a rotation of the light's polarization, or equivalently a translation in phase space, that affects only the first order moments of observables. Because this term does not alter the variances or covariances, the entanglement between the light and ensemble is unaffected. The second term can, therefore, be discarded without any impact on spin squeezing. The Faraday interaction takes the same form whether or not $\expect{\hat{f}_z}_\uparrow$, $\expect{\hat{f}_z}_\downarrow$, and $\expect{\hat{f}_z}_\wr$ are zero, producing identical dynamics.
\section{Differential Form of QND Measurement}
Previously in Sections \ref{sec::QNDmeas}, and \ref{sec::GaussSim}, we formulated the squeezing by QND measurement protocol through a series of updates on the covariance matrix of the light and ensemble. As previously noted, this treatment is only valid for fiducial states that satisfy \erf{eq::ModeCondition}. The dynamics of the ensemble state under QND measurement can also be written in the form of a stochastic master equation (SME) that is valid for any fiducial state. Combined with the generalizations of the coupled and transfer states from the previous section, this formalism can treat an ensemble prepared in any fiducial state.
Under continuous measurement of $\hat{F}_z$, the SME describing the evolution of the ensemble state is given by
\begin{align}\label{eq::planeSME}
d\hat{\rho}\big|_{\text{QND}}=\sqrt{\frac{\kappa}{4}}\mathcal{H}(\hat{\rho})dW+\frac{\kappa}{8}\mathcal{L}(\hat{\rho})dt,
\end{align}
where $\kappa=\chi^2\dot{N_L}$ is the measurement strength. Measurement backaction on the ensemble is taken into account by the superoperator
\begin{align}
\mathcal{H}(\hat{\rho})=\hat{F}_z\hat{\rho}+\hat{\rho}\hat{F}_z-2\expect{\hat{F}_z}\hat{\rho}.
\end{align}
The Lindblad dissipator,
\begin{align}
\mathcal{L}(\hat{\rho})=[\hat{F}_z,[\hat{\rho},\hat{F}_z]],
\end{align}
describes the decoherence of the ensemble from light that goes unmeasured after interacting with the atoms.
From the SME, we can determine the evolution of a covariance $\expect{\Delta\hat{O}\Delta\hat{A}}_S$ under continuous QND measurement of $\hat{F}_z$.
We first examine the evolution of the covariance due to measurement backaction,
\begin{align}\label{eq::backaction}
d\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\mathcal{H}}=&\frac{\sqrt{\kappa}}{4}\expect{\mathcal{H}(\hat{O}\hat{A}+\hat{A}\hat{O})}dW\\\notag
&-\sqrt{\frac{\kappa}{4}}\expect{\mathcal{H}(\hat{O})}\expect{\hat{A}}dW-\sqrt{\frac{\kappa}{4}}\expect{\hat{O}}\expect{\mathcal{H}(\hat{A})}dW\\\notag
&-\frac{\kappa}{4}\expect{\mathcal{H}(\hat{O})}\expect{\mathcal{H}(\hat{A})}dt.
\end{align}
Note that by the rules of It$\overline{\text{o}}$ calculus, differentials must be taken to second order \cite{JacSte06}. When $\hat{F}_z$ is written in the operator basis of collective pseudo-spins and populations in Eqs. (\ref{eq::Xdownup}) through (\ref{eq::PopWr}), all terms involving the populations vanish from this expression since they are treated as c-numbers. When $\hat{O}$ and $\hat{A}$ are collective pseudo spins, the first term in \erf{eq::backaction} contains third order moments of the collective pseudo spins. For an ensemble initially prepared in $\ket{\uparrow}^{\otimes N_A}$, the collective pseudo spins $\hat{X}_{\downarrow\uparrow}$, $\hat{Y}_{\downarrow\uparrow}$, $\hat{X}_{\uparrow\wr}$ and $\hat{Y}_{\uparrow\wr}$ are Gaussianly distributed. The collective pseudo spins $\hat{X}_{\wr\downarrow}$ and $\hat{Y}_{\wr\downarrow}$ are approximately Gaussianly distributed after population accumulates in the coupled state. Because the third order moments contain only Gaussian operators, they can be decomposed in terms of first and second order moments as
\begin{align}\label{eq::3rdGaussDecomp}
\frac{1}{6}\sum_{\text{perm}}\expect{\hat{O}\hat{A}\hat{Q}}=&\expect{\Delta\hat{O}\Delta\hat{Q}}_S\expect{\hat{A}}+
\expect{\Delta\hat{A}\Delta\hat{Q}}_S\expect{\hat{O}}+\expect{\Delta\hat{O}\Delta\hat{A}}_S\expect{\hat{Q}}
\\\notag&+\expect{\hat{O}}\expect{\hat{A}}\expect{\hat{Q}}.
\end{align}
The sum in the left hand side of this expression is taken over all permutations of $\hat{O}$, $\hat{A}$ and $\hat{Q}$. When the third order moments are decomposed under the Gaussian approximation, the equation of motion for the covariance reduces to the relatively simple expression
\begin{align}\label{eq::finalEoMQND7}
d\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\mathcal{H}}=&-\kappa\left(v(\uparrow)\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{O}}_S
+w(\uparrow)\expect{\Delta\hat{X}_{\wr\downarrow}\Delta\hat{O}}_S\right)\\\notag&
\times\left(v(\uparrow)\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{A}}_S
+w(\uparrow)\expect{\Delta\hat{X}_{\wr\downarrow}\Delta\hat{A}}_S\right)dt.
\end{align}
This differential equation nonlinearly couples the covariances of the collective pseudo-spins.
We next turn our attention to the evolution of the covariance due to the dissipative term in the SME,
\begin{align}
d\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\mathcal{L}}=&\frac{\kappa}{16}\expect{\mathcal{L}(\hat{O}\hat{A}+\hat{A}\hat{O})}dt-\frac{\kappa}{8}\expect{\mathcal{L}(\hat{O})}\expect{\hat{A}}dt\\\notag&-\frac{\kappa}{8}\expect{\hat{O}}\expect{\mathcal{L}(\hat{A})}dt\\\notag
=&\frac{\kappa}{8}\expect{\{[\hat{F}_z,\hat{O}],[\hat{A},\hat{F}_z]\}}dt-\frac{\kappa}{8}\expect{\Delta\hat{O}\Delta[\hat{F}_z,[\hat{A},\hat{F}_z]]}dt\\\notag
&-\frac{\kappa}{8}\expect{\Delta[\hat{F}_z,[\hat{O},\hat{F}_z]]\Delta\hat{A}}dt
\end{align}
When $\hat{O}$ and $\hat{A}$ are collective pseudospins, the final two terms in this expression involve either commutators or covariances of populations. Because the populations are treated as c-numbers, these terms are negligible.
To obtain the full evolution of the covariance under QND measurement, we combine the contributions from measurement backaction and dissipation,
\begin{align}
d\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\text{QND}}=&d\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\mathcal{H}}+d\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\mathcal{L}}\\\notag
=&-\kappa\left(v(\uparrow)\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{O}}_S
+w(\uparrow)\expect{\Delta\hat{X}_{\wr\downarrow}\Delta\hat{O}}_S\right)\\\notag&
\times\left(v(\uparrow)\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{A}}_S
+w(\uparrow)\expect{\Delta\hat{X}_{\wr\downarrow}\Delta\hat{A}}_S\right)dt\\\notag
&+\frac{\kappa}{8}v(\uparrow)^2\expect{\{[\hat{X}_{\downarrow\uparrow},\hat{O}],[\hat{A},\hat{X}_{\downarrow\uparrow}]\}}dt\\\notag&
+\frac{\kappa}{8}v(\uparrow)w(\uparrow)\expect{\{[\hat{X}_{\wr\downarrow},\hat{O}],[\hat{A},\hat{X}_{\downarrow\uparrow}]\}}dt\\\notag
&+\frac{\kappa}{8}v(\uparrow)w(\uparrow)\expect{\{[\hat{X}_{\downarrow\uparrow},\hat{O}],[\hat{A},\hat{X}_{\wr\downarrow}]\}}dt\\\notag&+\frac{\kappa}{8}w(\uparrow)^2\expect{\{[\hat{X}_{\wr\downarrow},\hat{O}],[\hat{A},\hat{X}_{\wr\downarrow}]\}}dt.
\end{align}
For the complete evolution of the covariance, we must also include decoherence due to optical pumping,
\begin{align}\label{eq::Covariance QNDop}
\frac{d}{dt}\expect{\Delta\hat{O}\Delta\hat{A}}_S=\frac{d}{dt}\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\text{QND}}
+\frac{d}{dt}\expect{\Delta\hat{O}\Delta\hat{A}}_S\big|_{\text{op}},
\end{align}
where the equation of motion for the covariances under optical pumping was derived previously in \erf{eq::CovarEvol2}. When the dynamics due to QND measurement and optical pumping are combined, we obtain a closed set of differential equations that couple the covariances between all pairs of collective pseudospins with the populations.
\section{Optimal Fiducial State}
The numerical results in Sec. \ref{sec::HPResults} demonstrate that the choice of fiducial state has a dramatic impact on the performance of Faraday-based squeezing protocols. Both the coherent squeezing and decoherence of the ensemble are highly dependent on the choice of fiducial state. Of the state preparations studied in Sec. \ref{sec::HPResults} in the presence of optical pumping, the $m_x=0$ preparation generated the most squeezing. A natural question is what fiducial state maximizes squeezing in the presence of optical pumping? Through the equations of motion for the covariances derived in the previous section, we can utilize a numerical search to find the fiducial state that maximizes the squeezing when the ensemble is subject to QND measurement and optical pumping.
\subsection{Optimization}
In the optimization, we seek to minimize the peak squeezing parameter over all possible fiducial states. For an ensemble of spin-$f$ atoms, we parametrize the fiducial state in terms of $2(2f+1)$ real numbers as $\ket{\uparrow}=\sum_{m_z=-f}^{f}(p_{2m_z-1}+ip_{2m_z})\ket{f,m_z}$, where $\sum_{m_z=-f}^{f}(p_{2m_z-1}^2+p_{2m_z}^2)=1$. The choice of fiducial state generates a unique set of coupled differential equations describing the evolution of the variances and populations. The evolution of the covariances under QND measurement and optical pumping is given by \erf{eq::Covariance QNDop}. There are 21 equations of motion describing the covariances, corresponding to each pair of collective pseudospins plus the variances of each collecective pseudospin. Because QND measurement negligibly effects the populations, we consider only the evolution of the populations due to optical pumping, which is given in \erf{eq::FirstOrderPopulations2}. For each fiducial state, the set of differential equations giving the evolution of the covariances and populations can be solved to determine the squeezing parameter after post-processing. In the case where the fiducial state does not support a transfer state, such as for $\ket{\uparrow_{\text{cat}}}$, the squeezing parameter is given by the expression in \erf{eq::noTransfer}, which is then minimized over all $t$ to determine the peak squeezing. When the fiducial state does support a transfer state, both squeezing parameters in Eqs. (\ref{eq::unnormSqParam2}) and (\ref{eq::noTransfer}) are minimized over all $t$ with the smaller of the two representing the peak squeezing. This ensures that the transfer state is only preserved if it is beneficial to spin squeezing.
For integer and half integer spins from $f=1$ to $f=5$, we minimize the squeezing parameter over all fiducial states and time using an interior point algorithm in MATLAB \cite{InteriorPt}. For the post-processing partial isometry, we choose the map to the SCS preparation. The exact form of the squeezing parameter that is minimized is given in \erf{eq::SCSSqParamOP}. In the case that the transfer state is discarded, \erf{eq::SCSSqParamOP} is minimized with $\Delta\hat{X}_{\wr\downarrow}^{2}\rightarrow N_\downarrow/2$, $\expect{\Delta\hat{X}_{\downarrow\uparrow}\Delta\hat{X}_{\wr\downarrow}}_S\rightarrow0$ and $N_\wr\rightarrow 0$.
Because the landscape of the squeezing parameter as a function of time and the fiducial state has not been studied, little is known about the presence of local minima or saddle points. The existence of numerous local minima is likely, as the convergence of the algorithm is highly dependent upon the initial fiducial state and time seeded to the interior point algorithm. To compensate for this, over 100 randomly chosen fiducial states are selected for each $f$ and seeded to the interior point algorithm. The minimal value of the squeezing parameter that results from this large set of initial seeds is taken to be the optimal solution. It should be emphasized that because so little about the optimization landscape is known, we cannot assert that the minimum squeezing parameters found by our algorithm are absolute minima.
\subsection{Results}
The fiducial states found by the numerical optimization outperform the state preparations we have considered thus far, as shown in Fig. \ref{fig::Optf}. Figure \ref{fig::Optf} (a) depicts the peak squeezing determined by the numerical optimization for each $f$ along with the peak squeezing generated by the SCS, cat, and $m_x=0$ state preparations. The numerical optimization significantly outperforms the state preparations for smaller $f$, but for $f\geq 2$ it performs only slightly better than $m_x=0$. Figure \ref{fig::Optf} (b) shows the squeezing generated by the fiducial state found by the numerical optimization for $f=4$ along with the state preparations. While the numerical optimization only improves upon the $m_x=0$ preparation by .3 dB, the fiducial state determined by numerical optimization appears more robust to optical pumping with its squeezing decaying more slowly with time.
\begin{figure}
\centering
\includegraphics[scale=.47]{Optf.pdf}
\caption{Performance of the fiducial states found by numerical optimization compared to the other state preparations. Plot (a) shows the peak squeezing generated by various state preparations for each $f$. Plotted are the state preparations found by numerical optimization for each $f$ (red), the SCS preparation (green), the cat preparation (black) and the $m_x=0$ (blue). Plot (b) shows the performance of the state preparations for $f=4$ versus time.} \label{fig::Optf}
\end{figure}
We now examine the fiducial states found by the numerical optimization, which generate the squeezing depicted in Fig. \ref{fig::Optf}. Figure \ref{fig::OptStates} graphically represents the fiducial states that minimize the squeezing parameter for both integer and half-integer $f$. Interestingly, the fiducial states found through the numerical optimization all take a similar form. For integer $f$, the fiducial states are approximately of the form
\begin{align}
\ket{\uparrow_\text{int}}=c_f\ket{f,m_z=f}+c_0\ket{f,m_z=0}+c_{f}\ket{f,m_z=-f}.
\end{align}
For half-integer $f$, the fiducial states are approximately of the form
\begin{align}
\ket{\uparrow_\text{half}}=&c_f\ket{f,m_z=f}+c_{1/2}\ket{f,m_z=1/2}\\\notag&+c_{1/2}\ket{f,m_z=-1/2}+c_{f}\ket{f,m_z=-f}.
\end{align}
Here, $c_f$, $c_0$ and $c_{1/2}$ are real constants.
\begin{figure}
\centering
\includegraphics[scale=.4]{OptStates.pdf}
\caption{Fiducial states found by numerical optimization for (a) integer $f$ and (b) half integer $f$. Each bar in position $(f, m_z)$ corresponds to the $\hat{f}_z$ eigenstate $\ket{f,\,m_z}$. The $z$-axis gives the weight, $|\bra{f,\,m_z}\psi\rangle|$, on each eigenstate for a fiducial state $\ket{\psi}$.}\label{fig::OptStates}
\end{figure}
In both cases, the fiducial states are the superposition between a state with a large variance in $\hat{f}_z$ and one with less or no variance in $\hat{f}_z$. For integer $f$, these states are $\ket{\uparrow_\text{cat}}$ and $\ket{f,m_z=0}$, the superposition of which yield a variance $(\Delta f_z^2)_{\uparrow_{\text{int}}}=2|c_f|^2f^2$. For half-integer $f$, the superposition consists of $\ket{\uparrow_\text{cat}}$ and $(\ket{f,m_z=1/2}\\+\ket{f,m_z=-1/2})/\sqrt{2}$, resulting in a variance of $(\Delta f_z^2)_{\uparrow_{\text{half}}}=2|c_f|^2f^2+|c_{1/2}|^2/2$. This superposition is likely caused by competition between the coherent squeezing interaction and the spin flip rate. The strength of the coherent squeezing interaction is governed by the collective spin coupling constant, given in \erf{eq::2ndXI}, which is proportional to $(\Delta f_z^2)_\uparrow$. From the master equation in \erf{eq::MasterRotating}, the spin flip rate also depends upon $(\Delta f_z^2)_\uparrow$,
\begin{align}
\Gamma_{\text{flip}}&=\gamma_s\bra{\downarrow}\mathcal{D}\left(\ket{\uparrow}\bra{\uparrow}\right)\ket{\downarrow}\\\notag
&=\frac{g_f\gamma_s}{9}\left((\Delta f_z^2)_\uparrow+\frac{1}{2}|\bra{\uparrow}\hat{f}_y\ket{\downarrow}|^2
+\frac{1}{2}|\bra{\uparrow}\hat{f}_x\ket{\downarrow}|^2\right).
\end{align}
Both the coherent squeezing interaction and spin flip rate increase with the variance of the fiducial state in $\hat{f}_z$. The superposition between states with large and small variances in $\hat{f}_z$ seem to balance the coherent and incoherent dynamics. The state with large variance boosts squeezing, while the state with smaller variance minimizes decoherence due to spin flips.
\begin{figure}
\centering
\includegraphics[scale=.49]{OptTransfer.pdf}
\caption{Performance of fiducial states found by numerical optimization for (a) $f=4$ and (b) $f=7/2$. The squeezing generated by the state preparations is plotted versus time with the transfer state preserved (solid) and with the transfer state eliminated (dashed).}\label{fig::OptTransfer}
\end{figure}
Another interesting property of these fiducial states is the existence of a beneficial transfer state. For simplicity, we focus on the case where $f$ is an integer. From \erf{eq::coupledDef}, the coupled state is
\begin{align}
\ket{\downarrow_{\text{int}}}=\frac{1}{\sqrt{2}|c_f|}\left(c_f\ket{f,m_z=f}-c_f\ket{f,m_z=-f}\right).
\end{align}
The coupled state enables us to deduce the transfer state from \erf{CoherenceState},
\begin{align}
\ket{\wr_{\text{int}}}=&\frac{c_f|c_0|}{\sqrt{2}|c_f|}\left(\ket{f,m_z=f}+\ket{f,m_z=-f}\right)\\\notag
&-\frac{\sqrt{2}|c_f|c_0}{|c_0|}\ket{f,m_z=0}.
\end{align}
Recall that the transfer state is beneficial to spin squeezing if the conditions in Eqs. (\ref{eq::TofUp}) and (\ref{eq::up2squiggle}) are satisfied by the fiducial, coupled and coherence states. For the master equation in the rotating frame, the first condition is equivalent to
\begin{align}\label{eq::NewT}
T(\uparrow)=\text{Re}[\bra{\wr}\mathcal{D}\left(\ket{\downarrow}\bra{\uparrow}\right)\ket{\downarrow} >0.
\end{align}
This ensures that negative correlations are preserved due to the entanglement involving the transfer and the coupled states. The second condition is equivalent to
\begin{align}\label{eq::NewN}
N(\uparrow)=\bra{\wr}\mathcal{D}(\ket{\uparrow}\bra{\uparrow})\ket{\wr}=0.
\end{align}
This ensures the transfer state does not contribute to the noise injection. For $f>1$, the states $\ket{\uparrow_\text{int}}$, $\ket{\downarrow_\text{int}}$ and $\ket{\wr_\text{int}}$ satisfy these conditions, demonstrating a beneficial transfer of coherence. Note that when $c_0=0$, $\ket{\uparrow_\text{int}}$ is a cat state with no transfer of coherence. The presence of a term with weight on the eigenstate $\ket{f,m_z=0}$ ensures that a beneficial transfer state exists, making $\ket{\uparrow_\text{int}}$ more robust to decoherence than the cat state.
For $f>3/2$, $\ket{\uparrow_\text{half}}$, $\ket{\downarrow_\text{half}}$ and $\ket{\wr_\text{half}}$ satisfy conditions (\ref{eq::NewT}) and (\ref{eq::NewN}), implying that a transfer of coherence likewise exists in the half-integer case. The impact of the transfer state on the squeezing generated by both $\ket{\uparrow_\text{int}}$ and $\ket{\uparrow_\text{half}}$ is shown in Fig. \ref{fig::OptTransfer} for (a) $f=4$ and (b) $f=7/2$. Preserving the transfer state increases the squeezing produced by both preparations by 2.6 dB. The presence of the transfer state additionally slows the decay of the squeezing due to optical pumping. Transfers of coherence are a significant factor in the performance of the fiducial states found through numerical optimization.
\chapter{Squeezing with Paraxial Beams}\label{paraxial}
In the previous discussion and in most work on spin squeezing involving free space atomic ensembles, the light is assumed to be a plane wave that couples identically to all atoms. While it is possible to create plane-like waves in a laboratory when the beam area is large compared to the spatial extent of the ensemble, this results in poor mode matching between their respective radiation patterns. Because generating interatomic entanglement requires the atoms to be as indistinguishable as possible, mode matching is essential for maximizing spin squeezing. In this chapter, we replace plane waves with focused paraxial beams and derive equations of motion for the ensemble observables. Because paraxial beams, unlike plane waves, are spatially inhomogeneous, the geometries of both the atomic ensemble and probe beam are crucial factors in determining the degree of atom-light coupling. As we have seen in the preceding chapters, the peak squeezing that we can achieve depends on the balance between coherent squeezing and decoherence. Since decoherence is also spatially inhomogeneous, the optimal geometries for squeezing depend both on spatial mode matching and optimizing this balance. Using our formalism, we find the optimal ensemble and beam geometries for maximal mode matching and generation of spin squeezing.
\section{Paraxial Beams}
In a realistic implementation of spin squeezing in a free space atomic ensemble, the light emerging from the probe laser is well approximated as a Gaussian beam in the TEM$_{00}$ spatial mode. The intensity of a beam in the TEM$_{00}$ mode is spatially varying and proportional to the square of the mode function,
\begin{align}\label{eq::TEM00mode}
u_{00}(\mathbf{r}_\bot,z)= \frac{w_0}{w(z)}e^{-\frac{\left|\mathbf{r}_\bot\right|^2}{[w(z)]^2}}e^{\frac{ik_0 \left|\mathbf{r}_\bot\right|^2}{2 R(z)}}e^{-i\Phi(z)}.
\end{align}
Here, the beam waist, radius of curvature, and Gouy phase are given by
\begin{align}
w(z) &= w_0 \sqrt{1+(z/z_R)^2} , \\
R(z) &= z\left(1+(z_R/z)^2\right)\;\;\;\text{and} \\
\Phi(z) &= \tan^{-1}(z/z_R),
\end{align}
for Rayleigh range $z_R \equiv k_0 w_0^2/2$ and minimum beam waist $w_0$.
The field of the probe beam in the TEM$_{00}$ mode satisfies the paraxial wave equation. For a spatially elongated ensemble, the light coherently scattered by the atoms is also paraxial. With this motivation, we partition the electric field scattered by the ensemble into two components,
\begin{align} \label{Eq::ModeDecomp}
\hat{\mathbf{E}}_{\text{scat}}^{(+)}(\mathbf{r}, t) = \hat{\mathbf{E}}_{\rm para}^{(+)}(\mathbf{r}, t) + \hat{\mathbf{E}}_{\rm diff}^{(+)}(\mathbf{r}, t),
\end{align}
where $\hat{\mathbf{E}}_{\rm para}^{(+)}(\mathbf{r}, t)$ is the field of the coherently scattered paraxial light and $\hat{\mathbf{E}}_{\rm diff}^{(+)}(\mathbf{r}, t)$ is the field of the scattered non-paraxial light, which includes light spontaneously emitted by the atoms. Solutions to the paraxial wave equation can be decomposed into sums of orthogonal Gaussian mode functions, $u_{pl} (\mbf{r})$. Employing such a decomposition, the positive frequency component of the paraxial field becomes
\begin{equation}\label{eq::PosFreqComp}
\hat{\mbf{E}}^{(+)}_{\text{para}}(\mbf{r},t) = \sum_{pl, a=x, y} \sqrt{\frac{2 \pi \hbar \omega_0}{c A}}\, \mbf{e}_a \, \hat{a}_{pl, a}(z,\tau) \, u_{pl}(\mbf{r},z) e^{i(k_0 z - \omega_0 t)},
\end{equation}
where $\tau = t-z/c$ is the retarded time along the propagation direction, $A$ is the transverse beam area and $a$ denotes the beam's polarization. As illustrated by the positive frequency component, each mode function describes the field amplitude of an orthogonal transverse spatial mode, which we denote by the indices $pl$. The operator $\hat{a}_{pl,a}(z_i,t)$ represents the annihilation of a photon in the transverse spatial mode $pl$ and the polarization mode $a$.
One of the most valuable features of the mode functions $u_{pl} (\mbf{r})$ is that they form a orthogonal basis over functions of the transverse coordinate, $\mbf{r}_\perp$, at a fixed $z$. Here, we have partitioned the spatial coordinate $\mathbf{r}$ into $\mathbf{r}_\bot$ and $z$, where $z$ is the longitudinal coordinate parallel to the beam's axis of propagation. The mode functions are chosen to be dimensionless, obeying the orthogonality and completeness relations
\begin{align} \label{Eq::TransverseOrthogonality}
\int d^2 \mbf{r}_\perp u^*_{pl} (\mbf{r}_\perp , z) u_{p'l'} &(\mbf{r}_\perp , z) = A \, \delta_{p, p'} \delta_{l, l'}
\end{align}
and
\begin{align}\label{Eq::TransverseCompSameZ}
\sum_{p,l} u_{pl}(\mbf{r}_\perp , z) u^*_{pl}(\mbf{r}_\perp' , z) &= A \, \delta^{(2)}(\mbf{r}_\perp-\mbf{r}_\perp').
\end{align}
These conditions imply that any function of $\mbf{r}_\perp$ at fixed $z$ can be expressed as a sum of mode functions. This property will prove vital in deriving the equations of motion for the system.
\section{Three-Dimensional Atomic Ensemble}
The spatial distribution of the ensemble is critical in determining properties of the atom-light interface. As we will later discuss, it is the geometry of the ensemble that determines the decomposition of the coherently scattered field into transverse spatial modes. Additionally, because the intensity of the probe varies throughout the ensemble, properties such as the strength of the Faraday interaction, the rate of squeezing and the rate of optical pumping are all position dependent. Whereas in the plane wave case we can ignore the position degrees of freedom of the atoms, in the paraxial case we must associate to each atom $i$ a spatial coordinate $\mathbf{r}_i=(\mbf{r}_{\bot i},z_i)$. To simplify calculations, we will often approximate the discrete distribution of atoms in the ensemble as a continuous density, $\eta(\mathbf{r})$. For instance, in the simulations presented in Sec. \ref{sec::MultimodeResults}, we treat the density of the ensemble as a cylindrically symmetric Gaussian cloud with
\begin{align} \label{Eq::AtomicDistribution}
\eta(\mathbf{r}) = \eta_0 \exp \left( - 2\frac{\rho^2}{\sigma_\perp^2} - 2\frac{z^2}{\sigma_z^2} \right).
\end{align}
In this expression, $\sigma_\perp^2$ and $\sigma_z^2$ are the transverse and longitudinal $1/e^2$ variances and $\eta_0$ is the peak density at the center of the cloud. This approximates the density of a large cloud of cold atoms confined in a dipole trap. As in the classical propagation of a wave through a gas, it is this continuous density that is responsible for the index of refraction; the density fluctuations associated with the discrete positions are responsible for diffuse scattering.
\section{The Multimode Faraday Interaction}\label{sec::MMFaraday}
As in the case of the Faraday interaction with plane waves, the collective spin of the atomic ensemble causes the polarization of the light to rotate from $x$ to $y$. Unlike the plane wave case, however, the ensemble radiates the light into a superposition of spatial modes outside the TEM$_{00}$ mode of the probe. These effects are described by the multimode Faraday interaction \cite{Baragiola14}
\begin{align}\label{Eq::multiFaraday}
\hat{H} \!= \! -i \frac{\hbar \sqrt{\kappa}}{2} \! \sum_{i,p,l} \! \Big[ & \beta^*_{pl}(\mbf{r}_{\perp i}, z_i ) \hat{a}_{pl,y}(z_i,t) \!- \! \mbox{h.c.} \Big] \! \hat{f}_z^{(i)},
\end{align}
where $\beta_{pl}(\mbf{r}_\perp, z)$ is a product of mode functions given by
\begin{align} \label{Eq::Beta}
\beta_{pl}(\mbf{r}_\perp, z) \equiv u^*_{pl}(\mbf{r}_\perp, z) u_{00}(\mbf{r}_\perp, z).
\end{align}
This Hamiltonian corresponds to annihilation of a photon in the fundamental mode with x-polarization and creation in another paraxial mode with y-polarization. The final result
follows from a Holstein-Primakoff approximation treating the fundamental mode as a macroscopically occupied c-number. Equation (\ref{Eq::multiFaraday}) is summed over all transverse modes $pl$ and atoms in the ensemble, $i$, with spatial coordinates $(\mbf{r}_{\perp i}, z_i )$. Note that rather than being proportional to the collective spin operator $\hat{F}_z$, like the plane wave Faraday interaction in \erf{eq::FaradayDef}, the multimode Faraday interaction depends upon a sum of the internal spin operators weighted by the functions $\beta_{pl}(\mbf{r}_\perp, z)$. These weighted sums, referred to as spin waves, are given by
\begin{align}\label{eq::SpinWaveDef}
\hat{F}_z^{pl}=\sum_{i} \beta_{pl}(\mbf{r}_{\perp i}, z_i)\hat{f}_z^{(i)}.
\end{align}
Each spin wave is the effective collective spin of atoms absorbing photons in the TEM$_{00}$ spatial mode of the probe beam and radiating into the transverse mode $pl$. A particularly important spin wave is the so-called fundamental spin wave, $\hat{F}_z^{00}$, which represents the effective collective spin of the atoms that re-radiate into the TEM$_{00}$ mode of the probe.
As in the plane wave case, squeezing of the ensemble can be created by QND measurement of the light. The presence of transverse spatial modes adds subtlety to this process, however. At the plane of the polarimeter, the probe light in the TEM${_{00}}$ mode interferes with the coherently scattered light, which is in a superposition of transverse spatial modes. The probe light destructively interferes with all spatial mode components of $\hat{\mathbf{E}}_{\rm para}^{(+)}(\mathbf{r}, t)$, except for $pl=00$. Consequently, the detector only measures scattered light in the TEM$_{00}$ mode. This light carries information about the atomic ensemble just as it did in the plane wave case. The polarimetry signal takes the form,
\begin{equation} \label{Eq::XplOut}
\hat{X}_{00}(\Delta t) = \hat{X}_{00}(0) +\sqrt{\frac{\kappa}{2}} \hat{F}^{00}_z(0) .
\end{equation}
Because the TEM$_{00}$ mode only couples to the fundamental spin wave, the signal depends upon $\hat{F}^{00}_z$ alone. As in the plane wave case, a measurement of $\hat{X}_{00}$ extracts information about the fundamental spin wave and creates squeezing by measurement backaction. The strength of the measurement backaction depends on the entanglement between the light and ensemble created by the Faraday interaction. Analogous to the plane wave case discussed in Sec. \ref{sec::EntangleFaraday}, the entanglement between the light and ensemble increases with the projection noise fluctuations of the fundamental spin wave. Thus, we define a paraxial collective spin coupling constant analogous to $\xi$ from \erf{eq::2ndXI},
\begin{equation}\label{Eq::CouplingStrength}
\xi_{\text{para}} = \frac{\int_0^{\Delta t}dt(\Delta X_{00}^2)_{PN}}{\int_0^{\Delta t}dt(\Delta X_{00}^2)_{SN}}= \left( \Delta F_z^{00}(0)\right)^2 \kappa \Delta t.
\end{equation}
Here, $(\Delta X_{00}^2)_{PN}$ and $(\Delta X_{00}^2)_{SN}$ are the projection noise variance and shot noise variance of the signal. In the definition of the paraxial collective spin coupling constant, these signal variances in the detector are integrated over a time $\Delta t$. As in the plane wave case, backaction is maximized when the projection noise of the ensemble dominates over the shot noise of the light.
From the definition of the paraxial collective spin coupling constant, measurement backaction increases with the initial projection noise fluctuations of the fundamental spin wave. In a departure from the plane wave case, these fluctuations depends not just upon the fiducial state, but upon the probe beam and the density of the atomic ensemble. For an ensemble with each atom prepared in the fiducial state $\ket{\uparrow}$,
\begin{align}
(\Delta F_z^{00}(0))^2&=\sum_{i,j}\beta_{00}(\mbf{r}_i)\beta_{00}(\mbf{r}_j)\bra{\uparrow_i\uparrow_j}\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}\ket{\uparrow_i\uparrow_j}\\&=\sum_i\beta_{00}(\mbf{r}_i)^2(\Delta f_z^2)_\uparrow.
\end{align}
By treating the atomic ensemble as a continuous density distribution, this expression becomes
\begin{align}
(\Delta F_z^{00}(0))^2=\int d^3\mbf{r} \, \eta (\mbf{r})\beta_{00}(\mbf{r})^2(\Delta f_z^2)_\uparrow=N_{\text{eff}}^{00\,(2)}(\Delta f_z^2)_\uparrow.
\end{align}
We refer to $N_{\text{eff}}^{00\,(2)} $ as an effective atom number in the fundamental mode.
The general expression for the $K$th effective atom number in the transverse mode $pl$ is given by
\begin{align}
N_{\text{eff}}^{pl\,(K)} =\int d^3\mbf{r} \, \eta (\mbf{r})\beta_{pl}(\mbf{r})^K.
\end{align}
The effective atom numbers are associated with different physical quantities. The initial variances of the spin waves are related to the $K=2$ effective atom numbers, while the means are related to $K=1$. The mean of the spin wave operator
$\hat{O}^{pl}=\sum_i\beta_{pl}(\mbf{r}_i)\hat{o}^{(i)}$ is given by
\begin{align}\label{eq::MeanSpinWaveN1}
\expect{\hat{O}^{pl}}=\int d^3\mbf{r} \, \eta (\mbf{r})\beta_{pl}(\mbf{r})\expect{\hat{o}}=N_{\text{eff}}^{pl\,(1)}\expect{\hat{o}}.
\end{align}
Because the collective spin coupling constant is proportional to $N_{\text{eff}}^{00\,(2)}$, this effective atom number has special significance. The parameter $N_{\text{eff}}^{00\,(2)}$ quantifies the effective number of atoms that are radiating light back into the probe mode. When $N_{\text{eff}}^{00\,(2)}$ is large, the ensemble is well mode matched and the atoms are less distinguishable, leading to enhanced interatomic entanglement and spin squeezing. In a parallel sense, the effective atom numbers $N_{\text{eff}}^{pl\,(2)}$ for $pl\neq 00$ quantify the strength of the scattered field in modes outside of the TEM$_{00}$ mode of the probe. Having large values of $N_{\text{eff}}^{pl\,(2)}$ for $pl\neq 00$ is a symptom of poor mode matching.
In analogy with the plane wave case, we define the effective optical density in terms of $N_{\text{eff}}^{00\,(2)}$ as
\begin{align}\label{eq::ODeff}
O\!D_{\text{eff}}=N_{\text{eff}}^{00\,(2)}\frac{\sigma_0}{A}.
\end{align}
This quantifies the strength of the mode matched coupling between the ensemble and probe. Note that the values of all effective atom numbers, and thus $O\!D_{\text{eff}}$, are governed by the density of the atomic ensemble and the geometry of the probe.
\section{Paraxial Spin Squeezing Parameter}
For a Gaussian probe beam in the TEM$_{00}$ mode, QND measurement creates squeezing in the associated spin wave $\hat{F}_z^{00}$. Because we are interested in utilizing this spin wave squeezing for applications in metrology, we quantify its strength through the angular resolution of a magnetometer, $\Delta\phi$, introduced in Sec. \ref{sec::QuantSqueezing}. In a situation analogous to the plane wave case, the ensemble is initially polarized along $x$. We wish to deduce the magnitude of a rotation, $\phi$, about $y$ by measuring the collective spin of the ensemble along $z$. This can be accomplished by the Faraday effect, i.e. shining a linearly polarized probe through the ensemble and measuring the rotation angle of the polarization. For a plane wave probe propagating in the $z$ direction, the rotation of the light's polarization is proportional to $\hat{F}_z$ by the Faraday interaction. For a Gaussian probe in the TEM$_{00}$ mode propagating along $z$, the rotation of the light's polarization is proportional to the fundamental spin wave, $\hat{F}_z^{00}$. The precision with which we can determine $\phi$, thus, depends on the uncertainty $\Delta F_z^{00}$. The precision also depends on a mean spin wave component orthogonal to $z$, analogous to $\expect{\hat{F}_x}$ in the plane wave case. For an ensemble initially polarized along $x$, the signal is the mean spin of the effective number of atoms addressed by the probe light in the TEM$_{00}$ mode,
\begin{align}
\expect{\hat{F}_x^{00}}=\sum_i\beta_{00}(\mbf{r}_i)\expect{\hat{f}_x^{(i)}}=N_{\text{eff}}^{00\,(1)}\expect{\hat{f}_x^{(i)}}.
\end{align}
Similar to Sec. \ref{sec::QuantSqueezing}, the complete expression for the angular resolution of the magnetometer is
\begin{align}\label{angRes00}
\Delta\phi=\frac{\Delta F_z^{00}}{\expect{\hat{F}_x^{00}}}.
\end{align}
For an ensemble prepared in a spin coherent state, the resolution is
\begin{align}
\Delta\phi_{SCS}=\frac{1}{N_{\text{eff}}^{00\,(1)}}\sqrt{\frac{N_{\text{eff}}^{00\,(2)}}{2f}}.
\end{align}
The Wineland squeezing parameter compares the performance of the state we wish quantify with the performance of a spin coherent state. For the paraxial case, the Wineland squeezing parameter becomes
\begin{equation} \label{Eq::SqueezingParam}
\zeta_{\text{para}} \equiv \left(\frac{\Delta \phi}{\Delta \phi_{\text{SCS}}}\right)^2 = 2f \frac{ \big(N^{(1)}_\text{eff} \big)^2 }{N_\text{eff}^{(2)}} \frac{\left(\Delta F_z^{00}\right)^2}{\expect{\hat{F}_x^{00}}^2}.
\end{equation}
This expression quantifies the degree to which QND measurement improves the angular resolution of a magnetometer over an initially spin coherent ensemble. Note that unlike the squeezing parameter for the plane wave case in \erf{eq::SqParameter}, the paraxial squeezing parameter depends on the geometry of the ensemble as well as the state of the collective spin due to the presence of the different effective atom numbers.
\section{Equation of Motion for Spin Waves}
To determine the paraxial squeezing parameter, we must track the evolution of the spin waves. We follow a procedure similar to Chapters \ref{sec::ModHPCovar} and \ref{sec::Beyond}, deriving a set of coupled differential equations that describe the behavior of the ensemble observables or spin waves, in this case. The spin waves evolve due to both optical pumping of the ensemble and continuous QND measurement. For simplicity, in the text we present only the equations of motion for the spin waves in the $f=1/2$ case, where the ensemble is prepared in a SCS. The fully generalized case for $f\geq1/2$ and arbitrary fiducial states is given in Appendix \ref{sec::fSpinWaves}.
As mentioned previously, decoherence from optical pumping occurs when light from the probe beam is diffusely scattered. The optical pumping induced by a paraxial beam on the ensemble acts locally on each atom. It is described by the master equation
\begin{align}\label{eq::FullMasterParaxial}
\frac{d\hat{\rho}}{dt}\Big|_{\text{op}}=\sum_i\gamma_s(\mathbf{r}_i)\mathcal{D}^{(i)}(\hat{\rho}),
\end{align}
where $\mathcal{D}^{(i)}$ is the superoperator given in \erf{eq::MasterD} in the rotating frame defined by the bias magnetic field along the $z$-axis. This master equation is almost identical to the master equation defined in \erf{eq::MasterRotating} for the plane wave case, except that the photon scattering rate is not uniform throughout the atomic ensemble. This is a consequence of the spatially inhomogeneous intensity of the probe, $I_{00}(\mathbf{r})$. The photon scattering rate, which increases with the intensity of the probe, is given by
\begin{align} \label{Eq::LocalScatRate}
\gamma_s(\mathbf{r}) = I_{00}(\mathbf{r}) \frac{\sigma_0}{\hbar \omega} \frac{\Gamma^2}{4 \Delta^2} = \gamma_{0}\beta_{00}(\mathbf{r}),
\end{align}
where $\gamma_0$ is the peak scattering rate at the ensemble's center. As a result of the inhomogeneous probe, atoms at positions in the ensemble with different intensities undergo different rates of optical pumping.
Continuous QND measurement of the fundamental spin wave is described by the stochastic master equation,
\begin{align} \label{Eq::HomodyneSME}
d \hat{\rho}&= \sqrt{ \frac{\kappa}{4} } \mathcal{H}_{00}(\hat{\rho}) \, dW + \frac{\kappa}{4} \sum_{p,l} \mathcal{L}_{pl}(\hat{\rho}) \, dt,
\end{align}
which takes a form similar to the SME in the plane wave case in \erf{eq::planeSME}. A detailed derivation of the paraxial SME is provided in Ref. \cite{Baragiola14}. The effect of measurement backaction on the ensemble is taken into account by the superoperator $\mathcal{H}_{00}(\hat{\rho}) $, where
\begin{align} \label{Eq::HSuperoperator}
\mathcal{H}_{00}(\hat{\rho}) = \hat{F}^{00}_z \hat{\rho} + \hat{\rho} \hat{F}^{00 \dag}_z - \text{Tr}((\hat{F}^{00}_z + \hat{F}^{00\dag}_z) \hat{\rho} ) \hat{\rho}.
\end{align}
Because the fundamental spin wave is the measured observable, $\mathcal{H}_{00}$ depends upon $\hat{F}^{00}_z$ rather than $\hat{F}_z$, as in the plane wave case. The Lindblad superoperator,
\begin{align} \label{Eq::LSuperoperator}
\mathcal{L}_{pl}(\hat{\rho}) = \hat{F}_z^{pl} \hat{\rho} \hat{F}_z^{pl\dag} - \frac{1}{2} \hat{F}_z^{pl\dag} \hat{F}_z^{pl} \hat{\rho} - \frac{1}{2} \hat{\rho} \hat{F}_z^{pl\dag} \hat{F}_z^{pl},
\end{align}
describes decoherence of the ensemble arising from coherent scattering of the light into all transverse spatial modes. In the case of $pl\neq 00$, the light is unmeasured and carries away information about the ensemble state.
\subsection{Evolution of the Mean Spin Waves}
We first consider the evolution of $\expect{\hat{F}_x^{pl}}$, the mean of a spin wave in spatial mode $pl$, defined as
\begin{align}\label{MeanFxpl}
\hat{F}_x^{pl} = \sum_i \beta_{pl}(\mbf{r}_i) \hat{f}_x^{(i)}.
\end{align}
Because collective scattering and measurement backaction negligibly affect the dynamics of the mean spin, to good approximation the evolution of $\expect{\hat{F}_x^{pl}}$ is dominated by optical pumping. The spatially varying nature of the photon scattering rate makes deriving the equations of motion for the ensemble observables slightly more complicated than in Sec. \ref{sec::covUpdate}. The master equation describing the evolution of the density operator, $\hat{\rho}^{(i)}$, of a single atom takes the form
\begin{align}\label{eq::ParaxRhoi}
\frac{d\hat{\rho}^{(i)}}{dt}\Big|_{\text{op}}=\gamma_s(\mathbf{r}_i)\mathcal{D}^{(i)}(\hat{\rho}).
\end{align}
From this master equation, we can compute the equation of motion for the spin component $\hat{f}_x^{(i)}$ of a single atom,
\begin{align}\label{eq::fxParax}
\frac{d}{dt}\hat{f}_x^{(i)}\Big|_{\text{op}}=\gamma_s(\mathbf{r}_i)\mathcal{D}^{(i)}(\hat{f}_x^{(i)}).
\end{align}
For $f=1/2$, the superoperator in this expression simplifies to ${\mathcal{D}_i}(\hat{f}_x^{(i)}) =-\hat{f}_x^{(i)}/3$. From \erf{MeanFxpl}, we obtain the equation of motion for the spin wave,
\begin{align}\label{Eq::SimplifiedMeanFx1}
\frac{d}{dt}\expect{\hat{F}_x^{pl}}=-\frac{\gamma_0}{3}\sum_i \beta_{00}(\mbf{r}_i) \beta_{pl}(\mbf{r}_i)\expect{\hat{f}_x^{(i)}}.
\end{align}
By decomposing $\beta_{00}(\mbf{r}) \beta_{pl}(\mbf{r})$ in terms of orthogonal mode functions, the right hand side of \erf{Eq::SimplifiedMeanFx1} can be expressed as a sum of spin wave operators. In terms of the mode functions,
\begin{align}\notag
\beta_{00}(\mbf{r}_\perp,z) \beta_{pl}(\mbf{r}_\perp,z)& =|u_{00}(\mbf{r}_\perp,z)|^2u_{pl}^*(\mbf{r}_\perp,z)u_{00}(\mbf{r}_\perp,z) \\
&= \sum_{p',l'} c^{pl}_{p'l'}(z) \beta_{p'l'}(\mbf{r}_\perp,z), \label{Eq::ProjCoeff_proto}
\end{align}
where we have made use of the orthogonality and completeness conditions in Eqs. (\ref{Eq::TransverseOrthogonality}) and (\ref{Eq::TransverseCompSameZ}) to define projection coefficients,
\begin{align} \label{Eq::ProjCoeff2}
c^{pl}_{p'l'}(z) \equiv \frac{1}{A} \int d^2 \mathbf{r}_\perp \left[ u_{00}(\mathbf{r}_\perp, z)\right]^2 u^*_{pl}(\mathbf{r}_\perp, z) u_{p'l'}(\mathbf{r}_\perp, z).
\end{align}
Because the projection coefficients depend on the longitudinal coordinate, $z$, we coarse grain the ensemble into longitudinal slices along $z$. When restricted to a coarse-grained slice $k$ of thickness $\delta z$ centered at longitudinal coordinate $z_k$, \erf{Eq::SimplifiedMeanFx1} becomes
\begin{align}\label{Eq::SimplifiedMeanFx2}
\frac{d}{dt}\expect{\hat{F}_x^{pl}(z_k)}=-\frac{\gamma_0}{3}\sum_{i_k} \beta_{00}(\mbf{r}_{i_k}) \beta_{pl}(\mbf{r}_{i_k})\expect{\hat{f}_x^{(i_k)}},
\end{align}
where $i_k$ is an index over all atoms in slice $k$. By performing the projection in \erf{Eq::ProjCoeff_proto}, we obtain an infinite hierarchy of differential equations that couple mean spin waves in a given slice to one another,
\begin{align}\label{Eq::ZkSliceFx}
\frac{d}{dt}\expect{\hat{F}_x^{pl}(z_k)}=-\frac{\gamma_0}{3}\sum_{p',l'}c^{pl}_{p'l'}(z_k)\expect{\hat{F}_x^{p'l'}(z_k)}.
\end{align}
An approximate solution to \erf{Eq::ZkSliceFx} is found for each slice by choosing a width, $\delta z$, and truncating the number of spin waves at some index $p_{max}, \,l_{max}$. The result is a finite system of coupled differential equations describing mean spins, $\expect{\hat{F}_x^{pl}(z_k)}$, in each slice $k$, where $0\leq l\leq l_{max}$ and $0\leq p\leq p_{max}$. Solving this system of coupled differential equations requires the initial conditions of the mean spin waves in each slice. Using $\expect{\hat{f}_x(0)}=1/2$ for the initial SCS state of the ensemble,
\begin{align}\label{Eq::meanSlice}
\expect{\hat{F}_x^{pl}(z_k, t=0)} = \sum_{i_k}\beta_{pl}(\mbf{r}_{i_k})\expect{\hat{f}_x^{(i_k)}(0)} =\frac{1}{2}\sum_{i_k}\beta_{pl}(\mbf{r}_{i_k}),
\end{align}
where $i_k$ is an index over all atoms in slice $k$. For an average atomic density, $\eta(\mathbf{r})$, the sum becomes an integral,
\begin{align}\label{Eq::meanSlice2}
\expect{\hat{F}_x^{pl}(z_k, t=0)} = \frac{\delta z}{2}\int d^2\mathbf{r} \,\eta(\mathbf{r},z_k)\beta_{pl}(\mathbf{r},z_k).
\end{align}
After solving the system of coupled differential equations, summing over the solutions at each slice gives the mean of the fundamental spin wave,
\begin{align}\label{Eq::FundSpinwave}
\expect{\hat{F}_x^{00}(t)}=\sum_{k}\expect{\hat{F}_x^{00}(z_k,t)}.
\end{align}
Equation (\ref{Eq::FundSpinwave}) is the fundamental mean spin in the definition of the paraxial squeezing parameter.
\subsection{Evolution of the Spin Wave Variances}
To solve for the variance of the fundamental spin wave, we follow a similar procedure. We start with the covariance between spin waves on different transverse modes $pl$ and $p'l'$,
\begin{align} \label{Eq::GenCovariance}
\expect{\Delta\hat{F}_z^{pl}\Delta\hat{F}_z^{p'l'}} =\expect{\hat{F}_z^{pl}\hat{F}_z^{p'l'}}-\expect{\hat{F}_z^{pl}}\expect{\hat{F}_z^{p'l'}} .
\end{align}
Unlike the mean spin, we cannot neglect the effects of continuous QND measurement. In this section, we solve for the equation of motion of the covariance under both optical pumping and continuous measurement.
We first consider optical pumping.
Decomposing the spin waves in \erf{Eq::GenCovariance} via \erf{eq::SpinWaveDef} shows that the covariance is the sum of both single atom and pairwise atomic correlations functions,
\begin{align} \label{eq::CovarCorrFuncts}
\expect{\Delta\hat{F}_z^{pl}\Delta\hat{F}_z^{p'l'}} =
&\sum_i\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_i)\expect{\Delta\hat{f}_z^{(i)\;2}}\\\notag
&+\sum_{i\neq j}\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_j)\expect{\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}}.
\end{align}
The equation of motion for the single atom correlation function, $\expect{\Delta\hat{f}_z^{(i)\;2}}$, can be determined through the master equation in \erf{eq::ParaxRhoi}, which gives the evolution of a single atom. From \erf{eq::ParaxRhoi},
\begin{align}\label{eq::1stOrderCorrSW}
\frac{d}{dt}\sum_i\expect{\Delta\hat{f}_z^{(i)\;2}}\big|_{\text{op}}=\sum_{i} \gamma_s(\mbf{r}_i) \Big\{ \big\langle\mathcal{D}^{(i)} \big( \hat{f}_z^{(i)2} \big) \big\rangle -2 \big\langle \mathcal{D}^{(i)} \big( \hat{f}_z^{(i)} \big) \big\rangle \big\langle \hat{f}_z^{(i)} \big\rangle \Big\}.
\end{align}
Deriving the equation of motion for the pairwise correlation function, $\expect{\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}}_{i\neq j}$, in \erf{eq::CovarCorrFuncts} requires the master equation describing the evolution of any two atoms in the ensemble. This master equation is given by
\begin{align}\label{Eq::rhoij2}
\frac{d}{dt}\hat{\rho}^{(i,j)}\Big|_{\rm op} =\gamma_s(\mbf{r}_i)\mathcal{D}^{(i)}(\hat{\rho}^{(i,j)})+ \gamma_s(\mbf{r}_j) \mathcal{D}^{(j)}(\hat{\rho}^{(i,j)}).
\end{align}
From \erf{Eq::rhoij2},
\begin{align} \label{eq::correlationDecay2}
&\frac{d}{dt}\sum_{i\neq j} \big\langle \Delta \hat{f}_z^{(i)}\Delta \hat{f}_z^{(j)} \big\rangle \big|_{\rm op} =\\\notag
& \;\;\;\;\; \sum_{i\neq j} \! \Big\{ \! \gamma_s(\mbf{r}_i) \big\langle \Delta\mathcal{D}_i \big[ \hat{f}_z^{(i)} \big]\Delta \hat{f}_z^{(j)}\big\rangle \! +\! \gamma_s(\mbf{r}_j) \big\langle \Delta \hat{f}_z^{(i)}\Delta \mathcal{D}_j \big[\hat{f}_z^{(j)}\big] \big\rangle \! \Big\}.
\end{align}
Combining Eqs. (\ref{eq::CovarCorrFuncts}), (\ref{eq::1stOrderCorrSW}) and (\ref{eq::correlationDecay2}) yields the equation of motion for the covariance,
\begin{align} \label{eq::CovarEvolParax}
\frac{d}{dt}&\expect{\Delta\hat{F}_z^{pl}\Delta\hat{F}_z^{p'l'}}\big|_{\text{op}} =\\\notag
&\sum_i\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_i) \gamma_s(\mbf{r}_i) \big\{ \big\langle\mathcal{D}^{(i)} \big( \hat{f}_z^{(i)2} \big) \big\rangle -2 \big\langle \mathcal{D}^{(i)} \big( \hat{f}_z^{(i)} \big) \big\rangle \big\langle \hat{f}_z^{(i)} \big\rangle \big\}
\\\notag
&+\sum_{i\neq j}\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_j)\big\{ \! \gamma_s(\mbf{r}_i) \big\langle \Delta\mathcal{D}^{(i)} \big( \hat{f}_z^{(i)} \big)\Delta \hat{f}_z^{(j)}\big\rangle \! +\! \gamma_s(\mbf{r}_j) \big\langle \Delta \hat{f}_z^{(i)}\Delta \mathcal{D}^{(j)} \big(\hat{f}_z^{(j)}\big) \big\rangle \! \big\}.
\end{align}
After some algebra, this expression becomes
\begin{align} \label{eq::CovarEvolParax23}
\frac{d}{dt}&\expect{\Delta\hat{F}_z^{pl}\Delta\hat{F}_z^{p'l'}}\big|_{\text{op}} =\\\notag&\gamma_s\!\sum_{i, j}\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_j)\big\{ \! \beta_{00}(\mbf{r}_i) \big\langle \Delta\mathcal{D}^{(i)} \big( \hat{f}_z^{(i)} \big)\Delta \hat{f}_z^{(j)}\big\rangle \! +\! \beta_{00}(\mbf{r}_j) \big\langle \Delta \hat{f}_z^{(i)}\Delta \mathcal{D}^{(j)} \big(\hat{f}_z^{(j)}\big) \big\rangle \! \big\}\\\notag
&+\gamma_s\sum_i\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_i)\beta_{00}(\mathbf{r}_i)\big\{ \big\langle\mathcal{D}^{(i)} \big( \hat{f}_z^{(i)2} \big) \big\rangle -\big\langle \{\mathcal{D}^{(i)} \big( \hat{f}_z^{(i)} \big), \hat{f}_z^{(i)}\} \big\rangle \big\}.
\end{align}
For $f=1/2$, the terms in this equation of motion can be simplified by ${\mathcal{D}^{(i)}}(\hat{f}_z^{(i)})= -2\hat{f}_z^{(i)}/9$ and ${\mathcal{D}^{(i)}}(\hat{f}_z^{(i)\, 2})= {\mathcal{D}^{(i)}}(\mathbb{I}^{(i)}/4)=0$, where the latter equality is the consequence of ${\mathcal{D}^{(i)}}$ being trace preserving when $f=1/2$. Equation (\ref{eq::CovarEvolParax23}) becomes
\begin{align} \label{eq::CovarEvolParax2}
\frac{d}{dt}&\expect{\Delta\hat{F}_z^{pl}\Delta\hat{F}_z^{p'l'}}\big|_{\text{op}} =\\\notag&-\frac{2\gamma_s}{9}\sum_{i}\big( \beta_{pl}(\mathbf{r}_i)\beta_{00}(\mbf{r}_i)\expect{\Delta\hat{f}_z^{(i)}\Delta\hat{F}_z^{p'l'}}
+ \beta_{p'l'}(\mathbf{r}_i)\beta_{00}(\mbf{r}_i)\expect{\Delta\hat{F}_z^{pl}\Delta\hat{f}_z^{(i)}} \big)
\\\notag
&+\frac{\gamma_s}{9}\sum_i\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_i)\beta_{00}(\mathbf{r}_i).
\end{align}
By coarse graining each spin wave into longitudinal slices in the same manner as the mean spin, we can use the projection coefficients in \erf{Eq::ProjCoeff2}. When the spin wave $\hat{F}_z^{pl}$ is restricted to slice $k$ and the spin wave $\hat{F}_z^{p'l'}$ is restricted to slice $k'$, the equation of motion becomes
\begin{align} \label{eq::CovarEvolParax3}
\frac{d}{dt}\expect{\Delta\hat{F}_z^{pl}(z_k)&\Delta\hat{F}_z^{p'l'}(z_{k'})}\big|_{\text{op}} =\\\notag&-\frac{2\gamma_s}{9}\sum_{i_k} \beta_{pl}(\mathbf{r}_{i_k})\beta_{00}(\mbf{r}_{i_k})\expect{\Delta\hat{f}_z^{(i_k)}\Delta\hat{F}_z^{p'l'}(z_{k'})}
\\\notag&-\frac{2\gamma_s}{9}\sum_{i_{k'}} \beta_{p'l'}(\mathbf{r}_{i_{k'}})\beta_{00}(\mbf{r}_{i_{k'}})\expect{\Delta\hat{F}_z^{pl}(z_k)\Delta\hat{f}_z^{({i_{k'}})}} \big)
\\\notag
&+\delta_{k,k'}\frac{\gamma_s}{9}\sum_i\beta_{pl}(\mathbf{r}_i)\beta_{p'l'}(\mathbf{r}_i)\beta_{00}(\mathbf{r}_i)\end{align}
By performing the projection in \erf{Eq::ProjCoeff_proto}, we obtain
\begin{align}\label{eq::CovarEvolParaxfinal}
&\frac{d}{dt} \expect{\Delta\hat{F}_z^{pl}(z_k) \Delta\hat{F}_z^{p'l'}(z_{k'})}\big|_{\text{op}}=\\\notag
&-\!\frac{2\gamma_s}{9}\!\sum_{p''l''}\Big[ c^{pl}_{p''l''}(z_k)\expect{\Delta\hat{F}_z^{p''l''}(z_k)\Delta\hat{F}_z^{p'l'}(z_{k'})} \!+\!c_{p''l''}^{p'l'}(z_{k'})\expect{\Delta\hat{F}_z^{pl}(z_k)\Delta\hat{F}_z^{p''l''}(z_{k'})}\Big] \\\notag&+\frac{\gamma_s}{9}N_{p'l'}^{pl}(z_k)\delta_{k,k'}. \nonumber
\end{align}
In the equation of motion, $N_{p'l'}^{pl}(z_k)$ arises from the sum in the final term of \erf{eq::CovarEvolParax3}, which can be expressed as an integral over the density of the atomic cloud,
\begin{align}
N_{p'l'}^{pl}(z_k)= \delta z \int d^2\mathbf{r}\eta(\mathbf{r},z_k)\beta_{00}(\mathbf{r},z_k)\beta_{pl}(\mathbf{r},z_k)\beta_{p'l'}(\mathbf{r},z_k).
\end{align}
Note that for the fundamental mode, $p,p',l,l'=0$ and $N_{p'l'}^{pl}(z_k)$ is $N^{(3)}_{\text{eff}}$ in slice $z_k$.
We now turn our attention to the evolution of the covariances under continuous QND measurement. As demonstrated by \erf{eq::CovarEvolParaxfinal}, optical pumping couples the covariances $\expect{\Delta\hat{F}_z^{pl}(z_k)\Delta\hat{F}_z^{p'l'}(z_{k'})}$ between spin waves in slices $z_k$ and $z_{k'}$ to one another. From the SME in \erf{Eq::HomodyneSME}, we can find the equations of motion for these covariances as the fundamental spin wave is measured. The SME includes decoherence from collective scattering into transverse modes other than the fundamental mode, which is described by the map $\mathcal{L}_{pl}$ in \erf{Eq::LSuperoperator}. Note that this map has no effect on the covariances since the spin waves $\hat{F}_z^{pl}$ commute with one another. The evolution of the covariances, thus, depends entirely on measurement backaction, which is described by the map $\mathcal{H}_{00}$ in \erf{Eq::HSuperoperator}. The equation of motion for the covariances is given by
\begin{align}
d\expect{ \Delta&\hat{F}_z^{pl}(z_k) \Delta\hat{F}_z^{p'l'}(z_{k'})}\Big|_{QND} =\label{Eq::CovHomodyne} \\\notag
&\sqrt{ \frac{\kappa}{4} } \bigg\{ \big\langle \mathcal{H}_{00}[\hat{F}_z^{pl}(z_k)\hat{F}_z^{p'l'}(z_{k'})]\big\rangle- \big\langle\mathcal{H}_{00}[\hat{F}_z^{pl}(z_k)]\big\rangle \big\langle\hat{F}_z^{p'l'}(z_{k'})\big\rangle \\\notag
& - \big\langle\hat{F}_z^{pl}(z_k)\big\rangle \big\langle\mathcal{H}_{00}[\hat{F}_z^{p'l'}(z_{k'})]\big\rangle
\bigg\} dW -\frac{\kappa}{4}\big\langle\mathcal{H}_{00}[\hat{F}_z^{pl}(z_k)]\big\rangle\big\langle\mathcal{H}_{00}[\hat{F}_z^{p'l'}(z_{k'})]\big\rangle dt.
\end{align}
The final term in the equation of motion arises from the rule of It\={o} calculus that differentials must be taken to second order \cite{JacSte06}, i.e. $d(XY) = (dX) Y + X (dY) + (dX)(dY)$. The map $\mathcal{H}_{00}$ couples the first and second order moments of the spin waves to higher order moments. For the initial SCS along $x$ and during its subsequent evolution, the spin waves $\hat{F}_z^{pl}$ are Gaussian distributed, both over the entire cloud and within each coarse-grained slice $k$. The third order moments of the spin waves can, therefore, be expressed in terms of first and second order moments through the relation \cite{JacSte06}
\begin{align}
\expect{\hat{X}\hat{Y}\hat{Z}} =& \expect{\Delta\hat{X}\Delta\hat{Y}}\expect{\hat{Z}} + \expect{\Delta\hat{X}\Delta\hat{Z}}\expect{\hat{Y}} + \expect{\Delta\hat{Y}\Delta\hat{Z}}\expect{\hat{X}} \\\notag&+\expect{\hat{X}}\expect{\hat{Y}}\expect{\hat{Z}}.
\end{align}
The decomposition in \erf{eq::3rdGaussDecomp} reduces to this expression when the operators commute.
In this regime, all stochastic terms in \erf{Eq::CovHomodyne} cancel, leaving the deterministic equation:
\begin{align}\label{Eq::CovarianceBackaction}
\frac{d}{dt}\expect{\Delta\hat{F}_z^{pl}(z_k)& \Delta\hat{F}_z^{p'l'} (z_{k'})}\Big|_{QND} = -\kappa\big\langle\Delta\hat{F}_z^{pl}(z_k)\Delta\hat{F}_z^{00}\big\rangle\big\langle\Delta\hat{F}_z^{p'l'}(z_{k'})\Delta\hat{F}_z^{00}\big\rangle \\\label{Eq::CovarianceBackaction2}
&=-\kappa\sum_{k'',k'''}\big\langle\Delta\hat{F}_z^{pl}(z_k)\Delta\hat{F}_z^{00}(z_{k''})\big\rangle\big\langle\Delta\hat{F}_z^{p'l'}(z_{k'})\Delta\hat{F}_z^{00}(z_{k'''})\big\rangle.
\end{align}
These dynamics, which arise from continuous polarimetry measurements, serve to generate the correlations that produce spin squeezing.
By combining \erf{eq::CovarEvolParaxfinal} and \erf{Eq::CovarianceBackaction2}, we obtain the full equation of motion for the covariances under both optical pumping and continuous measurement,
\begin{align}
\frac{d}{dt}\expect{&\Delta\hat{F}_z^{pl}(z_k) \Delta\hat{F}_z^{p'l'} (z_{k'})}=
\frac{d}{dt}\expect{\Delta\hat{F}_z^{pl}(z_k) \Delta\hat{F}_z^{p'l'} (z_{k'})}\Big|_{\text{op}}\\\notag&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+
\frac{d}{dt}\expect{\Delta\hat{F}_z^{pl}(z_k) \Delta\hat{F}_z^{p'l'} (z_{k'})}\Big|_{QND}\\\label{eq::FullEOMCOV}
= &-\kappa\sum_{k'',k'''}\big\langle\Delta\hat{F}_z^{pl}(z_k)\Delta\hat{F}_z^{00}(z_{k''})\big\rangle\big\langle\Delta\hat{F}_z^{p'l'}(z_{k'})\Delta\hat{F}_z^{00}(z_{k'''})\big\rangle\\\notag&\!-\!\frac{2\gamma_s}{9}\!\sum_{p''l''}\!\!\Big[ c^{pl}_{p''l''}\!(z_k)\!\expect{\Delta\hat{F}_z^{p''l''}\!(z_k)\Delta\hat{F}_z^{p'l'}(z_{k'})}\!+\!c_{p''l''}^{p'l'}(z_{k'})\!\expect{\Delta\hat{F}_z^{pl}\!(z_k)\Delta\hat{F}_z^{p''l''}\!(z_{k'})}\Big] \\\notag &+\frac{\gamma_s}{9}N_{p'l'}^{pl}(z_k)\delta_{k,k'}.
\end{align}
This is an infinite set of differential equations that nonlinearly couples all covariances between spin waves in slices $k$ and $k'$ to one another.
As in the case of the mean spin waves, the solution to this set of equations is approximated by truncating \erf{eq::FullEOMCOV} at some $p_{\text{max}}$ and $l_{\text{max}}$. Solving the resulting finite set of differential equations requires the initial values of the covariances, which are given by
\begin{align}\notag
\expect{\Delta\hat{F}_z^{pl}(z_k)&\Delta\hat{F}_z^{p'l'}(z_{k'})}(0)= \!\! \int \!\!d^2\mathbf{r} \, \eta(\mathbf{r},z_k) \beta_{pl}(\mathbf{r},z_k)\beta_{p'l'}(\mathbf{r},z_k)\expect{\hat{f}_z^{(i_{k})}(0) \hat{f}_z^{(i_{k'})}(0)} \\
&= \delta_{k,k'}\frac{\delta z}{4}\int d^2\mathbf{r} \, \eta(\mathbf{r},z_k) \beta_{pl}(\mathbf{r},z_k)\beta_{p'l'}(\mathbf{r},z_k).
\end{align}
The second equality follows because $\expect{\hat{f}_z^{(i_{k})}(0) \hat{f}_z^{(i_{k'})}(0)} =\delta_{k,k'}/4$ for the initial SCS of the ensemble. With these initial conditions and the equations of motion, we can solve for the evolution of all covariances under both QND measurement and decoherence by optical pumping. In particular, we can solve for the covariances between the fundamental spin waves in different slices. Summing these covariances over all slices $k$ and $k'$ yields the variance of the fundamental spin wave,
\begin{align}
(\Delta F_z^{00})^2=\sum_{k,k'}\expect{\Delta\hat{F}_z^{00}(z_k)\Delta\hat{F}_z^{00}(z_{k'})}.
\end{align}
From the variance of the fundamental spin wave, we can determine the paraxial squeezing parameter.
\section{Results}\label{sec::MultimodeResults}
In this section, we use the equations of motion for the spin waves to determine the optimal geometries of the atomic ensemble and probe beam for both mode matching and spin squeezing. In an experimental setting, the probe beam is tuned by varying the beam waist, $\omega_0$. Because the Rayleigh range increases with the square of the beam waist, $\omega_0$ determines both the longitudinal and transverse extent of the ensemble that lies within the region of high field intensity. We consider an experimental implementation in which the atoms are cooled and confined in a crossed beam dipole trap. In this case, the geometry of the atomic ensemble is well approximated by the continuous Gaussian density distribution in \erf{Eq::AtomicDistribution}. Adjusting the angle between the trapping beams creates atomic ensembles with different transverse and longitudinal $1/e^2$ variances, which we denote by $\sigma_\perp^2$ and $\sigma_z^2$. The geometry of the ensemble is described concisely with the aspect ratio, defined as $AR=\sigma_z/\sigma_\perp$.
For different geometries of the probe beam and ensemble, we numerically solve the equations of motion for the mean spin waves and the spin wave covariances to obtain the squeezing parameter. The infinite set of differential equations describing the evolution of the ensemble observables are truncated at some
transverse mode $p_{\text{max}},\,l_{\text{max}}$ once convergence in the set of numerical solutions is achieved. To define the region of maximum mode matching, we also solve for $OD_{\text{eff}}$ for different probe and ensemble geometries. The $OD_{\text{eff}}$ is determined through the formula in \erf{eq::ODeff}, where the beam area $A=\pi\omega_0^2/2$ for the Gaussian probe. Note that unlike the spin squeezing, the effective optical density is a purely geometric quantity.
\subsection{Ensemble Geometry and Optical Pumping}\label{sec::EnGeo}
First, we examine how the geometry of the ensemble affects decoherence due to optical pumping. We focus on the case where the ensemble is prepared in an SCS for $f=1/2$. Figure \ref{fig::ODeff50} shows the dynamics of various spin wave observables for two different ensemble geometries with fixed $OD_{\text{eff}}=50$ and $\omega_0=20\mu\text{m}$. To obtain a fixed optical density at different cloud geometries, we vary $\eta_0$, the peak density of the Gaussian density function in \erf{Eq::AtomicDistribution}. Because the effective optical density is held constant, the strength of the coherent squeezing interaction is identical for each of the ensemble geometries. The effect of decoherence on the different ensemble geometries solely accounts for the discrepancies in the behavior of the observables depicted in Fig. \ref{fig::ODeff50}. Figure \ref{fig::ODeff50} (a) shows the squeezing generated for an ensemble with a ``pancake" geometry ($AR=.1$) and an ensemble with a ``pencil" geometry ($AR=316$). In this plot, the pencil geometry generates substantially more squeezing than the pancake. For reference, the squeezing generated by QND measurement in the absence of decoherence is also depicted. Because the magnitude of this squeezing depends only upon the size of the fixed $OD_{\text{eff}}$, it represents the squeezing generated by either ensemble geometry in absence of decoherence. By comparing the performance of both geometries to the solution without decoherence, it is evident that the pancake geometry is far more susceptible to decoherence due to optical pumping.
To explain the relative robustness of the pencil geometry to optical pumping as compared to the pancake, we also examine the dynamics of the fundamental spin wave variance and mean spin wave for these geometries. Fig. \ref{fig::ODeff50} (b) shows that the behavior of the fundamental spin wave variance for the pancake and pencil geometries is comparable. There is a substantial difference in the decay of the fundamental mean spin wave for these geometries, however, as shown in Fig. \ref{fig::ODeff50} (c). The faster decay of the mean spin for the pancake geometry explains its poor performance compared to the pencil. The mean spin of the pencil decays more favorably because a larger fraction of the atoms are spread out longitudinally, far from the beam waist. Because of the reduced probe intensity farther away from the beam waist, these atoms have a lower rate of optical pumping. To achieve the same $OD_\text{eff}$ in the pancake geometry, the atoms must be concentrated at the waist, where they are more likely to undergo optical pumping. Consequently, the pancake is much more susceptible to optical pumping.
\begin{figure}[H]
\centering
\includegraphics[scale=.37]{ODeff50.pdf}
\caption{Ensemble observables as a function of time for different ensemble geometries, fixed $OD_{\text{eff}}=50$ and fixed beam waist $\omega_0=20 \mu\text{m}$. (a) Squeezing versus time for an ensemble with $AR=0.1$ (green solid line) and $AR=316$ (red dashed line). Also shown are the dynamics of the squeezing without decoherence (black dotted line). (b) The fundamental spin wave variance, normalized by $N_{\text{eff}}^{(2)}/4$, versus time for $AR=0.1$ (green solid line), $AR=316$ (red dashed line) and without decoherence (black dotted line). (c) Fundamental mean spin, normalized by $N_{\text{eff}}^{(1)}/2$, versus time for $AR=0.1$ (green solid line) and $AR=316$ (red dashed line). }\label{fig::ODeff50}
\end{figure}
\subsection{Optimal Ensemble and Beam Geometries for Fixed Atom Number }
In the previous section, we analyzed the effect of the ensemble geometry on the squeezing of the fundamental spin wave. We now investigate optimal geometries of both the ensemble and the beam for a fixed atom number, peak intensity and ensemble volume. The number of atoms and peak intensity are held constant at $N_A=9.8\times 10^6$ and $\eta_0=5\times 10^{11} \text{cm}^{-3}$ , respectively. Figure \ref{fig::Multimode} (a) shows contours of peak squeezing as a function of the aspect ratio and the beam waist for a $f=1/2$ ensemble initially prepared in an SCS. The maximum value of the peak squeezing, $\zeta_{m}^{-1} = 10.0$ dB, occurs at AR $= 256$ at a beam waist of $w^{\rm}_0=31$ $\mu$m. The ensemble geometry is a pencil at the maximum peak squeezing with its length extending over several Rayleigh ranges, $\sigma_z/z^{\rm opt}_R = 2.42$. The transverse width of the cloud at the maximum is slightly larger than the beam waist with $\sigma_\perp/w^{\rm opt}_0 = 1.09$.
To understand the optimal region for squeezing, we plot similar contours of $OD_\text{eff}$ as a function of the aspect ratio and the beam waist in Fig. \ref{fig::Multimode} (b). Comparison of Figs. \ref{fig::Multimode} (a) and (b) shows that the maximum peak squeezing occurs in a region of high effective optical density, but does not perfectly coincide with the maximum of $OD_{\text{eff}}$. Because the effective optical density quantifies the strength of the coherent squeezing interaction, this discrepancy is due to decoherence. The maximum peak squeezing occurs in a region where the beam waist is smaller than in the region of maximum $OD_{\text{eff}}$.
Maximum squeezing occurs at smaller beam waists because the region of the beam with greatest intensity, the Rayleigh range, is smaller. Because the scattering rate $\gamma_s(\mathbf{r})$ is proportional to the local intensity, atoms outside the Rayleigh range experience a decreased rate of optical pumping. Although a smaller Rayleigh range implies a decreased $OD_{\text{eff}}$, the reduction of the decoherence rate dominates in this regime. This is a direct analogy to Sec. \ref{sec::EnGeo}, in which pencil-shaped clouds were more robust to decay due to a large number of atoms farther away from the beam waist.
\begin{figure}[H]
\centering
\includegraphics[scale=.43]{HalfSqODeff.pdf}
\caption{Contours of (a) peak squeezing for $f=1/2$ and (b) $OD_\text{eff}$ as a function of the aspect ratio of the ensemble and the beam waist of the probe. The ensemble is prepared in a SCS. Note that the effective optical density depends only upon the ensemble and probe geometries, making it independent of the spin size $f$. For both plots, the volume of the cloud and total atom number, $N_A=9.84\times10^6$, are held constant. The maximum peak squeezing achieved in (a) is $\zeta_m^{-1}=10.0$ dB, occurring at $AR=256$ and $\omega_0=31\mu m$.}\label{fig::Multimode}
\end{figure}
The equations of motion for spin wave observables of ensembles with $f\geq 1/2$ are derived in Appendix \ref{sec::fSpinWaves}. In Fig. \ref{fig::Multimode4}, we show a preliminary result related to the higher spin case. This figure plots contours of peak squeezing as a function of the aspect ratio and the beam waist for a $f=4$ ensemble initially prepared in an SCS. Like the $f=1/2$ case described above, the number of atoms and the peak intensity are held constant at $N_A=9.8\times 10^6$ and $\eta_0=5\times 10^{11} \text{cm}^{-3}$ , respectively. The volume of the cloud is also fixed. For $f=4$, the maximum peak squeezing is $\zeta_m^{-1}=7.8$ dB, which is smaller than the maximum when $f=1/2$. This occurs because the coupling strength between the light and ensemble, quantified by $\xi_{\text{parax}}$ in \erf{Eq::CouplingStrength}, decreases with increasing $f$.
The maximum for $f=4$ occurs at $AR=300$, which is a pencil, similar to the optimal geometry for $f=1/2$. Also like $f=1/2$, the beam waist of the maximum squeezing, $\omega_0=28\mu m$, is smaller than the beam waist of maximum $OD_{\text{eff}}$.
\begin{figure}[H]
\centering
\includegraphics[scale=.4]{SqContour4.pdf}
\caption{Peak squeezing for an $f=4$ ensemble prepared in an SCS. Contours of the peak squeezing are plotted as a function of the aspect ratio of the ensemble and the beam waist of the probe. Like the contour plots in Fig. \ref{fig::Multimode}, the volume of the cloud and total atom number, $N_A=9.84\times10^6$, are held constant. The maximum peak squeezing is $\zeta_m^{-1}=7.8$ dB, occurring at $AR=300$ and $\omega_0=28\mu m$, as indicated by the ``x". }\label{fig::Multimode4}
\end{figure}
\chapter{Conclusion and Outlook}\label{conclusion}
This dissertation explores how internal spin control affects spin squeezing and decoherence in large ensembles of alkali atoms with hyperfine spin $f$. While most studies of spin squeezing in atomic ensembles have been restricted to the $f=1/2$ case, we demonstrate that higher spin atoms offer substantial advantages. First, the number of internal degrees of freedom is greater, offering numerous options for internal spin control. By using internal spin control to prepare the ensemble in states with larger projection noise, we can enhance the entangling power of the Faraday interaction when $f>1/2$. Post-processing via internal spin control converts this increased interatomic entanglement into metrologically relevant spin squeezing. Post-processing can also squeeze the internal spin of the atoms, producing gains in spin squeezing that increase with $f$. Higher spin atoms also offer advantages due to their robustness to decoherence. Transfers of coherence can preserve interatomic entanglement under optical pumping when $f>1/2$. Harmful optical pumping processes, such as spin flips, are also suppressed at larger $f$. The initial state preparation of the ensemble determines the enhancement in the entangling power of the Faraday interaction and the susceptibility of the ensemble to optical pumping. For the appropriate choice of a fiducial state, higher spin atoms outperform $f=1/2$, even in absence of internal spin squeezing. The peak squeezing achieved by the optimization protocol in Chapter \ref{sec::Beyond} occurs for $f=2$, rather than $f=1/2$.
In this dissertation, we have also introduced new ways of modeling dissipative dynamics in large atomic ensembles. Even for large $f$, we have shown that restricting the hyperfine spin of each atom to an embedded qutrit captures the ensemble dynamics relevant to spin squeezing and optical pumping. For certain fiducial states, we have shown that the multilevel Holstein-Primakoff approximation can be modified to accommodate dissipative dynamics. For these fiducial states, the ensemble of embedded qutrits becomes a Gaussian state on two effective bosonic modes. We formulate optical pumping as a update on the covariance matrix of the Gaussian ensemble state. Using the covariance matrix update formalism, optical pumping can be easily combined with coherent squeezing dynamics. For any fiducial state, the combined effects of squeezing by QND measurement and optical pumping can be modeled on the ensemble through a Stochastic master equation. Using the SME, we derive a system of differential equations describing the ensemble observables relevant to spin squeezing. The squeezing parameter can by optimized through these differential equations to reveal fiducial states that maximize spin squeezing.
These computational methods can also be extended to the case of a three-dimensional atomic ensemble interacting with a paraxial probe beam. This enables us to obtain parameter regimes that maximize mode matching and spin squeezing in the presence of optical pumping. This is the first treatment of optical pumping in a three dimensional atom-light interface.
\section{Future Directions}
There are several future directions for the research presented in this dissertation. A natural project to pursue relates to Chapter \ref{sec::Beyond}, in which we found fiducial states that optimize the squeezing generated by QND measurement. Because we have not studied the landscape of the squeezing parameter as a function of time and the fiducial state, we cannot assert that the states we obtain via numerical search are global optima. A detailed study of the optimization landscape of the squeezing parameter can determine whether these states are global optima and, if not, the states that are. Extending the optimization procedure to squeezing protocols besides QND measurement would also be useful. The optimization procedure requires expressing a squeezing protocol in differential form, which is not accomplished as easily for the double pass protocols. An expression for the double pass squeezing protocols in differential form would also be valuable for the three-dimensional atom light interface presented in Chapter \ref{paraxial}, since we solve for the evolution of the spin wave observables using a truncated set of differential equations.
Another possible extension of this work concerns generating spin squeezing suited for particular metrological applications, such as atomic clocks. The Wineland squeezing parameter that we have used to quantify squeezing throughout this text was originally proposed in the context of Ramsey spectroscopy for atomic clocks. The precision of an atomic clock is dictated by the fluctuations in the measured frequency, $\Delta\omega=\Delta J_z/\expect{\hat{J}_x}$. Here, $\mathbf{J}$ is a collective spin composed of two level systems or qubits with energy splitting $\hbar\omega_0$, where $\omega_0$ is the clock frequency. When the collective spin is composed of qudits, $\Delta\omega^2$ is equivalent to the angular resolution $\Delta\phi^2$ presented in \erf{AngRes}, which determines the squeezing parameter. Atomic clocks composed of alkali ensembles typically treat each atom as a qubit formed by the clock states $\ket{f_\pm,m=0}$, where $f_\pm$ are the angular momenta of the ground hyperfine manifolds. The frequency resolution can be improved by squeezing the collective spin $\mathbf{J}$, which is composed of the ``clock qubits". Many protocols create squeezing in atomic clocks by probing on the clock transition of the atoms, $\ket{f_-,m=0}\rightarrow\ket{f_+,m=0}$ \cite{Oblak05,SchleierSmith09,CHen14}. Using internal spin control, we can prepare each atom in the ensemble in any fiducial state $\ket{\uparrow}$ in the $f$ manifold. A squeezing protocol based on the Faraday interaction can create squeezing in the ensemble of embedded qubits consisting of the fiducial state and coupled state, $\ket{\downarrow}$. Internal spin control can then be used to map the fiducial and coupled states to the clock states, $\ket{\uparrow}\rightarrow\ket{f_+,m=0}$ and $\ket{\downarrow}\rightarrow\ket{f_-,m=0}$. For some choice of fiducial state, can this protocol generate more squeezing than probing directly on the clock transition? This is another avenue for investigation.
Lastly, the continuous tomography protocol described Refs. \cite{Silberfarb05} and \cite{Riofrio11} can reconstruct the spin state of an alkali atom prepared in one of the ground hyperfine manifolds. This is accomplished by evolving an ensemble of identical alkalis through a set of informationally complete observables by internal spin control while the ensemble simultaneously undergoes QND measurement. Rather than reconstructing the state of a single alkali atom $i$, finding a way to reconstruct the second order correlation functions
\begin{align}\label{corr9}
\expect{\Delta\hat{f}_z^{(i)}\Delta\hat{f}_z^{(j)}}_{i\neq j}
\end{align}
would offer a novel way of measuring spin squeezing and other phenomena related to interatomic entanglement in the ensemble. Because \erf{corr9} depends upon covariances between the qutrit operators in Sec. \ref{sec::NewObs}, the differential methods presented in this dissertation might be a natural way of reconstructing the second order correlation function. Tracking the evolution of the covariances under QND measurement, control and optical pumping requires that we use a basis of fiducial, coupled and transfer states that evolve with the internal spin control Hamiltonian. This ensures that the populations remain approximately c-numbers and that the Gaussian approximation used to derive the equation of motion for the covariances under QND measurement in \erf{eq::finalEoMQND7} remains valid. The internal spin control Hamiltonian, however, cannot be expressed in the basis of fiducial, coupled and transfer states for $f>1$. Finding the complete evolution of the covariances requires some adaptation of the procedure.
\chapter*{Appendices}
\addcontentsline{toc}{chapter}{Appendices}
|
1,116,691,498,430 | arxiv | \section{Introduction}
In recent years a number of unusual superconducting (SC) states have been
discovered in different new materials \cite{Bennemann04}. Most of these
materials are strongly type-II superconductors, possessing highly
anisotropic or even quasi-two-dimensional (2D) electronic structures. Of
special interest in the present paper are SC materials showing peculiar
clean-limit features at high magnetic fields and low temperatures, notably
the recently discovered family of heavy-fermion compounds CeRIn$_{5}$ (R =
Rh, Ir and Co) \cite{Petrovic01}, and some of the organic charge transfer
salts of the type (BEDT-TTF)$_{2}$X \cite{Ishiguro90},\cite{Wosnitza96}.
The heavy-fermion compound CeCoIn$_{5}$, for example, which is believed to
be an unconventional ( $d$-wave) superconductor \cite{Izawa01} similar to
the high-Tc cuprates, exhibits the highest Tc ($\sim 2.3$ K) among the
Ce-based heavy-Fermion compounds. This material is characterized by
exceptionally strong Pauli paramagnetic pair-breaking \cite{Clogston62},\cite%
{Chandrasekhar62} due to its extremely large electron effective mass and
small Fermi velocity, which could lead to discontinuous (first-order) SC
phase transitions at sufficiently high magnetic fields \cite{Sarma63},\cite%
{Maki64},\cite{Fulde69}.
Recently Bianchi \textit{et.al.}\cite{Bianchi0203} have observed a dramatic
changeover of the second-order SC phase transition to a first-order
transition in specific heat measurements performed on this material as the
magnetic field is increased above some critical values for both parallel and
perpendicular field orientations with respect to the easy conducting planes.
\ Similar effect has been very recently observed by Lortz \textit{et.al. }%
\cite{Lortz07} in the nearly 2D organic superconductor $\kappa $-(BEDT-TTF)$%
_{2}$Cu(NCS)$_{2}$, but only for magnetic field orientation parallel to the
superconducting layers, where the orbital (diamagnetic) pair-breaking is
completely suppressed. Under these conditions the usual (uniform) SC state
is expected to be unstable with respect to formation of a nonuniform SC
state, predicted more than 40 years ago by Fulde and Ferrel \cite{FF64}, and
by Larkin and Ovchinikov \cite{LO64} (FFLO). The corresponding SC order
parameter is spatially-modulated along the field direction with a
characteristic wavenumber, $q$ , whose kinetic energy cost is compensated by
the Pauli pair-breaking energy. The critical temperature, $T_{fflo}$, for
the appearance of the FFLO phase is found to equal $0.56T_{c}$. At the
corresponding tricritical point the normal, the uniform and non-uniform SC
phases are all met.
The possibility of a changeover to first-order transitions can be
effectively investigated within the Ginzburg-Landau (GL) theory of
superconductivity since for the uniform SC phase (i.e. for $q=0$) the
coefficient (usually denoted by $\beta $) of the quartic term in the GL
expansion changes sign at a temperature $T^{\ast }$, which coincides with $%
T_{fflo}$ \cite{Saint-James69}. The identity of $T^{\ast }$ with $T_{fflo}$
is peculiar to the clean limit of a superconductor with no orbital
pair-breaking. In conventional s-wave superconductors electron scattering by
non-magnetic impurities shifts $T_{fflo}$ below the critical temperature $%
T^{\ast }$ \cite{Agterberg01}, allowing discontinuous phase transitions at
temperatures $T_{fflo}<T\leq T^{\ast }$, since (following Anderson's
theorem) $\beta $ is not influenced by nonmagnetic impurities. In
superconductors with unconventional electron pairing, where $\beta $ is
strongly influenced by non-magnetic impurity scattering, the situation is
reversed, i.e. $T^{\ast }<T_{fflo}$.
The interplay between orbital and spin depairing in a pure $s$-wave
isotropic 3D superconductor was first discussed by Gruenberg and Gunther\cite%
{gruenberg66}, who conjectured (i.e. without presenting any result for the
coefficient $\beta $ ) that for $T<0.56T_{c}$ the N-SC transition is of the
second order whereas at lower field there should be a first-order transition
to a uniform SC phase. \ Houzet and Buzdin\cite{houzet01} have essentially
confirmed this picture by exploiting order-parameter and gradient expansions
in the GL theory to find that $T^{\ast }<T_{fflo}$ so that at temperatures $%
T^{\ast }<T<T_{fflo}$, there are second order transitions to either the LO
or FF phase. It should be noted, however, that the orbital effect was
treated there by using gradient expansions, which is a valid approximation
only at very low magnetic fields.
In contrast to all the works outlined above, Adachi and Ikeda have recently
found \cite{adachi03} that, in a clean, $d$-wave, 2D (layered)
superconductor, the orbital effect always shifts $T_{fflo}$ below $T^{\ast }$%
. In this work the authors have used order parameter expansion in the Gorkov
Green's function approach to the GL theory up to six order, avoiding the
restrictions of gradient expansion by exploiting the lowest Landau level
(LLL) approximation for the condensate of Cooper-pairs. Accounting for
impurity scattering destroys the FFLO phase and, in contrast to the pure
paramagnetic situation, somewhat reduces $T^{\ast }$. The effect of SC
thermal fluctuations was found in this work to broaden the discontinuous
mean-field transition at $T^{\ast }$ into a crossover. The reliance on
(FFLO) wavenumber expansion and on extensive numerical computations in this
work has saved formidable analytical efforts, leaving however, interesting
questions unanswered. In particular, the origin of the relative shift of $%
T_{fflo}$ below $T^{\ast }$ by the orbital effect found in this work, in
contrast to all the other works, remains unknown.
In the present paper we develop a formalism based on order parameter
expansion within the Gorkov theory for a strongly type-II superconductor,
with both $s$- and $d_{x^{2}-y^{2}}$-wave electron pairing at high magnetic
fields, which is sufficiently simple to yield useful analytical expressions
for the SC free energy to any desired order in the expansion. The
fundamental interplay between spin induced paramagnetic and orbital
diamagnetic effects at an arbitrary magnetic field is studied, within a
model of anisotropic electron systems covering the entire 3D-2D crossover
range, without relying on gradient or wavenumber expansions. These
advantages enable us to shed new light on the yet undecided debate
concerning the order of the SC phase transitions in the presence of strong
Zeeman spin splitting, and to push our investigation into the unexplored
region of very low temperatures, where quantum magnetic oscillations have
been shown to be observable in the heavy fermion compounds \cite%
{Settai02,Shishido03}.
Specifically, it is found that the relevant parameter controlling the
relative position of $T_{fflo}$ with respect to $T^{\ast }$ is the
dimensionality of the electronic orbital motion in the crystal lattice,
through its influence on the orbital (diamagnetic)\ pair-breaking effect.
For a 3D Fermi surface (isotropic or anisotropic), where the electron motion
along the magnetic field direction reduces the cyclotron kinetic energy, the
shift of $T^{\ast }$ to low temperatures is larger than that of $T_{fflo}$.
In this case the kinetic energy of Cooper-pairs associated with their motion
along the field can compensate the spin-splitting effect, and thus leading
to an increase of $\beta $ and disappearance of the first-order transition.
The corresponding phase diagram is similar to that suggested in Ref. \cite%
{gruenberg66}, where the N-SC transition is of second order, whereas the
transition between non-uniform and uniform (along the field) SC states is of
first order. In the quasi-2D limit (i.e. for quasi-cylindrical Fermi
surfaces) the enhanced orbital pair-breaking shifts $T_{fflo}$ below $%
T^{\ast }$, in agreement with Adachi and Ikeda \cite{adachi03}.
\section{Order parameter expansion in the presence of spin-splitt Landau
levels}
\vspace{1pt}Our starting point is an effective BCS-like Hamiltonian with a $%
d_{x^{2}-y^{2}}$-wave pairing interaction similar to that exploited,e.g. by
Agterberg and Yang \cite{Agterberg01}. The conventional $s$-wave situation
can be similarly worked out and so will not be presented in detail here. The
thermodynamical potential (per unit volume) for the corresponding $d$-wave
superconductor, as expanded in the order parameter with nonlocal normal
electron kernels, may be written as:
\begin{equation}
\Omega =\frac{\Delta _{0}^{2}}{V}+\sum\limits_{m=1}\frac{\left( -1\right)
^{m}}{m}\widetilde{\Omega }_{2m}\left\{ \Delta \left( \mathbf{R},\mathbf{r}%
\right) \right\} \label{OPExpan}
\end{equation}%
where $\widetilde{\Omega }_{2m}\left\{ \Delta \left( \mathbf{R},\mathbf{r}%
\right) \right\} $ is a functional of the SC order parameter, $\Delta \left(
\mathbf{R},\mathbf{r}\right) $, having a power-low dependence $\sim |\Delta
_{0}|^{2m}$ on the global amplitude, $\Delta _{0}$ , of the order parameter,
and $V$ is a BCS coupling constant (given in units of energy$\times $
volume). The corresponding $d$-wave order parameter depends on both the
center of mass ($\mathbf{R}$) and relative ($\mathbf{r}$) coordinates of a
condensate of electron pairs: $\Delta (\mathbf{R},\mathbf{r})=\Delta (%
\mathbf{R})\varphi (\mathbf{r})$. It should be determined self-consistently
from the corresponding pair-correlation functions. Only stationary solutions
are considered, neglecting quantum and thermal fluctuations. In addition the
order parameter in the mean field approximation is selected as a hexagonal
vortex lattice. Actually this assumption is not very important since the
second order term in the order parameter expansion does not depend on the
vortex lattice structure whereas the lattice structure dependence of the
quartic term is very weak (for reviews see \cite{Rasolt-Tesanovic92},\cite%
{MRVW92}).
For the underlying system of normal electrons we assume a simple model of
quadratic energy dispersion $\varepsilon \left( k_{x},k_{y},k_{z}\right)
=\hbar ^{2}\left( k_{x}^{2}+k_{y}^{2}\right) /2m^{\ast }+\hbar
^{2}k_{z}^{2}/2m_{z}^{\ast }$ and anisotropic effective mass tensor: $%
m^{\ast }\leq m_{z}^{\ast }$. A quasi-2D situation is characterized by a
sufficiently large anisotropy parameter $\chi _{a}=\sqrt{m_{z}^{\ast
}/m^{\ast }}$, corresponding to an elongated Fermi surface with a Fermi
momentum $k_{F}$ and Fermi energy $\varepsilon _{F}\equiv \hbar
^{2}k_{F}^{2}/2m^{\ast }$, which is truncated by the Brillouin zone (BZ)
face at $k_{z,\max }=\pi /d$ , where $d$ is the lattice constant
perpendicular to the easy planes. A parameter determining the dimensionality
of the Fermi surface may be defined by: $v_{0}=\sqrt{\frac{\varepsilon
_{z,\max }}{\varepsilon _{F}}}$ , where $\varepsilon _{z,\max }\equiv \hbar
^{2}k_{z,\max }^{2}/2m_{z}^{\ast }$ is the maximal value of the electron
energy along the field. Thus, in the 2D limit, $\ k_{z,\max }\ll k_{F}$ , we
have $v_{0}\rightarrow 0$ , while the system may be regarded 3D (isotropic
or anisotropic ) if $\ k_{z,\max }\simeq k_{F}$ for which the Fermi surface
is contained entirely within the first BZ, namely for $v_{0}=1$. \
At any order of the expansion, Eq.(\ref{OPExpan}), the nonlocal electronic
kernel of the corresponding functional (see e.g. Eqs. (\ref{Omega_gen}),(\ref%
{Kernel_2}), and Eqs. (\ref{Omega4_def}),(\ref{K4_def})) consists of a
product of $m=1,2,...$ pairs of normal electron Green's functions in a
constant magnetic field,$\mathbf{H}=H\widehat{z}$ (i.e. perpendicular to the
easy conducting layers), which are written in the form: \ $G_{\uparrow
\downarrow }\left( \mathbf{R}_{1},\mathbf{R}_{2},\omega _{\nu }\right)
=G_{0\uparrow \downarrow }\left( \mathbf{R}_{2}-\mathbf{R}_{1},\omega _{\nu
}\right) g\left( \mathbf{R}_{1},\mathbf{R}_{2}\right) $, where the gauge
factor is given by$\ g\left( \mathbf{R}_{1},\mathbf{R}_{2}\right) =e^{-\frac{%
i}{2a_{H}}\left[ \mathbf{R}_{1}\times \mathbf{R}_{2}\right] \cdot \widehat{z}%
}$, $a_{H}=\sqrt{c\hbar /eH}$ , and the gauge invariant part can be
calculated by the well known expression\cite{Bychkov62}
\begin{widetext}
\begin{equation}
G_{0\uparrow \downarrow }\left( \mathbf{R}_{2}-\mathbf{R}_{1},\omega _{\nu
}\right) =\frac{1}{2\pi a_{H}^{2}}\int \frac{dk_{z}}{2\pi }%
e^{ik_{z}(Z_{2}-Z_{1})}\sum_{n}\dfrac{e^{-\rho ^{2}/4}L_{n}(\rho ^{2}/2)}{%
\mu -\varepsilon _{nk_{z}\uparrow \downarrow }+i\hbar \omega _{\nu }+i%
\mathrm{sign}\left( \omega _{\nu }\right) \hbar \Gamma } \label{GreenFunc}
\end{equation}
Here $\omega _{\nu }=\pi k_{B}T\left( 2\nu +1\right) /\hbar $ with $\nu
=0,\pm 1,...$ is Matzubara frequency, $\varepsilon _{nk_{z}\uparrow }=\hbar
\omega _{c}\left( n+1/2+x^{2}-g/2\right) $ , \ $\varepsilon
_{nk_{z}\downarrow }=\omega _{c}\left( n+1/2+x^{2}+g/2\right) $, the
spin-split normal electron energy levels, $\omega _{c}=eH/m^{\ast }$-the
in-plane electronic cyclotron frequency, $x^{2}=\xi ^{2}k_{z}^{2}\equiv
\frac{k_{z}^{2}}{2m_{z}^{\ast }\omega _{c}}$is a dimensionless longitudinal
(parallel to the magnetic field) kinetic energy, $\omega _{c}g\equiv
eH/m_{0} $, is the Zeeman spin splitting energy, $\Gamma $- the impurity
scattering relaxation rate, and $\mu =\hbar \omega _{c}\left(
n_{F}+1/2\right) \approx \varepsilon _{F}=\hbar ^{2}k_{F}^{2}/2m^{\ast }$ is
the chemical potential. The spatial variables are dimensionless in-plane
(perpendicular to the magnetic field) coordinates, $\mathbf{\rho }=\frac{%
\mathbf{R}_{2\bot }-\mathbf{R}_{1\bot }}{a_{H}}$, and longitudinal
coordinates:$\ Z_{1}=\mathbf{R}_{1}\cdot \widehat{z}$ , and $Z_{2}=\mathbf{R}%
_{2}\cdot \widehat{z}$. \ \
\subsection{The quadratic term}
In the expansion, Eq.(\ref{OPExpan}), the second order term, which describes
the SC condensation energy of spin-singlet electron-pairs, propagating from
initial ($i=1$) to final ($i=2$) coordinates $\mathbf{R}_{i}\pm \frac{%
\mathbf{r}_{i}}{2}$, is given by:
\begin{equation}
\Omega _{2}=\frac{\Delta _{0}^{2}}{V}-\frac{1}{\mathcal{V}_{0}}\int d^{3}%
\mathbf{R}_{1}d^{3}\mathbf{R}_{2}\tilde{\Gamma}_{2}\left( \mathbf{R}_{1},%
\mathbf{R}_{2}\right) \tilde{K}_{2}\left( \mathbf{R}_{1},\mathbf{R}%
_{2}\right) \equiv \frac{\Delta _{0}^{2}}{V}-A_{0}\Delta _{0}^{2}
\label{Omega_gen}
\end{equation}%
where $\mathcal{V}_{0}=SL_{z}$ is the volume of the system. The vertex part,
$\tilde{\Gamma}_{2},$ is a product of two order parameters multiplied by the
gauge factors, $g\left( \mathbf{R}_{2},\mathbf{R}_{1}\right) $, which are
functions of the center of mass coordinates only, due to cacellation by the
corresponding phase factors of the order parameters, namely:%
\begin{equation}
\tilde{\Gamma}_{2}\left( \mathbf{R}_{1},\mathbf{R}_{2}\right) =g^{\ast
}\left( \mathbf{R}_{1},\mathbf{R}_{2}\right) g\left( \mathbf{R}_{2},\mathbf{R%
}_{1}\right) \Delta \left( \mathbf{R}_{1}\right) \Delta ^{\star }\left(
\mathbf{R}_{2}\right) \label{Vertex_def}
\end{equation}%
The kernel $\tilde{K}_{2}$ is a product of two translational invariant
Green's functions, convoluted with the corresponding factors of the order
parameters, which depend only on the relative pair coordinates, namely:
\begin{eqnarray}
&&\tilde{K}_{2}\left( \mathbf{R}_{1},\mathbf{R}_{2}\right) =k_{B}T\sum_{\nu
}\int d^{3}\mathbf{r}_{1}d^{3}\mathbf{r}_{2}\varphi \left( \mathbf{r}%
_{1}\right) \varphi ^{\ast }\left( \mathbf{r}_{2}\right) \label{Kernel_2} \\
&&\times G_{0\uparrow }^{\ast }\left( \mathbf{R}_{2}-\mathbf{R}_{1}+\frac{%
\mathbf{r}_{2}-\mathbf{r}_{1}}{2},\omega _{\nu }\right) G_{0\downarrow
}\left( \mathbf{R}_{1}-\mathbf{R}_{2}+\frac{\mathbf{r}_{2}-\mathbf{r}_{1}}{2}%
,\omega _{\nu }\right) \notag
\end{eqnarray}
The factor of the order parameter which depends on the pair center of mass
coordinates is written as\cite{RMP01}:%
\begin{equation}
\Delta \left( \mathbf{R}\right) =c(Z)\sum\limits_{n}e^{i\pi n^{2}/2}\phi
_{n}\left( \mathbf{R}_{\bot }\right) ;\ \ \ \ \ \ \ \ \phi _{n}\left(
\mathbf{R}_{\bot }\right) =e^{i\frac{2\pi n}{a_{x}}X-\left( Y-\frac{\pi n}{%
a_{x}}\right) ^{2}};\ \ \ \ \ \ a_{x}= \label{OrderParam}
\end{equation}%
where $c(Z)=c_{0}e^{iqZ}$ is the Fulde-Ferrell modulation factor. Exploiting
the fact that the kernel $\tilde{K}_{2}\left( \mathbf{R}_{1},\mathbf{R}%
_{2}\right) $ depends only on the difference $\mathbf{R}_{1}-\mathbf{R}_{2}$%
, one may carry out the integration in Eq.(\ref{Omega_gen}) first over the
in-plane mean coordinates $\mathbf{R}_{\perp }\mathbf{=}\left( \mathbf{R}%
_{\bot ,1}+\mathbf{R}_{\bot ,2}\right) /2$ to get the following average
vertex part \cite{Stephen92}
\begin{equation}
\left\langle \tilde{\Gamma}_{2}\right\rangle =\frac{1}{V}\int \tilde{\Gamma}%
_{2}\left( \mathbf{R}_{1},\mathbf{R}_{2}\right) d^{2}R_{\bot }=|c_{0}|^{2}%
\frac{a_{x}}{\sqrt{2\pi }}e^{-\rho ^{2}/2-iq(Z_{2}-Z_{1})}=\Delta
_{0}^{2}e^{-\rho ^{2}/2-iq(Z_{2}-Z_{1})} \label{Vertex_22}
\end{equation}%
where $\Delta _{0}^{2}=\frac{1}{\mathcal{V}_{0}}\int d^{3}R|\Delta \left(
\mathbf{R}\right) |^{2}$ ($\mathcal{V}_{0}=SL_{z}$), and then integrate over
the rest of the coordinates $\mathbf{\rho =}\left( \mathbf{\mathbf{R}_{\bot
,2}}-\mathbf{R}_{\bot ,1}\right) /a_{H}$\ and $\rho _{z}=\left(
Z_{2}-Z_{1}\right) /a_{H}$.
Since, among other things, we are interested in the effect of quantum
magnetic oscillations, we apply a technique of exact summation over LLs
suggested in Ref.\cite{hait}. It is similar to the Poisson summation
formula, which transforms the summation over LLs into summation over
harmonics of the inverse magnetic field, and allows to deal seperately with
the uniform (quasi-classical) contribution and the various quantum
corrections. This technique can be briefly described as follows. Let us
consider the integral representation of the Green's functions, $\left[
n_{F}-n-x^{2}\pm i\omega \right] ^{-1}=\int_{0}^{\infty }d\tau e^{\pm i\tau %
\left[ n_{F}-n-x^{2}\pm i\omega \right] }$ and perform the summation over
LLs using the well known identity, $\sum_{n=0}^{\infty }z^{n}L_{n}\left(
t\right) =\left( 1-z\right) ^{-1}\exp \left( \frac{tz}{z-1}\right) $ with $%
z=e^{\pm i\tau }$ and $t=\rho ^{2}/2$. Taking advantage of these relations
the gauge invariant part of the Green's function for $\omega _{\nu }\geq 0$
can be transformed to:%
\begin{equation}
G_{0\uparrow \downarrow }\left( \mathbf{R}_{2}-\mathbf{R}_{1};\omega _{\nu
}\right) =\frac{1}{2\pi a_{H}^{2}\hbar \omega _{c}}\int \frac{dk_{z}}{2\pi }%
e^{ik_{z}(Z_{2}-Z_{1})}\int_{0}^{\infty }d\tau \frac{e^{i\tau \left[
n_{F}-x^{2}+g+i\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }\right] }}{%
\left( 1-e^{-i\tau }\right) }\exp \left( \frac{\rho ^{2}}{4}\frac{1+e^{i\tau
}}{1-e^{i\tau }}\right) \label{GreenFunc_tau}
\end{equation}%
where $\widetilde{\omega }_{\nu }\equiv \omega _{\nu }/\omega _{c}$, $%
\widetilde{\Gamma }\equiv \Gamma /\omega _{c}$. For $\omega _{\nu }<0$ one
should replace $\tau $ with $-\tau $ (or $\omega _{\nu }$ with $-|\omega
_{\nu }|$).
The scattering of electrons by non-magnetic impurities is taken into account
here as a self-energy correction to the single electron Green's functions
using the standard relaxation time approximation. Vertex corrections to the
quadratic kernel $\tilde{K}_{2}\left( \mathbf{R}_{1},\mathbf{R}_{2}\right) $
(as well as to higher order ones), which are known to exactly cancel the
self-energy insertions in the very weak magnetic field regime of
convensional s-wave superconductors (see e.g. \cite{Werthamer69}), are not
so crucial in the strong magnetic field regime of both the s and d-wave
situations investigated here, and will be therefore neglected in our
calculations, as done, e.g. in Refs.\cite{adachi03,Mineev}. In any event,
for the high magnetic field and relatively clean superconductors considered
here, the length scale, $a_{H}$, corresponding to the diamagnetic
pair-breaking is much smaller than the electron mean free path $v_{F}/\Gamma
$, and the effect of impurity scattering is marginal.
Utilizing this approximation we rewrite the kernel in the following form:
\begin{eqnarray}
\tilde{K}_{2}\left( \mathbf{\rho },\rho _{z}\right) &=&\frac{k_{B}T}{\left(
2\pi a_{H}^{2}\hbar \omega _{c}\right) ^{2}}\sum_{\nu }\int dz_{1}dz_{2}\int
\frac{dk_{z,1}}{2\pi }e^{ik_{z,1}(\rho _{z}+\sigma _{z}/2)}\int \frac{%
dk_{z,2}}{2\pi }e^{ik_{z,2}(\rho _{z}-\sigma _{z}/2)} \label{Kernel2} \\
&&\int_{0}^{\infty }d\tau _{1}e^{i\tau _{1}\left[ n_{F}-\xi
^{2}k_{z,1}^{2}+g+i\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }\right]
}\int_{0}^{\infty }d\tau _{2}e^{-i\tau _{2}\left[ n_{F}-\xi
^{2}k_{z,2}^{2}-g-i\widetilde{\omega }_{\nu }-i\widetilde{\Gamma }\right]
}J\left( \tau _{1},\tau _{2,}\mathbf{\rho }\right) \notag
\end{eqnarray}%
where $\ $ $\mathbf{\sigma =}\left( \mathbf{r}_{\perp ,2}-\mathbf{r}_{\perp
,1}\right) /a_{H}$ , $\sigma _{z}=\left( z_{2}-z_{1}\right) /a_{H}$,\ and%
\begin{eqnarray}
J\left( \tau _{1},\tau _{2,}\mathbf{\rho }\right) &=&\int d^{2}\mathbf{r}%
_{\perp ,1}d^{2}\mathbf{r}_{\perp ,2}f^{\ast }\left( \mathbf{r}_{2}\right)
\left( 1-e^{-i\tau _{1}}\right) ^{-1}\exp \left( \frac{\left( \mathbf{\rho }+%
\mathbf{\sigma }/2\right) ^{2}}{4}\frac{1+e^{i\tau _{1}}}{1-e^{i\tau _{1}}}%
\right) \notag \\
&&\times \left( 1-e^{i\tau _{2}}\right) ^{-1}\exp \left( \frac{\left(
\mathbf{\rho }-\mathbf{\sigma }/2\right) ^{2}}{4}\frac{1+e^{-i\tau _{2}}}{%
1-e^{-i\tau _{2}}}\right) \label{Jdef}
\end{eqnarray}
In Eq.(\ref{Kernel2}) we use the representation $\varphi \left( \mathbf{r}%
\right) =\delta \left( z\right) f\left( \mathbf{r}_{\perp }\right) $ , where
$f\left( \mathbf{r}_{\perp }\right) =\int \frac{d^{2}k}{\left( 2\pi \right)
^{2}}f_{\mathbf{k}}e^{i\left( \mathbf{k\cdot r}_{\perp }\right) }$ describes
the two types of the electron pairing, the symmetric $s$-wave pairing with%
\begin{equation}
f_{sk}=\frac{1}{2}\left( \cos \left( k_{x}d\right) +\cos \left(
k_{y}d\right) \right) \label{s_pair}
\end{equation}%
and $d_{x^{2}-y^{2}}$-wave pairing,%
\begin{equation}
f_{dk}=\frac{1}{2}\left( \cos \left( k_{x}d\right) -\cos \left(
k_{y}d\right) \right) \label{d_pair}
\end{equation}%
producing nodes in the order parameter along the $k_{x}=\pm k_{y}$
directions. The $\delta $-dependence on $z$ enables us to readily perfom the
first two integrations in Eq. (\ref{Kernel2}).
Using the resulting expression for $\tilde{K}_{2}\left( \mathbf{\rho },\rho
_{z}\right) $, and the vertex function $\left\langle \tilde{\Gamma}%
_{2}\right\rangle $ one can calculate the nontrivial coefficient $A_{0}$ in
Eq.(\ref{Omega_gen}) by performing the integrals over $\mathbf{\rho }$,and $%
\rho _{z}$
\begin{equation}
A_{0}=\int d^{2}\mathbf{\rho }d\rho _{z}\tilde{K}_{2}\left( \mathbf{\rho }%
,\rho _{z}\right) e^{-\rho ^{2}/2-iq\rho _{z}} \label{A0_rho}
\end{equation}%
with the other integrals incorporated in the kernel as appearing in Eq.(\ref%
{Kernel2}). It is convenient to perform the integration over $\mathbf{\rho }$
first since both the function $J\left( \tau _{1},\tau _{2,}\mathbf{\rho }%
\right) $and the vertex part have a gaussian dependence on $\mathbf{\rho }$
which can be readily carried out with the result:%
\begin{equation}
J\left( \tau _{1},\tau _{2}\right) =\int d^{2}\mathbf{\rho }e^{-\rho
^{2}/2}J\left( \tau _{1},\tau _{2,}\mathbf{\rho }\right) =\frac{2\pi
J_{p}\left( \tau _{1},\tau _{2}\right) }{2-e^{-i\tau _{1}}-e^{i\tau _{2}}}
\label{Jp1}
\end{equation}%
where%
\begin{equation}
J_{p}\left( \tau _{1},\tau _{2}\right) =\int d^{2}\mathbf{r}_{\perp ,1}d^{2}%
\mathbf{r}_{\perp ,2}f\left( \mathbf{r}_{1}\right) f^{\ast }\left( \mathbf{r}%
_{2}\right) e^{-\frac{\gamma _{\tau }}{8}\mathbf{\sigma }^{2}} \label{Jp2}
\end{equation}%
and $\gamma _{\tau }=\frac{2+e^{-i\tau _{1}}+e^{i\tau _{2}}}{2-e^{-i\tau
_{1}}-e^{i\tau _{2}}}$. \ It should be noted here that the type of pairing
influences the SC condensation energy through the functional dependence of $%
J_{p}$ on the pairing function $f_{\mathbf{k}}$.
Performing the straightforward calculation of $J_{p}\left( \tau _{1},\tau
_{2}\right) $ for both functions one obtains,%
\begin{equation}
J_{sp}\left( \tau _{1},\tau _{2}\right) =\frac{1}{4}\left( 1+e^{-\frac{1}{4}%
\gamma _{\tau }d^{2}}\right) ^{2}\simeq 1 \label{Jsp}
\end{equation}%
\begin{equation}
J_{dp}\left( \tau _{1},\tau _{2}\right) =\frac{1}{4}\left( 1-e^{-\frac{1}{4}%
\gamma _{\tau }d^{2}}\right) ^{2}\simeq \frac{1}{4}\left( \frac{\gamma
_{\tau }d^{2}}{4}\right) ^{2} \label{Jdp}
\end{equation}%
where the last approximate step is obtained in the limit $\frac{\gamma
_{\tau }}{4}d^{2}\ll 1.$ \ This can be justified by noting that the scale of
the function $\gamma _{\tau }$ is of the order unity whereas $d$ (in units
of the magnetic length) is much smaller than one. In the opposite limit: $%
J_{sp}\left( \tau _{1},\tau _{2}\right) =J_{dp}\left( \tau _{1},\tau
_{2}\right) =\frac{1}{4}$\textbf{.}
Thus, noting that the integration of\textbf{\ }$A_{0}$\textbf{\ }over the
center of mass coordinates yields just the total volume of the system, and
performing the integration\ over the relative coordinate $\rho _{z}$,%
\begin{eqnarray}
&&\int d\rho _{z}\int \frac{dk_{z,1}}{2\pi }e^{ik_{1,z}\rho _{z}}\int \frac{%
dk_{z,2}}{2\pi }e^{ik_{z,2}\rho _{z}}\ e^{i\tau _{1}\left[ n_{F}-\xi
^{2}k_{z,1}^{2}+g+i\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }\right]
}e^{-i\tau _{2}\left[ n_{F}-\xi ^{2}k_{z,2}^{2}-g-i\widetilde{\omega }_{\nu
}-i\widetilde{\Gamma }\right] }e^{-iq\rho _{z}} \notag \\
&=&\int \frac{dk_{z}}{2\pi }\ e^{i\tau _{1}\left[ n_{F}-\xi ^{2}\left(
k_{z}+q/2\right) ^{2}+g+i\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }%
\right] }e^{-i\tau _{2}\left[ n_{F}-\xi ^{2}\left( k_{z}-q/2\right) ^{2}-g-i%
\widetilde{\omega }_{\nu }-i\widetilde{\Gamma }\right] }, \label{Kz_integ}
\end{eqnarray}%
one obtaines for a $d$-wave superconductor:%
\begin{eqnarray}
A_{0}^{\left( d\right) } &=&\frac{6k_{B}T}{\left( \hbar ^{2}k_{z,\max
}^{2}/2m^{\ast }\right) ^{2}a_{H}^{2}}\sum_{\nu }\int \frac{dk_{z}}{2\pi }%
\int_{-\infty }^{\infty }d\tau _{1}d\tau _{2}\ \label{A0_d1} \\
&&\times \frac{e^{i\tau _{1}\left[ n_{F}-\xi ^{2}\left( k_{z}+q/2\right)
^{2}+g+i\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }\right] }e^{-i\tau
_{2}\left[ n_{F}-\xi ^{2}\left( k_{z}-q/2\right) ^{2}-g-i\widetilde{\omega }%
_{\nu }-i\widetilde{\Gamma }\right] }}{\left( 2-e^{-i\tau _{1}}-e^{i\tau
_{2}}\right) ^{3}}. \notag
\end{eqnarray}
A similar expression can be derived for an $s$-wave superconductor. Below we
will present only the final result for this case (\emph{\ }see Eqs.\textbf{\
}\ref{A0_fin}\textbf{,}\ref{Theta_s}\textbf{)}.
Eq. (\ref{A0_d1}) is an \textit{exact} representation for the coefficient of
the quadratic term in the order parameter expansion, Eq.\ref{Omega_gen},
which includes low temperature quantum corrections and quantum magnetic
oscillations. It can be written as a sum of contributions from poles at the
2D lattice: $\tau _{1}=2\pi n_{1}$ and $\tau _{2}=2\pi n_{2}$ , with $%
n_{1,2}=0,1,...$. The dominant (zero harmonic) quasiclassical contribution
arises from the pole at $n_{1}=n_{2}=0$ whereas the quantum corrections are
associated with the poles at $n_{1}=n_{2}\not=0$. It is easy to see that the
oscillating terms correspond to the off-diagonal poles, $n_{1}\not=n_{2}$,
in the $\left( \tau _{1},\tau _{2}\right) $-plane.
In the present paper we are interested mainly in the quasiclassical
contribution for which a further simplification can be achieved. Changing to
new variables: $\tau _{2}=\rho _{0}+\frac{\tau }{2};$ \ \ \ $\tau _{1}=\rho
_{0}-\frac{\tau }{2}$, and exploiting the expansion $2-e^{-i\tau
_{1}}-e^{i\tau _{2}}\simeq \rho _{0}^{2}-i\tau $ near the "quasiclassical"
pole $\tau _{1}=\tau _{2}=0$ , one carries out the integral over $\tau $ in
Eq. (\ref{A0_d1}) to have:%
\begin{eqnarray}
A_{0}^{\left( d\right) } &=&\frac{6k_{B}T}{\left( \hbar ^{2}k_{z,\max
}^{2}/2m^{\ast }\right) ^{2}a_{H}^{2}}\sum_{\nu }\int dk_{z}\int_{0}^{\infty
}d\rho _{0}e^{2i\rho _{0}\left[ \xi ^{2}qk_{z}+g+i\widetilde{\omega }_{\nu
}+i\widetilde{\Gamma }\right] } \notag \\
&&\times \left[ n_{F}-\xi ^{2}\left( k_{z}^{2}+\left( q/2\right) ^{2}\right) %
\right] ^{2}e^{-\rho _{0}^{2}\left[ n_{F}-\xi ^{2}\left( k_{z}^{2}+\left(
q/2\right) ^{2}\right) \right] } \label{A0_d2}
\end{eqnarray}%
Note that the lowest order expansion of the denominator in Eq.(\ref{A0_d1})
about $\tau _{1}=\tau _{2}=0$ is kept under the entire range of integration
since the important integration inerval is of the order $\tau \sim \rho
_{0}^{2}\ll \rho _{0}\sim \frac{1}{\sqrt{n_{F}}}\ll 1$. Note also that
throughout this paper we assume that $n_{F}\gg 1$.
It is convenient to rescale variables as%
\begin{equation}
u\equiv \sqrt{n_{F}}\rho _{0};\ \ \ x_{0}\equiv \xi q;\ \ \ v\equiv \frac{%
\xi k_{z}}{\sqrt{n_{F}}} \label{scale_var}
\end{equation}%
and neglect the energy of an electron pair along the $z$-axis, $\left( \xi
q\right) ^{2}$, with respect to Fermi energy, $n_{F}$. Perfoming the
explicit summation over Matsubara frequencies one obtains, in terms of the
new variables, the following result:%
\begin{equation}
A_{0}^{\left( d\right) }=N\left( 0\right) \lambda _{d}\frac{2\pi k_{B}T}{%
\sqrt{\mu \hbar \omega _{c}}}\int_{0}^{\infty }du\frac{1-e^{-\frac{2\omega
_{D}}{\sqrt{\mu \hbar \omega _{c}}}u}}{\sinh \left( \frac{2\pi k_{B}T}{\sqrt{%
\mu \hbar \omega _{c}}}u\right) }e^{-\frac{2\widetilde{\Gamma }}{\sqrt{n_{F}}%
}u}\cos \left( \frac{2g}{\sqrt{n_{F}}}u\right) \Theta _{2}^{\left( d\right)
}\left( u,x_{0}\right) \label{A0_fin}
\end{equation}%
where $\lambda _{d}=3\left( \frac{k_{F}d}{\pi }\right) ^{4}$, $N(0)$ is the
electron density of states per spin at the Fermi energy\texttt{( }$N(0)=%
\frac{\sqrt{m^{\ast }m_{z}^{\ast }}k_{F}}{2\pi \hbar ^{3}}\ $), and:
\begin{equation}
\Theta _{2}^{\left( d\right) }\left( u,x_{0}\right) =\int_{0}^{1}d\nu \left(
1-v^{2}\right) ^{2}\cos \left( 2x_{0}uv\right) e^{-u^{2}\left(
1-v^{2}\right) }. \label{Theta_d}
\end{equation}
A similar result is obtained for an $s$-wave superconductor. In this case $%
\lambda _{d}\rightarrow \lambda _{s}=1$ and
\begin{equation}
\Theta _{2}^{\left( s\right) }\left( u,x_{0}\right) =\int_{0}^{1}d\nu \cos
\left( 2x_{0}uv\right) e^{-u^{2}\left( 1-v^{2}\right) }. \label{Theta_s}
\end{equation}%
For $g=x_{0}=\widetilde{\Gamma }=0$ Eqs. (\ref{A0_fin})(\ref{Theta_s})
reduces to the quadratic term derived by Helfand-Werthammer \cite{WHH}.
\subsection{The quartic term}
The quartic term in the perturbation expansion, Eq.(\ref{OPExpan}), which
corresponds to a closed loop diagram with four vertices, is given by:%
\begin{equation}
\Omega _{4}^{(s,d)}=\frac{1}{\mathcal{V}_{0}}\int d^{3}\mathbf{R}_{1}d^{3}%
\mathbf{R}_{2}d^{3}\mathbf{R}_{3}d^{3}\mathbf{R}_{4}\widetilde{\Gamma }%
_{4}\left( \mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},\mathbf{R}%
_{4}\right) \tilde{K}_{4}\left( \mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},%
\mathbf{R}_{4}\right) , \label{Omega4_def}
\end{equation}%
where the kernel, containing the gauge invariant factors of the four
electron Green's functions, is:
\begin{equation*}
\tilde{K}_{4}\left( \mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},\mathbf{R}%
_{4}\right) =k_{B}T\sum_{\nu }\int d^{3}\mathbf{r}_{1}d^{3}\mathbf{r}%
_{2}d^{3}\mathbf{r}_{3}d^{3}\mathbf{r}_{4}\varphi \left( \mathbf{r}%
_{1}\right) \varphi ^{\ast }\left( \mathbf{r}_{2}\right) \varphi \left(
\mathbf{r}_{3}\right) \varphi ^{\ast }\left( \mathbf{r}_{4}\right)
\end{equation*}%
\begin{eqnarray}
&&\times G_{0\uparrow }^{\ast }\left( \mathbf{R}_{2}-\mathbf{R}_{1}+\frac{%
\mathbf{r}_{2}-\mathbf{r}_{1}}{2},\omega _{\nu }\right) G_{0\downarrow
}\left( \mathbf{R}_{3}-\mathbf{R}_{2}-\frac{\mathbf{r}_{3}-\mathbf{r}_{2}}{2}%
,\omega _{\nu }\right) \notag \\
&&\times G_{0\uparrow }^{\ast }\left( \mathbf{R}_{4}-\mathbf{R}_{3}+\frac{%
\mathbf{r}_{4}-\mathbf{r}_{3}}{2},\omega _{\nu }\right) G_{0\downarrow
}\left( \mathbf{R}_{1}-\mathbf{R}_{4}-\frac{\mathbf{r}_{1}-\mathbf{r}_{4}}{2}%
,\omega _{\nu }\right) \label{K4_def}
\end{eqnarray}%
and the vertex part:
\begin{equation}
\widetilde{\Gamma }_{4}\left( \mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},%
\mathbf{R}_{4}\right) =g^{\ast }\left( \mathbf{R}_{1},\mathbf{R}_{2}\right)
g\left( \mathbf{R}_{2},\mathbf{R}_{3}\right) g^{\ast }\left( \mathbf{R}_{3},%
\mathbf{R}_{4}\right) g\left( \mathbf{R}_{4},\mathbf{R}_{1}\right) \Delta
\left( \mathbf{R}_{1}\right) \Delta ^{\ast }\left( \mathbf{R}_{2}\right)
\Delta \left( \mathbf{R}_{3}\right) \Delta ^{\ast }\left( \mathbf{R}%
_{4}\right) \label{Gamma4_def}
\end{equation}%
which consists of the gauge factors $g(\mathbf{R}_{i},\mathbf{R}_{j})$ and
the order parameter values at the four center of mass positions for two
electron pairs. \
Since the dependence of the order parameter on the relative pair coordinates
is separable from that of the center of mass coordinates, the latter
dependence is selected to have the usual Abrikosov lattice structure,%
\begin{equation}
\Delta \left( \mathbf{R}\right) =c_{0}\ e^{iqZ}e^{-\frac{1}{2}\left(
|u|^{2}-u^{2}\right) }\sum_{n=0,\pm 1,\pm 2,...}e^{iq_{n}u-q_{n}^{2}/4},
\label{Delta_comp}
\end{equation}%
with $q_{n}=2\pi n/a_{x}$ \ and $u=X+iY$, . To simplfy the calculation of
the vertex part we exploit several assumptions. Substituting Eq.(\ref%
{Delta_comp}) to Eq.(\ref{Gamma4_def}), one may keep only diagonal terms
with $q_{n1}=q_{n2}=q_{n3}=q_{n4}=p$ , since all off-diagonal terms are
small by the gaussian factor $\sim \exp \left[ -\left( q_{n4}-q_{n1}\right)
^{2}-\left( q_{n4}-q_{n1}\right) ^{2}\right] $. Furthermore, we may replace
summation over $p$ with an appropriate integration. Both of these
assumptions are equivalent to neglecting particular vortex lattice
structures, corresponding to replacement of the Abrikosov structure
parameter, $\beta _{A}$, with $\frac{\sqrt{\pi }}{a_{x}}$ \cite{RMP01},
which yields only a small error.
With the above assumptions the vertex part reduces to:
\begin{equation}
\widetilde{\Gamma }_{4}\left( \mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},%
\mathbf{R}_{4}\right) =\frac{a_{x}\sqrt{\pi }}{2\pi }|c_{0}|^{4}e^{iq\left(
Z_{1}-Z_{2}+Z_{3}-Z_{4}\right) }e^{-\frac{1}{4}\sum_{l=1}^{3}|\rho
_{l}|^{2}}e^{\frac{1}{4}\left[ \left( u_{1}-u_{3}\right) ^{2}+\left(
u_{2}^{\ast }-u_{4}^{\ast }\right) ^{2}\right] } \label{Gamma4_comp}
\end{equation}%
where $\mathbf{\rho }_{l}=u_{l+1}-u_{l}$. Since the dominant contribution to
the quartic term arises from small propagation distances, $|u_{l}|\ \leq 1$%
\cite{MRVW92,ZM97}, one may expand the last exponential on the RHS of Eq.(%
\ref{Gamma4_comp}), up to leading order, under the integrals over angular
variables in Eq.(\ref{Omega4_def}). Additional angular dependence is due to
the kernel, $K_{4}$, through its dependence on the absolute values of linear
combinations of "external", $\mathbf{R}_{l+1}-\mathbf{R}_{l}$, and
"internal", $\mathbf{r}_{l+1}-\mathbf{r}_{l}$ ($l=1,..,4$), coordinates (see
Eq. \ref{K4_def}). Since the characteristic size of $\left\vert \mathbf{r}%
_{l+1}-\mathbf{r}_{l}\right\vert \sim d$ is much smaller than the scale of $%
\left\vert \mathbf{R}_{l+1}-\mathbf{R}_{l}\right\vert \sim a_{H}$ , the
dependence of the kernel on $\mathbf{r}_{l+1}-\mathbf{r}_{l}$ (and
consequently its dependence on the angular variables) may be neglected at
large $\left\vert \mathbf{R}_{l+1}-\mathbf{R}_{l}\right\vert $. \ Therefore,
the integration over angular variables in this region involves only the last
exponential in Eq.(\ref{Gamma4_comp}), resulting in:
\begin{eqnarray*}
\left\langle e^{\frac{1}{4}\left[ \left( u_{1}-u_{3}\right) ^{2}+\left(
u_{2}^{\ast }-u_{4}^{\ast }\right) ^{2}\right] }\right\rangle &\approx &1+%
\frac{1}{4}\left\langle \left( u_{1}-u_{3}\right) ^{2}+\left( u_{2}^{\ast
}-u_{4}^{\ast }\right) ^{2}\right\rangle =1, \\
\text{since \ }\left\langle u_{l}^{2}\right\rangle &=&\left\langle
u_{l}u_{k}^{\ast }\right\rangle =0\ ,\ \left( l\not=k\right)
\end{eqnarray*}%
whereas for small values of $\left\vert \mathbf{R}_{l+1}-\mathbf{R}%
_{l}\right\vert $ this exponential is always close to $1$ and the remaining
integration over angular variables can be perfomed in closed form (see
below). Thus, one can approximate the vertex part by the following simple
expression:%
\begin{equation}
\widetilde{\Gamma }_{4}\left( \mathbf{R}_{1},\mathbf{R}_{2},\mathbf{R}_{3},%
\mathbf{R}_{4}\right) =\frac{a_{x}\sqrt{\pi }}{2\pi }|c_{0}|^{4}e^{iq\left(
Z_{1}-Z_{2}+Z_{3}-Z_{4}\right) }e^{-\frac{1}{4}\sum |\rho _{l}|^{2}}
\label{Gamma4_fin}
\end{equation}%
which depends only on nearest neighboring coordinates.
Making use of Eq.(\ref{Gamma4_fin}), the remaining calculation of the
quartic term is similar to that used for the quadratic term, but
considerably massier. Below we present only an outline of the derivation.
Since integrations over $z_{i}$ are trivial we shall use from now on only 2D
vector notations with integrations over $Z_{i}$ written explicitly.
Combining Eqs.(\ref{Omega4_def}),(\ref{K4_def}),(\ref{Gamma4_fin}) our
starting expression for the quartic term is given by:%
\begin{equation*}
\Omega _{4}^{(s,d)}=\frac{k_{B}T}{V_{0}}\left( \frac{1}{2\pi \hbar \omega
_{c}}\right) ^{4}\frac{a_{x}\sqrt{\pi }}{2\pi L_{z}}|c_{0}|^{4}\sum_{\nu
}\int dZ_{1}dZ_{2}dZ_{3}dZ_{4}
\end{equation*}%
\begin{equation*}
\times \int \prod_{i=1}^{4}\frac{dk_{z,i}}{2\pi }%
e^{-ik_{z,1}(Z_{2}-Z_{1})}e^{ik_{z,2}(Z_{3}-Z_{2})}e^{-ik_{z,3}(Z_{4}-Z_{3})}e^{ik_{z,4}(Z_{1}-Z_{4})}e^{iq\left( Z_{1}-Z_{2}+Z_{3}-Z_{4}\right) }
\end{equation*}%
\begin{equation}
\ \ \times \int d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2}d^{2}\mathbf{r}%
_{3}d^{2}\mathbf{r}_{4}f\left( \mathbf{r}_{1}\right) f^{\ast }\left( \mathbf{%
r}_{2}\right) f\left( \mathbf{r}_{3}\right) f^{\ast }\left( \mathbf{r}%
_{4}\right) \times \Theta _{4}\left( \mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}%
_{3},\mathbf{r}_{4};\left\{ k_{z,i}\right\} ;\omega _{\nu }\right)
\label{Omega4_int}
\end{equation}%
where the function $\Theta _{4}\left( \mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r%
}_{3},\mathbf{r}_{4};\left\{ k_{z,i}\right\} ;\omega _{\nu }\right) $
includes integration over all electron pair coordinates: \emph{\ }%
\begin{equation*}
\Theta _{4}\left( \mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3},\mathbf{r}%
_{4};\left\{ k_{z,i}\right\} ;\omega _{\nu }\right) =\frac{1}{L_{x}L_{y}}%
\int d^{2}\mathbf{R}_{1}d^{2}\mathbf{R}_{2}d^{2}\mathbf{R}_{3}d^{2}\mathbf{R}%
_{4}e^{-\frac{1}{4}\sum |\rho _{l}|^{2}}\times
\end{equation*}%
\begin{equation*}
\int_{0}^{\infty }d\tau _{1}e^{-i\tau _{1}\left[ n_{F}-x_{1}^{2}-g-i%
\widetilde{\omega }_{\nu }-i\widetilde{\Gamma }\right] }\frac{\exp \left(
\frac{R_{12}^{2}}{4}\frac{1+e^{-i\tau _{1}}}{1-e^{-i\tau _{1}}}\right) }{%
1-e^{i\tau _{1}}}\int_{0}^{\infty }d\tau _{2}e^{i\tau _{2}\left[
n_{F}-x_{2}^{2}+g+i\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }\right] }%
\frac{\exp \left( \frac{R_{23}^{2}}{4}\frac{1+e^{i\tau _{2}}}{1-e^{i\tau
_{2}}}\right) }{1-e^{-i\tau _{2}}}\times
\end{equation*}%
\begin{equation}
\int_{0}^{\infty }d\tau _{3}e^{-i\tau _{3}\left[ n_{F}-x_{3}^{2}-g-i%
\widetilde{\omega }_{\nu }-i\widetilde{\Gamma }\right] }\frac{\exp \left(
\frac{R_{34}^{2}}{4}\frac{1+e^{-i\tau _{3}}}{1-e^{-i\tau _{3}}}\right) }{%
1-e^{i\tau _{3}}}\int_{0}^{\infty }d\tau _{4}e^{i\tau _{4}\left[
n_{F}-x_{4}^{2}+g+i\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }\right] }%
\frac{\exp \left( \frac{R_{41}^{2}}{4}\frac{1+e^{i\tau _{4}}}{1-e^{i\tau
_{4}}}\right) }{1-e^{-i\tau _{4}}} \label{Theta4_def}
\end{equation}%
Here the coordinates, $\mathbf{R}_{i,i+1}$, in Eq.(\ref{Theta4_def}) are the
linear combinations of $\mathbf{\rho }_{l}=\mathbf{R}_{l+1}-\mathbf{R}_{l}$
and $\mathbf{\eta }_{l}=\mathbf{r}_{l+1}-\mathbf{r}_{l}$ :%
\begin{eqnarray}
\mathbf{R}_{12} &=&\mathbf{\rho }_{1}+\frac{1}{2}\mathbf{\eta }_{1};\ \ \ \
\ \ \ \ \ \ \mathbf{R}_{23}=\mathbf{\rho }_{2}-\frac{1}{2}\mathbf{\eta }_{2}
\notag \\
\mathbf{R}_{34} &=&\mathbf{\rho }_{3}+\frac{1}{2}\mathbf{\eta }_{3};\ \ \ \
\ \ \ \ \ \ \mathbf{R}_{41}=\mathbf{\rho }_{4}-\frac{1}{2}\mathbf{\eta }_{4}
\label{rel_coord}
\end{eqnarray}%
The gaussian integration over $\mathbf{\rho }_{l}$ reduces Eq. (\ref%
{Theta4_def}) to:
\begin{equation*}
\Theta _{4}\left( \mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3},\mathbf{r}%
_{4};\left\{ k_{z,i}\right\} ;\omega _{\nu }\right) =\int_{0}^{\infty }d\tau
_{1}d\tau _{2}d\tau _{3}d\tau _{4}\frac{\left( 2\pi \right) ^{3}}{\varkappa }%
\exp \left[ -\frac{1}{8\varkappa }\left( \mathbf{\eta }_{1}-\mathbf{\eta }%
_{2}+\mathbf{\eta }_{3}-\mathbf{\eta }_{4}\right) ^{2}\right]
\end{equation*}%
\begin{equation}
e^{-i\tau _{1}\left[ n_{F}-x_{1}^{2}-g-i\widetilde{\omega }_{\nu }-i%
\widetilde{\Gamma }\right] }e^{i\tau _{2}\left[ n_{F}-x_{2}^{2}+g+i%
\widetilde{\omega }_{\nu }+i\widetilde{\Gamma }\right] }e^{-i\tau _{3}\left[
n_{F}-x_{3}^{2}-g-i\widetilde{\omega }_{\nu }-i\widetilde{\Gamma }\right]
}e^{i\tau _{4}\left[ n_{F}-x_{4}^{2}+g+i\widetilde{\omega }_{\nu }+i%
\widetilde{\Gamma }\right] } \label{Theta4}
\end{equation}%
where $\varkappa =4-e^{i\tau _{1}}-e^{-i\tau _{2}}-e^{i\tau _{3}}-e^{-i\tau
_{4}}$. It should be noted here that Eq.(\ref{Theta4}) has been obtained by
exploiting the fact that the dominant contributions to the integrals
originate in the regions where $\tau _{i}\ll 1$.
Furthermoe, noting that in the above equation the $\mathbf{\eta }_{i}$- and $%
k_{z}$-dependences are factorized, one can perfom the integrations over both
sets of variables separetely. For a $d$-wave superconductor we obtain:
\begin{equation*}
\int d^{2}\mathbf{r}_{1}d^{2}\mathbf{r}_{2}d^{2}\mathbf{r}_{3}d^{2}\mathbf{r}%
_{4}f\left( \mathbf{r}_{1}\right) f^{\ast }\left( \mathbf{r}_{2}\right)
f\left( \mathbf{r}_{3}\right) f^{\ast }\left( \mathbf{r}_{4}\right) \times
\exp \left[ -\frac{1}{8\varkappa }\left( \mathbf{\eta }_{1}-\mathbf{\eta }%
_{2}+\mathbf{\eta }_{3}-\mathbf{\eta }_{4}\right) ^{2}\right]
\end{equation*}%
\begin{equation}
=\frac{1}{4^{3}}\left( 1-e^{-\frac{d^{2}}{\varkappa }}\right) ^{4}\left(
3+2e^{-\frac{d^{2}}{\varkappa }}+e^{-2\frac{d^{2}}{\varkappa }}\right)
^{2}\simeq \frac{9}{16}\frac{d^{8}}{\varkappa ^{4}} \label{r_int}
\end{equation}%
where the last appoximation is valid under the same conditions discussed in
the derivation of the quadratic term. Thus the quartic term is transformed
to:%
\begin{equation*}
\Omega _{4}^{(d)}=\frac{k_{B}T}{\left( \hbar \omega _{c}\right) ^{4}}\frac{%
a_{x}\sqrt{\pi }}{\left( 2\pi \right) ^{3}}|c_{0}|^{4}\frac{9d^{8}}{%
16a_{H}^{8}}\sum_{\nu }\int dk_{z}\int_{0}^{\infty }\frac{d\tau _{1}d\tau
_{2}d\tau _{3}d\tau _{4}}{\left( 4-e^{i\tau _{1}}-e^{-i\tau _{2}}-e^{i\tau
_{3}}-e^{-i\tau _{4}}\right) ^{5}}
\end{equation*}%
\begin{equation}
e^{-i\left( \tau _{1}-\tau _{2}+\tau _{3}-\tau _{4}\right) \left( n_{F}-\xi
^{2}k_{z}^{2}-\xi ^{2}\left( \frac{q}{2}\right) ^{2}\right) }e^{-\left( \tau
_{1}+\tau _{2}+\tau _{3}+\tau _{4}\right) \left( \widetilde{\omega }_{\nu }+%
\widetilde{\Gamma }\right) }e^{i\left( \tau _{1}+\tau _{2}+\tau _{3}+\tau
_{4}\right) \left( g+\xi ^{2}k_{z}q\right) }, \label{Omega4_int2}
\end{equation}%
where an additional integration over $\zeta =\left( \tau _{1}-\tau _{2}+\tau
_{3}-\tau _{4}\right) /2$ for small $\tau _{i}$\ can be performed. Rescaling
variables as%
\begin{equation}
\varrho =\sqrt{n_{F}}\frac{\tau _{1}+\tau _{2}+\tau _{3}+\tau _{4}}{2};\ \ \
\ s=\frac{\sqrt{n_{F}}}{2}\left( \tau _{3}-\tau _{1}\right) ;\ \ \ \ \notag
\end{equation}%
and summing up over $\nu $ one obtain the final result for the quartic term:%
\begin{equation}
\Omega _{4}^{(d)}=B^{(d)}\int_{0}^{\infty }d\varrho \frac{1-e^{-\frac{%
2\omega _{D}}{\sqrt{\mu \hbar \omega _{c}}}\varrho }}{\sinh \left( \frac{%
2\pi k_{B}T}{\sqrt{\mu \hbar \omega _{c}}}\varrho \right) }e^{-\frac{2%
\widetilde{\Gamma }}{\sqrt{n_{F}}}\varrho }\cos \left( 2\varrho g_{0}\right)
\Theta _{4}^{\left( d\right) }\left( \varrho ,q\right) \label{Omega4_fin}
\end{equation}%
where $B^{(d)}=c_{4}^{(d)}B_{0}$ with $c_{4}^{(d)}=\frac{3}{16}\left( \frac{%
k_{F}d}{\pi }\right) ^{8}$, $B_{0}=\left( \frac{\sqrt{\pi }}{a_{x}}\right)
\frac{\pi k_{B}T\Delta _{0}^{4}}{\left( \mu \hbar \omega _{c}\right) ^{3/2}}%
N\left( 0\right) $, and
\begin{equation}
\Theta _{4}^{\left( d\right) }\left( \varrho ,q\right) =\int_{0}^{1}dv\left(
1-v^{2}\right) ^{4}e^{-\frac{1}{2}\varrho ^{2}\left( 1-v^{2}\right) }\cos
\left( 2vq\varrho \right) \left( \int_{0}^{\varrho }dse^{-s^{2}\left(
1-v^{2}\right) }\right) ^{2} \label{Theta4_fin}
\end{equation}%
\end{widetext}
The result for an $s$-wave superconductor can be obtained from Eq.(\ref%
{Omega4_fin}) by replacing the factor $\left( 1-v^{2}\right) ^{4}$ in the
definition of $\Theta _{4}^{\left( d\right) }$ and the factor $c_{4}^{(d)}$
in the normalization coefficient $B^{(d)}$ with unity. The $s$-wave quartic
term for zero spin splitting is equivalent to that obtained in Ref.\cite%
{MRVW92}.
\section{Results and Discussion}
The analysis presented in the previous sections enables us to write a
GL-like expansion of the SC contribution to the thermodynamic potential for
an $s$ or $d_{x^{2}-y^{2}}$- wave pairing up to second order in $\Delta
_{0}^{2}$:%
\begin{eqnarray}
\Omega ^{\left( s,d\right) } &=&\alpha ^{\left( s,d\right) }\left(
t,b,q\right) \Delta _{0}^{2}+\frac{1}{2}\beta ^{\left( s,d\right) }\left(
t,b,q\right) \Delta _{0}^{4} \notag \\
&+&\frac{1}{3}\gamma ^{\left( s,d\right) }\left( t,b,q\right) \Delta
_{0}^{6}+... \label{omega}
\end{eqnarray}
For the quadratic term we have:
\begin{eqnarray}
\alpha ^{\left( s,d\right) }\left( t,b,q\right) &=&\frac{1}{\lambda }-\frac{%
c_{2}^{\left( s,d\right) }}{\varsigma \left( T\right) }\int_{0}^{\infty
}d\rho \frac{\left( 1-e^{-\frac{2\rho }{\varsigma \left( T_{D}\right) }%
}\right) }{\sinh \left( \frac{2\rho }{\varsigma \left( T\right) }\right) }
\notag \\
&&e^{-2\rho /l}\cos \left( \frac{2g}{r_{F}}\rho \right) \Theta _{2}^{\left(
s,d\right) }\left( \rho ,q\right) \notag \\
\Theta _{2}^{\left( s,d\right) }\left( \rho ,q\right)
&=&\int_{0}^{v_{0}}d\nu \vartheta _{2}^{\left( s,d\right) }\left( v\right)
\cos \left( q\rho v/\chi _{a}\right) \notag \\
&&\exp \left[ -\left( 1-v^{2}\right) \rho ^{2}/2a_{H}^{2}\right]
\label{alpha}
\end{eqnarray}%
where $\lambda =N(0)V$ , $\ \varsigma \left( T_{D}\right) \equiv \hbar
v_{F}/\pi k_{B}T_{D}$ , with $T_{D}$ the Debye temperature, and $v_{F}=\sqrt{%
2\varepsilon _{F}/m^{\ast }}$-the inplane Fermi velocity, $\varsigma \left(
T\right) \equiv \hbar v_{F}/\pi k_{B}T$- the thermal mean-free path, $r_{F}=%
\sqrt{2n_{F}}a_{H}$- the electronic cyclotron radius at the Fermi energy,
and $l$ is the mean-free path due to impurity scattering.
The differences between $s$-wave and $d$-wave SCs are given by $%
c_{2}^{\left( s\right) }=1$, $\ c_{2}^{\left( d\right) }=3\left( \frac{k_{F}d%
}{\pi }\right) ^{4}$, and $\vartheta _{2}^{\left( s\right) }\left( v\right)
=1$, $\vartheta _{2}^{\left( d\right) }\left( v\right) =\left(
1-v^{2}\right) ^{2}$.
The quartic term has a similar structure:%
\begin{eqnarray}
&&\beta ^{\left( s,d\right) }\left( t,b,q\right) =B_{0}c_{4}^{\left(
s,d\right) }\int_{0}^{\infty }d\rho \frac{\left( 1-e^{-\frac{2\rho }{%
\varsigma \left( T_{D}\right) }}\right) }{\sinh \left( \frac{2\rho }{%
\varsigma \left( T\right) }\right) } \notag \\
&&e^{-2\rho /l}\cos \left( \frac{2g}{r_{F}}\rho \right) \Theta _{4}^{\left(
s,d\right) }\left( \rho ,q\right) \label{beta} \\
&&\Theta _{4}^{\left( s,d\right) }\left( \rho ,q\right)
=\int_{0}^{v_{0}}d\nu \vartheta _{2}^{\left( s,d\right) }\left( v\right)
\cos \left( q\rho v/\chi _{a}\right) \notag \\
&&\exp \left[ -\left( 1-v^{2}\right) \rho ^{2}/4a_{H}^{2}\right] \left(
\int_{0}^{\rho /\sqrt{2}a_{H}}dse^{-s^{2}\left( 1-v^{2}\right) }\right) ^{2}
\notag
\end{eqnarray}%
where \ $B_{0}=\left( \frac{\sqrt{\pi }}{a_{x}}\right) \frac{N\left(
0\right)\pi k_{B}T}{\left( \varepsilon _{F}\hbar \omega _{c}\right) ^{3/2}}$
and $c_{4}^{\left( s\right) }=1$, $c_{4}^{(d)}=\frac{3 }{16} \left( \frac{%
k_{F}d}{\pi }\right) ^{8}$, $\vartheta _{4}^{\left( s\right) }\left(
v\right) =1$, and $\vartheta _{4}^{\left( d\right) }\left( v\right) =\left(
1-v^{2}\right) ^{4}$. \
On the basis of the above formulas we discuss below the H-T phase diagram
for different values of the relevant parameters. Three independent
dimensionless parameters:$\ \left( 2a_{H}/r_{F}\right) g$ , $%
2a_{H}/\varsigma \left( T\right) $ and $qa_{H}/\chi _{a}$, control the basic
integrals in these equations. The first two parameters measure the strength
of the spin and thermal pair-breaking mechanisms, respectively, relative to
the orbital (diamagnetic) depairing. The third parameter determines the
relative strength of the compensating FFLO mechanism. The value of the spin
pair-breaking parameter, $\ \sigma \equiv g\left( 2a_{H}/r_{F}\right)
_{H=H_{c20}^{orb}}$ , where $H_{c20}^{orb}$ is the upper critical field at $%
T=0$ in the absence of spin pair-breaking, is related to the well known Maki
parameter \cite{Maki64}, $\alpha _{M}=\frac{\left( \hbar e/m_{0}c\right)
H_{c20}^{orb}}{1.76k_{B}T_{c0}}$, by: $\sigma =1.1\alpha _{M}$. Here $T_{c0}$
is the transition temperature at zero magnetic field.
\begin{figure}[tbp]
\includegraphics[width=7cm]{fig1.eps}
\caption{Phase transition lines for a 3D system with s-wave pairing : N-SC
transitions for order parameters with (solid line) or without (doted line)
FFLO modulation, and FFLO-BCS transitions obtained for GL free energy with
(dashed line) or without (doted-dashed line) quartic correction. The N-SC
transition is of second order whereas the FFLO-BCS transition is of first
order. The value of the spin splitting parameter is $\protect\sigma=1.8$.}
\label{fig:1}
\end{figure}
\begin{figure}[tbp]
\includegraphics[width=7cm]{fig2.eps}
\caption{ The GL coefficients (in arbitrary units ) $\protect\beta \left(
t,b,q=0\right) $ (dashed lines), $\frac{d\protect\alpha \left(
t,b,q=0\right) }{dq^{2}}$(solid lines) ,$\frac{d\protect\beta \left(
t,b,q=0\right) }{dq^{2}}$(doted lines) as functions of the parameter $\frac{%
2a_{H}}{\protect\varsigma \left( T\right) }$ for (a) $v_{0}=.4$ \ and (b) $%
v_{0}=1$. }
\label{fig:2}
\end{figure}
As we shall show below, the situation $T_{fflo}>T^{\ast }$, where $T^{\ast }$
is the temperature at which $\beta \left( t,b_{c2},q=0\right) =0$ , is
realized in 3D systems (corresponding to $v_{0}=1$), regardless of the
spin-splitting strength and the type of electron pairing. \ A typical phase
diagram is shown in Fig. 1 for $s$-wave pairing and spin pair-breaking
parameter $\sigma =3$.
\begin{figure}[tbp]
\includegraphics[width=6cm]{fig4a.eps} \\
\includegraphics[width=6cm]{fig4b.eps} \\
\includegraphics[width=6cm]{fig4c.eps} \\
\hspace{0cm} $\widetilde{q}$
\caption{(a) The dependence of the GL coefficient $\protect\alpha $ (orange)
and the mean field SC free energy, $-\protect\theta \left( \protect\alpha %
\right) \frac{\protect\alpha ^{2}}{2\protect\beta _{A}\protect\beta }$
(blue), on the modulation paramer, \ $\widetilde{q}\equiv \left( \frac{2}{%
\protect\sigma }\right) \protect\sqrt{\frac{m^{\ast }}{m_{z}^{\ast }}}\left(
qa_{H_{c20}^{orb}}\right) $, in a 3D system ($v_{0}=1$) , at $t=0.4$, $%
b=0.1142$, i.e. near the tricritical point, just below the Normal-nonuniform
SC transition. It is seen that $\protect\alpha <0$ in a small region around $%
\ \widetilde{q}=0.6$ where the SC free energy has a minimum.{\protect\large %
\ }The value of the spin splitting parameter is $\protect\sigma =3$. (b):
The same as in (a) but for a slightly lower field, $b=0.114$, where a
uniform ($q=0$) metastable SC state is present. (c): The same as in (b) but
for a slightly lower field, $b=0.1139$, where a uniform ($q=0$)\ equilibrium
SC state is present, while a metastable SC state exists at $q\not=0$.}
\label{fig:3}
\end{figure}
As long as $T>T^{\ast }$ (so that $\beta \left( t,b_{c2},q=0\right) >0$) the
normal to SC (N-SC) phase transition is of second-order and the (reduced)
critical field, $b_{c2}\left( t\right) $, can be determined as the maximal
value of $b\equiv H/H_{c20}^{orb}$ obtained from the equation $\alpha \left(
t,b,q\right) =0$ for all values of $q$ , at the (reduced) temperature $%
t\equiv T/T_{c0}$. The solution of this equation for $q=0$ yields a
transition line, $b=b_{c2}^{\left( 0\right) }\left( t\right) $, ignoring the
possibility of a FFLO state. The tricritical point, $T_{fflo}$, is defined
as the maximal temperature at which $b_{c2}\left( t\right) >b_{c2}^{\left(
0\right) }\left( t\right) $. It can alternatively be determined from the
equation $\frac{d\alpha \left( t,b_{c2},q=0\right) }{dq^{2}}=0$ , which is
equivalent to the condition for vanishing of the coefficient of $\left\vert
\nabla \Delta \right\vert ^{2}$ in a gradient expansion of the SC\ free
energy\cite{houzet01}.
For $T<T^{\ast }$ and sufficiently strong spin pair-breaking there can be a
changeover to first-order SC transitions, but since $\beta \left(
t,b_{c2},q\neq 0\right) >\beta \left( t,b_{c2},q=0\right) $ (see Fig.2 ),
the segment of the $b_{c2}\left( t\right) $-line with first order
transitions arises only at very low temperatures. For moderate\ $\sigma $
values\ the coefficient\ $\beta \left( t,b_{c2},q\right) $\ at optimal\ $q$\
is always positive and the N-SC transition is of the second order at
arbitrarily low temperature.
The transition within the SC region from the nonuniform (FFLO) to uniform
(BCS) phase at $T>T^{\ast }$ can not be obtained just by analyzing the
quadratic term $\alpha \left( t,b,q\right) $ since the SC order parameter is
finite there. It can be obtained by minimizing the SC free energy (including
both quadratic and quartic terms) with respect to the modulation wave number
$q$. \ Neglecting the sixth and higher order terms in the expansion, the
corresponding (standard) GL free energy, $\Omega \left( q\right) \simeq
-\theta \left( \alpha \right) \frac{\alpha ^{2}}{2\beta }$, ( $\theta \left(
\alpha \right) $\ being the Heaviside step function), which has a single
minimum at $q\not=0$ for field near $b_{c2}$ (see Fig. 3a), developes a
double-well structure (see Fig. 3b) as a function of $q$ upon decreasing the
field below $b_{c2}$ at a given temperature $T$ (due to the symmetry $%
q\leftrightarrow -q$ only positive values may be considered). One of these
minima is always at $q=0$, and it becomes energetically favorable at a
critical field for a first order phase transition from the FFLO to the
uniform BCS phase. The second (metastable) minimum at $q\not=0$\ disappears
completely upon further field decrease (see Fig. 3c).\ \ \
\begin{figure}[th]
\includegraphics[width=6cm]{fig5a.eps} \\
\includegraphics[width=6cm]{fig5b.eps} \\
$\widetilde{q}$
\caption{The GL coefficients, $\protect\alpha $\ (green) and $\protect\beta $%
\ (blue), and the mean field SC free energy (orange) as functions of the
modulation parameter, $\widetilde{q}$, in a 3D system ($v_{0}=1$) at a
relatively low temperature $t=.25$\ and decreasing field values (a) $b=.118$%
\ and (b) $b=.117$. Note the vanishing of $\protect\beta $ inside the region
where $\protect\alpha <0$ , around which the used approximation, $-\protect%
\theta \left( \protect\alpha \right) \frac{\protect\alpha ^{2}}{2\protect%
\beta _{A}\protect\beta }$, for $\Omega (q)$ breaks down (dashed sector of
the orange line ). }
\label{fig:4}
\end{figure}
At temperatures $T$ below $T^{\ast }$ the first two terms in the expansion
of the thermodynamic potential are not sufficient to correctly describe the
uniform SC state since for negative $\beta $-\ values the scale of the SC
free energy is determined by the sixth order term.\texttt{\ }In contrast,
the free energy of the nonuniform state, where $\beta \left(
t,b,q\not=0\right) >0$, can be obtained from the stantard GL functional
(with the assumption that the contribution of the sixth order term is small
compared to that of the quartic term). The characteristic $q$-dependences of
the GL coefficients, $\alpha $\ and $\beta $, and the mean field free energy
$-\theta \left( \alpha \right) \frac{\alpha ^{2}}{2\beta _{A}\beta }$, for $%
T<T^{\ast }$\ are illustrated in Fig. 4.\texttt{\ }Whereas at high fields
(Fig. 4a) the minimum of the SC energy occurs in a region where $\beta >0$,
at lower fields (see Fig. 4b) it approaches the expanding temperature domain
of negative $\beta $.\texttt{\ }Thus, even for moderate spin splitting and
low temperature the transition line from the nonuniform to uniform SC state
cannot be determined without knowing the sixth-order term. It is clear,
however, that this transition is of the first order.
It should be noted that if one attempts to determine the FFLO-BCS phase
boundary from the equation $\frac{d\alpha \left( t,b,q=0\right) }{dq^{2}}=0$
it will greatly overestimate the size of the FFLO phase as compared to that
obtained by minimizing $-\frac{\alpha ^{2}}{2\beta }$ (see Fig. 1). This
remarkable difference is due to the strong $q^{2}$-dependence of the quartic
coefficient $\beta $ (see Fig.2).
The suppression of the orbital effect in the considered 3D systems, with
ellipsoidal Fermi surfaces contained entirely within the BZ, is due to the
factor $1-v^{2}$ appearing in the Gaussian exponents of Eqs. (\ref{alpha}),(%
\ref{beta}). The recovery of this effect in quasi-2D systems with truncated
ellipsoidal Fermi surface, where $v_{0}<1$, can reverse the relation between
$T_{fflo}$ and $T^{\ast }$. Fig. 2b, where the GL coefficients are shown for
$\sigma =1.8$, $v_{0}=.4$ and $s$-wave pairing, illustrates the situation
with $T_{fflo}<T^{\ast }$, which occurs for all values of $v_{0}$ below a
critical dimensionality $v_{0,cr}\approx 0.44$ (see Fig. 5), and depends
only weakly on the spin-splitting parameter $\sigma $.
\begin{figure}[th]
\includegraphics[width=6cm]{fig3.eps}
\caption{Solutions$\ \frac{2a_{H}}{\protect\varsigma \left( T\right) }%
\propto T$ of the equation $\protect\beta \left( q=0\right) =0$,
corresponding to $T^{\ast }$ (dashed line), and the equation $\frac{d\protect%
\alpha \left( q=0\right) }{dq^{2}}=0$, corresponding to $T_{fflo}$ (solid
line) vs the dimensionality crossover parameter $v_{0}$ for $\protect\sigma %
=2.5$.}
\label{fig:5}
\end{figure}
\begin{figure}[tbp]
\includegraphics[width=7cm]{fig6.eps}
\caption{Schematic phase diagram for a quasi 2D system. The shaded area
corresponds to a nonuniform SC (FFLO) phase, the dashed line corresponds to $%
\protect\alpha \left( t,b,q=0\right) =0$ , and the dotted-dashed line can be
obtained from $\frac{d}{dq^{2}}\protect\alpha \left( t,b,q=0\right) =0$.}
\label{fig:6}
\end{figure}
The corresponding phase diagram (see Fig. 6 ) for $v_{0}$ below this
crossing point is quite different from that found for the 3D systems shown
in Fig.1. First of all, since $\beta <0$ , one may use Eqs. (\ref{alpha}),(%
\ref{beta})\ to determine the phase diagram only under the assumption that
the sixth order coefficient $\gamma $ is positive (see Ref.\cite{adachi03}).
In this case a discontinuous SC transition occurs at $\Omega \left( \Delta
_{0}^{2}\right) =0$\ with $\Delta _{0}^{2}=\left( \frac{3\left\vert \beta
\right\vert }{4\gamma }\right) $ and $\alpha =\frac{3\beta ^{2}}{16\gamma }%
>0 $ , and the corresponding critical field, $b_{c2}(t)$ , should be larger
than $b_{c2}^{0}\left( t\right) $ , obtained from the equation $\alpha
(q=0)=0$. \ Thus, at a temperature below $T^{\ast }$, the N-SC phase
boundary includes a segment of first order transitions, which may end at
zero temperature, or at a finite temperature, depending on the
spin-splitting strength. This dependence appears because of the competition
between the decreasing explicit dependence of $\beta $ on decreasing
temperature and its increasing implicit dependence through $q\left( T\right)
$ at the FFLO state. \ The boundary between the BCS and FFLO states should
be determined by minimizing the free energy, Eq. (\ref{omega}), with respect
to $q$. This may be restricted to the explicit dependence on $q$ since the
order parameter is determined by: $\frac{\partial \Omega }{\partial \Delta
^{2}}=0$. Consequently the positive sign of $\frac{d\beta \left(
t,b,q=0\right) }{dq^{2}}$ (see Fig. 2b) results in partial cancellation of
the leading contribution to $\frac{\partial \Omega }{\partial q^{2}}$, which
is proportional to $\frac{d\alpha \left( t,b,q=0\right) }{dq^{2}}$\ and
negative in the FFLO part of the phase diagram.\ Moreover, since for the
discontinuous transition the order parameter is finite just below the
transition the higher order terms in $\Delta _{0}^{2}$ (Eq. (\ref{omega}))
should be taken into account. As a result $T_{fflo}$ should be smaller than $%
T_{fflo}^{0}$- the temperature obtained from the equation $\frac{d\alpha
\left( t,b,q=0\right) }{dq^{2}}$ $=0$ , as schematically shown in Fig. 6.
\section{ Conclusions}
It is shown that the expected changeover to first-order SC transitions in
clean, strongly type-II superconductors in the Pauli paramagnetic limit can
take place only in materials with quasi-cylindrical Fermi surfaces,
regardless of the type of the electron (s or d-wave) pairing interaction
which leads to superconductivity. This finding clarifies the confusing
current literature on this topic\cite{gruenberg66},\cite{houzet01},\cite%
{adachi03}.
The observation of such a changeover in the heavy fermion compound CeCoIn$%
_{5}$ for magnetic field orientation perpendicular to the easy conducting
plane\cite{Bianchi0203} is consistent with the quasi-2D character of its
electronic band structure \cite{Settai01}. The interesting situation of a 2D
superconductor under a magnetic field parallel to the conducting plane, for
which a changeover to discontinuous SC transitions was reported very recently%
\cite{Lortz07}, is more subtle since the vanishingly small cyclotron
frequency characterizing this case does not allow utilization of the Landau
orbitals approach employed here (for a recent review see, e.g.\cite%
{Matsuda-Shimahara07}).
\textbf{Acknowledgements}: This research was supported by the Israel Science
Foundation founded by the Academy of Sciences and Humanities, by the
Argentinian Research Fund at the Technion, and by EuroMagNET under the EU
contract RII3-CT-2004-506239.\vspace{1pt}
|
1,116,691,498,431 | arxiv | \section*{Appendix}
\subsection*{Solubility of thiadiazole derivative (1-[5-(3-chloro-4-methyl-phenylamino)-1,2,4-thiadiazol-3-yl]-propan-2-ol) in scCO$_2$}
In this work we measured the solubility (concentration of a saturated solution) of thiadiazole derivative (structural formula shown at Fig. \ref{Fig01supp}) in scCO$_2$ as a function of temperature in the temperature range of 313.15--393.15 K along the isochore, corresponding to CO$_2$ density equal to 1.3 of its critical value ($\rho_{cr}=10.6249$ mol/l). Here we used a self-consistent approach developed by us (see e.g. refs. \cite{oparin2016new,kalikin2020carbamazepine}). Within this approach, based on the Beer-Lambert law, we use the integral extinction coefficient $\varepsilon_{int}$ (molar absorption coefficient) value of a chosen analytical spectral band.
\begin{figure}[h!]
\center{\includegraphics[width=0.7\linewidth]{thiad_struct.jpg}}
\caption{Molecular structure of the studied thiadiazole derivative.}
\label{Fig01supp}
\end{figure}
On the first stage, in order to define the temperature dependence of $\varepsilon_{int}$ we measured the IR spectra of thiadiazole derivative in its solution in chloroform (CHCl$_3$) with concentration of the solute ($c$) equal to $1.2549\cdot10^{-2}$ mol/l. The spectra were measured on FT-IR spectrometer Bruker Vertex 80v using special high pressure high temperature (HPHT) optical cell with a constant volume developed by us. This cell as well as the experimental setup are described in details in our previous works (see e.g. refs. \cite{oparin2014dynamic,oparin2019polymorphism}). The spectra were measured in the wavenumber range of 1000--4000 cm$^{-1}$ with a resolution of 1 cm$^{-1}$, the optical path length ($l$) was 0.140 mm. Set of the experimental spectra in the wavenumber range of 1000--1650 cm$^{-1}$ is presented in Fig. \ref{Fig1supp}a. The spectral band in the domain of 1030--1070 cm$^{-1}$ that is related to deformation vibrations of Cl - methyl - substituted benzene ring in molecule of thiadiazole derivative (see Fig. \ref{Fig01supp}) was chosen as analytical. To calculate the integral intensity of this band ($A$) we applied standard procedure of spectra fitting based on the non-linear curve approximation using Fityk software package \cite{wojdyr2010fityk}. Example of the fitting of the analytical spectral band is shown in the Fig. \ref{Fig1supp}b. The dependence of $A=f(T)$ presented in Fig. \ref{Fig1supp}c can be described by a linear equation with high accuracy, confirming the correctness of the analytical spectral band choice (see e.g. refs. \cite{oparin2016new,kalikin2020carbamazepine,oparin2014dynamic}). Then, the data obtained by the linear fit were used to calculate temperature dependence of $\varepsilon_{int}$ according to following equation:
\begin{equation}
\varepsilon_{int}=\frac{A(T)}{l\cdot c}
\end{equation}
The dependence of $\varepsilon_{int}$ is presented in Fig. \ref{Fig1supp}d. All these values as well as the values of the intensity and data obtained by their linear fit are tabulated in Table \ref{table_supp_1}.
In order to define the concentration of a saturated solution of thiadiazole derivative in scCO$_2$, on the second stage we measured the IR spectra of thiadizazole derivative in its saturated solution in scCO$_2$ being in permanent contact with the excess of the crystalline thiadiazole derivative in the same temperature range along the same isochore. For these measurements we used the same HPHT cell but with the optical path length of 1.127 mm. The spectra were also measured in the wavenumbers range of 1000--4000 cm$^{-1}$ with a resolution of 1 cm$^{-1}$ and presented in Fig. \ref{Fig2supp}a. We used the same procedure of spectra fitting to define the temperature dependence of the integral intensity of the analytical spectral band (see Fig. \ref{Fig2supp}b). The dependence of $A=f(T)$ is presented in Fig. \ref{Fig2supp}c. As would be expected, this dependence can be fitted with the high accuracy by the exponential equation, which is typical for such slightly soluble in scCO$_2$ organic substances (see e.g. refs. \cite{oparin2016new,kalikin2020carbamazepine,oparin2014dynamic}).
Then, we used these data to calculate the concentration of a saturated solution of the thiadiazole derivative in scCO$_2$ solving the inverse task:
\begin{equation}
c(T)=\frac{A(T)}{l\cdot\varepsilon_{int}}
\end{equation}
The dependence of the thiadiazole derivative solubility in scCO$_2$ ($c=f(T)$) is presented in Fig. \ref{Fig2supp}d. We also tabulated these values along with the values of $A$ and data obtained by their exponential fit (see Table \ref{table_supp_1}). In this table we also presented the molar fraction values ($X$) of the thiadiazole derivative in scCO$_2$ as a function of temperature. $X$ values were calculated following next equation:
\begin{equation}
X=\frac{c_{solute}}{c_{solute}+\rho_{solvent}},
\end{equation}
where $\rho_{solvent}$ is CO$_2$ density equal to 1.3$\cdot\rho_{cr}$.
\begin{figure}[h!]
\center{\includegraphics[width=0.7\linewidth]{Suppl.1.Extinction.jpg}}
\caption{a. Experimental IR spectra of thiadiazole derivative dissolved in CHCl$_3$ (solute concentration 1.2549$\cdot 10^{-2}$ mol/l), area limited with rectangular corresponds to the analytical spectral band, insert corresponds to the enlarges analytic spectral band area); b. Example of the fitting of the analytical spectral band at $T=383.15$ K (dots -- experimental spectrum, blue line -- superposition, red line -- modeling spectral band, green line -- base line); c. Temperature dependence of the modeling spectral band integral intensity, obtained from the fitting procedure (dots -- data, line -- fitting line); d. Temperature dependence of the calculated extinction coefficient of analytical spectral band.}
\label{Fig1supp}
\end{figure}
\begin{figure}[h!]
\center{\includegraphics[width=0.7\linewidth]{Suppl.2.Solubility.jpg}}
\caption{a. Experimental IR spectra of saturated solution of thiadiazole derivative in scCO2, area limited with rectangular corresponds to the analytical spectral region, insert corresponds to the enlarges analytic spectral band are); b. Sample of the fitting of the analytical spectral band at $T=383.15$ K (dots -- experimental spectrum, blue line -- superposition, red line -- spectral profile corresponding to analytical spectral band (see Fig. \ref{Fig1supp}a), dark yellow lines -- other spectral contributions, green line -- base line); c. Temperature dependence of the integral intensity of the modeling spectral profile corresponding to the analytical spectral band (dots -- data, line -- fitting line). d. Temperature dependence of the calculated concentration of saturated solution of thiadiazole derivative in scCO$_2$.}
\label{Fig2supp}
\end{figure}
\begin{table}[h!]
\centering
\caption{Extinction coefficient ($\varepsilon_{int}$) of the analytical spectral band calculated on the basis of integral intensity $A^*$, $c$ -- concentration of a saturated solution of thiadiazole derivative in scCO$_2$ calculated on the basis of the integral intensity $A^{**}$ and ($\varepsilon_{int}$), $X$ -- molar fraction of thiadiazole derivative in the solution corresponding to the concentration $c$. $A$ -- integral intensity of analytical spectral band, $A^*$ -- integral intensity of analytical spectral band fitted by linear equation, $A^{**}$ -- integral intensity of analytical spectral band fitted by the exponential equation.}
\begin{tabular}{c|c|c|c|c|c|c|c}
\footnotesize
& \multicolumn{3}{c}{Thiadiazole derivative in CHCl$_3$} & \multicolumn{4}{|c}{Thiadiazole derivative in scCO$_2$} \\
\hline
$T$,K & $A$,[cm$^{-1}$] & $A^*$,cm$^{-1}$ & $\varepsilon_{int}$,cm$\cdot$ mol$^{-1}$ & $A$,cm$^{-1}$ & $A^{**}$,cm$^{-1}$ & $c$,mol/l & $X$,m.f. \\
\hline
313.15 & 0.389 & 0.3872 & 2.2041$\cdot10^6$ & 0.161 & 0.1357 & 5.4644$\cdot10^{-4}$ & 3.9561$\cdot10^{-5}$ \\
323.15 & 0.381 & 0.3818 & 2.1731$\cdot10^6$ & 0.200 & 0.2101 & 8.5771$\cdot10^{-4}$ & 6.2095$\cdot10^{-5}$ \\
333.15 & 0.378 & 0.3763 & 2.1422$\cdot10^6$ & 0.301 & 0.3034 & 1.2566$\cdot10^{-3}$ & 9.0972$\cdot10^{-5}$ \\
343.15 & 0.370 & 0.3709 & 2.1112$\cdot10^6$ & 0.396 & 0.4205 & 1.7674$\cdot10^{-3}$ & 1.2795$\cdot10^{-4}$ \\
353.15 & 0.367 & 0.3655 & 2.0802$\cdot10^6$ & 0.560 & 0.5676 & 2.4211$\cdot10^{-3}$ & 1.7526$\cdot10^{-4}$ \\
363.15 & 0.359 & 0.3600 & 2.0492$\cdot10^6$ & 0.750 & 0.7522 & 3.2572$\cdot10^{-3}$ & 2.3577$\cdot10^{-4}$ \\
373.15 & 0.356 & 0.3546 & 2.0182$\cdot10^6$ & 1.000 & 0.9840 & 4.3263$\cdot10^{-3}$ & 3.1313$\cdot10^{-4}$ \\
383.15 & 0.348 & 0.3491 & 1.9872$\cdot10^6$ & 1.300 & 1.2750 & 5.6932$\cdot10^{-3}$ & 4.1202$\cdot10^{-4}$ \\
393.15 & 0.344 & 0.3437 & 1.9562$\cdot10^6$ & 1.621 & 1.6404 & 7.4406$\cdot10^{-3}$ & 5.3841$\cdot10^{-4}$ \\
\end{tabular}
\label{table_supp_1}
\end{table}
\subsection*{Comparison of experimental solubility data with that obtained using DFT calculation.}
Computational solubility approach is based on the equilibrium condition between the solution phase and solute's solid phase
\begin{equation}
X\approx \frac{p^{sat}}{\rho_b k_BT}\exp(\beta\nu^s[p-p^{sat}]-\beta\Delta G_{solv}).
\label{slblt}
\end{equation}
In the framework of the proposed methodology we extract the solute's vapor pressure $p^{sat}$ from the experimental literature data, molar volume of the solute $\nu^s$ is determined on the basis of the group contribution methods \cite{immirzi1977prediction,cao2008use} and the solvation free energy $\Delta G_{solv}$ we compute basing on the classical density functional theory (cDFT), where particles of the solute and solvent are modeled as coarse-grained hard spheres, interacting through the effective Lennard-Jones (LJ) potential. The solvent-solvent and solute-solute LJ parameters are determined by the fitting of the corresponding critical points via the equation of state. The solvent-solute parameters are determined using the standard Berthelot-Lorentz mixing rules. Critical parameters of the CO$_2$ are taken from NIST \cite{nist}, naproxen, ibuprofen, aspirin and dislunisal critical parameters are taken from the ref.\cite{garlapati2009temperature}, carbamazepine parameters -- from the ref. \cite{li2013new}, and the parameters of the thiadiazole derivative (1-[5-(3-chloro-4-methyl-phenylamino)-1,2,4-thiadiazol-3-yl]-propan-2-ol) were calculated, using group contribution methods \cite{tu1995group, lydersen1955estimation, klincewicz1984estimation}. The experimentally measured data of the vapor pressure temperature dependence are taken from the refs. \cite{perlovich2004naproxen,perlovich2004ibuprofen,perlovich2004aspirin,perlovich2003diflunisal,drozd2017novel,bui2014phycsico}. All values of the obtained parameters are represented in the Table \ref{table_supp_2}.
\begin{table}[h!]
\centering
\caption{Solute's critical temperature and pressure, molar volume, parameters of the solute-solute LJ interaction potential and coefficients of the empirical dependence of the solute's vapor pressure on temperature: $\ln(p^{sat})=A-B/T$.}
\begin{tabular}{l|c|l|l|c|c|c|c}
solute & $T_c$,K & $P_c$,bar & $v^s$, cm$^3\cdot$mol$^{-1}$ & $\sigma_{ss}$, $\si {\angstrom}$ & $\varepsilon_{ss},K $ & $A$ & $B$ \\
\hline
naproxen & 807.00 & 24.52 & 179.0 & 7.356 & 580.418 & 39.7 & 15431 \\
ibuprofen & 749.70 & 23.0 & 182.1 & 7.565 & 571.387 & 40.4 & 13927 \\
aspirin & 762.90 & 32.8 & 129 & 6.553 & 548.700 & 38.2 & 13190 \\
carbamazepine & 786.83 & 25.71 & 180.48 & 7.180 & 565.911 & 32.7 & 13343 \\
diflunisal & 869.80 & 32.11 & 125.5 & 6.894 & 625.585 & 36.4 & 14400 \\
thiadiazole derivative & 895.83 & 23.67 & 202.16 & 7.708 & 644.306 & 33.2 & 13989 \\ \end{tabular}
\label{table_supp_2}
\end{table}
Validation of the solubility values correctness computed via the cDFT-based approach was provided by the comparison with the experimental solubility data, taken from the literature when available or obtained, basing on the experimental approach, described above. At the Fig. \ref{Fig3supp} we present the results of the isotherms comparison with the available in literature results. The corresponding experimental data are taken from: ibuprofen -- ref.\cite{kuznetsova2013solubility}, aspirin -- ref.\cite{ravipaty2008polar}, naproxen -- ref.\cite{garmroodi2004solubilities}, diflunisal -- ref.\cite{coimbra2008solubility} and carbamazepine -- ref.\cite{yamini2001solubilities}. One can see a reasonable divergence in the results for several compounds at low pressure values, but starting from the pressure around 150 bars the agreement is decent. Although the calculation then overestimate the solubility values at high pressures, we should note that these discrepancies do not exceed more than a half of the order magnitude. We find such agreement satisfactory as the originally proposed cDFT-based approach is supposed to be considered as a tool for the fast and sufficient estimation of the solute's solubility, rather than a technique used to obtain the high-accuracy solubility values.
\begin{figure}[h!]
\center{\includegraphics[width=0.7\linewidth]{Suppl.3.Comparison-isotherms.jpg}}
\caption{Comparison of the experimental solubility isotherms (colored circles) available in the literature with the ones obtained using cDFT-based approach (dashed lines).}
\label{Fig3supp}
\end{figure}
Experimentally measured thiadiazole derivative solubility values in comparison with the calculated ones are presented at the Fig. \ref{Fig4supp} alongside the same comparison for the carbamazepine, experimental data for which were taken from the ref.\cite{kalikin2020carbamazepine}. Once again the same trend can be observed but now regarding the temperature dependence, namely the underestimation at low temperatures and overestimation at high ones. Overall, starting from the 343 K the agreement is rather satisfactory.
\begin{figure}[h!]
\center{\includegraphics[width=0.7\linewidth]{Suppl.4.Comparison-isochores.jpg}}
\caption{Comparison of the experimental solubility isochores (dots) with the ones obtained using cDFT-based approach (dashed line). Experimental data for carbamazepine were taken from the ref.\cite{kalikin2020carbamazepine}}
\label{Fig4supp}
\end{figure}
\section*{Acknowledgments}
The research was supported by The Ministry of Science and Higher Education of the Russian Federation (grant no. RFMEFI61618$\times$0097). This research was supported through resources of supercomputer facilities provided by NRU HSE.
|
1,116,691,498,432 | arxiv | \section{Introduction}
Optimization is a vast field and is arguably one of the most useful tools for scientists and engineers.
With applications in almost any industry, from operations research to climate analysis to process control to robotics, the need to further our understanding of optimization and develop efficient algorithms to solve optimization problems is clear.
The mathematical structure and geometric interpretations of optimization make it an exciting academic research area.
It is interesting for its own sake.
So it is fortunate that optimization also happens to be extremely useful in solving real problems and developing real technology.
Another fortunate feature of optimization is that it has a rich history of remarkable leaps in understanding.
One discovery of particular importance was the realization that the distinction between complex and easy optimization problems does not hinge on linearity, but rather, convexity \cite{optimality-rockafellar}.
Rockafellar published this historical paper in 1993.
The date of his seminal discovery is interesting to note when put into context.
Humans first stepped on the moon in 1969.
So it wasn't until over 20 years later that we realized the fundamental importance of convexity in optimization problems.
Nowadays, we are consistently sending rockets to space and back, which would not be possible without numerical optimization, in particular, convex optimization \cite{rocket-landing}.
How many more discoveries of the same magnitude as Rockafellar's are left to make?
Currently, it seems that the theory behind convex optimization is nearly complete.
So what developments are necessary to further our understanding of optimization and increase its utility?
Optimization that includes uncertainty is a research frontier that is ripe for research.
In this survey paper, we review deterministic optimization and optimization under both aleatoric and epistemic uncertainty. From our past research about modeling uncertainty and optimization under uncertainty, we start applying them to the Artificial Intelligence (AI) domain. We have realized that optimization under uncertainty is one of the important filed in AI. We start to first do a literature study and explain what (important) methods have been used in optimization through this survey paper.
The structure of this paper is as follows:
In Section 2 we briefly review optimization without uncertainty, convex and nonconvex.
In Section 3, we review the state-of-the-art methods for optimization under aleatoric and epistemic uncertainty.
In Section 4, we discuss optimization under uncertainty broadly and compare the different approaches.
In Section 5, we conclude with a brief summary and possible research directions.
Throughout the paper, we provide specific applications of optimization where many of the applications are focused on optimal control.
It is important to remember, however, that these applications are just one of many use cases for the techniques discussed in this survey paper.
We provide these applications for concreteness.
For a complete survey of optimization as applied to optimal control specifically, we refer the reader to the excellent review paper \cite{trajectory-generation}.
\section{Deterministic Optimization}
\textcolor{black}{In this section, we review the main two classes of deterministic Optimization, namely convex and non-convex optimization, respectively.}
\subsection{Convex Optimization}
A generic optimization problem can be written as:
\begin{mini}|s|[0]
{x \in X}{J(x)}
{}
{\label{eq:minimizationProblem}}{}
\addConstraint{g(x)}{ \le 0}
\addConstraint{h(x)}{ = 0}
\end{mini}
where $x$ is the decision variable vector, $J$ is the objective function, $g$ are inequality constraints, and $h$ are equality constraints.
If $g$ and $h$ are removed, the problem becomes an unconstrained optimization problem.
The goal in constrained optimization is to minimize $J$ while satisfying the constraints imposed by $g$ and $h$. \textcolor{black}{A wide variety of problems in the real world can be transcribed into the above form.}
In convex optimization, $J$ is a convex function, $g$ creates a convex set, and $h$ is an affine function.
For formal definitions of convex functions and convex sets we refer the reader to the classical book, \cite{convex-optimization-book}.
The key feature of convex optimization problems is that global conclusions can be made from local function evaluations.
This property is what makes convex optimization an important area of study and is why it can be effectively applied to real problems.
This allows solvers to quickly and efficiently find the true, globally optimal solution to the problem up to arbitrary precision (discounting floating point precision limits imposed by computers).
There are four main classes of convex optimization problems: linear programs, quadratic programs, second-order cone programs, and semidefinite programs \cite{convex-optimization-book}.
In linear programs, a linear objective is optimized over a polyhedron, which is the shape of the feasible set of linear programs.
Written in standard form, the feasible space of a linear program is the intersection of an affine subspace and the nonnegative orthant.
\begin{mini}|s|[0]
{x \in X}{c'x + d}
{}
{\label{eq:minimizationProblem}}{}
\addConstraint{x}{ \ge 0}
\addConstraint{Ax}{ = b}
\end{mini}
A nice property of linear programs is that an optimal solution (if one exists) can always be found at one of the vertices of the polyhedron.
This enables solution methods such as the simplex method to give algebraic solutions which can then be used for sensitivity analysis.
Quadratic programs have the same feasible region description as linear programs. The difference is the objective takes a quadratic form.
\begin{mini}|s|[0]
{x \in X}{(1/2)x'Qx + c'x + d}
{}
{\label{eq:minimizationProblem}}{}
\addConstraint{Gx}{ \le h}
\addConstraint{Ax}{ = b}
\end{mini}
For this reason, optimal solutions to quadratic programs are not always found on a vertex of the polyhedron feasible region.
Quadratic programs are often used for model predictive controllers where the constraints define the system dynamics and control limits and the objective specifies a cost that penalizes state error and control effort.
Second-order cone programs optimize a linear objective over a feasible region that is specified by the intersection of an affine subspace and the second-order cone, also referred to as the Lorentz cone \cite{Lorentz-cone}.
\begin{mini}|s|[0]
{x \in X}{q'x}
{}
{\label{eq:minimizationProblem}}{}
\addConstraint{|| Gx + h ||_2}{ \le c'x + d}
\addConstraint{Ax}{ = b}
\end{mini}
Robust linear programs can be cast as second-order cone programs
\cite{second-order-cone-programming-applications}.
Semidefinite programs optimize a linear objective over the intersection of an affine subspace and the cone of symmetric positive semidefinite matrices.
The feasible space of a semidefinite program is a spectrahedra \cite{semidefinite-optimization-convex-algebraic-geometry}.
\begin{mini}|s|[0]
{X \in S^n}{\langle C, x \rangle}
{}
{\label{eq:minimizationProblem}}{}
\addConstraint{\langle A, X \rangle}{= b}
\addConstraint{X}{ \succeq 0}
\end{mini}
where $\langle X, Y \rangle := \Tr(X^TY)$.
A unifying trait of the four convex optimization problems discussed above is that the feasible region lies within a proper cone.
Problems that optimize over the intersection of an affine subspace and a convex cone are referred to as conic programs, which subsume the four types of optimization problems mentioned above.
\subsection{Nonconvex Optimization}
Many of the real-world problems that we care about have nonconvexities either in the objective or in the constraints.
We briefly discuss two different approaches to solving nonconvex optimization problems in this section.
Lossless convex relaxations: It is sometimes possible to remove acute nonconvexities in a lossless manner, meaning that the problem can be reconfigured to be a convex program where the optimal solution to the convexified problem is the same as the optimal solution to the original problem.
\textcolor{black}{\cite{convex-approach-mars-powered-descent} shows that the nonconvex thrust vector $T(t) \in \mathbb{R}^3$ constraint of a powered descent space vehicle (described in \eqref{18}) can be removed in a lossless way by introducing additional decision variables, given in \eqref{19}.}
\begin{equation} \label{18}
\rho_{min} \leq || T(t) ||_2 \leq \rho_{max}, \forall t \in [0, t_f]
\end{equation}
\begin{equation} \label{19}
\rho_{min} \leq \sigma(t) \leq \rho_{max}, || T(t) ||_2 \leq \sigma(t), \forall t \in [0, t_f]
\end{equation}
It is proofed via the maximum principle that the optimal solution to the lifted problem \eqref{19} can always be projected down to the feasible region defined by the original coordinates in \eqref{18}.
Sequential convex programming: Methods for solving nonconvex problems to local optimality have been widely studied.
The general approach is to iteratively approximate the original problem with linearizations and other convex relaxations, solve the approximate convex problem, project the solution from the convex subproblem back to the feasible space of the nonconvex problem, and repeat the process starting from the new projected point until some convergence criteria.
See \cite{convex-optimization-for-trajectory-generation} for a detailed review of a couple of specific algorithms for sequential convex programming.
Also worth mentioning are mixed-integer programs which are problems with decision variables that are restricted to be integer-valued.
These problems are nonconvex due to the integer variables but if the rest of the problem is specified by a convex objective and convex constraints, then these problems can still be efficiently solved to global optimality via branch-and-bound algorithms \cite{branch-and-bound}.
Mixed-integer programs have the ability to combine discrete, combinatorial aspects of problems with smooth constraints.
Mixed-integer programs are especially interesting when used within a learning framework \cite{lvis-contact-aware-controllers}.
\section{Optimization under uncertainty}
\textbf{Aleatoric} uncertainty is the uncertainty resulting from true randomness in a given process.
This uncertainty cannot be reduced by further experimentation.
Aleatoric uncertainty is typically modeled by probability distributions.
One of the most common models for aleatoric uncertainty is the gaussian probability distribution.
\textbf{Epistemic} uncertainty stems from a lack of knowledge about the system or process of interest.
This type of uncertainty can be reduced by obtaining further information.
One canonical example of a model for epistemic uncertainty is the probability box or p-box \cite{shariatmadar2019pbox}.
The p-box is defined by lower and upper cumulative distribution functions.
The true distribution function is located within these lower and upper bounds.
In this section, we review the main classes of optimization methods that handle uncertainty.
\subsection{Robust optimization}
Robust optimization optimizes for the worse case.
It does not require any specified probability distributions of the uncertain data.
Instead, in robust optimization, one bounds the uncertain variables to a set of possible values and then optimizes for the worst possible realization of the uncertain variables from those sets.
A generic formulation of the robust optimization problem can be written as follows:
\begin{mini}|s|[0]
{x} {f(x)}
{\label{0}}{}
\addConstraint{g(x, \Delta)}{\leq 0}{, \; \forall \Delta \in \bm{\Delta}}
\addConstraint{h(x)}{= 0}
\end{mini}
where $\bm{\Delta}$ is the set of all possible uncertainties in the problem.
The structure of $\bm{\Delta}$ significantly impacts the solution approach and overall tractability of the problem.
See \cite{optimization-uncertainty-survey} for a condensed list of common uncertainty sets used in robust optimization problems.
In the context of optimal control, the uncertainty typically comes from uncertainty in the parameters of the system dynamics.
A simple model of uncertainty in dynamic systems is bounded external additive disturbance:
\begin{equation} \label{uncertain_linear_system}
\dot{x}(t) = A(t)x(t) + B(t)u(t) + Gw(t), \; w(t) \in \bm{W}
\end{equation}
where the disturbance set $\bm{W}$ is usually a convex, compact set.
Using this model, the robust optimal control problem can be solved via a minimax formulation:
\begin{equation*}
\begin{aligned}
& \underset{x, u}{\text{min}} \; \underset{w}{\text{max}}
& & \int_{0}^{T} l(x(t), u(t)) \; dt + V_f(x(T))\\
& \text{subject to}
& & u(t) \in U, \; \forall w \in \bm{W}\\
& & & x(t) \in X, \; \forall w \in \bm{W}\\
\end{aligned}
\end{equation*}
Another popular approach is to assume the state space matrices $A$ and $B$ come from polytopic sets:
\begin{equation} \label{uncertain_linear_system}
\dot{x}(t) = A(t)x(t) + B(t)u(t)
\end{equation}
\begin{equation} \label{uncertain_linear_system}
A(t) \in Co(A_1, ..., A_n), \; B(t) \in Co(B_1, ..., B_n)
\end{equation}
where $Co$ denotes the convex hull.
See \cite{robust-design-survey} for an in-depth review of robust optimization.
\subsubsection{Sum of squares (SOS) optimization}
Sum of squares optimization is an active research area with applications in machine learning, control theory, and several other disciplines.
It can be seen as a particular type of robust optimization when applied to systems analysis and control.
An important problem in mathematics is checking the global nonnegativity of a function of multiple variables:
\begin{equation} \label{1}
F(x) \geq 0 \ \forall x
\end{equation}
In the general case, the problem can be shown to be undecidable.
To make the problem tractable yet still useful, it is constructive to consider the class of polynomial functions and for polynomial functions, a sufficient condition to show global nonnegativity is to construct a sum of squares decomposition of the polynomial:
\begin{equation} \label{2}
F(x) = \sum_{i}f_i^2(x)
\end{equation}
A polynomial that is the sum of squares can also be expressed in the quadratic form:
\begin{equation} \label{3}
F(x) = z'Qz
\end{equation}
where $Q$ is positive semi-definite and $z$ is a basis of monomials of degree less than or equal to half of the degree of $F$ \cite{parrilo_thesis}.
After selecting a monomial basis vector, searching for a positive semi-definite $Q$ can be done via semidefinite programming for which known efficient algorithms exist \cite{boyd-semidefinite-programming}.
One example of how the sum of squares optimization is used for optimization under uncertainty is in the analysis and synthesis of Lyapunov stable systems with bounded uncertainty in either the system dynamics or operating environment.
The authors in \cite{funnel-libraries} utilize the sum of squares programming to generate trajectories for a fixed-wing aircraft that are guaranteed to succeed while taking into account the possibility of bounded disturbances, uncertainty in the environment, and uncertainty in the parametric model.
Their approach focuses on computing tight approximations of the reachable sets that the system may evolve to over the course of a trajectory.
So for a closed loop time-varying system defined in error coordinates around a trajectory and using uncertainty $w(t)$ from a semi-algebraic set $\{w | g_{w,j}(w) \geq 0, \forall j = 1,...,N_w\}$ to model external disturbances or parametric model uncertainties (aleatoric and epistemic uncertainty):
\begin{equation} \label{4}
\dot{x} = f(t, x(t), w(t))
\end{equation}
they parameterize the reachable sets $F(t)$ by the sublevel sets of nonnegative time-varying functions $V$:
\begin{equation} \label{5}
F(t) = \{x(t) | V(t, x) \leq \rho(t)\}
\end{equation}
and thus the constraint:
\begin{equation} \label{6}
V(t, x) = \rho(t) \Rightarrow \dot{V}(t, x, w) < \dot{\rho}(t), \forall t \in [0, T]
\end{equation}
is sufficient for approximating the reachable set over the course of a trajectory.
By selecting polynomial expressions for $\dot{x}$, $V$, and $\rho$, the constraint in \eqref{6} can be written as:
\begin{equation} \label{7}
\dot{\rho}(t) - \dot{V}(t,x,w) + \lambda_1(t,x,w)[\rho(t) - V(t,x)] + \lambda_2(t,x,w)[t(t-T)] + \sum_{j=1}^{N_w}\lambda_3(t,x,w)g_{w,j}(w)\ \text{is SOS}
\end{equation}
\begin{equation} \label{8}
\lambda_2, \lambda_3\ \text{is SOS}
\end{equation}
where $\lambda_1$, $\lambda_2$, $\lambda_3$ are polynomials with coefficients that are decision variables in the SOS program.
To sustain convexity and handle the bilinear constraints, the algorithm used alternates between two SOS programs: one with decision variables ($\lambda_1, \lambda_2, \lambda_3$) and another with ($V, \rho, \lambda_2, \lambda_3$). This review of the methodology used in \cite{funnel-libraries} is simplified and deliberately leaves out other details such as the decision variables associated with the cost function.
Sum of square optimization can also be used in the performance analysis of black-box algorithms that are widely used in machine learning with large datasets.
The authors in \cite{tan2021analysis} utilize the sum of squares programming to provide convergence rate bounds for first-order optimization algorithms.
\subsection{Multiparametric Programming}
Multiparametric programming is a powerful methodology to compute the solution sets to optimization problems with parametric uncertainty in the right-hand side of the constraints or in the objective function without making any assumptions about the underlying data distributions.
Multiparametric programming can be used to solve optimal solutions to problems under uncertainty and also problems with feasible regions where the solutions to all sub-regions of the feasible regions are desired \cite{multiparametric-programming-process-systems}.
The general multiparametric programming problem is formulated by:
\begin{mini}|s|[0]
{x \in \mathbb{R}^n, \theta \in \mathbb{R}^m}{f(x,\theta)}
{\label{9}}{}
\addConstraint{g(x,\theta)}{\leq 0}
\addConstraint{h(x, \theta)}{= 0}
\end{mini}
where $x$ is the vector of decision variables and $\theta$ is the vector of uncertain parameters.
Multiparametric programming builds off of the Basic Sensitivity Theorem and uses the result that the active set is constant in the neighborhood of a realization of the uncertain parameter vector $\theta$.
From this, critical regions with constant active sets within the parameter space of the uncertain vector can be constructed.
An explicit solution to each critical region can be derived from the associated unique set of KKT conditions \cite{multiparametric-programming-process-systems}.
\begin{equation}
x^*(\theta) = \left\{
\begin{aligned}
&x_1(\theta) \; \text{if} \; \theta \in \theta^1 \\
&x_2(\theta) \; \text{if} \; \theta \in \theta^2 \\
&\;\;\;\;\;\;\;\;\;\;\; \vdots \\
&x_3(\theta) \; \text{if} \; \theta \in \theta^3 \\
\end{aligned}
\right\}
\end{equation}
One of the most famous applications of multiparametric programming is in the development of explicit MPC \cite{explicit-mpc}. The generic MPC problem is to solve the following finite horizon regulation problem with every control tick:
\begin{mini}|s|[0]
{x \in X, u \in U}{\sum_{k=0}^{N-1}l(x_k, u_k) + F(x_n)}
{\label{10}}{}
\addConstraint{x_{k+1}}{= f(x_k, u_k)}
\addConstraint{x_0}{= x(t)}
\addConstraint{u_k}{= K(x_k)}
\end{mini}
where $X \subseteq \mathbb{R}^n$ and $U \subseteq \mathbb{R}^m$ are closed sets containing the origin.
$f$ represents the system dynamics, $l$ is the optimal cost-to-go, $F$ is the terminal cost, and $K$ is some state feedback gain \cite{Alessio2009}.
\cite{explicit-mpc} showed that the MPC problem \eqref{10} with quadratic cost and linear time-invariant system dynamics has an explicit solution that is continuous piecewise affine in decision variables, continuous piecewise quadratic in the objective function, and has polytopic critical regions.
This allows for the offline computation of the optimal control law for the entire bounded state space. The online computation becomes simplified and is just a matter of determining what critical region of the state space the current state is in.
\subsection{Stochastic optimization}
Stochastic optimization allows a user to specify the probability distributions that uncertain parameters come from.
Probability distributions can be placed on variables in the constraints, objective, or both.
A generic formulation for a stochastic optimization program that optimizes for the expected values of random variables can be written as:
\begin{mini}|s|[0]
{x} {\mathop{\mathbb{E}}[f(x, w)]}
{\label{0}}{}
\addConstraint{\mathop{\mathbb{E}}[g(x, w)]}{\leq 0}
\addConstraint{\mathop{\mathbb{E}}[h(x,w)]}{= 0}
\end{mini}
where $w$ represents the uncertain variables.
Depending on the probability distributions that are used in the problem, stochastic optimization can struggle with tractability.
However, for certain types of probability distributions, expectations are relatively cheap and the optimization problem can be efficiently solved.
We now discuss a couple of ways in which stochastic optimization has been effectively utilized for trajectory planning of dynamical systems with uncertain dynamics and/or uncertainty in the environment.
The first method proposed in \cite{gaussian-planning} assumes all uncertainty is Gaussian and linear dynamics:
\begin{equation} \label{14}
x_{t+1} = Ax_t + Bu_t + w_t + \nu_t
\end{equation}
where $w_t \sim N(0, Q)$ represents model uncertainty and $\nu_t \sim N(0, R)$ represents external disturbances.
Under the assumptions shown in \eqref{14} and that the initial state is a Gaussian distribution $N(x_0, P_0)$, the mean and covariance of the Gaussian distributions of future states can be represented as:
\begin{equation} \label{15}
\mu_t = \sum_{i=0}^{t-1}A^{t-i-1}Bu_i + A^tx_0
\end{equation}
\begin{equation} \label{16}
\Sigma_{x_t,y_t} = \sum_{i=0}^{t-1}A^{t-i-1}Q(A^T)^{t-i-1} + \sum_{i=0}^{t-1}A^{t-i-1}R(A^T)^{t-i-1} + A^tP_0(A^T)^t
\end{equation}
From \eqref{15} and \eqref{16} we can see that the future state means is a linear function of control inputs and the future state covariance is independent of control inputs and is known a priori.
This allows the stochastic obstacle-free trajectory planning problem with chance constraints on hitting an obstacle to be written as a deterministic mixed integer linear program.
To handle non-Gaussian uncertainty, \cite{nongaussian-sampling} proposes a sampling scheme for systems with future states that depend explicitly on the initial state $x_0$, control inputs $u$, and additive uncertainty $
\nu$:
\begin{equation} \label{17}
x_t = \sum_{i=0}^{t-1}A^{t-i-1}B(u_i + \nu_i) + A^tx_0
\end{equation}
The general approach is to sample N pairs where each pair is composed of an initial state $x_0$ and vector of additive uncertainty \{$\nu_0, ..., \nu_{T-1}$\}. Using these N pairs, the intractable stochastic optimization problem is turned into a tractable deterministic one where the percentage of the N pairs that succeed approximates the success rate of the optimized control input trajectory.
To handle non-Gaussian probability distributions without sampling, it has been shown that moments of probability distributions can be utilized to model the uncertainty and formulate convex trajectory optimization programs \cite{risk-bounded-trajectories}.
We refer the reader to the survey papers \cite{optimization-under-uncertainty-soa} and \cite{stochastic-programming-process-systems} for more comprehensive overviews of stochastic optimization.
\subsection{Loop formulations}
Perhaps the simplest model for epistemic uncertainty is the interval model \cite{shariatmadar2021interval}, where one only provides two values to capture the uncertainty of an uncertain parameter: the lower and upper bound. Therefore, optimization problems utilizing intervals on some of the variables is a form of optimization under epistemic uncertainty. One approach to optimizing with interval variables is a nested loop formulation:
\begin{equation}
\label{nested-loop}
\begin{aligned}
& \underset{x}{\text{min}} \; \underset{w}{\text{max}}
& & f(x, w) \\
& \text{subject to}
& & g(x, w) \leq 0, \\
& & & h(x, w) = 0, \\
& & & x \in X, \\
& & & w_{lb} \leq w \leq w_{ub} \\
\end{aligned}
\end{equation}
where the inner loop is a search over the epistemic variables $w$ for the upper bound on the objective and the outer loop is the true optimization problem at hand for chosen values of the epistemic variables \cite{portfolio-optimization-epistemic}.
$x$ is the vector of decision variables, $g(.)$ and $h(.)$ represent generic inequality and equality constraints, respectively.
A nested loop formulation for optimization with interval variables is a form of robust optimization and the similarities with minimax MPC as mentioned earlier in this paper should be apparent.
A nested loop optimization problem is computationally intensive due to the need to solve the inner optimization problem at every step of the outer optimization problem. Decoupled loop methods such as the sequential optimization reliability assessment (SORA) method aim to reduce the computational intensity of nested loop methods \cite{reliability-based-design}.
The decoupled loop formulation of \eqref{nested-loop} can be written as:
\begin{equation}
\label{decoupled-one}
\begin{aligned}
& x^* = \underset{x}{\text{argmin}}
& & f(x, w^*) \\
& \text{subject to}
& & g(x, w^*) \leq 0, \\
& & & h(x, w^*) = 0, \\
& & & x \in X, \\
\end{aligned}
\end{equation}
\begin{equation}
\label{decoupled-two}
\begin{aligned}
& w^* = \underset{x}{\text{argmax}}
& & f(x^*, w) \\
& \text{subject to}
& & g(x^*, w) \leq 0, \\
& & & h(x^*, w) = 0, \\
& & & w_{lb} \leq w \leq w_{ub} \\
\end{aligned}
\end{equation}
In this decouple formulation, the two optimization problems are solved iteratively with the epistemic variables $w$ being fixed in the optimization problem \eqref{decoupled-one} and the decision variables $x$ being fixed in the optimization problem \eqref{decoupled-two} \cite{portfolio-optimization-epistemic}.
In decoupled loop methods, there are still two optimization loops but since the loops are not nested, the comparative computational efficiency can be significant.
Decouple loop methods are popular for robustness and reliability-based design optimization problems \cite{robustness-bases-design} \cite{reliability-based-design}.
Sometimes it is possible to convert a double loop formulation, whether it be nested or decoupled, into a single loop formulation.
\cite{likelihood-representation-of-epistemic-uncertainty} proposed a single loop formulation to robust optimization problems with interval uncertainty by estimating a unique distribution for the random variables via a worst-case maximum likelihood-based estimation.
\cite{portfolio-optimization-epistemic} extended the work in \cite{likelihood-representation-of-epistemic-uncertainty} to incorporate the correlation between input random variables.
The approach is to first obtain a unique distribution for the random variables via a nested optimization problem:
\begin{equation}
\label{single-loop-one}
\begin{aligned}
& \underset{p}{\text{max}} \; \underset{w}{\text{min}}
& & \text{log}(L(p;w)) \\
& \text{subject to}
& & w_{lb} \leq w \leq w_{ub} \\
\end{aligned}
\end{equation}
where $p$ is the parameters of a multivariate normal distribution, $\mu$ and $\Sigma$. $\text{log}(L(p, w))$ is the log-likelihood function for the multivariate normal distribution of the random variable $w$.
After solving the above optimization problem, the resulting PDF that represents the random variables under interval uncertainty can then be used in a single loop optimization formulation, such as \eqref{decoupled-one}, where $w^*$ is chosen to be worst-case maximum likelihood estimates.
\subsection{Bayesian inference}
Although Bayesian methods typically only apply to uncertainty distributions that are completely known, Bayesian inference has been used within optimization methods that incorporate epistemic uncertainty \cite{reliability-based-design-mixed-aleatory-epistemic}.
An important application of epistemic optimization is reliability-based design.
In reliability-based design, the reliability for a particular constraint can be written as:
\begin{equation}
\begin{aligned}
R = Pr[g(X, P) > 0]
\end{aligned}
\end{equation}
where $X$ and $P$ are vectors that contain both aleatoric and epistemic variables.
Due to the epistemic variables, the reliability $R$ is uncertain and can be modeled using Bayesian inference.
See \cite{reliability-based-design-mixed-aleatory-epistemic} for a review of Bayesian inference as it applies to reliability.
In \cite{bayesian-reliability-optimization} a confidence measure is defined $\zeta(\mu_x)$ and proposes a multi-objective optimization problem for reliability-based optimization under epistemic uncertainty:
\begin{equation}
\begin{aligned}
& \underset{\mu_x}{\text{min}} \; f(\mu_x, \mu_p)\; \underset{\mu_x}{\text{max}} \; \zeta(\mu_x) \\
& \text{subject to} \; 0 \leq \zeta(\mu_x) \leq 1
\end{aligned}
\end{equation}
Solving this problem results in a set of designs with different confidence values for the desired reliability value $R$.
To make the above problem more tractable, one usually selects a confidence value and then the reliability is computed from the reliability distribution produced via Bayesian inference \cite{reliability-based-design-mixed-aleatory-epistemic}.
This leads to the Bayesian reliability-based design optimization formulation:
\begin{equation}
\begin{aligned}
& \underset{d, \mu_x}{\text{min}}
& & f(d, \mu_x, \mu_p) \\
& \text{subject to}
& & Pr(g(d, X, P) \leq 0) \leq Pr_{target} \\
& & & h(d) \geq 0 \\
& & & d_L \leq d \leq d_u, \; \mu_{xL} \leq \mu_x \leq \mu_{xU} \\
\end{aligned}
\end{equation}
where $d$ is a vector of deterministic variables, $X$ is a vector of aleatoric variables, $P$ is a vector or epistemic variables, and $g(d, X, P) \leq 0$ defines a failure region \cite{reliability-based-design-mixed-aleatory-epistemic}.
\subsection{Upper and lower expectations}
When optimizing with variables that are modeled by epistemic models such as the p-boxes, a typical approach is to take a lower or upper expectation of those epistemic variables and then optimize the resulting deterministic problem.
Whether to use the lower or upper expectation is decided by if the objective is to minimize or maximize and whether one wants to optimize for the worst or best-case scenario.
A lower expectation can be formulated as:
\begin{equation}
\begin{aligned}
E_l = \underset{p \in P}{\text{min}} \;
& \int_{W} (h(w) \in H) p(w) \; dw\\
\end{aligned}
\end{equation}
where $h(w)$ is the value of interest that depends on the uncertain variable $w$ from the space of uncertain variables $W$.
$p$ is a probability distribution within the set of distributions $P$.
\cite{epistemic-trajectory} presents a method to generate optimal trajectories that are robust against epistemic uncertainty where the uncertainty is modeled with p-boxes.
They utilize a surrogate model and take the lower expectation of the objective and constraints to solve the epistemic optimization problem.
We refer the reader to \cite{linear-programming-p-box} for a theoretical treatment of the solution to linear programs with p-box uncertainty models.
Here, the approach is to convert the uncertain optimization problem to a deterministic one using imprecise decision theory.
\subsection{Transform to deterministic problem}
The general approach of taking an uncertain optimization problem and transforming it into a deterministic one accounts for many of the methods in epistemic optimization.
The key distinguishing feature of these different approaches to epistemic optimization lies in the method of selecting fixed variables/functions in place of epistemic ones.
Here we discuss a few of these approaches that are apparent in recent literature.
Perhaps the most simple is to approximate the epistemic parameters through some statistical function \cite{dealing-with-epistemic-multiobjective} \cite{objective-penalty-function}:
\begin{equation}
g(f(x, w))
\end{equation}
where $g(.)$ is some statistical function, $f(.)$ is the function to approximate, and $w$ is an epistemic variable.
The simplicity of this approach may come at the cost of producing inaccurate solutions.
For generic optimization problems, it is not surprising that reducing an epistemic parameter down to a fixed parameter via a statistical function may lead to an optimization problem that uses many inaccurate parameters.
Another approach is to define a robustness criterion $R$ and maximize this criterion.
$R$ is defined by the variation of the associated uncertain function $f(x)$ which has uncertainty in $x$ \cite{robustness-multi-objective}.
Defining a robustness criterion is popular in the field of robust design and we refer the reader to the comprehensive survey paper \cite{robust-design-survey} to learn more about optimization under uncertainty as it applies to design.
Another category is interval-based approaches where the uncertainty is captured by an upper and lower bound:
\begin{equation}
f(x, w) = [y_l, y_u]
\end{equation}
\cite{multiobjective-fuzzy-system} presents a genetic algorithm for optimization problems where uncertainty in the objective is represented by intervals of fuzzy sets.
See \cite{dealing-with-epistemic-multiobjective} for a survey on epistemic multi-objective optimization.
\subsection{Fuzzy optimization}
\subsubsection{Main Approaches to Fuzzy Optimization}
The fuzzy optimization problems considered as $A=\{a\}$ is a set of possible outcomes, and an objective function is defined as $f:A\longrightarrow\mu(B)$, where $\mu(B)$ is the fuzzy sets defined in $B$, the real line. In other words, $f(a)$ is a fuzzy value which illustrates a fuzzy evaluation of the possible outcome $a\in A$. The set of possible options is defined by a fuzzy set $F$ in $A$ such that $F(a)\in[0,1]$ is called the degree of possibility. For fully possible the degree is 1 and 0 otherwise, through all values.
\subsubsection{Mathematical form of the fuzzy optimization problem}
The fuzzy optimization problem is defined as
\begin{equation}\label{fuzzy-eq}
\begin{aligned}
& \underset{x\widetilde{\in} F}{\widetilde{\max}}
& & f(x)\\
\end{aligned}
\end{equation}
where it is about finding a possibly maximum value of $f$ ($\widetilde{\max}$) over the $x$'s “possibly belonging” ($\widetilde{\in}$) to the fuzzy feasible set $F$. The problem \eqref{fuzzy-eq} are discussed in various formats, e.g., given by Bellman and Zadeh’s \cite{fuzzy-environment}.
A major drawback of fuzzy optimization is that implementation is not efficient. Currently one must use fuzzy arithmetic or fuzzy operators which transforms the problem to be solved under fuzzy logic \cite{fuzzy-arithmetic-industrial} \cite{lp-interval-fuzzy-sets}. This defuzzification process is not efficient enough yet for online use and is a possible research direction.
\subsection{Constrained optimization under uncertainty using decision theory}
One of the most general models of epistemic uncertainty, even more, general than interval and belief function models, is imprecise probability. Generally, an optimization problem under (imprecise) uncertainty is defined as follows
\begin{align}\label{COUU}
&\underset{x{\in} \mathcal{X}}{{\max}}\quad f(x,Y)\notag\\
&\text{s.t. }\quad xRZ,
\end{align}
where $x$ is the optimization vector in any set $\mathcal{X}$, $Y$ and $Z$ are random vectors in $\mathcal{Y}$ and $\mathcal{Z}$, and $R$ is a relation in the set $\mathcal{X}\times\mathcal{Z}$. A simple case of constrained optimization under an uncertainty problem is a linear programming problem under uncertainty which is defined as
\begin{align}\label{LPUU}
&\underset{x\in\mathbb{R}^n_{\ge 0}}{{\max}}\quad U^Tx\notag\\
&\text{s.t. }\quad Yx\ge Z,
\end{align}
where $x$ is the optimization vector in $\mathbb{R}^n_{\ge 0}$, $(Y,Z,U)$ are random vectors in $\mathbb{R}^{m\times n}\times\mathbb{R}^m\times\mathbb{R}^n$, and $R:=\ge$ is the relation in the set $\mathbb{R}^m\times\mathbb{R}^m$. Many kinds of research and works, like \cite{shariatmadar2019pbox, shariatmadar_contamination_2020,shariatmadar2021interval} have been done to solve the optimization under uncertainty problems \eqref{COUU} and \eqref{LPUU} at our group in KU Leuven. We first convert the problem to a decision problem and use decision criteria to provide solutions based on the criteria.
\section{Discussion}
\subsection{Comparison}
In this review, a variety of different approaches to optimization under aleatoric and epistemic uncertainty were discussed.
When the distributions of the random variables are precisely known, the uncertainty is classified as aleatoric.
Aleatoric uncertainty is inherent to the process under consideration.
If an optimization problem only considers aleatoric uncertainty then the optimization problem can typically be categorized as a stochastic optimization program.
Stochastic optimization leverages knowledge of the underlying probability distributions of the random variables in the problem.
Once probability distributions for the random variables are obtained, one can use those distributions to take the expectations of the variables and optimize for the average scenario.
If sampling from the distributions is cheap, one can also optimize over a set of samples.
Furthermore, the uncertainty moments of the random variables can be utilized to introduce chance constraints or constraints that must be satisfied with a given level of confidence \cite{chance-constrained-programming}.
When the distributions of the random variables are not precisely known, the uncertainty is classified as epistemic.
Epistemic uncertainty comes from a lack of knowledge and can, in principle, be reduced to aleatoric uncertainty after sufficient experimentation.
The type of epistemic uncertainty model used in an optimization problem influences the procedure to solve the problem.
A simple and popular method for modeling epistemic uncertainty is through a bounded set where the random variable is known to lie within the set but no probabilities are associated with the set.
Robust optimization uses bounded sets to optimize for the worst-case scenario, leading to conservative behavior.
Multiparametric programming also uses bounded sets and computes offline the optimal solutions as a function of all possible realizations of uncertain variables.
Multiparametric programming suffers from the curse of dimensionality and has thus limited multiparametric programming to relatively simple systems operating in more controlled environments.
The interval model is a bounded set in one dimension and when using interval models, one can utilize double or single-loop formulations.
When using the more informative p-box model, it is most common to take the lower/upper expectations of the random variables and then optimize using those expectations.
Many epistemic optimization methods optimize for the worst-case realization of the epistemic variables but one can also use Bayesian inference methods to enforce levels of reliability/robustness into the problem.
A separate class of optimization problems, fuzzy optimization, exists for optimization problems using fuzzy sets.
The main drawback of fuzzy optimization is that the current solution methods are not efficient enough for real-time use.
\subsection{(Constrained) optimization under uncertainty in machine learning}
Generally, when there is data available or a repeatable task is running one question is that could we learn a pattern from the data or the tasks. Mainly, in machine learning, we use this data as input to learn or train a model. This has been done via several machine learning techniques such as supervised or unsupervised learning, reinforcement or inverse reinforcement learning, and so on. In almost all of these techniques, we use optimization theory to solve machine learning problems. For instance, in regression problems where the goal is to fit a model to data, one solution is to use a Bayesian neural network \cite{kononenko1989bayesian}, in which the parameters are represented by probability distributions. In this method, a back-propagation method is used to minimize a loss function. In one of our ongoing research, we solve an interval neural network \cite{ishibuchi1993, oala2021} problem via a constrained optimization problem under uncertainty. By doing so, we find the best model with the highest accuracy to fit the data using the worst-case scenario optimization technique. This work is in progress and will be our next publication.
\section{Conclusion}
The field of optimization has made tremendous strides in progress over the last half-century.
Theoretical developments have improved our understanding of the structure of optimization problems.
Advancements in computer technology continue to make optimization more applicable to everyday engineering.
With the inclusion of uncertainty into the optimization problems, the resulting solutions are more robust to the inevitable discrepancies between the modeled parameters and the real system.
However, simply assigning probability distributions to unknown parameters often assumes more knowledge than what one really has.
By accepting the lack of knowledge about certain parameters, the problem then becomes one of epistemic uncertainty.
Using epistemic models often results in more realistic models, however, it complicates the problem significantly.
Incorporating epistemic uncertainty can immediately make a tractable optimization problem intractable.
The most popular model for epistemic uncertainty is the bounded set.
It is apparent that the theory of optimization under epistemic uncertainty with more advanced models than the bounded set needs to be further developed.
There is a disproportionate amount of literature around advanced epistemic uncertainty modeling and the associated optimization problems.
A unified mathematical framework for optimizing under epistemic uncertainty beyond the bounded set is of paramount importance and is a research direction with many opportunities.
Furthermore, the need to develop efficient algorithms to solve optimization under epistemic uncertainty will naturally follow once the theory is in place.
\section{Acknowledgment}
This work is supported by the FETOPEN European Union's Horizon 2020 research and innovation programme under grant agreement No. 964505 (\href{https://www.epistemic-ai.eu/}{Epistemic AI}).
\medskip
\printglossary
\printbibliography
\end{document} |
1,116,691,498,433 | arxiv | \section{INTRODUCTION}
The low temperature properties of many physical systems, such as
quantum tunneling between flux states in a SQUID, \cite{Leggett84},
two-level atoms coupled to the electromagnetic field in quantum optics,
\cite{LeClair}, or tunneling dislocations and point defects in
solids\cite{disl} can be described by the dissipative two-state system model
\cite{leggett.87,weiss.99}. All these systems
have the common feature that at low energies the subsystem
investigated (the flux states of the SQUID, the atom, or the crystal
defect in the examples above) can occupy only two distinct quantum states,
which are coupled to a continuum of excitations
(the electromagnetic field, the phonons or the conduction electrons),
leading to the appearance of dissipation in the system
\cite{caldeira+leggett}.
In the dissipative two state system (DTSS) model
the two distinct quantum states above are described in terms of a
pseudospin $\sigma_i$, with $\sigma_z = \pm 1$ corresponding to the two
states of the subsystem. Generally, these two states have slightly
different energy with an energy difference $\varepsilon$ (also called asymmetry
energy) and the decoupled two-state system (TSS) can tunnel
between them with a tunneling amplitude $\Delta$.
The heat bath is modeled by a continuum of independent quantum
oscillators with density $\varrho(\omega)$ coupled linearly
to $\sigma_z$ with a frequency dependent coupling $g(\omega)$.
A detailed analysis shows that the dynamical properties of
the two-state system are uniquely determined by the environment's
spectral function,
$J(\omega) \sim \varrho(\omega) g^2(\omega)\sim \omega^s$ \cite{weiss.99}.
Here, we only concentrate on Ohmic dissipation
corresponding to $s=1$, i.e. $J(\omega) \sim
\omega$. This includes, for instance, the important case of a tunneling defect
in a metal, where the low-energy bosonic excitations are
electron-hole pairs close to the Fermi surface. These excitations
have a linear dispersion, and in a first approximation their
coupling to the defect is energy-independent, leading to $J(\omega)
= 2\pi \alpha \omega$ for $\omega < \omega_c$ with $\omega_c$ a
high-energy cutoff in the model of the order of the Fermi energy
$E_F$. The dynamical behaviour of the TSS
as a function of the coupling strength $\alpha$ has been the subject of
extensive studies during the past two decades and it is
well understood by now \cite{general-dynamics}. To distinguish between the
different cases it is useful to introduce the zero temperature
spin correlation function $S(t) \equiv
\Im \langle\sigma_z(0)\sigma_z(t)\rangle$.
{\it (a)} For $\alpha < 1/2 $ the TSS oscillates between the two
states $\sigma_z= \pm 1$ and $S(t)$ has an oscillatory behaviour.
However, the environment introduces
some decoherence in the system, reflected in the
exponential decay of the envelope of $S(t)$. It also renormalizes the
tunneling amplitude: $\Delta \to \Delta_r < \Delta$.
{\it (b)} In the parameter range
$1/2<\alpha < 1$ the coherent oscillations\cite{note-coherence}
become completely
suppressed, and $S(t)$ shows an exponential behaviour without a change
of sign. {\it (c)} Finally, for $\alpha>1$ and a finite level asymmetry
the TSS becomes localized in the lowest quantum state (at $T=0$)
\cite{localization}. In this case $S(t)$ tends to a {\it finite}
value as $t\to \infty$.
It is important to note that the localized state
obtained is immediately destroyed once assisted tunneling or assisted pair
tunneling is included in the DTSS model \cite{Zawa,MF}.
In the present paper we study the {\it thermodynamics} of a
dissipative TSS model in the parameter range $0<\alpha < 1$.
To this purpose we exploit a mapping between the spin anisotropic
Kondo model (AKM) describing an impurity spin coupled to the spin density of
the conduction electrons via an anisotropic exchange interaction
(see Sec.~IIb) and the dissipative TSS model, \cite{leggett.87,weiss.99},
and study the Bethe Ansatz equations \cite{tsvelik.83} for the former
both numerically and analytically. As discussed in Sec.~\ref{ss:equiv} and
Appendix~\ref{app:equiv}, within this mapping the tunneling amplitude
maps to the spin flip scattering amplitude, $\Delta \leftrightarrow
J_\perp$, the asymmetry energy $\varepsilon$ corresponds to a
local magnetic field $h$ applied to the impurity spin, and the
dissipation strength $\alpha$ is related to the coupling $J_z$ in the
Kondo model. It is very remarkable that the $\alpha$ values
separating the three different regions of the DTSS model are mapped to
some special points in the parameter space of the Kondo model.
The point $\alpha=1$ turns out to correspond to the case $J_z=0$,
separating the ferromagnetic ($J_z < 0 \Leftrightarrow\alpha > 1$) and the
antiferromagnetic ($J_z > 0 \Leftrightarrow \alpha < 1$) regimes in the
Kondo model. While in the first case the Kondo model scales
to a finite fixed point, in the second the Kondo fixed point
turns out to be at infinite coupling. The crossover to the 'strong
coupling' regime happens at the so-called Kondo energy, $T_K$,\cite{Hewson,GrunerZawa}
which can be identified with the
renormalized tunneling amplitude in the DTSS model, $\Delta_r \sim T_K$.
The other special point, $\alpha = 1/2$, can be shown to be equivalent
to the Toulouse line\cite{toulouse.69} of the AKM. Along this line the
Bethe Ansatz (BA) equations simplify enormously, and the model
can be described by a simple resonant level model without interaction.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig1a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig1b.eps}}}
\vspace{0.1cm}
\caption{(a) The specific heat $C(T)$
on a linear temperature scale for the symmetric case at weak and
strong dissipations. The position of the maximum of the Schottky
peak in $C(T)$ is of order $\Delta_{r}$ but
its exact location changes with $\alpha$, as does the peak height.
(b) The quantity $\Delta_{r}C(T)/k_{B}T$ shows non-monotonic behaviour
for weak dissipations, and monotonic behaviour for $\alpha \ge 1/3$.
}
\label{linear-scale-heats}
\end{figure}
We give a detailed description of the thermodynamics of the DTSS model,
solving the BA equations for arbitrary asymmetry and temperature
for both $\alpha > 1/2$ and $\alpha < 1/2$.
For the sake of completeness and clarity, we also included a
detailed analysis of the two models to demonstrate, how the
different concepts such as scaling, strong coupling limit, energy
scales, etc. appear in the AKM and the DTSS model.
As we shall see, there are clear indicators in the specific heat, $C(T)$,
for distinguishing weak from strong dissipation limits.
This is not evident in $C(T)$ directly, which shows a Schottky anomaly at
$k_{B}T\sim \Delta_{r}$ for all dissipation strengths $\alpha<1$ as depicted
in Fig.\,\ref{linear-scale-heats}a . However, as seen in
Fig.\,\ref{linear-scale-heats}b, and as we shall discuss in detail
later, the quantity $C(T)/T$, shows quite different behaviour at weak
and strong dissipations. For dissipations $\alpha < 1/3$,
with no asymmetry, $C(T)/T$ is found to have a peak at
$k_{B}T\sim \Delta_{r}$ indicating the expected tendency towards
activated behaviour of the two-level system as $\alpha\rightarrow 0$.
A quite different behaviour is found for $\alpha\ge 1/3$ where we find
that $C(T)/T$ is monotonically decreasing with increasing temperature.
The tendency towards activated behaviour, signaled by a peak at approximately
$\sqrt{\Delta_{r}^{2}+\varepsilon^{2}}$ in $C(T)/T$, is also found at
all dissipation strengths for sufficiently large (typically of order
$\Delta_{r}$) asymmetries $\varepsilon$.
In the Kondo language this corresponds to the Zeeman splitting of the
Kondo resonance due to a local magnetic field. In contrast, the
dielectric susceptibility,
$\chi_{sb} = -\partial^{2} F/ \partial \varepsilon^{2}$, with $F(T)$ the
two-level system free energy, shows
only a monotonically decreasing behaviour with increasing temperature
for all dissipation strengths $0<\alpha<1$ in the symmetric case (see
Fig.\,\ref{linear-scale-chi} and for further details Sec.~\ref{sec-susc}).
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig2.eps}}}
\vspace{0.1cm}
\caption{
The dielectric susceptibility, $\chi_{sb}$, on a linear temperature scale
for the symmetric case at weak and strong dissipations.
}
\label{linear-scale-chi}
\end{figure}
A peak in $\chi_{sb}(T)$ at finite temperature only appears for a
sufficiently large level asymmetry. The low temperature behaviour corresponds
to that of a renormalized Fermi liquid at all $\alpha < 1$ with the
renormalizations increasing with $\alpha$. Beyond these
overall features we also discuss the detailed form of the universal
scaling functions of the dissipative two-state system for all $\alpha<1$
and $\varepsilon$.
The only previous detailed studies of the thermodynamics of the
Ohmic two-state system which we are aware of are, (a), the numerical
renormalization group study \cite{costi.98} and, (b), the work
of G\"{o}rlich and Weiss \cite{goerlich.88}.
The latter authors used a path integral method \cite{leggett.87} to calculate
the partition function of the dissipative two-state system for both Ohmic
and non-Ohmic dissipation. Their results are restricted to weak level
asymmetries $\varepsilon\ll \Delta_{r}$ and no results are presented for
the finite temperature dielectric susceptibility. For the Ohmic case
and $\alpha\ll 1$ they recover the linear $T$ behaviour
of the specific heat at low temperature $k_{B}T\ll \Delta_{r}$ and the
correct high temperature behaviour at all $\alpha$, however reliable results
for strong dissipation $1/2<\alpha<1$ could not be obtained within their
perturbative approach. The numerical renormalization group calculations in
\cite{costi.98} are non-perturbative and gave the specific heat accurately
for all temperatures at both weak and strong dissipations. A drawback of
this method, however, is the logarithmic discretization
\cite{wilson.75,kww.80} of
the fermionic environment. This limits the ability of the method to resolve
finite temperature features, such as the peak in $C(T)/T$ at
$k_{B}T\sim \Delta_{r}$ for $\alpha\ll 1$. The calculation of
the dielectric susceptibility by this method \cite{costi.98} is also
problematical at sufficiently low temperatures $k_{B}T \leq \Delta_{r}$
\cite{problematical-nrg-chi}.
As explained in \cite{costi.98}, accurate results for $\chi_{sb}$ at $T=0$
required an analysis of the strong-coupling fixed point Hamiltonian
together with the leading irrelevant deviations. Thus, this method
gave accurate results for $\chi_{sb}$ at $T=0$ (from the fixed point analysis)
and for $k_{B}T\ge \Delta_{r}$, but it was not possible within this method
to determine equally accurately the bahaviour of $\chi_{sb}$ for
$0<k_{B}T\leq \Delta_{r}$. As we shall see the Bethe Ansatz method we use
in this paper overcomes all the above difficulties.
The paper is organized as follows: in Sec.~\ref{models-sec} we introduce the
model of the dissipative two-state system and outline its equivalence to
the anisotropic Kondo model for the case of Ohmic dissipation.
Some implications of the Anderson-Yuval scaling picture
of the anisotropic Kondo model for the Ohmic two-state system are briefly
discussed. This gives a qualitative understanding of the physics of the
latter in both the tunneling and localized regimes in terms of the fixed
points, their stability and their associated low energy scales. Finally
we show the connection between the scaling picture and the renormalization
group flow obtained from the exact solution of the AKM via the Bethe Ansatz.
The correspondence between the models via bosonization, described in
Appendix~\ref{app:equiv},
is then used to translate the thermodynamic Bethe Ansatz
equations for the anisotropic Kondo model, derived by Tsvelik and Wiegman,
into the language of the Ohmic two-state system in Sec.~\ref{tba-sec}
for both strong dissipation, $\alpha>1/2$, (or weak anisotropy in
the Kondo model) and weak dissipation, $\alpha < 1/2$
(or large anisotropy in the Kondo model) and at any level
asymmetry $\varepsilon$ (or local magnetic field in the Kondo model).
Analytic results are then presented for the specific heat and dielectric
susceptibility of the two-state system at high and low temperatures and
arbitrary dissipation in Sec.~\ref{sec-asymptotic} and at all
temperatures at the Toulouse point ($\alpha=1/2$) in Sec.~\ref{sec:Toulouse}.
The Wilson ratio for the Ohmic two-state system is discussed
in Sec.~\ref{Wilsonr}. Sec.~\ref{num-sec} gives the numerical solution of the
thermodynamic Bethe Ansatz equations at all temperatures for both weak
and strong dissipation and for both symmetric and asymmetric two-level
systems. Our conclusions are summarized in Sec.~\ref{sec-conclusions}.
Appendix~\ref{tba-derivation} contains
some details on the Bethe Ansatz solution of the AKM and the corresponding
thermodynamic Bethe Ansatz (TBA) equations which we solved in this paper.
Appendix~\ref{num-procedure}
gives details of the numerical procedure used to solve the TBA
equations and Appendix~\ref{wd-univ-eq} contains the
universal TBA equations for weak
dissipation (large anisotropies in the AKM), with some corrections made to
those found originally in Ref.~\onlinecite{tsvelik.83}
\section{MODELS}
\label{models-sec}
\subsection{The dissipative two-state system}
\label{sec-dtss}
The model of the dissipative two-state system is given by
\begin{eqnarray}
H_{SB} & = &-\frac{1}{2}\hbar\Delta \sigma_{x}+\frac{1}{2}\varepsilon\sigma_{z}
+\sum_{i} \omega_{i}(a_{i}^{\dagger}a_{i}+\frac{1}{2})\nonumber\\
&+&\frac{1}{2}q_{0}\sigma_{z}\sum_{i}
\frac{C_{i}}{\sqrt{2m_{i}\omega_{i}}}(a_{i}+a_{i}^{\dagger})\label{eq:SB}.
\end{eqnarray}
Here $\sigma_{i},i=x,y,z$ are Pauli spin matrices, the two states of the
system correspond to $\sigma_{z}=\uparrow$ and $\sigma_{z}=\downarrow$.
$\Delta$ is the bare tunneling matrix element and $\varepsilon$ is a bias.
The environment is represented by an infinite set of
harmonic oscillators (labeled by the index $i$)
with masses $m_{i}$ and
frequency spectrum $\omega_{i}$ coupling
linearly to the coordinate $Q=\frac{1}{2}q_{0}\sigma_{z}$
of the two-level system via a
term characterized by the couplings $C_{i}$.
The environment spectral
function is given in terms of these couplings, oscillator masses
and frequencies by
$J(\omega)=\frac{\pi}{2}
\sum_{i}(\frac{C_{i}^{2}}{m_{i}\omega_{i}})
\delta(\omega-\omega_{i})$.
In the case of an Ohmic heat bath, of interest to us here, we have
$J(\omega)=2\pi\alpha\omega$, for $\omega << \omega_{c}$, where $\omega_{c}$
is a high energy cut-off and $\alpha$ is a dimensionless parameter
characterizing the strength of the dissipation.
The Ohmic two-state model (also called the Ohmic spin-boson model)
has been intensively studied (for reviews we
refer the reader to \cite{leggett.87,weiss.99}). The model has
a low energy scale, $\Delta_r<\Delta$ for $\Delta << \omega_c$,
which depends on the dissipation
strength $\alpha$, and which may be interpreted as a renormalized tunneling
{\em amplitude}. For $\alpha=0$ the two-level system is decoupled from the
environment and $\Delta_r=\Delta$, whereas with increasing coupling to the
environment, this energy scale is strongly renormalized:
$\Delta_{r}/\omega_c \sim (\Delta/\omega_c)^{1/(1-\alpha)}$.
Another scale, the frequency of tunneling oscillations,
$\Omega(\alpha,\Delta_{r})=Q(\alpha)\Gamma(\alpha,\Delta_{r})$, is relevant
for time dependent quantities.
Here $\Gamma(\alpha,\Delta_{r})\sim \Delta_{r}$ is the
decay rate and $Q(\alpha)=\cot(\frac{\pi}{2}\frac{\alpha}{1-\alpha})$ is the
quality factor of the oscillations \cite{leggett.87}. The latter vanishes
at the Toulouse point $\alpha=\frac{1}{2}$, where the tunneling oscillations
vanish (the ``coherence-decoherence'' crossover).
For $0<\alpha<1/2$ the dynamics corresponds to damped
oscillations of frequency $\Omega(\alpha,\Delta_{r})$
\cite{leggett.87,weiss.99,lesage.98}.
This is sometimes called the ``coherent''
\cite{note-coherence} regime. The system exhibits phase coherence throught
this regime, albeit with damped oscillatory contributions to
real time dynamical quantities. A smooth crossover to
``incoherent''behaviour occurs at $\alpha=1/2$. The
tunneling amplitude remains finite in the ``incoherent''
regime $1/2\le \alpha < 1$, but there is no phase coherence in time dependent
dynamical quantities: $\Omega(\alpha,\Delta_{r})=0$ for $\alpha\ge 1/2$.
Another physically relevant value of the dissipation
strength is $\alpha=1/3$, where an inelastic peak, present
in the neutron scattering cross-section for $\alpha<1/3$, vanishes and gives
rise to a quasielastic peak for $\alpha>1/3$
\protect{\cite{costi.96,lesage.96,voelker.98}} (see also the discussion in
Sec.~\ref{sec-symmetric-heats}). Finally, for sufficiently strong
dissipation $\alpha\rightarrow 1^{-}$, the
renormalized tunneling amplitude vanishes
giving rise to the phenomenon of ``localization'' or ``self-trapping'' for
$\alpha > \alpha_{c}\approx 1$ ($\alpha_{c}$ depends also on the precise
value of $\Delta$). In this paper we will be interested only in the
thermodynamics of the dissipative two-state system. For such quantities
the exact solution shows that, in the tunneling regime
($0<\alpha<1$) \cite{note-tunneling}, the only relevant scale
is $\Delta_{r}$.
\subsection{Equivalence to the anisotropic Kondo model}
\label{ss:equiv}
The equivalence of the Ohmic two-state system to the anisotropic Kondo
model (AKM) has been shown at the Hamiltonian level via bosonization
\cite{guinea.85b} as outlined in Appendix~\ref{app:equiv}.
This equivalence was believed to be valid in the region $\alpha > 1/2$, which
corresponds (see below for the precise statement of the equivalence)
to the region in the parameter space of the AKM between weak-coupling
($\rho J_{\parallel} \ll 1$) and the Toulouse point
($\rho J_{\parallel}\approx 1$).
Recent work \cite{costi.96} shows
that the equivalence extends beyond the Toulouse point into the region
describing weak dissipation $0<\alpha<1/2$
(or large antiferromagnetic $J_{\parallel}$ in the AKM, see also
\cite{kotliar.96}).
The AKM is given by \cite{anderson.70}
\begin{eqnarray}
H &=& \sum_{k,\sigma} \varepsilon_{k}c_{k\sigma}^{\dagger}c_{k\sigma} +
\frac{J_{\perp}}{2}\sum_{kk'}
(c_{k\uparrow}^{\dagger}c_{k'\downarrow}S^{-} +
c_{k\downarrow}^{\dagger}c_{k'\uparrow}S^{+})\nonumber\\
&+& \frac{J_{\parallel}}{2}\sum_{kk'}
(c_{k\uparrow}^{\dagger}c_{k'\uparrow} -
c_{k\downarrow}^{\dagger}c_{k'\downarrow})S^{z}
+ g\mu_{B}hS_{z}.\label{eq:AKM}
\end{eqnarray}
The first term represents non-interacting conduction electrons and the
second and third terms represent an exchange interaction between a localized
spin $1/2$ and the conduction electrons with strength
$J_{\perp},J_{\parallel}$. The last term in Eq.~(\ref{eq:AKM})
is a local magnetic field, $h$, coupling only
to the impurity spin. The correspondence
between $H$ and $H_{SB}$, outlined in Appendix~\ref{app:equiv}, requires that
\begin{eqnarray}
\varepsilon&=&g\mu_{B}h\\
\frac{\Delta}{\omega_{c}}&=& \rho J_{\perp}\\
\alpha&=&(1+ \frac{2 \delta}{ \pi})^{2},
\end{eqnarray}
where $\tan{\delta}= -\frac{ \pi \rho J_{\parallel}}{4}$, $\delta$
is the phase shift for scattering of electrons from a
potential $J_{\parallel}/4$ and $\rho=1/2D$ is the conduction
electron density of states per spin at the Fermi level for a
flat band of width $2D=\omega_{c}$ \cite{leggett.87,costi.96}.
We note that weak dissipation ($\alpha\rightarrow 0$) in the Ohmic two-state
model corresponds to large antiferromagnetic coupling
($J_{\parallel}\rightarrow\infty$) in the Kondo model whereas dissipation
strength $\alpha>1$ in the Ohmic two-state model corresponds to ferromagnetic
coupling $J_{\parallel}<0$ in the Kondo model.
\subsection{Renormalization group flow}
\label{sec-rgflow}
The renormalization group flow of the Ohmic two-state system can be obtained
by making use of the above equivalence and the Anderson-Yuval scaling
equations\cite{anderson.70} for the AKM. These equations hold to lowest
order in $\rho J_{\perp}$ but for all $\rho J_{\parallel}$:
$-\infty<\rho J_{\parallel}<+\infty$. They are therefore valid for all
$0\le \alpha \le 4$ provided $\Delta/\omega_{c}\ll 1$. The Anderson-Yuval
scaling equations extend the validity of the well known Poor Man's scaling
equations to the whole $J_{\parallel}$ axis and reduce to those when
$\rho J_{\parallel}\ll 1$. In terms of the dimensionless quantities
$\rho J_\perp$ and
\begin{equation}
\tilde{\epsilon}=-8\frac{\delta}{\pi}
(1+\frac{\delta}{\pi})
\end{equation} where $\delta$ was defined above,
the Anderson-Yuval scaling equations read
\cite{anderson.70,leggett.87}
\begin{eqnarray}
\frac{d\tilde{\epsilon}}{d\ln D} & = & (\tilde{\epsilon}-2)(\rho J_{\perp})^{2}
+ {\cal O}(\rho J_{\perp})^{4}
\nonumber\\
\frac{d\rho J_{\perp}}{d\ln D} & = & -\frac{\tilde{\epsilon}}{2}\rho J_{\perp}
+ {\cal O}(\rho J_{\perp})^{3}
\label{eq.YuvalAnd}
\end{eqnarray}
By using the correspondence between the models given above, and noting that
$\tilde{\epsilon}=2(1-\alpha)$, we obtain the following scaling equations for
the Ohmic two-state system
\begin{eqnarray}
\frac{d\alpha}{d\ln \omega_{c}} & = & \alpha(\frac{\Delta}{\omega_{c}})^{2}
+ {\cal O}(\frac{\Delta}{\omega_{c}})^{4}\label{eq:scaling1}\\
\frac{d(\Delta/\omega_{c})}{d\ln \omega_{c}} & = & -(1-\alpha)
(\frac{\Delta}{\omega_{c}}) + {\cal O}(\frac{\Delta}{\omega_{c}})^{3}\label{eq:scaling2}
\end{eqnarray}
Note that in these equations $\alpha$ and $\Delta$ are running variables
which are functions of the running cut-off, $\omega_{c}$. The equations have
to be supplemented by specifying initial conditions
\begin{eqnarray*}
\alpha(\omega_{c}&=&\omega_{0})=\alpha_{0}\nonumber\\
\Delta(\omega_{c}&=&\omega_{0})=\Delta_{0}.
\end{eqnarray*}
where $\alpha_{0},\Delta_{0},\omega_{0}$ are now the parameters appearing
in the bare Hamiltonian (where they appeared
as $\alpha,\Delta,\omega_{c}$).
We shall use this notation for the remainder of this section.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm {\epsffile{new-fig3.eps}}}
\vspace{0.1cm}
\caption{
The scaling trajectories of the Ohmic two-state system obtained
from the Anderson-Yuval scaling equations for the AKM. Only the region
$0<\alpha<3$ is shown. The left and right separatrices at
$\alpha=1,\Delta/\omega_{c}=0$ define the regions labeled $A$, $B$ and $C$ and
the arrows indicate the direction of decreasing $\omega_{c}$.
}
\label{scaling-trajectories}
\end{figure}
From (\ref{eq:scaling1}--\ref{eq:scaling2}) there is a line
of fixed points at $\Delta_{0}/\omega_{0}=0$ for $\alpha_{0} \ge 0$. Their
stability to a finite $\Delta_{0}/\omega_{0}$
follows from (\ref{eq:scaling2}),
which states that $\Delta/\omega_{c}$ is relevant, marginal or irrelevant
depending on whether the dissipation strength $\alpha_{0}$ is less than,
equal to or larger than 1 \cite{note1}. Hence, the line of fixed points
at $\Delta_{0}/\omega_{0}=0$ for $\alpha_{0}>1$ are stable low energy fixed
points, whereas the line of fixed points at $\Delta_{0}/\omega_{0}=0$ for
$\alpha_{0}\le 1$ are unstable high energy fixed points. The scaling
trajectories can be calculated by dividing the
two equations (\ref{eq:scaling1}--\ref{eq:scaling2}) and integrating
the resulting equation from $\omega_{0}$ down to $\omega_{c}$:
\begin{equation}
\frac{1}{2}\left[(\frac{\Delta}{\omega_{c}})^{2}-
(\frac{\Delta_{0}}{\omega_{0}})^{2}\right] = -((\ln\alpha - \alpha) -
(\ln \alpha_{0}-\alpha_{0}))\label{eq:trajectories}
\end{equation}
They are shown in Fig.(\ref{scaling-trajectories}).
The arrows indicate the direction of decreasing
$\omega_c$. When the flow is to strong coupling, the scaling trajectories
will be quantitatively correct only for $\Delta/\omega_{c}\ll 1$.
The scaling diagram is divided into three regions
by two separatrices meeting at $\alpha=1,\Delta/\omega_{c}=0$.
The regime A ($\alpha_{0}>1$) corresponds to the localized regime
of the Ohmic two-state system (or the ferromagnetic sector of the
AKM). The dimensionless tunneling amplitude $\Delta/\omega_{c}$ is
irrelevant and the flow is to a line of fixed
points $(\alpha=\alpha^{\ast}$ and $\Delta/\omega_{c}=0)$.
This case is easily analyzed since the scaling
equations remain valid as $\omega_{c}\rightarrow 0$. Since $\Delta/\omega_{c}$
decreases as $\omega_{c}\rightarrow 0$, it follows from (\ref{eq:scaling1})
that $\alpha$ remains almost unrenormalized:
$\alpha\rightarrow \alpha^{*}\approx \alpha_{0}$ as $\omega_{c}\rightarrow 0$.
Integrating (\ref{eq:scaling2}) gives a renormalized tunneling amplitude
$\Delta_{r}\equiv\Delta(\omega_{c})=
\Delta_{0}(\omega_{c}/\omega_{0})^{\alpha_{0}}$
which vanishes at $T=0$ at low energies. Quantum mechanical tunneling is
absent for $\alpha_{0}>1$ at $T=0$ and for sufficiently small
$\Delta_{0}/\omega_{0}$.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm {\epsffile{new-fig4.eps}}}
\vspace{0.1cm}
\caption{Flow of running coupling constants in the localized
region $A$ of Fig.\,\ref{scaling-trajectories}, for $\alpha_{0}=1.4$
and $\Delta_{0}/\omega_{0}=0.01$.
}
\label{running-couplings-localized}
\end{figure}
At finite temperature the low energy
cut-off $\omega_{c}$ is replaced by $k_{B}T$ resulting in the well known
temperature temperature dependent tunneling amplitude
$\Delta_{r}(T)=\Delta_{0}(k_{B}T/\omega_{0})^{\alpha_{0}}$ in the strong
dissipation limit.
Fig.\,\ref{running-couplings-localized}
shows the flow of the dimensionless coupling constants for a typical
case in the localized regime. These were obtained by
integrating (\ref{eq:scaling1}--\ref{eq:scaling2}) using the Runge-Kutta
algorithm for 1st order differential equations.
The regime B ($\alpha<1$) corresponds to the
tunneling regime of the Ohmic two-state system (or the antiferromagnetic
sector of the AKM): in this regime $\Delta/\omega_{c}$ is relevant
and the flow for $0< \alpha_{0}\le 1$ is away from the line of high
energy fixed points at $\Delta/\omega_{c}=0$ towards the strong coupling
fixed point at $\alpha=0$ and $\Delta/\omega_{c}=\infty$.
This is shown in the numerical solution of
(\ref{eq:scaling1}--\ref{eq:scaling2}), in
Fig.\,\ref{running-couplings-tunneling}a.
The scaling analysis, of course, breaks down when $\Delta/\omega_{c}=O(1)$,
however, other methods, such as the numerical renormalization group and
the Bethe Ansatz, show that the low energy fixed point is at
$\Delta/\omega_{c}=\infty$ and $\alpha=0$. In this regime,
$\Delta(\omega_c)$ tends to a finite renormalized tunneling amplitude,
$\Delta_{r}$ as $\omega_{c}\rightarrow 0$. In the AKM this low energy scale
is the Kondo scale, generalized to the anisotropic case.
We can estimate the $T=0$ renormalized tunneling amplitude as the crossover
scale separating weak ($\Delta/\omega_{c}\ll1$) and strong coupling
($\Delta/\omega_{c}\gg1$) regimes of the model.
Define $\Delta_{r}=\Delta(\tilde{\omega}_{c})$ where $\tilde{\omega}_{c}$ is
the crossover scale such that
$\Delta(\tilde{\omega}_{c})/\tilde{\omega}_{c}=1$.
Integrating (\ref{eq:scaling2}) down to this crossover scale
\begin{equation}
\int_{1}^{\frac{\Delta_{0}}{\omega_{0}}}
\frac{d(\Delta/\omega_{c})}{\Delta/\omega_{c}} =
-\int_{\tilde{\omega}_{c}}^{\omega_{0}}(1-\alpha) d\ln \omega_{c}\nonumber
\end{equation}
and approximating $\alpha$ by $\alpha_{0}$ over this energy range gives
\begin{equation}
\Delta_{r}/\omega_{0}=
(\Delta_{0}/\omega_{0})^{\frac{1}{1-\alpha_{0}}},
\end{equation}
the correct low energy scale for the Ohmic two-state system, up to
prefactors depending on $\alpha_{0}$.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm {\epsffile{new-fig5a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm {\epsffile{new-fig5b.eps}}}
\vspace{0.1cm}
\caption{Flow of running coupling constants for (a) $\alpha_{0}=0.1$
and $\Delta_{0}/\omega_{0}=0.01$ corresponding to the tunneling
region $B$ in Fig.\,\ref{scaling-trajectories},
and (b) $\alpha_{0}=1.3$ and $\Delta_{0}/\omega_{0}=0.4$
corresponding to the tunneling region $C$ in Fig.\,\ref{scaling-trajectories}.
In (b), $\Delta_{c}$ is a crossover scale at which $\Delta$ becomes of O(1).
}
\label{running-couplings-tunneling}
\end{figure}
Finally, there is a region C, of strong dissipation, which also corresponds
to the tunneling regime of the Ohmic two-state system. However, the flow
to the strong coupling fixed point is such that, for $\alpha_{0}>1$,
$\Delta/\omega_{c}$ initially decreases with decreasing $\omega_{c}$, signaling
a tendency to localization for strong dissipation, but eventually, as a
result of a strong renormalization of $\alpha$ to below 1, $\Delta/\omega_{c}$
becomes relevant and then increases to strong coupling.
The flow is to the same strong coupling fixed point as for region B and
therefore region C belongs to the tunneling regime of the model.
Due to the strong renormalization of $\alpha$,
Fig.\,\ref{running-couplings-tunneling}b, it is difficult
to estimate the form of the low energy scale $\Delta_{r}$ in this case.
The renormalization group flow described above consists of a one parameter
family of scaling trajectories labeled by a parameter $C=C(\alpha,\Delta)$
which takes a constant value along each trajectory. This constant of the
motion is called a scaling invariant and can be found from
(\ref{eq:trajectories}):
\begin{equation}
C(\alpha,\Delta)=-1-\frac{1}{2}(\frac{\Delta}{\omega_{c}})^{2}
-(\ln\alpha - \alpha)\label{eq:constant-of-motion}
\end{equation}
Corresponding to this one parameter family of scaling trajectories we expect
the scaling functions for physical quantities to consist of a
one parameter family labeled by $C$, with different scaling functions for each
scaling trajectory. The scaling invariant $C$ is not unique but depends on the
cut-off scheme of the theory. A scaling trajectory may
be specified differently (i.e. by a different function $C$) depending on
the cut-off scheme . One then has to identify the scaling trajectories
within the different schemes if one wishes to compare results.
The Bethe Ansatz solution of the AKM, which
we want to use in the next section, is an example where such a different
scheme is used. We state here how we identify the scaling trajectories of
the Bethe Ansatz solution with those we discussed above for the Ohmic
two-state system (or equivalently the AKM with a finite bandwidth
$2D=\omega_{c}$) and leave the details to Appendix~\ref{tba-derivation}.
As discussed in the appendix, the Bethe Ansatz solution
yields a renormalization
group flow depending on two functions, $\mu$ and $f$, of the dimensionless
couplings of the AKM. The function $\mu$ is the scaling invariant
and specifies the scaling trajectories and the function $f$ sets the
low energy scale
\begin{equation}
T_{K}={2D}\exp(-f/(\mu/\pi)). \label{eq:tk-definition}
\end{equation}
Comparing the high-temperature behaviour of the two models we find
the correspondence
\begin{equation}
\mu/\pi = 1-\alpha_{0}\;.
\end{equation}
We shall show in Sec.~\ref{Wilsonr} that the static susceptibility,
$\chi_{BA}$, calculated from the Bethe Ansatz solution is equal
to the dielectric susceptibility, $\chi_{sb}$, of the Ohmic two-state
system and that, at $T=0$, these are related to the Kondo temperature, as
defined above, by
\begin{equation}
\chi_{BA}=\chi_{sb}=
\frac{1}{2\pi(1-\mu/\pi)T_{K}}=\frac{1}{2\pi\alpha_{0}T_{K}}.
\label{eq:chisb-tk}
\end{equation}
This suggests that we define the renormalized tunneling
amplitude $\Delta_{r}$ in terms of $T_{K}$ by \cite{note-factor-alpha}
\begin{equation}
\Delta_{r} = \alpha_{0}T_{K}\label{eq:deltar-tk}
\end{equation}
so that the local $T=0$ dielectric susceptibility, $\chi_{sb}$, of the Ohmic
two-state system is given in terms of $\Delta_{r}$ by:
\begin{equation}
\chi_{sb}=\frac{1}{2\pi\Delta_{r}}.\label{eq:chi-deltar}
\end{equation}
The above relations fix $f$ in terms of $\mu$ and $T_{K}$
(or equivalently in terms of $\alpha_{0}$ and $\Delta_{r}$) and will prove
useful in translating the Bethe Ansatz results of Sec III into results
for the Ohmic two-state system.
\section{THERMODYNAMIC BETHE ANSATZ EQUATIONS}
\label{tba-sec}
\subsection{Thermodynamic Bethe Ansatz equations for $\alpha=1/\nu$ and
$\alpha=1-1/\nu$}
\label{tba-section}
The thermodynamic Bethe Ansatz equations for the anisotropic Kondo model have
been derived by Tsvelik and Wiegman \cite{tsvelik.83}
(for a short overview of the derivation see Appendix~\ref{tba-derivation}).
For a general anisotropy they consist of an infinite set,
$n=1,2,\dots$, of coupled integral equations for the ``excitation
energies''
$\epsilon_{n}(\lambda)$ (defined in Appendix~\ref{tba-derivation}).
The thermodynamics is calculated from the impurity
free energy which depends explicitly only on $\epsilon_{1}$. As discussed
in \cite{tsvelik.83}, the infinite set of integral equations for the
$\epsilon_{n}$ decouple to a finite set for values of the anisotropy
corresponding to rational values of the scaling invariant $\mu/\pi$ of the
anisotropic Kondo model, and hence to rational values of the dissipation
strength $\alpha=1-\mu/\pi$ in the Ohmic two-state system. In particular
for anisotropies given by $\mu/\pi=1/\nu$ and
$\mu/\pi=1-1/\nu$ with $\nu=3,4,\dots$, there are only $\nu$ coupled integral
equations for the $\nu$ quantities $\epsilon_{1},\epsilon_{2},\dots,
\epsilon_{\nu}$. These two cases allow us to study both
the weak dissipation ($\alpha=1/\nu < 1/2$) and the strong dissipation
($\alpha=1-1/\nu > 1/2$) limits of the Ohmic two-state system. Explicitly,
the $\nu$ coupled integral equations in Eq.6.2.11 of \cite{tsvelik.83} are
\begin{eqnarray}
\frac{\epsilon_{j}(\lambda)}{T} & = &
s\ast\bigl[\ln(1+\exp{\frac{\epsilon_{j-1}}{T}})
(1+\exp{\frac{\epsilon_{j+1}}{T}})\;,\nonumber\\
& + & \delta_{j,\nu-2}\ln(1+\exp{-\frac{\epsilon_{\nu}}{T}})\bigr]
+ \delta_{j,1}D(\lambda)
\nonumber\\
\frac{\epsilon_{\nu-1}(\lambda)}{T} & = & x_{0} +
s\ast\ln(1+\exp{\frac{\epsilon_{\nu-2}}{T}})\;,
\label{eq:BAepsilonint}\\
\frac{\epsilon_{\nu}(\lambda)}{T} & = & x_{0} -
s\ast\ln(1+\exp{\frac{\epsilon_{\nu-2}}{T}})\;.\nonumber
\end{eqnarray}
Here $x_{0}=\frac{1}{2}\frac{g\nu\varepsilon}{T}$, $\varepsilon$ is the bias,
$g=1$ is a $g$-factor in the Kondo problem and $D(\lambda)$ is the
driving term which is given explicitly by
\begin{equation}
D(\lambda) = -\mbox{sign}[\alpha-\frac{1}{2}]
\frac{\omega_c}{T}\arctan(\exp(\pi\lambda))\;.
\label{eq:driving}
\end{equation}
with $\omega_c$ a high energy cut-off (corresponding to the band width
cutoff, $D$, in the Kondo model).
The integral operator $s\ast$ is defined as in \cite{tsvelik.83}:
\begin{equation}
s\ast f(\lambda) = \int_{-\infty}^{+\infty}\frac1
{2\cosh(\pi(\lambda-\lambda'))}{f(\lambda')}\;d\lambda'.
\label{eq:integral-operator}
\end{equation}
The $\epsilon_{j}$'s ($j=1,\dots,\nu$) satisfy boundary
conditions at $\lambda=\pm\infty$ that can be obtained easily
from Eq.~(\ref{eq:BAepsilonint}).
The impurity contribution to the free energy can be simply expressed as
\begin{equation}
F = - T \int_{-\infty}^{+\infty} s(\lambda + \frac f\mu) \;\ln\bigl(1 +
e^{\varepsilon_1(\lambda)/T}\bigr)\;d\lambda\;.
\label{eq.impfreeen}
\end{equation}
Eqs.~(\ref{eq:BAepsilonint}) and (\ref{eq.impfreeen}) describe the
complete thermodynamics of the model.
Note that the field term $x_0$ in Eq.~(\ref{eq:BAepsilonint})
represents a {\it global} magnetic field in the AKM coupling to both the
impurity and conduction electrons.
However, as we discuss in Sec.~\ref{Wilsonr}, due to electron-electron
interactions introduced to assure integrability , the impurity susceptibility
of the BA solution, $-\partial^{2}F/\partial \varepsilon^{2}$, coincides
with the susceptibility of the Ohmic two-state system (i.e. with the
impurity susceptibility of the AKM with a field coupling {\em only} to the
impurity). Therefore the dielectric susceptibility,
$\chi_{sb}(T,\varepsilon)$, and specific heat, $C(T,\varepsilon)$,
of the Ohmic two-state system can be simply calculated as:
\begin{eqnarray}
\chi_{sb}(T,\varepsilon) &=& -\partial^{2}F/\partial \varepsilon^{2}\;,\\
C(T,\varepsilon)&=&-T\partial^{2}F/\partial T^{2}\;.
\end{eqnarray}
\subsection{Analytic results}
\subsubsection{Scaling and universality}
A careful analysis of Eqs.~(\ref{eq:BAepsilonint}) and (\ref{eq.impfreeen})
makes immediately transparent the meaning of universality.
For $\mu < \pi/2$ ($\alpha > 1/2$) the impurity free energy
can be shown to be dominated by contributions from the region
$\lambda \ll 0$. In this limit the driving term can be approximated as
$D(\lambda)\approx -\frac{\omega_{c}}T e^{\pi\lambda}$ and the explicit
cutoff dependence can be transformed out of the equations by a simple
shift, $\lambda\to \lambda + \frac1\pi \ln \frac T{\omega_c}$.
In this way one arrives at the following universal equations for the
quantities $\varphi_{j}(\lambda)\equiv \epsilon_{j}(\lambda +
\frac1\pi \ln \frac T{\omega_c}) /T$:
\begin{eqnarray}
&&{\varphi_{j}(\lambda)} =
s\ast\bigl[\ln(1+e^{\varphi_{j-1}})
(1+e^{\varphi_{j+1}})\nonumber\\
&&\phantom{nnnn} + \delta_{j,\nu-2}\ln(1+e^{-\varphi_{\nu}})\bigr]
- \delta_{j,1}e^{\pi\lambda}
\nonumber \\
&&\varphi_{\nu-1}(\lambda) = x_{0} +
s\ast\ln(1 + e^{\varphi_{\nu-2}})
\label{eq:univ} \\
&&\varphi_{\nu}(\lambda) = x_{0} -
s\ast\ln(1+e^{\varphi_{\nu-2}}),
\nonumber\;, \\
&& F = - T \int_{-\infty}^{+\infty}
s(\lambda + \frac 1 \pi \ln \frac T {T_K})
\;\ln\bigl(1 +
e^{\varphi_1(\lambda)}\bigr)\;d\lambda\;,
\label{eq:impfree}
\end{eqnarray}
where the Kondo temperature has been introduced as
\begin{eqnarray}
T_K & = & \omega_c \exp(-\pi f/ \mu) \nonumber \\
& \equiv & \Delta_r/\alpha \;.
\label{eq:T_K}
\end{eqnarray}
For $\mu > \pi/2$ ($\alpha<1/2$) the seemingly innocent sign change
of the driving term alters the structure of the solutions completely.
The derivation of the universal equations becomes much more
complicated in this case, but is still possible \cite{tsvelik.83}.
These were not used
in obtaining the numerical results described below, but they served as a
useful check on the simpler set of equations (\ref{eq:BAepsilonint}).
We reproduce them, correcting some minor typos in
\cite{tsvelik.83}, in Appendix~\ref{wd-univ-eq},
together with a comparison of
numerical results obtained from them and the simpler set of equations
(\ref{eq:BAepsilonint}).
The equations above clearly show that the thermodynamic quantities depend
only on the ratios $\varepsilon/T$ and $T/T_K \sim T/\Delta_r$. Note,
however, that while the parameter $f$ only influences $T_K$,
for each $\mu$ ($\alpha$) one obtains a different set of equations
and therefore different thermodynamic behaviour. Thus two
models have essentially the same universal behaviour if and only if
their parameter $\mu$ ($\alpha$) is the same. From these considerations
immediately follows that the usual RG scaling trajectories
correspond to the lines $T_K={\rm const}$ and $\mu = {\rm const}$.
(The latter
requirement also follows from the fact that $2\mu$ turns out to
be the anomalous dimension characterizing the high temperature
behaviour, which should be scale invariant.) In the small coupling
limit one immediately obtains the usual
leading logarithmic scaling equations by expanding $f$ and $\mu$
in $\rho J_\perp\approx I_\perp$ and $\rho J_z\approx I_z$:
\begin{eqnarray}
{d {\rho J_z} \over d \ln \omega_c } & =& - (\rho J_\perp)^2\;,\\
{d {\rho J_\perp} \over d \ln \omega_c } &=& - \rho J_\perp \rho J_z \;,
\end{eqnarray}
in agreement with Eq.~(\ref{eq.YuvalAnd}).
\subsubsection{Asymptotic properties}
\label{sec-asymptotic}
The asymptotic behaviour of various physical quantities can be
determined by analyzing Eq.~(\ref{eq:BAepsilonint}).
Rewriting Eq.~(\ref{eq:BAepsilonint}) in terms of the quantities
$\xi_{j}$, $j=1,\dots,\nu$ defined by
\begin{eqnarray}
\xi_{j} & = &\ln[1+\exp(\frac{\epsilon_{j}}{T})]\;\;\;\;\;(j=1,\dots,\nu-1),
\nonumber\\
\xi_{\nu} & = &\ln[1+\exp(-\frac{\epsilon_{\nu}}{T})]\;,
\label{eq:xi}
\end{eqnarray}
one can easily show that in the $\lambda\to-\infty$ limit the
asymptotic solution of the BA equations behaves as
\begin{eqnarray}
\xi_j(\lambda \to -\infty)& =& \xi^{-}_j(x_0) + b^{-}_j(x_0)
e^{ \tau \lambda} \;.
\end{eqnarray}
with $\tau = 2\mu$. On the other hand, for $\lambda\gg \ln(T/\omega_c)$
$ \xi_1$ vanishes extremely fast and
can be approximated by 0.
Substituting these into Eq.~(\ref{eq:impfree}) and using
$\mu/\pi=1-\alpha$ and $\Delta_{r}\sim T_{K}$ one can
immediately extract the leading behaviour of the impurity free energy.
In the high temperature limit one obtains:
\begin{eqnarray}
&&F(T\gg \Delta_{r}, \varepsilon) \approx -T \Bigl\{\ln\frac{\sinh(\varepsilon g/T)}
{\sinh(\varepsilon g/2T)} + \nonumber \\
&& \phantom{nnnnn} - \bigl(\frac{\Delta_{r}}T\bigr)^{2-2\alpha}
\Bigl( A + B \bigl({\varepsilon \over T }\bigr)^2\Bigr) \Bigr\}\;,
\\
&&\chi_{sb}(T\gg \Delta_{r};\;\varepsilon=0)
= -\frac{\partial^{2}F}{\partial
\epsilon^{2}}\approx \nonumber \\
&& \phantom{nnnnn} \approx\frac1T \bigl( {g^2 \over 4 } - 2 B
\bigl(\frac{\Delta_{r}}T\bigr)^{2-2\alpha} \bigr)\;,\label{eq:chi-hightemp}
\\
&&C (T\gg \Delta_{r};\;\varepsilon=0) \sim \bigl(
\frac{\Delta_{r}}T\bigr)^{2-2\alpha}\;,\label{eq:c-hightemp}
\end{eqnarray}
where the constants $A$ and $B$ depend only on $\mu$. From these equations
it is clear that in order to recover the free
impurity spin at high temperatures --- unlike the choice $g =
(\nu-1)/\nu$ of Ref.~\onlinecite{tsvelik.83} --- we have to take the bare
value $g=1$ for the electronic g-factor. This special choice will
influence the Wilson ratio discussed below.
Similarly, for the low temperature regime we obtain:
\begin{eqnarray}
&&F(\varepsilon \ll T \ll \Delta_{r}) \approx {T^2\over \Delta_{r}}
\Bigl( \frac{\alpha\pi} 6 + {g^2\over 4\pi}{\varepsilon^2 \over T^2 }
\Bigr)\;,
\label{eq:fimp}
\\
&&\chi_{sb}(T\ll \Delta_{r};\;\varepsilon=0) \approx \frac1{\Delta_{r}}
{g^2\over 2\pi}\;,
\label{eq:chi-fliq}
\\
&&C(T\ll \Delta_{r};\;\varepsilon=0) \approx
{\pi\over 3}\frac {T\alpha}{\Delta_{r}}\;,
\label{eq:c-fliq}
\end{eqnarray}
where the numerical constants have been calculated following the
same lines as in Ref.~\onlinecite{tsvelik.83}.
Thus at low temperatures the well-known Fermi liquid behaviour is
recovered \cite{Nozieres,Yamada}.
\subsubsection{Susceptibility and Wilson ratio for the Ohmic two-state system}
\label{Wilsonr}
As discussed by Wiegmann and Tsvelik,\cite{tsvelik.83} to ensure
integrability, an artificial electron-electron interaction has to
be introduced. While this interaction has no effect in the course
of the solution of the isotropic model, in the anisotropical model it
renormalizes the electronic and impurity $g$-factors:
\begin{equation}
g =1 \to \tilde g = {1\over \sqrt{\alpha}} = {1\over \sqrt{1-\mu/\pi}}\;.
\end{equation}
This can be most easily checked by calculating the host susceptibility of the
AKM , $\chi_{host}= -\partial^2 F_{host}/\partial h^2$,
following the lines of Ref.~\onlinecite{tsvelik.83}.
(Here $F_{host}$ is the free energy of the electrons, and is
given by an expression
similar to Eq.~(\ref{eq.impfreeen}) with $f=0$.) After a tedious calculation
one obtains the result that:
\begin{eqnarray}
\chi_{host} = {\chi_{free}\over1-\mu/\pi} = {\chi_{free}\over
\alpha}\;,
\label{chi_host}
\end{eqnarray}
where $\chi_{free} = L/(4\pi N)$ denotes
the free electron susceptibility ($L$ is the length
of the system, $N$ the number of electrons, and $v_F = k_B = 2\mu_B =
1$). Note that the specific heat, $C_{host} = C_{free} = TL\pi /N 3$,
is completely unaffected by the electron-electron interaction.
We now prove the statement made in Sec.~\ref{tba-section}, that the
impurity contribution to the global susceptibility of the AKM,
{\em obtained from the Bethe Ansatz calculations}, is identical to the
susceptibility of the Ohmic two-state system. We denote the bare
g-factors of the impurity and conduction electrons in the AKM
by $g_{i}$ and $g_{e}$ and the corresponding susceptibility of the AKM by
$\chi(g_{i},g_{e})$ \cite{note-SCREAM}.
Now in the Bethe Ansatz solution, starting
with bare values $g_{i}=g_{e}=1$, the renormalizations discussed above
imply that the BA susceptibility,
$\chi_{BA}\equiv -\partial^{2}F/\partial\varepsilon^{2}$,
is given by
\begin{eqnarray}
\chi_{BA}=\chi(\tilde{g}_{i}=\tilde{g}_{e}=1/\sqrt{\alpha})\equiv
\chi(g_{i}=g_{e}=1)/\alpha.\label{eq:chi-ba}
\end{eqnarray}
The dielectric susceptibility of the Ohmic two-state system, $\chi_{sb}$,
measures the response to a local electric field and is equal to the impurity
susceptibility of the AKM, $\chi(g_{i}=1,g_{e}=0)$, with the magnetic
field coupling only to the impurity spin:
\begin{equation}
\chi_{sb}=\chi(g_{i}=1,g_{e}=0).
\label{eq:chi-sb}
\end{equation}
This follows from the equivalence of the two models discussed in
Appendix~\ref{app:equiv}.
We now make use of Eq.21 of Ref.~\onlinecite{vigmann.78}
(valid for arbitrary $T$), connecting the impurity susceptibilities of the
AKM with arbitrary g-factors. This states that
\begin{eqnarray}
\chi(g_i = g_e = 1) & = & (1 + 2{\delta\over \pi})^2\chi(g_i = 1;g_e = 0)\;,
\label{eq:w+f}
\end{eqnarray}
where the phase shifts have been defined in
Appendix~\ref{app:equiv} (note the sign change
with respect to Ref.~\onlinecite{vigmann.78}).
Hence, using (\ref{eq:chi-ba}-\ref{eq:chi-sb}) and Eq.~(\ref{eq:alpha})
$\alpha= (1 + 2{\delta\over \pi})^2$, we find that the BA susceptibility
is just the susceptibility of the Ohmic two-state system,
\begin{equation}
\chi_{BA}(T,\varepsilon) \equiv -\partial^{2}F/\partial
\varepsilon^{2}
= \chi(\tilde g_i = \tilde g_e = {1/\sqrt{\alpha}}) = \chi_{sb}\;,
\end{equation}
as stated earlier. This can be further checked by calculating
the high temperature Curie susceptibility and the Wilson ratio
(from Eqs.~(\ref{eq:chi-fliq}) and (\ref{eq:c-fliq}))
for which we find
\begin{eqnarray}
\chi_{sb} = \chi_{BA} (T\gg T_K\sim \Delta_{r}) \approx {1\over 4T}\;, \\
R_{sb} = R_{BA} \equiv \lim_{T\to0}{C_{free}(T)\over \chi_{free}}
{\chi_{BA}\over C(T)} = 2/\alpha\;,
\end{eqnarray}
in agreement with exact results obtained for the spin-boson
model.\cite{sassetti.90}
Furthermore, one can easily check by a bosonization procedure
along the Toulouse line\cite{JanZar} that Eq.~(\ref{Toulouse}) is identical
with the impurity contribution to the free energy in the presence
of a {\em local} magnetic field applied at the impurity.
We note that the above result for the Wilson ratio of the Ohmic two-state
system holds for all level asymmetries (local magnetic field in the AKM).
Finally, in order to prevent confusion, we state the connection of the above
Wilson ratio for the Ohmic two-state system to that usually encountered
in the Kondo model. The former is defined using the susceptibility of
the Ohmic two-state system $\chi_{sb}$ which we showed was equal to
the susceptibility $\chi_{BA}$ resulting from the BA calculation on the
AKM (with an electron-electron interaction to ensure integrability). The
susceptibility used in defining the Wilson ratio for magnetic impurities
is, however, not $\chi_{BA}=\chi_{sb}=\chi(g_{i}=1,g_{e}=0)$ but
$\chi(g_{i}=1,g_{e}=1)$. Therefore, in terms of this susceptibility,
and using Eq.~(\ref{eq:w+f}), the corresponding Wilson ratio, $R_{akm}$,
is given by
\begin{equation}
R_{akm} \equiv \lim_{T\to0}{C_{free}(T)\over \chi_{free}}
{\chi(g_{i}=g_{e}=1)\over C(T)} = 2\;.
\end{equation}
The enhancement over the non-interacting value $R=1$, indicates
that the quasiparticles are strongly interacting at low temperatures
\cite{Hewson}.
\subsubsection{The Toulouse point: $\alpha=1/2$}
\label{sec:Toulouse}
The AKM possesses a so-called Toulouse
line\cite{toulouse.69,vigmann.78,JanZar}.
Along this line the model can be mapped by a simple unitary
transformation to a resonant level model without interaction
and can be solved by refermionization. For a dissipative two-state
system this line has been shown to correspond to the special
value $\alpha = 1/2$ ($\mu = \pi/2$, $\nu=2$ in the BA solution)
separating the coherent and incoherent tunneling
regimes.
Along this line the BA solution simplifies enormously too: For
$\nu=2$ only 'one-strings' with parity $v=\pm$ are allowed, and
as one can immediately check by using Eq.~(\ref{eq:spinBA}), the rapidities
of the spin-excitations are completely decoupled. Therefore,
in the first of Eqs.~(\ref{eq:BAepsilonint}) only the driving term remains
and one obtains in the scaling limit:
\begin{eqnarray}
\frac1T\epsilon_1(\lambda)& = & -\frac{\omega_c}T \;e^{\pi\lambda}
+ {g\varepsilon\over T}\;,\nonumber \\
\frac1T\epsilon_2(\lambda)& = & \frac{\omega_c}T \;e^{\pi\lambda}
+ {g\varepsilon\over T}\;.
\end{eqnarray}
Substituting these expressions into Eq.~(\ref{eq.impfreeen}) one
immediately arrives at
\begin{eqnarray}
F &=& -{T\over \pi} \int_{0}^{\infty} dk {T_K\over
k^2 + T_K^2} \ln \bigl\{1 + 2\cosh(g\varepsilon/T) e^{-k/T}
\nonumber \\
&+&e^{-2k/T}\bigr\}\;,
\label{Toulouse}
\end{eqnarray}
which coincides with the resonant level result
(note the slight difference with respect to the formula 6.2.15 of
Ref.~\cite{tsvelik.83}).
It is straightforward to verify that in the limit $T\rightarrow 0$
$$
\chi_{sb} = g^2 / (\pi T_K)\equiv g^{2}/2\pi\Delta_{r},
\;\;\;\alpha=1/2
$$
and
$$
C = \pi T /(3 T_K)\equiv \pi T/6\Delta_{r},\;\;\;\alpha=1/2
$$
giving the expected Wilson ratio, $ R_{sb}=2/\alpha=4$, for the
Ohmic two-state system at $\alpha=1/2$, with $R_{sb}$ as
defined above. The high temperature limits
$S=\ln 2$ and $\chi_{BA}(T\gg T_{K}\sim \Delta_{r})=g^{2}/4T$
are also easily verified.
\section{NUMERICAL RESULTS AT ALL TEMPERATURES}
\label{num-sec}
\subsection{Numerical procedure}
\label{subsec-num-proc}
The closure of the infinite set of thermodynamic Bethe Ansatz equations to
a finite set at rational values of the dissipation strength $\alpha$ can
be used to obtain highly accurate results for the thermodynamics. This avoids
the truncation errors associated with solving these equations at other
values of $\alpha$. In particular
at $\alpha=1/\nu$ and $\alpha=1-1/\nu$ we have $\nu$ equations. In the
numerical procedure we found it more convenient to set up integral equations
for new quantities $\xi_{j}, j=1,\dots,\nu$ defined by eq.~(\ref{eq:xi})
\begin{eqnarray}
\xi_{j} & = &\ln[1+\exp(\frac{\epsilon_{j}}{T})]=\ln[1+\kappa_{j}],\;\; j=1,\dots,\nu-1\nonumber\\
\xi_{\nu} & = &\ln[1+\exp(-\frac{\epsilon_{\nu}}{T})]=\ln[1+\kappa_{\nu}],
\label{eq:kappa}
\end{eqnarray}
where the functions $\kappa_{j},j=1,\dots,\nu$ are introduced for later
convenience.
The TBA equations then take the form,
\begin{eqnarray}
\xi_{j}(\lambda) & = & \ln[1+\exp(\delta_{j,1}D(\lambda)\nonumber\\
&+& s\ast(\xi_{j-1}+\xi_{j+1} + \delta_{j,\nu-2}\xi_{\nu}))],\nonumber\\
\xi_{\nu-1}(\lambda) & = & \ln[1+\exp(x_{0} + s\ast\xi_{\nu-2})],\;
\label{eq:tba-ksi}\\
\xi_{\nu}(\lambda) & = & \ln[1+\exp(-x_{0} + s\ast\xi_{\nu-2})].\nonumber
\end{eqnarray}
The impurity free energy is given by
\begin{equation}
F(T,\varepsilon) = -k_{B}T\int_{-\infty}^{+\infty}s(\lambda + f/\mu)
\xi_{1}(\lambda,T,\varepsilon)d\lambda
\end{equation}
where $s(\lambda)=(2\cosh(\pi\lambda))^{-1}$ and $f/\mu$ is
related to the low energy scale, $T_{K}$, of the AKM by Eq.~(\ref{eq:T_K})
The Kondo temperature, $T_{K}$, is related to the renormalized tunneling
amplitude, $\Delta_{r}$, by $\Delta_{r}=\alpha T_{K}$ as discussed in Sec.IIc.
The entropy, specific heat and dielectric susceptibility can be obtained
by numerically differentiating the Free energy:
\begin{eqnarray*}
S(T,\varepsilon) &= & - \frac{\partial F}{\partial T}
=\int s(\lambda + f/\mu)\;\frac{\partial\; T\xi_{1}
(\lambda,T,\varepsilon)}{\partial T}\;d\lambda\nonumber\\
C(T,\varepsilon) &= & - T\frac{\partial^{2} F}{\partial T^{2}}
=\int s(\lambda + f/\mu)\;\frac{\partial^{2}\; T\xi_{1}
(\lambda,T,\varepsilon)}{\partial T^{2}}\;d\lambda\nonumber\\
\chi(T,\varepsilon) &= &-\frac{\partial^{2}
F}{\partial \varepsilon^{2}} =
\frac{g^{2}}{4T}\int s(\lambda + f/\mu)
\frac{\partial^{2} \xi_{1}(\lambda,T,\varepsilon)}{\partial x_{0}^{2}}
d\lambda\nonumber
\end{eqnarray*}
A more accurate procedure is to set up integral
equations for the derivatives $\partial\;(T\xi_{j})/\partial T$,
$\partial\xi_{j}/\partial x_{0}$ and $\partial^{2}\xi_{j}/\partial x_{0}^{2}$.
More precisely, we set up integral equations for a new set of functions,
$E_{j},F_{j}$ and $G_{j}=\partial\;(T\xi_{j})/\partial T =
\xi_{j}+T\partial\xi_{j}/\partial T$
where $E_{j}$ and $F_{j}$ are the first and second field derivatives of
$\xi_{j}$,
\begin{eqnarray}
E_{j} &\equiv &\frac{\partial\xi_{j}}{\partial x_{0}} = \frac{\partial\kappa_{j}/
\partial x_{0}}{1+\kappa_{j}},\label{ej-definition}\\
F_{j} &\equiv &
\frac{\partial^{2} \xi_{j}}{\partial x_{0}^{2}}=
\frac{\partial^{2}\kappa_{j}/\partial x_{0}^{2}}{1+\kappa_{j}} -
\left[\frac{\partial\kappa_{j}/
\partial x_{0}}{1+\kappa_{j}}\right]^{2},\label{fj-definition}
\end{eqnarray}
and the functions $\kappa_{j}$ where defined in (\ref{eq:kappa}).
Each set of functions $E_{j},F_{j},G_{j},j=1,\dots,\nu$ then obey
coupled linear inhomogeneous integral equations.
The equations are,
\begin{eqnarray}
E_{j} & = & (1-e^{-\xi_{j}})\,
s*(E_{j-1}+E_{j+1}+\delta_{j,\nu-2}E_{\nu})\nonumber\\
E_{\nu-1} & = & (1-e^{-\xi_{\nu-1}})\,(s*E_{\nu-2}+1),\;\label{eq:ejba}\\
E_{\nu} & = & (1-e^{-\xi_{\nu}})\,(s*E_{\nu-2}-1)\nonumber
\end{eqnarray}
for the $E_{j}$, with the inhomogeneity appearing in the last two equations.
For the $F_{j}$ we have
\begin{eqnarray}
F_{j} & = & Q_{j}+
(1-e^{-\xi_{j}})\,s*(F_{j-1}+F_{j+1}+\delta_{j,\nu-2}F_{\nu})\nonumber\\
F_{\nu-1} & = & Q_{\nu-1} + (1-e^{-\xi_{\nu-1}})\,s*F_{\nu-2},\;\label{eq:fjba}\\
F_{\nu} & = & Q_{\nu} + (1-e^{-\xi_{\nu}})\,s*F_{\nu-2},\nonumber
\end{eqnarray}
with an inhomogeneous term,
$$
Q_{j} = E_{j}^{2}e^{-\xi_{j}}(1-e^{-\xi_{j}})^{-1}.\nonumber
$$
and for the $G_{j}$ we find
\begin{eqnarray}
G_{j} & = & S_{j}+(1-e^{-\xi_{j}})\,
s*(G_{j-1}+G_{j+1}+\delta_{j,\nu-2}G_{\nu})\nonumber\\
G_{\nu-1} & = & S_{\nu-1}+ (1-e^{-\xi_{\nu-1}})\,s*G_{\nu-2},\;\label{eq:gjba}\\
G_{\nu} & = & S_{\nu} + (1-e^{-\xi_{\nu }})\,s*G_{\nu-2}\nonumber
\end{eqnarray}
with an inhomogeneous term,
$$
S_{j} = -e^{-\xi_{j}}\ln e^{-\xi_{j}}-(1-e^{-\xi_{j}})\ln (1-e^{-\xi_{j}}).
\nonumber
$$
The numerical procedure is to first solve iteratively
for the $\xi_{j},j=1,\dots,\nu$. Using this solution one then
iteratively solves
for the functions $E_{j},F_{j},G_{j},j=1,\dots,\nu$ in turn.
Fig.\,\ref{tba-graph} shows
a graphical representation of these integral equations.
\begin{figure}[t]
\centerline{\epsfysize 4.0cm {\epsffile{new-fig6.eps}}}
\vspace{0.1cm}
\caption{
Graphical representation of the integral equations for the
$\xi_{j},E_{j},F_{j},G_{j},j=1,\dots,\nu$.
}
\label{tba-graph}
\end{figure}
The entropy
and susceptibility are thereby obtained without the need to take any numerical
derivatives. Only one derivative is required to obtain the specific heat from
the entropy, so we did not set up separate integral equations for the second
temperature derivative of $T\xi_{1}$. Such a procedure has been used for zero
field static susceptibilities in\cite{desgranges.85}. This approach also
overcomes difficulties at large fields (level asymmetries) and low
temperatures found in early treatments of similar TBA equations for
Kondo systems \cite{rajan.82} provided one deals with the exponential
decrease in the $j=1$ ($j=\nu$) functions for strong (weak) dissipation
respectively. This and further details of the numerical
procedure and its accuracy are given in Appendix~\ref{num-procedure}.
\subsubsection{Choice of parameters}
The TBA equations were solved for weak dissipation at $\alpha=1/6,1/5,1/4,1/3$,
and for strong dissipation at $\alpha=2/3,3/4,4/5$. The exact closed solution
was used to obtain results at the Toulouse point $\alpha=1/2$. The
thermodynamics was calculated at temperatures
$t_{m}=\alpha k_{B}T_{m}/\Delta_{r}=2^{m/2}$ with
$m=-20,-19,\dots,+19,+20$, and for level asymmetries
$\tilde{\varepsilon}_{n}=\alpha \varepsilon_{n}/\Delta_{r}=2^{n}$
with $n=-4,-3,\dots,+3,+4$ and for the symmetric case $\varepsilon=0$.
\subsection{Entropy and Specific heat}
\subsubsection{Symmetric case: $\varepsilon=0$}
\label{sec-symmetric-heats}
The entropy of the symmetric two-state system is shown in
Fig.\,\ref{s+c-symmetric-all-alpha}a as a
function of temperature for several values of the dimensionless dissipation
strength, $\alpha$, ranging from weak to strong dissipation.
The correct value of the entropy, $S=\ln 2$, is recovered at
high temperature for all $\alpha$.
Fig.\,\ref{s+c-symmetric-all-alpha}b
shows the universal specific heat curves for the dissipative two-state
system. As in other strongly correlated systems \cite{vollhardt.97}
we observe a characteristic crossing point for the specific heat curves
at a temperature $k_{B}T^{+}/\Delta_{r}=0.66\pm 0.02$.
At low temperature
the specific heat is given by
$C(T)=\alpha\tilde{\gamma} (T/\Delta_{r}) +b(\alpha)(T/\Delta_{r})^{3}+\dots$
with a linear coefficient of specific heat
$\gamma=\alpha\tilde{\gamma}/\Delta_{r}$
which vanishes as $\alpha\rightarrow 0$.
From the definition of the low
energy scale $\Delta_{r}$ in terms of the zero temperature susceptibility and
the Wilson ratio (to be discussed below) it follows that
$\tilde{\gamma}=\pi/3$ for all $\alpha$, a useful check on the
numerical results.
The coefficient $b(\alpha)$ of
the $T^{3}$ term is negative for $\alpha\geq 1/3$, a special point in the
parameter space of the
dissipative two-state system.
\label{subsec-entropy+heat-symm}
\begin{figure}[t]
\centerline{\epsfysize 6.1cm {\epsffile{new-fig7a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm {\epsffile{new-fig7b.eps}}}
\vspace{0.1cm}
\caption{
(a) Entropy, $S(T)$, and (b) specific heats, $C(T)/T$,
for the symmetric two-state
system ($\varepsilon=0$) for weak ($\alpha<1/2$) and strong ($\alpha>1/2$)
dissipation cases. The $T^{3}$ coefficient in $C(T)/T$ is negative for
$\alpha>1/3$ and positive for $\alpha<1/3$.
}
\label{s+c-symmetric-all-alpha}
\end{figure}
The significance of $\alpha=1/3$ is best seen
in the context of the dynamics of the two-state system, where it corresponds
to the value of the dissipation strength at which the frequency,
$\Omega(\alpha)$, of tunneling oscillations (manifested in real time
correlation functions) becomes equal to the decay rate, $\Gamma(\alpha)$,
of these oscillations, i.e. $\Omega (\alpha=1/3)=\Gamma (\alpha=1/3)$ (or
the quality factor $Q(\alpha)=\Omega(\alpha)/\Gamma(\alpha)$ becomes unity)
\cite{leggett.87,voelker.98}.
For dissipation $\alpha < 1/3$ we
have $\Omega (\alpha)>\Gamma (\alpha)$ and
the well defined oscillatory mode appears to be reflected in the
characteristic peak in the specific heat $C(T)/T$ \cite{note-ct-peak}.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm {\epsffile{new-fig8a.eps}}}
\vspace{0.2cm}
\centerline{\epsfysize 6.1cm {\epsffile{new-fig8b.eps}}}
\vspace{0.1cm}
\caption{
(a) Specific heat, $C(T)/(\alpha T/\Delta_{r})$, showing
the development of the peak at $k_{B}T\approx \Delta_{r}$ for
$\alpha < 1/3$ and $\varepsilon=0$. $\lim_{T\rightarrow 0}
C(T)/(\alpha T/\Delta_{r})=\tilde{\gamma}=\pi/3 = 1.04719755$
is recovered to 5 decimal places. (b) the
logarithmic derivative, $d\log\; C(T,\varepsilon)/d\log\;T$, at $\alpha=2/3$
for the symmetric ($\varepsilon=0$) and asymmetric
($\varepsilon/\Delta_{r}=8$) cases. The approach to the expected power
law $C(T)\sim (\Delta_{r}/k_{B}T)^{\delta}$ with $\delta={2\alpha-2}=2/3$ for
$\alpha=2/3$ is found at high temperatures for both cases. For the asymmetric
system the power law arises at higher temperature corresponding to the
higher low energy scale behaving as
$\sqrt{\varepsilon^{2}+\Delta_{r}^{2}}>\Delta_{r}$ for $\epsilon\gg
\Delta_{r}$.
}
\label{schottky-peak+high-temp-limit}
\end{figure}
The peak in $C(T)/T$ for $\alpha< 1/3$ is shown in more detail in
Fig.\,\ref{schottky-peak+high-temp-limit}a and is reminiscent of the
activated behaviour seen in non-interacting two-level systems.
Since the excitation
spectrum of the Ohmic two-state system is gapless for all
$\alpha>0$, there is no exponential suppression of $C(T)/T$ at low
$k_{B}T< \Delta_{r}$ as with non-interacting two-level systems. The
linear specific heat persists at low temperature and the system remains
strongly interacting down to $T=0$ as is also clear from the value of
the Wilson ratio (see later). At high temperatures $k_{B}T \gg \Delta_{r}$
the specific heat vanishes according to a power law,
$C(T)\sim (\Delta_{r}/k_{B}T)^{2-2\alpha}$ with an $\alpha$ dependent power in
accordance with the asymptotic result given by Eq.(\ref{eq:c-hightemp}) and
in \cite{goerlich.88}.
This is shown in Fig.\,\ref{schottky-peak+high-temp-limit}b
and we see that, as for low temperatures, the behaviour
at high temperatures is again drastically different to the behaviour of
a non-interacting two-level system which shows
$C(T) \sim (\Delta_{0}/k_{B}T)^{2}$ for $k_{B}T\gg \Delta_{0}$ with $\Delta_{0}$
the bare tunneling matrix element. The limit $\alpha\rightarrow 1^{-}$
corresponds to the weak coupling Kondo model. For weak coupling,
$\rho J_{\perp}, \rho J_{\parallel}\ll 1$, the exchange operators in the AKM
become marginal (see above discussion of the
Anderson-Yuval scaling equations)
and consequently the high temperature properties acquire logarithmic
corrections leading to a slow approach of the thermodynamic quantities to
their high temperature limits \cite{costi.98}
( this can be seen in the entropy in Fig.\,\ref{s+c-symmetric-all-alpha}a
and more clearly in the results for the susceptibility
to be described in the next section).
\subsubsection{Asymmetric case: $\varepsilon>0$}
\label{subsec-entr+heat-asymm}
\begin{figure}[b]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig9a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig9b.eps}}}
\vspace{0.1cm}
\caption{
Entropy, $S(T)$, for the asymmetric two-state
system for a range of asymmetries at (a) $\alpha=1/5$ and (b) $\alpha=4/5$.
}
\label{entropy-asymmetric}
\end{figure}
We now turn to the asymmetric two-state system.
Fig.\,\ref{entropy-asymmetric}
shows the temperature dependence of the entropy
for different level asymmetries at $\alpha=1/5$ and $\alpha=4/5$.
The correct high temperature limit $S=\ln 2$ is recovered for all
level asymmetries and dissipation strengths. As for
the symmetric case, we see again that the entropy
approaches its high temperature
limit more slowly for strong dissipation than for weak dissipation, again
a result of increasing marginality of the interactions with
increasing $\alpha$ at the high energy fixed point.
In Fig.\,\ref{specific-heats-asymmetric-weak}--
\ref{specific-heats-asymmetric-strong}
we show the specific heats for different level asymmetries and
dissipation strengths. The specific heat remains linear at low temperature,
$C(T,\varepsilon)\sim \gamma T$, for all level asymmetries. The
linear coefficient $\gamma\sim 1/\Delta_{r}$ is reduced with
increasing asymmetry $\varepsilon$, a consequence of the increasing low
energy scale with increasing $\varepsilon$, $\Delta_{r}\rightarrow \sqrt{
\Delta_{r}^{2}+\varepsilon^{2}}$. We see that a sufficiently large asymmetry
eventually leads to a peak in $C(T)/T$ for all dissipation strengths, but
that for such a peak to form requires a sizeable asymmetry for $\alpha>1/3$.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig10a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig10b.eps}}}
\vspace{0.1cm}
\caption{
Specific heats, $C(T)/T$, for the asymmetric two-state
system for a range of asymmetries, and some typical cases for weak
dissipation, (a) $\alpha=1/5$, (b) $\alpha=1/3$.
}
\label{specific-heats-asymmetric-weak}
\end{figure}
It is important to note that even for large asymmetries, the shape of the
specific heat curves is still different to those of a non-interacting
two-level system: at low temperature the specific heat remains linear rather
than exponential and at high temperature the asymptotic behaviour of $C(T)$
is not the non-interacting $1/T^{2}$ but instead behaves as
$1/T^{2-2\alpha}$ as shown analytically for $\varepsilon=0$ and numerically
for $\varepsilon>0$ in Fig.\,\ref{schottky-peak+high-temp-limit}.
Only for $\alpha\ll 1$ do we expect the
specific heat to be reasonably described by the non-interacting result, and
then only outside the Fermi liquid regime $k_{B}T> \Delta_{r}$.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig11a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig11b.eps}}}
\vspace{0.1cm}
\caption{
Specific heats, $C(T)/T$, for the asymmetric two-state
system for a range of asymmetries, and some typical cases for strong
dissipation, (a) $\alpha=2/3$, (b) $\alpha=4/5$.
}
\label{specific-heats-asymmetric-strong}
\end{figure}
\subsection{Dielectric Susceptibility}
\label{sec-susc}
\subsubsection{Symmetric case: $\varepsilon=0$}
\label{susec-susc-symm}
Fig.\,\ref{susc-symmetric-all-alpha}
shows that the dielectric susceptibility of the dissipative two-state
system remains finite down to $T=0$ for all dissipation strengths
$\alpha< 1$. This is also shown in Fig.\,\ref{sus-symmetric-low-t-fermi-liquid}
together with the Fermi liquid $T^{2}$ corrections at $k_{B}T\ll \Delta_{r}$
given by Eq.(\ref{eq:chi-fliq}), $\chi_{sb}(T)=
\chi_{sb}(0)(1-c(\alpha)(k_{B}T/\Delta_{r})^{2})$.
By our definition $\chi_{sb}(T=0)=1/2\pi\Delta_{r}$, we have
that $\Delta_{r}\chi_{sb}(T=0)=1/2\pi =0.1591549$ which is reproduced by
our numerical solution to 5 decimal places in all cases (Fig.\,
\ref{sus-symmetric-low-t-fermi-liquid}).
In contrast to the specific heat, $C(T)/T$, the susceptibility is a
monotonically decreasing function of temperature for all dissipation
strengths. There is no signature of the onset of activated behaviour in
the susceptibility as there was for $\alpha<1/3$ in $C(T)/T$. As we shall see
below, a finite temperature peak in $\chi_{sb}$ only arises when there is
a finite level asymmetry. The dielectric susceptibility looks, superficially,
like that for a non-interacting system, however the universal scaling
curves depend sensitively on $\alpha$, as can be seen in Fig.\,
\ref{susc-symmetric-all-alpha}, so that this resemblance is misleading.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig12a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig12b.eps}}}
\vspace{0.1cm}
\caption{
Dielectric susceptibility, $\chi_{sb}(T)$, for the symmetric two-state
system ($\varepsilon=0$) for weak ($\alpha<1/2$) and strong ($\alpha>1/2$)
dissipation cases. The susceptibility is finite at $T=0$ with
$\Delta_{r}(\alpha)\chi(T=0)=1/2\pi$ in all cases as seen in (a) and
attains its free-spin value of $1/4T$ at high temperatures
$k_{B}T \gg \Delta_{r}$, as seen in (b).
}
\label{susc-symmetric-all-alpha}
\end{figure}
The strong renormalization of the tunneling amplitude
$\Delta_{r}/\omega_{c} \sim (\Delta_{0}/\omega_{c})^{1/(1-\alpha)}$ as
$\alpha\rightarrow 1^{-}$ gives rise to strongly renormalized dielectric
susceptibilities at low temperatures and strong dissipation
($\chi_{sb}(T=0)=1/2\pi\Delta_{r}$). The approach of the
susceptibility to its free spin value of
$1/4$ at $k_{B}T\gg \Delta_{r}$ at high temperatures
(Fig.\,\ref{susc-symmetric-all-alpha}b ) is governed
by power laws with exponents which are functions of the dissipation strength
as given by Eq.(\ref{eq:chi-hightemp}), $k_{B}T\chi_{sb}(T)\approx
1/4 -2B(\Delta_{r}/k_{B}T)^{2-2\alpha}$, and verified in
Fig.\,\ref{chi-sd3-high-temp-corrections}.
The approach to the free spin value becomes slower as
$\alpha\rightarrow 1^{-}$ and eventually logarithmic corrections to the
susceptibility set in (see Fig.\,\ref{susc-symmetric-all-alpha}b ).
\begin{figure}[t]
\centerline{\epsfysize 6.1cm {\epsffile{new-fig13.eps}}}
\vspace{0.1cm}
\caption{The $T^{2}$ Fermi liquid corrections to the dielectric
susceptibility at low temperatures for the symmetric case and
$\alpha=1/6,1/5,\dots,3/4,4/5$.
}
\label{sus-symmetric-low-t-fermi-liquid}
\end{figure}
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig14.eps}}}
\vspace{0.1cm}
\caption{
High temperature corrections to the dielectric susceptibility for
$\alpha=2/3$ for $\tilde{\varepsilon}=0$ and $\tilde{\varepsilon}=8$.
$\lim_{T\rightarrow \infty}\log_{10}[(k_{B}T)^{2\alpha-2}(1/4 -
k_{B}T\chi_{sb}(T))]=\log_{10}(2B)$
}
\label{chi-sd3-high-temp-corrections}
\end{figure}
\subsubsection{Asymmetric case: $\varepsilon>0$}
\label{susec-susc-asymm}
The dielectric susceptibility in the presence of a level asymmetry is shown
in Fig.\,\ref{susc-asymmetric-all-alpha-weak}--\ref{susc-asymmetric-all-alpha-strong}. For all dissipation strengths we see that a sizeable asymmetry
of the order of $\Delta_{r}$ is required to give a finite temperature
peak in $\chi_{sb}$. The approach of the susceptibility to its high
temperature limit of $1/4T$ is governed by the same power laws as those
found for the symmetric case and verified
in Fig.\,\ref{chi-sd3-high-temp-corrections}.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig15a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig15b.eps}}}
\vspace{0.1cm}
\caption{
Dielectric susceptibility, $\chi_{sb}(T)$, for a range of asymmetries
and some typical cases of weak dissipation, (a) $\alpha=1/5$,
(b) $\alpha=1/3$.
}
\label{susc-asymmetric-all-alpha-weak}
\end{figure}
\subsubsection{Wilson ratio}
The Wilson ratio for the Ohmic two-state model, $R_{sb}$, was defined earlier
together with the usual Wilson ratio, $R_{akm}$, for the AKM. These take
the values $2/\alpha$ and $2$ respectively and are valid for both
the symmetric and asymmetric cases (i.e. in both zero and finite fields for
the Kondo model). The Wilson ratio served as a useful check on our numerical
solution, which recovered it with an accuracy of not less than 4 decimal
places for all $\alpha$ and $\varepsilon$.
\begin{figure}[t]
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig16a.eps}}}
\vspace{0.1cm}
\centerline{\epsfysize 6.1cm
{\epsffile{new-fig16b.eps}}}
\vspace{0.1cm}
\caption{
Dielectric susceptibility, $\chi_{sb}(T)$, for a range of asymmetries
and some typical cases of strong dissipation, (a) $\alpha=2/3$,
(b) $\alpha=4/5$.
}
\label{susc-asymmetric-all-alpha-strong}
\end{figure}
\section{SUMMARY AND DISCUSSION}
\label{sec-conclusions}
In the present paper we studied the thermodynamics of a TSS
with Ohmic dissipation by exploiting a mapping between the DTSS and
the anisotropic Kondo model, and solving the Bethe Ansatz equations
derived by Wiegman and Tsvelik for the latter.
Treating the BA equations in a careful way we were able to
calculate essentially exactly
the specific heat and the susceptibility of the DTSS for all
temperatures and level asymmetries in the delocalized phase of the DTSS,
characterized by dissipation strengths $0<\alpha<1$. The Bethe Ansatz
solution makes the universal properties of the DTSS clear: thermodynamic
quantities are universal functions of two variables,
$k_{B}T/\Delta_{r}$ and $\varepsilon/k_{B}T$ for all $\alpha<1$. In the
limit $\alpha\rightarrow 1^{-}$ these functions reduce to those of
the usual isotropic $S=1/2$ Kondo model, which is seen as a special point
in the parameter space of the DTSS \cite{costi.98}. The well known logarithmic
corrections to physical quantities for $\alpha\rightarrow 1^{-}$
at high temperatures
$k_{B}T\gg \Delta_{r}$ give way to power law corrections away from $\alpha=1$.
We determined these power laws both analytically and numerically for all
$\alpha<1$ at finite $\varepsilon$. In the context of the RG
the change from logarithmic to power law corrections to physical quantities
at high temperature indicates that the tunneling term in the Hamiltonian
changes from being marginally relevant (about the high energy fixed point)
at $\alpha=1$ to relevant at $\alpha<1$ as discussed in
Sec.~\ref{sec-rgflow}. At low temperature $k_{B}T\ll \Delta_{r}$ the
thermodynamics of the DTSS is that of a renormalized local Fermi liquid
with an enhanced linear specific heat $C(T)\sim \alpha k_{B}T/\Delta_{r}$,
and an enhanced, but finite, dielectric
susceptibility at $T=0$, $\chi_{sb}(T=0)=1/2\pi\Delta_{r}$.
The renormalizations increase dramatically as
$\alpha\rightarrow 1^{-}$ due to the strong renormalization of the low
energy scale $\Delta_{r}/\omega_{c}\sim (\Delta/\omega_{c})^{1/(1-\alpha)}$.
We have shown that the characteristic thermodynamic properties
of the DTSS change smoothly as one increases the dissipation
strength from $\alpha \ll 1/2$ to $ 1 > \alpha > 1/2$,
corresponding to weak and strong dissipations, respectively.
In the former case, where the DTSS displays coherent oscillations between
its two positions \cite{weiss.99}, we find the expected tendency towards
activated behaviour in the specific heat. A clear signal of this
behaviour is the appearance of a peak in $C(T)/T$ at the renormalized
tunneling amplitude $\Delta_{r}$. Such a peak is absent for dissipations
$\alpha>1/3$ in the symmetric case. A finite level asymmetry accentuates
the tendency towards activated behaviour and always gives rise to a finite
temperature peak in $C(T)/T$ at temperature
$k_{B}T=\tilde{\Delta}_{r} \sim \sqrt{\Delta_r^2 + \varepsilon^2}$ provided
$\varepsilon \ge \Delta_{r}$. For strong dissipation $\alpha>1/2$, the specific
heat is qualitatively similar to that of isotropic $S=1/2$ Kondo systems and
shows a monotonically decreasing $C(T)/T$ with increasing temperature.
The dielectric susceptibility of the DTSS was calculated at all dissipation
strengths, level asymmetries and temperatures for the first time.
For the symmetric case we found that this quantity decreases monotonically
with increasing temperature for all dissipation strengths and develops
a finite temperature peak only for sufficiently large level asymmetries
(of the order of $\Delta_{r}$).
The Ohmic two-state system is a generic model capable of describing
a large number of different physical systems. Previous theoretical and
experimental work on such systems has, however, largely focussed on the
dynamic properties. Given the detailed understanding which we now have
of the thermodynamics, it may be worthwhile to consider also
thermodynamic measurements on Ohmic two-state systems. We therefore
briefly mention below some possibilities where our results could be
directly tested.
One of the possible physical realizations is provided by two-level
systems in metals. Amorphous metals are not the best candidates,
however, since in these materials a broad distribution
of DTSS's with very different physical parameters occur. Therefore,
to calculate their contribution to the thermodynamic
properties one should average over them. This averaging is
not impossible, but would require solving the BA equations for
arbitrary values of $\alpha$, something which could be implemented by
using further results of Ref~\cite{tsvelik.83}. In addition
one would require the form of the TSS distribution. Better candidates
are ${\rm H}$ tunneling in ${\rm Nb}$ \cite{wipf.84} and
metallic materials with tunneling centers
formed by substitutional impurities, such as $Pb_{1-x} Ge_{x}
Te$ \cite{PbGeTe}. Good quality single
crystals can be produced from the latter alloy.
The Germanium ions form identical eight-state
systems, which couple to the conduction electrons.
Applying external stress on the sample one can reduce
the degeneracy of the lowest lying states to two.
Since in this case the DTSS's have approximately identical parameters
for weak concentrations their individual thermodynamic properties can be
observed by measuring the sample's macroscopic properties.
Another possible candidate for thermodynamic measurements is
provided by a SQUID. In this case the energy difference between the
two flux states of the SQUID can be easily tuned by an
external magnetic field. Measuring the average flux
$ \langle\Phi\rangle$ as a function of
the asymmetry energy one can readily determine the susceptibility
$\chi \sim \partial \langle\Phi\rangle/\partial B$ that can directly
be compared to our calculations.
Finally, the results we have obtained for the thermodynamics of the
anisotropic Kondo model may have some relevance to the problem of
an isotropic $S=1/2$ Kondo impurity in a Luttinger liquid. Schiller
and Ingersent \cite{schiller.95} have shown that a $S=1/2$ Kondo
impurity in a modified Luttinger liquid consisting only of left-moving
spin-down electrons and right-moving spin-up electrons can be mapped
exactly onto the AKM. It is clear that in a more realistic description
of the Luttinger liquid that this exact mapping will only be approximate,
nevertheless we expect that some of the general trends in the thermodynamic
properties of a $S=1/2$ Kondo impurity in an interacting system should be
captured by such a mapping onto the AKM or equivalently onto the Ohmic
two-state system. The strength, $U$, of the Coulomb interaction can then
be related \cite{schiller.95} to $J_{\parallel}$ in the AKM and hence to
the dissipation strength $\alpha$ in the Ohmic two-state system.
The non-interacting system with $U=0$ corresponds to $\alpha=1$
and $U\rightarrow \infty$ corresponds to $\alpha=0$. Within such a
picture we therefore expect, from the results of this paper, that the
thermodynamic scaling functions of an isotropic $S=1/2$ Kondo impurity
in a Luttinger liquid will change continuously with increasing Coulomb
interaction $U$ (or decreasing dissipation strength $\alpha$). At low
temperature the thermodynamics of such systems should be similar to that
of a local Fermi liquid, as recently discussed in \cite{wang.98}. For
sufficiently large $U$ (i.e. sufficiently small $\alpha$),
$C(T)/T$ may develop a finite temperature peak at
$k_{B}T\sim \Delta_{r}\sim T_{K}$ in analogy to our findings for the Ohmic
two-state system. Such a peak could then be taken as a signature of the
Kondo effect in a strongly interacting system. The unusual heavy fermion
behaviour of ${\rm Nd_{2-x}Ce_{x}CuO_{4}}$, e.g. the non-monotonic
behaviour of $C(T)/T$ \cite{brugger.93},
may be consistent with such an interpretation. Strong interactions have
also been invoked to explain this behaviour in Ref.~\cite{fulde.93}.
Other, more conventional explanations, such as lattice coherence effects,
cannot be ruled out however.
The approach we have developed in this paper can be extended to more
complicated models of two-level systems coupled Ohmically to an
environment. The effect of indirect or electron-assisted tunneling
processes \cite{Zawa,muramatsu.86} on the thermodynamics of two-level
systems in metals will be studied in a future publication.
\acknowledgments
The authors are grateful to N. Andrei and A.M. Tsvelik for
useful discussions. We are grateful also to C. Roth for help with
Bosonization in Appendix~\ref{app:equiv}. This research has been supported by
the Hungarian Grant Nos. OTKA~T026327, OTKA~F016604, and the
U.S-Hungarian Joint Fund No.~587. G. Z. has been
supported by the Magyary Zolt\'an Foundation and grant
No.~DE-FG03-97ER45640 of the U.S DOE Office of Science, Division of
Materials Research. T. A. C. thanks the
Deutsche Forschungsgemeinschaft for financial support and the
hospitality of the Max Planck Institut f\"{u}r Physik Komplexe Systeme,
Dresden and the Centre for Electronic Correlations and Magnetism,
University of Augsburg, Germany, where part of this research was carried out.
|
1,116,691,498,434 | arxiv | \section{Introduction}
The purpose of this work is to establish the existence of a distributional
corrector in the deterministic homogenization theory for a family of second
order elliptic equations in divergence form with rapidly oscillating
coefficients, and find an approximation scheme for the homogenized
coefficients, without smoothness assumption on the coefficients. Under
additional condition, we also study the convergence rates in the asymptotic
almost periodic setting. We start with the statement of the problem (\ref%
{1.1}).
Let $\mathcal{A}$ be an algebra with mean value on $\mathbb{R}^{d}$, that
is, a closed subalgebra of the $\mathcal{C}^{\ast }$-algebra of bounded
uniformly continuous real-valued functions on $\mathbb{R}^{d}$, $\mathrm{BUC}%
(\mathbb{R}^{d})$, which contains the constants, is translation invariant
and is such that any of its elements possesses a mean value in the following
sense: for every $u\in \mathcal{A}$, the sequence $(u^{\varepsilon
})_{\varepsilon >0}$ ($u^{\varepsilon }(x)=u(x/\varepsilon )$) weakly$\ast $%
-converges in $L^{\infty }(\mathbb{R}^{d})$ to some real number $M(u)$
(called the mean value of $u$) as $\varepsilon \rightarrow 0$. The mean
value expresses as
\begin{equation}
M(u)=\lim_{R\rightarrow \infty }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}}u(y)dy\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }u\in \mathcal{A} \label{0.1}
\end{equation}%
where we have set $%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ }
\vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}}=\frac{1}{\left\vert B_{R}\right\vert }\int_{B_{R}}$.
For $1\leq p<\infty $, we define the Marcinkiewicz space $\mathfrak{M}^{p}(%
\mathbb{R}^{d})$ to be the set of functions $u\in L_{loc}^{p}(\mathbb{R}%
^{d}) $ such that
\begin{equation*}
\underset{R\rightarrow \infty }{\lim \sup }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}}\left\vert u(y)\right\vert ^{p}dy<\infty .
\end{equation*}%
Then $\mathfrak{M}^{p}(\mathbb{R}^{d})$ is a complete seminormed space
endowed with the seminorm
\begin{equation*}
\left\Vert u\right\Vert _{p}=\left( \underset{R\rightarrow \infty }{\lim
\sup }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}}\left\vert u(y)\right\vert ^{p}dy\right) ^{1/p}.
\end{equation*}%
We denote by $B_{\mathcal{A}}^{p}(\mathbb{R}^{d})$ ($1\leq p<\infty $) the
closure of $\mathcal{A}$ in $\mathfrak{M}^{p}(\mathbb{R}^{d})$. Then for any
$u\in B_{\mathcal{A}}^{p}(\mathbb{R}^{d})$ we have that
\begin{equation}
\left\Vert u\right\Vert _{p}=\left( \lim_{R\rightarrow \infty }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}}\left\vert u(y)\right\vert ^{p}dy\right) ^{\frac{1}{p}%
}=(M(\left\vert u\right\vert ^{p}))^{\frac{1}{p}}. \label{0.2}
\end{equation}%
Consider the space $B_{\mathcal{A}}^{1,p}(\mathbb{R}^{d})=\{u\in B_{\mathcal{%
A}}^{p}(\mathbb{R}^{d}):{\Greekmath 0272} _{y}u\in (B_{\mathcal{A}}^{p}(\mathbb{R}%
^{d}))^{d}\}$ which is a complete seminorned space with respect to the
seminorm
\begin{equation*}
\left\Vert u\right\Vert _{1,p}=\left( \left\Vert u\right\Vert
_{p}^{p}+\left\Vert {\Greekmath 0272} _{y}u\right\Vert _{p}^{p}\right) ^{\frac{1}{p}}.
\end{equation*}%
The Banach counterpart of the previous spaces are defined as follows. We set
$\mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d})=B_{\mathcal{A}}^{p}(\mathbb{R}%
^{d})/\mathcal{N}$ where $\mathcal{N}=\{u\in B_{\mathcal{A}}^{p}(\mathbb{R}%
^{d}):\left\Vert u\right\Vert _{p}=0\}$. We define $\mathcal{B}_{\mathcal{A}%
}^{1,p}(\mathbb{R}^{d})$ mutatis mutandis: replace $B_{\mathcal{A}}^{p}(%
\mathbb{R}^{d})$ by $\mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d})$ and $%
\partial /\partial y_{i}$ by $\overline{\partial }/\partial y_{i}$, where $%
\overline{\partial }/\partial y_{i}$ is defined by
\begin{equation}
\frac{\overline{\partial }}{\partial y_{i}}(u+\mathcal{N}):=\frac{\partial u%
}{\partial y_{i}}+\mathcal{N}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }u\in B_{\mathcal{A}}^{1,p}(\mathbb{R%
}^{d}). \label{0.3}
\end{equation}%
It is important to note that $\overline{\partial }/\partial y_{i}$ is also
defined as the infinitesimal generator in the $i$th direction coordinate of
the strongly continuous group $\mathcal{T}(y):\mathcal{B}_{\mathcal{A}}^{p}(%
\mathbb{R}^{d})\rightarrow \mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d});\
\mathcal{T}(y)(u+\mathcal{N})=u(\cdot +y)+\mathcal{N}$. Let us denote by $%
\varrho :B_{\mathcal{A}}^{p}(\mathbb{R}^{d})\rightarrow \mathcal{B}_{%
\mathcal{A}}^{p}(\mathbb{R}^{d})=B_{\mathcal{A}}^{p}(\mathbb{R}^{d})/%
\mathcal{N}$, $\varrho (u)=u+\mathcal{N}$, the canonical surjection. Remark:
$u\in B_{\mathcal{A}}^{1,p}(\mathbb{R}^{d})$ implies $\varrho (u)\in
\mathcal{B}_{\mathcal{A}}^{1,p}(\mathbb{R}^{d})$ and observing (\ref{0.3}), $%
\frac{\overline{\partial }\varrho (u)}{\partial y_{i}}=\varrho \left( \frac{%
\partial u}{\partial y_{i}}\right) $.
We assume in the sequel that the algebra $\mathcal{A}$ is ergodic, that is,
any $u\in \mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d})$ that is invariant
under $(\mathcal{T}(y))_{y\in \mathbb{R}^{d}}$ is a constant in $\mathcal{B}%
_{\mathcal{A}}^{p}(\mathbb{R}^{d})$, i.e., if $\left\Vert \mathcal{T}%
(y)u-u\right\Vert _{p}=0$ for every $y\in \mathbb{R}^{d}$, then $\left\Vert
u-c\right\Vert _{p}=0$, $c$ a constant. Let us also recall the following
property \cite{CMP, NA}:
\begin{itemize}
\item[(\textbf{1)}] The mean value $M$ viewed as defined on $\mathcal{A}$,
extends by continuity to a non negative continuous linear form (still
denoted by $M$) on $B_{\mathcal{A}}^{p}(\mathbb{R}^{d})$. For each $u\in B_{%
\mathcal{A}}^{p}(\mathbb{R}^{d})$ and all $a\in \mathbb{R}^{d}$, we have $%
M(u(\cdot +a))=M(u)$, and $\left\Vert u\right\Vert _{p}=\left( M(\left\vert
u\right\vert ^{p})\right) ^{1/p}$.
\end{itemize}
To the space $B_{\mathcal{A}}^{p}(\mathbb{R}^{d})$ we also attach the
following \textit{corrector} space
\begin{equation*}
B_{\#\mathcal{A}}^{1,p}(\mathbb{R}^{d})=\{u\in W_{loc}^{1,p}(\mathbb{R}%
^{d}):{\Greekmath 0272} u\in B_{\mathcal{A}}^{p}(\mathbb{R}^{d})^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }%
M({\Greekmath 0272} u)=0\}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}%
In $B_{\#\mathcal{A}}^{1,p}(\mathbb{R}^{d})$ we identify two elements by
their gradients: $u=v$ in $B_{\#\mathcal{A}}^{1,p}(\mathbb{R}^{d})$ iff $%
{\Greekmath 0272} (u-v)=0$, i.e. $\left\Vert {\Greekmath 0272} (u-v)\right\Vert _{p}=0$. We equip $%
B_{\#\mathcal{A}}^{1,p}(\mathbb{R}^{d})$ with the gradient norm $\left\Vert
u\right\Vert _{\#,p}=\left\Vert {\Greekmath 0272} u\right\Vert _{p}$ and obtain a
Banach space \cite[Theorem 3.12]{Casado} containing $B_{\mathcal{A}}^{1,p}(%
\mathbb{R}^{d})$.
We recall the $\Sigma $-convergence. A sequence $(u_{\varepsilon
})_{\varepsilon >0}\subset L^{p}(\Omega )$ ($1\leq p<\infty $) is said to:
\begin{itemize}
\item[(i)] \emph{weakly }$\Sigma $\emph{-converge} in $L^{p}(\Omega )$ to $%
u_{0}\in L^{p}(\Omega ;\mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d}))$ if,
as $\varepsilon \rightarrow 0$,
\begin{equation}
\int_{\Omega }u_{\varepsilon }(x)f\left( x,\frac{x}{\varepsilon }\right)
dx\rightarrow \int_{\Omega }M(u_{0}(x,\cdot )f(x,\cdot ))dx \label{3.1}
\end{equation}%
for any $f\in L^{p^{\prime }}(\Omega ;\mathcal{A})$ ($p^{\prime }=p/(p-1)$)$%
; $
\item[(ii)] \emph{strongly }$\Sigma $\emph{-converge} in $L^{p}(\Omega )$ to
$u_{0}\in L^{p}(\Omega ;\mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d}))$ if (%
\ref{3.1}) holds and further $\left\Vert u_{\varepsilon }\right\Vert
_{L^{p}(\Omega )}\rightarrow \left\Vert u_{0}\right\Vert _{L^{p}(\Omega ;%
\mathcal{B}_{\mathcal{A}}^{p}(\mathbb{R}^{d}))}$.
\end{itemize}
We denote (i) by "$u_{\varepsilon }\rightarrow u_{0}$ in $L^{p}(\Omega )$%
-weak $\Sigma $", and (ii) by "$u_{\varepsilon }\rightarrow u_{0}$ in $%
L^{p}(\Omega )$-strong $\Sigma $".
The main properties of the above concept are:
\begin{itemize}
\item Every bounded sequence in $L^{p}(\Omega )$ ($1<p<\infty $) possesses a
subsequence that weakly $\Sigma $-converges in $L^{p}(\Omega )$.
\item If $(u_{\varepsilon })_{\varepsilon \in E}$ is a bounded sequence in $%
W^{1,p}(\Omega )$, then there exist a subsequence $E^{\prime }$ of $E$ and a
couple $(u_{0},u_{1})\in W^{1,p}(\Omega )\times L^{p}(\Omega ;B_{\#\mathcal{A%
}}^{1,p}(\mathbb{R}^{d}))$ such that
\begin{align*}
u_{\varepsilon }& \rightarrow u_{0}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }W^{1,p}(\Omega )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-weak} \\
\frac{\partial u_{\varepsilon }}{\partial x_{j}}& \rightarrow \frac{\partial
u_{0}}{\partial x_{j}}+\frac{\partial u_{1}}{\partial y_{j}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }%
L^{p}(\Omega )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-weak }\Sigma \ \ (1\leq j\leq d)
\end{align*}
\item If $u_{\varepsilon }\rightarrow u_{0}$ in $L^{p}(\Omega )$-weak $%
\Sigma $ and $v_{\varepsilon }\rightarrow v_{0}$ in $L^{q}(\Omega )$-strong $%
\Sigma $, then $u_{\varepsilon }v_{\varepsilon }\rightarrow u_{0}v_{0}$ in $%
L^{r}(\Omega )$-weak $\Sigma $, where $1\leq p,q,r<\infty $ and $\frac{1}{p}+%
\frac{1}{q}=\frac{1}{r}$.
\end{itemize}
Our aim is to study the following problem
\begin{equation}
-{\Greekmath 0272} \cdot \left( A\left( x,\frac{x}{\varepsilon }\right) {\Greekmath 0272}
u_{\varepsilon }\right) =f\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\ in }\Omega \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }u_{\varepsilon }\in
H_{0}^{1}(\Omega ) \label{1.1}
\end{equation}%
where $\varepsilon >0$ is a small parameter, $f\in L^{2}(\Omega )$, $\Omega $
is an open bounded set of $\mathbb{R}^{d}$ (integer $d\geq 1$) with smooth
boundary $\partial \Omega $, and $A\in \mathcal{C}(\overline{\Omega }%
;L^{\infty }(\mathbb{R}^{d})^{d\times d})$ is a symmetric matrix satisfying
\begin{equation}
\alpha \left\vert \lambda \right\vert ^{2}\leq A(x,y)\lambda \cdot \lambda
\leq \beta \left\vert \lambda \right\vert ^{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for all }(x,\lambda
)\in \overline{\Omega }\times \mathbb{R}^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and a.e. }y\in \mathbb{R}%
^{d}; \label{1.2}
\end{equation}%
\begin{equation}
A(x,\cdot )\in (B_{\mathcal{A}}^{2}(\mathbb{R}^{d}))^{d\times d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for
all }x\in \overline{\Omega } \label{1.3}
\end{equation}%
where $\alpha $ and $\beta $ are two positive real numbers.
It is well-known that under assumptions (\ref{1.2}), problem (\ref{1.1})
uniquely determines a function $u_{\varepsilon }\in H_{0}^{1}(\Omega )$.
Under the additional assumption (\ref{1.3}), the following result holds.
\begin{theorem}
\label{t1.1}There exists $u_{0}\in H_{0}^{1}(\Omega )$ such that $%
u_{\varepsilon }\rightarrow u_{0}$ weakly in $H_{0}^{1}(\Omega )$ and
strongly in $L^{2}(\Omega )$ (as $\varepsilon \rightarrow 0$) and $u_{0}$
solves uniquely the problem
\begin{equation}
-{\Greekmath 0272} \cdot (A^{\ast }(x){\Greekmath 0272} u_{0})=f\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\Omega , \label{1.4}
\end{equation}%
$A^{\ast }$ being the homogenized matrix defined by
\begin{equation}
A^{\ast }(x)=M\left( A(x,\cdot )(I_{d}+{\Greekmath 0272} _{y}\chi (x,\cdot ))\right)
\label{1.5}
\end{equation}%
where, $\chi =(\chi _{j})_{1\leq j\leq d}\in \mathcal{C}(\overline{\Omega }%
;B_{\#\mathcal{A}}^{1,2}(\mathbb{R}^{d})^{d})$ is such that, for any $x\in
\Omega $, $\chi _{j}(x,\cdot )$ is the unique solution (up to an additive
constant depending on $x$) of the problem
\begin{equation}
{\Greekmath 0272} _{y}\cdot \left( A(x,\cdot )(e_{j}+{\Greekmath 0272} _{y}\chi _{j}(x,\cdot
))\right) =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}. \label{1.6}
\end{equation}%
If we set $u_{1}(x,y)={\Greekmath 0272} u_{0}(x)\chi (x,y)=\sum_{i=1}^{d}\frac{\partial
u_{0}}{\partial x_{i}}(x)\chi _{i}(x,y)$ and assume that $u_{1}\in
H^{1}(\Omega ;\mathcal{A}^{1})$ ($\mathcal{A}^{1}=\{v\in \mathcal{A}:{\Greekmath 0272}
_{y}v\in (\mathcal{A})^{d}\}$), then, as $\varepsilon \rightarrow 0$,
\begin{equation}
u_{\varepsilon }-u_{0}-\varepsilon u_{1}^{\varepsilon }\rightarrow 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
in }H^{1}(\Omega )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ strongly} \label{1.7}
\end{equation}%
where $u_{1}^{\varepsilon }(x)=u_{1}(x,x/\varepsilon )$ for a.e. $x\in
\Omega $.
\end{theorem}
\begin{remark}
\label{r1.1'}\emph{Problem (\ref{1.6}) is the }corrector problem\emph{. It
helps to obtain a first order approximation }$u_{\varepsilon }(x)\approx
u_{0}(x)+\varepsilon u_{1}(x,x/\varepsilon )$\emph{\ of }$u_{\varepsilon }$%
\emph{\ as seen in (\ref{1.7}). Its solvability is addressed in the
following result, which is the first main result of this work.}
\end{remark}
\begin{theorem}
\label{t4.1}Let $\xi \in \mathbb{R}^{d}$ and $x\in \overline{\Omega }$ be
fixed. There exists a unique (up to an additive function of $x$) function $%
v_{\xi }\in \mathcal{C}(\overline{\Omega };H_{loc}^{1}(\mathbb{R}^{d}))$
such that ${\Greekmath 0272} _{y}v_{\xi }\in \mathcal{C}(\overline{\Omega };B_{\mathcal{%
A}}^{2}(\mathbb{R}^{d})^{d})$ and $M({\Greekmath 0272} _{y}v_{\xi }(x,\cdot ))=0$,
which solves the equation
\begin{equation}
{\Greekmath 0272} _{y}\cdot \left( A(x,\cdot )(\xi +{\Greekmath 0272} _{y}v_{\xi }(x,\cdot
))\right) =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}. \label{4.1}
\end{equation}
\end{theorem}
The proof of Theorem \ref{t4.1} will be obtained as a consequence of Lemma %
\ref{l4.1} in Section 2 below. The progress compared to the previously known
results exists in the solution of the corrector problem: it is obtained by
approximation with distributional solutions of partial differential
equations in sufficiently large balls. Since the approximation can be
quantitatively controlled, this method also provides a basis for the
numerical calculation. Theorem \ref{t4.1} is well known in the random
stationary ergodic environment. However for the general deterministic
setting, we believe that a detailed proof must be provided since it also
covers the non ergodic algebras framework.
The next step consists in finding an approximation scheme for the
homogenized matrix $A^{\ast }$ (see (\ref{1.5})). This problem has been
solved (for (\ref{1.1})) in the periodic setting, since under the periodic
assumption, the corrector problem is posed on a bounded domain (namely the
periodic cell $Y=(0,1)^{d}$) since in that case, the solution $\chi _{j}$ is
periodic. A huge contrast between the periodic setting and the general
deterministic setting (as considered in this work) is that in the latter,
the corrector problem is posed on the whole space $\mathbb{R}^{d}$, and
cannot be reduced (as in the periodic framework) to a problem on a bounded
domain. As a result, the solution of the corrector problem (\ref{1.6}) (and
hence the homogenized matrix which depends on this solution) can not be
computed directly. Therefore, as in the random setting (see e.g. \cite%
{BP2004}), truncations of (\ref{1.6}) must be considered, particularly on
large domains $(-R,R)^{d}$ with appropriate boundary conditions, and the
homogenized coefficients will therefore be captured in the asymptotic
regime. This is done in Theorem \ref{t3.1} (see Section 3). We then find the
rate of convergence for the approximation scheme (see Theorem \ref{t3.2}).
It is natural to determine the convergence rates for the approximation (\ref%
{1.7}) setting in two cases:
\begin{itemize}
\item[1)] the asymptotic periodic one represented by the algebra $\mathcal{A}%
=\mathcal{C}_{0}(\mathbb{R}^{d})+\mathcal{C}_{per}(Y)$;
\item[2)] the asymptotic almost periodic one represented by the algebra $%
\mathcal{A}=\mathcal{C}_{0}(\mathbb{R}^{d})+AP(\mathbb{R}^{d})$.
\end{itemize}
In case 1), the corrector function $\chi _{j}(x,\cdot )$ (solution of (\ref%
{1.6})) belongs to the Sobolev-Besicovitch space $B_{\mathcal{A}}^{1,2}(%
\mathbb{R}^{d})$ associated to the algebra $\mathcal{A}$ and is bounded in $%
L^{\infty }(\mathbb{R}^{d})$. As a result, we proceed as in the well-known
periodic setting. In contrast with case 1), the corrector function in case
2) does not (in general) belong to the associated Sobolev-Besicovitch space $%
B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$, but rather to $B_{\#\mathcal{A}%
}^{1,2}(\mathbb{R}^{d})$. So information is available mainly for the
gradient of the corrector. To address this issue, we use the approximate
corrector $\chi _{T,j}$, distributional solution to $-{\Greekmath 0272} \cdot
A(e_{j}+{\Greekmath 0272} \chi _{T,j})+T^{-2}\chi _{T,j}=0$ in $\mathbb{R}^{d}$, which
belongs to $B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$ as shown in Section 2.
This leads to the following result, which is one of the main result of the
work.
\begin{theorem}
\label{t1.4}Let $\Omega $ be a $\mathcal{C}^{1,1}$ bounded domain in $%
\mathbb{R}^{d}$. Suppose that the matrix $A(x,y)\equiv A(y)$ and is
asymptotic almost periodic. Assume that $A$ satisfies \emph{(\ref{1.2})}.
For $f\in L^{2}(\Omega )$, let $u_{\varepsilon }$ and $u_{0}$ be the weak
solutions of Dirichlet problems \emph{(\ref{1.1})} and \emph{(\ref{1.4})}
respectively. Then there exists a function $\eta :(0,1]\rightarrow \lbrack
0,\infty )$ depending on $A$ with $\lim_{t\rightarrow 0}\eta (t)=0$ such
that
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272}
u_{0}\right\Vert _{H^{1}(\Omega )}\leq C\eta (\varepsilon )\left\Vert
f\right\Vert _{L^{2}(\Omega )} \label{Eq03}
\end{equation}%
and
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}\right\Vert _{L^{2}(\Omega )}\leq C\left[
\eta (\varepsilon )\right] ^{2}\left\Vert f\right\Vert _{L^{2}(\Omega )}
\label{Eq02}
\end{equation}%
where $T=\varepsilon ^{-1}$ and $\chi _{T}$ is the approximate corrector
defined by \emph{(\ref{11.5})}, and $C=C(\Omega ,A,d)$.
\end{theorem}
The precise convergence rates in case 1) are presented in the following
result.
\begin{theorem}
\label{t5.1}Suppose that $A$ is asymptotic periodic and satisfies
ellipticity conditions \emph{(\ref{1.2})} and \emph{(\ref{2.2})}. Assume $%
\Omega $, $f$, $u_{\varepsilon }$ and $u_{0}$ are as in Theorem \emph{\ref%
{t1.4}}. Denoting by $\chi $ the corrector defined by \emph{(\ref{1.6})},
there exists $C=C(\Omega ,A,d)>0$ such that
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi ^{\varepsilon }{\Greekmath 0272}
u_{0}\right\Vert _{H^{1}(\Omega )}\leq C\varepsilon ^{\frac{1}{2}}\left\Vert
f\right\Vert _{L^{2}(\Omega )} \label{5.8}
\end{equation}%
and
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}\right\Vert _{L^{2}(\Omega )}\leq
C\varepsilon \left\Vert f\right\Vert _{L^{2}(\Omega )}. \label{1.14}
\end{equation}
\end{theorem}
Theorem \ref{t5.1} can be obtained as a special case of Theorem \ref{t1.4}.
However we provide an independent proof since we do not need the approximate
corrector in this special situation. Estimate (\ref{1.14}) is optimal.
The above results generalize the well known ones in the periodic and the
uniformly almost periodic settings as considered in \cite{Shen}. In Theorem %
\ref{t5.1} we assume that the matrix $A$ has the form $A=A_{0}+A_{per}$
where $A_{0}$ has entries in $L^{2}(\Omega )$ and $A_{per}$ is periodic. In
Theorem \ref{t1.4}, we do not make any restriction on $A_{0}$ as above.
Also, the estimate (\ref{Eq02}) is near optimal. The assumptions will be
made precise in the latter sections.
The problem considered in Theorems \ref{t1.4} and \ref{t5.1} has been
firstly addressed in the periodic framework by Avellaneda and Lin \cite{AL87}
(see also \cite{Jikov}), and in the random setting (that is, for second
order linear elliptic equations with random coefficients) by Yurinskii \cite%
{Yu86}, Pozhidaev and Yurinskii \cite{Po-Yu89}, and Bourgeat and Piatnitski
\cite{BP2004} (see also a recent series of works by Gloria and Otto \cite%
{Gloria, GNO14, GNO15}, and the recent monograph \cite{Armstrong1}).
Although it is shown in \cite{24'''} that deterministic homogenization
theory can be seen as a special case of random homogenization theory at
least as far as the qualitative study is concerned, we can not expect to use
this random formulation to address the issues of rate of convergence in the
deterministic setting. Indeed, in the random framework, the rate of
convergence relies systematically on the \emph{uniform mixing} property (see
e.g. \cite{BP2004, Po-Yu89, Yu86}) of the coefficients of the equation. As
proved by Bondarenko et al. \cite{BBMM05}, the almost periodic operators do
not satisfy the uniform mixing property. As a result, we can not use the
random framework to address the issue in the general deterministic setting.
We therefore need to elaborate a new framework for solving the underlying
problem. Beyond the periodic (but non-random) setting Kozlov \cite{Koz79}
determined the rates of convergence in almost periodic homogenization by
using almost periodic coefficients satisfying a \textit{frequency condition}
(see e.g. (\ref{FC})). In the same vein, Bondarenko et al. \cite{BBMM05}
derived the rates of convergence by considering a perturbation of periodic
coefficients (in dimension $d=1$). The very first works that use the general
almost periodicity assumption are a recent series of work by Shen et al.
\cite{AS2016, Shen, Shen1} in which they treated second order linear
elliptic systems in divergence form. They used approximate correctors to
derive the rates of convergence. A reason to use approximate correctors is
the lack of sufficient knowledge on the corrector itself. Indeed in that
case it is known that the gradient of the corrector is almost periodic.
However it is not known in general whether the corrector itself is almost
periodic. Under certain conditions, it is shown in \cite{Armstrong, Shen1}
that the corrector is almost periodic. But the approximate corrector is in
general almost periodic together with its gradient.
It seems necessary to compare ours results in Theorems \ref{t1.4} and \ref%
{t5.1} with the existing ones in the literature. First of all, it is worth
noting that the algebra of continuous asymptotic almost periodic functions
is included in the Banach space of Weyl almost periodic functions; see e.g.
\cite{Besicovitch}. Thus the results obtained in \cite{Shen1} can be seen as
generalizing those in Theorems \ref{t1.4} and \ref{t5.1}. However it is not
exactly the case. Indeed in \cite{Shen1}, the rates of convergence are found
in terms of the modulus of Weyl-almost periodicity of the matrix $A$, that
is, in terms of the function
\begin{equation*}
\rho _{A}^{1}(R,L)=\sup_{y\in \mathbb{R}^{d}}\inf_{\left\vert z\right\vert
\leq R}\left( \sup_{x\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{L}(x)}\left\vert A(t+y)-A(t+z)\right\vert ^{2}dt\right) ^{\frac{1%
}{2}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }R,L>0
\end{equation*}%
where $B_{L}(x)$ stands for the open ball \ in $\mathbb{R}^{d}$ centered at $%
x$ and of radius $L>0$. In our work, we distinguish two cases: 1) the
asymptotic periodic case in which we show that the rate of convergence is
optimal, that $\left\Vert u_{\varepsilon }-u_{0}\right\Vert _{L^{2}(\Omega
)}=O(\varepsilon )$; 2) In the general continuous asymptotic almost periodic
setting, we show as in \cite{Shen1}, that the rate of convergence depends on
the modulus of asymptotic almost periodicity defined by
\begin{equation*}
\rho _{A}(R,L)=\sup_{y\in \mathbb{R}^{d}}\inf_{\left\vert z\right\vert \leq
R}\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{L}(0))}.
\end{equation*}%
As it is easily seen, the comparison between $\rho _{A}^{1}(R,L)$ and $\rho
_{A}(R,L)$ is not straightforward. So our result in Theorem \ref{t5.1} does
not follows directly from its counterpart Theorem 1.4 in \cite{Shen1}.
Our work combines the framework of \cite{Shen} with the general
deterministic homogenization theory introduced by Zhikov and Krivenko \cite%
{Zhikov4} and Nguetseng \cite{Hom1}. Furthermore, numerical simulations
based on finite volume method are provided to sustain our main theoretical
results.
The further investigation is organized as follows. Section 2 is devoted to
the proof of Theorems \ref{t1.1} and \ref{t4.1}. Section 3 deals with the
approximation of the homogenized coefficients. In Section 4, we prove
Theorems \ref{t1.4} while in Section 5 we prove Theorem \ref{t5.1}. In
Section 6, we provide some examples of concrete algebras and functions for
which the results, in particular those of Theorems \ref{t3.2}, \ref{t1.4}
and \ref{t5.1} apply. Finally, in Section 7 we present numerical results
illustrating the method and supporting the proposed procedure.
\section{Existence result for the corrector equation}
Let the matrix $A$ satisfy (\ref{1.2}) and (\ref{1.3}). Our aim is to solve
the corrector problem (\ref{1.6}). Let $B_{\mathcal{A}}^{2,\infty }(\mathbb{R%
}^{d})=B_{\mathcal{A}}^{2}(\mathbb{R}^{d})\cap L^{\infty }(\mathbb{R}^{d})$,
which is a Banach space under the $L^{\infty }(\mathbb{R}^{d})$-norm.
\begin{lemma}
\label{l4.1}Let $h\in \mathcal{C}(\overline{\Omega };B_{\mathcal{A}%
}^{2,\infty }(\mathbb{R}^{d}))$ and $H\in \mathcal{C}(\overline{\Omega };B_{%
\mathcal{A}}^{2,\infty }(\mathbb{R}^{d})^{d})$. For any $T>0$, there exists
a unique function $u\in \mathcal{C}(\overline{\Omega };B_{\mathcal{A}}^{1,2}(%
\mathbb{R}^{d}))$ such that
\begin{equation}
-{\Greekmath 0272} _{y}\cdot \left( A(x,\cdot ){\Greekmath 0272} _{y}u(x,\cdot )\right)
+T^{-2}u(x,\cdot )=h(x,\cdot )+{\Greekmath 0272} _{y}\cdot H(x,\cdot )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }%
\mathbb{R}^{d} \label{4.2}
\end{equation}%
for any fixed $x\in \overline{\Omega }$. The solution $u$ satisfies further
\begin{equation}
\sup_{z\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}\left( T^{-2}\left\vert u(x,y)\right\vert ^{2}+\left\vert
{\Greekmath 0272} u(x,y)\right\vert ^{2}\right) dy\leq C\sup_{z\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}(\left\vert H(x,y)\right\vert ^{2}+T^{2}\left\vert
h(x,y)\right\vert ^{2})dy \label{4.3}
\end{equation}%
for any $R\geq T$ and all $x\in \overline{\Omega }$, where the constant $C$
depends only on $d$, $\alpha $ and $\beta $.
\end{lemma}
\begin{proof}
Since the variable $x$ in (\ref{4.2}) behaves as a parameter, we drop it
throughout the proof of the existence and uniqueness. Thus, in what follows,
we keep using the symbol ${\Greekmath 0272} $ instead of ${\Greekmath 0272} _{y}$ to denote the
gradient with respect to $y$, if there is no danger of confusion.\medskip
1. \textit{Existence}. Fix $R>0$ and define $v_{T,R}\equiv v_{R}\in
H_{0}^{1}(B_{R})$ as the unique solution of
\begin{equation*}
-{\Greekmath 0272} \cdot A{\Greekmath 0272} v_{R}+T^{-2}v_{R}=h+{\Greekmath 0272} \cdot H\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }B_{R}.
\end{equation*}%
Extending $v_{R}$ by $0$ off $B_{R}$, we obtain a sequence $(v_{R})_{R}$ in $%
H_{loc}^{1}(\mathbb{R}^{d})$. Let us show that the sequence $(v_{R})_{R}$ is
bounded in $H_{loc}^{1}(\mathbb{R}^{d})$. We proceed as in \cite{Gloria}
(see also \cite{Po-Yu89}). In the variational formulation of the above
equation, we choose as test function, the function $\eta _{z}^{2}v_{R}$,
where $\eta _{z}(y)=\exp (-c\left\vert y-z\right\vert )$ for a fixed $z\in
\mathbb{R}^{d}$, $c>0$ to be chosen later. We get
\begin{align*}
\int_{B_{R}}\eta _{z}^{2}A{\Greekmath 0272} v_{R}\cdot {\Greekmath 0272}
v_{R}+T^{-2}\int_{B_{R}}\eta _{z}^{2}v_{R}^{2}& =-2\int_{B_{R}}\eta
_{z}v_{R}A{\Greekmath 0272} v_{R}\cdot {\Greekmath 0272} \eta _{z}-2\int_{B_{R}}\eta
_{z}v_{R}H\cdot {\Greekmath 0272} \eta _{z} \\
& -\int_{B_{R}}\eta _{z}^{2}H\cdot {\Greekmath 0272} v_{R}+\int_{B_{R}}h\eta
_{z}^{2}v_{R} \\
& =I_{1}+I_{2}+I_{3}+I_{4}.
\end{align*}%
The left-hand side of the above equality is bounded from below by
\begin{equation*}
\alpha \int_{B_{R}}\eta _{z}^{2}\left\vert {\Greekmath 0272} v_{R}\right\vert
^{2}+T^{-2}\int_{B_{R}}\eta _{z}^{2}v_{R}^{2},
\end{equation*}%
while for the right-hand side, we have the following bounds (after using the
Young's inequality and the bounds on $A$):
\begin{align*}
\left\vert I_{1}\right\vert & \leq \frac{\alpha \beta T^{-2}}{k}%
\int_{B_{R}}v_{R}^{2}\left\vert {\Greekmath 0272} \eta _{z}\right\vert ^{2}+\frac{%
T^{2}\beta k}{\alpha }\int_{B_{R}}\eta _{z}^{2}\left\vert {\Greekmath 0272}
v_{R}\right\vert ^{2}, \\
\left\vert I_{2}\right\vert & \leq \frac{\alpha \beta T^{-2}}{k}%
\int_{B_{R}}v_{R}^{2}\left\vert {\Greekmath 0272} \eta _{z}\right\vert ^{2}+\frac{T^{2}k%
}{\alpha \beta }\int_{B_{R}}\eta _{z}^{2}\left\vert H\right\vert ^{2}, \\
\left\vert I_{3}\right\vert & \leq \frac{T^{2}\beta k}{\alpha }%
\int_{B_{R}}\eta _{z}^{2}\left\vert {\Greekmath 0272} v_{R}\right\vert ^{2}+\frac{%
T^{-2}\alpha }{4k}\int_{B_{R}}\eta _{z}^{2}\left\vert H\right\vert ^{2}, \\
\left\vert I_{4}\right\vert & \leq \frac{\alpha \beta T^{-2}c^{2}}{k}%
\int_{B_{R}}v_{R}^{2}\eta _{z}^{2}+\frac{T^{2}k}{4\alpha \beta c^{2}}%
\int_{B_{R}}\eta _{z}^{2}\left\vert h\right\vert ^{2}
\end{align*}%
where $k>0$ is to be chosen later. Noticing that $\left\vert {\Greekmath 0272} \eta
_{z}\right\vert =c\eta _{z}$, we readily get after using the series of
inequalities above,
\begin{eqnarray*}
&&\int_{B_{R}}\eta _{z}^{2}\left( \alpha -2\frac{T^{2}\beta k}{\alpha }%
\right) \left\vert {\Greekmath 0272} v_{R}\right\vert ^{2}+T^{-2}\int_{B_{R}}\eta
_{z}^{2}\left( 1-3\frac{\alpha \beta c^{2}}{k}\right) v_{R}^{2} \\
&\leq &\int_{B_{R}}\left[ \left( \frac{T^{2}k}{\alpha \beta }+\frac{%
T^{-2}\alpha }{4\beta k}\right) \left\vert H\right\vert ^{2}+\frac{kT^{2}}{%
4\alpha \beta c^{2}}\left\vert h\right\vert ^{2}\right] \eta _{z}^{2}.
\end{eqnarray*}%
Choosing therefore $k=\frac{\alpha ^{2}}{4\beta T^{2}}$ and $c=\frac{1}{%
2\beta T}\left( \frac{\alpha }{6}\right) ^{1/2}$, we obtain the estimate
\begin{equation}
\alpha \int_{B_{R}}\eta _{z}^{2}\left\vert {\Greekmath 0272} v_{R}\right\vert
^{2}+T^{-2}\int_{B_{R}}\eta _{z}^{2}v_{R}^{2}\leq \int_{B_{R}}\left[ \left(
\frac{\alpha }{4\beta ^{2}}+\frac{1}{\alpha }\right) \left\vert H\right\vert
^{2}+\frac{3}{2}T^{2}\left\vert h\right\vert ^{2}\right] \eta _{z}^{2}.
\label{4.5}
\end{equation}%
The inequality (\ref{4.5}) above shows that the sequence $(v_{R})$ is
bounded in $H_{loc}^{1}(\mathbb{R}^{d})$; indeed, for any compact subset $K$
in $\mathbb{R}^{d}$, the left-hand side of (\ref{4.5}) is bounded from below
by $c_{K}(\alpha \int_{B_{R}}\left\vert {\Greekmath 0272} v_{R}\right\vert
^{2}+T^{-2}\int_{B_{R}}v_{R}^{2})$ where $c_{K}=\min_{K}\eta _{z}^{2}>0$
while the right-hand side is bounded from above by $C\int_{\mathbb{R}%
^{d}}\eta _{z}^{2}$ where
\begin{equation*}
C=\left( \frac{\alpha }{4\beta ^{2}}+\frac{1}{\alpha }\right) \left\Vert
H\right\Vert _{\mathcal{C}(\overline{\Omega };L^{\infty }(\mathbb{R}%
^{d}))}^{2}+\frac{3}{2}T^{2}\left\Vert h\right\Vert _{\mathcal{C}(\overline{%
\Omega };L^{\infty }(\mathbb{R}^{d}))}^{2}.
\end{equation*}%
Hence there exist a subsequence of $(v_{R})$ and a function $v\in
H_{loc}^{1}(\mathbb{R}^{d})$ such that the above mentioned subsequence
weakly converges in $H_{loc}^{1}(\mathbb{R}^{d})$ to $v$, and it is easy to
see that $v$ is a distributional solution of (\ref{4.2}) in $\mathbb{R}^{d}$%
. Taking the $\lim \inf_{R\rightarrow \infty }$ in (\ref{4.5}) yields
\begin{equation}
\alpha \int_{\mathbb{R}^{d}}\eta _{z}^{2}\left\vert {\Greekmath 0272} v_{R}\right\vert
^{2}+T^{-2}\int_{\mathbb{R}^{d}}\eta _{z}^{2}v_{R}^{2}\leq \int_{\mathbb{R}%
^{d}}\left[ \left( \frac{\alpha }{4\beta ^{2}}+\frac{1}{\alpha }\right)
\left\vert H\right\vert ^{2}+\frac{3}{2}T^{2}\left\vert h\right\vert ^{2}%
\right] \eta _{z}^{2}. \label{4.6}
\end{equation}%
We infer from (\ref{4.6}) that
\begin{equation}
\sup_{z\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}(\left\vert {\Greekmath 0272} v\right\vert ^{2}+T^{-2}v^{2})\leq C
\label{e2.4}
\end{equation}%
where $C$ does not depend on $z$, but on $T$. Estimate (\ref{4.3}) (for $R=T$%
) follows from \cite{Po-Yu89} while the case $R>T$ is a consequence of
Caccioppoli's inequality; see \cite[Lemma 3.2]{Shen1}.
Let us show that $v\in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$. It suffices
to check that $v$ solves the equation
\begin{equation}
M(A(\xi +{\Greekmath 0272} v)\cdot {\Greekmath 0272} \phi +T^{-2}v\phi )=M(h\phi -H\cdot {\Greekmath 0272}
\phi )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, all }\phi \in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d}).
\label{4.7}
\end{equation}%
To this end, let $\varphi \in \mathcal{C}_{0}^{\infty }(\mathbb{R}^{d})$ and
$\phi \in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$. Define (for fixed $%
\varepsilon >0$), $\psi (y)=\varphi (\varepsilon y)\phi (y)$. Choose $\psi $
as test function in the variational form of (\ref{4.2}) and get
\begin{align*}
& \int_{\mathbb{R}^{d}}\left[ A{\Greekmath 0272} u\cdot (\varepsilon \phi {\Greekmath 0272}
\varphi (\varepsilon \cdot )+\varphi (\varepsilon \cdot ){\Greekmath 0272} \phi
)+T^{-2}u\varphi (\varepsilon \cdot )\phi \right] dy \\
& =\int_{\mathbb{R}^{d}}\left[ h\varphi (\varepsilon \cdot )\phi -H\cdot
(\varepsilon \phi {\Greekmath 0272} \varphi (\varepsilon \cdot )+\varphi (\varepsilon
\cdot ){\Greekmath 0272} \phi )\right] dy.
\end{align*}%
The change of variables $t=\varepsilon y$ leads (after multiplication by $%
\varepsilon ^{d}$) to
\begin{align*}
& \int_{\mathbb{R}^{d}}\left[ A^{\varepsilon }({\Greekmath 0272} _{y}u)^{\varepsilon
}\cdot (\varepsilon \phi ^{\varepsilon }{\Greekmath 0272} \varphi +\varphi ({\Greekmath 0272}
_{y}\phi )^{\varepsilon })+T^{-2}u^{\varepsilon }\varphi \phi ^{\varepsilon }%
\right] dt \\
& =\int_{\mathbb{R}^{d}}\left[ h^{\varepsilon }\phi ^{\varepsilon }\varphi
-H^{\varepsilon }\cdot (\varepsilon \phi ^{\varepsilon }{\Greekmath 0272} \varphi
+\varphi ({\Greekmath 0272} _{y}\phi )^{\varepsilon })\right] dt
\end{align*}%
where $w^{\varepsilon }(t)=w(t/\varepsilon )$ for a given $w$. Letting $%
\varepsilon \rightarrow 0$ above yields
\begin{align*}
\int_{\mathbb{R}^{d}}M(A{\Greekmath 0272} u\cdot {\Greekmath 0272} \phi +T^{-2}u\phi )\varphi dt&
=\int_{\mathbb{R}^{d}}M(h\phi -H\cdot {\Greekmath 0272} \phi )\varphi dt \\
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{for all }\varphi & \in \mathcal{C}_{0}^{\infty }(\mathbb{R}^{d})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
and }\phi \in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d}).
\end{align*}%
which amounts to (\ref{4.7}). So, we have just shown that, if $v\in
H_{loc}^{1}(\mathbb{R}^{d})$ solves (\ref{4.2}) in the sense of
distributions in $\mathbb{R}^{d}$, then it satisfies (\ref{4.7}). Before we
proceed any further, let us first show that (\ref{4.7}) possesses a unique
solution in $B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$ up to an additive
function $w\in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$ satisfying $%
M(\left\vert w\right\vert ^{2})=0$. First and foremost, we recall that the
space $\mathcal{B}_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})=B_{\mathcal{A}}^{1,2}(%
\mathbb{R}^{d})/\mathcal{N}$ (where $\mathcal{N}=\{u\in B_{\mathcal{A}%
}^{1,2}(\mathbb{R}^{d}):\left\Vert u\right\Vert _{1,2}=0\}$) is a Hilbert
space with inner product
\begin{equation*}
(u+\mathcal{N},v+\mathcal{N})_{1,2}=M(uv+{\Greekmath 0272} u\cdot {\Greekmath 0272} v)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }%
u,v\in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d}).
\end{equation*}%
If $w\in \mathcal{N}$ then $M(w)=0$, since $\left\vert M(w)\right\vert \leq
M(\left\vert w\right\vert )\leq (M(\left\vert w\right\vert
^{2}))^{1/2}=\left\Vert w\right\Vert _{2}=0$, so that $\left( ,\right)
_{1,2} $ is well defined. Now, (\ref{4.7}) is equivalent to $a(v,\phi )=\ell
(\phi ) $ for all $\phi \in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$ where
\begin{equation*}
a(v,\phi )=M(T^{-2}v\phi +A{\Greekmath 0272} v\cdot {\Greekmath 0272} \phi ),\ \ell (\phi
)=M(h\phi -H\cdot {\Greekmath 0272} \phi ).
\end{equation*}%
$a(\cdot ,\cdot )$ defines a continuous coercive bilinear form on $\mathcal{B%
}_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$; $\ell $ is a continuous linear form
on $\mathcal{B}_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$. Lax-Milgram theorem
implies that $v+\mathcal{N}$ is a unique solution of (\ref{4.7}). This
yields $v\in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$.\medskip
2. \textit{Uniqueness}. The uniqueness of the solution amounts to consider (%
\ref{4.2}) with $h=0$ and $H=0$. We derive from (\ref{4.6})
\begin{equation*}
\alpha \int_{\mathbb{R}^{d}}\eta _{z}^{2}\left\vert {\Greekmath 0272} v\right\vert
^{2}+T^{-2}\int_{\mathbb{R}^{d}}\eta _{z}^{2}v^{2}=0,
\end{equation*}%
so that $v=0$ for the corresponding equation.\medskip
3. \textit{Continuity}. To investigate the continuity of $v$ with respect to
$x$, we fix $x_{0}\in \overline{\Omega }$ \ and we let $w(x)=v(x,\cdot
)-v(x_{0},\cdot )$. Then $w(x)\in B_{\mathcal{A}}^{1,2}(\mathbb{R}^{d})$ and
\begin{eqnarray*}
-{\Greekmath 0272} \cdot A(x,\cdot ){\Greekmath 0272} w(x)+T^{-2}w(x) &=&h(x,\cdot )-h(x_{0},\cdot
)+{\Greekmath 0272} \cdot (H(x,\cdot )-H(x_{0},\cdot )) \\
&&+{\Greekmath 0272} \cdot (A(x,\cdot )-A(x_{0},\cdot )){\Greekmath 0272} v(x_{0},\cdot ),
\end{eqnarray*}%
so that, using estimate (\ref{4.3}), we find (for any $R\geq T$)
\begin{eqnarray*}
\sup_{z\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}\left( T^{-2}\left\vert w(x)\right\vert ^{2}+\left\vert
{\Greekmath 0272} w(x)\right\vert ^{2}\right) dy &\leq &CT^{2}\sup_{z\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}\left\vert h(x,y)-h(x_{0},y)\right\vert ^{2}dy \\
&&+C\sup_{z\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}\left\vert H(x,y)-H(x_{0},y)\right\vert ^{2}dy \\
&&+C\sup_{z\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}\left\vert A(x,y)-A(x_{0},y)\right\vert ^{2}\left\vert
{\Greekmath 0272} v(x_{0},y)\right\vert ^{2}dy \\
&\leq &CT^{2}\left\Vert h(x,\cdot )-h(x_{0},\cdot )\right\Vert _{L^{\infty }(%
\mathbb{R}^{d})}^{2} \\
&&+C\left\Vert H(x,\cdot )-H(x_{0},\cdot )\right\Vert _{L^{\infty }(\mathbb{R%
}^{d})}^{2} \\
&&+C\left\Vert A(x,\cdot )-A(x_{0},\cdot )\right\Vert _{L^{\infty }(\mathbb{R%
}^{d})}^{2}.
\end{eqnarray*}%
Continuity is a consequence of the following estimate
\begin{eqnarray*}
&&T^{-2}\left\Vert v(x,\cdot )-v(x_{0},\cdot )\right\Vert
_{2}^{2}+\left\Vert {\Greekmath 0272} v(x,\cdot )-{\Greekmath 0272} v(x_{0},\cdot )\right\Vert
_{2}^{2} \\
&\equiv &\lim_{R\rightarrow \infty }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}T^{-2}\left\vert w(x)\right\vert ^{2}+\left\vert {\Greekmath 0272}
w(x)\right\vert ^{2}dy \\
&\leq &CT^{2}\left\Vert h(x,\cdot )-h(x_{0},\cdot )\right\Vert _{L^{\infty }(%
\mathbb{R}^{d})}^{2}+C\left\Vert H(x,\cdot )-H(x_{0},\cdot )\right\Vert
_{L^{\infty }(\mathbb{R}^{d})}^{2} \\
&&+C\left\Vert A(x,\cdot )-A(x_{0},\cdot )\right\Vert _{L^{\infty }(\mathbb{R%
}^{d})}^{2}.
\end{eqnarray*}
\end{proof}
\begin{proof}[Proof of Theorem \protect\ref{t4.1}]
1. \textit{Existence and continuity}. Let us denote by $(\chi _{T,j}(x,\cdot
))_{T\geq 1}$ (for fixed $1\leq j\leq d$) the sequence constructed in Lemma %
\ref{l4.1} and corresponding to $h=0$ and $H=Ae_{j}$, $e_{j}$ being denoting
the $j$th vector of the canonical basis of $\mathbb{R}^{d}$. It satisfies (%
\ref{4.3}), so that by the weak compactness, the sequence $({\Greekmath 0272} \chi
_{T,j}(x,\cdot ))_{T\geq 1}$ weakly converges in $L_{loc}^{2}(\mathbb{R}%
^{d})^{d}$ (up to extraction of a subsequence) to some $V_{j}(x,\cdot )\in
L_{loc}^{2}(\mathbb{R}^{d})^{d}$. From the equality $\partial ^{2}\chi
_{T,j}(x,\cdot )/\partial y_{i}\partial y_{l}=\partial ^{2}\chi
_{T,j}(x,\cdot )/\partial y_{l}\partial y_{i}$, a limit passage in the
distributional sense yields $\partial V_{j,i}(x,\cdot )/\partial
y_{l}=\partial V_{j,l}(x,\cdot )/\partial y_{i}$, where $V_{j}=(V_{j,i})_{1%
\leq i\leq d}$. This implies $V_{j}(x,\cdot )={\Greekmath 0272} \chi _{j}(x,\cdot )$
for some $\chi _{j}(x,\cdot )\in H_{loc}^{1}(\mathbb{R}^{d})$. Using the
boundedness of $(T^{-1}\chi _{T,j}(x,\cdot ))_{T\geq 1}$ in $L_{loc}^{2}(%
\mathbb{R}^{d})$, we pass to the limit in the variational formulation of (%
\ref{4.2}) (as $T\rightarrow \infty $) to get that $\chi _{j}$ solves (\ref%
{4.1}). Arguing exactly as in the proof of (\ref{4.7}) (in Lemma \ref{l4.1}%
), we arrive at $V_{j}(x,\cdot )\in B_{\mathcal{A}}^{2}(\mathbb{R}^{d})^{d}$%
. Also, since $\chi _{T,j}(x,\cdot )\in B_{\mathcal{A}}^{1,2}(\mathbb{R}%
^{d}) $, we have $M({\Greekmath 0272} \chi _{T,j}(x,\cdot ))=0$, hence $M({\Greekmath 0272} \chi
_{j}(x,\cdot ))=0$. We repeat the proof of the Part 3. in the previous lemma
to find that ${\Greekmath 0272} _{y}\chi _{j}\in \mathcal{C}(\overline{\Omega };B_{%
\mathcal{A}}^{2}(\mathbb{R}^{d})^{d})$.\medskip
2. \textit{Uniqueness} (of ${\Greekmath 0272} _{y}\chi _{j}$). Fix $x\in \overline{%
\Omega }$ and assume that $\chi _{j}(x,\cdot )\in H_{loc}^{1}(\mathbb{R}%
^{d}) $ is such that $-\func{div}(A(x,\cdot ){\Greekmath 0272} _{y}\chi _{j}(x,\cdot
))=0$ in $\mathbb{R}^{d}$ and ${\Greekmath 0272} _{y}\chi _{j}(x,\cdot )\in B_{\mathcal{%
A}}^{2}(\mathbb{R}^{d})^{d}$. Then it follows from \cite[Property (3.10)]%
{Shen} that, given $0<\sigma <1$, there exists $C_{\sigma }>0$ independent
from $r$ and $R$\ such that
\begin{equation}
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}}\left\vert {\Greekmath 0272} _{y}\chi _{j}(x,y)\right\vert ^{2}dy\leq
C_{\sigma }\left( \frac{r}{R}\right) ^{\sigma }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}}\left\vert {\Greekmath 0272} _{y}\chi _{j}(x,y)\right\vert ^{2}dy\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
for all }0<r<R. \label{04}
\end{equation}%
Next, since $-\func{div}(A(x,\cdot ){\Greekmath 0272} _{y}\chi _{j}(x,\cdot ))=0$ in $%
\mathbb{R}^{d}$ and ${\Greekmath 0272} _{y}\chi _{j}(x,\cdot )\in B_{\mathcal{A}}^{2}(%
\mathbb{R}^{d})^{d}$, we show as for (\ref{4.7}) that
\begin{equation}
M(A(x,\cdot ){\Greekmath 0272} _{y}\chi _{j}(x,\cdot )\cdot {\Greekmath 0272} _{y}\phi )=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
for all }\phi \in B_{\#\mathcal{A}}^{1,2}(\mathbb{R}^{d}). \label{05}
\end{equation}%
Choosing $\phi =\chi _{j}(x,\cdot )$ in (\ref{05}), and using the
ellipticity of $A$, it emerges $M(\left\vert {\Greekmath 0272} _{y}\chi _{j}(x,\cdot
)\right\vert ^{2})=0$, that is, $\lim_{R\rightarrow \infty }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}}\left\vert {\Greekmath 0272} _{y}\chi _{j}(x,y)\right\vert ^{2}dy=0$.
Coming back to (\ref{04}) and letting there $R\rightarrow \infty $, we are
led to $\int_{B_{r}}\left\vert {\Greekmath 0272} _{y}\chi _{j}(x,y)\right\vert ^{2}dy=0$
for all $r>0$. This gives ${\Greekmath 0272} _{y}\chi _{j}(x,\cdot )=0$.
\end{proof}
We can now prove Theorem \ref{t1.1}.
\begin{proof}[Proof of Theorem \protect\ref{t1.1}]
Let $\Phi _{\varepsilon }=\psi _{0}+\varepsilon \psi _{1}^{\varepsilon }$
with $\psi _{1}^{\varepsilon }(x)=\psi _{1}(x,x/\varepsilon )$ ($x\in \Omega
$), where $\psi _{0}\in \mathcal{C}_{0}^{\infty }(\Omega )$ and $\psi
_{1}\in \mathcal{C}_{0}^{\infty }(\Omega )\otimes \mathcal{A}^{\infty }$, $%
\mathcal{A}^{\infty }=\{u\in \mathcal{A}:D^{\alpha }u\in \mathcal{A}$ for
all $\alpha \in \mathbb{N}^{d}\}$. Taking $\Phi _{\varepsilon }$ (wich
belongs to $\mathcal{C}_{0}^{\infty }(\Omega )$) as a test function in the
variational formulation of (\ref{1.1}) yields
\begin{equation}
\int_{\Omega }A^{\varepsilon }{\Greekmath 0272} u_{\varepsilon }\cdot {\Greekmath 0272} \Phi
_{\varepsilon }dx=\int_{\Omega }f\Phi _{\varepsilon }dx. \label{4.6'}
\end{equation}%
It is not difficult to see that the sequence $(u_{\varepsilon
})_{\varepsilon >0}$ is bounded in $H_{0}^{1}(\Omega )$, so that,
considering an ordinary sequence $E\subset \mathbb{R}_{+}^{\ast }$, there
exist a couple $(u_{0},u_{1})\in H_{0}^{1}(\Omega )\times L^{2}(\Omega ;B_{\#%
\mathcal{A}}^{1,2}(\mathbb{R}^{d}))$ and a subsequence $E^{\prime }$ of $E$
such that, as $E^{\prime }\ni \varepsilon \rightarrow 0$,
\begin{equation*}
u_{\varepsilon }\rightarrow u_{0}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }H_{0}^{1}(\Omega )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-weak
and in }L^{2}(\Omega )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-strong}
\end{equation*}%
\begin{equation}
{\Greekmath 0272} u_{\varepsilon }\rightarrow {\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}u_{1}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }%
L^{2}(\Omega )^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-weak }\Sigma .\ \ \ \ \ \ \ \ \ \ \label{4.7'}
\end{equation}%
On the other hand
\begin{equation}
{\Greekmath 0272} \Phi _{\varepsilon }={\Greekmath 0272} \psi _{0}+({\Greekmath 0272} _{y}\psi
_{1})^{\varepsilon }+\varepsilon ({\Greekmath 0272} \psi _{1})^{\varepsilon
}\rightarrow {\Greekmath 0272} \psi _{0}+{\Greekmath 0272} _{y}\psi _{1}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }L^{2}(\Omega
)^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-strong }\Sigma . \label{4.8'}
\end{equation}%
This yields in (\ref{4.6'}) the following limit problem
\begin{equation}
\int_{\Omega }M\left( A({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}u_{1})\cdot ({\Greekmath 0272} \psi
_{0}+{\Greekmath 0272} _{y}\psi _{1})\right) dx=\int_{\Omega }f\psi _{0}dx\ \ \forall
(\psi _{0},\psi _{1})\in \mathcal{C}_{0}^{\infty }(\Omega )\times (\mathcal{C%
}_{0}^{\infty }(\Omega )\otimes \mathcal{A}^{\infty }). \label{4.9'}
\end{equation}%
Problem (\ref{4.9'}) above is equivalent to the system
\begin{equation}
\int_{\Omega }M\left( A({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}u_{1})\cdot {\Greekmath 0272} \psi
_{0}\right) dx=\int_{\Omega }f\psi _{0}dx\ \ \forall \psi _{0}\in \mathcal{C}%
_{0}^{\infty }(\Omega ) \label{4.10'}
\end{equation}%
\begin{equation}
\int_{\Omega }M\left( A({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}u_{1})\cdot {\Greekmath 0272} _{y}\psi
_{1}\right) dx=0\ \ \forall \psi _{1}\in \mathcal{C}_{0}^{\infty }(\Omega
)\otimes \mathcal{A}^{\infty }. \label{4.11'}
\end{equation}%
Taking in (\ref{4.11'}) $\psi _{1}(x,y)=\varphi (x)v(y)$ with $\varphi \in
\mathcal{C}_{0}^{\infty }(\Omega )$ and $v\in \mathcal{A}^{\infty }$, we get
\begin{equation}
M\left( A(x,\cdot )({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}u_{1})\cdot {\Greekmath 0272} _{y}v\right)
=0\ \ \forall v\in \mathcal{A}^{\infty },x\in \overline{\Omega },
\label{4.12'}
\end{equation}%
which is, thanks to the density of $\mathcal{A}^{\infty }$ in $B_{\mathcal{A}%
}^{1,2}(\mathbb{R}^{d})$, the weak form of
\begin{equation}
{\Greekmath 0272} _{y}\cdot \left( A(x,\cdot )({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}u_{1})\right) =0%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ (for all fixed }x\in \overline{\Omega }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
),} \label{4.13'}
\end{equation}%
with respect to the duality defined by (\ref{4.12'}). So fix $\xi \in
\mathbb{R}^{d}$ and consider the problem
\begin{equation}
{\Greekmath 0272} _{y}\cdot \left( A(x,\cdot )(\xi +{\Greekmath 0272} _{y}v_{\xi }(x,\cdot
))\right) =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d};\ v_{\xi }(x,\cdot )\in B_{\#\mathcal{A%
}}^{1,2}(\mathbb{R}^{d}). \label{4.14'}
\end{equation}%
Thanks to Theorem \ref{t4.1}, Eq. (\ref{4.14'}) possesses a unique solution $%
v_{\xi }$ (up to an additive constant depending on $x$) in $\mathcal{C}(%
\overline{\Omega };B_{\#\mathcal{A}}^{1,2}(\mathbb{R}^{d}))$. Choosing there
$\xi ={\Greekmath 0272} u_{0}(x)$, the uniqueness of the solution implies $%
u_{1}(x,y)=\chi (x,y)\cdot {\Greekmath 0272} u_{0}(x)$ where $\chi =(\chi _{j})_{1\leq
j\leq d}$ with $\chi _{j}=v_{e_{j}}$, $e_{j}$ the $j$th vector of the
canonical basis of $\mathbb{R}^{d}$. Replacing in (\ref{4.10'}) $u_{1}$ by $%
\chi \cdot {\Greekmath 0272} u_{0}$, we get
\begin{equation*}
\int_{\Omega }(M(A(I+{\Greekmath 0272} _{y}\chi ){\Greekmath 0272} u_{0})\cdot {\Greekmath 0272} \psi
_{0}dx=\int_{\Omega }f\psi _{0}dx\ \ \forall \psi _{0}\in \mathcal{C}%
_{0}^{\infty }(\Omega ),
\end{equation*}%
that is, $-{\Greekmath 0272} \cdot A^{\ast }(x){\Greekmath 0272} u_{0}=f$ in $\Omega $.
It remains to verify (\ref{1.7}). Define $\Phi _{\varepsilon
}(x)=u_{0}(x)+\varepsilon u_{1}(x,x/\varepsilon )$. Then using (\ref{1.2})
we obtain
\begin{align*}
\alpha \int_{\Omega }\left\vert {\Greekmath 0272} u_{\varepsilon }-{\Greekmath 0272} \Phi
_{\varepsilon }\right\vert ^{2}dx& \leq \int_{\Omega }A^{\varepsilon }{\Greekmath 0272}
(u_{\varepsilon }-\Phi _{\varepsilon })\cdot {\Greekmath 0272} (u_{\varepsilon }-\Phi
_{\varepsilon })dx \\
& =\int_{\Omega }f(u_{\varepsilon }-\Phi _{\varepsilon })dx-\int_{\Omega
}A^{\varepsilon }{\Greekmath 0272} \Phi _{\varepsilon }\cdot {\Greekmath 0272} (u_{\varepsilon
}-\Phi _{\varepsilon })dx.
\end{align*}%
Since $u_{1}\in L^{2}(\Omega ;\mathcal{A}^{1})$, we have that $\int_{\Omega
}f(u_{\varepsilon }-\Phi _{\varepsilon })dx\rightarrow 0$. Indeed $\Phi
_{\varepsilon }\rightarrow u_{0}$ in $L^{2}(\Omega )$ (and hence $%
u_{\varepsilon }-\Phi _{\varepsilon }\rightarrow 0$ in $L^{2}(\Omega )$).
Next observe that ${\Greekmath 0272} \Phi _{\varepsilon }\rightarrow {\Greekmath 0272}
u_{0}+{\Greekmath 0272} _{y}u_{1}$ in $L^{2}(\Omega )$-strong $\Sigma $; in fact, $%
{\Greekmath 0272} \Phi _{\varepsilon }={\Greekmath 0272} u_{0}+\varepsilon ({\Greekmath 0272}
u_{1})^{\varepsilon }+({\Greekmath 0272} _{y}u_{1})^{\varepsilon }$, and since ${\Greekmath 0272}
_{y}u_{1}\in L^{2}(\Omega ;\mathcal{A})$, we obtain $({\Greekmath 0272}
_{y}u_{1})^{\varepsilon }\rightarrow {\Greekmath 0272} _{y}u_{1}$ in $L^{2}(\Omega )$%
-strong $\Sigma $. One gets readily ${\Greekmath 0272} u_{\varepsilon }-{\Greekmath 0272} \Phi
_{\varepsilon }\rightarrow 0$ in $L^{2}(\Omega )$-weak $\Sigma $. Using $A$
as a test function, $\int_{\Omega }A^{\varepsilon }{\Greekmath 0272} \Phi _{\varepsilon
}\cdot {\Greekmath 0272} (u_{\varepsilon }-\Phi _{\varepsilon })dx\rightarrow 0$. We
have just shown that $u_{\varepsilon }-u_{0}-\varepsilon u_{1}^{\varepsilon
}\rightarrow 0$ in $L^{2}(\Omega )$ and ${\Greekmath 0272} (u_{\varepsilon
}-u_{0}-\varepsilon u_{1}^{\varepsilon })={\Greekmath 0272} u_{\varepsilon }-{\Greekmath 0272}
\Phi _{\varepsilon }\rightarrow 0$ in $L^{2}(\Omega )$. This proves (\ref%
{1.7}) and completes the proof of Theorem \ref{t1.1}.
\end{proof}
We assume henceforth that the matrix $A$ does not depend on $x$, that is, $%
A(x,y)=A(y)$. Let $\chi _{T}=(\chi _{T,j})_{1\leq j\leq d}$ be defined by (%
\ref{e01}).
\begin{lemma}
\label{l11.1}Let $T\geq 1$ and $\sigma \in (0,1)$. Assume that $A\in (%
\mathcal{A})^{d\times d}$. There exist positive numbers $C=C(A,d)$ and $%
C_{\sigma }=C_{\sigma }(d,\sigma ,A)$ such that
\begin{equation}
T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq C,
\label{e5.6}
\end{equation}%
\begin{equation}
\sup_{x\in \mathbb{R}^{d}}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}(x)}\left\vert {\Greekmath 0272} \chi _{T}\right\vert ^{2}dy\right) ^{%
\frac{1}{2}}\leq C_{\sigma }\left( \frac{T}{r}\right) ^{\sigma }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }%
0<r\leq T, \label{e5.7}
\end{equation}%
\begin{equation}
\left\vert \chi _{T}(x)-\chi _{T}(y)\right\vert \leq C_{\sigma }T^{1-\sigma
}\left\vert x-y\right\vert ^{\sigma }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }\left\vert x-y\right\vert
\leq T. \label{e5.8}
\end{equation}
\end{lemma}
\begin{proof}
Let us first check (\ref{e5.6}). From the inequality (\ref{4.3}), we deduce
that
\begin{equation}
\sup_{z\in \mathbb{R}^{d},R\geq T}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{R}(z)}\left\vert \chi _{T}\right\vert ^{2}\right) ^{\frac{1}{2}%
}\leq CT \label{e5.9}
\end{equation}%
where $C$ depends only on $d$, $\alpha $ and $\beta $. Now fix $%
z=(z_{i})_{1\leq i\leq d}$ in $\mathbb{R}^{d}$ and define
\begin{equation}
u(y)=\chi _{T,j}(y)+y_{j}-z_{j}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }y\in \mathbb{R}^{d}. \label{e5.11}
\end{equation}%
Then $u$ solves the equation
\begin{equation}
{\Greekmath 0272} \cdot (A{\Greekmath 0272} u)=T^{-2}\chi _{T,j}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}.
\label{e5.12}
\end{equation}%
Using the De Giorgi-Nash estimates, we obtain
\begin{eqnarray*}
\sup_{B_{T}(z)}\left\vert u\right\vert &\leq &C\left[ \left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}(z)}\left\vert u\right\vert ^{2}\right) ^{\frac{1}{2}%
}+T^{2}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}(z)}\left\vert T^{-2}\chi _{T,j}\right\vert ^{2}\right) ^{%
\frac{1}{2}}\right] \\
&\leq &CT+C\sup_{x\in \mathbb{R}^{d}}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}(x)}\left\vert \chi _{T,j}\right\vert ^{2}\right) ^{\frac{1}{2}%
}\leq CT
\end{eqnarray*}%
where $C=C(d,A)$. It follows that $\left\vert \chi _{T,j}(z)\right\vert \leq
CT$. Whence (\ref{e5.6}). Now, concerning (\ref{e5.8}), one uses Schauder
estimates: if $v\in H_{loc}^{1}(\mathbb{R}^{d})$ is a weak solution of $%
-{\Greekmath 0272} \cdot (A{\Greekmath 0272} v)=h+{\Greekmath 0272} \cdot H$ in $B_{2R}(x_{0})$, then for
each $\sigma \in (0,1)$ and for all $x,y\in B_{R}(x_{0})$,
\begin{eqnarray}
\left\vert v(x)-v(y)\right\vert &\leq &C\left\vert x-y\right\vert ^{\sigma }
\left[ R^{-\sigma }\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2R}(x_{0})}\left\vert v\right\vert ^{2}\right) ^{\frac{1}{2}%
}+\sup _{\substack{ z\in B_{R}(x_{0}) \\ 0<r<R}}r^{2-\sigma }\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}(z)}\left\vert h\right\vert ^{2}\right) ^{\frac{1}{2}}\right.
\label{e5.10} \\
&&\left. +\sup_{\substack{ z\in B_{R}(x_{0}) \\ 0<r<R}}r^{1-\sigma }\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}(z)}\left\vert H\right\vert ^{2}\right) ^{\frac{1}{2}}\right]
\notag
\end{eqnarray}%
where $C=C(\sigma ,A)$ (see e.g. \cite{Giaquinta} or \cite[Theorem 3.4]{Shen}%
). Assume $x,y\in \mathbb{R}^{d}$ with $\left\vert x-y\right\vert \leq T$.
Applying (\ref{e5.10}) with $2R=T$, $h=T^{-2}\chi _{T,j}$, $H=Ae_{j}$, $%
v=\chi _{T,j}$ and $x_{0}=0$,
\begin{eqnarray*}
\left\vert \chi _{T,j}(x)-\chi _{T,j}(y)\right\vert &\leq &C\left\vert
x-y\right\vert ^{\sigma }(T^{-\sigma }\left\Vert \chi _{T,j}\right\Vert
_{L^{\infty }}+T^{2-\sigma }\left\Vert T^{-2}\chi _{T,j}\right\Vert
_{L^{\infty }}+T^{1-\sigma }\left\Vert A\right\Vert _{L^{\infty }}) \\
&\leq &CT^{1-\sigma }\left\vert x-y\right\vert ^{\sigma },
\end{eqnarray*}%
where we have used (\ref{e5.6}) for the last inequality above. To obtain (%
\ref{e5.7}), we use Caccioppoli's inequality for $-{\Greekmath 0272} \cdot (A{\Greekmath 0272}
\chi _{T,j})+T^{-2}\chi _{T,j})={\Greekmath 0272} \cdot (Ae_{j})$ in $B_{2r}(x)$ and (%
\ref{e5.8}) to get
\begin{eqnarray*}
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}(x)}\left\vert {\Greekmath 0272} \chi _{T,j}(y)\right\vert ^{2}dy &\leq
&Cr^{-2}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2r}(x)}\left\vert \chi _{T,j}(y)-\chi _{T,j}(x)\right\vert
^{2}dy+C%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2r}(x)}\left\vert A\right\vert ^{2}dy \\
&\leq &Cr^{-2}(T^{1-\sigma }r^{\sigma })^{2}+C\leq C\left( \frac{T^{1-\sigma
}}{r^{1-\sigma }}\right) ^{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ since }0<r\leq T\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.}
\end{eqnarray*}%
(\ref{e5.7}) follows by replacing $\sigma $ by $1-\sigma $. This finishes
the proof.
\end{proof}
The next result will be used in the forthcoming sections. It involves
Green's function $G:\mathbb{R}^{d}\times \mathbb{R}^{d}\rightarrow \mathbb{R}
$ solution of
\begin{equation}
-{\Greekmath 0272} _{x}\cdot \left( A(x){\Greekmath 0272} _{x}G(x,y)\right) =\delta _{y}(x)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
in }\mathbb{R}^{d}. \label{10.1}
\end{equation}%
The properties of the function $G$ require the definition of the weak-$L^{2}$
space denoted by $L^{2,\infty }(\mathbb{R}^{d})$ (see \cite[Chapter 1]{BL76}
for its definition) together with its topological dual denoted by $L^{2,1}(%
\mathbb{R}^{d})$ (see \cite{Tar07} for its definition).
\begin{proposition}
\label{p10.1}Assume the matrix $A\in L^{\infty }(\mathbb{R}^{d})^{d\times d}$
is uniformly elliptic (see \emph{(\ref{1.2})}) and symmetric. Then equation
\emph{(\ref{10.1})} has a unique solution in $L^{\infty }(\mathbb{R}%
_{y}^{d};W_{loc}^{1,1}(\mathbb{R}_{x}^{d}))$ satisfying:
\begin{itemize}
\item[(i)] $G(\cdot ,y)\in W_{loc}^{1,2}(\mathbb{R}^{d}\backslash \{y\})$
for all $y\in \mathbb{R}^{d};$
\item[(ii)] There exists $C=C(d)>0$ such that%
\begin{equation}
\left\Vert {\Greekmath 0272} _{y}G(x,\cdot )\right\Vert _{L^{2,\infty }(\mathbb{R}%
^{d})}\leq C, \label{10.2}
\end{equation}%
\begin{equation}
\left\vert G(x,y)\right\vert \leq \left\{
\begin{array}{l}
C(1+\left\vert \log \left\vert x-y\right\vert \right\vert )\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ if }d=2 \\
C\left\vert x-y\right\vert ^{2-d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ if }d\geq 3%
\end{array}%
\right. \RIfM@\expandafter\text@\else\expandafter\mbox\fi{, all }x,y\in \mathbb{R}^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }x\neq y, \label{10.3}
\end{equation}%
\begin{equation}
\int_{B_{2R}(x)\backslash B_{R}(x)}\left\vert {\Greekmath 0272} _{y}G(x,y)\right\vert
^{q}dy\leq \frac{C}{R^{N(q-1)-q}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for all }R>0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }1\leq q\leq
2. \label{2.8}
\end{equation}%
\noindent If $A$ has H\"{o}lder continuous entries, then for $d\geq 3$ and
for all $x,y\in \mathbb{R}^{d}$ with $x\neq y,$%
\begin{equation}
\left\vert {\Greekmath 0272} _{y}G(x,y)\right\vert \leq C\left\vert x-y\right\vert
^{1-d}. \label{10.4}
\end{equation}
\end{itemize}
\end{proposition}
Properties (\ref{10.3}) and (\ref{10.4}) are classical; see e.g. \cite[%
Theorems 1.1 and 3.3]{Widman}. (\ref{2.8}) is proved in \cite[Lemma 4.2]%
{Lebris}.
\section{Approximation of homogenized coefficients: quantitative estimates}
To simplify the presentation of the results, we assume from now on that $%
A(x,y)=A(y)$. We henceforth denote the mean value by $\left\langle \cdot
\right\rangle $.
\subsection{Approximation by Dirichlet problem}
In the preceding section, we saw that the corrector problem is posed on the
whole of $\mathbb{R}^{d}$. However, if the coefficients of our problem are
periodic (say the function $y\mapsto A(y)$ is $Y$-periodic ($%
Y=(-1/2,1/2)^{d} $), then this problem reduces to another one posed on the
bounded subset $Y$ of $\mathbb{R}^{d}$, and this yields coefficients that
are computable. Contrasting with the periodic setting, the corrector problem
in the general deterministic framework cannot be reduced to a problem on a
bounded domain. Therefore, truncations must be considered, particularly on
large domains like $Q_{R}$ (the closed cube centered at the origin and of
side length $R$) with appropriate boundary conditions. We proceed exactly as
in the random setting (see \cite{BP2004}). We consider the equation
\begin{equation}
-{\Greekmath 0272} _{y}\cdot \left( A(e_{j}+{\Greekmath 0272} _{y}\chi _{j,R})\right) =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }%
Q_{R},\ \ \chi _{j,R}\in H_{0}^{1}(Q_{R}), \label{3.3}
\end{equation}%
which possesses a unique solution satisfying
\begin{equation}
\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}\left\vert {\Greekmath 0272} _{y}\chi _{j,R}\right\vert ^{2}dy\right) ^{%
\frac{1}{2}}\leq C\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for any }R\geq 1 \label{i}
\end{equation}%
where $C$ is independent of $R$. Set $\chi _{R}=(\chi _{j,R})_{1\leq j\leq
d} $. We define the effective and approximate effective matrices $A^{\ast }$
and $A_{R}^{\ast }$ respectively, as follows
\begin{equation}
A^{\ast }=\left\langle A(I+{\Greekmath 0272} _{y}\chi )\right\rangle \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }%
A_{R}^{\ast }=%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}A(y)(I+{\Greekmath 0272} _{y}\chi _{R}(y))dy. \label{eq5}
\end{equation}
\begin{theorem}
\label{t3.1}The generalized sequence of matrices $A_{R}^{\ast }$ converges,
as $R\rightarrow \infty $, to the homogenized matrix $A^{\ast }$.
\end{theorem}
\begin{proof}
We set, for $x\in Q_{1}$, $w_{j}^{R}(x)=\frac{1}{R}\chi _{j,R}(Rx)$, $%
A_{R}(x)=A(Rx)$ and consider the re-scaled version of (\ref{3.3}) whose $%
w_{j}^{R}$ is solution. It reads as
\begin{equation}
-{\Greekmath 0272} \cdot (A_{R}(e_{j}+{\Greekmath 0272} w_{j}^{R}))=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }Q_{1}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, \ }%
w_{j}^{R}=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ on }\partial Q_{1}. \label{3.6}
\end{equation}%
Then (\ref{3.6}) possesses a unique solution $w_{j}^{R}\in H_{0}^{1}(Q_{1})$
satisfying the estimate
\begin{equation}
\left\Vert {\Greekmath 0272} w_{j}^{R}\right\Vert _{L^{2}(Q_{1})}\leq C\ \ \ (1\leq
j\leq d) \label{3.7}
\end{equation}%
where $C>0$ is independent of $R>0$. Proceeding as in the proof of Theorem %
\ref{t1.1}, we derive the existence of $w_{j}\in H_{0}^{1}(Q_{1})$ and $%
w_{j,1}\in L^{2}(Q_{1};B_{\#\mathcal{A}}^{1,2}(\mathbb{R}^{d}))$ such that,
up to a subsequence not relabeled,
\begin{equation}
w_{j}^{R}\rightarrow w_{j}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }H_{0}^{1}(Q_{1})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-weak and }{\Greekmath 0272}
w_{j}^{R}\rightarrow {\Greekmath 0272} w_{j}+{\Greekmath 0272} _{y}w_{j,1}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }%
L^{2}(Q_{1})^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-weak }\Sigma \label{4.00}
\end{equation}%
and the couple $(w_{j},w_{j,1})$ solves the equation
\begin{equation}
\int_{Q_{1}}\left\langle A(e_{j}+{\Greekmath 0272} w_{j}+{\Greekmath 0272} _{y}w_{j,1})\cdot
({\Greekmath 0272} \psi _{0}+{\Greekmath 0272} _{y}\psi _{1})\right\rangle dx=0\ \ \forall (\psi
_{0},\psi _{1})\in \mathcal{C}_{0}^{\infty }(Q_{1})\times (\mathcal{C}%
_{0}^{\infty }(Q_{1})\otimes \mathcal{A}^{\infty }), \label{4.100}
\end{equation}%
which can be rewritten in the following equivalent form (\ref{4.101})-(\ref%
{4.102})
\begin{equation}
\int_{Q_{1}}\left\langle A(e_{j}+{\Greekmath 0272} w_{j}+{\Greekmath 0272}
_{y}w_{j,1})\right\rangle \cdot {\Greekmath 0272} \psi _{0}dx=0\ \ \forall \psi _{0}\in
\mathcal{C}_{0}^{\infty }(Q_{1}) \label{4.101}
\end{equation}%
and
\begin{equation}
\left\langle A(e_{j}+{\Greekmath 0272} w_{j}+{\Greekmath 0272} _{y}w_{j,1})\cdot {\Greekmath 0272}
_{y}v\right\rangle dx=0\ \ \forall v\in \mathcal{A}^{\infty }. \label{4.102}
\end{equation}%
To solve (\ref{4.102}), we consider its weak distributional form
\begin{equation}
{\Greekmath 0272} _{y}\cdot \left( A(e_{j}+{\Greekmath 0272} w_{j}+{\Greekmath 0272} _{y}w_{j,1})\right) =0%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \label{4.103}
\end{equation}%
So fix $\xi \in \mathbb{R}^{d}$ and consider the problem
\begin{equation}
{\Greekmath 0272} _{y}\cdot \left( A(e_{j}+\xi +{\Greekmath 0272} _{y}\pi _{j}(\xi )\right) =0%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d};\ \pi _{j}(\xi )\in B_{\#\mathcal{A}}^{1,2}(%
\mathbb{R}^{d}). \label{4.104}
\end{equation}%
Then $\pi _{j}(\xi )$ has the form $\pi _{j}(\xi )=\chi _{j}+\theta _{j}(\xi
)$ where $\chi _{j}$ is the solution of the corrector problem (\ref{1.6})
and $\theta _{j}(\xi )$ solves the equation
\begin{equation}
{\Greekmath 0272} _{y}\cdot \left( A(\xi +{\Greekmath 0272} _{y}\theta _{j}(\xi )\right) =0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
in }\mathbb{R}^{d};\ \theta _{j}(\xi )\in B_{\#\mathcal{A}}^{1,2}(\mathbb{R}%
^{d}), \label{4.105}
\end{equation}%
that is, $\theta _{j}(\xi )=\xi \cdot \chi $ where $\chi =(\chi _{k})_{1\leq
k\leq d}$ with $\chi _{k}$ being the solution of (\ref{1.6}) corresponding
to $j=k$ therein. It follows that $\pi _{j}(\xi )=\chi _{j}+\xi \cdot \chi $%
, so that the function $w_{j,1}$, which corresponds to $\pi _{j}({\Greekmath 0272}
w_{j})$, has the form $w_{j,1}=\chi _{j}+\chi \cdot {\Greekmath 0272} w_{j}$. Coming
back to (\ref{4.101}) and replacing there $w_{j,1}$ by $\chi _{j}+\chi \cdot
{\Greekmath 0272} w_{j}$, we obtain
\begin{equation}
\int_{Q_{1}}\left\langle A(I+{\Greekmath 0272} _{y}\chi )\right\rangle (e_{j}+{\Greekmath 0272}
w_{j})\cdot {\Greekmath 0272} \psi _{0}dx=0\ \ \forall \psi _{0}\in \mathcal{C}%
_{0}^{\infty }(Q_{1})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \label{4.106}
\end{equation}%
This shows that $w_{j}\in H_{0}^{1}(Q_{1})$ solves uniquely the equation
\begin{equation}
-{\Greekmath 0272} \cdot (A^{\ast }(e_{j}+{\Greekmath 0272} w_{j}))=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }Q_{1}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,}
\label{3.9}
\end{equation}%
and further we have, as $R\rightarrow \infty $,
\begin{equation}
A_{R}(e_{j}+{\Greekmath 0272} w_{j}^{R})\rightarrow A^{\ast }(e_{j}+{\Greekmath 0272} w_{j})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
in }L^{2}(Q_{1})^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{-weak.} \label{3.10}
\end{equation}%
To see (\ref{3.10}), we observe that the sequence $(A_{R}(e_{j}+{\Greekmath 0272}
w_{j}^{R}))_{R}$ is bounded in $L^{2}(Q_{1})^{d}$ and we choose a test
function $\Phi \in \mathcal{C}_{0}^{\infty }(Q_{1})^{d}$; then by the
sigma-convergence (where we take $A(y)\Phi (x)$ as a test function) we have
from the second convergence result in (\ref{4.00}) that
\begin{eqnarray*}
\int_{Q_{1}}A_{R}(e_{j}+{\Greekmath 0272} w_{j}^{R})\cdot \Phi dx &\rightarrow
&\int_{Q_{1}}\left\langle A(e_{j}+{\Greekmath 0272} w_{j}+{\Greekmath 0272} _{y}w_{j,1})\cdot \Phi
\right\rangle dx \\
&=&\int_{Q_{1}}\left\langle A(e_{j}+{\Greekmath 0272} w_{j}+{\Greekmath 0272}
_{y}w_{j,1})\right\rangle \cdot \Phi dx.
\end{eqnarray*}%
But according to (\ref{4.106}), we see that
\begin{equation*}
\left\langle A(e_{j}+{\Greekmath 0272} w_{j}+{\Greekmath 0272} _{y}w_{j,1})\right\rangle
=\left\langle A(I+{\Greekmath 0272} _{y}\chi )\right\rangle (e_{j}+{\Greekmath 0272}
w_{j})=A^{\ast }(e_{j}+{\Greekmath 0272} w_{j}).
\end{equation*}%
Now, since (\ref{3.9}) has the form $-{\Greekmath 0272} \cdot (A^{\ast }{\Greekmath 0272} w_{j})=0$
in $Q_{1}$, ($A^{\ast }$ has constant entries) we infer from the ellipticity
property of $A^{\ast }$ and the uniqueness of the solution to $-{\Greekmath 0272} \cdot
(A^{\ast }{\Greekmath 0272} w_{j})=0$ in $H_{0}^{1}(Q_{1})$ that $w=(w_{1},...,w_{d})=0$%
. Hence the whole sequence $(w_{j}^{R})_{R}$ weakly converges towards $0$ in
$H_{0}^{1}(Q_{1})$. Therefore, integrating (\ref{3.10}) over $Q_{1}$, we
readily get (denoting $w^{R}=(w_{1}^{R},...,w_{d}^{R})$)%
\begin{equation*}
A_{R}^{\ast }=%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{1}}A(I+{\Greekmath 0272} w^{R})dx\rightarrow
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{1}}A^{\ast }(I+{\Greekmath 0272} w)dx=A^{\ast }
\end{equation*}%
as $R\rightarrow \infty $, where $I$ is the $d\times d$ identity matrix.
This completes the proof.
\end{proof}
\subsection{Quantitative estimates}
We study the rate of convergence for the approximation scheme of the
previous subsection, under the assumption that the corrector lies in $B_{%
\mathcal{A}}^{2}(\mathbb{R}^{d})$. To this end, instead of considering the
corrector problem (\ref{1.6}) we rather consider its regularized version (%
\ref{4.2}) which we recall here below:
\begin{equation*}
-{\Greekmath 0272} \cdot A(y)(e_{j}+{\Greekmath 0272} \chi _{T,j})+T^{-2}\chi _{T,j}=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }%
\mathbb{R}^{d}.
\end{equation*}%
We define the regularized homogenized matrix by
\begin{equation}
A_{T}^{\ast }=\left\langle A(I+{\Greekmath 0272} \chi _{T})\right\rangle ,\ \ \chi
_{T}=(\chi _{T,j})_{1\leq j\leq d} \label{3.11}
\end{equation}%
Recalling that the homogenized matrix has the form $A^{\ast }=\left\langle
A(I+{\Greekmath 0272} \chi )\right\rangle $, we show in (\ref{3.19}) below that $%
\left\vert A^{\ast }-A_{T}^{\ast }\right\vert \leq CT^{-1}$, so that $%
A_{T}^{\ast }\rightarrow A^{\ast }$ as $T\rightarrow \infty $.
With this in mind, we define the approximate regularized coefficients
\begin{equation}
A_{R,T}^{\ast }=%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}A(I+{\Greekmath 0272} \chi _{T}^{R}),\ \ \chi _{T}^{R}=(\chi
_{T,j}^{R})_{1\leq j\leq d} \label{3.12}
\end{equation}%
where $\chi _{T,j}^{R}$ (the regularized approximate corrector) solves the
problem
\begin{equation}
-{\Greekmath 0272} \cdot A(e_{j}+{\Greekmath 0272} \chi _{T,j}^{R})+T^{-2}\chi _{T,j}^{R}=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
in }Q_{R},\ \chi _{T,j}^{R}\in H_{0}^{1}(Q_{R}). \label{3.13}
\end{equation}%
Then
\begin{equation*}
A_{R,T}^{\ast }\underset{(\ast )}{\overset{R\rightarrow \infty }{\rightarrow
}}A_{T}^{\ast }\underset{(\ast \ast )}{\overset{T\rightarrow \infty }{%
\rightarrow }}A^{\ast }.
\end{equation*}%
Convergence ($\ast \ast $) will result from (\ref{3.19}) below, while for
convergence ($\ast $), we proceed exactly as in the proof of Theorem \ref%
{t3.1}.
The aim here is to estimate the expression $\left\vert A^{\ast
}-A_{R,T}^{\ast }\right\vert $ in terms of $R$ and $T$, and next take $R=T$
to get the suitable rate of convergence. The following theorem is the main
result of this section.
\begin{theorem}
\label{t3.2}Suppose $\chi \in B_{\mathcal{A}}^{2}(\mathbb{R}^{d})^{d}$. Let $%
\delta \in (0,1)$. There exist $C=C(d,\delta ,A)$ and a continuous function $%
\eta _{\delta }:[1,\infty )\rightarrow \lbrack 0,\infty )$, which depends
only on $A$ and $\delta $, such that $\lim_{t\rightarrow \infty }\eta
_{\delta }(t)=0$ and
\begin{equation}
\left\vert A^{\ast }-A_{T,T}^{\ast }\right\vert \leq C\eta _{\delta }(T)%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for all }T\geq 1. \label{3.17}
\end{equation}
\end{theorem}
The proof breaks down into several steps which are of independent interest.
\begin{lemma}
\label{l3.2}Let $u\in B_{\mathcal{A}}^{2}(\mathbb{R}^{d})$. For any $%
0<R<\infty $,
\begin{equation}
\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}u-\left\langle u\right\rangle \right\vert \leq \sup_{y\in
\mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}\left\vert u(t+y)-u(t)\right\vert dt. \label{3.18}
\end{equation}
\end{lemma}
\begin{proof}
Let $u\in B_{\mathcal{A}}^{2}(\mathbb{R}^{d})$. We know that, for any $y\in
\mathbb{R}^{d}$,
\begin{equation*}
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}(y)}u-%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}u=%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}\left( u(t+y)-u(t)\right) dt.
\end{equation*}%
Now, let $k>1$ be an integer; we have $Q_{kR}=\cup
_{i=1}^{k^{d}}Q_{R}(x_{i}) $ for some $x_{i}\in \mathbb{R}^{d}$, so that
\begin{equation*}
\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$
} \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{kR}}u-%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}u\right\vert \leq \frac{1}{k^{d}}\sum_{i=1}^{k^{d}}\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}(x_{i})}u-%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}u\right\vert \leq \sup_{y\in \mathbb{R}^{d}}\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}(y)}u-%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{R}}u\right\vert .
\end{equation*}%
Letting $k\rightarrow \infty $ we are led to (\ref{3.18}).
\end{proof}
The next result evaluates the difference between $A^{\ast }$ and $%
A_{T}^{\ast }$.
\begin{lemma}
\label{l3.3}Assume that $\chi _{j}$ (defined by \emph{(\ref{1.6})}) belongs
to $B_{\mathcal{A}}^{2}(\mathbb{R}^{d})$. There exists $C=C(d,A)$ such that
\begin{equation}
\left\vert A^{\ast }-A_{T}^{\ast }\right\vert \leq CT^{-1}. \label{3.19}
\end{equation}
\end{lemma}
\begin{proof}
First, let us set $v=\chi _{T,j}-\chi _{j}$. Then $v$ solves the equation $%
-{\Greekmath 0272} \cdot (A{\Greekmath 0272} v)+T^{-2}v=-T^{-2}\chi _{j}$ in $\mathbb{R}^{d}$. It
follows from Lemma \ref{l4.1} that
\begin{equation*}
\sup_{x\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(x)}\left( \left\vert {\Greekmath 0272} v\right\vert ^{2}+T^{-2}\left\vert
v\right\vert ^{2}\right) \leq CT^{-2}\sup_{x\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(x)}\left\vert \chi _{j}\right\vert ^{2}\leq CT^{-2}.
\end{equation*}%
In the last inequality above, we have used the fact that $\chi _{j}\in B_{%
\mathcal{A}}^{2}(\mathbb{R}^{d})$, so that
\begin{equation*}
\sup_{x\in \mathbb{R}^{d},T>0}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(x)}\left\vert \chi _{j}\right\vert ^{2}\leq C.
\end{equation*}%
The above inequality stems from the fact that $\lim_{T\rightarrow \infty }%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(x)}\left\vert \chi _{j}\right\vert ^{2}$ exists uniformly in $%
x\in \mathbb{R}^{d}$. We infer
\begin{equation}
\sup_{x\in \mathbb{R}^{d}}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(x)}\left\vert A{\Greekmath 0272} (\chi _{T,j}-\chi _{j})\right\vert
^{2}\right) ^{\frac{1}{2}}\leq \left\Vert A\right\Vert _{\infty }\sup_{x\in
\mathbb{R}^{d}}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle
-}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(x)}\left\vert {\Greekmath 0272} (\chi _{T,j}-\chi _{j})\right\vert
^{2}\right) ^{\frac{1}{2}}\leq CT^{-1}. \label{3.20}
\end{equation}%
Now, using Lemma \ref{l3.2} with $u=A{\Greekmath 0272} (\chi _{T}-\chi )$, we obtain
\begin{equation}
\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$
} \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}}A{\Greekmath 0272} (\chi -\chi _{T})-(A^{\ast }-A_{T}^{\ast })\right\vert
\leq \sup_{y\in \mathbb{R}^{d}}%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}}\left\vert A{\Greekmath 0272} (\chi -\chi _{T})(t+y)-A{\Greekmath 0272} (\chi -\chi
_{T})(t)\right\vert dt. \label{3.21}
\end{equation}%
However, from the equality
\begin{equation*}
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}}A{\Greekmath 0272} (\chi -\chi _{T})(t+y)dt=%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(y)}A{\Greekmath 0272} (\chi -\chi _{T})(t)dt
\end{equation*}%
associated to the inequality
\begin{equation*}
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(y)}\left\vert A{\Greekmath 0272} (\chi -\chi _{T})(t)\right\vert dt\leq
\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(y)}\left\vert A{\Greekmath 0272} (\chi -\chi _{T})\right\vert ^{2}\right)
^{\frac{1}{2}},
\end{equation*}%
we deduce that the right-hand side of (\ref{3.21}) is bounded by $%
2\sup_{y\in \mathbb{R}^{d}}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}(y)}\left\vert A{\Greekmath 0272} (\chi -\chi _{T})\right\vert ^{2}\right)
^{\frac{1}{2}}$. Taking into account (\ref{3.20}), we get immediately
\begin{equation*}
\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$
} \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}}A{\Greekmath 0272} (\chi -\chi _{T})-(A^{\ast }-A_{T}^{\ast })\right\vert
\leq CT^{-1}.
\end{equation*}%
It follows that
\begin{equation*}
\left\vert A^{\ast }-A_{T}^{\ast }\right\vert \leq \left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}}A{\Greekmath 0272} (\chi -\chi _{T})-(A^{\ast }-A_{T}^{\ast })\right\vert
+%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{Q_{T}}\left\vert A{\Greekmath 0272} (\chi -\chi _{T})\right\vert \leq CT^{-1}.
\end{equation*}
\end{proof}
We are now in a position to prove the theorem.
\begin{proof}[Proof of Theorem \protect\ref{t3.2}]
We decompose $A^{\ast }-A_{R,T}^{\ast }$ as follows:%
\begin{equation*}
A^{\ast }-A_{R,T}^{\ast }=(A^{\ast }-A_{T}^{\ast })+(A_{T}^{\ast
}-A_{R,T}^{\ast }).
\end{equation*}%
We consider each term separately.
Lemma \ref{l3.3} yields $\left\vert A^{\ast }-A_{T}^{\ast }\right\vert \leq
CT^{-1}$. As regard the term $A_{T}^{\ast }-A_{R,T}^{\ast }$, we observe
that $v=\chi _{T,j}-\chi _{T,j}^{R}$\ solves the equation
\begin{equation*}
-{\Greekmath 0272} \cdot A{\Greekmath 0272} v+T^{-2}v=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }Q_{R}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }v=\chi _{T,j}%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ on }\partial Q_{R},
\end{equation*}%
so that, proceeding exactly as in \cite[Proof of Lemma 1]{BP2004} we obtain
\begin{equation}
\left\vert A_{T}^{\ast }-A_{R,T}^{\ast }\right\vert ^{2}\leq C\left(
T^{2}\exp (-c_{1}TR^{\delta })+R^{\delta -1}\right) \label{3.22}
\end{equation}%
where $0<\delta <1$, and $C$ and $c_{1}>0$ are independent of $R$ and $T$.
We emphasize that in \cite{BP2004}, the above inequality has been obtained
without any help stemming from the random character of the problem. It
relies only on the bounds of the Green function of the operator $-{\Greekmath 0272}
\cdot A{\Greekmath 0272} +T^{-2}$ and on the bounds of the regularized corrector $\chi
_{T}$.
Choosing $R=T$ in (\ref{3.22}), we define the function
\begin{equation*}
\eta _{\delta }(t)=\frac{1}{t}+t\exp \left( -\frac{c_{1}}{2}t^{1+\delta
}\right) +t^{\frac{1}{2}(\delta -1)}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }t\geq 1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}%
Then $\eta _{\delta }$ is continuous with $\lim_{t\rightarrow \infty }\eta
_{\delta }(t)=0$. We see that
\begin{equation*}
\left\vert A^{\ast }-A_{T,T}^{\ast }\right\vert \leq C\eta _{\delta }(T)%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for any }T\geq 1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}%
This concludes the proof of the theorem.
\end{proof}
\section{Convergence rates: the asymptotic periodic setting}
\subsection{Preliminary results}
Let us consider the corrector problem (\ref{1.6}) in which $A$ satisfies in
addition the assumptions (\ref{2.1}) and (\ref{2.2}) below:
\begin{equation}
A=A_{0}+A_{per}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }A_{per}\in L_{per}^{2}(Y)^{d\times d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and
}A_{0}\in L^{2}(\mathbb{R}^{d})^{d\times d}; \label{2.1}
\end{equation}%
The matrix $A_{per}$ is symmetric and further
\begin{equation}
\alpha \left\vert \lambda \right\vert ^{2}\leq A_{per}(y)\lambda \cdot
\lambda \leq \beta \left\vert \lambda \right\vert ^{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for all }%
\lambda \in \mathbb{R}^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and a.e. }y\in \mathbb{R}^{d}. \label{2.2}
\end{equation}%
Let $H_{\infty ,per}^{1}(\mathbb{R}^{d})=\{u\in L_{\infty ,per}^{2}(\mathbb{R%
}^{d}):{\Greekmath 0272} u\in L_{\infty ,per}^{2}(\mathbb{R}^{d})^{d}\}$ where $%
L_{\infty ,per}^{2}(\mathbb{R}^{d})=L_{0}^{2}(\mathbb{R}^{d})+L_{per}^{2}(Y)$
and $L_{0}^{2}(\mathbb{R}^{d})$ is the completion of $\mathcal{C}_{0}(%
\mathbb{R}^{d})$ with respect to the seminorm (\ref{0.2}).
\begin{proposition}
\label{l2.1}Let $H$ be a function such that $H\in L^{2}(\mathbb{R}^{d})^{d}$
for $d\geq 3$ and $H\in (L^{2}(\mathbb{R}^{d})\cap L^{2,1}(\mathbb{R}%
^{d}))^{d}$ for $d=2$. Assume $A$ satisfies \emph{(\ref{1.2})}. Then there
exists $u_{0}\in L^{p}(\mathbb{R}^{d})$ with ${\Greekmath 0272} u_{0}\in L^{2}(\mathbb{R%
}^{d})^{d}$ such that $u_{0}$ solves the equation
\begin{equation}
-{\Greekmath 0272} \cdot A{\Greekmath 0272} u_{0}={\Greekmath 0272} \cdot H\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}
\label{8.1}
\end{equation}%
where $p=2^{\ast }\equiv 2d/(d-2)$ for $d\geq 3$ and $p=\infty $ for $d=2$.
\end{proposition}
\begin{proof}
1) We first assume that $d\geq 3$. Let $Y^{1,2}=\{u\in L^{2^{\ast }}(\mathbb{%
R}^{d}):{\Greekmath 0272} u\in L^{2}(\mathbb{R}^{d})^{d}\}$ (where $2^{\ast }=2d/(d-2)$%
), and equip $Y^{1,2}$ with the norm $\left\Vert u\right\Vert
_{Y^{1,2}}=\left\Vert u\right\Vert _{L^{2^{\ast }}(\mathbb{R}%
^{d})}+\left\Vert {\Greekmath 0272} u\right\Vert _{L^{2}(\mathbb{R}^{d})}$, which makes
it a Banach space. By the Sobolev's inequality (see \cite[Theorem 4.31, page
102]{Adams}), there exists a positive constant $C=C(d)$ such that
\begin{equation}
\left\Vert u\right\Vert _{L^{2^{\ast }}(\mathbb{R}^{d})}\leq C\left\Vert
{\Greekmath 0272} u\right\Vert _{L^{2}(\mathbb{R}^{d})}\ \ \forall u\in Y^{1,2}.
\label{6.3}
\end{equation}%
We deduce from (\ref{6.3}) that (\ref{8.1}) possesses a unique solution in $%
Y^{1,2}$ satisfying the inequality
\begin{equation}
\left\Vert u_{0}\right\Vert _{Y^{1,2}}\leq C\left\Vert H\right\Vert _{L^{2}(%
\mathbb{R}^{d})}. \label{6.4}
\end{equation}
2) Now assume that $d=2$. We use $G(x,y)$ defined by (\ref{10.1}) to express
$u_{0}$ as
\begin{equation}
u_{0}(x)=-\int_{\mathbb{R}^{d}}{\Greekmath 0272} _{y}G(x,y)\cdot H(y)dy. \label{8.6}
\end{equation}%
The expression (\ref{8.6}) makes sense since we may proceed by approximation
by assuming first that $H\in \mathcal{C}_{0}^{\infty }(\mathbb{R}^{2})^{2}$
and next using the density of $\mathcal{C}_{0}^{\infty }(\mathbb{R}^{2})$ in
$L^{2,1}(\mathbb{R}^{2})$ together with property (\ref{10.3}) to conclude.
So, using the generalized H\"{o}lder inequality, we get
\begin{equation}
\left\Vert u_{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}\leq \sup_{x\in
\mathbb{R}^{2}}\left\Vert {\Greekmath 0272} _{y}G(x,\cdot )\right\Vert _{L^{2,\infty }(%
\mathbb{R}^{2})}\left\Vert H\right\Vert _{L^{2,1}(\mathbb{R}^{2})}.
\label{8.2}
\end{equation}%
This completes the proof.
\end{proof}
\begin{lemma}
\label{l1.2}Assume that $A=A_{0}+A_{per}$ where $A$ and $A_{per}$ are
uniformly elliptic (see \emph{(\ref{1.2}) }and \emph{(\ref{2.2})}) with $%
A_{0}$ (resp. $A_{per}$) having entries in $L^{2}(\mathbb{R}^{d})$ (resp. $%
L_{per}^{\infty }(Y)$). Assume further that $A_{per}$ and $A$ are H\"{o}lder
continuous. Let the number $p$ be as in Proposition \emph{\ref{l2.1}}. Let $%
\chi _{j,per}\in H_{per}^{1}(Y)$ be the unique solution of
\begin{equation}
-{\Greekmath 0272} _{y}\cdot \left( A_{per}(e_{j}+{\Greekmath 0272} _{y}\chi _{j,per})\right) =0%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }Y,\ \ \int_{Y}\chi _{j,per}dy=0. \label{2.3}
\end{equation}%
Then \emph{(\ref{1.6})} possesses a unique solution $\chi _{j}\in H_{\infty
,per}^{1}(Y)$ (in the sense of Theorem \emph{\ref{t4.1}}) satisfying $\chi
_{j}=\chi _{j,0}+\chi _{j,per}$ where $\chi _{j,0}\in L^{p}(\mathbb{R}^{d})$
with ${\Greekmath 0272} _{y}\chi _{j,0}\in L^{2}(\mathbb{R}^{d})^{d}$, and
\begin{equation}
\left\Vert \chi _{j}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq C
\label{*2}
\end{equation}%
where $C=C(d,A)$.
\end{lemma}
\begin{proof}
First, we notice that if $\chi _{j,per}$ solves (\ref{2.3}) then $\chi
_{j,0} $ solves
\begin{equation*}
-{\Greekmath 0272} _{y}\cdot \left( A{\Greekmath 0272} _{y}\chi _{j,0}\right) ={\Greekmath 0272} _{y}\cdot
\left( A_{0}(e_{j}+{\Greekmath 0272} _{y}\chi _{j,per})\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}%
^{d}.
\end{equation*}%
Assuming that $A_{per}$ is H\"{o}lder continuous, we get ${\Greekmath 0272} _{y}\chi
_{j,per}\in L^{\infty }(Y)$. Since $A_{0}\in L^{2}(\mathbb{R}^{d})^{d\times
d}$ it follows that $g=A_{0}(e_{j}+{\Greekmath 0272} _{y}\chi _{j,per})\in L^{2}(%
\mathbb{R}^{d})^{d}$. Proposition \ref{l2.1} implies that $\chi _{j,0}\in
L^{p}(\mathbb{R}^{d})$ with ${\Greekmath 0272} _{y}\chi _{j,0}\in L^{2}(\mathbb{R}%
^{d})^{d}$. Hence $\left\langle \chi _{j,0}\right\rangle =0$ and $%
\left\langle {\Greekmath 0272} _{y}\chi _{j,0}\right\rangle =0$. This proves that $\chi
_{j}=\chi _{j,per}+\chi _{j,0}\in H_{\infty ,per}^{1}(Y)$ for $d\geq 3$. For
$d=2$, $\chi _{j,0}\in L_{0}^{2}(\mathbb{R}^{2})$ since $\chi _{j,0}$
vanishes at infinity. Indeed, we use (\ref{8.2}) to get
\begin{equation*}
\left\Vert \chi _{j,0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}\leq
\sup_{x\in \mathbb{R}^{d}}\left\Vert {\Greekmath 0272} _{y}G(x,\cdot )\right\Vert
_{L^{2,\infty }(\mathbb{R}^{2})}\left\Vert g\right\Vert _{L^{2,1}(\mathbb{R}%
^{2})}
\end{equation*}%
and proceed as in \cite[Section 3, page 14]{Lebris} (first approximate $g$
by smooth functions in $\mathcal{C}_{0}^{\infty }(\mathbb{R}^{2})^{2}$) to
show that $\chi _{j,0}\in L_{0}^{2}(\mathbb{R}^{2})$.
Let us now verify (\ref{*2}). We drop for a while the index $j$ and just
write $\chi =\chi _{0}+\chi _{per}$, where the couple $(\chi _{per},\chi
_{0})$ solves the system
\begin{equation}
-{\Greekmath 0272} _{y}\cdot \left( A_{per}(e_{j}+{\Greekmath 0272} _{y}\chi _{per})\right) =0%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }Y, \label{2.4}
\end{equation}%
\begin{equation}
-{\Greekmath 0272} _{y}\cdot \left( A{\Greekmath 0272} _{y}\chi _{0}\right) ={\Greekmath 0272} _{y}\cdot
\left( A_{0}(e_{j}+{\Greekmath 0272} _{y}\chi _{per})\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}.
\label{2.5}
\end{equation}%
It is well known that $\chi _{per}$ is bounded in $L^{\infty }(\mathbb{R}%
^{d})$. Let us first deal with $\chi _{0}$. Let $g=A_{0}(e_{j}+{\Greekmath 0272}
_{y}\chi _{per})$ and use the Green function defined in Proposition \ref%
{p10.1} to express $\chi _{0}$ as
\begin{equation}
\chi _{0}(y)=-\int_{\mathbb{R}^{d}}{\Greekmath 0272} _{x}G(y,x)g(x)dx. \label{2.7}
\end{equation}%
We recall that $G$ satisfies the inequality (\ref{10.4}) for $d\geq 3$\ and (%
\ref{10.2}) for $d=2$, respectively.
We first assume that $d\geq 3$. Let $y\in \mathbb{R}^{d}$ and choose $\gamma
\in \mathcal{C}_{0}^{\infty }(B_{2}(y))$ such that $\gamma =1$ on $B_{1}(y)$
and $0\leq \gamma \leq 1$. We write $\chi _{0}$ as
\begin{align*}
\chi _{0}(y)& =-\int_{\mathbb{R}^{d}}{\Greekmath 0272} _{x}G(y,x)\cdot g(x)\gamma
(x)dx-\int_{\mathbb{R}^{d}}{\Greekmath 0272} _{x}G(y,x)\cdot g(x)(1-\gamma (x))dx \\
& =v_{1}(y)+v_{2}(y).
\end{align*}%
As for $v_{1}$, owing to (\ref{10.4}), we have
\begin{equation*}
\left\vert v_{1}(y)\right\vert \leq C\left\Vert g\right\Vert _{L^{\infty }(%
\mathbb{R}^{d})}\int_{B_{2}(y)}\left\vert x-y\right\vert ^{1-d}dx\leq
C\left\Vert g\right\Vert _{L^{\infty }(\mathbb{R}^{d})}
\end{equation*}%
where $C=C(d)$. As for $v_{2}$, (\ref{10.4}) and H\"{o}lder's inequality
imply,
\begin{equation*}
\left\vert v_{2}(y)\right\vert \leq C\left\Vert g\right\Vert _{L^{2}(\mathbb{%
R}^{d})}\left( \int_{\mathbb{R}^{d}\backslash B_{2}(y)}\left\vert
x-y\right\vert ^{2-2d}dx\right) \leq C\left\Vert g\right\Vert _{L^{2}(%
\mathbb{R}^{d})}
\end{equation*}%
since $2d-2>d$ for $d\geq 3$.
When $d=2$, we use (\ref{10.2}) and the continuous embedding $L^{2}(\mathbb{R%
}^{d})\hookrightarrow L^{2,1}(\mathbb{R}^{d})$ to get
\begin{equation*}
\left\Vert \chi _{0}\right\Vert _{L^{\infty }(\mathbb{R}^{2})}\leq
\sup_{x\in \mathbb{R}^{2}}\left\Vert {\Greekmath 0272} _{y}G(x,\cdot )\right\Vert
_{L^{2,\infty }(\mathbb{R}^{2})}\left\Vert g\right\Vert _{L^{2,1}(\mathbb{R}%
^{2})}\leq C\left\Vert g\right\Vert _{L^{2}(\mathbb{R}^{d})}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}
\end{proof}
\begin{lemma}
\label{l1.4}\emph{(i)} Let $g\in L^{2}(\mathbb{R}^{d})+L_{per}^{2}(Y)$ be
such that $\left\langle g\right\rangle =0$. Then there exists at least one
function $u\in H_{\infty ,per}^{1}(Y)$ such that
\begin{equation}
\Delta u=g\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d},\ \left\langle u\right\rangle =0.
\label{2.9}
\end{equation}%
\emph{(ii)} Assume further that $g\in L^{\infty }(\mathbb{R}^{d})$ and $u$
is bounded; then $u,{\Greekmath 0272} u\in \mathcal{B}_{\infty ,per}(\mathbb{R}^{d})$
and
\begin{equation}
\left\Vert {\Greekmath 0272} u\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq
C\left\Vert g\right\Vert _{L^{\infty }(\mathbb{R}^{d})}, \label{2.10}
\end{equation}%
where $C>0$ depends only on $d$.
\end{lemma}
\begin{proof}
(i) We write $g=g_{0}+g_{per}$ with $g_{0}\in L^{2}(\mathbb{R}^{d})$ and $%
g_{per}\in L_{per}^{2}(Y)$. Since $\left\langle g\right\rangle =0$, we have $%
\left\langle g_{per}\right\rangle =0$. So let $v_{per}\in H_{per}^{1}(Y)$ be
the unique solution of
\begin{equation*}
\Delta v_{per}=g_{per}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }Y\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }\left\langle v_{per}\right\rangle
=0.
\end{equation*}%
We observe that if $u$ solves (\ref{2.9}), then $u$ has the form $%
u=v_{0}+v_{per}$ where $v_{0}\in H^{1}(\mathbb{R}^{d})$ solves the problem
\begin{equation*}
\Delta v_{0}=g_{0}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }v_{0}(x)\rightarrow 0%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }\left\vert x\right\vert \rightarrow \infty .
\end{equation*}%
Since $g_{0}\in L^{2}(\mathbb{R}^{d})$, $v_{0}$ easily expresses as
\begin{equation*}
v_{0}(x)=\int_{\mathbb{R}^{d}}\Gamma _{0}(x-y)g_{0}(y)dy
\end{equation*}%
where $\Gamma _{0}$ denotes the fundamental solution of the Laplacian in $%
\mathbb{R}^{d}$ (with pole at the origin). This shows the existence of $u$
in $H^{1}(\mathbb{R}^{d})+H_{per}^{1}(Y)\subset H_{\infty ,per}^{1}(\mathbb{R%
}^{d})$.
Let us check (ii). First, since (\ref{2.9}) is satisfied, $u$ is thus the
Newtonian potential of $g$ in $\mathbb{R}^{d}$, and by \cite[page 71,
Problem 4.8 (a)]{Gilbarg}, ${\Greekmath 0272} u\in \mathcal{C}_{loc}^{1/2}(\mathbb{R}%
^{d})$. Using therefore the continuity of ${\Greekmath 0272} u$ together with the fact
that ${\Greekmath 0272} u$ also lies in $L_{\infty ,per}^{2}(\mathbb{R}^{d})$, we infer
that ${\Greekmath 0272} u\in \mathcal{B}_{\infty ,per}(\mathbb{R}^{d})=\mathcal{C}_{0}(%
\mathbb{R}^{d})\oplus \mathcal{C}_{per}(Y)$. We then proceed as in the proof
of Lemma \ref{l1.2} to obtain $u\in \mathcal{B}_{\infty ,per}(\mathbb{R}%
^{d}) $. This completes the proof.
\end{proof}
The following result is a mere consequence of the preceding lemma. Its proof
is therefore left to the reader.
\begin{corollary}
\label{c1.1}Let $\mathbf{g}$\ be a solenoidal vector in $(L^{2}(\mathbb{R}%
^{d})+L_{per}^{2}(Y))^{d}$\ (i.e. ${\Greekmath 0272} \cdot \mathbf{g}=0$) with $%
\left\langle \mathbf{g}\right\rangle =0$. Then there exists a skew symmetric
matrix $G$\ with entries in $L_{\infty ,per}^{2}(Y)$\ such that $\mathbf{g}%
={\Greekmath 0272} \cdot G$. If further $\mathbf{g}$\ belongs to $L^{\infty }(\mathbb{R}%
^{d})^{d}$, then $G$\ has entries in $\mathcal{B}_{\infty ,per}(\mathbb{R}%
^{d})$ and
\begin{equation}
\left\Vert G\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq C\left\Vert
\mathbf{g}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}. \label{*5}
\end{equation}
\end{corollary}
\subsection{Convergence rates: proof of Theorem \protect\ref{t5.1}}
Let $u_{\varepsilon }$, $u_{0}\in H_{0}^{1}(\Omega )$ be the weak solutions
of (\ref{1.1}) and (\ref{1.4}) respectively. Assume further that $u_{0}\in
H^{2}(\Omega )$. We suppose in addition that $\Omega $ is sufficiently
smooth. For any function $h\in L_{loc}^{2}(\mathbb{R}^{d})$ and $\varepsilon
>0$ we define $h^{\varepsilon }$ by $h^{\varepsilon }(x)=h(x/\varepsilon )$
for $x\in \mathbb{R}^{d}$. We define the first order approximation of $%
u_{\varepsilon }$ by $v_{\varepsilon }=u_{0}+\varepsilon \chi ^{\varepsilon
}{\Greekmath 0272} u_{0}$. Let $w_{\varepsilon }=u_{\varepsilon }-v_{\varepsilon
}+z_{\varepsilon }$ where $z_{\varepsilon }\in H^{1}(\Omega )$ is the weak
solution of the following problem
\begin{equation}
-{\Greekmath 0272} \cdot A^{\varepsilon }{\Greekmath 0272} z_{\varepsilon }=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\Omega
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }z_{\varepsilon }=\varepsilon \chi ^{\varepsilon }{\Greekmath 0272} u_{0}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
on }\partial \Omega . \label{5.3}
\end{equation}%
$z_{\varepsilon }$ will be used to approximate the difference of $%
u_{\varepsilon }$ and its first order approximation $v_{\varepsilon }$.
\begin{lemma}
\label{l5.1}The function $w_{\varepsilon }$ solves the problem
\begin{equation}
\left\{
\begin{array}{l}
-{\Greekmath 0272} \cdot \left( A^{\varepsilon }{\Greekmath 0272} w_{\varepsilon }\right) ={\Greekmath 0272}
\cdot \left( A^{\varepsilon }({\Greekmath 0272} u_{0}+({\Greekmath 0272} _{y}\chi )^{\varepsilon
}{\Greekmath 0272} u_{0}-\left\langle A({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}\chi {\Greekmath 0272}
u_{0})\right\rangle \right) \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\varepsilon
{\Greekmath 0272} \cdot \left( A^{\varepsilon }{\Greekmath 0272} ^{2}u_{0}\chi ^{\varepsilon
}\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\Omega \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ w_{\varepsilon }=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ on }\partial
\Omega .%
\end{array}%
\right. \label{5.4}
\end{equation}
\end{lemma}
\begin{proof}
Let $y=x/\varepsilon $. Then
\begin{equation*}
A(y){\Greekmath 0272} w_{\varepsilon }=A(y)({\Greekmath 0272} u_{\varepsilon }-{\Greekmath 0272} u_{0}-{\Greekmath 0272}
_{y}\chi (y){\Greekmath 0272} u_{0}-\varepsilon ({\Greekmath 0272} ^{2}u_{0})\chi (y)+{\Greekmath 0272}
z_{\varepsilon }),
\end{equation*}%
hence
\begin{align*}
{\Greekmath 0272} \cdot A\left( y\right) {\Greekmath 0272} w_{\varepsilon }& ={\Greekmath 0272} \cdot
A(y){\Greekmath 0272} u_{\varepsilon }-{\Greekmath 0272} \cdot A(y){\Greekmath 0272} u_{0}-{\Greekmath 0272} \cdot
A(y)({\Greekmath 0272} _{y}\chi (y){\Greekmath 0272} u_{0}) \\
& -\varepsilon {\Greekmath 0272} \cdot (A(y)({\Greekmath 0272} ^{2}u_{0})\chi (y)) \\
& ={\Greekmath 0272} \cdot A^{\ast }{\Greekmath 0272} u_{0}-{\Greekmath 0272} \cdot A(y){\Greekmath 0272} u_{0}-{\Greekmath 0272}
\cdot A(y)({\Greekmath 0272} _{y}\chi (y){\Greekmath 0272} u_{0}) \\
& -\varepsilon {\Greekmath 0272} \cdot (A(y)({\Greekmath 0272} ^{2}u_{0})\chi (y)).
\end{align*}%
But
\begin{equation*}
A^{\ast }{\Greekmath 0272} u_{0}=\left\langle A({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}\chi {\Greekmath 0272}
u_{0})\right\rangle \equiv \left\langle A(I+{\Greekmath 0272} _{y}\chi ){\Greekmath 0272}
u_{0}\right\rangle .
\end{equation*}
Thus
\begin{align*}
-{\Greekmath 0272} \cdot A^{\varepsilon }{\Greekmath 0272} w_{\varepsilon }& ={\Greekmath 0272} \cdot \left[
A\left( y\right) ({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}\chi {\Greekmath 0272} u_{0})-\left\langle
A({\Greekmath 0272} u_{0}+{\Greekmath 0272} _{y}\chi {\Greekmath 0272} u_{0})\right\rangle \right] \\
& +\varepsilon {\Greekmath 0272} \cdot (A(y)({\Greekmath 0272} ^{2}u_{0})\chi (y)),
\end{align*}%
which is the statement of the lemma.
\end{proof}
Set
\begin{equation*}
a_{ij}(y)=b_{ij}(y)+\sum_{k=1}^{d}b_{ik}(y)\frac{\partial \chi _{j}}{%
\partial y_{k}}(y)-b_{ij}^{\ast }
\end{equation*}%
where $A^{\ast }=(b_{ij}^{\ast })_{1\leq i,j\leq d}$ is the homogenized
matrix, and let $a_{j}=(a_{ij})_{1\leq i\leq d}$. Then $a_{j}\in \lbrack
L^{\infty }(\mathbb{R}^{d})\cap L_{\infty ,per}^{2}(Y)]^{d}$ with ${\Greekmath 0272}
\cdot a_{j}=0$ and $\left\langle a_{j}\right\rangle =0$. Hence by Corollary %
\ref{c1.1}, there is a skew-symmetric matrix $G_{j}$ with entries in $%
\mathcal{A}=\mathcal{B}_{\infty ,per}(Y)$ such that $a_{j}={\Greekmath 0272} _{y}\cdot
G_{j}$. Moreover in view of (\ref{*5}) in Corollary \ref{c1.1}, we have
\begin{equation*}
\left\Vert G_{j}\right\Vert _{\infty }\leq C\left\Vert a_{j}\right\Vert
_{\infty }.
\end{equation*}%
With this in mind and recalling that $G_{j}$ is skew-symmetric, Eq. (\ref%
{5.4}) becomes
\begin{equation}
-{\Greekmath 0272} \cdot A\left( \frac{x}{\varepsilon }\right) {\Greekmath 0272} w_{\varepsilon
}=\varepsilon {\Greekmath 0272} \cdot \left( r_{1}^{\varepsilon }+r_{2}^{\varepsilon
}\right) \label{5.5}
\end{equation}%
where
\begin{equation*}
r_{1}^{\varepsilon }(x)=\sum_{j=1}^{d}G_{j}(y){\Greekmath 0272} \frac{\partial u_{0}}{%
\partial x_{j}}(x)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }r_{2}^{\varepsilon }(x)=A(y){\Greekmath 0272}
^{2}u_{0}(x)\chi (y)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }y=\frac{x}{\varepsilon }.
\end{equation*}%
Now, since $w_{\varepsilon }\in H_{0}^{1}(\Omega )$, it follows from the
ellipticity of $A$ (see (\ref{1.2})) that
\begin{align*}
\alpha \left\Vert {\Greekmath 0272} w_{\varepsilon }\right\Vert _{L^{2}(\Omega )}& \leq
\varepsilon \left( \left\Vert r_{1}^{\varepsilon }\right\Vert _{L^{2}(\Omega
)}+\left\Vert r_{2}^{\varepsilon }\right\Vert _{L^{2}(\Omega )}\right) \\
& \leq C\varepsilon \left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}
\end{align*}%
where $C=C(d,A,\Omega )$.
We have just proved the following result.
\begin{proposition}
\label{p5.1}Let $\Omega $ be a smooth bounded domain in $\mathbb{R}^{d}$.
Suppose that $A=A_{0}+A_{per}$ and $A$ and $A_{per}$ are uniformly elliptic
(see \emph{(\ref{1.2})} and \emph{(\ref{2.2})}). For $f\in L^{2}(\Omega )$,
let $u_{\varepsilon }$, $u_{0}$ and $v_{\varepsilon }$ be weak solutions of
Dirichlet problems \emph{(\ref{1.1})}, \emph{(\ref{1.4})} and \emph{(\ref%
{5.3})}, respectively. Assume $u_{0}\in H^{2}(\Omega )$. There $%
C=C(d,A,\Omega )$ such that
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi ^{\varepsilon }{\Greekmath 0272}
u_{0}+z_{\varepsilon }\right\Vert _{H_{0}^{1}(\Omega )}\leq C\varepsilon
\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}. \label{5.6}
\end{equation}
\end{proposition}
The estimate of the deviation of $u_{\varepsilon }$ and $v_{\varepsilon }$
is a consequence of the following lemma whose proof is postponed to the next
section and is obtained as a special case of the proof of a general result
formulated as Lemma \ref{l5.3}. Observe that in Lemma \ref{l5.3} we replace $%
T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}$ by $%
\varepsilon $ (see Remark \ref{r5.2}).
\begin{lemma}
\label{l5.2'}Assume $u_{0}\in H^{2}(\Omega )$. Let $z_{\varepsilon }$ be the
solution of problem \emph{(\ref{5.3})}. There exists $C=C(d,A,\Omega )$ such
that
\begin{equation}
\left\Vert z_{\varepsilon }\right\Vert _{H^{1}(\Omega )}\leq C\varepsilon ^{%
\frac{1}{2}}\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}. \label{5.7}
\end{equation}
\end{lemma}
\begin{proof}[Proof of Theorem \protect\ref{t5.1}]
Since $\Omega $ is a $\mathcal{C}^{1,1}$-bounded domain in $\mathbb{R}^{d}$
and the matrix $A^{\ast }$ has constant entries, it is known that $u_{0}$
satisfies the inequality
\begin{equation}
\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}\leq C\left\Vert f\right\Vert
_{L^{2}(\Omega )},\ C=C(d,\alpha ,\Omega )>0. \label{5.7'}
\end{equation}%
Using (\ref{5.6}) together with (\ref{5.7}) and (\ref{5.7'}), we arrive at
\begin{align*}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi ^{\varepsilon }{\Greekmath 0272}
u_{0}\right\Vert _{H^{1}(\Omega )}& \leq \left\Vert u_{\varepsilon
}-u_{0}-\varepsilon \chi ^{\varepsilon }{\Greekmath 0272} u_{0}+z_{\varepsilon
}\right\Vert _{H_{0}^{1}(\Omega )}+\left\Vert z_{\varepsilon }\right\Vert
_{H^{1}(\Omega )} \\
& \leq C\varepsilon ^{\frac{1}{2}}\left\Vert u_{0}\right\Vert _{H^{2}(\Omega
)}\leq C\varepsilon ^{\frac{1}{2}}\left\Vert f\right\Vert _{L^{2}(\Omega )},
\end{align*}%
and derive the statement of (\ref{5.8}) in Theorem \ref{t5.1}. As for (\ref%
{1.14}) we proceed exactly as in the proof of (\ref{Eq02}) in the proof of
Theorem \ref{t1.4}; see in particular Remark \ref{r5.3} in the next section.
This concludes the proof of Theorem \ref{t5.1}.
\end{proof}
\section{Convergence rates: the asymptotic almost periodic setting}
\subsection{Preliminaries}
We treat the asymptotic almost periodic case in a general way, dropping
restrictions (\ref{2.1}) and (\ref{2.2}). The results in this section extend
those of the preceding section as well as those in the almost periodic
setting obtained in \cite{Shen}.
We recall that a bounded continuous function $u$ defined on $\mathbb{R}^{d}$
is asymptotically almost periodic if there exists a couple $(v,w)\in AP(%
\mathbb{R}^{d})\times \mathcal{C}_{0}(\mathbb{R}^{d})$ such that $u=v+w$. We
denote by $\mathcal{B}_{\infty ,AP}(\mathbb{R}^{d})=AP(\mathbb{R}^{d})+%
\mathcal{C}_{0}(\mathbb{R}^{d})$ the Banach algebra of such functions. We
denote by $H_{\infty ,AP}^{1}(\mathbb{R}^{d})$ the Sobolev-type space
attached to the Besicovitch space $B_{\mathcal{A}}^{2}(\mathbb{R}^{d})\equiv
L_{\infty ,AP}^{2}(\mathbb{R}^{d})=L_{0}^{2}(\mathbb{R}^{d})+B_{AP}^{2}(%
\mathbb{R}^{d})$: $H_{\infty ,AP}^{1}(\mathbb{R}^{d})=\{u\in L_{\infty
,AP}^{2}(\mathbb{R}^{d}):{\Greekmath 0272} u\in L_{\infty ,AP}^{2}(\mathbb{R}%
^{d})^{d}\} $. Here $L_{0}^{2}(\mathbb{R}^{d})$ is the completion of $%
\mathcal{C}_{0}(\mathbb{R}^{d})$ with respect to the seminorm (\ref{0.2})
while $B_{AP}^{2}(\mathbb{R}^{d})$ is the Besicovitch space associated to
the algebra $AP(\mathbb{R}^{d})$. We also denote by $\mathcal{C}_{b}(\mathbb{%
R}^{d})$ the algebra of real-valued bounded continuous functions defined on $%
\mathbb{R}^{d}$.
The following characterization of $\mathcal{B}_{\infty ,AP}(\mathbb{R}^{d})$
is a useful tool for the considerations below.
\begin{proposition}
\label{p11.1}Let $u\in \mathcal{C}_{b}(\mathbb{R}^{d})$. Then $u\in \mathcal{%
B}_{\infty ,AP}(\mathbb{R}^{d})$ if and only if
\begin{equation}
\sup_{y\in \mathbb{R}^{d}}\inf_{z\in \mathbb{R}^{d},\left\vert z\right\vert
\leq L}\left\Vert u(\cdot +y)-u(\cdot +z)\right\Vert _{L^{\infty }(\mathbb{R}%
^{d}\backslash B_{R})}\rightarrow 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }L\rightarrow \infty \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and
}R\rightarrow 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \label{11.1}
\end{equation}
\end{proposition}
\begin{proof}
A set $E$ in $\mathbb{R}^{d}$ is relatively dense if there exists $L>0$ such
that $\mathbb{R}^{d}=E+B_{L}$ (where we recall that $B_{L}=B(0,L)$), that
is, any $x\in \mathbb{R}^{d}$ expresses as a sum $y+z$ with $y\in E$ and $%
z\in B_{L}$. This being so, it is known that $u\in \mathcal{C}_{b}(\mathbb{R}%
^{d})$ lies in $\mathcal{B}_{\infty ,AP}(\mathbb{R}^{d})$ if and only if for
any $\varepsilon >0$, there is $R=R(\varepsilon )>0$ such that the set
\begin{equation*}
\left\{ \tau \in \mathbb{R}^{d}:\left\vert u(t+\tau )-u(t)\right\vert
<\varepsilon \ \ \forall \left\vert t\right\vert \geq R\right\}
\end{equation*}%
is relatively dense; see e.g. \cite{Zaidman}. But this is shown to be
equivalent to (\ref{11.1}).
\end{proof}
\begin{remark}
\label{r11.1}\emph{We notice that, for any }$u\in \mathcal{C}_{b}(\mathbb{R}%
^{d})$\emph{, }%
\begin{equation*}
\lim_{R\rightarrow \infty }\left( \sup_{\left\vert y\right\vert \leq
R}\left\vert u(y)\right\vert \right) =\lim_{R\rightarrow 0}\left(
\sup_{\left\vert y\right\vert \geq R}\left\vert u(y)\right\vert \right) .
\end{equation*}%
\emph{In view of the above equality we may replace (\ref{11.1}) by }%
\begin{equation}
\sup_{y\in \mathbb{R}^{d}}\inf_{\left\vert z\right\vert \leq L}\left\Vert
u(\cdot +y)-u(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}\rightarrow 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
\emph{\ as }}L,R\rightarrow \infty \label{11.2}
\end{equation}%
\emph{since the limits in (\ref{11.1}) and (\ref{11.2}) are the same. In
practice we will rather use (\ref{11.2}).}
\end{remark}
\begin{definition}
\label{d5.1}\emph{For a function }$u\in \mathcal{B}_{\infty ,AP}(\mathbb{R}%
^{d})$\emph{\ we define the modulus of asymptotic almost periodicity of }$u$%
\emph{\ by }%
\begin{equation}
\rho _{u}(L,R)=\sup_{y\in \mathbb{R}^{d}}\inf_{\left\vert z\right\vert \leq
L}\left\Vert u(\cdot +y)-u(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
\emph{\ for }}L,R>0. \label{11.3}
\end{equation}%
\emph{In particular we set }%
\begin{equation}
\rho (L,R)=\sup_{y\in \mathbb{R}^{d}}\inf_{\left\vert z\right\vert \leq
L}\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,
}L,R>0. \label{11.4}
\end{equation}
\end{definition}
\begin{remark}
\label{r5.1}\emph{Observe that if }$R=\infty $\emph{\ (that is, }$B_{R}=%
\mathbb{R}^{d}$\emph{) in (\ref{11.3}), then }$u\in \mathcal{B}_{\infty ,AP}(%
\mathbb{R}^{d})$\emph{\ is almost periodic if and only if }$\rho
_{u}(L,\infty )\rightarrow 0$\emph{\ as }$L\rightarrow \infty $\emph{.}
\end{remark}
\subsection{Estimates of approximate correctors}
First we recall that the approximate corrector $\chi _{T}=(\chi
_{T,j})_{1\leq j\leq d}$ is defined as the distributional solution of
\begin{equation}
-{\Greekmath 0272} \cdot \left( A(e_{j}+{\Greekmath 0272} \chi _{T,j})\right) +T^{-2}\chi _{T,j}=0%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d},\ \ \chi _{T,j}\in H_{\infty ,AP}^{1}(\mathbb{R}%
^{d}) \label{11.5}
\end{equation}%
where $A\in (L_{\infty ,AP}^{2}(\mathbb{R}^{d})\cap L^{\infty }(\mathbb{R}%
^{d}))^{d\times d}$ is symmetric and uniformly elliptic.
In all that follows in this section we assume that $A\in (\mathcal{B}%
_{\infty ,AP}(\mathbb{R}^{d}))^{d\times d}$.
\begin{theorem}
\label{t11.1}Let $T\geq 1$. Then $\chi _{T}\in \mathcal{B}_{\infty ,AP}(%
\mathbb{R}^{d})$ and for any $x_{0},y,z\in \mathbb{R}^{d}$,
\begin{equation}
\left\Vert \chi _{T}(\cdot +y)-\chi _{T}(\cdot +z)\right\Vert _{L^{\infty
}(B_{R}(x_{0}))}\leq CT\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert
_{L^{\infty }(B_{R}(x_{0}))} \label{11.9}
\end{equation}%
for any $R>2T$, where $C=C(d,A)$.
\end{theorem}
\begin{proof}
Fix $R>2T$. We need to show that, for any $x_{0},y,z\in \mathbb{R}^{d}$ and $%
t\in B_{R}(x_{0})$,
\begin{equation*}
\left\vert \chi _{T}(t+y)-\chi _{T}(t+z)\right\vert \leq CT\left\Vert
B(\cdot +y)-B(\cdot +z)\right\Vert _{L^{\infty }(B_{R}(x_{0}))}.
\end{equation*}%
We follow the same approach as in the proof of \cite[Theorem 6.3]{Shen}.
Without restriction, assume $x_{0}=0$. We choose $\varphi \in \mathcal{C}%
_{0}^{\infty }(B_{\frac{7}{4}T})$ such that $\varphi =1$ in $B_{\frac{3}{2}%
T} $, $0\leq \varphi \leq 1$ and $\left\vert {\Greekmath 0272} \varphi \right\vert \leq
CT^{-1}$. We also assume that $d\geq 3$ (the case $d=2$ follows from the
case $d=3$ by adding a dummy variable). Define $u(x)=\chi _{T,j}(x+y)-\chi
_{T,j}(x+z)$ ($x\in \mathbb{R}^{d}$) and note that $u$ solves the equation
\begin{eqnarray*}
-{\Greekmath 0272} \cdot (A(\cdot +y){\Greekmath 0272} u)+T^{-2}u &=&{\Greekmath 0272} \cdot (A(\cdot
+y)-A(\cdot +z))e_{j} \\
&&+{\Greekmath 0272} \cdot \lbrack (A(\cdot +y)-A(\cdot +z)){\Greekmath 0272} v]\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{%
R}^{d}
\end{eqnarray*}%
where $v(x)=\chi _{T,j}(x+z)$. We have
\begin{eqnarray}
-{\Greekmath 0272} \cdot (A(\cdot +y){\Greekmath 0272} u) &=&-T^{-2}u\varphi +{\Greekmath 0272} \cdot
(\varphi (A(\cdot +y)-A(\cdot +z))e_{j}) \label{11.10} \\
&&+{\Greekmath 0272} \cdot (\varphi (A(\cdot +y)-A(\cdot +z)){\Greekmath 0272} v) \notag \\
&&-(A(\cdot +y)-A(\cdot +z))e_{j}{\Greekmath 0272} \varphi -A(\cdot +y){\Greekmath 0272} u\cdot
{\Greekmath 0272} \varphi \notag \\
&&-{\Greekmath 0272} \cdot (uA(\cdot +y){\Greekmath 0272} \varphi ). \notag
\end{eqnarray}
Denoting by $G^{y}$ the fundamental solution of the operator $-{\Greekmath 0272} \cdot
(A(\cdot +y){\Greekmath 0272} )$ in $\mathbb{R}^{d}$, we use the representation formula
in (\ref{11.10}) to get, for $x\in B_{T}$,
\begin{eqnarray*}
u(x) &=&-T^{-2}\int_{\mathbb{R}^{d}}G^{y}(x,t)u(t)\varphi (t)dt-\int_{%
\mathbb{R}^{d}}{\Greekmath 0272} _{t}G^{y}(x,t)\varphi (t)(A(t+y)-A(t+z))e_{j}dt \\
&&-\int_{\mathbb{R}^{d}}{\Greekmath 0272} _{t}G^{y}(x,t)\varphi
(t)(A(t+y)-A(t+z)){\Greekmath 0272} v(t)dt \\
&&-\int_{\mathbb{R}^{d}}G^{y}(x,t)(A(t+y)-A(t+z))e_{j}{\Greekmath 0272} \varphi (t)dt \\
&&-\int_{\mathbb{R}^{d}}G^{y}(x,t)A(t+y){\Greekmath 0272} u(t)\cdot {\Greekmath 0272} \varphi (t)dt
\\
&&+\int_{\mathbb{R}^{d}}{\Greekmath 0272} _{t}G^{y}(x,t)A(t+y)u(t){\Greekmath 0272} \varphi (t)dt.
\end{eqnarray*}%
It follows that
\begin{eqnarray}
\left\vert u(x)\right\vert &\leq &CT^{-2}\int_{B_{2T}}\left\vert
G^{y}(x,t)\right\vert \left\vert u(t)\right\vert dt+ \label{e11.10} \\
&&+C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}\int_{B_{2T}}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert dt \notag
\\
&&+C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}\int_{B_{2T}}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert \left\vert
{\Greekmath 0272} v(t)\right\vert dt \notag \\
&&+C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}\int_{B_{2T}}\left\vert G^{y}(x,t)\right\vert \left\vert {\Greekmath 0272}
\varphi (t)\right\vert dt \notag \\
&&+C\left( \int_{B_{2T}}\left\vert G^{y}(x,t)\right\vert ^{2}\left\vert
{\Greekmath 0272} \varphi (t)\right\vert ^{2}dt\right) ^{\frac{1}{2}}\left(
\int_{B_{2T}}\left\vert {\Greekmath 0272} u\right\vert ^{2}\right) ^{\frac{1}{2}}
\notag \\
&&+C\left( \int_{B_{2T}}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert
^{2}\left\vert {\Greekmath 0272} \varphi (t)\right\vert ^{2}dt\right) ^{\frac{1}{2}%
}\left( \int_{B_{2T}}\left\vert u\right\vert ^{2}\right) ^{\frac{1}{2}}.
\notag
\end{eqnarray}%
Let us first deal with the last two terms in (\ref{e11.10}). Let $0<\tau <1$
be such that $B_{\tau T}(x)\subset B_{T}$ (recall that $x\in B_{T}$). Then $%
B_{2T}\backslash B_{\tau T}(x)\subset B_{3T}(x)\backslash B_{\tau T}(x)$ and
since ${\Greekmath 0272} \varphi =0$ in $B_{T}$ (and hence in $B_{\tau T}(x)$), it
holds that
\begin{eqnarray*}
\left( \int_{B_{2T}}\left\vert G^{y}(x,t)\right\vert ^{2}\left\vert {\Greekmath 0272}
\varphi (t)\right\vert ^{2}dt\right) ^{\frac{1}{2}} &\leq &CT^{-1}\left(
\int_{B_{3T}(x)\backslash B_{\tau T}(x)}\frac{dt}{\left\vert x-t\right\vert
^{2(d-2)}}\right) ^{\frac{1}{2}} \\
&\leq &CT^{1-\frac{d}{2}};
\end{eqnarray*}%
\begin{eqnarray*}
\left( \int_{B_{2T}}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert
^{2}\left\vert {\Greekmath 0272} \varphi (t)\right\vert ^{2}dt\right) ^{\frac{1}{2}}
&\leq &CT^{-1}\left( \int_{B_{3T}(x)\backslash B_{\tau T}(x)}\left\vert
{\Greekmath 0272} _{t}G^{y}(x,t)\right\vert ^{2}\right) ^{\frac{1}{2}} \\
&\leq &CT^{-1}\left( \sum_{i=\left[ \frac{\ln \tau }{\ln 2}\right]
}^{2}\int_{B_{2^{i+1}T}(x)\backslash B_{2^{i}T}(x)}\left\vert {\Greekmath 0272}
_{t}G^{y}(x,t)\right\vert ^{2}dt\right) ^{\frac{1}{2}} \\
&\leq &CT^{-1}\left( \sum_{i=\left[ \frac{\ln \tau }{\ln 2}\right]
}^{2}(2^{i}T)^{2-d}\right) ^{\frac{1}{2}}\leq CT^{-\frac{d}{2}},
\end{eqnarray*}%
where $\left[ \frac{\ln \tau }{\ln 2}\right] $ stands for the integer part
of $\frac{\ln \tau }{\ln 2}$. We infer that the last two terms in (\ref%
{e11.10}) are bounded from above by $T\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert {\Greekmath 0272} u\right\vert ^{2}\right) ^{\frac{1}{2}%
}+\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert u\right\vert ^{2}\right) ^{\frac{1}{2}}$. Next we
notice that the inequality (\ref{4.3}) in Lemma \ref{l4.1} is still valid
without taking the sup over $\mathbb{R}^{d}$; indeed this is a mere
consequence of Caccioppoli's inequality. Rewriting this inequality applied
to (\ref{11.10}) yields
\begin{eqnarray*}
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left( \left\vert {\Greekmath 0272} u\right\vert ^{2}+T^{-2}\left\vert
u\right\vert ^{2}\right) &\leq &C%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert A(t+y)-A(t+z)\right\vert ^{2}dt \\
&&+C%
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert A(t+y)-A(t+z)\right\vert ^{2}\left\vert {\Greekmath 0272}
v\right\vert ^{2}dt \\
&\leq &C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}^{2}
\end{eqnarray*}%
where we have used the facts that $R>2T$ and
\begin{equation*}
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert {\Greekmath 0272} v\right\vert ^{2}dt\leq C\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ (see (\ref%
{4.3}) in Lemma \ref{l4.1}).}
\end{equation*}%
It follows at once that
\begin{equation}
T\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert {\Greekmath 0272} u\right\vert ^{2}\right) ^{\frac{1}{2}%
}+\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert u\right\vert ^{2}\right) ^{\frac{1}{2}}\leq
CT\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}.
\label{e1.1}
\end{equation}
Concerning the second term in the right-hand side of (\ref{e11.10}), we have
\begin{eqnarray}
\int_{B_{2T}}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert dt &\leq
&C\int_{B_{3T}(x)}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert dt
\label{e2.1} \\
&\leq &C\sum_{i=-\infty }^{1}\int_{B_{2^{i+1}T}(x)\backslash
B_{2^{i}T}(x)}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert dt\leq
C\sum_{i=-\infty }^{1}2^{i}T\leq CT, \notag
\end{eqnarray}%
where we have used for the first inequality in (\ref{e2.1}), the fact that $%
B_{2T}\subset B_{3T}(x)$ (recall that $x\in B_{T}$), and for the last
inequality, (\ref{2.8}) (for $q=1$). It follows that
\begin{equation*}
C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}\int_{B_{2T}}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert dt\leq
CT\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}.
\end{equation*}%
As for the third term in the right-hand side of (\ref{e11.10}) is concerned,
we concentrate on the control of the integral
\begin{equation*}
I=\int_{B_{2T}}\left\vert {\Greekmath 0272} _{t}G^{y}(x,t)\right\vert \left\vert {\Greekmath 0272}
v(t)\right\vert dt.
\end{equation*}%
First, we note that the function $v$ solves the equation
\begin{equation*}
-{\Greekmath 0272} \cdot (A(\cdot +z){\Greekmath 0272} v)+T^{-2}v={\Greekmath 0272} \cdot (A(\cdot +z)e_{j})%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d}
\end{equation*}%
so that appealing to (\ref{4.3}),
\begin{equation}
\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert {\Greekmath 0272} v\right\vert ^{2}\right) ^{\frac{1}{2}}\leq
C\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \label{e11.15}
\end{equation}
Next, H\"{o}lder inequality and (\ref{e11.15}) lead to
\begin{eqnarray*}
I &\leq &CT^{\frac{d}{2}}\left( \int_{B_{2T}}\left\vert {\Greekmath 0272}
_{t}G^{y}(x,t)\right\vert ^{2}dt\right) ^{\frac{1}{2}}\leq CT^{\frac{d}{2}%
}\left( \int_{B_{3T}(x)\backslash B_{\tau T}(x)}\left\vert {\Greekmath 0272}
_{t}G^{y}(x,t)\right\vert ^{2}\right) ^{\frac{1}{2}} \\
&\leq &CT^{\frac{d}{2}}\left( \sum_{i=\left[ \frac{\ln \tau }{\ln 2}\right]
}^{2}\int_{B_{2^{i+1}T}(x)\backslash B_{2^{i}T}(x)}\left\vert {\Greekmath 0272}
_{t}G^{y}(x,t)\right\vert ^{2}dt\right) ^{\frac{1}{2}}\leq CT^{\frac{d}{2}%
}\left( \sum_{i=\left[ \frac{\ln \tau }{\ln 2}\right] }^{2}(2^{i}T)^{2-d}%
\right) ^{\frac{1}{2}} \\
&\leq &CT^{\frac{d}{2}}T^{1-\frac{d}{2}}=CT.
\end{eqnarray*}%
For the fourth term in the right-hand side of (\ref{e11.10}), we have
\begin{equation*}
\int_{B_{2T}}\left\vert G^{y}(x,t)\right\vert \left\vert {\Greekmath 0272} \varphi
(t)\right\vert dt\leq CT^{-1}\int_{B_{3T}(x)}\frac{dt}{\left\vert
x-t\right\vert ^{d-2}}dt\leq CT.
\end{equation*}%
We have therefore shown that
\begin{equation}
\left\vert u(x)\right\vert \leq CT^{-2}\int_{B_{2T}}\frac{\left\vert
u(t)\right\vert }{\left\vert x-t\right\vert ^{d-2}}dt+CT\left\Vert A(\cdot
+y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}. \label{e11.17}
\end{equation}%
Using the well known fractional integral estimates, (\ref{e11.17}) yields
\begin{equation*}
\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{T}}\left\vert u\right\vert ^{q}\right) ^{\frac{1}{q}}\leq C\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert u\right\vert ^{p}\right) ^{\frac{1}{p}%
}+CT\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}
\end{equation*}%
where $1<p<q\leq \infty $ with $\frac{1}{p}-\frac{1}{q}<\frac{2}{d}$.
However from (\ref{e1.1}) we derive the estimate
\begin{equation*}
\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{2T}}\left\vert u\right\vert ^{2}\right) ^{\frac{1}{2}}\leq
CT\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{R})},
\end{equation*}%
so that by an iteration argument, we are led to
\begin{equation*}
\left\Vert u\right\Vert _{L^{\infty }(B_{T})}\leq CT\left\Vert A(\cdot
+y)-A(\cdot +z)\right\Vert _{L^{\infty }(B_{R})}.
\end{equation*}%
This yields (recalling that $x_{0}=0$)
\begin{equation*}
\left\vert u(0)\right\vert \leq CT\left\Vert A(\cdot +y)-A(\cdot
+z)\right\Vert _{L^{\infty }(B_{R})}.
\end{equation*}%
Recalling that $0$ may be replaced by any $t\in B_{R}$, this completes the
proof.
\end{proof}
\begin{theorem}
\label{t11.2}Let $T\geq 1$ and $R>2T$. For any $0<L\leq T$ and $\sigma \in
(0,1)$, there is $C_{\sigma }=C_{\sigma }(\sigma ,A)$ such that
\begin{equation}
T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq
C_{\sigma }\left( \rho (L,R)+\left( \frac{L}{T}\right) ^{\sigma }\right) .
\label{11.11}
\end{equation}
\end{theorem}
\begin{proof}
Let $y,z\in \mathbb{R}^{d}$ with $\left\vert z\right\vert \leq L\leq T$.
Then
\begin{equation*}
\left\vert \chi _{T}(y)\right\vert \leq \left\vert \chi _{T}(y)-\chi
_{T}(0)\right\vert +\left\vert \chi _{T}(0)\right\vert
\end{equation*}%
and
\begin{eqnarray*}
\left\vert \chi _{T}(y)-\chi _{T}(0)\right\vert &\leq &\left\vert \chi
_{T}(y)-\chi _{T}(z)\right\vert +\left\vert \chi _{T}(z)-\chi
_{T}(0)\right\vert \\
&=&\left\vert \chi _{T}(0+y)-\chi _{T}(0+z)\right\vert +\left\vert \chi
_{T}(z)-\chi _{T}(0)\right\vert \\
&\leq &\sup_{x\in B_{R}}\left\vert \chi _{T}(x+y)-\chi _{T}(x+z)\right\vert
+\left\vert \chi _{T}(z)-\chi _{T}(0)\right\vert \\
&\leq &CT\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}+C_{\sigma }T^{1-\sigma }L^{\sigma }
\end{eqnarray*}%
where for the last inequality above we have used (\ref{e5.8}) (in Lemma \ref%
{l11.1}) and (\ref{11.9}) (in Theorem \ref{t11.1}). It follows readily that
\begin{equation}
\sup_{y\in \mathbb{R}^{d}}\left\vert \chi _{T}(y)-\chi _{T}(0)\right\vert
\leq T\left( C\rho (L,R)+C_{\sigma }\left( \frac{L}{T}\right) ^{\sigma
}\right) . \label{11.12}
\end{equation}%
On the other hand, observing that
\begin{eqnarray*}
\left\vert \chi _{T}(0)\right\vert &\leq &\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}}(\chi _{T}(t)-\chi _{T}(0))dt\right\vert +\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}}\chi _{T}(t)dt\right\vert \\
&\leq &\sup_{y\in \mathbb{R}^{d}}\left\vert \chi _{T}(y)-\chi
_{T}(0)\right\vert +\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}}\chi _{T}(t)dt\right\vert
\end{eqnarray*}%
and letting $r\rightarrow \infty $, we use the fact that $\left\langle \chi
_{T}\right\rangle =0$ to get
\begin{equation*}
\left\vert \chi _{T}(0)\right\vert \leq \sup_{y\in \mathbb{R}^{d}}\left\vert
\chi _{T}(y)-\chi _{T}(0)\right\vert .
\end{equation*}%
The above inequality associated to (\ref{11.12}) yield (\ref{11.11}).
\end{proof}
Now, we set (for $T\geq 1$ and $\sigma \in (0,1]$)
\begin{equation}
\Theta _{\sigma }(T)=\inf_{0<L<T}\left( \rho (L,3T)+\left( \frac{L}{T}%
\right) ^{\sigma }\right) \label{11.13}
\end{equation}%
where $\rho (L,R)$ is given by (\ref{11.4}). Then $T\mapsto \Theta _{\sigma
}(T)$ is a continuous decreasing function satisfying $\Theta _{\sigma
}(T)\rightarrow 0$ when $T\rightarrow \infty $ (this stems from the
asymptotic almost periodicity of $A$, so that $\rho (L,3T)\rightarrow 0$ as $%
T\rightarrow \infty $). We infer from (\ref{11.11}) that
\begin{equation}
T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq
C_{\sigma }\Theta _{\sigma }(T) \label{11.14}
\end{equation}%
and hence
\begin{equation*}
T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}%
^{d})}\rightarrow 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ as }T\rightarrow \infty .
\end{equation*}
As in \cite{Shen} we state the following result.
\begin{lemma}
\label{l11.2}Let $g\in L_{\infty ,AP}^{2}(\mathbb{R}^{d})\cap L^{\infty }(%
\mathbb{R}^{d})$ with $\left\langle g\right\rangle =0$ and
\begin{equation}
\sup_{x\in \mathbb{R}^{d}}\left(
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}(x)}\left\vert g\right\vert ^{2}\right) ^{\frac{1}{2}}\leq
C_{0}\left( \frac{T}{r}\right) ^{1-\sigma }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for }0<r\leq T
\label{11.15}
\end{equation}%
where $\sigma \in (0,1]$. Then there is a unique $u\in H_{\infty ,AP}^{1}(%
\mathbb{R}^{d})$ such that
\begin{equation}
-\Delta u+T^{-2}u=g\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d},\ \ \left\langle u\right\rangle
=0 \label{11.16}
\end{equation}%
and
\begin{equation}
T^{-2}\left\Vert u\right\Vert _{L^{\infty }(\mathbb{R}^{d})}+T^{-1}\left%
\Vert {\Greekmath 0272} u\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq C,
\label{11.17}
\end{equation}%
\begin{equation}
\left\vert {\Greekmath 0272} u(x)-{\Greekmath 0272} u(y)\right\vert \leq C_{\sigma }T^{1-\sigma
}\left\vert x-y\right\vert ^{\sigma }\ \ \forall x,y\in \mathbb{R}^{d}
\label{11.18}
\end{equation}%
where $C=C(d)$ and $C_{\sigma }=C_{\sigma }(d,\sigma )$. Moreover $u$ and $%
{\Greekmath 0272} u$ belong to $\mathcal{B}_{\infty ,AP}(\mathbb{R}^{d})$ with
\begin{equation}
T^{-2}\left\Vert u\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq C\Theta
_{1}(T) \label{11.19}
\end{equation}%
and
\begin{equation}
T^{-1}\left\Vert {\Greekmath 0272} u\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq
C\Theta _{\sigma }(T) \label{11.20}
\end{equation}%
where $\Theta _{\sigma }(T)$ is defined by \emph{(\ref{11.13})} and $%
C=C(d,\sigma ,g)$.
\end{lemma}
\begin{proof}
If we proceed as in the proof of Lemma \ref{l4.1}, we derive the existence
of a unique $u\in H_{\infty ,AP}^{1}(\mathbb{R}^{d})$ solving (\ref{11.16});
we may also refer to \cite{Po-Yu89} for another proof. Next using the
fundamental solution of $-\Delta +T^{-2}$, we easily get (\ref{11.17}). We
infer from (\ref{11.17}) that $u,{\Greekmath 0272} u\in \mathcal{B}_{\infty ,AP}(%
\mathbb{R}^{d})$. In order to obtain (\ref{11.18}) we use (\ref{11.15}) and
proceed as in \cite[Lemma 7.1]{Shen}. It remains to check (\ref{11.19}) and (%
\ref{11.20}). To that end, we apply (\ref{11.17}) to the function
\begin{equation*}
\frac{u(\cdot +y)-u(\cdot +z)}{\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert
_{L^{\infty }(B_{R})}}
\end{equation*}%
with $u$ solution of (\ref{11.16}). Then
\begin{equation}
T^{-2}\left\Vert u(\cdot +y)-u(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}\leq C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})} \label{11.21}
\end{equation}%
and
\begin{equation}
T^{-1}\left\Vert {\Greekmath 0272} u(\cdot +y)-{\Greekmath 0272} u(\cdot +z)\right\Vert
_{L^{\infty }(B_{R})}\leq C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert
_{L^{\infty }(B_{R})}. \label{11.22}
\end{equation}%
Using the boundedness of the gradient (see (\ref{11.17})), we obtain
\begin{equation}
\left\vert u(x)-u(t)\right\vert \leq CT\left\vert x-t\right\vert \ \ \forall
x,t\in \mathbb{R}^{d}. \label{11.23}
\end{equation}%
Next assuming that $\left\vert z\right\vert \leq L\leq T$, we have
\begin{eqnarray*}
T^{-2}\left\vert u(y)-u(0)\right\vert &\leq &T^{-2}\left\vert
u(y)-u(z)\right\vert +T^{-2}\left\vert u(z)-u(0)\right\vert \\
&\leq &C\left\Vert A(\cdot +y)-A(\cdot +z)\right\Vert _{L^{\infty
}(B_{R})}+CT^{-1}L
\end{eqnarray*}%
where we used (\ref{11.21}) and (\ref{11.23}). Hence
\begin{equation}
\sup_{y\in \mathbb{R}^{d}}T^{-2}\left\vert u(y)-u(0)\right\vert \leq C(\rho
(L,R)+T^{-1}L) \label{11.24}
\end{equation}%
for any $R>2T$ and $L>0$. Also, using the inequality
\begin{eqnarray*}
T^{-2}\left\vert u(0)\right\vert &\leq &T^{-2}\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}}(u(t)-u(0))dt\right\vert +T^{-2}\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}}u(t)dt\right\vert \\
&\leq &T^{-2}\sup_{y\in \mathbb{R}^{d}}\left\vert u(y)-u(0)\right\vert
+T^{-2}\left\vert
\mathchoice {{\setbox0=\hbox{$\displaystyle{\textstyle -}{\int}$ } \vcenter{\hbox{$\textstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\textstyle{\scriptstyle -}{\int}$ } \vcenter{\hbox{$\scriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptstyle{\scriptscriptstyle -}{\int}$
} \vcenter{\hbox{$\scriptscriptstyle -$
}}\kern-.6\wd0}}{{\setbox0=\hbox{$\scriptscriptstyle{\scriptscriptstyle
-}{\int}$ } \vcenter{\hbox{$\scriptscriptstyle -$ }}\kern-.6\wd0}}%
\!\int_{B_{r}}u(t)dt\right\vert
\end{eqnarray*}%
together with the fact that $\left\langle u\right\rangle =0$, we get (after
letting $r\rightarrow \infty $)
\begin{equation}
T^{-2}\left\vert u(0)\right\vert \leq C(\rho (L,R)+T^{-1}L)\ \ \forall \
0<L\leq T \label{11.25}
\end{equation}%
where we have also used (\ref{11.24}). Putting together (\ref{11.24}) and (%
\ref{11.25}), and choosing in the resulting inequality $R=3T$, and finally
taking the $\inf_{0<L<T}$, we are led to (\ref{11.19}).
Proceeding as above using this time (\ref{11.18}) and (\ref{11.22}) we
arrive at (\ref{11.20}).
\end{proof}
\begin{lemma}
\label{l5.2}Let $\chi _{T,j}$ be defined by \emph{(\ref{11.5})}, and let $%
\Omega $ be an open bounded set of class $\mathcal{C}^{1,1}$ in $\mathbb{R}%
^{d}$. Then
\begin{equation}
\int_{\Omega }\left\vert \left( {\Greekmath 0272} _{y}\chi _{T,j}\right) \left( \frac{x%
}{\varepsilon }\right) w(x)\right\vert ^{2}dx\leq C\int_{\Omega }(\left\vert
w\right\vert ^{2}+\delta ^{2}\left\vert {\Greekmath 0272} w\right\vert ^{2})dx,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\
all }w\in H^{1}(\Omega ) \label{5.30}
\end{equation}%
where $\delta =T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}%
^{d})}$ with $T=\varepsilon ^{-1}$, and $C=C(A,\Omega ,d)>0$.
\end{lemma}
\begin{proof}
By a density argument, it is sufficient to prove (\ref{5.30}) for $w\in
\mathcal{C}_{0}^{\infty }(\Omega )$. We recall that $\chi _{T,j}$ solves the
equation
\begin{equation}
-{\Greekmath 0272} \cdot (A(e_{j}+{\Greekmath 0272} \chi _{T,j}))+T^{-2}\chi _{T,j}=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }%
\mathbb{R}^{d}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.} \label{5.32}
\end{equation}%
Testing (\ref{5.32}) with $\psi (y)=\varphi (\varepsilon y)$ where $\varphi
\in H_{loc}^{1}(\mathbb{R}^{d})$ with compact support, and next making the
change of variable $x=\varepsilon y$, we get
\begin{equation*}
\int_{\mathbb{R}^{d}}\left[ (A^{\varepsilon }(e_{j}+({\Greekmath 0272} _{y}\chi
_{T,j})^{\varepsilon })\cdot {\Greekmath 0272} \varphi +T^{-2}\chi _{T,j}^{\varepsilon
}\varphi \right] dx=0
\end{equation*}%
where $u^{\varepsilon }(x)=u(x/\varepsilon )$ for $u\in H_{loc}^{1}(\mathbb{R%
}^{d})$. Choosing $\varphi (x)=\chi _{T,j}(x/\varepsilon )\left\vert
w(x)\right\vert ^{2}$ with $w\in \mathcal{C}_{0}^{\infty }(\Omega )$, we
obtain
\begin{equation*}
\int_{\Omega }\left[ (A^{\varepsilon }(e_{j}+({\Greekmath 0272} _{y}\chi
_{T,j})^{\varepsilon })\cdot \left( \frac{1}{\varepsilon }({\Greekmath 0272} _{y}\chi
_{T,j})^{\varepsilon }\left\vert w\right\vert ^{2}+2w\chi
_{T,j}^{\varepsilon }{\Greekmath 0272} w\right) +T^{-2}\left\vert \chi
_{T,j}^{\varepsilon }\right\vert ^{2}\left\vert w\right\vert ^{2}\right]
dx=0,
\end{equation*}%
or
\begin{eqnarray}
\int_{\Omega }A^{\varepsilon }({\Greekmath 0272} _{y}\chi _{T,j})^{\varepsilon }w\cdot
({\Greekmath 0272} _{y}\chi _{T,j})^{\varepsilon }wdx &=&-2\varepsilon \int_{\Omega
}A^{\varepsilon }({\Greekmath 0272} _{y}\chi _{T,j})^{\varepsilon }w\cdot \chi
_{T,j}^{\varepsilon }{\Greekmath 0272} wdx \label{5.31} \\
&&-\int_{\Omega }w(A^{\varepsilon }e_{j})\cdot ({\Greekmath 0272} _{y}\chi
_{T,j})^{\varepsilon }wdx \notag \\
&&-2\varepsilon \int_{\Omega }w(A^{\varepsilon }e_{j})\cdot \chi
_{T,j}^{\varepsilon }{\Greekmath 0272} wdx \notag \\
&&-\varepsilon T^{-2}\int_{\Omega }\left\vert \chi _{T,j}^{\varepsilon
}\right\vert ^{2}\left\vert w\right\vert ^{2}dx \notag \\
&=&I_{1}+I_{2}+I_{3}+I_{4}. \notag
\end{eqnarray}%
The left hand-side of (\ref{5.31}) is estimated from below by $\alpha
\int_{\Omega }\left\vert ({\Greekmath 0272} _{y}\chi _{T,j})^{\varepsilon }w\right\vert
^{2}dx$ while, for the respective terms of the right hand-side of (\ref{5.31}%
) we have, after the use of H\"{o}lder and Young inequalities,
\begin{equation*}
\left\vert I_{1}\right\vert \leq \frac{\alpha }{3}\int_{\Omega }\left\vert
({\Greekmath 0272} _{y}\chi _{T,j})^{\varepsilon }w\right\vert ^{2}dx+C\varepsilon
^{2}\int_{\Omega }\left\vert \chi _{T,j}^{\varepsilon }\right\vert
^{2}\left\vert {\Greekmath 0272} w\right\vert ^{2}dx;
\end{equation*}%
\begin{equation*}
\left\vert I_{2}\right\vert \leq \frac{\alpha }{3}\int_{\Omega }\left\vert
({\Greekmath 0272} _{y}\chi _{T,j})^{\varepsilon }w\right\vert ^{2}dx+C\int_{\Omega
}\left\vert w\right\vert ^{2}dx;
\end{equation*}%
\begin{equation*}
\left\vert I_{3}\right\vert \leq C\int_{\Omega }\left\vert w\right\vert
^{2}dx+C\varepsilon ^{2}\int_{\Omega }\left\vert \chi _{T,j}^{\varepsilon
}\right\vert ^{2}\left\vert {\Greekmath 0272} w\right\vert ^{2}dx\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }\left\vert
I_{4}\right\vert \leq C\int_{\Omega }\left\vert w\right\vert ^{2}dx.
\end{equation*}%
It follows that
\begin{eqnarray*}
\int_{\Omega }\left\vert ({\Greekmath 0272} _{y}\chi _{T,j})^{\varepsilon }w\right\vert
^{2}dx &\leq &C\varepsilon ^{2}\int_{\Omega }\left\vert \chi
_{T,j}^{\varepsilon }\right\vert ^{2}\left\vert {\Greekmath 0272} w\right\vert
^{2}dx+C\int_{\Omega }\left\vert w\right\vert ^{2}dx \\
&\leq &C\varepsilon ^{2}\left\Vert \chi _{T,j}\right\Vert _{L^{\infty }(%
\mathbb{R}^{d})}^{2}\int_{\Omega }\left\vert {\Greekmath 0272} w\right\vert
^{2}dx+C\int_{\Omega }\left\vert w\right\vert ^{2}dx.
\end{eqnarray*}%
Since $T=\varepsilon ^{-1}$, we get (\ref{5.30}), taking into account that $%
T^{-1}\left\Vert \chi _{T},j\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq
T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}$.
\end{proof}
\begin{remark}
\label{r5.2}I\emph{n the case of asymptotic periodic functions, we replace }$%
\chi _{T,j}$\emph{\ by }$\chi _{j}\in H_{\infty ,per}^{1}(Y)$\emph{\
solution of the corrector problem (\ref{1.6}) and we have (in view of Lemma %
\ref{l1.2}) }$\left\Vert \chi _{j}\right\Vert _{L^{\infty }(\mathbb{R}%
^{d})}\leq C$\emph{. It follows that }%
\begin{equation*}
\int_{\Omega }\left\vert \left( {\Greekmath 0272} _{y}\chi _{j}\right) \left( \frac{x}{%
\varepsilon }\right) w(x)\right\vert ^{2}dx\leq C\int_{\Omega }(\left\vert
w\right\vert ^{2}+\varepsilon ^{2}\left\vert {\Greekmath 0272} w\right\vert ^{2})dx,%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{\emph{\ for all }}w\in H^{1}(\Omega )
\end{equation*}%
\emph{where }$C=C(A,\Omega ,d)$\emph{.}
\end{remark}
Let $u_{0}\in H_{0}^{1}(\Omega )$ be the weak solution of (\ref{1.4}). Let $%
z_{\varepsilon }\in H^{1}(\Omega )$ be the unique weak solution of
\begin{equation}
-{\Greekmath 0272} \cdot (A^{\varepsilon }{\Greekmath 0272} z_{\varepsilon })=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\Omega
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }z_{\varepsilon }=\varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272} u_{0}%
\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ on }\partial \Omega \label{6.31}
\end{equation}%
where $\Omega $ is as in Lemma \ref{l5.2}. Then we have
\begin{lemma}
\label{l5.3}Let $z_{\varepsilon }$ be as in \emph{(\ref{6.31})} with $%
T=\varepsilon ^{-1}$. Then there exists $\varepsilon _{0}\in \lbrack 0,1)$
such that
\begin{equation}
\left\Vert z_{\varepsilon }\right\Vert _{H^{1}(\Omega )}\leq C\left(
T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\right)
^{\frac{1}{2}}\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )},\ 0<\varepsilon
\leq \varepsilon _{0}, \label{6.38}
\end{equation}%
where $C=C(A,\Omega )>0$.
\end{lemma}
It follows from (\ref{6.38}) that for any $\sigma \in (0,1)$, there exists $%
C_{\sigma }=C_{\sigma }(\sigma ,A,\Omega )>0$ such that
\begin{equation}
\left\Vert z_{\varepsilon }\right\Vert _{H^{1}(\Omega )}\leq C_{\sigma
}(\Theta _{\sigma }(\varepsilon ^{-1}))^{\frac{1}{2}}\left\Vert
u_{0}\right\Vert _{H^{2}(\Omega )},\ 0<\varepsilon \leq \varepsilon _{0}
\label{6.39}
\end{equation}%
where $\Theta _{\sigma }$ is defined by (\ref{11.13}).
For the proof of Lemma \ref{l5.3}, we need the following result whose proof
can be found in \cite{Suslina}.
\begin{lemma}[{\protect\cite[Lemma 5.1]{Suslina}}]
\label{l5.4}Let $\Omega $ be as in Lemma \emph{\ref{l5.2}}. Then there
exists $\delta _{0}\in (0,1]$ depending on $\Omega $ such that, for any $%
u\in H^{1}(\Omega )$,
\begin{equation}
\int_{\Gamma _{\delta }}\left\vert u\right\vert ^{2}dx\leq C\delta
\left\Vert u\right\Vert _{L^{2}(\Omega )}\left\Vert u\right\Vert
_{H^{1}(\Omega )}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }0<\delta \leq \delta _{0} \label{5.34}
\end{equation}%
where $C=C(\Omega )$ and $\Gamma _{\delta }=\Omega _{\delta }\cap \Omega $
with $\Omega _{\delta }=\{x\in \mathbb{R}^{d}:\mathrm{dist}(x,\partial
\Omega )<\delta \}$.
\end{lemma}
\begin{proof}[Proof of Lemma \protect\ref{l5.3}]
We set $w={\Greekmath 0272} u_{0}$ and $u=z_{\varepsilon }$. Assuming $u_{0}\in
H^{2}(\Omega )$, we have that $w\in H^{1}(\Omega )^{d}$. Since $\delta
:=T^{-1}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}%
^{d})}\rightarrow 0$ as $T\rightarrow \infty $, we may assume that $0<\delta
\leq \delta _{0}$ where $\delta _{0}$ is as in Lemma \ref{l5.4}. Let $\theta
_{\delta }$ be a cut-off function in a neighborhood of $\partial \Omega $
with support in $\Omega _{2\delta }$ (a $2\delta $-neighborhood of $\partial
\Omega $), $\Omega _{\rho }$ being defined as in Lemma \ref{l5.4}:
\begin{equation}
\theta _{\delta }\in \mathcal{C}_{0}^{\infty }(\mathbb{R}^{d})\RIfM@\expandafter\text@\else\expandafter\mbox\fi{,
\textrm{supp}}\theta _{\delta }\subset \Omega _{2\delta }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }0\leq
\theta _{\delta }\leq 1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }\theta _{\delta }=1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ on }\Omega
_{\delta }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, }\theta _{\delta }=0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ on }\mathbb{R}^{d}\backslash
\Omega _{2\delta }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }\delta \left\vert {\Greekmath 0272} \theta _{\delta
}\right\vert \leq C. \label{5.35}
\end{equation}%
We set $\Phi _{\varepsilon }(x)=\varepsilon \theta _{\delta }(x)\chi
_{T}(x/\varepsilon )w(x)$. Then
\begin{equation*}
\left\Vert u\right\Vert _{H^{1}(\Omega )}\leq C\varepsilon \left\Vert \chi
_{T}^{\varepsilon }w\right\Vert _{H^{1/2}(\partial \Omega )}\leq C\left\Vert
\Phi _{\varepsilon }\right\Vert _{H^{1}(\Omega )}.
\end{equation*}%
So we need to estimate $\left\Vert {\Greekmath 0272} \Phi _{\varepsilon }\right\Vert
_{L^{2}(\Omega )}$. But
\begin{eqnarray*}
{\Greekmath 0272} \Phi _{\varepsilon } &=&\varepsilon \chi _{T}^{\varepsilon }w{\Greekmath 0272}
\theta _{\delta }+({\Greekmath 0272} _{y}\chi _{T})^{\varepsilon }w\theta _{\delta
}+\varepsilon \chi _{T}^{\varepsilon }\theta _{\delta }{\Greekmath 0272} w \\
&=&J_{1}+J_{2}+J_{3}.
\end{eqnarray*}%
We have
\begin{eqnarray*}
\left\Vert J_{1}\right\Vert _{L^{2}(\Omega )}^{2} &\leq &C\varepsilon
^{2}\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}^{2}\delta
^{-2}\int_{\Gamma _{2\delta }}\left\vert w\right\vert ^{2}dx \\
&\leq &C\int_{\Gamma _{2\delta }}\left\vert w\right\vert ^{2}dx\leq C\delta
\left\Vert w\right\Vert _{L^{2}(\Omega )}\left\Vert w\right\Vert
_{H^{1}(\Omega )}
\end{eqnarray*}%
where we have used (\ref{5.34}) for the last inequality above. For $J_{2}$,
we have (using (\ref{5.30}) and (\ref{5.34}))
\begin{eqnarray*}
\left\Vert J_{2}\right\Vert _{L^{2}(\Omega )}^{2} &\leq &\int_{\Omega
}\left\vert ({\Greekmath 0272} _{y}\chi _{T})^{\varepsilon }w\theta _{\delta
}\right\vert ^{2}dx\leq C\int_{\Omega }\left( \left\vert w\theta _{\delta
}\right\vert ^{2}+\delta ^{2}\left\vert {\Greekmath 0272} (w\theta _{\delta
})\right\vert ^{2}\right) dx \\
&\leq &C\int_{\Gamma _{2\delta }}\left\vert w\right\vert ^{2}dx+C\delta
^{2}\int_{\Omega }\left\vert {\Greekmath 0272} (w\theta _{\delta })\right\vert ^{2}dx \\
&\leq &C\delta \left\Vert w\right\Vert _{L^{2}(\Omega )}\left\Vert
w\right\Vert _{H^{1}(\Omega )}+C\delta ^{2}\int_{\Omega }\left\vert {\Greekmath 0272}
(w\theta _{\delta })\right\vert ^{2}dx.
\end{eqnarray*}%
But ${\Greekmath 0272} (w\theta _{\delta })=w{\Greekmath 0272} \theta _{\delta }+\theta _{\delta
}{\Greekmath 0272} w$, and
\begin{eqnarray*}
\int_{\Omega }\left\vert {\Greekmath 0272} (w\theta _{\delta })\right\vert ^{2}dx &\leq
&C\int_{\Gamma _{2\delta }}\left\vert {\Greekmath 0272} \theta _{\delta }\right\vert
^{2}\left\vert w\right\vert ^{2}dx+C\int_{\Omega }\left\vert \theta _{\delta
}{\Greekmath 0272} w\right\vert ^{2}dx \\
&\leq &C\delta ^{-1}\left\Vert w\right\Vert _{L^{2}(\Omega )}\left\Vert
w\right\Vert _{H^{1}(\Omega )}+C\int_{\Omega }\left\vert {\Greekmath 0272} w\right\vert
^{2}dx.
\end{eqnarray*}%
Hence
\begin{equation*}
\left\Vert J_{2}\right\Vert _{L^{2}(\Omega )}^{2}\leq C\delta \left\Vert
w\right\Vert _{L^{2}(\Omega )}\left\Vert w\right\Vert _{H^{1}(\Omega
)}+C\delta ^{2}\left\Vert w\right\Vert _{H^{1}(\Omega )}^{2}.
\end{equation*}%
As for $J_{3}$,
\begin{equation*}
\left\Vert J_{3}\right\Vert _{L^{2}(\Omega )}^{2}\leq C\varepsilon
^{2}\int_{\Omega }\left\vert \chi _{T}^{\varepsilon }\right\vert
^{2}\left\vert {\Greekmath 0272} w\right\vert ^{2}dx\leq C\delta ^{2}\left\Vert
w\right\Vert _{H^{1}(\Omega )}^{2}.
\end{equation*}%
Finally, using Young's inequality together with the fact that $\delta
^{2}\leq \delta $ we are led to
\begin{eqnarray}
\left\Vert {\Greekmath 0272} \Phi _{\varepsilon }\right\Vert _{L^{2}(\Omega )}^{2}
&\leq &C\delta \left\Vert w\right\Vert _{L^{2}(\Omega )}\left\Vert
w\right\Vert _{H^{1}(\Omega )}+C\delta ^{2}\left\Vert w\right\Vert
_{H^{1}(\Omega )}^{2} \label{5.35'} \\
&\leq &C\delta \left\Vert w\right\Vert _{H^{1}(\Omega )}^{2}+C\delta
^{2}\left\Vert w\right\Vert _{H^{1}(\Omega )}^{2} \notag \\
&\leq &C\delta \left\Vert w\right\Vert _{H^{1}(\Omega )}^{2}. \notag
\end{eqnarray}%
So we choose $\varepsilon _{0}$ such that $0<\delta \leq \delta _{0}$ for $%
0<\varepsilon \leq \varepsilon _{0}$ (recall that $0<\delta \rightarrow 0$
as $0<\varepsilon \rightarrow 0$). We thus derive (\ref{6.38}) since $%
\left\Vert \Phi _{\varepsilon }\right\Vert _{L^{2}(\Omega )}^{2}\leq \delta
\left\Vert w\right\Vert _{H^{1}(\Omega )}^{2}$.
\end{proof}
\subsection{Convergence rates: proof of Theorem \protect\ref{t1.4}}
Assume that $\Omega $ is of class $\mathcal{C}^{1,1}$. Let $u_{\varepsilon }$%
, $u_{0}\in H_{0}^{1}(\Omega )$ be the weak solutions of (\ref{1.1}) and (%
\ref{1.4}) respectively. Let $\chi _{T}^{\varepsilon }(x)=\chi
_{T}(x/\varepsilon )$ for $x\in \Omega $ and define
\begin{equation}
w_{\varepsilon }=u_{\varepsilon }-u_{0}-\varepsilon \chi _{T}^{\varepsilon
}{\Greekmath 0272} u_{0}+z_{\varepsilon } \label{6.30}
\end{equation}%
where $T=\varepsilon ^{-1}$ and $z_{\varepsilon }\in H^{1}(\Omega )$ is the
weak solution of (\ref{6.31}).
\begin{theorem}
\label{t6.1}Suppose that $A$ is as in the preceding subsection. Assume that $%
u_{0}\in H^{2}(\Omega )$. Then for any $\sigma \in (0,1)$ there exists $%
C_{\sigma }=C_{\sigma }(\sigma ,A,\Omega )$ such that
\begin{equation}
\left\Vert w_{\varepsilon }\right\Vert _{H^{1}(\Omega )}\leq C_{\sigma
}\left( \left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon ^{-1}}\right\Vert
_{2}+\Theta _{\sigma }(\varepsilon ^{-1})\right) \left\Vert u_{0}\right\Vert
_{H^{2}(\Omega )}. \label{6.37}
\end{equation}
\end{theorem}
\begin{proof}
Set
\begin{equation*}
A_{T}=A+A{\Greekmath 0272} _{y}\chi _{T}-A^{\ast }
\end{equation*}%
where $A^{\ast }$ is the homogenized matrix and where we have taken $%
T=\varepsilon ^{-1}$. Then by simple computations as in Lemma \ref{l5.1} we
get
\begin{equation*}
-{\Greekmath 0272} \cdot \left( A^{\varepsilon }{\Greekmath 0272} w_{\varepsilon }\right) ={\Greekmath 0272}
\cdot \left( A_{T}^{\varepsilon }{\Greekmath 0272} u_{0}\right) +\varepsilon {\Greekmath 0272}
\cdot (A^{\varepsilon }{\Greekmath 0272} ^{2}u_{0}\chi _{T}^{\varepsilon }).
\end{equation*}%
This implies that
\begin{equation}
\left\Vert {\Greekmath 0272} w_{\varepsilon }\right\Vert _{L^{2}(\Omega )}\leq
C\left\Vert A_{T}^{\varepsilon }{\Greekmath 0272} u_{0}\right\Vert _{L^{2}(\Omega
)}+C\varepsilon \left\Vert A^{\varepsilon }{\Greekmath 0272} ^{2}u_{0}\chi
_{T}^{\varepsilon }\right\Vert _{L^{2}(\Omega )}. \label{10.6}
\end{equation}%
We use (\ref{11.14}) to get
\begin{eqnarray}
\varepsilon \left\Vert A^{\varepsilon }{\Greekmath 0272} ^{2}u_{0}\chi
_{T}^{\varepsilon }\right\Vert _{L^{2}(\Omega )} &\leq &C\varepsilon
\left\Vert \chi _{T}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\left\Vert
{\Greekmath 0272} ^{2}u_{0}\right\Vert _{L^{2}(\Omega )} \label{10.7} \\
&\leq &C\Theta _{\sigma }(T)\left\Vert {\Greekmath 0272} ^{2}u_{0}\right\Vert
_{L^{2}(\Omega )}. \notag
\end{eqnarray}%
Concerning the term $\left\Vert A_{T}^{\varepsilon }{\Greekmath 0272} u_{0}\right\Vert
_{L^{2}(\Omega )}$, we need to replace $A_{T}$ by a matrix $\mathcal{A}_{T}$
whose mean value is zero. So, we let $\mathcal{A}_{T}=A_{T}-\left\langle
A_{T}\right\rangle $ so that $\left\langle \mathcal{A}_{T}\right\rangle =0$
and $A_{T}^{\varepsilon }{\Greekmath 0272} u_{0}=\mathcal{A}_{T}^{\varepsilon }{\Greekmath 0272}
u_{0}+\left\langle A_{T}\right\rangle {\Greekmath 0272} u_{0}$. The inequality $%
\left\vert \left\langle A_{T}\right\rangle \right\vert \leq C\left\Vert
{\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{T}\right\Vert _{2}$ yields readily
\begin{equation}
\left\Vert \left\langle A_{T}\right\rangle {\Greekmath 0272} u_{0}\right\Vert
_{L^{2}(\Omega )}\leq C\left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{T}\right\Vert
_{2}\left\Vert {\Greekmath 0272} u_{0}\right\Vert _{L^{2}(\Omega )}. \label{10.8}
\end{equation}%
It remains to estimate $\left\Vert \mathcal{A}_{T}^{\varepsilon }{\Greekmath 0272}
u_{0}\right\Vert _{L^{2}(\Omega )}$. We denote by $a_{T,ij}$ the entries of $%
\mathcal{A}_{T}$: $a_{T,ij}=b_{T,ij}-\left\langle b_{T,ij}\right\rangle
\equiv a_{ij}$ where
\begin{equation*}
b_{T,ij}(y)=b_{ij}(y)+\sum_{k=1}^{d}b_{ik}(y)\frac{\partial \chi _{T,j}}{%
\partial y_{k}}(y)-b_{ij}^{\ast }.
\end{equation*}%
In view of Lemma \ref{l11.2}, let $f_{T,ij}\equiv f_{ij}\in H_{\infty
,AP}^{1}(\mathbb{R}^{d})$ be the unique solution of
\begin{equation*}
-\Delta f_{ij}+T^{-2}f_{ij}=a_{ij}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\mathbb{R}^{d},\ \ \left\langle
f_{ij}\right\rangle =0.
\end{equation*}%
Owing to (\ref{e5.7}), we see that $a_{ij}$ verifies (\ref{11.15}), so that (%
\ref{11.19}) and (\ref{11.20}) are satisfied, that is:
\begin{equation}
T^{-2}\left\Vert f_{ij}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\leq
C\Theta _{1}(T)\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }T^{-1}\left\Vert {\Greekmath 0272} f_{ij}\right\Vert
_{L^{\infty }(\mathbb{R}^{d})}\leq C\Theta _{\sigma }(T). \label{11.26}
\end{equation}%
We set $\mathbf{f}=(f_{ij})_{1\leq i,j\leq d}$. Then writing (formally)
\begin{equation*}
a_{ij}=-\sum_{k=1}^{d}\left( \frac{\partial }{\partial y_{k}}\left( \frac{%
\partial f_{ij}}{\partial y_{k}}-\frac{\partial f_{kj}}{\partial y_{i}}%
\right) +\frac{\partial }{\partial y_{i}}\left( \frac{\partial f_{kj}}{%
\partial y_{k}}\right) \right) +T^{-2}f_{ij}
\end{equation*}%
and using the fact that
\begin{equation*}
\sum_{i,k=1}^{d}\frac{\partial ^{2}}{\partial y_{i}\partial y_{k}}\left(
\frac{\partial f_{ij}}{\partial y_{k}}-\frac{\partial f_{kj}}{\partial y_{i}}%
\right) =0,
\end{equation*}%
we readily get
\begin{align}
-{\Greekmath 0272} \cdot (\mathcal{A}_{T}^{\varepsilon }{\Greekmath 0272} u_{0})& ={\Greekmath 0272} \cdot
\left( (\Delta \mathbf{f})^{\varepsilon }{\Greekmath 0272} u_{0}\right) -T^{-2}{\Greekmath 0272}
\cdot (\mathbf{f}^{\varepsilon }{\Greekmath 0272} u_{0}) \label{11.27} \\
& =\sum_{i,j,k=1}^{d}\frac{\partial }{\partial x_{i}}\left( \frac{\partial }{%
\partial x_{k}}\left( \frac{\partial f_{ij}}{\partial x_{k}}-\frac{\partial
f_{kj}}{\partial x_{i}}\right) \left( \frac{x}{\varepsilon }\right) \frac{%
\partial u_{0}}{\partial x_{j}}\right) \notag \\
& +\sum_{i,j,k=1}^{d}\frac{\partial }{\partial x_{i}}\left( \frac{\partial
^{2}f_{kj}}{\partial x_{k}\partial x_{i}}\left( \frac{x}{\varepsilon }%
\right) \frac{\partial u_{0}}{\partial x_{j}}\right) -T^{-2}{\Greekmath 0272} \cdot (%
\mathbf{f}^{\varepsilon }{\Greekmath 0272} u_{0}) \notag \\
& =-\sum_{i,j,k=1}^{d}\frac{\partial }{\partial x_{i}}\left( \varepsilon
\left( \frac{\partial f_{ij}}{\partial x_{k}}-\frac{\partial f_{kj}}{%
\partial x_{i}}\right) \left( \frac{x}{\varepsilon }\right) \frac{\partial
^{2}u_{0}}{\partial x_{k}\partial x_{j}}\right) \notag \\
& +\sum_{i,j,k=1}^{d}\frac{\partial }{\partial x_{i}}\left( \frac{\partial
^{2}f_{kj}}{\partial x_{k}\partial x_{i}}\left( \frac{x}{\varepsilon }%
\right) \frac{\partial u_{0}}{\partial x_{j}}\right) -T^{-2}{\Greekmath 0272} \cdot (%
\mathbf{f}^{\varepsilon }{\Greekmath 0272} u_{0}). \notag
\end{align}%
Testing (\ref{11.27}) with $\varphi \in H_{0}^{1}(\Omega )$, we obtain
\begin{eqnarray}
\left\Vert \mathcal{A}_{T}^{\varepsilon }{\Greekmath 0272} u_{0}\right\Vert
_{L^{2}(\Omega )} &\leq &C\varepsilon \left( \int_{\Omega }\left\vert {\Greekmath 0272}
\mathbf{f}\left( \frac{x}{\varepsilon }\right) \right\vert ^{2}\left\vert
{\Greekmath 0272} ^{2}u_{0}\right\vert ^{2}dx\right) ^{\frac{1}{2}} \label{6.35} \\
&&+C\sum_{j=1}^{d}\left( \int_{\Omega }\left\vert {\Greekmath 0272} h_{T,j}\left( \frac{%
x}{\varepsilon }\right) \right\vert ^{2}\left\vert {\Greekmath 0272} u_{0}\right\vert
^{2}dx\right) ^{\frac{1}{2}}+\left\vert \left\langle A_{T}\right\rangle
\right\vert \left\Vert {\Greekmath 0272} u_{0}\right\Vert _{L^{2}(\Omega )} \notag \\
&&+CT^{-2}\left( \int_{\Omega }\left\vert \mathbf{f}^{\varepsilon
}\right\vert ^{2}\left\vert {\Greekmath 0272} u_{0}\right\vert ^{2}dx\right) ^{\frac{1}{%
2}} \notag \\
&=&I_{1}+I_{2}+I_{3}+I_{4} \notag
\end{eqnarray}%
where $h_{T,j}=\sum_{k=1}^{d}\frac{\partial f_{kj}}{\partial y_{k}}\in
L_{\infty ,AP}^{2}(\mathbb{R}^{d})$. We estimate each term above separately.
Let us first deal with $I_{2}$. Observe that $h_{T,j}=\func{div}f_{.j}$
where $f_{.j}=(f_{kj})_{1\leq k\leq d}$. It follows from the definition of $%
f_{ij}$ that
\begin{equation*}
-\Delta f_{.j}+T^{-2}f_{.j}=A(e_{j}+{\Greekmath 0272} \chi _{T,j})-\left\langle
A(e_{j}+{\Greekmath 0272} \chi _{T,j})\right\rangle ,
\end{equation*}%
so that, owing to the definition of $\chi _{T,j}$,
\begin{equation}
-\Delta h_{T,j}+T^{-2}h_{T,j}=T^{-2}\chi _{T,j}. \label{6.36}
\end{equation}%
Next, since the function $g=T^{-1}\chi _{T,j}$ satisfies assumption (\ref%
{11.15}) of Lemma \ref{l11.2} with $\sigma =1$, it follows that $h_{T,j}$
satisfies estimate (\ref{11.20}), that is,
\begin{equation*}
T^{-1}\left\Vert {\Greekmath 0272} h_{T,j}\right\Vert _{L^{\infty }(\mathbb{R}%
^{d})}\leq C_{\tau }\Theta _{\tau }(T)\ \ \forall \tau \in (0,1).
\end{equation*}%
Therefore
\begin{equation*}
\left\vert I_{2}\right\vert \leq C\varepsilon \left\Vert {\Greekmath 0272}
h_{T,j}\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\left\Vert {\Greekmath 0272}
u_{0}\right\Vert _{L^{2}(\Omega )}\leq C\Theta _{\sigma }(T)\left\Vert
{\Greekmath 0272} u_{0}\right\Vert _{L^{2}(\Omega )}.
\end{equation*}
As regard $I_{1}$, we infer from (\ref{11.26}) that
\begin{equation*}
\left\vert I_{1}\right\vert \leq C\varepsilon \left\Vert {\Greekmath 0272} \mathbf{f}%
\right\Vert _{L^{\infty }(\mathbb{R}^{d})}\left\Vert {\Greekmath 0272}
^{2}u_{0}\right\Vert _{L^{2}(\Omega )}\leq C\Theta _{\sigma }(T)\left\Vert
{\Greekmath 0272} ^{2}u_{0}\right\Vert _{L^{2}(\Omega )}.
\end{equation*}%
Concerning $I_{4}$, we use the first inequality in (\ref{11.26}) to get
\begin{equation*}
\left\vert I_{4}\right\vert \leq C\Theta _{1}(T)\left\Vert {\Greekmath 0272}
u_{0}\right\Vert _{L^{2}(\Omega )}
\end{equation*}%
where we have put $T=\varepsilon ^{-1}$. Finally, using the inequality $%
\left\vert \left\langle A_{T}\right\rangle \right\vert \leq C\left\Vert
{\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{T}\right\Vert _{2}$ we get
\begin{equation*}
\left\vert I_{3}\right\vert \leq C\left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi
_{T}\right\Vert _{2}\left\Vert {\Greekmath 0272} u_{0}\right\Vert _{L^{2}(\Omega )}.
\end{equation*}%
The result follows thereby.
\end{proof}
We are now in a position to prove Theorem \ref{t1.4}.
\begin{proof}[Proof of Theorem \protect\ref{t1.4}]
Using (\ref{6.37}) together with (\ref{6.39}) we get, for any $\sigma \in
(0,1)$,
\begin{equation*}
\begin{array}{l}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi _{T=\varepsilon
^{-1}}^{\varepsilon }{\Greekmath 0272} u_{0}\right\Vert _{H^{1}(\Omega )} \\
\leq \left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi _{T=\varepsilon
^{-1}}^{\varepsilon }{\Greekmath 0272} u_{0}+z_{\varepsilon }\right\Vert _{H^{1}(\Omega
)}+\left\Vert z_{\varepsilon }\right\Vert _{H^{1}(\Omega )} \\
\leq C\left( \left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon
^{-1}}\right\Vert _{2}+\Theta _{\sigma }(\varepsilon ^{-1})\right)
\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}+C_{\sigma }(\Theta _{\sigma
}(\varepsilon ^{-1}))^{\frac{1}{2}}\left\Vert u_{0}\right\Vert
_{H^{2}(\Omega )} \\
\leq C\left( \left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon
^{-1}}\right\Vert _{2}+\left( \Theta _{1}(\varepsilon ^{-1})\right) ^{\sigma
}+\left( \Theta _{1}(\varepsilon ^{-1})\right) ^{\frac{\sigma }{2}}\right)
\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )} \\
\leq C\left( \left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon
^{-1}}\right\Vert _{2}+\left( \Theta _{1}(\varepsilon ^{-1})\right) ^{\sigma
}\right) ^{\frac{1}{2}}\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )},%
\end{array}%
\end{equation*}%
the last inequality above stemming from the fact that $\left\Vert {\Greekmath 0272}
\chi -{\Greekmath 0272} \chi _{\varepsilon ^{-1}}\right\Vert _{2}+\left( \Theta
_{1}(\varepsilon ^{-1})\right) ^{\sigma }\rightarrow 0$ when $\varepsilon
\rightarrow 0$, so that we may assume
\begin{equation*}
\left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon ^{-1}}\right\Vert
_{2}+\left( \Theta _{1}(\varepsilon ^{-1})\right) ^{\sigma }<1\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for
sufficiently small }\varepsilon .
\end{equation*}%
Choosing $\sigma =\frac{1}{2}$, we obtain
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi _{T=\varepsilon
^{-1}}^{\varepsilon }{\Greekmath 0272} u_{0}\right\Vert _{H^{1}(\Omega )}\leq C\left(
\left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon ^{-1}}\right\Vert
_{2}+\left( \Theta _{1}(\varepsilon ^{-1})\right) ^{\frac{1}{2}}\right) ^{%
\frac{1}{2}}\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}. \label{5.37''}
\end{equation}%
We recall that, since $\Omega $ is a $\mathcal{C}^{1,1}$-bounded domain in $%
\mathbb{R}^{d}$ and the matrix $A^{\ast }$ has constant entries, it holds
that
\begin{equation}
\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}\leq C\left\Vert f\right\Vert
_{L^{2}(\Omega )},\ C=C(d,\alpha ,\Omega )>0. \label{5.37'}
\end{equation}%
Next, set for $\varepsilon \in (0,1]$,
\begin{equation*}
\eta (\varepsilon )=\left( \left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon
^{-1}}\right\Vert _{2}+\left( \Theta _{1}(\varepsilon ^{-1})\right) ^{\frac{1%
}{2}}\right) ^{\frac{1}{2}}.
\end{equation*}%
Since $\eta (\varepsilon )\rightarrow 0$ as $\varepsilon \rightarrow 0$, we
obtain from (\ref{5.37''}) and (\ref{5.37'}), the statement of (\ref{Eq03})
in Theorem \ref{t1.4}.
It remains to check the near optimal convergence rates result (\ref{Eq02}).
We proceed in two parts.\medskip
\textit{Part I}. We first check that
\begin{equation}
\left\Vert u_{\varepsilon }\right\Vert _{H^{1}(\Gamma _{2\delta })}\leq
C\eta (\varepsilon )\left\Vert f\right\Vert _{L^{2}(\Omega )}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ where }%
\delta =\left( \eta (\varepsilon )\right) ^{2}. \label{5.37}
\end{equation}%
Indeed, we have $u_{\varepsilon }=(u_{\varepsilon }-u_{0}-\varepsilon \chi
_{T}^{\varepsilon }{\Greekmath 0272} u_{0})+u_{0}+\varepsilon \chi _{T}^{\varepsilon
}{\Greekmath 0272} u_{0}$, so that
\begin{equation*}
\left\Vert u_{\varepsilon }\right\Vert _{H^{1}(\Gamma _{2\delta })}\leq
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272}
u_{0}\right\Vert _{H^{1}(\Gamma _{2\delta })}+\left\Vert u_{0}\right\Vert
_{H^{1}(\Gamma _{2\delta })}+\left\Vert \varepsilon \chi _{T}^{\varepsilon
}{\Greekmath 0272} u_{0}\right\Vert _{H^{1}(\Gamma _{2\delta })}.
\end{equation*}%
It follows from (\ref{Eq03}) and (\ref{5.37'}) that
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272}
u_{0}\right\Vert _{H^{1}(\Gamma _{2\delta })}\leq C\eta (\varepsilon
)\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}\leq C\eta (\varepsilon
)\left\Vert f\right\Vert _{L^{2}(\Omega )}. \label{Eq04}
\end{equation}%
Using (\ref{5.34}) we obtain
\begin{equation}
\left\Vert u_{0}\right\Vert _{H^{1}(\Gamma _{2\delta })}\leq C\delta ^{\frac{%
1}{2}}\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}\leq C\delta ^{\frac{1}{2}%
}\left\Vert f\right\Vert _{L^{2}(\Omega )}. \label{Eq05}
\end{equation}%
To estimate $\left\Vert \varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272}
u_{0}\right\Vert _{H^{1}(\Gamma _{2\delta })}$, we consider a cut-off
function $\theta _{2\delta }$ of the same form as in (\ref{5.35}), but with $%
\delta $ replaced there by $2\delta $. Letting $w={\Greekmath 0272} u_{0}$, we observe
that $\varepsilon \chi _{T}^{\varepsilon }w=\varepsilon \theta _{2\delta
}\chi _{T}^{\varepsilon }w$ on $\Gamma _{2\delta }$, so that
\begin{equation*}
{\Greekmath 0272} (\varepsilon \chi _{T}^{\varepsilon }w)=\varepsilon \chi
_{T}^{\varepsilon }w{\Greekmath 0272} \theta _{2\delta }+({\Greekmath 0272} _{y}\chi
_{T})^{\varepsilon }w\theta _{2\delta }+\varepsilon \chi _{T}^{\varepsilon
}\theta _{2\delta }{\Greekmath 0272} w\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ on }\Gamma _{2\delta }.
\end{equation*}%
Following the same procedure as in the proof of Lemma \ref{l5.3}, we get
\begin{equation}
\left\Vert \varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272} u_{0}\right\Vert
_{H^{1}(\Gamma _{2\delta })}\leq C\delta ^{\frac{1}{2}}\left\Vert
u_{0}\right\Vert _{H^{2}(\Omega )}\leq C\delta ^{\frac{1}{2}}\left\Vert
f\right\Vert _{L^{2}(\Omega )}. \label{Eq06}
\end{equation}%
Choosing $\delta =\left( \eta (\varepsilon )\right) ^{2}$ in (\ref{Eq05})
and (\ref{Eq06}), and taking into account (\ref{Eq04}), we readily get (\ref%
{5.37}).\medskip
\textit{Part II}. Note that (\ref{6.37}) implies
\begin{equation}
\left\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272}
u_{0}+z_{\varepsilon }\right\Vert _{L^{2}(\Omega )}\leq C\left( \eta
(\varepsilon )\right) ^{2}\left\Vert f\right\Vert _{L^{2}(\Omega )}.
\label{5.38}
\end{equation}%
Thus, using the inequality
\begin{equation}
\left\Vert \varepsilon \chi _{T}^{\varepsilon }{\Greekmath 0272} u_{0}\right\Vert
_{L^{2}(\Omega )}\leq C\left( \Theta _{1}(\varepsilon ^{-1})\right) ^{\frac{1%
}{2}}\left\Vert u_{0}\right\Vert _{H^{2}(\Omega )}\leq C\left( \eta
(\varepsilon )\right) ^{2}\left\Vert f\right\Vert _{L^{2}(\Omega )},
\label{5.39}
\end{equation}%
we see that proving (\ref{Eq02}) amounts to prove that
\begin{equation}
\left\Vert z_{\varepsilon }\right\Vert _{L^{2}(\Omega )}\leq C\left( \eta
(\varepsilon )\right) ^{2}\left\Vert f\right\Vert _{L^{2}(\Omega )}
\label{5.40}
\end{equation}%
where $C=C(d,A,\Omega )$. To that end, we consider the function
\begin{equation}
v_{\varepsilon }=z_{\varepsilon }-\Phi _{\varepsilon }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{, where }\Phi
_{\varepsilon }=\varepsilon \theta _{\delta }\chi _{T}^{\varepsilon }{\Greekmath 0272}
u_{0}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }\delta =\left( \eta (\varepsilon )\right) ^{2}.
\label{5.41}
\end{equation}%
Then $v_{\varepsilon }\in H_{0}^{1}(\Omega )$ and $-{\Greekmath 0272} \cdot
(A^{\varepsilon }{\Greekmath 0272} v_{\varepsilon })=F_{\varepsilon }\equiv {\Greekmath 0272}
\cdot (A^{\varepsilon }{\Greekmath 0272} \Phi _{\varepsilon })$ in $\Omega $. As shown
in (\ref{5.35'}) (where we use the inequality (\ref{5.7'})), we have
\begin{equation}
\left\Vert {\Greekmath 0272} \Phi _{\varepsilon }\right\Vert _{L^{2}(\Omega )}\leq
C\delta ^{\frac{1}{2}}\left\Vert f\right\Vert _{L^{2}(\Omega )}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }%
\left\Vert \Phi _{\varepsilon }\right\Vert _{L^{2}(\Omega )}\leq C\delta
\left\Vert f\right\Vert _{L^{2}(\Omega )}. \label{5.42}
\end{equation}%
Now, let $F\in L^{2}(\Omega )$ be arbitrarily fixed, and let $t_{\varepsilon
}\in H_{0}^{1}(\Omega )$ be the solution of
\begin{equation}
-{\Greekmath 0272} \cdot (A^{\varepsilon }{\Greekmath 0272} t_{\varepsilon })=F\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ in }\Omega .
\label{5.43}
\end{equation}%
Following the homogenization process of (\ref{1.1}) (see the proof of
Theorem \ref{t1.1} in Section 2), we deduce the existence of a function $%
t_{0}\in H_{0}^{1}(\Omega )$ such that $t_{\varepsilon }\rightarrow t_{0}$
in $H_{0}^{1}(\Omega )$-weak and $t_{0}$ solves uniquely the equation $%
-{\Greekmath 0272} \cdot (A^{\ast }{\Greekmath 0272} t_{0})=F$ in $\Omega $. It follows from (\ref%
{5.37}) that
\begin{equation}
\left\Vert {\Greekmath 0272} t_{\varepsilon }\right\Vert _{L^{2}(\Gamma _{2\delta
})}\leq C\eta (\varepsilon )\left\Vert F\right\Vert _{L^{2}(\Omega )}.
\label{5.44}
\end{equation}%
Taking in the variational form of (\ref{5.43}) $v_{\varepsilon }$ test
function, we obtain
\begin{eqnarray}
\int_{\Omega }Fv_{\varepsilon }dx &=&\int_{\Omega }A^{\varepsilon }{\Greekmath 0272}
t_{\varepsilon }\cdot {\Greekmath 0272} v_{\varepsilon }dx=\int_{\Omega }{\Greekmath 0272}
t_{\varepsilon }\cdot A^{\varepsilon }{\Greekmath 0272} v_{\varepsilon }dx=\left(
F_{\varepsilon },t_{\varepsilon }\right) \label{5.45} \\
&=&-\int_{\Omega }A^{\varepsilon }{\Greekmath 0272} \Phi _{\varepsilon }\cdot {\Greekmath 0272}
t_{\varepsilon }dx=-\int_{\Gamma _{2\delta }}A^{\varepsilon }{\Greekmath 0272} \Phi
_{\varepsilon }\cdot {\Greekmath 0272} t_{\varepsilon }dx \notag
\end{eqnarray}%
where in (\ref{5.45}), the second equality stems from the fact that the
matrix $A$ is symmetric, and in the last equality we have used the
definition and properties of $\Phi _{\varepsilon }$. Hence, using together
(the first inequality in) (\ref{5.42}) and (\ref{5.44}), we are led to
\begin{eqnarray*}
\left\vert \int_{\Omega }Fv_{\varepsilon }dx\right\vert &\leq &C\left\Vert
{\Greekmath 0272} \Phi _{\varepsilon }\right\Vert _{L^{2}(\Omega )}\left\Vert {\Greekmath 0272}
t_{\varepsilon }\right\Vert _{L^{2}(\Gamma _{2\delta })}\leq C\delta ^{\frac{%
1}{2}}\left\Vert f\right\Vert _{L^{2}(\Omega )}\delta ^{\frac{1}{2}%
}\left\Vert F\right\Vert _{L^{2}(\Omega )} \\
&\leq &C\delta \left\Vert f\right\Vert _{L^{2}(\Omega )}\left\Vert
F\right\Vert _{L^{2}(\Omega )}.
\end{eqnarray*}%
Since $F$ is arbitrary, it emerges
\begin{equation}
\left\Vert v_{\varepsilon }\right\Vert _{L^{2}(\Omega )}\leq C\delta
\left\Vert f\right\Vert _{L^{2}(\Omega )}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }\delta =(\eta
(\varepsilon ))^{2}. \label{5.46}
\end{equation}%
Combining (\ref{5.46}) with the second estimate in (\ref{5.42}) yields (\ref%
{5.40}). This concludes the proof of Theorem \ref{t1.4}.
\end{proof}
\begin{remark}
\label{r5.3}\emph{In the asymptotic periodic setting of the preceding
section, we replace }$\chi _{T}$\emph{\ by }$\chi $\emph{\ so that }$%
\left\Vert {\Greekmath 0272} \chi -{\Greekmath 0272} \chi _{\varepsilon ^{-1}}\right\Vert _{2}=0$%
\emph{. Moreover, if we look carefully at the proof of (\ref{Eq02}), we
notice that, in view of Remark \ref{r5.2}, we may replace }$\eta
(\varepsilon )$\emph{\ by }$\varepsilon ^{1/2}$\emph{, so that (\ref{Eq02})
becomes }%
\begin{equation*}
\left\Vert u_{\varepsilon }-u_{0}\right\Vert _{L^{2}(\Omega )}\leq
C\varepsilon \left\Vert f\right\Vert _{L^{2}(\Omega )}
\end{equation*}%
\emph{where }$C=C(d,\alpha ,\Omega )$\emph{. This shows the optimal }$L^{2}$%
\emph{-rates of convergence in Theorem \ref{t5.1}.}
\end{remark}
\section{Some examples}
\subsection{Applications of Theorem \protect\ref{t3.2}}
Theorem \ref{t3.2} has been proved under the assumption that the corrector $%
\chi _{j}$ lies in $B_{\mathcal{A}}^{2}(\mathbb{R}^{d})$ for each $1\leq
j\leq d$. We provide some examples in which this hypothesis is fulfilled.
\subsubsection{\textbf{The almost periodic setting}}
We assume here that the entries of the matrix $A$ are almost periodic in the
sense of Besicovitch \cite{Besicovitch}. Then this falls into the scope of
Theorem \ref{t1.1} by taking there $\mathcal{A}=AP(\mathbb{R}^{d})$.
Now, we distinguish two special cases.
\textbf{Case 1}. The entries of $A$ are continuous quasi-periodic functions
and satisfy the frequency condition (see \cite{Jikov2}). We recall that a
function $b$ defined on $\mathbb{R}^{d}$ is quasi-periodic if $b(y)=\mathcal{%
B}(\omega _{1}\cdot y,...,\omega _{m}\cdot y)$ where $\mathcal{B}\equiv
\mathcal{B}(z_{1},...,z_{m})$ is a $1$-periodic function with respect to
every argument $z_{1}$,..., $z_{m}$. The $\omega ^{1},...,\omega ^{m}$ are
the frequency vectors, and $\omega _{j}\cdot y=\sum_{i=1}^{d}\omega
_{j}^{i}y_{i}$ is the inner product of vectors in $\mathbb{R}^{d}$. The
frequency condition on the vectors $\omega ^{1},...,\omega ^{m}\in \mathbb{R}%
^{d}$ amounts to the following assumption:
\begin{itemize}
\item[(\textbf{FC})] There is $c_{0},\tau >0$ such that
\begin{equation}
\left\vert \sum_{j=1}^{m}k_{j}\omega _{i}^{j}\right\vert \geq
c_{0}\left\vert k\right\vert ^{-\tau }\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for all }k\in \mathbb{Z}%
^{m}\backslash \{0\}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and }1\leq i\leq d. \label{FC}
\end{equation}
\end{itemize}
It is clear that if (\textbf{FC}) is satisfied, then the vectors $\omega
^{1},...,\omega ^{m}$ \ are rationally independent, that is,
\begin{equation*}
\sum_{j=1}^{m}k_{j}\omega _{i}^{j}\neq 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ for every }1\leq i\leq d\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
and all }k\in \mathbb{Z}^{m}\backslash \{0\}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{.}
\end{equation*}%
Then as shown in \cite[Lemma 2.1]{Jikov2}, the corrector problem (\ref{1.6})
possesses a solution which is quasi-periodic. So, it belongs to $B_{AP}^{2}(%
\mathbb{R}^{d})$ (the space $B_{\mathcal{A}}^{2}(\mathbb{R}^{d})$ with $%
\mathcal{A}=AP(\mathbb{R}^{d})$) since any quasi-periodic function is almost
periodic. We may hence apply Theorem \ref{t3.2}.
\textbf{Case 2}. The entries of $A$ are continuous almost periodic
functions. In \cite[Theorem 1.1]{Armstrong} are formulated the assumptions
implying the existence of bounded almost periodic solution to the problem (%
\ref{1.6}). Hence the conclusion of Theorem \ref{t3.2} holds. Notice that
this class of solutions contains continuous quasi-periodic ones (provided
that the assumptions of \cite[Theorem 1.1]{Armstrong} are satisfied) but
also some other almost periodic functions that are not quasi-periodic as
shown in \cite[Section 4]{Armstrong}.
\subsubsection{\textbf{The asymptotic periodic setting}}
We assume that $A=A_{0}+A_{per}$ where $A_{0}\in L^{2}(\mathbb{R}%
^{d})^{d\times d}$ and $A_{per}\in L_{per}^{2}(Y)^{d\times d}$. We are here
in the framework of asymptotic periodic homogenization corresponding to $%
\mathcal{A}=\mathcal{B}_{\infty ,per}(\mathbb{R}^{d})=\mathcal{C}_{0}(%
\mathbb{R}^{d})\oplus \mathcal{C}_{per}(Y)$. In the proof of Lemma \ref{l1.2}%
, we showed that the corrector lies in $L_{\infty ,per}^{2}(Y)=L_{0}^{2}(%
\mathbb{R}^{d})+L_{per}^{2}(Y)$, which is nothing else but the space $B_{%
\mathcal{A}}^{2}(\mathbb{R}^{d})$ with $\mathcal{A}=\mathcal{B}_{\infty
,per}(\mathbb{R}^{d})$. So Theorem \ref{t3.2} applies to this setting.
\begin{remark}
\label{r3.1}\emph{Assume (i) }$A=A_{0}+A_{ap}$ \emph{with }$A_{0}\in
\mathcal{C}_{0}(\mathbb{R}^{d})^{d\times d}$\emph{\ and }$A_{ap}\in AP(%
\mathbb{R}^{d})^{d\times d}$\emph{, (ii) the entries of }$A_{ap}$\emph{\
either are quasi-periodic and satisfy the frequency condition, or fulfill
the hypotheses of \cite[Theorem 1.1]{Armstrong}. we may use the same trick
as in Lemma \ref{l1.2} to show that the corrector lies, in each of these
cases, in }$B_{\infty ,AP}^{2}(\mathbb{R}^{d})=L_{0}^{2}(\mathbb{R}%
^{d})+B_{AP}^{2}(\mathbb{R}^{d})$\emph{. Therefore the conclusion of Theorem %
\ref{t3.2} holds true.}
\end{remark}
\subsection{Applications of Theorems \protect\ref{t1.4} and \protect\ref%
{t5.1}}
Here we give some concrete examples of functions for which Theorems \ref%
{t1.4} and \ref{t5.1} hold. Let $I_{d}$ denote the identity matrix in $%
\mathbb{R}^{d\times d}$.
\subsubsection{\textbf{The asymptotic periodic setting}}
We assume that $A=A_{0}+A_{per}$ where $A_{0}=b_{c}I_{d}$ with $%
b_{c}(y)=\exp (-c\left\vert y\right\vert ^{2})$ for any fixed $c>0$. $%
A_{per} $ is any continuous periodic symmetric matrix function satisfying
the ellipticity condition (\ref{2.2}). In the special $2$-dimension setting,
we may take $A_{0}=b_{1}I_{2}$ and
\begin{equation*}
A_{per}=\left(
\begin{array}{ll}
a_{1} & 0 \\
0 & a_{2}%
\end{array}%
\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }a_{1}(y)=4+\cos (2\pi y_{1})+\sin (2\pi y_{2}),\
a_{2}(y)=3+\cos (2\pi y_{1})+\cos (2\pi y_{2}).
\end{equation*}%
This special example is used for numerical tests in the next section.
\subsubsection{\textbf{The asymptotic almost periodic setting}}
As in the preceding subsection, we take $A_{0}=b_{c}I_{d}$ with $%
b_{c}(y)=\exp (-c\left\vert y\right\vert ^{2})$. We assume that $%
A=A_{0}+A_{ap}$ with $A_{ap}$ being any matrix with continuous almost
periodic entries such that $A$ satisfies hypothesis (\ref{1.2}). In the
special $2$-dimension setting used for numerical tests below, we take $%
A_{0}=b_{1}I_{2}$ and
\begin{equation*}
A_{ap}=\left(
\begin{array}{ll}
a_{1} & 0 \\
0 & a_{2}%
\end{array}%
\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }a_{1}(y)=4+\sin (2\pi y_{1})+\cos (\sqrt{2}\pi y_{2}),\
a_{2}(y)=3+\sin (\sqrt{3}\pi y_{1})+\cos (\pi y_{2}).
\end{equation*}
\section{Numerical simulations}
Our goal in this section is to check numerically the theoretical results
derived in the previous sections. We will consider the finite volume method
with two-point flux approximation. Of course multi-point flux approximation
can be considered when the matrix $A$ is non-diagonal. Even we will not
provide similar results for the discrete problem from numerical
approximation, similar results should normally be observed when the space
discretization step is small enough (fine grid) as the convergence of the
finite volume method for such elliptic problems is well known \cite{FV}.
\subsection{Finite volume methods}
The finite volume methods are widely applied when the differential equations
are in divergence form. To obtain a finite volume discretization, the domain
$\Omega $ is subdivided into subdomains $(K_{i})_{i\in \mathcal{I}},\;%
\mathcal{I}$ being the corresponding set of indices, called control volumes
or control domains such that the collection of all those subdomains forms a
partition of $\Omega $. The common feature of all finite volume methods is
to integrate the equation over each control volume $K_{i},\;i\in \mathcal{I}$
and apply Gauss's divergence theorem to convert the volume integral to a
surface integral. An advantage of the two-point approximation is that it
provides monotonicity properties, under the form of a local maximum
principle. It is efficient and mostly used in industrial simulations. The
main drawback is that finite volume method with two-point approximation is
applicable in the so called admissible mesh \cite{FV, Antonio3} and not in a
general mesh. This drawback has been filled by finite volume methods with
multi-point flux approximations \cite{B, H} which allow to handle anisotropy
in more general geometries.
For illustration, we consider the problem find $u\in H_{0}^{1}(\Omega )$
\begin{equation}
-{\Greekmath 0272} \cdot (A(x){\Greekmath 0272} u)=f\,\,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{in}\,\,\,\Omega . \label{pb1}
\end{equation}%
We assume that $f\in L^{2}(\Omega )$ and that $A$ is diagonal, so a
rectangular grid should be an admissible mesh \cite{FV, Antonio3}. Consider
an admissible mesh $\mathcal{T}$ with the corresponding control volume $%
(K_{i})_{i\in \mathcal{I}}$, we denote by $\mathcal{E}$ the set of edges of
control volumes of $\mathcal{T},\;\mathcal{E}_{int}$ the set of interior
edges of control volume of $\mathcal{T}$, $u_{i}$ the approximation of $u$
at the center (or at any point) of the control volume $K_{i}\in \mathcal{T}$
and $u_{\sigma }$ the approximation of $U$ at the center (or at any point)
of the edge $\sigma \in \mathcal{E}$. For a control volume $K_{i}\in
\mathcal{T}$, we denote by $\mathcal{E}_{i}$ the set of edges of $K_{i}$, so
that $\partial K_{i}=\underset{\sigma \in \mathcal{E}_{i}}{\bigcup }\sigma $.
We integrate \eqref{pb1} over any control volume $K_{i}\in \mathcal{T}$, and
use the divergence theorem to convert the integral over $K_{i}$ to a surface
integral,
\begin{equation*}
-\int_{\partial K_{i}}A(x){\Greekmath 0272} u\cdot \mathbf{n}_{i,\sigma
}ds=\int_{K_{i}}f(x)dx.
\end{equation*}%
To obtain the finite volume scheme with two-point approximation, the
following finite difference approximations are needed
\begin{eqnarray}
\underset{\sigma \in \mathcal{E}_{i}}{\sum }F_{i,\sigma } &\approx
&\int_{\partial K_{i}}A(x){\Greekmath 0272} u\cdot \mathbf{n}_{i,\sigma }ds \\
F_{i,\sigma } &=&-\mathrm{meas}(\sigma )\;C_{i,\sigma }\dfrac{u_{\sigma
}-u_{i}}{d_{i,\sigma }} \\
C_{i,\sigma } &=&|C_{K_{i}}\,\mathbf{n}_{i,\sigma }|,\quad A_{K_{i}}=\dfrac{1%
}{\mathrm{meas}(K_{i})}\int_{K_{i}}A(x)dx
\end{eqnarray}%
Here \ $\mathbf{n}_{i,\sigma }$ is the normal unit vector to $\sigma $
outward to $K_{i}$,
$\mathrm{meas}(\sigma )$ is the Lebesgue measure of the edge $\sigma \in
\mathcal{E}_{i}$ and $d_{i,\sigma }$ the distance between the center of $%
K_{i}$ and the edge $\sigma $. Since the flux is continuous at the interface
of two control volumes $K_{i}$ and $K_{j}$ (denoted by $i\mid j$) we
therefore have $F_{i,\sigma }=-F_{j,\sigma }$ for $\sigma =i\mid j$\footnote{%
interface of the control volumes $K_{i}$ and $K_{j}$}, which yields
\begin{equation*}
\left\{
\begin{array}{l}
F_{i,\sigma }=-\tau _{\sigma }\left( u_{j}-u_{i}\right) =-\dfrac{\mu
_{\sigma }\,\mathrm{meas}(\sigma )}{d_{i,j}}\left( u_{j}-u_{i}\right)
,\,\sigma =i\mid j\quad \newline
\\
\tau _{\sigma }=\mathrm{meas}(\sigma )\dfrac{C_{i,\sigma }C_{j,\sigma }}{%
C_{i,\sigma }d_{i,\sigma }+C_{j,\sigma }d_{j,\sigma }}\quad (\RIfM@\expandafter\text@\else\expandafter\mbox\fi{%
transmissibility through}\,\sigma )%
\end{array}%
\right.
\end{equation*}%
with
\begin{equation*}
\mu _{\sigma }=d_{i,j}\dfrac{C_{i,\sigma }C_{j,\sigma }}{C_{i,\sigma
}d_{i,\sigma }+C_{j,\sigma }d_{j,\sigma }},
\end{equation*}%
where $d_{i,j}$ is the distance between the center of $K_{i}$ and center of $%
K_{j}$. We will set $d_{i,j}=d_{i,\sigma }$ for $\sigma =\mathcal{E}_{i}\cap
\partial \Omega $. For $\sigma \subset \partial \Omega $ ($\sigma \notin
\mathcal{E}_{int}$ ), we also write
\begin{eqnarray*}
F_{i,\sigma } &=&-\tau _{\sigma }\left( u_{\sigma }-u_{i}\right) \\
&=&-\dfrac{\mathrm{meas}(\sigma )\mu _{\sigma }}{d_{i,\sigma }}\left(
u_{\sigma }-u_{i}\right) .
\end{eqnarray*}%
The finite volume discretization is therefore given by
\begin{eqnarray}
\underset{\sigma \in \mathcal{E}_{i}}{\sum }F_{i,\sigma } &=&f_{K_{i}}
\label{ode} \\
f_{K_{i}} &=&\int_{K_{i}}f(x)dx
\end{eqnarray}%
Let $h=~$size$(\mathcal{T})=\underset{i\in \mathcal{I}}{\sup }\underset{%
(x,y)\in K_{i}^{2}}{\sup }|x-y|$ be the maximum size of $\mathcal{T}$. We
set $u_{h}=(u_{i})_{i\in \mathcal{I}}$, $N_{h}=|\mathcal{I}|$ and $%
F=(f_{K_{i}})_{i\in \mathcal{I}}+bc$ , $bc$ being the contribution of the
boundary condition \footnote{%
Here $bc$ is null as we are looking for solution in $H_{0}^{1}(\Omega )$}.
Applying \eqref{ode} through all control volumes, the corresponding finite
volume scheme is given by
\begin{equation}
A_{h}u_{h}=F, \label{fv}
\end{equation}%
where $A_{h}$ is an $N_{h}\times N_{h}$ matrix. The structure of $A_{h}$
depends of the dimension $d$ and the geometrical shape of the control
volume. For diagonal $A$, if $\Omega $ is a rectangular or parallelepiped
domain, any rectangular grid ($d=2$) or parallelepiped grid ($d=3$) is an
admissible mesh and yields a 5-point scheme ($d=2$) or 7-point scheme ($d=3$%
) for the problem \eqref{pb1}. To solve efficiently the linear system %
\eqref{fv}, we have used the Matlab linear solver bicgstab with ILU(0)
preconditioners.
\subsection{Simulations in dimension 2}
\subsubsection{The Asymptotic periodic setting}
\label{ssect5} For the numerical tests, we consider problems (\ref{1.1}) and
(\ref{1.4}) in dimension $d=2$ with the finite volume method scheme %
\eqref{fv}. We denote by $I_{d}$ the square identity matrix in $\mathbb{R}%
^{d\times d}$. We take $A=A_{0}+A_{per}$ with
\begin{align*}
A_{0}& =b_{0}I_{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }b_{0}(x_{1},x_{2})=\exp
(-(x_{1}^{2}+x_{2}^{2}))\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and } \\
A_{per}& =\left(
\begin{array}{ll}
b_{1} & 0 \\
0 & b_{2}%
\end{array}%
\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }b_{1}=4+\cos (2\pi x_{1})+\sin (2\pi x_{2}),\
b_{2}=3+\cos (2\pi x_{1})+\cos (2\pi x_{2}).
\end{align*}%
The right-hand side function $f$ is given by $f=1$.
The computational domain is $\Omega =(-1,1)^{2}$.
We take $\varepsilon =1/N$ for some integer $N$. We will choose $N$ in the
set $\{2,3,4,5,6\}$.
The aim in this section is to compute numerically the \textquotedblright
exact solution\textquotedblright\ $u_{\varepsilon }$ (for a fixed $%
\varepsilon >0$) coming from the finite volume scheme with small $h$, and
compare it with its first order asymptotic periodic approximation $%
v_{\varepsilon }(x)=u_{0}(x_{1},x_{2})+\varepsilon \chi (\frac{x_{1}}{%
\varepsilon },\frac{x_{2}}{\varepsilon })\cdot {\Greekmath 0272} u_{0}(x_{1},x_{2})$.
For this purpose, the strategy is carried out as follows:
\begin{enumerate}
\item We compute the exact solution of (\ref{1.1}) with our finite volume
scheme on a rectangular fine mesh of size $h>0$, with $h$ sufficiently small
to ensure that the discretization error is much smaller than $\varepsilon$,
which is the order of the error associated to the homogenization
approximation (see either Proposition \ref{p5.1} or Theorem \ref{t5.1}).
\item We compute the corrector functions $\chi_{1}$ and $\chi_{2}$
associated to the respective directions $e_{1}=(1,0)$ and $e_{2}=(0,1)$. To
this end, we rather consider their approximations by the finite volume
scheme \eqref{fv}, which are solutions to Eq. (\ref{3.3}), and we perform
this computation on the domain $Q_{6}=(-6,6)^{2}$ with Dirichlet boundary
conditions (as in (\ref{3.3})). We also compute their gradients $%
{\Greekmath 0272}\chi_{1}$ and ${\Greekmath 0272}\chi_{2}$. Here we take the mesh size $h=8\times
10^{-3}$ independent of $\varepsilon$.
\item With ${\Greekmath 0272} \chi _{1}$ and ${\Greekmath 0272} \chi _{2}$ computed as above, we
compute the homogenized matrix $A_{6}^{\ast }$ as in (\ref{eq5}), namely
\begin{equation*}
A_{6}^{\ast }=\left( \frac{1}{12}\right) ^{2}\int_{Q_{6}}A(x)(I_{2}+{\Greekmath 0272}
\chi (x))dx
\end{equation*}%
where here, $\chi =(\chi _{1},\chi _{2})$ so that ${\Greekmath 0272} \chi $ is the
square matrix with entries $c_{ij}=\frac{\partial \chi _{j}}{\partial x_{i}}$%
.
\item With $A_{6}^{\ast }$ now being denoted by $A^{\ast }$, we compute the
exact solution $u_{0}$ of (\ref{1.4}).
\item Finally we compute the first order approximation $v_{%
\varepsilon}(x)=u_{0}(x)+\varepsilon \chi (x/\varepsilon )\cdot {\Greekmath 0272}
u_{0}(x)$ and we compare it to the exact solution $u_{\varepsilon }$, which
has been computed at step 1.
\end{enumerate}
The goal is to check the convergence result in Theorem \ref{t5.1} given by %
\eqref{5.8}, but with the numerical solution using finite volume method.
Indeed we want to evaluate the following error
\begin{equation}
Err(\varepsilon )=\dfrac{\Vert u_{\varepsilon }-u_{0}-\varepsilon \chi
^{\varepsilon }{\Greekmath 0272} u_{0}\Vert _{H^{1}(\Omega )}}{\Vert u_{0}\Vert
_{H^{2}(\Omega )}}=\dfrac{\Vert u_{\varepsilon }-v_{\varepsilon }\Vert
_{H^{1}(\Omega )}}{\Vert u_{0}\Vert _{H^{2}(\Omega )}}. \label{error}
\end{equation}%
As we already mentioned, $u_{0}$, $u_{\varepsilon }$ and $v_{\varepsilon }$
are computed numerical using the finite volume scheme for a fixed $h=8\times
10^{-3}$ independent of a fixed $\varepsilon $. All the norms involved in %
\eqref{error} are computed using their discrete forms \cite{FV, Antonio3}.
The coefficients of $A$ and $f$ are $\mathcal{C}^{\infty }(\Omega )$, so the
corresponding solutions $u_{0}$, $u_{\varepsilon }$ and $v_{\varepsilon }$
should be regular enough. Their graphs are given in Figure \ref{FIG03}. As
we can observe in Table \ref{phi}, the error decreases when $\varepsilon $
decreases, and therefore the convergence of $u_{\varepsilon }$ and $%
v_{\varepsilon }$ towards $u_{0}$ when $\varepsilon \rightarrow 0$ is
ensured. We can also observe that the corrector plays a key role as graph of
$u_{\varepsilon }$ is close to the one of $v_{\varepsilon }$. The numerical
value of $A_{6}^{\ast }\equiv A^{\ast }$ obtained and used for $u_{0}$ and $%
v_{\varepsilon }$ is given by
\begin{equation*}
A_{6}^{\ast }=\left(
\begin{array}{ll}
3.895923 & 0.00001 \\
0 & 2.849959%
\end{array}%
\right) .
\end{equation*}
\begin{table}[h!]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$1/\varepsilon $ & 2 & 3 & 4 & 5 & 6 \\ \hline
Err$(\varepsilon )$ & 0.5298 & 0.1382 & 0.0620 & 0.0577 & 0.0573 \\ \hline
\end{tabular}%
\end{center}
\caption{ $Err(\protect\varepsilon )$ with the corresponding $1/\protect%
\varepsilon $ for a fixed $h=2\times 10^{-3}$ independent of a fixed $%
\protect\varepsilon $.}
\label{phi}
\end{table}
\begin{figure}[h!]
\subfigure[]{
\label{FIG03a}
\includegraphics[width=0.45\textwidth]{nue.jpg}} \hskip 0.01\textwidth
\subfigure[]{
\label{FIG03b}
\includegraphics[width=0.45\textwidth]{nu0.jpg}} \hskip 0.01\textwidth
\subfigure[]{
\label{FIG03c}
\includegraphics[width=0.45\textwidth]{nve.jpg}} \hskip 0.01\textwidth
\subfigure[]{
\label{FIG03d}
\includegraphics[width=0.45\textwidth]{nuemu0.jpg}}
\subfigure[]{
\label{FIG03e}
\includegraphics[width=0.45\textwidth]{nuemve.jpg}}
\caption{The graphs of $u_{\protect\varepsilon }$, $u_0$, $v_{\protect%
\varepsilon }$, $\vert u_{\protect\varepsilon}-u_0\vert $ and $\vert u_{%
\protect\varepsilon}-v_{\protect\varepsilon} \vert$ in the asymptotic
periodic setting, are shown in (a), (b), (c), (d) and (e) respectively for $%
\protect\varepsilon=1/6$ and $h=2\times10^{-3}$.}
\label{FIG03}
\end{figure}
\subsubsection{The asymptotic almost periodic setting}
Here we take $A=A_{0}+A_{ap}$ with
\begin{align*}
A_{0}& =b_{0}I_{2}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }b_{0}(x_{1},x_{2})=\exp
(-(x_{1}^{2}+x_{2}^{2}))\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and } \\
A_{ap}& =\left(
\begin{array}{ll}
b_{1} & 0 \\
0 & b_{2}%
\end{array}%
\right) \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ with }b_{1}=4+\sin (2\pi x_{1})+\cos (\sqrt{2}\pi x_{2}),\
b_{2}=3+\sin (\sqrt{3}\pi x_{1})+\cos (\pi x_{2}).
\end{align*}%
The right-hand side function $f$ is given by $f(x_{1},x_{2})=\cos (\pi
x_{1})\cos (\sqrt{5}\pi x_{2})$. The computational domain is as above, that
is, $\Omega =(-1,1)^{2}$. We follow the same steps as above. The
corresponding value of $A_{6}^{\ast }$ is
\begin{equation*}
A_{6}^{\ast }=\left(
\begin{array}{ll}
4.0118 & 0.0002 \\
0.0032 & 3.0206%
\end{array}%
\right) .
\end{equation*}%
We solve (\ref{1.4}) using finite volume method with multi-point flux
approximation \cite{B, H}. From Table \ref{phi2} and Figure \ref{FIG033}, we
can draw the same conclusion as in Section \ref{ssect5}.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$1/\varepsilon $ & 2 & 3 & 4 & 5 & 6 \\ \hline
Err$(\varepsilon )$ & 0.24 & 0.1520 & 0.1284 & 0.0768 & 0.0265 \\ \hline
\end{tabular}%
\end{center}
\caption{ $Err(\protect\varepsilon )$ with the corresponding $1/\protect%
\varepsilon $ for a fixed $h=2\times 10^{-3}$ independent of a fixed $%
\protect\varepsilon $.}
\label{phi2}
\end{table}
\begin{figure}[h!]
\subfigure[]{
\label{FIG03a}
\includegraphics[width=0.45\textwidth]{naue.jpg}} \hskip 0.01\textwidth
\subfigure[]{
\label{FIG03b}
\includegraphics[width=0.45\textwidth]{nau0.jpg}} \hskip 0.01\textwidth
\subfigure[]{
\label{FIG03b}
\includegraphics[width=0.45\textwidth]{nave.jpg}} \hskip 0.01\textwidth
\subfigure[]{
\label{FIG03b}
\includegraphics[width=0.45\textwidth]{nauemu0.jpg}}
\subfigure[]{
\label{FIG03b}
\includegraphics[width=0.45\textwidth]{nauemve.jpg}}
\caption{The graphs of $u_{\protect\varepsilon }$, $u_0$, $v_{\protect%
\varepsilon }$ $\vert u_{\protect\varepsilon}-u_0\vert $ and $\vert u_{%
\protect\varepsilon}-v_{\protect\varepsilon} \vert$ in the asymptotic almost
periodic setting, are shown in (a), (b), (c), (d) and (e) respectively for $%
\protect\varepsilon=1/6$ and $h=2\times10^{-3}$.}
\label{FIG033}
\end{figure}
\begin{acknowledgement}
\emph{The work of the second author has been supported by Robert Bosch
Stiftung through the AIMS ARETE chair programme (Grant No 11.5.8040.0033.0)
while the work of the third author has been carried out under the support of
the Alexander von Humboldt Foundation. They gratefully acknowledge the two
Foundations.}
\end{acknowledgement}
\section*{Abstract (Not appropriate in this style!)}%
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\[email protected]
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|
1,116,691,498,435 | arxiv | \section{Kitaev Chain Example}
We study the model as introduced in the main text, with $V = e^{-3}$. We drive the system with the unitary $P_X = \prod_i \sigma^i_x$. In the majorana formalism, this corresponds to the operator
\begin{equation}
P_X = \gamma_{2} \gamma_{3} \gamma_{6} \gamma_{7} ... = \prod_i \gamma_{4i-2} \gamma_{4i-1}
\end{equation}
One can easily confirm that the Hamiltonian $H_{0} = (H + P_X H P_X)/2$ which is invariant under $P_X$ exactly has weak and strong alternating bonds between nearest-neighbor majorana fermions. It is important to note that an imperfect weakening of the odd set of bonds is \emph{sufficient} to drive the majorana system into the Kitaev phase (provided interactions are weak enough). In other words, provided the $\mathcal{Z}_2$ fermion parity is perfectly conserved, one does not need to introduce additional symmetries to enter the Kitaev phase.
\begin{center}
\begin{figure}[ht]
\includegraphics[width=3.5in]{kitaev_vs_L.pdf}
\caption{Coherence of majorana fermion at the edge of the wire improves exponentially with increasing $L$ as expected in a Kitaev chain, but not with increasing $n_f$. Solid (dotted) lines correspond to $n_f = 1 (2)$.}
\label{fig:appfig1}
\end{figure}
\end{center}
\begin{center}
\begin{figure}[ht]
\includegraphics[width=3.5in]{kitaev_x.pdf}
\caption{Coherence of $P_X$, $x(t)$ improves with increasing $n_f$ in the same model as Fig.~\ref{fig:appfig2}.}
\label{fig:appfig2}
\end{figure}
\end{center}
The simplest demonstration of the above fact comes from numerical results of Figs.~\ref{fig:appfig1} and~\ref{fig:appfig2}. In the first figure, we examine the coherence of the majorana at one end of the system. There is initial relaxation that occurs on the microscopic time scales followed by a long plateau where coherence is maintained, before an eventual relaxation. As seen in Fig.~\ref{fig:appfig1}, the relaxation of the coherence occurs on a time scale that depends exponentially on length $L$ (equal linear displacements of the relaxation time scale can be seen on the log scale as $L$ is increased in fixed steps). In the same plot, dotted lines correspond to the protocol carried out at $n_f = 2$, while solid lines correspond to $n_f = 1$. In Fig.~\ref{fig:appfig2}, we plot the coherence $x(t)$ corresponding to the operator $P_X$. Clearly, this coherence is improved as $n_f$ is increased. It is important to recall that the protocol engineers a particular symmetry (in this case $P_X$) and there is an optimal number of fractal layers $n_f$ associated with it. Phases that rely on the preservation of said symmetry will be more robust the better this symmetry is implemented. In this particular instance, the creation of isolated majorana modes relies on the system being in the correct phase (with alternating weak odd and strong even bonds) and not on the engineering of additional global symmetries (besides the fermion parity which we assume to be conserved). Thus, the coherence of the majoranas only weakly depends on the fractal layers $n_f$.
\section{Comments on Numerical Methods}
Time evolution has been performed using either exact diagonalization or recursive multiplication to get the unitary for time-evolution at exponentially long times.
In the first method, we diagonalize $U(T_f)$. Subsequently, all operators are represented in the basis that diagonalizes $U(T_f)$. In this basis, the phase picked up by each individual Floquet eigenstate can be determined at time easily, and we do so at times that are uniformly spread out on a log scale. Coherences are evaluated as mentioned in the main text, averaging over a complete basis of initial states, except in the case of heating, where the initial state is the ground state of the Hamiltonian at zeroth order of the Magnus expansion, $H_{\vec{0}}$. This method usually works well except in instances where there is massive degeneracy in the spectrum; this makes precise basis rotation hard to compute when diagonalizing $U(T_f)$. In the example studied in the main text, this unfortunately limits us to studying time-evolution for $L = 4, 8, 12$ as $L = 6, 10, ...$ are degenerate, which further limits our ability to perform finite size scaling in a meaningul way with system sizes accessible.
In the second method, which we use in instances where the spectrum has degeneracies (applicable to the Kitaev chain considered in the previous section), we simply compute unitary matrices for time-evolution at longer times by the recursion relation $U(2t) = U(t) \cdot U(t)$, for $t > T_f$. This resolves the issues with exact diagonalization in the presence of degeneracies, but is slower and suffers from the issue of the time-evolution matrix at late times becoming less unitary. This limits the dynamics to shorter times.
\end{document}
|
1,116,691,498,436 | arxiv | \section{INTRODUCTION}
Although fission barrier heights $B_{f}$ are not directly measurable
quantities, i.e. are not quantum observables, they are very useful
in estimating nuclear fission rates. As the activation energy $E_a$ (per mole)
in chemistry gives a rate $k$ of a chemical reaction at temperature $T$ via
the Arrhenius law: $k=Ae^{-E_a/RT}$ ($R$ - the gas constant; $A$ - the
frequency factor) \cite{tHoff,Arrhe}, the fission barrier gives the fission
rate $\Gamma_f$ of an excited (as they usually are in nuclear reactions)
nucleus via:
$\Gamma_f\sim e^{-B_f/kT_{eff}}$, where $T_{eff}$ is an effective temperature
derived from the excitation energy, and $k$ - the Boltzman constant.
For example, knowing fission barriers of possible fusion products
helps predicting a cross section for a production of a given evaporation
residue in a heavy ion reaction: one can figure out whether
neutron or alpha emission wins a competition with fission at each
stage of the deexcitation of a compound nucleus.
Moreover, one can try to understand the experimentally established,
intriguing growth of the total cross sections around Z=118; for its
correlation with $B_f$, see e.g. Fig. 6 and the related discussion in
\cite{O1}.
On the other hand, the prediction of the spontaneous or low energy (i.e. from
a weakly excited state) fission rates, governed by the regime of the
collective quantum tunneling, requires an additional knowledge of
the barrier shape and mass parameters.
A non-observable status of the fission barrier, again in analogy to that of
the activation energy in chemistry, is reflected in its possible dependence
on a reaction type and/or the excitation energy (effective temperature) range.
This leads to some uncertainty in calculations of fission barriers. In
particular, it is not clear whether intrinsic configurations should be
conserved along the level crossings, which increases $B_f$, or the adiabatic
state should be followed. This is especially relevant for odd-$A$ and
odd-odd nuclei, in which sharp crossings of levels occupied by the
odd particle exclude the strictly adiabatic scenario. It is known that if the
projection of the single-particle angular momentum on the symmetry axis of a
nucleus $\Omega$ is conserved, the diabatic effect on the fission barrier can
be huge, see e.g. \cite{Kisomers}. As there is no accepted formula for a
barrier correction due to the non-adiabaticity, it is usually ignored, even in
odd-$N$ and/or odd-$Z$ nuclei.
A general idea is that at the excitation energies close to, and higher than
the barrier, but still not inducing sizable dissipative corrections,
the adiabatic barrier could be used for calculating fission rates.
Since calculations of potential energy surfaces (PES's) for odd-$A$ and
odd-odd nuclei involve a repetition of calculations for many low-lying
quasiparticle states which multiplies the effort (especially in odd-odd
systems), systematic studies of their fission barriers are rather scarce.
Up to now, they were provided mainly by the Los Alamos microscopic-macroscopic
(MM) model and recently by some self-consistent models \cite{Gorbar}.
The current state of theoretical predictions in fission of even - even nuclei
(with Z$\geq$ 100) has been discussed recently in \cite{nucl2015}.
In the present paper we extend our MM model based
on the deformed Woods-Saxon potential, which up to now was applied mainly
to even-even nuclei \cite{Kow}, to odd-$A$ and odd-odd SH systems.
We study a wide range of isotopes which, perhaps, may be of some use for
astrophysical purposes. The fission barriers are calculated using the
adiabatic assumption, i.e. they are the smallest possible.
Since the model has been quite reasonable, in particular in reproducing
first \cite{Kow} and second \cite{IIbarriers} fission barriers in actinides,
as well as super- \cite{kowskal} and hyper-deformed \cite{IIIbarriers1,IIIbarriers2} minima, we prefer to keep its
parameters unchanged. The shell and pairing correction for an odd nucleon
system is done by blocking the lowest-lying quasiparticle states.
The modification of the macroscopic energy by including the average
pairing energy contribution which we introduced for nuclear masses
in \cite{JachKowSkal2014} is irrelevant for fission barriers.
The other motivation of our study is to improve the predictions for the
fission saddles. This requires simultaneously taking into account
a large number of shape variables \cite{IIbarriers,IIIbarriers2} and relying
on an {\it in principle} exact method for finding saddles to escape
errors inherent in the mostly used constrained minimization method, see
\cite{Moller2009,Dubray}.
As usual, to make the involved computational effort manageable one has to
make some compromises which will be discussed in detail.
The need for a simultaneous consideration of many shape variables in
PES's calculations is common to all nuclear models, including self-consistent
theories based on some effective interactions \cite{Schunck}.
The results on fission saddles obtained up to now in the SH region clearly
show the great importance of triaxial deformation, neglected in many
published work.
A recent study \cite{BroSkal} of barriers within both the MM Woods-Saxon and
Skyrme SLy6 Hartree-Fock plus BCS models shows that triaxiality
is even more crucial beyond $Z =$ 126.
A description of our method of calculations is given in section II. The
results, details of the additional calculations, and comparisons with
other calculated barriers are presented and discussed in section III.
Finally, the conclusions are summarized in section IV.
\section{The Method}
Multidimensional energy landscapes are calculated within the
MM model besed on the deformed Woods-Saxon potential \cite{WS}.
The Strutinski shell and pairing correction \cite{STRUT67}
is taken for the microscopic part.
For the macroscopic part we used the Yukawa plus exponential model \cite{KN}
with parameters specified in \cite{MUNPATSOB}. Thus, all parameter values
are kept exactly the same as in all recent applications of the
model to heavy and superheavy nuclei
The main point in fission barrier calculations is its reliability which,
once the model for calculating energy of a nucleus as a function of
deformation is fixed, hangs on two main
ingredients: 1) the kind and dimension of the admitted deformation space and
2) a method applied to the search for saddles.
Mononuclear shapes can be parameterized via spherical harmonics
${\rm Y}_{lm}(\vartheta ,\varphi)$ (for brevity we will just use
the symbol ${\rm Y}_{\lambda\mu}$ ) by the following equation of the nuclear surface:
\begin{equation}
R(\vartheta ,\varphi)= c(\{\beta\}) R_0
\{ 1+ \sum _{\lambda=1}^{\infty}\sum _{\mu=-\lambda}^{+\lambda} \beta_{\lambda\mu}{\rm Y}_{\lambda\mu}\},
\label{eq:radius}
\end{equation}
where $c(\{\beta\})$ is the volume-fixing factor and $R_0$ is the radius of a spherical nucleus.
This parameterization has its limitations; certainly, it is not suitable for
too elongated shapes.
However, for moderately deformed saddle points in superheavy nuclei
it excellently reproduces all shapes generated by other parametrizations,
e.g. by \cite{Mol2000}, as we checked in numerous tests.
For nuclear ground states it is possible to confine analysis to
axially-symmetric shapes, with the expansion truncated at $\beta_{80}$:
\begin{eqnarray}
R(\vartheta ,\varphi) &=& c(\{\beta\})R_0 \{ 1 + \beta_{20} {\rm Y}_{20}+
\beta_{30} {\rm Y}_{30} + \beta_{40} {\rm Y}_{40} \nonumber\\
&+& \beta_{50} {\rm Y}_{50} + \beta_{60} {\rm Y}_{60} +
\beta_{70} {\rm Y}_{70} + \beta_{80} {\rm Y}_{80} \}. \nonumber\\
&&
\end{eqnarray}
Thus, a seven dimensional minimization is performed using the gradient method.
For odd systems, the additional minimization over configurations is
performed at every step of the gradient procedure. Considered configurations
consist of the odd particle occupying one of the levels close to the Fermi
level and the rest of the particles forming a paired BCS state on the
remaining levels. Ten states above and ten states below the Fermi level have
been blocked and energy minimized over these configurations.
The main problem in a search for saddle points is that, since they are neither
minima nor maxima, one has to know energy on a multidimensional grid of
deformations (the often used and much simpler method of minimization with
imposed constraints may produce invalid results
\cite{Moller2009,Dubray,IIbarriers,Schunck}.
To find saddles on a grid we used the Imaginary Water Flow (IWF) technique.
This conceptually simple and at the same time very efficient (from a numerical
point of view) method was widely used and discussed before
\cite{Luc91,Mam98,Hayes00,Moeler04,Moller2009,IIbarriers}. The
number of numerically tractable deformation parameters
$\{\beta_{\lambda \mu}\}$ is practically limited.
More than five-dimensional grids, keeping in mind a subsequent interpolation,
are intractable in calculations for many ($\sim$ 1000) nuclei. Including
mass- and axially-symmetric deformations ($\beta_{20}$, $\beta_{40}$,
$\beta_{60}$, $\beta_{80}$ - see \cite{Patyk1,Patyk2,Patyk3,Smol1} together with both,
mass-asymmetry ($\beta_{30}$, $\beta_{50}$, $\beta_{70}$) and triaxiality
(at least $\beta_{22}$) would mean at least an eight-dimensional mesh
and was impossible at present.
Based on our previous results showing that triaxial saddles are abundant in SH nuclei \cite{Kow}, we consider that quadrupole
triaxial shapes have to be necessarily included.
We treated the effects of mass-asymmetry and nonaxial higher multipoles
as corrections and analysed them at the second stage of calculations.
A rationale for a lesser importance of mass-asymmetric saddles is that,
while they constitute a second, more deformed ($\beta_{20}\approx
0.7 - 0.8)$,
prominent barrier peak in actinides, their heights are much reduced
in SH nuclei where they become irrelevant. In the remaining, less deformed
saddles the mass asymmetry occurs less frequently.
As to the nonaxial multipoles of higher order, they are less important
for saddles with small to moderate $\gamma$ [where $\gamma$ is the
Bohr's quadrupole nonaxiality parameter, cf. Eq. (\ref{betgam})].
They become important for $\gamma$ closer to $60^o$
where they are needed to produce oblate shapes having $x$ as the symmetry
axis.
Thus, they should be included for nuclei with a large
oblate g.s. deformation and a short triaxial barrier.
The additional studies of the mass-asymmetry and higher nonaxial multipoles
are described in the proper subsections of the Results section.
Thus, at the first stage, for all 1305 investigated nuclei the saddle points
were searched in a five dimensional deformation space spanned by:
$\beta_{20}$, $\beta_{22}$, $\beta_{40}$, $\beta_{60}$, $\beta_{80}$, using
the IWF technique.
The appropriate nuclear radius expansion has the form:
\begin{eqnarray}
\label{maingrid}
R(\vartheta ,\varphi) &=& c(\{\beta\})R_0 \{ 1 + \beta_{20} {\rm Y}_{20}
+\frac{\beta_{22}}{\sqrt{2}}\left[{{\rm Y}_{22} +
{\rm Y}_{2 - 2}}\right] \nonumber\\ &+&\beta_{40} {\rm Y}_{40} +
\beta_{60} {\rm Y}_{60} + \beta_{80} {\rm Y}_{80} \}.
\end{eqnarray}
The five-dimensional calculations are performed on the following deformation
mesh:
\begin{eqnarray}
\beta_{20} & = & \ \ 0.00 \ (0.05) \ 0.60 \nonumber \\
\beta_{22} & = & \ \ 0.00 \ (0.05) \ 0.45 \nonumber \\
\beta_{40} & = & \ \ -0.20 \ (0.02) \ 0.20 \nonumber \\
\beta_{60} & = & \ \ -0.10 \ (0.02) \ 0.10 \nonumber\\
\beta_{80} & = & \ \ -0.10 \ (0.02) \ 0.10 \nonumber\\
\end{eqnarray}
This makes a grid of 29250 points which was subsequently interpolated
to a fivefold denser grid of 50735286 points with the step 0.01 in
each dimension. On the latter, the saddle point, or rather several saddle
points - if there were a few of comparable heights within the 0.5 MeV
energy window - were searched for by means of the IWF procedure.
For odd or odd-odd nuclei, at each grid point we were looking for low-lying
configurations by blocking particles on levels from the 10-th below to the
10-th above the Fermi level (in neutrons or/and protons).
The fact that searches for ground states and for saddles are separated -
performed using different deformation spaces - allows saving some
number of deformation parameters in Eq. (\ref{maingrid}).
This is equivalent to
assuming that the fission saddles have mostly prolate deformations large
enough to make nonaxial deformations of multipolarity $\lambda\geq 3$ less
important. One has to check this assumption afterwards and separately
treat nuclei in which the inclusion of nonaxial deformations with
$\lambda\geq 4$ is necessary.
Although, as mentioned before, in SH nuclei the second barriers at large
deformations are usually smaller than the first one or do not exist at all,
for $Z=98$-101 the mesh (4) was extended to $\beta_{20}=1.5$ and the second
saddles were searched for by the IWF technique. It turned out that these
more deformed barriers are indeed mostly smaller than the first ones and
decrease with increasing $Z$.
Only in Cf isotopes with $N=134-160$ there were some second saddles
(at $\beta_{20}\approx 0.9$) higher than the first one by at most 0.5 MeV.
However, even those saddles were lowered by at least 1 MeV after including
the mass-asymmetry. Therefore, we have reasons to believe that the range of
$\beta_{20}$ in (4) is sufficient for knowing the height of the fission
barrier in the whole studied region.
\section{Results and discussion}
In the present paper we have systematically calculated fission-barrier
heights $B_{f}$ as the energy difference between the saddle point and
the ground state. The saddle point is defined as the minimum over possible
fission paths of the maximal energy along the path.
Let us emphasize that the calculations presented here have
been performed without adding any zero-point vibration energy.
We have included 1305 heavy and superheavy nuclei with
proton numbers $98\leq Z \leq 126$ and neutron numbers
in the range $134\leq N \leq 192$, with the smallest $N$ for a given $Z$
increasing by one with every step in $Z$. All obtained barriers have been
collected in Table \ref{bartot}.
On all PES's presented here, energy is normalized in such a way that its
macroscopic part is set to zero at the spherical shape.
\subsection{Potential Energy surfaces}
Some idea about the positions of ground states, secondary minima and saddles
may be gained from PES's. Chosen examples are shown in figures for:
$^{252}$Lr - Fig. \ref{figb103_149},
$^{270}$Db - Fig. \ref{figb105_165}, $^{276}$Mt - Fig. \ref{figb109_167},
$^{280}$Cn - Fig. \ref{figb112_168}, and $^{297}$119 - Fig. \ref{figb119_178}.
Overall evolution of ground states with increasing $Z$ from prolate to
spherical can be seen there. In some nuclei one can see multiple saddles
of which the one defining the fission barrier should be properly chosen.
Sometimes the saddles between competing minima can be important,
therefore the determination of all saddles on the map is necessarily needed.
The energy landscapes Fig. 1-5 were obtained by minimizing energy on the
5D grid (\ref{maingrid}) with respect to $\beta_{40}$, $\beta_{60}$ and
$\beta_{80}$. One should be aware of two related circumstances: 1)
As the grid Eq. (\ref{maingrid}) does not include nonaxial deformations
$\lambda\geq 4$, the axial deformations $\lambda= 4, 6, 8$ {\it with respect
to the $x$-axis} cannot be reproduced, so the landscapes are inexact
around the oblate $\gamma=60^o$ axis. 2)
A reduction of a $n$-dimensional grid of energy values via the minimization
over $n-2$ deformations sometimes leads to an energy surface composed from
disconnected patches, corresponding to multiple minima in the auxiliary
(those minimized over) dimensions. This can distort the picture of the
barrier (actually, a reduction of multi-dimensional data to a
two-dimensional map is a general problem).
With these reservations in mind, one can still explore some of the details
shown in the maps. In particular, the prolate g.s. minimum with
strongly nonaxial first saddle point at $\beta_{20} =0.41$ and
$\beta_{22} =0.18$ is visible in $^{252}$Lr. One can notice that the
axially symmetric saddle lies more than 2 MeV higher.
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{103_149.EPS}
\caption{Energy surface, $E-E_{mac}(sphere)$, for $Z=103$ and $N=149$.}
\label{figb103_149}.
\end{figure}
A slightly less steep, prolate g.s. minimum and a gently emerging
second minimum is visible in Fig \ref{figb105_165} for $^{270}$Db.
The triaxial saddle at $\beta_{20} =0.52$ and $\beta_{22} =0.13$ has
a smaller triaxiality $\gamma$ than the saddle in $^{252}$Lr.
A decrease in barrier height due to triaxiality is $\approx$ 2 MeV,
Fig. \ref{figb105_165}.
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{105_165.EPS}
\caption{The same as in \ref{figb103_149} but for $Z=105$ and $N=165$.}
\label{figb105_165}.
\end{figure}
In a heavier nucleus $^{276}$Mt, a prolate deformation of the g.s. is clearly
smaller than in $^{252}$Lr, see Fig \ref{figb109_167}. The second minimum,
which was barely outlined in $^{270}$Db, is more pronounced here,
giving the fission barrier a double-hump structure.
The deformation $\beta_{20}\approx 0.5$ of the second saddle is
much smaller than that of the second barriers in actinides.
Thus, a two-peak structure of the barrier in SH nuclei may be viewed as
a result of a division (split) of the first barrier, occurring with growing
$Z$. The higher second axial saddle is lowered by triaxiality
by $\approx$ 1.5 MeV, but eventually is still higher than the first axial
saddle.
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{109_167.EPS}
\caption{The same as in \ref{figb103_149} but for $Z=109$ and $N=167$.}
\label{figb109_167}.
\end{figure}
For $^{280}$Ds a topology of the PES is even more complicated.
We see several minima: prolate - the g.s. and a superdeformed one, and a
shallow oblate. The map shows also a few saddles. The axially deformed saddle
point at $\beta_{20} =0.3$ has a similar height as the nonaxial saddle at
$\beta_{20} =0.54$ and $\beta_{22} =0.12$. It follows from the IWF calculation
that the second fission barrier is nonaxial in this case. The axial
second saddle is lowered by $\approx 1$ MeV owing to the nonaxiallity.
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{112_168.EPS}
\caption{The same as in \ref{figb103_149} but for $Z=112$ and $N=168$.}
\label{figb112_168}.
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{119_178.EPS}
\caption{The same as in \ref{figb103_149} but for $Z=119$ and $N=178$.}
\label{figb119_178}.
\end{figure}
The nucleus $Z=119$, $N=178$ is spherical in its g.s. - Fig. \ref{figb119_178}.
There is a secondary oblate minimum (whose depth is underestimated in the map
due to omission of nonaxial $\lambda=4, 6$ deformations). There is a low
triaxial second saddle at $\beta_{20}\approx 0.5$ and two "first saddles"
with different triaxiality, of which the one with a larger $\gamma$ is the
fission saddle.
Still another type of PES, typical of nuclei with the superdeformed oblate
g.s., is presented in Fig.\ref{figb4} in the subsection C.
\subsection{Role of the mass asymmetry}
To study the effect of the reflection (mass) - asymmetry on the
fission barriers, a two-step procedure has been performed.
At the first stage, we have checked the stability of all the saddles found
on the basic 5D mesh (the first, the second, ..., axially symmetric or
triaxial, of energy within 0.5 MeV of the highest saddle) against the
mass-asymmetry. This was done by a 3D energy minimization with respect to
$\beta_{30}$, $\beta_{50}$ and $\beta_{70}$ around each saddle. Since most
of the saddles are non-axial, the most general version of our Woods-Saxon
code had to be used. In this case, when both symmetries (axial and mass
symmetry) are broken simultaneously, the nuclear shapes are defined by the
following equation of the nuclear surface:
\begin{eqnarray}
R(\vartheta ,\varphi)= R_0 c(\{\beta\})\left\{ 1 \right.&+& \beta_{20}
{\rm Y}_{20}+\frac{\beta_{22}}{\sqrt{2}} \left[{{\rm Y}_{22} + {\rm Y}_{2 - 2}}\right] \nonumber\\
&+& \beta_{30} {\rm Y}_{30} + \beta_{40} {\rm Y}_{40} + \beta_{50} {\rm Y}_{50} \nonumber\\
&+& \beta_{60} {\rm Y}_{60} + \beta_{70} {\rm Y}_{70} + \beta_{80} {\rm Y}_{80} \}.
\end{eqnarray}
It turned out that this minimization lowers energy of only those saddles in
which: i) there is no triaxiality, ii) deformation $\beta_{20}\approx 0.3$.
This supports an often expressed conventional "wisdom", that the
mass-asymmetry and triaxiality effects on fission saddle are decoupled.
This is why, at the second step of the procedure, we could carry out a full
IWF analysis on a grid including only axially-symmetric deformations:
$\beta_{20},\beta_{30},\beta_{40},\beta_{50},\beta_{60},\beta_{70},\beta_{80}$,
with $\beta_{20}$ restricted to a quite short interval $0.25 - 0.40$:
\begin{eqnarray}
\beta_{20} & = & \ \ 0.25 \ (0.05) \ 0.40 \nonumber \\
\beta_{30} & = & \ \ 0.00 \ (0.05) \ 0.25 \nonumber \\
\beta_{40} & = & \ \ -0.15 \ (0.05) \ 0.20 \nonumber \\
\beta_{50} & = & \ \ 0.00 \ (0.05) \ 0.15 \nonumber\\
\beta_{60} & = & \ \ -0.10 \ (0.05) \ 0.10 \nonumber\\
\beta_{70} & = & \ \ 0.00 \ (0.05) \ 0.15 \nonumber\\
\beta_{80} & = & \ \ -0.10 \ (0.05) \ 0.10. \nonumber\\
\end{eqnarray}
This seven-dimensional grid, composed of 76800 deformations,
was subject to the fivefold interpolation in all directions before it was
used in the IWF procedure. This means that the IWF calculations have been
performed on the grid containing 1 690 730 496(!) points.
We have made such 7-dimensional analysis for more than 100 nuclei, for which
the effect of minimization was greater than 300 keV. Results for these nuclei
are shown
in Table \ref{Octu}. The rest of 127 cases shown in Table \ref{Octu} are the
test nuclei, in which the effect of the minimization was smaller than 0.3 MeV.
The results for these additional nuclei allow to appreciate whether the
(in principle exact) IWF method could produce a greater effect that the
(inexact) minimization.
As one can see, the adopted procedure allowed to omit the problem of
searching for a saddle by using the (inexact) minimization method which
is not always reliable \cite{IIbarriers,Moller2009}. For example, for Z=118
and N=165, the discussed effect resulting from the minimization
amounts to 0.44 MeV, which, just in this case, is quite similar to 0.46 MeV
obtained from the IWF technique; however, in Z=113 and N=163 one obtains
$\approx 0.5$ MeV difference between saddles obtained by both methods.
In this particular nucleus, the $\approx 0.77$ MeV barrier
lowering by the mass-asymmetry is the largest among all studied nuclei.
It should be also noted that for the isotopes of Z = 113 the
effect of the mass-asymmetry is particularly large, see the the top panel
in Fig. \ref{oct}.
In the bottom panel of Fig. \ref{oct}, we show the difference between the
results of the both methods - the minimization - (MIN) and
"Imaginary Water Flow" - (IWF).
One can see that this difference increases with the neutron number.
In particular,
there is practically no effect derived from the mass-asymmetry in $^{281}113$
when IWF is used. On the contrary, the approach based on minimization
suggests still a quite substantial (spurious) effect (0.55 MeV).
One might notice that our conclusion concerning decoupling of the variables
describing the axial and reflection asymmetries is in a delicate
contradiction with the studies \cite{Lu}.
\begingroup
\begin{table*}
\caption{\label{Octu} Mass(reflection)-asymmetry effect on the fission barrier
from the minimization - MIN and from the Imaginary Water Flow method - IWF
(in MeV).}
\begin{ruledtabular}
\begin{tabular}{|cccccccccc|}
& N & IWF & MIN & N & IWF & MIN & N & IWF & MIN \\
\noalign{\smallskip}\hline\noalign{\smallskip}
& & $\mathbf{Z=109}$ & & & $\mathbf{Z=114}$ & & & $\mathbf{Z=117}$ & \\
& 157 & 0.39 & 0.81 & 155 & 0.28 & 0.59 & 157 & 0.24 & 0.34 \\
& 158 & 0.22 & 0.42 & 156 & 0.14 & $<$0.30 & 158 & 0.28 & $<$0.30 \\
& 159 & 0.54 & 0.45 & 157 & 0.72 & 0.83 & 159 & 0.24 & 0.34 \\
& 160 & 0.31 & 0.54 & 158 & 0.46 & 0.46 & 160 & 0.12 & $<$0.30 \\
& & $\mathbf{Z=110}$& & 159 & 0.67 & 0.68 & 161 & 0.26 & $<$0.30 \\
& 157 & 0.41 & 0.69 & 160 & 0.45 & 0.66 & 165 & 0.36 & 0.39 \\
& 158 & 0.19 & 0.31 & 161 & 0.53 & 0.79 & 166 & 0.23 & $<$0.30 \\
& 159 & 0.52 & 0.46 & 162 & 0.42 & 0.64 & 167 & 0.19 & 0.50 \\
& 160 & 0.50 & 0.40 & 163 & 0.58 & 0.65 & 168 & 0.07 & $<$0.30 \\
& 161 & 0.43 & 0.47 & 164 & 0.40 & 0.63 & 169 & 0.05 & 0.37 \\
& 162 & 0.35 & 0.31 & 165 & 0.42 & 0.65 & & $\mathbf{Z=118}$ & \\
& & $\mathbf{Z=111}$ && 166 & 0.38 & 0.53 & 163 & 0.30 & 0.32 \\
& 157 & 0.49 & 0.97 & 167 & 0.11 & 0.68 & 164 & 0.23 & $<$0.30 \\
& 158 & 0.36 & 0.78 & 168 & 0.06 & 0.41 & 165 & 0.46 & 0.44 \\
& 159 & 0.61 & 0.83 & & $\mathbf{Z=115}$ && 166 & 0.28 & 0.31 \\
& 160 & 0.67 & 0.85 & 157 & 0.28 & 0.64 & 167 & 0.20 & 0.63 \\
& 161 & 0.87 & 0.89 & 158 & 0.25 & 0.50 & 168 & 0.15 & 0.39 \\
& 162 & 0.66 & 0.80 & 159 & 0.34 & 0.49 & & $\mathbf{Z=119}$ & \\
& 163 & 0.56 & 0.83 & 160 & 0.39 & 0.38 & 165 & 0.46 & 0.57 \\
& 164 & 0.58 & 0.68 & 161 & 0.56 & 0.58 & 166 & 0.33 & 0.37 \\
& 166 & 0.48 & 0.49 & 162 & 0.42 & 0.39 & 167 & 0.34 & 0.49 \\
& & $\mathbf{Z=112}$ && 163 & 0.46 & 0.54 & 168 & 0.27 & 0.32 \\
& 157 & 0.57 & 0.83 & 164 & 0.49 & 0.45 & 169 & 0.31 & 0.57 \\
& 158 & 0.32 & 0.45 & 165 & 0.47 & 0.60 & 170 & 0.24 & 0.38 \\
& 159 & 0.58 & 0.55 & 166 & 0.53 & 0.54 & 171 & 0.23 & 0.32 \\
& 160 & 0.60 & 0.49 & 167 & 0.42 & 0.80 & &$\mathbf{Z=120}$ & \\
& 161 & 0.51 & 0.60 & 168 & 0.20 & 0.55 & 165 & 0.39 & 0.38 \\
& 162 & 0.53 & 0.48 & 169 & 0.13 & 0.31 & 166 & 0.17 & $<$0.30 \\
& 163 & 0.56 & 0.64 & 170 & 0.07 & 0.30 & 167 & 0.20 & 0.49 \\
& 164 & 0.44 & 0.43 & & $\mathbf{Z=116}$ && 168 & 0.15 & $<$0.30 \\
& 165 & 0.33 & 0.48 & 155 & 0.40 & 0.41 & 169 & 0.10 & 0.46 \\
& 166 & 0.34 & 0.34 & 156 & 0.19 & $<$0.30 & & $\mathbf{Z=121}$ & \\
& 167 & 0.20 & 0.35 & 157 & 0.36 & 0.52 & 165 & 0.25 & 0.40 \\
& & $\mathbf{Z=113}$ && 158 & 0.26 & 0.34 & 166 & 0.23 & $<$0.30 \\
& 155 & 0.14 & 0.49 & 159 & 0.35 & 0.44 & 167 & 0.38 & 0.52 \\
& 156 & 0.24 & 0.34 & 160 & 0.28 & 0.49 & 168 & 0.31 & 0.34 \\
& 157 & 0.80 & 0.98 & 161 & 0.40 & 0.44 & 169 & 0.36 & 0.60 \\
& 158 & 0.50 & 0.75 & 162 & 0.33 & 0.37 & 170 & 0.30 & 0.43 \\
& 159 & 0.56 & 0.91 & 163 & 0.48 & 0.54 & & $\mathbf{Z=122}$ & \\
& 160 & 0.61 & 0.88 & 164 & 0.40 & 0.38 & 164 & 0.00 & $<$0.30 \\
& 161 & 0.72 & 1.06 & 165 & 0.46 & 0.50 & 165 & 0.21 & $<$0.30 \\
& 162 & 0.57 & 0.93 & 166 & 0.33 & 0.40 & 166 & 0.12 & $<$0.30 \\
& 163 & 0.76 & 1.25 & 167 & 0.30 & 0.38 & 167 & 0.19 & 0.31 \\
& 164 & 0.49 & 0.89 & 168 & 0.11 & $<$0.30 & 168 & 0.11 & $<$0.30 \\
& 165 & 0.54 & 0.98 & 169 & 0.09 & 0.32 & 169 & 0.10 & 0.45 \\
& 166 & 0.40 & 0.86 & & & & & $\mathbf{Z=123}$ & \\
& 167 & 0.19 & 0.78 & & & & 166 & 0.06 & $<$0.30 \\
& 168 & 0.10 & 0.55 & & & & 167 & 0.08 & 0.35 \\
& & & & & & & &$\mathbf{Z=124}$ & \\
& & & & & & & 165 & 0.23 & 0.31 \\
& & & & & & & 166 & 0.06 & $<$0.30 \\
& & & & & & & 167 & 0.10 & 0.32 \\
& & & & & & & & & \\
\noalign{\smallskip}
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup
\begin{figure}[h!]
\includegraphics[width=1.3\linewidth,height=5.0in]{b3b5b7efect.EPS}
\caption{ Top panel:
The fission barrier lowering by the mass-asymmetry obtained by the
(in principle exact) Imaginary Water Flow method - IWF and by the
(easier, but sometimes misleading) minimization method - MIN.
Bottom panel: The difference between both methods in MeV (in principle -
the error in the barrier height due to the minimization method). }
\label{oct}
\end{figure}
\subsection{Role of the triaxiality}
The importance of including triaxiality in a calculation of fission barrier
heights was indicated many times before \cite{CWIOK92,CWIOK94,CWIOK96,GHERGH99,DUTTA00,DECHARGE2003,BONNEAU04,Cwiok05,DOB2007,KOWAL2009}.
In particular, it was shown that the effect of {\it both}
quadrupole and a general hexadecapole nonaxiality, when accounted for
within the {\it nonexact} method of constrained minimization (used generally
in all selfconsistent studies), may reach 2.5 MeV
for some superheavy even-even nuclei, see Fig. 5 in \cite{Kow}.
Here, we extend our previous discussion of its role to
the odd and odd-odd nuclei and, at the same time, improve the treatment
by employing the exact IWF method in potentially most interesting cases.
By using the original 5D mesh (4) we have obtained saddles with
{\it quadrupole} nonaxiality for about 900 nuclei, what constitutes more than
70 \% of all fission barriers. We illustrate this conspicuous effect in Fig.
\ref{nonaxial} on the example of two isotopic chains, Z=103 and 113.
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.5in]{nonaxial.EPS}
\caption{ Effect of the non-axiallity on the fission barrier heights (see
text for further explanations).}
\label{nonaxial}
\end{figure}
We show the difference between axial and nonaxial barriers in these nuclei.
One can see that for lighter Lawrencium isotopes the effect of nonaxiality is
quite considerable. Starting with $N=$164, it is weakening quickly and finally
vanishes for $N\geq$ 176. Somewhat different dependence of the effect on the
neutron number occurs in $Z=113$ isotopes. The maximum lowering of the barrier
of more than 1.5 MeV occurs for $N\approx 165$, there is a second maximum at
$N=179$, and the effect becomes large again at $N=192$. Inbetween, for
$N\approx 154$ and $N\approx 174$, there is no effect at all.
Thus, the effect of nonaxiallity has to be studied carefully, indeed.
Another task is to consider the influence of the hexadecapole nonaxiality,
namely: $\beta_{42}, \beta_{44}$ in Eq. \ref{eq:radius}, on the fission
barriers. The unconstrained inclusion of these shapes would lead to a 7D grid
which is too much for now.
To evaluate the effect without increasing the grid
dimension we constrained $\beta_{42}$ and $\beta_{44}$ to be
functions of the quadrupole nonaxial deformation $\beta_{22}$, or actually
$\gamma$, and $\beta_{40}$, in a well known manner \cite{geom}.
Using the conventional notation:
\begin{eqnarray}
\label{betgam}
\beta& = &\sqrt{\beta_{20}^2+\beta_{22}^2} , \nonumber\\
\gamma& = & {\rm arctg}\frac{\beta_{22}}{\beta_{20}},
\end{eqnarray}
the following form of Eq. \ref{eq:radius} was used:
\begin{eqnarray}
\label{b4}
R(\vartheta ,\varphi)= c({\beta})R_0 \left\{ 1 \right.&+& \beta\cos{(\gamma)} {\rm Y}_{20} \nonumber\\
&+& \frac{\beta \sin{(\gamma)}}{\sqrt{2}}\left[{{\rm Y}_{22} + {\rm Y}_{2 - 2}}\right] \nonumber\\
&+&\beta_{40} \frac{1}{6}(5 \cos^{2}{(\gamma)} +1 ){\rm Y}_{40} \nonumber\\
&-&\beta_{40} \frac{1}{6}\sqrt{\frac{15}{2}} \sin{(2 \gamma)} \left[{{\rm Y}_{42} + {\rm Y}_{4 - 2}}\right] \nonumber\\
&+&\beta_{40} \frac{1}{6}\sqrt{\frac{35}{2}} \sin^{2}{(\gamma)} \left[{{\rm Y}_{44} + {\rm Y}_{4 - 4}}\right] \nonumber\\
&+& \beta_{60} {\rm Y}_{60} + \beta_{80} {\rm Y}_{80} \}.
\end{eqnarray}
On this 5D grid, the hexadecapole nonaxiality (but not the $\beta_{60}$ and
$\beta_{80}$ terms) preserves the modulo-60$^o$
invariance in $\gamma$, so, in particular, the parameter $\beta_{40}$
describes a deformation which is axially symmetric around the $z$ axis
at $\gamma=0^o$ and around the $x$ axis at $\gamma=60^o$, which allows
to better approximate energy at oblate shapes. For this reason, while the
original mesh Eq. (\ref{maingrid}) may be expected more reliable
for barriers at
small $\gamma$, the one of Eq. (\ref{b4}) is better for saddles closer to
$\gamma=60^o$, like those in nuclei with well- or super-deformed oblate
ground states.
Our method of proceeding is analogous to that used in the study of the
mass-asymmetry. The difference is that we do not have to perform the
first step: a minimization with respect to $\beta_{42}$ and $\beta_{44}$
at the saddles found from the grid Eq. (\ref{maingrid}).
Such calculations were already
done in the previous studies of the effect of nonaxial deformations of higher
multipolarity on the fission barrier in heaviest nuclei
\cite{KowSob1,KowSob2,KowSob3,SobJachKow}. We know that the minimization
gave the largest effect in the following four regions of nuclei,
see Fig. 2 in \cite{SobJachKow}: (I) $Z \approx 122$, $N \approx 160$ -
up to 1.5 MeV, and a $\sim$ 3 times smaller effect for nuclei with larger
$N$ and $Z>120$, (II) $Z \approx 110$, $N \approx 146$ - up to 1 MeV,
(III) $Z\approx114$, $N\approx184$ - up to 1 MeV, and (IV)
$Z\approx104$, $N\approx170$ - up to 0.4 MeV.
By applying the IWF method on the mesh Eq. (\ref{b4}) we have found
the saddles for a dozen of nuclei from the last three regions,
for which the effect of
minimization was the largest. It turned out that, compared to saddles
found on the original grid Eq. (\ref{maingrid}), they were lowered by
less than 150 keV in the region (II), by less then 100 keV in the region
(III), and even increased by $\sim$ 100 keV in the region (IV).
On this basis we conclude that the lowering of the fission saddles
found by the minimization in \cite{Kow,KowSob3} in these three regions
is in a large measure a spurious effect which mostly vanishes when saddles
are fixed by a proper method.
On the contrary, the substantial effect (up to $\approx 1$ MeV) of the
nonaxial hexadecapole in the region (I), although smaller than found by
the minimization, survives in the exact IWF treatment. This might be expected
as these are very heavy $Z\geq 119$ nuclei with short barriers and oblate
(also superdeformed) ground states, so $\beta_{42}$ and $\beta_{44}$
are necessary to reproduce energy in the vicinity of the oblate axis.
Therefore, in the whole region of nuclei with $Z\geq 118$ we calculated
triaxial barriers by the IWF method using the mesh Eq. (\ref{b4}) and then
selected the proper fission barriers from two 5D calculations.
Three types of saddles in nuclei from the region (I) are shown
for a very heavy and exotic nucleus $^{285}122$ in Fig. \ref{figb4}.
The landscape was created from the 5D mesh Eq. (\ref{b4}).
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.2in]{122_163new.EPS}
\caption{Energy surface, $E-E_{mac}(sphere)$, for the nucleus $Z=122$,
$N=163$, resulting from the calculation according to Eq. (\ref{b4}).}
\label{figb4}
\end{figure}
This nucleus has a global superdeformed oblate (SDO) minimum with
the quadrupole deformation $\beta_{20} = -0.455$ (spheroid with the axis ratio $\approx$ 3:2).
It represents a neutron-deficient area of superheavy nuclei according to
recent predictions \cite{SDO}. These intriguing SDO minima were already
confirmed, as the global ones, by various self-consistent models
\cite{staszproch,skal2015}.
There is a saddle close to the oblate axis, separating the SDO g.s.
from the wide minimum near the spherical shape - type a);
the axially symmetric saddle is designated as b).
One fission path may go through the saddles a) and b), the higher of which
would define the barrier along this path. The second fission path goes through
a triaxial saddle of type c) at $\beta_{20}\approx 0.4$, $\gamma\approx 35^o$.
The fission barrier of $B_{f}=3.6$ MeV corresponds to the saddle c) as found
by using the grid Eq. (\ref{b4}). It turns out that saddles of type a) and c)
are much lowered by including $\beta_{42}$, $\beta_{44}$, the first usually
more than the second.
Table \ref{hexeffect} summarizes the effect of nonaxial hexadecapole
on the barriers in the region (I). It contains 75 nuclei in which the barrier
lowering is greater than 300 keV.
The most frequent saddle type in the region (I), on both grids, is c), but
there are also more complicated cases in which the saddle type changes when
$\beta_{42}$ and $\beta_{44}$ are included. The largest effect of 1.167 MeV
occurs in the nucleus $Z=125$, $N=163$.
Let us remark that the difference between the results of the constrained
minimization and the IWF method for the nonaxial hexadecapole is the main
source of the discrepancy between the current fission barriers and those
published in \cite{Kow} for even-even nuclei.
\begingroup
\begin{table*}
\caption{\label{hexeffect} The barrier lowering (in MeV) greater than 0.3 MeV
in nuclei $Z\geq 118$, in particular in those with SDO ground states,
from the IWF calculations on the 5D mesh including $\beta_{42}$ and
$\beta_{44}$ according to Eq. (\ref{b4}). Also reported is the associated
change in the saddle type (for a description of saddle types see text);
no entry means that a c-type saddle results from both grids,
Eq. (\ref{maingrid}) and (\ref{b4}).}
\begin{ruledtabular}
\begin{tabular}{|cccccccccc|}
& N & $\Delta B_f$ & saddle & N & $\Delta B_f$ & saddle & N & $\Delta B_f$ & saddle \\
\noalign{\smallskip}\hline\noalign{\smallskip}
& & $\mathbf{Z=119}$ & & & $\mathbf{Z=122}$ & & & $\mathbf{Z=125}$ \\
& 155 & 0.597 & & 158 & 0.779 & & 161 & 1.083& \\
& 156 & 0.482 & & 159 & 0.959 & & 162 & 0.958 & \\
& 157 & 0.472 & & 160 & 0.807 & & 163 & 1.167 & \\
& 158 & 0.566 & & 161 & 0.731
& & 164 & 0.936 & \\
& 159 & 0.585 & & 162 & 0.690 & & 165 & 0.439 & a$\rightarrow$c \\
& 160 & 0.508 & & 163 & 0.469 & a$\rightarrow$c & 166 & 0.806 &b$\rightarrow$c \\
& 161 & 0.315 &b$\rightarrow$c& 164 & 0.364 &b$\rightarrow$c& 167 & 0.806 & \\
& 162 & 0.471 &a$\rightarrow$c& 169 & 0.403 &b$\rightarrow$c& 168 & 0.800 & \\
& 170 & 0.343 &b$\rightarrow$c& 170 & 0.365 & & 169 & 0.714 & \\
& 172 & 0.480 &b$\rightarrow$c& & $\mathbf{Z=123}$ & & 170 & 0.551 & \\
& 173 & 0.501 & & 159 & 0.831 & & &$\mathbf{Z=126}$& \\
& 174 & 0.400 & & 160 & 0.821 & & 162 & 0.995 & \\
& & $\mathbf{Z=120}$ & & 161 & 0.863 & & 163 & 1.099 & \\
& 156 & 0.613 & & 162 & 0.924 & & 164 & 1.034 & \\
& 157 & 0.731 & & 163 & 0.496 &a$\rightarrow$c& 165 & 0.802 & \\
& 158 & 0.652 & & 164 & 0.480 &a$\rightarrow$c& 166 & 0.912 & \\
& 159 & 0.778 & & 168 & 0.357 &b$\rightarrow$c& 167 & 0.807 & \\
& 160 & 0.696 & & 169 & 0.300 &b$\rightarrow$c& 168 & 0.845 & \\
& 161 & 0.658 & & &$\mathbf{Z=124}$ & & 169 & 0.911 & \\
& 162 & 0.581 &a$\rightarrow$c& 160 & 0.819 & & 170 & 0.735 & \\
& 163 & 0.323 &a$\rightarrow$b& 161 & 0.868 & & 171 & 0.534 & \\
& &$\mathbf{Z=121}$& & 162 & 0.896 & & 172 & 0.434 & \\
& 157 & 0.747 & & 163 & 0.741 & & & & \\
& 158 & 0.774 & & 164 & 0.739 &a$\rightarrow$c& & & \\
& 159 & 0.690 & & 165 & 0.333 &b$\rightarrow$c& & & \\
& 160 & 0.830 & & 166 & 0.334 &b$\rightarrow$c& & & \\
& 161 & 0.688 & & 167 & 0.455 &b$\rightarrow$c& & & \\
& 162 & 0.633 &b$\rightarrow$c& 168 & 0.519 &b$\rightarrow$c& & & \\
& & & & 169 & 0.459 & & & & \\
& & & & 170 & 0.328 & & & & \\
\noalign{\smallskip}
\end{tabular}
\end{ruledtabular}
\end{table*}
\endgroup
\subsection{Isotopic dependence}
Calculated fission barriers given in Table \ref{bartot} are illustrated along
isotopic chains in Figures: \ref{figb98-100} - \ref{figb125-126}.
Generally, it can be seen that:
i) in the whole region $Z=$98 - 126 the fission barrier heights are limited
by: $B_f\leq 8.06$ MeV;
ii) there are characteristic maxima of fission barriers at $Z\approx 100$,
$N\approx 150$, near $Z=108$, $N=162$
(deformed magic shells) and $Z=114$, $N=178$ (not 184); high barriers
occur also at the border of the studied region, for $Z=98$, $N\approx 183$;
iii) over intervals of $N$ where $B_f(N)$ increase or are on average constant,
the fission barriers in a neighboring system $N_{even}+1$ are higher
than $B_f(N_{even})$; it may the opposite over intervals where $B_f(N)$
strongly decrease; the same behaviour can be seen when comparing barriers for
isotones - see Fig. \ref{Gp169}. This quite pronounced odd-even staggering in
barriers is related to a decrease in the pairing gap due to blocking as it
will be discussed in the next subsection.
In the isotopic dependence of the fission barriers for Cf, Es and Fm nuclei,
shown in Fig. \ref{figb98-100}, there are two
peaks of a similar size, at $N =$ 152 and
$N =$ 184. The minima of $B_f(N)$ occur at $N\approx 170$. Odd-even staggering
in $B_f$ for Es is stronger around $N=152$, while for Cf
it is stronger near $N=184$.
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{98_100.eps}
\caption{Isotopic dependance of fission barriers for $Z=98,99$ and $Z=100$.}
\label{figb98-100}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{101_103.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=101,102$ and $Z=103$.}
\label{figb101-103}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{104_106.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=104,105$ and $Z=106$.}
\label{figb104-106}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{107_109.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=107,108$ and $Z=109$.}
\label{figb107-109}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{110_112.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=110,111$ and $Z=112$.}
\label{figb110-112}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{113_115.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=113,1143$ and $Z=115$.}
\label{figb113-115}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{116_118.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=116,117$ and $Z=118$.}
\label{figb116-118}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{119_121.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=119,120$ and $Z=121$.}
\label{figb119-121}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{122_124.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=122,123$ and $Z=124$.}
\label{figb122-124}
\end{figure}
\begin{figure}
\includegraphics[width=1.2\linewidth,height=3.2in]{125_126.eps}
\caption{The same as in Fig.\ref{figb98-100} but for $Z=125$ and $Z=126$.}
\label{figb125-126}
\end{figure}
For Md, No and Lr isotopes (Fig. \ref{figb101-103}), the second
maximum around $N=$184 is weakening. A maximum associated with the semi-magic
deformed shell at $N=$162 appears. As before, the minima of $B_f(N)$ are
located at $N\approx 170$.
For Rf, Db, Sg, Bh and Hs nuclei (Fig. \ref{figb104-106} and
\ref{figb107-109}), previously distinct maximum
at $N=$152 becomes more flat, and a kind of plateau forms
between $N=$152 and 162. For Mt isotopes
this plateau changes into a local minimum in the isotopic dependence $B_f(N)$,
located around $N=$155.
The highest barriers in Bh, Hs and Mt isotopic sets occur at $N\approx 162$.
For Ds, Rg and Cn nuclei (Fig. \ref{figb110-112}), with the
increasing proton number the $N=$184 spherical shell starts to dominate.
However, not much lower barriers are obtained near the deformed gap $N=$162.
For nuclei: $Z=$ Nh, Fl, Mc (Fig. \ref{figb113-115}), one can see
one region with high barriers, around $N=$180. One can notice that the
maxima in $B_f(N)$ are already shifted toward $N < 184$. Slight residues
of the formerly observed shells at $N=$152 and $N=$162 can be spotted.
For nuclei: $Z=$ Lv, Ts, Og (Fig. \ref{figb116-118}), the main
maximum in $B_f$ progresses further towards smaller $N$, reaching finally
$N\approx 175$. The minima in $B_f(N)$, observed before at $N=172$,
gradually disappear.
For nuclei: $Z=$119, Z=120, Z=121 (Fig. \ref{figb119-121}),
the situation is similar to that described above.
Barriers in nuclei $Z=$122, 123, 124 (Fig. \ref{figb122-124}),
compared to the previous set, are clearly lower.
The maximum is even more shifted towards smaller $N$.
For nuclei: $Z=$125, 126 (Fig. \ref{figb125-126}) the fission barriers
are still lower. Their maxima occur at $N=171$ and 173.
All calculated fission barriers heights are collected together and shown as
a map $B_f(Z,N)$ in Fig. \ref{Bftot}.
One can see three areas with clearly raised barriers: around $N\approx$152,
$N=$162 and $N\approx 180$, and the region of low barriers around $N=170$,
as discussed above. The effect of the odd particle, i.e.
an often (but not always) higher barrier in a neighboring odd-particle
system can be also seen in Fig. \ref{Bftot}.
\onecolumngrid
\hspace{-10cm}
\begin{figure}[t!]
\includegraphics[width=1.2\linewidth,height=5.2in]{1.eps}
\caption{ Calculated fission barrier heights $B_{f}$ for superheavy nuclei.}
\label{Bftot}
\end{figure}
\twocolumngrid
\subsection{Role of the pairing interaction and the odd-even barrier
staggering}
It is known that the blocking procedure often causes an excessive reduction
of the pairing gap in systems with an odd particle number. This effect
is much more pronounced in the g.s. than in the fission saddle, as the
pairing gap is never small in the latter. One device to avoid
an excessive even-odd staggering in nuclear binding was to assume
a stronger (typically by $\sim$ 5\%) pairing interaction for
odd-particle-number systems, see \cite{Gor0,Gor1,Gor2,Gor3}.
Here, instead of performing another grid calculation with modified pairing
strengths, we tested the magnitude of their effect on fission barriers by
increasing them by 5 and 10 percent for odd particle numbers (neutrons
or protons) at previously found ground states and saddle points.
The results of this test are presented in Fig. \ref{Gp169} for the N=169
isotones and in Fig. \ref{Gn109} for the Z=109 isotopic chain.
Both the isotopic and isotonic dependence show that increasing
the intensity of pairing leads to a reduction of the fission barrier by a
variable amount.
When the pairing strengths are increased by 5\% for odd particle numbers,
the fission barriers decrease in odd-even, even-odd and odd-odd systems
by up to 0.5 MeV; the 10\% increase in the pairing strengths
can decrease the barriers at most by about 1 MeV.
The same pairing change leads to the suppression, and then the inversion
of the staggering effect.
The even-odd barrier staggering related to pairing is convoluted with the
isotopic or isotonic dependence related to the mean-field.
With the original pairing, when one separates a linear part of the latter
by calculating: $B_f(Z_{odd},N)-1/2[B_f(Z_{odd}+1,N)+
B_f(Z_{odd}-1,N)]$, and an analogous quantity for odd neutron numbers,
one obtains numbers between 1.053 and $-0.947$ MeV, with the average
of $\approx 0.22$ MeV for protons and $\approx 0.26$ MeV for neutrons.
As shown by black points in Fig. \ref{Gp169}, \ref{Gn109}, the effect is
indeed irregular and, when present, typically at the level of several
hundred keV.
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.5in]{GpN169.EPS}
\caption{ Effect of the pairing strength increase
(while keeping fixed the g.s. and saddle deformations)
in N=169 isotones:
standard $G_n$ and $G_p$ - black points, $G_n$ and $G_p$ increased
by 5\% (10\%) for odd-$Z$ and odd-$N$ nuclei - red (green) points. }
\label{Gp169}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.5in]{GnZ109.EPS}
\caption{ Effect of the pairing strength increase
(while keeping fixed the g.s. and saddle deformations)
in Z=109 isotopes:
standard $G_n$ and $G_p$ - black points, $G_n$ and $G_p$ increased
by 5\% (10\%) for odd-$Z$ and odd-$N$ nuclei - red (green) points. }
\label{Gn109}
\end{figure}
The 5\% increase in pairing for odd particle numbers
reduces the staggering in $N=169$ isotones and nearly cancels it in
$Z=109$ isotopes (red points in Fig.\ref{Gp169} and \ref{Gn109}).
The important point is that the 10\% increase in pairing for odd number of
particles {\it inverts} the staggering, at least locally: near $Z=120$ in
$N=169$ isotones and near $N=153$, $N=162$ and $N=180$ in Mt isotopes
(green points in Fig.\ref{Gp169} and \ref{Gn109}).
Although the spontaneous fission rates of odd-particle number
nuclei are smaller by 3-5 orders of magnitude than those of their even
neighbors, the experimental fission barriers in actinides show only
a moderate odd-even staggering, c.f. \cite{SMIR93,OBN}. Still, it is
inconceivable that the fission barriers in odd-$Z$ or odd-$N$ systems should
be on average {\it smaller} than in their even neighbors.
This indicates that the 10\% increase in pairing strengths in odd-$N$
or odd-$Z$ systems would be too large.
A qualitative argument which follows is that even if the blocking method
overestimates the pairing decrease, the fission barriers of odd-$Z$ or/and
odd-$N$ nuclei should fall in a strip between the black and red points in
Fig.\ref{Gp169} and \ref{Gn109}. Thus, the test of
the pairing influence on barriers points that a possible overestimate of
barriers in odd-$A$ and odd-odd nuclei, induced by the blocking, should not
be much larger than 0.5 MeV. One may add in this context that
the barriers from the FRLDM model do not show any even-odd staggering
due to the way the pairing was included there.
\subsection{Comparison with other theoretical calculations and some empirical
data}
Let us discuss the results in Table \ref{bartot} in relation to
available empirical data and to the other theoretical estimates.
As an empirical check of our model, one can use the barriers
in the actinide region. We have reported quite a spectacular agreement
of the calculated first \cite{Kow} and second \cite{IIbarriers} fission
barriers in even-even actinides with the data \cite{SMIR93,OBN}, with root mean square
deviation 0.5 MeV and 0.7 MeV, respectively.
The heaviest nucleus in which the fission barrier height has been measured
recently is $^{254}$No.
The value $B_f$=6.0 $\pm$ 0.5 MeV at spin 15$\hbar$, giving by
extrapolation $B_f$=6.6 $\pm$ 0.9 MeV at the spin 0$\hbar$, has been
deduced from the measured distribution of entry points in the excitation
energy vs. angular momentum plane \cite{seweryniak}. This result perfectly
agrees with our evaluation: $B_f$=6.88 MeV (at spin 0$\hbar$ ) and with the
MM model \cite{FRLDM} which gives: $B_f$=6.76 MeV.
The selfconsistent calculations, mainly based on the Skyrme interaction,
overestimate this barrier significantly \cite{Bonneau,Duguet,staszczak2006}
(9.6 and 8.6 or 12.5 MeV, respectively).
There are experimental estimates of barriers in a few SH nuclei,
based on observed ER production probabilities \cite{ITKIS}, which again
well agree with our barriers, see \cite{Kow}. Apart from those,
fission barriers in the SH region are generally unknown.
As a supplementary insight, one can crosscheck barriers evaluated
within various models. Quite recently we noted a dramatic divergence in
calculated fission barriers \cite{japan2015}.
Since, as it was discussed previously, the inclusion of traxiality is
absolutely necessary in the SH region, we have
chosen only models which take this into account.
In fact, there is only one systematic calculation, including triaxiality and
odd-particle-number nuclei - the Finite Range Liquid Drop Model
\cite{mol95,Moller2009,FRLDM} (FRLDM) developed by Los Alamos group.
It can be noted though, that the inner fission barrier is fixed there
in only three-dimensional deformation space, what is certainly not enough.
The first conclusion from the comparison between our results and those of
FRLDM is that a conspicuous barrier staggering between odd- and even-particle
number nuclei is obtained in the Woods-Saxon model. As mentioned before, this
results from the blocking treatment of pairing. At present it is not certain
how large this staggering should be.
One can include more models for comparison if one confines it to
even - even nuclei. We take the covariant density functional model \cite{RMF}
with the nonlinear meson-nucleon coupling, represented by the NL3*
parametrization of the relativistic mean-field (RMF) Lagrangian and
the Hartree-Fock-Bogoliubov (HFB) approach with the SkM* Skyrme energy
density functional \cite{SKM}.
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.5in]{108th.eps}
\caption{ Fission barriers predicted by various models for Hassium isotopes: black - WS model, green – FRLDM \cite{FRLDM}, blue – SkM* \cite{SKM}, red – RMF with NL3 parametrization \cite{RMF}. Experimental data taken from \cite{ITKIS}. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)}
\label{108th}
\end{figure}
As can be seen in Fig. \ref{108th}, fission barriers in Hassium nuclei are
quite similar in all models. The values of $B_f$
differ up to 2 MeV, but never more. Regarded as a function of $N$, they show
a maximum close to the semi- magic number $N=$162 while the second
maximum is related with the $N=$184 spherical gap. In the FRLDM this
maximum is barely outlined and slightly shifted to the neutron
deficient side. The minimum in barriers is obtained in both MM
models at the similar place ($N=$170), while the RMF gives the smallest
barriers at $Z=$174.
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.5in]{114th.eps}
\caption{ The same as in Fig \ref{108th} but for Z=114.}
\label{114th}
\end{figure}
As one can see in Fig. \ref{114th}, for Flerovium isotopes the barriers
calculated here are in agreement with the experimental (empirical) estimates
\cite{ITKIS} and with the self-consistent calculations \cite{SKM} based on
the SKM* interaction. The FRLDM \cite{FRLDM} overestimates these
quasi-empirical barriers \cite{ITKIS} significantly.
Although only the lower limit for the barrier height has been estimated in
\cite{ITKIS}, which would reproduce the known cross sections on the picobarn
level, such a high barrier seems problematic, see discussion in
\cite{Wilczynska1,Wilczynska2}.
On the other hand, with extremely small barriers obtained within the RMF model
one cannot explain experimentally known millisecond fission half-life
in $^{284}$Fl.
One should note, however, that a slight tuning of the RMF model \cite{RMF2015}
gives higher barriers, thus, closer to ours. This is true, especially in Cn
and Fl isotopes, see details in Fig. 5 in \cite{RMF2015} and discussion
included there.
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.5in]{120th.eps}
\caption{ The same as in Fig \ref{108th} but for Z=120.}
\label{120th}
\end{figure}
For $Z=$120 our results, shown in Fig. \ref{120th}, are very close to those
obtained within the RMF model.
The results of \cite{FRLDM} are systematically higher by $\approx$ 1 MeV.
This is in an evident contrast with the Skyrme SkM* prediction \cite{SKM} of
the highest barriers for $Z=120$ \cite{SKM} - related to the proton magic gap.
Three models: FRLDM, RMF and ours converge at N=182-184 to
$B_{f} \simeq 5$ MeV. The nucleus $^{302}$120 is particulary interesting,
as two unsuccessful attempts to produce it have already taken place in GSI,
providing a cross-section limit of 560 fb \cite{120GSI1} or 90 fb in
\cite{120GSI2}, and in Dubna \cite{120DUBNA}, providing the limit of 400 fb.
The cross-section estimates \cite{Wilczynska3} do not support a possibility
of an easy production of this SH isotope in the laboratory.
It seems that with the barrier of the order of 10 MeV, as obtained in the
frame of the self-consistent theory, producing superheavy Z=120 nuclei
should not pose any difficulties.
\begin{figure}[h!]
\includegraphics[width=1.2\linewidth,height=3.5in]{126th.eps}
\caption{ The same as in Fig \ref{108th} but for Z=126.}
\label{126th}
\end{figure}
In the case of $Z=$126, shown in Fig. \ref{126th}, both MM models give
significantly smaller barriers than the model based on the SKM* force.
For example, the barrier $B_{f}\approx 9$ MeV for $^{310}126$, calculated with
this Skyrme interaction, is still impressively large. This might induce
thoughts on the ways of synthesis of such superheavy systems, but one has to
remember that the predicted half-lives with respect to $\alpha$ decay
are below the present-day $10^{-5}$ s time-limit for the experimental
identification.
On the contrary, $B_f\approx 2$ MeV obtained in the MM
approach does not induce any hopes; it only points to a quite striking
disagreement between models.
\section{Conclusions}
We have determined fission barriers for 1305 heavy and superheavy nuclei,
including odd-$A$ and odd-odd systems, within the macroscopic-microscopic
method by following the adiabatic configuration in each nucleus.
The applied Woods-Saxon model was widely used for heavy nuclei
and well reproduces experimental fission barriers in actinides.
For odd-$Z$ or/and odd-$N$ nuclei pairing was included within the blocking
procedure.
Triaxial and mass-asymmetric deformations were included and
the IWF method used for finding the saddles which allowed to
escape errors inherent in the constrained minimization approach.
To find saddles, energy for each nucleus was calculated on a 5D deformation
grid and then 5-fold interpolated in each dimension for
the IWF search. Two additional energy grids: a second 5D and another 7D,
were calculated in order to include nonaxial hexadecapole and
mass-asymmetry effects on fission barriers.
The following conclusions can be drawn from our investigation:
i) Global calculations confirm the existence of two physically important
areas in the $Z$-$N$ plane with prominent barriers: one located around
the semi-magic quantum numbers $Z=100-108$ and $N=150-162$ (connected with
deformed closed shells) and the second - of nearly spherical nuclei around
$Z=114$ and $N=176-180$.
The highest fission barrier among the studied nuclei occurs in very exotic
Es$^{250}$.
ii) The well-known effect of the mass asymmetry on the second barrier
in actinides is not very relevant for the heaviest nuclei since very
deformed barriers at $\beta_{20}\approx 0.8$ decrease with increasing $Z$
and fission barriers are fixed by the less deformed saddles.
However, in some nuclei with $Z\geq 109$ the mass(reflection) asymmetry
effect lowers the first saddles which are sometimes split into two humps.
It seems that this concerns only axially-symmetric saddles. The largest
barrier lowering (by 0.8 MeV) has been observed for $Z=113$ and $N=157$.
iii) It has been demonstrated that the inclusion of triaxial shapes
significantly reduces the fission barriers by up to 2.5 MeV;
about 70\% of the found fission barriers correspond to triaxial
saddles.
Besides the quadrupole nonaxiality we checked also the effect of
hexadecapole nonaxiality which significantly lowers the fission
barrier in $Z\geq 119$ nuclei, especially neutron-deficient ones.
iv) Rather strong, irregular odd-even $Z$ or $N$ barrier staggering effect
resulted from the blocking formalism used for pairing.
The barrier of an odd nucleus $Z_{even}+1$ or $N_{even}+1$ is typically
by several hundred keV higher than that of its even neighbor.
v) The existing theoretical evaluations of fission barriers differ
significantly. Even the results of the two models based on the
microscopic-macroscopic approach differ dramatically for some nuclei.
Our calculations indicate, in contrast to the self-consistent mean-field
studies, that fission barriers, still quite substantial for
some $Z=118$ nuclei, become lower than 5.5 MeV for $Z=126$.
\section{ACKNOWLEDGMENT}
M.K. and J.S. were co-financed by the National Science Centre under Contract No. UMO-2013/08/M/ST2/00257 (LEA COPIGAL). One of the authors
(P.J.) was cofinanced by Ministry of Science and Higher Education:
„Iuventus Plus” grant Nr IP2014 016073.
This research was also supported by an allocation
of advanced computing resources provided by the Swierk Computing Centre (CIS)
at the National Centre for Nuclear Research (NCBJ) (http://www.cis.gov.pl).
|
1,116,691,498,437 | arxiv | \section{Introduction}
Quantum information is a rapidly expanding field of research because of its theoretical advances in fast algorithms, superdence quantum coding, quantum error correction, teleportation, cryptography and so forth [3,4,5]. Most of these applications are based on entanglement in quantum states. Although entanglement in pure state systems is relatively well understood, its understanding in the so called mixed quantum states [6], which are statistical mixtures of pure quantum states, is at a primitive level. Recently, Braunstein, Ghosh and Severini [2, 7], have initiated a new approach towards the mixed state entanglement by associating graphs with density matrices and understanding their classification using these graphs. Hildebrand, Mancini and Severini [8] testified that the degree condition is equivalent to the PPT-criterion. They also considered the concurrence of density matrices of graphs and pointed out that there are examples on four vertices whose concurrence is a rational number. In this paper we generalize the work of these authors and give a method to associate a graph with the density matrix (real or complex), of an arbitrary density operator, and also to associate a graph with the matrix representing hermitian operator (observable) of the quantum system, both with respect to a standard orthonormal basis in Hilbert space. We define a modified tensor product of graphs and use it to give Werner's definition for the separability of a $m$-partite quantum system, in $\mathbb{R}^{q_1} \otimes \mathbb{R}^{q_2} \otimes \cdots \otimes \mathbb{R}^{q_m}$, as well as $\mathbb{C}^{q_1} \otimes \mathbb{C}^{q_2} \otimes \cdots \otimes \mathbb{C}^{q_m}$ in terms of graphs. We also deal with classification of pure and mixed states and related concepts like Von-Neumann entropy in terms of graphs.
The paper is organized as follows. In Section 2, we define weighted graphs and their generalized Laplacians which correspond to density matrices, and discuss the permutation invariance of this association. We also deal with pure and mixed states in terms of graphs. Section 3 deals with Von-Neumann entropy. Section 4 is concerned with separability issues as mentioned above. In Section 5, we deal with graph operations which correspond to quantum operations [5, 9, 10]. In Section 6, we present a method to associate a graph with a general hermitian matrix, having complex off-diagonal elements. We define the modified tensor product for complex weighted graphs and express the separability of mixed quantum states in a complex Hilbert space in terms of graphs, using Werner's definition. In section 7, we present some graphical criteria for the associated Laplacian to be positive semidefinite. Finally, we close with a summary and some general comments. Sections 2 to 5 deal with graphs with real weights , that is, quantum states living in real Hilbert space. Graphs with complex weights, corresponding to density operators with complex off diagonal elements are treated in section 6. However, a large part of the results obtained for real Hilbert space in sections 2 to 5, go over to the case of complex Hilbert space (see section 8 (ix)).
\section{Density matrix of a weighted graph}
\subsection{Definitions}
A graph $G = (V, E)$ is a pair of a nonempty and finite set called vertex set $V(G)$, whose elements are called vertices and a set $E(G)\subseteq V^2(G)$ of unordered pairs of vertices called edges. A loop is an edge of the form $\{ v_i, v_i\}$ for some vertex $v_i$. A graph $G$ is on $n$ vertices if $|V(G)| = n.$ We call the graph as defined above a simple graph. $|E(G)| = m+s$, where $m$ : number of edges joining vertices, $s$ : number of loops in $G$ [11].
A {\it weighted graph} $(G, a)$ is a graph together with a {\it weight function} [12]
$$ a : V(G) \times V(G) \rightarrow I\!\!R$$
which associates a real number (weight) $a(\{u, v\})$ to each pair $\{u, v\}$ of vertices. The function $a$ satisfies the following properties:
\begin{description}
\item(i) $a(\{u, v\}) \ne 0$ if $\{u, v\} \in E(G, a)$ and $a(\{u, v\}) = 0$ if $\{u, v\} \not\in E(G, a)$.
\item(ii) $a(\{u, v\}) = a(\{v, u\})$
\item(iii) $a(v, v) \ne 0$ if $\{v,v\}\in E(G,a)$ and is zero otherwise.
\end{description}
If $e = \{u, v\}$ is an edge in $E(G, a)$, property (ii) allows us to write $a(e)$ or $a_{uv}$ for $a(\{u, v\})$. A simple graph can be viewed as a weighted graph with all nonzero weights equal to 1.
In the case of simple graphs the degree $d_v$ of a vertex $v \in V(G)$ is defined as the number of edges in $E(G)$ incident on $v$. For a weighted graph we set
\begin{equation}
d_{(G,a)} (v) =d_v = \sum_{u \in V(G,a)} a_{uv}. \label{try}
\end{equation}
The {\it adjacency matrix} of a weighted graph with $n$ vertices $M(G, a) = [a_{uv}]_{u, v \in V(G, a)}$ is a $n \times n$ matrix whose rows and columns are indexed by vertices in $V(G,a)$ and whose $uv$-th element is $a_{uv}$. Obviously the adjacency matrix $M(G, a)$ is a real symmetric matrix with diagonal element $vv$ equal to the weight of the loops on vertex $v$ (i.e $a_{vv})$.
The {\it degree matrix} for the weighted graph $\Delta(G, a)$ is a $n \times n$ diagonal matrix, whose rows and columns are labeled by vertices in $V(G, a)$ and whose diagonal elements are the degrees of the corresponding vertices.
$$\Delta(G, a) = diag [d_v; v \in V(G, a)]. \eqno{(2)}$$
The {\it combinatorial Laplacian} of a weighted graph is defined to be
$$ L(G, a) = \Delta(G, a) - M(G, a). \eqno{(3)}$$
The {\it degree sum} of $(G, a)$ is defined as
$$d_{(G, a)} = \sum_{v \in V(G, a)} d_v = Tr \Delta(G, a) . \eqno{(4)}$$
The Laplacian defined by (3) has no record of loops in the graph. Therefore we define the {\it generalized Laplacian} of a graph $(G, a)$, which includes loops, as
$$ Q(G, a) = \Delta(G, a) - M(G, a) + \Delta_0(G, a) \eqno{(5)}$$
where $\Delta_0(G, a)$ is a $n \times n$ diagonal matrix with diagonal elements equal to the weights of the loops on the corresponding vertices
$$ [\Delta_0(G, a)]_{vv} = a_{vv} . \eqno{(6)}$$
We call $\Delta_0(G, a)$ the {\it loop matrix} of the graph $(G, a)$ .
For a given weighted graph $(G, a)$, the generalized Laplacian, defined by (5), is not necessarily a positive semidefinite matrix. When, for a given graph $(G, a)$, the generalized Laplacian $Q(G,a)$ is positive semidefinite, we can define the density matrix of the corresponding graph $(G, a)$ as
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} Q(G, a) = \frac{1}{d_{(G, a)}} [L(G, a) + \Delta_0(G, a)] \eqno{(7)}$$
where $Tr(\sigma(G, a)) = 1$.
Note that, this definition of the density matrix of a weighted graph $(G, a)$ reduces to that of the density matrix for a simple graph without loops [7].
Whenever we can define the density matrix for a graph $(G, a)$ we say that the graph $(G, a)$ has density matrix.
For any density matrix $\sigma$, we can obtain the corresponding graph as follows:
\begin{description}
\item(i) Determine the number of vertices of the graph from the size $(n \times n)$ of the density matrix. The number of vertices = $n$. Label the vertices from 1 to $n$.
\item(ii) If the $ij$-th element of $\sigma$ is not zero draw an edge between vertices $v_i$ and $v_j$ with weight $- \sigma_{ij}$.
\item(iii) Ensure that $d_{v_i} = \sigma_{ii}$ by adding loop of appropriate weight to $v_i$ if necessary.
\end{description}
{\it Example}(1): For the following three matrices,we find the corresponding graphs.
\begin{description}
\item(1) $ \sigma = \frac{1}{2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \ea \right] $
in $\mathbb{R}^2$.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=3cm,height=.5cm]{fig1.eps}
Figure 1
\end{center}
\end{figure}
\item(2) $\sigma = \frac{1}{16} \left[ \begin{array}{rrrr} 9 & -1 & -1 & 1 \\ -1 & 3 & -1 & -1 \\ -1 & -1 & 3 & -1 \\ 1 & -1 & -1 & 1 \ea \right] $
in $\mathbb{R}^2 \otimes \mathbb{R}^2$.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=4cm,height=3cm]{fig2.eps}
Figure 2
\end{center}
\end{figure}
\item(3) $\sigma = \frac{1}{4} \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \ea \right] $
in $\mathbb{R}^2 \otimes \mathbb{R}^2$.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=4cm,height=4cm]{fig3.eps}
Figure 3
\end{center}
\end{figure}
\end{description}
\subsection{Invariance under isomorphism}
Two weighted graphs $(G, a)$ and $(G', a')$ are isomorphic if there is a bijective map [13]
$$ \phi : V(G, a) \longmapsto V(G', a')$$
such that
$$ \{\phi(v_i), \phi(v_j)\} \in E(G' a') ~ \mbox{iff}~~ \{v_i, v_j\} \in E(G, a), i, j = 1, 2, \cdots, n$$
and
$$ a_{\phi(v_i) \phi(v_j)}' = a_{v_iv_j} ~~ i, j = 1, 2, \cdots, n.$$
We denote isomorphism of $(G, a)$ and $(G', a')$ by $(G, a) \cong (G' a')$.
Equivalently, two graphs $(G, a)$ and $(G', a')$ are isomorphic if there exists a permutation matrix $P$ such that
$$ P^TM(G, a) P = M(G', a').$$
Note that,
$$ P^T \Delta(G, a)P = \Delta(G', a'); ~~P^T \Delta_0(G, a)P = \Delta_0(G', a')$$
Therefore we have
$$ P^TQ(G, a) P = Q(G', a'). \eqno{(8)}$$
This means that $Q(G, a)$ and $Q(G', a')$ are similar and have the same eigenvalues. Therefore, if $Q(G, a)$ is positive semidefinite then so is $Q(G', a')$. Therefore, if $(G, a)$ has the density matrix so does $(G', a')$. We have proved
\noindent {\bf Theorem 2.1 :} The set of all weighted graphs having density matrix is closed under isomorphism.\hspace{\stretch{1}}$ \blacksquare$
Since isomorphism is an equivalence relation, this set is partitioned by it, mutually isomorphic graphs forming the partition.
\subsection{Correspondence with quantum mechanics}
Henceforth, we consider only the graphs having density matrix unless stated otherwise. The basic correspondence with QM is defined by the density matrix of the graph. For a graph with $n$ vertices the dimension of the Hilbert space of the corresponding quantum system is $n$. To establish the required correspondence we fix an orthonormal basis in the Hilbert space $\mathbb{R}^{q_1} \otimes \mathbb{R}^{q_2} \otimes \cdots \otimes \mathbb{R}^{q_m}$ of the system, which we call the standard basis and denote it by $\{ | ijk\ell \cdots \rangle\}, i, j, k, \ell \cdots = 1, 2, \cdots, n = q_1 q_2 \cdots q_m$, or by $\{ |v_i\rangle\}, i = 1, \cdots, n = q_1 q_2 \cdots q_m$. We label $n$ vertices of the graph $(G, a)$ corresponding to the given density matrix by the $n$ basis vectors in the standard basis. We say that the graph $(G, a)$ corresponds to the quantum state (density operator) whose matrix in the standard basis is the given density matrix. Finally, we set up a procedure, by associating appropriate projection operators with edges and loops of $(G, a)$ to reconstruct this quantum state from the graph $(G, a)$. (See Theorem 2.7). In view of Theorem 2.1 , if $(G,a)$ has density matrix $\sigma$ and $(G,a)\cong(G',a')$ with the corresponding permutation matrix $P$,then $(G',a')$ has the density matrix $P^T \sigma P$. All of this paragraph applies to the complex weighted graph (section 6).
\subsubsection{Pure and mixed states}
A density matrix $\rho$ is said to be pure if $Tr(\rho^2) = 1$ and mixed otherwise. Theorem 2.3 gives a necessary and sufficient condition on a graph $(G, a)$ for $\sigma(G, a)$ to be pure. For a graph $(G, a)$ having $k$ components $(G_1, a_1) , (G_2, a_2), \cdots, (G_k, a_k)$ we write $(G, a) = (G_1, a_1) \uplus (G_2, a_2) \uplus \cdots \uplus (G_k, a_k)$ where $a_i, i = 1, \cdots, k$ are the restrictions of the weight function of the graph $(G, a)$ to the components. We can order the vertices such that $M(G, a) = \oplus^k_{i=1} M(G_i, a_i)$. When $k = 1$, $(G, a)$ is said to be connected. From now on we denote by $\lambda_1(A), \lambda_2(A), \cdots, \lambda_k(A)$ the $k$ different eigenvalues of the Hermitian matrix $A$ in the nondecreasing order. The set of the eigenvalues of $A$ together with their multiplicities is called the spectrum of $A$ [13, 14, 15].
\noindent {\bf Lemma 2.2 :} The density matrix of a graph $(G, a)$ without loops has zero eigenvalue with multiplicity greater than or equal to the number of components of $(G, a)$ with equality applying when weight function $a=$constant $> 0$,
\noindent {\bf Proof :} Let $(G, a)$ be a graph with $n$ vertices and $m$ edges. Since $Q(G, a)$ is positive semidefinite, for $x \in I\!\!R^n$ we must have [12]
$$ x^TQ(G,a)x = \sum^m_{k=1} a_{i_kj_k} (x_{i_k} - x_{j_k})^2 + \sum^s_{t=1} a_{i_ti_t} x^2_{i_t} \ge 0.$$
For the graph without loops the above inequality becomes
$$ x^TQ(G, a)x = \sum a_{i_k j_k} (x_{i_k} - x_{j_k})^2 \ge 0. \eqno{(9)}$$
For $x^T = (1~ 1 ~ \cdots 1)$ we can see $x^T Qx= 0$. This means that $x^T = (1~~ 1~~ 1\cdots 1)$ is an unnormalized eigenvector belonging to the eigenvalue 0 [13]. If there are two components $(G_1, a)$ and $(G_2, a)$ of $(G, a)$, with $n_1, m_1$ and $n_2, m_2$ as the number of vertices and edges in $(G_1, a)$ and $(G_2, a)$ respectively, we can decompose the sum in equation (9) as
$$ x^TQ(G, a)x = \sum^{m_1}_{k_1=1} a_{i_{k_1}j_{k_1}} (x_{i_{k_1}} - x_{j_{k_1}})^2 + \sum^{m_2}_{k_2=1} a_{i_{k_2}j_{k_2}} (x_{i_{k_2}} - x_{j_{k_2}})^2 . \eqno{(10)}$$
For $x^T = (1~1~1 \cdots 1)$ both the terms in (10) vanish. Now consider two vectors $x_1^T = ( 0 ~ 0 ~ \cdots 0 1 ~ 1 \cdots 1)$ with first $n_1$ components zero and last $n_2$ components 1 and $x_2^{T} = (1~~ 1\cdots 1~~0~~0 \cdots 0)$ with first $n_1$ components 1 and last $n_2$ components zero, $(n_1 + n_2 = n)$. Obviously the RHS of (10) vanishes for both $x_1$ and $x_2$. This implies $x_1$ and $x_2$ are two orthogonal eigenvectors with eigenvalue zero. This means multiplicity of zero eigen value is at least 2 (number of components in $(G, a)).$
The equality condition for $a_{uv}=$constant$ > 0 ~~ \forall ~~ \{ u, v\} \in E(G, a)$ is proved in [7]. \hspace{\stretch{1}}$ \blacksquare$
\noindent {\bf Theorem 2.3 :} The necessary and sufficient condition for the state given by a graph $(G,a )$ to be pure is
$$ \sum^n_{i=1} d^2_i + 2 \sum^m_{k=1} a^2_{i_kj_k} = d^2_{(G, a)} \eqno{(11)}$$
where $d_i$ is the degree of the vertex $v_i$ , $a_{i_kj_k}$ is the weight of the edge $\{ v_{i_k}, v_{j_k}\},(v_{i_k} \ne v_{j_k})$ and $d_{(G, a)}$ is the degree sum $d_{(G, a)} = \sum\limits^n_{i=1} d_i$.
\noindent {\bf Proof :} Equation (11) is just the restatement of the requirement $Tr(\sigma^2(G, a)) = 1$, which is the necessary and sufficient condition for the state $\sigma(G,a)$ to be pure.\hspace{\stretch{1}}$ \blacksquare$
\noindent {\bf Lemma 2.4 :} The graph $(G,a)$ for a pure state $\sigma(G,a)$ has the form $(K_\ell, b) \uplus v_{\ell +1} \uplus v_{\ell +2} \uplus \cdots \uplus v_n$ for some $1 \le \ell \le n$.
\noindent {\bf Proof :} Since the state is pure, it has the form
$$ | \psi \rangle = \sum^\ell_{k=1} c_{i_k} |v_{i_k}\rangle, 1 \le i_k \le n.$$
We can permute the basis vectors to transform this sum to $|\psi\rangle = \sum\limits^\ell_{i=1} c_i|v_i\rangle$. That is, the $\ell$ basis kets contributing to the sum in the above equation become the vectors $|v_1\rangle, |v_2\rangle, \cdots, |v_\ell\rangle$ under this permutation. The resulting density matrix $|\psi\rangle \langle \psi|$ has a block of first $\ell$ rows and first $\ell$ columns all of whose elements are nonzero, while all the other elements of density matrix are zero. The graph corresponding to this density matrix is just the required graph. \hspace{\stretch{1}}$ \blacksquare$
{\it Example (2)} : We now give important cases of pure state graphs in $\mathbb{R}^2$ which we use later.
\begin{figure}[!h]
\includegraphics[width=3cm,height=0.7cm]{fig4.eps}
Figure 4
\end{figure}
(i)$\sigma(K_2, a) = \frac{1}{2a_{12}} \left[ \begin{array}{cc} a_{12} & - a_{12} \\ - a_{12} & a_{12} \ea \right] = \frac{1}{2} \left[ \begin{array}{cc} 1 & -1 \\ -1 & 1 \ea \right] = P[ \frac{1}{\sqrt 2} (|v_1\rangle - |v_2\rangle]$, the corresponding graph is as shown in Figure 4.
\begin{figure}[!h]
\includegraphics[width=3cm,height=1cm]{fig5.eps}
Figure 5
\end{figure}
(ii) $ \sigma(K_1, a) = \frac{1}{a} \left[ \begin{array}{cc} a & 0 \\ 0 & 0 \ea \right] = \left[\begin{array}{cc} 1 & 0 \\ 0 & 0 \ea \right] = P[|v_1\rangle]$, the corresponding graph is as shown in Figure 5.
\begin{figure}[!h]
\includegraphics[width=3cm,height=1cm]{fig6.eps}
Figure 6
\end{figure}
(iii)$ a_{12} > 0, \sigma(K_2, -a) = \frac{1}{2a_{12}} \left[ \begin{array}{cc} a_{12} & a_{12} \\ a_{12} & a_{12} \ea \right] = \frac{1}{2} \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \ea \right] = P[ \frac{1}{\sqrt 2} (|v_1\rangle + |v_2\rangle)], a > 0$. The corresponding graph is as shown in Figure 6.
\begin{figure}[!h]
\includegraphics[width=3cm,height=3cm]{fig7.eps}
Figure 7
\end{figure}
(iv) $$ \sigma(G, a) = \frac{1}{4} \left[ \begin{array}{cccc} 1 & -1 & -1 & 1 \\ -1 & 1 & 1 & -1 \\ -1 & 1& 1 & -1\\ 1 & -1 & -1 & 1 \ea \right]= P[ (|-\rangle |-\rangle)]$$, where $ |-\rangle=\frac{1}{\sqrt 2}(|1\rangle - |2\rangle)$, the corresponding graph is as shown in Figure 7.
\begin{figure}[!h]
\includegraphics[width=4cm,height=4cm]{fig8.eps}
Figure 8
\end{figure}
(v) $$ \sigma(G, a) = \frac{1}{4} \left[ \begin{array}{cccccc} 1 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 &0 \\1 & 1& 0 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 \ea \right]= P[ (|+\rangle |+\rangle)],$$ where $ |+\rangle=\frac{1}{\sqrt 2}(|1\rangle + |2\rangle)$
in $\mathbb{R}^2 \otimes \mathbb{R}^3$, the corresponding graph is as shown in Figure 8.
It may be seen that in each of the cases in example (2), same density matrix on the standard basis corresponds to infinite family of graphs as the nonzero weight on each edge or loop is multiplied by a constant. But this is a false alarm because any weight $a \ne 1$ only changes the length of the corresponding state vector in the Hilbert space (i.e. state becomes unnormalized) which does not have any physical significance. Another example pertaining to this situation is the random mixture (see Lemma (3.1 )).
$$ \sigma(G, a) = \frac{1}{an} \left[ \begin{array}{cccc} a \\ & a & & 0 \\ & & \ddots \\ 0 & & & a \ea \right] = \frac{1}{n} \left[ \begin{array}{cccc} 1 & & & 0 \\ & 1 & & \\ & & 1 \\ & & \ddots \\ 0 & & & 1 \ea \right] = \frac{1}{n} I_n.$$
However, this does not lead to any contradiction because of the uniqueness of the random mixture [6].
All the density matrices in (i), (ii), (iii), (iv), (v) above represent pure states.
\noindent{\bf Remark 2.5 :} Any graph with weight function $a=$ constant $>0$ has the same density matrix for all $a>0$. This infinite family of graphs corresponds to the same quantum state (density operator).
\noindent {\bf Definition 2.6 :} A graph $(H, b)$ is said to be a factor of graph $(G, a)$ if $V(H, b) = V(G, a)$ and there exists a graph $(H', b')$ such that $V(H', b') = V(G, a)$ and $M(G, a) = M(H, b) + M(H', b')$. Thus a factor is only a spanning subgraph. Note that
$$ a_{v_iv_j} = \left\{ \begin{array}{lll} b_{v_iv_j} & \mbox{if} & \{v_i,v_j\} \in E(H, b) \\ b'_{v_iv_j} & \mbox{if} & \{v_i,v_j\} \in E(H', b') \ea \right. $$
Now let $(G, a)$ be a graph on $n$ vertices $v_1, \cdots, v_n$ having $m$ edges\\ $\{v_{i_1}, v_{j_1}\}, \cdots, \{v_{i_m}, v_{j_m}\}$ and $s$ loops $\{v_{i_1}, v_{i_1}\} \cdots \{v_{i_s}, v_{i_s}\}$ where $1 \le i_1j_1, \cdots, i_m j_m \le n, 1 \le i_1 i_2 \cdots i_s \le n$.
Let $(H_{i_kj_k}, a_{i_kj_k})$ be the factor of $(G, a)$ such that
$$ [M(H_{i_kj_k}, a_{i_kj_k})]_{u,w} = \left\{ \begin{array}{l} a_{i_kj_k} ~~ \mbox{if}~~ u = i_k~~ \mbox{and}~~ w = j_k ~~\mbox{or}~~ u = j_k, w = i_k \\ 0 ~~ \mbox{otherwise} \ea \right. \eqno{(12)}$$
Let $(H_{i_t,i_t}, a_{i_t i_t})$ be a factor of $(G, a)$ such that
$$ [M(H_{i_ti_t}, a_{i_t i_t})]_{uw} = \left\{ \begin{array}{l} a_{i_t i_t}~~ \mbox{when}~~ u = i_t = w \\ 0 ~~ \mbox{otherwise} \ea \right. \eqno{(13)}$$
\noindent {\bf Theorem 2.7 :} The density matrix of a graph $(G, a)$ as defined above with factors given by equation (12) and (13) can be decomposed as
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} \sum^m_{k=1} 2a_{i_kj_k} \sigma(H_{i_kj_k}, a_{i_kj_k}) + \frac{1}{d_{(G, a)}} \sum^s_{t=1} a_{i_ti_t} \sigma(H_{i_ti_t}, a_{i_ti_t}) \eqno{(14)}$$
or
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} \sum^m_{k=1} 2a_{i_kj_k} P[\frac{1}{\sqrt 2}(|v_{i_k}\rangle - |v_{j_k}\rangle)] + \frac{1}{d_{(G, a)}} \sum^s_{t=1} a_{i_ti_t} P[|v_{i_t}\rangle]\eqno{(15)}$$
\noindent {\bf Proof :} From equation (12), (13) and Theorem 2.3 and Lemma 2.4, the density matrix
$$\sigma (H_{i_kj_k}, a_{i_kj_k}) = \frac{1}{2a_{i_kj_k}} [ \Delta(H_{i_kj_k}, a_{i_kj_k}) - M(H_{i_kj_k},a_{i_kj_k})]$$
is a pure state. Also,
$$ \sigma (H_{i_ti_t}, a_{i_ti_t}) = \frac{1}{a_{i_ti_t}} [ \Delta_0 (H_{i_t, i_t}, a_{i_ti_t})]$$
is a pure state. Now
$$ \Delta(G, a) = \sum^m_{k=1} \Delta(H_{i_kj_k}, a_{i_kj_k}) + \sum^s_{t=1} \Delta_0(H_{i_ti_t}, a_{i_ti_t})$$
$$M(G, a) = \sum^m_{k=1} M(H_{i_kj_k}, a_{i_kj_k}) + \sum^s_{t=1} \Delta_0(H_{i_ti_t}, a_{i_ti_t}).$$
Therefore
$$\sigma(G, a) = \frac{1}{d_{(G, a)}} \left[ \sum^m_{k=1} \Delta(H_{i_kj_k}, a_{i_kj_k}) - \sum^m_{k=1} M(H_{i_kj_k}, a_{i_kj_k})\right]\\
+ \frac{1}{d_{(G, a)}} \left[ \sum^s_{t=1} \Delta_0 (H_{i_ti_t}, a_{i_ti_t})\right]$$
$$= \frac{1}{d_{(G, a)}} \sum^m_{k=1} [\Delta(H_{i_kj_k}, a_{i_kj_k}) - M(H_{i_kj_k}, a_{i_kj_k})] \\
+ \frac{1}{d_{(G, a)}} \sum^s_{t=1} \Delta_0(H_{i_ti_t}, a_{i_ti_t})$$
$$ = \frac{1}{d_{(G, a)}} \sum_k 2a_{i_kj_k} \sigma(H_{i_kj_k}, a_{i_kj_k}) \\
+ \frac{1}{d_{(G, a)}} \sum_t a_{i_ti_t} \sigma(H_{i_ti_t}, a_{i_ti_t}) ~~~~~~~~~~\mbox{(14)}$$
In terms of the standard basis, the $uw$-th element of matrices $\sigma(H_{i_kj_k}, a_{i_kj_k})$ and $\sigma(H_{i_ti_t}, a_{i_ti_t})$ are given by $\langle v_u | \sigma(H_{i_kj_k} , ,a_{i_kj_k}) | v_w \rangle$ and $\langle v_u | \sigma (H_{i_ti_t} a_{i_ti_t} | v_w\rangle$ respectively. In this basis
$$ \sigma(H_{i_kj_k}, a_{i_kj_k}) = P[ \frac{1}{\sqrt 2} ( | v_{i_k} \rangle - | v_{j_k} \rangle )]$$
$$ \sigma(H_{i_ti_t}, a_{i_ti_t}) = P[| v_{i_t} \rangle ] .$$
Therefore equation (14) becomes
$$\sigma(G, a) = \frac{1}{d_{(G, a)}} \sum^m_{k=1} 2a_{i_kj_k} P[\frac{1}{\sqrt 2} (| v_{i_k}\rangle - | v_{j_k} \rangle) + \frac{1}{d_{(G, a)}} \sum ^s_{t=1} a_{i_ti_t} P[ | v_{i_t} \rangle]~~~~~~~~~~\mbox{(15)}$$
$\hspace{\stretch{1}} \blacksquare$
\noindent {\bf Remark 2.8 :} If all weights $a_{i_kj_k} > 0$ then equations (14), (15) give $\sigma(G, a)$ as a mixture of pure states. However, in the next subsection we show that any graph $(G, a)$ having density matrix can be decomposed into graphs (spanning subgraphs) corresponding to pure states.
\subsubsection{Convex combination of density matrices}
Consider two graphs $(G_1, a_1) $ and $(G_2, a_2)$ each on the same $n$ vertices, having $\sigma(G_1, a_1)$ and $\sigma(G_2, a_2)$ as their density matrices respectively. We give an algorithm to construct the graph $(G, a)$ whose density matrix is
$$ \sigma(G, a) = \lambda \sigma(G_1, a_1) + (1 - \lambda) \sigma(G_2, a_2)$$
$0 \le \lambda \le 1, \lambda = \alpha/\beta, \alpha, \beta > 0 $ are real.
We use the symbol $\sqcup$ to denote the union of the edge sets of two graphs $(G_1, a_1) $ and $(G_2, a_2)$ on the same set of vertices to give $(G,a).$ If $\{v_i,v_j\} \in E(G_1,a_1)$ and $\{v_i,v_j\} \in E(G_2,a_2)$ then $a(\{v_i,v_j\}) = a_1(\{v_i,v_j\}) + a_2(\{v_i,v_j\})$. We write $(G,a)= (G_1, a_1) \sqcup(G_2, a_2).$
If $E(G_1,a_1)$ and $E(G_2,a_2)$ are disjoint sets, then we call the resulting graph $(G,a)$ the disjoint edge union of $(G_1,a_1)$ and $(G_2,a_2)$, we write $(G,a)= (G_1, a_1) \dotplus(G_2, a_2).$
The algorithm is as follows :
\noindent {\bf Algorithm 2.9 :}
\begin{description}
\item(i) Put $\lambda = \alpha/\beta$ so that $(1 - \lambda) = \frac{\beta - \alpha}{\beta}$, where $\alpha>0$ ,$\beta>0$ are real.
\item(ii) Write $\sigma(G, a) = \frac{1}{\beta} (\alpha \sigma(G_1,a_1) + (\beta - \alpha) \sigma(G_2, a_2))$.
\item(iii) Modify the weight functions of the two graphs $(G_1, a_1)$ and $(G_2, a_2)$ to get $a_1' = \alpha a_1$ and $a_2' = (\beta - \alpha)a_2$.
\item(iv) The graph $(G, a)$ corresponding to $\sigma$ in step (ii) is
$$ (G, a) = (G_1, a_1') \sqcup (G_2, a_2') \eqno{(16)}$$
such that
$$ a_{v_iv_j} = (a_1')_{v_iv_j} + (a_2')_{v_iv_j} \eqno{(16a)}$$
$$ a_{v_iv_i} = (a_1')_{v_iv_i} + (a_2')_{v_iv_i} \eqno{(16b)}$$
where we take $(a_{1,2}')_{v_iv_j} = 0 = (a_{1,2}')_{v_iv_i}$ if $\{v_i,v_j\},\{v_i,v_i\} \not\in E(G_1, a_1)$ or $E(G_2, a_2)$
\end{description}
$\hspace{\stretch{1}}$ $\blacksquare$
We can apply this algorithm to any convex combination of more than two density matrices $\sigma(G, a)= \sum\limits^k_{i=1} p_i \sigma(G_i, a_i),\; \sum\limits_i p_i = 1$, by writing $p_i = \alpha_i/\beta, \alpha_i,\beta >0$ and real$, i = 1, \cdots, k$.
{\it Example(3)} : consider the density matrices
\begin{description}
\item(i) $\sigma(G_1, a_1) = | + + \rangle \langle + + | = \frac{1}{4} \left[ \begin{array}{cccc} 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 &1 \\ 1 & 1 & 1 & 1 \ea \right]$\\
whose graph is shown in Figure 9\\
\begin{figure}[!h]
\includegraphics[width=4cm,height=3cm]{fig9.eps}
Figure 9
\end{figure}
\begin{eqnarray*}
\sigma(G_2, a_2) & = & \frac{1}{2} | 11 \rangle \langle 11 | + \frac{1}{2} | \psi^+ \rangle \langle \psi^+ | \\
& = & \frac{1}{4} \left[ \begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \ea \right].
\end{eqnarray*}
whose graph is shown in Figure 10
\begin{figure}[!h]
\includegraphics[width=4cm,height=3cm]{fig10.eps}
Figure 10
\end{figure}
The graph corresponding to
\begin{eqnarray*}
\sigma(G, a) & = & \frac{1}{3} \sigma(G_1, a_1) + \frac{2}{3} \sigma(G_2, a_2) \\
& = & \frac{1}{12} \left[ \begin{array}{cccc} 5 & 1 & 1 & 1 \\ 1 & 3 & 3 & 1 \\ 1 & 3 & 3 & 1 \\ 1 & 1 & 1 &1 \ea \right].
\end{eqnarray*}
is given in Figure 11
\begin{figure}[!h]
\includegraphics[width=4cm,height=3cm]{fig11.eps}
Figure 11
\end{figure}
\end{description}
\noindent {\bf Lemma 2.10 :} Let $(G_1, a_1)$, $(G_2, a_2)$ and $(G, a)$ satisfy
$$(G, a) = (G_1, a_1) \sqcup (G_2, a_2) $$ or,
$$(G,a)= (G_1, a_1) \dotplus(G_2, a_2).$$
Then
$$ Q(G, a) = Q(G_1, a_1) + Q(G_2, a_2) $$
and
$$ \sigma(G, a) = \frac{d_{(G_1,a_1)}}{d_{(G, a)}} \sigma(G_1, a_1) + \frac{d_{(G_2,a_2)}}{d_{(G, a)}} \sigma(G_2, a_2).$$
\noindent {\bf Proof :} For two factors of $(G, a), (G_1, a_1)$ and $(G_2, a_2)$ we have
\begin{eqnarray*}
M(G, a) & = & M(G_1, a_1) + M(G_2, a_2) \\
\Delta(G, a) & = & \Delta(G_1, a_1) + \Delta(G_2, a_2)\\
\Delta_0(G, a) & = & \Delta_0(G_1, a_1) + \Delta_0(G_2, a_2) \\
L(G, a) & = & \Delta(G, a) - M(G, a)\\
Q(G, a) & = & L(G, a) + \Delta_0(G, a)
\end{eqnarray*}
Substitute $M(G, a), \Delta(G, a), \Delta_0(G, a)$ and $L(G, a)$ in $Q(G, a)$ as above to get
$$Q(G, a) = Q(G_1, a_1) + Q(G_2, a_2)$$
and also
$$ \sigma(G, a) = \frac{d_{(G_1, a_1)}}{d_{(G, a)}} \sigma(G_1, a_1) + \frac{d_{(G_2, a_2)}}{d_{(G, a)}} \sigma(G_2, a_2).$$\hspace{\stretch{1}}$ \blacksquare$
\noindent {\bf Remark 2.11 :} Obviously, the operation $\sqcup$ is associative. We can apply Lemma 2.10 for more than two graphs,
$$(G, a) = \sqcup_i (G_i, a_i) \Rightarrow Q(G, a) = \sum_i Q(G_i, a_i)$$
and
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} \sum_i d(G_i, a_i) \sigma(G_i, a_i). $$
\noindent {\bf Theorem 2.12 :} Every graph $(G, a)$ having density matrix $\sigma(G, a)$ can be decomposed as $(G, a) = \sqcup_i (G_i, a_i)$ where $\sigma(G_i, a_i)$ is a pure state.
\noindent {\bf Proof :} Every density matrix can be written as the convex combination of pure states $\sigma(G, a) = \sum\limits^k_{i=1} p_i | \psi_i \rangle \langle \psi_i|$.
By applying Algorithm (2.9), Lemma 2.10 and Remark 2.11, we get the result.\hspace{\stretch{1}}$ \blacksquare$
\subsubsection{Tracing out a part}
Consider a bipartite system with dimension $pq$. Let $\sigma(G,a)$ be a state of the system with graph $(G,a)$ having $pq$ vertices labeled by $(ij), i=1,\cdots, p $ and $j=1, \cdots, q$. If we trace out the second part with dimension $q$, we get the state of the first part which is $p \times p$ reduced density matrix of $\sigma(G,a)$. The corresponding graph $(G',a')$ has $p$ vertices indexed by $(i)$ and its weight function $a'$ is given by $$ a'_{ij}=\sum_{k=1}^q a_{ik,jk} , i\neq j$$ and $$a'_{ii}=\sum_{k=1}^q d_{ik}-\sum_{l\in V(G',a')} a'_{il}, l\neq i.$$ Where $d_{ik}$ is the degree of vertex $(ik)$ in original graph.
{\it Example (4)} : Consider a graph $(G,a)$ as shown in Figure 12a in $\mathbb{R}^2 \otimes \mathbb{R}^2$. The corresponding density matrix is $$\sigma^{AB}(G, a) = \frac{1}{16} \left[ \begin{array}{cccc} 9 & -1 & -1 & 1\\ -1 & 3 & -1 & -1 \\ -1 & -1 & 3 & -1 \\ 1 & -1 & -1 & 1 \ea \right]$$\\
after tracing out the second particle the graph $(G',a')$ on two vertices becomes as in Figure 12b with corresponding density matrix $$\sigma^A(G', a') = \frac{1}{16} \left[ \begin{array}{cc} 12 & -2\\ -2 & 4 \ea \right]=\frac{1}{8} \left[\begin{array}{cc} 6 & -1\\ -1 & 2\ea \right]$$\\
which is the same as the reduced density matrix $\sigma^A$ of $\sigma^{AB}$.
\begin{figure}[!h]
\includegraphics[width=4cm,height=4cm]{fig12a.eps}
Figure 12a
\end{figure}
\begin{figure}[!h]
\includegraphics[width=4cm,height=1cm]{fig12b.eps}
Figure 12b
\end{figure}
\section{Von Neumann entropy}
The Von Neumann entropy of $n \times n$ density matrix $\sigma$ is
$$ S(\sigma) = - \sum^n_{i=1} \lambda_i(\sigma) \log_2 \lambda_i(\sigma) $$
It is conventional to define $0 \log 0 = 0$. The Von Neumann entropy is a measure of mixedness of the density matrix. For a pure state $\sigma, S(\sigma)= 0$.
\subsection{Maximum and minimum}
Let
$$ (G, a) = \uplus^n_{i=1} (K^i_1, a_i) \eqno{(17)}$$
where $(K^i_1, a_i)$ is the graph on $i$-th vertex with a loop having weight $a_i > 0$.
\noindent {\bf Lemma 3.1 :} Let $(G, a)$ be given by (17) with the additional constraint that $a_i = c = \frac{1}{n}, i = 1, 2, \cdots, n$. The density matrix of the graph $(G, a)$ is the random mixture of pure states with $\sigma(G, a) = \frac{1}{n} I_n$.
\noindent {\bf Proof :} For the graph $(G, a)$, the first term in (14) vanishes. Then
$$ \sigma(G, a) = \frac{1}{d_{(G,a)}} \sum^n_{t=1} \Delta_0(H_{i_ti_t}, a)$$
where $\Delta_0(H_{i_ti_t}, a)$ is the $n \times n$ matrix with all elements zero except $(i_t, i_t)th$ element which is equal to $a$. This means
$$ \sigma(G, a) = \frac{a}{d_{(G,a)}} \left[ \begin{array}{cccc} 1 & & & 0 \\ & 1 & & \\ & & \ddots \\ 0 & & & 1 \ea \right] = \frac{1}{n} I_n,$$
because $d_{(G,a)}=na$.\hspace{\stretch{1}}$ \blacksquare$
\noindent {\bf Theorem 3.2 :} Let $(G, a)$ be a graph on $n$ vertices. Then
\begin{description}
\item(i) $\max_{(G, a)} S(\sigma(G, a)) = \log_2 n$
\item(ii) $\min_{(G,a)} s(\sigma(G, a)) = 0$, and this value is attained if $\sigma(G, a)$ is pure.
\end{description}
\noindent {\bf Proof :} (i) By Lemma 3.1 $\sigma(G, a)$ defined in the Lemma has eigenvalue $1/n$ with multiplicity $n$. The corresponding Von Neumann entropy is $\log_2 n$. Since $(G, a)$ is on $n$ vertices, the support of $\sigma(G, a)$ has dimension $\le n$. Any matrix having dimension of support $\le n$ cannot have Von Neumann entropy $> \log_2 n$.
(ii) For pure state $S(\sigma(G,a))=0$ and $S(\sigma(G,a))\nless 0$ .\hspace{\stretch{1}}$ \blacksquare$
\section{Separability}
In this section we primarily deal with the graphs representing a bipartite quantum system with Hilbert space $\mathbb{R}^p \otimes \mathbb{R}^q$ of dimension $pq$. Obviously, the corresponding graph has $n = pq$ vertices. We label the vertices using standard (product) basis $\{| v_i \rangle = | u_{s+1} \rangle \otimes |w_t \rangle \}, 0 \le s \le p-1, 1 \le t \le q, i = sq + t$.
\subsection{Tensor product of weighted graphs}
The tensor product of two graphs $(G, a)$ and $(H, b)$ denoted $(G, a) \otimes (H, b)$ is defined as follows.
The vertex set of $(G, a) \otimes (H, b)$ is $V(G, a) \times V(H, b)$. Two vertices $(u_1, v_1)$ and $(u_2, v_2)$ are adjacent if $\{u_1, u_2\} \in E(G, a)$ and $\{v_1, v_2\} \in E(H, b)$. The weight of the edge $\{ (u_1, v_1), (u_2, v_2)\}$ given by $a_{\{u_1, u_2\}} b_{\{v_1,v_2\}}$ and is denoted by $c(\{ (u_1, v_1), (u_2, v_2)\})$. Note that either $u_1$ and $u_2$ or $v_1$ and $v_2$ or both can be identical, to include loops.
The adjacency, degree and the loops matrices of $(G, a) \otimes (H, b)$ are given by
$$M((G, a) \otimes (H, b)) = M(G, a) \otimes M(H, b) \eqno{(18a)}$$
$$ \Delta((G, a) \otimes (H, b)) = \Delta(G, a) \otimes \Delta(H, b) \eqno{(18b)}$$
$$ \Delta_0((G, a) \otimes (H, b)) = \Delta_0(G, a) \otimes \Delta_0(H, b) \eqno{(18c)}$$
Note that
$$ L((G, a) \otimes (H, b)) \ne L(G, a) \otimes L(H, b) $$
$$Q((G, a) \otimes (H, b)) \ne Q(G, a) \otimes Q(H, b).$$
In fact, in general, the tensor product of two graphs having density matrix may not have density matrix.
For two simple graphs $G$ and $H$ we know that [16, 7]
$$d_{G \otimes H} = d_G \cdot d_H.$$
This result is also satisfied by the tensor product of the weighted graphs.
$$ d_{(G, a) \otimes (H, b)} = d_{(G, a)} \cdot d_{(H, b)} . \eqno{(19)}$$
\subsection{Modified tensor product}
We modify the tensor product of graphs in order to preserve positivity of the generalized Laplacian of the resulting graph.
Given a graph $(G, a)$ we define $(G^\phi, a)$ by
$$ V(G^\phi, a) = V(G, a)$$
$$E[(G^\phi, a)] = E(G, a)\setminus \{ \{v_i,v_i\} : \{v_i, v_i\} \in E(G, a)\}$$
That is $(G^\phi, a)$ is obtained from $(G, a)$ by removing all loops.
Given a graph $(G, a)$ we define $(\widetilde{G}, a)$ by
$$ V(\widetilde{G}, a) = V(G, a)$$
$$E(\widetilde{ G}, a) = E(G, a) \setminus \{ \{v_i, v_j\} : i \ne j, \{v_i, v_j\} \in E(G, a)\}.$$
That is, $(\widetilde{ G}, a)$ is obtained by removing all edges connecting neighbors and keeping loops.
Note that in both $(G^\phi, a)$ and $(\widetilde{ G}, a)$, weight function $a$ remains the same, only its support is restricted.
Given a graph $(G, a)$ we define $(-G, a) = (G, - a)$. Given a graph $(G, a)$ we define $(G^\#, a')$
$$V(G^\#,a')= V(G,a)$$
$$(G^\#,a')=\uplus_i^n(K_i,a'_i)$$
where $K_i$ is the graph consisting of $i$th vertex with a loop and $a'_i$ is the weight of the loop on the $i$th vertex. If $a'_i=0$ then there is no loop on the $i$th vertex. $a'_i, i=1,2,\cdots n$ are given by $$a'_i=\sum_{v_k \in V(G,a)}a(\{v_i,v_k\})\eqno{(20a)}$$
Note that the term $v_k=v_i$ is also included in the sum.
We now define the graph operators on the set of graphs
$$ \left. \begin{array}{ll} \mbox{(i)} & \eta : (G, a) \rightarrow (- G, a) = (G ,-a) \\ \mbox{(ii)} & {\cal L} : (G, a) \rightarrow (G^\phi, a) \\ \mbox{(iii)} & {\cal N} : (G, a) \rightarrow (G^\#, a') \\ \mbox{(iv)} & \Omega : (G, a) \rightarrow (\widetilde{ G}, a) \ea \right\} \eqno{(20b)}$$
Some properties of the graph operators defined in (20b) are,
$$ \begin{array}{ll} \mbox{(i)} & M(\eta(G, a)) = - M(G, a) \\ & \Delta(\eta(G, a)) = - \Delta(G, a) \\ & \Delta_0(\eta(G, a)) = - \Delta_0(G, a) \ea \eqno{(21)}$$
$$ d_{\eta(G, a)} = - d_{(G, a)} $$
$$ \begin{array}{ll} \mbox{(ii)} & M({\cal L}(G, a)) = M(G, a) - \Delta_0(G, a) \\ & \Delta({\cal L}(G, a)) = \Delta(G, a) - \Delta_0(G, a) \ea \eqno{(22)}$$
$$ \Delta_0({\cal L}(G, a)) = [0] $$
$$ d_{{\cal L}(G, a)} = Tr(\Delta(G, a)) - Tr(\Delta_0(G, a)) = d_{(G^\phi, a)}$$
$$ \begin{array}{ll} \mbox{(iii)} & M({\cal N}(G, a)) = \Delta(G, a) \\ & \Delta({\cal N}(G, a)) = \Delta(G, a) \\ & \Delta_0({\cal N}(G, a)) = \Delta(G, a) \\ & d_{{\cal N}(G, a)} = Tr(\Delta(G, a)) = d_{(G, a)} \ea \eqno{(23)}$$
$$ \begin{array}{ll} \mbox{(iv)} & M(\Omega(G, a)) = \Delta_0(G, a) \\& \Delta(\Omega(G, a)) =\Delta_0(G, a) \\& \Delta_0(\Omega(G, a)) = \Delta_0(G, a) \\ & d_{\Omega(G, a)} = Tr(\Delta_0(G, a)) \ea \eqno{(24)}$$
{\it Example (5)} :
Given a graph $(G,a)$ as shown in Figure 13a, if we act by $\eta$,
${\cal L}$ ,${\cal N}$ and $\Omega$ on $(G,a)$, we get the graphs $\eta(G,a)$, ${\cal L}(G,a)$, ${\cal N}(G,a)$
and $\Omega(G,a)$
as shown in Figures 13b, 13c, 13d and 13e respectively.
\begin{figure}
\includegraphics[width=12cm,height=5cm]{fig13.eps}
Figure 13
\end{figure}
\noindent {\bf Definition 4.1 :} Let $(G, a)$ and $(H, b)$ be two graphs with $p$ and $q (> p)$ vertices respectively. Then their modified tensor product is defined by
\begin{eqnarray*}
(G, a) \boxdot (H, b) & = &\{ {\cal L}(G, a) \otimes {\cal L} \eta (H, b)\} \dotplus \{{\cal L}(G, a) \otimes {\cal N}(H, b)\}\\
& & \dotplus \{ {\cal N}(G, a) \otimes {\cal L}(H, b)\} \dotplus \{ \Omega(G, a) \otimes \Omega(H, b)\}~~~~~~~~~\mbox{(25)}
\end{eqnarray*}
$$ V\{ (G, a) \boxdot (H, b)\} = V(G, a) \times V(H, b) $$
whose cardinality is $pq$.
$E\{ (G, a) \boxdot (H, b) \} $ = Disjoint union of the edge set of each term in (25).
\noindent {\bf Lemma 4.2 :} (i) $\Delta((G, a) \boxdot (H, b)) = \Delta(G, a) \otimes \Delta(H, b)$.
(ii) $\Delta_0((G, a) \boxdot (H, b)) = \Delta_0(G, a) \otimes \Delta_0(H, b)$.
\noindent {\bf Proof :} Consider the degree matrix of the modified tensor product we have
\begin{eqnarray*}
\Delta((G, a) \boxdot (H, b))& =& \Delta({\cal L}(G, a) \otimes {\cal L}\eta(H, b)) + \Delta({\cal L}(G,a) \otimes {\cal N}(H, b))\\
& & + \Delta({\cal N}(G, a) \otimes {\cal L}(H, b)) + \Delta(\Omega(G, a) \otimes \Omega(H, b)) \\
\end{eqnarray*}
This follows from Lemma 2.10. Using equation (18b) and equations (21) to (24) to the terms on the RHS of the above equation we get
$$\Delta((G, a) \boxdot (H,b)) = \Delta(G, a) \otimes \Delta(H, b).$$
Equations (ii) can be proved similarly.\hfill $\blacksquare$
\noindent {\bf Corollary 4.3 :} $d_{(G,a) \boxdot (H,b)}(v_1,v_2)\; =\;d_{(G,a)}(v_1) \cdot d_{(H,b)}(v_2)$
\noindent {\bf Proof :} This follows directly from equation (i) in Lemma 4.2.\hfill $\blacksquare$
\noindent {\bf Remark 4.4 :} From corollary we get $d_{(G,a)\boxdot(H,b)}\; =\;d_{(G,a)} \cdot d_{(H,b)}$
\noindent {\bf Theorem 4.5 :} Consider a bipartite system in $\mathbb{R}^p \otimes \mathbb{R}^q$ in the state $\sigma$. Then $\sigma = \sigma_1 \otimes \sigma_2$ if and only if $\sigma$ is the density matrix of the graph $(G, a) \boxdot (H, b)$, where $(G, a)$ and $(H, b)$ are the graphs having density matrices $\sigma_1$ and $\sigma_2$ respectively.
\noindent {\bf Proof :} \noindent {\bf If part :} Given $(G, a), (H, b)$ we want to prove
$$\sigma((G, a) \boxdot (H, b)) = \sigma_1(G, a) \otimes \sigma_2(H, b).$$
From the definition of the modified tensor product we can write
$$ \sigma((G,a) \boxdot (H, b)) = \frac{1}{d_{(G, a) \boxdot (H, b)}} \{Q[{\cal L}(G, a) \otimes {\cal L} \eta(H, b)$$
$$ \dotplus {\cal L}(G, a) \otimes {\cal N}(H, b)\dotplus {\cal N}(G, a) \otimes {\cal L}(H, b) \dotplus \Omega(G, a) \otimes \Omega(H, b)]\}$$
Using Lemma 2.10, Remark 2.11 and Remark 4.4 we get
\begin{eqnarray*}
\sigma((G, a) \boxdot (H, b)) & = & \frac{1}{d_{(G, a)} \cdot d_{(H, b)}} [ Q({\cal L}(G, a) \otimes {\cal L} \eta(H, b)) \\
& & + Q({\cal L}(G, a) \otimes {\cal N}(H, b)) + Q({\cal N}(G, a)\\
& & \otimes {\cal L}(H, b)) + Q(\Omega(G, a) \otimes \Omega(H, b))] ~~~~~~~~~~~~~~~~~~~~~~ \mbox{(26)}
\end{eqnarray*}
We can calculate every term in (26) using (21) - (24) and substitute in (26) to get
$$ \sigma((G, a) \boxdot (H, b)) = \sigma(G, a) \otimes \sigma(H, b) .$$
\noindent {\bf Only if part :} Given $\sigma = \sigma_1 \otimes \sigma_2$, consider the graphs $(G, a)$ and $(H, b)$ for $\sigma_1$ and $\sigma_2$ respectively. Then the graph of $\sigma$ has the generalized Laplacian
\begin{eqnarray*}
& & [L(G, a) + \Delta_0(G, a)] \otimes [L(H, b) + \Delta_0(H, b)] \\
& & = L(G, a) \otimes L(H, b) + L(G, a) \otimes \Delta_0(G, a) + \Delta_0(G, a) \otimes L(H, b) + \Delta_0(G, a) \otimes \Delta_0(H, b)
\end{eqnarray*}
Now it is straightforward to check that the graphs corresponding to each term are given by the corresponding terms in the definition of $(G, a) \boxdot (H, b)$. \hspace{\stretch{1}}$ \blacksquare$
\noindent {\bf Remark 4.6 :} Note that the proof of Theorem 4.5 does not depend in any way on the positivity or the hermiticity of the associated generalized Laplacians. Therefore we have $$Q((G,a) \boxdot(H,b))\;=\;Q(G,a)\otimes Q(H,b)$$ for any two graphs $(G,a)$ and $(H,b)$
\noindent {\bf Corollary 4.7 :} The modified tensor product is associative and distributive with respect to the disjoint edge union $\dotplus$.
\noindent {\bf Proof :} Let $(G_1,a_1), (G_2,a_2)$ and $(G_3,a_3)$ be any graphs. Using Theorem 4.5 and Remark 4.6 , we can write
$$Q(((G_1,a_1) \boxdot(G_2,a_2))\boxdot(G_3,a_3))\;$$
$$=\;Q((G_1,a_1) \boxdot(G_2,a_2)) \otimes Q(G_3,a_3) $$
$$=\;(Q(G_1,a_1) \otimes Q(G_2,a_2))\otimes Q(G_3,a_3) $$
$$=\;Q(G_1,a_1) \otimes (Q(G_2,a_2)\otimes Q(G_3,a_3)) $$
$$=\;Q(G_1,a_1) \otimes Q((G_2,a_2)\boxdot (G_3,a_3) )$$
$$=\;Q((G_1,a_1)\boxdot ((G_2,a_2)\boxdot (G_3,a_3) )$$
Therefore ,
$$((G_1,a_1) \boxdot(G_2,a_2))\boxdot(G_3,a_3)\;=\;(G_1,a_1)\boxdot ((G_2,a_2)\boxdot(G_3,a_3) )$$
Similarly, using Lemma 2.10 and distributive property of the matrix tensor product we get
$$Q((G_1,a_1) \boxdot((G_2,a_2))\dotplus (G_3,a_3)))\;$$
$$=\;Q((G_1,a_1) \boxdot(G_2,a_2)) \dotplus ((G_1,a_1) \boxdot (G_3,a_3)) $$
Which gives
$$(G_1,a_1) \boxdot ((G_2,a_2)) \dotplus (G_3,a_3))\;=\; (G_1,a_1) \boxdot (G_2,a_2) \dotplus (G_1,a_1) \boxdot (G_3,a_3) $$
\noindent {\bf Definition 4.8 :} The cartesian product of two weighted graphs $(G,a)$ and $(H,b)$ is denoted $(G,a) \square (H,b)$ with weight function $c$ defined as follows.
$$V(G, a) \times V(H, b).$$
$E((G,a ) \square (H,b))=\{\{(u,v),(x,y)\} |\; u=x$ and $\{v,y\} \in E(H,b),\; v \ne y,\; c(\{(u,v),(u,y)\}) = d_u \cdot b(\{v,y\}) $ or $v = y$ and $\{u,x\} \in E(G,a), \; u \ne x,\; c(\{(u,v),(x,v)\})= d_v \cdot a(\{u,x\}).$
Where $d_u$ and $d_v$ are the degrees of the vertices $u \in E(G,a)$ and $v \in E(H,b)$ respectively. It is straightforward to check that
$$(G,a) \square (H,b)\;=\; {\cal L}(G, a) \otimes {\cal N}(H, b) \dotplus {\cal N}(G, a) \otimes {\cal L}(H, b)$$
Which can be taken to be the definition of the cartesian product of graphs in terms of the operators ${\cal L}$ and $ {\cal N}$ . We also note that
\begin{eqnarray*}
(G, a) \boxdot (H, b) & = &\{ {\cal L}(G, a) \otimes {\cal L} \eta (H, b)\} \dotplus \{(G,a) \square (H,b)\} \dotplus \{ \Omega(G, a) \otimes \Omega(H, b)\}
\end{eqnarray*}
Note that the isolated vertices in $(G,a)$ or $(H,b)$ do not contribute to $(G,a) \square (H,b)$ as their degree is zero.
{\it Example(6)} : Consider $(G,a), (H,b)$ where $V(G,a)=\{1,2\}, E(G,a)=\{ \{1,2\}\} $ and $V(H,b)=\{1,2,3,4\}, E(H,b)=\{\{1,2\}, \{2,3\}, \{3,4\}\}$ with weight functions $a=b=1$, as shown in Figure $14a,14b$. The modified tensor product of these graphs is given by Figures 15a, 15b, 15c, 15d, for each term in (25) and the resulting graph is as shown in Figure 15e. The corresponding density matrix of the graph $(G,a)\boxdot(H,b)$ is
$$\sigma((G, a)\boxdot (H,b)) \; = \; \frac{1}{12} \left[ \begin{array}{cccccccc} 1 & -1 & 0 & 0 & -1 & 1 & 0 & 0 \\-1 & 2 & -1 & 0 & 1 & -2 & 1 & 0 \\ 0 & -1 & 2 & -1 & 0 & 1 & -2 & 1 \\ 0 & 0 & -1 & 1 & 0 & 0 & 1 & -1 \\ -1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 \\ 1 & -2 & 1 & 0 & -1 & 2 & -1 & 0 \\ 0 & 1 & -2 & 1 & 0 & -1 & 2 & -1 \\ 0 & 0 & 1 & -1 & 0 & 0 & -1 & 1 \ea \right].$$
which is the same as $\sigma(G,a)\otimes \sigma (H,b)$ .
\begin{figure}[!h]
\includegraphics[width=6cm,height=3cm]{fig14.eps}
Figure 14
\end{figure}
\begin{figure}[!h]
\includegraphics[width=8cm,height=6cm]{fig15a.eps}
\includegraphics[width=4cm,height=3cm]{fig15b.eps}
Figure 15
\end{figure}
\noindent {\bf Corollary 4.9 :} The density matrix of the modified tensor product of two graphs is separable.
\noindent {\bf Proof :} From Theorem 4.5 we see that $\sigma((G, a) \boxdot (H, b))$ is actually a product state. \hspace{\stretch{1}}$ \blacksquare$
\noindent {\bf Corollary 4.10 :} $\sigma = \sigma_1 \otimes \sigma_2 \otimes \cdots \otimes \sigma_k$ for a $k$-partite system if and only if the graph of $\sigma$ is the modified tensor product of the graphs of $\sigma_1, \cdots, \sigma_k$.
\noindent {\bf Proof :} Apply Theorem 4.2 successively to $(\sigma_1 \otimes \sigma_2), ((\sigma_1 \otimes \sigma_2) \otimes \sigma_3)\cdots $ and then use the associativity of the modified tensor product corollary 4.7. \hspace{\stretch{1}}$ \blacksquare$
\noindent {\bf Corollary 4.11 :} A state $\sigma$ of a $k$-partite system is separable if and only if the graph (G,a) for $\sigma$ has the form
$$ (G,a)\;=\; \sqcup_i \boxdot^k_{j=1} (G^j_i, a^j_i). $$
\noindent {\bf Proof :} Let $\sigma$ be separable i.e.
$$ \sigma = \sum_i w_i \sigma_i^{(1)} \otimes \sigma_i^{(2)} \otimes \cdots \otimes \sigma_i^{(k)}, ~~ \sum_i w_i = 1.$$
By Algorithm 2.9 and Corollary 4.10 the graph of $\sigma$ has the form
$$(G, a) = \sqcup_i \boxdot^k_{j=1} (G^j_i, a^j_i).$$
Now let the graph of a $k$-partite state be
$$(G, a) = \sqcup_i \boxdot^k_{j=1} (G^j_i, a^j_i).$$
Then by Lemma 2.10 , Remark 2.11 and the above corollary to Theorem 4.5
$$ \sigma(G, a) = \sum_i w_i \sigma_1^{(1)} \otimes \sigma_1^{(2)} \otimes \cdots \otimes \sigma_i^{(k)}.$$ \hspace{\stretch{1}}$ \blacksquare$
Corolary 4.11 says that Werner's definition [1] of a separable state in $\mathbb{R}^{q_1} \otimes \mathbb{R}^{q_2} \otimes \mathbb{R}^{q_3} \otimes \cdots \otimes \mathbb{R}^{q_k}$ system, can be expressed using corresponding graphs.
\noindent {\bf Lemma 4.12 :} For any $n = pq$ the density matrix $\sigma(K_n, a)$ is separable in $\mathbb{R}^p \otimes \mathbb{R}^q$ if the weight function is constant $> 0$.
\noindent {\bf Proof :} The proof is same as that given for corollary 4.3 in [7], for simple graph.\hspace{\stretch{1}}$ \blacksquare$
{\it Example (7)} : Consider the graph $(K_4, a)$. The vertices of $(K_4, a)$ are denoted by 1, 2, 3, 4, where weight function is constant, say , $a = 3 > 0$ and has loops in vertices 1, 2. We associate to these vertices the orthonormal basis $\{ |1\rangle = |1\rangle |1\rangle, |2\rangle = |1\rangle |2\rangle, |3\rangle = |2\rangle |1\rangle, |4\rangle = |2\rangle |2\rangle\}$. In terms of this basis $\sigma(K_4, a)$ can be written as
$$ \sigma(K_4, a) = \frac{1}{42} \left[ \begin{array}{rrrr} 12 & -3 & -3 & -3 \\ -3 & 12 & -3 & -3 \\ -3 & -3 & 9 & -3 \\ -3 & -3 & -3 & 9 \ea \right] = \frac{1}{14} \left[ \begin{array}{rrrr} 4 & -1 & -1 & -1 \\ -1 & 4 & -1 & -1 \\ -1 & -1 & 3 & -1 \\ -1 & -1 & -1 & 3 \ea \right] $$
and from equation (15) we can write $\sigma(K_4, a)$ as
\begin{eqnarray*}
\sigma(K_4, a) & = & \frac{1}{42} [6P[|1\rangle \frac{1}{\sqrt 2} (|1\rangle - |2\rangle)] + 6P[ \frac{1}{\sqrt 2} (|1\rangle - |2\rangle)|1\rangle]\\
& & + 6P[ \frac{1}{\sqrt 2} (|1\rangle - |2\rangle) |2\rangle] + 6P[|2\rangle \frac{1}{\sqrt 2} (|1\rangle - |2\rangle)] + 6P[\frac{1}{\sqrt 2} (|11\rangle - |22\rangle)] \\
& & 6P[ \frac{1}{\sqrt 2} (|12\rangle - |21\rangle)] + 3P[|11\rangle] + 3P[|12\rangle]\}
\end{eqnarray*}
\begin{eqnarray*}
\sigma(K_4, a) & = & \frac{1}{7} P[|1\rangle \frac{1}{\sqrt 2} (|1\rangle - |2\rangle)] + \frac{1}{7} P[ \frac{1}{\sqrt 2}(|1\rangle - |2\rangle) |1\rangle] \\
& & + \frac{1}{7} P[ \frac{1}{\sqrt 2} ( |1\rangle - |2\rangle)|2\rangle] + \frac{1}{7} P[|2\rangle \frac{1}{\sqrt 2} (|1\rangle - |2\rangle)] \\
& & + \frac{2}{7} \{ \frac{1}{2} P[ \frac{1}{\sqrt 2} (|11\rangle - |22\rangle)] + \frac{1}{2} P[ \frac{1}{\sqrt 2} (|12\rangle - |21\rangle)]\} \\
& & \frac{1}{14} P[|11\rangle] + \frac{1}{14} P[|12\rangle].
\end{eqnarray*}
Each of the first four terms in the above expression is a projector on a product state, and also the last two terms are projectors, while the fifth and sixth terms give rise to the separable density matrix $\frac{1}{2} p[\mid - \rangle \mid + \rangle] + \frac{1}{2} p[| + \rangle | - \rangle]$, where $| \pm\rangle \stackrel{def}{=} \frac{1}{\sqrt 2} (|1\rangle \pm |2\rangle)$ [7]. Thus $\sigma(K_4, a)$ , $a$ constant, is separable in $\mathbb{R}^2 \otimes \mathbb{R}^2$.
Note that there exists a graph which is complete with a real weight function, which is entangled as the following graph shows in
\begin{figure}[!h]
\includegraphics[width=2cm,height=2cm]{fig16.eps}
Figure 16
\end{figure}
\noindent {\bf Remark 4.13 :} Separability of $\sigma(K_n, a)$ with constant weight function $> 0$ does not depend upon the labeling of $V(K_n, a)$ provided every vertex has a loop or there are no loops. Given a graph, an isomorphism from $(G, a) \longmapsto (G, a)$ is called automorphism. Under composition of maps, the set of automorphisms of $(G, a)$ form a group denoted $Aut(G, a)$. If $\sigma(K_n, a)$ is separable, and if the $Aut(K_n, a) = S_n ,\; (G, a) \cong (K_n, a)$ is also separable. Note that $Aut(K_n, a) = S_n$ provided all weights are equal and either every vertex has a loop or there are no loops.
\noindent {\bf Lemma 4.14 :} The complete graph $(K_n, a)$ on $n \geq 2$ vertices corresponding to a separable state with weight function $ constant > 0$ is not a modified tensor product of two graphs.
\noindent {\bf Proof :} It is clear that, if $n$ is prime then $(K_n, a)$ is not a modified tensor product of graphs. We then assume that $n$ is not a prime. Suppose that there exist graphs $(G, b)$ and $(H, c)$ respectively on $p$ and $s$ vertices such that $(K_{ps}, a) = (G, b) \boxdot (H, c)$ where $ b$ and $c$ are constants . From the definition of the modified tensor product
$$ a(\{ (u_1,v_1), (u_2, v_2)\}) = b(\{ u_1, u_2\}) \cdot c(\{ v_1, v_2\}),$$
the degree sum is
$$ d_{(G, b)} = \sum_{u \in V(G, b) } d_u = \sum_{u \in V(G, b)} \sum_{w \in V(G, b)} b_{wv} = 2b|E(G, b)|.$$
We know that
$ d_{(G, b)} \le b(p(p-1))$ and also $d_{(H, c)} = 2c|E(H, b)| \le cs(s-1)$ and
$$ d_{(G, b)}\cdot d_{(H, c)} \le bcps(p-1) (s-1) = bcps(ps - p - s + 1) .\eqno{(27)}$$
Now observe that $V((G, b) \boxdot (H, c)) = ps$ and,
$$ d_{(G, b) \boxdot (H, c)} = aps(ps -1) \eqno{(28)}$$
because $(G, b) \boxdot (H, c) = (K_{ps}, a)$.
We know that
$$ d_{(G, b) \boxdot (H, c)} = d_{(G, b)}\cdot d_{(H, c)}. \eqno{(29)}$$
Substituting from (27) and (28) we see that (29) is satisfied only when $ p = 1 = s $ , i.e. $n = ps = 1$. \hspace{\stretch{1}}$ \blacksquare$
Lemma 4.12, Lemma 4.14 and Theorem 4.5 together imply that a complete graph $(K_n,a)$ on $n \ge2$ vertices with $a=$ constant $>0$ is a separable state but not a product state.
{\bf Definition 4.15 :} Consider a graph $(G,a)$, without loops, pertaining to a bipartite system of dimension $pq$ . The partial transpose of $(G,a)$, denoted $(G^{\Gamma_B},a')$ , is a graph defined as $V(G^{\Gamma_B},a')=V(G,a)$ , $\{il,kj\}\in E(G^{\Gamma_B},a') \Longleftrightarrow \{ij,kl\}\in E(G,a)$ and $a'(\{il,kj\}=a(\{ij,kl\}.$
{\bf Lemma 4.16 :} Consider a bipartite separable state $\sigma(G,a)$ with the associated graph $(G,a)$ without loops. Then $\Delta(G,a)=\Delta(G^{\Gamma_B},a'),$ where $(G^{\Gamma_B},a')$ is the partial transpose of $(G,a)$.
{\bf Proof :} Let $Q(G,a)$ be the Laplacian of a graph $(G,a)$ with real weights without loops, on $n$ vertices. Let $D$ be any $n\times n$ real diagonal matrix in the standard orthonormal basis $\{|v_i\rangle\};i=1,2,\dots,n$, such that $D\neq 0$ and $Tr(D)=0$. This means that there is at least one negative entry in the diagonal of $D$. Denote this element by $D_{ii}=b_i$. Let $|\psi_0 \rangle= \sum_j|v_j\rangle$ and $|\phi\rangle=\sum_j \chi_j|v_j \rangle$ where,
\begin{displaymath}
\chi_j =
\left\{ \begin{array}{ll}
0 & \textrm{if $j\neq i$}\\
k \in R & \textrm{if $j=i$}
\end{array} \right.
\end{displaymath}
Let $|\chi \rangle=|\psi\rangle+|\phi\rangle=\sum_{i=1}^n(1+\chi_j)|v_j\rangle$. Then
\begin{eqnarray*}
\langle\chi|Q(G,a)+D|\chi\rangle& = & \langle\psi_0|Q(G,a)|\psi_0\rangle+
\langle\psi_0|Q(G,a)|\phi\rangle+\langle\phi|Q(G,a)|\psi_0\rangle+\\
& & \langle\phi|Q(G,a)|\phi\rangle+\langle\psi_0|D|\psi_0\rangle+\langle\psi_0|D|\phi\rangle+\langle\phi|D|\psi_0\rangle+\langle\phi|D|\phi\rangle
\end{eqnarray*}
Since $|\psi_0\rangle$ is (unnormalized) vector having all components equal unity, from equation (9) it follows that $\langle\psi_0|Q(G,a)|\psi_0\rangle=0$. Also $\langle\psi_0|D|\psi_0\rangle=Tr(D)=0$.
We have
$$\langle\phi|Q(G,a)|\phi\rangle=k^2(Q(G,a))_{ii}=k^2d_i$$
$$\langle\psi_0|Q(G,a)|\phi\rangle=\langle\phi|Q(G,a)|\psi_0\rangle = 0.$$
Finally, the remaining terms in the above equation are given by
$$\langle\phi|D|\phi\rangle=b_ik^2$$
$$\langle\psi_0|D|\phi\rangle = b_ik=\langle\phi|D|\psi_0\rangle.$$
Thus
$$\langle\chi|Q(G,a)+D|\chi\rangle = k^2(b_i+d_i)+ 2kb_i$$
So we can then always choose a positive $k$ , such that
$$\langle\chi|Q(G,a)+D|\chi\rangle < 0.$$
It then follows $Q(G,a)+D \ngeq 0.$
This expression is identical with that obtained in [2].
For any graph $G$ on $n=pq$ vertices $$v_1=u_1w_1, v_2=u_1w_2, \dots, v_{pq}=u_pw_q,$$ consider the degree condition $\Delta(G) = \Delta(G^{\Gamma_B}).$ Now $$(L(G))^{\Gamma_B}=(\Delta(G) - \Delta(G^{\Gamma_B}))+ L(G^{\Gamma_B}).$$
Let $$D=\Delta(G) - \Delta(G^{\Gamma_B}).$$
Then $D$ is an $n\times n$ real diagonal matrix with respect to the orthonormal basis $$|v_i\rangle = |u_1\rangle\otimes |w_1\rangle, \dots, |v_{pq}\rangle = |u_p\rangle\otimes |w_q\rangle.$$
Also $$Tr(D)=Tr(\Delta(G))-Tr(\Delta(G^{\Gamma_B}))= 0.$$
We have two possible cases : $D\neq 0$ or $D = 0$. If $D\neq 0$, that is the degree condition is not satisfied $(i.e. \Delta(G)\neq \Delta(G^{\Gamma_B}))$ we have seen that $L(G)+ D\ngeq 0$. As a consequence, $L(G^{\Gamma_B})+D\ngeq 0$ and then $(L(G))^{\Gamma_B}\ngeq 0.$ Hence $\rho(G) $ is entangled.\hspace{\stretch{1}}$ \blacksquare$
{\bf Lemma 4.17 :} A graph $(G,a)$ for a bipartite state corresponds to a separable state if $\{ij,kl\}$ $(i\neq k,j\neq l) \in E(G,a)\Longrightarrow \{il,kj\}\in E(G,a)$ and $a_{ij,kl}=a_{il,kj}$.
{\bf Proof :} Suppose $a_{ij,kl}=a_{il,kj}=a,i\neq k,j\neq l.$ The contribution of the corresponding two edges is $$a \{ P[\frac{1}{\sq2}(|ij\rangle-|kl\rangle)]+P[\frac{1}{\sq2}(|il\rangle-|kj\rangle)]\}$$ which is a separable state. Thus all such pairs contribute separable states. Any other edge $\{ij,kl\}$ with $i=k$ or $ j=l$ has the contribution $a_{ij,kl}P[|i\rangle \otimes (\frac{1}{\sq2}(|j\rangle-|l\rangle))]$ which is separable. Loops contribute the product states $P[|ii\rangle]$.
\hspace{\stretch{1}}$ \blacksquare$
The reverse implication is not true in general. The counter-example is the graph (Figure 12a) in example (4) which is separable.
\section{Graph Operators}
A graph operation is a map that takes a graph to another graph [17]. We deal with four cases namely deleting and adding an edge and deleting and adding a vertex.
Deleting an edge $(\{ v_i, v_j\}, a_{v_iv_j})]$ from a graph $(G, a)$ results in a graph\\ $(G, a) - ( \{ v_i, v_j\}, a_{v_iv_j}) \stackrel{def}{=} (V(G, a), E(G, a) \setminus \{ v_i, v_j\})$ with $a_{v_iv_j} = 0$. Note the possibility $v_i = v_j$ corresponding to the edge being a loop. Addition of an edge $(\{ u_i, v_j\}, a_{ij})$ maps $(G, a)$ to the graph $(G, a) + (\{ v_i, v_j\}, a_{ij}) \stackrel{def}{=} [V(G, a), E(G, a) \cup \{ v_i, v_j\}]$ with $a_{v_iv_j} = a_{ij}$. Deletion of a vertex $v_i$ maps $(G,a)$ to $(G, a) - \{ v_i\} \stackrel{def}{=} [V(G, a) \setminus \{ v_i\}, E(G, a) \setminus E_i]$ where $E_i$ is the set of all edges incident to $v_i$ (including the loop on $v_i)$ with the weight function zero for the edges in $E_i$. Adding a vertex $v_i$ to $(G, a)$ maps $(G, a)$ to $(G, a) + \{ v_i\} \stackrel{def}{=} (V(G, a) \cup \{ v_i\}, E(G, a))$.
A very important point is that, in general, the set of graphs having density matrix is not closed under these operations. Addition of an edge with positive weight and deletion of an edge with negative weight preserves the positivity of the generalized Laplacian resulting in the graph having density matrix. However, addition (deletion) of an edge with negative (positive) weight may lead to a graph which does not have density matrix. In the next section, we give a method for addition and deletion of an edge which preserves the positivity of the generalized Laplacian. Deletion and addition of vertices always preserves the positivity of the generalized Laplacian.
Let ${\cal B}({\cal H}^n)$ be the space of all bounded linear operators on ${\cal H}^n$. A linear map $\Lambda : {\cal B}({\cal H}^n) \rightarrow {\cal B}({\cal H}^m)$ is said to be hermiticity preserving if for every hermitian operator $O \in {\cal B}({\cal H}^n), \Lambda(O)$ is an hermitian operator in ${\cal B}({\cal H}^m)$. A hermiticity preserving map $\Lambda : {\cal B}({\cal H}^n) \rightarrow {\cal B}({\cal H}^m)$ is said to be positive if for any positive operator $O \in {\cal B}({\cal H}^n), \Lambda(O)$ is a positive operator in ${\cal B}({\cal H}^m)$. A positive map $\Lambda : {\cal B}({\cal H}^n) \rightarrow {\cal B}(H^m)$ is said to be completely positive if for each positive integer $k, (\Lambda \otimes I_{k^2}) : {\cal B}({\cal H}^n \otimes {\cal H}^k) \rightarrow {\cal B}(H^m \otimes {\cal H}^k)$ is again a positive map. A completely positive map $\Lambda : {\cal B}({\cal H}^n) \rightarrow {\cal B}({\cal H}^m)$ is said to be trace preserving if $Tr(\Lambda(O)) = Tr(O)$, for all $O \in {\cal B}({\cal H}^n)$. A quantum operation is a trace preserving completely positive map (for short, TPCP) [9, 5]. In standard Quantum Mechanics, any physical transformation of a quantum mechanical system is described by a quantum operation [6]. We are going to use the following result:
\noindent {\bf (Kraus representation Theorem)} [10] : Given a quantum operation $\Lambda : {\cal B}({\cal H}^n) \rightarrow {\cal B}({\cal H}^m)$, there exist $m \times n$ matrices $A_i$, such that $\Lambda(\rho) = \sum\limits_i A_i \rho A_i^\dagger$, where $\rho$ is any density matrix acting on ${\cal H}^n$ and $\sum\limits_i A^\dagger_i A_i = I_m$ (The converse is true). The matrices $A_i$'s are called Kraus operators.
A projective measurement ${\cal M} = \{ P_i; i = 1, 2, \cdots, n\}$, on a quantum mechanical system $S$ whose state is $\rho$, consists of pairwise orthogonal projectors $P_i : {\cal H}_s \rightarrow {\cal H}_s$, such that $\sum\limits^n_{i=1} P_i = I_{dim({\cal H}_s)}$. The $i$-th outcome of the measurement occurs with probability $Tr(P_i \rho)$ and the post-measurement state of $S$ is $\frac{P_i\rho P_i}{tr(P_i \rho)}.$ Whenever the $i$-th outcome of the measurement occurs, we say that $P_i$ clicks. Last two paragraphs apply to complex Hilbert space and so also to real Hilbert space.
\subsection{Deletion and addition of an edge for a weighted graph with all weights $> 0$ }
Here we describe how to delete or add an edge by means of TPCP. Our method of deleting an edge from a weighted graph with all positive weights is a simple generalization of the method in [7]. Let $(G, a)$ be a graph on $n$ vertices $v_1, \cdots, v_n$ and $m$ edges $\{v_{i_1} v_{j_1}\} \cdots \{v_{i_m} v_{j_m}\}, i_k \ne j_k, k = 1, \cdots, m$ and $s$ loops $\{ v_{i_1} v_{i_1} \} \cdots \{ v_{i_s}v_{i_s}\}$. Our purpose is to delete the edge $\{ v_{i_k} v_{j_k}\}, i_k \ne j_k$. Then we have
\begin{eqnarray*}
\sigma(G,a) = \frac{1}{d_{(G,a)}} \{ \sum^m_{\ell=1} 2a_{i_\ell j_\ell} P[\frac{1}{\sqrt 2} (| v_{i_\ell} \rangle - | v_{j_\ell} \rangle)] + \sum^s_{t=1} a_{i_ti_t} P[|v_{i_t} \rangle ]\}
\end{eqnarray*}
and
\begin{eqnarray*}
\sigma((G, a) - \{ v_{i_k} v_{j_k}\}) = \frac{1}{d_{(G,a)} - 2a_{i_kj_k}}
\left\{ \sum^m_{\begin{array}{c} \ell =1 \\ \ell \ne k \ea} 2a_{i_\ell j_{\ell}} P [ \frac{1}{\sqrt 2} (|v_{i_\ell} \rangle - | v_{j_\ell} \rangle)] + \sum^s_{t=1} a_{i_ti_t} P[|v_{i_t}\rangle ]\right\}.
\end{eqnarray*}
A measurement in the basis ${\cal M} = \{ \frac{1}{\sqrt 2} (| v_{i_k} \rangle \pm | v_{j_k} \rangle), |v_i\rangle : i \ne i_k, j_k$ and $i = 1, 2, \cdots, n\}$ is performed on the system prepared in the state $\sigma(G, a)$.
The probability that $P_+ = P[\frac{1}{\sqrt 2} (|v_{i_k} \rangle + | v_{j_k} \rangle )]$ clicks is
\begin{eqnarray*}
& & Tr[P_+ \sigma(G, a)] = \sum^n_{i=1} \langle v_i | P_+ \sigma(G, a) | v_i \rangle \\
& & = \frac{1}{2d_{(G, a)}} \{ \sum^m_{\begin{array}{c} \ell = 1 \\ \ell \ne k \ea} a_{i_\ell j_\ell}[\delta_{i_k i_\ell} - \delta_{i_k j_\ell} + \delta_{j_ki_\ell} - \delta_{j_kj_\ell}]^2 + \sum^s_{t=1} a_{i_t i_t}(\delta_{i_ti_k} +\delta_{i_tj_k})^2\}~~~~~~~~~~~ {(30)}
\end{eqnarray*}
The state after the measurement is $P [ \frac{1}{\sqrt 2} (|v_{i_k} \rangle + | v_{j_k} \rangle]$. Let $U^+_{k \ell} $ and $U^+_{kt}$ be $n \times n$ unitary matrices such that $U^+_{k\ell} [ \frac{1}{\sqrt 2} (| v_{i_k} \rangle + | v_{j_k} \rangle)] = \frac{1}{\sqrt 2} (|v_{i_\ell} \rangle - | v_{j_\ell} \rangle)$ for $\ell = 1, \cdots, k -1, k+1, \cdots, m$ and $U^+_{kt} [ \frac{1}{\sqrt 2} (| v_{i_k} \rangle + | v_{j_k} \rangle )] = | v_{i_t} \rangle, t = 1, \cdots, s$. Now, with probability $2a_{i_\ell j_\ell}/(d_{(G, a)} - 2a_{i_kj_k})$
we apply $U^+_{k \ell}$ on $P [ \frac{1}{\sqrt 2} (|v_{i_k} \rangle + | v_{j_k} \rangle]$ for each $\ell = 1, \cdots, k -1, k+1, \cdots, m$ ,and with probability $a_{i_ti_t}/(d_{(G, a)} - 2a_{i_kj_k})$ we apply $U^+_{k t} $ on $P [ \frac{1}{\sqrt 2} (|v_{i_k} \rangle + | v_{j_k} \rangle]$ for each $t = 1, \cdots, s$. Finally we obtain $\sigma(( G, a) - \{ v_{i_k} v_{j_k} \})$ with probability given by $(30).$ The probability that $ P[ \frac{1}{\sqrt 2} (|v_{i_k} \rangle - | v_{j_k} \rangle )]$ clicks is
\begin{eqnarray*}
& & \frac{1}{2d_{(G, a)}} \{ \sum^m_{\begin{array}{c} \ell = 1 \\ \ell \ne k \ea} a_{i_\ell j_\ell}[\delta_{i_k i_\ell} - \delta_{i_k j_\ell} - \delta_{j_ki_\ell} + \delta_{j_kj_\ell}]^2 + \sum^s_{t=1} a_{i_t i_t}(\delta_{i_ti_k} - \delta_{i_tj_k})^2\} ~~~~~~~ (31)
\end{eqnarray*}
the state after measurement is $ P[ \frac{1}{\sqrt 2} (|v_{i_k} \rangle - | v_{j_k} \rangle )]$. Let $U^-_{k \ell}$ and $U^-_{k t}$ $n \times n$ unitary matrices such that
$$ U^-_{k \ell} \frac{1}{\sqrt 2} (|v_{i_k} \rangle - | v_{j_k}\rangle) = \frac{1}{\sqrt 2} (|v_{i_\ell} \rangle - | v_{j_\ell} \rangle)$$
for $\ell = 1, \cdots, k-1, k+1, \cdots, m$ and
$$ U^-_{k t} \frac{1}{\sqrt 2} (|v_{i_k} \rangle - | v_{j_k}\rangle) = |v_{i_t} \rangle$$ for $t=1,.....\cdot,s$. With probability $2a_{i_\ell j_\ell}/(d_{(G, a)} - 2a_{i_kj_k})$
we apply $U^-_{k\ell}$ on $P[\frac{1}{\sqrt 2} (| v_{i_k} \rangle - | v_{j_k} \rangle )]$ for each $\ell = 1, \cdots, k-1, k+1, \cdots, m$ and with probability $a_{i_ti_t}/(d_{(G, a)} - 2a_{i_kj_k})$ we apply $U^-_{kt}$ on $P[ \frac{1}{\sqrt 2} (| v_{i_k} \rangle - | v_{j_k} \rangle)]$ for each $t = 1, 2, \cdots, s$. Finally we obtain $\sigma(( G, a) - \{ v_{i_k} v_{j_k} \})$ with probability given by $(31)$.
The probability that $P[| v_i\rangle]$ where $i \ne i_k,j_k$ and $i = 1, \cdots, n$ clicks is
$$ \frac{1}{d_{(G, a)}}\left\{ \sum^m_{\begin{array}{c} \ell =1 \\ \ell \ne k\ea} a_{i_\ell j_\ell} (\delta_{i i_\ell} - \delta_{ij_\ell})^2 + \sum^s_{t=1} a_{i_ti_t}(\delta_{ii_t})^2 \right\} \eqno{(32)}$$
and the state after measurement is $P[|v_i\rangle]$. Let $U_{i \ell} $ and $U_{it}$ be $n \times n$ unitary matrices such that $U_{i \ell}[ | v_i\rangle] = \frac{1}{\sqrt 2} (| v_{i_\ell} \rangle - | v_{j_\ell}\rangle]$ for $\ell = 1, \cdots k-1, k+1, \cdots, m$ and $U_{it} [| v_{i} \rangle] = | v_{i_t} \rangle$ for $t = 1, \cdots, s$. With probability $2a_{i_\ell j_\ell} /(d_{(G, a)} - 2a_{i_k j_k})$ we apply $U_{i\ell}$ on $P[| v_i\rangle]$ for each $\ell = 1, \cdots, k-1, k+1, \cdots, m$ and with probability $a_{i_t i_t}/(d_{(G, a)} - 2a_{i_k j_k})$ we apply $U_{it}$ on $P[|v_i\rangle]$ for each $t = 1, \cdots, s$.
We obtain $\sigma((G, a) - \{ v_{i_k}, v_{j_k} \})$ with probability given by (32). This completes the process.
The set of Kraus operators that realizes the TPCP for deleting the edge $\{ v_{i_k}, v_{j_k}\}$ is then
\begin{eqnarray*}
& & \{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} - 2a_{i_kj_k}}} U^+_{k\ell} P[ \frac{1}{\sqrt 2} (|v_{i_k} \rangle + | v_{j_k}\rangle)] ;
\ell = 1, \cdots, k-1, k+1, \cdots, m \} \\
& & \cup \{\sqrt{\frac{a_{i_ti_t}}{d_{(G,a)} - 2a_{i_kj_k}}} U^+_{kt} P[\frac{1}{\sqrt 2} (|v_{i_k} \rangle + | v_{j_k}\rangle)]: t = 1, \cdots, s \} \\
& & \cup \{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} - 2a_{i_kj_k}}} U^-_{k\ell} P[\frac{1}{\sqrt 2} (|v_{i_k} \rangle - | v_{j_k}\rangle)]: \ell = 1, \cdots, k-1, k+1, \cdots m\}\\
& & \cup \{ \sqrt{\frac{a_{i_ti_t}}{d_{(G, a)} - 2a_{i_kj_k}}} U^-_{kt} P[\frac{1}{\sqrt 2} (|v_{i_k} \rangle - | v_{j_k}\rangle)] : t = 1, \cdots, s\}\\
& & \cup \{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} - 2a_{i_k j_k}}} U_{i\ell} P[ | u_i\rangle] : i = 1, \cdots, n, i \ne i_k, j_k ; \ell = 1, \cdots, k-1, k+1, \cdots, m\} \\
& & \cup \{ \sqrt{\frac{a_{i_t i_t}}{d_{(G, a)} - 2a_{i_kj_k}}} U_{it} P[|v_i\rangle] : i = 1, \cdots, n, i \ne i_k, j_k; t = 1, \cdots, s \}
\end{eqnarray*}
The set of Kraus operators that realizes TPCP for adding back edge $\{ v_{i_k}, v_{j_k} \}$ to $(G, a) - \{ v_{i_k} v_{j_k} \}$ is.
\begin{eqnarray*}
& & \left\{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} + 2a_{i_kj_k}}} V^+_{k\ell} P[\frac{1}{\sqrt 2} (|v_{i_k}\rangle + |v_{j_k}\rangle)] : \ell = 1, 2, \cdots, m\right\} \\
& &\cup \left\{ \sqrt{\frac{a_{i_ti_t}}{d_{(G, a)} + 2a_{i_kj_k}}} V^+_{kt} P[\frac{1}{\sqrt 2} (|v_{i_k}\rangle + | v_{j_k}\rangle)] : t = 1, \cdots, s\right\} \\
& & \cup \left\{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} + 2a_{i_kj_k}}} V^-_{k\ell} P[\frac{1}{\sqrt 2} (|v_{i_k} \rangle - |v_{j_k}\rangle)] : \ell = 1, 2, \cdots, m\right\} \\
& & \cup \left\{ \sqrt{\frac{a_{i_ti_t}}{d_{(G, a)} + 2a_{i_kj_k}}} V^-_{kt} P[ \frac{1}{\sqrt 2} (|v_{i_k}\rangle - |v_{j_k}\rangle)] : t = 1, \cdots, s\right\}\\
& & \cup \left\{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} + 2a_{i_kj_k}}} V_{i\ell} P[|v_i\rightarrow] : i = 1, 2, \cdots, n, i \ne i_k, j_k, \ell = 1, 2, \cdots, m\right\}\\
& & \cup \left\{ \sqrt{\frac{a_{i_ti_t}}{d_{(G, a)} + 2a_{i_kj_k}}} V_{it} P[|v_i\rangle] : i = 1, 2, \cdots, n, i \ne i_k, j_k, t = 1, 2, \cdots, s\right\}
\end{eqnarray*}
where $V^+_{k\ell}, V^-_{k\ell}, V^-_{kt}, V_{i\ell}, V_{it}$ are $n \times n$ unitary matrix defined as follows:
\begin{eqnarray*}
& & V^+_{k\ell} \frac{1}{\sqrt 2}(|v_{i_k} \rangle + | v_{j_k}\rangle) = \frac{1}{\sqrt 2} (|v_{i_\ell}\rangle - |v_{j_\ell}\rangle), ~~ \mbox{for}~~ \ell = 1, 2, \cdots, m\\
& & V^+_{k t} \frac{1}{\sqrt 2} (|v_{i_k}\rangle + |v_{j_k}\rangle) = |v_{i_t}\rangle,~~ \mbox{for}~~ t = 1, \cdots, s, \\
& & V^-_{k \ell} \frac{1}{\sqrt 2} (|v_{i_k} \rangle - |v_{j_k}\rangle) = \frac{1}{\sqrt 2} (|v_{i_\ell} \rangle - |v_{j_\ell}\rangle), ~ \mbox{for}~~ \ell = 1, 2, \cdots, m\\
& & V^-_{kt} \frac{1}{\sqrt 2} (|v_{i_k} \rangle - |v_{j_k} \rangle) = |v_{i_t}\rangle) ,~~ \mbox{for}~~ t = 1, \cdots, s \\
& &V_{i\ell} |v_i\rangle = \frac{1}{\sqrt 2} (|v_{i_\ell}\rangle - |v_{j_\ell}\rangle),~~ \mbox{for}~~ i = 1, 2, \cdots, n, i \ne i_k, j_k, \ell = 1, \cdots, m\\
& & V_{it} |v_i\rangle = |v_{i_t}\rangle,~~ \mbox{for}~~ i = 1, \cdots, n, i \ne i_k, j_k, t = 1, \cdots, s.
\end{eqnarray*}
For deleting a loop $\{v_{i_{t'}}, v_{i_{t'}}\}$ a measurement in the basis $\{ | v_i \rangle, i = 1, \cdots,n \}$ is performed on the system prepared in the state $\sigma(G, a)$. Then the probability that $P[| v_i \rangle]$ clicks for $i = 1, \cdots n$ is
$$ \frac{1}{d_{(G, a)}} \left\{ \sum^m_{\ell =1} a_{i_\ell j_\ell}(\delta_{ii_\ell} - \delta_{ij_\ell} )^2 + \sum^s_{\begin{array}{c} t = 1 \\ t \ne t' \ea} a_{i_ti_t} [\delta_{ii_t}]^2 \right\}. \eqno{(33)}$$
The state after the measurement is $P[| v_i \rangle]$. Let $U_{i\ell}$ be $n \times n$ unitary matrices such that $U_{i\ell} [|v_i\rangle] = \frac{1}{\sqrt 2} (| v_{i_\ell} \rangle - | v_{j_\ell} \rangle).$ For $ i = 1, \cdots, m$ and $U_{it}[| v_i \rangle] = | v_{i_t} \rangle$, for $ t = 1, \cdots, t' -1, t' + 1, \cdots, s$. With probability $2a_{i_\ell j_\ell} /(d_{(G, a)} - a_{i_{t'}, i_{t'}})$ we apply $U_{i\ell}$ on $P[|v_i\rangle]$ for each $\ell = 1, \cdots, m$ and with probability $a_{i_{t}i_{t}} / (d_{(G, a)} - a_{i_{t'}i_{t'}})$ we apply $U_{it}$ on $P[| v_i \rangle]$ for each $t = 1, \cdots, t'-1, t'+1, \cdots s$. We obtain $\sigma((G, a) - \{ v_{i_{t'}} v_{i_{t'}}\})$ with probability given by (33).
The set of Kraus operators that realizes the TPCP for deleting the loop $\{ v_{i_{t'}}, v_{i_{t'}}\}$ is
$$ \left\{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} - a_{i_{t'}, i_{t'}}}} U_{i\ell} P[ |v_i \rangle ] ~~ i = 1, \cdots, m ,~~ \ell = 1, \cdots, m\right\}$$
$$ \cup \left\{ \sqrt{\frac{2a_{i_t i_t}}{d_{(G, a)} - a_{i_{t'}, i_{t'}}}} U_{it} P[ |v_i \rangle ] ~~ i = 1, \cdots, m ,~~ t = 1, \cdots, t' -1, t'+1, \cdots s\right\}$$
The set of Kraus operators that realizes the TPCP for adding the loop $\{ v_{i_{t'}} v_{i_{t'}}\}$
$$ \left\{ \sqrt{\frac{2a_{i_\ell j_\ell}}{d_{(G, a)} + a_{i_{t'}i_{t'}}}} V_{i\ell} P[|v_i\rangle] : i = 1, \cdots, n, \ell = 1, \cdots, m\right\}$$
$$ \cup\left\{ \sqrt{\frac{a_{i_ti_t}}{d_{(G, a)} + a_{i_{t'}i_{t'}}}} V_{it} P[|v_i\rangle] : i = 1, \cdots, n, t = 1, \cdots, s \right\}$$
where $V_{i\ell}, V_{it}$ are $n \times n$ unitary matrices define as follows :
$$V_{i\ell} |v_i\rangle = \frac{1}{\sqrt 2} (|v_{i_\ell}\rangle - |v_{j_\ell}\rangle),~~ \mbox{for}~~ \ell = 1, \cdots, m, i = 1, \cdots, n $$
$$V_{it} |v_i\rangle = |v_{i_t}\rangle,~~ \mbox{for}~~ t = 1, \cdots, s, i = 1, \cdots, n.$$
\subsection{Deletion and addition of an edge with real weight, which preserves the positivity of the generalized Laplacian}
Let $(G, a)$ be a graph with real weights on its edges not necessarily positive. We are basically concerned here with the deletion of $\{ v_i, v_j\}$ with $a_{v_iv_j} > 0$ and the addition of $\{ v_i,v_j\}$ with $a_{v_iv_j} < 0$, because in other cases the positivity of the Laplacian is preserved. We define the sets
$$ E^+ = \{ \{v_i, v_j\} \in E(G, a),a_{v_iv_j} > 0 \}, \eqno{(34)}$$
$$ E^- = \{ \{v_i, v_j\} \in E(G, a) , a_{v_iv_j} < 0 \} \eqno{(35)}$$
and $E = E^+ \cup E^-$.
We define a graph operator $\Xi$ as
$$ \Xi [E] = E \cup \{ \{v_i,v_i\}, \{v_j,v_j\} : a_{v_iv_i} = a_{v_jv_j} = 2|a_{v_iv_j}|~~\mbox{and}~~ \{v_i, v_j\} \in E^- \} \eqno{(36)}$$
Suppose we wish to delete a positive weighted edge $\{ v_{i_k}, v_{j_k}\} \in E^+$ then we define the resulting graph as
$$ \Xi {\cal L} ((G, a) - \{ v_{i_k}, v_{j_k} \} ) $$
where the graph operator ${\cal L}$ is defined in (20b).
For adding a negative weighted edge between $v_i$ and $v_j, i \ne j$, we act on $E(G, a)$ by the appropriate element of the set of operators $\{ \in_{ij}\}, i, j = 1, \cdots, n, i \ne j$ defined as
$$\in_{ij} [E] = E \cup \{ \{ v_i, v_j\}, \{ v_i, v_i\}, \{ v_j, v_j\} : a_{v_iv_j} < 0, a_{v_iv_i} = 2|a_{v_iv_j}| = a_{v_jv_j}\} \eqno{(37)}$$
To obtain the set of the corresponding TPCP operators we decompose the resulting graph, $(G', a')$ given by $\Xi {\cal L}((G, a) - \{ v_{i_k}, v_{j_k}\}) (a_{v_{i_k }v_{j_k}} > 0)$ (eq. (36) ) or by $\in_{ij}((G, a) + \{ v_i, v_j \}) (a_{v_iv_j} < 0)$ (eq. (37)) or by $((G, a) - \{ u_{i_k} v_{j_k} \}) (a_{v_{i_k} v_{j_k}} < 0)$ or by $(G, a) + \{ v_i v_j \}) (a_{v_iv_j} > 0)$ into spanning subgraphs determined by the sets $E^+$ and $E^-$ and treat the spanning subgraph corresponding to $E^-$ replace the weights $a_{u_iv_j}$ of edges $\{v_i, v_j\} \in E^-$ by $- a_{v_iv_j}$, so that both the spanning subgraphs have only positive weights. For getting the Kraus operators we go through the following steps.
(a) First we determine the degree sums for the resulting graphs $(G', a')$ in four cases.
\begin{description}
\item(i) Deletion of a positive weighted edge $\{ u_{i_k}, v_{j_k} \}$
$$ d_{(G',a')} = d_{(G, a)} - 2a_{v_{i_kj_k}} - \sum_i a_{ii} + 2 \sum_{\{u_i,v_j\} \in E^-} |a_{v_iv_j}| \eqno{(38)}$$
\item(ii) Addition of a positive weighted edge $\{ v_i, v_j\}$.
$$ d_{(G', a')} = d_{(G, a)} + 2a_{v_iv_j} \eqno{(39)} $$
\item(iii) Deletion of a negative weighted edge $\{ v_{i_k}, v_{j_k}\}$
$$ d_{(G', a')} = d_{(G, a)} - 2a_{v_{i_k} v_{j_k}} \eqno{(40)}$$
\item(iv) Addition of a negative weighted edge $\{ v_i, v_j\}$
$$ d_{(G', a')} = d_{(G, a)} + 2a_{v_i, v_j} + 4|a_{v_i,v_j}| \eqno{(41)} $$
\end{description}
(b) We construct the Kraus operators separately for $G^+$ and $G^-$ for deleting the same edge $\{ v_i, v_j\}$ from $G^\pm \sqcup \{ v_i, v_j \}$ or adding the edge $\{ v_i, v_j\}$ to $G^\pm$, using the method given in Section 5.1. However, the probability of applying various unitary operator $U^\pm_{k\ell}$ and $U^\pm_{k\ell}, U_{i\ell}$ and $U_{it}$ is determinate using $d_{(G', a')}$ as in step (a) above.
(c) Let $\{ A_i\}$ and $\{ B_i\}$ denote the sets of Krous operators for the graph operations on $G^+$ and $G^-$ as described in (b). Then
$$ \sigma(G', a') = \sum_i A_i \sigma(G, a) A^\dagger_i - \sum_j B_j \sigma(G, a) B^\dagger_j \eqno{(42)}$$
and
$$ \sum_i A_i^\dagger A_i - \sum_j B_j^\dagger B_j = I \eqno{(43)}$$
which can be justified by construction.
We comment here that it is possible to modify the graph, after deleting a positive edge or adding a negative edge, which can preserve positivity in different ways, leading to different sets of Kraus operators. The basic idea is to add new loops. In our method we try to minimize the addition of loops. Further, in our method we cannot reverse the graph operation for deleting a positive edge or adding a negative edge. But this is not a problem since the quantum operations given by super operators are, in general, irreversible.
\subsection {\bf Deleting Vertices}
In order to delete a vertex $v_i$ from a graph (G,a),
\begin{enumerate}
\item[(i)] Delete edges, including loops, on $v_i$, one by one, by successively
applying the procedure in 5.2. The resulting graph $(G', a')$ has density matrix with $i$-th row and $i$-th column containing all zeroes.
\item[(ii)] We now perform, on $\sigma(G', a')$, the projective measurement $M = \{ I_n - P[|v_i\rangle], P[|v_i\rangle]\}$. Since $P[|v_i\rangle]$ is the matrix with all elements zero except the $i$-th diagonal element, while $\sigma(G', a')$ as all zeros in $i$-th row and column,the probability that $P[|v_i\rangle]$ clicks $ = Tr(\sigma P[| v_i\rangle]) = 0$. Thus when $M$ is performed on $\sigma(G', a')', I_n - P[|v_i\rangle]$ clicks with probability one and the state after measurement is $\sigma(G', a') - \{ v_i\})$ and is the same as $\sigma(G', a')$ without $i$-th row and $i$-th column.
\end{enumerate}
\noindent {\bf Adding a vertex :} Let $(G, a)$ be a graph on $n$ vertices $v_1, \cdots, v_n$ and $m$ edges $\{ v_{i_k} v_{j_k} \}, k = 1, \cdots, m, i_k \ne j_k$ and $s$ loops $\{ v_{i_t} v_{i_t}\}, t = 1, \cdots, s; \le i_k, j_k, i_t \le n$. Consider the following density operator
$$ \rho = \left( \frac{1}{2} \sum^2_{i=1} b_{ii} P[|u_i\rangle]) \otimes (\sigma (G, a)\right)$$
where $\{ |u_1\rangle, |u_2 \rangle\}$ form an orthonormal basis of $\mathbb{C}^2$. We associate vertices $u_i, i = 1, 2$ to the state $|u_i\rangle$. Consider the graph $H = (\{ u_1, u_2\}, \{ \{ u_1,u_1\}, \{ u_2, u_2\}\})$ with associated weights $b > 0$. It is easy to check that $\sigma(H, b) = \frac{1}{2} \sum\limits^2_{i=1} b_{ii} P[|u_i\rangle]$. Also observe that
$$\rho = \sigma((H, b) \boxdot (G, a)) = \sigma(H, b) \otimes \sigma(G, a).$$
Thus $(H, b) \boxdot (G, a)$ is the graph on $2n$ vertices labeled by $u_1 v_1, \cdots,u_1v_n, u_2v_1, \cdots, u_2v_n$ and with $2m$ edges and $2s$ loops (see Section 4.2) $\{ u_1v_{i_1}, u_1v_{j_1}\} \cdots \{ u_1v_{i_m}, u_1 v_{j_m}\}\\ \{ u_2v_{i_1}, u_2v_{j_1}\} \cdots \{ u_2v_{i_m}, u_2v_{j_m}\}$ and loops $\{ u_1v_{i_t}, u_1 v_{i_t}\}, \{ u_2v_{i_t}, u_2v_{i_t}\}, t = 1, \cdots, s$. So $(H, b) \boxdot (G, a) = (H_1,a_1) \uplus (H_2, a_2)$ where
$$ (H_1,a_1) = (\{ u_1v_1 \cdots u_1v_n\}, \{ \{ u_1v_{i_1} ,u_1v_{j_1}\} \cdots \{ u_1v_{i_m}, u_1v_j\} \})$$
$$ (H_2, a_2) = ( \{ u_2v_1 \cdots u_2v_n\}, \{ \{ u_2v_{i_1}, u_2v_{j_1}\} \cdots \{u_2v_{i_m}, u_2 v_{j_m}\} \})$$
We first delete all edges and loops of $(H, b) \boxdot (G, a)$ which are incident to the vertex $u_2v_1 \in V(H_2, a_2)$ as in Section 5.2. Now we perform the following projective measurement on $\sigma((H, b) \boxdot (G, a))$
$$ M = \{ I_{2n} - \sum^n_{i=2} P[|u_2 v_i\rangle], \sum^n_{i=2} P[| u_2 v_i\rangle] \}$$
The probability that $I_{2n} - \sum\limits^n_{i=2} P[v_2 v_i\rangle ]$ is one and the state after the measurement is $\sigma((H_1, a_1) + \{ u_2v_1\})$.
{\it Example (8)} : Consider the graph as given in the Figure (16) , we want to delete the edge \{1,2\} with positive weight by means of TPCP.
calculate the Kraus operators for $G^+$ and $G^-$ as in section 5.2 ,where $A_i$ for $i=1,...,\cdots 24$ and $B_i$ for $G^-$ ,$i=1,.....,4$
and substitute in the the equation
$$ \sigma(G', a') = \sum_i A_i \sigma(G, a) A^\dagger_i - \sum_j B_j \sigma(G, a) B^\dagger_j$$
where
$$ \sigma(G, a) = \frac{1}{8} \left[ \begin{array}{cccc} 1 & -1 & -1 & 1 \\ -1 & 3 & -1 & -1 \\ -1 & -1 &3 & -1 \\ 1 & -1 & -1 & 1 \ea \right] .$$
we get
$$ \sigma(G, a) = \frac{1}{10} \left[ \begin{array}{cccc} 2 & 0 & -1 & 1 \\ 0 & 2 & -1 & -1 \\ -1 & -1 &3 & -1 \\ 1 & -1 & -1 & 3 \ea \right] .$$
and
$$ \sum_{i=1}^{24} A_i^\dagger A_i - \sum_{j=1}^4 B_j^\dagger B_j = I$$
\begin{figure}[!h]
\includegraphics[width=2cm,height=2cm]{fig16.eps}
Figure 16
\end{figure}
\section{Representation of a general hermitian operator by a graph}
In this section, we generalize sections 2 - 4, to quantum states in a complex Hilbert space, that is, to the density matrices with complex off-diagonal elements. We have also given rules to associate a graph to a general hermitian operator. We believe that any further advance in the theory reported in this paper will prominently involve graph operators and graphs associated with operators.
\subsection{Representation of a general density matrix with complex off diagonal elements}
Consider a $n \times n$ density matrix with complex off-diagonal elements. We associate with this density matrix an oriented graph $(G, a)$ on $n$ vertices, $m$ edges and $s$ loops with weight function
$$ a : V(G) \times V(G) \rightarrow \mathbb{C}.$$
The weight function $a$ has the following properties:
(i) $a(\{ u, v\}) \ne 0$ if $\{ u, v\} \in E(G, a)$ and $0$ otherwise.
(ii) $a(\{ u, v \}) = a^*(\{ v, u \})$
we write $a(\{u,v\})\; =\; |a(\{u,v\})| \; e^{i\phi_{uv}}, \phi_{vv} = 0$.
Note that, when $\phi_{ij}=l \pi , l\;=\; 0,1, \cdots$, i.e. $a(\{u,v\})$ is real, positive when $l$ is even and real negative when $l$ is odd.
The degree $d_v$ of vertex $v$ is given by
$$d_{(G,a)}(v)\;=\; d_v = \sum_{u \in V(G, a) , \\ u \ne v} |a(\{u,v\})| \;+\; a(\{v,v\}) \eqno{(44)}$$
$$ d_{(G,a)}\;=\; \sum_{v \in V(G, a) } d_v$$
The adjacency matrix $M(G, a)$ of a complex weighted graph with $n$ vertices is a $n \times n$ matrix whose rows and columns are indexed by vertices in $V(G,a)$ .
$$M_{uv}\;=\; a(\{u,v\})\;=\;a^*(\{v,u\})\;=\;(M_{vu})^*.$$
The degree matrix $\Delta(G, a)$ of the complex weighted graph is a $n \times n$ real diagonal matrix, whose rows and columns are labeled by vertices in $V(G, a)$ and whose diagonal elements are the degrees of the corresponding vertices.
$$\Delta(G, a) = diag [d_v; v \in V(G, a)]$$
where $d_v$ is given by equation (44).
The loop matrix $\Delta_0(G, a)$ of a graph $(G,a)$ is a $n \times n$ real diagonal matrix with diagonal elements equal to the weights of the loops on the corresponding vertices
$$ [\Delta_0(G, a)]_{vv} = a_{vv} . $$
The generalized Laplacian of a graph $(G, a)$, which includes loops, is
$$ Q(G, a) = \Delta(G, a)\;+\; M(G, a) \;-\; \Delta_0(G, a) \eqno{(45)}$$
Note that $Q(G, a)$ is hermitian matrix . If the generalized Laplacian $Q(G,a)$ is positive semidefinite, we can define the density matrix of the corresponding graph $(G, a)$ as
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} \; Q(G, a) \eqno{(46)}$$
where $Tr(\sigma(G, a)) = 1$.
For any $n \times n$ density matrix $\sigma$ with complex off diagonal elements we can obtain the corresponding graph as follows:
\noindent {\bf Algorithm 6.1 :}
\begin{description}
\item(i) Label the $n$ vertices of the graph by the kets from the standard orthonormal basis.
\item(ii) For every nonzero $ij$th element with $j > i$ given by $ a(\{i,j\}) $ draw an edge between vertices labeled $|v_i\rangle$ and $|v_j\rangle$, with weight $a(\{i,j\})$.
\item(iii) Ensure that $d_{v_i} = \sigma_{ii}$ by adding loop of appropriate weight to $v_i$ if necessary.
\end{description}
{\it Example (9)} :
(1)
\begin{figure}[!h]
\includegraphics[width=10cm,height=4cm]{fig17.eps}
Figure 17
\end{figure}
$$ P[|y, +\rangle] = \frac{1}{2} \left[ \begin{array}{cc} 1 & -i \\ i & 1 \ea \right] = \frac{1}{2} \left[ \begin{array}{cc} 1 & e^{-i\pi/2}\\ e^{-i\pi/2} & 1 \ea \right] $$
where $|y, +\rangle = \frac{1}{\sqrt 2} (|1\rangle + i|2\rangle)$ and the corresponding graph is as shown in Figure 17a
(2)
$$ P[|y, - \rangle] = \frac{1}{2} \left[ \begin{array}{cc} 1 & i \\ -i & 1 \ea \right] = \frac{1}{2} \left[ \begin{array}{cc} 1 & e^{i\pi/2} \\ e^{-i\pi/2} & 1 \ea \right] $$
where $|y, -\rangle = \frac{1}{\sqrt 2} (|1\rangle - i|2\rangle)$ and the corresponding graph is as shown in Figure 17b
(3)
$$ P[|y, + \rangle|y,+\rangle] = \frac{1}{4} \left[ \begin{array}{cccc} 1 & -i & -i & -1 \\ i & 1 & 1 & -i \\item & 1 & 1 & -i\\-1 & i & i & 1\ea \right] $$
The corresponding graph is as shown in Figure 17c.
Note that Remark 2.1 is valid also for complex weighted graphs.
\textbf{Remark 6.2 :} Theorem 2.3 applies to complex weighted graphs with equation (11) changed to
$$ \sum^n_{i=1} d^2_i + 2 \sum^m_{k=1} |a_{i_kj_k}|^2 = d^2_{(G, a)} \eqno{(47)}$$
also Lemma 2.4 applies to complex weighted graphs.
\noindent {\bf Definition 6.3 :} A graph $(H, b)$ is said to be a factor of graph $(G, a)$ if $V(H, b) = V(G, a)$ and there exists a graph $(H', b')$ such that $V(H', b') = V(G, a)$ and $M(G, a) = M(H, b) + M(H', b')$. Thus a factor is only a spanning subgraph. Note that
$$ a_{v_iv_j} = \left\{ \begin{array}{lll} b_{v_iv_j} & \mbox{if} & \{v_i,v_j\} \in E(H, b) \\ b'_{v_iv_j} & \mbox{if} & \{v_i,v_j\} \in E(H', b') \ea \right. $$
Now let $(G, a)$ be a graph on $n$ vertices $v_1, \cdots, v_n$ having $m$ edges\\ $\{v_{i_1}, v_{j_1}\}, \cdots, \{v_{i_m}, v_{j_m}\}$ and $s$ loops $\{v_{i_1}, v_{i_1}\} \cdots \{v_{i_s}, v_{i_s}\}$ where $1 \le i_1j_1, \cdots, i_m j_m \le n, 1 \le i_1 i_2 \cdots i_s \le n$.
Let $(H_{i_kj_k}, a_{i_kj_k})$ be the factor of $(G, a)$ such that
$$ [M(H_{i_kj_k}, a_{i_kj_k})]_{u,w} = \left\{ \begin{array}{l} a_{i_kj_k} ~~ \mbox{if}~~ u = i_k~~ \mbox{and}~~ w = j_k ~~\mbox{or}~~ a^*_{i_kj_k} \mbox{if}~~u = j_k, w = i_k \\ 0 ~~ \mbox{otherwise} \ea \right. \eqno{(48)}$$
$$ [\Delta(H_{i_kj_k}, a_{i_kj_k})]_{u,w} = \left\{ \begin{array}{l} |a_{i_kj_k}| ~~ \mbox{if}~~ u = i_k= w ~~\mbox{or} ~~u = j_k= w \\ 0 ~~ \mbox{otherwise} \ea \right. \eqno{(49)}$$
Let $(H_{i_t,i_t}, a_{i_t i_t})$ be a factor of $(G, a)$ such that
$$ [M(H_{i_ti_t}, a_{i_t i_t})]_{u,w} = [\Delta(H_{i_ti_t}, a_{i_ti_t})]_{u,w} = \left\{ \begin{array}{l} a_{i_t i_t}~~ \mbox{when}~~ u = i_t = w \\ 0 ~~ \mbox{otherwise} \ea \right. \eqno{(50)}$$
\noindent {\bf Theorem 6.4 :} The density matrix of a graph $(G, a)$ as defined above with factors given by equation (48), (49) and (50) can be decomposed as
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} \sum^m_{k=1} 2 |a(\{i_k,j_k\})| \sigma(H_{i_kj_k}, a_{i_kj_k}) + \frac{1}{d_{(G, a)}} \sum^s_{t=1} a_{i_ti_t} \sigma(H_{i_ti_t}, a_{i_ti_t}) \eqno{(51)}$$
or
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} \sum^m_{k=1} 2|a(\{i_k,j_k\})| P[\frac{1}{\sqrt 2}(|v_{i_k}\rangle -e^{i\phi_{i_kj_k}} |v_{j_k}\rangle)] + \frac{1}{d_{(G, a)}} \sum^s_{t=1} a_{i_ti_t} P[|v_{i_t}\rangle]\eqno{(52)}$$
Where $\phi_{i_kj_k} = \pi$ for any edge $\{i_k,j_k\}$ with real positive weight and $\phi_{i_kj_k} = 0$ for any real negative weight.
\noindent {\bf Proof :} From equation (48), (49), (50) and Remark 6.2, the density matrix
$$\sigma (H_{i_kj_k}, a_{i_kj_k}) = \frac{1}{2|a_{i_kj_k}|} [ \Delta(H_{i_kj_k}, a_{i_kj_k}) + M(H_{i_kj_k},a_{i_kj_k})]$$
is a pure state. Also,
$$ \sigma (H_{i_ti_t}, a_{i_ti_t}) = \frac{1}{a_{i_ti_t}} [ \Delta_0 (H_{i_t, i_t}, a_{i_ti_t})]$$
is a pure state. Now
$$ \Delta(G, a) = \sum^m_{k=1} \Delta(H_{i_kj_k}, a_{i_kj_k}) + \sum^s_{t=1} \Delta_0(H_{i_ti_t}, a_{i_ti_t})$$
$$M(G, a) = \sum^m_{k=1} M(H_{i_kj_k}, a_{i_kj_k}) + \sum^s_{t=1} \Delta_0(H_{i_ti_t}, a_{i_ti_t}).$$
Therefore , from eq. (46)
$$\sigma(G, a) = \frac{1}{d_{(G, a)}} \left[ \sum^m_{k=1} \Delta(H_{i_kj_k}, a_{i_kj_k}) + \sum^m_{k=1} M(H_{i_kj_k}, a_{i_kj_k})\right]\\
+ \frac{1}{d_{(G, a)}} \left[ \sum^s_{t=1} \Delta_0 (H_{i_ti_t}, a_{i_ti_t})\right]$$
$$= \frac{1}{d_{(G, a)}} \sum^m_{k=1} [\Delta(H_{i_kj_k}, a_{i_kj_k}) + M(H_{i_kj_k}, a_{i_kj_k})] \\
+ \frac{1}{d_{(G, a)}} \sum^s_{t=1} \Delta_0(H_{i_ti_t}, a_{i_ti_t})$$
$$ = \frac{1}{d_{(G, a)}} \sum_k 2|a(\{i_k,j_k\})| \sigma(H_{i_kj_k}, a_{i_kj_k}) \\
+ \frac{1}{d_{(G, a)}} \sum_t a_{i_ti_t} \sigma(H_{i_ti_t}, a_{i_ti_t}) \eqno{(51)}$$
In terms of the standard basis, the $uw$-th element of matrices $\sigma(H_{i_kj_k}, a_{i_kj_k})$ and $\sigma(H_{i_ti_t}, a_{i_ti_t})$ are given by $\langle v_u | \sigma(H_{i_kj_k} , ,a_{i_kj_k}) | v_w \rangle$ and $\langle v_u | \sigma (H_{i_ti_t} a_{i_ti_t} | v_w\rangle$ respectively. In this basis
$$ \sigma(H_{i_kj_k}, a_{i_kj_k}) = P[ \frac{1}{\sqrt 2} ( | v_{i_k} \rangle -e^{i\phi_{i_kj_k}} | v_{j_k} \rangle )]$$
$$ \sigma(H_{i_ti_t}, a_{i_ti_t}) = P[| v_{i_t} \rangle ] .$$
Therefore equation (51) becomes
$$\sigma(G, a) = \frac{1}{d_{(G, a)}} \sum^m_{k=1} 2|a(\{i_k,j_k\})| P[\frac{1}{\sqrt 2} (| v_{i_k}\rangle -e^{i\phi_{i_kj_k}} | v_{j_k} \rangle) + \frac{1}{d_{(G, a)}} \sum ^s_{t=1} a_{i_ti_t} P[ | v_{i_t} \rangle]~~~~~~~~~~\eqno{(52)}$$
Where $\phi_{i_kj_k} = \pi$ for any edge $\{i_k,j_k\}$ with real positive weight and $\phi_{i_kj_k} = 0$ for any real negative weight.
$\hspace{\stretch{1}} \blacksquare$
{\it Example (10)} :
(1) For a graph given in Figure 17b, the density matrix is
\begin{eqnarray*}
\sigma(G, a) & = & \frac{1}{2}\{2 P[\frac{1}{\sqrt 2} (|1\rangle - e^{i\pi/2} |2\rangle)]\}=P[\frac{1}{\sqrt 2} (|1\rangle - i|2\rangle)]
\end{eqnarray*}
(2) For a graph given in Figure 17c, the density matrix is
\begin{eqnarray*}
\sigma(G, a) & = & \frac{1}{4} \{ 2P[\frac{1}{\sqrt 2} (|11\rangle - e^{-i\pi/2} |12\rangle)] + 2P[\frac{1}{\sqrt 2}(|11\rangle - e^{-i\pi/2} |21\rangle)]\\
& & + 2 P[\frac{1}{\sqrt 2} (|11\rangle - |22\rangle)] + 2 P[\frac{1}{\sqrt 2} (|12\rangle + |21\rangle)] \\
& & + 2P[\frac{1}{\sqrt 2} (|12\rangle - e^{-i\pi/2} |22\rangle)] + 2P[\frac{1}{\sqrt 2} (|21\rangle - e^{-i\pi/2} |22\rangle)] \\
& & - 2P[|11\rangle] - 2P[|22\rangle] -2P[|12\rangle]-2P[|21\rangle]\}
\end{eqnarray*}
$$ \sigma(G, a) = \frac{1}{4} \left[ \begin{array}{cccc} 1 & -i & -i & -1 \\ i & 1 & 1 & -i \\ i & 1 &1 & -i \\ -1 & i & i & 1 \ea \right] .$$
{\it Example (11)} : Consider the state $$ \sigma = \frac{1}{3} P[|y, +\rangle |y, + \rangle] + \frac{2}{3} P[|y, + \rangle | \psi \rangle]$$
where $ |y, +\rangle = \frac{1}{\sqrt 2} (|1\rangle + i|2\rangle)$ and $|\psi\rangle = \frac{1}{\sqrt 3} (|1\rangle + i\sqrt 2|2\rangle)$
\begin{eqnarray*}
\sigma & = & \frac{1}{36} \left[ \begin{array}{cccc} 7 & -(3 + 4\sqrt 2)i & -7i & -(3 + 4\sqrt 2)\\ (3 + 4\sqrt 2)i & 11 & 3 + 4\sqrt 2 & -11 i \\ 7i & 3 + 4\sqrt 2 & 7 & -(3 + 4\sqrt 2)i \\ -(3 + 4\sqrt 2) & 11i & (3 + 4\sqrt 2)i & 11 \ea \right] \\
& = & \frac{1}{36} \left[ \begin{array}{cccc} 7 & (3 + 4\sqrt 2)e^{-i\pi/2} & 7e^{-i\pi/2} & -(3 + 4\sqrt 2)\\ (3 + 4\sqrt 2)e^{i\pi/2} & 11 & 7 & 11e^{-i\pi/2} \\ 7e^{i\pi/2} & 3 + 4\sqrt 2 & 7 & (3 + 4\sqrt 2)e^{-\pi/2} \\ -(3 + 4\sqrt 2) & 11e^{i\pi/2} & (3 + 4\sqrt 2)e^{i\pi/2} & 11 \ea \right]
\end{eqnarray*}
The corresponding graph is as shown in Figure 18 ,
\begin{figure}[!h]
\includegraphics[width=8cm,height=4cm]{fig18.eps}
Figure 18
\end{figure}
and using the equation (45) to get the matrix from graph in Figure 18,
\begin{eqnarray*}
\sigma(G, a) & = & \frac{1}{36} \{ 2(3 + 4 \sqrt 2)P[\frac{1}{\sqrt 2} (|11\rangle - e^{-i\pi/2} |12\rangle)] \\
& & + 2 \times 7 P[\frac{1}{\sqrt 2} (|11\rangle - e^{-i\pi/2} |21\rangle)] + (3 + 4\sqrt 2) P[\frac{1}{\sqrt 2}(|11\rangle - |22\rangle)] \\
& & + 2(3 + 4\sqrt 2) P[ \frac{1}{\sqrt 2} (|12\rangle + |21 \rangle)] + 2 \times 11 P[\frac{1}{\sqrt 2} (|12\rangle - e^{-i\pi/2} |22\rangle)] \\
& & + 2(3 + 4\sqrt 2) P[\frac{1}{\sqrt 2} (|21\rangle - e^{-i\pi/2} |22\rangle)] -(6+8 \sqrt{2})P[|11\rangle] - (6 + 8 \sqrt 2) P[|22\rangle]\\
& & - (6 + 8 \sqrt 2)P[|12\rangle] - (6 + 8 \sqrt 2) P[|21\rangle]\}.
\end{eqnarray*}
We can check that
$$ \sigma(G, a) = \frac{1}{36} \left[\begin{array}{cccc} 7 & -(3 + 4\sqrt 2)i & - 7i & -(3 + 4\sqrt 2) \\ (3 + 4\sqrt 2)i & 11 & 3 + 4 \sqrt 2 & -11i \\ 7i & 3 + 4 \sqrt 2 & 7 & -(3 + 4\sqrt 2)i \\ -(3 + 4\sqrt 2) & 11i & (3 + 4\sqrt 2)i & 11 \ea \right]$$
We can also check that this state is not pure by applying Remark 6.2 on the graph.
\subsection{ Separability}
\textbf{Remark 6.6:} The definition of the tensor product $(G,a) \otimes (H,b)$ of two complex weighted graphs $(G,a)$ and $(H,b)$ is the same as given before. However note that $\{v_1,v_2\} \in E(G,a), \{w_1,w_2\} \in E(H,b)$ implies
$$c(\{(v_1,w_1),(v_2,w_2)\})\;=\; a(\{v_1,v_2\}) b(\{w_1,w_2\})$$ and $$c(\{v_1,w_2),(v_2,w_1)\})\;=\; a(\{v_1,v_2\}) b(\{w_2,w_1\})=a(\{v_1,v_2\}) b^*(\{w_1,w_2\}).$$
\textbf{Remark 6.7 :} Equations (18a) and (18c) are valid for the tensor product of complex weighted graphs. Also, $Q((G,a) \otimes (H,b)) \ne Q(G,a) \otimes Q(H,b)$. Equation (18b) holds good only for graphs without loops, for graphs with only loops or when one factor has no loops and other factor has only loops.
For such graphs equation (18b) immediately gives $$ d_{(G,a) \otimes (H,b)} (v,w) = d_{(G,a)} (v) \cdot d_{(H,b)} (w).$$
\\
\textbf{6.2.1 Modified tensor product}
\\
The modified tensor product of two complex weighted graphs requires the operator ${\cal N}$ to be redefined in the following way. We replace the equation (20a) by
$$a'_i=\sum_{\begin{subarray}{I}
{v_k \in V(G,a)}\\
\hskip .4cm {v_k \ne v_i}
\end{subarray}}
|a(\{v_i,v_k\})| + a(\{ v_i,v_i\}) \eqno{(53)}$$
The definitions of the operators $\eta, {\cal L}$ and $\Omega$ remain the same. Equations (21) to (24) are satisfied by these operators on the complex weighted graphs. We further have
$$ \begin{array}{ll} \mbox{(i)} & M({\cal N} {\cal L}(G, a)) = \Delta(G, a) - \Delta_0(G, a) \\ & \Delta({\cal N} {\cal L}(G, a)) = \Delta(G, a) - \Delta_0(G, a) \\ & \Delta_0({\cal N} {\cal L}(G, a)) = \Delta(G, a) - \Delta_0(G, a)\\ & Q({\cal N} {\cal L}(G,a))= \Delta(G, a) - \Delta_0(G, a) \ea \eqno{(54)}$$
The modified tensor product of two complex weighted graphs $(G,a)$ and $(H,b)$ with $p$ and $q(> p)$ vertices respectively is
\begin{eqnarray*}
(G,c)=(G, a) \boxdot (H, b) & = & {\cal L}(G, a) \otimes {\cal L} (H, b) \dotplus {\cal L}(G, a) \otimes {\cal N}(H, b) \dotplus {\cal N}(G, a) \otimes {\cal L}(H, b)\\
& & \dotplus \{ \Omega(G, a) \otimes \Omega(H, b) \sqcup 2 {\cal N} {\cal L}(G,a) \otimes {\cal N} {\cal L} \eta (H,b)\} ~~~~~~~~~~~~~~{(55)}
\end{eqnarray*}
The weight function $c$ of $(G, a) \boxdot (H, b)$ is obtained via the definition of tensor product and the disjoint edge union.
\textbf{Lemma 6.8 :} $\Delta((G, a) \boxdot (H, b)) = \Delta(G, a) \otimes \Delta(H, b)$.
\textbf{Proof :} Since Lemma 2.10 applies to disjoint edge union of complex weighted graphs,
$$ \Delta((G, a) \boxdot (H, b) ) = \Delta( {\cal L}(G, a) \otimes {\cal L} (H, b)) + \Delta({\cal L}(G, a) \otimes {\cal N}(H, b)) +\Delta( {\cal N}(G, a) \otimes {\cal L}(H, b))$$
$$+ \Delta( \Omega(G, a) \otimes \Omega(H, b)) + \Delta( 2 {\cal N} {\cal L}(G,a) \otimes {\cal N} {\cal L} \eta (H,b))$$
The last two terms are justified because the graphs involved are real weighted graphs. Using Remark 6.7 we get
$$ \Delta((G, a) \boxdot (H, b) ) = \Delta( {\cal L}(G, a)) \otimes \Delta( {\cal L} (H, b)) + \Delta({\cal L}(G, a)) \otimes\Delta( {\cal N}(H, b)) +\Delta( {\cal N}(G, a)) \otimes \Delta({\cal L}(H, b))$$
$$+ \Delta( \Omega(G, a) ) \otimes \Delta(\Omega(H, b)) + 2 \Delta( {\cal N} {\cal L}(G,a)) \otimes \Delta({\cal N} {\cal L} \eta (H,b))$$
Using equations (21) to (24) and (54), we get, after some simplification,
$$\Delta((G, a) \boxdot (H, b) ) = \Delta((G, a)) \otimes \Delta( (H, b))$$
\\
\textbf{Corollary 6.9 :} $d_{(G,a) \boxdot (H,b)}(v,w)\; =\;d_{(G,a)}(v) \cdot d_{(H,b)}(w)$
and$$ d_{(G,a) \boxdot (H,b)}\; =\; d_{(G,a)} \cdot d_{(H,b)}$$
\noindent {\bf Proof :}
The first result follows directly from Lemma 6.8. For the second note that $$ Tr(\Delta((G, a) \boxdot (H, b) )) = Tr (\Delta(G, a) \otimes \Delta (H, b))= Tr( \Delta(G, a)) \cdot Tr(\Delta( H, b))$$ where $Tr$ denotes the trace. \hfill $\blacksquare$
\textbf{Theorem 6.10 :} Consider a bipartite syatem in $\mathbb{C}^p \otimes \mathbb{C}^q$ in the state $ \sigma $. Then $ \sigma = \sigma_1 \otimes \sigma_2$ if and only if $\sigma $ is the density matrix of the graph $(G,a) \boxdot (H,b)$ where $(G,a)$ and $(H,b)$ are the graphs having density matrices $\sigma_1$ and $\sigma_2 $ respectively.
\textbf{Proof :} \noindent {\bf If part :} Given $(G, a), (H, b)$ we want to prove
$$\sigma((G, a) \boxdot (H, b)) = \sigma_1(G, a) \otimes \sigma_2(H, b).$$
From the definition of the modified tensor product we can write
$$ \sigma((G,a) \boxdot (H, b)) = \frac{1}{d_{(G, a) \boxdot (H, b)}} \{Q[{\cal L}(G, a) \otimes {\cal L} (H, b)$$
$$ \dotplus {\cal L}(G, a) \otimes {\cal N}(H, b)\dotplus {\cal N}(G, a) \otimes {\cal L}(H, b) \dotplus \{\Omega(G, a) \otimes \Omega(H, b)) \sqcup 2 {\cal N} {\cal L}(G,a) \otimes {\cal N} {\cal L} \eta (H,b) ]\}$$
Using Remark 6.5 and corollary 6.9 we get \\
$\sigma((G, a) \boxdot (H, b)) = \frac{1}{d_{(G, a)} \cdot d_{(H, b)}} [ Q({\cal L}(G, a) \otimes {\cal L} (H, b))+ Q({\cal L}(G, a) \otimes {\cal N}(H, b))$
$$+ Q({\cal N}(G, a) \otimes {\cal L}(H, b)) + Q(\Omega(G, a) \otimes \Omega(H, b) \sqcup 2 {\cal N} {\cal L}(G,a) \otimes {\cal N} {\cal L} \eta (H,b))]\eqno{(56)}$$
We can calculate every term in (56) using (21) to (24) and (54) and substitute in (56) to get
$$ \sigma((G, a) \boxdot (H, b)) = \sigma(G, a) \otimes \sigma(H, b) .$$
\textbf{Only if part :} Given $ \sigma = \sigma_1 \otimes \sigma_2$ consider the graphs $(G,a)$ and $(H,b)$ for $\sigma_1$ and $\sigma_2$ respectively. Then the graph of $\sigma$ has the generalized Laplacian $$ Q(G,a) \otimes Q(H,b) = ( \Delta(G,a) + M(G,a)- \Delta_0(G,a)) \otimes (\Delta(H,b)+M(H,b)- \Delta_0(H,b))$$
$$ = \Delta(G,a) \otimes \Delta(H,b)+ \Delta(G,a) \otimes (M(H,b)- \Delta_0(H,b)) +( M(G,a)- \Delta_0(G,a)) \otimes \Delta(H,b)+ $$
$$( M(G,a)- \Delta_0(G,a)) \otimes (M(H,b)- \Delta_0(H,b)) \eqno{(57)} $$
Using equation (21) to (24) and (54) we see that RHS of equation (57) is the generalized Laplacian for $(G,a) \boxdot (H,b)$
\hfill $\blacksquare$
\textbf{Remark 6.13 :} The proof that the modified tensor product is associative and distributive with respect to the disjoint edge union is the same as that for the case of real weighted graphs (corollary 4.7).
\\
\textbf{Remark 6.14 :} The definition of the cartesian product of graphs is the same as given in definition 4.8.
\\
\textbf{Remark 6.15 :} Corollaries 4.9 and 4.10 apply to complex weighted graphs without any change.
\subsection{Convex combination of density matrices}
Consider two graphs $(G_1, a_1) $ and $(G_2, a_2)$ each on the same $n$ vertices, having $\sigma(G_1, a_1)$ and $\sigma(G_2, a_2)$ as their density matrices respectively, where $a_1$ and $a_2$ are complex weight functions. Let $(G,a)$ be the graph of the density matrix $\sigma(G, a)$ which is a convex combination of $\sigma(G_1, a_1) $ and $\sigma(G_2, a_2)$,
$$ \sigma(G, a) = \lambda \sigma(G_1, a_1) + (1 - \lambda) \sigma(G_2, a_2), 0 \le \lambda \le 1.$$
It is straightforward, using the definitions of the operators ${\cal N} ,{\cal L}$ and $\eta$, to varify that
$$ (G,a)=[\lambda {\cal N}(G_1,a_1)\sqcup (1-\lambda){\cal N}(G_2,a_2)]\sqcup [\lambda {\cal L}(G_1,a_1)\sqcup (1-\lambda){\cal L}(G_2,a_2)]$$
$$\sqcup \eta {\cal L}[\lambda {\cal L}(G_1,a_1)\sqcup (1-\lambda){\cal L}(G_1,a_2)] \eqno{(58)}$$
We can apply this equation to any convex combination of density matrices. Let $$ \sigma(G, a) = \sum_i p_i \sigma(G_i, a_i) , \sum_i p_i =1$$
Then,
$$ (G,a)=[\sqcup_i p_i{\cal N}(G_i,a_i)]\sqcup [\sqcup_i p_i {\cal L}(G_i,a_i)] $$
$$\sqcup \eta {\cal L}[\sqcup_i p_i {\cal L}(G_i,a_i)] \eqno{(59)}$$
where $a$ and $\{a_i\}$ are complex weight functions, $a{(\{v_l,v_k\})}=\sum_i a'_i{(\{v_l,v_k\})}$ and $a{(\{v_l,v_l\})}=\sum_i a'_i{(\{v_l,v_l\})}$ with $a'_i=p_i a_i.$
\textbf{Lemma 6.11 :} Let $(G_1,a_1)$, $(G_2,a_2)$ and $(G,a)$ satisfy eq.(58). Then
$$ \sigma(G, a) = \frac{d_{(G_1,a_1)}}{d_{(G, a)}} \sigma(G_1, a_1) + \frac{d_{(G_2,a_2)}}{d_{(G, a)}} \sigma(G_2, a_2).$$
\textbf{Proof :} Similar to that of Lemma 2.10.\hfill $\blacksquare$
In general, if $(G,a)$ satisfies eq.(59) for some set of graphs $\{(G_i,a_i)\}$, we have,
$$ \sigma(G, a) = \frac{1}{d_{(G, a)}} \sum_i d_{(G_i,a_i)} \sigma(G_i, a_i).\eqno{(60)}$$
\textbf{Theorem 6.12 :} Every graph $(G,a)$ having density matrix $\sigma(G,a)$ can be decomposed as in eq.(59), where $\sigma(G_i,a_i)$ is a pure state.
\textbf{Proof :} Same as that of Theorm 2.12.
\hfill $\blacksquare$
\textbf{Corollary 6.16 :} A state of a $k$-partite system is separable if and only if the graph $(G,a)$ for $\sigma$ has the form $$ (G,a)=[\sqcup_i {\cal N} \boxdot^k_{j=1} (G^j_i, a^j_i)]\sqcup [\sqcup_i {\cal L}\boxdot^k_{j=1} (G^j_i, a^j_i)] $$
$$\sqcup \eta {\cal L}[\sqcup_i {\cal L} \boxdot^k_{j=1} (G^j_i, a^j_i)] \eqno{(61)}$$
\textbf{Proof :} Same as of Corollary 4.11, where we refer to Theorem 6.4 instead of Theorem 4.5 and Lemma 6.11 instead of Lemma 2.10 and eq.(60) instead of Remark 2.11.\hfill $\blacksquare$
Corolary 6.16 says that Werner's definition [1] of a separable state in $\mathbb{C}^{q_1} \otimes \mathbb{C}^{q_2} \otimes \mathbb{C}^{q_3} \otimes \cdots \otimes \mathbb{C}^{q_k}$ system, can be expressed using corresponding graphs.
\subsection{Representation of a hermitian operator (observable) by a graph}
In order to represent a general hermitian matrix corresponding to a quantum observable $A$ we lift the requirement that the Laplacian be positive semidefinite and $Tr[A]=1$. In other words we take the generalized Laplacian as the matrix for the graph.
Given a Hermitian matrix $A$, the algorithm 6.1 can be implemented to get its graph $(G, a)$. The corresponding observable $\hat A$ of a graph $(G, a)$ is
$$ \hat A = \sum^m_{k=1} 2a_{i_kj_k} P[\frac{1}{\sqrt 2} (|v_{i_k}\rangle - e^{i\phi_{i_kj_k}}|v_{j_k}\rangle)] + \sum^s_{t=1} a_{i_ti_t} P[|v_{i_t}\rangle] \eqno{(58)}$$
{\it Example (12)} : Give the graph of $\sigma_x$ and $\sigma_y$.
(1)~~~~ $\sigma_x = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \ea \right] $.
The corresponding graph of $\sigma_x$ is shown in Figure 19a
(2)$ \sigma_y = \left[ \begin{array}{cc} 0 & -i \\ i & 0 \ea \right] = \left[ \begin{array}{cc} 0 & e^{-i\pi/2} \\ e^{i\pi/2} & 0 \ea \right].$
The corresponding graph of $\sigma_y$ is shown in Figure 19b
\begin{figure}[!h]
\includegraphics[width=8cm,height=1cm]{fig19.eps}
Figure 19
\end{figure}
Using Equation (46) to get the operators from graphs
$$ \sigma_x = -2P[\frac{1}{\sqrt 2} (|1\rangle - |2\rangle)] + P[|1 \rangle] + P[|2\rangle] = |1\rangle \langle 2| + |2\rangle \langle 1| = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \ea \right] $$
$$ \sigma_y = 2P[\frac{1}{\sqrt 2} (|1\rangle - e^{-i\pi/2} |2\rangle)] - P[|1 \rangle] - P[|2\rangle] = -i|1\rangle \langle 2| + i|2\rangle \langle 1| = \left[ \begin{array}{cc} 0 & -i\\ i& 0 \ea \right] $$
\section{Some graphical criteria for the positive semidefiniteness of the associated Laplacian}
In this section we address the following question. Given a graph, can the positive semidefiniteness of the associated Laplacian be determined using the topological properties of the graph? A general answer to this question seems to be difficult because the theory of weighted graphs, with negative and complex weights is almost unavailable. Many results obtained for simple graphs do not apply to the weighted graphs with real or complex weights. Nevertheless, we give here the above mentioned criteria in some special cases.
{\bf Lemma 7.1 :} Let $(G,a)$ be a graph with real or complex weights, having one or more nonisolated vertices with degree zero. Then the Laplacian of $(G,a)$ is not positive semidefinite.
\noindent {\bf Proof :} Such a graph $(G,a)$ has one or more zeroes along the diagonal of its Laplacian with nonzero entries in the corresponding rows. However, a hermitian matrix with one or more zeros in its diagonal has at least one negative eigenvalue unless all the elements in the corresponding rows and columns are zero [18].\hspace{\stretch{1}}$ \blacksquare$
{\bf Lemma 7.2 :} Let $(G,a)$ be a $n$ vertex graph with real weights, having at least one loop and let the weights on all the loops be negative. Then the Laplacian of $(G,a)$ is not positive semidefinite.
\noindent {\bf Proof :} For the given $(G,a)$ and some $x$ in $\mathbb{R}^n$ we have
$$x^T[Q(G,a)]x=\sum_k a_{i_kj_k}(x_{i_k}-x_{j_k})^2-\sum_t|a_{i_ti_t}|x^2_{i_t}$$
where the first sum is over edges and the second sum is over loops. It is easy to check that
$x^T[Q(G,a)]x<0$ for $x=(1~1~1~\dots ~1)^T$.\hspace{\stretch{1}}$ \blacksquare$
{\bf Lemma 7.3 :} Let $(G,a)$ be a graph without loops satisfying $a(u,v)=a_{uv}e^{i\phi_{uv}}, (\phi_{uv}\neq 2\pi n)$. Then the associated Laplacian is positive semidefinite.
\noindent {\bf Proof :} This follows directly from Theorem 6.4.\hspace{\stretch{1}}$ \blacksquare$
{\bf Observation 7.4 :} Let $(G,a)$ be a graph satisfying $a(u,v)=a_{uv}e^{i\phi_{uv}}$ and $a(\{v,v\})\geq 0$ for all vertices in $V(G,a)$. Then the associated Laplacian is positive semidefinite.
\noindent {\bf Proof :} The Laplacian is a hermitian matrix which is diagonally dominant. Therefore, by Gersgoin circle criterian [14, 15, 19] it is positive semidefinite\hspace{\stretch{1}}$ \blacksquare$
On a $n$ vertex graph $(G,a)$, we define a new graph operator $\Theta(u_i)$ which deletes the vertex $u_i$ and rolls the edges incident on $u_i$ into loops with same weights on the edges connecting neighbors of $u_i$ as shown in Figure 20. We call the resulting subgraph {\it principal subgraph}. The Laplacian of the principal subgraph obtained by operating $\Theta(u_i)$ on $(G,a)$ is the principal submatrix of the Laplacian of $(G,a)$ obtained by deleting $i$th row and $i$th column.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=8cm,height=3cm]{fig20.eps}
Figure 20
\end{center}
\end{figure}
{\bf Lemma 7.5 :} If one or more principal subgraphs of $(G,a)$ are not positive semidefinite, then $(G,a)$ is not positive semidefinite.
\noindent {\bf Proof :} This follows from the result that all the principal submatrices of a positive semidefinite matrix are positive semidefinite[15].\hspace{\stretch{1}}$ \blacksquare$
{\bf Lemma 7.6 :} Let $(G,a)$ be either a $n$ vertex tree $(n\geq 2)$ or a $n$ vertex cycle $(n\geq 4)$. We assume that there are no loops in $(G,a)$ and that $a(u,v)$ is real for all $\{u,v\} \in E(G,a)$. Then $(G,a)$ has a positive semidefinite Laplacian if and only if $a(\{u,v\})>0$ for all $(u,v)\in E(G,a)$.
\noindent {\bf Proof :} \textbf{Only if part :} We prove that $a(\{u,v\})<0$ for any one $\{u,v\} \in E(G,a) \Longrightarrow Q(G,a)\ngeq 0$. Let $(T,a)$ be a tree with $v_1,\dots,v_n$ vertices, and let $\{v_i,v_{i+1}\}$ be an edge in $(T,a)$ with negative weight $a(\{v_i,v_{i+1}\})<0$. We operate on $(T,a)$ by $\Theta(v_{i+1})$. There are two possibilities. If $v_{i+1}$ is a leaf , we get only one component with a negative weighted loop on $v_i$. By Lemma 7.2, the Laplacian of this principal subgraph is not positive semidefinite and by Lemma 7.5 the Laplacian of $(T,a)$ is also not positive semidefinite. If $v_{i+1}$ is not a leaf then $\Theta(v_{i+1})$ will result in two or more principal subgraphs. The principal subgraph containing vertex $v_i$ is a graph having one loop with negative weight. By Lemma 7.2 the Laplacian of this principal subgraph is not positive semidefinite and from Lemma 7.5 $Q(T,a)\ngeq 0.$
Let $C_n$ be an $n$-cycle and let $a(\{v_i,v_{i+1}\})=a<0$. We operate by $\Theta(v_{i+1})$ which results in a $n-1$ vertex path , say $P_{n-1}$ with $v_i$ having a negative loop and $v_{i+2}$ having a positive or negative loop. If both the loops are negative we can use Lemma 7.2 and Lemma 7.5 in succession to show that $Q(C_n,a)\ngeq 0.$
Suppose the loop on $v_{i+2}$ is positive . Then for some $x\in R^{n-1}$ we have
$$x^TQ(P_{n-1},a)x=\sum_{k=1}^{n-2}a_{i_kj_k}(x_{i_k}-x_{j_k})^2+a(v_{i+1},v_{i+2})x_{v_{i+2}}^2-|a|x_{v_i}^2$$
It is straightforward to check that $x^TQ(P_{n-1},a)x<0$ for $x^T=(1~1~\dots 0 ~ 1~\dots ~1)$, that is a vector $x$ with all components $1$ except $v_{i+2}th$ component which is zero. Thus $Q(P_{n-1},a')\ngeq 0$. By Lemma 7.5 $Q(G,a)\ngeq 0$.
\textbf{If part :} Assume $Q(G,a)\ngeq 0$. This implies that there exists at least one $x\in R^n$ satisfying
$$x^TQ(G,a)x=\sum_k a(i_k,j_k)(x_{i_k}-x_{j_k})^2<0$$
Since $(x_{i_k}-x_{j_k})^2\geq 0$ for all $k$, the above inequality is satisfied only when $a(i_k,j_k)<0$ for some $k$. This proves the if part.\hspace{\stretch{1}}$ \blacksquare$
We observe that the proof of if part applies to all graphs as it should.
{\bf Lemma 7.7 :} Let all loops on a graph $(G,a)$ have real positive weights. Let every edge $\{u,v\}\in E(G,a)$ having $a(u,v)<0$ be common to pair of $C_3$. Let all such pairs of $C_3$, each containing a negative edge be disjoint. Let all the edges in each pair of $C_3$, other than the contained negative edge have positive weights satisfying $a(u,v)$ greater than the absolute value of the weight on the negative edge. Then the Laplacian of $(G,a)$ is positive semidefinite.
\noindent {\bf Proof :} Consider a negative edge common to two $C_3$'s as shown in the Figure 21.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=3cm,height=3cm]{fig21.eps}
Figure 21
\end{center}
\end{figure}
By hypothesis $b_i>a,i=1,2,3,4$. We can write $b_i=a+c_i,c_i>0,i=1,2,3,4$. We can decompose this graph as the edge union as shown in Figure 22.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=12cm,height=7cm]{fig22.eps}
Figure 22
\end{center}
\end{figure}
The first graph on RHS has all positive weights and hence has a positive semidefinite Laplacian. It is straighforword to check, that the second and the third graphs on RHS correspond to the projectors $P[\frac{1}{\sqrt2}(|j\rangle-2|l\rangle+|k\rangle)]$ and $P[\frac{1}{\sqrt2}(|j\rangle-2|i\rangle+|k\rangle)]$ respectively. Hence they have positive semidefinite Laplacians. The Lapalcian of the graph on LHS is the the sum of the Laplacian of the graphs on RHS (Lemma 2.10), each of which is positive semidefinite. But we Know that the sum of positive semidefinite matrices is a positive semidefinite matrix [15]. Now the graph $(G,a)$ can be written as edge union of the factors (spanning subgraph) as Figure 21 (possibly more than once) and the remaining factor which has all positive weights. The Laplacian of each factor is positive semidefinite and the Laplacian of the given graph, being the sum of positive semidefinite matrices, is positive semidefinite.\hspace{\stretch{1}}$ \blacksquare$
{\bf Lemma 7.8 :} If all the negative edges of a real weighted graph $(G,a)$ occur as in the following subgraph as shown in Figure (23), where $c_i>b\; ;\; i= 1, 2, \dots, 8$ and $b>a>0$ then the associated Laplacian is positive semidefinite.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=8cm,height=6cm]{fig23.eps}
Figure 23
\end{center}
\end{figure}
\noindent {\bf Proof :} We can decompose the above graph into factors as shown in Figure(24)
\begin{figure}[!h]
\begin{center}
\includegraphics[width=14cm,height=8cm]{fig24.eps}
Figure 24
\end{center}
\end{figure}
From the graphical equation in Figure (24) we see that the first factor on RHS corresponds to $P[|-\rangle|-\rangle]$, second factor has a positive semidefinite Laplacian from Lemma 7.7 and the third factor has a positive semidefinite Laplacian as it has all positive weights. Since this graph occurs (once or more) as disjoint subgraphs of $(G,a)$ it can be written as edge union of one or more of these subgraphs and the remaining graph containing only positive or complex edges. Since each of these has a positive semidefinite Laplacian, $(G,a)$ also has a positive semidefinite Laplacian.\hspace{\stretch{1}}$ \blacksquare$
{\bf Lemma 7.9 :} Let $(G^{2^n},a)$ be a complete signed graph with weight function $a_{ij}\in \{-1,1\}$
without loops on $2^n$ vertices $n\geq 1$. Let $E_i$ denote the set of edges incident on $i$th vertex $(|E_i|=2^n-1)$ and let $E_i^+, E_i^-$ denote the sets of edges incident on the the $i$th vertex with weight $+1$ and $-1$ respectively, $(E_i=E_i^+ +E_i^-)$. Let $(G^{2^n},a)$ satisfy the following condition
(i) $|E_i^-|=2^{n-1}-1, i=1, 2, \dots, 2^n$, so that the degree of every vertex$=1$.
Then $(G^{2^n},a)$ corresponds to a pure state in $2^n$ dimensional Hilbert space.
\noindent {\bf Proof :} We need to prove that condition (i) in the statement of the Lemma can be realized for all $n$ and that the resulting signed graph corresponds to a pure state for all $n$. We use induction on $n$. It is clear that the assertion is true for $n=1$ with the corresponding pure state given by $P[\frac{1}{\sqrt2}(|1\rangle-|2\rangle)]$.
Now assume that assertion (that is condition (i) and purity of the corresponding state) is true for $n=k$. For the graph corresponding to $n=k+1$ with $|V(G,a)|=2^{k+1}$ consider the modified tensor product
\begin{eqnarray*}
(G^2,a)\boxdot (G^{2^k},a)=(G^{2^{k+1}},a)=
\{ {\cal L}(G^2, a) \otimes {\cal L} \eta (G^{2^k}, a)\} \dotplus \{{\cal L}(G^2, a) \otimes {\cal N}(G^{2^k}, a)\}\\
\dotplus \{ {\cal N}(G^2, a) \otimes {\cal L}(G^{2^k}, a)\} \dotplus \{ \Omega(G^2, a) \otimes \Omega(G^{2^k}, a)\} ~~\mbox{(59)}
\end{eqnarray*}
where ${\cal L}, \eta, {\cal N}$ and $\Omega$ are graph operators defined in equation (20b) and $G^2, G^{2^k}$ are graphs with number of vertices $2$ and $2^k$ respectively.
Note that the last term corresponds to an empty graph as $G^{2^k}$ does not have any loops. Since the modified tensor product of two complete graphs is also a complete graph, $(G^{2^{k+1}},a)$ is a complete graph. Therefore $|E_i(G^{2^{k+1}},a)|=2^{k+1}-1, i=1, 2, \dots, 2^{k+1}$.
To show that condition (i) is realized for $(G^{2^{k+1}},a)$ given the induction hypothesis, we note that the first term in equation (59) contributes $|E^+_i(G^{2^k},a)|$ negative edges to $(1,i)$th vertex in $(G^{2^{k+1}})$ and the third term contributes $|E^-_i(G^{2^k},a)|$ negative edges, while the other two terms have no contribution. Therefore
$$|E^-_{1i}(G^{2^{k+1}},a)|=|E^+_i(G^{2^k},a)|+|E^-_i(G^{2^k},a)|=2^k-1$$
Similarly, the first three terms contribute $|E^-_i(G^{2^k},a)|, 1$ and $|E^+_i(G^{2^k},a)|$ positive edges to $(1,i)$th vertex. Therefore,
$$|E^+_{1i}(G^{2^{k+1}},a)|=|E^-_i(G^{2^k},a)|+|E^+_i(G^{2^k},a)|+1=2^{k-1}-1+2^{k-1}+1=2^k$$
and similarly for $E_{2i}.$
That $(G^{2^{k+1}},a)$ corresponds to pure state follows from the fact that the state corresponding to the modified tensor product of two graphs is the tensor product of the states corresponding to the factors. Since the state for $(G^{2^k},a)$ is pure by induction hypothesis and $(G^2,a)$ is pure, the preceding statement means that $(G^{2^{k+1}},a)$ is a pure product state.\hspace{\stretch{1}}$ \blacksquare$\\
\section{Summary and Comments}
Following is a brief summary of the main features of the paper
\begin{verse}
(i) We have given rules to associate a graph to a quantum state and a quantum state to a graph, with positive semidefinite generalized Laplacian, for states in real as well as complex Hilbert space(2.1 and 6.1).
(ii) We have shown that projectors involving states in the standard basis are associated with the edges of the graph (Theorems 2.7 and 6.4)
(iii) We have given graphical criteria for a state being pure. In particular we have shown that a pure state must have a graph which is a clique plus isolated vertices (Theorem 2.3, 2.4, Remark 6.2)
(iv) For states in a real Hilbert space , we have given an algorithm to construct graph corresponding to a convex combination of density matrices, in terms of the graphs of these matrices (2.3.2).
(v) We have defined a modified tensor product of two graphs in terms of the graph operators ${\cal L}, \eta, {\cal N}, \Omega$ and obtained the properties of these operators (4.2, 6.2). We have shown that this product is associative and distributive with respect to the disjoint edge union of graphs (corollary 4.7, Remark 6.11).
(vi) We have proved that the density matrix of the modified tensor product of two graphs is the tensor product matrices of the factors. (Theorem 4.5, 6.10 ). For real density matrices, we show that a convex combination of the products of density matrices has a graph which is the edge union of the modified tensor products of the graphs for these matrices (corollary 4.11). Thus we can code werner's definition of separability in terms of graphs.
(vii) We have generalized the separability criterion given by S. L. Braunstein, S. Ghosh,T. Mansour, S. Severini, R.C. Wilson [2] to the real density matrices having graphs without loops Lemma (4.16).
(viii) We have found the quantum superoperators corresponding to the basic operations on graphs, namely addition and deletion of edges and vertices. it is straightforward to see that all quantum operations on states result in the addition / deletion of edges and/ or vertices , or redistribution of weights. However, addition / deletion of edges / vertices correspond to quantum operations which are irreversible, in general. Hence it seems to be difficult to encode a unitary operator, which has to be reversible, in terms of the operations on graphs. Further , graphs do not offer much advantage for quantum operations which only redistribute the weights, without changing the topology of the graph, as in this case the graph is nothing more than a clumsy way of writing the density matrix.
(ix) In section 6, we generalize the results obtained in sections 2 - 4, to quantum states in a complex Hilbert space, that is, to the density matrices with complex off-diagonal elements. In fact, all the results previous to section 5 go over to the complex case, except Lemma 4.16. Many of these results have been explicitly dealt with (e.g. Theorem 6.4, Remark 6.2, section 6.2 etc). We have also given rules to associate a graph to a general hermitian operator. We believe that any further advance in the theory reported in this paper will prominently involve graph operators and graphs associated with operators.
(x) Finally, we have given several graphical criteria for the positive semidefiniteness of the generalized Laplacian associated with a graph. Note that by Lemma 7.3 and observation 7.4 all graphs with complex weights, either without loops or with positive weighted loops have positive semidefinite generalized Laplacians. This characterizes a large class of graphs coding quantum states.
\end{verse}
This paper is essentially a generalization of the work by Braunstein, Ghosh and Severini [7] in which the idea of coding quantum mechanics of multipartite quantum systems in terms of graphs was implemented. The motivation in both Braunstein, Ghosh, Severini and this paper is to explore the possibility of facilitating the understanding of mulipartite and mixed state bipartite entanglement using graphs and various operations on them. Whether such a goal can be reached is too early to say. In order to code arbitrary quantum states and observables in terms of graphs, we have to deal with weighted graphs with real or complex weights. Unfortunately, a mathematical theory of such graphs is lacking. Many results pertaining to simple graphs are not available for such weighted graphs. We hope that, through the need of understanding entanglement and related issues the mathematical structure of weighted graphs gets richer and in turn gives a feedback to our understanding of entanglement.
\vspace{0.3cm}
\begin{center}
{\bf Acknowledgement} \\
\end{center}
It is a pleasure to acknowledge Sibasish Ghosh for useful discussions and Guruprasad Kar and Prof. R. Simon for their encouragement. One of us (ASMH) wishes to acknowledge the Government of Yemen for financial support. We thank Bhalachandra Pujari for his help with Letex and Figures.
\begin{center}
{\bf References} \\
\end{center}
\begin{verse}
[1] Werner R F 1989 {\it Phys.Rev.A}{\bf 40} 4277.\\
[2] Braunstein S L, Ghosh S, Mansour T, Severini S, Wilson R C 2006 {\it Phys. Rev. A}{\bf73} 012320\\
[3] Alber G, Beth T, Horodecki M, Horodecki P, Horodecki R, Rotteler M, Weinfurter H, Werner R and Zeilinger A 2001 {\it Quantum Information : An Introduction to Basic Theoretical Concepts and Experiments} (Springer Verlag).\\
[4] Bouwmeester D, Ekert A and Zeilinger A 2000 {\it The Physics of Quantum Information} (Springer Verlag).\\
[5] Nielsen A and Chuang I 2000 {\it Quantum Computation and Quantum Information} (Cambridge University Press).\\
[6] Peres A 1993 {\it Quantum Theory : Concepts and Methods} (Kluwer Academic Publishers).\\
[7] Braunstein S, Ghosh S, severini S 2006 {\it Ann. of combinatorics} {\bf 10} No. 3. e-print quant-ph/0406165.\\
[8] Hildebrand R, Mancini S and Severini S arXiv: cs. CC/0607036 (Accepted in Mathematical Structure in Computer Scinence).\\
[9] Kraus K 1983 {\it States, Effects and Operators : Fundamental Notions of Quantum Theory} (Lecture Notes in Physics, Vol. 190, Springer Verlag).\\
[10] Preskill, http://www.theory.caltech.edu/people/preskil/ph2291/. \\
[11] West D 2002 {\it Introduction to graph theory} (Prentice Hall India).\\
[12] Mohar B 1991 {\it The Laplacian spectrum of graphs} (Graph theory combinotorics and Applications vol.II, wiley).\\
[13] Godsil C and Royle G 2001 {\it Algebraic Graph Theory} (Springer-Verlag). \\
[14] Lancaster P and Tismenetsky M 1985 {\it The Theory of Matrices}(Academic Press Inc.).\\
[15] Horn R and Johnson C 1990 {\it Matrix Analysis} (Cambridge University Press).\\
[16] Imrich W and Klavzar S 2000 {\it Product Graphs, Structure and Recognition} (With a forward by Peter Winkler, Wiley - Interscience Series in Discrete Mathematics and Optimization, Wiley - Interscience, New York). \\
[17] Prisner E 1995 {\it Graph Dynamics} (Pitman Research Notes in Mathematics Series, 338, Longman, Harlow).\\
[18] Satake I. 1975 {\it Linear Algebra} (Marcel Dekker, INC. New York).\\
[19] Marcus M, Minc H 1992 {\it A survy of matrix theory and matrix inequalities} (Dover).
\end{verse}
\end{document} |
1,116,691,498,438 | arxiv | \section{Introduction}
The variations of wind speed, in a certain site, are strictly related to the economic aspects of a wind farm, such as maintenance operations, especially in the off shore farms, pitch angle control on new wind turbines and evaluation of a new site. Many researchers are working proposing new models that can allow the prediction of wind speed, minutes, hours or days ahead. Many of these models are based on neural networks \cite{14,03}, autoregressive models \cite{05,07,753}, Markov chains \cite{sha05,nfa04,06,01,13,794}, hybrid models where the previous mentioned models are combined \cite{you03,01,08,10,11,16,480,555,fore4,fore6,fore8,fore13,f1,f2} and other less used models \cite{fore1,fore2,fore3,fore7,fore9,fore11}. Often, these models are either focused on specific time scale forecasting, or synthetic time series generation. Instead, our model can be used both for time series generation and for forecasting at different time scales.
The approach we propose here is based on indexed semi-Markov chain (ISMC) model that was advanced by the same authors in \cite{wind2} and applied to the generation of synthetic wind speed time series. In \cite{wind2} we showed that our model is able to reproduce correctly the statistical behavior of wind speed. The ISMC model is a nonparametric model because it does not require any assumption on the form of the distribution function of wind speed.
In this work we use the same model, slightly modified by adding a daily deterministic component, to forecast future values of wind speed. We will show that this model performs better than a simple persistence model, by comparing the root mean square errors. The ISMC model is able to forecast wind speed at different time scale without loosing the goodness of forecasting which is almost independent from the time horizon. Another important aspect addressed by this work is the number of data needed to have a good forecast. With this aim we will show the root mean square error as a function of the data used to calibrate the model.
The paper is organized as follows. First of all, in Section 2, we describe the database used for the analysis. In Section 3, we present the model and its validation. Then, in Section 4, we present results of the wind speed forecasting through an indicator of goodness and comparison with the persistence model. Finally in Section 5 we present some concluding remarks.
\section{Database}
The database used for the analysis in this work is freely available from \cite{data} and is composed of more than 230000 data of wind speed collected every 10 minutes. The weather station of L.S.I. -Lastem is situated in Italy at N 45$°$ 28' 14,9'' $-$ E 9$°$ 22' 19,9'' and at 107 $m$ of altitude. The station uses a combined speed-direction anemometer at 22 $m$ above the ground. It has a measurement range that goes from 0 to 60 $m/s$, a threshold of 0,38 $m/s$ and a resolution of 0,05 $m/s$. The database and its empirical probability density function are represented in Figure \ref{fig1}.
\begin{figure}
\centering
\includegraphics[height=10cm]{fig1.pdf}
\caption{Database and its probability density distribution.}\label{fig1}
\end{figure}
We discretized wind speed into 8 states (see Table \ref{st}) chosen to cover all the wind speed distribution. Table \ref{st} shows the wind speed states with their related wind speed values.
\begin{table}
\begin{center}
\begin{tabular}{|c|*{2}{c|}|}
\hline
Sate & Wind speed range $m/s$ \\ \hline
1 & 0 to 1 \\ \hline
2 & 1 - 2 \\ \hline
3 & 2 - 3 \\ \hline
4 & 3 - 4 \\ \hline
5 & 4 - 5 \\ \hline
6 & 5 - 6 \\ \hline
7 & 6 - 7 \\ \hline
8 & $>$7 \\ \hline
\end{tabular}
\caption{Wind speed discretization}
\label{st}
\end{center}
\end{table}
In order to analyze the behavior at different time scales, we resampled the data at different sampling frequencies: namely 30 minutes, 1 hour and 2 hours.
\section{Model}
\subsection{The indexed semi-Markov chain model}
The general formulation of the ISMC as developed in references \cite{dami11a}, \cite{dami11b}, \cite{dami12b} and \cite{wind2} is here discussed informally.
Semi-Markov processes have similar idea as those that generate Markov processes. The processes are both described by a set of finite states $v_n$ whose transitions are ruled by a transition probability matrix. The semi-Markov process differs from the Markov process because the transition times $T_n$ are generated according to random variables. Indeed, the time between transitions $T_{n+1}-T_n$ is random and may be modeled by means of any type of distribution functions.
In studies concerning wind speed modeling the states $v_n$ indicates discretized wind speed at the nth transition and $T_n$ the time in which the nth change of wind speed occurs.
In \cite{dami12,wind4}, different semi-Markov models were applied to the wind speed modeling and it was shown that the semi-Markov models over perform the Markov models and therefore they are to be preferred in the modeling of wind speed to Markovian models.
In order to better represent the statistical characteristics of wind speed, in a recent article, the idea of an ISMC was advanced in the field of wind speed, see \cite{wind2}.
The novelty, with respect to the semi-Markov case, consists in the introduction of a third random variable defined as follow:
\begin{equation}
U_{n}^{m}= \sum_{k=0}^{m} v_{n-1-k} \cdot \frac{T_{n-k}-T_{n-1-k}}{T_{n}-T_{n-1-m}}.
\end{equation}
This variable can be interpreted as a moving average of order $m+1$ executed on the series of the past wind speed values $(v_{n-1-k})$ with weights given by the fractions of sojourn times in that wind speed $(T_{n-k}-T_{n-1-k})$ with respect to the interval time on which the average is executed $(T_n-T_{n-1-m})$.
Also the process $U^m$ has been discretized, Table \ref{Um} shows the states of the process and their values.
\begin{table}
\begin{center}
\begin{tabular}{|c|*{2}{c|}|}
\hline
Sate & $U^m$ range $m/s$ \\ \hline
1 & 0 to 2.1 \\ \hline
2 & 2.1 - 2.6 \\ \hline
3 & 2.6 - 3.4 \\ \hline
4 & 3.4 - 6 \\ \hline
5 & $>$6 \\ \hline
\end{tabular}
\caption{$U^m$ processes discretization}
\label{Um}
\end{center}
\end{table}
The parameter $m$ must be optimized as a function of the specific database. The optimization is made by finding the value of $m$ that realize the minimum of the root mean square error (RMSE) between the autocorrelation functions (ACF) of real and simulated data, see \cite{wind2}. In our analysis $m=7$.
The reasons to introduce this index of memory are found in the presence of a strong autocorrelation that characterize the wind speed process.
In the same work we have shown that if a too small memory is used, the autocorrelation is already persistent but decreases faster than real data. With a longer memory the autocorrelation remain high for a very long period and also its value is very close to that of real data. If $m$ is increased further the autocorrelation drops again to small values. This behavior suggests the existence of an optimal memory $m$. In our opinion one can justify this behavior by saying that short memories are not enough to identify in which status (low, medium low, medium, medium high, high, see Table 2) is the index $U^m$, too long memories mix together different status and then much of the information is lost in the average.
The one step transition probability matrix can be evaluated by considering the counting transition between the three random variables considered before. Then, the probability $p_{i,j}(t,u)$ represents the transition probability from the actual wind speed state $i$, to the wind speed state $j$, given that the sojourn time spent in the state $i$ is equal to $t$ and the value of the process $U^m$ is $u$. These probabilities can be computed as:
\begin{equation}
p_{i,j}(t,u)= \frac{ n_{i,j} (t,u) }{\sum\limits_{j} n_{i,j}(t,u)},
\end{equation}
\label{pri}
\noindent where $n_{ij}(t,u)$ is the total number of transitions observed in the database from state $i$ to state $j$ in next period having a sojourn time spent in the wind speed $i$ equal to $t$ and the value of the index process equal to $u$.
\indent The ISMC model revealed to be particularly efficient in reproducing together the probability density function of wind speed and the autocorrelation function, see \cite{wind2}.
\subsection{Deterministic wind speed component}
\label{deter}
The speed of wind shows a diurnal behavior due to the alternation between night and day. In Figure \ref{acfp}, in which are plotted the ACF of real and simulated data, it is possible to note this sinusoidal trend (see Section \ref{nodo} for a better explanation of the figure). To model this seasonality we add a deterministic component given by a sine wave to the indexed semi-Markov model:
\begin{equation}
v_d = A \cdot sin \left( \frac{2 \pi}{24} h \right) \;\;\;\; h=1,2,...,n.
\end{equation}
\label{det}
The value of the parameter $A$ has to be optimized according to the database used for the analysis. In our case $A=0.41$ and it has been obtained by minimizing the RMSE between the ACF of real and synthetic data by using a genetic algorithm.
\subsection{Transition probability matrix}
We computed the transition probability matrix by using equation $(2)$ to the wind speed database. Two examples of the estimated matrices are given in Tables \ref{m1} and \ref{m2}. As described above, in the model the transition matrix do depend from initial and arrival states but also from the sojourn time and the value of the random variable U. In the example given here we show the transition matrices for $U^m=2$ and $t=2$ and for $U^m=4$ and $t=2$ respectively, evaluated from the original database with the sampling frequency of 10 minutes.
A first comparison between Table \ref{m1} and Table \ref{m2} reveals that the value of the index process affects seriously the transition probability to the next wind speed value. As a matter of example if $i=1$, $t=2$, $U_{n}^{m}=2$, the probability to have a wind speed $j=1$ in next period is equal to $0.7065$, see Table \ref{m1}. On the contrary, if $i=1$, $t=2$, $U_{n}^{m}=4$, the probability to have a wind speed $j=1$ in next period becomes $0.4900$, see Table \ref{m2}. The differences in the one step transition probabilities are significant and confirm the hypothesis that next wind speed depends also on the value of the index process. This fact shows that the index process should be used when dealing with wind speed data.
\begin{table}
\begin{center}
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
$P_{ij}$&\textbf{j=1}&\textbf{2}&\textbf{3}&\textbf{4}&\textbf{5}&\textbf{6}&\textbf{7}&\textbf{8}\\\hline
\textbf{i=1}&0.7065&0.2856&0.0074&0.0001&0.0001&0.0001&0.0001&0.0000\\\hline
\textbf{2}&0.1546&0.7095&0.1310&0.0042&0.0004&0.0002&0.0001&0.0001\\\hline
\textbf{3}&0.0064&0.2779&0.6300&0.0800&0.0045&0.0008&0.0003&0.0003\\\hline
\textbf{4}&0.0005&0.0170&0.3227&0.5764&0.0773&0.0044&0.0011&0.0005\\\hline
\textbf{5}&0.0000&0.0054&0.0349&0.3737&0.4919&0.0753&0.0134&0.0054\\\hline
\textbf{6}&0.0000&0.0000&0.0000&0.0238&0.4048&0.3929&0.1786&0.0000\\\hline
\textbf{7}&0.0000&0.0000&0.0357&0.0357&0.0714&0.3571&0.2857&0.2143\\\hline
\textbf{8}&0.0000&0.0000&0.0000&0.0000&0.0435&0.0000&0.2174&0.7391\\\hline
\end{tabular}
\caption{Transition matrix for $U^m=2$ and $t=2$.}\label{m1}
\end{center}
\end{table}
\begin{table}
\begin{center}
\centering
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline
$P_{ij}$&\textbf{j=1}&\textbf{2}&\textbf{3}&\textbf{4}&\textbf{5}&\textbf{6}&\textbf{7}&\textbf{8}\\\hline
\textbf{i=1}&0.4900&0.4300&0.0700&0.0000&0.0100&0.0000&0.0000&0.0000\\\hline
\textbf{2}&0.1002&0.6171&0.2488&0.0242&0.0048&0.0016&0.0000&0.0032\\\hline
\textbf{3}&0.0048&0.1456&0.6323&0.1975&0.0185&0.0000&0.0014&0.0000\\\hline
\textbf{4}&0.0000&0.0154&0.2270&0.5886&0.1553&0.0113&0.0018&0.0006\\\hline
\textbf{5}&0.0000&0.0009&0.0268&0.2763&0.5638&0.1220&0.0092&0.0009\\\hline
\textbf{6}&0.0000&0.0000&0.0060&0.0301&0.3414&0.5120&0.1004&0.0100\\\hline
\textbf{7}&0.0000&0.0000&0.0000&0.0000&0.0467&0.3400&0.4600&0.1533\\\hline
\textbf{8}&0.0000&0.0000&0.0116&0.0233&0.0000&0.0465&0.2674&0.6512\\\hline
\end{tabular}
\caption{Transition matrix for $U^m=4$ and $t=2$.}\label{m2}
\end{center}
\end{table}
\subsection{Model validation}
\label{nodo}
We compute the ACF of real and synthetic data in order to assess the ability of the model to reproduce statistical properties of real wind speed data. We generate a synthetic time series by means of Monte Carlo simulation. The specific algorithm used for the generation of the trajectory can be found in \cite{wind2}.
If $v$ indicates wind speed, the time lagged $(\tau)$ autocorrelation of wind speed is defined as:
\begin{equation}
\label{autosquare}
\Sigma(\tau)=\frac{Cov(v(t+\tau),v(t))}{Var(v(t))}.
\end{equation}
The time lag $\tau$ was made to run from 10 minutes up to 100 hours.
The ACF of real and the synthetic data are plotted in Figure \ref{acfp}. As it is possible to note, the ACF has a sinusoidal trend with a period of 24 hours. This behavior is reproduced by our model with the introduction of the deterministic wind speed component evaluated by the equation $(\ref{det})$.
\begin{figure}
\centering
\includegraphics[height=8cm]{acf.pdf}
\caption{Autocorrelation function of real and synthetic data}\label{acfp}
\end{figure}
To asses the differences between the ACF of real and synthetic data we used the root mean square error (RMSE) which is defined as follows:
$$ RMSE = \sqrt{ \frac{1}{n} \sum_{i=1}^n \left( v_{i}^{r} - v_{i}^{s} \right)^2 } , $$
where $v^{r}$ and $v^{s}$ represent real data and synthetic one respectively, while $n$ is the length of the two series. For the ACF plotted in Figure \ref{acfp} we obtained a RMSE equal to $0.0223$.
\section{Results}
\subsection{Wind speed forecasting}
In this section the ISMC model is used to forecast future wind speed states by using a one step ahead forecasting procedure, for different time horizons and for various time scales. Particularly, we tested our model using the previously described databases with a sampling frequency of 10 minutes, 30 minutes, 1 hour and 2 hours.
For each one of the sampling frequencies, the database is divided into two subsets: the first part is used to find the transition probability matrix (as described in the previous section), we will call this part the setting period; the second part is used to compare the model forecasting with real data (called testing period).
As a first attempt to verify the model performance, we used two years of data as setting period and one year as testing. We will show in the paragraph \ref{nod} how to find the best setting period.
Once the transition matrix is set, the forecasted states are computed as follows:
\begin{equation}
v^f=\sum_{j=1}^{k} j \, p_{i,j}(t,u),
\end{equation}
where $k$ is the number of states in which wind speed is discretized and $p_{i,j}(t,u)$ is the transition probability matrix. The formula represents the expected value of the next transition given that the present wind speed value is $i$, the sojourn time spent in the state $i$ is equal to $t$ and the value of the index process $U^{m}$ is $u$.
In Figure \ref{figone} we show the results obtained using our model for the four different time scales. In the figure the black continuous line represents real data while the dashed red line is the predicted series. In this figure the predicted series are long 100 time horizon (specific time depending on the sampling frequency).
\begin{figure}
\centering
\includegraphics[height=10cm]{figurona.pdf}
\caption{Wind speed forecasting one step ahead for 100 time horizon. (a) 10 minutes database, (b) 30 minutes database, (c) 1 hour database, (d) 2 hours database.}\label{figone}
\end{figure}
Already from this figure, it is possible to note that the goodness of the prediction does not fall down at the increasing of the length of the forecasted series.
To better verify this point, in Table \ref{tabE} we show quantitative results of our forecasting model for all the considered time scales and for different time horizons. Particularly, we show mean and standard deviation of the RMSE between real and predicted data tested on 50 different forecasted series. Table \ref{tabE} shows that the goodness of prediction remains almost constant even varying time scales and time horizons.
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|c|c|}\hline
\diaghead{\theadfont Diag ColumnmnHead II}%
{Time\\Scale}{Time\\Horizon}&
\thead{50}&\thead{100}&\thead{500}& \thead{1000}\\
\hline
10 minutes & 0.44 $\pm $ 0.02 & 0.44 $\pm $ 0.02 & 0.48 $\pm $ 0.02 & 0.52 $\pm $ 0.02 \\
\hline
30 minutes & 0.48 $\pm $ 0.01 & 0.50 $\pm $ 0.01 & 0.56 $\pm $ 0.01 & 0.62 $\pm $ 0.01 \\
\hline
1 hour & 0.54 $\pm $ 0.01 & 0.54 $\pm $ 0.01 & 0.61 $\pm $ 0.01 & 0.64 $\pm $ 0.01 \\
\hline
2 hour & 0.56 $\pm $ 0.01 & 0.59 $\pm $ 0.01 & 0.65 $\pm $ 0.01 & 0.69 $\pm $ 0.01 \\
\hline
\end{tabular}
\end{center}
\caption{RMSE between real data and forecasted series for different time scale and time horizon.}
\label{tabE}
\end{table}
We compare our model with a simple persistence model. This simple method is often used, still today, in industry for its simplicity and for its efficiency for very short-term predictions. It assumes that the wind speed at time $t+ \Delta t$ is equal to the wind speed at time $t$. Commonly this method is used to compare the behavior of new forecasting models \cite{pers}. Overall our model has a higher efficiency in the forecast for all the time scales and time horizons. The persistence model do not change its goodness of forecasting at varying of the time horizon. Then we compare our results with the persistence model at different time scales. For the frequency of 10 minutes, 30 minutes, 1 hour and 2 hours we have respectively an RMSE between the true series and the forecasted one generated through the persistence model of $0.59 \pm 0.05$, $0.63 \pm 0.08$, $0.73 \pm 0.09$ and $0.85 \pm 0.11$. As is possible to note the persistence model has less precision on the forecasting of the wind speed with respect to our model and the standard deviation increases at the increasing of the time scale in contrast to our model that has a reduction of the variability at the increasing of the time scale.
\subsection{Number of data optimization}\label{nod}
A serious problem to deal with in applying a nonparametric model is that of data availability. An important point is that of establishing the dimension of the setting period needed for a correct implementation of the model. From one part, reducing the setting period may determine the goodness of prediction to drop down; on the other hand the availability of large database is time consuming and consequently not economically efficient and sometimes not statistically necessary. To fix this point as related to the ISMC model we computed the RMSE between real data and a forecasted time series of 1000 time horizon.
We show, in Figure \ref{mincc}, the results obtained for the 30 minutes sampling frequency. It can be noted that the RMSE, plotted as a function of the logarithm of the setting period length, after about 3000 data (corresponding roughly to 2 months) remains almost constant, suggesting that the use of a larger setting period is not necessary.
\begin{figure}
\centering
\includegraphics[height=8cm]{min_cc.pdf}
\caption{RMSE between real wind speed and forecasted series as a function of the logarithm of the number of data.}\label{mincc}
\end{figure}
We repeated the same analysis for all the sampling frequency used obtaining: 20000 data (roughly 6 months) for 10 minutes sampling frequency, 2500 data (roughly 3 months) for 1 hour, and 2000 (roughly 5 months) for 2 hours. The decreasing in the number of data need to have a good forecasting is mainly due to the reduction of noise when the sampling frequency increases.
\section{Discussion and conclusion}
In previous works we presented new stochastic models, all based on a semi-Markov approach, to generate synthetic time series of wind speed.
We showed that all the models perform better than corresponding Markov chain based models in reproducing statistical features of wind speed. Using these results, here, we tried to apply the model which we recognized to be the best among those, namely the indexed semi-Markov chain (ISMC) model, to forecast future wind speed in a specific site.
The ISMC model is a nonparametric model and because of this it does not need any assumption on the distribution of wind speed and on wind speed variations.
In previous papers we showed that the ISMC model is able to reproduce correctly, and at the same time, both the probability distribution function of wind speed and the autocorrelation function.
The results presented in this paper show that the model can be efficiently used to forecast wind speed at different horizon times. The forecast performance is almost independent from the time horizon used to forecast; the model can be used without degradation during the considered horizon time, at different time scales (we showed this for time scales ranging from 10 minutes to 2 hours).
The number of data needed to reach a good forecast performance do depend on the time scale used for forecasting; the model always works better than a simple persistence model.
All these characteristics suggest that the advanced ISMC model may be used both for modeling wind speed data and for wind speed prediction. Therefore, it may be utilized as input data for any wind energy system.
\section*{References}
|
1,116,691,498,439 | arxiv | \section{Introduction}
The {\it Linearization Theorem} for proper Lie groupoids is a cornerstone of the theory. It generalizes classic results such as Ehresmann's theorem for submersions, Reeb stability for foliations, and the tube theorem of compact group actions. It also serves as a key tool in establishing local models for Poisson geometry. The original source \cite{w} proves the regular case and made important contributions, such as a reduction to the fixed-point case. A first complete proof was given in \cite{z},
with some confusion in the statements and the extra assumption of {\it source-locally trivial}.
The hypothesis and variants were later clarified in \cite{cs}.
Given $G\rightrightarrows M$ a Lie groupoid, the {\it linear model} around an orbit $O\subset M$,
which we review in Prop. \ref{prop:local-model}, is the groupoid-theoretic normal bundle $\nu(G,G\r{O})\rightrightarrows \nu(M,O)$.
The Linearization Theorem
\cite[Thm.1-Cor.2]{cs} claims that the groupoid is locally isomorphic to its linear model if it is {\it (s-)proper at a point}. This can be replaced by global (s-)proper, as explained in \cite[Rmk.5.1.4]{dh1}, by restricting the attention to an {\it invariant neighborhood}, namely one that contains every orbit that it meets.
We can then restate the theorem as follows:
\begin{theorem}[{\cite{cs}}]\label{thm:linearization}
If $G\rightrightarrows M$ is a proper Lie groupoid and $O\subset M$ is an orbit, then there are open neighborhoods $O\subset U\subset \nu(M,O)$ and $O\subset V\subset M$ and a {\it linearization} isomorphism:
$$\phi:(\nu(G,G\r{O})\r{U}\rightrightarrows U)\cong (G\r{V}\rightrightarrows V).$$
If $G\rightrightarrows M$ is source-proper then the linearization is {\it invariant}, $U,V$ can be taken to be invariant.
\end{theorem}
In \cite{cs} the authors give a simple proof of the fixed-point case, clarify several other aspects, and propose as an open problem the characterization of invariant linearization. As they posed it, while the above theorem implies a large number of related {\it classic results}, it is intriguing that it does not cover the linearization for proper actions of non-compact groups \cite[Thm.2.4.1]{dk}.
They propose as a possible solution the notion of {\it source inv-trivial} groupoid, which we can replace by source trivial, again by restricting the attention to an invariant neighborhood.
\begin{question}[cf. {\cite[Probl.0.1]{cs}}]\label{q:open}
Does a proper Lie groupoid $G\rightrightarrows M$ whose source map is trivial admit an invariant linearization around its orbits?
\end{question}
Our first contribution here is a negative answer to this question in Ex. \ref{ex:counter}. It turns out that \cite[Ex.10.1]{w}, which combines exotic structures in ${\mathbb R}^4$ and the results on smooth fibrations from \cite{m}, is already a counter-example. We made a slight simplification, and use the main result in \cite{dh2} to insure that a locally trivial submersion over a contractible manifold is trivial.
A new approach to the linearization of groupoids
was developed in \cite{dhf}.
Given $G\rightrightarrows M$ a Lie groupoid, denote by $G_2=G\times_M G$ the manifold of pairs of composable arrows, which identify with commutative triangles.
A {\it 2-metric} is a metric $\eta_2$ on $G_2$ that is fibered for the multiplication $m:G_2\rightarrow G$ and invariant under the action $S_3\curvearrowright G_2$ permuting the vertices of a triangle. Such an $\eta_2$ induces a 1-metric $\eta_1$ on $G$, and a {\it 0-metric} $\eta_0$ on $M$, which is invariant under the normal representation \cite{ppt}.
The main results in \cite{dhf} show a recipe to cook up 2-metrics on proper groupoids, called the {\it gauge trick}, and show that 2-metrics give linearizations via the exponential maps around full invariant subgroupoids, in particular around orbits.
\begin{theorem}[\cite{dhf}]\label{thm:riem-grpd-lin}
If $G\rightrightarrows M$ is proper and $\eta_2$ is a 2-metric
then there are open neighborhoods $O\subset U\subset \nu(M,O)$ and $O\subset V\subset M$ such that the following is an isomorphism:
$$\exp=(\exp^{\eta_1},\exp^{\eta_0}):(\nu(G,G\r{O})\r{U}\rightrightarrows U)\cong (G\r{V}\rightrightarrows V)$$
If $G\rightrightarrows M$ is source-proper then we can take $U$ and $V$ to be invariant.
\end{theorem}
In this paper we build over the Riemannian theory of groupoids and stacks \cite{dhf,dhf2,dhdm} to characterize proper groupoids that are invariantly linearizable. In Thm. \ref{thm:sufficient} we give a sufficient condition for invariant linearization in terms of completeness of groupoid metrics. Then we show in Cor. \ref{coro:tube} how our criterion easily implies the tube theorem for proper actions of non-compact groups. And in Thm. \ref{thm:main2} we cook up complete 0-metrics on proper invariantly linearizable groupoids. This can be seen as (i) a partial converse for Thm. \ref{thm:sufficient}, (ii) a fixed version of \cite[Prop.3.14]{ppt} which is one of the main results there, and (iii) a stacky version of \cite[Thm.5]{dh2}, for we build a complete metric on $M$ which is fibered with respect to the stacky projection $M\rightarrow[M/G]$. Our proof in fact adapts the ideas presented in \cite{dh2}. We can summarize our main contributions as follows:
\begin{theorem2}[\ref{thm:sufficient}, \ref{thm:main2}]\label{thm:main-intro}
Let $G\rightrightarrows M$ be a proper groupoid. Then:
\begin{itemize}
\item[(i)] If it admits a 2-metric $\eta_2$ such that $\eta_0$ is complete, then $G\rightrightarrows M$ is invariantly linearizable.
\item[(ii)] If $G\rightrightarrows M$ is invariantly linearizable then it admits a complete 0-metric $\eta_0$.
\end{itemize}
\end{theorem2}
Note that (ii) is not the exact converse of (i), for we do not know if there is a 0-metric $\eta_0$ which actually extends to a 2-metric. The extension problem for metrics may not have a positive answer in general, see \cite{dhf}, and keeping track of completeness along the gauge trick is delicate.
Anyway, we conjecture that the converse of (i) holds, and we prove it for regular groupoids in Cor. \ref{cor:regular}, where this extension problem has always a solution.
\medskip
\medskip
{\bf Acknowledgments.}
We thank M. Alexandrino, H. Bursztyn, R. Fernandes, P. Frejlich and I. Struchiner for many stimulating talks.
MdH was partially supported by National Council for Scientific and Technological Development — CNPq grants 303034/2017-3 and 429879/2018-0, and by FAPERJ grant 210434/2019.
MdM was supported by FAPESP grant 2019/14777-3.
\section{Linearization and source-triviality}
We review various constructions for the linear model of a Lie groupoid around an orbit, provide examples and basic facts about invariantly linearizable groupoids, and we present the Example \ref{ex:counter} of a source-trivial groupoid that is not invariantly linearizable, hence giving a partial answer to the open question proposed by \cite[Probl.0.1]{cs}.
\medskip
Given $G\rightrightarrows M$ a Lie groupoid and given $O\subset M$ an orbit, we denote by $G_O\subset G$ the submanifolds of arrows within objects of $O$, so $G_O=s^{-1}(O)=t^{-1}(O)$.
The restriction $G_O\rightrightarrows O$ becomes a Lie subgroupoid, and the {\bf linear model} of $G\rightrightarrows M$ around $O$ can be defined in any of the following equivalent ways (see \cite{w,cs,dh1}):
\begin{proposition}\label{prop:local-model}
The following groupoids are canonically isomorphic:
\begin{itemize}
\item[A)] The Lie groupoid-theoretic normal bundle $\nu(G,G_O)\rightrightarrows \nu(M,O)$, whose objects and arrows are the normal bundles $\nu(M,O)=TM\r{O}/TO$ and $\nu(G,G_O)=TG\r{G_O}/TG_O$, and whose structure maps are induced by those of the tangent groupoid $TG\rightrightarrows TM$;
\item[B)] The action groupoid $G_O\times_O \nu(M,O)\rightrightarrows \nu(M,O)$ of the normal representation, which is the linear action of the restriction $G_O\rightrightarrows O$ over the normal bundle $\nu(M,O)\rightarrow O$ given by $g\cdot [v]=[\partial_\epsilon y_\epsilon\r{\epsilon=0} ]$, where $y_\epsilon\xleftarrow{g_\epsilon} x_\epsilon$ is any curve satisfying $g_0=g$ and $[\partial_\epsilon x_\epsilon\r{\epsilon=0} ]=[v]$;
\item[C)]
The quotient $[(P_x\times P_x\rightrightarrows P_x)\times(N_x\rightrightarrows N_x)]/G_x$ of the pair groupoid of a source-fiber $P_x=G(-,x)=s^{-1}(x)$ times the unit groupoid of the normal vector space $N_x=T_xM/T_xO$, by the group action
$G_x\curvearrowright P_x\times P_x\times N_x$, $\lambda\cdot(h,h',v)=(h\lambda^{-1},h'\lambda^{-1},\lambda v)$.
\end{itemize}
\end{proposition}
\begin{proof}
Construction A) is better understood by thinking on the category of VB-groupoids over the restriction $G_O\rightrightarrows O$ \cite[3.4]{dh1}. The linear model is just the cokernel of the VB-groupoid inclusion
$(TG_O\rightrightarrows TO)\rightarrow(TG\r{G_O}\rightrightarrows TM\r{O})$. Since both normal bundles have the same rank, the core of this VB-groupoid is trivial, and therefore it is the action groupoid of a representation, namely the normal representation of B). Finally, since $G_O\rightrightarrows O$ is a transitive groupoid, it is Morita equivalent to the isotropy group $G_x\rightrightarrows x$, for $x\in O$, and then there is a 1-1 correspondence between their representations.
The bibundle realizing this Morita equivalence is $P_x$, and the correspondence between representations follow a universal formula specialized in C).
\end{proof}
The restriction $G_O\rightrightarrows O$ embeds into the linear model as the 0-section. The Lie groupoid $G\rightrightarrows M$ is {\bf linearizable}
around $O$ if there are opens neighborhoods $O\subset U\subset \nu(M,O)$ and $O\subset V\subset M$ and a linearization isomorphism
$$\phi:(\nu(G,G_O)\r{U}\rightrightarrows U)\cong (G\r{V}\rightrightarrows V)$$
The linearization is {\bf invariant} if $U$ and $V$ are so, namely if they contain every orbit they meet.
\begin{examples}
\begin{enumerate}
\item Source-proper groupoids, namely those whose source map $s:G\rightarrow M$ is a proper map, are invariantly linearizable, this is part of Thm. \ref{thm:linearization}.
\item Submersion groupoids arising from fiber bundles (locally trivial submersions) are invariantly linearizable, and they are source-proper if and only if the fiber is compact.
\item Action groupoids from proper actions of Lie groups are also invariantly linearizable \cite[Thm. 2.4.1]{dk}, and they are source-proper if and only if the group is compact.
\end{enumerate}
\end{examples}
The source-local triviality, which holds in the above examples, is in fact a necessary condition.
\begin{lemma}\label{lemma:necessary}
If $G\rightrightarrows M$ is invariantly linearizable then it is source-locally trivial.
\end{lemma}
\begin{proof}
Working locally, it is enough to show that the linear model is source-locally trivial. And this follows easily from the description C) in Prop. \ref{prop:local-model}.
\end{proof}
The main goal of the present paper is to understand which proper groupoids are invariantly linearizable besides the source-proper ones. A first remark in this direction is the following:
\begin{lemma}\label{lemma:orbit-type}
Let $G\rightrightarrows M$ be proper with connected orbit space $M/G$. If $G$ is invariantly linearizable, then either it is source-proper, or none of its orbits are compact.
\end{lemma}
\begin{proof}
For each $x\in M$ the isotropy bundle $G(-,x)\rightarrow O_x$ is a principal bundle with structure group $G_x$, which is compact for $G$ is proper. Then the source-fiber $G(-,x)$ is compact if and only if the corresponding orbit $O_x$ is so.
Since $G$ is invariantly linearizable the source map $s:G\rightarrow M$ is locally trivial and a source-fiber is compact if and only if the nearby ones are, and this is the case if and only if the groupoid is source-proper.
This shows that both the compact and the non-compact orbits define opens in the orbit space $M/G$, hence the result.
\end{proof}
Source-local triviality is a necessary condition for invariant linearization, and as expressed by Weinstein in \cite{w}, it is tempting to think that it is also sufficient,
but \cite[Ex.10.1]{w} already showed a counter-example.
Looking for a sufficient condition to ensure invariant linearization,
\cite[Def.4.12]{cs} proposes the notion of {\bf source inv-trivial}, see also Q. \ref{q:open}.
We show here that this is also not enough, by relating local and global triviality with the following result.
\begin{lemma}\label{lemma:bundles}
A smooth fiber bundle $p:E\rightarrow B$ over a contractible base $B$ must be trivial.
\end{lemma}
\begin{proof}
Let $\{0,1\}\subset I\subset{\mathbb R}$ be an open interval and let $h: B\times I \rightarrow B$ be a smooth homotopy between $h_0(x)=x$ and $h_1(x)=b$ constant.
The pullback bundle $h^*E=(B\times I)\times_B E \rightarrow B\times I$ admits a complete Ehresmann connection $H$ by \cite[Thm.3]{dh2}. The horizontal lift of $(0,\partial/\partial t )$ on $B\times I$ with respect to $H$ is a vector field $X$ whose flow $\varphi_1$ gives an isomorphism between
$h^*E\r{B\times 0}\cong h_0^*E=E$ and $h^*E\r{B\times 1}\cong h_1^*E=E_b\times B$, proving that $E$ is indeed trivial.
\end{proof}
As a counter-example to Q. \ref{q:open} we propose the submersion groupoid associated to a map in \cite[Ex.40]{m}, which is similar to the one originally given by Weinstein \cite[Ex.10.1]{w}.
\begin{example}\label{ex:counter}
Let $V\subset {\mathbb R}^4$ be an open subset homeomorphic but not diffeomorphic to ${\mathbb R}^4$, and let
$E=\{(v,t)\in {\mathbb R}^4\times {\mathbb R} : v\in V\text{ or }t\neq 0\}$.
Then $E$ is a 5-dimensional Euclidean open, contractible, and simply connected at infinity, and therefore diffeomorphic to ${\mathbb R}^5$ \cite[Ex.37]{m}. The projection $\pi:E\rightarrow{\mathbb R}$ is not locally trivial, for the fiber $E_0\cong V$ is not diffeomorphic to the others.
Consider the product between the submersion groupoid of $\pi$ and the group $SU(2)$.
$$(G\rightrightarrows M)=(E\times_{\mathbb R} E\rightrightarrows E)\times(SU(2)\rightrightarrows\ast)$$
It is shown in \cite[Ex.40]{m} that the source map $s:G\rightarrow M$ is locally trivial, and it follows from our Lemma \ref{lemma:bundles} that is globally trivial too, in particular source inv-trivial. But there is not an invariant linearization of $G\rightrightarrows M$, for its orbits identify with the fibers $E_t$ of $\pi$, and the orbits of the linear model $\nu(G,G_{E_0})$ are all diffeomorphic to $E_0=V$.
\end{example}
\section{Completeness as a sufficient condition}
We recall Riemannian submersions and Riemannian groupoids and discuss some preliminaries. Then we present a sufficient condition for invariant linearization, and derive the tube theorem for proper actions of non-compact groups as a corollary.
\medskip
\def\overline{\overline}
Given $\pi: E \rightarrow B$ a submersion, a Riemannian metric $\eta^E$ on $E$ is {\bf fibered} if for all $e,e'$ in the same fiber $E_b=\pi^{-1}(b)$ the composition
$T_eE_b^{\perp} \rightarrow T_b B \leftarrow T_e'E_b^{\perp}$ is an isometry. Equivalently,
$\eta^E$ is fibered if it induces a metric $\eta^B$ on $B$ so that $\pi$ becomes a {\bf Riemannian submersion}, namely $\overline{d\pi_e}:\nu_e(E,E_b)\cong T_eE_b^{\perp} \rightarrow T_b B$ is an isometry for every $e\in E$.
In such a Riemannian submersion, if a geodesic $\gamma$ on $E$ is orthogonal to a fiber then it is orthogonal to every fiber, and its projection is a geodesic \cite[Cor.2]{o}. In this case we say that $\gamma$ is a {\bf horizontal geodesics}.
If $S\subset B$ is embedded and $S'=\pi^{-1}(S)$, the exponential maps yield a commutative square,
$$\xymatrix{
\nu(E,S')\supset U' \ar[r]^(.8){\exp^{E}} \ar[d]_{\overline{d\pi}} & E \ar[d]^\pi \\
\nu(B,S) \supset U \ar[r]_(.7){\exp^{B}} & B
}$$
where $\nu(E,S')\cong TS'^\bot\subset TE$, $\overline{d\pi}=d\pi\r{\nu(E,S')}$, and $U'$ and $U$ are opens around the 0-sections satisfying $\overline{d\pi}(U')\subset U$. The following is a sharper vesion of \cite[Prop.5.9]{dhf}:
\begin{lemma}
\label{lemma:admissible}
\begin{enumerate}[i)]
\item If $\exp^{B}\r{U}$ is \'etale then so does $\exp^{E}\r{U'}$, and the converse holds if $\overline{d\pi}(U')=U$.
\item If $\exp^{B}\r{U}$ is injective then so does $\exp^{E}\r{U'}$, and the converse holds if $\eta^E$ is complete and $U'=\overline{d\pi}^{-1}(U)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Given $(e,w)\in U'$, writing $\overline{d\pi}(e,w)=(b,v)$, $\exp^E(e,w)=\tilde e$ and $\exp^B(b,v)=\tilde b$, we have the following map of short exact sequences:
$$
\xymatrix{0\ar[r]& \ker_{(e,w)} d({\overline{d\pi}})\ar[r]\ar[d] & T_{(e,w)}U' \ar[r]^{d({\overline{d\pi}})}\ar[d]^{d\exp^E} & T_{(b,v)}U \ar[d]^{d\exp^B} \ar[r] &0\\
0 \ar[r] & \ker_{\tilde e} d\pi \ar[r] & T_{\tilde e}E \ar[r]^{d\pi} & T_{\tilde b} B \ar[r] & 0}$$
The first vertical arrow is an isomorphism, it identifies with the differential of the parallel transport
\smash{$E_b\cong \overline{d\pi}^{-1}(b,v)\xrightarrow{\exp^E} E_{\tilde b}$}
over the geodesic $\exp^B(b,\epsilon v)$. It follows that the second vertical arrow is an isomorphism if and only if the third one is, hence i).
Suppose that $\exp^{E}(e,w)=\exp^E(e',w')$ with $(e,w),(e',w')\in U'$. Then
$\exp^B(\pi(e),\overline{d\pi}(w))=\exp^B(\pi(e'),\overline{d\pi}(w'))$ and, if
$\exp^{B}\r{U}$ is injective,
$(\pi(e),\overline{d\pi}(w))=(\pi(e'),\overline{d\pi}(w'))$. The geodesics $\exp^E(e,\epsilon w)$ and $\exp^E(e',\epsilon w')$ have then the same projection and the same value at $1$, so they are equal and $(e,w)=(e',w')$, proving that $\eta^E\r{U'}$ is injective. Next we prove the converse.
Consider $(b,v),(b',v')\in U$ with $\exp^{B}(b,v)=\exp^B(b',v')$. Picking $(e,w)\in U'$ such that $\overline{d\pi}(e,w)=(b,v)$, the geodesic $\exp^E(e,\epsilon w)$ is a horizontal lift of $\exp^B(b,\epsilon v)$. If $\eta^E$ is complete we can lift $\exp^B(b',\epsilon v')$ to a horizontal geodesic $\gamma$ such that $\gamma(1)=\exp^E(e,w)$. Then $(\gamma(0),\dot\gamma(0))\in \overline{d\pi}^{-1}(U)=U'$ and satisfy $\exp^E(\gamma(0),\dot\gamma(0))=\gamma(1)=\exp^E(e,w)$. Since $\exp^E\r{U'}$ is injective we conclude that $(\gamma(0),\dot\gamma(0))=(e,w)$ and that
$(b',v')=\overline{d\pi}(\gamma(0),\dot\gamma(0))=\overline{d\pi}(e,w)=(b,v)$.
\end{proof}
Given a Lie groupoid $G\rightrightarrows M$, write $G_2=G\times_M G$ for the manifold whose points are pairs of composable arrows, or equivalently commutative triangles. There is a canonical action $S_3\curvearrowright G_2$ which permutes the vertices of a triangle. A {\bf 2-metric} is a metric $\eta_2$ on $G_2$ that is fibered for the multiplication $m:G_2\rightarrow G$ and invariant under the $S_3$-action.
A 2-metric $\eta_2$ induces metrics $\eta_1$ on $G$ and $\eta_0$ on $M$ such that the following hold \cite{dhf}:
\begin{itemize}
\item $m,\pi_1,\pi_2:G_2\rightarrow G$ and $s,t:G\rightarrow M$ are Riemannian submersions;
\item $u(M)\subset G$ is totally geodesic;
\item $\eta_0$ is a {\bf 0-metric}, namely it is invariant under the normal representation, in the sense that for every $y\xleftarrow g x$ the linear map $\rho_g:\nu(M,O)_x\rightarrow\nu(M,O)_y$ is an isometry; and
\item $\eta^M$ makes the foliation by orbits a singular Riemannian foliation.
\end{itemize}
The main theorems on groupoid metrics in \cite{dhf} are: (i) every proper groupoid admits a 2-metric, and (ii) the exponential maps of $\eta_1,\eta_0$ yield groupoid linearizations around orbits and, more generally, invariant submanifolds.
The normal vectors in $\nu(G,G_O)$ and $\nu(M,O)$ give rise to {\bf normal geodesics}, namely geodesics on $G$ that are both horizontal for the source and target, and geodesics on $M$ that are orthogonal to the orbits.
We will show here that if a groupoid metric is complete then the resulting linearization is invariant.
Given $(M,\eta)$ a Riemannian manifold, $S\subset M$ a submanifold and $r>0$,
write $B'(S,r)\subset\nu(M,S)$ for the {\bf infinitesimal tube} around $S$ of radius $r$, namely the normal vectors with norm smaller than $r$, and write $B(S,r)\subset M$ for the {\bf tube} around $S$ of radius $r$, namely the points whose distance at $S$ is smaller than $r$.
If $\eta$ is complete and $S$ is closed, then the distance from a point $s$ to $S$ can be realized by a geodesic orthogonal to $S$, and therefore $\exp(B'(S,r))=B(S,r)$.
\begin{proposition}\label{prop:tube}
Given $G\rightrightarrows M$ a Lie groupoid and $O\subset M$ an orbit, then:
\begin{enumerate}[i)]
\item If $\eta_0$ is a 0-metric then the infinitesimal tubes $B'(O,r)$ are invariant in the linear model;
\item If $\eta_2$ is a 2-metric then $B'(G_O,r)=\overline{ds}^{-1}(B'(O,r))=\overline{dt}^{-1}(B'(O,r))=\nu(G,G_O)\r{B'(O,r)}$;
\item If $\eta_2$ is a 2-metric, $G\rightrightarrows M$ is proper, and the induced 0-metric $\eta_0$ is complete, then the tubes $\exp^{\eta_0}(B'(O,r))=B(O,r)$ are invariant in $G\rightrightarrows M$.
\end{enumerate}
\end{proposition}
\begin{proof}
i) is immediate, for the normal representation preserves the norm of the normal vectors. ii) is also easy, for in this case the source and target map are Riemannian submersions, and therefore the norm of normal vectors are preserved. Regarding iii), let $r>0$ and let $O'$ be another orbit of $G$ that intersects $B(O,r)$ in some $x\in M$. Let us show that
for any $y\xleftarrow g x$ the point $y$ is also in the tube $B(O,r)$.
Since $G$ is proper the orbit $O'$ is closed, and since $\eta_0$ is complete there is a normal geodesic $a$ realizing the distance between $O$ and $x$, so $a(0)\in O$, $a(1)=x$ and $\nrm{\dot a(0)}<r$.
By \cite[Prop.10]{dhdm} the normal geodesic $a$ admits a global lift through the source $\tilde a$ starting at $g$. Then the projection via the target $t\circ \tilde a$ is a normal geodesic of length smaller than $r$ and connecting $O$ and $y$, hence proving that $y\in B(S,r)$.
\end{proof}
We are now ready to prove our first main result, giving a sufficient condition for invariant linearization, in terms of completeness of compatible metrics.
\begin{theorem}\label{thm:sufficient}
Let $G\rightrightarrows M$ be a proper groupoid and $\eta_2$ a 2-metric such that the induced 0-metric $\eta_0$ is complete. Then $G\rightrightarrows M$ is invariantly linearizable around its orbits.
\end{theorem}
\begin{proof}
Let $O\subset M$ be an orbit.
Fix $x_0\in O$, take $x_0\in W\subset O$ a relatively compact neighborhood, and pick $0<r<\frac{1}{2}d(x_0,O\setminus W)$ such that
$\exp^{\eta_0}\r{B'(W,r)}$ is an embedding. We will show now that $\exp^{\eta_0}$ is then an embedding over the whole infinitesimal tube $B'(O,r)$.
{\it $\exp^{\eta_0}\r{B'(O,r)}$ is \'etale:} Given $(x,v)\in B'(O,r)$, take
$(x_0,v_0)\xleftarrow {(g,w)} (x,v)$ in $\nu(G,G_O)\r{B'(O,r)}=B'(G_O,r)$, see Prop. \ref{prop:tube}.
The normal geodesics are defined for all time by \cite[Prop.10]{dhdm}, and $\exp^{\eta_0}:B'(W,r)\rightarrow M$ is an open embedding. It follows from Lemma \ref{lemma:admissible} applied to the target map that $\exp^{\eta_1}: \overline{dt}^{-1}(B'(W,r))\rightarrow G$ is also an open embedding. The same Lemma, now applied to the source map, says that $\exp^{\eta_0}:\overline{ds}(\overline{dt}^{-1}(B'(W,r)))\rightarrow M$ is at least \'etale, and therefore $d\exp^{\eta_1}:T_{(g,w)}\nu(G,G\r{O})\rightarrow T_{\exp^{\eta_1}(g,w)}G$ is a linear isomorphism.
{\it $\exp^{\eta_0}\r{B'(O,r)}$ is injective:} Let $(x,v),(x',v')\in B'(O,r)$ be such that $\exp^{\eta_0}(x,v)=\exp^{\eta_0}(x',v')$. As before, take
\smash{$(x_0,v_0)\xleftarrow {(g,w)} (x,v)$} in $B'(G_O,r)$.
Since $s(\exp^{\eta_1}(g,w))=\exp^{\eta_0}(x',v')$, we can lift the geodesic $\exp^{\eta_0}(x',\epsilon v')$ through the source to a normal geodesic $\gamma$ such that $\gamma(1)=\exp^{\eta_1}(g,w)$.
We claim that the projection $t\circ\gamma$ starts in $W$. This is because $\exp^{\eta_0}(x_0,v_0)=t(\exp^{\eta_1}(g,w))=t(\gamma(1))=\exp^{\eta_0}((t\circ\gamma)(0),\dot{(t\circ\gamma)}(0))$, and therefore
$$d(x_0,(t\circ\gamma)(0))\leq d(x_0,\exp^{\eta_0}(x_0,v_0))+d((t\circ\gamma)(1),(t\circ\gamma)(0))\leq \n{v_0}+\n{\dot{(t\circ\gamma)}(0)}<2r.$$
By construction $\exp^{\eta_0}$ is injective over $B'(W,r)$, from where $(x_0,v_0)=((t\circ\gamma)(0),\dot{(t\circ\gamma)}(0))$ and $\gamma(\epsilon)=\exp^{\eta_1}(g,\epsilon w)$.
Finally, projecting via the source, we conclude that
$\exp^{\eta_0}(x',\epsilon v')=s(\gamma(\epsilon))=s(\exp^{\eta_1}(g,\epsilon w))=\exp^{\eta_0}(x,\epsilon v)$ and that $(x',v')=(x,v)$.
We have that $(M,\eta_0)$ is complete, that the geodesics normal to the orbits in $G$ are defined for all time, see Prop. \cite[Prop.10]{dhdm}, and we have just seen that $\exp^{\eta_0}\r{B'(O,r)}$ is an open embedding. It follows from the proof in \cite[Thm.5.11]{dhf} that
we can actually take $U=B'(O,r)$ and get a linearization isomorphism
$$\exp:(\nu(G,G\r{O})\r{B'(O,r)}\rightrightarrows B'(O,r))\rightarrow (G\r{B(O,r)}\rightrightarrows B(O,r))$$
By Prop. \ref{prop:tube} $B'(G_O,r)$ and $\exp^{\eta_0}(B'(O,r))=B(O,r)$ are both invariant, hence the result.
\end{proof}
The criterion presented in the previous theorem allows us to derive the linearization of proper actions of non-compact groups as a corollary of the linearization of groupoids. We need the following simple remark on completeness of pushed forward metrics (cf. \cite[Thm.1]{h}).
\begin{lemma}\label{lemma:quotient}
If $\pi:(E,\eta^E)\rightarrow (B,\eta^B)$ is a Riemannian submersion and $\eta^E$ is complete then so does $\eta^B$.
In particular, if $K$ is a Lie group, $(M,\eta^M)$ is complete and $K \curvearrowright M$ is a free and proper isometric action then the quotient $M/K$ inherits a complete metric.
\end{lemma}
\begin{proof}
Given a geodesic $a$ in $B$, if $\tilde a$ is some local horizontal lift, then it can be extended to every time because $E$ is complete, and the projection gives an extension of $a$.
\end{proof}
We will now cook up a complete 2-metric on an action groupoid coming from a proper action, using the {\bf gauge-trick} from \cite{dhf,dhf2}, which combined with Thm. \ref{thm:sufficient}, frames the Tube theorem into the theory.
\begin{corollary}\label{coro:tube}
If $K\curvearrowright M$ is a proper Lie group action, then the action groupoid
$K\times M\rightrightarrows M$ is invariantly linearizable around its orbits.
\end{corollary}
\begin{proof}
Let $\eta^K$ be a right invariant metric on $K$, which is of course complete, as the right translations of geodesics are again geodesics, and regard $\eta^K\times\eta^K\times\eta^K$ as a 2-metric on the pair groupoid $K\times K\rightrightarrows K$.
Let $\eta^M$ be a complete $K$-invariant metric on $M$, see e.g. \cite[Lemma.4.3.6]{dm}, and regard it as a 2-metric on the unit groupoid $M\rightrightarrows M$.
Then $(\eta^K\times\eta^K\times\eta^K,\eta^M)$ is a complete 2-metric on the product and it is fibered for the canonical groupoid fibration
$$(K\times K\rightrightarrows K)\times(M\rightrightarrows M) \rightarrow (K\times M\rightrightarrows M)$$
given on objects, arrows and pairs of composable arrows by the following formulas:
$$(k,x)\mapsto kx \qquad (k_2,k_1,x)\mapsto (k_2k_1^{-1},k_1x) \qquad
(k_3,k_2,k_1,x)\mapsto (k_3k_2^{-1},k_2k_1^{-1},k_1x)$$
The pushforward 2-metric is complete by Lemma \ref{lemma:quotient} and the result follows from Thm. \ref{thm:sufficient}.
\end{proof}
Our Thm. \ref{thm:sufficient} is a groupoid version of the classic result\cite[Thm.1]{h}, asserting that a complete Riemannian submersion is locally trivial, whose converse was later shown in \cite[Thm.5]{dh2}. Our result should also be compared with \cite[Thm.1]{mr}, where a complete singular Riemannian foliation $(M,F,\eta)$ is shown to be isomorphic to a linear model over a tube around a leaf -- the complete hypothesis is missing in their statement but used along the proof. When the foliation is induced by a complete Riemannian groupoid then the invariant linearization gives a similar result. The problem of comparing both models is left to be explored elsewhere.
\section{Cooking up complete invariant metrics}
We review here the Morita invariance of 2-metrics and the result metrics on stacks \cite{dhf2}. Then we prove our second main result, which shows the existence of complete 0-metrics on proper invariantly linearizable groupoids. We finally show that in the regular case this 0-metric can be extended to a 2-metric, and pose the question for the general case.
\medskip
Given a Lie groupoid $G\rightrightarrows M$, we denote by $[M/G]$ it Morita equivalence, or equivalently its orbit {\bf differentiable stacks} (see e.g. \cite{dh1,dhf2}). Two 2-metrics $\eta_2,\eta'_2$ on $G\rightrightarrows M$ are {\bf equivalent} if for every $x\in M$ they induce the same inner product on $\nu(M,O)_x$. The class $[\eta_2]$ is a Morita invariant by \cite[Thm.6.3.3]{dhf2}, hence it defines a {\bf stacky metric}. Stacky geodesics were then introduced and studied in \cite{dhdm}. Next we provide a quick review of the concepts that we need, and refer there for further details and examples.
A {\bf stacky curve} $\alpha:I\rightarrow[M/G]$ is described by a sequence of curves of objects $a_i:I_i\rightarrow M$, where $I_i\subset I\subset{\mathbb R}$ are connected opens, together with curves of arrows $a_{i+1,i}:I_{i+1}\cap I_i\rightarrow G$ linking them by $ta_{i+1,i}=a_{i+1}$ and $sa_{i+1,i}=a_i$.
Two collections $(a_i,a_{i+1,i})$ define the same stacky curve if they induce isomorphic maps $(\coprod_i I_{i+1,i}\rightrightarrows \coprod_i I_i)\rightarrow (G\rightrightarrows M)$ over a common refinement.
If $\eta$ is a 2-metric on $G\rightrightarrows M$, a {\bf stacky geodesic} $\alpha$ is a stacky curve which can be represented by geodesics $(a_i,a_{i+1,i})$ normal to the orbits.
If every stacky geodesic can be extended to every time we say that $[\eta_2]$ is a {\bf complete} stacky metric on $[M/G]$.
\begin{proposition}\label{prop:complete-various}
Let $G\rightrightarrows M$ be a proper groupoid.
\begin{enumerate}[a)]
\item If a 2-metric $\eta_2$ induces a complete 0-metric $\eta_0$ then $[\eta_2]$ is complete;
\item There may not exists a complete 0-metric $\eta_0$;
\item There always exists a 2-metric $\eta_2$ such that $[\eta_2]$ is complete.
\end{enumerate}
\end{proposition}
\begin{proof}
A stacky geodesic $\alpha:I\rightarrow[M/G]$ is locally represented by a geodesic $a:I_i\rightarrow M$ normal to the orbits, this extends to the whole line $\bar a:{\mathbb R}\rightarrow M$, giving a stacky global extension $[\bar a]$ of $\alpha$. This proves a).
Regarding b), let $G\rightrightarrows M$ be the submersion groupoid arising from the first projection $\pi_1:{\mathbb R}^2\setminus\{0\}\rightarrow{\mathbb R}$. The existence of a complete 0-metric $\eta_0$ would imply that $\eta_0\times\eta_0\times\eta_0$ is a 2-metric extending it, that $G\rightrightarrows M$ is invariantly linearizable by Thm. \ref{thm:sufficient}, and that $\pi$ is locally trivial by Lemma \ref{lemma:necessary}, which is clearly not the case.
Finally, c) appears as \cite[Cor.21]{dhdm}, where a product $f\eta$ of an arbitrary metric $\eta$ and a conformal factor measuring the distance to $\infty$ is considered.
\end{proof}
Note that a) is a stacky version of Lemma \ref{lemma:quotient} and \cite[Thm.1]{h} applied to $M\rightarrow[M/G]$.
The counter-example in b) is already presented in \cite[Rmk.5]{dhdm} and shows that the completeness in \cite[Prop.3.14]{ppt} is not always possible to obtain. We will now correct this result by adding up the key hypothesis of invariantly linearizable, hence acquiring a partial converse for Thm. \ref{thm:sufficient}. In light of Lemma \ref{lemma:orbit-type}, we can split the problem in the source-proper case and in the case where the orbits are non-compact. The first case easily follows from the following result.
\begin{proposition}\label{prop:cooking-sproper}
Let $G\rightrightarrows M$ be a $s$-proper groupoid. A 2-metric $\eta_2$ is complete if and only if the stacky metric $[\eta_2]$ on $[M/G]$ is complete.
\end{proposition}
\begin{proof}
If $\eta_2$ is complete then we have just proved that $[\eta_2]$ is also complete in Prop. \ref{prop:complete-various}. Suppose now that $[\eta]$ is complete. Let $a: (p,q)\rightarrow M$ be a maximal geodesic and suppose that $q<\infty$.
Given $q_n\nearrow q$, by \cite[Thm.3]{dhdm} we get a Cauchy sequence $([a(q-\frac{1}{n})])$ in $M/G$. By the stacky Hopf-Rinow Theorem \cite[Thm.19]{dhdm} there is $x\in M$ such that $[a(q-\frac{1}{n})]\rightarrow[x]$. It follows that $a(t)$ sits inside some compact tube $\overline{B(O_x,\epsilon)}$ for $t$ close to $q$, hence $a$ is extendable and we reach a contradiction. The proof $p=-\infty$ is analogous.
\end{proof}
Given $G\rightrightarrows M$ an invariantly linearizable proper groupoid with non-compact orbits, our strategy to cook up an invariant complete metric on it will be the following: (i) set a complete stacky metric on $[M/G]$, (ii) lift it locally to invariantly linearizable opens via the stacky submersion $U\rightarrow [U/G_{U}]$, and (ii) patch the local pieces together in way inspired by \cite[Thm.5]{dh2}. Step (i) was recalled in Prop. \ref{prop:complete-various}. Let us address now the Step (ii).
\begin{lemma}\label{lem:cooking-localmodel}
Let $\eta_2$ be a 2-metric on the
the infinitesimal tube $B'(G_0,2r)\rightrightarrows B'(O,2r)$, viewed as a subgroupoid of the linear model $\nu(G,G_O)\rightrightarrows\nu(M,O)$. Then there is a new 2-metric $\eta'_2$ on $B'(G_0,2r)\rightrightarrows B'(O,2r)$ such that $\eta'_0$ is complete and such that $\eta_2\r{B'(G_0,r)}\sim\eta'_2\r{B'(G_0,r)}$.
\end{lemma}
\begin{proof}
Pick $x\in O$, write $K=G_x$ and $N=B(0,2r)\subset\nu_x(G,G_O)$. Then the infinitesimal tube is Morita equivalent to the action groupoid $K\times N\rightrightarrows N$ via the following two maps,
$$(B'(G_0,2r)\rightrightarrows B'(O,2r))\leftarrow (P\times P\rightrightarrows P)\times (K\times N\rightrightarrows N)\rightarrow (K\times N\rightrightarrows N)$$
where the second is just the projection, and the first is the quotient by the subgroupoid $K\rightrightarrows\ast$.
Let $\eta'$ be a 2-metric on $K\times N\rightrightarrows N$ corresponding to $\eta$ by this Morita equivalence \cite[Thm.6.3.3]{dhf2}.
Let $f:N\rightarrow{\mathbb R}$ be a positive $K$-invariant function such that $f\r{B'(0,r)}\equiv 1$ and $f\eta'_0$ is complete. The composition $f_2=f\pi:K\times K\times N\rightarrow{\mathbb R}$ satisfy that $f_2\eta'_2$ is a 2-metric on $K\times N\rightrightarrows N$ and that
$(f\eta')\r{B'(0,r)}$ equals $\eta\r{B'(0,r)}$.
Let $\eta''_0$ be a complete $K$-invariant metric on $P$ \cite[Lemma.4.3.6]{dm}. Then the product $\eta''_0\times\eta''_0\times\eta''_0\times f_2\eta'_2$ is a complete 2-metric on the middle groupoid that is $K$-invariant, and its quotient via the first map is the desired 2-metric.
\end{proof}
We are finally in conditions to prove our second main theorem,
\begin{theorem}\label{thm:main2}
Let $G\rightrightarrows M$ be a proper groupoid that is invariantly linearizable around its orbits. Then it admits a complete 0-metric $\eta_0$.
\end{theorem}
\begin{proof}
By Prop. \ref{prop:complete-various} we can consider a 2-metric $\eta_2$ on $G\rightrightarrows M$ such that $[\eta_2]$ is complete.
The problem now consists of showing that $[\eta_2]$ can be lifted to a complete metric on $M$ along the stacky submersion $M\rightarrow[M/G]$.
We can work on each connected component of $M/G$ independently.
It follows from Lemma \ref{lemma:orbit-type} that we can either assume that it is source-proper or that none of its orbits are compact. In the first case, it follows from Prop. \ref{prop:cooking-sproper} that $\eta_2$ is already complete and we are done. In the second case, we will show how to replace $\eta_2$ by an equivalent metric $\eta_2'$ such that $\eta'_0$ is complete.
For each $x$ in $M$, since $G\rightrightarrows M$ is invariantly linearizable around $O_x$, there exist $r>0$ and $O_x\subset V_x\subset M$ such that
$\exp:(B(G_{O_x},2r_x)\rightrightarrows B(O_x,2r_x))\cong (G\r{V_x}\rightrightarrows V_x)$ is an isomorphism.
Write $W_x=\exp(B(O_x,r_x))$.
Using Lemma \ref{lem:cooking-localmodel} we can build a new metric $\eta^x_2$ on $G\r{V_x}\rightrightarrows V_x$ such that $\eta^x_2\r{W_x}\sim \eta_2\r{W_x}$ and that $\eta^x_0$ is complete. The next step will be to merge the several $\eta_2^x$ using a smart partition of 1 emulating what is done in \cite[Thm.5]{dh2}.
Extract a countable covering $\{W_{x_i}\}_{i\in{\mathbb N}}$ from $\{W_x\}_{x\in M}$, and fix $f:M\rightarrow [0,+ \infty)$ a smooth proper function. Note that
$\overline W_{x_i}\cap\{f>n\}\supset O_{x_i}\cap\{f>n\} \neq\emptyset$ for all $n$ because we are assuming that the orbits are non-compact.
For each pair $i,n$ such that $\overline W_{x_i}\cap\{f\leq n\}\neq\emptyset$ the set $\overline{B(\overline W_{x_i}\cap\{f\leq n\},1)}$ is compact
and therefore we can pick $l(i,n)>0$ satisfying $d(\overline W_{x_i}\cap\{f\leq n\}, \overline W_{x_i}\cap\{f>n+l(i,n)\})>1$.
We define an {\bf $f$-tube} within $V_{x_i}$ with inner radius $n$ to be a set of the form $T_i(n)=\overline W_{x_i}\cap \{n\leq f\leq n+l(i,n)\}$. Using these compact $f$-tubes we will merge the 0-metrics $\eta^{x_i}_0$ into a complete 0-metric $\eta'_0$ equivalent to $\eta_0$.
We first construct a sequence of $f$-tubes $\{T_i(n(i,j))\}_{i\leq j}$ with inner radius defined inductively as follows. We start by setting $n(1,1)$ so that $T_1(n(1,1))\neq\emptyset$. After choosing $n(i,j)$, if $i<j$, we set $n(i+1,j)$ so that the new tube $T_{i+1}(n(i+1,j))$ is not empty and does not meet any of the previous tubes, which is possible because $f$ has a maximum over the union of them. If $i=j$ we proceed similarly, setting $n(1,j+1)$ so that $T_{1}(n(1,j+1))$ is not empty and does not meet the previous tubes.
This way we end up with a sequence $\{T_i(n(i,j))\}_{i\leq j}$
such that (i) it contains infinitely many $f$-tubes within $V_{x_i}$, and (ii) the terms of the sequence are pairwise disjoint.
Let $T_i=\bigcup_{j} T_i(n(i,j))$ be the union of the $f$-tubes in of the sequence within $V_{x_i}$.
Let $\{\varphi_i\}_{i\in{\mathbb N}}$ be a partition of unity subordinated to $V_{x_i}\setminus \bigcup_{i\neq k} T_{k}$.
Finally, take the contangent average of the metrics $\eta^{x_i}_0$
$$\eta'_0= \left(\sum_i \varphi_i (\eta_0^{x_i})^{*}\right)^{*}.$$
Since each $\eta^{x_i}_0$ induce the same inner product on the normal vector spaces $\nu(M,O)_x$ as $\eta_0$, the same holds for $\eta_0$, and therefore $\eta_0$ is a 0-metric. It only remains to show that $\eta'_0$ is complete.
Let $a:(p,q)\rightarrow M$ be a maximal unit-speed geodesic, suppose $q\leq\infty$ and take $q_n\nearrow q$.
By the relation between the stacky metric and the distance on $M/G$ established in \cite[Thm.3]{dhdm} we have $d([a(q_n)],[a(q_m)])\leq \m{q_n-q_m}$, so $([a(q_n)])$ is a Cauchy sequence. By the stacky Hopf-Rinow \cite[Thm.19]{dhdm} the space $(M/G,d)$ is complete and we get $x\in M$ such that $[a(q_n)]\rightarrow[x]$.
Let $i$ be such that $x\in W_{x_i}$, and hence $a(q-\delta, q)\subset W_{x_i}$ for small $\delta$.
Since $a(q-\delta,q)$ cannot be extended, it cannot be contained in any compact, and therefore it must go through infinitely many tubes $T_i(n(i,j))$. But over each of these tubes the metric $\eta'_0$ agrees with $\eta^{x_i}_0$, and therefore $a$ needs at least time 1 to go through each of them. This leads to a contradiction proving that $q=\infty$. The proof of $p=-\infty$ is analogous.
\end{proof}
It should be noted that Thm. \ref{thm:main2} is not the precise converse of Thm. \ref{thm:sufficient}, for a priori the constructed 0-metric is not induced by a 2-metric. The problem of extending a 0-metric to a 2-metric is subtle and we refer to \cite{dhf} for several examples.
We believe that a proper invariantly linearizable groupoid may indeed admit a 2-metric $\eta_2$ with $\eta_0$ complete, but we have not found yet a proof.
When working with (Hausdorff) regular groupoids things get simpler:
\begin{lemma}\label{prop:extension}
If $G\rightrightarrows M$ is regular then every 0-metric $\eta_0$ extends to a 2-metric.
\end{lemma}
\begin{proof}
Let $F\subset TM$ be the foliation by orbits and let $\eta'$ be an auxiliary metric on $G$. Writing $I=\ker ds\cap \ker dt$, we get the following vector bundle orthogonal decomposition:
$$TG= I \oplus (\ker ds \cap I^\bot) \oplus (\ker dt \cap I^\bot) \oplus
(\ker ds+\ker dt)^\bot$$
The maps $ds,dt:(\ker ds+\ker dt)^\bot\rightarrow F^\bot$ are fiberwise isomorphism, and the pullbacks of $\eta\r{F^\bot}$ along these two maps agree, for $\eta$ is invariant. We endow $(\ker ds+\ker dt)^\bot$ with this metric.
We also endow $\ker ds \cap I^\bot$ and $\ker dt\cap I^\bot$ with the pullback metrics of $\eta_0\r{F}$ along $ds$ and $dt$, respectively, equipped $I$ with an arbitrary metric, and declare the four terms to be orthogonal. The resulting metric $\eta'$ on $G$ is fibered with respect to the source and the target, and therefore, the cotangent average $\frac{1}{2}(\eta'^* +i^*\eta'^*)$ is a 1-metric $\eta'_1$ \cite[Prop.2.2]{dhf}.
We can use a similar argument to extend $\eta'_1$ to a 2-metric, or alternatively, we can directly apply the gauge trick to $\eta'_1$, for $\eta'_0$ is in this case preserved, see \cite[Lemma.3.1.5]{dhf2}.
\end{proof}
\begin{corollary}\label{cor:regular}
A regular proper groupoid $G\rightrightarrows M$ is invariantly linearizable if and only if it admits a 2-metric $\eta_2$ with $\eta_0$ complete.
\end{corollary}
{\small
|
1,116,691,498,440 | arxiv | \section{Introduction}
SN~1006 is the supernova remnant (SNR) for which X-ray synchrotron
emission from diffusive shock accelerated electrons was first proposed
(Reynolds \& Chevalier 1981) and detected in X-rays with {\it ASCA}
(Koyama et al.\ 1995). It remains an unrivaled laboratory for
studying these phenomena because of its large size ($\sim30^\prime$
diameter; Winkler \& Long 1997) and low interstellar absorption
(6.8$\times$10$^{20}$\,cm$^{-2}$; Dubner et al.\ 2002). Very
recently, the HESS team reported the firm detection of TeV
$\gamma$-ray emission from this SNR (Acero et al.\ 2010).
One of {\it Chandra}'s great discoveries in particle acceleration
physics was that rims of SN~1006 and other young SNRs are very narrow,
much narrower than the 1/12 shock radii expected for a strong shock
with a compression ratio of 4 (e.g., Long et al.\ 2003; Bamba et al.\
2003; 2005). There is a general consensus that these narrow filaments
are indirect evidence for strongly amplified magnetic fields at or
upstream of the shock. However, the origin of the narrowness has been
debated; two interpretations proposed so far have clearly different
scenarios (e.g., Cassam-Chena\"i et al.\ 2007). One considers the
effect of a rapid decay of the amplified magnetic field downstream, so
that bright narrow magnetic filaments are formed behind the shock
(Pohl et al.\ 2005). The other assumes a relatively constant, strong
magnetic field downstream of the shock. In this case, accelerated
electrons quickly lose their energy through synchrotron radiation,
resulting in narrow synchrotron X-ray filaments (e.g., V\"olk et al.\
2005 and references therein). The nature of the magnetic-field
amplification is not well understood at this time; the post-shock
evolution of the field holds important clues to the process, so
settling the question of the mechanism limiting filament widths has
considerable significance.
Meanwhile, in RX~J1713.7-3946 and Cas~A SNRs, several
synchrotron-dominated knotty features, whose size is about
10$^{\prime\prime}$, were found to show year-scale time variations
(e.g., Uchiyama et al.\ 2007; Patnaude \& Fesen 2009). The rapid
variations may reflect fast acceleration or cooling of accelerated
electrons in strongly amplified magnetic fields up to the level of mG.
On the other hand, more diffuse regions in
these SNRs do not show rapid variations, which leads the authors to
consider that these regions have somewhat weaker magnetic fields.
Thus, it is now possible to roughly estimate magnetic-field strengths,
$B$, in SNRs, when time variations in synchrotron emission can be
measured. Synchrotron X-ray flux variation can be produced
stochastically, even in the absence of variations in the electron
distribution, in the presence of a stochastic magnetic field (Bykov,
Uvarov, \& Ellison 2008). Somewhat smaller rms values
of magnetic field are required, but substantial amplification is still
necessary. (It should be noted that absence of variation does
not demand low magnetic-field strengths; systematic, smooth
steady-state magnetic-field amplification could produce steady
emission varying only on overall SNR dynamical timescales, decades to
centuries.)
In this paper, we investigate time variations of discrete features
in the northeastern (NE) limb of SN~1006, using two {\it Chandra}
observations taken in 2000 and 2008. We have recently measured proper
motions of the shock fronts in the NE limb to be almost uniform at
0.$^{\prime\prime}$5\,yr$^{-1}$ (Katsuda et al.\ 2009; hereafter
Paper I). By correcting for the proper motion, we can track the same
regions away from the shock front (or loosely, the same fluid
elements) in two epochs. We also reveal detailed spatial
structures of the synchrotron emission in the NE limb. Based on the
results, we discuss why the synchrotron filaments are so narrow and
the mechanism that limits the maximum energy of accelerated electrons.
\section{Observations}
We use two {\it Chandra} observations taken in 2000 (ObsID.\ 732) and
2008 (ObsID.\ 9107) that are the same data presented in our previous
proper-motion measurements (Paper I). We use the reduced data
products described in Paper I. We note that the
second observation was specifically intended to allow a proper-motion
measurement, and therefore the pointing direction, roll angle, and
exposure time are the same as those in the first observation. This
configuration was chosen to allow as precise a comparison in the two
epochs as possible because the same physical regions are seen at
almost the same detector position with the same effective area and
spatial resolution.
\section{Analysis and Results}
Figure~\ref{fig:image} (a) shows a three-color {\it Chandra} image of
SN~1006, where red, green, and blue correspond to 0.5--0.8\,keV
(mostly, K-shell lines of O), 0.8--2.0\,keV (mostly, Ne, Mg, and Si K
lines), and 2.0--5.0\,keV (mostly, synchrotron continuum) bands,
respectively. Regions seen in white are dominated by nonthermal
synchrotron emission, while those in red or green are dominated by
thermal emission. In this paper, we focus on the nonthermal emission,
for which we will investigate time variations as well as detailed
spatial structures.
As shown in Fig.~\ref{fig:image} (b), we extract spectra from a number
of small regions that are annular sectors covering the
nonthermally-dominated area in the NE limb. We use the SNR center of
[(ra, dec) = ($15^\mathrm{h}$02$^\mathrm{m}$54$^\mathrm{s}$.9,
$-41^{\circ}56^{\prime}08^{\prime\prime}.9$) (J2000)] determined from
the {\it ROSAT} HRI image (Paper I). The sizes of the regions range
from 10$^{\prime\prime}\times30^{\prime\prime}$ to
15$^{\prime\prime}\times115^{\prime\prime}$ to assure that
each regions contains about 3000 counts. There are 175 regions in
all. For simplicity of our spectral analysis, we exclude boundary
regions between the front- and back-illuminated chips (the boundaries
are indicated as dashed lines in Fig.~\ref{fig:image} (b)). Since we
know that forward shocks in the NE limb are moving at
$0^{\prime\prime}.5$\,yr$^{-1}$ (Paper I), we simply shift all the
regions by 4$^{\prime\prime}$ outward for the second observation. In
this way, we extract two spectra (taken in 2000 and 2008) from each
region. We subtract background emission from the source-free areas in
the identical chip of the same observation. (The background never
amounts to more than 15\% of the total counts in a region, and is
normally much less, so the use of $\chi^2$ statistics is a reasonable
approximation.) In addition to the X-ray data, we also use a VLA
image at 1.37\,GHz (Dyer et al.\ 2009) to constrain the normalization
of nonthermal emission. We calculate radio fluxes from co-spatial
regions, i.e., the small regions shifted by 2$^{\prime\prime}$ outward
compared with those for the first observations, since the radio image
was taken in 2004. The radio resolution is $14^{\prime\prime} \times
6^{\prime\prime}$ (long axis N-S), so the effect of this correction
should be small.
Although most of the regions show featureless spectra, some regions
exhibit K lines from metals such as O, Ne, Mg, or Si. We thus employ
an absorbed nonthermal plus thermal components model, where we employ
the {\tt tbabs} model (Wilms et al.\ 2000) for absorption, the {\tt
srcut} model, which describes synchrotron emission from a power-law
distribution of electrons with an exponential cut-off (Reynolds \&
Keohane 1999), with correction described in Reynolds 2008 for the
nonthermal component, and the {\tt vpshock} model, which describes
thermal emission from a non-equilibrium ionization (NEI) plane-shock
plasma, in conjunction with NEI version 2.0 (Borkowski et al.\ 2001)
for the thermal component. In the fitting, photons in an energy range
of 0.4--8.0\,keV are used. We fix the intervening hydrogen column
density, $N_\mathrm{H}$, to be 6.8$\times$10$^{20}$cm$^{-2}$ (Dubner
et al.\ 2002). In the {\tt vpshock} component, we fix the abundances
of O, Ne, Mg, and Si to be 4.4, 1.5, 15, and 50 times solar values
(Anders \& Grevesse 1989), respectively, following the most recent
{\it XMM-Newton} results (Miceli et al.\ 2009). Other elemental
abundances are fixed to solar values. The electron temperature,
$kT_\mathrm{e}$, and the ionization timescale, $n_\mathrm{e}t$, are
fixed to 0.5\,keV and $1\times10^{10}$\,cm$^{-3}$\,sec, respectively
(Miceli et al.\ 2009), where $n_\mathrm{e}t$ is the electron density
times the elapsed time after shock heating and the {\tt vpshock} model
assumes a range of $n_\mathrm{e}t$ from zero up to
$1\times10^{10}$\,cm$^{-3}$\,sec. Note that Miceli et al.'s results
actually show that $kT_\mathrm{e} \sim 0.4$\,keV and $n_\mathrm{e}t
\sim 1.5 \times 10^{10}$\,cm$^{-3}$\,sec in the NE limb, but this
parameter set does not affect our spectral-fit parameters presented
below. The only free parameter we set in the {\tt vpshock} model is
the volume emission measure (VEM; VEM $=\int n_\mathrm{e}n_\mathrm{H}
dV$, where $n_\mathrm{H}$ is the number density of protons, and $V$ is
the X-ray--emitting volume). For the {\tt srcut} component, we let
the cut-off frequency and the mean spectral index (the $\alpha$
parameter) inferred from the X-ray spectrum be free parameters,
whereas the normalization (the flux at 1\,GHz) is fixed to the value
extrapolated from the radio flux at 1.37\,GHz (Dyer et al.\ 2009)
assuming a photon index of 0.55. In the
initial fits, we allowed cut-off frequencies to vary freely in the two
(2000 and 2008) data sets, but we found them to be consistent with
each other. We thus simultaneously fit the 2000 and 2008 spectra, by
linking all the spectral-fit parameters except for an additional
parameter, the relative intensity of the {\tt srcut} component between
2000 and 2008, which is allowed to vary freely so that we can measure
time variations in its flux.
Figure~\ref{fig:spec1} shows example spectra extracted from regions A
(with no significant thermal emission) and B (with significant thermal
emission) indicated in Fig.~\ref{fig:image} (b). Black and red
correspond to 2000 and 2008, respectively. The spectral difference
between the two colors, which is clearly seen below 1\,keV, shows the
accumulation of molecular contaminants on the ACIS-S optical blocking
filter. Also shown in the figure are the best-fit models and the
residuals. Since the evolution of the contaminants is accounted for in
the response files, the same model (with slightly adjusted intensity
of the {\tt srcut} component) fits both 2000 and 2008 data well.
Spectral-fit parameters and fit statistics for the example spectra are
summarized in Table~\ref{tab:param}. In this fitting procedure, a
number of parameters for the thermal component are assumed and fixed.
Although these values are plausible, it is worth checking the
sensitivity of the fit results to varying these parameters. Before
investigating, we first note that fit results for most of the regions
are not sensitive to the treatments of thermal parameters since these
regions are dominated by nonthermal emission like Region A
(Fig.~\ref{fig:spec1} left). On the other hand, the rest of the
regions including Region B (Fig.~\ref{fig:spec1} right), where
contributions of the thermal emission are relatively large, could be
affected by the assumptions in the fitting. We tried fitting the
spectrum from Region B with $\pm$10\% different values for
$N_\mathrm{H}$, $kT_\mathrm{e}$, and $n_\mathrm{e}t$. We found that
the fit results, i.e., the best-fit parameters in the {\tt srcut}
component, are not significantly changed from the original results
listed in Table~\ref{tab:param}. Thus, variations of up to 10\% in
these parameters would not affect our results. We have also checked
different sets of metal abundances. We examined three cases: (1)
C=N=O=4.4, (2) Si=S=50, and (3) Fe=(Ni=)20 times the solar values,
where the Fe abundance is based on the Fe-rich ejecta measured in the
southeastern portion of SN~1006 with {\it Suzaku} (Yamaguchi et al.\
2008). The first case yields a slightly better fit than the original
fit, but the best-fit parameters are consistent with those in
Table~\ref{tab:param}. The second case gave us almost the same
results as those in Table~\ref{tab:param}, since S K lines are
negligible for the assumed plasma conditions of ($kT_\mathrm{e}$,
$n_\mathrm{e}t$)=(0.5\,keV, 1$\times10^{10}$\,cm$^{-3}$\,sec). In the
third case, we find significantly different results: the photon index
and the cut-off frequency are found to be 0.534$\pm0.007$ and
2.3($\pm0.2)\times10^{16}$\,Hz, respectively. However, the fit level
($\chi^2$=158) is not as good as that in Table~\ref{tab:param}.
Therefore, we believe that the Fe abundance in the NE limb is more
likely to be closer to the solar values rather than 20 times the solar
values, and that our fitting procedure with the solar abundance for Fe
is robust.
Maps of the the reduced $\chi^{2}$s, best-fit parameters (mean
spectral index and cut-off frequencies), fluxes in the {\tt srcut}
component (2000 and 2008), and the flux ratios (2008/2000) are shown
in Fig.~\ref{fig:spatial} (a)--(f), where fluxes are calculated in the
0.4--8.0\,keV band after correcting for interstellar absorption.
Figure~\ref{fig:spatial} (a) shows that the fits are fairly good for
all the regions: the reduced $\chi^2$s are derived to be less than
1.5. We see relatively worse fits at the southern regions. This
is because of the simplicity of the thermal model. Relatively large
residuals are found around 0.7\,keV energies where spectral modeling
of the thermal emission is quite hard due to either missing K lines of
O or inadequate atomic data for Fe L-shell lines (e.g., Yamaguchi et
al.\ 2008). Also, such residuals are particularly evident in the
southern regions where the contributions of thermal emission are
relatively large compared with the rest of the regions. It is highly
likely that this discrepancy does not affect the spectral-fit
parameters in the nonthermal component. Therefore, we are confident
of the best-fit parameters shown in Fig.~\ref{fig:spatial}.
As shown in Fig.~\ref{fig:spatial} (b), mean spectral index are
inferred to be around 0.5, which is consistent with the recent results
from {\it Chandra} (Allen et al.\ 2008) and {\it XMM-Newton} (Miceli
et al.\ 2009), but is slightly flatter than the radio value of
0.60 (0.51--0.68 for 90\% C.L.) reported for the integrated spectrum
(Allen et al.\ 2008). Our inferred mean spectral indices depend on
flux ratios between radio and X-rays. Thus, systematic uncertainties
of radio fluxes, which could originate from relatively worse spatial
resolution of the radio image than that of the X-ray image, are
subject to additional uncertainties of the mean spectral index. We
check for the Region~A spectrum that a 50\% larger radio flux would
yield a $\sim$5\% larger photon index. Therefore, together with the
relatively large uncertainty ($\sim$15\%) of the radio value from the
integrated spectrum (Allen et al.\ 2008), we do not formally find
significant inconsistency of the mean spectral index. Nonetheless,
the best-estimated values show discrepancy, which would suggest a
curved nonthermal spectrum expected in a nonlinear theory of diffusive
shock acceleration (Reynolds \& Ellison 1992), as previously noted by
others (e.g., Allen et al.\ 2008). Cut-off frequencies in
Fig.~\ref{fig:spatial} (c) show strong variations in both radial and
azimuthal direction. These spatial variations are also generally
consistent with previous studies (Rothenflug et al.\ 2004; Allen et
al.\ 2008; Miceli et al.\ 2009). We find a correlation between the
cut-off frequency and the flux, for the first time. This will be
briefly discussed in the next section. Detailed discussion about these
spatial structures will be published elsewhere.
We find flux maps of 2000 and 2008 to be quite similar to each
other. This is confirmed by a flux ratio map in
Fig.~\ref{fig:spatial} (f), where most of the regions are in red
(i.e., constant fluxes). In the figure, we see that a few northern
regions show somewhat higher values (yellow color in
Fig.~\ref{fig:spatial} (f)) than the others. This is because
of imperfect correction for the proper motions there; these regions
have slightly larger proper motions than assumed here (see Paper
I) and we checked that, if we choose spectral extraction regions more
explicitly, we do not see flux variations. To study the flux
variation more quantitatively, we plot the flux ratio (2008/2000)
as a function of the flux in 2000 in Fig.~\ref{fig:hist1} left. We
can see that the data points are clustered tightly about a line
representing no time variation. This means that the fluxes in 2000
and 2008 are quite similar with each other for most of the regions.
On the other hand, it should be noted that there are a few data points
showing relatively large variations of $\sim$20\%. We can not rule
out dramatic changes in those otherwise completely undistinguished
regions, but we believe that the fluxes in these regions are not really
changing. This is because some of them are expected to be due to
imperfect corrections for the proper motions as mentioned above. And
others are just statistical fluctuations, because (1) we have checked
that these regions are randomly scattered (Fig.~\ref{fig:spatial} (f))
and (2) the histogram of the flux ratio is well represented by a
Gaussian function as shown in Fig.~\ref{fig:hist1} right. In fact,
if we apply a constant model to the flux ratios in
Fig.~\ref{fig:hist1} left, we obtain a fairly good fits of
$\chi^2$/dof = 157/174 with a null hypothesis probability of 0.82.
Thus, from a statistical point of view, we cannot reject the
possibility that the fluxes are constant everywhere in the NE limb.
Moreover, by looking at spectra from these regions, we see
nothing special in their spectral features nor particular spectral
parameters. In this context, we conclude that there are no peculiar
regions showing strong time variations, and that all the regions show
little or no time variations. We notice that the center of the
histogram in Fig.~\ref{fig:hist1} right is apparently shifted from
unity: the Gaussian center is measured to be 0.977$\pm$0.006 (90\%
C.L.).
The apparent decline in the global synchrotron flux is interesting,
but falls within the calibration uncertainties for the effective area
(3\%\footnote{http://web.mit.edu/iachec/IACHEC\_2\_talks/IACHEC\_II\_chandra\_summary.pdf})
determined by the CXC. Therefore, we have explored a variety of
possibilities for improving the relative calibration of our
measurement. There are relatively few point sources in the field, and
these are not expected to be constant with time in any event. Indeed,
the brightest source, QSO1 in Winkler et al.\ (2005) was brighter in
2008 by 50\% than in 2000. A more promising alternative is to search
for a change in the thermal emission and compare this to changes in
the nonthermal emission.
Our strategy to estimate changes in nonthermal and thermal emission is
as follows. We first estimate nonthermal flux variations from the
nonthermally-dominated regions (i.e., the outer regions elongated in
the azimuthal direction in Fig.~\ref{fig:image} (b)) as we have
already analyzed. In this process, we assume no time variation for
thermal emission to avoid possible degeneracy in separating thermal
and nonthermal components; it is difficult to estimate the
contribution of thermal emission correctly in these regions, and
incorrect intensity ratios between the two epochs for thermal emission
would affect those for nonthermal emission more or less. In any case,
thermal emission makes only a small contribution in those regions.
Next, we estimate thermal flux variations from thermally-dominated
regions (i.e., interior regions elongated in the radial direction in
Fig.~\ref{fig:image} (b)). (Here, we assume the flux variation of
nonthermal emission has the value measured above.) Spectra of
these thermally-dominated regions are clearly distinct from those of
the nonthermally-dominated regions (Fig.~\ref{fig:spec1}),
resulting in different mean photon energies between them. To compare
fluxes in these different kinds of spectra as accurately as possible,
it may be important to consider the energy dependence of effective area
and quantum efficiency. Therefore, we divide the spectra into three
energy bands: 0.4--0.8\,keV (K lines of O), 0.8--1.0\,keV (K lines of
Ne), and 1.0--8.0\,keV. Finally, we compare flux variations between
nonthermal and thermal emission in the three energy bands.
To measure flux variations in synchrotron emission of the three
energy bands separately, we re-fit spectra from nonthermally-dominated
regions. We employ the same model used above (i.e., {\tt vpshock}
plus {\tt srcut}). We also treat the spectral-fit parameters in the
same manner as above. The only exception is that the mean spectral
indices are fixed to 0.5 which is typical in the NE limb (typical, if
no spectral curvature is assumed) for relatively narrow energy bands
of 0.4--0.8\,keV and 0.8--1.0\,keV where we cannot constrain both the
photon index and the cut-off frequency (Note that, when fitting the
1.0--8.0\,keV band, we allow the mean spectral indices to vary freely,
since we can constrain them). As mentioned above, we assume no time
variations for the thermal component. In this way, we fit all the
spectra in the three energy bands, and derive statistically acceptable
fits for them. Example spectra from region A indicated in
Fig.~\ref{fig:image} right are shown in Fig.~\ref{fig:spec2} left.
Then, fluxes of the nonthermal component are calculated from the
best-fit models. Figure~\ref{fig:hist_all} (the first row) shows
histograms of the flux ratios (2008/2000) for the three energy bands
together with their best-fit Gaussian functions. The best-fit values
of the Gaussian center are summarized in Table~\ref{tab:flux_ratio},
from which we can see the energy dependence of the flux variations.
Next, we investigate flux variations in thermal emission. Since we
were concerned that the thermally-dominated regions might not have the
same proper motion as the synchrotron dominated regions, we examine
four cases: 0$^{\prime\prime}$, 1$^{\prime\prime}$,
2$^{\prime\prime}$, and 3$^{\prime\prime}$ shifts in radial direction
between the two epochs.
It is also difficult to satisfactorily reproduce the thermal emission
by plasma models (e.g., Yamaguchi et al.\ 2008). Therefore, we
alternatively apply a phenomenological model consisting of several
Gaussian components in addition to two bremsstrahlung components plus
an {\tt srcut} component. The use of a phenomenological model is also
justified by the fact that we are not trying to draw inferences from
the model parameters but are just trying to get a good flux measurement.
For the 0.4--0.8\,keV band, we include five Gaussians at
$\sim$0.44\,keV (N He$\alpha$), $\sim$0.5\,keV (N Ly$\alpha$),
$\sim$0.57\,keV (O He$\alpha$), $\sim$0.66\,keV (O Ly$\alpha$ and O
He$\beta$), and $\sim$0.7\,keV (O He$\gamma$ and/or Fe L). For the
0.8--1.0\,keV band, we include two Gaussians at $\sim$0.71\,keV (O
He$\delta$) and $\sim$0.91\,keV (Ne He$\alpha$). For the
1.0--8.0\,keV band, we include three Gaussians at $\sim$1.35\,keV (Mg
He$\alpha$), $\sim$1.8\,keV (Si He$\alpha$), and $\sim$2.4\,keV (S
He$\alpha$). Center energies, widths, and normalizations in the
Gaussian components are treated as free parameters, but for those at
0.7\,keV and 0.71\,keV, only normalizations are allowed to vary freely
with fixed center energies and fixed widths at zero. We fix
$kT_\mathrm{e}$s in the two
bremsstrahlung components to 0.5\,keV and 2.0\,keV, based on recent
X-ray analyses from {\it Suzaku} (Yamaguchi et al.\ 2008) and {\it
XMM-Newton} (Miceli et al.\ 2009). In the {\tt srcut} model, the
mean spectral index is fixed to 0.5. The normalization is also fixed to
the value estimated from the 1.37\,GHz image. The cut-off frequency
is left as a free parameter. The relative intensity of the {\tt
srcut} model between the two epochs is fixed to those derived in the
previous paragraph (see, Table~\ref{tab:flux_ratio}), whereas that of
the thermal component (i.e., the sum of all the components excluding
the {\tt srcut} component) is allowed to vary freely so that we can
obtain its flux variation. This model yields statistically
acceptable fits for all the spectra in the three energy bands.
Example spectra from region C indicated in Fig~\ref{fig:image} right
are shown in Fig.~\ref{fig:spec2} right. Similarly to the
nonthermally-dominated regions, we generate flux-ratio histograms of
the thermal component as shown in Fig.~\ref{fig:hist_all}, where the
second, third, fourth, and fifth rows are responsible for
0$^{\prime\prime}$, 1$^{\prime\prime}$, 2$^{\prime\prime}$, and
3$^{\prime\prime}$ shifted cases, respectively. The values of the
best-fit Gaussian centers are summarized in
Table~\ref{tab:flux_ratio}.
Looking at Table~\ref{tab:flux_ratio}, we see that there are flux
changes in both the thermal and nonthermal emission and that the
changes in the two components. In fact, ratios of the flux variations
between nonthermal and thermal emission are calculated to be about
unity at all the three energy bands as also shown in
Table~\ref{tab:flux_ratio}. This is strong evidence that the changes
in flux are due to calibration effects.
Can the flux changes in both the thermal and nonthermal emission be
understood? The fluxes increase by $\sim$3\% at low energies whereas
they decrease by $\sim$4\% at high energies. Also notable in the
table is energy dependence of flux variations: the fluxes increase by
$\sim$3\% at low energies, whereas they decrease by $\sim$4\% at high
energies. Since the contaminants on the optical blocking filter could
influence spectra below 1\,keV, the increasing flux at low energies
could be due to this effect. On the other hand, it cannot fully
explain the decreasing flux at high energies.
There are two possibilities for the variations at high energies. One
is that some calibration effects cause the apparent changes for both
thermal and nonthermal emission, i.e., the fluxes are actually almost
constant with time. In this case, the time variation of nonthermal
emission would be less than 1\% over 8\,yrs, based on the
time-variation ratio between nonthermal and thermal measured in
1--8\,keV (see, Table~\ref{tab:flux_ratio}). This interpretation is
supported by the fact that flux variations of thermal emission are in
good agreement with those of nonthermal emission; it is likely that
the agreements are not just coincidence but that they have the same
underlying origin. As an additional check of calibration effects, we
compared two observations of clusters of galaxies, since they are not
expected to change over the time period of interest (i.e.,
$\sim$10\,yrs). We chose the Fornax cluster and HCG62, since they
were observed twice over this time period with the same chip (i.e.,
chip7) on the ACIS-S array. We found that both of them are apparently
declining: $\sim$7\% between 2000 and 2009 for the Fornax cluster and
$\sim$3\% between 2000 and 2008 for HCG62. This result implies the
presence of calibration effects. Given that we measure relative
fluxes between 2000 and 2008, we need time-dependent calibration
effects to explain the flux variations seen. These effects includes
the buildup of the ACIS contaminant, the increase in charge transfer
inefficiency which could result in $\lesssim$1\% uncertainty in flux
measurements, and the variable particle background which could result
in $\sim$1\% uncertainty in flux measurements (a private communication
with Paul Plucinsky). To estimate the flux uncertainty from
contaminants, we use the {\tt acisabs} model in XSPEC with response
files without corrections for the effects of contaminants. This model
allows us to examine various amounts of contaminants by specifying
various time since the launch of {\it Chandra} in its parameter. We
find that a 10\% variation of the quantum efficiency at 0.67\,keV
would result in a 1\% variation of the flux in 1--8\,keV for a
nonthermally-dominated spectrum. Therefore, calibration uncertainties
in relative fluxes could be as large as $\sim$3\%, consistent with our
measurements. We conclude that the time variation in the flux from
synchrotron emission is most likely constant to 1\%, and certainly
less than 1\%.
However, we cannot fully rule out the other possibility that both
thermal and nonthermal emission are declining at similar rates by
chance. Therefore, it is interesting to investigate the time
variation from a theoretical point of view. Simple models for the
evolution of synchrotron brightness of SNRs (e.g., Reynolds \&
Chevalier 1981) predict the rate at which synchrotron flux
should be dropping. As shown in the Appendix A, assuming as in that
paper that both magnetic-field energy density and
relativistic-electron energy density scale with postshock pressure $P
\propto \rho u_s^2$, but generalizing from the assumption of Sedov
evolution made there to the observed expansion rate $R \propto t^m$
with $m = 0.54$ (Paper I), we predict that above the cut-off frequency
$\nu_{\rm cutoff}$ the synchrotron intensity should drop off at (0.2
-- 0.25)\% yr$^{-1}$ (1.6\% -- 2.0\% between 2000 and 2008), of the
same order as the small variation we find. The thermal emission
should change, too. However, as it depends on NEI effects, etc.,
modeling its time variation is much harder and is beyond the scope of
this paper. Without estimates of the time variation for thermal
emission, we leave it open whether the rate of the time
variation over 8 yrs is at $\sim$4\% or less than 1\%, or whether
the entire effect is due to calibration uncertainties.
\section{Discussion}
We have investigated time variations of discrete regions in the NE
limb of SN~1006, using two {\it Chandra} observations taken in 2000
and 2008. We found that there are no particular features showing
strong time variations, and that the synchrotron emission stays at
constant within 4\% and probably with 1\% over the time span. This
result distinguishes SN~1006 from core-collapse SNRs such as
RX~J1713.7-3946 and Cas~A in which several hot spots show year-scale
time variations of a factor $\sim$2 or more (e.g., Uchiyama et al.\
2007). To understand the cause of the difference between SN~1006 and
others, it should be noted that there are no knotty features in the
SN~1006 NE limb. In fact, diffuse regions in RX~J1713.7-3946 and
Cas~A do not show fast time variations, either. This suggests that
rapid time variations are only observed in bright knotty features.
Such a situation is indeed predicted by a recently proposed theory
that interactions between SNR shocks and ambient small-scale cloudlets
amplify magnetic fields through plasma instabilities, and resultant
strongly magnetized features (which appears as knots or filaments)
show rapid brightness changes (Giacalone \& Jokipii 2007; Inoue et
al.\ 2009). In this view, the fact that we do not find knotty
features which could show rapid time variations in SN~1006 is
reasonably interpreted as it is located at high Galactic latitude
where small-scale cloudlets are not present. Additionally, SN 1006
as a Type Ia remnant is interacting with undisturbed ISM instead of
the stellar wind of the progenitor as is likely for the other two
objects, and massive-star winds may be quite clumpy. Further
investigations for the rest of the limb of SN~1006 and other SNRs will
be good opportunities to test the scenario for the origin of rapid
time variations in terms of amplification of magnetic fields.
Our failure to find strong time variability in the synchrotron
emission from SN~1006 is consistent with the absence of small
structures in its morphology. While significant brightness changes on
a timescale of a few years may be explained as electron acceleration
or synchrotron-loss timescales, requiring magnetic field strengths of
0.1 -- 1 mG (e.g., Uchiyama et al.\ 2007), the absence of such changes
does not require that the magnetic fields be weak. Steady-state
particle acceleration at the shock, followed by downstream convection
in the presence of energy losses, would result in synchrotron flux
varying only on the timescales estimated in the Appendix A, which are
independent of $B$ and depend only on the shock deceleration rate.
The absence of strong variability in SN~1006 may then be explained by
its being a remnant of a Type Ia supernova, expanding into relatively
uniform material. The high magnetic fields estimated assuming filament
thicknesses are set by synchrotron losses ($B \sim 100 \ \mu$G, e.g.,
Vink \& Laming 2003; Morlino et al.~2010; Ksenofontov et al.~2005)
are not in contradiction with our result of little flux variability.
We also revealed spatial structures of the synchrotron emission in
unprecedented detail, and found a correlation between the flux and the
cut-off frequency. Given that the flux likely depends on the magnetic
field, the simplest explanation is that the cut-off frequency depends
on the magnetic field as well, so that the magnetic field controls
spatial structures of both the flux and the cut-off frequency. This
is important in understanding the mechanism limiting the maximum
energy, $E_\mathrm{max}$, of accelerated particles in SN~1006. If the
SNR age and/or escape of particles limit $E_\mathrm{max}$, then
$E_\mathrm{max} \propto B$ (Reynolds 2008). In this case, the cut-off
frequency, which is proportional to $E_\mathrm{max}^2 B$, goes as
$B^3$. On the other hand, if radiative losses limit $E_\mathrm{max}$,
then $E_\mathrm{max} \propto B^{-0.5}$, canceling the $B$-dependence
of the cut-off frequency. Therefore, the possible $B$-dependence of
the cut-off frequency we found suggests that synchrotron radiative
losses do not limit $E_\mathrm{max}$ in the SN~1006 NE limb. This
would mean that the observed $E_\mathrm{max}$ of electrons would apply
to ions as well. Using the highest cut-off frequency of
$\sim2\times10^{17}$Hz at the outermost regions and a magnetic field
just behind the shock of 90$\mu$G (Morlino et al.\ 2010), we estimate
$E_\mathrm{max}$ to be
$\sim$12($\nu_\mathrm{cutoff}/2\times10^{17}$Hz)$^{0.5}$($B$/90$\mu$G)$^{-0.5}$
TeV. We note that it is also possible that some additional physical
effect, for instance a dependence on the obliquity angle between the
shock velocity and upstream magnetic field, affects both electron
injection and acceleration rate. In the presence of such an effect,
radiative losses might still be the operative limitation on the
electron spectrum. Another interpretation for the
correlation between the flux and the cut-off frequency is discussed
in the Appendix B.
\section{Conclusion}
We tracked time variations in synchrotron flux of discrete regions in
the SN~1006 NE limb from two {\it Chandra} observations in 2000 and
2008. Unlike core-collapse SNRs RX~J1713.7-3946 and Cas~A where
year-scale variations were found in small-scale knotty structures
(e.g., Uchiyama et al.\ 2007; Patnaude \& Fesen 2009), we found that
the X-ray emission from the SN~1006 NE limb is quite steady. We set
the upper limit of global time variations in the NE limb to be 4\% and
most likely 1\% over 8\,yrs. While simple considerations lead to a
prediction of a decline of 1 -- 2\% over this period, calibration
uncertainties are also of comparable size. We also revealed detailed
spatial structures of the synchrotron emission. We found a
correlation between the flux and the cut-off frequency, which suggests
that the maximum energy of accelerated electrons is not limited by
synchrotron losses. If this is the case, the maximum energy for
electrons, which we calculate to be $\sim$12\,TeV, would be the same
as that for ions. The correlation might also point to new physical
effects on electron injection or acceleration. In conclusion, we
found no indications of particle acceleration or synchrotron losses in
discrete features in the SN~1006 NE limb.
\acknowledgments
We acknowledge helpful scientific discussions with Una Hwang. We are
grateful to Paul Plucinsky and Alexey Vikhlinin for discussion of the
{\it Chandra} ACIS calibration. S.K.\ is supported by a JSPS Research
Fellowship for Research Abroad, and in part by the NASA grant under
the contract NNG06EO90A. P.F.W.\ acknowledges the support of the NSF
through grant AST 0908566.
\section{Appendix A}
We can estimate the expected rate of change of X-ray synchrotron flux
from SN 1006 with a very simple model, emission from a homogeneous
region just behind the shock whose synchrotron radiation is produced
by a power-law distribution of electrons with an exponential cutoff at
an energy $E_{\rm max}$. We shall assume that the shock puts a
constant fraction of post-shock energy density into relativistic
electrons and another constant fraction into magnetic-field energy.
As the shock decelerates, these energies decrease, resulting in a
decrease in the synchrotron emissivity at low energies but also a drop
in $E_{\rm max}$, giving a faster rate of decrease at photon energies
produced by electrons with $E > E_{\rm max}$. As the remnant radius
$R$ increases, however, the intensity along a line of sight $I_\nu$
grows as $R$. Of course the true situation is much more complex,
but these simple considerations allow for an estimate.
The synchrotron emissivity from an exponentially truncated power-law
distribution of electrons $N(E) = KE^{-s} e^{-E/E_{\rm max}}$ between
$E_l$ and $E_h > E_{\rm max}$ is given approximately by
\begin{equation}
j_\nu = c_j(\alpha) K B^{1 + \alpha} \nu^{-\alpha} \exp(-\sqrt{\nu/\nu_c})
\end{equation}
where $\alpha = (s - 1)/2)$ and $\nu_c \equiv c_1 E_{\rm max}^2 B$
($c_1 \equiv 1.82 \times 10^{18}$ cgs; $c_j(0.6) = 3.48 \times 10^{-12}$).
In general, $c_j \equiv c_5(\alpha) (2c_1)^\alpha$ in the notation of
Pacholczyk (1970), with $c_5(0.6) = 1.17 \times 10^{-23}$.
The intensity along a line of sight is $I_\nu = \int j_\nu\, dl
\cong j_\nu L \propto j_\nu R$. We take $\alpha = 0.6$, roughly
the radio value, although the results are not highly sensitive
to $\alpha$.
We consider a spherical evolving supernova remnant of radius $R
\propto t^m$ and shock speed $u_s \equiv dR/dt = mR/t \propto t^{m -
1}$, expanding into a uniform medium of density $\rho$. We assume
that the shock puts a constant fraction of post-shock thermal energy
($\propto \rho u_s^2$) into relativistic electrons:
\begin{equation}
u_e \equiv \int_{E_l}^{E_h} N(E) dE
\cong {K \over {s - 1}} \left( E_l^{1-s} - E_{\rm max}^{1 - s} \right)
\cong {K \over {s - 1}} E_l^{1 - s}
\end{equation}
where we have assumed $E_l \ll E_{\rm max}$. Then if $E_l$ = const.,
$K \propto u_e \propto u_s^2 \propto t^{2m - 2}$.
Next we assume that the magnetic energy density $B^2/8\pi$ is
amplified to a (probably different) constant fraction of $\rho u_s^2$:
$B^2 \propto u_s^2 \Rightarrow B \propto u_s \propto t^{m - 1}$.
For energies far below the cutoff energy $E_{\rm max}$ (i.e., at observing
frequencies $\nu \ll \nu_c$), we can find the time-dependence of the
intensity $I_\nu$ along any given line of sight:
\begin{equation}
I_\nu \propto j_\nu R \propto t^{2m - 2} t^{(m - 1)(1 + \alpha)} t^m
= t^{(m - 1)(3 + \alpha) + m}.
\end{equation}
For SN 1006, in the NE, $m = 0.54$ (Paper I).
Then for $\alpha = 0.6$,
\begin{equation}
I_\nu (\nu \ll \nu_c) \propto t^{(-0.46)(3.6) + m} = t^{-1.12} \equiv t^p.
\end{equation}
Then the prediction for the decay of synchrotron emission below
$\nu_c$, for instance in the radio, is
\begin{equation}
{1 \over I_\nu} {dI_\nu \over dt} = {p \over t} = -0.11\% \ {\rm yr}^{-1},
\end{equation}
or a total drop of 0.89\% in 8 years.
However, as we are considering the 1--8 keV continuum, and our fitted
values for $h\nu_c$ are typically below 1 keV ($\nu_c < 2.4 \times
10^{17}$ Hz), we need to consider the time-dependence of $\nu_c$,
i.e., of $E_{\rm max}$. If acceleration is limited by synchrotron
losses, $E_{\rm max} \propto B^{-1/2}u_s$, and $\nu_c \propto E_{\rm
max}^2 B \propto u_s^2 \propto t^{2m - 2}.$ In any case, let $\nu_c =
\nu_0 (t /t_0)^n$.
Then, writing $I_\nu = I_0 (t/t_0)^p e^{-\sqrt{\nu/\nu_c(t)}}$,
\begin{equation}
{dI_\nu \over dt}
= I_\nu \left( {p \over t} + {n \over 2t} \sqrt{\nu \over \nu_c}\right).
\end{equation}
For SN 1006, $\nu_c \sim 2 \times 10^{17}$ Hz in the synchrotron-bright NE,
so taking a mean photon energy of about 4 times that (3.3 keV), and
using $n = 2m - 2 = -0.92$ and $p = -1.12$ as above,
\begin{equation}
{1 \over I_\nu} {d I_\nu \over dt} = {1 \over t} (p + n) = {-2.04 \over 1000}
\Rightarrow -0.20\% \ {\rm yr}^{-1}
\end{equation}
or about --1.6\% over 8 years. A more careful integration over the
curved spectrum between 1 and 8 keV shouldn't change this estimate by
much.
If, alternatively, the acceleration is limited by the finite age of
SN~1006, we have
$E_{\rm max} \propto B u_s2 t \Rightarrow \nu_c \propto B3 u_s4 t2
\propto t^{7(m - 1) + 2} = t^{-1.22}$ and
\begin{equation}
{1 \over I_\nu} {d I_\nu \over dt} = {-2.34 \over 1000} \Rightarrow
-0.23\% \ {\rm yr}^{-1}
\end{equation}
or about --1.9\% over 8 years.
For completeness, a third alternative is the escape of particles above
some energy, perhaps due to absence of MHD waves to scatter them.
Then, if waves disappear above some wavelength $\lambda_m$,
$E_{\rm max} \propto \lambda_m B \Rightarrow \nu_c \propto \lambda_m^2
B^3 \Rightarrow n = 3(m - 1)$ (ignoring possible evolutionary changes
to $\lambda_m$). This gives $n = -1.38$ and
\begin{equation}
{1 \over I_\nu} {d I_\nu \over dt} = {-2.5 \over 1000} \Rightarrow
0.25\% \ {\rm yr}^{-1}
\end{equation}
or about --2.0\% over 8 years.
\section{Appendix B}
Now the acceleration rate is proportional to the
diffusion coefficient $\kappa = \lambda_{\rm mfp}c/3$, where the mean
free path $\lambda_{\rm mfp}$ is normally taken to be proportional to
the gyroradius, $\lambda_{\rm mfp} = \eta r_g = \eta E/eB$ (the last
equality applying in the extreme-relativistic limit). In the
``quasi-linear'' approximation, $\eta = (\delta B/B)^{-2}$, where
$\delta B$ is the magnitude of resonant MHD fluctuations. (The ``Bohm
limit'' is $\eta = 1$ or $\lambda_{\rm mfp} = r_g$; constant $\eta$ at
some other value $> 1$ is termed ``Bohm-like.'' Constant $\eta$
corresponds to a ``white noise'' spectrum of MHD waves, equal energy in
all decades of wavenumber: if $I(k) dk$ is the energy in waves with
wavenumbers in $dk$, $I \propto k^{-1}$.) While most workers assume
Bohm-like or Bohm-limit diffusion (e.g., Berezhko, Ksenofontov, \&
V\"olk 2009), one could imagine a departure from this assumption; to
produce the correlation we observe, it would be necessary to have
$\eta$ decrease with $B$ (i.e., one needs more rapid acceleration
where the field is stronger). To our knowledge, there is at present
no theoretical prediction of such an effect, but it might exist.
However, it can be shown (Reynolds 2004) that the most straightforward
generalization, in which $\eta$ depends on $E$ because the turbulent
spectrum of MHD waves has a different slope, $I(k) \propto k^{-n}$
with $n \ne 1$, produces the wrong correlation.
In this case, the acceleration time to energy $E$, $\tau(E)$, obeys
$\tau(E) \propto E^\beta/B$ where $\beta = 2 - n$. Then $\beta = 1$
is the Bohm limit. The value of $\beta$ depends on the nature of the
scattering medium; for scattering by MHD waves with a Kolmogorov
spectrum, $ n = 5/3$ so $\beta = 1/3$, and normal turbulent spectra
are expected to be steep, $n > 1 \Rightarrow \beta < 1$.
Equating $\tau(E)$ to the synchrotron loss time gives the maximum
energy of loss-limited acceleration $E_{\rm max}({\rm loss}) \propto
B^{-[1/(1 + \beta)]}$, and corresponding cut-off frequency $\nu_c
\propto B^{(\beta - 1)/(\beta + 1)}$. (So for Bohm-like diffusion,
$\eta$ = constant or $\beta = 1$, there is no $B$-dependence. Models
in which turbulence is generated by the cosmic rays themselves
typically produce Bohm-like diffusion.) If $\beta < 1$, higher $B$
lowers the cut-off frequency, producing the opposite correlation to
the one we observe. However, it is still possible that some as yet
unknown effect produces turbulence with highest power at short
wavelengths, $n < 1 \Rightarrow \beta > 1$. In this unlikely case,
brighter regions would be expected to have higher cut-off frequencies
as observed, due to variations in $B$. That, or some other alteration
to the standard diffusion picture, could allow acceleration to be
loss-limited. It must be noted, however, that increasing $\eta$ above
1 in some parts of the shock lowers the maximum energy to which
particles of any species can be accelerated there, and may impact the
ability of shocks to produce the highest-energy Galactic cosmic ray
ions.
\begin{deluxetable}{lcccc}
\tabletypesize{\scriptsize}
\tablecaption{Spectral-fit parameters for example spectra in regions A
and B}
\tablewidth{0pt}
\tablehead{
\colhead{Parameters}&\colhead{Region A}
&\colhead{Region B}
}
\startdata
$N_\mathrm{H}$ (10$^{20}$\,cm$^{-2}$)&6.8 (fixed)&6.8 (fixed) \\
\hline
\multicolumn{3}{c}{{\tt srcut} component}\\
Constant factor (2008/2000)& 1.00$\pm$0.05 & 0.91$\pm$0.08 \\
Mean spectral index & 0.502$^{+0.004}_{-0.005}$ &
0.503$^{+0.007}_{-0.006}$ \\
Cut-off frequency (10$^{16}$\,Hz) & 19$^{+2}_{-4}$ & 1.6$\pm$0.1 \\
Flux at 1\,GHz (Jy\,arcmin$^{-2}$) & 0.236 (fixed)& 0.052 (fixed)\\
\hline
\multicolumn{3}{c}{{\tt vpshock} component}\\
$kT_\mathrm{e}$ (keV) & 0.5 (fixed) & 0.5 (fixed) \\
log($n_\mathrm{e}t/\mathrm{cm}^{-3}$\,sec)&10 (fixed)&10 (fixed) \\
O & 4.4 (fixed)& 4.4 (fixed)\\
Ne & 1.5 (fixed)& 1.5 (fixed) \\
Mg & 15 (fixed) & 15 (fixed) \\
Si & 50 (fixed) & 50 (fixed) \\
$\int n_\mathrm{e}n_\mathrm{H}dl$$^\mathrm{a}$ (10$^{16}$\,cm$^{-5}$)
& 0.5$\pm$0.3 & 12$\pm$1 \\
\hline
$\chi^2$/d.o.f. & 132/169 & 141/104 \\
\enddata
\tablecomments{Errors indicate the 90\% confidence ranges. Abundances
not listed are fixed to the solar values. $^\mathrm{a}$VEM
normalized by the region area; $dl$ is the plasma depth.
}
\label{tab:param}
\end{deluxetable}
\begin{deluxetable}{lccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Flux ratios (2008/2000)}
\tablewidth{0pt}
\tablehead{
\colhead{}&\colhead{0.4--0.8\,keV}
&\colhead{0.8--1.0\,keV} & \colhead{1.0--8.0\,keV}
}
\startdata
Nonthermal & 1.028$\pm$0.012 & 0.962$\pm$0.011 & 0.964$\pm$0.006\\
\hline
Thermal (no shift) & 1.021$\pm$0.010 & 0.968$\pm$0.022 & 0.947$\pm$0.016\\
Thermal (1$^{\prime\prime}$ shift) & 1.026$\pm$0.010 & 0.975$\pm$0.021 & 0.956$\pm$0.014\\
Thermal (2$^{\prime\prime}$ shift) & 1.031$\pm$0.010 & 0.991$\pm$0.016
& 0.966$\pm$0.013\\
Thermal (3$^{\prime\prime}$ shift) & 1.035$\pm$0.011 & 0.988$\pm$0.022
& 0.972$\pm$0.012\\
Thermal (mean) & 1.028$\pm$0.010 & 0.982$\pm$0.020
& 0.961$\pm$0.016\\
\hline
Nonthermal / Thermal & 0.997$\pm$0.015 & 0.980$\pm$0.023
& 1.003$\pm$0.018\\
\enddata
\tablecomments{Errors indicate the 90\% confidence ranges.
}
\label{tab:flux_ratio}
\end{deluxetable}
\begin{figure}
\plottwo{f1a.eps}{f1b.eps}
\caption{(a) {\it Chandra} three-color image after vignetting
effects are corrected. Red, green, and blue correspond to
0.5--0.8\,keV (mostly, K lines of O), 0.8--2.0\,keV (mostly, K lines
of Ne, Mg, and Si), and 2.0--5.0\,keV (mostly, synchrotron emission)
bands, respectively. The image is
binned by 1$^{\prime\prime}$.97 and has been smoothed by a Gaussian
kernel of $\sigma = 5^{\prime\prime}.90$. The intensity scale is
square root. The field of view of the {\it Chandra} observations
of the NE limb (ObsIDs 732 and 9107) are shown as a white box.
(b) Same as Fig.~\ref{fig:image} (a), but focused on the NE limb.
The field of view the {\it Chandra} observations are within two
white lines. We extract spectra from white (and red) pie-shaped
regions. Example spectra for red regions indicated as letters A--C
are shown in Figs.~\ref{fig:spec1} and \ref{fig:spec2}. The SNR
center of [(ra, dec) =
$15^\mathrm{h}$02$^\mathrm{m}$54$^\mathrm{s}$.9,
$-41^{\circ}56^{\prime}08^{\prime\prime}.9$ (J2000)] is taken from
Paper I.
}
\label{fig:image}
\end{figure}
\begin{figure}
\plottwo{f2a.eps}{f2b.eps}
\caption{Left: Example spectra extracted from region A indicated in
Fig.~\ref{fig:image} along with the best-fit models and the
residuals. Black and red correspond to the 2000 and 2008 data,
respectively. Components of nonthermal and thermal emission are
separately illustrated. Right: Same as left but for region B
indicated in Fig.~\ref{fig:image}.
}
\label{fig:spec1}
\end{figure}
\begin{figure}
\includegraphics[angle=0,scale=0.65]{f3.eps}\hspace{1cm}
\caption{Results from the spatially-resolved spectral analysis for the
nonthermally-dominated regions. Panels (a)--(f) show distributions of
reduced $\chi^2$s, mean spectral indices inferred from the X-ray spectra,
cut-off frequencies, fluxes corrected for the interstellar absorption
in 0.4--8.0\,keV for 2000 and 2008, and the flux ratios,
respectively. Values of the cut-off frequencies and fluxes are in
units of 1$\times$10$^{16}$\,Hz and
1$\times$10$^{-13}$\,ergs\,cm$^{-2}$\,sec$^{-1}$\,arcmin$^{-2}$,
respectively.
}
\label{fig:spatial}
\end{figure}
\begin{figure}
\includegraphics[angle=0,scale=0.65]{f4.eps}\hspace{1cm}
\caption{Left: Flux ratio in the srcut component as a function of the
flux in 2000. Errors of the vertical axis are 90\% confidence
ranges of the constant parameters for the {\tt srcut} component,
while those of the horizontal axis are square root of photon
numbers in the srcut component. A horizontal solid line drawn in the
figure represent a constant flux line.
Right: Histogram of flux ratios (2008/2000). The best-fit Gaussian
function is shown together.
}
\label{fig:hist1}
\end{figure}
\begin{figure}
\plottwo{f5a.eps}{f5b.eps}
\caption{Left: Same as Fig.~\ref{fig:spec1} left, but the spectra are
fitted separately in energy bands of 0.4--0.8\,keV, 0.8--1.0\,keV,
and 1.0--8.0\,keV.
Right: Same as left but for a thermally-dominated region,
region C indicated in Fig.~\ref{fig:image}.
}
\label{fig:spec2}
\end{figure}
\begin{figure}
\includegraphics[angle=0,scale=0.5]{f6.eps}\hspace{1cm}
\caption{Histograms of flux ratios (2008/2000) for energy bands of
0.4--0.8\,keV (left column), 0.8--1.0\,keV (middle column), and
1.0--8.0\,keV (right column). The first row is responsible for
nonthermally-dominated regions, while others are responsible for
thermally-dominated regions.
}
\label{fig:hist_all}
\end{figure}
\clearpage
|
1,116,691,498,441 | arxiv | \section{Introduction}
\label{sec:intro}
Among the numerous open questions in contemporary high-energy physics, the origin of cosmic dark matter (DM) and that of the baryon asymmetry in the Universe occupy a pivotal place. Not only do they constitute two of the most striking and fundamental pieces of evidence for the existence of physics beyond the Standard Model (BSM) of particle physics, but they do so only once the latter is placed within a cosmological framework. Resolving either (let alone both) of these questions most likely requires particle physics to be viewed from a cosmological standpoint and, conversely, cosmology to be analyzed in terms of the behavior of the fundamental constituents of matter and their interactions. And, indeed, in doing so during the past decades model builders have not been short of ideas concerning the nature of dark matter and the mechanism through which matter came to dominate over antimatter in the Universe.
Among the various dark matter candidates that have been proposed we can mention axions \cite{Preskill:1982cy, Abbott:1982af, Dine:1982ah}, primordial black holes \cite{Hawking:1971ei, Chapline:1975ojl, Green:2020jor}, a vast number of incarnations of weakly or feebly interacting massive particles (WIMPs or FIMPs; for reviews \textit{cf e.g.} \cite{Arcadi:2017kky, Bernal:2017kxu}), gravitationally produced dark matter \cite{Kolb:1998ki}, and asymmetric dark matter (for reviews \textit{cf e.g.} \cite{Davoudiasl_2012, Petraki:2013wwa}), just to name a few. Similarly, the matter-antimatter asymmetry of the Universe has been explained in terms of different mechanisms of baryogenesis, like baryogenesis based on grand unified theories (GUTs) \cite{Yoshimura:1978ex}, electroweak baryogenesis \cite{Kuzmin:1985mm,Cohen:1993nk}, the Affleck-Dine mechanism \cite{Affleck:1984fy}), and leptogenesis \cite{Fukugita:1986hr}. Interestingly, during the past decade, there have also been several attempts to actually link the two questions, most notably in the contexts of WIMP baryogenesis \cite{McDonald_2011,Cui_2012,Cui_2013} or asymmetric dark matter \cite{Hall:2010jx,Unwin:2014poa,Cui:2020dly}.
Recently, the main idea behind Akhmedov-Rubakov-Smirnov (ARS) leptogenesis \cite{Akhmedov:1998qx}, namely, that of a lepton number asymmetry being generated through $CP$-violating sterile neutrino oscillations, was exploited in \cite{Shuve_2020}, where the authors proposed that a baryon asymmetry could, instead, be also generated by augmenting the SM with exotic scalars and fermions directly coupling to quarks. The fermions, which were taken to be singlets under the SM gauge group can, moreover, play the role of viable dark matter candidates through the freeze-in mechanism \cite{McDonald:2001vt, Hall:2009bx}, whereas their $CP$-violating oscillations, in the presence of electroweak sphaleron transitions, can generate the observed baryon asymmetry of the Universe.
In this paper, we propose a mechanism which borrows ideas both from Dirac leptogenesis \cite{Dick:1999je,Murayama_2002,Cerdeno_2006,Gonzalez:2009} and from this scenario of ``freeze-in baryogenesis''. As we will describe in detail in the following, we consider a heavy particle species which is charged under (parts of) the SM gauge group and which can decay through feeble interactions into SM fermions along with a neutral particle. The latter is our dark matter candidate, produced upon the decays of the heavy particle through the freeze-in mechanism, along the lines presented in \cite{Shuve_2020}. However, in our case the decays \textit{themselves} violate $CP$, in a similar manner as in leptogenesis models. Then, in the presence of electroweak sphalerons, we will see that an asymmetry can be generated between SM fermions and antifermions. The relevant processes proceed through feeble couplings, preventing them from ever reaching equilibrium and, thus, satisfying the third Sakharov condition \cite{Sakharov:1967dj}. The first condition, namely, baryon number violation, is satisfied due to the active sphaleron processes in a way that resembles neither GUT baryogenesis (explicit $B$ violation in decays) nor leptogenesis (explicit $L$ violation in decays), although we will also comment on the possibility of direct $B/L$ violation as well.
The paper is structured as follows: in Section \ref{sec:GeneralFramework} we discuss the general features of our mechanism, namely, the way through which the required dark matter abundance and the baryon/lepton asymmetries can be generated, without adopting any concrete microscopic model. In Section \ref{sec:Model} we propose a simple model as a proof-of-concept that concrete incarnations of our mechanism can, indeed, be constructed. We compute the predicted dark matter abundance and baryon asymmetry, quantify the effects of processes that wash out the latter and briefly comment on the phenomenological perspectives of our model, notably in relation to searches for long-lived particles (LLPs) at the Large Hadron Collider (LHC). Finally, in Section \ref{sec:conclusions} we summarize our main findings and conclude. Some more technical aspects are left for the Appendix.
\section{Dark matter and baryogenesis from freeze-in: General framework}\label{sec:GeneralFramework}
Before presenting a concrete realization of our take on the freeze-in baryogenesis idea, let us briefly summarize a few key notions that will be useful for the discussion that follows: frozen-in dark matter and how the freeze-in framework can enable us to satisfy the three Sakharov conditions that are necessary for successful baryogenesis. A concrete implementation of these ideas will be presented in detail in Section \ref{sec:Model}.
\subsection{Freeze-in DM abundance}\label{Freeze-in DM abundance}
The freeze-in mechanism for dark matter production relies on two basic premises:
\begin{itemize}
\item The initial DM abundance is zero.
\item Dark matter interacts only extremely weakly (``feebly'') with the Standard Model particles (along with any other particles that are in thermal equilibrium with them).
\end{itemize}
Under these assumptions, and further assuming that dark matter production takes place in a radiation-dominated Universe, dark matter never reaches thermal equilibrium with the plasma. Instead, it is produced through the out-of-equilibrium decays or annihilations of bath particles and all dark matter depletion processes, the rate of which typically scales as $\left\langle \sigma v \right\rangle \times n_{DM}^2$ (where $\sigma$ is the dark matter annihilation cross section, $v$ its velocity, and $n_{DM}$ its number density), can be ignored.
\subsection{Baryon asymmetry abundance $Y_{B}$}\label{sec:Basymmetrygeneral}
In general, the decays and/or annihilations that are responsible for dark matter production can also violate both the baryon number $B$ and $C/CP$.\footnote{Very similar remarks can be made if the decay violates, instead, the lepton number. This is also the option that we will adopt later in this work.} Then, as long as we stick to the freeze-in framework, these processes occur out-of-equilibrium with the thermal plasma, thus fulfilling all three Sakharov conditions.
Intuitively, and ignoring all washout processes, if we denote the measure of $CP$ violation by $\epsilon_{CP}$, we would expect the generated asymmetry in the SM fermion $Y_{\Delta f}$ to be connected to the dark matter abundance $Y_{DM}$ through a relation of the type
\begin{equation}\label{eq:BasymmetryDM}
Y_{\Delta f}\,\sim\,\epsilon_{CP}\,Y_{DM}
\end{equation}
In reality, this limit cannot be attained given that some amount of washout is almost inevitable, whereas concrete realizations typically require the introduction of additional particles and decay channels. In this respect this relation may be viewed as an upper limit to the asymmetry that can be generated through decays that simultaneously produce dark matter.
In fact, in the following, when studying a concrete incarnation of our decay-induced freeze-in baryogenesis idea, we will see the following.
\begin{itemize}
\item Since we will be starting with non-self-conjugate initial states $F_i$ (Dirac fermions), $CPT$ conservation and unitarity impose the existence of multiple decay channels of the $F_i$'s for a non-vanishing $CP$ asymmetry to be generated. To this goal, we will exploit possible decays of the heavy fermions into different Standard Model fermions (leptons), \textit{i.e.} flavor effects.
\item The freeze-in framework will necessitate extremely small values for the couplings involved in the decay process. The predicted $CP$ violation, being an effect that arises from the interference of tree-level and one-loop processes is, then, even further suppressed, which will lead us to consider resonantly enhanced configurations in self-energy-type diagrams \cite{Pilaftsis_2004, Pilaftsis:2005rv, Pilaftsis_1999, Anisimov_2006}. Therefore, at least two heavy fermions $F_i$ must be added.
\item The baryon and/or lepton number need not be violated by the decay processes. As proposed, \textit{e.g.} in \cite{Dick:1999je}, a $CP$ asymmetry can be translated to a baryon asymmetry by the electroweak sphalerons. If, additionally, the BSM heavy fermions are immune to the action of sphalerons, in the end a net asymmetry will be generated in the baryon and lepton sectors.
\end{itemize}
\section{A concrete realization}\label{sec:Model}
Let us now elaborate the previous considerations through a concrete, simple model. We extend the SM by two heavy vectorlike leptons $F_i$ which are singlets under $SU(3)_{\text{c}}\times SU(2)_{\LH}$ but carry hypercharge and a real gauge-singlet scalar $S$ which is our freeze-in DM candidate. We moreover impose a discrete $Z_2$ symmetry on the Lagrangian, under which all exotic states are taken to be odd while the SM particles are even. Under these assumptions, the Lagrangian reads
\begin{equation}\label{eq:Lgeneral}
\mathcal{L}=\mathcal{L}_{\text{SM}}+\mathcal{L}_{S}+\mathcal{L}_{SF}
\end{equation}
\noindent
where $\mathcal{L}_{\text{SM}}$ is the SM Lagrangian,
\begin{equation}\label{eq:LS}
\mathcal{L}_{S} = \partial_\mu S ~ \partial^\mu S - \frac{\mu_S^2}{2} S^2 + \frac{\lambda_S}{4!} S^4 + \lambda_{Sh} S^2 \left(H^\dagger H\right)
\end{equation}
\noindent
describes interactions of dark matter with itself and with the Standard Model Higgs doublet and
\begin{align}
\mathcal{L}_{SF}\, =\, \sum_{i} \left( \bar{F}_i \left(i\slashed{D}\right) F_i - M_i \bar{F_i} F_i \right)
& -\, \sum_{\alpha, i} \left( \lambda_{\alpha i}\,S\,\bar{F}_i\,\PRH\,e_{\alpha}\,+\,\lambda_{\alpha i}^*\,S\,\bar{e}_{\alpha}\,\PLH\,F_i \right)
\end{align}
\noindent
where $e_{\alpha}$ are the right-handed SM charged leptons of flavor $\alpha=\{e,\mu,\tau\}$ and $\lambda_{\alpha i}$ denote the feeble couplings. The heavy fermions $F_i$ are assumed to carry the same lepton number with the SM leptons, so their interactions are lepton number conserving. Note that, without loss of generality, we have neglected potential off-diagonal couplings among the heavy fermions $F_i$. For simplicity in what follows, we will also set the Higgs portal coupling, $\lambda_{Sh}$, to zero.
\\
\\
The tree-level proper decay width $\Gamma_{F_i\rightarrow e_{\alpha}\,S}\big|_0$ in the limit $M_i\gg m_{e_{\alpha}}+m_{S}$ is given by
\begin{equation}\label{eq:decay width}
\Gamma_{F_i\rightarrow e_{\alpha}\,S}\big|_0\,\simeq\,\frac{\left|\lambda_{\alpha i}\right|^2}{16\pi\,g_{F_i}}\,M_i
\end{equation}
\noindent
where $g_{F_i}=2$ are the internal degrees of freedom of species $F_i$. The equilibrium decay rate density $\gamma_{F_i\rightarrow e_{\alpha}\,S}$ evaluated in the decaying particle rest frame is \cite{Hall:2009bx}
\begin{align}\label{eq:decay rate density}
\gamma_{F_i\rightarrow e_{\alpha}\,S}\,&\equiv\,\int\text{d}\Pi_{F_i}\,\text{d}\Pi_{e_{\alpha}}\,\text{d}\Pi_{S}\,\left(2\pi\right)^4\,\delta^{(4)}\left(p_{F_i}-p_{e_{\alpha}}-p_{S}\right)\,f^{\text{eq}}_{F_i}\,\left|\mathcal{M}\right|^2_{F_i\rightarrow e_{\alpha}\,S}\nonumber
\\
&=\,\frac{g_{F_i}}{2\pi^2}\,M_i^2\,\Gamma_{F_i\rightarrow e_{\alpha}\,S}\,T\,K_1\left(\frac{M_i}{T}\right)
\end{align}
\begin{figure}[t]
\centering
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Decay_Tree_Level.pdf}
\caption{}
\label{fig:tree-level}
\end{subfigure}\hspace*{\fill}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Decay_1Loop.pdf}
\caption{}
\label{fig:loop}
\end{subfigure}\hspace*{\fill}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Gauge_Scattering_1_schannel.pdf}
\caption{}
\label{fig:gauge scattering 1-s}
\end{subfigure}\hspace*{\fill}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Gauge_Scattering_1_uchannel.pdf}
\caption{}
\label{fig:gauge scattering 1-u}
\end{subfigure}
\medskip
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Gauge_Scattering_2_schannel.pdf}
\caption{}
\label{fig:gauge scattering 2-s}
\end{subfigure}\hspace*{\fill}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Gauge_Scattering_2_uchannel.pdf}
\caption{}
\label{fig:gauge scattering 2-u}
\end{subfigure}\hspace*{\fill}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Gauge_Scattering_3_tchannel.pdf}
\caption{}
\label{fig:gauge scattering 3-t}
\end{subfigure}\hspace*{\fill}
\begin{subfigure}{0.24\textwidth}
\includegraphics[width=\linewidth]{Figures/Gauge_Scattering_3_uchannel.pdf}
\caption{}
\label{fig:gauge scattering 3-u}
\end{subfigure}
\caption{Feynman diagrams that contribute to the generation of the baryon asymmetry.}
\label{fig:Feynman diagrams}
\end{figure}
\noindent
where $\text{d}\Pi_k=\text{d}^3\textbf{p}_k/\left(2\pi\right)^32E_k$ is the elementary Lorentz invariant phase space volume of species $k$, $\left|\mathcal{M}\right|^2$ denotes the squared matrix element summed, but not averaged, over the internal degrees of freedom of the initial states, $K_1$ is the modified Bessel function of the second kind of order one and the distribution function of $F_i$ approximately follows the Maxwell-Boltzmann distribution, $f_{F_i}^{\text{eq}}= e^{-E_{F_i}/T}$. The corresponding thermally averaged equilibrium decay rate density $\Braket{\gamma_{F_i\rightarrow e_{\alpha}\,S}}$ reads
\begin{equation}
\Braket{\gamma_{F_i\rightarrow e_{\alpha}\,S}}\,\equiv\,\frac{\gamma_{F_i\rightarrow e_{\alpha}\,S}}{n_{F_i}^{\text{eq}}}\,=\,\frac{K_1\left(M_i/T\right)}{K_2\left(M_i/T\right)}\,\Gamma_{F_i\rightarrow e_{\alpha}\,S}\,\simeq\,\left\{\,
\begin{array}{@{}ll@{}}
\frac{M_i}{2T}\,\Gamma_{F_i\rightarrow e_{\alpha}\,S},\quad &T\,\gg\,M_i
\\\\
\Gamma_{F_i\rightarrow e_{\alpha}\,S},\quad &T\,\ll\,M_i
\end{array}
\right.
\end{equation}
\noindent
where $n_{F_i}^{\text{eq}}$ is the equilibrium number density of $F_i$ assuming zero chemical potential
\begin{equation}
n_{F_i}^{\text{eq}}\,\equiv\,g_{F_i}\int\frac{\text{d}^3\textbf{p}}{\left(2\pi\right)^3}\,f_{F_i}^{\text{eq}}\,=\,\frac{g_{F_i}}{2\pi^2}\,M_i^2\,T\,K_2\left(\frac{M_i}{T}\right)
\end{equation}
\\
\noindent
The time dilation factor $K_1\left(M_i/T\right)/K_2\left(M_i/T\right)$ implies that decays are inhibited at temperatures higher than the decaying state mass $M_i$.
The dominant $2\leftrightarrow 2$ scattering processes modifying the abundance of $S$ are those which involve the $U(1)_Y$ gauge boson as an external state, \textit{i.e.} $F_iB\leftrightarrow e_{\alpha}S$, $F_iS\leftrightarrow e_{\alpha}B$ and $F_i\bar{e}_{\alpha}\leftrightarrow SB$, which will be henceforth referred to as gauge scatterings. The corresponding matrix elements depend on the product of a feeble and a gauge coupling, whereas all other scattering processes involve higher powers of feeble couplings and are therefore subleading. All relevant Feynman diagrams are shown in Fig.~\ref{fig:Feynman diagrams}.
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{Figures/Plot_RateDensities.pdf}
\caption{Tree-level decay and gauge scattering rate densities involving $F_1$ and $e_e$ as external states, normalized to the Hubble parameter $H$ and the equilibrium number density of photons $n_{\gamma}$. The feeble coupling of the interactions is $\lambda_{e1}\simeq 2.15\times 10^{-8}$ and the densities have been evaluated at tree-level and in the limit $M_1\gg m_{e_{\alpha}}+m_S$. }
\label{fig:Rate Densities}
\end{figure}
The equilibrium interaction rate density of a generic scattering process $ab\rightarrow cd$ is \cite{Gondolo:1990dk}
\begin{align}
\gamma_{ab\rightarrow cd}\,&\equiv\,\int\text{d}\Pi_a\,\text{d}\Pi_b\,\text{d}\Pi_c\,\text{d}\Pi_d\,\left(2\pi\right)^4\,\delta^{(4)}\left(p_{a}+p_{b}-p_{c}-p_{d}\right)\,f^{\text{eq}}_{a}\,f^{\text{eq}}_{b}\,\left|\mathcal{M}\right|^2_{ab\rightarrow cd}\nonumber
\\
&=\,\frac{T}{512\pi^6}\,\int_{\tilde{s}_{\text{min}}}^{\infty}\,\text{d}\tilde{s}\,\frac{1}{\sqrt{\tilde{s}}}\,\left|\textbf{p}_{\text{in}}\right|\,\left|\textbf{p}_{\text{fin}}\right|\,K_1\left(\frac{\sqrt{\tilde{s}}}{T}\right)\int\,\text{d}\Omega\,\left|\mathcal{M}\right|^2_{ab\rightarrow cd}
\end{align}
\\
\noindent
where $\textbf{p}_{\text{in}}$, $\textbf{p}_{\text{fin}}$ and $\sqrt{\tilde{s}}$ are the initial and final momenta and the energy in the center-of-momentum frame respectively, with $\tilde{s}_{\text{min}}=\max\big\{\left(m_a+m_b\right)^2\,,\,\left(m_c+m_d\right)^2\big\}$. In Fig.~\ref{fig:Rate Densities}, we present the decay and gauge scattering rate densities involving $F_1$ and the right-handed electron $e_e$ as external states in terms of the dimensionless parameter $z\equiv M_1/T$. For the feeble coupling, we use the value $\lambda_{e1}= 2.145\times 10^{-8}$ and work at lowest order in perturbation theory and in the limit $M_1\gg m_{e_{\alpha}}+m_S$. In order to study their effect on the Boltzmann equations, it is convenient to normalize them to the Hubble parameter $H$ and to the number density of photons $n_{\gamma}=2\zeta(3)T^3/\pi^2$, where $\zeta$ is the Riemann zeta function \cite{Kolb:1990vq}. The integrals appearing in the various scattering rate densities have been computed with the Cuba numerical library \cite{Cuba_2005}.
The s-channel resonance that appears in the scattering $F_iB\leftrightarrow e_{\alpha}S$ has been regularized by the finite decay width of $F_i$. Also, the IR-induced resonances due to the exchange of massless SM leptons that appear in the u-channel of $F_iB\leftrightarrow e_{\alpha}S$ and in the t-channel of $F_i\bar{e}_{\alpha}\leftrightarrow SB$ have both been regulated by the thermal mass of the right-handed SM leptons $m_e^2\left(T\right)=g_Y^2T^2/8$ (see \cite{Cline_1994} and references therein). Note that, as thermal effects are irrelevant at the temperature of a few $\TeV$ that is of interest to us, we include the thermal mass only when it acts as a regulator of the IR-divergences. For a few examples in which finite temperature effects can become important \textit{cf e.g.} \cite{Baker:2017zwx,Dvorkin:2019zdi,Darme:2019wpd}.
\subsection{Out-of-equilibrium decays}\label{subsec:Out-of-Equilibrium Decays}
\noindent
As we already mentioned, one of the crucial ingredients of our setup is that all processes involving dark matter (and, for that matter, $CP$ violation) must never attain chemical equilibrium. In order to fulfill this condition, an upper bound on the magnitude of the freeze-in couplings can be obtained by requiring the total thermally averaged decay rate to be smaller than the Hubble expansion rate at the characteristic temperature $T=M_i$. Because of the feeble nature of the $\lambda_{\alpha i}$ couplings, successful baryogenesis requires a resonant enhancement of the asymmetry, which, in turn, implies that the masses of the two heavy fermions have to be very close to each other. Then, for $M_1\simeq M_2$ the out-of-equilibrium condition reads
\begin{equation}\label{eq:out-of-equilibrium condition}
\Big(\sum_{\alpha,i}\Braket{\gamma_{F_i\rightarrow e_{\alpha}\,S}}\,\lesssim\,H\,\Big)\big|_{T\,=\,M_1}
\end{equation}
\noindent
The Hubble parameter is given by
\begin{equation}\label{eq:Hubble parameter}
H\left(T\right)\,\simeq\,\frac{1.66\,\sqrt{g_{*\rho}}}{M_{Pl}}\,T^2
\end{equation}
\noindent
where $M_{Pl}\simeq 1.22\times 10^{19}\,\GeV$ is the Planck mass and $g_{*\rho}\simeq 106.75$ are the effective degrees of freedom related to the energy density. At leading order and in the limit $M_i\gg m_{e_{\alpha}}+m_{S}$, the out-of-equilibrium condition \eqref{eq:out-of-equilibrium condition} reads ($g_{F_i}=2$)
\begin{equation}
\sum_{\alpha,i}\Gamma_{F_i\rightarrow e_{\alpha}S}\,\lesssim\,2H\,\big|_{T\,=\,M_1}\;\Rightarrow\;\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^2\,\lesssim\,2.83\,\times\,10^{-13}\,\left(\frac{M_1}{\TeV}\right)
\end{equation}
\noindent
Thus, for heavy leptons at the $\TeV$ range, the couplings have to be smaller than $\sim 10^{-7}$ for the decays to proceed out-of-equilibrium.
\subsection{Freeze-in DM abundance}\label{subsec:Freeze-in DM abundance - Decays}
\noindent
The freeze-in DM abundance $Y_S$ that is produced from decays and $2\rightarrow 2$ scattering processes in our model follows the Boltzmann equation
\begin{align}\label{eq:Boltzmann DM full}
s\,\frac{\text{d}Y_S}{\text{d}t}\,&=\,\sum_{\alpha,i}\left\{F_i\,\leftrightarrow e_{\alpha}S\right\}\,+\,\sum_{\alpha,i}\left\{F_iB\leftrightarrow e_{\alpha}S\right\}\,-\,\sum_{\alpha,i}\left\{F_iS\leftrightarrow e_{\alpha}B\right\}\,+\,\sum_{\alpha,i}\left\{F_i\bar{e}_{\alpha}\leftrightarrow SB\right\}\nonumber
\\
&+\,2\sum_{i,j}\left\{F_i\bar{F}_j\leftrightarrow SS\right\}\,+\,2\sum_{\alpha,\beta}\left\{\bar{e}_{\alpha}e_{\beta}\leftrightarrow SS\right\} \ .
\end{align}
\noindent
In writing \eqref{eq:Boltzmann DM full}, we have used the notations
\begin{subequations}
\begin{alignat}{3}
\left\{a\,b\leftrightarrow c\,d\right\}\,&\equiv\,\left(a\,b\leftrightarrow c\,d\right)\,+\,\left(\bar{a}\,\bar{b}\leftrightarrow \bar{c}\,\bar{d}\right)
\\
\left[a\,b\leftrightarrow c\,d\right]\,&\equiv\,\left(a\,b\leftrightarrow c\,d\right)\,-\,\left(\bar{a}\,\bar{b}\leftrightarrow \bar{c}\,\bar{d}\right)
\\
\left(a\,b\leftrightarrow c\,d\right)\,&\equiv\,\int\text{d}\Pi_a\text{d}\Pi_b\text{d}\Pi_c\text{d}\Pi_d\left(2\pi\right)^4\delta^{(4)}\Big[\left|\mathcal{M}\right|^2_{ab\rightarrow cd}f_af_b\left(1\pm f_c\right)\left(1\pm f_d\right)\nonumber
\\
&\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\,\left|\mathcal{M}\right|^2_{cd\rightarrow ab}f_cf_d\left(1\pm f_a\right)\left(1\pm f_b\right)\Big]
\end{alignat}
\end{subequations}
\noindent
where $\delta^{\left(4\right)}$ is an abbreviation for $\delta^{\left(4\right)}\left(p_a+p_b-p_c-p_d\right)$, $f_k$ is the distribution function of species $k$ and $s$ denotes the entropy density. We will make the following assumptions:
\begin{itemize}
\item The initial DM abundance is zero. Combined with the feeble couplings, this allows us to ignore the inverse decays $e_{\alpha}S\rightarrow F_i$, \textit{i.e.} $f_S\simeq 0$.
\item The DM production takes place during the radiation dominated era. At this epoch, time and temperature are related by $\dot{T}\simeq -HT$, which is valid for $\partial g_{*\rho}/\partial T\simeq 0$. Using this relation, we may switch variables and write
\begin{equation}
s\frac{\text{d}}{\text{d}t}Y_{S}\,=\,sHz\,\frac{\text{d}}{\text{d}z}Y_{S}
\end{equation}
\noindent
The entropy density during the radiation dominated era is given by
\begin{equation}\label{eq:entropy density}
s\,=\,\frac{2\pi^2}{45}\,g_{*s}\,T^3
\end{equation}
\noindent
where $g_{*s}$ are the effective degrees of freedom with respect to the entropy density.
\item The distribution functions of the visible sector species obey the Maxwell-Boltzmann statistics, \textit{i.e.} we neglect Bose-enhancement and Pauli-blocking factors. Hence, we can write, in general,
\begin{equation}
\left(a\,b\leftrightarrow c\,d\right)\,=\,\gamma_{ab\rightarrow cd}\,\frac{Y_a}{Y_a^{\text{eq}}}\,\frac{Y_b}{Y_b^{\text{eq}}}\,-\,\gamma_{cd\rightarrow ab}\,\frac{Y_c}{Y_c^{\text{eq}}}\,\frac{Y_d}{Y_d^{\text{eq}}}
\end{equation}
\end{itemize}
Let us first focus on the heavy lepton decays $F_i\rightarrow e_{\alpha}S$ and the corresponding $CP$-conjugate processes, which provide the dominant contribution to DM production for $z\gtrsim 1$ \cite{Hall:2009bx}. Under the aforementioned assumptions, the Boltzmann equation for the DM abundance can be written as
\begin{align}\label{eq:Boltzmann equation DM decays}
sHz\frac{\text{d}}{\text{d}z}Y_{S}\,&\simeq\,\sum_{\alpha,i}\left\{F_i\,\leftrightarrow e_{\alpha}S\right\}\nonumber
\\
&=\,2\,\sum_{\alpha, i}\gamma^{F_i}_{e_{\alpha}S}\,\frac{Y_{F_i+\bar{F}_i}}{Y^{\text{eq}}_{F_i+\bar{F}_i}}\,+\,\mathcal{O}\left(\epsilon^2\right)\nonumber
\\
&=\,\frac{g_{F_1}}{\pi^2}\,M_1^2\,T\,K_1\left(\frac{M_1}{T}\right)\sum_{\alpha}\Gamma_{F_1\rightarrow e_{\alpha}S}\,+\,\frac{g_{F_2}}{\pi^2}\,M_2^2\,T\,K_1\left(\frac{M_2}{T}\right)\sum_{\alpha}\Gamma_{F_2\rightarrow e_{\alpha}S}
\end{align}
\noindent
where $\gamma^{F_i}_{e_{\alpha}S}$ is the equilibrium decay rate density at tree-level, $\epsilon$ denotes the $CP$ asymmetry and we have used $Y_{F_i+\bar{F}_i}\simeq Y_{F_i+\bar{F}_i}^{\text{eq}}$.\footnote{Gauge scatterings keep $F_i\left(\bar{F}_i\right)$ close to thermal equilibrium down to $z\sim 25$, when they eventually freeze-out. Since the baryon asymmetry is generated prior to their freeze-out (when sphalerons become inactive), this is a valid approximation.} If we consider $M_1\simeq M_2$ (resonant case) and substitute the decay width \eqref{eq:decay width}, the Hubble parameter \eqref{eq:Hubble parameter} and the entropy density \eqref{eq:entropy density} in Eq.~\eqref{eq:Boltzmann equation DM decays}, then at tree-level and in the limit $M_i\gg m_{e_{\alpha}}+m_{S}$ the DM abundance simplifies to
\begin{equation}\label{eq:DM abundance from decays}
Y_{S}\left(z\right)\,=\,\frac{45M_{Pl}}{32\times 1.66\,\pi^5\,g_{*s}\,\sqrt{g_{*\rho}}}\,\frac{\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^2}{M_1}\,\int_{z_{RH}}^{z}\text{d}z^{\prime}\,{z^{\prime}}^3\,K_1\left(z^{\prime}\right)
\end{equation}
\\
\noindent
where $z_{RH}\equiv M_1/T_{RH}$ and we have considered for simplicity that $\partial g_{*s}/\partial T\simeq 0$. In our analysis, $T_{RH}$ will be set to $10^{12}\,\GeV$. At the present day $T=T_0\simeq 2.73\,$K, so $T_0\ll M_1\ll T_{RH}$ and the integral contributes a factor of $3\pi/2$, yielding
\begin{equation}
Y_{S}\left(z_0\right)\,=\,\frac{135M_{Pl}}{64\times 1.66\,\pi^4\,g_{*s}\,\sqrt{g_{*\rho}}}\,\frac{\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^2}{M_1}
\end{equation}
\\
\noindent
The experimentally observed DM abundance is
\begin{equation}\label{eq:DM_Observed_Abundance}
Y_{S}\left(z_0\right)\,=\,\frac{\Omega_{DM}\,\rho_c}{m_{S}\,s_0}
\end{equation}
\noindent
where $\Omega_{\text{DM}}h^2= 0.1200\pm 0.0012$, $\rho_c\equiv 3H_0^2/8\pi\,G\simeq 10.537\,h^2\,\GeV m^{-3}$ is the critical density and $s_0\simeq 2.9\times 10^{9}\,m^{-3}$ is the entropy density at the present day \cite{2020}. If we ignore dark matter production through scattering processes, the DM mass $m_{S}$ can be related to the heavy lepton mass $M_{1}$ and to the feeble couplings as
\begin{equation}\label{eq:DM mass}
m_{S}\,=\,\frac{64\times 1.66\,\pi^4\,g_{*s}\,\sqrt{g_{*\rho}}\,\Omega_{DM}\,\rho_c}{135M_{Pl}\,s_0}\,\frac{M_1}{\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^2}
\end{equation}
\noindent
Lyman-$\alpha$ forest observations can be used in order to extract a lower bound on the DM mass, the exact value of which depends on the underlying mechanism of DM genesis. For DM candidates that freeze-out the current bound is $m_{DM}\gtrsim\left(1.9-5.3\right)\keV$ at $95\%$ C.L. \cite{garzilli2019warm,Palanque_Delabrouille_2020,Ir_i__2017}. This limit can be mapped onto the case of freeze-in-produced DM yielding $m_S\gtrsim\left(4-16\right)\keV$ \cite{Ballesteros_2021,Boulebnane_2018,DEramo:2020gpr,Decant:2021mhj}. This, in turn, imposes an upper limit on the feeble couplings
\begin{equation}\label{eq:Lyman-a bound}
\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^2\,\lesssim\,7.55\times 10^{-16}\left(\frac{M_1}{\TeV}\right)
\end{equation}
in order not to overclose the Universe, where we have used the more conservative bound $m_S\gtrsim 4\keV$. Thus, we see that the Lyman-$\alpha$ forest sets a more severe constraint than the out-of-equilibrium condition of Eq.~\eqref{eq:out-of-equilibrium condition}, forcing the feeble couplings to lie in the $10^{-8}$ range and below for heavy lepton masses at the $\TeV$ scale.
For a more rigorous treatment one should take into account the impact of scattering processes, which can modify the predicted DM abundance at $z\lesssim 1$, an epoch when scatterings are dominant. Including such scattering processes will be essential for the calculation of the baryon asymmetry presented in the next section. We focus only on scattering processes which involve gauge bosons as external states and ignore the subleading ones (second row of Eq.~\eqref{eq:Boltzmann DM full}). In this case, the Boltzmann equation takes the form
\begin{align}\label{eq:Boltzmann equation DM scatterings}
sHz\,\frac{\text{d}Y_S}{\text{d}z}\,&=\,\sum_{\alpha,i}\left\{F_i\,\leftrightarrow e_{\alpha}S\right\}\,+\,\sum_{\alpha,i}\left\{F_iB\leftrightarrow e_{\alpha}S\right\}\,-\,\sum_{\alpha,i}\left\{F_iS\leftrightarrow e_{\alpha}B\right\}\,+\,\sum_{\alpha,i}\left\{F_i\bar{e}_{\alpha}\leftrightarrow SB\right\}\nonumber
\\
&=\,2\,\sum_{\alpha,i}\Big(\gamma^{F_i}_{e_{\alpha}S}\,+\,\gamma^{F_iB}_{e_{\alpha}S}\,+\,\gamma^{F_i\bar{e}_{\alpha}}_{SB}\,+\,\gamma^{F_iS}_{e_{\alpha}B}\Big)\,+\,\mathcal{O}\left(\epsilon^2\right)
\end{align}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{Figures/Plot_DM_Abundance.pdf}
\caption{DM abundance generated solely by decays, as well as by decays and scatterings for heavy leptons masses $M_1\simeq M_2 = 1.2\TeV$ and couplings $\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^2\simeq 6.25\times 10^{-16}$.}
\label{fig:DM Abundance}
\end{figure}
\noindent
where we have made use of the freeze-in approximation $f_S\simeq 0$ and considered $Y_{F_i+\bar{F}_i}\simeq Y_{F_i+\bar{F}_i}^{\text{eq}}$. In Figure \ref{fig:DM Abundance} we fix the masses of the heavy leptons $M_1\simeq M_2=1.2\TeV$ and the couplings $\sum_{\alpha,i}\left|\lambda_{\alpha i}\right|^2\simeq 6.25\times 10^{-16}$, and we compare the predicted DM abundance as estimated if we take into account only decays of the heavy leptons (lower, green line) with the result obtained through a full-blown numerical solution of Eq.~\eqref{eq:Boltzmann equation DM scatterings} (upper, yellow line). We observe that, as expected, the inclusion of the scattering processes does not drastically modify the predicted amount of dark matter in the Universe. Note that we have also cross-checked all of our results by implementing our model in {\tt FeynRules} \cite{Alloul:2013bka} and computing the predicted freeze-in DM abundance with {\tt CalcHEP/micrOMEGAs 5} \cite{Belyaev:2012qa, Belanger:2018ccd}. Given our findings, we conclude that the analytic estimate of the DM mass given by Eq.~\eqref{eq:DM mass} constitutes a reliable approximation. For the values of the physical parameters used in Figure \ref{fig:DM Abundance}, the value of the dark matter mass in order to reproduce the observed DM abundance in the Universe is $m_S\simeq 5.81\keV$, which is compatible with the Lyman-$\alpha$ forest bounds discussed previously.
\subsection{$CP$ asymmetry}\label{subsec:CP Asymmetry}
The $CP$ asymmetry generated through the decays of $F_{1,2}$ arises, at lowest order, due to the interference of the tree-level and 1-loop self-energy Feynman diagrams (wave-part contribution) as shown in Figures \ref{fig:tree-level} and \ref{fig:loop}. It can be defined in terms of the decay widths as
\begin{equation}\label{eq:CP Asymmetry def}
\epsilon_{\alpha\,i}\,\equiv\,\frac{\Gamma\left(F_i\rightarrow e_{\alpha}S\right)\,-\,\Gamma\left(\bar{F}_i\rightarrow\bar{e}_{\alpha}S\right)}{\sum_{\alpha}\Gamma\left(F_i\rightarrow e_{\alpha}S\right)\,+\,\Gamma\left(\bar{F}_i\rightarrow\bar{e}_{\alpha}S\right)}\, = \,\frac{\Gamma\left(F_i\rightarrow e_{\alpha}S\right)\,-\,\Gamma\left(\bar{F}_i\rightarrow\bar{e}_{\alpha}S\right)}{2\,\Gamma_i}
\end{equation}
\noindent
where $\Gamma_i$ is the total decay width of $F_i$. If we separate the matrix element $\left.\mathcal{M}\right|_k$, with $k$ being the loop order $\left(k=0,1,\ldots\right)$, into a coupling constant part $c_k$ (denoting a collection of coupling constants) and an amplitude part $\mathcal{A}_k$, \textit{i.e.} $\left.\mathcal{M}\right|_k\equiv\sum_{i=0}^{k}\,c_i\mathcal{A}_i$, we can rewrite the $CP$ asymmetry as \cite{Davidson:2008bu}
\begin{equation}\label{eq:CP asymmetry 1}
\epsilon_{\alpha\,i}\,=\,-2\,\frac{\text{Im}\left\{c_0c_1^*\right\}}{\sum_{\alpha}\left|c_0\right|^2}\frac{\int\text{d}\Pi_{e_{\alpha}}\text{d}\Pi_{S}\left(2\pi\right)^4\delta^{(4)}\text{Im}\left\{\mathcal{A}_0\mathcal{A}_1^*\right\}}{\int\text{d}\Pi_{e_{\alpha}}\text{d}\Pi_{S}\left(2\pi\right)^4\delta^{(4)}\left|\mathcal{A}_0\right|^2}
\end{equation}
\noindent
The imaginary part of the amplitudes is related to the discontinuity of the corresponding graph as $2i\text{Im}\left\{\mathcal{A}_0\mathcal{A}_1^*\right\}=\text{Disc}\{\mathcal{A}_0\mathcal{A}_1^*\}$, which can be computed using the Cutkosky cutting rules. For a non-degenerate $F_i$ spectrum, $M_j-M_i\gg\Gamma_{j}$, we obtain \cite{PackageX}
\begin{align}\label{eq:CP asymmetry}
\epsilon_{\alpha i}\,=\,-\frac{1}{16\pi}\,\frac{1}{1\,-\,x_j}\,\frac{\text{Im}\Big\{\lambda_{\alpha i}^*\lambda_{\alpha j}\big[\lambda^{\dagger}\lambda\big]_{ji}\Big\}}{\left[\lambda^{\dagger}\lambda\right]_{ii}}
\end{align}
\noindent
where $x_j\equiv M_j^2/M_i^2$ and we have used the standard notation $\left[\lambda^{\dagger}\lambda\right]_{ii}\equiv\left|\lambda_{ei}\right|^2+\left|\lambda_{\mu i}\right|^2+\left|\lambda_{\tau i}\right|^2$ and $\left[\lambda^{\dagger}\lambda\right]_{ji}\equiv\lambda_{ej}^* \lambda_{ei}+\lambda_{\mu j}^* \lambda_{\mu i}+\lambda_{\tau j}^* \lambda_{\tau i}$. This is half the result obtained in standard leptogenesis (see \textit{e.g.} Eq.~(5.13) in \cite{Davidson:2008bu}), because we deal with Dirac fermions where only a charged lepton propagates in the loop, whereas Majorana neutrinos can decay to both a charged lepton or a light neutrino accompanied by the Higgs field.
From $CPT$ invariance and unitarity, we know that the total decay width of a state and its $CP$-conjugate are equal
\begin{equation}\label{eq:total decay width}
\sum_{\alpha}\Gamma\left(F_i\rightarrow e_{\alpha}S\right)\,=\,\sum_{\alpha}\Gamma\left(\bar{F}_i\rightarrow\bar{e}_{\alpha}S\right)\,\equiv\,\Gamma_i
\end{equation}
\noindent
Hence, the $CP$ asymmetry vanishes when summed over the flavors, $\epsilon_i\,\equiv\,\sum_{\alpha}\epsilon_{\alpha\,i}\,=\,0$. This can be seen by summing Eq.~\eqref{eq:CP asymmetry} over flavors, a case in which the argument of the imaginary part becomes real and it vanishes identically:
\begin{equation}
\sum_{\alpha}\text{Im}\Big\{\lambda_{\alpha i}^*\lambda_{\alpha j}\big[\lambda^{\dagger}\lambda\big]_{ji}\Big\}\,=\,\text{Im}\left\{\big[\lambda^{\dagger}\lambda\big]_{ij}\big[\lambda^{\dagger}\lambda\big]_{ji}\right\}\,=\,0
\end{equation}
\noindent
However, flavor effects \textit{can} lead to a non-vanishing baryon/lepton asymmetry, since washout processes are flavor dependent and, therefore, the lepton asymmetries in each flavor are washed out in a different way \cite{Abada_2006,Nardi_2006,Barbieri_2000,Blanchet_2007}.
Because of the feeble nature of the $\lambda_{\alpha i}$ couplings, a resonant enhancement of the $CP$ asymmetry is needed in order to generate a sufficiently large baryon asymmetry. This occurs when the mass difference between the heavy leptons is of the order of their decay widths and is related to the wave-part contribution to the $CP$ asymmetry (Figure \ref{fig:loop}). The resulting $CP$ asymmetry is given by
\begin{align}\label{eq:CP asymmetry regularized}
\epsilon_{\alpha i}\,=\,-\frac{1}{16\pi}\,\frac{1\,-\,x_j}{\left(1\,-\,x_j\right)^2\,+\,g_j^2}\,\frac{\text{Im}\Big\{\lambda_{\alpha i}^*\lambda_{\alpha j}\big[\lambda^{\dagger}\lambda\big]_{ji}\Big\}}{\left[\lambda^{\dagger}\lambda\right]_{ii}}
\end{align}
\noindent
where $g_j\equiv\Gamma_j/M_i$. In deriving \eqref{eq:CP asymmetry regularized}, we have regularized the divergence for small $x_j$ by applying the resummation procedure presented in \cite{Pilaftsis_1999}, with the regulator given by $M_1\Gamma_2$. Note that a more complete treatment of the resonance enhancement requires the employment of non-equilibrium techniques, which yield, in general, different regulators. Such an analysis has been performed \emph{e.g.} in \cite{Dev:2017wwc}, where the nearly degenerate states are considered to be out-of-equilibrium with the bath, in contrast to our model, where the $F_i$'s are kept close to equilibrium and the required deviation occurs for the DM states. Such an analysis is beyond the scope of this paper and is left for future work.
If we also express the coupling constants in terms of their magnitude and phase, $\lambda_{\alpha i}\,=\,\left|\lambda_{\alpha i}\right|e^{i\phi_{\alpha i}}$, Eq.~\eqref{eq:CP asymmetry regularized} can be rewritten as
\begin{align}
\epsilon_{\alpha i}\,&=\,-\frac{1}{16\pi}\,\frac{1\,-\,x_j}{\left(1\,-\,x_j\right)^2\,+\,g_j^2}\,\frac{\left|\lambda_{\alpha i}\right|\left|\lambda_{\alpha j}\right|}{\big[\lambda^{\dagger}\lambda\big]_{ii}}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta i}\right|\left|\lambda_{\beta j}\right|\sin\left(-\,\phi_{\alpha i}\,+\,\phi_{\alpha j}\,-\,\phi_{\beta j}\,+\,\phi_{\beta i}\right)\nonumber
\\
&\equiv\,-\frac{1}{16\pi}\,\frac{1\,-\,x_j}{\left(1\,-\,x_j\right)^2\,+\,g_j^2}\,\frac{\left|\lambda_{\alpha i}\right|\left|\lambda_{\alpha j}\right|}{\big[\lambda^{\dagger}\lambda\big]_{ii}}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta i}\right|\left|\lambda_{\beta j}\right|\,p^{ij}_{\alpha\beta}
\end{align}
\noindent
where $p^{ij}_{\alpha\beta}=-p^{ji}_{\alpha\beta}=-p^{ij}_{\beta\alpha}$. The resonance condition reads $M_2-M_1\sim \Gamma_2/2$ \cite{Pilaftsis_1999} and in this case the $CP$ asymmetry can be maximally enhanced to
\begin{subequations}
\begin{alignat}{2}
\epsilon_{\alpha 1}^{\text{res}}\,&=\,\frac{g_{F_1}}{2}\,\frac{\left|\lambda_{\alpha 1}\right|\left|\lambda_{\alpha 2}\right|}{\big[\lambda^{\dagger}\lambda\big]_{11}\big[\lambda^{\dagger}\lambda\big]_{22}}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta 1}\right|\left|\lambda_{\beta 2}\right|\,p^{12}_{\alpha\beta}
\\
\epsilon_{\alpha 2}^{\text{res}}\,&=\,g_{F_2}\,\frac{\left|\lambda_{\alpha 1}\right|\left|\lambda_{\alpha 2}\right|}{\big[\lambda^{\dagger}\lambda\big]_{11}^2\,+\,\big[\lambda^{\dagger}\lambda\big]_{22}^2}\,\sum_{\beta\neq\alpha}\left|\lambda_{\beta 1}\right|\left|\lambda_{\beta 2}\right|\,p^{12}_{\alpha\beta}
\end{alignat}
\end{subequations}
These are the expressions that we will be using in the numerical analysis that follows in order to compute $\epsilon_{\alpha i}$.
\subsection{Baryon asymmetry}
In the previous sections, we described how our model can satisfy two out of the three Sakharov conditions, namely, $C/CP$ violation and out-of-equilibrium dynamics. The last condition to be fulfilled in order to generate a baryon asymmetry is the violation of the baryon/lepton number. In the case of Majorana heavy states, a conserved lepton number cannot be consistently assigned in the presence of interaction and mass terms in the Lagrangian and, therefore, $L$ is violated. This is not the case in our model, where the heavy states are of Dirac nature. In this case, we may rely on sphaleron departure from equilibrium during the epoch that the lepton asymmetry is generated, along the lines described in \cite{Gonzalez:2009}.
In the model we propose, the heavy leptons $F_i$ carry the same lepton number as the SM leptons and the total lepton asymmetry is $Y_L=Y_{L_{\text{SM}}}+Y_{L_F}$, where $Y_{L_{\text{SM}}}\equiv \sum_{\alpha}Y_{L_{\alpha}}$ and $Y_{L_F}=\sum_i Y_{L_{F_i}}$. All processes conserve the combination $Y_{B-L}\equiv Y_B-Y_{L_{\text{SM}}}-Y_{L_F}$, \textit{i.e.} $\text{d}Y_{B-L}/\text{d}z=0$. We also assume that the Universe is initially totally symmetric, $Y_{B-L_{\text{SM}}}|_{z_{RH}}=Y_{L_F}|_{z_{RH}}=0$ and, therefore, at any $z$ it holds $Y_{B-L_{\text{SM}}}=Y_{L_F}$.
Since sphalerons are insensitive to the lepton asymmetry $Y_{L_F}$, as $F_i$ are $SU(3)_{\text{c}}\times SU(2)_{\LH}$-singlets, they affect only the non-zero lepton asymmetry stored in the SM lepton sector $Y_{L_{\text{SM}}}$. In particular, they convert it to a baryon asymmetry by imposing certain relations among the chemical potentials of the various species (see the Appendix). Once sphalerons depart from equilibrium, which occurs at $T_{sph}= 131.7\pm 2.3\GeV$ \cite{D_Onofrio_2014}, the baryon and lepton numbers are separately conserved. When the heavy leptons decay away, the total baryon asymmetry, being proportional to $Y_{B-L_{\text{SM}}}$, vanishes. However, if sphalerons become inactive during the decay epoch of the heavy leptons, then the baryon asymmetry freezes at $Y_B\propto Y_{B-L_{\text{SM}}}|_{T_{sph}}$, which, in general, is not null.
Taking into consideration all the decay and $2\rightarrow 2$ scattering processes, the full Boltzmann equations of the asymmetries read
\begin{align}\label{eq:Boltzmann Asymmetry 0}
-sHz\frac{\text{d}Y_{\Delta F_i}}{\text{d}z}\,&=\,\sum_{\alpha}\left[F_i\,\leftrightarrow e_{\alpha}S\right]\,+\,\sum_{\alpha}\left[F_iB\leftrightarrow e_{\alpha}S\right]\,+\,\sum_{\alpha}\left[F_iS\leftrightarrow e_{\alpha}B\right]\,+\,\sum_{\alpha}\left[F_i\bar{e}_{\alpha}\leftrightarrow SB\right]\nonumber
\\
&+\,\sum_{\alpha,\beta,j}\left[F_i\bar{e}_{\alpha}\leftrightarrow\bar{F}_je_{\beta}\right]\,+\,\sum_{\alpha,\beta}\left[F_i\bar{e}_{\alpha}\leftrightarrow F_j\bar{e}_{\beta}\right]\,+\,\sum_{\alpha,\beta}\left[F_ie_{\beta}\leftrightarrow F_je_{\alpha}\right]\nonumber
\\
&+\,\sum_{\alpha,\beta}\left[F_i\bar{F}_{j\neq i}\leftrightarrow e_{\alpha}\bar{e}_{\beta}\right]\,+\,\left[F_i\bar{F}_{j\neq i}\leftrightarrow SS\right]\,+\,\sum_{\alpha,\beta,j}\left[F_iF_j\leftrightarrow e_{\alpha}e_{\beta}\right]\nonumber
\\
&+\,\left[F_iS\leftrightarrow F_{j\neq i}S\right]
\end{align}
\begin{align}\label{eq:Boltzmann Asymmetry 00}
-sHz\frac{\text{d}Y_{\Delta_{\alpha}}}{\text{d}z}\,&=\,\sum_{i}\left[F_i\,\leftrightarrow e_{\alpha}S\right]\,+\,\sum_{i}\left[F_iB\leftrightarrow e_{\alpha}S\right]\,+\,\sum_{i}\left[F_iS\leftrightarrow e_{\alpha}B\right]\,+\,\sum_{i}\left[F_i\bar{e}_{\alpha}\leftrightarrow SB\right]\nonumber
\\
&+\,\sum_{i,j,\beta}\left[F_i\bar{e}_{\alpha}\leftrightarrow \bar{F}_je_{\beta}\right]\,+\,\sum_{i,j,\beta\neq\alpha}\left[F_i\bar{e}_{\alpha}\leftrightarrow F_j\bar{e}_{\beta}\right]\,+\,\sum_{i,j,\beta\neq\alpha}\left[F_ie_{\beta}\leftrightarrow F_je_{\alpha}\right]\nonumber
\\
&+\,\sum_{i,j,\beta\neq\alpha}\left[F_i\bar{F}_j\leftrightarrow e_{\alpha}\bar{e}_{\beta}\right]\,+\,\sum_{\beta\neq\alpha}\left[\bar{e}_{\alpha}e_{\beta}\leftrightarrow SS\right]\,+\,\sum_{i,j,\beta}\left[F_iF_j\leftrightarrow e_{\alpha}e_{\beta}\right]\nonumber
\\
&+\sum_{\beta\neq\alpha}\big[e_{\beta}S\leftrightarrow e_{\alpha}S\big]^{\prime}
\end{align}
\noindent
where $Y_{\Delta F_i}\equiv Y_{F_i}-Y_{\bar{F}_i}$ and $Y_{\Delta_{\alpha}}\equiv Y_B/3-Y_{L_{\text{SM}{\alpha}}}$. The primed term indicates that one has to subtract the contribution due to the on-shell propagation of $F_i$, usually referred to as real intermediate state subtraction (RISS), which is already taken into account by the successive decays $e_{\alpha}S\leftrightarrow F_i\leftrightarrow e_{\alpha}S$.
The various terms in the Boltzmann equations can be expressed in terms of the $CP$ asymmetry $\epsilon_{\alpha i}$, the tree-level rate densities and the asymmetric abundances. As is typically done, we linearize in the SM chemical potentials \cite{Kolb:1979qa}
\begin{equation}
\frac{Y_{e_{\alpha}\left(\bar{e}_{\alpha}\right)}}{Y_{e_{\alpha}}^{\text{eq}}}\,\equiv\,e^{\pm\mu_{e_{\alpha}}/T}\,\simeq\,1\,\pm\,\frac{\mu_{e_{\alpha}}}{T}\,=\,1\,\pm\,\frac{Y_{\Delta e_{\alpha}}}{2Y_{\gamma}}
\end{equation}
\noindent
All non-gauge interactions are subleading in comparison to the gauge interactions and can be safely discarded. We include only the $CP$-violating part of the RISS term, which ensures that the source term (proportional to $\epsilon_{\alpha i}$) takes the correct form, \textit{i.e.} it vanishes when all species are in chemical equilibrium. The Boltzmann equations can be rewritten as\footnote{In deriving Eq.~\eqref{eq:Boltzmann equation DM scatterings}, we dropped dark matter annihilation processes altogether. In baryo- and leptogenesis, inverse reactions can be crucial and need to be taken into account. In order to do so in an efficient manner, in Eqs.~\eqref{eq:Boltzmann Asymmetry 1} and \eqref{eq:Boltzmann Asymmetry 2} we will approximate $f_S \simeq f_S^{\rm eq} \frac{Y_S}{Y_S^{\rm eq}}$ -- a relation which is rigorously applicable to bath particles. We have checked that -- as expected, since we have restricted ourselves to regions of the parameter space in which the condition of Eq.~\eqref{eq:out-of-equilibrium condition} is satisfied -- this is, indeed, a good approximation which leads to only a small ($\sim 2\%$) reduction of the predicted dark matter abundance for large values of $z$.}
\begin{align}
-sHz\frac{\text{d}Y_{\Delta F_i}}{\text{d}z}\,&=\,\sum_{\alpha}\Big(y_{F_i}\,-\,\frac{Y_S}{Y_S^{\text{eq}}}\,y_{e_{\alpha}}\Big)\,\Big(\gamma^{F_i}_{e_{\alpha}S}\,+\,\gamma^{F_iB}_{e_{\alpha}S}\Big)\,+\,\sum_{\alpha}\Big(\frac{Y_S}{Y_S^{\text{eq}}}\,y_{F_i}\,-\,y_{e_{\alpha}}\Big)\,\gamma^{F_iS}_{e_{\alpha}B}\nonumber
\\
&+\,\sum_{\alpha}\Big(y_{F_i}\,-\,y_{e_{\alpha}}\Big)\,\gamma^{F_i\bar{e}_{\alpha}}_{SB}\label{eq:Boltzmann Asymmetry 1}
\\\nonumber
\\
-sHz\frac{\text{d}Y_{\Delta_{\alpha}}}{\text{d}z}\,&=\,2\,\sum_i\epsilon_{\alpha i}\,\Big[\,\Big(1\,-\,\frac{Y_S}{Y_S^{\text{eq}}}\Big)\,\sum_{\rho}\Big(\gamma^{F_i}_{e_{\rho}S}\,+\,\gamma^{F_iB}_{e_{\rho}S}\,+\,\gamma^{F_i\bar{e}_{\rho}}_{SB}\,-\,\gamma^{F_iS}_{e_{\rho}B}\Big)\Big]\nonumber
\\
&+\,\sum_i\Big[y_{F_i}\,-\,\frac{Y_S}{Y_S^{\text{eq}}}\,\sum_{\beta}\left(\text{B}^{F_i}_{e_{\beta}S}\,y_{e_{\beta}}\right)\Big]\,\gamma^{F_i}_{e_{\alpha}S}\,+\,\sum_i\Big(y_{F_i}\,-\,\frac{Y_S}{Y^{\text{eq}}_S}\,y_{e_{\alpha}}\Big)\,\gamma^{F_iB}_{e_{\alpha}S}\nonumber
\\
&+\,\sum_i\Big(\frac{Y_S}{Y_S^{\text{eq}}}\,y_{F_i}\,-\,y_{e_{\alpha}}\Big)\,\gamma^{F_iS}_{e_{\alpha}B}\,+\,\sum_i\Big(y_{F_i}\,-\,y_{e_{\alpha}}\Big)\,\gamma^{F_i\bar{e}_{\alpha}}_{SB}\label{eq:Boltzmann Asymmetry 2}
\end{align}
\noindent
where $y_{F_i}\equiv Y_{\Delta F_i}/Y_{F_i}^{\text{eq}}$, $y_{e_{\alpha}}\equiv Y_{\Delta e_{\alpha}}/Y_{\gamma}$, $\text{B}^{F_i}_{e_{\beta}S}$ denotes the branching ratio of the decay $F_i\rightarrow e_{\beta}S$ and we have used $Y_{F_i+\bar{F}_i}\simeq Y_{F_i+\bar{F}_i}^{\text{eq}}$. In deriving the equations above, we have taken into account that the $CP$ asymmetry in scattering processes stemming from self-energy diagrams (Figure \ref{fig:loop}) is always equal to the $CP$ asymmetry from decays \cite{Nardi_2007}. As described in \cite{Davidson:2008bu} one must also include the contributions from the RISS $2\rightarrow 3$ scatterings involving gauge bosons, in order to obtain the correct form for the scattering source terms. These processes can, however, be neglected as subleading with regards to their impact on washout. Note that the source term in the Boltzmann equation of the asymmetry in $F_i$ vanishes as a consequence of $CPT$ and unitarity, as explained in Section \ref{subsec:CP Asymmetry}. Lastly, these equations are not totally independent, as they are related through the conservation of $Y_{B-L}$, resulting in $\sum_{i}Y_{\Delta F_i}=\sum_{\alpha}Y_{\Delta\alpha}$.
One can express the asymmetries in each SM flavor $y_{e_{\alpha}}$ in terms of $Y_{\Delta\alpha}$, solving a system of algebraic equations which relate the chemical potentials and abundances in equilibrium of the various species (see the Appendix). For the temperatures of interest we find
\begin{equation}
\begin{pmatrix}
y_{e_e} \\
y_{e_{\mu}} \\
y_{e_{\tau}}
\end{pmatrix}
\,=\,
-\frac{1}{711}\,
\left[\begin{array}{ccc}
230 & -7 & -7 \\
-7 & 230 & -7 \\
-7 & -7 & 230
\end{array}\right]\,
\begin{pmatrix}
Y_{\Delta_e} \\
Y_{\Delta_{\mu}} \\
Y_{\Delta_{\tau}}
\end{pmatrix}\frac{1}{Y_{\gamma}}
\end{equation}
\\
\noindent
In the same way, we have calculated the amount of the SM lepton asymmetry converted to baryon asymmetry by the sphaleron transitions to be
\begin{equation}
Y_{B}\,=\,\frac{22}{79}\,\sum_{\alpha}Y_{\Delta\alpha}
\end{equation}
\begin{figure}[t]
\centering
\includegraphics[width=0.75\linewidth]{Figures/Plot_Asymmetries.pdf}
\caption{The generated DM abundance $Y_S$, the asymmetries $Y_{\Delta\alpha}$, $Y_{\Delta F_1}$ and the baryon asymmetry $Y_B$ as a function of the dimensionless parameter $z=M_1/T$, for the heavy leptons masses and feeble couplings given in Eq.~\eqref{eq:parameter values}. The dashed line illustrates the evolution that $Y_B$ would follow in the absence of sphaleron decoupling, while the vertical line denotes the temperature at which sphalerons depart from equilibrium.}
\label{fig:Asymmetries}
\end{figure}
The system of the five coupled Boltzmann equations of the asymmetries \eqref{eq:Boltzmann Asymmetry 1} and \eqref{eq:Boltzmann Asymmetry 2} has been solved numerically, using the DM abundance generated by decays and scattering processes of Eq.~\eqref{eq:Boltzmann equation DM scatterings}. We confirm that the proposed model can indeed generate the observed baryon asymmetry $Y_{\text{B}}= 8.71\pm 0.06\times 10^{-11}$ \cite{2020} in the resonant regime. As an illustration, we present in Figure \ref{fig:Asymmetries} an explicit example, using the following values of the parameters:
\begin{equation}\label{eq:parameter values}
\begin{split}
M_1\,&=\,1.2\TeV,\\
\left|\lambda_{e1}\right|\,&=\,2.145\times 10^{-8},\\
\left|\lambda_{\mu 1}\right|\,&=\,\left|\lambda_{\tau 1}\right|\,=\,9\times 10^{-9},\\
p^{12}_{e\mu}\,&=\,p^{12}_{e\tau}\,=\,p^{12}_{\mu\tau}\,=\,1
\end{split}
\qquad
\begin{split}
M_2\,&=\,M_1\,+\,\Gamma_2/2\\
\left|\lambda_{e2}\right|\,&=\,\left|\lambda_{\tau 2}\right|\,=\,9\times 10^{-10}\\
\left|\lambda_{\mu 2}\right|\,&=\,8\times 10^{-10}\\
&
\end{split}
\end{equation}
\\
\noindent
The corresponding resonant $CP$ asymmetries $\epsilon_{\alpha i}$ turn out to be: $\epsilon_{e1}\simeq 2.1\times 10^{-1},\,\epsilon_{\mu 1}\simeq -5.7\times 10^{-2},\,\epsilon_{\tau 1}\simeq -1.53\times 10^{-1}$ and $\epsilon_{e2}\simeq 1.53\times 10^{-3},\,\epsilon_{\mu 2}\simeq -4.2\times 10^{-4},\,\epsilon_{\tau 2}\simeq -1.11\times 10^{-3}$. We have explicitly verified that $Y_{B-L}$ is conserved for all values of $z$ or, equivalently, that the relation $\sum_{i}Y_{\Delta F_i}=\sum_{\alpha}Y_{\Delta\alpha}$ holds. Similarly, the baryon asymmetry vanishes in the absence of flavor effects, \textit{i.e.} when $|\lambda_{e1}|=|\lambda_{\mu 1}|=|\lambda_{\tau 1}|$ and $|\lambda_{e2}|=|\lambda_{\mu 2}|=|\lambda_{\tau 2}|$.
Note that, contrary to the standard leptogenesis scenarios, the SM flavor asymmetries $Y_{\Delta\alpha}$ are constantly generated, since the $CP$-violating processes always occur out-of-equilibrium. They eventually attain their final value as soon as the DM state freezes-in at $z\sim 3-5$. On the other hand, as the heavy leptons decay, their asymmetries $Y_{\Delta F_i}$ decrease and eventually vanish at high $z$. The asymmetry in $F_2$ is many orders of magnitude smaller than the one in $F_1$, due to its smaller couplings $\lambda_{\alpha 2}\ll\lambda_{\alpha 1}$, and is not shown in Figure \ref{fig:Asymmetries}. Once the $F_i$'s have decayed away, conservation of $Y_{B-L}$ implies that also $Y_{B-L_{SM}}\rightarrow 0$. Thus, the model predicts an equal amount of baryon $Y_B$ and SM lepton $Y_{L_{\text{SM}}}$ asymmetries left in the Universe \cite{Gonzalez:2009}.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\linewidth]{Figures/Plot_Baryon_Asymmetry_Scan.pdf}
\caption{Combinations of the feeble couplings and the heavy lepton mass which can generate the observed baryon asymmetry, for the set of parameters depicted in Table \ref{table: scenarios parameters}. The dashed lines depict representative DM masses $m_S=\{2,4,6,16\}\keV$ for which the observed DM abundance of the Universe can be reproduced. Masses below $4\keV$ are excluded from current Lyman-$\alpha$ forest observations (gray shaded area).}
\label{fig:Plot_Baryon_Asymmetry_Scan}
\end{figure}
Our numerical analysis reveals that at least one of the feeble couplings must have a magnitude larger than $\sim 10^{-8}$ in order to account for the observed baryon asymmetry. Our main results are summarized in Figure \ref{fig:Plot_Baryon_Asymmetry_Scan}, where we show the combinations of the feeble couplings and the heavy lepton mass for which a viable baryon asymmetry can be generated, for the two illustrative scenarios depicted in Table \ref{table: scenarios parameters}.
\begin{table}[h]
\centering
\begin{tabular}{l*{5}{c}r}\hline\hline
& $\left|\lambda_{\mu 1}\right|$ & $\left|\lambda_{\tau 1}\right|$ & $\left|\lambda_{e2}\right|$ & $\left|\lambda_{\mu 2}\right|$ & $\left|\lambda_{\tau 2}\right|$ \\
\hline
Scenario 1 & $10^{-9}$ & $2\times 10^{-9}$ & $9\times 10^{-10}$ & $8\times 10^{-10}$ & $9\times 10^{-10}$ \\
Scenario 2 & $9\times 10^{-9}$ & $9\times 10^{-9}$ & $9\times 10^{-10}$ & $8\times 10^{-10}$ & $9\times 10^{-10}$ \\
\hline\hline
\end{tabular}
\caption{Values of the parameters used for the two scenarios shown in Figure \ref{fig:Plot_Baryon_Asymmetry_Scan}.}
\label{table: scenarios parameters}
\end{table}
\noindent
In both cases, the $CP$ asymmetry is resonantly enhanced, \textit{i.e.} the mass of $F_2$ is $M_2=M_1+\Gamma_2/2$ and we consider $p^{12}_{e\mu}=p^{12}_{e\tau}=p^{12}_{\mu\tau}=1$ for the phases of the feeble couplings. The $\lambda_{e1}$ coupling is being treated as a free parameter and lies in the $10^{-8}$ ballpark.
The dark matter relic density constraint can be satisfied throughout the parameter ranges depicted in the figure, for an appropriate choice of the mass $m_S$. However, in the shaded region, the required dark matter mass turns out to be in conflict with the Lyman-$\alpha$ bound of Eq.~\eqref{eq:Lyman-a bound}.
We observe that the first scenario (upper, blue line) is excluded from Lyman-$\alpha$ constraints for all values of the heavy lepton mass $M_1$. On the contrary, the second scenario (lower, orange line) is able to account for both the baryon asymmetry and the DM abundance for a wide range of $M_1$, $550\GeV\lesssim M_1\lesssim 2800\GeV$. Note that, interestingly, we find that for any combination of parameter values the relic density constraint is satisfied for DM masses that do not exceed $\sim 6\keV$. This implies that if Lyman-$\alpha$ constraints become stronger in the future, they may be able to fully exclude the model's viable parameter space.
\subsection{Phenomenological aspects}\label{subsec:pheno}
In the previous paragraphs, we saw that our simple model, described by the Lagrangian of Eq.~\eqref{eq:Lgeneral}, can provide a common explanation for the observed dark matter content and baryon asymmetry of the Universe. As we showed, this can be achieved by assuming highly degenerate heavy fermions $F_i$ with masses $M_{i} \sim 1$ TeV and Yukawa-like couplings of the order of $\lambda_{\alpha 1} \sim 10^{-8}-10^{-7}$ and $\lambda_{\alpha 2} \sim 10^{-10}$ for $F_1$ and $F_2$, respectively.
Our model is intended to serve mainly as a proof-of-concept for the freeze-in baryogenesis mechanism that we propose. In this spirit, a full-blown analysis of its phenomenological predictions goes well beyond the scope of the present work, especially since alternative constructions assuming, \textit{e.g.}, different quantum numbers for the various particles involved, can lead to wildly different phenomenological signatures. Still, despite its simplicity, this model does exhibit an interesting phenomenology, which we will briefly comment upon. Note that a simpler variant of this model involving a single vectorlike heavy fermion $F$ was already studied, \textit{e.g.}, in \cite{Belanger:2018sti, Brooijmans:2020yij}, mainly from the viewpoint of its predicted dark matter abundance as well as its signatures in new physics searches at the LHC.
First, the heavy fermions can mediate lepton flavor-violating decays of the type $\ell_\beta \rightarrow \ell_\alpha \gamma$. In particular, given the constraint $Br(\mu \rightarrow e \gamma) < 4.2\times 10^{-13}$ \cite{MEG:2016leq} and assuming that (as was the case throughout the analysis presented in the previous sections) $\lambda_{e1} \sim \lambda_{\mu 1}$, the analysis presented in \cite{Brooijmans:2020yij} restricts the product of the Yukawa-like couplings of $F_{1}$ to the first two generation leptons as
\begin{equation}
\frac{\sqrt{\lambda_{e1} \lambda_{\mu 1}}}{M_1} \lesssim 3.6\times 10^{-5} \ {\rm GeV}^{-1}
\end{equation}
which is easily satisfied in the cosmologically relevant part of our parameter space. In deriving this bound, we have neglected the interference between Feynman diagrams involving different species of heavy fermions running in the loop, since the couplings $\lambda_{\alpha 2} \ll \lambda_{\alpha 1}$. Besides, constraints stemming from measurements of the muon lifetime turn out to be subleading \cite{Brooijmans:2020yij}.
On the side of collider searches, let us first point out the fact that, given the feeble nature of the $\lambda_{\alpha i}$ couplings, the mean proper decay length of $F_1$ is of the order of a few centimeters, whereas that of $F_2$ is of the order of several meters. Given these lifetime ranges, both $F_{1}$ and $F_2$ can be looked for in searches for long-lived particles at the LHC. In particular, as shown in \cite{Belanger:2018sti, Brooijmans:2020yij}, searches for displaced leptons accompanied by missing energy can target the production and decay of $F_1$ and, for the values of $\lambda_{\alpha 1}$ that we assume, already exclude $F_1$ masses up to $\sim 400-500$ GeV. $F_2$, on the other hand, survives long enough to escape the detector and can be probed by searches for heavy stable charged particles, with masses smaller than $\sim 500-600$ GeV being already excluded. The corresponding searches at the High-Luminosity Run of the LHC will probe $F_{1,2}$ masses reaching up to $\sim 800$ GeV and $\sim 1.5$ TeV, respectively \cite{Belanger:2018sti}.
In summary, if our model is to simultaneously explain the dark matter abundance and the baryon asymmetry of the Universe while remaining consistent with Lyman-$\alpha$ bounds, then it predicts interesting signatures for LLP searches at the LHC. In conjunction with the fact that, as we showed, explaining the matter-antimatter asymmetry of the Universe tends to favor $F_{1,2}$ masses lying in the range of a few TeV, we can hope that a substantial part of the cosmologically favored part of the parameter space will be probed by the LHC within the next few years.
\section{Conclusions and Outlook}\label{sec:conclusions}
In this paper, we discussed a mechanism for the simultaneous generation of the dark matter density and the baryon asymmetry of the Universe. To this goal, we relied on the out-of-equilibrium decays of heavy bath particles into a (feebly coupled) dark matter state along with Standard Model charged fermions, which leads to dark matter production via the freeze-in mechanism. If, moreover, $CP$ is violated by these same decay processes, a viable matter-antimatter asymmetry can also be generated either directly (if the decays also lead to baryon/lepton number violation) or through the interplay of the generated $CP$ asymmetry with the SM electroweak sphalerons.
As a proof-of-concept, we employed a simple model in which the role of the heavy bath particles was played by two $SU(3)_c\times SU(2)_{\LH}$-singlet vectorlike fermions with a non-zero hypercharge, and dark matter was identified with a gauge singlet real scalar field. We showed that, indeed, such a simple construction can lead to both dark matter production and successful baryogenesis and we briefly discussed the phenomenological prospects of such a construction, particularly in relation to searches for long-lived particles at the LHC.
Our proposal draws inspiration from the idea presented in \cite{Shuve_2020}: First, the dark matter content and the baryon asymmetry of the Universe are generated simultaneously through the freeze-in mechanism. Second, at least within the framework of our concrete incarnation of this proposal, in both cases the electroweak sphalerons are used in order to convert a $CP$ asymmetry into a baryonic one. In our case, however, instead of considering oscillations -- as in ARS leptogenesis -- we rather relied on decays in order to generate this initial $CP$ asymmetry -- a process which is somewhat reminiscent of GUT baryogenesis. Moreover, we presented a general thermodynamical treatment that can also be applied in cases in which baryon/lepton number is violated already at the decay level.
There are several topics which we did not expand upon in this paper, and which would merit much more detailed investigation. First, more elaborate models can certainly be developed in this context of freeze-in baryogenesis, potentially establishing connections with theoretically well-motivated UV completions of the Standard Model. As an example, we could mention the fact that we assumed rather \textit{ad hoc} values for the couplings of the heavy bath particles to dark matter and to the SM fermions. More realistic flavor structures can be envisaged in quite a straightforward manner, whereas nothing forbids dark matter itself to be asymmetric, along the lines described in \cite{Hall:2010jx}. Second, albeit related to the previous point, our discussion of more phenomenological aspects has been fairly elementary. Concrete, well-motivated constructions can have important implications for flavor physics (\textit{e.g.} flavor-violating decays of or contributions to electric/magnetic moments of SM fermions), collider physics (\textit{e.g.} new resonances and/or final states in LHC searches, potentially involving long-lived particles \cite{Alimena:2019zri}), or cosmological measurements (\textit{e.g.} direct detection experiments or cosmic microwave background observations). Such considerations, which are typically fairly model dependent, are left for future work.
\acknowledgments
The authors acknowledge fruitful exchanges with Genevi\`eve B\'elanger, Marcos A. G. Garcia, Kalliopi Petraki and Alexander Pukhov. We are particularly grateful toward Sacha Davidson for enlightening discussions and comments on an earlier version of this manuscript and toward Dimitrios Karamitros for useful comments on our findings. This research is co-financed by Greece and the European Union (European Social Fund - ESF) through the Operational Programme \textquote{Human Resources Development, Education and Lifelong Learning} in the context of the project \textquote{Strengthening Human Resources Research Potential via Doctorate Research - 2nd Cycle} (MIS-5000432), implemented by the State Scholarships Foundation (IKY). This research work was supported by the Hellenic Foundation for Research and Innovation (H.F.R.I.) under the ``First Call for H.F.R.I. Research Projects to support Faculty members and Researchers and the procurement of high-cost research equipment grant'' (Project Number: 824).
\begin{appendix}
\section{Spectator processes}\label{app:Spectator Processes}
During the baryon and/or lepton number violation era, there are various processes which can modify the number densities of the particles species. They are called "spectator processes" because they affect the baryon/lepton number indirectly by distributing the asymmetry among all SM fermions and Higgs fields. These processes include the gauge interactions, the Yukawa interactions and the electroweak and QCD non-perturbative sphaleron transitions. If these processes are in thermal equilibrium with the cosmic plasma, then their effect is to impose certain relations among the chemical potentials of the various particle species. Recall that the asymmetry in the particle and antiparticle equilibrium number densities of species $k$, denoted as $n_{\Delta k}\equiv n_{k}-n_{\bar{k}}$, is given in the ultra-relativistic limit $m_{k}\ll T$ and $\mu_{k}\ll T$ by
\begin{align}\label{eq:number_density_asymmetry}
n_{\Delta k}\,\equiv\,n_{k}-n_{\bar{k}}\,=\,\left\{\,
\begin{array}{@{}ll@{}}
g_{k}\mu_{k}\frac{T^2}{3},\qquad &k:\text{boson}
\\\\
g_{k}\mu_{k}\frac{T^2}{6},\qquad &k:\text{fermion}
\end{array}
\right.
\end{align}
\\
\noindent
where $\mu_{k}$ is the chemical potential and $g_{k}$ are the independent degrees of freedom of species $k$. The total baryon $n_B$ and lepton $n_L$ asymmetries can be written as
\begin{equation}
n_B\,=\,\sum_{k}\,B_{k}\,n_{\Delta k},\qquad n_L\,=\,\sum_{k}\,L_{k}\,n_{\Delta k}
\end{equation}
\noindent
where $B_{k}$ and $\,L_{k}$ are the baryon and lepton numbers of species $k$, respectively.
In our case, the lepton asymmetry is mainly generated during $T_{EW}\leq T\ll 10^{8}\,\GeV$ when all spectator processes are active, including the sphaleron processes. These violate the $n_{B+L}\equiv n_B+n_L$ asymmetry but conserve the orthogonal combination $n_{B-L}\equiv n_B-n_L$. A rough estimation is that sphalerons will erase the $n_{B+L}$ asymmetry, resulting in
\begin{subequations}
\begin{alignat}{2}
n_B\,=\,\frac{n_{B+L}}{2}\,+\,\frac{n_{B-L}}{2}\,&\overset{\text{sphalerons}}{\longrightarrow}\,n_B\,=\,\frac{n_{B-L}}{2}
\\
n_L\,=\,\frac{n_{B+L}}{2}\,-\,\frac{n_{B-L}}{2}\,&\overset{\text{sphalerons}}{\longrightarrow}\,n_L\,=\,-n_B\,=\,-\frac{n_{B-L}}{2}
\end{alignat}
\end{subequations}
\\
\noindent
A more careful treatment shows that, in spite of the fact that sphalerons violate $n_{B+L}$, thermodynamic equilibrium requires a non-zero value, \textit{i.e.} $n_{B+L}\neq 0$. To quantify the asymmetry distribution, one can always relate the $n_{B-L}$ number density asymmetry with $n_{B}$ and $n_{L}$ as follows:
\begin{equation}
n_{B-L}\,\equiv\,n_{B}\,-\,n_{L}\,=\,c\,n_{B-L}\,-\,\left(c-1\right)\,n_{B-L}\,\Rightarrow\,\left\{\,\begin{array}{@{}ll@{}}
n_{B}\,=\,c\,n_{B-L}\\\\
n_{L}\,=\,\left(c-1\right)\,n_{B-L}
\end{array}
\right.
\end{equation}
\noindent
where $c\lesssim 1$ is the constant of proportionality. To evaluate it, we assign chemical potentials to the particle spectrum of the model. This consists of all the SM species with three families and one Higgs doublet, together with the two BSM vectorlike fermions $F_i$.
\subsubsection{SM}
Let us first consider only the SM spectrum and assign the following non-vanishing chemical potentials \cite{Harvey:1990qw,Buchmuller:2005eh}
\begin{subequations}\label{eq:SM_chemical_potentials}
\begin{alignat}{5}
\mu_{q_{\alpha}}\qquad&\left(\text{LH quark doublets}\right)
\\
\mu_{u_{\alpha}}\qquad&\left(\text{RH up-quark singlets}\right)
\\
\mu_{d_{\alpha}}\qquad&\left(\text{RH down-quark singlets}\right)
\\
\mu_{{\ell}_{\alpha}}\qquad&\left(\text{LH lepton doublets}\right)
\\
\mu_{e_{\alpha}}\qquad&\left(\text{RH charged lepton singlets}\right)
\\
\mu_H\qquad&\left(\text{Higgs doublet,}\,H=\left(h^+,h^0\right)^T\right)
\end{alignat}
\end{subequations}
\noindent
where $\alpha=\{1,2,3\}$ is a SM flavor index.\footnote{Note that in this Appendix we are introducing a slightly different notation with respect to the rest of the paper, denoting the SM right-handed leptons by $\{e_1,e_2,e_3\}$ instead of $\{e_e,e_{\mu},e_{\tau}\}$.} The corresponding antiparticle states have opposite chemical potentials, \textit{e.g.} $\mu_{\tilde{H}}=-\mu_{H}$, where $\tilde{H}=\left(h^{0*},-h^{-}\right)^T$. Note that the electrically chargeless gauge bosons have vanishing chemical potentials, as they carry no conserved quantum number, while the chemical potential of the $W^{\pm}$ gauge bosons vanishes, because the third - component of the weak isospin is zero at $T>T_{EW}$ \cite{Harvey:1990qw}. Hence, the number density asymmetries of the various components can be written, according to Eq.~\eqref{eq:number_density_asymmetry}, as
\begin{subequations}\label{eq:SM_asymmetric_densities}
\begin{alignat}{6}
n_{\Delta q_{\alpha}}\,&\equiv\,n_{q_{\alpha}}\,-\,n_{\bar{q}_{\alpha}}\,=\,6\,\mu_{q_{\alpha}}\,\frac{T^2}{6}
\\
n_{\Delta u_{\alpha}}\,&\equiv\,n_{u_{\alpha}}\,-\,n_{\bar{u}_{\alpha}}\,=\,3\,\mu_{u_{\alpha}}\,\frac{T^2}{6}
\\
n_{\Delta d_{\alpha}}\,&\equiv\,n_{d_{\alpha}}\,-\,n_{\bar{d}_{\alpha}}\,=\,3\,\mu_{d_{\alpha}}\,\frac{T^2}{6}
\\
n_{\Delta {\ell}_{\alpha}}\,&\equiv\,n_{{\ell}_{\alpha}}\,-\,n_{\bar{\ell}_{\alpha}}\,=\,2\,\mu_{{\ell}_{\alpha}}\,\frac{T^2}{6}
\\
n_{\Delta e_{\alpha}}\,&\equiv\,n_{e_{\alpha}}\,-\,n_{\bar{e}_{\alpha}}\,=\,\mu_{e_{\alpha}}\,\frac{T^2}{6}
\\
n_{\Delta H}\,&\equiv\,n_H\,-\,n_{\tilde{H}}\,=\,2\,\mu_H\,\frac{T^2}{3}\,=\,4\,\mu_H\,\frac{T^2}{6}
\end{alignat}
\end{subequations}
\noindent
The baryon and lepton number density asymmetries for each flavor $\left(n_{B_{\alpha}},\,n_{L_{\alpha}}\right)$, as well as the corresponding total ones $\left(n_B,\,n_L\right)$, can be written in terms of the chemical potentials in Eq.~\eqref{eq:SM_chemical_potentials} as follows:
\begin{subequations}
\begin{alignat}{4}
n_{B_{\alpha}}\,&\equiv\,\sum_{k}\,B_{k_{\alpha}}\,n_{\Delta k_{\alpha}}\,=\,\frac{1}{3}\left(n_{\Delta q_{\alpha}}\,+\,n_{\Delta u_{\alpha}}\,+\,n_{\Delta d_{\alpha}}\right)\,=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_B\,&\equiv\,\sum_{\alpha=1}^3\,n_{B_{\alpha}}\,=\,\sum_{\alpha=1}^3\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_{L_{\alpha}}\,&\equiv\,\sum_{k}\,L_{k_{\alpha}}\,n_{\Delta k_{\alpha}}\,=\,n_{\Delta\ell_{\alpha}}\,+\,n_{\Delta e_{\alpha}}\,=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_{L}\,&\equiv\,\sum_{\alpha=1}^3\,n_{L_{\alpha}}\,=\,\sum_{\alpha=1}^3\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}
\end{alignat}
\end{subequations}
\noindent
These 16 SM chemical potentials are not totally independent; they are related through the processes that attain chemical equilibrium during the era of the asymmetry generation. At $T\ll 10^8$ these are
\begin{itemize}
\item $SU(2)_L$ sphaleron transitions which induce an effective 12-fermion effective operator $\mathcal{O}_{(B-L)_{\text{SM}}}\,=\,\prod_{i=1}^{N_f}\left(q_iq_iq_i\ell_i\right)$, which implies
\begin{equation}\label{eq:SU(2)_sphaleron_constraint}
\sum_{\alpha=1}^3\left(3\mu_{q_{\alpha}}\,+\,\mu_{\ell_{\alpha}}\right)\,=\,0
\end{equation}
\item $SU(3)_c$ sphaleron transitions which give rise to
\begin{equation}\label{eq:SU(3)_sphaleron_constraint}
\sum_{\alpha=1}^3\left(2\mu_{q_{\alpha}}\,-\,\mu_{u_{\alpha}}\,-\,\mu_{d_{\alpha}}\right)\,=\,0
\end{equation}
\item Yukawa interactions which imply
\begin{subequations}\label{eq:Yukawa_constraints}
\begin{alignat}{3}
\mu_{q_{\alpha}}\,-\,\mu_H\,-\,\mu_{d_{\alpha}}\,=\,0,\qquad&\left(\,q_{\alpha}\,\tilde{H}\,\bar{d}_{\alpha}\,+\,\text{h.c.}\,\right)\label{eq:Yukawa_1}
\\
\mu_{q_{\alpha}}\,+\,\mu_H\,-\,\mu_{u_{\alpha}}\,=\,0,\qquad&\left(\,q_{\alpha}\,H\,\bar{u}_{\alpha}\,+\,\text{h.c.}\,\right)\label{eq:Yukawa_2}
\\
\mu_{{\ell}_{\alpha}}\,-\,\mu_H\,-\,\mu_{e_{\alpha}}\,=\,0,\qquad&\left(\,{\ell}_{\alpha}\,\tilde{H}\,\bar{e}_{\alpha}\,+\,\text{h.c.}\,\right)\label{eq:Yukawa_3}
\end{alignat}
\end{subequations}
\end{itemize}
Out of the 11 constraints of Eqs.~\eqref{eq:SU(2)_sphaleron_constraint} - \eqref{eq:Yukawa_constraints}, only ten of them are linearly independent, as the QCD sphaelron-induced relation \eqref{eq:SU(3)_sphaleron_constraint} can be obtained by adding Eqs.~\eqref{eq:Yukawa_1} and \eqref{eq:Yukawa_2}. An additional independent constraint can be obtained from the hypercharge $\left(\Upsilon=Q-t_3\right)$ neutrality condition, which implies
\begin{itemize}
\item Hypercharge constraint
\begin{align}\label{eq:hypercharge_constraint}
n_{\Upsilon}\,&\equiv\,\sum_{k}\,\Upsilon_{k}\,n_{\Delta k}\,=\,0\nonumber
\\
&\Rightarrow\,\sum_{\alpha=1}^3\Big(\frac{1}{6}n_{\Delta q_{\alpha}}\,+\,\frac{2}{3}n_{\Delta u_{\alpha}}\,-\,\frac{1}{3}n_{\Delta d_{\alpha}}\,-\,\frac{1}{2}\,n_{\Delta\ell_{\alpha}}\,-\,n_{\Delta e_{\alpha}}\Big)\,+\,\frac{1}{2}n_{\Delta H}\,=\,0\nonumber
\\
&\Rightarrow\,\sum_{\alpha=1}^3\left(\mu_{q_{\alpha}}\,+\,2\mu_{u_{\alpha}}\,-\,\mu_{d_{\alpha}}\,-\,\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,+\,2\mu_H\,=\,0
\end{align}
\end{itemize}
\noindent
where $\Upsilon_{k}$ is the hypercharge of species $k$. Two more constraints are imposed by the equality of the baryon flavor asymmetries \cite{Davidson:2008bu}.
\begin{itemize}
\item Baryon flavor asymmetry equality
\begin{align}\label{eq:baryon_flavor_constraints}
& n_{B_1}\,=\,n_{B_2}\,=\,n_{B_3}\nonumber
\\\nonumber
\\
\Rightarrow\,& 2\,\mu_{q_3}\,+\,\mu_{u_3}\,+\,\mu_{d_3}\,=\,2\,\mu_{q_2}\,+\,\mu_{u_2}\,+\,\mu_{d_2}\,=\,2\,\mu_{q_1}\,+\,\mu_{u_1}\,+\,\mu_{d_1}
\end{align}
\end{itemize}
\noindent
Hence, there are in total 13 independent constraints (Eqs.~\eqref{eq:SU(2)_sphaleron_constraint}, \eqref{eq:Yukawa_constraints}, \eqref{eq:hypercharge_constraint} and \eqref{eq:baryon_flavor_constraints}) and, therefore, one can express the 16 SM chemical potentials \eqref{eq:SM_chemical_potentials} in terms of three chemical potentials, that we chose to be $\{\mu_{\ell_{\alpha}}\}$. Solving the system of equations, one obtains
\begin{subequations}\label{eq:SM_chemical_potentials_constraints}
\begin{alignat}{5}
\mu_{q_{\alpha}}\,&=\,-\frac{1}{9}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)
\\
\mu_{u_{\alpha}}\,&=\,\frac{5}{63}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)
\\
\mu_{d_{\alpha}}\,&=\,-\frac{19}{63}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)
\\
\mu_{e_1}\,&=\,\frac{1}{21}\left(17\mu_{\ell_1}\,-\,4\mu_{\ell_2}\,-\,4\mu_{\ell_3}\right)
\\\mu_{e_2}\,&=\,\frac{1}{21}\left(-4\mu_{\ell_1}\,+\,17\mu_{\ell_2}\,-\,4\mu_{\ell_3}\right)
\\
\mu_{e_3}\,&=\,\frac{1}{21}\left(-4\mu_{\ell_1}\,-\,4\mu_{\ell_2}\,+\,17\mu_{\ell_3}\right)
\\
\mu_{H}\,&=\,\frac{4}{21}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)
\end{alignat}
\end{subequations}
\\
\noindent
Now the baryon and lepton number density asymmetries can be written in terms of the chemical potentials $\mu_{\ell_{\alpha}}$ as
\begin{subequations}
\begin{alignat}{4}
n_{B_{\alpha}}\,&=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^2}{6}\,=\,-\frac{4}{9}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)\,\frac{T^2}{6}
\\
n_B\,&=\,3\,n_{B_{\alpha}}\,=\,-\frac{4}{3}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)\,\frac{T^2}{6}
\\
n_{L_{\alpha}}\,&=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_{L}\,&=\,\sum_{\alpha=1}^3\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}\,=\,\frac{17}{7}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)\,\frac{T^2}{6}
\end{alignat}
\end{subequations}
\noindent
The flavor asymmetry combination $n_{\Delta_{\alpha}}\equiv n_{B_{\alpha}}-n_{L_{\alpha}}=n_{B}/3-n_{L_{\alpha}}$ can be written as
\begin{flalign*}
n_{\Delta_{\alpha}}\,&=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6} &
\\\nonumber
\\
& \Rightarrow\,
\begin{pmatrix}
n_{\Delta_1} \\
n_{\Delta_2} \\
n_{\Delta_3}
\end{pmatrix}
\,=\,
-\frac{1}{63}\,
\left[\begin{array}{ccc}
205 & 16 & 16 \\
16 & 205 & 16 \\
16 & 16 & 205
\end{array}\right]\,
\begin{pmatrix}
\mu_{\ell_1} \\
\mu_{\ell_2} \\
\mu_{\ell_3}
\end{pmatrix}\,\frac{T^2}{6} &
\end{flalign*}
\\
\noindent
Inverting the expression above, one finds
\begin{equation}
\begin{pmatrix}
\mu_{\ell_1} \\
\mu_{\ell_2} \\
\mu_{\ell_3}
\end{pmatrix}
\,=\,
-\frac{1}{711}\,
\left[\begin{array}{ccc}
221 & -16 & -16 \\
-16 & 221 & -16 \\
-16 & -16 & 221
\end{array}\right]\,
\begin{pmatrix}
n_{\Delta_1} \\
n_{\Delta_2} \\
n_{\Delta_3}
\end{pmatrix}\,\frac{6}{T^2}
\end{equation}
\\
\noindent
Using Eq.~\eqref{eq:SM_asymmetric_densities}, we can express the chemical potentials $\mu_{\ell_{\alpha}}$ in terms of the flavor asymmetries $n_{\Delta\ell_{\alpha}}$ and relate them to $n_{\Delta\alpha}$ as \footnote{Note that our result agrees with reference \cite{Nardi_2006}, while the authors of reference \cite{Gonzalez:2009} define $n_{\Delta\ell_{\alpha}}$ per gauge degree of freedom and, therefore, their result is smaller by a factor of 2.}
\begin{equation}
\begin{pmatrix}
n_{\Delta\ell_1} \\
n_{\Delta\ell_2} \\
n_{\Delta\ell_3}
\end{pmatrix}
\,=\,
-\frac{2}{711}\,
\left[\begin{array}{ccc}
221 & -16 & -16 \\
-16 & 221 & -16 \\
-16 & -16 & 221
\end{array}\right]\,
\begin{pmatrix}
n_{\Delta_1} \\
n_{\Delta_2} \\
n_{\Delta_3}
\end{pmatrix}
\end{equation}
\\
\noindent
The total $B-L$ asymmetry is
\begin{equation}
n_{B-L}\,\equiv\,n_B\,-\,n_L\,=\,-\frac{79}{21}\left(\mu_{\ell_1}\,+\,\mu_{\ell_2}\,+\,\mu_{\ell_3}\right)\,\frac{T^2}{6}
\end{equation}
\noindent
Hence, in the SM and for $T_{sph}\ll T\ll 10^{8}\,\GeV$, the total baryon $n_B$ and the $n_L$ asymmetries are related to $n_{B-L}$ by
\begin{align}
n_B\,=\,\frac{28}{79}\,n_{B-L}\,=\,\frac{28}{79}\,\sum_{\alpha=1}^3n_{\Delta\alpha}
\\
n_L\,=\,-\frac{51}{79}\,n_{B-L}\,=\,-\frac{51}{79}\,\sum_{\alpha=1}^3n_{\Delta\alpha}
\end{align}
\\
\subsubsection{Our model}
We extend the SM to include also two heavy vectorlike leptons $F_1$ and $F_2$ with chemical potentials $\mu_{F_1}$ and $\mu_{F_2}$, respectively, and asymmetry
\begin{equation}
n_{\Delta F_i}\,=\,\mu_{F_i}\,\frac{T^2}{6}
\end{equation}
\noindent
Since the BSM fermions are gauged under $U(1)_{\Upsilon}$, with $\Upsilon_{F_i}=\Upsilon_{e}=-1$, the hypercharge constraint \eqref{eq:hypercharge_constraint} is modified to
\begin{align}\label{eq:hypercharge_constraint_modified}
n_{\Upsilon}\,&\equiv\,\sum_{k}\,\Upsilon_{k}\,n_{\Delta k}\,=\,0\nonumber
\\
&\Rightarrow\,\sum_{\alpha=1}^3\Big(\frac{1}{6}n_{\Delta q_{\alpha}}\,+\,\frac{2}{3}n_{\Delta u_{\alpha}}\,-\,\frac{1}{3}n_{\Delta d_{\alpha}}\,-\,\frac{1}{2}\,n_{\Delta\ell_{\alpha}}\,-\,n_{\Delta e_{\alpha}}\Big)\,+\,\frac{1}{2}n_{\Delta H}\,-\,\sum_{i=1}^2\,n_{\Delta F_i}\,=\,0\nonumber
\\
&\Rightarrow\,\sum_{\alpha=1}^3\left(\mu_{q_{\alpha}}\,+\,2\mu_{u_{\alpha}}\,-\,\mu_{d_{\alpha}}\,-\,\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,+\,2\mu_H\,-\,\left(\mu_{F_1}\,+\,\mu_{F_2}\right)\,=\,0
\end{align}
\noindent
The theory conserves the total $n_{B-L}\equiv n_B-n_L= n_B-n_{L_{SM}}-n_{L_F}$ asymmetry, where
\begin{subequations}
\begin{alignat}{4}
n_{B_{\alpha}}\,&\equiv\,\sum_{k}\,B_{k_{\alpha}}\,n_{\Delta k_{\alpha}}\,=\,\frac{1}{3}\left(n_{\Delta q_{\alpha}}\,+\,n_{\Delta u_{\alpha}}\,+\,n_{\Delta d_{\alpha}}\right)\,=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_B\,&\equiv\,\sum_{\alpha=1}^3\,n_{B_{\alpha}}\,=\,\sum_{\alpha=1}^3\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^2}{6}\,=\,3\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_{L_{\alpha}}\,&\equiv\,\sum_{k:\,SM}\,L_{k_{\alpha}}\,n_{\Delta k_{\alpha}}\,=\,n_{\Delta\ell_{\alpha}}\,+\,n_{\Delta e_{\alpha}}\,=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_{L_{SM}}\,&\equiv\,\sum_{\alpha=1}^3\,n_{L_{\alpha}}\,=\,\sum_{\alpha=1}^3\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_{L_F}\,&\equiv\,\sum_{i=1}^2\,n_{\Delta F_i}\,=\,\sum_{i=1}^2\,\mu_{F_i}\,\frac{T^2}{6}
\\
n_L\,&\equiv\,n_{L_{SM}}\,+\,n_{L_F}\,=\,\left(\sum_{\alpha=1}^3\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,+\,\sum_{i=1}^2\,\mu_{F_i}\right)\,\frac{T^2}{6}
\end{alignat}
\end{subequations}
\noindent
If we also assume that the early Universe is totally symmetric, $n_{B-L_{SM}}|_0=0$ and $n_F|_0=0$, then we obtain an additional constraint to $\mu_{F_1}+\mu_{F_2}$, that is
\begin{align}\label{eq:B-L_conservation_constraint}
& n_B\,-\,n_{L_{SM}}\,-\,n_{L_F}\,=\,\sum_{\alpha=1}^3\,n_{\Delta\alpha}\,-\,\sum_{i=1}^2\,n_{\Delta F_i}\,=\,0\nonumber
\\
\Rightarrow &\sum_{\alpha=1}^3\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,-\,\left(\mu_{F_1}\,+\,\mu_{F_2}\right)\,=\,0
\end{align}
\noindent
Replacing this in Eq.~\eqref{eq:hypercharge_constraint_modified}, we obtain the modified hypercharge constraint expressed in terms of the SM chemical potentials.
\begin{itemize}
\item Modified hypercharge constraint
\begin{equation}\label{eq:hypercharge_constraint_modified_2}
\sum_{\alpha=1}^3\left(-\,\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,-\,2\mu_{d_{\alpha}}\,+\,\mu_{\ell_{\alpha}}\right)\,+\,2\mu_H\,=\,0
\end{equation}
\end{itemize}
\noindent
Now the 16 SM chemical potentials are constrained under the $SU(2)$ sphaleron \eqref{eq:SU(2)_sphaleron_constraint} and Yukawa \eqref{eq:Yukawa_constraints} processes (10 constraints), as well as under the baryon flavor asymmetry equality \eqref{eq:baryon_flavor_constraints} and the modified hypercharge \eqref{eq:hypercharge_constraint_modified_2} conditions (3 constraints). Solving the system of equations, we may express all chemical potentials in terms of $\mu_{e_{\alpha}}$
\begin{subequations}\label{eq:BSM_chemical_potentials_constraints}
\begin{alignat}{5}
\mu_{q_{\alpha}}\,&=\,-\frac{11}{144}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)
\\
\mu_{u_{\alpha}}\,&=\,-\frac{13}{72}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)
\\
\mu_{d_{\alpha}}\,&=\,\frac{1}{36}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)
\\
\mu_{\ell_1}\,&=\,\frac{1}{48}\left(43\mu_{e_1}-5\mu_{e_2}-5\mu_{e_3}\right)
\\\mu_{\ell_2}\,&=\,\frac{1}{48}\left(-5\mu_{e_1}+43\mu_{e_2}-5\mu_{e_3}\right)
\\
\mu_{\ell_3}\,&=\,\frac{1}{48}\left(-5\mu_{e_1}-5\mu_{e_2}+43\mu_{e_3}\right)
\\
\mu_{H}\,&=\,-\frac{5}{48}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)
\end{alignat}
\end{subequations}
\\
\noindent
Now the baryon and lepton number density asymmetries can be written in terms of the chemical potentials $\mu_{e_{\alpha}}$ as
\begin{subequations}
\begin{alignat}{4}
n_{B_{\alpha}}\,&=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\right)\,\frac{T^2}{6}\,=\,-\frac{11}{36}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)\,\frac{T^2}{6}
\\
n_B\,&=\,3\,n_{B_{\alpha}}\,=\,-\frac{11}{12}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)\,\frac{T^2}{6}
\\
n_{L_{\alpha}}\,&=\,\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}
\\
n_{L_{SM}}\,&=\,\sum_{\alpha=1}^3\left(2\mu_{\ell_{\alpha}}\,+\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}\,=\,\frac{19}{8}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)\,\frac{T^2}{6}
\\
n_{L_F}\,&=\,\left(\mu_{F_1}\,+\,\mu_{F_2}\right)\,\frac{T^2}{6}\,=\,\sum_{\alpha=1}^3\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6}\nonumber
\\
& =\,-\frac{79}{24}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)\,\frac{T^2}{6}\,=\,n_B\,-\,n_{L_{SM}}
\\
n_L\,&\equiv\,n_{L_{SM}}\,+\,n_{L_F}\,=\,-\frac{11}{12}\left(\mu_{e_1}\,+\,\mu_{e_2}\,+\,\mu_{e_3}\right)\,\frac{T^2}{6}\,=\,n_B
\end{alignat}
\end{subequations}
\\
\noindent
The flavor asymmetry combination $n_{\Delta_{\alpha}}\equiv n_{B_{\alpha}}-n_{L_{\alpha}}=n_{B}/3-n_{L_{\alpha}}$ can be written as
\begin{flalign*}
n_{\Delta_{\alpha}}\,&=\,\left(2\mu_{q_{\alpha}}\,+\,\mu_{u_{\alpha}}\,+\,\mu_{d_{\alpha}}\,-\,2\mu_{\ell_{\alpha}}\,-\,\mu_{e_{\alpha}}\right)\,\frac{T^2}{6} &
\\\nonumber
\\
& \Rightarrow\,
\begin{pmatrix}
n_{\Delta_1} \\
n_{\Delta_2} \\
n_{\Delta_3}
\end{pmatrix}
\,=\,
-\frac{1}{72}\,
\left[\begin{array}{ccc}
223 & 7 & 7 \\
7 & 223 & 7 \\
7 & 7 & 223
\end{array}\right]\,
\begin{pmatrix}
\mu_{e_1} \\
\mu_{e_2} \\
\mu_{e_3}
\end{pmatrix}\,\frac{T^2}{6} &
\end{flalign*}
\\
\noindent
Inverting the expression above, one finds
\begin{equation}
\begin{pmatrix}
\mu_{e_1} \\
\mu_{e_2} \\
\mu_{e_3}
\end{pmatrix}
\,=\,
-\frac{1}{711}\,
\left[\begin{array}{ccc}
230 & -7 & -7 \\
-7 & 230 & -7 \\
-7 & -7 & 230
\end{array}\right]\,
\begin{pmatrix}
n_{\Delta_1} \\
n_{\Delta_2} \\
n_{\Delta_3}
\end{pmatrix}\,\frac{6}{T^2}
\end{equation}
\\
\noindent
Using Eq.~\eqref{eq:SM_asymmetric_densities}, we can express the chemical potentials $\mu_{e_{\alpha}}$ in terms of the flavor asymmetries $n_{\Delta e_{\alpha}}$ and relate them to $n_{\Delta\alpha}$ as
\begin{equation}
\begin{pmatrix}
n_{\Delta e_1} \\
n_{\Delta e_2} \\
n_{\Delta e_3}
\end{pmatrix}
\,=\,
-\frac{1}{711}\,
\left[\begin{array}{ccc}
230 & -7 & -7 \\
-7 & 230 & -7 \\
-7 & -7 & 230
\end{array}\right]\,
\begin{pmatrix}
n_{\Delta_1} \\
n_{\Delta_2} \\
n_{\Delta_3}
\end{pmatrix}
\end{equation}
\\
\noindent
Hence, in this model and for $T_{sph}\ll T\ll 10^{8}\,\GeV$, the total baryon $n_B$ and the SM lepton $n_{L_{\text{SM}}}$ asymmetries are related to $n_{B-L_{\text{SM}}}$ by
\begin{align}\label{eq:baryon asymmetry FIBG}
n_B\,=\,\frac{22}{79}\,n_{B-L_{SM}}\,=\,\frac{22}{79}\,\sum_{\alpha=1}^3n_{\Delta\alpha}
\\
n_{L_{SM}}\,=\,-\frac{57}{79}\,n_{B-L_{SM}}\,=\,-\frac{57}{79}\,\sum_{\alpha=1}^3n_{\Delta\alpha}
\end{align}
\noindent
This result agrees with that obtained in reference \cite{Shuve_2020}.
\\
\end{appendix}
\bibliographystyle{JHEP}
|
1,116,691,498,442 | arxiv | \section{Introduction}
In this work we examine the effects of varying single particle energies in a shell model calculation using the GXFP1A interaction \cite{1}. We will focus on the yrast spectrum of $^{48}$Cr $J=0, 2, 4,..., 16$. We perform calculations with the GXFP1A interaction \cite{1}. There have been many publications using these interactions in the PF shell with the shell model program ANTOINE \cite{3}, notably the works of E. Caurier et al. \cite{4}\cite{5} and more recently of V. Kumar et al. \cite{6}. Other works related to $^{48}$Cr include those of K. Hara et al. \cite{7}, F.Brandolini et al. \cite{8}, E.Caurier et al. \cite{9}, Z.C.Gao et al. \cite{10} and R.A. Herrera et al. \cite{11}.
\clearpage
\restoregeometry
\section{Arrangement of Tables and Figures}
The original single particle energies are given in the second column of Table \ref{t1}. In the third column we show the corresponding values for the FPD6 interaction \cite{2}. The orbits are 0$f_{7/2} , 1p_{3/2}, 0f_{5/2}$ and $1p_{1/2}$. We will then alter the single particle energies as indicated in Tables \ref{t2}, \ref{t3}, \ref{t4}, and \ref{t5}. We first perform calculations with what we call CASE-1.
\begin{table}[H]
\centering
\caption{Single particle energies (MeV) of GXPF1A and FPD6.}
\vspace{0.2cm}
\begin{tabular}{|c|c|c|}
\hline
Orbit & GXPF1A & FPD6\tabularnewline
\hline
\hline
$0f_{7/2}$ & 0 (-8.6240) & 0 (-8.3876)\tabularnewline
\hline
$1p_{3/2}$ & 2.9447 & 1.8942\tabularnewline
\hline
$0f_{5/2}$ & 7.2411 & 6.4910\tabularnewline
\hline
$1p_{1/2}$ & 4.4870 & 3.9093\tabularnewline
\hline
\end{tabular}
\label{t1}
\end{table}
In Table \ref{t2} we keep the splitting of the pair A ($p_{3/2} - f_{7/2}$) constant and likewise pair B ($p_{1/2} - f_{5/2}$). We then shift the pair B energy by a positive amount $\Delta$ relative to the first pair. We then study the dependence of the energies of the even $J$ states in $^{48}$Cr. We further study the effects on B(E2)'s $J$ to $J-2$. In Table \ref{t3} we extend the calculation to negative $\Delta$.
Next we perform calculations which what we call CASE-2. In Tables \ref{t4} and \ref{t5} do a similar analysis as was done in CASE-1, but now with pair C ($f_{5/2} -f_{7/2}$) constant and we pair D ($p_{1/2}-p_{3/2}$) shifted by an amount $\Delta$. In the latter case the spin orbit splittings are held constant. Again we study the effects on energies and B(E2)s.
\begin{sidewaystable}[H]
\caption{Energy spectra of $^{48}$Cr using GXPF1A interaction. Here we have keep the single-particle energies of $0f_{7/2}$
and $1p_{3/2}$ as the original one, and changed the single-particle
energies of $0f_{5/2}$ and $1p_{1/2}$ moved up by original plus $\Delta$.}
\label{t2}
\resizebox{\textwidth}{6cm}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Energy & GXPF1A & $\Delta$ = 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 20 & 40 & 60 & 80 & 100 \tabularnewline
\hline
$0^{+}$ & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \tabularnewline
\hline
$2^{+}$ & 0.788 & 0.767 & 0.750 & 0.737 & 0.727 & 0.718 & 0.711 & 0.705 & 0.700 & 0.696 & 0.692 & 0.669 & 0.654 & 0.646 & 0.644 & 0.642 \tabularnewline
\hline
$4^{+}$ & 1.717 & 1.687 & 1.664 & 1.646 & 1.632 & 1.620 & 1.610 & 1.601 & 1.594 & 1.588 & 1.582 & 1.547 & 1.519 & 1.508 & 1.501 & 1.498 \tabularnewline
\hline
$6^{+}$ & 3.229 & 3.152 & 3.090 & 3.039 & 2.998 & 2.963 & 2.934 & 2.908 & 2.886 & 2.867 & 2.849 & 2.747 & 2.671 & 2.639 & 2.622 & 2.610 \tabularnewline
\hline
$8^{+}$ & 4.753 & 4.649 & 4.553 & 4.478 & 4.415 & 4.361 & 4.314 & 4.274 & 4.239 & 4.208 & 4.179 & 4.001 & 3.881 & 3.827 & 3.796 & 3.778 \tabularnewline
\hline
$10^{+}$ & 6.429 & 6.238 & 6.080 & 5.952 & 5.846 & 5.758 & 5.684 & 5.621 & 5.565 & 5.517 & 5.474 & 5.219 & 5.033 & 4.957 & 4.915 & 4.889 \tabularnewline
\hline
$12^{+}$ & 7.722 & 7.479 & 7.296 & 7.153 & 7.037 & 6.941 & 6.860 & 6.791 & 6.731 & 6.679 & 6.632 & 6.356 & 6.153 & 6.070 & 6.024 & 5.996 \tabularnewline
\hline
$14^{+}$ & 9.701 & 9.432 & 9.227 & 9.063 & 8.929 & 8.818 & 8.724 & 8.693 & 8.572 & 8.511 & 8.456 & 8.129 & 7.887 & 7.788 & 7.733 & 7.699 \tabularnewline
\hline
$16^{+}$ & 12.805 & 12.411 & 12.115 & 11.845 & 11.699 & 11.546 & 11.417 & 11.308 & 11.213 & 11.130 & 11.057 & 10.623 & 10.305 & 10.156 & 10.105 & 10.061 \tabularnewline
\hline
\hline
B(E2: J $\rightarrow$ J-2 ) ($e^{2}fm^{4}$) & & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & 249 & 237 & 228 & 222 & 217 & 212 & 209 & 206 & 204 & 202 & 200 & 191 & 184 & 180 & 179 & 178 \tabularnewline
\hline
$4^{+}$ & 336 & 321 & 310 & 302 & 296 & 291 & 287 & 283 & 280 & 278 & 276 & 262 & 252 & 248 & 245 & 244 \tabularnewline
\hline
$6^{+}$ & 336 & 317 & 304 & 296 & 290 & 285 & 280 & 277 & 274 & 271 & 270 & 256 & 247 & 244 & 241 & 240 \tabularnewline
\hline
$8^{+}$ & 306 & 288 & 276 & 268 & 262 & 257 & 253 & 250 & 248 & 245 & 244 & 232 & 223 & 222 & 218 & 217 \tabularnewline
\hline
$10^{+}$ & 212 & 195 & 192 & 186 & 186 & 185 & 185 & 184 & 184 & 184 & 183 & 181 & 180 & 179 & 179 & 178 \tabularnewline
\hline
$12^{+}$ & 162 & 163 & 162 & 162 & 162 & 161 & 161 & 160 & 160 & 160 & 159 & 157 & 156 & 155 & 154 & 154 \tabularnewline
\hline
$14^{+}$ & 126 & 125 & 124 & 123 & 123 & 123 & 123 & 122 & 122 & 121 & 121 & 120 & 118 & 118 & 118 & 117 \tabularnewline
\hline
$16^{+}$ & 62 & 65 & 66 & 67 & 68 & 68 & 68 & 68 & 68 & 68 & 68 & 68 & 68 & 67 & 67 & 67 \tabularnewline
\hline
\hline
Q$(eb)$ & & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & -0.30 & -0.29 & -0.29 & -0.28 & -0.28 & -0.27 & -0.27 & -0.27 & -0.27 & -0.27 & -0.26 & -0.26 & -0.25 & -0.24 & -0.25 & -0.25 \tabularnewline
\hline
$4^{+}$ & -0.40 & -0.39 & -0.38 & -0.38 & -0.36 & -0.36 & -0.36 & -0.35 & -0.35 & -0.34 & -0.34 & -0.33 & -0.31 & -0.31 & -0.30 & -0.29 \tabularnewline
\hline
$6^{+}$ & -0.40 & -0.38 & -0.37 & -0.36 & -0.35 & -0.34 & -0.33 & -0.33 & -0.32 & -0.32 & -0.32 & -0.29 & -0.28 & -0.28 & -0.27 & -0.27 \tabularnewline
\hline
$8^{+}$ & -0.41 & -0.38 & -0.36 & -0.34 & -0.33 & -0.32 & -0.31 & -0.30 & -0.29 & -0.28 & -0.28 & -0.26 & -0.22 & -0.24 & -0.22 & -0.22 \tabularnewline
\hline
$10^{+}$ & -0.21 & -0.17 & -0.16 & -0.15 & -0.13 & -0.13 & -0.12 & -0.12 & -0.12 & -0.11 & -0.11 & -0.10 & -0.09 & -0.09 & -0.09 & -0.09 \tabularnewline
\hline
$12^{+}$ & -0.03 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 \tabularnewline
\hline
$14^{+}$ & -0.05 & -0.04 & -0.03 & -0.03 & -0.03 & -0.03 & -0.03 & -0.03 & -0.02 & -0.02 & -0.02 & -0.02 & -0.01 & -0.01 & -0.01 & -0.01 \tabularnewline
\hline
$16^{+}$ & -0.09 & -0.08 & -0.07 & -0.07 & -0.06 & -0.06 & -0.06 & -0.06 & -0.05 & -0.05 & -0.05 & -0.04 & -0.04 & -0.04 & -0.04 & -0.04 \tabularnewline
\hline
\end{tabular} }
\end{sidewaystable}
\begin{sidewaystable}[H]
\caption{Energy spectra of $^{48}$Cr using GXPF1A interaction. Here
we have keep the single-particle energies of $0f_{7/2}$
and $1p_{3/2}$ as the original one, and changed the single-particle
energies of $0f_{5/2}$ and $1p_{1/2}$ moved up by original minus $\Delta$.}
\label{t3}
\resizebox{\textwidth}{6cm}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Energy & $\Delta$ = -1 & -2 & -3 & -4 & -5 & -6 & -7 & -8 & -9 & -10 & -20 & -40 & -60 & -80 & -100 \tabularnewline
\hline
$0^{+}$ & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \tabularnewline
\hline
$2^{+}$ & 0.816 & 0.845 & 0.865 & 0.872 & 0.878 & 0.882 & 0.876 & 0.857 & 0.842 & 0.910 & 0.490 & 0.385 & 0.359 & 0.349 & 0.340 \tabularnewline
\hline
$4^{+}$ & 1.757 & 1.808 & 1.861 & 1.915 & 1.968 & 1.995 & 1.973 & 1.890 & 1.760 & 1.710 & 1.424 & 1.134 & 1.061 & 1.025 & 1.007 \tabularnewline
\hline
$6^{+}$ & 3.316 & 3.376 & 3.352 & 3.359 & 3.433 & 3.496 & 3.494 & 3.410 & 3.234 & 1.864 & 2.650 & 2.153 & 2.036 & 1.9165 & 1.931 \tabularnewline
\hline
$8^{+}$ & 4.877 & 4.970 & 4.987 & 5.056 & 5.177 & 5.261 & 5.260 & 5.147 & 4.626 & 4.058 & 2.966 & 2.433 & 2.297 & 2.237 & 2.202 \tabularnewline
\hline
$10^{+}$ & 6.621 & 6.663 & 6.633 & 6.760 & 6.965 & 7.102 & 7.114 & 6.998 & 6.439 & 5.770 & 7.675 & 6.865 & 6.651 & 6.552 & 6.495 \tabularnewline
\hline
$12^{+}$ & 8.060 & 8.539 & 9.099 & 9.397 & 9.469 & 9.449 & 9.373 & 9.131 & 8.195 & 7.487 & 15.588 & 35.059 & 54.929 & 74.871 & 94.838 \tabularnewline
\hline
$14^{+}$ & 10.060 & 10.547 & 11.133 & 11.585 & 11.648 & 11.589 & 11.512 & 11.397 & 10.959 & 10.434 & 26.249 & 65.601 & 105.436 & 145.363 & 185.319 \tabularnewline
\hline
$16^{+}$ & 13.361 & 14.034 & 14.169 & 13.720 & 13.424 & 13.263 & 13.134 & 12.965 & 12.778 & 12.854 & 28.411 & 67.637 & 107.439 & 147.349 & 187.297 \tabularnewline
\hline
\hline
B(E2: J $\rightarrow$ J-2 ) ($e^{2}fm^{4}$) & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & 268 & 295 & 326 & 348 & 359 & 362 & 360 & 356 & 343 & 306 & 141 & 128 & 124 & 123 & 122 \tabularnewline
\hline
$4^{+}$ & 362 & 400 & 449 & 484 & 499 & 501 & 492 & 457 & 388 & 242 & 182 & 165 & 161 & 158 & 156 \tabularnewline
\hline
$6^{+}$ & 369 & 421 & 448 & 466 & 491 & 511 & 516 & 501 & 429 & 172 & 151 & 137 & 114 & 135 & 130 \tabularnewline
\hline
$8^{+}$ & 344 & 418 & 487 & 523 & 552 & 568 & 571 & 540 & 205 & 199 & 8 & 8 & 21 & 6 & 7 \tabularnewline
\hline
$10^{+}$ & 249 & 317 & 382 & 428 & 476 & 511 & 517 & 400 & 223 & 191 & 0.12 & 0.06 & 0.06 & 0.05 & 0.05 \tabularnewline
\hline
$12^{+}$ & 155 & 141 & 236 & 360 & 412 & 440 & 446 & 15 & 198 & 173 & 0.008 & 0.009 & 0.009 & 0.009 & 0.009 \tabularnewline
\hline
$14^{+}$ & 131 & 147 & 212 & 324 & 390 & 403 & 389 & 0.45 & 6 & 2 & 0.028 & 0.02 & 0.02 & 0.02 & 0.02 \tabularnewline
\hline
$16^{+}$ & 56 & 24 & 98 & 114 & 209 & 245 & 255 & 246 & 75 & 46 & 0.23 & 0.73 & 0.96 & 0.94 & 1.093 \tabularnewline
\hline
\hline
Q$(eb)$ & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & -0.32 & -0.32 & -0.31 & -0.29 & -0.27 & -0.26 & -0.26 & -0.26 & -0.26 & -0.18 & +0.24 & +0.22 & +0.22 & +0.23 & +0.22 \tabularnewline
\hline
$4^{+}$ & -0.42 & -0.43 & -0.43 & -0.41 & -0.41 & -0.41 & -0.42 & -0.45 & -0.49 & -0.51 & +0.30 & +0.29 & +0.28 & +0.28 & +0.28 \tabularnewline
\hline
$6^{+}$ & -0.42 & -0.42 & -0.37 & -0.34 & -0.34 & -0.35 & -0.37 & -0.41 & -0.54 & -0.61 & +0.29 & +0.26 & +0.13 & +0.27 & +0.24 \tabularnewline
\hline
$8^{+}$ & -0.43 & -0.43 & -0.40 & -0.39 & -0.38 & -0.38 & -0.38 & -0.46 & -0.70 & -0.68 & -0.40 & -0.38 & -0.38 & -0.38 & -0.38 \tabularnewline
\hline
$10^{+}$ & -0.27 & -0.37 & -0.41 & -0.42 & -0.42 & -0.42 & -0.41 & -0.42 & -0.74 & -0.72 & -0.01 & -0.005 & -0.003 & -0.002 & -0.0019 \tabularnewline
\hline
$12^{+}$ & -0.05 & -0.10 & -0.30 & -0.42 & -0.42 & -0.41 & -0.40 & -0.75 & -0.74 & -0.72 & -0.48 & -0.46 & -0.45 & -0.45 & -0.45 \tabularnewline
\hline
$14^{+}$ & -0.08 & -0.16 & -0.33 & -0.44 & -0.43 & -0.42 & -0.41 & -0.40 & -0.49 & -0.48 & -0.42 & -0.39 & -0.39 & -0.39 & -0.39 \tabularnewline
\hline
$16^{+}$ & -0.01 & -0.32 & -0.36 & -0.43 & -0.44 & -0.44 & -0.43 & -0.43 & -0.44 & -0.49 & -0.51 & -0.48 & -0.48 & -0.47 & -0.47 \tabularnewline
\hline
\end{tabular}}
\end{sidewaystable}
\begin{sidewaystable}[H]
\caption{Energy spectra of $^{48}$Cr using GXPF1A interaction. Here
we have keep the single-particle energies of $0f_{7/2}$
and $0f_{5/2}$ as the original one, and changed the single-particle
energies of $1p_{3/2}$ and $1p_{1/2}$ moved up by original plus $\Delta$.}
\label{t4}
\resizebox{\textwidth}{6cm}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Energy & $\Delta$ = 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 20 & 40 & 60 & 80 & 100 \tabularnewline
\hline
$0^{+}$ & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \tabularnewline
\hline
$2^{+}$ & 0.918 & 1.008 & 1.068 & 1.107 & 1.133 & 1.152 & 1.165 & 1.175 & 1.182 & 1.187 & 1.205 & 1.207 & 1.206 & 1.206 & 1.205 \tabularnewline
\hline
$4^{+}$ & 1.803 & 1.864 & 1.907 & 1.937 & 1.960 & 1.976 & 1.989 & 1.999 & 2.007 & 2.014 & 2.041 & 2.051 & 2.053 & 2.054 & 2.054 \tabularnewline
\hline
$6^{+}$ & 3.293 & 3.312 & 3.311 & 3.303 & 3.293 & 3.283 & 3.274 & 3.265 & 3.257 & 3.250 & 3.205 & 3.171 & 3.157 & 3.150 & 3.145 \tabularnewline
\hline
$8^{+}$ & 4.729 & 4.699 & 4.671 & 4.647 & 4.625 & 4.607 & 4.591 & 4.578 & 4.565 & 4.555 & 4.491 & 4.444 & 4.425 & 4.416 & 4.409 \tabularnewline
\hline
$10^{+}$ & 6.198 & 6.040 & 5.940 & 5.864 & 5.807 & 5.762 & 5.725 & 5.696 & 5.670 & 5.649 & 6.352 & 5.455 & 5.425 & 5.409 & 5.399 \tabularnewline
\hline
$12^{+}$ & 7.396 & 7.108 & 6.948 & 6.833 & 6.747 & 6.680 & 6.626 & 6.582 & 6.544 & 6.513 & 6.346 & 6.241 & 6.201 & 6.179 & 6.167 \tabularnewline
\hline
$14^{+}$ & 9.244 & 8.954 & 8.757 & 8.614 & 8.507 & 8.423 & 8.355 & 8.300 & 8.253 & 8.213 & 8.002 & 7.869 & 7.819 & 7.792 & 7.776 \tabularnewline
\hline
$16^{+}$ & 12.280 & 11.939 & 11.702 & 11.529 & 11.396 & 11.292 & 11.207 & 11.137 & 11.078 & 11.027 & 10.754 & 10.578 & 10.511 & 10.475 & 10.453 \tabularnewline
\hline
\hline
B(E2: J $\rightarrow$ J-2 ) ($e^{2}fm^{4}$) & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & 213 & 189 & 173 & 161 & 153 & 147 & 142 & 138 & 135 & 133 & 121 & 115 & 112 & 111 & 111 \tabularnewline
\hline
$4^{+}$ & 276 & 234 & 205 & 186 & 171 & 161 & 153 & 148 & 143 & 139 & 122 & 114 & 110 & 109 & 109 \tabularnewline
\hline
$6^{+}$ & 268 & 224 & 196 & 178 & 166 & 156 & 150 & 145 & 140 & 137 & 122 & 115 & 111 & 111 & 111 \tabularnewline
\hline
$8^{+}$ & 255 & 223 & 203 & 190 & 180 & 173 & 168 & 164 & 160 & 157 & 143 & 135 & 132 & 131 & 131 \tabularnewline
\hline
$10^{+}$ & 184 & 170 & 161 & 155 & 150 & 147 & 143 & 140 & 138 & 135 & 127 & 121 & 119 & 118 & 117 \tabularnewline
\hline
$12^{+}$ & 153 & 145 & 139 & 134 & 131 & 127 & 125 & 123 & 121 & 119 & 112 & 107 & 105 & 103 & 103 \tabularnewline
\hline
$14^{+}$ & 120 & 114 & 110 & 107 & 104 & 102 & 100 & 99 & 98 & 97 & 90 & 87 & 85 & 85 & 84 \tabularnewline
\hline
$16^{+}$ & 60 & 59 & 57 & 56 & 55 & 55 & 54 & 54 & 53 & 53 & 50 & 48 & 48 & 47 & 47 \tabularnewline
\hline
\hline
Q$(eb)$ & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & -0.26 & -0.22 & -0.18 & -0.16 & -0.13 & -0.12 & -0.10 & -0.09 & -0.08 & -0.08 & -0.04 & -0.01 & -0.0091 & -0.0068 & -0.0047 \tabularnewline
\hline
$4^{+}$ & -0.36 & -0.34 & -0.32 & -0.30 & -0.28 & -0.27 & -0.27 & -0.26 & -0.25 & -0.25 & -0.22 & -0.19 & -0.19 & -0.18 & -0.18 \tabularnewline
\hline
$6^{+}$ & -0.32 & -0.26 & -0.22 & -0.19 & -0.17 & -0.15 & -0.14 & -0.13 & -0.12 & -0.12 & -0.08 & -0.06 & -0.05 & -0.05 & -0.05 \tabularnewline
\hline
$8^{+}$ & -0.33 & -0.27 & -0.23 & -0.20 & -0.18 & -0.17 & -0.15 & -0.15 & -0.13 & -0.13 & -0.09 & -0.08 & -0.07 & -0.070 & -0.06 \tabularnewline
\hline
$10^{+}$ & -0.15 & -0.12 & -0.10 & -0.09 & -0.08 & -0.08 & -0.07 & -0.07 & -0.06 & -0.06 & +0.01 & -0.04 & -0.04 & -0.041 & -0.04 \tabularnewline
\hline
$12^{+}$ & -0.02 & -0.02 & -0.02 & -0.02 & -0.01 & -0.01 & -0.01 & -0.01 & -0.01 & -0.01 & -0.01 & -0.009 & -0.009 & -0.008 & -0.008 \tabularnewline
\hline
$14^{+}$ & -0.04 & -0.04 & -0.03 & -0.03 & -0.03 & -0.03 & -0.03 & -0.03 & -0.03 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 & -0.02 \tabularnewline
\hline
$16^{+}$ & -0.06 & -0.05 & -0.04 & -0.03 & -0.03 & -0.03 & -0.03 & -0.03 & -0.02 & -0.02 & -0.02 & -0.02 & -0.01 & -0.01 & -0.01 \tabularnewline
\hline
\end{tabular}}
\end{sidewaystable}
\begin{sidewaystable}[H]
\caption{Energy spectra of $^{48}$Cr using GXPF1A interaction. Here
we have keep the single-particle energies of $0f_{7/2}$
and $0f_{5/2}$ as the original one, and changed the single-particle
energies of $0p_{3/2}$ and $1p_{1/2}$ moved up by original minus $\Delta$.}
\label{t5}
\resizebox{\textwidth}{6cm}{%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Energy & $\Delta$ = -1 & -2 & -3 & -4 & -5 & -6 & -7 & -8 & -9 & -10 & -20 & -40 & -60 & -80 & -100 \tabularnewline
\hline
$0^{+}$ & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \tabularnewline
\hline
$2^{+}$ & 0.638 & 0.532 & 0.540 & 0.795 & 1.232 & 1.294 & 1.317 & 1.333 & 1.344 & 1.352 & 1.367 & 1.361 & 1.358 & 1.355 & 1.353 \tabularnewline
\hline
$4^{+}$ & 1.612 & 1.539 & 1.584 & 1.881 & 2.738 & 3.838 & 4.334 & 4.446 & 4.501 & 4.529 & 4.552 & 4.520 & 4.505 & 4.497 & 4.491 \tabularnewline
\hline
$6^{+}$ & 3.103 & 2.991 & 3.061 & 3.577 & 5.074 & 7.084 & 8.712 & 9.843 & 10.860 & 11.862 & 21.830 & 41.799 & 61.787 & 81.781 & 101.778 \tabularnewline
\hline
$8^{+}$ & 4.753 & 4.745 & 4.863 & 5.389 & 6.785 & 8.699 & 10.701 & 12.712 & 14.683 & 15.949 & 25.960 & 45.866 & 65.829 & 85.809 & 105.797 \tabularnewline
\hline
$10^{+}$ & 6.728 & 6.848 & 6.935 & 7.594 & 9.491 & 11.692 & 13.547 & 15.467 & 17.414 & 19.376 & 39.219 & 79.143 & 119.119 & 159.105 & 199.098 \tabularnewline
\hline
$12^{+}$ & 8.341 & 8.942 & 8.984 & 9.706 & 11.763 & 14.550 & 17.441 & 20.075 & 22.152 & 24.139 & 43.927 & 83.780 & 123.727 & 163.699 & 203.682 \tabularnewline
\hline
$14^{+}$ & 10.459 & 11.392 & 11.312 & 12.011 & 14.324 & 17.686 & 21.251 & 24.414 & 27.380 & 30.332 & 60.070 & 119.925 & 179.874 & 239.848 & 299.832 \tabularnewline
\hline
$16^{+}$ & 13.655 & 14.610 & 14.605 & 15.296 & 17.590 & 20.955 & 24.656 & 28.478 & 32.358 & 36.270 & 75.918 & 155.754 & 235.698 & 315.670 & 395.653 \tabularnewline
\hline
\hline
B(E2: J $\rightarrow$ J-2 ) ($e^{2}fm^{4}$) & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & 302 & 359 & 388 & 363 & 257 & 190 & 170 & 158 & 151 & 145 & 123 & 112 & 109 & 107 & 106 \tabularnewline
\hline
$4^{+}$ & 417 & 496 & 523 & 413 & 65 & 13 & 222 & 188 & 167 & 154 & 116 & 104 & 100 & 99 & 98 \tabularnewline
\hline
$6^{+}$ & 425 & 500 & 508 & 425 & 273 & 189 & 22 & 18 & 44 & 51 & 54 & 54 & 54 & 55 & 54 \tabularnewline
\hline
$8^{+}$ & 385 & 454 & 453 & 368 & 234 & 118 & 8 & 0.04 & 0.60 & 3 & 2 & 2 & 2 & 2 & 1 \tabularnewline
\hline
$10^{+}$ & 293 & 391 & 392 & 319 & 195 & 3 & 2 & 2 & 4 & 3 & 0.45 & 0.32 & 0.29 & 0.28 & 0.27 \tabularnewline
\hline
$12^{+}$ & 171 & 316 & 341 & 325 & 282 & 24 & 21 & 18 & 13 & 11 & 10 & 10 & 10 & 9 & 9 \tabularnewline
\hline
$14^{+}$ & 139 & 232 & 267 & 261 & 210 & 153 & 136 & 56 & 22 & 15 & 8 & 7 & 7 & 7 & 7 \tabularnewline
\hline
$16^{+}$ & 66 & 132 & 160 & 159 & 156 & 154 & 125 & 43 & 23 & 17 & 9 & 8 & 7 & 7 & 7 \tabularnewline
\hline
\hline
Q$(eb)$ & & & & & & & & & & & & & & & \tabularnewline
\hline
$2^{+}$ & -0.35 & -0.39 & -0.40 & -0.35 & +0.13 & +0.25 & +0.26 & +0.25 & +0.25 & +0.25 & +0.23 & +0.22 & +0.22 & +0.21 & +0.21 \tabularnewline
\hline
$4^{+}$ & -0.45 & -0.50 & -0.53 & -0.55 & -0.53 & -0.50 & +0.29 & +0.32 & +0.32 & +0.32 & +0.30 & +0.28 & +0.28 & +0.28 & +0.27 \tabularnewline
\hline
$6^{+}$ & -0.49 & -0.56 & -0.61 & -0.63 & -0.61 & -0.49 & -0.14 & +0.01 & +0.07 & +0.09 & +0.12 & +0.13 & +0.13 & +0.13 & +0.13 \tabularnewline
\hline
$8^{+}$ & -0.51 & -0.60 & -0.67 & -0.70 & -0.69 & -0.67 & -0.65 & -0.63 & -0.54 & -0.13 & -0.09 & -0.09 & -0.09 & -0.09 & -0.08 \tabularnewline
\hline
$10^{+}$ & -0.38 & -0.61 & -0.69 & -0.69 & -0.66 & -0.35 & -0.34 & -0.33 & -0.32 & -0.32 & -0.29 & -0.27 & -0.27 & -0.26 & -0.26 \tabularnewline
\hline
$12^{+}$ & -0.08 & -0.59 & -0.69 & -0.69 & -0.67 & -0.63 & -0.58 & -0.37 & -0.28 & -0.26 & -0.24 & -0.24 & -0.24 & -0.23 & -0.24 \tabularnewline
\hline
$14^{+}$ & -0.08 & -0.57 & -0.70 & -0.68 & -0.66 & -0.64 & -0.58 & -0.48 & -0.46 & -0.45 & -0.41 & -0.40 & -0.39 & -0.39 & -0.39 \tabularnewline
\hline
$16^{+}$ & -0.14 & -0.59 & -0.71 & -0.70 & -0.68 & -0.66 & -0.65 & -0.64 & -0.63 & -0.62 & -0.59 & -0.58 & -0.57 & -0.57 & -0.57 \tabularnewline
\hline
\end{tabular}}
\end{sidewaystable}
\begin{table}[H]
\centering
\caption{Ratio E(J)$_{\Delta}$/E(J) with GXPF1A for $\Delta$=1,10, and 20.}
\label{t6}
\vspace{0.3cm}
\begin{tabular}{|c|c|c|c|}
\hline
$J/\Delta$ & 1 & 10 & 20\tabularnewline
\hline
\hline
2 & .973 & .878 & .849\tabularnewline
\hline
4 & .983 & .921 & .901\tabularnewline
\hline
6 & .976 & .882 & .851\tabularnewline
\hline
8 & .978 & .879 & .842\tabularnewline
\hline
10 & .977 & .851 & .812\tabularnewline
\hline
12 & .968 & .859 & .823\tabularnewline
\hline
14 & .972 & .872 & .838\tabularnewline
\hline
16 & .969 & .863 & .829\tabularnewline
\hline
Q(2$^{+})$ & .967 & .867 & .867\tabularnewline
\hline
$\sqrt{B(E2)}$ & .975 & 0.896 & .876\tabularnewline
\hline
\end{tabular}.
\end{table}
\begin{table}[H]
\centering
\caption{Comparison of original energies with normalized ones for $\Delta=$20 using GXPF1A. Renormalization factor = 1.2.}
\label{t7}
\vspace{0.3cm}
\begin{tabular}{|c|c|c|}
\hline
$J$ & Original Spectrum & $\Delta$=20 Renormalized\tabularnewline
\hline
\hline
0 & 0.000 & 0.000\tabularnewline
\hline
2 & 0.788 & 0.802\tabularnewline
\hline
4 & 1.717 & 1.854\tabularnewline
\hline
6 & 3.279 & 3.294?\tabularnewline
\hline
8 & 4.752 & 4.801\tabularnewline
\hline
10 & 6.420 & 6.268?\tabularnewline
\hline
12 & 7.722 & 7.627\tabularnewline
\hline
14 & 9.701 & 9.755\tabularnewline
\hline
16 & 12.805 & 12.748\tabularnewline
\hline
\end{tabular}
\end{table}
\clearpage
We now discuss the arrangements of the figures. There are 4 sets, each with 4 subsets:
\subsection{SET 1}
\subsection*{CASE - 1}
Pair B ($f_{5/2}-p_{1/2}$) shifted by an amount $\Delta$ relative to Pair A ($p_{3/2}-f_{7/2}$).
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigXA.jpg}
\caption{E2, E4, E6, E8.}
\label{f11}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigXB.jpg}
\caption{E10, E12, E14, E16.}
\label{f12}
\end{minipage}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigA.jpg}
\caption{B(E2) 2, 4, 6, 8.}
\label{f13}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigB.jpg}
\caption{B(E2) 10, 12, 14, 16.}
\label{f14}
\end{minipage}
\end{figure}
\clearpage
\subsection{SET 2}
\subsection*{CASE - 2}
Pair D ($p_{1/2}-p_{3/2}$) shifted by an amount $\Delta$ relative to pair D ($f_{5/2}-f_{7/2}$).
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigXC.jpg}
\caption{E2, E4, E6, E8.}
\label{f21}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigXD.jpg}
\caption{E10, E12, E14, E16.}
\label{f22}
\end{minipage}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigC.jpg}
\caption{B(E2) 2, 4, 6, 8.}
\label{f23}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigD.jpg}
\caption{B(E2) 10, 12, 14, 16.}
\label{f24}
\end{minipage}
\end{figure}
\clearpage
\subsection{SET 3}
\subsection*{CASE - 1}
Global view: Pair B ($f_{5/2}-p_{1/2}$) shifted by an amount $\Delta$ relative to Pair A ($p_{3/2}-f_{7/2}$).
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigGE.jpg}
\caption{E2, E4, E6, E8.}
\label{f31}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigGF.jpg}
\caption{E10, E12, E14, E16.}
\label{f32}
\end{minipage}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigE.jpg}
\caption{B(E2) 2, 4, 6, 8.}
\label{f33}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigF.jpg}
\caption{B(E2) 10, 12, 14, 16.}
\label{f34}
\end{minipage}
\end{figure}
\clearpage
\subsection{SET 4}
\subsection*{CASE - 2}
Global view: Pair D ($p_{1/2}-p_{3/2}$) shifted by an amount $\Delta$ relative to pair D ($f_{5/2}-f_{7/2}$).
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigGG.jpg}
\caption{E2, E4, E6, E8.}
\label{f41}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigGH.jpg}
\caption{E10, E12, E14, E16.}
\label{f42}
\end{minipage}
\end{figure}
\begin{figure}[H]
\centering
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigG.jpg}
\caption{B(E2) 2, 4, 6, 8.}
\label{f43}
\end{minipage}%
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{FigH.jpg}
\caption{B(E2) 10, 12, 14, 16.}
\label{f44}
\end{minipage}
\end{figure}
\clearpage
\section{Overview of All the Results}
A casual view of all the figures gives on the impassion that all the curves look very similar, be it energy levels or B(E2)'s, be it CASE - 1 or CASE - 2. There are gradual changes with increasing positive $\Delta$ in most cases. For $\Delta$ becoming more negative things change smoothly for a while but then there are sudden turn overs. This is undoubtedly due to the fact that there will be crossovers of single particle energies.
For CASE - 1 we note with GXPF1A the 0$f_{5/2}$ is 4.4870 MeV above 0$f_{7/2}$ so with $\Delta$ lowers that -4.4870 we expect strange things to happen. In the first 4 figures we indeed see down turnovers at about that value of $\Delta$.
For CASE - 2 we note that 0 1$p_{3/2}$ is 2.9447 MeV above 0$f_{7/2}$ so we we expect some sudden changes for $\Delta$ about -3 MeV. In contrast to CASE - 1 here we see sudden upturns as we further decrease $\Delta$. For the B(E2)'s there is a downturn for CASE - 2.
We will give a more detailed discussion in the next subsection. We will focus on positive $\Delta$.
\subsection{SET 1. CASE - 1}
In Fig \ref{f11} where we shift pair B ($p_{1/2}-f_{5/2}$) relative to pair A we note that the excitation energies E2, E4, E6, and E8 slowly come down was we go from $\Delta$ = 0 to 20. However the overall spectrum seems not to change very much.
In Fig \ref{f12} we have E10, E12, E16 and E16 also coming down with increasing $\Delta$ but at a faster rate than for the lower J's in Fig 1. Again, the overall spectrum seems not to change very much.
In Fig \ref{f13} The B(E2)s also come down with an increase in $\Delta$ but at a faster rate than the energies. This is probably due to the fact that when $\Delta$ increases there is less configuration mixing and hence less collectivity.
In Fig \ref{f14} We show a striking difference in the B(E2) behavior
for $J=10, 12, 14, 16$ when compared with the case in Fig \ref{f13} for $J=$2, 4, 6, and 8. Now the curves for positive $\Delta$ are much
flatter-much less variation with increasing $\Delta$. A flat curve suggests configuration mixing , which is on the decease as
$\Delta$ is increasing , is not so important. The higher the spin
the more important are the single particle orbitals with high J
e.g. $f_{7/2}$.
There have been previous discussion of the band structure
of $^{48}$Cr and the fact the the higher J states do not belong to
the same "band" as the lower ones. The general consensus is
that for the higher states there is an alignment of some of the
nucleons along the rotation axis. This shows up more
dramatically in the B(E2)s rather than the energies.
\subsection{SET 2. CASE - 2}
As shown in Fig \ref{f21} when we shift ($p_{1/2}-p_{3/2}$) to more
positive values we find that the excitation energies go slightly
up although the entire spectrum does not change very much. This is in contrast to the case of Section1 where the energies
went slightly down. AN exception is $J=8^{+}$which is surprisingly flat.
In Fig \ref{f22} we go to higher spin and on the whole the energies
decrease with increasing $\Delta$. This is another indication that
perhaps the high J states do not belong to the same band as the
lower ones. For the first tile we see a crossover near $\Delta$ = 20
with $J=12^{+}$starting to come down below $J= 10^{+}$.
In Fig \ref{f23} we come back to B(E2)s. They decree with
increasing $\Delta$ . This is not so different from The qualitative
behavior in Fig \ref{f13}. This is again due to the fact that for
increasing $\Delta$ there is less configuration mixing and hence
less collectivity.
In Fig \ref{f24}. We look at B(E2)s for $J=$10, 12, 14 and 16. There is a
decrease with increasing $\Delta$ but not as severe as for $J=$2, 4, and 6. For the $J= 16^{+}$ to 14$^{+}$transition the curve is very flat.
For such a high spin you need high $J$ single orbits to construct
the state, so the reducing of contributions from $p_{3/2}$ and $p_{1/2}$
is not so important.
\subsection{Global view SET 3. CASE - 1}
Here we will put more emphasis on negative $\Delta$. In Fig \ref{f31}
we see large increase in the excitation energies as we say decrease $\Delta$ form +5 to -5. We could argue that at $\Delta$ =-5
we have a lot of single particle orbitals close to together and
this increases the pairing so that J=0 drops down a lot relative
to $J =$ 2, 4, 6, and 8. For $\Delta$ even more negative the low $J$
states are dominated by the $f_{5/2}$ and $p_{1/2}$ orbitals. As one
makes $\Delta$ more negative configuration for ($f_{7/2},p_{3/2}$)
becomes less important so the curve flattens out.
In Fig \ref{f32} ($J = 10, 12, 14, 16$) the behavior for negative $\Delta$ is
completely different than for low $J$. Except for $J=10^{+}$, the
other energy levels got up in a linear fashion with increasing
negative $\Delta$. This is not difficult to understand. On needs
the high $J$ $f_{7/2}$ orbital to construct these states and moving say
$f_{5/2}$ below $f_{7/2}$ is equivalent to putting $f_{7/2}$ above $f_{5/2}$.
For the B(E2)s in Figs \ref{f33} for $J =$ 2, 4, 6 and 8 the flatness is due
to the reduced configuration mixing due to the wide
separation of the $f_{7/2}$ form $f_{5/2}$.
\subsection{Global view SET 4. CASE - 2}
In this case the lowest orbits have the lowest spins i.e. $p_{3/2}$
and $p_{1/2}$. It is even harder to make low lying high spin states
in this case. It is therefore not surprising to see in Fig \ref{f41} that
even lower $J$ state energies i.e. $J=$6 and 8 have a linear rise
with increasing negative $\Delta$.
In more detail with he $p_{3/2}$ and $p_{1/2}$ orbitals the maximum spin we can have for 4 protons is $J_p$ = 2. via the configuration $p_{3/2}$ $p_{3/2}$ $p_{3/2}$ $p_{1/2}$, which is equivalent to $p_{3/2}^{-1} p_{1/2}$. The same is true for 4 neutrons. So with $J_p$ = 2, $J_n$ = 2 we can only make states up to $J = 4$. This explains why only $J=0, 2, 4$ remain at low wineries as we increase negative $\Delta$.
And of course in Fig \ref{f42} we see for the same reason that all
high $J$ state excitation energies (10, 12, 14, and 16) rise linearly
with increasing negative $\Delta$.
The flattening of the B(E2)
curves in Fig \ref{f43} and Fig \ref{f44} is due to the reduction in
configuration mixing as the space between SET 3. and SET 4.
single particle energies widens. The enhancement near $\Delta$
=-5 is due to the fact that all single particle orbitals are close
to each other so there is an enhancement of collectivity due to
the increase of configuration mixing.
\section{Closing Remarks - Scaling Behavior}
In Table \ref{t6} we show the ratio E(J)$\Delta$/E(J) for $\Delta$=1, 10 and 20. Note that although there are some
fluctuations the ratios are similar. If the ratios for a given $\Delta$ were all the same we would have
perfect scaling. In that idealized situation we would get identical spectra for any finite $\Delta$ with that
of $\Delta$=0 by multiplying the entire matrix for that $\Delta$ for that by a constant. Thus, is a
phenomenological approach. If we limited ourselves to fitting the spectra of the 16 yrast states of
$^{48}$Cr, we would have an infinite number of choices of combinations of 2 body matrix elements
and single particle energies which would yield the same results. In truth as seen in Table \ref{t6} the
ratios are not exactly the same but they are close enough to the idealized situation so that a large
range of choices would lead to equally good results for these spectra. Of course if we expanded
the data i.e. included other states. the result would be different. In Table \ref{t7} we compare the
original spectrum of GXPF1A with that for $\Delta$=20 multiplied by a renormalization factor 1.2. This
renormalization factor multiplies the entire matrix including the $\Delta$=20 single particle energies.
We see that the spectra are reasonably close- it would be hard to prefer one to the other. However
the single particle energies are vastly different. Originally the 0$f_{5/2}$ and 1$p_{1/2}$ are 7.241 and
4.487 MeV above 0$f_{7/2}$. Now they are 27.241 and 24.487 MeV above the 0$f_{7/2}$ orbit.
When one makes truncation in the PF shell by dropping orbits it is more natural to drop the spin
orbit partners 0$f_{5/2}$ and 1$p_{1/2}$ than it is to drop the 2 p-shell orbits. This was done by Zamick et
al. \cite{13} in the context of quadrupole moments and B(E2)'s. They studied the effects of dropping
spin-orbit partners 0$f_{5/2}$ and 1 $p_{1/2}$ on these electromagnetic properties. In the present context
this is equivalent to setting $\Delta$to infinity. To a large extent the results of the truncated calculations
could be put into line with the full calculations by enlarging the effective charges in the former
when the FPD6 interaction is used. The ratio full to truncated for Q(2$^+$) 2 , Q(4$^+$) 2 ,B(E2, 2$^+$ $\rightarrow$
0$^+$) and B(E2 , 4$^+$ $\rightarrow$ 2$^+$) were all very close to 1.4.
The possibility of scaling behavior is intriguing and will be further investigated in the near
future.
\clearpage
\section*{ACKNOWLEDGEMENTS}
P C Srivastava acknowledges a research grant from SERB (India),
CRG/2019/000556 and Kalam cluster at Physics Department, IIT-Roorkee.
C. Fan acknowledges supports from Aresty Research Center.
\vspace{0.8cm}
|
1,116,691,498,443 | arxiv | \section{Introduction}
Let $(G;A,B)$ be a double Lie group \cite{WeLu} i.e. $G$ is a Lie group, $A,B$
are closed subgroups and any element of $G$ has a unique decomposition:
$g=a b\,,\,a\in A\,,\,b\in B$. Any double Lie group leads to a Manin group
and hence a pair of Poisson-Lie groups in duality (we do not require that
$G^*$ is simply connected). Let us recall that a Manin
group $(M;P,Q)$ is a double Lie group, where $M$ is equipped with invariant,
non-degenerate scalar product, vanishing on $TP$ and $TQ$.
We briefly sketch the way from double Lie groups to Manin groups\cite{SZ3}.
Having a double Lie group we can define two compatible differential groupoid
structures on $G$ with $A$ and $B$ as sets of identities. This forms a
$D^*$-group (in terminology of \cite{SZ3}). Applying the phase functor we get
two compatible symplectic groupoid structure on $T^*G$ ($S^*$-group). Then
using the symplectic form one can define invariant, non degenerate, scalar
product on $T^*G$. The sets of identities for both structures (namely
$P:=(TA)^0,Q:=(TB)^0$ ) are then Poisson-Lie groups, dual to each other.
The infinitesimal version of a double Lie group is a double Lie algebra and
that of a Manin group is a Manin triple (cf. section 2). It is therefore
clear, that there should be a procedure which assigns a Manin triple (or a Lie
bialgebra) to each double Lie algebra. We present it in section 2.
In section 3 we relate some decompositions of $so(p+1,q)$ or $so(p,q+1)$
algebras with series of Lie bialgebra structures on $iso(p,q)$, among them
the so-called $\kappa$-deformation.
The main application of this study is to make a step towards a construction
on the $C^*$-algebra level of quantum groups corresponding to those Lie
bialgebras. By looking which double
Lie algebras give rise to (global) double Lie groups, we are in position to
distinguish between "good"(complete) and "bad"(non-complete)
cases \cite{SZ4}. In section 4 we show examples of decompositions from section
3 which do not give rise to a global decomposition of the corresponding Lie
groups. This is the case of the $\kappa$-deformation of Poincare group. It
strongly suggests that the $\kappa$-deformed Poincare group does not exist on
the $C^*$-algebra level. Contrary to this, the case corresponding to
$\kappa$-deformation of the Euclidean group is a "good" one: the global
decomposition is just the Iwasawa decomposition. Passing from groupoids to
their $C^*$ algebras \cite{Con}\cite{Ren} we get $\kappa$-deformed Euclidean
group on the $C^*$-algebra level. This will be described
elsewhere\cite{PS}.\vspace{1ex}
{\em Throughout this paper all vector spaces, Lie algebras, Lie
groups are real and $\oplus$ means (if used without any comment) direct sum
of vector spaces. }
\section{Double Lie algebras and Manin triples.}
Let $(G,\pi)$ be a Poisson-Lie group with Lie algebra $\frak g$. Then $\pi$
determines linear mapping $\delta\,:\,\frak g\longrightarrow \frak g\wedge\frak g $ which is a 1-cocycle
on $\frak g$ relative to adjoint representation on $\frak g\wedge\frak g$ and the dual map
$\frak g^*\wedge\frak g^*\longrightarrow \frak g^* $ is a Lie bracket. Conversely, if $G$ is connected and
simply connected then any such $\delta$ gives us a multiplicative Poisson
structure on $G$. Let us recall the following:
\begin{defi}
{\em \cite{WeLu} A pair $(\frak g,\delta)$ is said to be a }Lie bialgebra {\em if
$\frak g$ is a Lie algebra, $\delta\,:\,\frak g\longrightarrow\frak g\wedge\frak g$ is a 1-cocycle on $\frak g$
relative to the adjoint representation of $\frak g$ on $\frak g\wedge\frak g$ and
$\delta^*\,:\,\frak g^*\wedge\frak g^*\longrightarrow\frak g^*$ is a Lie bracket.}
\end{defi}
In this situation we also say that $\delta$ is a cobracket on $\frak g$.
\begin{tw}
{\em (Manin)\cite{Dr}}
Let $\frak g$ be a Lie algebra, $\frak g^*$ its dual space and let $<\,,\,>$ denote the
canonical symmetric bilinear form on $\frak g\oplus\frak g^*$. Let $\frak g^*$ be given a Lie
algebra structure. Then the dual map to the bracket on $\frak g^*$ is a cobracket
on $\frak g$ iff there exists a Lie algebra structure on $\frak g\oplus\frak g^*$ such that:
\begin{enumerate}
\item $\frak g\,,\,\frak g^*$ are subalgebras of $\frak g\oplus\frak g^*$.
\item The form $<,>$ on $\frak g\oplus\frak g^*$ is invariant.
\end{enumerate}
In this case the bracket on $\frak g\oplus\frak g^*$ is unique and is given by:\\
\mbox{$[X+\alpha,Y+\beta]=[X,Y]-ad^{\veee}_{\beta} X +ad^{\veee}_{\alpha}Y
+[\alpha,\beta]+ad^{\veee}_X\beta-ad^{\veee}_Y\alpha$} where $[\,,\,]$ denotes brackets
on $\frak g$ and $\frak g^*$ and $ad^{\veee}$ denotes the coadjoint representations of $\frak g$
and $\frak g^*$. The $\frak g\oplus\frak g^*$ with the above bracket will be denoted by
$\frak g\bowtie \frak g^*$.
\end{tw}
The Lie bialgebras are in one to one correspondence with Manin triples.
\begin{defi}
{\em \cite{Dr} Three Lie algebras $(\frak m;\frak p,\frak q)$ form a} Manin triple {\em
iff: \begin{enumerate}
\item $\frak p,\frak q$ are Lie subalgebras of $\frak m$ and $\frak m=\frak p\oplus\frak q$ as a vector
space sum.
\item $\frak m$ is equipped with invariant, non-degenerate scalar product such
that $\frak p,\frak q$ are isotropic.
\end{enumerate}}
\end{defi}
We need also the definition of a double Lie algebra:
\begin{defi}{\em \cite{WeLu}}
A double Lie algebra {\em is a triple $(\frak g;\frak a,\frak b)$ such that $\frak a,\frak b$ are
Lie subalgebras of $\frak g$ and $\frak g=\frak a\oplus\frak b$ as a vector space sum.}
\end{defi}
Now let $(\frak g;\frak a,\frak b)$ be a double Lie algebra. Consider the coadjoint action
semi-direct product $\frak g\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak g^*$ with natural bilinear, symmetric,
invariant form $<\,,\,>$.
Then $\frak a^0$ is $\frak a$ invariant and $\frak b^0$ is $\frak b$ invariant (where
$\frak a^0, \frak b^0$ denotes annihilators of $\frak a$ and $\frak b$ respectively). \\
Indeed if \mbox{$x,y\in\frak a$}, \mbox{$\alpha\in\frak a^0$} then
\mbox{$<[x,\alpha],y>$}=\mbox{$<\alpha,[y,x]>=0$}
since $\frak a$ is a subalgebra. Of course $\frak a\oplus\frak a^0$ and $\frak b\oplus\frak b^0$ are
isotropic. In this way we get that $(\frak g\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak g^*; \frak a\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak a^0,\frak b\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak b^0)$ is
a Manin triple and we have a bialgebra structure on $ \frak a\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak a^0.$
The cobraket $\delta$ satisfies: $\delta(\frak a)\subset \frak a^0\wedge\frak a$ and
$\delta(\frak a^0)\subset\frak a^0\wedge\frak a^0$. So \mbox{$(\frak a\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak a^0)\bowtie
(\frak a\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak a^0)^*= \frak g\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak g^*$} where we identified $\frak b\oplus\frak b^0$ with
$(\frak a\oplus\frak a^0)^*$. Thus we have a procedure which associates with each double
Lie algebra a Lie bialgebra $(\frak h,\delta)$ with the following properties:
1. $\frak h=\frak a\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V$ ($V$-abelian ideal), 2. $\delta(\frak a)\subset\frak a\wedge V,
\delta(V)\subset V\wedge\ V.$
Conversely, let $(\frak h:=\frak a\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V,\delta)$ be a semidirect product with abelian
ideal $V$ and cobraket $\delta$ such that $\delta(\frak a)\subset V\wedge \frak a$ and
$\delta(V)\subset V\wedge V$. $\delta$ defines a Lie algebra structure on
$\frak h^*=\frak a^0\oplus V^0$. Let us show, that $\frak h^*=\frak a^0\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V^0$ with abelian ideal
$V^0$. We adopt the following notation: capital letters $A,B,...$ are elements
of $\frak a$, $A^0,B^0,...$ are elements of $\frak a^0$, small $x,y,...$ are elements of
$V$ and $x^0,y^0,...$ are elements of $V^0$. We have:
\begin{itemize}
\item $<C,[A^0,B^0]>=<\delta(C),A^0\wedge B^0 >=0$, since $\delta(\frak a)\subset V\wedge
\frak a$, so $\frak a^0$ is a subalgebra;
\item $<z,[x^0,y^0]>=<\delta(z),x^0\wedge y^0 >=0 $, since $\delta(V)\subset V\wedge
V \\
<C,[x^0,y^0]>=<\delta(C),x^0\wedge y^0>=0$, so $V^0$ is an abelian subalgebra;
\item $<z,[A^0,x^0]>=<\delta(z),A^0\wedge x^0>=0$ and $V^0$ is an ideal.
\end{itemize}
Now we prove that the Lie algebra $\frak h\oplus\frak h^*$ with bracket given in the
theorem 1 coincides with semidirect product $(\frak a\oplus \frak a^0)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}(\frak a\oplus \frak a^0)^*$
with coadjoint action if we identify $V\oplus V^0$ with $(\frak a\oplus \frak a^0)^*$ by
duality: $<<A+B^0,x+y^0>>:=<A,y^0>+<x,B^0>$. \\
For the coadjoint representation of $\frak h$ we have:
\begin{itemize}
\item $ad^{\veee}_{\frak a}(V^0)\subset V^0$ : $<z,ad^{\veee}_A(y^0)>=<[z,A],y^0>=0$ (since
$[\frak a,V]\subset V$);
\item $ad^{\veee}_{\frak a}(\frak a^0)\subset \frak a^0 $ : $<C,ad^{\veee}_A(B^0)>=<[C,A],B^0>=0$ (since
$[\frak a,\frak a]\subset \frak a$ );
\item $ad^{\veee}_V(V^0)=0 $ : $<z,ad^{\veee}_x(y^0)>=<[z,x],y^0>=<0,y^0>=0\,,\\
<A,ad^{\veee}_x(y^0)>=<[A,x],y^0>=0$ (since $[\frak a,V]\subset V$);
\item $ad^{\veee}_V(\frak a^0)\subset V^0$ : $<z,ad^{\veee}_x(A^0)>=<[z,x],A^0>=<A^0,0>=0$.
\end{itemize}
And for the coadjoint representation of $\frak h^*$:
\begin{itemize}
\item $ad^{\veee}_{\frak a^0}(\frak a)\subset \frak a$ : $<ad^{\veee}_{A^0}(B),C^0>=<B,[C^0,A^0]>=0$ (since
$[\frak a^0,\frak a^0]\subset \frak a^0$);
\item $ad^{\veee}_{\frak a^0}(V)\subset V$ : $<ad^{\veee}_{A^0}(y), z^0>=<y,[z^0,A^0]>=0$
(since $[\frak a^0,V^0]\subset V^0)$;
\item $ad^{\veee}_{V^0}(\frak a)\subset V$ : $<ad^{\veee}_{y^0}(A),z^0>=<A,[z^0,y^0]>=0$ (since
$[V^0,V^0]=0$);
\item $ad^{\veee}_{V^0}(V)=0$ : $<ad^{\veee}_{x^0}(y),z^0>=<y,[z^0,x^0]>=<y,0>=0\,,\\
<ad^{\veee}_{x^0}(y),A^0>=<y,[A^0,x^0]>=0$ (since $[\frak a^0,V^0]\subset V^0$).
\end{itemize}
From this it follows that $V \oplus V^0$ is an abelian ideal and $\frak a\oplus \frak a^0$ is a
subalgebra. If we identify $V\oplus V^0$ with $(\frak a\oplus \frak a^0)^*$ we are in the
situation in Theorem 1. and it follows that the action of $\frak a\oplus \frak a^0$ is a
coadjoint action. In this way we have proved the following:
\begin{prop}
Any double Lie algebra $(\frak g;\frak a,\frak b)$ leads to a Lie bialgebra $(\frak h,\delta)$
such that $\frak h=\frak a\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V$ is a semi-direct product with abelian ideal $V$ and
the cobracket $\delta$ satisfies: $\delta(\frak a)\subset \frak a\wedge V$ and
$\delta(V)\subset V\wedge V$. Conversely, any Lie bialgebra of this type is
obtained in this way.
\end{prop}
\section{Iwasawa-type decompositions of $so(p,q)$ and
bialgebra structures on $iso(p-1,q)\,,\,iso(p,q-1)$.}
\subsection{Inhomogenous $so(p,q)$ algebras and $b$-type Poisson structures.}
Let $(V,\eta)$ be a n+1 dimensional, real vector space with symmetric,
nondegenerate, billinear form $\eta$ of signature $(p,q)$. By $\eta$ we also
denote isomorphism $V \longrightarrow V^*$ given by $\eta(x)(y):=\eta(x,y)$.
Let $iso(p,q):=so(p,q)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V$ be an inhomogenous $so(p,q)$ Lie algebra.
$so(p,q)$ is isomorphic to $V\wedge V$ by: $x\wedge
y\mapsto \la{x}{y}:=x\otimes\eta(y) -y\otimes\eta(x)$. If $(e_i)$ is an orthonormal
basis of $V$: $(\la{i}{j}:=\la{e_i}{e_j}\,,\,i<j) $ form a basis of
$so(p,q)$, with commutators: $[\la{i}{j},\la{k}{l}]=\eta_{il}
\la{j}{k}+\eta_{jk}\la{i}{l}-\eta_{ik}\la{j}{l}- \eta_{jl}\la{i}{k}$.\\
Let $K(A,B):=-\frac{1}{2} Tr(A B)$. This is $ad$ invariant, non degenerate
scalar product on $so(p,q)$ and $(\la{i}{j}\,,\,i<j)$ form an orthonormal
basis. \\
Let $(\la{i}{j}^*\,,\,e_k^*\,:\,i <j)$ be a
basis in $iso(p,q)^*$ given by: \\
$<\la{k}{l},\la{i}{j}^*>:=K(\la{i}{j},\la{k}{l})=\eta_{ik}\eta_{jl}-\eta_{il}
\eta_{jk}$ and $<e_l,e_k^*>:=\eta_{kl}$. \\
In this basis the coadjoint representation of $iso(p,q)$ has the following
form: \\
$ad^{\veee}_{\la{a}{b}}(\la{c}{d}^*)=\eta_{ad}\la{b}{c}^*+\eta_{bc}\la{a}{d}^*-
\eta_{ac}\la{b}{d}^*-\eta_{bd}\la{a}{c}^*\;,\;
\;ad^{\veee}_{\la{a}{b}}(e_k^*)=\eta_{kb} e_a^* - \eta_{ka} e_b^*\,,\\
ad^{\veee}_{e_a}(\la{c}{d}^*)=0\;,\;\; ad^{\veee}_{e_a}(e_b^*)=-\la{a}{b}^*. $
Let $\frak g:=iso(p,q)$ and $\frak h:=so(p,q)$, so $\frak g=\frak h\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V$.
It is known \cite{SZ1} that all bialgebra structures on $\frak g$ for $p+q>2$
are coboundary i.e. are of the form $\delta=\partial r$ for some $r\in
\frak g\wedge\frak g$ ($\delta(x)=\partial r (x):=ad_x(r)$) where $r$ satisfies the
generalized classical Yang-Baxter equation: $[r,r]\in (\frak g\wedge\frak g\wedge\frak g)_{inv}$.
Since $\frak g\wedge\frak g=(\frak h\wedge\frak h)\oplus(\frak h\wedge V)\oplus (V\wedge V)$ we can write :
$r=c+b+a\,,\, c\in \frak h\wedge\frak h\,,\,b\in\frak h\wedge V\,,\,a\in V\wedge V$. We say that $r$
is of $b$-type iff $r=b$. In this case $b$ satisfies $[b,b]=t\Omega\,,\,t\in
R$ where $\Omega:= \eta^{jl}\eta^{km}e_j\wedge e_k\otimes\la{l}{m}$ is the canonical
$\frak g$-invariant element of $\frak g\wedge\frak g\wedge\frak g$.
We will be interested in the following solutions of this equation:
\begin{enumerate}
\item $b_x:=\eta^{jk}e_j\wedge\la{x}{e_k}\,,\,x\in V\,$
is a solution with $t=-\eta(x,x)$ \cite{SZ1}.
\item $\tilde{b}_x:=b_x+x\wedge X$ where $X \in\frak h$ and $X x=0$ is a solution
with the same $t$ \cite{SZ1}.
\item Let $x\in V$ be a null vector and let $v_i\in V,\,X_i\in \frak h$ satisfy:
$X_i x=0,\,X_i v_j=-\delta_{ij} x,\,[X_i,X_j]=0$. Then $b:=b_x+x\wedge Y+\sum
v_i\wedge X_i$, where $Y:=\sum \alpha_i X_i,\,\alpha_i\in R$ is a solution
with $t=0$ \cite{PS1}.
\item $b:=\tilde{b}_x+v\wedge X$ where $Xv=v$ is a solution with $t=-\eta(x,x)$ \cite{PS1}.
\end{enumerate}
We will need the fact that $b$ is completely determined by
the bracket on $V^*$\cite{SZ2}.
Let $b=v_i\wedge h_i$ (sumation). We use the same letter for the mapping $b: V^*
\longrightarrow \frak h $ given by: $ b(\alpha):=<v_i,\alpha>h_i$. Then the bracket on $V^*$
can be expressed by $b:\,[\alpha,\beta]=b(\alpha)\beta-b(\beta)(\alpha)$,
where the action of $\frak h$ on $V^*$ is a coadjoint action.
Let $e_k^*:=\eta(e_k)$ and $b(e_k^*)=:{b_k}^{mn}\la{m}{n}$ with
${b_k}^{mn}=-{b_k}^{nm}$, $[e_i^*,e_j^*]=:{f_{ij}}^k
e_k^*,\,{f_{ij}}^k=-{f_{ji}}^k$. Then:
$[e_i^*,e_j^*]={b_i}^{kl}(\eta_{lj}e_k^*-\eta_{kj}e_l^*)-{b_j}^{kl}
(\eta_{il}e_k^*- \eta_{ik}e_l^*)=\\=2({b_i}^{kl}\eta_{lj}-{b_j}^{kl}\eta_{il})
e_k^*={f_{ij}}^k e_k^*$. From this it follows that
$f_{ijk}=2(b_{jik}-b_{ijk})$ (we used $\eta_{ij}$ to lower indeces).
This equation determines $b_{ijk}$: \mbox{$b_{ijk}=\frac{1}{4}
(f_{jki}-f_{ijk}-f_{kij})$} and $b$, since $b=b^{kmn}e_k\wedge\la{m}{n}$.\\
Let us also notice the following:
\begin{lem} Let $p+q>2$ and let $\delta=\partial r$ be a cobracket on
$iso(p,q)=so(p,q)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V=:\frak h\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V$ which satisfies: $\delta(\frak h)\subset \frak h\wedge V
,\,\delta(V)\subset V\wedge V$. Then $r=b$. \\
Proof: {\em $r=c+b+a\,,\, c\in \frak h\wedge\frak h\,,\,b\in\frak h\wedge V\,,\,a\in V\wedge V$. Then
the condition $\delta(\frak h)\subset \frak h\wedge V$ is equivalent to
$ad_h(c)=0$ and $ad_h(a)=0$ for $h\in\frak h$ and $\delta(V)\subset V\wedge V$ is
equivalent to $ad_v(c)=0$ for $v\in V$. But since $\frak h$ is semisimple, from the
first equality it follows that $c=0$. Also since isomorphism $V\wedge V\ni x\wedge
y\mapsto\la{x}{y}\in\frak h$ intertwines action of $\frak h$ on $V\wedge V$ with the
adjoint action on $\frak h$ we have $a=0$. $\Box$}
\end{lem}
\subsection{Iwasawa-type decomposition of $so(p,q)$.}
In this section we put: $p+q=:n+1\,,\;n>2;$\\
$so(p,q)\supset\frak h_1:=<\la{i}{j}\,:\,2\leq i < j\leq n+1>=so(p-1,q);\\
so(p,q)\supset\frak h_2:=<\la{i}{j}\,:\,1\leq i < j\leq n>=so(p,q-1)$;\\
$f:=\la{1}{n+1}\;,\;\;\;g_k:=\la{1}{k}+\la{k}{n+1}\,,\;2\leq k\leq n
\;,\;\frak n:=<g_2,...,g_n>$.\\
With this notation: $so(p,q)=\frak h_1\oplus<f>\oplus \frak n=\frak h_2\oplus<f>\oplus \frak n$.\\
Then $u:=<f>\oplus \frak n$ is a Lie subalgebra:
$[f,g_k]=\eta_{11}g_k=g_k\;,\;[g_k,g_l]=0$.
The corresponding annihilators in $so(p,q)^*$ are equal:
$\frak h_1^0=<\la{1}{l}^*\,,\,2\leq l\leq n+1>\,,
\,\,\frak h_2^0=<\la{l}{n+1}^*\,,\,1\leq l\leq n>\,,\,\,\\
u^0=<g_l^*\,,\,2\leq l\leq n>\oplus <\la{m}{s}^*\,,\,2\leq m,s\leq n>$ where
$g_l^*:=\la{1}{l}^*+\la{l}{n+1}^*$. \\
In this way we get two double Lie algebras: $(so(p,q);so(p-1,q),u)$ and
$(so(p,q);so(p,q-1),u)$.\vspace{1ex}
{\it The double Lie algebra $(so(p,q);so(p-1,q),u)$.}\\
We use $K$ to equip $\frak h_1^0$ with scalar product. With respect to this
product $(\la{1}{l}^*\,,\,2\leq l\leq n+1)$ form an orthonormal basis. The
signature is $(p-1,q)$.
The action of $\frak h_1$ is given by:\\
$\la{i}{j}(\la{1}{l}^*)=\eta_{jl}\la{1}{i}^* -\eta_{il}\la{1}{j}^*$, so
$\frak h_1\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak h_1^0 =iso(p-1,q)$. Let us put $v_l:=\la{1}{l}^*$. We already know
that $\delta=\partial b$ for some $b\in \frak h_1^0 \wedge \frak h_1$ and $b$ is determined
by the bracket on $u=(\frak h_1^0)^*$. We have: $f=v_{n+1}^*\,,\,g_l=v_l^*$ and
$[v_l^*,v_m^*]=\delta_{ln+1} v_m^*-\delta_{mn+1} v_l^*$, so
${f_{lm}}^s=\delta_{ln+1}\delta_m^s-\delta_{mn+1}\delta_l^s$. Using
formula from section 2:
$b_{ijk}=\frac{1}{2}(\delta_{jn+1}\eta_{ik}-\delta_{kn+1}\eta_{ij})$
and $b=\eta^{sl}v_s\wedge\la{n+1}{l}=b_{v_{n+1}}$.
In this way we have shown that the double Lie algebra $(so(p,q);so(p-1,q),u)$
leads to the cobracket on $iso(p-1,q)$ given by $b=b_x\,,\,\,\eta(x,x)<0$.
\vspace{1ex}
{\it The double Lie algebra $(so(p,q);so(p,q-1),u)$.}\\
Now we use $-K$ as a scalar product on $\frak h_2^0$. With respect to this product
product \mbox{$(\la{l}{n+1}^*,\,1\leq l\leq n)$} form an orthonormal basis. The
signature is $(p,q-1)$. The action of
$\frak h_2$ is given by:\\
$\la{i}{j}(\la{l}{n+1}^*)=\eta_{jl}\la{i}{n+1}^* -\eta_{il}\la{j}{n+1}^*$, so
$\frak h_2\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak h_2^0 =iso(p,q-1)$. Let us put $v_l:=\la{l}{n+1}^*$ then
\mbox{$f=-v_{1}^*$}, \mbox{$g_l=-v_l^*$.}
In the same way as above we get:
${f_{lm}}^s=\delta_{1m}\delta_l^s-\delta_{1l}\delta_m^s\,,\,\,
b_{ijk}=\frac{1}{2}(\delta_{1k}\eta_{ij}-\delta_{1j}\eta_{ik})$ and
$b=-\eta^{sm}v_s\wedge\la{1}{m}=-b_{v_1}$.
So we see that the double Lie algebra $(so(p,q);so(p,q-1),u)$
leads to the cobracket on \mbox{$iso(p,q-1)$} given by $b=b_x\,,\,\,\eta(x,x)>0$.
\vspace{1ex}
We can be a little bit more general and instead of $f$ take
$\tilde{f}:=\la{1}{n+1}+s$ where\\ $s:=s^{ij}\la{i}{j} \in
<\la{i}{j}\,,\,2\leq i,j\leq n>=so(p-1,q-1)\,,\,s^{ij}=-s^{ji}$.
Then $[\tilde{f},g_k]=g_k+s^{ij}(\eta_{jk}g_i- \eta_{ik}g_j)$ so again
$\tilde{u}:=<\tilde{f}>\oplus \frak n$ is a Lie subalgebra and $so(p,q)=\frak h_1\oplus\tilde{u}=\frak h_2\oplus\tilde{u}$.
Let us analyze the new cobrackets on $\frak h_1\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} \frak h_1^0$ and $\frak h_2\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}\frak h_2^0$.
\vspace{1ex}
{\it Double Lie algebra $(so(p,q);so(p-1,q),\tilde{u})$.}\\
Keeping the same notation as above: $v_k^*=g_k\,,\,v_{n+1}^*=\tilde{f}$.\\
\mbox{$[v_l^*,v_m^*]=[\delta_{ln+1}(\delta_m^s+s^{sj}\eta_{jm}-
s^{is}\eta_{im})
-\delta_{mn+1}(\delta_l^s+s^{sj}\eta_{jl}-s^{is}\eta_{il})]v_s^*$}.
From this it follows that:\\ \mbox{$b^{sml}=\frac{1}{2}(\eta^{mn+1}\eta^{sl}-
\eta^{ln+1}\eta^{sm})-\frac{1}{2}\eta^{n+1s}(s^{lm}-s^{ml})$} and\\
$b=\eta^{mp}v_m\wedge \la{n+1}{p}+v_{n+1}\wedge
s=b_{v_{n+1}}+v_{n+1}\wedge s$. \\
So we get the cobracket on $iso(p-1,q)$ given by $b=b_x+x\wedge X
\,,\,\,\eta(x,x)<0$.
\vspace{1ex}
{\it Double Lie algebra $(so(p,q);so(p,q-1),\tilde{u})$.}\\
Now
\mbox{$[v_l^*,v_m^*]=[\delta_{1m}(\delta_l^s+s^{sj}\eta_{jl}-s^{is}\eta_{il})
-\delta_{1l}(\delta_m^s+s^{sj}\eta_{jm}-s^{is}\eta_{im})]v_s^*$}. From this
equation: $b^{sml}=\frac{1}{2}(\eta^{1l}\eta^{sm}-
\eta^{1m}\eta^{sl})-\frac{1}{2}\eta^{1s}(s^{lm}-s^{ml}) $ and
$b=-b_{v_1}-v_1\wedge s$.\\
This is the cobracket on $iso(p,q-1)$ given by $b=b_x+x\wedge X
\,,\,\,\eta(x,x)>0$.
In this way we have shown that bialgebra structures on $iso(p,q)$ of type 1
and 2 for non null vectors come from the double Lie algebra structures on
$so(p+1,q)$ or $so(p,q+1)$.
\vspace{1ex}
Now is time for type 4. Let $\tilde{f}=f+s$ be as above and let us assume that $s$
has d-dimensional eigenspace with eigenvalue 1. Then this is null subspace
and one can choose an orthonormal basis \mbox{$(e_i\,,\,2\leq i\leq n)$}
such that this eigenspace is equal:\\
\mbox{$<e_{m_1}-e_{n_1},...,e_{m_d}-e_{n_d}>$} with $2~\leq~m_k~<~n_k~\leq
~n\,,\,\eta_{m_km_k}=1\,,\,\eta_{n_kn_k}=-1$ for $k=1,...,d$. Let
$D:=\{m_1,n_1,...,m_d,n_d\}$ and $\chi^D$ be the characteristic function of
$D$. A short calculation shows that in this situation:
$s^{ij}\eta_{jp}\chi^D(i)-s^{ij}\eta_{ip}\chi^D(j)=-\chi^D(p)$. \\
Let us define: $\tilde{g}_k:=\chi^D(k) f+g_k\,,\,k=2,...,n$ and
$\tilde{U}:=<\tilde{f}>\oplus <\tilde{g}_2,...,\tilde{g}_n>.$
\begin{lem}
$so(p,q)=\frak h_1\oplus\tilde{U}=\frak h_2\oplus\tilde{U}$ and
$\,\tilde{U}$ is a Lie subalgebra.\\
Proof:
$[s,\tilde{g}_p]=[s,g_p]=s^{ij}(\eta_{jp}g_i-\eta_{ip}g_j)=
s^{ij}(\eta_{jp}(g_i+\chi^D(i) f)-\eta_{ip}(g_j+\chi^D(j)
f))+\\-(s^{ij}\eta_{jp}\chi^D(i) -s^{ij}\eta_{ip}\chi^D(j))f=
s^{ij}(\eta_{jp}\tilde{g}_i-\eta_{ip}\tilde{g}_j)+\chi^D(p)f.$\\
{\em From this it follows:} $[\tilde{f},\tilde{g}_p]=[f+s,\chi^D(p) f+g_p]=
\tilde{g}_p+s^{ij}(\eta_{jp}\tilde{g}_i-\eta_{ip}\tilde{g}_j)$.\\
$[\tilde{g}_k,\tilde{g}_l]=\chi^D(k) g_l-\chi^D(l)g_k=\chi^D(k) \tilde{g}_l-\chi^D(l)\tilde{g}_k$\\
{\em In this way $\tilde{U}$ is a subalgebra and simple
calculations show that it is complementary to $\frak h_1$ and $\frak h_2.\; \Box$}
\end{lem}
{\it Double Lie algebra $(so(p,q);so(p-1,q),\tilde{U})$.}\\
We have: $\tilde{f}=v_{n+1}^*\,,\,\tilde{g}_l=g_l=v_l^*$ for $l\not\in D$ and
$\tilde{g}_l-\tilde{f}=v_l^*$ for $l\in D$.
From the bracket on $\tilde{U}$: $[v_{n+1}^*,v_k^*]=v_k^*+
s^{ij}(\eta_{jk}v_i^*-\eta_{ik}v_j^*)$ and
$[v_k^*,v_l^*]=2(\chi^D(l)\eta_{ki}s^{mi}-\chi^D(k)\eta_{li}s^{mi})v_m^*$.\\
So $f_{ijk}=F_{ijk}+2(\chi^D(j)s_{ki}-\chi^D(i)s_{kj})$ where we put
$F_{ijk}$ - the structure constants for\\ \mbox{$(so(p,q);so(p-1,q),\tilde{u})$}.
In this way: $b=b_{v_{n+1}}+v_{n+1}\wedge s-
((v_{m_1}-v_{n_1})+\dots +(v_{m_d}-v_{n_d}))\wedge s$ and this is $b$ of type 4
for $\eta(x,x)<0.$
\vspace{1ex}
{\it Double Lie algebra $(so(p,q);so(p,q-1),\tilde{U})$.}\\
This is completely analogous to the above case and we get solution of type 4
with $\eta(x,x)>0$.
\begin{re}{\em The solutions of type 1 and 2 for null vectors and of type 3
are the special cases of the following double Lie algebras. \\
Let $\overline{f}:=e_1+\lambda \la{1}{n+1} +s +g$ where $\lambda\in R\,,\,s$-as
above$,\, g\in<\la{1}{k}+\la{k}{n+1}\,,\,k=2,...,n>$ \\
$w:=e_1-e_{n+1}.$ For any basis $(x_k)$ of $<e_2,...,e_n>$ let
$g_k:=\la{1}{x_k}+\la{x_k}{n+1}.$ Then $<\overline{f},w,x_k>$ is a subalgebra of
$iso(p,q)$ complementary to $so(p,q)$, the same holds for
$<\overline{f},w,x_k+\lambda g_k>.$ Moreover if $<e_2,...,e_n>$ is a direct sum
of $s$-invariant subspaces we can make above choice on each subspace
separately. This leads to the family of double Lie algebras of form
$(iso(p,q);so(p,q),\frak a)$ and a family of cobrackets on $so(p,q)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}
so(p,q)^0\subset iso(p,q)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} (iso(p,q))^*$ which we can identify with
$iso(p,q)$. }\end{re}
\section{Global decompositions}
Let $G$ be a Poisson-Lie group. Then $\frak g^*$ and $\frak m:=\frak g\bowtie\frak g^*$ are Lie
algebras. We consider the following problem: to find a connected Lie group
$M$ with Lie algebra $\frak m$ such that:
\begin{enumerate}
\item $G$ is a Lie subgroup of $M$
\item $M=G G^*$ (or at least $G G^*$ is dense in $M$), where $G^*$ is the
analytic subgroup of $M$ with Lie algebra $\frak g^*$.
\end{enumerate}
We study this problem for two of the Poisson-Lie groups obtained in the last
section: $E(n):=SO(n)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} R^n$ and $P_0(n):=SO_0(1,n-1)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} R^n$ coming from the
double Lie algebras $(so(1,n);so(n),\tilde{u})$ and $(so(1,n);so(1,n-1),\tilde{u})$
(notation as in section 3).
Let $V$ be a finite dimensional, real vector space and $K\subset GL(V)$ a
closed, connected subgroup which acts on V without fixed points (except 0).
Let $G:=K\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} V$ be a semidirect product and $\frak g=\frak k\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} v$ its Lie algebra. In
this situation the center of $G$ is trivial and $G=Int(\frak g)$ ($Int(\frak g)$ is the
adjoint group of $\frak g$\cite{He}.)
Suppose we are given a bialgebra structure on $\frak g$ as in
section 2. Then we know that $\frak g^*=\frak k^0\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} v^0$ with $v^0$-abelian ideal
and $\frak m:=\frak g\bowtie \frak g^*=(\frak k\oplus \frak k^0)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} (v\oplus v^0)=:\frak h\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} \frak h^*$ and the
action is a coadjoint action. We assume that $\frak h$ is semisimple.
Then the center of $\frak m$ is trivial.
Let $M$ be a connected Lie group with Lie
algebra $\frak m$; $H,H^*,G^*,K^0,V^0$ be analytic subgroups with Lie algebras
$\frak h,\frak h^*,\frak g^*,\frak k^0,v^0$ respectively. Moreover let us assume that $G$ is
contained in $M$, so $G,K,V$ are identified with analytic subgroups of $M$
with Lie algebras $\frak g,\frak k,v$ and $K,V$ are closed in $G$. Since $\frak h$ is
semisimple $\tilde{H}:=Int(\frak h)$ is a closed \cite{He} (and by definition
connected)
subgroup of $GL(\frak h)$ and $\tilde{H}$ acts on $\frak h^*$ without fixed points
(except 0). Let $\tilde{M}:=\tilde{H}\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} \frak h^*$ then $\tilde{M}$ has
trivial center and the same Lie algebra as $M$, so $\tilde{M}=Int(\frak m)$ and
$M$ is a covering group of $\tilde{M}$, the covering homomorphism is given by
$\phi:=Ad_M$. We have $\tilde{H}=\phi(H)$ and let
$\tilde{G^*},\tilde{K^0},...$ denote the images by $\phi$ of $G^*,K^0,...$.
These are analytic subgroups of $\tilde{M}$ with Lie algebras
$\frak g^*,\frak k^0,...$. We are going to prove the following:
\begin{prop} If $\,G G^*$ is dense in $M$ then $\tilde{K} \tilde{K^0}$ is
dense in $\tilde{H}$.
Proof: {\em Let $\tilde{h}\in\tilde{H}$ and $\tilde{h}=\phi(m)$ for some
$m\in M$.
$G G^*$ is dense in $M$, so $m=\lim g_n g_n^*$ for $g_n\in G,\,g_n^*\in
G^*$. But $g_n=k_n v_n,\,k_n\in K,\,v_n\in V$ and since $\frak g^*=\frak k^0\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} v^0$
any element of $G^*$ has, possibly non unique, decomposition $g_n^*=k_n^0
v_n^0,\,k_n^0\in K^0,\,v_n^0\in V^0$. Because $\frak h^*$
is an ideal in $\frak m$, $H^*$ is normal subgroup and $v_nk_n^0=k_n^0 x_n$ for
some $x_n\in H^*$. In this way $\tilde{h}=\lim \phi(k_n)\phi(k_n^0)\phi(x_n)
\phi(v_n^0)=\lim \phi(k_n)\phi(k_n^0)\phi(x_n v_n^0)$ with
$\phi(k_n)\phi(k_n^0)\in \tilde{K}\tilde{K^0}\subset \tilde{H}$ and $\phi(x_n v_n^0) \in
\frak h^*$. Now, the convergence
in $\tilde{M}$ is convergence along "coordinates" in $\tilde{H}$ and $\frak h^*$
so $\tilde{h}=\lim \phi(k_n) \phi(k_n^0)$ and $\tilde{K}\tilde{K^0}$ is dense
in $\tilde{H}$. $\Box$ }
\end{prop}
In the following we need the {\it Iwasawa decomposition of
$\,SO_0(1,n)$.} \cite{He} \\
Let $so(1,n)=\frak k\oplus\frak a\oplus\frak n$ where $\frak k:=<\la{i}{j}\,,\,2\leq i,j\leq
n+1>=so(n) \,,\; \frak a:=<\la{1}{n+1}>\,,\;\\
\frak n:=<\la{1}{k}+\la{k}{n+1}\,,\,2\leq
k\leq n>$ be the Iwasawa decomposition of $so(1,n)$. To this corresponds
decomposition of a connected component of the identity: $SO_0(1,n)=K A N$
where $K,A,N$ are analytic subgroups of $SO_0(1,n)$ with algebras $\frak k,\frak a,\frak n$
respectively. \vspace{1ex}
\\In our case these subgroups look as follows:
$K=\left\{\left( \begin{array}{cc} 1 & 0\\0 & T \end{array} \right)\;,\,T\in
SO(n)\right\}$ \\
$A$ is one parameter subgroup: $A(t)=\left( \begin{array}{ccc} \cosh t & 0
&\sinh t\\ 0 & I &0\\ \sinh t & 0 &\cosh t \end{array} \right) \,,\, t\in R\,,\, I$
is $(n-1)\times (n-1)$ identity matrix; \vspace{1ex}\\
and elements of $N$: $N(x)= \left( \begin{array}{ccc}
1+\frac{1}{2} |x|^2 & -x & \frac{1}{2}|x|^2 \\ -x^t &
I & -x^t \\
-\frac{1}{2}|x|^2 & x & 1-\frac{1}{2}|x|^2
\end{array} \right)\,,\; x:=(x_2,...,x_n)\in
R^{n-1},\,|x|^2~=~\sum_{i=2}^n x_i^2$.\vspace{1ex}\\
Moreover $N$ is commutative
and $A(t) N(x)=N(e^{-t}x) A(t)$.\vspace{1ex}
Now we pass to Poisson-Lie groups
$E(n):=SO(n)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} R^n$ and $P_0(n):=SO_0(1,n-1)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} R^n$.
In both cases $\tilde{H}=SO_0(1,n)\,,\,
\tilde{M}=SO_0(1,n)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex} so(1,n)^*$ ;
$\frak k^0=\tilde{u}=<\tilde{f}>\oplus\frak n\,,\\
\tilde{K^0}=F N$ where
$F$ is one parameter subgroup of elements: $\exp (t \tilde{f})=:F(t)= A(t) S(t)
=S(t) A(t)$,\\
$S(t):=\left(\begin{array}{ccc} 1 & 0 & 0\\
0 &\exp (t s)&0\\0&0&1\end{array} \right)\;\;,\,
A(t),\,N$-as above.
\vspace{1ex}
{\it The Euclidean group.} \\
$\;\frak k:=so(n)=<\la{i}{j}\;,\,2\leq i,j\leq n+1>\,,\,\tilde{K}=\left\{\left(
\begin{array}{cc} 1 & 0\\0 & T \end{array} \right)\;,\,T\in SO(n)\right\}$. In this case we will show that the
global decomposition $SO_0(1,n)=\tilde{K} F N$ holds and this is
sligthly modified Iwasawa decomposition.\\
Let $SO_0(1,n)\ni g=k A(p) N(y)\,,\,\,k\in K\,,\,A(p)\in
A\,,\,N(y)\in N$ be a decomposition of $g$. We seek for
$\tilde{k}\in\tilde{K}=K\,,\, F(t)\,,\,N(x)$ such that:
$\tilde{k} F(t) N(x)=g= k A(p) N(y)$. Using the definition of $F$
: $\tilde{k} S(t) A(t) N(x)= k A(p) N(y)$. Since $S(t)\in K$ and
the decomposition is unique we have: $\tilde{k} S(t)=k\,;\;
t=p\,;\,x=y$ and $\tilde{k}=k S(-p)$.
This proves that the decomposition is global and if $s=0$ this is just the
Iwasawa decomposition. Hence the Manin group for $E(n)$ is $SO_0(1,n)\hspace{0.3ex}\tiny{\rhd\mbox{\hspace{-2ex}}<}\hspace{0.3ex}
so(1,n)^*$.
{\it The Poincare group.}\\ $\;\frak k:=so(1,n-1)=<\la{i}{j}\;,\,1\leq i,j\leq n>
\,,\,\tilde{K}=\left\{\left( \begin{array}{cc} \tilde{T} & 0\\0 & 1\end{array}
\right) \;,\,\tilde{T}\in SO_0(1,n-1)\right\}$.\\ We will show that
$SO_0(1,n)\setminus \tilde{K} F N$ contains an open subset.\\
Let $W$ be a function on $SO_0(1,n)$ defined by:
$W(g):=\eta(g(e_1-e_{n+1}),e_{n+1})$. This function is obviously continous,
and is easy to see that if $g=\tilde{k} f n\,,\,\tilde{k}\in\tilde{K},f\in
F,n\in N$ then $W(g)>0$.
But for $SO_0(1,n)\ni k_0:=\left(\begin{array}{cc} I_{n-1} &0\\
0 & -I_2\end{array}\right)\,,\,I_l$-is $l\times l$ identity matrix we have
$W(k_0)<0$ what proves the assertion.
Next we show that one can find non-connected extension of $P_0(n)$ for which
there exists connected (in fact simply connected) Poisson dual group $G^*$
and the set $G G^*$ is dense in $M$.
\begin{lem} $\tilde{K} F N=\{k a n\,:\,k\in K,a\in A,n\in
N,k_{n+1 n+1}>0\}.$\\
Proof: {\em We try to solve:
$\tilde{k} F(t) N(x)= k A(p) N(y) \;,\,\tilde{k}\in
\tilde{K}\,,\,k\in K$.
Using the commutation relation between $A$ and $N$ we
get:\\ $\tilde{k} S(t)=k A(p) N(y-x)A(-t)=k A(p-t)
N(e^{-t}(y-x))$.\\ Let $z:=e^{-t}(y-x)$ and
$w:=p-t$, then:\vspace{1ex}\\
$A(w) N(z)=\left( \begin{array}{ccc} \cosh w+\frac{1}{2}|z|^2
e^{-w} & -z e^{-w} & \sinh w+\frac{1}{2} |z|^2 e^{-w}\\
-z^t & I & -z^t\\ \sinh w-\frac{1}{2}|z|^2 e^{-w} &z
e^{-w} & \cosh w-\frac{1}{2}|z|^2 e^{-w} \end{array} \right).$\vspace{1ex}\\
Now we look at the last row of the equality $\tilde{k} S(t)=k A(w) N(z)$
: \\
$(n+1,1):\,-\sum_{j=2}^n k_{n+1j}z_j+k_{n+1n+1}(\sinh w-\frac{1}{2}|z|^2
e^{-w})=0 \\
(n+1,j):\,k_{n+1j}+k_{n+1n+1}z_j e^{-w}=0\;,\,j=2,..,n\\
(n+1,n+1):\,-\sum_{j=2}^n k_{n+1j}z_j+k_{n+1n+1}(\cosh w-\frac{1}{2}|z|^2
e^{-w})=1 .$\\
It follows that $k_{n+1 n+1} e^{-w}=1$ and $z_j=-k_{n+1 j}$.
This determines $t$ and $x$. So for any $k\in K$ such that $k_{n+1
n+1}>0$ and any $a\in A,n\in N$ we can find $\tilde{k}\in\tilde{K},f\in F,
m\in N$ such that $k a n=\tilde{k} f m$.} $\Box$
\end{lem}
If $k_0$ is as above it is clear that $k_0\tilde{K} k_0=\tilde{K}$ so the set
$X:=\tilde{K}\cup k_0\tilde{K}$ is a (non connected) closed Lie subgroup of
$SO_0(1,n)$. Moreover from the lemma above any element of $SO_0(1,n)\ni g=k a
n$ such that $k_{n+1 n+1}\neq 0$ can be uniquely decomposed $g=x f m$ with
$x\in X,f\in F,m\in N$.\vspace{2ex}
\noindent{\bf Aknowledgments\hspace{1em}} I woud like to thank to Dr. S.
Zakrzewski for inspiration of this work and to Dr. A. Strasburger for
dicsussions. \\
The research was supported by Polish KBN grant No. 2 P301 020 07.
|
1,116,691,498,444 | arxiv | \section{Introduction}
\label{sec1}
Internetwork (IN) magnetic fields are dynamic magnetic structures that populate the interior of supergranular cells \citep{LivingstonHarvey, Smithson1975}. They are spread all over the Sun \citep{Wang}, maintain the photospheric network \citep{Gosicetal2014}, and may hold a significant fraction of the total magnetic energy stored at the solar surface \citep{TrujilloBueno2004}. For these reasons IN fields are considered to be the main building blocks of the quiet Sun (QS) magnetism (see \citealt{BellotRubioOrozcoSuarez2019} for a review).
Recent Hinode observations showed that IN fields mainly appear in the form of magnetic bipoles in the photosphere \citep{2022ApJ...925..188G}, likely generated by small-scale surface dynamo \citep{Rempel2014}. According to some numerical models \citep{Isobeetal2008, Amarietal2015, MorenoInsertisetal2018}, and observations \citep[e.g.,][]{MartinezGonzalezBellotRubio2009, MartinezGonzalezetal2010, 2021ApJ...911...41G}, these fields may upon appearance in the photosphere rise through the lower atmospheric layers, and locally heat the chromosphere and transition region.
Considering the magnetic and energy budget of IN fields, it is therefore important to determine the global contribution of the emerging IN fields to the dynamics and energetics of the chromosphere and the atmospheric layers above. This open question has not yet been addressed in detail, using high resolution observations that simultaneously cover the solar atmosphere from the photospheric to coronal heights. The main reason for this was the lack of suitable observations and the need for sophisticated analysis that allows to identify footpoints of magnetic loops, and determine their history in a reliable way. Such observations at the photospheric level are provided by the Narrowband Filter Imager \citep[NFI;][]{Tsuneta} aboard the Hinode satellite \citep{2007SoPh..243....3K}, and at the chromospheric/transition region and coronal levels by the Interface Region Imaging Spectrograph \citep[IRIS;][]{DePontieuetal2014} and the Atmospheric Imaging Assembly \citep[AIA;][]{Lemenetal2012} onboard the Solar Dynamics Observatory \citep[SDO;][]{Pesnelletal2012}. Furthermore, to understand the impact of newly emerging IN fields on the chromospheric energy balance, one would need to determine the thermodynamic properties from chromospheric lines, considering the non-local thermodynamic equilibrium (non-LTE) radiative transfer. Diagnosing chromospheric conditions requires inversion codes that are typically slow and difficult to use. In this work we will take advantage of a new approach that solves these issues through a combination of machine learning and classical inversion techniques to speed up and facilitate the recovery of thermodynamical information from the solar spectra \citep[e.g.,][]{SainzDaldaetal2019}.
The main goal in this letter is to establish the global impact of the newly emerging IN fields on the lower solar atmosphere. We address this open question by employing coordinated, multi-wavelength IRIS, Hinode and SDO observations. These instruments allow us to study the spatio-temporal evolution of the QS fields at high spatial, spectral, and temporal resolution, while observing the solar atmosphere from the photosphere up to the transition region and corona.
The observations used in this paper are described in Section \ref{sec2}. The
identification, classification and tracking of IN bipolar flux features is explained in Section \ref{sec3}. Section \ref{sec4} provides the results, while conclusions are given in Section \ref{sec5}.
\section{Observations and data processing}
\label{sec2}
The observations used in this work were obtained on 2013 March 23. IRIS measurements start at 07:09:49~UT and end at 12:05:37~UT. Hinode data set covers this interval from 08:04:38~UT to 10:59:36~UT. The observations show the spatial and temporal evolution of a QS region at the disk center.
IRIS data set is a medium-sit-and-stare raster, taking spectra in the near ultraviolet (NUV) band\footnote{IRIS also takes spectra in the two far ultraviolet domains, but these were not used in this letter.} in the wavelength range from 2790~\AA\ to 2835~\AA\/. The NUV spectroscopic measurements sample the solar atmosphere from the photosphere to the upper chromosphere. The spectra are recorded every 5 seconds along a slit length of $60$\arcsec. Slit-jaw images (pixel size is $0\farcs16$) were taken using the \ion{C}{2} 1330~\AA\ (SJI 1330), \ion{Si}{4} 1400~\AA\ (SJI 1400), \ion{Mg}{2} k 2796~\AA\ (SJI 2796), and \ion{Mg}{2} h wing at 2832~\AA\ (SJI 2832) filters, compensating for the solar rotation. The cadence of the slit-jaw images are 18~s, 15~s, 15~s, and 89~s, respectively. The IRIS data were corrected for dark current, flat-field, geometric distortion, and scattered light \citep{2018SoPh..293..149W}.
Using the IRIS$^{2}$ inversion code\footnote{The IRIS$^{2}$ code is publicly available in the IRIS tree of SolarSoft. For more details about the code and the installation see \url{https://iris.lmsal.com/iris2}.} \citep{SainzDaldaetal2019} we derived the thermodynamical properties of the observed QS atmosphere as a function of the optical depth. The code employs the k-means clustering method to build a database of the representative IRIS \ion{Mg}{2} h and k spectral profiles (RP) and the corresponding atmospheric models. These RPs were inverted with the STiC code\footnote{STiC is publicly available to the community and can be downloaded from the author’s web site at \url{https://github.com/jaimedelacruz/stic}.} \citep{delaCruzRodriguezetal2016, delaCruzRodriguezetal2019}. For each observed \ion{Mg}{2} h and k pair, IRIS$^{2}$ assigns the model atmosphere resulting from the inversion of the closest RP to the observed profiles.
The NFI was employed in shutterless mode to obtain the full Stokes vector in the photospheric \ion{Fe}{1} 5250~\AA\ line at 2 wavelength positions from the line center. These observations provide circular and linear polarization maps, showing photospheric activity of the vertical (loop footpoints) and horizontal (loop tops) components of magnetic fields, respectively. After the data reduction process and co-alignment of Hinode and IRIS observations, the effective field of view (FOV) was reduced to $35\arcsec \times 60\arcsec$ (the pixel size is $0\farcs16$), which is sufficient to capture the evolution of at least two supergranular cells for two hours and 40 minutes at a cadence of $\sim60$~s. This allows us to track the temporal evolution of IN fields in a magnetogram sequence that considerably exceeds the mean lifetime of IN magnetic structures on granular scales. Magnetograms $M$ were calculated in the standard way using the Stokes $I$ and $V$ filtergrams:
\begin{equation}
\label{mag_eq}
M=\frac{1}{2}\left(
\frac{V_{\text{blue}}}{I_{\text{blue}}}-\frac{V_{\text{red}}}{I_{\text{red}}}
\right),
\end{equation}
\noindent where ``blue'' indicates the measurements in the blue wing of the line and ``red'' in the red wing. The linear polarization maps $\rm LP$ are computed as ${\rm LP}= \sum_{i=1}^{2} \sqrt{Q(\lambda_i)^2 + U(\lambda_i)^2}/I(\lambda_i)/2$. All magnetogram and LP maps were smoothed using a $3\times3$ Gaussian-type spatial kernel to reduce the noise, and the five-minute oscillations were removed from the maps by applying a subsonic filter \citep{1989ApJ...336..475T, 1992A&A...256..652S}.
\begin{figure*}[!t]
\centering \includegraphics[width=0.99\textwidth]{fig1.png}
\caption{\textit{From left to right:} Hinode/NFI magnetograms and linear polarization maps, IRIS SJI 1400 and IRIS SJI 2796 slit-jaw images. The detected bipoles are enclosed with contours of different colors. Regions 1, 2, and 3 (red ellipses) show the largest emerging cluster of magnetic elements, one small-scale IN loop, and a NW patch that originate from a previously emerged IN bipolar system. The animation of this figure runs from $\Delta t$=0:00:00 to $\Delta t$=2:36:45.\newline
({\em An animation of this figure is available.})}
\label{fig1}
\end{figure*}
We also make use of the Helioseismic and Magnetic Imager \citep[HMI;][]{Scherreretal2012} and AIA observations. This allows us to determine the evolution of the observed QS region before the IRIS and Hinode observations started and to examine emission at coronal heights.
The alignment of the datasets was carried out by compensating for solar rotation and scaling all images to match the IRIS pixel size. All IRIS, Hinode and SDO sequences were interpolated in time applying the nearest neighbor method of interpolation to match the cadence of the SJI 2796 images (15~s). Images are then aligned comparing prominent network (NW) features and bright points in IRIS SJI 2832 images with the Hinode intensity filtergrams and AIA 1600 \AA\/ channel.
\section{Identification and tracking of internetwork bipoles}
\label{sec3}
To detect IN magnetic bipoles (loops and clusters) and separate them from the unipolar fields (isolated flux concentrations), we first identified all individual magnetic features in the magnetograms and LP maps. We consider loops to represent two circular polarization patches (positive and negative polarity footpoints) moving away from each other, while flux clusters consist of two or more patches that emerge within a short time interval in a relatively small area.
Using the YAFTA code\footnote{YAFTA (Yet Another Feature Tracking Algorithm) is an automatic tracking code written in IDL and can be downloaded from the author's website at http://solarmuri.ssl.berkeley.edu/\(\sim\)welsch/public/software/YAFTA.} and the downhill identification method \citep[]{WelschLongcope}, we automatically tracked all the detected flux patches to determine their spatio-temporal evolution. This process includes identifications of all merging, fragmentation and cancellation events that magnetic patches may undergo during their lifetimes. In this way, we can determine the history of every detected magnetic feature. Real features in the magnetograms are separated from the background signal by setting a flux density threshold to $2\sigma$ (10~Mx~cm$^{-2}$). This allows as to detect more faint magnetic elements, considering all of them to be real if their minimum size is at least 4 pixels and they live two frames or more. Magnetic features that appear and disappear in situ and are visible in only one frame are discarded. The reason is that those flux patches may just represent intrinsic flux fluctuations around the threshold level.
The appearance of loop footpoints and clusters in the photosphere is preceded by LP signal between footpoints that move away from each other. Therefore, magnetic bipoles are identified by searching for all LP signals (loop tops) that are followed by pairs of opposite-polarity flux features that appear in situ (loop footpoints). To be selected, these footpoints have to appear within 6 minutes after the first one becomes visible in a magnetogram, and they have to move away from each other. Although clusters bring numerous magnetic patches to the solar surface, they follow the same pattern of the spatio-temporal evolution as loops, i.e., flux features move away from each other with respect to their common center of appearance. Usually, there are multiple LP patches within clusters. The tracking and identification of IN bipoles in this work is similar to the method described in \cite{2022ApJ...925..188G}, the main difference being that here we use the NFI LP maps instead of extrapolations of the magnetic field lines to identify the loop tops.
For the strongest flux patches visible in the first frame, we cannot determine their history from the NFI magnetograms. Thus, we used HMI data to determine if they appear as unipolar or bipolar structures. This is important because the strongest magnetic elements may have a considerable impact on the lower solar atmosphere. Since HMI is not sensitive to the weakest IN patches, we classified as bipoles only those flux patches that are clearly resolved and appear in situ, following the expected pattern of flux emergence.
\begin{figure*}[!t]
\centering \includegraphics[width=0.99\textwidth]{fig2.png}
\caption{Temperature spatio-temporal map from the IRIS$^{2}$ inversions at $\log_{10}\tau_{500}=-5.8$. The white boxes indicate locations and times when the emerging IN bipoles were under the IRIS slit.}
\label{fig2}
\end{figure*}
\begin{figure*}[!t]
\centering \includegraphics[width=0.99\textwidth]{fig3.png}
\caption{\textit{From left to right:} Hinode/NFI magnetogram, AIA 304 \AA\/, AIA 171 \AA\/, and AIA 193 \AA\/ images. The detected IN bipoles are enclosed with contours having different colors. Regions 1, 2, and 3 marked with the red ellipses show the same as in Figure~\ref{fig1}. The animation of this figure runs from $\Delta t$=0:00:00 to $\Delta t$=2:36:45.\newline
({\em An animation of this figure is available.})}
\label{fig3}
\end{figure*}
\section{Results}
\label{sec4}
Using HMI and Hinode/NFI magnetograms we identified and tracked the spatio-temporal evolution of individual magnetic elements representing footpoints of $161$ bipolar structures (IN loops and clusters of magnetic elements). This translates into an emergence rate of $\sim0.038$~bipoles per hour and arcsec$^{2}$, which is in agreement with the results reported by \cite{2022ApJ...925..188G}. If we take into account the total area occupied by the footpoints, only $2\%$ of the available FOV at any given time is covered by bipolar IN fields. Note that this ratio depends on spatial resolution, magnetic sensitivity, and intrinsic fluctuations of the total instantaneous unsigned IN flux \citep[e.g.,][]{2022ApJ...925..188G}.
The two largest bipoles whose footpoints are visible in the first NFI magnetogram are identified employing HMI data. They are marked in Figure~\ref{fig1} with violet contours inside the encircled region 1 (red ellipse) and at the bottom of the FOV at $(x, y) = (16\arcsec, 0\arcsec)$. By the time Hinode/NFI started to observe, most of the footpoints of the two clusters either transformed into NW features or canceled with the opposite polarity NW elements. The tracking results can be evaluated with the animation accompanying Figure~\ref{fig1}, which shows all the magnetic bipoles (left panel) detected in our Hinode/NFI magnetograms. Flux patches belonging to the same bipole have the same colors. The corresponding contours are overplotted on the LP maps, SJI 1400 and SJI 2796 images, from left to right, respectively.
To detect chromospheric activity related to the newly emerging IN fields, we used SJI 1400 and 2796 images. As can be seen from Figure~\ref{fig1} and the accompanying movie, the strongest emission in both filtergram sequences is co-spatial with strong magnetic elements, i.e., large clusters and the positive- and negative- polarity NW elements centered at $(x, y) = (16\arcsec, 0\arcsec)$ and $(x, y) = (10\arcsec, 56\arcsec)$, respectively. The rest of the FOV is overwhelmed by bright features. We determined that $\sim90\%$ of the detected bipoles overlap with IRIS brightenings in SJI 1400 that are identified considering all the pixels with the signal above a threshold level of 60 counts per pixel, as in \cite{Gosicetal2018}. We remind the reader that the SJI 1400 filter is sensitive to emission from the transition region \ion{Si}{4} 1394/1403~\AA\ lines and continuum formed in the upper photosphere/lower chromosphere. Therefore, most of the SJI 1400 bright patches in the QS regions are formed in the chromosphere due to upward propagating acoustic waves in non-magnetic environment \citep{MartinezSykoraetal2015}. When we apply a subsonic filter to remove those short-lived (two minutes) bright grains, the overlap between the IN bipoles and IRIS SJI 1400 brightenings drops to $\sim60\%$, suggesting that many small-scale IN loops and clusters do not perturb the chromospheric layers.
By visual inspection of IRIS SJI features above the detected bipoles, we determined that the bipoles are either embedded in regions with already ongoing activities in the chromosphere or the overlapping IRIS SJI 1400 brightenings ($\sim60\%$) above them do not seem to be different from the background activity (for example the loop inside region 2 in the SJI 1400 panel in Figures \ref{fig1} and \ref{fig3}). This scenario applies to all other clusters, except for the strongest clusters (regions 1 and 3). The southern positive polarity magnetic element ($(x, y) = (16\arcsec, 0\arcsec)$) locally impacts the chromosphere through interactions with the surrounding opposite-polarity flux features. Numerous dynamic and episodic bright loops and grains can be seen around this footpoint, probably energized by reconnection of the magnetic field lines of the footpoint and the surrounding flux patches.
The cluster within region 1 exhibits the temporal and spatial evolution similar to the cluster described in \cite{2021ApJ...911...41G}. The footpoints expanded with time and started interacting with the negative polarity NW patches in the north. Eventually the region produced a surge-like event around $\Delta t$=00:25:30, (onset at $\Delta t$=00:22:30), which is expected to be observed when new and preexisting fields reconnect \citep{Guglielminoetal2018, NobregaSiverioetal2017}.
Figure~\ref{fig2} shows the temperature map derived from IRIS$^{2}$ inversions. The IRIS slit covered seven emerging bipoles. Five of them are embedded in the background activity and do not perturb the chromosphere. They do not produce any excess emission neither in the IRIS NUV nor FUV spectral lines. The bipole labeled as B4, is co-spatial with increased temperature, but this is likely due to cancellation with the opposite polarity magnetic features in its vicinity \citep{Gosicetal2018}. Only a few bipoles can be associated with the chromospheric activity. For example, the negative polarity footpoint (B5), clearly shows an increase of the chromospheric temperature. This magnetic element eventually becomes an NW element. Bipole B1 emerges next to an ongoing cancellation event (hence a higher temperature before the bipole emerged), with which the positive footpoint starts interacting and eventually completely disappears. This cancellation maintained an increased temperature in that region for the next 26 minutes. Another intriguing event is B2 loop that shows a slight temperature increase towards the end of the white box. This is probably the result of an upward propagating wave because we do not see any activity above the footpoints in the filtered IRIS slit-jaw images. In addition, this is a small, short-lived loop (8 minutes), so it is unlikely that in such a short time this loop can reach the chromosphere.
Very limited activity in the lower solar atmosphere within the observed QS region is also apparent in the AIA filtergrams displayed in Fig~\ref{fig3}. The AIA 304~\AA\/, 171~\AA\/ and 193~\AA\/ channels show the chromospheric (304~\AA\/) and coronal (171~\AA\/ and 193~\AA\/) activity inside regions 1 and 3. The rest of the FOV looks very quiet with some long loops extended across the FOV that originate in an active region in the north (not visible in the Hinode and IRIS observations).
\section{Conclusions}
\label{sec5}
In this work we used Hinode/NFI, IRIS and SDO/AIA observations to detect newly emerging IN bipoles in the solar photosphere and estimate the global and direct contribution of emerging fields on the chromospheric dynamics and energetics. Our results suggest that the majority ($\sim98\%$ based on the SJI 1400 activity) of bipolar IN structures do not have enough magnetic buoyancy nor live long enough to rise through the solar atmosphere and directly affect the solar chromosphere and beyond. This result should be understood as a minimum. More active QS regions may generate stronger emerging fields capable of rising through the solar atmosphere. Also, our statistics could have been different had we identified more bipoles under the slit, which will be possible in future with multislit instrument such as MUSE \citep{2020ApJ...888....3D}. In the meantime, if we consider only the bipoles under the slit, despite of the small sample, then $40\%$ of the loops may heat the chromosphere either through reconnection with the overlying magnetic fields or through cancellation with the surrounding flux patches.
Based on our observations, only the strongest three detected bipoles produced local temperature increase in the chromosphere. They are also capable of generating surge-like phenomena through reconnection of their magnetic field lines with the preexisting fields. We conclude that newly emerging IN bipoles, at the sensitivity levels and spatial resolution of Hinode/NFI magnetograms, cannot globally maintain the chromospheric heating in a direct way through interaction with the ambient overlying magnetic fields. We either do not see a lot of evidence of heating, except for larger events, or the large events are too sporadic in space and time to considerably support the chromospheric heating. It would be interesting to study longer-duration events to increase the statistical sample that can be studied under the slit. We also note that our analysis has been focused on detecting changes in emission or chromospheric temperature as a result of the detected emergence of IN magnetic elements. We did not investigate a scenario in which the continual emergence of undetected or undetectable IN elements leads to a steady heating of the atmosphere or continuous background emission. The results presented in this letter also do not exclude a possibility that the footpoints of IN bipoles may possibly contribute to the chromospheric heating indirectly through other mechanisms such as magneto-acoustic waves and shocks, braiding of the magnetic field lines, swirls or cancellations with the opposite polarity fields in the intergranular lanes.
To better understand the smallest and weakest QS fields, we will need long duration observations with higher spatial resolution and sensitivity. Such observations are, for example, achievable with the Daniel K. Inouye Solar Telescope \citep[DKIST;][]{Elmoreetal2014}, and from space with the Solar Orbiter's Polarimetric and Helioseismic Imager \citep{2020A&A...642A..11S}. These instruments may detect and resolve fields that are not accessible to the currently available telescopes, but they may be continuously emerging and contributing directly to heating through interaction with preexisting fields.
A key aspect of this issue is also to study which processes determine whether emerging IN fields rise through the solar atmosphere and transfer mass and energy. This will be investigated in detail in our future work using radiative MHD Bifrost simulations (\citeauthor{2011A&A...531A.154G} \citeyear{2011A&A...531A.154G}; see also Hansteen et al. 2022, in prep.).
\begin{acknowledgments}
MG, BDP and ASD are supported by NASA contract NNG09FA40C (IRIS). We acknowledge the use of IRIS, Hinode and SDO/AIA/HMI data. IRIS is a NASA Small Explorer Mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by ESA and the Norwegian Space Centre. The Hinode data used here were acquired in the framework of the Hinode Operation Plan 243 {\em ``Effects of Quiet Sun Weak Fields on the Chromosphere and Transition Region.''}. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as a domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). AIA is an instrument on board the Solar Dynamics Observatory, a mission for NASA’s Living With a Star program. This research has made use of NASA’s Astrophysics Data System.
\end{acknowledgments}
|
1,116,691,498,445 | arxiv | \section{Introduction} \label{section:intro}
In \cite{Itai88}, Itai and Rodeh proposed a communication protocol, called the {\em $k$-tree protocol}, which allows all nodes of a network to communicate through a distinguished root node~$v$, even when some set of $k-1$ or fewer edges are removed from the network. The protocol requires the graph $G$ modelling the network to have two properties. First, the graph $G$ must remain connected when any $k-1$ edges are removed, so $k$ can be at most the edge connectivity of $G$. Second, it requires a collection of $k$ spanning trees for $G$, $\{T_1,\ldots,T_k\}$, with the following property (which they called the {\em $k$-tree condition for edges}): for all vertices $w$ distinct from $v$, and for any $i,j$ where $1\leq i < j \leq k$, the paths in $T_i$ and $T_j$ from $v$ to $w$ are internally disjoint.
\newpage
Clearly, if $G$ has $k$ edge-disjoint spanning trees, then it satisfies the $k$-tree condition. (This is not a requirement: for example, a cycle satisfies the $2$-tree condition, but does not have two edge-disjoint spanning trees.) In particular, if the number of edge-disjoint spanning trees (denoted $\sigma(G)$) is equal to the edge connectivity (denoted $\lambda(G)$), then Itai and Rodeh's protocol can be applied; we call such graphs {\em maximum spanning tree packable}, or {\em max-STP}.
In~\cite{ubbnet}, two of the present authors considered a related network protocol, where they require a collection $\mathcal{U}$ of spanning trees (not necessarily pairwise disjoint) for $G$ chosen so that for any $t$ edges (where $t<\lambda(G)$), there exists a spanning tree $T \in \mathcal{U}$ disjoint from those $t$ edges. Ideally, this collection of (not necessarily disjoint) spanning trees (called an {\em uncovering-by-bases}, or UBB, for $G$) should be as small as possible. The class of max-STP graphs with $\lambda(G)=\sigma(G)=k$ is also of interest here, as the $k$ edge-disjoint spanning trees form a UBB which is optimal in two ways: (i) the spanning trees are disjoint, so the UBB is as small as possible; and (ii) the number of edges which can be ``uncovered'' by the collection is as large as possible. (In fact, it was the study of UBBs which led the authors to the results in the present paper.) The notion of UBBs also generalizes to matroids: see \cite[Section~7]{btubb}. Another class of matroids where UBBs arise naturally are as follows.
A {\em base} for a group acting on a set is a subset of points whose pointwise stabilizer is trivial; equivalently, every group element is uniquely specified by its action on those points. (See~\cite{Cameron99} for more details.) A UBB for a permutation group is a collection of bases so that any $t$-subset of points is disjoint from some base in the collection; these have applications to the decoding of permutation codes (see~\cite{ecpg}). In~\cite{CameronFDF95}, Cameron and Fon-Der-Flaass investigated permutation groups whose bases form the bases of a matroid: such groups are known as {\em IBIS groups}. In the case of an IBIS group, a UBB for the group is also a UBB for the corresponding matroid. An important sub-class of IBIS groups are the {\em base-transitive} groups, where the bases lie in a single orbit of the group; constructions of UBBs for many examples of base-transitive groups are given in~\cite{btubb}.
A straightforward example of a max-STP graph $G$ with $\lambda(G)=\sigma(G)=k$ is the graph obtained from a tree $T$ with $n$ edges by replacing each edge of $T$ by $k$ parallel edges. A UBB for $G$ can be formed from disjoint copies of $T$. The cycle matroid of $G$ is obtained from a free matroid by replacing each point with a parallel class of size $k$ (regardless of the structure of $T$); alternatively, this is the transversal matroid of a uniform set-partition. This same matroid arises from a base-transitive group: if $H$ is a group acting regularly on a set $X$ of $k$ points, consider the wreath product $H\wr S_n$ acting on $n$ disjoint copies of $X$, labelled $X_1,\ldots,X_n$. A base for this group consists of a single point chosen from each of $X_1,\ldots,X_n$, and the corresponding matroid is again the transversal matroid of the set system $\{X_1,\ldots,X_n\}$. As a code, this group can correct $d=\lfloor (k-1)/2 \rfloor$ errors, and a UBB requires $d+1$ disjoint transversals; in the cycle matroid of $G$, this is equivalent to $d+1$ disjoint spanning trees. (This class of codes is discussed in more detail in~\cite{thomas}.)
\subsection{Graphs}
Let $G=(V,E)$ be a graph. Throughout this paper, we shall assume that graphs are connected, and we allow for the possibility of multiple edges.
An {\em edge cut} in $G$ is a partition $(V_1,V_2)$ of the vertex set of $G$ into two non-empty subsets. The corresponding set of edges are those with one endpoint in $V_1$ and the other endpoint in $V_2$. Removing the edges of an edge cut disconnects $G$; if the number of such edges is $k$, we call it a {\em $k$-edge cut}. In a mild abuse of terminology, we will use the term ``edge cut'' to refer both to the partition of the vertex set and the set of edges of the cut. The {\em edge-connectivity} of $G$, denoted $\lambda(G)$, is the least value of $k$ for which there exists a $k$-edge cut in $G$. We note that sometimes we will refer to a $k$-edge cut by the set of edges whose removal disconnects the graph, rather than the partition of $V$. Also, we say that $G$ is {\em $k$-edge connected} if $\lambda(G) \geq k$.
The {\em spanning tree packing number} of $G$, denoted $\sigma(G)$, is the maximum number of edge-disjoint spanning trees in $G$. (We usually shorten this to {\em STP number}.) A survey of results on STP numbers can be found in Palmer \cite{Palmer01}. In particular, graphs with given STP number were characterized independently by Nash-Williams \cite{NashWilliams61} and Tutte \cite{Tutte61}, both in 1961.
\begin{thm}[Nash-Williams; Tutte] \label{thm:TutteNW}
A connected graph $G$ has at least $k$ edge-disjoint spanning trees if and only if, for every partition of $V(G)$ into $r$ parts, there are at least $k(r-1)$ edges between the parts.
\end{thm}
It is a straightforward observation that $\sigma(G) \leq \lambda(G)$: clearly, to disconnect $G$ we must remove at least one edge from each of the $\sigma(G)$ disjoint spanning trees (and possibly some other edges as well). Also, in 1983 Gusfield \cite{Gusfield83} showed that it follows from Nash-Williams and Tutte's result that $\lambda(G) \leq 2\sigma(G)$ (see also Diestel \cite[Section 3.5]{Diestel00}). When presented with an inequality such as $\sigma(G) \leq \lambda(G)$, it seems natural to ask when equality is achieved, $\sigma(G) = \lambda(G)$.
\begin{defn} \label{defn:maxSTP}
A graph $G$ is said to be {\em maximum spanning tree-packable}, or {\em max-STP} for short, if $\lambda(G)=\sigma(G)$, i.e.\ the edge connectivity is equal to the spanning tree packing number.
\end{defn}
\begin{example} \label{example:easy}
The graph~$G$ in Figure~\ref{fig:maxSTPexample} is a max-STP graph with $\lambda(G)=\sigma(G)=2$. (In fact, as $G$ has $8$ vertices and $14$ edges, any pair of edge-disjoint spanning trees must contain all edges of $G$.)
\setlength{\unitlength}{7mm}
\begin{figure}[h]
\centering
\begin{picture}(7,2.2)
\put(-0.15,0){\complete}
\put(4.85,0){\complete}
\put(2,0){\line(1,0){3}}
\put(2,2){\line(1,0){3}}
\end{picture}
\caption{A max-STP graph with $\sigma(G)=\lambda(G)=2$. \label{fig:maxSTPexample}}
\end{figure}
\end{example}
In Section~\ref{section:graphdecomp}, we will present a structure theorem for max-STP graphs. The two parameters $\lambda(G)$ and $\sigma(G)$ both have straightforward analogues in matroid theory, so in Section~\ref{section:matroiddecomp} we prove an analogous theorem for matroids where the two parameters agree.
\subsection{Matroids}
A {\em matroid} $M$ consists of a (finite) ground set $E$ together with a family $\mathcal{I}$ of subsets of $E$, called {\em independent sets}, which satisfy the following three axioms:
\begin{itemize}
\item[I1.] $\mathcal{I} \neq \emptyset$;
\item[I2.] if $I\in \mathcal{I}$ and $J\subseteq I$, then $J\in \mathcal{I}$;
\item[I3.] if $I,J\in \mathcal{I}$ and $|J|<|I|$, then there exists $x \in I\setminus J$ such that $J\cup\{x\} \in \mathcal{I}$.
\end{itemize}
The maximal independent sets are called the {\em bases} of $M$; the collection of these is denoted $\mathcal{B}$. The bases are necessarily equicardinal; this size is called the {\em rank} of the matroid and is denoted by $\rank(M)$. The rank of an arbitrary subset $X \subseteq E$ is the size of the largest independent set contained in $X$ and is denoted by $\rank(X)$. For any subsets $X$ and $Y$, the rank satisfies the inequality $\rank(X \cap Y) \leq \rank(X)+\rank(Y)-\rank(X \cup Y)$. A {\em flat} of rank $k$ in a matroid is a maximal set of rank $k$. The intersection of flats is always a flat; a {\em hyperplane} is a flat of rank $\rank(M)-1$. A {\em cocircuit} is a minimal set that intersects every basis, i.e.\ a minimal subset $S \subseteq E$ for which $S\cap B_i \neq \emptyset$ for all $B_i \in \mathcal{B}$. Equivalently, the cocircuits of a matroid are exactly the complements of the hyperplanes of the matroid. If $M=(E,\mathcal{I})$ is a matroid and $X\subseteq E$, the matroid obtained by the {\em deletion} of $X$, denoted $M\setminus X$, has ground set $E\setminus X$, and its collection of independent sets is $\{ I\setminus X \, : \, I\in\mathcal{I} \}$. (For background material on matroids, we refer the reader to Oxley \cite[Chapter~1]{Oxley92}).
One of the motivating examples of matroids (and the one most relevant to this paper) is the {\em cycle matroid} of a graph $G=(V,E)$, where the ground set $E$ is indeed the edge set of $G$, and where the bases are the maximum spanning forests of $G$. We denote this matroid by $M(G)$. The independent sets of $M(G)$ are the subsets of $E$ which contain no cycles of $G$. In particular if $G$ is a connected graph with $n$ vertices, the bases of $M(G)$ are the spanning trees of $G$, and $M(G)$ has rank $n-1$. More generally, if $G$ has $c$ connected components, then $M(G)$ has rank $n-c$. A cocircuit of $M(G)$ is a minimal set of edges whose removal increases the number of components of $G$ by one, i.e., a minimal edge-cut.
The natural analogue of spanning tree packing number for matroids is as follows.
\begin{defn} \label{defn:bpn}
Let $M=(E,\mathcal{I})$ be a matroid. The {\em base packing number} of $M$, denoted $\sigma(M)$, is the size of the largest set of disjoint bases of $M$.
\end{defn}
Matroids with given base packing number were described by Edmonds \cite{Edmonds65} in 1965 (see also Oxley \cite{Oxley92}, Theorem 12.3.11), thereby generalizing the result of Nash-Williams and Tutte (Theorem~\ref{thm:TutteNW} above). Edmonds showed that a matroid $M=(E,\mathcal{I})$ has $k$ disjoint bases if and only if, for every subset $X \subseteq E$, the following inequality holds:
\[ k\cdot\rank(X) + |E\setminus X| \geq k\cdot\rank(M).\]
Edge-connectivity also has a natural analogue for matroids.
\begin{defn} \label{defn:cogirth}
Let $M=(E,\mathcal{I})$ be a matroid. The {\em cogirth} of $M$, denoted $\lambda(M)$, is the smallest size of a cocircuit in $M$.
\end{defn}
The inequality $\sigma(G)\leq\lambda(G)$ carries over to matroids in a straightforward way.
\begin{prop} \label{prop:ineq}
Let $M=(E,\mathcal{I})$ be a matroid. Then $\sigma(M)\leq \lambda(M)$.
\end{prop}
\proof Given any collection of $\sigma(M)$ disjoint bases, any cocircuit of $M$ must intersect each of these bases in at least one element. \endproof
As with graphs, when presented with the inequality in Proposition \ref{prop:ineq}, it seems natural to ask when equality is achieved. In Section~\ref{section:matroiddecomp}, we present a result which describes the structure of matroids for which $\sigma(M)=\lambda(M)$.
\subsection{Preliminary results} \label{subsection:prelims}
We begin with a brief discussion of matroid connectivity. A matroid $M=(E,\mathcal{I})$ is {\em disconnected} if there is a partition of $E$ into non-empty sets $E_1,E_2$, such that there are matroids $M_1=(E_1,\mathcal{B}_1)$ and $M_2=(E_2,\mathcal{B}_2)$ with the property that every base of $M$ is the union of a base for $M_1$ and a base for $M_2$. In this situation, we write $M= M_1\oplus M_2$. If there is no such partition, we say $M$ is {\em connected}.
Now, for the cycle matroid of a graph $G$, it is easy to see that if $G$ is a disconnected graph, then $M(G)$ is disconnected as a matroid. However, the converse is not true in general: if $G$ is connected but contains a cut vertex, $M(G)$ is disconnected. The appropriate matroid definition of being connected in the graphic matroid is in fact equivalent to the graph being 2-connected. The fact that the notions of connectivity do not coincide is the main reason why we present separate analyses for graphs and matroids in this paper. In particular, Theorem~\ref{thm:main} for graphs is not simply a corollary of Theorem~\ref{thm:decomp} for matroids. (For a further discussion of matroid connectivity, see Oxley~\cite[Chapter 4]{Oxley92}, and \cite[\S8.2]{Oxley92} for a comparison of this notion with that of graph connectivity.)
The following lemma is straightforward, but will prove to be crucial to us.
\begin{lemma} \label{lemma:disjointcocircuits}
Suppose that $M=(E,\mathcal{I})$ is a matroid for which $\sigma(M)=\lambda(M)=k$. Then any pair of distinct minimum cocircuits of $M$ are disjoint.
\end{lemma}
\proof Suppose for a contradiction that $C_1$ and $C_2$ are distinct minimum cocircuits of $M$ contain some element $x$ in common. Now, $E\setminus (C_1 \cup C_2) = (E\setminus C_1) \cap (E \setminus C_2)$ is the intersection of two distinct hyperplanes, and therefore has rank at most $\rank(M)-2$ ($\ast$).
Consider a collection of $k=\sigma(M)$ disjoint bases $B_1,\ldots,B_k$; note that $x\in B_i$ for some $i$, since each cocircuit intersects each basis non-trivially. Since $|C_1|=|C_2|=\lambda(M)=\sigma(M)=k$, it follows that each of the $k$ bases contains exactly one element of $C_1$ and of $C_2$. In particular, $B_i \cap (C_1 \cup C_2) = \{x\}$, so $B_i \setminus \{x\} \subseteq E\setminus (C_1 \cup C_2)$. But the rank of $B_i \setminus \{x\}$ is $\rank(M)-1$, contradicting ($\ast$).
Hence $C_1 \cap C_2 = \emptyset$. \endproof
In the case of graphs, Lemma~\ref{lemma:disjointcocircuits} states that in a max-STP graph, no pair of minimal $k$-edge cuts can have an edge in common. This can be shown directly by a straightforward counting argument, similar to that of Zhang~\cite[Lemma 2.2]{Zhang02}.
The next lemma is also not difficult.
\begin{lemma} \label{lemma:deletecocircuit}
Suppose that $\sigma(M)=k$. Then for any cocircuit $C$ of $M$ of size $k$, and where $C\neq E$, we have $\sigma(M\setminus C)\geq k$.
\end{lemma}
\proof Let $\{B_1,\ldots,B_k\}$ be $k$ disjoint bases of $M$. Each $B_i$ intersects $C$ in a unique element, say $x_i$. For $1 \leq i \leq k$, let $B_i'=B_i\setminus \{x_i\}$. Then it follows that $\{B_1',\ldots,B_k'\}$ is a set of disjoint bases for $M\setminus C$.
\endproof
In the case of graphs, Lemma~\ref{lemma:deletecocircuit} states that in a graph with $k$ edge-disjoint spanning trees, if an $k$-edge cut $C$ is deleted, then both connected components of the resulting graph $G\setminus C$ will also have at least $k$ edge-disjoint spanning trees (unless that component is an isolated vertex).
From now on, we will consider the cases of graphs and matroids separately.
\section{Decomposing graphs} \label{section:graphdecomp}
In this section, we will obtain our structural description of the max-STP graphs. To assist with this, we define the following ``joining'' operation.
\begin{defn} \label{defn:kjoin}
Let $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be connected graphs, where $V_1 \cap V_2 =\emptyset$, and let $K$ be a set of $k$ edges with one end in $V_1$ and one end in $V_2$ (for some integer $k$). Then the {\em $K$-join} of $G_1$ and $G_2$, denoted by $G_1 \ast_K G_2$, is the graph with vertex set $V_1 \cup V_2$ and edge set $E_1 \cup E_2 \cup K$.
\end{defn}
When the set $K$ of $k$ edges is not specified (or is not important), we speak of a {\em $k$-join} of $G_1$ and $G_2$, and denote it by $G_1 \ast_k G_2$.
We follow the definition with a couple of remarks. First, by construction $(V_1,V_2)$ is a $k$-edge cut of $G_1 \ast_K G_2$, so consequently the edge-connectivity of the $K$-join is at most $k$. Second, two $k$-joins will not, in general, be isomorphic (unless we have a special case, such as when $k=1$ and both $G_1$ and $G_2$ are vertex-transitive). Example~\ref{example:K4*2K4} below shows the kind of situation which may arise.
\begin{example} \label{example:K4*2K4}
There are three non-isomorphic possibilities for $K_4 \ast_2 K_4$ (the $2$-join of two copies of $K_4$), as shown in Figure \ref{fig:K4*2K4}.
\end{example}
\setlength{\unitlength}{7mm}
\begin{figure}[hbtp]
\centering
\begin{picture}(7,8)
\put(-0.15,6){\complete}
\put(4.85,6){\complete}
\put(2,6){\line(1,0){3}}
\put(2,8){\line(1,0){3}}
\put(-0.15,3){\complete}
\put(4.85,3){\complete}
\put(2,5){\line(1,0){3}}
\put(2,5){\line(3,-2){3}}
\put(-0.15,0){\complete}
\put(4.85,0){\complete}
\qbezier(2,2)(3.5,2.5)(5,2)
\qbezier(2,2)(3.5,1.5)(5,2)
\end{picture}
\caption{The three non-isomorphic possibilities for $K_4 \ast_2 K_4$. \label{fig:K4*2K4}}
\end{figure}
We are interested in $k$-joins because they preserve the property we are concerned with. First, it is a straightforward exercise to show that if a graph $G$ has the form $G=G_1 \ast_K G_2$, where $k = |K| \leq \sigma(G_i) \leq \lambda(G_i)$ (for $i=1,2$), then $\lambda(G)=\sigma(G)=k$. Conversely, if $G$ is a max-STP graph with $\lambda(G)=\sigma(G)=k$, then $G$ is necessarily a $k$-join of graphs $G_1$, $G_2$, and where $G_1$, $G_2$ each satisfy exactly one of the following:
\begin{itemize}
\item[(i)] $G_i$ has one vertex and no edges;
\item[(ii)] $k \leq \sigma(G_i) < \lambda(G_i)$;
\item[(iii)] $k < \sigma(G_i) = \lambda(G_i)$;
\item[(iv)] $\sigma(G_i) = \lambda(G_i) = k$.
\end{itemize}
\begin{defn} \label{defn:kirred}
We call a graph {\em $k$-irreducible} if it belongs to classes (i)--(iii) above; we call it {\em $k$-reducible} if it belongs to class (iv).
\end{defn}
We remark that these four classes (i)--(iv) partition the class of all graphs with $k$ or more edge-disjoint spanning trees (i.e.~the class determined by Nash-Williams and Tutte). Also, we observe that a graph $G$ in class (iii) will itself be a max-STP graph, but with a higher spanning tree packing number and edge-connectivity; such a graph will also be $k'$-reducible, where $k'=\sigma(G)=\lambda(G)>k$.
A complication arises when two or more $k$-joins are made. Suppose that we make a $k$-join $H=G_1 \ast_k G_2$, and then make $H \ast_k G_3$ (where $G_i=(V_i,E_i)$). If the $k$ edges in the second $k$-join are all attached to exactly one of $G_1$ or $G_2$ (assume without loss of generality that this is $G_2$), then in the resulting graph $(V_1,V_2\cup V_3)$ and $(V_1\cup V_2,V_3)$ will both be $k$-edge cuts, obtained by removing the edges of the first or second $k$-joins respectively. However, if the second $k$-join attaches $G_3$ to some vertices in each of $G_1$ and $G_2$, only $(V_1\cup V_2,V_3)$ is a $k$-edge cut. This phenomenon is demonstrated in Example \ref{example:orderindep} below.
\begin{example} \label{example:orderindep}
Figure \ref{figure:orderindep} shows two ways of forming 2-joins of three copies of $K_4$. The graph on the left has two 2-edge cuts, while the graph on the right has only one.
\end{example}
\setlength{\unitlength}{7mm}
\begin{figure}[hbtp]
\centering
\begin{picture}(15,6)
\put(-0.15,4){\complete}
\put(3.85,4){\complete}
\put(2,4){\line(1,0){2}}
\put(2,6){\line(1,0){2}}
\put(3.85,0){\complete}
\put(4,2){\line(0,1){2}}
\put(6,2){\line(0,1){2}}
\put(8.85,4){\complete}
\put(12.85,4){\complete}
\put(11,4){\line(1,0){2}}
\put(11,6){\line(1,0){2}}
\put(10.85,0){\complete}
\put(11,2){\line(0,1){2}}
\put(13,2){\line(0,1){2}}
\end{picture}
\caption{Two ways of forming $2$-joins of three copies of $K_4$. \label{figure:orderindep} }
\end{figure}
We call a sequence of $k$-joins {\em order-independent} if the joining edges of any one of them yield a $k$-edge cut in the resulting graph. Thus in Figure \ref{figure:orderindep}, the 2-joins in the graph on the left are order-independent, while those in the graph on the right are not.
The term order-independent refers to the fact that the final graph is invariant of different choices of which $k$-join is performed first, second, third, etc. However, which ``pieces'' are joined by a particular $k$-join remain fixed.
In the case where $G$ is a max-STP graph with $\lambda(G)=\sigma(G)=k$, and which has more than one $k$-edge cut, Lemma~\ref{lemma:disjointcocircuits} tells us that no pair of $k$-edge cuts can ``overlap'' (i.e.\ they can have no edge in common). Furthermore, by construction, if we have made two $k$-joins order-independently, then the two $k$-edge cuts arising from these must be non-overlapping. Following this, we are now able to state our decomposition theorem for max-STP graphs.
\begin{thm} \label{thm:main}
Suppose that $G$ is a max-STP graph satisfying $\lambda(G)=\sigma(G)=k$. Then we have the following.
\begin{itemize}
\item[(i)] There exists a unique set $\mathcal{A}$ of $k$-irreducible graphs $G_1,\ldots,G_m$ (for some $m$).
\item[(ii)] There exists a unique rooted tree $R$ with $m$ leaves labelled by $G_1,\ldots,G_m$, such that the root is labelled by $G$ and each node is labelled by an order-independent $k$-join of its children.
\item[(iii)] For each non-leaf, labelled by $H$ and its $d$ children labelled $H_1,\ldots,H_d$, there exists a unique tree $T_H$ with vertices $\{1,\ldots,d\}$ labelled by $H_1,\ldots,H_d$, such that for each edge $e=ij$ of $T_H$, there exists a $k$-edge cut $K_e$ of $H$ such that $H_i \ast_{K_e} H_j$ is an induced subgraph of $H$.
\end{itemize}
\end{thm}
We remark that Theorem~\ref{thm:main} implies that if $\lambda(G)=\sigma(G)=k$, then $G$ must be obtained by an iterated $k$-join of $k$-irreducible graphs.
\begin{proof}
We start with $G$ and build the rooted tree $R$ and trees $T_H$ recursively.
Suppose that a graph $\Gamma$ is the label of a node which has yet to be considered. If $\Gamma$ is $k$-irreducible, this node will be a leaf in $R$, and we add $\Gamma$ to $\mathcal{A}$.
If $\Gamma$ is $k$-reducible, then $\lambda(\Gamma)=\sigma(\Gamma)=k$, and so $\Gamma$ contains some collection of $k$-edge cuts. By Lemma \ref{lemma:disjointcocircuits}, these must be pairwise non-overlapping. Removing the edges from all of these $k$-edge cuts yields a graph with some number $d\geq 2$ of connected components; label these $\Gamma_1,\ldots,\Gamma_d$. We observe that each $\Gamma_i$ must be $k$-edge connected and contains at least $k$ edge-disjoint spanning trees.
We note that $\Gamma$ is therefore an order-independent $k$-join of $\Gamma_1,\ldots,\Gamma_d$, and we can build a tree to specify explicitly which pairs are joined, as follows. Define $T_\Gamma$ to be the graph obtained from $\Gamma$ by contracting each $\Gamma_i$ to a single vertex (and removing any multiple edges). Lemma \ref{lemma:disjointcocircuits} ensures that this graph is a tree. Now, each edge $e$ of $T_\Gamma$ corresponds to exactly one of the $k$-edge cuts of $\Gamma$, so we can label these $k$-edge cuts by the edges of $T_\Gamma$.
Finally, we add a child node of $\Gamma$ to $R$ labelled by $\Gamma_i$ for each $i \in \{1,\ldots,d\}$, and apply the recursion to each of these new nodes.
\end{proof}
\begin{defn} \label{defn:ingredients}
For a given max-STP graph $G$, the {\em max-STP decomposition} of $G$ is the triple $\mathcal{I}(G) = (\mathcal{A},R,\{T_H\})$.
\end{defn}
\begin{example} \label{example:maxSTP}
Consider the max-STP graph $G$ shown in Figure \ref{figure:maxSTP}, which has $\lambda(G)=\sigma(G)=2$. Now, $G$ possesses exactly one 2-edge cut, so the root vertex of $R$, labelled by $G$, has two descendants; the tree $T_G$ associated with it is the unique tree on two vertices. Now, one of the child nodes is labelled by a copy of $K_4$, which is 2-irreducible, so the node becomes a leaf. The other child node is labelled by a graph $H$ which is the order-independent 2-join of three copies of $K_4$. Thus this node has three child nodes, all leaves labelled by a copy of $K_4$, and the associated tree $T_H$ is the unique tree on 3 vertices.
Thus the max-STP decomposition of $G$ are $(\mathcal{A},R,\mathcal{T} )$, where $\mathcal{A}$ contains of four copies of $K_4$, $R$ is as shown in Figure \ref{figure:maxSTP}, and \mbox{$\mathcal{T}= \{$ {\setlength{\unitlength}{7mm}
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\put(0.1,0.2){\blob}
\put(0.1,0.2){\line(1,0){1}}
\put(1.1,0.2){\blob}
\end{picture} },{\setlength{\unitlength}{7mm}
\begin{picture}(2.2,0.3)
\put(0.1,0.2){\blob}
\put(0.1,0.2){\line(1,0){1}}
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\put(1.1,0.2){\line(1,0){1}}
\put(2.1,0.2){\blob}
\end{picture} } $\}$}.
\end{example}
\begin{figure}[hbtp]
\setlength{\unitlength}{7mm}
\centering
\begin{picture}(14,9)
\put(2.85,0){\complete}
\put(-0.15,4){\complete}
\put(2.85,7){\complete}
\put(5.85,4){\complete}
\put(3,2){\line(-1,2){1}}
\put(5,2){\line(1,2){1}}
\put(0,6){\line(1,1){3}}
\put(2,6){\line(1,1){1}}
\put(6,6){\line(-1,1){1}}
\put(8,6){\line(-1,1){3}}
\put(11,2){\blob}
\put(12,2){\blob}
\put(13,2){\blob}
\put(12,4){\blob}
\put(14,4){\blob}
\put(13,6){\blob}
\put(11,2){\line(1,2){1}}
\put(12,2){\line(0,1){2}}
\put(13,2){\line(-1,2){1}}
\put(12,4){\line(1,2){1}}
\put(14,4){\line(-1,2){1}}
\end{picture}
\caption{A max-STP graph $G$ with $\lambda(G)=\sigma(G)=2$, and the associated rooted tree $R$. \label{figure:maxSTP}}
\end{figure}
We remark that it is not possible to uniquely recover the original graph from a max-STP decomposition; the exact $k$-joins must be specified. For example, the two graphs in Figure~\ref{figure:orderindep} are non-isomorphic, yet their max-STP decompositions are the same. Moreover in any reconstruction of a graph from a max-STP decomposition, the collection of $k$-edge cuts of the graph at any node of $R$ must be precisely the set of $k$-joins applied to the graphs at the child nodes. For example, none of the graphs in Figure~\ref{figure:possibilities} are valid reconstructions using the elements
\mbox{$( \{K_4,K_4,K_4,K_4\},\, R,\, \{$
{\setlength{\unitlength}{7mm}
\begin{picture}(1.2,0.3)
\put(0.1,0.2){\blob}
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\end{picture} },{\setlength{\unitlength}{7mm}
\begin{picture}(2.2,0.3)
\put(0.1,0.2){\blob}
\put(0.1,0.2){\line(1,0){1}}
\put(1.1,0.2){\blob}
\put(1.1,0.2){\line(1,0){1}}
\put(2.1,0.2){\blob}
\end{picture} } $\})$}
of the max-STP decomposition from Example~\ref{example:maxSTP}; their actual max-STP decompositions are also given in Figure~\ref{figure:possibilities}.
\begin{figure}[hbtp]
\setlength{\unitlength}{7mm}
\centering
\subfigure[Leaving some previous 2-joins order-independent of the new 2-join]
\begin{picture}(15,6.7)
\put(-0.15,4){\complete}
\put(3.85,4){\complete}
\put(7.85,4){\complete}
\put(5.85,0){\complete}
\put(2,4){\line(1,0){2}}
\put(2,6){\line(1,0){2}}
\put(6,4){\line(1,0){2}}
\put(6,6){\line(1,0){2}}
\put(6,2){\line(0,1){2}}
\put(8,2){\line(0,1){2}}
\put(12,4.5){\blob}
\put(13,2.5){\blob}
\put(13,4.5){\blob}
\put(13,6.5){\blob}
\put(14,4.5){\blob}
\put(15,2.5){\blob}
\put(12,4.5){\line(1,2){1}}
\put(13,4.5){\line(0,1){2}}
\put(14,4.5){\line(-1,2){1}}
\put(13,2.5){\line(1,2){1}}
\put(15,2.5){\line(-1,2){1}}
\put(10,0.5){ ${\displaystyle \mathcal{T}=\left\{ \phantom{\int} \right.}$
\begin{picture}(3.0,0.3)
\put(-0.3,0.2){\blob}
\put(-0.3,0.2){\line(1,0){1}}
\put(0.7,0.2){\blob}
\put(0.7,0.2){\line(1,0){1}}
\put(1.7,0.2){\blob}
\put(1.95,0.1){,}
\put(2.4,0.2){\blob}
\put(2.4,0.2){\line(1,0){1}}
\put(3.4,0.2){\blob}
\end{picture} ${\displaystyle \left. \phantom{\int} \right\} }$ }
\end{picture} }
\subfigure[Leaving {\em all} previous 2-joins order-independent of the new 2-join]
\begin{picture}(15,6.7)
\put(-0.15,4){\complete}
\put(3.85,4){\complete}
\put(7.85,4){\complete}
\put(3.85,0){\complete}
\put(2,4){\line(1,0){2}}
\put(2,6){\line(1,0){2}}
\put(6,4){\line(1,0){2}}
\put(6,6){\line(1,0){2}}
\put(4,2){\line(0,1){2}}
\put(6,2){\line(0,1){2}}
\put(12,3){\blob}
\put(13,3){\blob}
\put(14,3){\blob}
\put(15,3){\blob}
\put(13.5,5){\blob}
\put(12,3){\line(3,4){1.5}}
\put(13,3){\line(1,4){0.5}}
\put(14,3){\line(-1,4){0.5}}
\put(15,3){\line(-3,4){1.5}}
\put(11,1){ ${\displaystyle \mathcal{T}=\left\{ \phantom{\int} \right.}$
\begin{picture}(1.4,1)
\put(-0.3,0.7){\blob}
\put(0.7,0.7){\blob}
\put(1.7,0.7){\blob}
\put(0.7,-0.3){\blob}
\put(-0.3,0.7){\line(1,0){1}}
\put(0.7,0.7){\line(1,0){1}}
\put(0.7,-0.3){\line(0,1){1}}
\end{picture} ${\displaystyle \left. \phantom{\int} \right\} }$ }
\end{picture} }
\subfigure[Same as (b), but with a different set $\mathcal{T}$]
\begin{picture}(14,5.7)
\put(-0.15,3){\complete}
\put(3.85,3){\complete}
\put(7.85,3){\complete}
\put(11.85,3){\complete}
\put(2,3){\line(1,0){2}}
\put(2,5){\line(1,0){2}}
\put(6,3){\line(1,0){2}}
\put(6,5){\line(1,0){2}}
\put(10,3){\line(1,0){2}}
\put(10,5){\line(1,0){2}}
\put(2.5,0){\blob}
\put(3.5,0){\blob}
\put(4.5,0){\blob}
\put(5.5,0){\blob}
\put(4,2){\blob}
\put(2.5,0){\line(3,4){1.5}}
\put(3.5,0){\line(1,4){0.5}}
\put(4.5,0){\line(-1,4){0.5}}
\put(5.5,0){\line(-3,4){1.5}}
\put(7,1){ ${\displaystyle \mathcal{T}=\left\{ \phantom{\int} \right.}$
\begin{picture}(2.5,0.3)
\put(-0.3,0.2){\blob}
\put(-0.3,0.2){\line(1,0){1}}
\put(0.7,0.2){\blob}
\put(0.7,0.2){\line(1,0){1}}
\put(1.7,0.2){\blob}
\put(1.7,0.2){\line(1,0){1}}
\put(2.7,0.2){\blob}
\end{picture} ${\displaystyle \left. \phantom{\int} \right\} }$ }
\end{picture} }
\caption{Three graphs constructed from the same max-STP decomposition, but each with different max-STP decompositions.
\label{figure:possibilities}}
\end{figure}
\section{Decomposing matroids} \label{section:matroiddecomp}
From now on, all matroids we consider will be connected. In the case of matroids with $k$ disjoint bases, we introduce the following terminology which is analogous to the notion of $k$-reducibility for graphs we saw above.
\newpage
\begin{defn} \label{defn:reducible}
Suppose that $M$ is a matroid with $\sigma(M) \geq k$. We call $M$ {\em $k$-reducible} if $\lambda(M)=\sigma(M)=k$; otherwise (i.e.~if either $\sigma(M)>k$ or $k=\sigma(M)<\lambda(M)$) we call $M$ {\em $k$-irreducible}.
\end{defn}
We note that this partitions the class of matroids with $k$ disjoint bases (as characterized by Edmonds \cite{Edmonds65}) into either being $k$-reducible or $k$-irreducible. We also remark that the case of isolated vertices does not arise here, as their cycle matroids are empty.
As the object obtained from a matroid $M$ by deleting all of its minimum cocircuits will be keep appearing, it is useful to give it a formal name.
\begin{defn} \label{defn:crux}
Let $M$ be a matroid whose minimum cocircuits are $C_1,\ldots,C_\ell$. Then the {\em crux} of $M$, denoted $\chi(M)$, is defined to be
\[ \chi(M) = M\setminus \bigcup_{i=1}^\ell C_i. \]
\end{defn}
If it happens that $\bigcup_{i=1}^\ell C_i = E$, then by abuse of notation we write $\chi(M)=\emptyset$ (as a shorthand for the matroid $(\emptyset,\{\emptyset\})$).
\begin{lemma} \label{lemma:cocircuits}
Suppose that $\lambda(M)=\sigma(M)=k$, and that $\{C_1,\ldots,C_\ell\}$ are the minimum cocircuits of $M$. Then for all $i \neq j$, $C_i$ is a minimum cocircuit of $M\setminus C_j$.
\end{lemma}
\proof Suppose that $C_i$ and $C_j$ are minimum cocircuits of $M$, and that $\rank(M)=r$. Their complements are hyperplanes, so the intersection of their complements is a flat of rank at most $r-2$ (since it is strictly contained in two distinct hyperplanes). Since $\lambda(M)=\sigma(M)$, we have a base that intersects $C_i$ and $C_j$ in exactly one element each, so by removing these elements we have an independent set of size $r-2$ in the intersection of the two hyperplanes. Therefore, $M\setminus C_j$ is a matroid of rank $r-1$, which contains a flat $A$ of rank $r-2$, whose complement is $C_i$. Consequently, $C_i$ is a cocircuit of $M\setminus C_j$.
To show that $C_i$ is minimum, it suffices to show that $\lambda(M\setminus C_j)\geq k$. Since $M$ has $k$ disjoint bases, each of which intersects $C_j$ in a single element, we have $\sigma(M\setminus C_j)\geq k$, and thus $\lambda(M\setminus C_j)\geq k$. Hence $C_i$ is a minimum cocircuit of $M\setminus C_j$.
\endproof
\begin{lemma} \label{lemma:cruxrank}
Suppose that $M$ is a matroid such that $\lambda(M)=\sigma(M)=k$, $\rank(M)=r$, and whose minimum cocircuits are $C_1,\ldots,C_\ell$. Then the crux of $M$ has rank $r-\ell$.
\end{lemma}
\proof By applying Lemma~\ref{lemma:cocircuits} repeatedly, we see that deleting each minimum cocircuit in turn will reduce the rank by~$1$. Once we have deleted all minimum cocircuits $C_1,\ldots,C_\ell$ to obtain the crux, we have a matroid of rank $r-\ell$. \endproof
Even if a matroid $M$ is assumed to be connected, its crux $\chi(M)$ is not necessarily connected. (For instance, in the case of graphic matroids, this is clear.) So the following parameter makes sense.
\begin{notation}
We let $\delta(M)$ denote the number of connected components of $\chi(M)$.
\end{notation}
Note that if $\chi(M)=\emptyset$, we define $\delta(M)=0$. We remark that if $\sigma(M)=\lambda(M)=k$ and $\chi(M)=\emptyset$, then the matroid $M$ is the disjoint union of the minimum cocircuits, the bases are precisely transversals of the partition into minimum cocircuits, and thus $M$ is disconnected. (These are precisely the transversal matroids discussed at the end of Section~\ref{section:intro}.) Hence, if we restrict ourselves to connected matroids, the possibility that the crux is empty does not arise.
The next definition allows us to encode how a matroid with $\sigma(M)=\lambda(M)=k$ is assembled from its cocircuits and the connected components of its crux.
\begin{defn} \label{defn:assembly}
Let $M$ be a connected matroid such that $\sigma(M)=\lambda(M)=k$, with~$\ell$ minimum cocircuits $C_1,\ldots,C_\ell$, and where $\chi(M)$ has $d$ connected components $K_1,\ldots,K_d$. For each minimum cocircuit $C_j$, let $\mathcal{V}_j$ denote the largest subset of $\{K_1,\ldots,K_d\}$ such that the restriction of $M$ to
\[ C_j \cup \left( \bigcup_{K\in\mathcal{V}_j} K \right) \]
is connected.
Then the {\em assembly hypergraph} of $M$, denoted $\mathcal{H}(M)$, is the non-uniform hypergraph whose vertices are labelled by the connected components $K_1,\ldots,K_d$, where the hyperedges are labelled by the cocircuits $C_1,\ldots,C_\ell$, and the vertices incident with $C_j$ are precisely the members of $\mathcal{V}_j$.
\end{defn}
In other words, the assembly hypergraph $\mathcal{H}(M)$ tell us which components of $\chi(M)$ are ``joined'' by each cocircuit $C_j$ in the matroid $M$.
\begin{thm} \label{thm:decomp}
Suppose that $M=(E,\mathcal{I})$ is a matroid for which $\sigma(M)=\lambda(M)=k$. Then we have the following.
\begin{enumerate}
\item There exists a unique set of $k$-irreducible matroids $\mathcal{M}=\{M_1,\ldots,M_m\}$ (for some integer $m$).
\item There exists a unique rooted tree $R$ with $m$ leaves labelled by $M_1,\ldots,M_m$, such that the root is labelled by $M$ and each non-leaf labelled by $K$ has $d=\delta(K)$ children, labelled by the connected components of $\chi(K)$.
\item For each non-leaf, labelled by $K$ and its $d$ children labelled $K_1,\ldots,K_d$, there exists a unique assembly hypergraph with $\ell$ hyperedges, and where $\sum_{i=1}^d \rank(K_i) = \rank(K)-\ell$.
\end{enumerate}
\end{thm}
\proof We build the rooted tree $R$ recursively, obtaining $\mathcal{M}$ and the collection of assembly matrices as we go along. We begin by assigning the root node to our matroid $M$, and declaring $\mathcal{M}=\emptyset$.
Suppose that a matroid $K$ is the label of a node which is yet to be considered. If $K$ is $k$-irreducible, then this node becomes a leaf in $R$, and we add $K$ to $\mathcal{M}$.
On the other hand, if $K$ is $k$-reducible, then $\lambda(K)=\sigma(K)=k$, and so $K$ has some minimum cocircuits of size $k$; suppose that there are $\ell$ of these, labelled $C_1,\ldots,C_\ell$. By Lemma \ref{lemma:disjointcocircuits}, these are all disjoint. Now consider the crux $\chi(K)$; by repeatedly applying Lemma \ref{lemma:deletecocircuit}, this must have at least $k$ disjoint bases. Now suppose that the crux has $d=\delta(K)$ connected components $K_1,\ldots,K_d$; since $K$ is connected, there must be at least one of these. Furthermore, since a base for $\chi(K)$ is the disjoint union of a base for each component, each $K_i$ has $\sigma(K_i)\geq k$, and thus each $K_i$ must have a cocircuit of size at least $k$. By Lemma \ref{lemma:cruxrank}, the rank of $\chi(K)$ is $\rank(K)-\ell$, so the ranks of its connected components $K_1,\ldots,K_d$ must sum to this.
For each $j\in\{1,\ldots,\ell\}$, we add a hyperedge to the assembly hypergraph $\mathcal{H}(M)$ as follows. Consider the matroid $L$ formed by restricting $K$ to $\chi(K)\cup C_j$. Now, the component of $L$ containing $C_j$ will be of the form
\[ C_j \cup \left( \bigcup_{i \in I} K_i \right), \]
where $I \subseteq \{1,\ldots,d\}$. Then we add the hyperedge $\{ K_i \mid i \in I\}$.
Finally, we add a child node of $K$ to $R$ labelled by $K_i$ for each $i \in \{1,\ldots,d\}$, and apply the recursion to each of these new nodes. \endproof
\section{Discussion} \label{section:discussion}
We conclude the paper with a few remarks about our main results (Theorem~\ref{thm:main} for graphs and Theorem~\ref{thm:decomp} for matroids). We also consider some issues related to computational complexity.
A general comment about Theorem~\ref{thm:main} is perhaps in order. The graph clearly determines the
decomposition, but the converse is not true. In general, it is not possible to recover the
graph from the decomposition, except in some special cases (for instance, if the original graph
is a tree). In general, even for $k=1$ we cannot recover the graph: if the $k$-irreducible
subgraphs have more than one vertex there are many different ways the edge-cuts could be added
back in. While of course one could keep additional information at each stage in order to
permit this reconstruction, we prefer to view these results as a general structural description
of the original graph rather than as a precise encoding of it. In particular, Theorem~\ref{thm:main}
shows that, at each stage of the decomposition, a max-STP graph $G$ is globally ``tree-like'', in that
it may be contracted to the tree $T_G$. A similar comment applies to Theorem~\ref{thm:decomp}.
We also observe that there are some constraints on the ingredients of a max-STP decomposition.
For the case of graphs (Theorem~\ref{thm:main}), consider for definiteness the root node of $R$ and
one of its child nodes. The root is labelled by $G$ and has an associated tree $T_G$. The
child is labelled by $H$ and has an associated tree $T_H$. One of the vertices of $T_G$, say $v_H$,
corresponds to $H$; let the degree of $v_H$ be $d$. Then there are $d$ $k$-edge cuts in
$G$ incident with $H$. This collection of edge-cuts must be sufficient so that no $k$-edge cut of
$H$ is an edge-cut of $G$.
One way of characterizing this condition is that for every edge $e$ of $T_H$ there must exist a pair of edges
from one of the $d$ $k$-joins connected to $H$ in $T_G$ such that each edge is incident to a different
connected component of $T_H \setminus e$.
This gives a constraint on the structure of $T_H$. For instance it is
not hard to see from these constraints that if $k=1$ then $R$ has a root node labelled by
$T_G$ and child nodes labelled by the $k$-irreducible subgraphs. Note also that if a node
of $R$ has an associated tree then the number of children of that node is the number of
vertices of the associated tree; otherwise that node of $R$ is a leaf labelled by a
$k$-irreducible graph.
It is important to note that Theorem~\ref{thm:main} is not a simple corollary of Theorem~\ref{thm:decomp}.
Given a graph $G$, we may apply Theorem~\ref{thm:main} directly to $G$ or we may apply
Theorem~\ref{thm:decomp} to the matroid $M(G)$; however, we get two different decompositions. In the
latter the ``connected'' components are now blocks of the underlying graph, not connected
components in the graph-theoretic sense.
Finally, we remark that $\sigma$ can be computed in polynomial time for both graphs and
matroids (see Schrijver \cite[Sections~51.4 and~42.3]{Schrijver03}). As $\lambda$ can also be
found in polynomial time for graphs (\cite[Section~15.3]{Schrijver03}) this means that we can
determine if a graph is max-STP in polynomial time. Furthermore, it is actually possible to
find the collection of $\sigma(G)$ edge-disjoint spanning trees in polynomial time; the best
algorithm known for doing this is due to Gabow and Westermann \cite{Gabow92}. However for
matroids in general, or even binary matroids, determining the girth (or cogirth) was shown to
be NP-hard by Cho {\em et al.}\ in \cite{Cho07}: this follows from the equivalence of the
problem to that of determining the minimum distance of a binary linear code, which was shown to
be NP-hard by Vardy \cite{Vardy97}. Thus, unless $\rm{P}=\rm{NP}$ there is no obvious polynomial time
algorithm for testing $\lambda(M)=\sigma(M)$ for an arbitrary matroid $M$.
\section*{Acknowledgements}
The authors acknowledge financial support from NSERC and the Ontario Ministry of Research and Innovation.
|
1,116,691,498,446 | arxiv | \section{Introduction}
The origin of the ultrahigh energy cosmic rays (UHECRs), i.e., with energies above $10^{18}$ eV,
is still unknown. The three main observables used to study their nature are the energy spectrum,
composition profile, and distribution of their arrival directions. In this energy range,
these studies are carried out by detecting the atmospheric air showers initiated by the UHECR
primaries that interact with molecules of the atmosphere. The most common detection systems
include arrays of surface detectors, which allow reconstructing the lateral development of the
showers by detecting secondary particles that reach the ground, and fluorescence telescopes,
which are used to study the longitudinal development of the showers. The two observatories
currently taking data are the Pierre Auger Observatory \cite{Auger:15}, situated in the southern
hemisphere in Malarg{\"u}e, Province of Mendoza, Argentina, and Telescope Array \cite{TA:03},
located in the northern hemisphere, in Utah, United States. Both observatories combine arrays of
surface detectors with fluorescence telescopes.
The ankle in the UHECR flux has been observed by several experiments \cite{Patrignani:16}.
The Pierre Auger Observatory observes this spectral feature at an energy
$E_{\textrm{ankle}}=10^{(18.705 \pm 0.005)}$ eV \cite{AugerICRC:17}. At this point, the spectral
index, assuming the differential flux to be given by a power law $J \propto E^{-\gamma}$, changes
from $\gamma_1=-3.293 \pm 0.002$ below the ankle to $\gamma_2=-2.53 \pm 0.02$
above the ankle \cite{AugerICRC:17}. Similarly, Telescope Array observes the ankle at
$E_{\textrm{ankle}}=10^{(18.71 \pm 0.02)}$ eV and reports a change in the spectral index from
$\gamma_1=-3.246 \pm 0.005$ below to $\gamma_2=-2.66 \pm 0.03$ above the break \cite{IvanovTA:15}.
The suppression of the flux is observed at $E_{\textrm{s}}=10^{(19.59 \pm 0.02)}$ eV in the case
of Auger \cite{AugerICRC:17} and at $E_{\textrm{s}}=10^{(19.75 \pm 0.05)}$ eV in the case of
Telescope Array \cite{IvanovTA:15}. Even though these two values have been obtained by fitting
the respective energy spectrum with different functions, it can be seen that the suppression of
the spectrum is observed by Auger at a smaller energy than Telescope Array. Also, the Auger
spectrum takes smaller values than the ones corresponding to Telescope Array. The discrepancies
between the two observations can be diminished by shifting the energy scales of both experiments
within their systematic uncertainties. However, some differences are still present in the
suppression region \cite{IvanovAugerTA:17}.
It is very well known that the most sensitive parameters to the nature of the primary particle
are the muon content of the showers and the atmospheric depth of the shower maximum, $X_{max}$
(see, for instance, Ref.~\cite{Supanitsky:08}). The $X_{max}$ parameter can be reconstructed from
the data taken by the fluorescence telescopes. The secondary charged particles of the showers
interact with the nitrogen molecules of the atmosphere producing fluorescence light. Part of
this light is detected by the telescopes that take data on clear and moonless nights. In this
way, it is possible to observe the longitudinal development of the showers, which in turn may be
analyzed to infer the $X_{max}$ parameter. As mentioned before, this technique is employed
by both Auger and Telescope Array.
The composition analyses are performed by comparing experimental data with simulations of the showers.
These simulations are subject to large systematic uncertainties because they are based on high energy
hadronic interaction models that extrapolate low energy accelerator data to the highest energies.
Recently, the high energy hadronic interaction models more frequently used in the literature have been
updated by using data taken by the Large Hadron Collider \cite{Pierog:17}. Although the differences
between the shower observables predicted by different models have been reduced, there still remain
some discrepancies (see Ref.~\cite{Pierog:17} for details).
The mean value of $X_{max}$ obtained by Auger \cite{AugerXmax:14}, interpreted by using the updated
versions of current hadronic interaction models, shows that the composition is light from $\sim 10^{18}$
up to $\sim 10^{18.6}$ eV. From $\sim 10^{18.3}$ eV, the composition becomes progressively heavier for
increasing values of the primary energy. This trend is consistent with the results obtained by using the
standard deviation of the $X_{max}$ distribution \cite{AugerXmax:14}. Therefore, if the shower predictions,
based on the current high energy hadronic interaction models, are not too far from the correct description,
it can be said that there is evidence of the existence of a light component that dominates the spectrum
below the ankle. On the other hand, the $X_{max}$ parameter reconstructed from the data taken by the
fluorescence telescopes of Telescope Array is also compatible with a light composition below the ankle,
when it is interpreted by using the current hadronic interaction models \cite{TA:15}. It is worth mentioning
that the $X_{max}$ distributions, as a function of primary energy, obtained by Auger and Telescope Array
are compatible within systematic uncertainties \cite{Souza:17}. However, the presence of heavier primaries
above the ankle cannot be confirmed by the Telescope Array data due to the limited statistics of the event
sample \cite{Souza:17}.
The Auger data show that the large scale distribution of the cosmic ray arrival directions is compatible
with an isotropic flux, in the energy range from $\sim 10^{18}$ eV up to the ankle \cite{Auger:12}. This
result is incompatible with a galactic origin of the light component that seems to dominate the flux in
this energy range \cite{Auger:12}. Therefore, the scenarios in which the ankle is interpreted as the point
in which the galactic to extragalactic transition takes place are incompatible with present data, assuming
that the $X_{max}$ predictions based on current high energy hadronic interaction models are close to the
real ones.
There are two main scenarios that can explain the experimental data. In the first one, the light component
below the ankle corresponds to a different population of sources than the ones that are responsible of the
flux above the ankle \cite{Aloisio:14}. In this model, the spectral index of the spectrum injected by the
sources that dominate the flux below the ankle is steeper than the one corresponding to the other population.
In the second scenario, the light component originates from the photodisintegration of high energy and heavier
nuclei in a photon field present in the environment of the source. This has been proposed as a general
mechanism \cite{Unger:15} that can take place in starburst galaxies \cite{Anchordoqui:17} and also in the
context of the UHECR acceleration in $\gamma$-ray bursts \cite{Globus:15a,Globus:15b}. Also, in
Ref.~\cite{Kachel:17} a model combining photodisintegration and hadronic interactions of UHECRs in the
photon and proton gases present in the central regions of active galaxies has been proposed.
In this work, we study the possibility of the formation of the extragalactic light component that seems to
dominate the UHECR flux below the ankle by the photodisintegration of heavier and more energetic nuclei
in the radiation field present in the central region of active galaxies. In this scenario, the UHECRs are
accelerated near the supermassive black hole present in this type of galaxy. In this work, the propagation
in the source environment is modeled in more detail than in Ref.~\cite{Kachel:17}, including a
three-dimensional simulation of the propagation of the UHECR nuclei in the random magnetic field and the
photon gas present in the source environment. Also, the conditions by which these types of models can
properly describe the present experimental results are discussed.
\section{Cosmic ray propagation in the source environment}
As mentioned before, the case in which the UHECRs are accelerated in the central
region of an active galaxy is considered. Therefore, after escaping from the
acceleration zone, the cosmic rays propagate through a region that is filled with
a low energy photon gas and a turbulent magnetic field.
The simulation of the propagation of the cosmic ray nuclei in the source
environment is performed by using a dedicated program specifically
developed for that purpose. The interactions with the low energy photon
gas implemented are photodisintegration and photopion production. This
implementation is based on the CRPropa 3 program \cite{CRPropa3:16}. The
photopion production is performed by using the SOPHIA code \cite{sophia}.
Nuclear and neutron decay are also included in the simulation. The
propagation of the particles is three dimensional, which includes the
deflection of the charged particles in the turbulent magnetic field (see
below).
The source environment is modeled as a spherical region of radius $R=10^{-7}$ kpc
$\cong 3.1\times10^{14}$ cm \cite{Kachel:09}. The numerical density of the low
energy photon gas is taken as a broken power law \cite{Szabo:94,Unger:15},
\begin{equation}
\frac{dn}{d \varepsilon}(\varepsilon) = n_b \left\{
\begin{array}{ll}
\left( \mathop{\displaystyle \frac{\varepsilon}{\varepsilon_b} } \right)^{\alpha} & \varepsilon\leq \varepsilon_b \\[0.4cm]
\left( \mathop{\displaystyle \frac{\varepsilon}{\varepsilon_b} } \right)^{\beta} & \varepsilon_b < \varepsilon_b \leq \varepsilon_{max} \\[0.4cm]
0 & \varepsilon > \varepsilon_{max}
\end{array}
\right.,
\label{dnde}
\end{equation}
where the spectral indexes are taken from Ref.~\cite{Szabo:94}, $\alpha=3/2$ and $\beta=-2$,
and the energy break and the maximum energy are determined following Ref.~\cite{Kachel:09},
$\varepsilon_b=0.2$ eV and $\varepsilon_{max}=5$ eV.
It is worth mentioning that the luminosity of the low energy photon gas is related to the
normalization $n_b$ through the following expression,
\begin{equation}
L=c\, \pi R^2 \varepsilon_b^2 \left[ \frac{2}{7}+\ln\left( \frac{\varepsilon_{max}}{\varepsilon_b} \right) \right] n_b,
\end{equation}
where $c$ is the speed of light.
The interaction length $\lambda_I$ of a nucleus propagating in a photon gas is given by
\begin{equation}
\frac{1}{\lambda_{I}(E)} = \frac{1}{2\, \gamma^2} \int_0^\infty d\varepsilon \, \frac{dn}{d\varepsilon}(\varepsilon)\, \varepsilon^{-2}%
\int_0^{2\, \gamma \varepsilon} d\varepsilon' \varepsilon' \sigma(\varepsilon'),
\end{equation}
where $\gamma$ is the Lorentz factor of the nucleus and $\sigma(\varepsilon')$ is the photo-nuclear
interaction cross section for a photon of energy $\varepsilon'$ in the rest frame of the nucleus.
The interaction lengths corresponding to the photon gas density of Eq.~(\ref{dnde}) are calculated
by using the tools developed for CRPropa 3, which are accessible at \cite{CRPropaTab}.
The propagation of the charged nuclei in the random magnetic field is performed following the method
developed in Ref.~\cite{Protheroe:02}. The propagation is described by a three-dimensional random
walk in which the directions of the particles change according to the scattering length $\lambda_{SL}=3D/c$,
where $D$ is the spatial diffusion coefficient. The distance traveled by the particles after being
scattered by the magnetic field is sampled from an exponential distribution with the mean value given
by $\bar{\ell}=\bar{\theta}^2 \lambda_{SL}$, where $\bar{\theta}$ is the mean value of the exponential
distribution from which the angular change in the direction of propagation is sampled. In
Ref.~\cite{Protheroe:02}, it is found that the method renders accurate results for
$\bar{\theta} < 0.09$ rad ($5^{\circ}$).
The diffusion coefficient used in the simulations is taken from Ref.~\cite{Harari:14},
which is given by
\begin{equation}
D(E)=\frac{c}{3} l_c \left[4 \left( \frac{E}{E_c} \right)^2 +a_I \left( \frac{E}{E_c} \right) + a_L \left( \frac{E}{E_c} \right)^{2-m} \right],
\end{equation}
where $l_c$ is the coherent length of the random magnetic field. Here $E_c=Z e B\, l_c$, where
$Z$ is the charge number of the nucleus, $e$ is the absolute value of the electron charge, and
$B=\sqrt{\langle B^2(x) \rangle}$ is the root mean square of the random magnetic field. The
parameter $m$ and the numerical values of the parameter $a_I$ and $a_L$ depend on the type of
turbulence, and for a Kolmogorov spectrum, which is the one considered in this work, these three
parameters take the following values \cite{Harari:14}: $m=5/3$, $a_L=0.23$, and $a_I=0.9$.
The propagation of charged particles in a random magnetic field depends on the distance
of the particles under the influence of the field. For traveled distances much smaller than the
scattering length $\lambda_{SL}$, the propagation is ballistic, and for traveled distances much
larger than $\lambda_{SL}$, the propagation is diffusive (see, for instance, Ref.~\cite{Harari:14}).
It is worth mentioning that by using the method for the propagation of charged particles in a
random magnetic field developed in Ref.~\cite{Protheroe:02}, the two different regimes of propagation
are included.
The simulation starts by injecting a nucleus of certain type and energy at the center of a sphere
of radius $R$ and ends when all particles leave the sphere. The propagation of a particle proceeds
as follows. Let us consider a particle in a given position $\vec{x}$ inside the sphere with a
given velocity $\vec{v}=c\, \hat{n}$, where $\hat{n}$ is a unit vector (it is assumed that all particles
move at the speed of light). In the next step, the position of the particle is
$\vec{x}'=\vec{x}+\Delta s\, \vec{n}$, where $\Delta s$ is obtained by sampling the exponential
distribution with mean $\bar{\ell}=\bar{\theta}^2 \lambda_{SL}$ for which $\bar{\theta}$ is chosen in
such a way that it fulfils two conditions: it is smaller than 0.09 rad (see above) and also it is small
enough in such way that $\Delta s \ll \lambda_{T}$, where
$\lambda_{T}^{-1}=\lambda_{PD}^{-1}+\lambda_{PP}^{-1}+\lambda_{D}^{-1}$. Here $\lambda_{PD}$ is the
photodisintegration interaction length, $\lambda_{PP}$ is the photopion production interaction length,
and $\lambda_{D}$ is the decay length of the nucleus. The particle at position $\vec{x}'$ can interact,
decay, or change its direction of motion. In order to decide the outcome, an integer number is taken at
random from a Poisson distribution with mean value $\mu=\Delta s /\lambda_T$. If this number is one, the
particle interacts or decays, if not its direction of motion is modified in such a way that the new
velocity vector forms an angle $\theta$ with the velocity vector at position $\vec{x}$. The $\theta$ angle
is obtained by sampling an exponential distribution with mean value $\bar{\theta}$ (see above). In the case
that the particle decays or interacts, three distances are sampled from three different exponential distributions
with mean values $\lambda_{PD}, \lambda_{PP}$, and $\lambda_{D}$, and thus the process undergone by the particle
is the one corresponding to the smallest distance. As mentioned before, the implementation of the
photodisintegration, photopion production, and nuclear and neutron decay are based on the CRPropa 3 program.
Figure \ref{SILengths} shows the interaction lengths and the scattering lengths in the random magnetic
field for five different types of nuclei: proton (p), helium (He), nitrogen (N), silicon (Si), and iron (Fe).
The interaction length includes the photodisintegration and photopion production processes. Note that these
curves present the ``L'' shape mentioned in Ref.~\cite{Unger:15}. The dotted lines in the plots correspond to
the radius of the sphere, $R$. The top panel of the figure corresponds to a luminosity of
$L=10^{41}$ erg s$^{-1}$, a random magnetic field of $B=1$ G, and coherence length $l_c=R/10$. In this case,
the interaction length of protons is larger than the radius of the sphere in the energy range under consideration.
Also, the interaction length of helium is larger than the radius of the sphere above $\sim 10^{19.5}$ eV. However,
the interaction lengths of the other three species considered are smaller than the radius of the sphere in the
whole energy range. The scattering lengths of all nuclear species considered are larger than $R$, and then the
propagation of all nuclei is mainly ballistic in the energy range under consideration. Therefore, proton and
high energy helium nuclei are less affected than the other nuclear species by photodisintegration and photopion
production processes.
\begin{figure}[!ht]
\includegraphics[width=8cm]{SILengths_B001G.eps}
\includegraphics[width=8cm]{SILengths_B100G.eps}
\caption{Interaction and scattering lengths of proton (p), helium (He), nitrogen (N), silicon (Si), and
iron (Fe) as a function of energy. In both cases, the curves are ordered from bottom to top by decreasing
primary mass. The dotted line corresponds to the radius of the sphere. (Top) $B=1$ G. (Bottom)
$B=100$ G. The coherence length of the random magnetic field is $l_c=R/10$. \label{SILengths}}
\end{figure}
The bottom panel of the figure shows the interaction and the scattering lengths, but for $B=100$ G and
$l_c=R/10$. As can be seen from the plot, the propagation in the random magnetic field of all nuclear
species considered is diffusive at low energies and ballistic at high energies. Therefore, in general,
the nuclei stay inside the sphere more time than in the case corresponding to $B=1$ G, and then the
composition of the nuclei that leave the sphere becomes lighter compared with the one corresponding
to that case.
It is assumed that the cosmic ray nuclei are accelerated in a region close to a supermassive black hole,
in such a way that the energy spectrum is given by a power law with an exponential cutoff,
\begin{equation}
\varphi(E) = \varphi_0\, E^{-\Gamma} \exp\left( -\frac{E}{Z\, E_{max}^{p}} \right),
\label{SpecInj}
\end{equation}
where $\varphi_0$ is a normalization constant, $\Gamma$ is the spectral index, $E_{max}^{p}$ is the
maximum energy for the proton component, and $Z$ is the charge number of the nucleus. Note that the
cutoff energy is proportional to the charge number, which is motivated by acceleration processes of
electromagnetic origin \cite{Allard:05,AugerFit:17}. The spectral index is taken as $\Gamma=1$, which
is motivated by acceleration mechanisms taking place in the accretion disks around massive black holes
\cite{Blandford:76} and also by the fit of the flux and mass composition data obtained by Auger and
reported in Ref.~\cite{AugerFit:17}.
Figure \ref{SpecSource} shows the energy spectra of the cosmic rays that leave the sphere corresponding
to the injection of silicon (top panel) and nitrogen (bottom panel) nuclei at the center of the sphere.
The injection spectrum of the nuclei is the one corresponding to Eq.~(\ref{SpecInj}) with
$E_{max}^{p}=10^{18.5}$ eV. The magnetic field is such that $B=100$ G and $l_c=R/10$ (see bottom panel
of Fig.~\ref{SILengths}).
\begin{figure}[!ht]
\includegraphics[width=8cm]{SpecSource_Si.eps}
\includegraphics[width=8cm]{SpecSource_N.eps}
\caption{Energy spectra of nuclei that leave the sphere corresponding to silicon (top) and nitrogen
(bottom). The magnetic field is such that $B=100$ G and $l_c=R/10$. The parameters of the injection
spectrum are: $\Gamma=1$ and $E_{max}^{p}=10^{18.5}$ eV. \label{SpecSource}}
\end{figure}
From the figure it can be seen that, for both nuclear species, a low energy light component is generated
due to the interactions undergone by the primary nuclei during propagation through the sphere.
\section{Flux at Earth}
The cosmic rays that leave the source environment are injected in the intergalactic medium and propagated
from a given position in the Universe to Earth. The propagation of the particles is performed by using
CRPropa 3. The simulations include photopion production and photodisintegration in the cosmic microwave
background (CMB) and in the extragalactic background light (EBL), pair production on the CMB and on the EBL,
nuclear decay, and the effects of the expansion of the Universe. The intergalactic magnetic field intensity
is assumed to be negligible and then the propagation is unidimensional. A uniform distribution of sources
in the Universe is assumed and the EBL model used in the simulations is the one developed in
Ref.~\cite{Kneiske:04}. The redshift range considered in the simulations starts at $z=0$ and ends at $z=5$.
The production of UHECRs over cosmological timescales is unknown. This source evolution is accounted by
a function of the redshift $z$, $S(z)$. In this work, two cases are considered. $S(z)=1$, which corresponds to
the case of no evolution of the sources and
\begin{equation}
S(z) = \left\{
\begin{array}{ll}
\left( 1+z \right)^5 & z \leq 1.7 \\[0.2cm]
2.7^{5} & 1.7 < z \leq 2.7 \\[0.2cm]
2.7^{5} \times 10^{2.7-z} & z > 2.7
\end{array}
\right.,
\label{Sz}
\end{equation}
which corresponds to the case of active galactic nuclei (AGN) of Ref.~\cite{Aloisio:15}.
In order to fit the cosmic ray energy spectrum above $E=10^{17.5}$ eV, a galactic low energy iron
component \cite{Unger:15} is assumed. The flux at Earth is supposed to be a power law with an
exponential cutoff \cite{Aloisio:14}, which is given by
\begin{equation}
J_{G}(E) = c_G\ E^{-\Gamma_G} \exp\left(-\frac{E}{E_{cut}} \right),
\end{equation}
where $\Gamma_G = 3.29$ is the spectral index for energies below the ankle \cite{AugerICRC:17},
$E_{cut}$ is the cutoff energy of the galactic component, and $c_G$ is a normalization constant.
The last two parameters are chosen in each model considered in order to fit the Auger spectrum.
Following Ref.~\cite{AugerFit:17}, five nuclear species that are accelerated in the sources are
considered: proton, helium, nitrogen, silicon, and iron.
The Auger energy spectrum reported in Ref.~\cite{AugerICRC:17} is fitted by minimizing the $\chi^2$
given by
\begin{equation}
\chi^2=\sum_{i=1}^N \frac{\left( j_i-J_G(c_G,E_i)-\sum_A c_A J_A(E_i) \right)^2}{\sigma_i^2},
\end{equation}
where $j_i$ and $\sigma_i$ are the measured flux and its uncertainty, respectively, corresponding to
the energy bin centered at energy $E_i$. Here $N$ is the number of energy bins considered in the fit
and $A=\{\textrm{p, He, N, Si, Fe}\}$. The free fit parameters are $c_G$ and $c_A$, i.e., just the
relative contributions of the different components are fitted. Note that the fitting parameters have
to be positive or zero. This condition is fulfilled during the minimization procedure.
Figure \ref{M1} shows the fit of the Auger spectrum and the predicted mean value of the natural logarithm
of the mass number $\langle \ln(A) \rangle$ and its variance $\textrm{Var}[\ln(A)]$ as a function of
the primary energy compared with the experimental data obtained by Auger. The data points corresponding
to the mean value of the natural logarithm of the mass number and its variance are obtained from the
mean value and the variance of the $X_{max}$ parameter, which is reconstructed in an event-by-event
basis from the data taken by the fluorescence telescopes of Auger \cite{AugerXmax:14}. Both quantities
are obtained in Ref.~\cite{AugerXmax:14} by using simulations of the showers with the high energy
hadronic interaction model EPOS-LHC \cite{EPOSLHC:15}. It is worth mentioning that,
$\langle \ln(A) \rangle$ as a function of primary energy, obtained by using EPOS-LHC, falls in between
the ones corresponding to the two other models more frequently used in the literature, QGSJET-II-04
\cite{QGII:11,QGII:11b} and Sibyll 2.3c \cite{Sibyll2.3c:15}. Moreover, $\langle \ln(A) \rangle$
obtained by using Sibyll 2.3c is above the one corresponding to EPOS-LHC, which in turn is above the
one corresponding to QGSJET-II-04.
\begin{figure}[!ht]
\includegraphics[width=8cm]{SpectrumRef_M1.eps}
\includegraphics[width=8cm]{lnA_M1.eps}
\includegraphics[width=8cm]{VlnA_M1.eps}
\caption{Fit of the UHECR spectrum (top) and the prediction for the mean value of the natural logarithm
of the mass number (middle) and its variance (bottom). The experimental data were obtained by
Auger \cite{AugerICRC:17,AugerXmax:14} and the high energy hadronic interaction model used in the composition
analysis is EPOS-LHC. The parameters of the model are $L=10^{41}$ erg s$^{-1}$, $B=100$ G, $l_c=R/10$,
$E_{max}^{p}=10^{18.5}$ eV, and $S(z)$ from Eq.~(\ref{Sz}). \label{M1}}
\end{figure}
The model of Fig.~\ref{M1} assumes a luminosity of the photon gas $L=10^{41}$ erg s$^{-1}$, a random
magnetic field of the source environment such that $B=100$ G and $l_c=R/10$, and the maximum energy of the
injected proton component $E_{max}^{p}=10^{18.5}$ eV, i.e., the parameters used to obtain Fig.~\ref{SpecSource}.
The source evolution function considered is the one in Eq.~(\ref{Sz}). In the best fit scenario, the injected
composition is dominated by iron nuclei with a small contribution of protons, as can be seen from the top panel
of the figure, in which the contribution of the two components, obtained after propagation in the source
environment and in the intergalactic medium, are shown. Note that, in this scenario, the galactic component
appears at energies below $10^{17.5}$ eV. As can be seen from the figure, this model is not compatible with the
Auger data. The reason for that is the very fast evolution of the sources at low redshift values, which increases
the light component below the ankle, making the flux steeper than the one observed. Also the composition becomes
progressively light for decreasing values of primary energy, which is inconsistent with the minimum in the
$\langle \ln(A) \rangle$ observed at $10^{18.25}$ eV.
Considering the same scenario as before but for the case in which the sources do not evolve with redshift,
i.e.~$S(z)=1$, and for $B=1$ G, a good fit of the energy spectrum is obtained. In this case, the propagation in
the source environment is not affected by the random magnetic field, as can be seen from the top panel of
Fig.~\ref{SILengths}. The results corresponding to this model are shown in Fig.~\ref{M2}. In this case, the
injection spectrum is dominated by helium, silicon, and iron; the proton and nitrogen contributions are
negligible. In the top panel of the figure, the contributions of these three different nuclear species are
shown, which are obtained after propagation in the source environment and in the intergalactic medium. In
this case, the galactic component is not negligible in the energy range considered, as can be seen from the
top panel of the figure (dashed line), and is such that $E_{cut}=10^{17.75}$ eV. Note that the discrepancies
between the model predictions for $\textrm{Var}[\ln(A)]$ and the experimental data are larger at low energies,
and this can be due to the too simple assumption for the composition of the galactic flux.
\begin{figure}[!ht]
\includegraphics[width=8cm]{SpectrumRef_M2.eps}
\includegraphics[width=8cm]{lnA_M2.eps}
\includegraphics[width=8cm]{VlnA_M2.eps}
\caption{Same as Fig.~\ref{M1}, but for $B=1$ G and $S(z)=1$. \label{M2}}
\end{figure}
In order to study the influence of the random magnetic field present in the source environment, a scenario
with the same parameters as the previous one (with $S(z)=1$) but for $B=100$ G is considered. As can be seen
from Fig.~\ref{M3}, also in this case a good fit of the spectrum is obtained. In this scenario, the injected
spectrum is dominated by silicon and iron nuclei, and as can be seen from the top panel of the figure, the
contribution of the other components is negligible. From the middle panel of the figure, it can be seen that
$\langle \ln(A) \rangle$ is smaller than the one corresponding to the previous scenario. This is due to the
fact that increasing the magnetic field intensity increases the number of nuclei that propagate diffusively
through the source environment (see Fig.~\ref{SILengths}). The particles that propagate in the diffusive
regime travel larger path lengths, which causes an increase of the number of interactions, mainly
photodisintegrations, undergone by them.
\begin{figure}[!ht]
\includegraphics[width=8cm]{SpectrumRef_M3.eps}
\includegraphics[width=8cm]{lnA_M3.eps}
\includegraphics[width=8cm]{VlnA_M3.eps}
\caption{Same as Fig.~\ref{M1}, but for $S(z)=1$. \label{M3}}
\end{figure}
Photons and neutrinos are produced as a consequence of the UHECR propagation through the Universe.
These secondary particles are generated by the decay of pions produced by the photopion production
process undergone by the nuclei that interact with the low energy photons of the CMB and EBL. There
is also a contribution to the neutrino component that comes from neutron decay. The only energy loss
undergone by the neutrinos is the one corresponding to the adiabatic expansion of the Universe. Unlike
what happens to the neutrinos, the high energy photons interact with the low energy photons of the CMB
and EBL, initiating electromagnetic cascades that develop in the intergalactic medium. As a result, the
photon flux at Earth spans from the ultrahigh energy region down to energies below 1 GeV. Therefore,
the UHECRs can contribute to the low energy diffuse photon background.
The photon and neutrino fluxes corresponding to the two models compatible with the Auger data are
calculated by using CRPropa 3. Figure \ref{GammaNu} shows the $\gamma$-ray and neutrino fluxes for the
model corresponding to Fig.~\ref{M3} ($B=100$ G). As can be seen from the plot, the photon flux is
smaller than the isotropic $\gamma$-ray background (IGRB) observed
by the \emph{Fermi} Large Aea Telescope (\emph{Fermi}-LAT) \cite{FermiLATIGRB:15}. Moreover, the integral
of the flux between 50 GeV and 2 TeV is $\sim 20$ times smaller than the 90\% C.L.~upper
limit of Ref.~\cite{Supanitsky:16}, which was obtained by using the \emph{Fermi}-LAT analysis reported in
Ref.~\cite{FermiLATPRL:16}. Also, from the figure it can be seen that the secondary neutrino flux is much
smaller than the upper limits obtained by IceCube \cite{IceCube:18} and by Auger \cite{AugerNu:17}. Similar
results are obtained for the model of Fig.~\ref{M2} ($B=1$ G). Low values of the $\gamma$-ray and neutrino
fluxes are expected because it is very well known that the production of secondary particles, in models in
which the high energy part of the cosmic ray flux is dominated by heavy nuclei, is much smaller than in
the ones dominated by protons \cite{Aloisio:11}, which are still compatible with the neutrinos and
$\gamma$-ray constraints in a region of the parametric space \cite{Supanitsky:16,Berezinsky:16}.
\begin{figure}[!ht]
\includegraphics[width=8.5cm]{GammaRaysNuB100G.eps}
\caption{$\gamma$-ray and all flavors neutrino flux expected for the model of Fig.~\ref{M3}. The data points
correspond to the IGRB obtained by \emph{Fermi}-LAT \cite{FermiLATIGRB:15}. Also shown are the upper limits
to the neutrino flux at 90\% C.L., obtained by IceCube \cite{IceCube:18} and by Auger \cite{AugerNu:17}.
\label{GammaNu}}
\end{figure}
Increasing the luminosity of the photon gas present in the source environment makes the composition
lighter; this is due to the decrease of the interaction lengths of the nuclei. In the limit of
$\lambda_{PD},\, \lambda_{PP} \ll R$ all nuclear species are disintegrated before leaving the sphere,
and then a light composition formed by protons and neutrons is obtained. It is found that models with
$L\geq5\times10^{41}$ erg s$^{-1}$ are not compatible with the experimental data. Therefore, preferred
models are such that $L \lesssim 10^{41}$ erg s$^{-1}$, which corresponds to low luminosity AGN (LLAGN)
\cite{Ho:97,Ho:99}. It is worth noting that the central regions of these types of galaxies have been
proposed as sources of the astrophysical neutrino flux observed by IceCube \cite{IceCubeNF:13}. In these
models the high energy neutrinos are produced as a by-product of accelerated protons
\cite{Kimura:15,Khiali:16}.
The redshift evolution of the sources is poorly known. In general, the source evolution of AGN is assumed
to increase very fast between $z=0$ and $z=1-2$ (like in Eq.~(\ref{Sz})) \cite{Hasinger:05}. However, in
Ref.~\cite{Kimura:15} a nonevolving luminosity function for LLAGN is assumed. This is the case of the
scenarios developed in this work, which are compatible with the Auger data. The argument in
Ref.~\cite{Kimura:15} for this assumption is that LLAGN are similar to BL Lac objects (they both have a
faint disk component), which have a luminosity function consistent with no evolution \cite{Ajello:14}.
It is worth mentioning that it is possible to fit the Auger spectrum assuming the evolution function
of Ref.~\cite{Kachel:17}, which corresponds to AGNs of $\log(L_X/\textrm{erg})=43.5$. However,
the composition predicted in this case is heavier at high energies ($\gtrsim 10^{18.8}$ eV) and
lighter at low energies ($\lesssim 10^{18.3}$ eV) than the one obtained by using EPOS-LHC to analyze
the Auger data. The behavior of the high energy part of the composition profile is consistent with the
results obtained in Ref.~\cite{Kachel:17}. Therefore, also in this case the interaction of the nuclei
with the protons present in a second region, surrounding the photon gas and filled with a proton gas,
would be an appropriated mechanism to obtain a lighter composition at high energies, as it proposed in
Ref.~\cite{Kachel:17}.
In Ref.~\cite{Kachel:17}, a one-dimensional approach for the propagation of the nuclei in the source
environment is considered. In that approximation, only the diffusive regime of propagation is taken
into account. The escape time used in these types of calculations is taken as
$\tau(E) =\tau_0\, [E/(Z\, E_0)]^{-\delta}$, where $\tau_0$ is a normalization constant, $E$ is the
energy of the nucleus, $Z$ is its charge number, $E_0$ is a reference energy, and $\delta$ is a
positive index. In the leaky box model approximation $\tau(E) \propto 1/D(E)$, where $D(E)$ is the
diffusion coefficient. Therefore, the index $\delta$ gives the energy dependence of the diffusion
coefficient. The escape time in these types of models is a decreasing function of the energy, which
is valid up to a distance of the order of the size of the source environment region. Therefore,
depending on the parameters used for the escape time in the one-dimensional calculation and the size of
the source environment used in the three-dimensional approach, the high energy nuclei can escape from
the source environment region before, compared to the case in which the particle propagates ballistically.
In this case, a larger light component is expected at low energies for the three-dimensional calculation
due to the larger number of photodisintegrations undergone by the high energy nuclei. This is the case
for the model in which the source evolution function of Ref.~\cite{Kachel:17} is considered. The larger
number of light nuclei at low energies makes the composition lighter than the one obtained by Auger, using
EPOS-LHC to interpret the data, and also a harder spectral index is required, $\Gamma=1$, to obtain a
good fit of the spectrum compared to the one considered in Ref.~\cite{Kachel:17}.
It should be noted that an independent composition analysis in the region of the spectrum below the
ankle will be possible, in the near future, by using the information of the muon detectors of AMIGA
(Auger Muons and Infill for the Ground Array) that are being installed at the Auger site
\cite{Figueira:17}. As mentioned before, the muon content of the shower is very sensitive to the nature
of the primary cosmic ray.
\section{Conclusions}
In this work, we have studied the possibility that the presumed extragalactic light component that dominates
the UHECR flux below the ankle originates from the photodisintegration of more energetic and heavier nuclei in the
photon gas present in the central regions of active galaxies. In this scenario, the UHECRs are accelerated near
the supermassive black hole present in the central region of these galaxies. Note that these types of models
require only one population of UHECR sources to explain the experimental data above $\sim 10^{18}$ eV.
We have found that low luminosity active galaxies with no source evolution are compatible with present composition
and flux data, within the systematic uncertainties introduced by the high energy hadronic interaction models. It is
worth mentioning that these types of astronomical objects have been proposed as the source of the high energy
neutrinos observed by IceCube. We have also found that increasing the intensity of the random magnetic field in
the source environment makes the composition observed at Earth lighter, as expected. However, we have proved that
models with larger values of luminosity of the photon gas or with a strong source evolution are incompatible with
present experimental data.
\begin{acknowledgments}
A.~D.~S.~and A.~E.~are member of the Carrera del Investigador Cient\'ifico of CONICET, Argentina. This work is
supported by ANPCyT PICT-2015-2752, Argentina. The authors thank the members of the Pierre Auger Collaboration
for useful discussions and R. Clay and C. Dobrigkeit Chinellato for reviewing the manuscript.
\end{acknowledgments}
|
1,116,691,498,447 | arxiv | \section{I. Introduction}
\label{Intro}
Strongly correlated electron systems in one dimension (1d) have become an area
of immense interest from the perspective of both fundamental and technological
aspects of nanophysics. Intense experimental effort has focused on such
realizations of quantum wires with a few or single conducting channels as
cleaved-edge \cite{auslaender02}, V-groove \cite{palevski05}, and
crystallized-in-a-matrix \cite{zaitsev-zotov00} semiconductor quantum wires,
coupled quantum Hall edges running in opposite directions
\cite{kang00,grayson05}, single-wall carbon nanotubes \cite{nt}, polymer
nanofibers \cite{aleshin04}, and metallic nanowires
\cite{slot04,venkataraman06}. Central to much of the fascinating physics of
the 1d systems is that electron-electron (e-e) interactions in 1d geometry can
have dramatic effects leading to the emergence of a Luttinger liquid (LL)
\cite{giamarchi04}. The latter constitutes a canonical example of a non-Fermi
liquid, in which quasiparticle fermionic excitations are inappropriate to
describe low-energy physics. At the foundation of the conventional LL theory
is a description in terms of bosonic elementary excitations (plasmons,
spinons) \cite{giamarchi04}. Following this approach, the ground-state
properties of a clean LL are well understood for arbitrary strength of
interaction. Much has also been learned about the LL in the presence of a
single compact scatterer \cite{kane92,giamarchi04}. However, as far as a
disordered LL is concerned, a number of important questions, even at the most
fundamental level, remained largely unanswered until very recently (for a
review see Ref.~\cite{gornyi07}).
In the presence of disorder, quantum interference of scattered electron waves
leads to the effects of Anderson localization \cite{anderson58}. Similarly to
e-e interactions, the lower the dimensionality, the stronger the localization
effects. In a 1d electron gas, all electron states are localized even for an
arbitrarily weak random potential and the localization length is the mean free
path. In the case of noninteracting electrons, the quantum localization in 1d
has been studied in great detail (see, e.g., Ref.~\cite{berezinskii73}). A
principal complication that arises in the disordered LL is that the quantum
interference phenomena yielding the Anderson localization are conventionally
treated in terms of fermions, by employing the concepts of interference and
dephasing of fermionic excitations. The question of to what extent the notion
of phase relaxation in the localization problem is applicable to the
(non-Fermi) LL is therefore of crucial importance. Recently, this problem was
addressed in Refs.~\cite{gornyi05,gornyi07}, where the interaction-induced
dephasing rate that governs the localization term in the conductivity of the
disordered LL was calculated.
Another conceptually nontrivial aspect of the interplay between disorder and
interaction concerns the nonequilibrium properties of the LL. In the
homogeneous case, the LL model is completely integrable and as such does not
exhibit any relaxation to equilibrium: an arbitrary excited state will never
decay to the equilibrium state characterized by temperature. Of central
importance is therefore the question of how the equilibration of fermionic
and/or bosonic excitations in the LL occurs in the presence of disorder.
This paper is primarily concerned with various relaxation processes associated
with inelastic interactions between electrons in the disordered LL.
Specifically, we focus on the rates of e-e scattering, phase relaxation, and
energy relaxation, with emphasis on the essential differences between them. In
Sec.~II
we begin with the formulation of the model. Section
III
highlights a few aspects of temperature-dependent screening of disorder in
1d. Section
\ref{deph}
IV
covers the problem of phase relaxation---this
discussion largely follows the results of
Refs.~\cite{gornyi05,gornyi07} and serves as the starting point for our
approach to nonequilibrium physics of the LL. In
Sec.~V
we consider
energy relaxation and introduce a general framework \cite{bagopo} for studying
the behavior of the disordered LL out of equilibrium.
\section{II. Model}
\label{mod}
Let us specify the model. By decomposing the electron operator into right- and
left-moving parts , $\psi(x)=\psi_+(x)+\psi_-(x)$, the Hamiltonian of a
disordered LL is written as
\begin{eqnarray}
H&=&H_{\rm kin}+H_{\rm ee}+H_{\rm dis}~,
\label{1}\\
H_{\rm kin}&=&
-v_F\sum_{\mu=\pm}\int \!dx\,
\psi^\dagger_{\mu}\left(i\mu\partial_x+k_F\right) \psi_{\mu}~,
\label{2}
\\
H_{\rm ee}&=&{1\over 2}\sum_{\mu=\pm}\int \!dx
\left( n_{\mu} \, V_f\,
n_{-\mu}
+ n_{\mu} \, \tilde{V}_f\, n_{\mu}
\right)~,
\label{3}
\\
H_{\rm dis}&=& \int \!dx
\left[ U_b(x)\ \psi^\dagger_-\psi_++{\rm H.c.}\right]~.
\label{4}
\end{eqnarray}
Here $n_\mu= \psi^\dagger_\mu \psi_\mu$ is the density of the right and left
movers and their dispersion relation is linearized about two Fermi points at
the wavevectors $\pm k_F$ with the velocity $v_F$. Throughout the paper we
consider spinless electrons (for spin-related effects see
Ref.~\cite{yashenkin}).
The e-e interaction, Eq.~(\ref{3}), is characterized by the Fourier components
of the short-range (screened) interaction potential with zero momentum
transfer $V_f$ (forward scattering between right and left movers) and
$\tilde{V}_f$ (forward scattering of electrons from the same chiral branch on
each other). Unless the right and left movers are spatially separated (as in
coupled quantum Hall edges), $\tilde{V}_f=V_f$. The Luttinger model {\it per
se} does not include backward scattering characterized by the Fourier
component $V_b$ with momentum transfer $\pm 2k_F$. For spinless electrons,
however, $V_b$ can be trivially incorporated by shifting $V_f\to V_f-V_b$,
since two types of scattering---due to $V_f$ and $V_b$---are then related to
each other as direct and exchange processes. The local interaction between
identical fermions $\tilde{V}_f$ yields no scattering, but, due to a quantum
anomaly in the LL model, generates a shift of the Fermi velocity $v_F\to
v_F^*=v_F+\tilde{V}_f/2\pi$. It is customary to parametrize the strength of
e-e interaction by means of the Luttinger constant $K$:
\begin{equation}
K=\left({1-\alpha\over 1+\alpha}\right)^{1/2}~,\quad \alpha={V_f\over 2\pi
v_F^*}~.
\label{5}
\end{equation}
The velocity of elementary excitations (plasmons) in a clean LL is given by
\begin{equation}
u=v_F^*(1-\alpha^2)^{1/2}~,
\label{5a}
\end{equation}
which transforms for $\tilde{V}_f=V_f$ into $u=v_F/K$.
The low-energy theory described by the Hamiltonian (\ref{1}) is only then
well-defined when supplemented by an ultraviolet energy cutoff $\Lambda$. The
latter depends on microscopic details of the problem and obeys
\begin{equation}
\Lambda=u/\pi\lambda~,
\end{equation}
where the length scale $\lambda$ is set by the lattice constant, the Fermi
wavelength, or the spatial range of interaction in the original microscopic
theory, whichever gives the smallest $\Lambda$. Thus the complete set of
parameters defining the LL model in the absence of disorder includes $v_F^*$,
$V_f$, and $\Lambda$. It is worth noting that the input parameters of the
low-energy theory include Fermi-liquid-type renormalizations coming from
energy scales larger than $\Lambda$; in particular, the ``bare" $v_F$ in
Eq.~(\ref{1}) in general is not an interaction-independent constant
if the interaction is strong ($1-K\sim 1$).
The term $H_{\rm dis}$, Eq.~(\ref{4}), describes backscattering of electrons
off a static random potential $U(x)$. The latter is taken to be of white-noise
type with the correlators of backscattering amplitudes
\begin{equation}
\overline{U_b(x)U_b^*(0)}=\overline{U(x)U(0)}=w\delta(x)
\label{6}
\end{equation}
and $\overline{U_b(x)U_b(0)}=0$. The forward-scattering amplitudes are omitted
in Eq.~(\ref{4}) since they can be gauged out in the calculation of the
conductivity.
\section{III. Elastic scattering}
\label{dru}
One of the characteristic features of a LL is a large renormalization of the
strength of disorder (\ref{6}) by e-e interaction. In particular, the
conductivity without any localization \cite{anderson58} or pinning
\cite{larkin70} effects included (``Drude
conductivity") is $\sigma_{\rm D}(\omega,T)=e^2v_F/\pi[-i\omega+M(\omega,T)]$,
where the disorder-induced scattering rate in the dc limit
\begin{equation}
{1\over \tau(T)}={\rm Re}\,M(0,T)
=a_K\,{1\over \tau_0}\left({\Lambda\over T}\right)^{2(1-K)}
\label{7}
\end{equation}
grows as a power law with decreasing temperature $T$ for repulsive interaction
($K<1$). The momentum relaxation rate in the absence of interaction is given
by $\tau_0^{-1}=2wv_F^{-1}$ with $w$ from Eq.~(\ref{6}). Calculating the Drude
conductivity at finite $\omega$ and sending $\omega\to 0$ afterwards allows to
unambiguously determine the coefficient \cite{mirlin07}
$a_K=\Gamma^2(1+K)/\Gamma (2K)$ in the relaxation rate. Here and below the
disorder is supposed to be weak in the sense that $\Lambda\tau\gg 1$.
The underlying physics of the renormalization (\ref{7}) can be described in
terms of the $T$-dependent screening of individual impurities; specifically,
in terms of scattering by Friedel oscillations which slowly decay in real
space and are cut off on the spatial scale of the thermal length. At this
level, the only peculiarity of the LL as compared to higher dimensionalities
is that the renormalization of $\tau$ is more singular and, most importantly
from the calculational point of view, necessitates going beyond the
Hartree-Fock approximation, even for weak interaction (see, e.g.,
Ref.~\cite{polyakov03}).
In general, not only the strength of disorder but also the strength of
interaction is subject to renormalization and depends on $T$, so that the
function $\tau(T)$ is not a simple power law. An important question,
therefore, is under what condition the exponent in Eq.~(\ref{7}) is given by
the bare interaction coupling constant. One of the approaches to the problem
was formulated in Ref.~\cite{giamarchi88} in terms of a bosonic
renormalization group (RG). The RG approach does not allow to obtain the
$K$-dependent prefactor $a_K$ in Eq.~(\ref{7}), but is particularly beneficial
in predicting the $T$ dependence of the Drude conductivity. For spinless
electrons, the one-loop RG equations read
\begin{eqnarray}
dK/d{\cal L}&=&f(K){\cal D}~,
\label{8a}
\\
d{\cal D}/ d{\cal L}&=&(3-2K){\cal D}~,
\label{8b}
\end{eqnarray}
where ${\cal L}=\ln L/\lambda$ and ${\cal D}=2w\lambda/\pi u^2$. For the Drude
conductivity (i.e., as long as the localization effects are not included, see
Sec.~IV), the spatial scale $L$ is given by the thermal length $u/T$. The
scattering rate $1/\tau (T)$ is then proportional to $T{\cal D}(T)$. The
function $f(K)=-K^2/2+(1+K^2)(3-2K)/4$ vanishes at $K=1$, so that interaction
is not generated by disorder (in the original equations of
Ref.~\cite{giamarchi88}, the coupling constant $K$ contains an admixture
of disorder and therefore the corresponding $f(1)\neq 0$, see
Ref.~\cite{gornyi07} for a discussion of this point); moreover, the
interaction (hence $1-K$) does not change sign in the course of
renormalization.
The RG flow (\ref{8a}),(\ref{8b}) is characterized by a separatrix which
behaves as ${\cal D}=8(K-3/2)^2/9$ for $K>3/2$ and terminates at $K=3/2$. For
the bare (taken at $L=\lambda$) values of $\cal D$ and $K$ that lie below the
separatrix (i.e., for the case of strong attractive interaction with $K>3/2$),
the disorder strength $\cal D$ renormalizes to zero, otherwise $\cal D$ grows
with increasing $L$ to a strong-coupling point with ${\cal D}\sim 1$. The
renormalization of the coupling constant $K$ by disorder is essential if the
RG trajectory is close to the parabola ${\cal D}=8(K-3/2)^2/9$. For example,
if the RG flow passes through the point $K=3/2$, the integration of
Eqs.~(\ref{8a}) and (\ref{8b}) gives for ${\cal D}\ll 1$:
\begin{equation}
{\cal D}-{\cal D}_0={\cal D}_0\,\tan^2\left({3{\cal D}_0^{1/2}\over
2^{3/2}}\ln {L\over L_0}\right)~,
\label{9}
\end{equation}
where ${\cal D}_0$ and $L_0$ are the values of ${\cal D}$ and $L$ at $K=3/2$
and the sign of $\ln (L/L_0)$ is positive for running $K<3/2$ and negative
otherwise. One sees that ${\cal D}$ grows with increasing $L$ for $K<3/2$ as
\begin{equation}
{\cal D}=8/9\ln^2(l/L)
\label{10}
\end{equation}
(for ${\cal D}_0\ll {\cal D}\ll 1$). Here the renormalized mean free path $l$
(the scale at which ${\cal D}\sim 1$) obeys $\ln (l/L_0)=2^{1/2}\pi/3{\cal
D}_0^{1/2}$. The logarithmic dependence of $\cal D$ on $L$ is precisely due to
the renormalization of $K$.
On the other hand, if the bare $K<3/2$, the RG trajectory follows
Eq.~(\ref{9}) with $L_0=\lambda$ and ${\cal D}_0$ understood as the bare value
of ${\cal D}$ only at ${\cal D}-{\cal D}_0\gg (K-3/2)^2$. Integrating
Eqs.~(\ref{8a}),(\ref{8b}) in the opposite limit
\begin{equation}
{\cal D}-{\cal D}_0\ll
(K-3/2)^2~,
\label{11}
\end{equation}
one gets
\begin{equation}
{\cal D}={\cal D}_0(L/\lambda)^{3-2K}~,
\label{12}
\end{equation}
which corresponds to Eq.~(\ref{7}). Equation (\ref{11}) thus answers the
question of when the renormalization of $K$ may be neglected. Notice that for
repulsive interaction ($K<1$) the condition (\ref{11}) is satisfied for the
whole range of ${\cal D}\ll 1$ [which is where the RG equations
(\ref{8a}),(\ref{8b}) are valid]. It follows that for the most relevant case
of direct Coulomb interaction the renormalization of interaction on ballistic
scales (${\cal D}\ll 1$) plays no role and the exponent in Eq.~(\ref{7}) is
$T$-independent and given by the bare value of $K$ (the one in a clean
system). In other words, the renormalization of disorder for repulsive
interaction reduces to the renormalization of an individual impurity. It is
worth emphasizing that this does not mean that the disorder-induced correction
to the bare value of $1-K$ is small: in fact, the correction is of the order
of $1-K$ itself when ${\cal D}\sim 1$. The point is that the exponent of
${\cal D}(L)$ and, correspondingly, of the renormalized scattering rate
$1/\tau(T)$ is not given by the running coupling constant $K$, but rather is
accumulated on the whole RG trajectory.
\section{IV. Phase relaxation}
\label{deph}
The renormalization of $\tau$ stops with decreasing $T$ at
\begin{equation}
T\tau(T)\sim 1~,
\label{13}
\end{equation}
since the long-range Friedel oscillations created by disorder are cut off even
at zero $T$ on the spatial scale of the disorder-induced mean free path. This
condition gives the zero-$T$ mean free path $l\propto \tau_0^{1/(3-2K)}$
[notice that Eq.~(\ref{13}) is also expressible as ${\cal D}(L)\sim 1$ with
$L=u/T$] and, correspondingly, the zero-$T$ localization length $\xi\sim
l$. It is important to stress, however, that the above condition does not
correctly predict the onset of localization with decreasing $T$---in contrast
to the argument, frequently stated in the literature (see, e.g.,
Ref.~\cite{giamarchi04} and references therein) and based on the RG equations
(\ref{10}),(\ref{11}), which treat scalings with the length scales $L$ and
$u/T$ as interchangeable. While substituting $u/T$ for $L$ is justified for
the ``elastic renormalization" [Eq.~(\ref{7})], the one-loop equations
(\ref{10}),(\ref{11}) miss, by construction, the interference effects
(coherent scattering on several impurities) that lead to localization. The
status of the RG \cite{giamarchi88} is thus that of the Drude formula for
interacting electrons. The $T$ dependence of the conductivity $\sigma(T)$,
however, comes not only from the $T$-dependent screening of disorder
[Eq.~(\ref{7})], but also from the localization term in $\sigma(T)$ whose
amplitude is governed by phase relaxation due to inelastic e-e scattering. The
temperature below which the localization effects become strong is, in contrast
to Eq.~(\ref{13}), determined by the condition
\begin{equation}
\tau(T)/\tau_\phi(T)\sim 1~,
\label{14}
\end{equation}
where $\tau_\phi$ is the weak-localization dephasing time. Notice that for
weak interaction ($1-K\simeq \alpha\ll 1$), Eq.~(\ref{14}) is satisfied at
much higher $T$ than Eq.~(\ref{13}). Below we introduce the notion of
dephasing of localization effects in the disordered LL and analyze the phase
relaxation in the limit of weak interaction.
The very applicability of the notion of dephasing, as we know it from the
studies of higher-dimensional Fermi-liquid systems, to the LL is not
altogether apparent. A subtle question concerns the nature of elementary
excitations in the LL, especially in the presence of disorder. The clean LL is
a completely integrable model which is represented in terms of noninteracting
(hence nondecaying) bosons; however, the phase relaxation in electron systems
is conventionally described in terms of interacting fermions. Physically, the
difficulty is related to the fact that the bosonized approach describes
propagation of density fluctuations, whereas the natural language for quantum
interference phenomena is that of quantum amplitudes. To study the
interference effects and their dephasing, one has therefore to either proceed
with the standard bosonization, poorly suited to describe the quantum
interference in the inhomogeneous case, or try to define the observables in
such a way that they can be expressible in terms of decaying fermionic
excitations. In what follows in this section, we take the latter path and give
a succinct analysis of the phase relaxation in the disordered LL, based on the
results obtained within the ``functional bosonization" formalism
\cite{gornyi07} and the quasiclassical formalism \cite{gornyi05}, both of
which combine the fermionic and bosonic approaches to the problem.
Let us first point out one of the subtleties of the LL model, which is crucial
to our discussion of the phase and energy relaxation. The Golden rule
expression for the e-e collision rate at equilibrium, as follows from the
Boltzmann kinetic equation, reads
\begin{eqnarray}
{1\over \tau_{\rm ee} (\epsilon)}&=&\int\! d\omega\int\! d\epsilon' \,
{\cal K}(\omega)\nonumber \\
&\times&
\left(f^h_{\epsilon-\omega}f_{\epsilon'}
f^h_{\epsilon'+\omega}+f_{\epsilon-\omega}
f^h_{\epsilon'}f_{\epsilon'+\omega}\right)~,
\label{15}
\end{eqnarray}
where $f_\epsilon$ is the Fermi distribution function and
$f^h_{\epsilon}=1-f_{\epsilon}$. Consider the {\it clean} case. Then the
scattering kernel ${\cal K}(\omega)={\cal K}^H_{++}(\omega)+{\cal
K}^H_{+-}(\omega)+{\cal K}^F(\omega)$ to second order in the interaction is
given by
\begin{eqnarray}
{\cal K}^H_{++}&=&{\tilde{V}_f^2\over \pi^3\rho}\int \!{dq\over 2\pi}\,
\left[\,{\rm Re} D_{+}(\omega,q)\,\right]^2~,
\label{16}
\\
{\cal K}^H_{+-}&=&
{V_f^2\over \pi^3\rho}\int \!{dq\over 2\pi} \,
{\rm Re} D_{+}(\omega,q)\,{\rm Re} D_{-}(\omega,q)~,
\label{17}
\end{eqnarray}
and ${\cal K}^F=-{\cal K}^H_{++}$. The Hartree terms ${\cal K}^H_{++}$ and
${\cal K}^H_{+-}$ are related to scattering of two electrons from the same or
different chiral spectral branches, respectively, ${\cal K}^F$ is the Fock
(exchange) counterpart of ${\cal K}^H_{++}$, the thermodynamic density of
states $\rho=1/\pi v_F$, and $D_\pm=i\pi \rho/(\omega\mp qv_F+i0)$ are the
two-particle propagators in the clean limit.
Substituting Eqs.~(\ref{16}),(\ref{17}) in Eq.~(\ref{15}), we obtain the
lowest-order result for the e-e scattering rate at the Fermi level
$(\epsilon=0)$ in terms of the corresponding contributions to the retarded
electronic self-energy $\Sigma_+$ defined by
$G^R_+(\epsilon,p)=[\,\epsilon-v_Fp-\Sigma_+(\epsilon,p)\,]^{-1}$, where
$G^R_+$ is the retarded Green's function for right-movers. Specifically,
$\tau^{-1}_{\rm ee}=-2{\rm Im}\,\Sigma_+(0,0)$ with
$\Sigma_+(0,0)=\Sigma^H_{++}+\Sigma^H_{+-}+\Sigma^F$, where
\begin{eqnarray}
{\rm Im}\Sigma^H_{+\pm}&=&-{\pi\over 2}\alpha^2v_F\!\int \!d\omega
\,\omega\left(\coth{\omega\over 2T}-\tanh{\omega\over 2T}\right)\nonumber \\
&\times&\int \! dq\, \delta(\omega-v_Fq)\delta(\omega\mp v_Fq)~,
\label{18}
\end{eqnarray}
$\Sigma^F=-\Sigma^H_{++}$, and we put $V_f=\tilde{V}_f$. One sees that the
contribution of $\Sigma^H_{++}$ is diverging. For spinless electrons, however,
the divergency is canceled by the exchange interaction. Indeed, as we have
discussed in Sec.~II, the $\tilde{V}_f$ interaction drops out of the
problem in this case, inducing only a shift of the velocity $v_F\to v_F^*$. It
is worth noting that the ``Hartree-Fock cancellation" is only exact for the
point-like interaction (when $\tilde{V}_f$ is independent of the transferred
momentum), otherwise $\tilde{V}_f$ yields a nonzero contribution
\cite{chalker07} to $\tau_{ee}^{-1}$. The latter is small and can be neglected
in the low-$T$ limit for $\tilde{V}_f=V_f$ but is the only one present for
$V_f=0$, which is the case, e.g., for an isolated quantum Hall edge. The
remaining term $\Sigma^H_{+-}$ gives
\begin{equation}
\tau_{ee}^{-1}=-2{\rm Im}\,\Sigma^H_{+-}=\pi\alpha^2T~.
\label{19}
\end{equation}
This may look very similar to the familiar $T^2$ or $T^2\ln T$ dependence of
the e-e scattering rate in clean three- or two-dimensional electron systems,
respectively. However, the nontrivial point---which demonstrates the
peculiarity of the LL model---is that $\tau_{ee}^{-1}$ in Eq.~(\ref{19}) is
determined by
\begin{equation}
\omega, q=0~,
\label{20}
\end{equation}
i.e., by scattering processes with infinitesimally small energy transfers, in
contrast to higher dimensions where the characteristic transfer is of order
$T$. On the other hand, it is worth emphasizing that $T\tau_{ee}\gg 1$ for
$\alpha\ll 1$, which in Fermi-liquid theory is commonly referred to as one of
the conditions for the existence of a Fermi liquid. In this respect, the
weakly interacting LL, while being a canonical example of a non-Fermi liquid,
reveals the typical Fermi-liquid property. The LL physics (e.g., the power-law
singularity of the tunneling density of states at the Fermi level) is in fact
encoded in the singular {\it real} part of the self-energy $\Sigma_+(\epsilon,
p)$ (for more details see Ref.~\cite{gornyi07}).
It is the property (\ref{20}) that actually makes the 1d case special as far
as the dephasing problem is concerned. Indeed, in the spirit of
Ref.~\cite{altshuler82}, soft inelastic scattering with $qv_F,\omega\ll
\tau_\phi^{-1}$ is not expected to contribute to the dephasing of the
localization effects. In higher dimensions, in the ballistic regime $T\tau\gg
1$, this infrared cutoff is of no importance and the dephasing rate
$\tau_\phi^{-1}$ is given \cite{narozhny02} by $\tau_{ee}^{-1}$. However, in
view of Eq.~(\ref{20}), $\tau_\phi^{-1}$ in 1d cannot possibly reduce to
$\tau_{ee}^{-1}$.
The dephasing rate $\tau_\phi^{-1}$ can be accurately defined by calculating
the weak-localization correction to the conductivity of the disordered LL as a
function of $T$ \cite{gornyi05,gornyi07}. The leading localization correction
$\Delta\sigma$ in the ballistic limit $\tau_\phi/\tau\ll 1$ is related to the
interference of electrons scattered by three impurities. One of the diagrams
contributing to $\Delta\sigma$ (for the complete set of diagrams at the
leading order see Ref.~\cite{gornyi05,gornyi07}) is given by a
``three-impurity Cooperon" (Fig.~\ref{f1}), which describes the propagation of
two electron waves along the path connecting three impurities (``minimal
loop") in time-reversed directions. In the absence of dephasing, quantum
interference processes involving a larger number of impurities sum up to
exactly cancel (similarly to the noninteracting case \cite{berezinskii73}) the
Drude conductivity $\sigma_{\rm D}=e^2v_F\tau/\pi$, where $\tau$ is given by
Eq.~(\ref{7}). For $\tau_\phi/\tau\ll 1$, they only yield subleading
corrections through a systematic expansion in powers of $\tau_\phi/\tau$.
Within the functional-bosonization description of the LL
\cite{fogedby76,lerner04}, extended in Ref.~\cite{gornyi07} to treat
disordered systems, the interaction can be exactly accounted for by performing
a local gauge transformation $\psi_\mu(x,\tau)\to \psi_\mu(x,\tau)\exp
[\,i\theta_\mu(x,\tau)\,]$, where the bosonic field $\theta_\mu(x,\tau)$ is
related to the Hubbard-Stratonovich decoupling field $\varphi(x,\tau)$ by
\begin{equation}
(\partial_\tau-i\mu
v_F\partial_x)\theta_\mu(x,\tau)=\varphi (x,\tau)~.
\label{20a}
\end{equation}
Here $\tau$ is the Matsubara time. The correlator
$\left<\varphi(x,\tau)\varphi(0,0)\right>=V(x,\tau)$ gives the dynamically
screened interaction, for which the random-phase approximation (RPA) in the LL
model is exact \cite{dzyaloshinskii74}. In the presence of impurities, the
interaction can thus be completely gauged out to the backscattering impurity
vertices---Eq.~(\ref{20a}) is then {\it exact} for any given realization of
the impurity potential. In Fig.~\ref{f1}, the fluctuating disorder-induced
gauge factors are denoted by the wavy lines attached to the backscattering
vertices: each impurity vertex contributes the factor $\exp [\pm
(\theta_+-\theta_-)]$ and the averaging over fluctuations of $\varphi$ pairs
all the fields $\theta_\pm$ with each other. The interaction thus induces the
factor
\begin{equation}
\exp (-S_C)=\left<\exp [\,i(\theta_f-\theta_b)\,]\right>
\label{21a}
\end{equation}
to the Cooperon loop, where $\theta_f$ and $\theta_b$ are the phases
accumulated by an electron propagating along the ``forward'' and ``backward''
paths and the averaging is performed over the fluctuations of the field
$\varphi$. Notice that the averaging couples with each other not only the
phases $\theta_\pm$ attached to the impurities shown in Fig.~\ref{f1} but also
those attached to impurities which yield damping of the dynamically screened
interaction. As shown in Refs.~\cite{gornyi05,gornyi07}, the boson
damping is crucially important for the dephasing (see below) and a
parametrically accurate approximation is to include impurity-induced
backscattering in the effective interaction at the level of the
disorder-dressed RPA (``dirty RPA"). The total Cooperon action
\begin{equation}
S_C=S+S_{\rm renorm}
\label{21b}
\end{equation}
accounts then for both the dephasing ($S$) and the elastic renormalization of
impurities ($S_{\rm renorm}$) and we refer the reader for technical details of
the formalism to Ref.~\cite{gornyi07}.
The leading localization correction to the conductivity can be represented in
the form \cite{gornyi05,gornyi07}
\begin{equation}
\Delta\sigma=-2\sigma_{\rm D}\!\int_0^\infty \!\!dt_c \!\int_0^\infty
\!\!dt_a P_2(t_c,t_a)\exp \left[-S(t_c,t_a)\right]~,
\label{21}
\end{equation}
where
$P_2(t_c,t_a)=(1/8\tau^2)\exp (-t_c/2\tau)\Theta(t_c-2t_a)$ is the probability
density of return to point $x=0$ after two reflections at points $x=ut_a$ and
$x=-u(t_c/2-t_a)$. Here $ut_c$ gives the total length of the Cooperon loop and
$ut_a$, being the distance between two rightmost impurities, parametrizes the
geometry of the loop. The classical trajectory for the Cooperon is
characterized by a {\it single} velocity \cite{gornyi07} and this is $u$ (the
difference between $u$ and $v_F$ can be ignored for $\alpha\ll 1$, but
uniformly on the whole trajectory). The phase relaxation is encoded in the
dephasing action $S$ in Eq.~(\ref{21}), which is a growing function of the
size of the Cooperon loop and cuts off the localization correction at $t_c\sim
\tau_\phi$. The dephasing rate $\tau_\phi^{-1}$ is thus defined by the
characteristic scale of $t_c$ on which the dephasing action $S\sim 1$.
\begin{figure}[ht]
\centerline{
\includegraphics[width=6cm]{f1.eps}
}
\caption{
Three-impurity Cooperon diagram with interaction effects encoded in the
fluctuating factors $\exp (\pm\theta_\mu)$ (denoted by the wavy lines)
attached to the backscattering vertices (marked by the crosses). The dashed
lines connect the backscattering vertices belonging to the same impurity
(e.g., 1 and $\bar{1}$ refer to two backscatterings off impurity 1 at two
different times).
}\label{f1}
\end{figure}
In the limit $t_c\ll\tau$ the action reads \cite{gornyi05,gornyi07}:
\begin{equation}
S(t_c,t_a)=2\pi\alpha^2 \,T\,t_a \left(t_c-2t_a\right)/\tau~.
\label{22}
\end{equation}
Substitution of Eq.~(\ref{22}) into Eq.~(\ref{21}) gives
\begin{equation}
\Delta\sigma = -{1\over 4}\,\sigma_{\rm D}\left(\tau_\phi\over
\tau\right)^2\!\ln{\tau\over \tau_\phi} \propto {1\over
\alpha^2T}\,\ln(\alpha^2T)~,
\label{23}
\end{equation}
where
\begin{equation}
\tau_\phi^{-1}=\alpha(\pi T/\tau)^{1/2}~.
\label{24}
\end{equation}
One sees that the phase relaxation in the disordered LL occurs on
time scales much longer than the lifetime $\tau_{ee}$:
\begin{equation}
\tau_\phi=(\tau_{ee}\tau)^{1/2}\gg \tau_{ee}~.
\label{25}
\end{equation}
Note that $\tau_\phi^{-1}$ vanishes in the clean limit, in contrast to the
total e-e scattering rate---in agreement with the observation (\ref{20}) and
the basic fact \cite{altshuler82} that scattering with energy transfers
smaller than $\tau_\phi^{-1}$ is not effective in dephasing the localization
effects.
The vanishing of the dephasing action at $\tau^{-1}\to 0$ can be made more
transparent from the technical point of view by representing $S$ as a
difference between the self-energy ($S_{\rm ff}+S_{\rm bb}$) and vertex
($S_{\rm fb}+S_{\rm bf}$) contributions. Here the terms $S_{ij}$, associated
with an inelastic interaction between electrons propagating along the paths
$x_i(t)$ and $x_j(t)$, where $i,j=({\rm f})$ and $({\rm b})$ stand for the
``forward" and ``backward" time-reversed paths in the Cooperon, are given by
\begin{eqnarray}
S_{ij}&=&-T\int
{d\omega\over 2\pi}\int {dq\over 2\pi}\int_0^{t_c} \!dt_1\!\int_0^{t_c} \!dt_2
\,{1\over \omega}\,{\rm Im}\,V_{\mu\nu}(\omega,q) \nonumber \\ &\times&\,\,\exp
\{iq\left[\,x_i(t_1)-x_j(t_2)\,\right]-i\omega(t_1-t_2)\}~.
\label{26}
\end{eqnarray}{eqnarray}
Equation (\ref{26}) is similar to that in higher dimensionalities (``AAK
action" \cite{altshuler82}) with one subtle and important distinction.
Because of the Hartree-Fock cancellation of the bare interaction $\tilde{V}_f$
between electrons from the same chiral branch [recall the discussion after
Eq.~(\ref{18})], the dynamically screened retarded interaction
$V_{\mu\nu}(\omega,q)$ acquires the indices $\mu,\nu$ denoting the direction
of motion of the interacting electrons: $\mu={\rm sgn}\,\dot{x}_i$, $\nu={\rm
sgn}\,\dot{x}_j$. If one would keep both $V_f$ and $\tilde{V}_f$ processes in
$V(\omega,q)$, the dephasing action in 1d could {\it not} be written in the
form of Eq.~(\ref{26})---in contrast to higher dimensionalities, where
$S_{ij}$ is given by Eq.~(\ref{26}) with the ``full" $V(\omega,q)$.
Neglecting the disorder-induced damping of the dynamically screened
interaction yields
\begin{equation}
S_{\rm ff}=S_{\rm fb}=t_c/2\tau_{ee}
\label{27}
\end{equation}
and the exact cancellation of the self-energy and vertex parts in the total
dephasing action, hence the vanishing of $\tau_\phi^{-1}$ (\ref{24}) in the
clean limit. It is thus only because of the small difference between $S_{\rm
ff}$ and $S_{\rm fb}$ produced by the dressing of $V_{\mu\nu}(\omega,q)$ by
impurities (``dirty RPA" \cite{gornyi05,gornyi07}) that the dephasing action
(\ref{22}) is not zero. The characteristic energy transfer $\omega$ in the
processes that lead to the dephasing (i.e., contribute to the difference
$S_{\rm ff}-S_{\rm fb}$) is much larger than $\tau^{-1}$ [more accurately,
$\omega$ is spread over the range between $\tau_\phi^{-1}$ and $\tau^{-1}$,
because of the logarithmic factor in Eq.~(\ref{23})], which justifies the
expansion of $S$ in powers of $\tau^{-1}$, while the condition $T\tau_\phi\gg
1$ justifies the quasiclassical treatment of the electromagnetic fluctuations
in Eq.~(\ref{26}). Substituting Eq.~(\ref{24}) into Eq.~(\ref{14}) gives the
temperature scale $T_1\sim 1/\alpha^2\tau$ below which the localization
effects become strong (for the behavior of the conductivity at $T\ll T_1$ see
Ref.~\cite{fleishman80}). Note that $T_1\tau\gg 1$ for weak interaction.
\section{V. Energy relaxation}
\label{erel}
We now turn to the nonequilibrium properties of the disordered LL
\cite{bagopo}. Here we are primarily interested in the equilibration rate at
which an excited state relaxes to equilibrium (other aspects of the
nonequilibrium relaxation will be discussed elsewhere \cite{bagopo}). As
mentioned in Sec.~I, the integrability of the clean LL model precludes energy
relaxation. The absence of inelastic scattering in the LL deserves additional
comment. For scattering of electrons from different chiral branches on each
other, the energy and momentum conservation laws for linear electronic
dispersion lead to two equalities: $\omega-v_Fq=0$ and $\omega+v_Fq=0$. These
combine to give $\omega,q=0$ and thus no energy exchange [cf.\
Eq.~(\ref{20})]. For scattering of electrons of the same chirality $\mu$, the
energy-momentum conservation laws give a single equation $\omega-\mu v_Fq=0$
and at first glance the energy relaxation is allowed. Moreover, the relaxation
might seem to be very strong since the Golden-rule expression for the
probability of scattering contains the delta function $\delta (\omega-\mu
v_F)$ squared. For the point-like interaction, the diverging Hartree and
exchange terms cancel each other; however, for a finite-range
interaction---despite the LL model being still completely integrable---the
cancellation is no longer exact. The energy relaxation, nonetheless, is absent
in the LL model for a generic shape of the interaction potential. The point is
that beyond the Golden rule the diverging terms sum up to produce the
dynamically screened interaction between electrons (exactly given by the RPA),
which propagates with velocity $u(q)\neq v_F$. As a result, the probability of
scattering contains a product of two delta functions $\delta (\omega-\mu
v_Fq)\delta [\omega-\mu u(q)q]$ with {\it different} velocities, which yields
$\omega,q=0$ for electrons from the same chiral branch as well.
The energy relaxation is thus only present if one goes beyond the clean LL
model. One possibility comes from three-particle scattering \cite{Lunde},
which occurs for a nonzero range of interaction provided that the electronic
dispersion is nonlinear. The three-particle collision rate is small in the
parameter $T/\epsilon_F\ll 1$, where $\epsilon_F$ is the Fermi energy. Another
possibility is to take into account impurity backscattering, which may lead to
a much stronger mechanism of energy relaxation.
It is important that the nonequilibrium state of the LL in general cannot be
described in terms of a single---either bosonic or fermionic---distribution
function. The simplest example to illustrate this point is that of the clean
LL in which the left and right movers, separately at equilibrium within
themselves, are characterized by the Fermi distribution functions
$f^\pm_\epsilon=f_F(\epsilon-\mu_\pm)$ with different chemical
potentials. Then the distribution functions $N^\pm(\omega)$ of the left and
right plasmon modes are constructed as the convolutions of the fermion
functions:
\begin{equation}
N^\pm(\omega)=\frac{1}{\omega}\int d\epsilon\,
f^\pm_\epsilon (1-f^\pm_{\epsilon-\omega})=N_B(\omega)~,
\label{Local}
\end{equation}
i.e., are given by the {\it equilibrium} Bose distribution, independent of
$\mu_\pm$. This observation shows that the purely bosonic description of the
clean LL at a finite bias voltage is not complete. Such a ``partial
nonequilibrium" setup, in which the bosons are still at equilibrium, has been
studied previously by employing the conventional bosonization (see, e.g.,
Ref.~\cite{grabert}). Furthermore, the nonequilibrium transport through a {\it
single} impurity between {\it equilibrium} leads shifted by the voltage
$\mu_+-\mu_-$ has been studied in Ref.~\cite{fendley95}. However, the standard
scheme of bosonization will break down if the nonequilibrium distributions of
the injected right- and left-moving electrons are not the Fermi distributions.
The challenge is thus to formulate a theoretical framework to describe a
genuinely nonequilibrium situation in which both the bosonic and fermionic
excitations are out of equilibrium. It is worth stressing that in the
inhomogeneous case the nonequilibrium distribution functions do not obey the
simple local relation (\ref{Local}), since the distribution functions of
plasmons and electrons evolve with different velocities ($u$ and $v_F$).
Notice also that the necessity of introducing both the bosonic and fermionic
distribution functions is not peculiar to 1d: for higher-dimensional systems
see Ref.~\cite{Catelani}.
Our approach to nonequilibrium phenomena in the LL uses as a base the
formalism of the ``functional bosonization", developed in Ref.~\cite{gornyi07}
for the treatment of disordered LL at equilibrium. A conceptually similar
formalism has been applied earlier for higher-dimensional disordered
conductors in the study of the single-particle density of states out of
equilibrium in Ref.~\cite{Gutman1}. The nonequilibrium tunneling density of
states in the clean LL has been considered within the functional bosonization
approach in Ref.~\cite{Gutman2}. Here we formulate the theory of the
disordered LL out of equilibrium, which builds on the approaches of
Refs.~\cite{Khmelnitskii,Kamenev} and Ref.~\cite{Catelani}, in terms of the
effective nonequilibrium real-time action. To account for the e-e interaction,
we introduce a dynamical field $\phi(x,t)$ which decouples the four-fermion
term in the action by means of the conventional Hubbard-Stratonovich
transformation.
The central object of our theory is the quasiclassical Green's function ${\hat
g}(x,t_1,t_2)$ for electrons in the LL, taken at coinciding spatial points
\cite{Shelankov}:
\begin{eqnarray}
\hat{g}(x, t_1, t_2)&=&\lim_{x'\to x}\Bigl[\,2iv_F g(x,x',t_1,t_2)
\nonumber \\
&-&{\rm sign}(x-x')\delta(t_1-t_2)\,\Bigr]~.
\end{eqnarray}
This function, which is a $4\times 4$ matrix in the Keldysh and channel
(right/left) space,
satisfies the Eilenberger equation \cite{Eilenberger}:
\begin{equation}
iv_F\partial_x\hat{g}+[\,i\partial_t\hat\tau_z-\hat{H}\,,\hat{g}\,]=0~,
\label{Eilen_Eq}
\end{equation}
where
\begin{equation}
\hat{H}=\hat\phi\hat\tau_z+\frac{1}{2}({U_b}
\hat\tau^++{U_b^*}\hat\tau^-)~.
\label{Eilen_Eq_1}
\end{equation}
Here and throughout this section below, $v_F$ means the renormalized velocity
$v_F^*$ [see the discussion around Eq.~(\ref{5})], so that the difference
between $u$ and $v_F$ is of order $\alpha^2$ for small $\alpha$.
Equation (\ref{Eilen_Eq}) describes the motion of an electron in the random
potential characterized by the backscattering amplitude $U_b(x)$
[Eq.~(\ref{6})] in the presence of the dynamic field $\hat\phi(x,t)={\rm
diag}\,(\hat\phi^+,\hat\phi^-)$, where
$\hat\phi^\mu(x,t)=\phi^\mu_1(x,t)+\hat\sigma_x\phi^\mu_2(x,t)$ and
$\phi_1^{\mu}$ and $\phi_2^{\mu}$ are the classical and quantum components of
the Hubbard-Stratonovich field with chirality $\mu$, respectively. The Pauli
matrices $\tau_z$ and $\tau^\pm=\tau_x\pm i\tau_y$ act in the channel
space. We also introduce the Pauli matrices $\hat\sigma_{x,y,z}$ that act in
the Keldysh space. The Hamiltonian $\hat{H}$ (\ref{Eilen_Eq_1}) is defined on
the direct product of the time, Keldysh, and channel spaces. Accordingly, the
commutator $[\,,\,]$ in Eq.~(\ref{Eilen_Eq}) is understood with respect to all
three (``discretized" time, Keldysh, and channel) indices. The operator
$\partial_t$ acts as $\overrightarrow\partial_{t_1}$ from the left and as
$(-\overleftarrow\partial_{t_2})$ from the right. For the case of linear
electronic dispersion, assumed in the LL model, the Eilenberger equation
(\ref{Eilen_Eq}) is {\it exact} for any given realization of the
backscattering amplitude $U_b(x)$.
The next step is to average Eq.~(\ref{Eilen_Eq}) over disorder. At this point
we disregard the localization effects \cite{gornyi05,gornyi07}, which limits
the applicability of the subsequent derivation to sufficiently high
(effective) temperatures; specifically, for the length of the quantum wire
larger than the mean free path to $T\gg T_1\sim 1/{\alpha^2\tau}$ (see the end
of Sec.~IV). Under this condition we can perform the disorder averaging at the
level of the self-consistent Born approximation, which gives
\begin{eqnarray}
i\mu v_F\partial_x\overline{\hat{g}^\mu}+\Bigl[i\partial_t-
\hat\phi +\frac{i}{4\tau}\overline{\hat{g}^{-\mu}},
\overline{\hat{g}^\mu}\Bigr]=0
\label{Eilen_av}
\end{eqnarray}
for the disorder-averaged Green's function $\overline{{\hat g}^\mu}$. In
what follows we only deal with the averaged propagators and therefore omit the
bar for brevity.
The Green's function $\hat g$ satisfies the normalization condition
\begin{equation}
{\hat g}\circ {\hat g}={\hat 1}\,\delta(t_1-t_2)~,
\label{Constrain}
\end{equation}
where the dot denotes the convolution in all three (time, Keldysh, and
channel) spaces. The constraint (\ref{Constrain}) enables us to formulate the
effective action that reproduces Eq.~(\ref{Eilen_av}) as its saddle point in
the form essentially combining the actions derived in Refs.~\cite{Khmelnitskii}
and \cite{Kamenev}:
\begin{eqnarray}
S[\,\hat{g},\hat\phi,{\hat A}\,]&=&-\frac{1}{2v_F}{\rm
Tr}\left[\,(i\partial_t-
\hat\phi)\hat\tau_z+v_F\hat{A}\,\right]\hat{g} \nonumber\\
&-&\frac{i}{2}{\rm Tr}\,\hat{g}_0T^{-1}\partial_xT-\frac{i}{8v_F\tau}{\rm
Tr}\,\hat{g}^+\hat{g}^-~. \nonumber\\
\label{ActionK}
\end{eqnarray}
The Green's function in Eq.~(\ref{ActionK}) is represented as
$\hat{g}=T\hat{g}_0T^{-1}={\rm diag}\,(\hat{g}^+,- \hat{g}^-)$, where
$\hat{g}_0= {\rm diag}\,(\hat{g}^+_0,-\hat{g}^-_0)$ corresponds to the saddle
point of the action of the noninteracting problem and the unitary
transformation $T$ (diagonal in the channel space) parametrizes possible
fluctuations around $g_0$ [satisfying the constraint (\ref{Constrain})],
induced by fluctuations of the field $\hat\phi(x,t)$. To generate the response
functions in the Keldysh formalism \cite{rammer86,kamenev04}, we have also
added the external-source term ${\hat
A}(x,t)=a_1(x,t)+\hat\sigma_xa_2(x,t)$. The trace operation includes the
summation over the Keldysh and space indices and the integration over
time. The Keldysh partition function of the system can now be expressed as a
functional integral over $\hat\phi$,
\begin{eqnarray}
{\cal Z}[A]&\sim&\int {\cal D}\phi^{\mu}_{1,2}(x,t)\,\exp\left\{ iS[\,\hat\phi,
{\hat g},{\hat A}\,] \right. \nonumber \\
&+& \left. \frac{i}{2} {\rm Tr}\,\hat\phi\left(
V_f^{-1}\hat\tau_x+\frac{1}{2\pi v_F}\right)\hat\sigma_x\hat\phi\right\}~,
\end{eqnarray}
where ${\hat g}(x,t_1,t_2;[\hat\phi(x,t)])$ is understood as minimizing the
action ({\ref{ActionK}}) for a given $\hat\phi (x,t)$ under the constraint
(\ref{Constrain}).
Having written the Eilenberger equation (\ref{Eilen_av}) and its action
(\ref{ActionK}) we now use the standard technique \cite{rammer86,kamenev04} to
derive the quantum kinetic equations. We proceed at one-loop order with
respect to the effective interaction, which is equivalent to the ``dirty RPA"
\cite{gornyi07}. The one-loop derivation is controlled by the small parameters
$1/T\tau_\phi \ll 1$ and $\alpha \ll 1$. More specifically, following the
framework of Ref.~\cite{Catelani}, we introduce three different distribution
functions for each $\mu$. The first one, $f^\mu(\epsilon,x,t)$, describes the
{\it bare} electrons, moving with velocity $v_F$. The other two,
$N_p^\mu(\omega,x,t)$ and $N_g^\mu(\omega,x,t)$, describe two types of bosons,
having velocities $u_p=v_F/K$ and $u_g=v_F$. The bosons of the first kind
represent the usual plasmons ($p$) of the LL, whereas those of the second kind
are ``ghosts'' ($g$) constructed from the bare electron-hole pairs, thus
preventing from a double-counting of the degrees of freedom in the system [see
the discussion around Eqs.~(\ref{Energy_e})-(\ref{joule}) below].
We first apply the gauge transformation
\begin{equation}
{\tilde g}^\mu (x,t_1,t_2)=e^{-i\hat\theta^\mu(x,t_1)}\hat{g}^\mu (x,t_1,t_2)
e^{i\hat\theta^\mu (x,t_2)}~,
\label{gauge}
\end{equation}
where $\hat\theta^\mu =\theta_1^\mu +\hat\sigma_x\theta_2^\mu$ has the same
Keldysh structure as the field $\hat\phi$. This transformation is similar to
that in Ref.~\cite{lerner04,eckern07}, but different in that the equation of
motion for the phase $\hat\theta$ in the field $\hat\phi$ will incorporate
disorder, see Eq.~(\ref{Linear_Rel}) below. The ``rotated" Green's functions
${\tilde g}^\mu$ are expressed in terms of the electron distributions
$f^\mu_\epsilon(x,t)$, written in the time domain, as
\begin{equation}
{\tilde g}^\mu =\left[
\begin{array}{cc}
\delta(t_1-t_2) & 2h^\mu (t_1,t_2,x) \\
0 & -\delta(t_1-t_2)
\end{array} \right]~,
\end{equation}
where $h^\mu =\delta(t_1-t_2)-2f^\mu (t_1,t_2,x)$,
\begin{equation}
f^\mu(t_1,t_2,x)=\int\!{d\epsilon\over 2\pi}\,e^{i\epsilon
(t_1-t_2)}f^\mu_\epsilon[x,(t_1+t_2)/2]~,
\end{equation}
and we impose the condition
\begin{equation}
f^\mu (t_1,t_2,x)\vert_{t_1\to t_2}=\frac{i}{2\pi(t_1-t_2+i0)}~.
\label{cond}
\end{equation}
The fast charge and current fluctuations are now encoded in the fluctuations
of the phase factors $e^{\pm i\hat\theta}$ in Eq.~(\ref{gauge}). The
gauge-transformed action reads
\begin{eqnarray}
S[\hat\theta,\hat\phi ,{\tilde g}]&=&S_e+S_b+S_{\rm int}+S_{\rm imp}~,
\label{ActionG} \\
S_e&=&-\frac{1}{2 v_F}{\rm Tr}\left(i\partial_t-{\hat L}_0\hat\theta
-\hat\phi\right)\hat\tau_z{\tilde g} \nonumber \\
&&-\frac{i}{2}{\rm Tr}\,\hat{g}_0T^{-1}\partial_xT~, \\
S_b&=&\frac{1}{2\pi v_F}{\rm Tr}\left[\,\frac{1}{2}(\partial_t\hat\theta)
\,{\hat L}_0\,\hat\sigma_x\hat\theta
+\hat\phi\,\hat\sigma_x\partial_t\,\hat\theta\right]~, \nonumber\\ \\
S_{\rm int}&=&\frac{1}{2}{\rm Tr}\,\hat\phi
\left(V_f^{-1}\hat\tau_x+\frac{1}{2\pi v_F}\right)
\hat\sigma_x\hat\phi~, \\
S_{\rm imp}&=&-\frac{i}{8v_F\tau}{\rm Tr}\,
e^{-i(\hat\theta^--\hat\theta^+)}{\tilde g}^+
e^{i(\hat\theta^--\hat\theta^+)}{\tilde g}^-~, \nonumber \\
\label{ActionG_imp}
\end{eqnarray}
where ${\hat L}_0=\partial_t+\hat\tau_zv_F\partial_x$.
We treat the fluctuations of $\hat\theta$ and $\hat\phi$ in the Gaussian
approximation by expanding Eq.~(\ref{ActionG}) around the saddle point of $S$
for a given $\hat{g}^\mu$. Optimizing then the action with respect to
$\hat\theta$ for a given $\hat\phi$, we get a linear relation between
$\hat\theta$ and $\hat\phi$:
\begin{equation}
{\hat D}^{-1}_g\theta=-\hat\sigma_x\phi~,
\label{Linear_Rel}
\end{equation}
where we introduce the vector notation $\theta
=(\theta_1^+,\theta_1^-,\theta_2^+,\theta_2^-)^T$,
$\phi=(\phi_1^+,\phi_1^-,\phi_2^+,\phi_2^-)^T$, $T$ stands for
transposition, and the particle-hole propagator $D_g$ is constructed as
\begin{equation}
{\hat D}^{-1}_g=(\partial_t+\hat\tau_zv_F\partial_x)\hat\sigma_x
+\frac{1}{2}\hat\gamma (1-\tau_x)
\label{D_g}
\end{equation}
with
\begin{eqnarray}
\hat\gamma&=&\frac{1}{\tau}\left(
\begin{array}{cc}
0 & -1\\
1 & 2B_\omega
\end{array}
\right)~,
\label{Gamma} \\
B_{\omega}&=&\frac{1}{2\omega}\sum_{\mu}\int d\epsilon
\left(1-h^{\mu}_{\epsilon}\,h^{-\mu}_{\epsilon-\omega}\right)~.
\label{B_def}
\end{eqnarray}
The scattering operator $\hat\gamma$ (\ref{Gamma}) describes the
decay/recombination of the collective electron-hole excitation into/from the
electron and hole moving in opposite directions, assisted by impurity
scattering. Note that the approximation (\ref{Linear_Rel}) is equivalent
\cite{footnote} to the ``dirty RPA" \cite{gornyi07} in Sec.~IV.
Substituting Eq.~(\ref{Linear_Rel}) back into the approximate quadratic
action, we obtain the ``dirty-RPA" propagator of the effective interaction
\begin{equation}
\langle\phi\phi^T\rangle=\frac{i}{2}\hat V
=\frac{i}{2}\left(\hat\sigma_x\hat\tau_xV_f^{-1}
-\hat\Pi\right)^{-1}~,
\label{Phi_Cor_1}
\end{equation}
where
\begin{equation}
\hat\Pi=\frac{1}{2\pi v_F}\left[\,\hat\sigma_x
\left(\partial_t{\hat D}_g\right)\hat\sigma_x-\hat\sigma_x\,\right]
\label{Pi_operator}
\end{equation}
is the polarization operator. By combining Eqs.~(\ref{Phi_Cor_1}) and
(\ref{Linear_Rel}) we get the correlator of the phases $\theta$
(cf.\ Ref.~\cite{levitov01})
\begin{equation}
\langle\theta\theta^T\rangle=\frac{i}{2}\,
{\hat D}_g\,\sigma_x\,{\hat V}\,\sigma_x\,{\hat D}_g
=-\frac{i\pi v_F}{\partial_t}\left({\hat D}_p-{\hat D}_g\right)~,
\label{Theta_Cor}
\end{equation}
where ${\hat D}_p$ is the renormalized particle-hole propagator corresponding
to the plasmon modes with velocity $u$ given by Eq.~(\ref{5a}):
\begin{equation}
{\hat D}^{-1}_p=\left({\partial_t\over
1+\alpha\hat\tau_x}+v_F\hat\tau_z\partial_x\right)\hat\sigma_x+{1\over
2}\hat\gamma (1-\hat\tau_x)~.
\end{equation}
As follows from Eq.~(\ref{ActionG_imp}), the only phase that is coupled to the
electron backscattering off disorder is $\Phi=\frac{1}{2}(\theta^--
\theta^+)$. Another observation is that the propagators of the fluctuations of
$\theta$ have two different types of poles: $\omega=\pm uq$ and $\omega=\pm
v_F q$, both smeared by disorder. It is thus convenient to define the
correlator of the phase $\Phi$ as a difference of two terms:
\begin{equation}
\langle\Phi\Phi^T\rangle =\frac{i}{2}\left(\hat{\cal L}_p
-\hat{\cal L}_g\right)~,
\label{Phi_Cor}
\end{equation}
where
\begin{equation}
\hat{\cal L}_b=-\frac{i\pi v_F}{2\partial_t}\sum_{\mu\nu}\mu\nu
{\hat D}_b^{\mu\nu}
\label{Phi_Cor_2}
\end{equation}
and $b=p,g$ denotes the plasmon and ghost modes, which differ from each other
in that the plasmon mode is characterized by velocity $u$, whereas the ghost
mode by velocity $v_F$. Then the Wigner-transform of the Keldysh correlator
$\langle\Phi\Phi^T\rangle_K$ has to be described by four different
distribution function, $N_p^\pm(\omega,x,t)$ and $N_g^\pm(\omega,x,t)$,
evolving with velocities $u$ and $v_F$ to the right and to the left:
\begin{eqnarray}
&&\langle\Phi\Phi^T \rangle_K(\omega,q\simeq\pm\omega/u,x,t)=
\label{Np}\\
&&\qquad\qquad\left[2N_p^\pm(\omega,x,t)+1\right]
{\rm Im}{\cal L}_p^A(\omega,q)~, \nonumber \\
&&\langle\Phi\Phi^T\rangle_K(\omega,q\simeq\pm\omega/v_F,x,t)=
\label{Ng}\\
&&\qquad\qquad-\left[2N_g^\pm(\omega,x,t)+1\right]
{\rm Im}{\cal L}_g^A(\omega,q)~. \nonumber
\end{eqnarray}
To derive the kinetic equation for the electron distribution function, the
next step is to write down the equation of motion for the gauge-transformed
Green's function ${\tilde g}^\mu$ (\ref{gauge}). The latter follows from
the relation
\begin{equation}
\frac{\delta}{\delta{\tilde g}^\mu}\left(S_e+S_{\rm imp}\right)=0~.
\end{equation}
Using Eq.~(\ref{Linear_Rel}) we represent $S_e$ as
\begin{equation}
S_e=-\frac{1}{2v_F}{\rm Tr}\left(i\partial_t\hat\tau_z
-\tilde\phi\right){\tilde g}~,
\end{equation}
where the shifted phase $\tilde\phi =\tilde\phi_1+\tilde\phi_2\,\hat\sigma_x$,
\begin{equation}
\tilde\phi_\alpha
=\sum_\beta (\hat\sigma_x\hat\gamma)_{\alpha\beta}\,\Phi_\beta~,
\end{equation}
and $\hat\gamma$ is given by Eq.~(\ref{Gamma}). Notice that the field
$\tilde\phi$ does not depend on the chiral index $\mu$. The Eilenberger
equation for ${\tilde g}^\mu$ thus reads
\begin{equation}
i\mu v_F\partial_x{\tilde g}^\mu+\left[i\partial_t-\mu\tilde\phi+
\frac{i}{4\tau}e^{2i\mu\hat\Phi}
{\tilde g}^{-\mu}e^{-2i\mu\hat\Phi},\,\,{\tilde g}^\mu\right]=0~.
\label{Eilen_G}
\end{equation}
Equation (\ref{Eilen_G}) has to be averaged over the fluctuations of the phase
$\Phi$ with the correlator given by Eq.~(\ref{Phi_Cor}). Within the
``dirty-RPA" it is sufficient to represent ${\tilde g}^\mu$ as a sum ${\tilde
g}^\mu=\langle{\tilde g}^\mu\rangle+\delta{\tilde g}^\mu$, where
$\langle{\tilde g}^\mu\rangle$ is the mean value, and take into
account only the term in $\delta{\tilde g}^\mu$ that is linear in the
fluctuations of $\Phi$, keeping in mind that the quadratic-in-$\Phi$
fluctuations of ${\tilde g}^\mu$ are incorporated in the mean value.
By linearizing Eq.~(\ref{Eilen_G}) around the average
$\langle{\tilde g}^\mu\rangle$, we obtain $\delta{\tilde
g}^\mu=-2(\delta f^\mu)\,\hat\sigma_+$, where the fluctuation of the
distribution function obeys
\begin{equation}
\sum_{\mu}\left[{\hat D}^{-1}_{g, R}(\omega)\right]^{\nu\mu}\,
\delta f^\mu(\epsilon_1,\epsilon_2)=
\frac{i\nu}{\tau}\,\sum_{\alpha=1,2}
\lambda^\nu_\alpha (\epsilon_1,\epsilon_2)\,\Phi_\alpha(\omega)~.
\label{dff}
\end{equation}
In Eq.~(\ref{dff}), $\delta f^\mu (\epsilon_1,\epsilon_2)$ is the Fourier
transform of $\delta f^\mu(t_1,t_2,x)$ and $\omega=\epsilon_1-\epsilon_2$. The
source terms $\lambda^\nu_\alpha$ are expressed through the averages
$h^\mu_\epsilon$ as
\begin{eqnarray}
\lambda^\nu_1(\epsilon_1,\epsilon_2)&=&\frac{\nu}{2}\sum_\mu\mu\,
\left(h^{\mu}_{\epsilon_2}-h^{\mu}_{\epsilon_1}\right)~, \\
\lambda^{\nu}_2(\epsilon_1,\epsilon_2)&=&1+B_{\omega}
\left( h^{\nu}_{\epsilon_2}-h^{\nu}_{\epsilon_1} \right) \\
&-&\frac{1}{2} h^{\nu}_{\epsilon_1}
h^{-\nu}_{\epsilon_2}-\frac{1}{2} h^{\nu}_{\epsilon_2}
h^{-\nu}_{\epsilon_1}~, \nonumber
\end{eqnarray}
where $h^\mu_\epsilon=1-2f^\mu_\epsilon$.
Notice that the general formalism of the ``nonequilibrium functional
bosonization" [Eq.~(\ref{Eilen_G})] allows, in principle, for a
nonperturbative treatment of both the elastic renormalization and the
inelastic scattering if the phases $\Phi$ are kept in the exponents (see, in
particular, the calculation of the tunneling density of states in the clean LL
out of equilibrium in Ref.~\cite{Gutman2}). For our purposes in this paper, it
is sufficient to expand the exponential factors to second order in $\Phi$.
The Eilenberger-type equation for the average $\langle{\tilde g}^\mu
\rangle$ can now be written in the form
\begin{equation}
i\mu v_F\partial_x\langle{\tilde g}^\mu\rangle+
\Bigl[i\partial_t+\frac{i}{4\tau}
\langle \tilde g^{-\mu}\rangle ,
\,\langle{\tilde g}^\mu\rangle\Bigr]=
\hat{\rm St}^\mu_{e-e}+\hat{\rm St}^\mu_{e-b}~.
\label{Eilen_av_1}
\end{equation}
The collision integrals in the right-hand side of Eq.~(\ref{Eilen_av_1}) come
from the averages of second order in the fluctuations of $\Phi$. There are two
types of the collision integrals. One, $\hat{\rm St}^{\mu}_{e-b}$, comes from
the second-order terms in the expansion of the phase factors $e^{\pm
2i\mu\hat\Phi}$ in Eq.~(\ref{Eilen_G}). The other, $\hat{\rm St}^{\mu}_{e-e}$,
stems from the contraction of the linear correction $\delta{\tilde g}^\mu$
with the fluctuations of $\hat\Phi$.
The kinetic equation for the electron distribution function is obtained by
taking the Keldysh part of Eq.~(\ref{Eilen_av_1}):
\begin{eqnarray}
&&[\partial_t+\mu v_F(\partial_x+e{\cal E}\partial_\epsilon)]f^\mu_\epsilon=
-\frac{1}{2\tau}(f^{\mu}_\epsilon-f^{-\mu}_\epsilon) \nonumber \\
&&\qquad +{\rm St}_{e-e}+\sum_{b=p,g}\left(\mu\,{\rm St}^{\rm el}_{e-b}
+{\rm St}^{\rm inel}_{e-b}\right)~.
\label{Fermions_KE}
\end{eqnarray}
The electron-boson collision integral ${\rm St}_{e-b}^{\mu}$ describes
emission and absorption of the bosons of type $b=p,g$ by the fermions. Its
inelastic part, which is symmetric with respect to the channel index $\mu$,
reads
\begin{eqnarray}
&&{\rm St}_{e-b}^{\rm inel}(\epsilon)=\pm\frac{1}{4}\sum_{\mu\nu}
\int_{-\omega_0}^{\omega_0}\!d\omega\,\omega {\cal K}_{e-b}(\omega)
\nonumber \\
&&\times\left[N^{\nu}_b(\omega)f^{-\mu}_{\epsilon-\omega}
(1-f^{\mu}_\epsilon)-[1+N^{\nu}_b(\omega)]f^\mu_\epsilon
(1-f^{-\mu}_{\epsilon-\omega})\right]~. \nonumber\\
\label{Bosons_inel}
\end{eqnarray}
The elastic (antisymmetric in $\mu$) part is given by
\begin{eqnarray}
{\rm St}_{e-b}^{\rm el}(\epsilon)&=&\pm\frac{1}{4}\sum_{\mu\nu}
\int_{-\omega_0}^{\omega_0}\!d\omega\,\omega {\cal K}_{e-b}(\omega)
\nonumber \\
&\times&\mu\left[N^{\nu}_b(\omega)( f^{\mu}_{\epsilon}
-f^{\mu}_{\epsilon-\omega})+f^\mu_\epsilon
f^{-\mu}_{\epsilon-\omega})\right]~,\nonumber\\
\label{Bosons_el}
\end{eqnarray}
where the signs $\pm$ refer to the plasmons (+) and ghosts $(-)$. In
Eqs.~(\ref{Fermions_KE})-(\ref{Bosons_el}), $\tau$ and $u$ are understood as
renormalized down to a scale $\omega_0$, so chosen that
$\Delta\epsilon\ll\omega_0\ll \Lambda$ but
$\alpha\ln\left(\omega_0/\Delta\epsilon\right)\ll 1$, where $\Delta\epsilon$
is a characteristic energy scale for the distribution function (for the
quantum wire biased by a voltage $V$ it is given by $\max\{T,eV\}$). The high
energy renormalization (see Sec.~III), coming from scales larger than
$\omega_0$ and independent of the details of the nonequilibrium state at low
energies, can thus be taken into account at the level of input parameters
($\tau$ and $u$) for the kinetic equations.
The e-e collision integral ${\rm St}_{e-e}$ describes inelastic fermionic
collisions due to the interaction via the plasmon waves:
\begin{eqnarray}
{\rm St}_{e-e}(\epsilon)&=&\frac{1}{4}\sum_{\mu\nu}\int_{-\omega_0}^{\omega_0}
d\omega d\epsilon'{\cal K}_{e-e}(\omega)
\nonumber\\
&\times&\Bigl[
f^{-\nu}_{\epsilon'}(1-f^{\nu}_{\epsilon'-\omega})
f^{-\mu}_{\epsilon-\omega}(1-f^{\mu}_{\epsilon}) \nonumber \\
&-&f^{\nu}_{\epsilon'-\omega}(1-f^{-\nu}_{\epsilon'})
f^{\mu}_{\epsilon}(1-f^{-\mu}_{\epsilon-\omega})\Bigr]~.
\end{eqnarray}
Note that ${\rm St}_{e-e}$ does not depend on the chiral indices. The
collision kernels are written as
\begin{eqnarray}
{\cal K}_{e-b}(\omega)&=&\pm\frac{v_F}{\omega^2\tau}\int\frac{dq}{2\pi}
{\rm Re}\sum_{\mu\nu}\mu\nu{D}_b^{\mu\nu}(\omega,q)~, \\
{\cal K}_{e-e}(\omega)&=&{\cal K}(\omega)+
{\cal K}_{e-g}(\omega)-{\cal K}_{e-p}(\omega)~,
\label{Kernels}
\end{eqnarray}
where
\begin{equation}
{\cal K}(\omega)=-\frac{1}{\pi\omega}\int\frac{dq}{2\pi}
\sum_{\mu\nu}\,
{\rm Re}\,D_g^{\mu\nu}(\omega,q)\,{\rm Im}\,V^{\nu\mu}_R(\omega,q)
\label{KernK}
\end{equation}
and the explicit
form of the propagators $D^{\mu\nu}_g$ and $V^{\mu\nu}$ can be found in
Ref.~\cite{gornyi07} [see Eqs.~(A16),(A17),(A21)-(A23) there; note that
$D^{\mu\nu}_g$ corresponds to $v_FD_{\mu\nu}$ in Eqs.~(A16),(A17)].
For the electron-boson terms we obtain in the ballistic limit of energy
transfers $\omega$ larger than $\tau^{-1}$ the simple expressions
\begin{equation}
{\cal K}_{e-p}(\omega)=\frac{2K}{\omega^2\tau}~,\quad
{\cal K}_{e-g}(\omega)=\frac{2}{\omega^2\tau},\quad \omega\gg \tau^{-1}~.
\end{equation}
The asymptotic behavior of ${\cal K}(\omega)$ in three parametrically
different ranges of $\omega$ in the limit of weak interaction $\alpha\ll 1$ is
as follows:
\begin{eqnarray}
{\cal K}(\omega) = \left\{
\begin{array}{cc}
2\alpha^2/\omega^2\tau, &\tau^{-1}\ll\omega\ll\alpha T_1~, \\
8\alpha^4\tau, &\alpha T_1\ll\omega\ll T_1~, \\
2/\omega^2\tau, &\omega\gg T_1~,
\end{array}\right.
\label{Komega}
\end{eqnarray}
where $T_1\sim 1/\alpha^2\tau$ is the characteristic temperature below which
the localization effects are strong (see the discussion at the end of
Sec.~IV). The log-log plot of ${\cal K}(\omega)$ in the whole range of
$\omega$ for a particular value of $\alpha$ (taken very small for the purpose
of illustration) is shown in Fig.~\ref{Kw_plot}.
An important feature of ${\cal K}(\omega)$ in Eq.~(\ref{Komega}) is its
nonperturbative behavior with respect to $\alpha$ at $\omega \gg \alpha
T_1\sim 1/\alpha\tau$. In particular, for $\omega\gg T_1$ the e-e collision
kernel does not at all depend on the e-e interaction strength. The origin of
the nonperturbative dependence on $\alpha$, despite $\alpha\ll 1$ being small,
is related to the analytical structure of ${\cal K}(\omega)$ in
Eq.~(\ref{KernK}). Specifically, the integrand of ${\cal K}(\omega)$ contains
eight poles, which in the limit of large $\omega$ are only slightly ``damped"
by disorder: $q \simeq \pm \omega(1\pm i/2\omega\tau)/v_F$ and $q \simeq \pm
\omega (1\pm i/2\omega\tau)/u$. As a result, the contour of integration in the
plane of $q$ is squeezed between two close poles ($u\to v_F$ for $\alpha\to
0$), one of which is in the upper-half plane and the other in the lower
one. At $\omega \gg T_1$ we thus have ${\cal
K}(\omega)\propto\alpha^2/|u-v_F|$ and $\alpha$ drops out altogether.
\begin{figure}[t]
\includegraphics[width=3.0in]{f2.eps}
\caption{
Frequency dependence of the collision kernel ${\cal K}(\omega)$
[Eq.~(\ref{KernK})] for $\alpha=0.05$. Three different types of
scaling behavior of the dimensionless product $\tau^{-1}{\cal K}(\omega)$ are
indicated, as well as the characteristic values of $\omega$ .
}
\label{Kw_plot}
\end{figure}
The kinetic equations for the bosonic distribution functions $N_{p,g}^\mu$
follow from Eqs.~(\ref{Phi_Cor}),(\ref{Np}),(\ref{Ng}):
\begin{equation}
\left(\partial_t+\mu\,u_b\,\partial_x\right)N^\mu_b(\omega)=
-\frac{1}{\tau}N^\mu_b(\omega)+{\rm St}_{b-e}(\omega)~,
\label{kineqbos}
\end{equation}
where
\begin{equation}
{\rm St}_{b-e}(\omega)=\frac{1}{2\omega\tau}\sum_{\mu}\int\ d\epsilon
f^{\mu}_{\epsilon}(1-f^{-\mu}_{\epsilon-\omega})
\label {Bosons_KE}
\end{equation}
describes the creation of the boson from an electron-hole pair, where the
electron and hole move in the opposite directions.
The role of the ghost modes can be further elucidated if one considers the
energy conservation law. The electronic and bosonic energy
densities $\rho^{\epsilon}_e$, $\rho^{\epsilon}_b$ and the energy current
densities $j^{\epsilon}_e$, $j^{\epsilon}_b$ are given by
\begin{eqnarray}
\rho^{\epsilon}_e&=&\frac{1}{2\pi v_F}
\int_{-\infty}^\infty d\epsilon\,\epsilon (f^+_\epsilon+f^-_\epsilon)~,
\label{Energy_e} \\
j^{\epsilon}_e&=&\frac{1}{2\pi}\int_{-\infty}^\infty d\epsilon \,
\epsilon (f^+_\epsilon-f^-_\epsilon)~, \\
\rho^\epsilon_b&=&\frac{1}{2\pi u_b}\int_0^\infty d\omega \,
\omega [N_b^+(\omega)+N_b^-(\omega)]~,
\label{Energy_b} \\
j^\epsilon_b&=&\frac{1}{2\pi}\int_0^\infty d\omega \,
\omega [N_b^+(\omega)-N_b^-(\omega)]~.
\end{eqnarray}
The prefactors in Eqs.~(\ref{Energy_e}) and (\ref{Energy_b}) represent the
density of states for electrons and bosons, respectively. Then the kinetic
equations (\ref{Fermions_KE}) and (\ref{kineqbos}) assure the conservation law
\begin{equation}
\partial_t(\rho^{\epsilon}_e+\rho^{\epsilon}_p-\rho^{\epsilon}_g)
+\partial_x(j^{\epsilon}_e+j^{\epsilon}_p-j^{\epsilon}_g)=j_e\,{\cal E}~,
\label{joule}
\end{equation}
where in the right-hand side $\cal E$ is the applied electric field
and $j_e$ is the induced current of charge. The kinetic energy without any
interaction is determined by the energy of the bare electrons. In the presence
of Coulomb interaction, the plasmon energy is given by a sum of the e-e
interaction energy and the kinetic energy of the bare electron-hole pairs, the
latter being the ghost energy by construction. The total energy is thus given
by the sum of the plasmon and electron systems with a subtraction of the
energy of the ghosts.
We now turn to the rate of energy relaxation $\tau_E^{-1}$ in the limit of
weak nonequilibrium by linearizing the kinetic equations (\ref{Fermions_KE})
and (\ref{Bosons_KE}). One sees that at large energy transfers the inelastic
e-e scattering dominates over electron-boson collisions: ${\cal K}(\omega)\gg
{\cal K}_{e-g}(\omega)-{\cal K}_{e-b}(\omega)$ for $\omega\gg
1/\alpha^{3/2}\tau$. Assuming that the large $\omega$ give the main
contribution to the energy relaxation in the limit $T\gg T_1$, where the
localization effects can be neglected (see Sec.~IV), the equilibration rate at
which the fermionic system relaxes to a locally equilibrium Fermi distribution
is estimated as
\begin{equation}
\frac{1}{\tau_E(T)}\sim {1\over T}\int_0^T\!d\omega\,\omega^2{\cal K}(\omega)
\sim T^2{\cal K}(T)\sim\frac{1}{\tau}~.
\label{Tau_E}
\end{equation}
Notice that the characteristic $\omega$ in Eq.~(\ref{Tau_E}) is of order $T$,
which justifies the use of ${\cal K}(\omega)$ only. On the other hand, this
makes it impossible to describe the equilibration in terms of the much simpler
Fokker-Planck equation in the energy space. Remarkably, the equilibration rate
(\ref{Tau_E}) does not depend on the strength of interaction and is given by
the elastic scattering rate. The interaction constant $\alpha$ enters only
through the condition $T\gg T_1$. The equilibration rate turns out to be much
smaller than the (clean) e-e collision rate~(\ref{19}) and also much smaller
than the dephasing rate~(\ref{24}):
\begin{equation}
\tau_E^{-1}\ll\tau_\phi^{-1}\ll\tau_{ee}^{-1}.
\end{equation}
It is also worth emphasizing that the characteristic energy transfers are
parametrically different in these three types of relaxation processes.
\section{VI. Summary}
\label{conc}
In this paper, we have studied the relaxation properties of interacting
spinless electrons in a disordered quantum wire within the Luttinger-liquid
model. We first review the basic concepts in the disordered Luttinger liquid
at equilibrium, including the elastic renormalization, dephasing, and
interference-induced localization. We have introduced the general framework
for describing the relaxation processes in the strongly correlated (non-Fermi)
electron system at nonequilibrium. Our main result is the coupled set of the
kinetic equations for the fermionic (\ref{Fermions_KE}) and bosonic
(\ref{kineqbos}) distribution functions. The peculiarity of the Luttinger
liquid model is that the electron-electron scattering rate (the inverse
lifetime of the fermionic excitations) is finite at nonzero temperature but
the energy exchange is exactly zero in the clean limit. The energy relaxation
occurs only due to the scattering off disorder. We have calculated the energy
equilibration rate that turns out to be independent of the strength of
electron-electron interaction at sufficiently high temperature, when the
Anderson localization effects are suppressed, and equal to the rate of elastic
scattering off disorder.
\section{Acknowledgments}
\label{acknow}
We thank D.~Gutman, A.~Kamenev, M.~Kiselev, Y. Nazarov, and A.~Yashenkin for interesting
discussions. The work was supported by the Center for Functional
Nanostructures of the Deutsche Forschungsgemeinschaft, by the Russian
Foundation for Basic Research, and by the Program ``Leading Russian Scientific
Schools". The work of DAB and IVG, conducted as part of the project ``Quantum
Transport in Nanostructures" made under the EUROHORCS/ESF EURYI Awards scheme,
was supported by funds from the Participating Organisations of EURYI and the
EC Sixth Framework Programme.
|
1,116,691,498,448 | arxiv | \section{Introduction}
\label{sec:intro}
In this paper we will derive continuous dependence estimates for the
following boundary value problem:
\begin{align}
\label{EE}
F(x,u,Du,D^2u)&=0\qquad\text{in}\quad \Omega \quad (\Omega\subset\ensuremath{\mathbb{R}}^N),\\
G(x,Du)&=0\qquad\text{on}\quad\partial\Omega,
\label{BV}
\end{align}
where $u$ is the scalar unknown function, $Du$ and $D^2u$ denote its the gradient
and Hessian, and $\Omega$ is a bounded, smooth ($W^{3,\infty}$) domain in
$\ensuremath{\mathbb{R}}^N$. Informally speaking, by continuous dependence
estimates we mean estimates of the type
$$\|u_1-u_2\|\leq \|F_1-F_2\| + \|G_1-G_2\|$$
where $u_1$ and $u_2$ are solutions of two different boundary value
problem with data $F_1,G_1$ and $F_2,G_2$. The exact statement is
given in Section \ref{sec:results}.
Equation \eqref{EE} is degenerate elliptic, (possibly)
non-linear, and increasing in $u$.
This means that the possibly non-linear function $F(x,r,p,X)$ satisfies
$$F(x,r,p,X)\leq F(x,s,p,Y)\quad\text{for all}\quad r\leq s,\quad X\geq Y,$$
where $x \in\Omega$, $r,s\in\ensuremath{\mathbb{R}}$, $p\in\ensuremath{\mathbb{R}}^N$, and $X,Y\in \mathbb
S^N$. Here $\mathbb S^N$ is the set of real symmetric $N\times N$ matrices and
$X\geq 0$ in $\mathbb S^N$ means that $X$ is positive semi-definite.
The boundary condition \eqref{BV}
satisfies the Neumann type condition that $G$ is strictly increasing in $p$ in
the normal direction:
\begin{align*}
G(x,p+tn(x))\geq G(x,p)+\nu t,
\end{align*}
for some $\nu>0$ and all $t\geq0$, $x\in \partial\Omega$, $p$, and outward
unit normal vectors $n(x)$ to $\partial\Omega$ at $x$. The
other assumptions on $F$ and $G$ will be specified later.
The class of boundary conditions $G$ we treat in this paper includes
the classical Neumann condition, $\frac{\partial u}{\partial n}=g(x)$ in
$\partial\Omega$, oblique derivative conditions, and non-linear
boundary conditions like the capillary condition
\begin{align*}
&\frac{\partial u}{\partial n}=\bar\theta(x)(1+|Du|^2)^{1/2}\hspace{-2cm}&
\text{in}\qquad
\partial\Omega,\\
\intertext{and the controlled reflection
condition}
&\sup_{\alpha\in\ensuremath{\mathcal{A}}}\{\gamma_\alpha(x)\cdot Du
-g_\alpha(x)\}=0\hspace{-2cm}&\text{in}\qquad \partial \Omega.
\end{align*}
In this paper we will assume that $g$, $\bar \theta$, $g_\alpha$,
$\gamma_\alpha$ are Lipschitz continuous
functions, that $|\bar \theta|\leq \omega<1$ and $\gamma_\alpha\cdot
n\geq \nu>0$, and that $A$ is a compact metric space.
The main class of equations that our framework can handle are
equations satisfying assumption (H2) in the next section. Loosely
speaking, this is the class of equations where the non-linearity
$F(x,r,p,X)$ is uniformly continuous in $r,p,X$ locally uniformly in $x$.
This case will be referred to as the ``standard case'' in the rest of
this paper. Assumption (H2) excludes most of quasilinear equations,
but contains fully-nonlinear equations like the Bellman-Isaacs
equations from optimal stochastic control and stochastic differential
games theory:
\begin{align*}
\inf_{\theta_1\in \Theta_1}\sup_{\theta_2\in \Theta_2}\left\{-
\mathrm{tr}[a^{\theta_1,\theta_2}(x) D^2u] - b^{\theta_1,\theta_2}(x) Du -
c^{\theta_1,\theta_2}(x) u - f^{\theta_1,\theta_2}(x)\right\}=0
\quad\text{in}\quad \Omega,
\end{align*}
where $\Theta_1,\Theta_2\subset\ensuremath{\mathbb{R}}^m$ are compact metric spaces,
$c^{\theta_1,\theta_2}\geq\lambda>0$, the matrices
$a^{\theta_1,\theta_2}\geq0$, and the
coefficients are Lipschitz continuous uniformly in $\theta_1,\theta_2$.
In Sections \ref{sec:ex} and \ref{sec:ext}, we give all the details,
more examples, and extensions to problems on unbounded domains,
time-dependent problems, and certain quasilinear equations like e.g.
\begin{align*}
-\mathrm{tr}\Big[\Big(I-\frac{Du\otimes
Du}{1+|Du|^2}\Big)D^2u\Big]+\lambda u=f(x)
\quad\text{in}\quad \Omega.
\end{align*}
Since these equations may be degenerate and non-linear, their solutions
will in general not be smooth. In this paper we work with a concept of
weak solutions called viscosity solutions, a precise definition is
given at the end of this introduction. Viscosity solutions are at
least continuous in the interior of $\Omega$. The boundary conditions will be
interpreted in the weak viscosity sense which essentially means that
either the boundary condition or the equation has to hold on the
boundary. This allow us to have well-posed problems even when the
boundary conditions are classically incompatible. The solutions can
realized by the vanishing viscosity method, and they will be
discontinuous at parts of the boundary where the boundary conditions
are classically incompatible.
An overview of the viscosity solution theory, including Neumann
boundary value problems, can be found in the User's Guide \cite{cil}.
The viscosity solution theory for Neumann type boundary value problems
was initiated by Lions \cite{Li:Neumann} in 1985 for first order
equations, and has been developed by many authors since, see
\cite{pls,I4,ba2,ba3,issa,bl} and references therein for various
aspects of this theory. Today there are two leading
approaches, one due to Ishii \cite{I4} and another one due to Barles
\cite{ba2,ba3}. They apply under slightly different assumptions and
will be discussed below.
Starting with the standard case, i.e. nonlinear equations
\eqref{EE} satisfying (H2), we prove under natural and standard
assumptions, that these boundary value problems have unique H\"older
continuous viscosity solutions. The H{\"o}lder regularity results are
new and extend the Lipschitz regularity result of Barles \cite{ba2},
and we give for the first time a complete proof of such a regularity
result. We note that these regularity results are global up to the boundary.
Local up to the boundary H{\"o}lder estimates have previously been
obtained by Barles-Da Lio \cite{bl} for a different class of
equations. Whereas our equations are degenererate but strictly
increasing in the $u$ argument (assumption (H3) in the next section),
their equations are weakly non-degenerate satisfying some sort of
``strong ellipticity condition" but are not necessarily increasing in
$u$. The arguments needed to prove the two types of results are also
different, except for some ideas on the construction of test
functions that are needed in some of the proofs.
Next we prove continuous dependence results comparing H\"older
continuous (sub and super) solutions of different boundary value problems.
The results we obtain include both continuous dependence on
non-linearities for the equation and the boundary condition. The
results concerning the dependence on the boundary condition are
completely new, at least in a viscosity solution setting, while the results
we obtain for the equations apply to much more general boundary
conditions (including non-linear ones) than earlier results.
Continuous dependence results for the type of equations
we consider in
this paper have previously been obtained by e.g. Cockburn
et.al. \cite{CockGripenLonden}, Jakobsen and Karlsen
\cite{JKContDep,JK:Ell,JK:CDIPDE}, and Gripenberg \cite{Gr:CDBC}.
In all these
papers viscosity solutions methods are used. In some
cases such results can also be obtained from probabilistic arguments,
see e.g. \cite{fs} for results for Bellman equations
set in $\ensuremath{\mathbb{R}}^N$. Papers \cite{JKContDep,JK:Ell,JK:CDIPDE} treat very
general classes of equations set in $\ensuremath{\mathbb{R}}^N$ or $\ensuremath{\mathbb{R}}^N\times[0,T)$,
\cite{CockGripenLonden} treats zero-Neumann boundary value problem
for $x$-independent equations, and \cite{Gr:CDBC} treats a
zero-Dirichlet boundary value problem.
In the two last papers the domain $\Omega$ is convex and possibly
unbounded and in the last paper further restrictions on the class of
equations are needed (because of the Dirichlet condition) and the
Dirichlet condition is taken in the classical sense. All
these papers treat more general quasilinear equations than we can
treat here, e.g. $p$-Laplace type equations for $p>2$.
The technical explanation for the differences between our continuous
dependence result and the above mentioned results lays in the choice of
test function we use. To
handle weakly posed Neumann boundary conditions, the idea is to use a
test function that will never satisfy the boundary condition. The
effect will be that the equation holds also at the
boundary, and that the classical viscosity solution comparison
argument can be used (see the following sections). To achieve this the
usual test function has to be modified and the extent of the
modifications depend on how smooth and non-linear the Neumann
condition is. To handle possibly non-linear boundary conditions
or H\"older continuous solutions in combination with
boundary reflection directions that are only Lipschitz functions in the
space variable, it seemed that the only available or at least the most
optimal test function to use, is the one constructed by Barles in
\cite{ba2,ba3}.
As opposed to the basic test function used in the other papers on continuous
dependence, the test function of Barles is not symmetric in its
arguments ($x$ and $y$) and therefore it does not have equal $x$ and
$y$ gradients. We loose a cancellation property in the comparison proof
and hence can not handle as general gradient dependence in the
equations as with the basic test function.
In this paper we consider the same class of non-singular(!)
equations as Barles in \cite{ba2,ba3}, and this excludes most of the
quasilinear equations considered in
\cite{CockGripenLonden,JKContDep,JK:Ell,Gr:CDBC}, including
$p$-Laplace equations for $p\neq2$ (see also remark
(\ref{noplalpacian}) in section 5.).
At this point we mentioned that a different test function has
been constructed by Ishii in \cite{I4}. Compared with Barles, Ishii
is able to treat less regular domains but with more regular (and less
non-linear) boundary conditions (e.g. $C^1$ domains and $W^{2,\infty}$
reflections), see \cite{ba2} for a more detailed comparison.
Using Ishii's test function, continuous dependence results could probably be
obtained under a different set of assumptions (see above). We have not
considered this case.
We also point out that we can handle $u$-depending boundary conditions
only through additional arguments involving transformations. This is in
contrast to the general {\it comparison} results obtained by Barles
\cite{ba3} under similar assumptions for $F$ and $G$. In \cite{ba3}
$u$-dependence is handled directly by a sort of localization argument
(Lemma 5.2 in \cite{ba3})
which only works when you send some parameter
of the test function to zero. In our continuous dependence arguments,
we will have to optimize with respect to this parameter and the
optimal choice will in general not be zero or even small. See the
treatment of parameter $\ensuremath{\varepsilon}$ at end of the proof of Theorem
\ref{main}.
One way to handle $u$-depending Neumann type boundary value problems,
is to transform them into problems with no $u$-dependence, then to use
our results, and finally to transform back.
We do
not consider such transformations in this paper, instead we refer to
\cite{bl} where such transformations have been considered in a rather
general setting.
Continuous dependence results have to do with well-posedness of the
equation. Typically the boundary value problem we consider model some
physical process, and the data is measured data. A continuous
dependence result then implies that small measurement errors only
produce small errors in the solutions. Any reasonable model should
satisfy such a requirement in particular in view of numerical computations.
Moreover, continuous dependence results have been used in many other
contexts. They play a key part in the shaking of coefficients approach
of Krylov to obtain error estimates for approximation schemes for
Bellman equations \cite{Kr:HJB1,Kr:HJB2,Kr:LipCoeff,BJ:Err1,BJ:Err2,BJ:Err3},
in Bourgoing \cite{Bo:C1a} and in \cite{JKContDep} they are used to
obtain regularity results, and they have been used to estimate
diffusion matrix projection errors
\cite{BOZ}, source term splitting errors \cite{JKR}, and errors
coming from the truncation of Levy measures \cite{JK:CDIPDE}. They
have also been used to derive the rate of convergence for the
vanishing viscosity method
\cite{CockGripenLonden,JKContDep,JK:Ell,Gr:CDBC}, see also
e.g. \cite{CK06}.
The paper is organized as follows: In the next section we state the
assumptions on the boundary value problem \eqref{EE} and \eqref{BV}
in
the standard case and give
well-posedness and H\"older regularity results. We state the main
result, the continuous dependence result, and as an immediate
corollary we derive an estimate on the rate of convergence for the
vanishing viscosity method. The proofs of the main result along with
the regularity result are proven in Section \ref{sec:pfs}, and in
Section \ref{sec:ex} we apply our main result to obtain new
continuous dependence results for boundary value problems involving
Bellman-Isaacs equations. We give several extensions of our results in
Section \ref{sec:ext}, to time-depending equations, equations set on
unbounded domains, and certain quasilinear equations. Finally, in
the Appendix we derive the test function used in the
proofs in Section \ref{sec:pfs} along with its properties.
\subsection*{Notations}
We let $|\cdot|$ denote the Euclidean norm both in $\ensuremath{\mathbb{R}}^m$ (vectors) and
$\ensuremath{\mathbb{R}}^{m\times p}$ (matrices) for $m,p\in\ensuremath{\mathbb{N}}$. We denote by
$\mathbb{S}^N$ the space of symmetric $N\times N$ matrices, $\mathrm{tr}$ and
$^T$ denote trace and transpose of matrices, and $\leq$ denote the
natural orderings of both numbers and square matrices.
For $a,b\in\ensuremath{\mathbb{R}}$ we define $a\vee b=\max(a,b)$ and $a\wedge
b=\min(a,b)$. We will also denote various constants by $K$ or $C$, and
their values may change from line to line.
Let $BUSC(U)$, $BLSC(U)$, $C(U)$, and $W^{p,\infty}(U)$ denote the spaces of
bounded upper and lower semicontinuous functions, continuous
functions, and functions with $p$ essentially bounded derivatives, all
functions defined on $U$.
If $f:\mathbb{R}^N\rightarrow \mathbb{R}^{m\times p}$ is a function
and $\alpha \in(0,1]$,
then define the following (semi) norms :
\begin{align*}
|f|_0=\sup_{x\in \bar{\Omega}}|f(x)|, \qquad
[f]_{\alpha}=\underset{x \neq y}{\sup_{ x,y \in \bar{\Omega}}}
\frac{|f(x)-f(y)|}{|x-y|^{\alpha}},
\qquad \text{and} \qquad
|f|_{\alpha}=|f|_0 + [f]_{\alpha}.
\end{align*}
By $C^{0,\alpha}(\bar{\Omega})$ we denote the set of functions $f:\to
\mathbb{R}$ with finite norm $|f|_{\alpha}$.
We end this section by recalling the definition of a viscosity
solution:
\begin{definition}
An upper semicontinuous function $u$ is a {\it viscosity subsolution}
of (\ref{EE}) and (\ref{BV}) if for all $\phi\in C^2(\bar{\Omega})$,
at each maximum point $x_0\in\bar{\Omega}$ of $u-\phi$,
\begin{align}
F(x_0,u(x_0),D\phi(x_0),D^2\phi(x_0))&\le 0\quad\text{if }
x_0\in \Omega,\\
\min (
F(x_0,u(x_0),D\phi(x_0),D^2\phi(x_0)),G(x_0,Du(x_0))&\le 0\quad\text{if } x_0\in \partial\Omega
\end{align}
An lower-semicontinuous function $u$ is a {\it viscosity supersolution} of
(\ref{EE}) and (\ref{BV}) if for all $\phi\in C^2(\bar{\Omega})$, at
each minimum point $x_0\in\bar{\Omega}$ of $u-\phi$,
\begin{align}
F(x_0,u(x_0),D\phi(x_0),D^2\phi(x_0))&\ge 0\quad\text{if }
x_0\in \Omega,\\
\max (
F(x_0,u(x_0),D\phi(x_0),D^2\phi(x_0)),G(x_0,Du(x_0))&\ge 0\quad\text{if } x_0\in \partial\Omega
\end{align}
Finally $u$ is a solution when it is both a super and a sub-solution.
\end{definition}
\section{The main results}
\label{sec:results}
In this section we consider the standard case (when assumption (H2)
below holds).
Following \cite{ba2,ba3} we state the
assumptions on the boundary value problem \eqref{EE} and
\eqref{BV} and give results on comparison, uniqueness, and existence
of solutions. Then we give new H\"older regularity results extending the
Lipschitz regularity result of \cite{ba2} in two ways: we allow
H\"older continuous data and small $\lambda$ (see assumption (H3) below).
We also give a complete proof. The main result of this
paper, the continuous dependence result, is then stated, and as an immediate
consequence we derive an explicit rate for the convergence of the
vanishing viscosity method.
Here is a list of the assumptions we will use, starting by the domain:
\medskip
\noindent ({\bf H0}) $\ \Omega$ is a bounded
domain in $\ensuremath{\mathbb{R}}^N$ with a $W^{3,\infty}$ boundary.
\medskip
\noindent For the equation we use the following standard assumptions:
\medskip
\noindent({\bf H1}) $\ F \in C(\bar\Omega\times \mathbb{R} \times
\mathbb{R}^N \times \mathbb{S}^N).$
\medskip
\noindent ({\bf H2}) \
\begin{minipage}[t]{11cm}
There exists a modulus $\omega_{R,K}$ (a
continuous,
non-decreasing function satisfying $\omega_{R,K}(0)=0$) such that
$$F(y,r,q,Y)-F(x,r,p,X)\leq \omega_{R,K}\Big(|x-y|+\frac
1{\ensuremath{\varepsilon}^2}|x-y|^2+\eta^2+\ensuremath{\varepsilon}^2+B\Big),$$
for $\ensuremath{\varepsilon},\eta\in(0,1]$, $B\geq0$, $x,y \in\bar\Omega$, $r\in\ensuremath{\mathbb{R}}$,
$|r|\leq R$, $p,q\in \ensuremath{\mathbb{R}}^N$ and $X,Y \in \mathbb{S}^N$ satisfying
$|x-y|\le K \eta\varepsilon$, $|p-q| \le K (\eta^2+\ensuremath{\varepsilon}^2+B)$, $|p|+|q|\leq
K(\frac{\eta}{\ensuremath{\varepsilon}}+\eta^2+\ensuremath{\varepsilon}^2+B)$, and
\begin{align}
\label{XY-ineq}
\begin{pmatrix}
X & 0 \\
0 & -Y
\end{pmatrix}
\le
\ \frac{K}{\varepsilon^2}
\begin{pmatrix}
Id& -Id \\
-Id & Id
\end{pmatrix}
+K (\eta^2+\ensuremath{\varepsilon}^2+B)
\begin{pmatrix}
Id& 0 \\
0 & Id
\end{pmatrix}.
\end{align}
\end{minipage}
\medskip
\noindent ($\overline{\mathrm{\bf H2}}$) \ \begin{minipage}[t]{11cm} There exists
$\alpha\in(0,1]$ and $K_R\geq 0$ such that
$$F(y,r,q,Y)-F(x,r,p,X)\leq
K_R\Big(|x-y|^{\alpha}+\frac 1{\ensuremath{\varepsilon}^2}|x-y|^2+\eta^2+\ensuremath{\varepsilon}^2+B\Big),$$
where $\ensuremath{\varepsilon},\eta,B,R,x,y,p,q,X,Y$ are as in (H2).
\end{minipage}
\medskip
\noindent ({\bf H3}) \ For every $x,p,X,$ and for any $R>0$, there is
$\lambda_R > 0$ such that
$$F(x,r,p,X)- F(x,s,p,X)\geq
\lambda_{R}(r-s) \quad\mbox{ for }\quad -R\leq s \leq r \leq R.$$
\smallskip
\noindent The possibly fully nonlinear Neumann type boundary condition
satisfies:
\medskip
\noindent ({\bf HB1}) \
\begin{minipage}[t]{11cm} There exists $\nu >0$ such that for all $\mu >0,
x\in \partial\Omega, p\in \ensuremath{\mathbb{R}}^N,$
$$G(x,p+\mu n(x))-G(x,p) \ge \nu \mu,$$
where $n(x)$ is the unit outward normal at $x$.
\end{minipage}
\medskip
\noindent ({\bf HB2}) \ There exists a constant $K$
such that for all $x,y\in\partial \Omega$ and all $p,q\in \ensuremath{\mathbb{R}}^N,$
$$|G(x,p)-G(y,q)| \le K\left [(1 +|p|+|q|) |x-y| +|p-q|\ \right].$$
\begin{remark}
In general there is a trade off between the regularity of the boundary
$\partial\Omega$ and the generality and smoothness of the boundary
condition $G$, see \cite{ba2} for a discussion. (H0) compensates for very
general non-smooth boundary conditions.
\end{remark}
\begin{remark}
Assumption (H2) plays the same role as (3.14) in the Users' Guide
\cite{cil}. By this assumption the equation is degenerate
elliptic. Moreover, it is a refined version of assumption (H5-1) in
\cite{ba3} containing also a new parameter $B$. In the proofs, this
parameter will be used to carry information from the boundary
conditions (which are never satisfied, see the introduction) over to
the equations. Assumption ($\overline{\mathrm{H2}}$) is a strengthening of
hypothesis (H2) which yields H\"{o}lder regularity results.
By (H3) the equation is strictly increasing in the $u$
argument. Assumption (HB1) is the Neumann assumption, saying that the
boundary condition $G$, contains non-vanishing and non-tangential (to
$\partial\Omega$) oblique derivatives and it is a natural condition to insure
the well-posedness of the problem.
\end{remark}
We now state a comparison, uniqueness, existence, and regularity
result for solutions of \eqref{EE} and \eqref{BV}.
\begin{theorem}
\label{WP}
If (H0), (H1), (H2), (H3), (HB1), and (HB2) hold, then the
following statements are true:
\smallskip
\noindent (a)
If $u$ is a $BUSC(\bar\Omega)$ subsolution and $v$ is a $BLSC(\bar\Omega)$
supersolution of \eqref{EE} and \eqref{BV}, then $u\leq v$ in $\bar\Omega$.
\smallskip
\noindent (b) If $\lambda_R$ in (H3) is independent of $R$, then there exists a
unique solution $u\in C(\bar\Omega)$ of \eqref{EE} and \eqref{BV}.
\smallskip
\noindent (c) Assume ($\overline{H2}$) also holds, $u\in
C(\bar\Omega)$ is the solution of \eqref{EE} and \eqref{BV}, and
$\lambda:=\lambda_{|u|_0}>0$. Then there are constants
$\beta\in(0,\alpha]$ and $K$ (only depending on the data and $\lambda$)
such that
$$|u(x)-u(y)|\leq K|x-y|^{\beta} \quad\text{in}\quad
\bar\Omega\times\bar\Omega.$$
Furthermore, there exists a constant $\bar\lambda>0$ (only depending
on the data) such that if $\lambda>\bar\lambda$ then $\beta =\alpha$
(the maximal regularity is attained).
\end{theorem}
The comparison principle in (a) correspond to Theorem 2.1 in
\cite{ba3}. The uniqueness part in (b) follow from (a), and existence
follows from Perrons method \cite{Is:PM} since $w(x):=M-Kd(x)$ is a
supersolution of \eqref{EE} and $-w$ is a subsolution of \eqref{BV},
if $M,K\geq 0$ are big enough, and $d$ is the $W^{3,\infty}$ extension
of the distance function defined in the Appendix, see
Section 4 in \cite{ba3} for similar results. The regularity result,
part (c), will be proved in Section \ref{sec:pfs}.
\begin{remark}
The regularity results in part c) are global up to the boundary. Local
up to the boundary H{\"o}lder estimates have been
obtained by Barles-Da Lio \cite{bl} using different techniques and
assumptions on the nonlinearity of the equation. See the introduction for a discussion.
\end{remark}
Now we proceed to the continuous dependence result. We will derive an
upper bound on the difference between a viscosity subsolution $u_1$ of
\begin{eqnarray}
\label{E}
F_1(x,u_1(x),Du_1(x),D^2u_1(x)) &=& 0\quad \text{ in }\quad
\Omega, \\
G_1(x, Du_1(x))&=&0 \quad \text{ on }\quad \partial \Omega,\nonumber
\end{eqnarray}
and a viscosity supersolution $u_2$ of
\begin{eqnarray}
\label{E_tmp}
F_2(x,u_2(x),Du_2(x),D^2u_2(x))& =& 0
\quad \text{ in } \quad\Omega,\\
G_2(x,Du_2(x)) &=& 0\quad \text{ on } \quad \partial
\Omega.\nonumber
\end{eqnarray}
We assume the following estimates on the differences of
the two equations and of the two boundary conditions.
\medskip
\noindent {\bf (D1)} There are $\delta_1$, $\delta_2\geq0$,
and $K_F(K)\geq 0$ such that for any $K\geq0$,
\begin{align*}
& F_2(y,r,q,Y) - F_1(x,r,p,X)
\leq K_F(K) \Big(\eta ^2+
\delta_1+ \frac{1}{\varepsilon^2}\delta_2^2
+B\Big),
\end{align*}
for $0<\ensuremath{\varepsilon}\leq\eta:=\ensuremath{\varepsilon}^{\frac{\bar\alpha}{2-\bar\alpha}}\leq 1$ with
$\bar\alpha=\alpha\wedge\beta$, $B\geq0$,
$x,y \in\bar\Omega$, $r\in\ensuremath{\mathbb{R}}$, $|r|\leq K$, $p,q\in \ensuremath{\mathbb{R}}^N$ and
$X,Y \in \mathbb{S}^N$ satisfying $|x-y|\le K \eta\varepsilon$, $|p-q| \le K
\eta^2 + K B$, $|p|+|q|\leq
K(\frac{\eta}{\ensuremath{\varepsilon}}+\eta^2+B)$, and
\begin{align*}
&\begin{pmatrix}
X & 0 \\
0 & -Y
\end{pmatrix}
\le
\ \frac{K}{\varepsilon^2}
\begin{pmatrix}
Id& -Id \\
-Id & Id
\end{pmatrix}
+K (\eta^2 +B)
\begin{pmatrix}
Id& 0 \\
0 & Id
\end{pmatrix}.
\end{align*}
\noindent {\bf (D2)} There are $\mu_1,\mu_2,K_G\geq 0$ such that for
all $x\in\partial\Omega$ and $p\in\ensuremath{\mathbb{R}}^N$,
\begin{equation*}
G_2(x,p)- G_1(x,p)\le K_G ( \mu_1 + \mu_2 |p|).
\end{equation*}
\begin{remark}
\label{rem1}
Assumption (D1) is a ``continuous dependence'' version of (H2)
and ($\overline{\mathrm{H2}}$) in this paper, and assumption (3.14) in the
Users' Guide \cite{cil}. A similar assumption is used in Theorem 2.1 in
\cite{JK:Ell}.
By $\beta$ and $\alpha$ we denote the H\"older
exponents of the solutions and data respectively. In general $\alpha\geq
\beta$, and equality only holds when $\lambda$ in (H3) is big enough.
Since $|x-y|\le K\ensuremath{\varepsilon}\eta$, $\eta=\ensuremath{\varepsilon}^{\frac{\beta}{2-\beta}}$
imply $\frac{|x-y|^2}{\ensuremath{\varepsilon}^2}\leq K\eta^2$ and
$|x-y|^{\beta}\leq K\eta^2$, the $F_1-F_2$ inequality in
(D1) will be implied by the following more standard inequality
\begin{align*}
& F_2(y,r,q,Y) - F_1(x,r,p,X)
\leq K\Big(|x-y|^{\alpha}+\frac{1}{\varepsilon^2} |x-y|^2 +
\delta_1+ \frac{1}{\varepsilon^2}\delta_2^2
+\eta^2+B\Big).
\end{align*}
\end{remark}
\begin{remark} on assumption (D2). In the case of oblique derivative boundary
conditions, $G_i(x,p)=\gamma_i(x)\cdot p-g_i(x)$, $i=1,2$, and
$$|G_2(x,p)- G_1(x,p)|\le |(g_1- g_2)^+|_0 + |\gamma_1-\gamma_2| _0
|p|.$$
\end{remark}
Our main result is stated in the following theorem:
\begin{theorem}[Continuous Dependence Estimate]
\label{main}
Assume (H0), (H1), (H3), (HB1), and (HB2) hold for $H_1,H_2,G_1,G_2$,
and $u_1,u_2 \in C^{0,\beta}(\bar{\Omega})$ for $\beta\in(0,1]$. Define
$\nu^2=(\nu_1\vee\nu_2)(\nu_1\wedge\nu_2)$ and
$\lambda=\lambda_{1,|u_1|_0}\vee \lambda_{2,|u_2|_0}$.
If (D1) and (D2) hold and $u_1$ and $u_2$ satisfy the boundary value
problems \eqref{E} and \eqref{E_tmp} respectively, then there exist
a constant $C>0$ (depending only on $K_F$, $K$, $K_G$
, $|u_1|_\beta$, $|u_2|_\beta$, $\alpha$, $\beta$)
Such that
\begin{equation*}
\lambda \max_{\bar{\Omega}} (u_1-u_2) \ \leq\ C\Big(\delta_1 +
\delta_2^{\alpha\wedge\beta}+\frac{\mu_1}{\nu}+\Big(\frac{\mu_2}{\nu}\Big)^{\alpha\wedge\beta}\Big).
\end{equation*}
\end{theorem}
\begin{remark}
As far as we know this is the first result giving continuous dependence on the
boundary condition. The result also extends the earlier continuous
dependence on the equation type of results of
\cite{CockGripenLonden,JKContDep,JK:Ell,Gr:CDBC} since
much more general boundary conditions are considered (but at the
expense of less general equations!).
\end{remark}
We prove Theorem \ref{main} in Section \ref{sec:pfs}. An immediate
consequence of this result is an estimate on the rate of
convergence for the vanishing viscosity method.
For $\mu>0$ we consider the solution $u_\mu$ of
\begin{align}
\label{VV}
F(x,u,Du,D^2u)&=\mu \Delta u \qquad\text{in}\quad \Omega,
\end{align}
with boundary condition \eqref{BV}. The result is the following:
\begin{theorem}
\label{VVthm}
Assume (H0), (H1), ($\overline{H2}$), (H3), (HB1), (HB2),
$\mu>0$, and that $u$ and $u_\mu$ solve
\eqref{EE}/\eqref{BV} and \eqref{VV}/\eqref{BV} respectively. Then $u$
and $u_\mu$ belong to $C^{0,\beta}(\bar\Omega)$ for some
$\beta\in(0,\alpha]$ and
$$|u-u_\mu|_0\leq C\mu^{\beta/2}.$$
\end{theorem}
\begin{proof}
Regularity follows from Theorem \ref{WP}. By assumption ($\overline{\mathrm{H2}}$)
$$[F(y,r,q,Y)-\mu \,\mathrm{tr} \,Y]-F(x,r,p,X)\leq
C(|x-y|^{\alpha}+\frac{|x-y|}{\ensuremath{\varepsilon}^2}+\eta^2+\ensuremath{\varepsilon}^2)-\mu\, \mathrm{tr}\, Y,$$
and inequality \eqref{XY-ineq} implies that $-\mathrm{tr} Y\leq C\frac1{\ensuremath{\varepsilon}^2}+\mathrm{small\ terms}$.
Theorem \ref{main} immediately gives $u-u_\mu\leq
C\mu^{\beta/2}$. A lower bound can be found in a similar way.
\end{proof}
\begin{remark}
This result seems to be the first such result for complicated boundary
condition. We refer to \cite{CockGripenLonden,Gr:CDBC} for results on
weak $0$-Neumann or classical $0$-Dirichlet problems, to \cite{PeSa}
for results on linear Neumann boundary value problems for first order
equations, and to \cite{JKContDep,JK:Ell} for result in $\ensuremath{\mathbb{R}}^N$ or
$(0,T)\times\ensuremath{\mathbb{R}}^N$.
\end{remark}
\begin{remark}
The vanishing viscosity method has been studied by many authors
dealing with weak solutions of nonlinear PDEs. The method has
been used to obtain existence (and uniqueness!) of solutions for
degenerate (e.g. first order) problems by taking the limit as
$\mu\ra0$ (see e.g. \cite{BiBr,soug}), and it is well-known
that it is strongly related to the problem of proving convergence
rates for numerical approximations of such problems (see
e.g. \cite{crli,PeSa}).
\end{remark}
\section{Proofs of Theorems \ref{main} and \ref{WP} (c)}
\label{sec:pfs}
\begin{proof}[Proof of Theorem \ref{main}]
First we assume without loss of generality that
$$\delta_1,\delta_2,\frac{\mu_1}{\nu},\frac{\mu_2}{\nu}\le 1.$$
If this is not the case then the theorem holds since
$$u_1-u_2\leq
(|u_1|_0+|u_2|_0)\Big(\delta_1+\delta_2^{\bar\alpha}+\frac{\mu_1}{\nu}+\Big(\frac{\mu_2}{\nu}\Big)^{\bar\alpha}\Big),$$
where $\bar\alpha=\alpha\wedge\beta$.
Then we double the variables and consider
\begin{align*}
\psi(x,y) &=
u_1(x)-u_2(y)-\phi(x,y) \quad\text{and}\quad M=\max_{x,y\in\bar\Omega}
\psi(x,y)=\psi(\bar x,\bar y),
\end{align*}
where for $A,B\geq 0$,
\begin{align*}
\begin{aligned}
\phi(x,y)&=\frac{1}{\varepsilon^2} |x-y|^2 +\frac{A}{\varepsilon^2}
\left(d(x)-d(y)\right)^2 -B(d(x)+d(y))\\
&\quad-\tilde C_2(\frac{x+y}{2},
\frac{2(x-y)}{\varepsilon^2}) (d(x)-d(y)),
\end{aligned}
\end{align*}
and $\tilde C_2(x,p)=C_{2,a}(x,p)$ with
$a=\eta\ensuremath{\varepsilon}=\ensuremath{\varepsilon}^{\frac{2}{2-\bar\alpha}}$
($\eta=\ensuremath{\varepsilon}^{\frac{\bar\alpha}{2-\bar\alpha}}$ by (D1)).
The functions $C_{2,a}$ and $d$ are defined in the Appendix,
and the smooth function $\phi$ was introduced by Barles in
\cite{ba3}. We refer to the Appendix for the proofs
of the properties of $\phi$.
The existence of a point $(\bar x,\bar y)$ follows from compactness of
$\bar{\Omega}$ and the continuity of all functions involved.
Since $(\bar x,\bar y)$ is a maximum point,
$$ 2\psi(\bar x,\bar y) \ge \psi(\bar x,\bar x) +\psi(\bar y, \bar y).$$
Moreover, if $A$ is big enough, Lemma \ref{lem_pos} of the Appendix
implies that
\begin{align}
\label{posA}
\phi(\bar x,\bar y)\geq \frac{1}{2\ensuremath{\varepsilon}^2}|\bar x-\bar y|^2-K_0\ensuremath{\varepsilon}^2-B(d(\bar x)+d(\bar y)),
\end{align}
and H\"older regularity of $u_1$ and $u_2$ combined with the last two
inequalities yield
$$\frac{1}{2\varepsilon^2} |\bar x-\bar y|^2\le K_1|\bar x-\bar
y|^{\bar\alpha}\vee \ensuremath{\varepsilon}^2
$$
for some constant $K_1$ depending on $K_0$ and the H\"older constants of $u_1$
and $u_2$ (but not on $B$). Equivalently, since
$\eta=\ensuremath{\varepsilon}^{\frac{\bar\alpha}{2-\bar\alpha}}$ by (D1) and $\ensuremath{\varepsilon}\leq\eta$,
\begin{equation}
\label{bxbyestim}
|\bar x-\bar y|\le \tilde K_1 \varepsilon^{\frac{2}{2-\bar\alpha}}\,=\tilde
K_1\eta\ensuremath{\varepsilon}\quad
\hbox{and}\quad \frac{1}{\varepsilon^2} |\bar x-\bar y|^2\le \tilde K_1
\varepsilon^{\frac{2\bar\alpha}{2-\bar\alpha}}\,=\tilde K_1\eta^2.
\end{equation}
Now we choose $A$ and $B$ in the test function $\phi$ to
insure that when $\bar x$ or $\bar y$
belong to the boundary $\partial\Omega$, then the boundary conditions can
not hold there. See Lemma \ref{lem_BC} of the Appendix. This
means that the {\it equations} always has to hold at $\bar x$ and
$\bar y$. The precise choices of $A$ and $B$ are
$$B = K(\eta^2+\ensuremath{\varepsilon}^2) + \frac
K{\nu}\Big(\mu_1+\mu_2\frac{\eta}{\ensuremath{\varepsilon}}\Big)
\quad \text {and}\quad A=K,$$
for some $K$ only depending on the data of the problem.
By the maximum principle for semicontinuous functions, Theorem 3.2 of
the "Users' guide" \cite{cil}, there are $(p,X)\in \bar{J}^{2,+}_{\bar \Omega}
u_1(\bar x)$ and $(q,Y)\in \bar{J}^{2,-}_{\bar \Omega}u_2(\bar y)$ such that
\begin{gather*}
p= D_x \phi (\bar x,\bar y), \qquad q =-D_y\phi(\bar x,\bar y),\\
\begin{pmatrix}
X & 0 \\
0 & -Y
\end{pmatrix}
\le
[Id+ \varepsilon^2 D^2 \phi (\bar x,\bar y)] D^2 \phi (\bar x,\bar y).
\end{gather*}
Using the definition of viscosity sub and super solutions at $\bar x$ and
$\bar y$ (and Lemma \ref{lem_BC}) we get
\begin{equation*}
F_1(\bar x, u_1(\bar x),p,X)\le 0\le F_2(\bar y,u_2(\bar y),q,Y).
\end{equation*}
We rewrite this as
\begin{equation}
\label{solg1}
F_1(\bar x, u_1(\bar x),p,X)-F_1(\bar x, u_2(\bar y),p,X)\le
F_2(\bar y,u_2(\bar y),q,Y))-F_1(\bar x, u_2(\bar y),p,X).
\end{equation}
By Lemma \ref{lem_deriv}, the definitions of $p,q,X,Y$, and
$\ensuremath{\varepsilon}\leq\eta\leq 1$, it follows that
\begin{gather*}
|p-q|\leq K\eta^2+2B, \\
\begin{pmatrix}
X & 0 \\
0 & -Y
\end{pmatrix}
\le
\ \frac{K}{\varepsilon^2}
\begin{pmatrix}
Id& -Id \\
-Id & Id
\end{pmatrix}
+K( \eta^2 +B)
\begin{pmatrix}
Id& 0 \\
0 & Id
\end{pmatrix},
\end{gather*}
again for some $K$ only depending on the data of the problem.
Since we also have \eqref{bxbyestim}, we are in a position to use
assumption (D1). So if $u_1(\bar x)-u_2(\bar y)\ge 0$, then (D1) and (H3)
applied to \eqref{solg1} yield
\begin{equation*}
\lambda_1 (u_1(\bar x)-u_2(\bar y)) \le K_F(K) \Big(\eta ^2+
\delta_1+ \frac{1}{\varepsilon^2}\delta_2^2
+B\Big).
\end{equation*}
By \eqref{posA} and the definition of $\psi$, it follows that
\begin{equation*}
\label{finalstep}
u_1(x)-u_2(x)\le \psi_\varepsilon(x,x)\le
\psi_\varepsilon(\bar x,\bar y) \le u_1(\bar x)-u_2(\bar y) + 2B (d(\bar x)+d(\bar y)).
\end{equation*}
Therefore the two previous inequalities and the choice of $B$ implies that
$$\lambda_1 (u_1(x)-u_2(x)) \le K \Big(\eta^2 +\delta_1 +
\frac{1}{\varepsilon^2}\delta_2^2 +\frac{[\mu_1+ \mu_2
\frac{\eta}{\varepsilon}]}{\nu} \Big).$$
Remember that $\eta=\eta(\ensuremath{\varepsilon})=\ensuremath{\varepsilon}^{\frac{\bar\alpha}{2-\bar\alpha}}$ and let
$\ensuremath{\varepsilon}_1$ and $\ensuremath{\varepsilon}_2$ be defined by
\begin{align*}
\eta(\ensuremath{\varepsilon}_1)^2&=\frac1{\ensuremath{\varepsilon}_1^2}\delta_2^2 &&\text{or}\quad
\eta(\ensuremath{\varepsilon}_1)^2=\delta_2^{\bar\alpha},\\
\eta(\ensuremath{\varepsilon}_2)^2&=\frac{\mu_2\frac{\eta(\ensuremath{\varepsilon}_2)}{\ensuremath{\varepsilon}_2}}{\nu} &&\text{or}\quad
\eta(\ensuremath{\varepsilon}_2)^2=\frac{\mu_2^{\bar\alpha}}{\nu^{\bar\alpha}} .
\end{align*}
Now with $\ensuremath{\varepsilon}=\ensuremath{\varepsilon}_1\vee\ensuremath{\varepsilon}_2$ ($\leq 1$ by assumption) it
follows that
\begin{equation*}
\lambda_1 (u_1(x)-u_2(x)) \le K \Big(\delta_1+ \delta_2^{\bar\alpha}
+\frac{\mu_1}{\nu}+\frac{\mu_2^{\bar\alpha}}{\nu^{\bar\alpha}}\Big).
\end{equation*}
A closer look at the proof reveals that we
may replace $\lambda_1$ by $\lambda_1\vee\lambda_2$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{WP} (c)]
We start by proving $\alpha$-H\"older regularity when $\lambda$ is big
(the last statement of Theorem \ref{WP} (c)).
The proof is similar to the proof of Theorem \ref{main} except that we have
to modify the test function and use a bootstrap argument. The modified
test function is
\begin{align*}
\begin{aligned}
\phi_a(x,y)&=\frac{1}{\varepsilon^2} e^{-K_e(d(x)+d(y))} |x-y|^2 +\frac{A}{\varepsilon^2}
\left(d(x)-d(y)\right)^2 \\
&\quad-C_a\Big(\frac{x+y}{2},
\frac{2e^{-K_e(d(x)+d(y))}(x-y)}{\varepsilon^2}\Big) (d(x)-d(y))\\
&\quad -K_B(a+\ensuremath{\varepsilon}^{\frac{2\alpha}{2-\alpha}})(d(x)+d(y)).
\end{aligned}
\end{align*}
We refer to the Appendix for the definitions of $C_a$ and
$d$. Playing with the parameter
$a$, we will use a bootstrap argument to prove that $u$ has the
right regularity.
The new test function satisfies similar estimates as the ones
given in Lemmas \ref{lem_pos} - \ref{lem_deriv}. The moral is that the new
terms coming from the exponential term are not worse than the
old terms. We refer to \cite{ba3} for such
estimates given in the full generality (but with a different choice
of $a$).
Now let $\ensuremath{\varepsilon}\le 1$ and double the variables defining
$$M:=\psi(\bar x,\bar y)=\sup\psi(x,y)\quad \text{where}\quad
\psi(x,y)=u(x)-u(y)-\phi_a(x,y).$$
If $A$ is big enough, (an easy extension of) Lemma \ref{lem_pos} and
the inequality
$2\psi(\bar x,\bar y)\geq \psi(\bar x,\bar x)+\psi(\bar y,\bar y)$ imply that
\begin{align}
\label{pospos}
\frac1{2\ensuremath{\varepsilon}^2}e^{-K_e(d(x)+d(y))}|\bar x-\bar y|^2\leq
2[u(\bar x)-u(\bar y)]+K_0\ensuremath{\varepsilon}^2
\leq 2|u|_0+K_0.
\end{align}
Define $\eta^2=K^{-1}\frac1{\ensuremath{\varepsilon}^2}|\bar x-\bar y|^2$ with
$K=e^{2K_e}(2|u|_0+K_0)$. By \eqref{pospos},
$$\eta^2\leq 1\quad \text{and}\quad |\bar x-\bar y|\leq K^{1/2}\eta\ensuremath{\varepsilon}.$$
We proceed as in the proof of Theorem \ref{main}.
By arguments similar to the ones in the proof of Lemma \ref{lem_BC},
if $A$, $K_e$ and $K_B$ are big enough (not depending on $\ensuremath{\varepsilon}$, $a$
or $B$), then the equation holds even if $(\bar x,\bar y)$ lies on
$\partial(\Omega\times\Omega)$. Compared with the proof of Theorem \ref{main}, the
exponential allows us to cancel at the boundary all terms of the form
$\frac{1}{\varepsilon^2}|\bar x-\bar y|^2=K\eta^2$ and use
$B=K_B(a+\ensuremath{\varepsilon}^{\frac{2\alpha}{2-\alpha}})$ at each step.
Note that $D\phi_a$ and $D^2\phi_a$ still satisfy inequalities
\eqref{pmqest} and \eqref{scnd} in Lemma \ref{lem_deriv}. We will choose $a$
such that inequality \eqref{scnd}
takes the form of \eqref{XY-ineq}, i.e. we choose $a$ such that $\frac
{\ensuremath{\varepsilon}}a \eta^3\leq K$. Since $\eta^2\leq1$ we choose $a=\ensuremath{\varepsilon}$. Again
we use the definition of
viscosity solutions and subtract the equations (inequalities) at $\bar
x$ and $\bar y$ using the maximum principle for semi continuous
functions. By the appropriate version of Lemma
\ref{lem_deriv} and the definition of $\eta^2$ and $B$ we can now use (H1)
and ($\overline{\mathrm{H2}}$) to get
$$\lambda(u(\bar x)-u(\bar y))\leq
K\Big(|\bar x-\bar y|^{\alpha}+\frac 1{\ensuremath{\varepsilon}^2}|\bar x-\bar y|^2
+\eta^2+\ensuremath{\varepsilon}^2+K_B(\ensuremath{\varepsilon}+\ensuremath{\varepsilon}^{\frac{2\alpha}{2-\alpha}})\Big).$$
By Young's inequality, the definition of $\eta^2$, and $\ensuremath{\varepsilon}\leq1$ we have
$$\lambda(u(\bar x)-u(\bar y))\leq K\Big(\frac 1{\ensuremath{\varepsilon}^2} |\bar x-\bar
y|^2 + \ensuremath{\varepsilon}^{1\wedge\frac{2\alpha}{2-\alpha}}\Big).$$
When $A$ is big enough, an appropriate version of Lemma \ref{lem_pos},
the definition of $M$, and $0\leq d\leq 1$, imply that
$$u(\bar x)-u(\bar y)= M+\phi(\bar x,\bar y)\geq M
+\frac1{2\ensuremath{\varepsilon}^2}e^{-2K_e}|\bar x-\bar y|^2-K_0\ensuremath{\varepsilon}^2
-K_B(\ensuremath{\varepsilon}+\ensuremath{\varepsilon}^{\frac{2\alpha}{2-\alpha}})(d(\bar x)+d(\bar y)).$$
Combining the two last inequalities and using that $\ensuremath{\varepsilon}\leq1$ leads to
$$\lambda M \leq
\big(K-\frac{\lambda}2e^{-2K_e}\big)\frac1{\ensuremath{\varepsilon}^2}|\bar x-\bar
y|^{2}+K\ensuremath{\varepsilon}^{1\wedge\frac{2\alpha}{2-\alpha}}$$
If $\lambda$ is big enough, $\lambda M\leq K
\ensuremath{\varepsilon}^{1\wedge\frac{2\alpha}{2-\alpha}}$,
and the definition of $M$ leads to
$$u(x)-u(y) - \phi_{\ensuremath{\varepsilon}}(x,y)\leq M \leq
\frac{K}{\lambda}\ensuremath{\varepsilon}^{1\wedge\frac{2\alpha}{2-\alpha}}$$
for every $x,y\in \bar\Omega$. Now by the definition of $\phi_a$, the
properties of the distance function, and Young's inequality, we have
$$u(x)-u(y) \leq
K\frac{1}{\ensuremath{\varepsilon}^2}|x-y|^2+K\ensuremath{\varepsilon}^{1\wedge\frac{2\alpha}{2-\alpha}}.$$
If $|x-y|\leq 1$ we may take $\ensuremath{\varepsilon}=|x-y|^{\frac23}$ when
$1<\frac{2\alpha}{2-\alpha}$ and
$\ensuremath{\varepsilon}^{\frac{2\alpha}{2-\alpha}}=|x-y|^{\alpha}$ otherwise, the result
(since we may also interchange $x$ and $y$) is that
\begin{align}
\label{Hest}
|u(x)-u(y)|\leq K|x-y|^{\frac23\wedge\alpha}.
\end{align}
If $|x-y|\geq 1$, the result still holds since then $|u(x)-u(y)|\leq
2|u|_0|x-y|^{\frac23\wedge\alpha}$. We are now done if $\alpha\leq \frac23$.
If $\alpha\in (\frac23,1]$, we restart the proof using the regularity estimate
\eqref{Hest} to get a better choice of $a$ such that $\frac{\ensuremath{\varepsilon}}a\eta^3\leq
K$. From \eqref{Hest} and the first inequality in
\eqref{pospos},
$$\eta^2=K^{-1}\frac{1}{\ensuremath{\varepsilon}^2}|\bar x-\bar y|^2\leq K |\bar x-\bar y|^{\frac23}\vee\ensuremath{\varepsilon}^2 \qquad\text{and
hence}\qquad \eta\leq K\ensuremath{\varepsilon}^{\frac{\frac23}{2-\frac23}}\vee\ensuremath{\varepsilon},$$
so the new choice of $a$ should be
$\ensuremath{\varepsilon}\eta^3=\ensuremath{\varepsilon}^{\frac52}\vee\ensuremath{\varepsilon}^4$. But this quantity is
less than $\ensuremath{\varepsilon}^2$ so we may instead take $a=\ensuremath{\varepsilon}^2$ which still
implies $\frac{\ensuremath{\varepsilon}}a\eta^3\leq K$. Now it is a simple exercise to
redo the proof and show that for $\lambda$ big,
\begin{align*}
|u(x)-u(y)|\leq K|x-y|^{\alpha} \qquad \text{for}\qquad x,y\in\bar\Omega,
\end{align*}
and this completes the proof of the last part of Theorem \ref{WP}.
Now we will prove the first part of Theorem \ref{WP} (c) using the
result we proved above and an iterative argument of Lions
\cite{Li:Ex}. Here we only sketch parts of the
argument, since the details can be found in \cite{JK:Ell} for similar
equations. The idea is to consider for $\mu>0$
$$F(x,u^{n+1},Du^{n+1},D^2u^{n+1})+\mu u^{n+1} = \mu u^n$$
with boundary conditions \eqref{BV} and noting that $u^n$ converge
uniformly to $u$. If $\mu$ is big enough the above proven
result applies, and a careful look at the above argument
reveals that when $\lambda+\mu>K(=\bar\lambda)$, then
\begin{align*}
|u^{n+1}(x)-u^{n+1}(y)|\leq \Big(\frac{\mu|u^n|_\alpha}{\lambda+\mu - K}+
\mathrm{other\ terms}\Big)|x-y|^{\alpha}, \quad
x,y\in\bar\Omega, |x-y|\leq 1.
\end{align*}
Furthermore the comparison principle yields
$$|u^{n+1}-u|_0\leq \frac{\mu}{\mu+\lambda}|u^n-u|_0\leq
\Big(\frac{\mu}{\mu+\lambda}\Big)^n|u^0-u|_0.$$
When $|x-y|\leq 1$, the rest of the proof is exactly as in
\cite{JK:Ell} and we omit it. When $|x-y|>1$ any H\"older estimate
holds since $u$ is bounded. The result is a H\"older estimate for
any $\lambda>0$, but with a H\"older exponent that is smaller than $\alpha$.
\end{proof}
\section{Bellman-Isaacs type boundary value problems}
\label{sec:ex}
In this section we apply our results in Section \ref{sec:results} to
Bellman-Isaacs equations and several different types of boundary conditions.
The Bellman-Isaacs equations are of the form
\begin{align}
\label{HJB}
\inf_{\theta_1\in\Theta_1}\sup_{\theta_2\in\Theta_2}\left\{-
\mathrm{tr}[(\sigma\sigma^T)^{\theta_1,\theta_2}(x) D^2u] -
b^{\theta_1,\theta_2}(x) Du -
c^{\theta_1,\theta_2}(x) u - f^{\theta_1,\theta_2}(x)\right\}=0
\end{align}
in $\Omega$. Assumptions (H1), (H2), ($\overline{\mathrm{H2}}$), and (H3)
are satisfied \cite{cil,JK:Ell} if we assume:
\begin{itemize}
\item[1.] $\sigma^{\theta_1,\theta_2}$ and $b^{\theta_1,\theta_2}$ are
Lipschitz continuous in $x$ uniformly in $\theta_1,\theta_2$,
\item[2.] $c^{\theta_1,\theta_2}$ and $f^{\theta_1,\theta_2}$ are $\alpha$-H\"{o}lder continuous in $x$ uniformly in
$\theta_1,\theta_2$,
\item[3.] $c^{\theta_1,\theta_2}(x)\geq\lambda>0$ for all
$x,\theta_1,\theta_2$, and
\item[4.] $\Theta_1,\Theta_2$ are compact metric spaces.
\end{itemize}
Next, we list some typical boundary conditions we can consider:
(a) The classical Neumann condition:
$$\frac{\partial u}{\partial n}=g(x) \quad \text{in}\quad \partial\Omega.$$
(b) The oblique derivative condition:
$$\frac{\partial u}{\partial \gamma}=g(x)\quad \text{in}\quad \partial\Omega.$$
(c) The capillary boundary condition:
\begin{align}
\label{cap}
\frac{\partial u}{\partial n}=\bar\theta(x)(1+|Du|^2)^{1/2}\quad \text{in}\quad
\partial\Omega\quad \text{ with } |\bar\theta(x)| \le \omega<1.
\end{align}
(d) The ``controlled'' reflection boundary condition:
\begin{align}
\label{ctrlBC}
\inf_{\alpha\in\Theta_1}\sup_{\beta\in\Theta_2}\{\gamma^{\theta_1,\theta_2}(x)
\cdot Du
-g^{\theta_1,\theta_2}(x)\}=0\quad \text{in}\quad \partial\Omega.
\end{align}
Here $n(x)$ is the outward unit normal to $\partial\Omega$. Assumptions (HB1)
and (HB2) hold in all cases if assumption 4 holds along with
\begin{itemize}
\item[5.] there exists $\nu>0$ such that
$$\gamma(x)\cdot
n(x)\geq\nu\qquad\text{and}\qquad \gamma^{\theta_1,\theta_2}(x)\cdot
n(x)\geq\nu\quad\text{uniformly in $\theta_1,\theta_2$},$$
\item[6.] $g,\gamma,\bar
\theta,\gamma^{\theta_1,\theta_2},g^{\theta_1,\theta_2}$
are Lipschitz continuous in $x$ uniformly
in $\theta_1,\theta_2$.
\end{itemize}
Now we state new continuous dependence results for the for the
Bellman-Isaacs equations \eqref{HJB} combined with the controlled reflection
boundary conditions \eqref{ctrlBC}:
\begin{theorem}
Assume $u_1$ and $u_2$ satisfy the boundary value problem \eqref{HJB}
and \eqref{ctrlBC} with coefficients
$\sigma_1,b_1,c_1,f_1,\gamma_1,g_1$ and
$\sigma_2,b_2,c_2,f_2,\gamma_2,g_2$ respectively, where both sets of
coefficients satisfy assumptions 1--6 above.
Then $u_1,u_2$ belong to $C^{0,\beta}(\bar\Omega)$ for some
$\beta\in(0,\alpha]$, and
\begin{align*}
\lambda |u_1-u_2|_0 \leq &\, C\sup_{\Theta_1 \times\Theta_2}\Big[
|\sigma_1^{\theta_1,\theta_2}-\sigma_2^{\theta_1,\theta_2}|_0^{\beta} +
|b_1^{\theta_1,\theta_2}-b_2^{\theta_1,\theta_2}|_0^{\beta} \Big] \\
& + C\sup_{\Theta_1 \times \Theta_2}\Big[
|c_1^{\theta_1,\theta_2}-c_2^{\theta_1,\theta_2}|_0 +
|f_1^{\theta_1,\theta_2}-f_2^{\theta_1,\theta_2}|_0\Big] \\
&
+
\frac{C}{\nu}\sup_{\Theta_1\times\Theta_2}|g^{\theta_1,\theta_2}_1
-g^{\theta_1,\theta_2}_2|_0+\frac{C}{\nu^{\alpha}}
\sup_{\Theta_1\times\Theta_2}|\gamma^{\theta_1,\theta_2}_1
-\gamma^{\theta_1,\theta_2}_2|_0^{\beta}.
\end{align*}
\end{theorem}
This result is a direct consequence of Theorems \ref{WP} and
\ref{main}. In this case $\delta_1$ correspond to the second line in
the estimate,
\begin{gather*}
\delta_2^2=C\sup_{\Theta_1 \times
\Theta_2}[|\sigma^{\theta_1,\theta_2}_1-\sigma^{\theta_1,\theta_2}_2|^2+|b^{\theta_1,\theta_2}_1-b^{\theta_1,\theta_2}_2|^2],\\
\mu_1=\sup_{\Theta_1 \times
\Theta_2}|g^{\theta_1,\theta_2}_1-g^{\theta_1,\theta_2}_2|_0,\qquad \mu_2=\sup_{\Theta_1 \times
\Theta_2}|\gamma^{\theta_1,\theta_2}_1-\gamma^{\theta_1,\theta_2}_2|_0.
\end{gather*}
The dependence on the equation is as in \cite{JKContDep,JK:Ell} and
the derivation of $\delta_1$ and $\delta_2$ is explained there.
By Theorem \ref{VVthm} we have for the first time the rate of
convergence of the vanishing viscosity method for the boundary value
problem \eqref{HJB} and \eqref{ctrlBC}, i.e.
\begin{align}
\label{mHJB}
\inf_{\theta_1\in\Theta_1}\sup_{\theta_2\in\Theta_2}\left\{-
\mathrm{tr}[(\sigma\sigma^T)^{\theta_1,\theta_2}(x) D^2u] -
b^{\theta_1,\theta_2}(x) Du -
c^{\theta_1,\theta_2}(x) u -
f^{\theta_1,\theta_2}(x)\right\}=\mu\Delta u
\end{align}
in $\Omega$, with \eqref{ctrlBC} as boundary conditions. The result is
the following:
\begin{theorem}
Assume $u$ and $u_{\mu}$ satisfy \eqref{HJB} and \eqref{mHJB}
respectively with boundary values \eqref{ctrlBC}, and that assumptions
1 -- 6 hold.
Then $u,u_{\mu}$ belong to $C^{0,\beta}(\bar\Omega)$ for some
$\beta\in(0,\alpha]$ and
$$|u-u_\mu|_0\leq C\mu^{\frac{\beta}{2}}.$$
\end{theorem}
\section{Extensions}
\label{sec:ext}
It is possible to consider many kinds of extensions of the results in this
paper. We will consider three cases: (i)
$\Omega$ unbounded, (ii) time dependent problems, and (iii)
quasilinear equations. In the two first cases the results cover
e.g. Bellman-Isaacs equations under natural assumptions on the data.
\subsection{Unbounded domains}
Let $\Omega$ be unbounded and let
(H0u) denote assumption (H0) without the boundedness assumption. If we
assume that our sub and supersolutions $u$ and $v$ are bounded, then we
will get continuous dependence and regularity results simply by
following the arguments in this paper replacing the test function
$\phi_a$ by the standard modification
$$\phi_a(x,y)+\gamma(|x|^2+|y|^2),\quad \gamma>0.$$
The new test function will insure existence of maximum points when we
double the variables, and at the end of the proof it turns out (as usual)
that all terms depending on $\gamma$ will vanish when
$\gamma\ra0$. In the proof $B$ will now depend also on the
$\gamma$-terms and the $\gamma$-terms will tend to zero as
$\gamma\ra0$ with a speed depending on $B$, see assumption (D1u)
below. By careful computations and fixing $\ensuremath{\varepsilon}$ before sending
$\gamma\ra0$ we can conclude as before. We refer to \cite{JK:Ell} for
the details when $\Omega=\ensuremath{\mathbb{R}}^N$.
The corresponding continuous dependence result will now be given without
further proof.
We modify assumption (D1) so it corresponds to our new test function, see also
\cite{JK:Ell}:
\medskip
\noindent {\bf (D1u)} There are $\delta_1$, $\delta_2\geq0$, a modulus
$\omega$, and $K_F(K)\geq 0$, such that for any $K\geq0$,
\begin{align*}
& F_2(y,r,q,Y) - F_1(x,r,p,X)
\leq K_F(K) \Big(\eta ^2+
\delta_1+ \frac{1}{\varepsilon^2}\delta_2^2
+B + \gamma(1+|x|^2+|y|^2)\Big),
\end{align*}
for $\ensuremath{\varepsilon},\gamma\in(0,1]$, $\eta:=\ensuremath{\varepsilon}^{\frac{\alpha}{2-\alpha}}$, $B\ge 0$,
$x,y \in\bar\Omega$, $r\in\ensuremath{\mathbb{R}}$, $|r|\leq K$, $p,q\in \ensuremath{\mathbb{R}}^N$ and
$X,Y \in \mathbb{S}^N$ satisfying $|x-y|\le K \eta\varepsilon$, $|x|+|y|\leq\gamma^{1/2}\omega(\gamma)(1+B)$, $|p-q| \le K(
\eta^2 + B+\gamma(|x|+|y|))$, $|p|+|q|\leq
K(\frac{\eta}{\ensuremath{\varepsilon}}+\eta^2+B+\gamma(|x|+|y|))$,
and
\begin{align*}
&\begin{pmatrix}
X & 0 \\
0 & -Y
\end{pmatrix}
\le
\ \frac{K}{\varepsilon^2}
\begin{pmatrix}
Id& -Id \\
-Id & Id
\end{pmatrix}
+K (\eta^2 +B+ \gamma)
\begin{pmatrix}
Id& 0 \\
0 & Id
\end{pmatrix}.
\end{align*}
\begin{theorem}[$\Omega$ unbounded]
\label{ThmUnbnd}
Assume (H0u), (H1), (H3), (HB1), and (HB2) hold for $H_1,H_2,G_1,G_2$,
and $u_1,u_2 \in C^{0,\beta}(\bar{\Omega})$ for $\beta\in(0,1]$. Define
$\nu^2=(\nu_1\vee\nu_2)(\nu_1\wedge\nu_2)$ and
$\lambda=\lambda_{1,|u_1|_0}\vee \lambda_{2,|u_2|_0}$.
If (D1u) and (D2) hold and $u_1$ and $u_2$ satisfy the boundary value
problems \eqref{E} and \eqref{E_tmp} respectively then there exist a
constant $C>0$ (depending only on $K_F$, $K$, $K_G$
, $|u_1|_\beta$, $|u_2|_\beta$, $\alpha$)
such that
\begin{equation*}
\lambda \max_{\bar{\Omega}} (u_1-u_2) \ \leq\ C\Big(\delta_1 +
\delta_2^{\alpha\wedge\beta}+\frac{\mu_1}{\nu}+\Big(\frac{\mu_2}{\nu}\Big)^{\alpha\wedge\beta}\Big).
\end{equation*}
\end{theorem}
\subsection{Time dependent case}
Consider a Cauchy-Neumann problem of the form:
\begin{align}
\label{tEE}
u_t+F(t,x,u,Du,D^2u)&=0&&\text{in}\quad (0,T)\times\Omega,\\
G(x,Du)&=0&&\text{on}\quad (0,T)\times\partial\Omega,\label{tBV}\\
u(0,x)&=u_0(x)&&\text{on}\quad \{0\}\times\Omega.\label{tIV}
\end{align}
In this case we get results by similar arguments as above by replacing
the test function $\phi_a$ by
$$\bar\sigma t+e^{Kt}\phi_a(x,y), \quad \bar\sigma>0.$$
We have to replace assumptions (H1) -- (H3) and (D1) by assumptions
(H1p) -- (H3p) and (D1p) depending on $t$. In (H1p) we assume in addition
continuity in $t$, in (H3p) we allow $\lambda_R\geq0$, and in the last
two assumptions ((H2p) and (D1p)) we simply assume that (H2) and (D1) hold uniformly in
$t$. Note that one can always reduce a problem with $\lambda_R\in\ensuremath{\mathbb{R}}$, via an
exponential scaling of $u$, to a problem with $\lambda_R\geq0.$
Now existence, uniqueness, and regularity results follows as before by
appropriately choosing the constants $\bar\sigma$ and $K$. Note
however that in the result corresponding to Theorem \ref{WP} (c) the
H\"older exponent is always $\alpha$ and ``maximal regularity'' is
achieved regardless of the value $\lambda$. We refer to
\cite{JKContDep} for such results in the case $\Omega=\ensuremath{\mathbb{R}}^N$. Now we
state the continuous dependence result without further proof.
\begin{theorem}[Time dependent case]
\label{ThmTime}
Assume (H0), (H1p), (H3p), (HB1), and (HB2) hold for $H_1,H_2,G_1,G_2$,
$u_1,u_2\in C([0,T]\times\bar\Omega)$, and $u_{1,0},u_{2,0}\in
C^{0,\alpha}(\bar\Omega)$ for some $\alpha\in(0,1]$. Define
$\nu^2=(\nu_1\vee\nu_2)(\nu_1\wedge\nu_2)$.
If (D1p) and (D2) hold and $u_1$ and $u_2$ are sub and supersolutions
of initial boundary value problems \eqref{tEE}, \eqref{tBV}, and
\eqref{tIV} respectively for $F_1,G_1,u_{1,0}$ and $F_2,G_2,u_{2,0}$,
then there exist a constant $C>0$ (depending only on $K_F$, $K$,
$K_G$, $|u_{1,0}|_\alpha$,$|u_{2,0}|_\alpha$, $T$, $\alpha$)
such that for $t\in(0,T)$,
\begin{equation*}
\max_{\bar{\Omega}} (u_1(t,\cdot)-u_2(t,\cdot))
\ \leq\ |(u_{1,0}-u_{2,0})^+|_0+Ct\Big(\delta_1 +
\delta_2^{\alpha}+\frac{\mu_1}{\nu}+\Big(\frac{\mu_2}{\nu}\Big)^{\alpha}\Big).
\end{equation*}
\end{theorem}
Note that we do not need to assume that $u_1$ and $u_2$ are H\"older
continuous (in $x$) a priori. In fact this regularity follows from the above
theorem! To understand why, and to see details about the derivation in
the case $\Omega=\ensuremath{\mathbb{R}}^N$, we refer to \cite{JKContDep}.
\subsection{Some quasilinear equations}
\label{sec:extq}
Consider equations of the form
\begin{align}
\label{QLin_2}
-\mathrm{tr}[\sigma(x,Du)\sigma(x,Du)^T D^2u] - f(x,u,Du)+\lambda u =0
\quad\text{in}\quad \Omega,
\end{align}
where $\lambda>0$, $f(x,r,p)$ continuous, increasing in $r$,
$a(x,p)=\sigma(x,p)\sigma(x,p)^T$, and
\begin{align*}
|\sigma(x,p)-\sigma(y,q)|&\leq K\left(|x-y|+\frac{|p-q|}{1+|p|+|q|}\right),\\
|f(x,r,p)-f(y,r,q)|&\leq K\big[(1+|p|+|q|)|x-y|+|p-q|\big].
\end{align*}
In this case (H1) and (H3) hold in addition to an assumption similar
to (H2). If we also assume (H0), (HB1), and (HB2), then existence and
comparison for the boundary value problem
\eqref{QLin_2} and \eqref{BV} was proved in \cite{ba3}.
More general fully non-linear equations with ``quasilinear'' gradient
dependence can also be considered. We omit this to get a shorter and
clearer presentation. For the same reasons we also restrict ourselves
to the case of Lipschitz continuous solutions and data,
i.e. $\alpha=\beta=1,\eta\equiv\ensuremath{\varepsilon}$.
In this case the quasilinear term in the equation gives rise to a term like
$$\frac1{\ensuremath{\varepsilon}^2}|\sigma(p)-\sigma(q)|^2$$
in the proof of the comparison result (when $\sigma$ does not
depend on $x$). By \eqref{pqpq} in Lemma
\ref{lem_deriv},
$$|p-q|\leq K|p|\wedge|q||x-y|+K(\ensuremath{\varepsilon}^2+B)$$
when $\ensuremath{\varepsilon}$ is small enough, and hence by the assumptions on $\sigma$,
\begin{align}
\nonumber
\frac1{\ensuremath{\varepsilon}^2}|\sigma(p)-\sigma(q)|^2&\leq \frac{K}{\ensuremath{\varepsilon}^2}|x-y|^2
+\frac K{\ensuremath{\varepsilon}^2}\frac {\ensuremath{\varepsilon}^4+B^2}{1+|p|^2+|q|^2}\\
&\leq \frac
K{\ensuremath{\varepsilon}^2}|x-y|^2+K\ensuremath{\varepsilon}^2+\frac K{\ensuremath{\varepsilon}^2}B^2.\label{p-q-term}
\end{align}
This computation motivates replacing assumption (D1) by:
\medskip
\noindent {\bf (D1q)} There are $\delta_1$, $\delta_2\geq0$,
and $K_F(K)\geq 0$ such that for any $K\geq0$,
\begin{align*}
& F_2(y,r,q,Y) - F_1(x,r,p,X)
\leq K_F(K) \Big(\eta ^2+
\delta_1+ \frac{1}{\varepsilon^2}\delta_2^2
+B +\frac1{\ensuremath{\varepsilon}^2}B^2\Big),
\end{align*}
for $0<\ensuremath{\varepsilon}\leq 1$, $B\ge 0$,
$x,y \in\bar\Omega$, $r\in\ensuremath{\mathbb{R}}$, $|r|\leq K$, $p,q\in \ensuremath{\mathbb{R}}^N$ and
$X,Y \in \mathbb{S}^N$ satisfying $|x-y|\le K \varepsilon^2$, $|p-q| \le
K|p|\wedge|q|\ensuremath{\varepsilon}^2 + K(\ensuremath{\varepsilon}^2 + B),$ $|p|+|q|\leq
K(1+\ensuremath{\varepsilon}^2+B)$, and
\begin{align*}
&\begin{pmatrix}
X & 0 \\
0 & -Y
\end{pmatrix}
\le
\ \frac{K}{\varepsilon^2}
\begin{pmatrix}
Id& -Id \\
-Id & Id
\end{pmatrix}
+K (\varepsilon^2 +B)
\begin{pmatrix}
Id& 0 \\
0 & Id
\end{pmatrix}.
\end{align*}
The continuous dependence result now becomes:
\begin{theorem}[Quasilinear equations, Lipschitz solutions]
\label{ThmQlin}
Assume (H0), (H1), (H3), (HB1), and (HB2) hold for $H_1,H_2,G_1,G_2$, and
$u_1,u_2\in C^{0,1}(\bar\Omega)$. Define
$\nu^2=(\nu_1\vee\nu_2)(\nu_1\wedge\nu_2)$.
If (D1q) and (D2) hold and $u_1$ and $u_2$ are sub
and supersolutions of the boundary value problems \eqref{E} and
\eqref{E_tmp} respectively, then there exist a constant $C>0$
(depending only on $K_F$, $K$, $K_G$) such that
\begin{equation*}
\lambda \max_{\bar{\Omega}} (u_1-u_2) \ \leq\ C\Big(\delta_1 +
\delta_2+\frac{\mu_1}{\nu}+\frac{\mu_2}{\nu}\Big).
\end{equation*}
\end{theorem}
\noindent{\em Proof. }
The proof is similar to the Proof of Theorem \ref{main} with two
exceptions:
\begin{itemize}
\item[(i)] Assume $\delta_1, \delta_2, \frac{\mu_1}{\nu},
\frac{\mu_2}{\nu}\leq \bar C^{-1}$ where $\bar C$ is big enough (the general
case follows since $u_1,u_2$ are bounded). Since $\ensuremath{\varepsilon}$
is chosen in terms of $\delta_1, \delta_2, \frac{\mu_1}{\nu},
\frac{\mu_2}{\nu}$, a suitable choice of $\bar C$ will ensure that
$\ensuremath{\varepsilon}$ is small enough such that \eqref{pqpq} of Lemma
\ref{lem_deriv} holds. This estimate is needed before one
can apply (D1q).
\item[(ii)] At the end of the proof the following estimate will appear
(remember $\eta=\ensuremath{\varepsilon}$)
$$\lambda_1 (u_1(x)-u_2(x)) \le K \Big(\eta^2 +\delta_1 +
\frac{1}{\varepsilon^2}\delta_2^2 +\frac{[\mu_1+ \mu_2
\frac{\eta}{\varepsilon}]}{\nu} +\frac1{\ensuremath{\varepsilon}^2}\Big(\frac{[\mu_1+ \mu_2
\frac{\eta}{\varepsilon}]}{\nu}\Big)^2 \Big),$$
where the {\em new final term} in the right hand side of the inequality
is a consequence of the
$\frac1{\ensuremath{\varepsilon}^2}B^2$ term of (D1q). Minimizing $\ensuremath{\varepsilon}$ like we did in
Theorem \ref{main} then gives the result. $\hfill\Box$
\end{itemize}
As an example we consider an anisotropic quasilinear equation with
capillary boundary condition. The type of non-linearity appearing here
is similar to the non-linearity appearing in the mean curvature of
graph equation.
\begin{align*}
-\mathrm{tr}\Big[\sigma\sigma^T\Big(I-\frac{Du\otimes
Du}{1+|Du|^2}\Big)D^2u\Big]+\lambda u+f(x,u,Du)&=0
\quad\text{in}\quad \Omega,\\
\frac{\partial u}{\partial n}-\bar\theta(x)(1+|Du|^2)^{1/2}&=0\quad \text{in}\quad
\partial\Omega,
\end{align*}
where $\bar\theta$ is Lipschitz continuous satisfying $|\bar\theta(x)|
\le \omega<1$ and $f$ satisfies the assumptions mentioned above.
Assume $u_1$ and $u_2$ are Lipschitz solutions of this boundary value problem
with different $\sigma_1,\sigma_2,\theta_1,\theta_2$ but with same $f$ and
$\lambda$. Then we may apply Theorem \ref{ThmQlin} with
$\delta_1=0=\mu_1$, $\mu_2=|\bar\theta_1-\bar\theta_2|_0$, and
$$\delta_2^2=\sup_{p\in\ensuremath{\mathbb{R}}^N}\left|\sqrt{\sigma_1\sigma_1^T\Big(I-\frac{p\otimes
p}{1+|p|^2}\Big)}-\sqrt{\sigma_2\sigma_2^T\Big(I-\frac{p\otimes
p}{1+|p|^2}\Big)}\right|^2,$$
to obtain
$$\lambda|u_1-u_2|\leq
C\Big(|\sigma_1-\sigma_2|+\frac
{1}{\nu}|\bar\theta_1-\bar\theta_2|_0\Big).$$
\begin{remark}
\label{noplalpacian}
Neither the above assumptions on $\sigma$ nor assumption (D1q) is satisfied
$p$-Laplacian equations.
\end{remark}
|
1,116,691,498,449 | arxiv | \section{Introduction}
Automatic control systems are essential system parts of many industrial
cyber-physical systems (CPSs) and their flawless operations are of elemental
importance for optimal system operation and high product quality. It is
therefore not surprising that automatic control systems are often immediate
targets of cyber-attacks on industrial CPSs. Driven by the rapidly
increasing industrial demands for higher cyber-security, detection of
cyber-attacks on automatic control systems has drawn incredible research
attention in the current decade. Excellent reviews of state of the art of
research in this thematic area can be found in the recent surveys published
in \cite%
{DHXGZ2018,Survey-attack-detection2018,DIBAJI2019-survey,YMA2019,TGXHV2020,ZHANG2021,Zhou2021IEEE-Proc}%
.
\bigskip
Among various types of cyber-attacks, integrity attacks are specially
directed to automatic control systems \cite{DIBAJI2019-survey,GWSOM2019}. By
injecting attack signals into system input and output channels, e.g. via I/O
and network interfaces, integrity attacks can lead to remarkable system
performance degradations and even catastrophic damages. An early and
reliable detection of integrity attacks is becoming a vital requirement on
cyber-security of industrial CPSs, for instance, for power control systems
\cite{MMM2020}. Thanks to its well-established theoretical framework in the
past three decades, observer-based fault detection technique \cite{Ding2013}
is widely accepted as an efficient method, among numerous ones, to deal with
detection of integrity attacks on control systems \cite%
{DIBAJI2019-survey,GWSOM2019,TGXHV2020}. Unfortunately, different from
technical faults, cyber-attacks are artificially created and can be designed
and generated by an adversary. It is particularly insidious, when
cyber-attacks are generated in such a way that they cannot be detected using
the known detection techniques. Such cyber-attacks are called stealthy. This
observation and some real examples with stealthy cyber-attacks have strongly
motivated researchers to improve the existing detection schemes and develop
alternative solutions. In this regard, a great number of results have been
reported about detecting the so-called replay, zero dynamics and covert
attacks, which are stealthy integrity attacks as the standard observer-based
fault detection technique cannot detect them without modifications on the
applied algorithms \cite{DIBAJI2019-survey,GWSOM2019,TGXHV2020}.
Representative solutions are the watermark detection scheme \cite%
{Mo2015-Watermarked-detection}, the moving target method \cite%
{MT-method-CDC2015} and the auxiliary system aided detection scheme \cite%
{Zhang-CDC2017}, just citing the initial works on these methods. Our work is motivated
by the above observation and in particular driven by the questions like: what
is the general form of stealthy integrity attacks? what are the existence
conditions for such stealthy integrity attacks? is it possible to develop a
general observer-based scheme applied to detecting integrity attacks in
automatic control systems? Satisfactory answers to these questions could
help us (i) to reveal possible weakness of observer-based detection
technique by dealing with integrity cyber-attacks, and thus (ii) to prevent
new variations of stealthy integrity attacks, and (iii) to develop new
detection schemes, in particular such ones that are able to detect major
types of integrity attacks. The main objective of our work is to investigate
possible answers to the above questions. Different from the reported
studies, our work will study the issues of stealthy integrity attacks in the
unified framework of control and detection.
\bigskip
Inspired by the work in \cite{ZR2001}, and based on the parameterisations of
observers and observer-based residual generators \cite{Ding2013}, Ding et
al., 2010 \nocite{DYZDJWS2009}proposed an observer-based realisation and
implementation of all stabilising (dynamic output) controllers whose core is
an observer-based residual generator, and demonstrated its successful
applications. In the recent decade, on the basis of this work, a new unified
framework of control and detection has been established, which generalises
the integrated design schemes for control and detection initiated by Nett et
al. (1988) \nocite{NJM88} and further developed in the past decades \cite%
{KRNS96,SGN97,KNS04,HenryAUTO05,WY-LPV-08}. It has been applied to fault
diagnosis in automatic control systems with uncertainties, fault-tolerant
control and, more recently, to control performance degradation monitoring,
detection and recovery \cite{Ding2020}. The basic idea behind the control
and detection unified framework is that any controller is indeed
residual-driven and can be implemented in form of an observer and an
observer-based residual generator. This allows to extend the residual-based
detection space to the overall measurement space spanned by the system
inputs and outputs. As a result, it can be expected that the system
capability for detecting cyber-attacks is (considerably) enhanced.
\bigskip
The intended contributions of our work are summarised as
\begin{itemize}
\item revealing that any attacks lying in the system kernel space cannot be
detected by an observer-based detection system. In this context, the concept
of kernel attacks is introduced, which provides us with a general expression
of all stealthy integrity attacks (with respect to the observer-based
detection technique);
\item presenting existence conditions that integrity attacks are stealthy in
the unified framework of control and detection, and based on them,
\item proposing two schemes for detecting the kernel attacks (thus
including detecting replay, zero dynamics and covert attacks). The first one
is a natural extension of the observer-based detection schemes to a unified
control and detection system, while the second one is dedicated to a
detection scheme with encrypted transmissions of control and monitoring
signals in the feedback control system under consideration. This is helpful
to prevent adversary to gain system knowledge by means of eavesdropping
attacks.
\end{itemize}
\bigskip
The paper is organised as follows. In Section 2, the unified framework of
control and detection is first presented together with the necessary control
theoretical and mathematical preliminaries. It is followed by a short review
of replay, zero dynamics and covert attacks. Section 3 is dedicated to the
study on stealthy integrity attacks and introduction of the concept of
kernel attacks as a general form of stealthy integrity attacks. In Section
4, existence conditions for stealthy integrity attacks are first investigated and
presented. They build the basis for the development of two schemes for
detecting kernel attacks. These two schemes are presented in Sections 4 and
5, respectively. Their capability of detecting the kernel attacks are
illustrated and demonstrated by examples and experimental results in Section
6.
\bigskip
Throughout this paper, standard notations known in linear algebra and
advanced control theory are adopted. In addition, $\mathcal{RH}_{\infty }$
is used to denote the set of all stable systems. In the context of
cyber-attacks, when signal $\xi $ is attacked, it is denoted by $\xi ^{a},$
and the corresponding (injected) attack signal by $a_{\xi },$ i.e. $\xi
^{a}=\xi +a_{\xi }$.
\section{Preliminaries of system models, the unified framework of control
and detection, and stealthy integrity attacks}
As the methodological basis of our work, we first introduce the unified
framework of control and detection. It is followed by a short review of
system descriptions of stealthy integrity cyber-attacks on feedback control
systems.
\subsection{System representations and controller parameterisation}
\subsubsection{System factorisations, observer-based residual generation and
kernel space}
Consider a nominal plant model%
\begin{equation}
y(z)=G_{u}(z)u(z),y(z)\in \mathcal{C}^{m},u(z)\in \mathcal{C}^{p}
\label{eq2-1}
\end{equation}%
with $u$ and $y$ as the plant input and output vectors. It is assumed that $%
G_{u}(z)$ is a proper real-rational matrix and its minimal state space
realisation is given by the following discrete-time linear time invariant
(LTI) system%
\begin{align}
x(k+1)& =Ax(k)+Bu(k),x(0)=x_{0}, \label{eq2-2a} \\
y(k)& =Cx(k)+Du(k), \label{eq2-2b}
\end{align}%
where $x\in \mathcal{R}^{n}$ is the state vector and $x_{0}$ is the initial
condition of the system. Matrices $A,B,C,D$ are appropriately dimensioned
real constant matrices. A coprime factorisation of a transfer function
matrix over $\mathcal{RH}_{\infty }$ gives a further system representation
form and factorises the transfer matrix into two stable and coprime transfer
matrices. The left and right coprime factorisations (LCF and RCF) of $%
G_{u}(z)$ are given by
\begin{equation}
G_{u}(z)=\hat{M}^{-1}(z)\hat{N}(z)=N(z)M^{-1}(z), \label{eq2-3}
\end{equation}%
where the state space realisations of the left and right coprime pairs (LCP
and RCP) $\left( \hat{M}(z),\hat{N}(z)\right) $ and $\left( M(z),N(z)\right)
$ are
\begin{align}
\hat{M}(z)& =\left( A-LC,-L,C,I\right) ,\hat{N}(z)=\left(
A-LC,B-LD,C,D\right) , \label{eq2-4a} \\
M(z)& =\left( A+BF,B,F,I\right) ,N(z)=\left( A+BF,B,C+DF,D\right) .
\label{eq2-4b}
\end{align}%
Correspondingly, there exist RCP and LCP $\left( \hat{X}(z),\hat{Y}%
(z)\right) $ and $\left( X(z),Y(z)\right) $ so that the so-called Bezout
identity holds%
\begin{equation}
\left[
\begin{array}{cc}
X(z) & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y(z) \\
-\hat{N}(z) & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{cc}
M(z) & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }-\hat{Y}(z) \\
N(z) & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{X}(z)%
\end{array}%
\right] =\left[
\begin{array}{cc}
I\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } & 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } \\
0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } & I\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }%
\end{array}%
\right] . \label{eq2-5}
\end{equation}%
The state space computation formulas for $\left( \hat{X}(z),\hat{Y}%
(z)\right) $ and $\left( X(z),Y(z)\right) $ are%
\begin{align}
\hat{X}(z)& =\left( A+BF,L,C+DF,I\right) ,\hat{Y}(z)=\left(
A+BF,-L,F,0\right) , \label{eq2-6a} \\
X(z)& =\left( A-LC,-(B-LD),F,I\right) ,Y(z)=\left( A-LC,-L,F,0\right) .
\label{eq2-6b}
\end{align}%
In (\ref{eq2-4a})-(\ref{eq2-6b}), (real) matrices $F$ and $L$ are selected
such that $A+BF$ and $A-LC$ are Schur matrices \cite{Zhou98,Ding2014}.
\bigskip
We now consider an observer-based residual generator
\begin{align}
\hat{x}(k+1)& =A\hat{x}(k)+Bu(k)+L\left( y(k)-\hat{y}(k)\right) ,
\label{eq2-7a} \\
r_{0}(k)& =y(k)-\hat{y}(k),\hat{y}(k)=C\hat{x}(k)+Du(k) \label{eq2-7b}
\end{align}%
with $r_{0}(k)$ being the primary form of a residual vector. It can be
equivalently written as%
\begin{gather}
\hat{x}(k+1)=\left( A-LC\right) \hat{x}(k)+\left( B-LD\right) u(k)+Ly(k),
\notag \\
\Longrightarrow r_{0}(z)=y(z)-\hat{y}(z)=\hat{M}(z)y(z)-\hat{N}(z)u(z).
\label{eq2-7}
\end{gather}%
Note that if there exists no uncertainty in the plant and $x(0)=\hat{x}(0),$
it holds
\begin{equation*}
r_{0}(z)=0\Longrightarrow y(z)=\hat{M}^{-1}(z)\hat{N}(z)u(z),
\end{equation*}%
which illustrates the interpretation of LCF as an observer-based residual
generator. It is well-known that given plant model (\ref{eq2-1}), all LTI
residual generators can be parameterised by%
\begin{equation}
r(z)=R(z)r_{0}(z)=R(z)\left( y(z)-\hat{y}(z)\right) ,R(z)\in \mathcal{RH}%
_{\infty }, \label{eq2-8}
\end{equation}%
where $R(z)$ is the parameterisation transfer function matrix \cite{Ding2013}%
.
\begin{Rem}
Hereafter, we may drop out the domain variable $z$ or $k$ when there is no
risk of confusion.
\end{Rem}
\subsubsection{Parameterisation of stabilising controllers and basics of the
unified control and detection framework}
Consider the feedback control loop
\begin{equation*}
y(z)=G_{u}(z)u(z),u(z)=K(z)y(z)
\end{equation*}%
with the plant model $G_{u}(z)$ and controller $K(z).$ It is a well-known
result that all stabilising controllers can be parameterised by%
\begin{align}
K(z)& =-\left( X(z)-Q(z)\hat{N}(z)\right) ^{-1}\left( Y(z)+Q(z)\hat{M}%
(z)\right) \label{eq2-12a} \\
& =-\left( \hat{Y}(z)+M(z)Q(z)\right) \left( \hat{X}(z)-N(z)Q(z)\right) ^{-1}
\label{eq2-12b}
\end{align}%
with the parameter system $Q(z)\in \mathcal{RH}_{\infty },$ where the four
coprime pairs $\left( \hat{M},\hat{N}\right) ,$ $\left( M,N\right) ,$ $%
\left( \hat{X},\hat{Y}\right) $ and $\left( X,Y\right) $ are given in (\ref%
{eq2-4a})-(\ref{eq2-6b}) and satisfy Bezout identity (\ref{eq2-5}). The
parameterisation expression (\ref{eq2-12a})-(\ref{eq2-12b}) is called Youla
parameterisation \cite{Zhou98}. It follows from (\ref{eq2-4a})-(\ref{eq2-6b}%
) and Bezout identity \cite{DYZDJWS2009,Ding2020} that any (stabilising)
output feedback controller
\begin{equation}
u(z)=K(z)y(z)+v(z) \label{eq2-12c}
\end{equation}%
with $v(z)$ being the reference signal can be equivalently written as%
\begin{align}
\hat{x}(k+1)& =A\hat{x}(k)+Bu(k)+Lr_{0}(k), \label{eq2-13a} \\
u(z)& =F\hat{x}(z)-Q(z)r_{0}(z)+\bar{v}(z), \label{eq2-13b} \\
\bar{v}(z)& =\left( X(z)-Q(z)\hat{N}(z)\right) v(z). \label{eq2-13c}
\end{align}%
In other words, any output feedback controller is an observer-based
controller and driven by the residual signal $r_{0}$.
\bigskip
Recall that the basis of an observer-based fault diagnosis is residual
generation and evaluation \cite{Ding2013}. Thus, (\ref{eq2-13a})-(\ref%
{eq2-13b}) reveal that both diagnosis and control are driven by the residual
signal and can be integratedly realised by sharing a common observer-based
residual generator as the information provider. By means of the observer
parameterisation \cite{Ding2013}, we gain a deeper insight into the
information aspect of a feedback controller that the control signal $u(k)$ in (\ref%
{eq2-13b}) is an estimate for $Fx(k)+\bar{v}(k)$ and satisfies
\begin{equation}
\forall x(0),u(k),\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\lim\limits_{k\rightarrow \infty }\left(
u(k)-Fx(k)-\bar{v}(k)\right) =0, \label{eq2-15}
\end{equation}%
when there exists no uncertainty in the plant. The observer-based
realisation of stabilising feedback controllers (\ref{eq2-13a})-(\ref%
{eq2-13b}) and the estimator interpretation (\ref{eq2-15}) of (any) output
feedback controllers are the basics of the unified control and detection
framework and build the basis for our study on attack detection schemes
presented in the subsequent work.
\subsection{Integrity attacks under consideration}
The system configuration under consideration in\ the first part of our study
is sketched in Figure 1, in which the controller and attack detection system
are networked with the plant (equipped with sensors, actuators and a
computation system like micro-controllers). Via the communication network,
the plant receives the control signal $u(k)$ and sends the sensor signal $%
y(k)$ to the control and monitoring system.
\begin{figure}[h]
\centering\includegraphics[width=11cm,height=5.5cm]{CPS-fig1.png}
\caption{System configuration under consideration}
\end{figure}
\bigskip
Recall that our major attention is paid to integrity cyber-attacks that are
injected into the system I/O interface via the network, cause (considerable)
changes in the system dynamics, but cannot be detected by a standard
observer-based detector. As reviewed in \cite{DIBAJI2019-survey}, such
cyber-attacks include zero dynamics, covert and replay attacks. Below is a
short description of these attack types.
\subsubsection{\textbf{Zero dynamics attacks}}
Roughly speaking, a zero dynamics attack is referred to an attack $a_{u}(k)$
on the actuators, which causes no response at the system output over the
detection time interval $\left[ k_{0},k_{0}+N\right] $ and thus cannot be
detected \cite{TEIXEIRA-zero-attack_2015}. The corresponding attack model is
\begin{align}
x(k+1)& =Ax(k)+B\left( u(k)+a_{u}(k)\right) , \\
y^{a}(k)& =Cx(k)+D\left( u(k)+a_{u}(k)\right)
\end{align}%
with $y^{a}(k)$ satisfying the condition
\begin{equation}
\forall k\in \left[ k_{0},k_{0}+N\right] ,y^{a}(k)=y(k). \label{eq2-11}
\end{equation}%
It is obvious that the existence condition of zero dynamics attacks can be
expressed by means of the LCF of the plant as%
\begin{equation}
\hat{N}\left( z\right) a_{u}(z)=0. \label{eq2-11b}
\end{equation}
\subsubsection{\textbf{Covert attacks}}
Introduced by \cite{Smith2015}, covert attacks\textbf{\ }are modelled by%
\textbf{\ }
\begin{align}
x(k+1)& =Ax(k)+B\left( u(k)+a_{u}(k)\right) , \\
y^{a}(k)& =Cx(k)+D\left( u(k)+a_{u}(k)\right) +a_{y}(k)
\end{align}%
with $a_{u}(k)$ and $a_{y}(k)$ denoting attacks on the actuators and
sensors, respectively, and satisfying
\begin{equation}
a_{y}(z)+G_{u}(z)a_{u}(z)=0\Longrightarrow \forall k\in \left[ k_{0},k_{0}+N%
\right] ,y^{a}(k)=y(k). \label{eq2-17}
\end{equation}%
It is straightforward that the existence condition for covert attacks is%
\begin{equation}
\hat{M}(z)a_{y}(z)+\hat{N}\left( z\right) a_{u}(z)=0. \label{eq2-17a}
\end{equation}
\subsubsection{\textbf{Replay attacks}}
As described in \cite{Mo2015-Watermarked-detection}, replay attacks are
performed on the assumption that the plant under attacks is operating in the
steady state, which yields%
\begin{equation*}
y(k)\approx y(k-i),i=1,\cdots .
\end{equation*}%
Consequently, the attacker can \textquotedblleft replay", e.g. over the time
interval $\left[ k,k+M\right] ,$ the sensor data collected in the past (for
instance, by means of an eavesdropping attack), and simultaneously inject
signals in the actuators. Denote by $y\left( k_{0}+i\right)
,k_{0}+M<k,i=0,1,\cdots ,M,$ the data collected and saved by the attacker.
Replay attacks can be modelled by\textbf{\ }
\begin{align*}
x(j+1)& =Ax(j)+B\left( u(j)+a_{u}(j)\right) ,j\in \left[ k,k+M\right] , \\
y^{a}(j)& =Cx(j)+D\left( u(j)+a_{u}(j)\right) +a_{y}(j), \\
a_{y}(j)& =y(k_{0}+j-k)-\left( Cx(j)+D\left( u(j)+a_{u}(j)\right) \right) .
\end{align*}%
As a result,
\begin{equation}
\forall j\in \left[ k,k+M\right] ,y^{a}(j)=y(j-k+k_{0})\approx
y(j)=Cx(j)+Du(j). \label{eq2-20}
\end{equation}%
Notice that the attack signal $a_{y}(j)$ depends on the plant state vector $%
x(j)$ and is, therefore, not a pure additive attack signal.
\bigskip
In summary, it can be seen that the above three types of attacks have one
thing in common that they do not cause changes in the measurement output and
hence cannot be traced by the output variables. As a result, these attacks
cannot be detected using an observer-based detection scheme. In this
context, they are called stealthy attacks \cite{DIBAJI2019-survey}.
\subsection{Problem formulation}
The goal of our work is to investigate cyber-attacking issues in the unified
control and detection framework. We will first deal with the following three
problems:
\begin{itemize}
\item study on general system structural conditions, under which the
above-mentioned three types of attacks cannot be detected using an
observer-based detector. Based on the achieved results, a general class of
stealthy integrity cyber-attacks, the so-called kernel cyber-attacks, are
then defined;
\item derivation of system structural conditions, under which any integrity
cyber-attacks, as sketched in Figure 1, can be (structurally) uniquely
detected, and based on them,
\item development of an alternative attack detection scheme that ensures a
reliable detection of the integrity cyber-attacks shown in Figure 1.
\end{itemize}
A major reason why an integrity cyber-attack could be performed stealthily
is that the attacker has knowledge of system dynamics. One potential tool to
gain such knowledge is to collect sufficient plant input and output data by
means of eavesdropping attacks, which enable, for instance, the
identification of the plant model and even controller parameters. Under this
consideration, we will, in the further part of our work, propose an
alternative system configuration that leads to an encrypted data
transmission aiming at preventing attackers to gain system knowledge.
\section{Kernel attacks: a general form of stealthy integrity attacks}
In this section, we investigate the existence conditions for stealthy
attacks and generalise the different types of stealthy integrity attacks,
including the three types of integrity attacks introduced in the previous
section, as the so-called kernel attacks. To this end, we consider, in the
sequel, the system configuration sketched in Figure 1.
\subsection{Observer-based attack detection Strategy}
For our purpose, we extend the nominal model (\ref{eq2-1})-(\ref{eq2-2b}) to
the following attack model,
\begin{align}
x\left( k+1\right) & =Ax\left( k\right) +B\left( u(k)+a_{u}(k)\right)
+\omega (k), \label{eq3-1a} \\
y^{a}(k)& =Cx(k)+D\left( u(k)+a_{u}(k)\right) +a_{y}(k)+\nu (k),
\label{eq3-1b}
\end{align}%
where $\omega (k),\nu (k)$ represent the process and measurement noise
vectors, and $a_{y}(k),a_{u}(k)$ denote the attack signals on the actuators
and sensors, respectively. With respect to the system configuration shown in
Figure 1, an observer-based attack detector consists of (i) a residual
generator as given in (\ref{eq2-7a})-(\ref{eq2-7b}) with the generated
residual vector $r_{0}(k),$%
\begin{equation*}
r_{0}(k)=y^{a}(k)-\hat{y}^{a}(k),\hat{y}^{a}(k)=C\hat{x}(k)+Du(k),
\end{equation*}%
(ii) a residual evaluation function
\begin{equation*}
J(k)=J\left( \left\Vert r_{0}(k)\right\Vert \right)
\end{equation*}%
with $\left\Vert r_{0}(k)\right\Vert $ denoting a certain norm of $r_{0}(k),$
and (iii) detection logic described by%
\begin{equation*}
\left\{
\begin{array}{l}
J(k)\leq J_{th}\Longrightarrow \RIfM@\expandafter\text@\else\expandafter\mbox\fi{attack-free,} \\
J(k)>J_{th}\Longrightarrow \RIfM@\expandafter\text@\else\expandafter\mbox\fi{attack is detected,}%
\end{array}%
\right.
\end{equation*}%
where $J_{th}$ is the threshold. In order to achieve an optimal attack
detection, the observer gain matrix $L$, the evaluation function $J(k)$ and
the threshold $J_{th}$ are designed taking into account of the statistic
properties of $\omega (k),\nu (k).$ Suppose that $\omega (k),\nu (k)$ are
uncorrelated with the state and input vectors and satisfy
\begin{gather}
\omega (k)\sim \mathcal{N}\left( 0,\Sigma _{\omega }\right) ,\nu (k)\sim
\mathcal{N}\left( 0,\Sigma _{\nu }\right) ,x\left( 0\right) \sim \mathcal{N}%
\left( 0,\Pi _{0}\right) , \label{eq3-2a} \\
\mathcal{E}\left( \left[
\begin{array}{c}
\omega (i) \\
\nu (i) \\
x\left( 0\right)%
\end{array}%
\right] \left[
\begin{array}{c}
\omega (j) \\
\nu (j) \\
x\left( 0\right)%
\end{array}%
\right] ^{T}\right) =\left[
\begin{array}{cc}
\left[
\begin{array}{cc}
\Sigma _{\omega } & S \\
S^{T} & \Sigma _{\nu }%
\end{array}%
\right] \delta _{ij} & 0 \\
0 & \Pi _{0}%
\end{array}%
\right] ,\delta _{ij}=\left\{
\begin{array}{l}
1,i=j, \\
0,i\neq j%
\end{array}%
\right. \label{eq3-2b}
\end{gather}%
with known matrices $\Sigma _{\omega },\Sigma _{\nu },S.$ In this case, the
observer gain matrix can be determined using the (steady) Kalman filter
algorithm,
\begin{gather}
L_{K}:=L=\left( APC^{T}+S\right) \Sigma _{r}^{-1},P=APA^{T}+\Sigma _{\omega
}-L_{K}\Sigma _{r}L_{K}^{T}, \label{eq3-3a} \\
\Sigma _{r}=CPC^{T}+\Sigma _{\nu }=\mathcal{E}\left(
r_{0}(k)r_{0}^{T}(k)\right) ,\mathcal{E}\left( r_{0}(i)r_{0}^{T}(j)\right)
=\Sigma _{r}\delta _{ij}, \label{eq3-3b}
\end{gather}%
the $\chi ^{2}$ test statistic is used as the evaluation function,
\begin{equation*}
J(k)=r_{0}^{T}(k)\Sigma _{r}^{-1}r_{0}(k)\sim \mathcal{\chi }^{2}\left(
m\right) ,
\end{equation*}%
and finally the threshold $J_{th}$ is determined by means of $\mathcal{\chi }%
_{\alpha }^{2}\left( m\right) $ for a given upper-bound of false alarm rate $%
\alpha $ \cite{Ding2014}.
\subsection{Kernel attacks}
We now study the generalisation of stealthy attacks and their existence
conditions. Corresponding to the above described observer-based attack
detection strategy, we introduce the following definition.
\begin{Def}
\label{Def3-1}Given system model (\ref{eq3-1a})-(\ref{eq3-1b}) with $\omega
(k)=0,\nu (k)=0,$ and observer-based attack detector (\ref{eq2-7a})-(\ref%
{eq2-7b}), an integrity attack is stealthy if
\begin{equation*}
\forall u,r_{0}(z)=y^{a}(z)-\hat{y}^{a}(z)=0.
\end{equation*}
\end{Def}
For our purpose, the following definition of the so-called kernel space is
introduced.
\begin{Def}
Given the plant model (\ref{eq2-1}) and a corresponding LCP $\left( \hat{M}%
(z),\hat{N}(z)\right) ,$ we call the $\mathcal{H}_{2}\times \mathcal{H}_{2}$
subspace $\mathcal{K}_{P}$ defined by
\begin{equation}
\mathcal{K}_{P}=\left\{ \left[
\begin{array}{c}
u \\
y%
\end{array}%
\right] :\left[
\begin{array}{cc}
-\hat{N} & \hat{M}%
\end{array}%
\right] \left[
\begin{array}{c}
u \\
y%
\end{array}%
\right] =0,\left[
\begin{array}{c}
u \\
y%
\end{array}%
\right] \in \mathcal{H}_{2}\hspace{-2pt}\right\} \label{eq2-9}
\end{equation}%
kernel space of the plant.
\end{Def}
It is evident that the kernel space $\mathcal{K}_{P}$ consists of all
(bounded) input and output pairs $(u,y)$ satisfying
\begin{equation*}
\left[
\begin{array}{cc}
-\hat{N}(z) & \hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u(z) \\
y(z)%
\end{array}%
\right] =0.
\end{equation*}%
$\mathcal{K}_{P}$ is a closed subspace in $\mathcal{H}_{2}$ \cite%
{Vinnicombe-book}.
\bigskip
We are now in the position to present the existence condition of stealthy
attacks defined in Definition \ref{Def3-1}.
\begin{Theo}
\label{Theo3-1}Given plant model (\ref{eq3-1a})-(\ref{eq3-1b}) with $\omega
(k)=0,\nu (k)=0,$ and an observer-based attack detector (\ref{eq2-7a})-(\ref%
{eq2-7b}), an integrity attack is stealthy if and only if%
\begin{equation}
\left[
\begin{array}{c}
u(z) \\
y^{a}(z)%
\end{array}%
\right] \in \mathcal{K}_{P}. \label{eq3-11}
\end{equation}
\end{Theo}
\begin{proof}
Without loss of generality, assume that the LCP $\left(\hat{M},\hat{N}\right)$ is
given as described in (\ref{eq2-4a}). Then, it follows from
the well-known
parameterisation of observer-based residual generators \cite%
{Ding2013}
that all observer-based residual generators (attack detectors) of
the form (%
\ref{eq2-7a})-(\ref{eq2-7b}) can be written as%
\begin{equation*}
y^{a}(z)-%
\hat{y}^{a}(z)=R(z)\left[
\begin{array}{cc}
-\hat{N}(z) & \hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u(z) \\
y^{a}(z)%
\end{%
array}%
\right] ,
\end{equation*}%
where $R(z)$ is a stable and \textit{%
invertible} dynamic post-filter.
Consequently, $y^{a}(z)=\hat{y}^{a}(z)$ if
and only if%
\begin{equation*}
\left[
\begin{array}{cc}
-\hat{N}(z) &
\hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u(z) \\
y^{a}(z)%
\end{array}%
\right] =0\Longleftrightarrow \left[
\begin{%
array}{c}
u(z) \\
y^{a}(z)%
\end{array}%
\right] \in \mathcal{K}_{P}.
\end{equation*}%
The theorem is proved.
\end{proof}
\bigskip
In this context, we introduce the definition of kernel attacks, which gives
a general form of integrity attacks that cannot be detected using an
observer detector. As will be demonstrated in the example given below, the
zero dynamics, covert and replay attacks are special forms of the kernel
attacks.
\begin{Def}
Given system model (\ref{eq3-1a})-(\ref{eq3-1b}), an integrity attack is
called kernel attack when condition (\ref{eq3-11}) holds.
\end{Def}
\begin{Exp}
We first check a zero dynamics attack. It is evident that for $\omega
(k)=0,\nu (k)=0,$%
\begin{equation*}
r_{0}(z)=y^{a}(z)-\hat{y}^{a}(z)=\left[
\begin{array}{cc}
-\hat{N}(z) & \hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u(z) \\
y^{a}(z)%
\end{array}%
\right] =-\hat{N}(z)a_{u}(z).
\end{equation*}%
It follows from (\ref{eq2-11b}) that
\begin{equation*}
\hat{N}(z)a_{u}(z)=0\Longrightarrow \left[
\begin{array}{cc}
-\hat{N}(z) & \hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u(z) \\
y^{a}(z)%
\end{array}%
\right] =0,
\end{equation*}%
i.e. the zero dynamics attack is a kernel attack.
\bigskip
Now, consider the residual dynamics under a covert attack, which is given by%
\begin{equation*}
r_{0}(z)=\left[
\begin{array}{cc}
-\hat{N}(z) & \hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u(z) \\
y^{a}(z)%
\end{array}%
\right] =\hat{N}(z)a_{u}(z)+\hat{M}(z)a_{y}(z).
\end{equation*}%
According to (\ref{eq2-17a}), it holds
\begin{equation*}
\hat{N}(z)a_{u}(z)+\hat{M}(z)a_{y}(z)=0\Longrightarrow \left[
\begin{array}{cc}
-\hat{N}(z) & \hat{M}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u(z) \\
y^{a}(z)%
\end{array}%
\right] =0.
\end{equation*}%
Therefore, the covert attack is obviously a kernel attack.
\bigskip
Concerning the residual dynamics under a replay attack, recall the relation (%
\ref{eq2-20}). It turns out, for $j\in \left[ k,k+M\right] ,$
\begin{align}
r_{0}(j)& =y^{a}(j)-\hat{y}^{a}(j)\approx Cx(j)+Du(j)-\left( C\hat{x}%
(j)+Du(j)\right) =0 \\
& \Longrightarrow \left[
\begin{array}{c}
u \\
y^{a}%
\end{array}%
\right] \in \mathcal{K}_{P}.
\end{align}%
Thus, the replay attack is a kernel attack.
\end{Exp}
Given the plant dynamics described by (\ref{eq3-1a})-(\ref{eq3-1b}) with $%
\omega (k)=0,\nu (k)=0,$ the dynamics of the observer-based attack detector (%
\ref{eq2-7a})-(\ref{eq2-7b}) is described by%
\begin{equation*}
r_{0}(z)=y^{a}(z)-\hat{y}^{a}(z)=\hat{M}(z)a_{y}(z)+\hat{N}(z)a_{u}(z).
\end{equation*}%
Consequently, if the attack pair $\left( a_{u},a_{y}\right) $ is constructed
satisfying
\begin{equation}
\hat{M}(z)a_{y}(z)+\hat{N}(z)a_{u}(z)=0, \label{eq3-13}
\end{equation}%
it cannot be detected. Hence, we have the following theorem.
\begin{Theo}
Given the plant model (\ref{eq3-1a})-(\ref{eq3-1b}) and an observer-based
attack detector (\ref{eq2-7a})-(\ref{eq2-7b}), the pair $\left(
a_{u},a_{y}\right) $ builds a kernel attack if it satisfies (\ref{eq3-13}).
\end{Theo}
\begin{Rem}
We would like to point out that a replay attack does not satisfy (\ref{eq3-13}),
since it is, in fact, not an additive type of attacks, as remarked in the
previous section.
\end{Rem}
\bigskip
Recall that the kernel space $\mathcal{K}_{P}$ is a structural property of
the plant and determined by the dynamics of the nominal plant. As a result,
if an attacker is in possession of knowledge of the plant dynamics, kernel
attacks could be constructed according to (\ref{eq3-13}) and injected in the
plant without being detected by the observer-based attack detector (\ref%
{eq2-7a})-(\ref{eq2-7b}). In fact, recall that any LTI observer-based
residual generators, including the parity relation based one and diagnosis
observer, can be parameterised by \cite{Ding2013}
\begin{equation}
r(z)=R(z)r_{0}(z)=R(z)\left( y(z)-\hat{y}(z)\right) ,R(z)\neq 0,R(z)\in
\mathcal{RH}_{\infty }. \label{eq3-30}
\end{equation}%
The following corollary is obvious.
\begin{Corol}
\label{Coro3-1}Given plant model (\ref{eq3-1a})-(\ref{eq3-1b}), any attack
cannot be detected by an LTI attack detector of the form (\ref{eq3-30}), if
and only if the signal pair $\left( u,y^{a}\right) $ satisfies (\ref{eq3-11}%
), or if the attack signal pair $\left( a_{u},a_{y}\right) $ is constructed
satisfying (\ref{eq3-13}).
\end{Corol}
At the end of this section, we would like to emphasise that the concept of
the kernel attacks and the associated existence conditions given in Theorems
1 - 2 and Corollary \ref{Coro3-1} are described in terms of the LCF or
kernel space of the system under consideration. They are the system
structural properties and independent of the observer design and the
evaluation schemes adopted by the attack-detector. Our subsequent
investigation on detecting stealthy integrity attacks will be carried out in
this context.
\section{Analysis and detection of kernel attacks}
This section deals with detecting kernel attacks on the feedback control
systems shown in Figure 2. Consider that the observer-based attack detector (%
\ref{eq2-7a})-(\ref{eq2-7b}) performs the online detection by means of the
(online) data $(u(k),y^{a}(k)),$ whose dimension is $p+m.$ On the other
hand, the parameterisation of observer-based residual generators and Theorem %
\ref{Theo3-1} as well as Corollary \ref{Coro3-1} reveal that the residual
signals that belong to the $m$-dimensional kernel space $\mathcal{K}_{P}$
are only effective in detecting attacks which do not belong to $\mathcal{K}%
_{P}.$ In other words, in order to detect kernel attacks successfully,
generating additional signals to cover the overall $\left( p+m\right) $%
-dimensional data space is an effective alternative solution. In fact, the
so-called moving target or auxiliary system schemes reported, for instance,
in \cite%
{MT-method-CDC2015,MT-method-IEEE-TAC2020,Zhang-CDC2017,DIBAJI2019-survey}
for detecting zero dynamics and covert attacks, are special realisations of
this idea.
\begin{figure}[h]
\centering\includegraphics[width=7cm,height=3.5cm]{CPS-fig2.png}
\caption{Schematic description of the control loop under attack}
\end{figure}
\subsection{Analysis of closed-loop dynamics under kernel attacks}
Consider the feedback control loop with the plant model (\ref{eq3-1a})-(\ref%
{eq3-1b}) and controller described by (\ref{eq2-12c}), where $K(z)$
satisfies (\ref{eq2-12a})-(\ref{eq2-12b}). For the sake of the structural
analysis, $\omega (k)$ and $\nu (k)$ are assumed to be zero. It yields,
under attacks,%
\begin{align*}
y^{a}(z)& =G_{u}(z)u^{a}(z)+a_{y}(z), \\
u^{a}(z)& =K(z)y^{a}(z)+v(z)+a_{u}(z),
\end{align*}%
and the control loop configuration can be equivalently sketched by Figure 2.
It turns out
\begin{gather}
\left[
\begin{array}{c}
u^{a}(z) \\
y^{a}(z)%
\end{array}%
\right] =\left[
\begin{array}{cc}
I & -K(z) \\
-G_{u}(z) & I%
\end{array}%
\right] ^{-1}\left[
\begin{array}{c}
a_{u}(z)+v(z) \\
a_{y}(z)%
\end{array}%
\right] \label{eq4-1a} \\
=\left[
\begin{array}{cc}
\hat{V}(z) & -\hat{U}(z) \\
-\hat{N}(z) & \hat{M}(z)%
\end{array}%
\right] ^{-1}\left[
\begin{array}{c}
\hat{V}(z)\left( a_{u}(z)+v(z)\right) \\
\hat{M}(z)a_{y}(z)%
\end{array}%
\right] , \label{eq4-1b}
\end{gather}%
where
\begin{equation*}
\hat{V}=X-Q\hat{N}\in \mathcal{RH}_{\infty },\hat{U}=-Y-Q\hat{M}\in \mathcal{%
RH}_{\infty },
\end{equation*}%
and $\hat{V},\hat{M}$ are invertible. Recall the Bezout identity (\ref{eq2-5}%
) and extend it to%
\begin{gather}
\left[
\begin{array}{cc}
X-Q\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y+Q\hat{M} \\
-\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{M}%
\end{array}%
\right] \left[
\begin{array}{cc}
M & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }-\hat{Y}-MQ \\
N & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{X}-NQ%
\end{array}%
\right] =\left[
\begin{array}{cc}
I\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } & 0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } \\
0\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } & I\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }%
\end{array}%
\right] \Longleftrightarrow \label{eq2-5a} \\
\left[
\begin{array}{cc}
X-Q\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y+Q\hat{M} \\
-\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{M}%
\end{array}%
\right] ^{-1}=\left[
\begin{array}{cc}
M & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }-\hat{Y}-MQ \\
N & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{X}-NQ%
\end{array}%
\right] \in \mathcal{RH}_{\infty }. \label{eq2-5b}
\end{gather}%
As a result, we have
\begin{Theo}
\label{Theo4-1}Given the plant model (\ref{eq3-1a})-(\ref{eq3-1b}) and
controller (\ref{eq2-12c}) with $K(z)$ satisfying (\ref{eq2-12a})-(\ref%
{eq2-12b}), it holds%
\begin{gather}
\left[
\begin{array}{c}
\bar{a}_{u} \\
\bar{a}_{y}%
\end{array}%
\right] =\left[
\begin{array}{cc}
X-Q\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y+Q\hat{M} \\
-\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{M}%
\end{array}%
\right] \left[
\begin{array}{c}
u^{a} \\
y^{a}%
\end{array}%
\right] -\left[
\begin{array}{c}
\bar{v} \\
0%
\end{array}%
\right] ,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ } \label{eq3-12} \\
\bar{v}=\hat{V}v,\bar{a}_{u}=\hat{V}a_{u},\bar{a}_{y}=\hat{M}a_{y}. \notag
\end{gather}
\end{Theo}
It follows immediately from Theorem \ref{Theo4-1} that, using signals $%
y^{a}(k),u^{a}(k)$ and $v(k),$
\begin{itemize}
\item the attack signals $a_{y}(k),a_{u}(k)$ could be structurally detected
in the sense that
\begin{gather*}
\left[
\begin{array}{c}
a_{u}(z) \\
a_{y}(z)%
\end{array}%
\right] \neq 0\Longleftrightarrow \left[
\begin{array}{c}
\bar{a}_{u}(z) \\
\bar{a}_{y}(z)%
\end{array}%
\right] \neq 0\Longleftrightarrow \\
\left[
\begin{array}{cc}
X-Q\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y+Q\hat{M} \\
-\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{M}%
\end{array}%
\right] \left[
\begin{array}{c}
u^{a} \\
y^{a}%
\end{array}%
\right] -\left[
\begin{array}{c}
\bar{v} \\
0%
\end{array}%
\right] \neq 0,\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ and}
\end{gather*}
\item if $\hat{V}^{-1}\in \mathcal{RH}_{\infty },\hat{M}^{-1}\in \mathcal{RH}%
_{\infty },$ i.e. both the plant and controller are stable, the attack pair $%
\left( a_{y},a_{u}\right) $ could also be (structurally) uniquely identified
according to%
\begin{equation*}
\left[
\begin{array}{c}
a_{u} \\
a_{y}%
\end{array}%
\right] =\left[
\begin{array}{cc}
I & -K \\
-G_{u} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }I%
\end{array}%
\right] \left[
\begin{array}{c}
u^{a} \\
y^{a}%
\end{array}%
\right] +\left[
\begin{array}{c}
-v \\
0%
\end{array}%
\right] ,
\end{equation*}%
except that the transfer matrix
\begin{equation}
\left[
\begin{array}{cc}
M & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }-\hat{Y}-MQ \\
N & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{X}-NQ%
\end{array}%
\right] \label{eq3-14}
\end{equation}%
has a transmission zero at $z=z_{0},$ i.e.
\begin{equation}
rank\left[
\begin{array}{cc}
M(z_{0}) & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }-\hat{Y}(z_{0})-M(z_{0})Q(z_{0}) \\
N(z_{0}) & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{X}(z_{0})-N(z_{0})Q(z_{0})%
\end{array}%
\right] <m+p, \label{eq3-13a}
\end{equation}%
and
\begin{equation}
\left[
\begin{array}{c}
a_{u}(z) \\
a_{y}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
a_{u}(z_{0}) \\
a_{y}(z_{0})%
\end{array}%
\right] . \label{eq3-13b}
\end{equation}%
Considering that transmission zeros of transfer matrix (\ref{eq3-14}) are
structural properties of the plant and the controller, and whose number is
limited, we will not address this class of possible attacks whose
realisation requires not only full knowledge of the plant and controller,
but also very special forms of attack signals.
\end{itemize}
\bigskip
It is worth noting that, according to the relations given in (\ref{eq2-5a})-(%
\ref{eq2-5b}), attacks $a_{y},a_{u}$ can also be (structurally) detected
using the relation%
\begin{gather}
\left[
\begin{array}{c}
u^{a} \\
y^{a}%
\end{array}%
\right] -\left[
\begin{array}{c}
M \\
N%
\end{array}%
\right] \bar{v}=\left[
\begin{array}{cc}
M & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }-\hat{Y}-MQ \\
N & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{X}-NQ%
\end{array}%
\right] \left[
\begin{array}{c}
\bar{a}_{u} \\
\bar{a}_{y}%
\end{array}%
\right] \label{eq3-16} \\
=\left[
\begin{array}{cc}
I & -\hat{Y}-MQ \\
0 & \hat{X}-NQ%
\end{array}%
\right] \left[
\begin{array}{c}
a_{u} \\
\hat{N}a_{u}+\hat{M}a_{y}%
\end{array}%
\right] . \notag
\end{gather}%
Before we continue our study on applying signals $v(k),y^{a}(k)$ and $%
u^{a}(k)$ for attack detection, we would like to discuss about relations (%
\ref{eq3-12}) and (\ref{eq3-16}), which is helpful to gain a deep insight
into our solutions and two different implementation forms of attack
detectors. To this end, we first check the transfer function matrix
\begin{equation*}
\left[
\begin{array}{cc}
X-Q\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y+Q\hat{M} \\
-\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{M}%
\end{array}%
\right] =\left[
\begin{array}{cc}
\hat{V} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }-\hat{U} \\
-\hat{N} & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\hat{M}%
\end{array}%
\right]
\end{equation*}%
on the right-hand side of (\ref{eq3-12}). While the LCP $\left( \hat{M},\hat{%
N}\right) $ builds the kernel space of the plant, the pair $\left( \hat{V},%
\hat{U}\right) $ is left coprime and spans the kernel space of the
controller (\ref{eq2-12c}). In other words, the signal $r_{u}(z)$ defined by%
\begin{equation*}
u(z)-v(z)=K(z)y(z)\Longrightarrow \hat{V}(z)\left( u(z)-v(z)\right) -\hat{U}%
(z)y(z)=:r_{u}(z)
\end{equation*}%
can be viewed as a residual vector generated based on the controller
configuration. Since
\begin{equation*}
\dim \left[
\begin{array}{c}
r_{0} \\
r_{u}%
\end{array}%
\right] =m+p,
\end{equation*}%
any changes (caused, for instance, by attacks) in the space spanned by the
plant input and output vectors, $\left( u,y\right) ,$ can be (structurally)
uniquely detected. It is of interest to notice that the residual vector $%
r_{u}(k)$ can be generated as well using an observer of the form%
\begin{align}
\hat{x}_{u}(k+1)& =\left( A-LC\right) \hat{x}_{u}(k)+(B-LD)\left(
u(k)-v(k)\right) +Ly(k), \label{eq3-22a} \\
\left[
\begin{array}{c}
r_{u,1}(k) \\
r_{u,2}(k)%
\end{array}%
\right] & =\left[
\begin{array}{c}
u(k)-v(k)-F\hat{x}_{u}(k) \\
y(k)-D\left( u(k)-v(k)\right) -C\hat{x}_{u}(k)%
\end{array}%
\right] , \\
r_{u}(z)& =r_{u,1}(z)+Q(z)r_{u,2}(z). \label{eq3-22b}
\end{align}%
We now consider the left-hand side of (\ref{eq3-16}). In the attack-free
case, it is indeed a residual generator based on the closed-loop dynamics,
\begin{equation}
\left[
\begin{array}{c}
r_{u,c}(z) \\
r_{y,c}(z)%
\end{array}%
\right] :=\left[
\begin{array}{c}
u^{a}(z) \\
y^{a}(z)%
\end{array}%
\right] -\left[
\begin{array}{c}
M(z) \\
N(z)%
\end{array}%
\right] \bar{v}(z), \label{eq3-23}
\end{equation}%
whose state space representation is given by%
\begin{align*}
\hat{x}_{v}(k+1)& =\left( A-LC\right) \hat{x}_{v}(k)+(B-LD)v(k), \\
\bar{v}(k)& =v(k)-F\hat{x}_{v}(k)-q(k),q(z)=Q(z)\left( C\hat{x}%
_{v}(z)+Dv(z)\right) , \\
\hat{x}_{c}(k+1)& =\left( A+BF\right) \hat{x}_{c}(k)+B\bar{v}(k), \\
\left[
\begin{array}{c}
r_{u,c}(k) \\
r_{y,c}(k)%
\end{array}%
\right] & =\left[
\begin{array}{c}
u^{a}(k) \\
y^{a}(k)%
\end{array}%
\right] -\left[
\begin{array}{c}
F \\
C+DF%
\end{array}%
\right] \hat{x}_{c}(k)-\left[
\begin{array}{c}
I \\
D%
\end{array}%
\right] \bar{v}(k).
\end{align*}%
Hence, using the closed-loop dynamics based residual vectors, $r_{u,c}$ and $%
r_{y,c},$ with
\begin{equation*}
\dim \left[
\begin{array}{c}
r_{u,c} \\
r_{y,c}%
\end{array}%
\right] =m+p,
\end{equation*}%
it is possible to detect attacks uniquely as well.
\bigskip
In summary, in order to detect all kernel attacks uniquely, we can use the
(online) data $v(k),y^{a}(k)$ and $u^{a}(k)$ to generate either the
observer-based residuals $r_{0}$ and $r_{u}$ or the closed-loop dynamics
based residuals $r_{u,c}$ and $r_{y,c}.$ Unfortunately, $u^{a}(k)$ is only
available on the side of the plant, as shown in Figures 1 and 2. \ This
motivates us to propose a detection scheme described in the next sub-section.
\subsection{A conceptual scheme for detecting kernel attacks}
For the realisation of the detection solution based on (\ref{eq3-12}) given
in Theorem \ref{Theo4-1}, we propose the following conceptual detection
scheme.
\bigskip
It follows from the relation
\begin{equation*}
\left( X(z)-Q(z)\hat{N}(z)\right) \left( u(z)-v(z)\right) =-\left( Y(z)+Q(z)%
\hat{M}(z)\right) y(z)
\end{equation*}%
in attack-free case that signal%
\begin{equation}
r_{u,0}(z):=X(z)u(z)+Y(z)y(z)-\left( X(z)-Q(z)\hat{N}(z)\right) v(z)
\label{eq4-10}
\end{equation}%
builds a residual signal satisfying
\begin{equation*}
r_{u,0}(z)=r_{u}(z)+Q(z)r_{0}(z).
\end{equation*}%
Recall that $v$ is available at the monitoring side, while the signals $y,u$
exist on the plant side with $u$ being corrupted by $a_{u}.$ For our
purpose, we propose to generate the residual signal $r_{u,0}$ using the
following algorithm:
\begin{itemize}
\item compute
\begin{equation}
r_{en}(z):=X(z)u^{a}(z)+Y(z)y(z); \label{eq4-9}
\end{equation}
\item transmit $r_{en}(k)$ to the monitoring and control side;
\item compute, on the monitoring and control side,%
\begin{equation}
r_{u,0}(z)=r_{en}^{a}(z)-\left( X(z)-Q(z)\hat{N}(z)\right) v(z).
\label{eq4-9a}
\end{equation}%
Here, it is supposed that $r_{en}(z)$ is attacked by the attack signal $%
a_{r_{en}}(k)$, i.e.%
\begin{equation*}
r_{en}^{a}(k)=r_{en}(k)+a_{r_{en}}(k).
\end{equation*}
\end{itemize}
Figure 3 shows the corresponding system configuration.
\begin{figure}[h]
\centering\includegraphics[width=11cm,height=6cm]{CPS-fig3.png}
\caption{Schematic description of kernel attack detection scheme}
\end{figure}
\bigskip
Next, we check the dynamics of $r_{u,0},r_{0}$ without considering noises.
Recall that%
\begin{gather*}
u(z)=K(z)y^{a}(z)+v(z)\Longleftrightarrow \\
\left( X(z)-Q(z)\hat{N}(z)\right) \left( u(z)-v(z)\right) =-\left( Y(z)+Q(z)%
\hat{M}(z)\right) y^{a}(z).
\end{gather*}%
It yields
\begin{gather}
\left[
\begin{array}{c}
r_{u,0} \\
r_{0}%
\end{array}%
\right] =\left[
\begin{array}{ccc}
0 & 0 & I \\
-\hat{N} & \hat{M} & 0%
\end{array}%
\right] \left[
\begin{array}{c}
u \\
y^{a} \\
r_{en}^{a}%
\end{array}%
\right] -\left[
\begin{array}{c}
\bar{v} \\
0%
\end{array}%
\right] \notag \\
=\left[
\begin{array}{ccc}
X-Q\hat{N} & -Y-Q\hat{M} & I \\
\hat{N} & \hat{M} & 0%
\end{array}%
\right] \left[
\begin{array}{c}
a_{u} \\
a_{y} \\
a_{r_{en}}%
\end{array}%
\right] . \label{eq4-6}
\end{gather}%
Consequently,
\begin{equation*}
\left[
\begin{array}{c}
r_{u,0}(z) \\
r_{0}(z)%
\end{array}%
\right] =0
\end{equation*}%
if and only if $a_{r_{en}},a_{u},a_{y}$ solve
\begin{gather}
\hat{N}(z)a_{u}(z)+\hat{M}(z)a_{y}(z)=0, \label{eq4-7} \\
a_{r_{en}}(z)=Y(z)a_{y}(z)-X(z)a_{u}(z). \label{eq4-8}
\end{gather}%
We summary the above results in the following theorem.
\begin{Theo}
\label{Theo4-2}Given the plant model (\ref{eq3-1a})-(\ref{eq3-1b}), the
controller%
\begin{equation*}
u(z)=K(z)y^{a}(z)+v(z),
\end{equation*}%
with $K(z)$ satisfying (\ref{eq2-12a})-(\ref{eq2-12b}) and the system
configuration shown in Figure 3, where $a_{r_{en}},$ $a_{u},$ $a_{y}$ are
the attack signals, $r_{en}^{a},y^{a},u,v$ are the system signals being
available at the monitoring side and used for the attack detection purpose,
then attacks $\left( a_{r_{en}},a_{u},a_{y}\right) $ are stealthy, i.e. they
cannot be detected using the available system signals, if and only if the
conditions (\ref{eq4-7})-(\ref{eq4-8}) are satisfied.
\end{Theo}
\subsection{\textbf{Design and construction of }residual generator $r_{u,0}$%
\label{Sec4-3}}
It follows from Theorem \ref{Theo4-2} that an attacker could design attack
signals $a_{r_{en}},a_{u},$ $a_{y}$ so that conditions (\ref{eq4-7})-(\ref%
{eq4-8}) are satisfied when the attacker is in possession of knowledge of
the plant dynamics (regarding to (\ref{eq4-7})) and the construction of
system (\ref{eq4-9}) (regarding to (\ref{eq4-8})). This demands that
knowledge of the residual generator (\ref{eq4-9}) and (\ref{eq4-10}) should
be protected from the attacker so that the attacker could not be able to
construct the attack signal $a_{r_{en}}$ according to (\ref{eq4-8}). To this
end, encryption of the concerning system dynamics is the major task of
designing and constructing the residual generator $r_{u,0}$ described by (%
\ref{eq4-9}) and (\ref{eq4-10}). This will be realised in two steps.
\bigskip
At first, the residual generator (\ref{eq4-10}) is constructed at two
different sides of the networked control system, as shown in (\ref{eq4-9})
and (\ref{eq4-9a}). In a certain sense, the dynamic system
\begin{equation*}
r_{en}(z)=\left[
\begin{array}{cc}
X(z) & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y(z)%
\end{array}%
\right] \left[
\begin{array}{c}
u^{a}(z) \\
y(z)%
\end{array}%
\right]
\end{equation*}%
can be interpreted as an encoding algorithm and thus is called encoder. The
residual signal $r_{u,0}$ is then generated by a decoding algorithm in the
form
\begin{equation*}
r_{u,0}(z)=r_{en}(z)-\left( X(z)-Q(z)\hat{N}(z)\right) v(z).
\end{equation*}%
Note that even if knowledge of the (encoding) system (\ref{eq4-9}) is
protected, the attacker could identify the system dynamics using possibly
eavesdropped signals $r_{en},u^{a}$ and $y.$ In order to protect the system
dynamics from being identified, the involved system $(X, Y)$ is further
encrypted in the next step.
\bigskip
Recall that the state space form of the encoder (\ref{eq4-9}) is given by
\begin{align}
\varsigma (k+1)& =\left( A-LC\right) \varsigma (k)+Ly(k)+(B-LD)u^{a}(k), \\
r_{en}(k)& =u^{a}(k)-F\varsigma (k).
\end{align}%
Moreover, the following lemma can be proved.
\begin{Le}
\label{Le4-1}Given $\left( \hat{M}_{i},\hat{N}_{i}\right) ,\left(
X_{i},Y_{i}\right) ,i=1,2,$ subject to
\begin{align*}
\hat{M}_{i}& =\left( A-L_{i}C,-L_{i},C,I\right) ,\hat{N}_{i}=\left(
A-L_{i}C,B-L_{i}D,C,D\right) , \\
X_{i}& =\left( A-L_{i}C,-(B-L_{i}D),F_{i},I\right) ,Y_{i}=\left(
A-L_{i}C,-L_{i},F_{i},0\right) ,
\end{align*}%
it holds%
\begin{align}
\left[
\begin{array}{cc}
X_{1}(z) & Y_{1}(z)%
\end{array}%
\right] & =R_{12}(z)\left[
\begin{array}{cc}
X_{2}(z) & Y_{2}(z)%
\end{array}%
\right] +\bar{Q}_{11}(z)\left[
\begin{array}{cc}
-\hat{N}_{1}(z) & \hat{M}_{1}(z)%
\end{array}%
\right] \label{eq4-20} \\
& =R_{12}(z)\left[
\begin{array}{cc}
X_{2}(z) & Y_{2}(z)%
\end{array}%
\right] +\bar{Q}_{12}(z)\left[
\begin{array}{cc}
-\hat{N}_{2}(z) & \hat{M}_{2}(z)%
\end{array}%
\right] , \label{eq4-20a} \\
R_{12}(z)& =R_{21}^{-1}(z)=I+\left( F_{2}-F_{1}\right) \left(
zI-A_{F_{2}}\right) ^{-1}B\in \mathcal{RH}_{\infty },A_{F_{i}}=A+BF_{i},
\notag \\
\bar{Q}_{11}(z)& =F_{1}\left( zI-A_{L_{2}}\right) ^{-1}\left(
L_{2}-L_{1}\right) -\bar{R}_{12}(z)Q_{21}(z)\in \mathcal{RH}_{\infty
},A_{L_{i}}=A-L_{i}C, \notag \\
\bar{Q}_{12}(z)& =F_{1}\left( zI-A_{L_{1}}\right) ^{-1}\left(
L_{2}-L_{1}\right) -\left( F_{1}-F_{2}\right) \left( zI-A_{F_{2}}\right)
^{-1}L_{2}\in \mathcal{RH}_{\infty }, \notag \\
\bar{R}_{12}(z)& =\left( F_{1}-F_{2}\right) \left( zI-A_{F_{2}}\right)
^{-1}L_{2}\in \mathcal{RH}_{\infty }, \\
Q_{21}(z)& =Q_{12}^{-1}(z)=I+C\left( zI-A_{L_{2}}\right) ^{-1}\left(
L_{1}-L_{2}\right) \in \mathcal{RH}_{\infty }.
\end{align}
\end{Le}
The proof is given in Appendix.
\bigskip
Lemma \ref{Le4-1} reveals that varying the gain matrices $F_{2}$ and $L_{2}$
in $\left( X,Y\right) $ to $F_{1}$ and $L_{1}$ is equivalent to adding (i) a
(stable and invertible) post-filter to $r_{en}(z)$ and (ii) additional
residual signal $r_{0}$. On the basis of this result, we propose to switch $%
F $ and $L,$ denoted by $F_{\sigma }$ and $L_{\sigma },$ among a set of
values as follows:%
\begin{align*}
F_{\sigma }& \in \mathcal{F}:=\left\{ F_{i}\in \mathcal{R}^{p\times
n},A+BF_{i}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ is Schur},i\in \mathcal{I}\right\} ,\mathcal{I}=\left\{
1,\ldots ,\kappa \right\} , \\
L_{\sigma }& \in \mathcal{L}:=\left\{ L_{i}\in \mathcal{R}^{n\times
m},A-L_{i}C\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ is Schur},i\in \mathcal{I}\right\} ,
\end{align*}%
where $\sigma \in \mathcal{I}$ is the switching law that is to be protected
so that it is unknown for the attacker. Let $F_{0}$ and $L_{0}$ denote the
gain matrices $F$ and $L$ adopted in the control law (\ref{eq2-12a})-(\ref%
{eq2-12b}), and
\begin{align*}
r_{0,\sigma }(z)& =\hat{M}_{\sigma }(z)y(z)-\hat{N}_{\sigma
}(z)u^{a}(z),\sigma \in \mathcal{I}, \\
r_{en,\sigma }(z)& =X_{\sigma }(z)u^{a}(z)+Y_{\sigma }(z)y(z),
\end{align*}%
where%
\begin{align*}
\hat{M}_{\sigma }& =\left( A-L_{\sigma }C,-L_{\sigma },C,I\right) ,\hat{N}%
_{\sigma }=\left( A-L_{\sigma }C,B-L_{\sigma }D,C,D\right) , \\
X_{\sigma }& =\left( A-L_{\sigma }C,-(B-L_{\sigma }D),F_{\sigma },I\right)
,Y_{\sigma }=\left( A-L_{\sigma }C,-L_{\sigma },F_{\sigma },0\right) .
\end{align*}%
Then, we have
\begin{Theo}
\label{Theo4-3}Given the plant model (\ref{eq3-1a})-(\ref{eq3-1b}), the
control law $K(z)$ satisfying (\ref{eq2-12a})-(\ref{eq2-12b}) and the system
configuration shown in Figure 3, it holds%
\begin{align}
r_{0,\sigma }(z)& =P_{0,\sigma }(z)r_{0,p}(z),r_{0,p}(z)=\hat{M}_{0}(z)y(z)-%
\hat{N}_{0}(z)u^{a}(z), \\
P_{0,\sigma }(z)& =I+C\left( zI-A_{L_{\sigma }}\right) ^{-1}\left(
L_{0}-L_{\sigma }\right) \in \mathcal{RH}_{\infty },A_{L_{\sigma
}}=A-L_{\sigma }C, \\
r_{en,\sigma }(z)& =P_{u,\sigma }(z)r_{en,0}(z)+Q_{\sigma
}(z)r_{0,p}(z),r_{en,0}(z)=X_{0}(z)u^{a}(z)+Y_{0}(z)y(z), \\
P_{u,\sigma }(z)& =I+\left( F_{0}-F_{\sigma }\right) \left(
zI-A_{F_{0}}\right) ^{-1}B\in \mathcal{RH}_{\infty },A_{F_{0}}=A+BF_{0}, \\
Q_{\sigma }(z)& =F_{\sigma }\left( zI-A_{L_{\sigma }}\right) ^{-1}\left(
L_{0}-L_{\sigma }\right) -\left( F_{\sigma }-F_{0}\right) \left(
zI-A_{F_{0}}\right) ^{-1}L_{0}\in \mathcal{RH}_{\infty },
\end{align}%
where%
\begin{align*}
\hat{M}_{0}& =\left( A-L_{0}C,-L_{0},C,I\right) ,\hat{N}_{0}=\left(
A-L_{0}C,B-L_{0}D,C,D\right) , \\
X_{0}& =\left( A-L_{0}C,-(B-L_{0}D),F_{0},I\right) ,Y_{0}=\left(
A-L_{0}C,-L_{0},F_{0},0\right) .
\end{align*}
\end{Theo}
The proof of this theorem follows immediately from Lemma \ref{Le4-1}.
\begin{Rem}
Although in the attack-free case
\begin{equation*}
r_{0}(z)=r_{0,p}(z)=\hat{M}_{0}(z)y(z)-\hat{N}_{0}(z)u(z)=r_{0,n}(z),
\end{equation*}%
we would like to call the reader's attention to the differences between the
residual signals $r_{0,p}$ and $r_{0}.$ While $r_{0}$ is realised on the
monitoring and control side, $r_{0,p}$ is generated on the plant side. Here, $r_{0,n}$ denotes the influence of the noises on the residual vector.
Moreover, in case of attacks,
\begin{align*}
r_{0,p}(z)& =\hat{M}_{0}(z)y(z)-\hat{N}_{0}(z)u^{a}(z)+r_{0,n}(z)=r_{0,n}(z),
\\
r_{0}(z)& =\hat{M}_{0}(z)y^{a}(z)-\hat{N}_{0}(z)u(z)=\hat{N}_{0}(z)a_{u}(z)+%
\hat{M}_{0}(z)a_{y}(z)+r_{0,n}(z).
\end{align*}%
\end{Rem}
Theorem \ref{Theo4-3} demonstrates that switching the gain matrices $\left(
F_{\sigma },L_{\sigma }\right) $ can be equivalently interpreted as (i)
switching post-filters $P_{u,\sigma }(z)$ and $P_{0,\sigma }(z)$ to $%
r_{en,0}(z)$ and $r_{0,p}(z),$
\begin{align*}
P_{u,\sigma }(z)& \in \left\{ P_{u,i}(z)\in \mathcal{RH}_{\infty
},P_{u,i}=I+\left( F_{0}-F_{i}\right) \left( zI-A_{F_{0}}\right) ^{-1}B,i\in
\mathcal{I}\right\} , \\
P_{0,\sigma }(z)& \in \left\{ P_{0,i}(z)\in \mathcal{RH}_{\infty
},P_{0,i}=I+C\left( zI-A_{L_{i}}\right) ^{-1}\left( L_{0}-L_{i}\right) ,i\in
\mathcal{I}\right\} ,
\end{align*}%
and (ii) adding additional residual signal $Q_{\sigma }(z)r_{0,p}(z)$ with a
switching post-filter $Q_{\sigma }(z),$%
\begin{equation}
Q_{\sigma }(z)\in \left\{
\begin{array}{c}
Q_{i}(z)\in \mathcal{RH}_{\infty },i\in \mathcal{I}, \\
Q_{i}=F_{0}\left( zI-A_{L_{i}}\right) ^{-1}\left( L_{0}-L_{i}\right)
P_{0,i}(z)-\left( F_{i}-F_{0}\right) \left( zI-A_{F_{0}}\right) ^{-1}L_{0}%
\end{array}%
\right\} .
\end{equation}%
Note that $Q_{\sigma }(z)r_{0,p}(z)$ is noise. In the remaining part of this
work, $F_{0}$ and $L_{0}$ are used to denote the gain matrices $F$ and $L$
adopted in the control law (\ref{eq2-12a})-(\ref{eq2-12b}), and
correspondingly the LCP $\left( \hat{M},\hat{N}\right) ,\left( X,Y\right) $
are denoted by $\left( \hat{M}_{0},\hat{N}_{0}\right) ,\left(
X_{0},Y_{0}\right) ,$ respectively.
\bigskip
\begin{Rem}
The encrypting effect of adding switched post filters and noises by
switching the gain matrices among different values is analogues to the
existing approaches, for instance, reported in \cite%
{MT-method-CDC2015,MT-method-IEEE-TAC2020,Zhang-CDC2017,DIBAJI2019-survey},
although in our proposed method both the design and (online) computations
are considerably less demanding. In addition, the signal $r_{en}$ is encoded
on the plant side before the transmission and the real residual signal $%
r_{u,0}$ is recovered by a decoding algorithm on the monitoring and control
side.
\end{Rem}
The switched encoder system plays a central role for detecting the kernel
attacks successfully. Their use is to prevent an attacker from identifying
the dynamics of encoding system (\ref{eq4-9}) so that the attack signal $%
a_{r_{en}}(k)$ is set to be%
\begin{equation*}
a_{r_{en}}(z)=Y_{\sigma }(z)a_{y}(z)-X_{\sigma }(z)a_{u}(z).
\end{equation*}%
On the other hand, the needed online computations for the implementation of
\begin{equation}
r_{en,\sigma }(z)=X_{\sigma }(z)u^{a}(z)+Y_{\sigma }(z)y(z) \label{eq4-30}
\end{equation}%
that is to be performed on the plant side should be considered and kept as
less as possible.
\bigskip
Let $\sigma \left( k_{s}\right) $ denote the switching law with $k_{s}$ as
switching time instant, $F_{\sigma \left( k_{s}\right) }$ and $L_{\sigma
\left( k_{s}\right) }$ be the operating mode of the gain matrices between
two successive switching time instants $k_{s}=k_{0},k_{1}$. On the
assumption that
\begin{itemize}
\item the attacker could access $y(k),u^{a}(k)$, even
\item have knowledge of $F_{i}$ and $L_{i}$ and so that $\left(
X_{i},Y_{i}\right) ,i=1,\cdots ,\kappa ,$ are known,
\item the switching law $\sigma \left( k_{s}\right) $ is shared only by the
monitoring system and the plant system but kept hidden from the attacker,
\end{itemize}
the LCP $\left( X_{\sigma \left( k_{0}\right) },Y_{\sigma \left(
k_{0}\right) }\right) $ (i.e. the encoder (\ref{eq4-30}) running over the
time interval $[k_{0},k_{1})$) should not be detected or identified by the
attacker using the data collected over $[k_{0},k_{1}).$ This can be
formulated as an inverse problem of fault isolation or identification. It is
well-known that if the time interval $[k_{0},k_{1})$ is sufficiently short
with respect to the complexity (e.g. the order) of $\left( X_{\sigma \left(
k_{0}\right) },Y_{\sigma \left( k_{0}\right) }\right) $ and the mode number $%
\kappa ,$ with high confidential $\left( X_{\sigma \left( k_{0}\right)
},Y_{\sigma \left( k_{0}\right) }\right) $ cannot be detected or identified.
On the other hand, in order to guarantee the stability of the switched
system, the switching law $\sigma \left( k_{s}\right) $ is to be designed to
satisfy the so-called average dwell time (ADT) condition \cite%
{HM-CDC1999,ZZSL-IEEE-TAC2012}. Recall that $r_{en,\sigma }$ is only used
for the detection purpose and $\left( X_{\sigma \left( k_{0}\right)
},Y_{\sigma \left( k_{0}\right) }\right) $ has, different from the existing
approaches, no influence on the system control performance. As a result, $%
\left( X_{\sigma \left( k_{0}\right) },Y_{\sigma \left( k_{0}\right)
}\right) $ together with the switching law $\sigma \left( k_{s}\right) $ can
be designed so that (i) $\left( X_{\sigma \left( k_{0}\right) },Y_{\sigma
\left( k_{0}\right) }\right) $ is not identifiable over the time interval,
(ii) the ADT condition is satisfied. Since the major focus of this work is
on detecting kernel attacks, we will not discuss about the design of the
switching issues for $F_{\sigma }$ and $L_{\sigma }$ in more details. The
reader can refer to, for instance, the approach of cryptographically secure
pseudo random number generator (PRNG) described in \cite%
{MT-method-IEEE-TAC2020} or the approach proposed by \cite{Zhang-CDC2017}.
\subsection{Realisation of the detection scheme\label{sub-sec4-4}}
In this sub-section, we describe the realisation of the detection scheme
proposed in the previous sub-section. To this end, two issues are to be
addressed: (i) real-time implementation of the residual generators, and (ii)
design of test statistic and threshold setting. Concerning the first issue,
the major tasks consist of
\begin{itemize}
\item computation on the plant side:%
\begin{equation}
r_{en,\sigma }(z)=X_{\sigma }(z)u^{a}(z)+Y_{\sigma }(z)y(z), \label{eq4-21}
\end{equation}
\item signal transmissions from the plant side to the monitoring side:%
\begin{equation*}
r_{en,\sigma }^{a}(k)=r_{en,\sigma }(k)+a_{r_{en}}(k),y^{a}=y(k)+a_{y}(k),
\end{equation*}
\item computation on the monitoring and control side:%
\begin{align}
r_{u,0}(z)& =r_{en,\sigma }^{a}(z)-P_{u,\sigma }(z)\bar{v}_{0}(z),\bar{v}%
_{0}(z)=\left( X_{0}(z)-Q(z)\hat{N}_{0}(z)\right) v(z), \label{eq4-22} \\
r_{0}(z)& =\hat{M}_{0}(z)y^{a}(z)-\hat{N}_{0}(z)u(z), \label{eq4-23}
\end{align}
\end{itemize}
and under consideration of the plant model (\ref{eq3-1a})-(\ref{eq3-1b})
with the process and sensor noises satisfying (\ref{eq3-2a})-(\ref{eq3-2b}).
It follows from (\ref{eq2-4a})-(\ref{eq2-6b}) that the state space
realisations of (\ref{eq4-21})-(\ref{eq4-23}) are described respectively by%
\begin{align}
\varsigma (k+1)& =\left( A-L_{\sigma }C\right) \varsigma (k)+L_{\sigma
}y(k)+(B-L_{\sigma }D)u^{a}(k), \\
r_{en,\sigma }(k)& =u^{a}(k)-F_{\sigma }\varsigma (k)
\end{align}%
as well as%
\begin{align}
\hat{x}(k+1)& =\left( A-L_{0}C\right) \hat{x}%
(k)+(B-L_{0}D)u(k)+L_{0}y^{a}(k), \label{eq3-19a} \\
x_{v}(k+1)& =\left( A-L_{0}C\right) x_{v}(k)+(B-L_{0}D)v(k), \label{eq3-19b}
\\
r_{0}(k)& =y^{a}(k)-\left( C\hat{x}(k)+Du(k)\right) , \\
\bar{v}_{0}(z)& =v(z)-Fx_{v}(z)-Q(z)\left( Cx_{v}(z)+Dv(z)\right) ,
\label{eq4-19c} \\
r_{u,0}(z)& =r_{en,\sigma }^{a}(z)-P_{u,\sigma }(z)\bar{v}_{0}(z).
\label{eq4-19d}
\end{align}%
Next, the influences of $\omega (k),\nu (k)$ on $r_{u,0}(k)$ and $r_{0}(k)$
during attack-free operations are analysed aiming at setting an optimal
threshold. It turns out%
\begin{gather}
e(k+1)=\left( A-L_{0}C\right) e(k)+\omega (k)-L_{0}\nu (k),e(k)=x(k)-\hat{x}%
(k), \label{eq4-2a} \\
r_{0}(k)=Ce(k)+\nu (k), \\
r_{u,0}(z)=r_{en,\sigma }^{a}(z)-P_{u,\sigma }(z)\bar{v}_{0}(z)=P_{u,\sigma
}(z)\left( r_{en,0}(z)-\bar{v}_{0}(z)\right) +Q_{\sigma }(z)r_{0}(z), \\
r_{en,0}(z)-\bar{v}_{0}(z)=-Q(z)r_{0}(z)\Longrightarrow r_{u,0}(z)=\left(
Q_{\sigma }(z)-P_{u,\sigma }(z)Q(z)\right) r_{0}(z), \label{eq4-2b}
\end{gather}%
which implies that the residual vector%
\begin{equation*}
\left[
\begin{array}{c}
r_{u,0}(z) \\
r_{0}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
\bar{Q}_{\sigma }(z) \\
I%
\end{array}%
\right] r_{0}(z),\bar{Q}_{\sigma }=Q_{\sigma }-P_{u,\sigma }Q
\end{equation*}%
is a normally distributed color noise vector. In order to achieve an optimal
attack detection, a post-filter $P(z)$ is added as follows%
\begin{gather}
r(z)=\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] :=P(z)\left[
\begin{array}{c}
r_{u,0}(z) \\
r_{0}(z)%
\end{array}%
\right] , \label{eq4-24} \\
P(z)=\left[
\begin{array}{cc}
I & -\bar{Q}_{\sigma }(z) \\
0 & Q_{K0}(z)%
\end{array}%
\right] ,Q_{K0}(z)=I+C\left( zI-A_{L_{K}}\right) ^{-1}\left(
L_{0}-L_{K}\right) \label{eq4-24a} \\
\Longrightarrow r_{u}(z):=r_{u,0}(z)-\bar{Q}_{\sigma }(z)r_{0}(z)=0, \notag
\\
e(k+1)=A_{L_{K}}e(k)+\omega (k)-L_{K}\nu (k),A_{L_{K}}=A-L_{K}C, \notag \\
r_{0,K}(k)=r_{K}(k)=Ce(k)+\nu (k),
\end{gather}%
where $L_{K}$ is the Kalman filter gain matrix satisfying (\ref{eq3-3a}).
Correspondingly, $r_{K}(k)\sim \mathcal{N}\left( 0,\Sigma _{r}\right) $ and
is white with $\Sigma _{r}$ given in (\ref{eq3-3b}). It is remarkable that
the residual vector $r_{u}$ is fully decoupled from the noises $\omega
(k),\nu (k).$ In order to define a practical and easily computing (scale)
test statistic, $r_{u}(z)$ is treated as a (quasi-) random vector with a
covariance matrix whose inverse is approximated by $\lambda I$, where $%
\lambda >0$ is a sufficiently large number. As a result, we set the test
statistic equal to
\begin{equation}
J(k)=\lambda r_{u}^{T}(k)r_{u}(k)+r_{0,K}^{T}(k)\Sigma
_{r}^{-1}r_{0,K}(k)\sim \mathcal{\chi }^{2}\left( m\right) , \label{eq4-4}
\end{equation}%
which is subject to $\mathcal{\chi }^{2}$ distribution with $m$ degrees of
freedom in the attack-free operation, and the threshold
\begin{equation}
J_{th}=\mathcal{\chi }_{\alpha }^{2}\left( m\right) \label{eq4-5}
\end{equation}%
for a given upper-bound of false alarm rate $\alpha $.
\begin{Rem}
It is noteworthy that detecting kernel attacks is in the foreground of our
study. In order to highlight the basic ideas and major results in this
regard clearly, only process and measurement noises are taken into account.
The above simplified handling of $r_{u}$ follows from the geometric
interpretation of the $\mathcal{\chi }^{2}$ text statistic \cite{Ding2020}.
If unknown inputs and model parameter variations are to be considered,
advanced fault detection methods could be applied \cite{Ding2020}.
\end{Rem}
When the control loop is attacked, the dynamics of the observer-based attack
detector (\ref{eq4-22})-(\ref{eq4-23}) is governed by%
\begin{gather}
r_{u,0}(z)=r_{en,\sigma }^{a}(z)-P_{u,\sigma }(z)\bar{v}_{0}(z)
\label{eq4-3a} \\
=X_{\sigma }(z)u^{a}(z)+Y_{\sigma }(z)y(z)+a_{r_{en}}(z)-P_{u,\sigma }(z)%
\bar{v}_{0}(z) \notag \\
=P_{u,\sigma }(z)\left( X_{0}(z)u^{a}(z)+Y_{0}(z)y(z)\right) +Q_{\sigma
}(z)r_{0,p}(z)+a_{r_{en}}(z)-P_{u,\sigma }(z)\bar{v}_{0}(z) \notag \\
=a_{1}(z)+\bar{Q}_{\sigma }(z)r_{0,n}(z), \label{eq4-25} \\
a_{1}(z)=P_{u,\sigma }(z)\left( X_{0}(z)a_{u}(z)-Y_{0}(z)a_{y}(z)\right)
+a_{r_{en}}(z), \notag \\
r_{0}(z)=\hat{M}_{0}(z)y^{a}(z)-\hat{N}_{0}(z)u(z)=a_{2}(z)+r_{0,n}(z),
\label{eq4-26} \\
a_{2}(z)=\hat{M}_{0}(z)a_{y}(z)+\hat{N}_{0}(z)a_{u}(z), \notag
\end{gather}%
where $r_{0,n}(z)$ describes the influence of the noises on the residuals $%
r_{0,p}(z)$ and $r_{0}(z)$ and is given by%
\begin{equation}
e(k+1)=\left( A-L_{0}C\right) e(k)+\omega (k)-L_{0}\nu
(k),r_{0,n}(k)=Ce(k)+\nu (k).
\end{equation}%
Hence,
\begin{gather}
r(z)=P(z)\left[
\begin{array}{c}
r_{u,0}(z) \\
r_{0}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
a_{1}(z)-\bar{Q}_{\sigma }(z)a_{2}(z) \\
Q_{K0}(z)\left( a_{2}(z)+r_{0,n}(z)\right)%
\end{array}%
\right] \Longrightarrow \notag \\
J(k)=\lambda r_{u}^{T}(k)r_{u}(k)+r_{0,K}^{T}(k)\Sigma _{r}^{-1}r_{0,K}(k)
\notag \\
=\lambda \bar{a}_{1}^{T}(k)\bar{a}_{1}(k)+\left( \bar{a}_{2}(k)+r_{K}(k)%
\right) ^{T}\Sigma _{r}^{-1}\left( \bar{a}_{2}(k)+r_{K}(k)\right) \sim
\mathcal{\chi }^{2}\left( \delta ,m\right) , \label{eq4-27} \\
\bar{a}_{1}(z)=a_{1}(z)-\bar{Q}_{\sigma }(z)a_{2}(z),\bar{a}%
_{2}(z)=Q_{K0}(z)a_{2}(z),r_{K}(z)=Q_{K0}(z)r_{0,n}(z), \notag
\end{gather}%
where $\mathcal{\chi }^{2}\left( \delta ,m\right) $ denotes a noncentral $%
\mathcal{\chi }^{2}$ distribution with%
\begin{equation*}
\delta =\lambda \bar{a}_{1}^{T}(k)\bar{a}_{1}(k)+\bar{a}_{2}^{T}(k)\Sigma
_{r}^{-1}\bar{a}_{2}(k)
\end{equation*}%
as the noncentrality parameter and $m$ the degree of freedom. As well-known
\cite{Ding2014}, the test statistic (\ref{eq4-4}) and the threshold (\ref%
{eq4-5}) lead to the maximal fault detectability and guarantee the FAR
bounded by $\alpha .$ Moreover, from (\ref{eq4-25}), (\ref{eq4-26}) and (\ref%
{eq4-27}) it can be evidently seen that all attacks, $a_{r_{en}},$ $a_{u},$ $%
a_{y},$ can be well detected as far as the dynamics of the encoded signal $%
r_{en,\sigma }$ or equivalently $\left( X_{\sigma },Y_{\sigma }\right) $ is
not identified.
\bigskip
As summary of the proposed detection scheme, the configuration of the
detection system including data transmissions is sketched in Figure 4.
\begin{figure}[h]
\centering\includegraphics[width=11cm,height=8cm]{CPS-fig4.png}
\caption{Schematic description of the proposed attack detection system}
\end{figure}
At the end of this section, we would like to underline the following points:
\begin{itemize}
\item the detection scheme proposed in this section and based on the
residual signals $r_{0},r_{0,u}$ can be analogously realised as well using
the alternative residual signals $r_{u,c},r_{y,c}$ defined in (\ref{eq3-23});
\item the test statistic (\ref{eq4-4}) and the threshold (\ref{eq4-5})
deliver the optimal attack detection only on the assumptions of (i) the
statistic features of the noises being specified by (\ref{eq3-2a})-(\ref%
{eq3-2b}) , and (ii) the additive character of the kernel attacks being
under consideration \cite{Ding2020}, and
\item in case that the noises cannot be described by (\ref{eq3-2a})-(\ref%
{eq3-2b}) or/and the cyber-attacks are presented e.g. in multiplicative form
like false data injection attacks \cite{LZLWD2017,GWSOM2019}, sophisticated
detection schemes are needed. Some of these methods are reported in \cite%
{LD-Automatica-2020,Ding2020}.
\end{itemize}
\section{An encrypted configuration of feedback control and detection systems%
}
The basis for the execution of kernel attacks is that attackers have
knowledge of plant dynamics. Among numerous possibilities to gain such
information, eavesdropping attacks enable collecting sufficient amount of
process data which can then be used for identifying the plant dynamics. It
is state of the art that in real industrial applications plant input and
output data, $u(k)$ and $y(k),$ are often transmitted between the control
and monitoring station and the plant via networks. Such system
configurations make an identification of the plant dynamics considerably
easy. In this section, we propose an encrypted configuration scheme of
feedback control systems. The core of the alternatively configured control
systems consists in the transmission of encoded system signals, instead of $%
u(k)$ and $y(k),$ from which a direct identification of the plant dynamics
without \textit{a priori} knowledge becomes almost impossible. The basis for
this encrypted configuration is the so-called functionalisation of dynamic
controllers introduced in the unified framework of control and detection.
\subsection{Functionalisation of all stabilising feedback controllers}
Recall the observer-based realisation of all stabilising controllers given
in (\ref{eq2-13a})-(\ref{eq2-13c}). It can be divided into several
functional modules:
\begin{itemize}
\item an observer and an observer-based residual generator,
\begin{gather*}
\hat{x}(k+1)=A\hat{x}(k)+Bu(k)+L_{0}r_{0}(k), \\
r_{0}(k)=y(k)-\hat{y}(k),\hat{y}(k)=C\hat{x}(k)+Du(k),
\end{gather*}%
which serve as an information provider for the controller and diagnostic
system, and deliver a state estimation, $\hat{x},$ and the primary residual,
$r_{0}=y-\hat{y},$
\item control law%
\begin{equation*}
u(z)=F_{0}\hat{x}(z)-Q(z)r_{0}(z)+\hat{V}(z)v(z),
\end{equation*}%
including
\begin{itemize}
\item a feedback controller: $F_{0}\hat{x}(z)-Q(z)r_{0}(z)$ and
\item a feed-forward controller: $\hat{V}(z)v(z),\hat{V}=X_{0}-Q\hat{N}_{0},$
and in addition, for the detection purpose,
\end{itemize}
\item detector $R(z)r_{0}(z)$ with $R(z)$ as a stable post-filter.
\end{itemize}
This modular structure provides us with a clear parameterisation of the
functional modules:
\begin{itemize}
\item the state observer is parameterised by $L_{0},$
\item the feedback controller by $F_{0},Q,$
\item the feed-forward controller by $\hat{V},$ and
\item the detector by $R.$
\end{itemize}
Although all five parameters listed above are available for the design and
online optimisation objectives, they have evidently different
functionalities, as summarised below:
\begin{itemize}
\item $F_{0},L_{0}$ determine the stability and eigen-dynamics of the
closed-loop,
\item $R,\hat{V}$ have no influence on the system stability, and $R$ serves
for the optimisation of the detectability, while $\hat{V}$ for the tracking
behavior, and
\item $Q$ is used to enhance the system robustness and control performance.
The design and update of $Q$ will have influence on the system dynamics and
stability, when parameter uncertainties or degradations are present in the
system.
\end{itemize}
It is evident that the above five parameters have to be, due to their
different functionalities, treated with different priorities. Recall that
system stability and eigen-dynamics are the fundamental requirement on an
automatic control system. This requires that the system stability should be
guaranteed, also in case of cyber-attacks. Differently, $Q,R,\hat{V}$ are
used to optimise control or detection performance. In case that a temporary
system performance degradation is tolerable, the real-time demand and the
priority for an online optimisation of $Q,R,\hat{V}$ are relatively lower.
Under these considerations, we propose in the next sub-section an encrypted
control system configuration based on the above controller functionalisation.
\subsection{An encrypted system configuration scheme}
To begin with, we would like to emphasise that the objective of the system
configuration proposed in the sequel is to prevent system knowledge from
attackers in the manner that the plant model cannot be identified using the
data possibly collected by attackers by means of eavesdropping attacks.
Moreover, the basic requirements on the system control performance like the
stability are to be met.
\bigskip
The proposed encrypted system configuration mainly consists of
\begin{itemize}
\item on the plant side, an observer-based state feedback controller and
residual generator,%
\begin{align}
\hat{x}(k+1)& =A\hat{x}(k)+Bu(k)+L_{0}r_{0,p}(k),r_{0,p}(k)=y(k)-\hat{y}(k),
\notag \\
u(k)& =F_{0}\hat{x}(k)+\gamma (k)\Longrightarrow \notag \\
\hat{x}(k+1)& =\left( A+BF_{0}\right) \hat{x}(k)+B\gamma (k)+L_{0}r_{0,p}(k)
\label{eq5-3a} \\
& =\left( A-L_{0}C\right) \hat{x}(k)+\left( B-L_{0}D\right) u(k)+L_{0}y(k),
\label{eq5-3b}
\end{align}%
where $\gamma $ is the signal (vector) received from the monitoring and
control side,
\item on the monitoring and control side,%
\begin{equation*}
\gamma (z)=\hat{V}(z)v(z)-Q(z)r_{0,p}(z),
\end{equation*}%
where $r_{0,p}$ is received from the plant side and $v$ is the reference
vector,
\item transmission from the plant side to the monitoring and control side, $%
r_{0,p}(k),$
\item transmission from the monitoring and control side to the plant side, $%
\gamma (k).$
\end{itemize}
Depending on applications, the following functional modules can be further
realised and integrated on the monitoring and control side, for instance,
\begin{itemize}
\item reconstructing $y(k),$%
\begin{gather}
\hat{x}(k+1)=\left( A+BF_{0}\right) \hat{x}(k)+B\gamma (k)+L_{0}r_{0,p}(k),
\label{eq5-1a} \\
y(k)=r_{0,p}(k)+\hat{y}(k)=\left( C+DF_{0}\right) \hat{x}(k)+D\gamma
(k)+r_{0,p}(k) \label{eq5-1b}
\end{gather}%
with $r_{0,p}(k)$ received from the plant side,
\item tuning $Q$ using $r_{0,p}(k)$ and $v(k)$ to enhance the stability
margin, as reported in \cite{LLDYP-2019}, or
\item recovering control performance degradation using $y(k)$ and $u(k),$ as
described in \cite{Ding2020}.
\end{itemize}
It is evident that, according to the observer-based realisation of all
stabilising controllers, the control input $u(k)$ acted on the actuators
(located on the plant side) is given by%
\begin{equation*}
u(k)=F_{0}\hat{x}(k)+\gamma (k)\Longleftrightarrow u(z)=K(z)y(z)+v(z)
\end{equation*}%
with $K$ satisfying (\ref{eq2-12a})-(\ref{eq2-12b}). Different from the
standard system configuration, for instance the one shown in Figure 1, the
observer-based state feedback controller and residual generator (\ref{eq5-3a}%
)-(\ref{eq5-3b}) running on the plant side serve as
\begin{itemize}
\item an encoder for an encrypted transmission of the plant measurement $%
y(k),$ i.e. $r_{0,p}(k)$ instead of $y(k),$
\item a decoder for control input $u(k)=F_{0}\hat{x}(k)+\gamma (k),$ and
\item a local controller guaranteeing the basic control performance like the
stability even if the communication between the both sides of the control
system is considerably attacked.
\end{itemize}
Simultaneously, the recovering algorithm (\ref{eq5-1a})-(\ref{eq5-1b})
running on the monitoring and control side acts (i) as a decorder for $y$
and (ii) $\gamma =\hat{V}v-Qr_{0,p}$ as an encoder for an encrypted
transmission of the control signal from the monitoring and control side to
the plant.
\bigskip
Considering that, during the attack-free operation, $r_{0,p}$ is noise (and
even white noise when $L_{0}$ is set to be the Kalman filter gain matrix),
it is obviously impossible to identify the plant model $G_{u}$ by means of $%
r_{0,p}$ and $\gamma $ that could be eavesdropped during their transmission.
As a result, it can be claimed that the encrypted control system
configuration proposed in this sub-section fully fulfills the design
requirements.
\subsection{The associated attack detection scheme}
Figure 5 sketches schematically the proposed encrypted system configuration. On the assumptions that
\begin{itemize}
\item the control loop under consideration is configurated as sketched in
Figure 5,
\item the attacker has no knowledge about the plant model $G_{u},$ and
\item both $\gamma $ and $r_{0,p}$ are corrupted by the attack signals $%
a_{\gamma }$ and $a_{r_{0}}$ respectively, i.e.
\begin{align*}
\gamma ^{a}(k)& =\gamma (k)+a_{\gamma }(k)\Longrightarrow u^{a}(k)=\gamma
(k)+a_{\gamma }(k)+F_{0}\hat{x}(k), \\
r_{0}^{a}(k)& =r_{0,p}(k)+a_{r_{0}}(k),
\end{align*}
\end{itemize}
we propose the following attack detection scheme performed on the monitoring
and control side. Similar to the controller, the attack detector is also
distributedly realised on the both sides of the control system.
\begin{figure}[h]
\centering\includegraphics[width=14cm,height=10cm]{CPS-fig5.png}
\caption{Schematic description of the encrypted control and detection system
configuration}
\end{figure}
\bigskip
Remember that in the attack-free case%
\begin{gather}
X_{0}(z)u(z)+Y_{0}(z)y(z)-\gamma (z) \notag \\
=X_{0}(z)u(z)+Y_{0}(z)y(z)-\left( \bar{v}(z)-Q(z)r_{0,p}(z)\right) \notag \\
=u(z)-\left( F_{0}\hat{x}(z)+\gamma (z)\right) =0. \label{eq5-5}
\end{gather}%
It motivates us to encrypt the detector as follows. At first, the encoded
signal $\beta (k)$ is generated on the plant side,%
\begin{equation}
\beta (k)=F_{0}\hat{x}(k)-F_{\sigma }\hat{x}(k), \label{eq5-7e}
\end{equation}%
where $F_{\sigma }$ is a switched feedback gain introduced in the previous
section. It follows from Lemma \ref{Le4-1} and Theorem \ref{Theo4-3} that
\begin{gather}
\beta (z)=\left( F_{0}-F_{\sigma }\right) \hat{x}(z)=u^{a}(z)-F_{\sigma }%
\hat{x}(z)-\left( u^{a}(z)-F_{0}\hat{x}(z)\right) \notag \\
=R_{0\sigma }(z)\left( X_{0}(z)u^{a}(z)+Y_{0}(z)y(z)\right) +Q_{0\sigma
}(z)r_{0,p}(z), \label{eq5-7} \\
R_{0\sigma }(z)=P_{u,\sigma }(z)-I=\left( F_{0}-F_{\sigma }\right) \left(
zI-A_{F_{0}}\right) ^{-1}B, \label{eq5-7d} \\
Q_{0\sigma }(z)=\left( F_{0}-F_{\sigma }\right) \left( zI-A_{F_{0}}\right)
^{-1}L_{0}. \notag
\end{gather}%
Here, $P_{u,\sigma }(z)$ is given in Theorem \ref{Theo4-3}. The encoded
signal $\beta $ is then sent to the monitoring and control side, at which a
residual signal is generated by decoding $\beta $ as follows%
\begin{equation}
r_{\beta }(z)=\beta ^{a}(z)-R_{0\sigma }(z)\gamma (z) ,
\label{eq5-8}
\end{equation}%
where
\begin{equation*}
\beta ^{a}(k)=\beta (k)+a_{\beta }(k)
\end{equation*}%
denotes the corrupted signal $\beta $ due to the cyber-attack $a_{\beta }.$
It turns out, remembering (\ref{eq5-5}),
\begin{align}
r_{\beta }(z)& =a_{\beta }(z)+R_{0\sigma }(z)\left(
X_{0}(z)u^{a}(z)+Y_{0}(z)y(z)-\gamma (z)\right) +Q_{0\sigma
}(z)r_{0,p}(z) \notag \\
& =a_{\beta }(z)+R_{0\sigma }(z)X_{0}(z)a_{\gamma }(z)+Q_{0\sigma
}(z)r_{0,n}(z).
\end{align}%
with $r_{0,n}$ denoting the influence of the noises on the residual vector.
Therefore, it holds, on the monitoring and control side,
\begin{equation}
\left[
\begin{array}{c}
r_{\beta }(z) \\
r_{0}^{a}(z)%
\end{array}%
\right] =\left[
\begin{array}{ccc}
I & R_{0\sigma }(z)X_{0}(z) & 0 \\
0 & 0 & I%
\end{array}%
\right] \left[
\begin{array}{c}
a_{\beta }(z) \\
a_{\gamma }(z) \\
a_{r_{0}}(z)%
\end{array}%
\right] +\left[
\begin{array}{c}
Q_{0\sigma }(z) \\
I%
\end{array}%
\right] r_{0,n}(z). \label{eq5-8a}
\end{equation}%
As a result, we have
\begin{Theo}
\label{Theo4-4}Given the plant model (\ref{eq3-1a})-(\ref{eq3-1b}), the
control law $K$ satisfying (\ref{eq2-12a})-(\ref{eq2-12b}) and residuals $%
r_{\beta }$ and $r_{0}^{a}$ that are realised in the encrypted system
configuration shown in Figure 5, the attacks $a_{\beta },a_{\gamma }$ and $%
a_{r_{0}}$ are stealthy, if and only if the conditions,%
\begin{equation}
a_{r_{0}}(k)=0,a_{\beta }(z)+R_{0\sigma }(z)X_{0}(z)a_{\gamma }(z)=0,
\label{eq5-10}
\end{equation}%
are satisfied.
\end{Theo}
Theorem \ref{Theo4-4} reveals that
\begin{itemize}
\item an (additive) attack on the residual signal $r_{0,p}$ can be
(structurally) directly detected, and
\item keeping $a_{\beta },a_{\gamma }$ stealthy is almost impossible, since
condition (\ref{eq5-8}) can hardly be satisfied, (i) without system
knowledge, (ii) without knowing the purpose of using and transmissing $\beta
$ and $\gamma ,$ and (iii) in particular when $R_{0\sigma }(z)$ is a
switched system.
\end{itemize}
\subsection{Implementation of the control and detection systems}
Now, we summarise the implementation issues of the proposed control and
detection systems.
\bigskip
On the plant side, the state observer (\ref{eq5-3a}) (equivalently (\ref%
{eq5-3b})) builds the core of the system implementation. Based on the state
estimate $\hat{x}(k),$ the control input $u^{a}(k),$ the residual signal $%
r_{0,p}(k)$ as well as the encoded signal $\beta (k)$ are formed,
\begin{align}
u^{a}(k)& =F_{0}\hat{x}(k)+\gamma ^{a}(k), \label{eq5-7a} \\
r_{0,p}(k)& =y(k)-C\hat{x}(k)-Du^{a}(k), \label{eq5-7b} \\
\beta (k)& =\left( F_{0}-F_{\sigma }\right) \hat{x}(k). \label{eq5-7c}
\end{align}%
For running the realisation algorithms, the system on the plant side
receives the signal $\gamma ^{a}$ from the monitoring and control side. It
sends the residual signal $r_{0,p}$ and encoded signal $\beta $ to the
system running on the monitoring and control side. It is of considerable
interest to remark that the state observer (\ref{eq5-3a}) serves both as a
decoder for the control signal, as given in (\ref{eq5-7a}), and as an
encoder for the controller and for the generation of residual signal $%
r_{\beta }$ (that are implemented on the monitoring and control side), as
described by (\ref{eq5-7b}) and (\ref{eq5-7c}).
\bigskip
On the monitoring and control side, $\gamma (k)$ is first computed as
follows
\begin{gather}
x_{v}(k+1)=\left( A-L_{0}C\right) x_{v}(k)+(B-L_{0}D)v(k), \label{eq5-9a} \\
\gamma (z)=v(z)-F_{0}x_{v}(z)-Q(z)\left( Cx_{v}(z)+Dv(z)-r_{0}^{a}(z)\right)
. \label{eq5-9b}
\end{gather}%
Then, $r_{\beta }$ is generated as%
\begin{align}
x_{\beta }(k+1)& =\left( A+BF_{0}\right) x_{\beta }(k)+B\gamma
(k) , \label{eq5-10a} \\
r_{\beta }(k)& =\beta ^{a}(k)-\left( F_{\sigma }-F_{0}\right) x_{\beta }(k).
\label{eq5-10b}
\end{align}%
It is worth emphasising that computation (\ref{eq5-9a})-(\ref{eq5-9b})
serves as an encoder for the control signal, while the system (\ref{eq5-10a}%
)-(\ref{eq5-10b}) acts as a decoder.
\bigskip
Next, for detecting attacks $a_{\beta },a_{\gamma }$ and $a_{r_{0}}$
optimally, the residual vector
\begin{equation}
r(z)=\left[
\begin{array}{cc}
I & -Q_{0\sigma }(z) \\
0 & Q_{K0}(z)%
\end{array}%
\right] \left[
\begin{array}{c}
r_{\beta }(z) \\
r_{0}^{a}(z)%
\end{array}%
\right] =:\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] \label{eq5-6}
\end{equation}%
and the test statistic
\begin{equation*}
J(k)=\lambda r_{u}^{T}(k)r_{u}(k)+r_{0,K}^{T}(k)\Sigma _{r}^{-1}r_{0,K}(k)
\end{equation*}%
are built with $Q_{K0}$ as given in (\ref{eq4-24a}), which is analogue to
the result described in Sub-section \ref{sub-sec4-4}. We have
\begin{equation}
J(k)=\lambda r_{u}^{T}(k)r_{u}(k)+r_{0,K}^{T}(k)\Sigma
_{r}^{-1}r_{0,K}(k)\sim \mathcal{\chi }^{2}\left( m\right) , \label{eq4-4a}
\end{equation}%
and thus the threshold is set to be%
\begin{equation}
J_{th}=\mathcal{\chi }_{\alpha }^{2}\left( m\right) \label{eq4-5a}
\end{equation}%
for a given upper-bound of false alarm rate $\alpha $. In case of attacks,
\begin{gather}
J(k)=\lambda r_{u}^{T}(k)r_{u}(k)+r_{0,K}^{T}(k)\Sigma _{r}^{-1}r_{0,K}(k)
\notag \\
=\lambda a_{1}^{T}(k)a_{1}(k)+\left( a_{2}(k)+r_{K}(k)\right) ^{T}\Sigma
_{r}^{-1}\left( a_{2}(k)+r_{K}(k)\right) \sim \mathcal{\chi }^{2}\left(
\delta ,m\right) \\
a_{1}(z)=a_{\beta }(z)+R_{0\sigma }(z)X_{0}(z)a_{\gamma }(z)-Q_{0\sigma
}(z)a_{r_{0}}(z), \notag \\
a_{2}(z)=Q_{K0}(z)a_{r_{0}}(z),r_{K}(k)\sim \mathcal{N}\left( 0,\Sigma
_{r}\right) ,
\end{gather}%
where $\mathcal{\chi }^{2}\left( \delta ,m\right) $ denotes a noncentral $%
\mathcal{\chi }^{2}$ distribution with%
\begin{equation*}
\delta =\lambda a_{1}^{T}(k)a_{1}(k)+a_{2}^{T}(k)\Sigma _{r}^{-1}a_{2}(k)
\end{equation*}%
as the noncentrality parameter and $m$ the degree of freedom.
\section{Examples and experimental study}
\subsection{Examples of detecting typical kernel attacks}
As examples, we will demonstrate that the zero dynamics, covert and replay
attacks as kernel attacks\ can be well detected using the detection schemes
proposed in Sections 4 and 5.
\begin{Exp}
Consider a zero dynamics attack satisfying (\ref{eq2-11b}). For our purpose
of detecting $a_{u},$ applying both detection schemes presented in
Sub-sections \ref{Sec4-3}-\ref{sub-sec4-4} and Section 5 results in
\begin{itemize}
\item by detector (\ref{eq4-24}) whose dynamics with respect to the
(possible) attack signals is described by (\ref{eq4-27}):%
\begin{equation}
r(z)=\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
P_{u,\sigma }(z)X_{0}(z)a_{u}(z)+a_{r_{en}}(z) \\
r_{K}(z)%
\end{array}%
\right] , \label{eq6-1}
\end{equation}%
where $a_{r_{en}}$ denotes the (possible) attack signal on the transmitted
signal $r_{en}$ that builds an (encoded) part of $r_{u},$
\item by detector (\ref{eq5-6}) whose dynamics with respect to the
(possible) attack signals is described by (\ref{eq5-8a}):
\begin{equation}
r(z)=\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
a_{\beta }(z)+R_{0\sigma }(z)X_{0}(z)a_{\gamma }(z) \\
r_{K}(z)%
\end{array}%
\right] ,a_{u}(z)=a_{\gamma }(z) \label{eq6-2}
\end{equation}%
with the (additional) attack signal $a_{\beta }$ on the encoded signal $%
\beta .$
\end{itemize}
It is evident that in the former case, $a_{u}$ can be detected using $r$ as
far as the attacker could not identify $P_{u,\sigma }$ or equivalently $%
X_{\sigma }$ and thus set $a_{r_{en}}$ equal to $-P_{u,\sigma }X_{0}a_{u}.$
For the latter case, as long as the switched system $R_{0\sigma }X_{0}$
could not be identified, it is impossible for the attacker to construct $%
a_{\beta }$ equal to $-R_{0\sigma }X_{0}a_{u}.$ Consequently, both $a_{\beta
}$ and $a_{u}$ can be detected. We would like to emphasise that in this case
it is impossible to identify the plant dynamics $\left( \hat{M}_{0},\hat{N}%
_{0}\right) $ using eavesdropped data $\gamma (k),r_{0,p}(k).$
\end{Exp}
\begin{Exp}
Now, consider the both detection systems under a covert attack satisfying (%
\ref{eq3-13}). It holds,
\begin{itemize}
\item by detector (\ref{eq4-24}):%
\begin{equation}
r(z)=\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
P_{u,\sigma }(z)\left( X_{0}(z)a_{u}(z)-Y_{0}(z)a_{y}(z)\right)
+a_{r_{en}}(z) \\
r_{K}(z)%
\end{array}%
\right] , \label{eq6-3}
\end{equation}%
with the (possible) additional attack $a_{r_{en}}$ on the transmitted signal
$r_{en},$
\item by detector (\ref{eq5-6}):
\begin{align}
r(z)& =\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] =\left[
\begin{array}{c}
a_{\beta }(z)+R_{0\sigma }(z)X_{0}(z)a_{\gamma }(z)-Q_{0\sigma
}(z)a_{r_{0}}(z) \\
Q_{K0}(z)a_{r_{0}}(z)%
\end{array}%
\right] , \label{eq6-4} \\
a_{u}(z)& =a_{\gamma }(z),a_{y}(z)=a_{r_{0}}(z) \notag
\end{align}%
where it is assumed that the attack signal $a_{y}$ is added to the
transmitted signal $r_{0,p},$ since $r_{0,p}$ instead of $y$ is transmitted
from the plant side to the monitoring and control side.
\end{itemize}
It is clear that in the first case, $a_{u}$ and $a_{y}$ can be detected as
far as the attacker could not identify $\left( X_{\sigma },Y_{\sigma
}\right) $. It is of considerable interest to notice the results in the
second case. Using the detector (\ref{eq5-6}), we can identify the attack $%
a_{y}$ ($a_{r_{0}}$)$,$%
\begin{equation*}
a_{r_{0}}(z)=a_{y}(z)=Q_{K0}^{-1}(z)r_{0,K}(z),Q_{K0}^{-1}(z)=I+C\left(
zI-A+L_{0}C\right) ^{-1}\left( L_{K}-L_{0}\right) ,
\end{equation*}%
and moreover estimate $a_{u}$ based on%
\begin{equation*}
\hat{M}_{0}(z)a_{y}(z)+\hat{N}_{0}(z)a_{u}(z)=0\Longleftrightarrow \hat{N}%
_{0}(z)a_{u}(z)=-\hat{M}_{0}(z)Q_{K0}^{-1}(z)r_{0,K}(z),
\end{equation*}%
when $\left( a_{y},a_{u}\right) $ is a covert attack. In this case, $%
a_{\beta }$ can also be estimated in terms of
\begin{equation*}
a_{\beta }(z)=-R_{0\sigma }(z)a_{u}(z)+Q_{0\sigma }(z)a_{y}(z).
\end{equation*}%
Finally, as far as $R_{0\sigma }(z),Q_{0\sigma }(z)$ are not identified, any
attacks of $a_{\beta },a_{y},a_{u}$ can be detected. This example clearly
demonstrates the advantage of the detector (\ref{eq5-6}) over the detector (%
\ref{eq4-24}) and other reported attack detectors.
\end{Exp}
\begin{Exp}
We now address the detection issue of replay attacks under the assumption of
steady operation, i.e.
\begin{equation}
y(k)\approx y(k-i),u(k)=u(k-i),i=1,\cdots . \label{eq6-5}
\end{equation}%
Since in our detection schemes proposed in the last two sections additional
signals, $r_{en}$ and $\beta ,$ are transmitted from the plant side to the
monitoring and control side, it is assumed that the attacker has collected
and saved the (attack-free) data $r_{en}(j),\beta (j),j\in \left[
k_{0},k_{0}+M\right] .$ When the data are replayed over the time interval $%
\left[ k,k+M\right] ,k>k_{0}+M,$ it holds
\begin{equation}
r_{en}^{a}(i)=r_{en}(i-(k-k_{0})),\beta ^{a}(i)=\beta (i-(k-k_{0})),i\in
\left[ k,k+M\right] . \label{eq6-6}
\end{equation}%
Moreover, an attack signal on the actuators is injected, for instance,%
\begin{equation*}
a_{u}(i)=a_{\gamma }(i),i\in \left[ k,k+M\right] .
\end{equation*}%
It turns out
\begin{itemize}
\item by detector (\ref{eq4-24}):%
\begin{gather}
r(z)=\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] \approx \left[
\begin{array}{c}
\Delta r_{en}(z)-\Delta r_{K}(z) \\
r_{K}(z)%
\end{array}%
\right] , \label{eq6-7} \\
\Delta r_{en}(i)=r_{en,\sigma \left( i-(k-k_{0})\right)
}(i-(k-k_{0}))-r_{en,\sigma \left( i\right) }(i), \notag \\
r_{en,\sigma \left( i-(k-k_{0})\right) }(z)=X_{\sigma \left(
i-(k-k_{0})\right) }(z)u(z^{-(k-k_{0})})+Y_{\sigma \left( i-(k-k_{0})\right)
}(z)y(z^{-(k-k_{0})}), \notag \\
r_{en,\sigma \left( i\right) }(z)=X_{\sigma \left( i\right)
}(z)u(z)+Y_{\sigma \left( i\right) }(z)y(z), \notag \\
\Delta r_{K}(i)=r_{K,\sigma \left( i-(k-k_{0})\right)
}(i-(k-k_{0}))-r_{K,\sigma \left( i\right) }(i), \label{eq6-7a} \\
r_{K,\sigma \left( i-(k-k_{0})\right) }(z)=\bar{Q}_{\sigma \left(
i-(k-k_{0})\right) }(z)r_{K}(z^{-(k-k_{0})}),r_{K,\sigma \left( i\right)
}(z)=\bar{Q}_{\sigma \left( i\right) }(z)r_{K}(z), \label{eq6-7b}
\end{gather}%
due to assumptions (\ref{eq6-5}) and (\ref{eq6-6}),
\item by detector (\ref{eq5-6}):
\begin{gather}
r(z)=\left[
\begin{array}{c}
r_{u}(z) \\
r_{0,K}(z)%
\end{array}%
\right] \approx \left[
\begin{array}{c}
\Delta _{\beta }(z)-\Delta r_{0,K}(z) \\
r_{K}(z)%
\end{array}%
\right] , \label{eq6-8} \\
\Delta _{\beta }(i)=\beta (i-(k-k_{0}))-\beta (i), \notag \\
\Delta _{\beta }(z)=R_{0\sigma (i-(k-k_{0}))}(z)\left( X_{0}(z)u(z^{-\left(
k-k_{0}\right) })+Y_{0}(z)y(z^{-\left( k-k_{0}\right) })\right) \notag \\
-R_{0\sigma (i)}(z)\left( X_{0}(z)u(z)+Y_{0}(z)y(z)\right) , \label{eq6-8a}
\\
\Delta r_{0,K}(z)=Q_{0\sigma (i-(k-k_{0}))}(z)r_{K}(z^{-\left(
k-k_{0}\right) })-Q_{0\sigma (i)}(z)r_{K}(z). \label{eq6-8b}
\end{gather}
\end{itemize}
Now, we study dynamics (\ref{eq6-7}) and (\ref{eq6-8}). In the first case,
since $\left( X_{\sigma \left( i-(k-k_{0})\right) },Y_{\sigma \left(
i-(k-k_{0})\right) }\right) $ and $\left( X_{\sigma \left( i\right)
},Y_{\sigma \left( i\right) }\right) $ as well as $\bar{Q}_{\sigma \left(
i-(k-k_{0})\right) }$ and $\bar{Q}_{\sigma \left( i\right) }$ are generally
different, which leads to
\begin{eqnarray*}
\Delta r_{en} &\approx &\left( X_{\sigma \left( i-(k-k_{0})\right)
}-X_{\sigma \left( i\right) }\right) u+\left( Y_{\sigma \left(
i-(k-k_{0})\right) }-Y_{\sigma \left( i\right) }\right) y\neq 0, \\
\Delta r_{K} &\approx &\left( \bar{Q}_{\sigma \left( i-(k-k_{0})\right) }-%
\bar{Q}_{\sigma \left( i\right) }\right) r_{K}\neq 0.
\end{eqnarray*}%
Consequently, both the mean and co-variance matrix of $r_{u}(z)$ will
change, which can be well detected using the generalised likelihood ratio
(GLR) method \cite{Ding2020}. The second case is similar to the first one so
that the replay attack can be detected in general, thanks to the fact that
\begin{align*}
R_{0\sigma (i-(k-k_{0}))}\left[
\begin{array}{cc}
X_{\sigma \left( i-(k-k_{0})\right) } & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y_{\sigma \left(
i-(k-k_{0})\right) }%
\end{array}%
\right] & \neq R_{0\sigma (i)}\left[
\begin{array}{cc}
X_{\sigma \left( i\right) } & \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }Y_{\sigma \left( i\right) }%
\end{array}%
\right] , \\
Q_{0\sigma (i-(k-k_{0}))}& \neq Q_{0\sigma (i)}.
\end{align*}
\end{Exp}
In comparison with the existing detection methods, it is clear that
\begin{itemize}
\item the two detection schemes proposed in this work guarantee structural
detection of any kernel attacks, while the most existing methods can be
generally applied to detecting a special type of kernel attacks;
\item in particular, both methods deliver reliable detection of replay
attacks without adding (additional) signals like a watermark in $u$ \cite%
{Mo2015-Watermarked-detection}. In fact, $\Delta r_{en}-\Delta r_{K}$ and $%
\Delta _{\beta }-\Delta r_{0,K}$ delivered by the detectors (\ref{eq4-24})
and (\ref{eq5-6}), respectively, act like a watermark but without any
influence on the control performance;
\item the design of both detectors are straightforward without complicated
computations, and
\item the required online computations are less demanding.
\end{itemize}
\subsection{Experimental study}
Experimental study on detecting cyber-attacks on a real three-tank control system is running and the achieved results will be reported.
\section{Conclusions}
In this work, we have studied issues of detecting stealthy integrity
cyber-attacks in the unified control and detection framework. The first
effort has been dedicated to the general form of integrity cyber-attacks
that cannot be detected using the well-established observer-based detection
technique. It has been demonstrated that any attacks lying in the system
kernel space cannot be detected by an observer-based detection system.
Correspondingly, the concept of kernel attacks has been introduced. The
replay, zero dynamics and covert attacks which are widely investigated in
the literature are the examples of kernel attacks. Our further effort has
been focused on the existence conditions of stealthy integrity attacks. To
this end, the unified framework of control and detection has applied. It has
been revealed that all kernel attacks can be structurally detected when
residual generation is extended to the space spanned by the control signal.
In other words, not only the observer-based residual, but also the control
signal based residual signals are needed for a reliable detection of kernel
attacks. As a result of this work, the necessary and sufficient conditions
for detecting kernel attacks are given.
\bigskip
Based on the analytical results in the first part of our study, we have
proposed two schemes for detecting kernel attacks. Using the known results
and methods of the unified control and detection framework, both schemes
result in reliable detection of kernel attacks without any loss of control
performance. While the first detector is configured similar to the existing
methods like the moving target method and auxiliary system aided detection
scheme \cite{MT-method-CDC2015,Zhang-CDC2017,DIBAJI2019-survey,GWSOM2019},
the second detector is realised with the encrypted transmissions of control
and monitoring signals in the feedback control system that prevent adversary
to gain system knowledge by means of eavesdropping attacks. The theoretical
basis for such detector configurations is the observer-based,
residual-driven realisation of all stabilising feedback controllers. In
particular, the functionalisation of controllers in the unified control and
detection framework plays an essential role in developing the second
detection scheme.
\bigskip
It should be remarked that our study in this work has been performed on the
assumptions that (i) the LTI system models are not corrupted with model
uncertainties, and (ii) the kernel attacks are presented in the additive
form (although the replay attack is a multiplicative signal, it is handled
as an additive one). In this context, the concept of kernel attacks and the
derived existence conditions are in fact the expressions of structural
properties of the feedback control system under consideration. So far, the
proposed detection schemes would work well in laboratory conditions, but
cannot be directly applied in real industrial applications without
modifications. This fact motivates our future work to deal with
cyber-attacks in the multiplicative form, for instance, false data injection
attacks \cite{LZLWD2017,GWSOM2019}, and on automatic control systems with
uncertainties. The unified control and detection framework and the
associated detection methods developed recently \cite%
{LD-Automatica-2020,Ding2020} could serve as efficient tools.
\bigskip
\textbf{Appendix Proof of Lemma 1}
\bigskip
Since
\begin{align*}
R_{12}(z)F_{2}& =F_{2}+\left( F_{2}-F_{1}\right) \left( zI-A_{F_{2}}\right)
^{-1}BF_{2} \\
& =F_{1}+\left( F_{2}-F_{1}\right) +\left( F_{2}-F_{1}\right) \left(
zI-A_{F_{2}}\right) ^{-1}BF_{2} \\
& =F_{1}+\left( F_{2}-F_{1}\right) \left( zI-A_{F_{2}}\right) ^{-1}\left(
zI-A\right) ,
\end{align*}%
it turns out%
\begin{align*}
R_{12}(z)Y_{2}(z)& =-\left( F_{1}+\left( F_{2}-F_{1}\right) \left(
zI-A_{F_{2}}\right) ^{-1}\left( zI-A\right) \right) \left(
zI-A_{L_{2}}\right) ^{-1}L_{2}, \\
R_{12}(z)X_{2}(z)& =R_{12}(z)-\left( F_{1}+\left( F_{2}-F_{1}\right) \left(
zI-A_{F_{2}}\right) ^{-1}\left( zI-A\right) \right) \left(
zI-A_{L_{2}}\right) ^{-1}(B-L_{2}D).
\end{align*}%
Moreover, the relation%
\begin{gather*}
\left( zI-A\right) \left( zI-A_{L}\right) ^{-1}L=\left( I+LC\left(
zI-A\right) ^{-1}\right) ^{-1}L \\
=L\left( I-C\left( zI-A+LC\right) ^{-1}L\right) =L\hat{M}(z)
\end{gather*}%
leads to%
\begin{gather}
R_{12}(z)Y_{2}(z)=-F_{1}\left( zI-A_{L_{2}}\right) ^{-1}L_{2}+\bar{R}_{12}(z)%
\hat{M}_{2}(z) \label{eq4-15a} \\
R_{12}(z)X_{2}(z)=R_{12}(z)-F_{1}\left( zI-A_{L_{2}}\right) ^{-1}(B-L_{2}D)-
\notag \\
\left( F_{2}-F_{1}\right) \left( zI-A_{F_{2}}\right) ^{-1}\left(
I-L_{2}C\left( zI-A+L_{2}C\right) ^{-1}\right) (B-L_{2}D) \notag \\
=I-F_{1}\left( zI-A_{L_{2}}\right) ^{-1}(B-L_{2}D)+\left( F_{2}-F_{1}\right)
\left( zI-A_{F_{2}}\right) ^{-1}L_{2}D \notag \\
+\left( F_{2}-F_{1}\right) \left( zI-A_{F_{2}}\right) ^{-1}L_{2}C\left(
zI-A+L_{2}C\right) ^{-1}(B-L_{2}D) \notag \\
=I-F_{1}\left( zI-A_{L_{2}}\right) ^{-1}(B-L_{2}D)-\bar{R}_{12}(z)\hat{N}%
_{2}(z). \label{eq4-15b}
\end{gather}%
Next, we consider $\left( zI-A_{L_{2}}\right) ^{-1}L_{2}$ and $\left(
zI-A_{L_{2}}\right) ^{-1}(B-L_{2}D).$ It is straightforward that
\begin{gather}
\left( zI-A_{L_{2}}\right) ^{-1}L_{2}-\left( zI-A_{L_{1}}\right) ^{-1}L_{1}
\notag \\
=\left( zI-A_{L_{2}}\right) ^{-1}\left( L_{2}-\left( zI-A+L_{2}C\right)
\left( zI-A_{L_{1}}\right) ^{-1}L_{1}\right) \notag \\
=\left( zI-A_{L_{2}}\right) ^{-1}\left( L_{2}\left( I-C\left(
zI-A_{L_{1}}\right) ^{-1}L_{1}\right) -\left( zI-A\right) \left(
zI-A_{L_{1}}\right) ^{-1}L_{1}\right) \notag \\
=\left( zI-A_{L_{2}}\right) ^{-1}L_{2}\hat{M}_{1}(z)-\left(
zI-A_{L_{2}}\right) ^{-1}L_{1}\hat{M}_{1}(z)\Longrightarrow \notag \\
\left( zI-A_{L_{2}}\right) ^{-1}L_{2}=\left( zI-A_{L_{1}}\right)
^{-1}L_{1}+\left( zI-A_{L_{2}}\right) ^{-1}\left( L_{2}-L_{1}\right) \hat{M}%
_{1}(z) \label{eq4-16}
\end{gather}%
as well as%
\begin{gather}
\left( zI-A_{L_{2}}\right) ^{-1}(B-L_{2}D)-\left( zI-A_{L_{1}}\right)
^{-1}(B-L_{1}D)= \notag \\
\left( zI-A_{L_{2}}\right) ^{-1}\left( I-\left( zI-A+L_{2}C\right) \left(
zI-A_{L_{1}}\right) ^{-1}\right) B-\left( \left( zI-A_{L_{2}}\right)
^{-1}L_{2}-\left( zI-A_{L_{1}}\right) ^{-1}L_{1}\right) D \notag \\
=\left( zI-A_{L_{2}}\right) ^{-1}\left( L_{1}-L_{2}\right) C\left(
zI-A_{L_{1}}\right) ^{-1}B-\left( zI-A_{L_{2}}\right) ^{-1}\left(
L_{2}-L_{1}\right) \hat{M}_{1}(z)D= \notag \\
\left( zI-A_{L_{2}}\right) ^{-1}\left( L_{1}-L_{2}\right) \left( \hat{M}%
_{1}(z)D+C\left( zI-A_{L_{1}}\right) ^{-1}B\right) =-\left(
zI-A_{L_{2}}\right) ^{-1}\left( L_{2}-L_{1}\right) \hat{N}_{1}(z) \notag \\
\Longrightarrow \left( zI-A_{L_{2}}\right) ^{-1}(B-L_{2}D)=\left(
zI-A_{L_{1}}\right) ^{-1}(B-L_{1}D)-\left( zI-A_{L_{2}}\right) ^{-1}\left(
L_{2}-L_{1}\right) \hat{N}_{1}(z). \label{eq4-17}
\end{gather}%
Furthermore, it is well-known that
\begin{align}
\left[
\begin{array}{cc}
-\hat{N}_{2}(z) & \hat{M}_{2}(z)%
\end{array}%
\right] & =Q_{21}(z)\left[
\begin{array}{cc}
-\hat{N}_{1}(z) & \hat{M}_{1}(z)%
\end{array}%
\right] \Longleftrightarrow \label{eq4-18} \\
\left[
\begin{array}{cc}
-\hat{N}_{1}(z) & \hat{M}_{1}(z)%
\end{array}%
\right] & =Q_{12}(z)\left[
\begin{array}{cc}
-\hat{N}_{2}(z) & \hat{M}_{2}(z)%
\end{array}%
\right] . \label{eq4-18a}
\end{align}%
Summarising (\ref{eq4-15a})-(\ref{eq4-18a}) leads to
\begin{gather*}
R_{12}(z)Y_{2}(z)=Y_{1}(z)-\left( F_{1}\left( zI-A_{L_{2}}\right)
^{-1}\left( L_{2}-L_{1}\right) -\bar{R}_{12}(z)Q_{21}(z)\right) \hat{M}%
_{1}(z), \\
R_{12}(z)X_{2}(z)=X_{1}(z)+\left( F_{1}\left( zI-A_{L_{2}}\right)
^{-1}\left( L_{2}-L_{1}\right) -\bar{R}_{12}(z)Q_{21}(z)\right) \hat{N}%
_{1}(z)\Longrightarrow \\
\left[
\begin{array}{cc}
X_{1}(z) & Y_{1}(z)%
\end{array}%
\right] =R_{12}(z)\left[
\begin{array}{cc}
X_{2}(z) & Y_{2}(z)%
\end{array}%
\right] +\bar{Q}_{11}(z)\left[
\begin{array}{cc}
-\hat{N}_{1}(z) & \hat{M}_{1}(z)%
\end{array}%
\right]
\end{gather*}%
as well as%
\begin{gather*}
R_{12}(z)Y_{2}(z)=Y_{1}(z)-F_{1}\left( zI-A_{L_{2}}\right) ^{-1}\left(
L_{2}-L_{1}\right) \hat{M}_{1}(z)+\bar{R}_{12}(z)\hat{M}_{2}(z), \\
R_{12}(z)X_{2}(z)=X_{1}(z)+F_{1}\left( zI-A_{L_{2}}\right) ^{-1}\left(
L_{2}-L_{1}\right) \hat{N}_{1}(z)-\bar{R}_{12}(z)\hat{N}_{2}(z)%
\Longrightarrow \\
\left[
\begin{array}{cc}
X_{1}(z) & Y_{1}(z)%
\end{array}%
\right] =R_{12}(z)\left[
\begin{array}{cc}
X_{2}(z) & Y_{2}(z)%
\end{array}%
\right] +\bar{Q}_{12}(z)\left[
\begin{array}{cc}
-\hat{N}_{2}(z) & \hat{M}_{2}(z)%
\end{array}%
\right] , \\
\bar{Q}_{12}(z)=F_{1}\left( zI-A_{L_{2}}\right) ^{-1}\left(
L_{2}-L_{1}\right) Q_{12}(z)-\bar{R}_{12}(z).
\end{gather*}%
Since
\begin{gather*}
\left( zI-A_{L_{2}}\right) ^{-1}\left( L_{2}-L_{1}\right) Q_{12}(z)=\left(
zI-A_{L_{2}}\right) ^{-1}\left( L_{2}-L_{1}\right) \left( I+C\left(
zI-A_{L_{1}}\right) ^{-1}\left( L_{2}-L_{1}\right) \right) \\
=\left( zI-A_{L_{2}}\right) ^{-1}\left( I+\left( L_{2}-L_{1}\right) C\left(
zI-A_{L_{1}}\right) ^{-1}\right) \left( L_{2}-L_{1}\right) =\left(
zI-A_{L_{1}}\right) ^{-1}\left( L_{2}-L_{1}\right) ,
\end{gather*}%
we finally have%
\begin{equation*}
\bar{Q}_{12}(z)=F_{1}\left( zI-A_{L_{1}}\right) ^{-1}\left(
L_{2}-L_{1}\right) -\left( F_{1}-F_{2}\right) \left( zI-A_{F_{2}}\right)
^{-1}L_{2}.
\end{equation*}%
The lemma is proved.
\bigskip
\section*{Abstract (Not appropriate in this style!)}%
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|
1,116,691,498,450 | arxiv |
\section{Derivation of Surface Loss Model}
In Eq.\,(1) of the main manuscript, the participation ratio for interface $i$ is given by \cite{Koch2007,Gao2008}
\begin{equation}
p_i=W^{-1}t_i\epsilon_i\int ds\,|E|^2,
\label{SuppEqParticipationDef}
\end{equation}
where the interface has a small thickness $t_i$, dielectric constant $\epsilon_i$, and length coordinate $s$ and where the resonator structure has an energy per unit length $W$.
The metal-air (ma) interface consists of the metal, a thin metal oxide with thickness $t_\text{ma}\simeq3$\,nm and dielectric constant $\epsilon_\text{ma}$, and the outer air (vacuum) with $\epsilon_\text{a} = 1$. The electric field must be perpendicular to the metal surface, and because the interface layer is thin, we also approximate it as perpendicular in the dielectric, so $E_\text{ma}=E_{\text{ma}\perp}$. The continuity of $\epsilon E$ at the metal-oxide and air interface requires $\epsilon_\text{ma} E_{\text{ma}\perp,t} = E_{\text{a}\perp,t}$. Since the oxide is thin, $E$ does not change significantly over the oxide thickness. Combining all these results gives $E_\text{ma}\approx E_{\text{a}\perp}/\epsilon_\text{ma}$, so the participation ratio of the metal-air oxide is
\begin{eqnarray}
p_\text{ma}W/t_\text{ma} &=& \epsilon_\text{ma} \int ds\ |E_\text{ma}|^2 \nonumber \\
&=& \epsilon_\text{ma} \int ds\ |E_{\text{a}\perp}/\epsilon_\text{ma}|^2 \nonumber \\
&=& \epsilon_\text{ma}^{-1} \int ds\ |E_{\text{a}\perp}|^2.\label{SuppEqMA}
\end{eqnarray}
For the metal-substrate interface, we assume a thin dielectric layer of unknown origin between the metal and substrate, which might arise from a chemical reaction of the metal to the substrate or chemi- or physi-sorbed water on the wafer surface. As before, the electric field is perpendicular to the metal and the continuity of the displacement field requires $\epsilon_\text{ms} E_{\text{ms}\perp,t} = E_{\text{s}\perp,t}$, where ms represents this dielectric and s the substrate. Thus, we find $E_\text{ms}\approx E_{\text{s}\perp}\epsilon_\text{s}/\epsilon_\text{ms}$, so the participation ratio of the metal-substrate layer is
\begin{eqnarray}
p_\text{ms}W/t_\text{ms} &=& \epsilon_\text{ms} \int ds\ |E_\text{ms}|^2 \nonumber \\
&=& \epsilon_\text{ms} \int ds\ |E_{\text{s}\perp}\epsilon_\text{s}/\epsilon_\text{ms}|^2 \nonumber \\
&=& (\epsilon_\text{s}^2/\epsilon_\text{ms}) \int ds\ |E_{\text{s}\perp}|^2.\label{SuppEqMS}
\end{eqnarray}
For the substrate-air interface, there can be a dielectric layer from surface water or other contaminants from the air, described by a subscript sa. In addition to the perpendicular electric field which obeys $E_\text{sa}\approx E_{\text{a}\perp}/\epsilon_\text{sa}$ as before, there are also parallel field components obeying the boundary condition $E_{\text{a}\parallel}= E_{\text{sa}\parallel}=E_{\text{s}\parallel}$, since the interface layer is thin. Hence, the participation ratio of the substrate-air interface layer is
\begin{eqnarray}
p_\text{sa}W/t_\text{sa} &=& \epsilon_\text{sa} \int ds\ \left( |E_{\text{sa}\parallel}|^2 +|E_{\text{sa}\perp}|^2 \right) \nonumber \\
&=& \epsilon_\text{sa} \int ds\ |E_{\text{a}\parallel}|^2 + \epsilon_\text{sa}^{-1} \smallint ds\ |E_{\text{a}\perp}|^2.\label{SuppEqSA}
\end{eqnarray}
\section{Simulation Approach}
The coplanar and microstrip structures were simulated using the electric quasi-statics component of the finite element solver COMSOL's AC/DC module \cite{COMSOL}. We simulated a two dimensional cross-section with half of the resonator, using symmetry to account for the other half. We used adaptive meshing as a starting point and then performed additional meshing around the edges and the corners.
\begin{figure}[b]
\begin{center}
\includegraphics[width=3.25in]{AreaSurfaceV3.eps}
\end{center}
\caption{Etched coplanar waveguide participation ratios. Participation ratios are given in parts per million (ppm) for the metal-air (ma), metal-substrate (ms), and substrate-air (sa) interfaces, as calculated with the surface fields approach with the metal-air-substrate corner (c) treated separately. Participation ratios for the area approach are shown separately. We assumed 3\,nm surface dielectrics with $\epsilon_\text{ma} = \epsilon_\text{ms} = \epsilon_\text{sa}=10$; the geometry is given in Table \ref{tab:loss} in cases c1 and c2. The participation ratios $p_\text{ma}+p_\text{c}$, $p_\text{ms}$, and $p_\text{sa}$ agree to within 15\%, validating the simulations against numerical errors.}
\label{FigAreaSurface}
\end{figure}
\begin{table*}[t]
\caption{\label{tab:loss} Simulation results for a variety of microwave resonators, obtained from the primary surface-based model [s], the area model [a], and a previous calculation \cite{Wang2009}. Coplanar waveguide and microstrip resonator dimensions are as indictated in Fig.\,\ref{FigDims}; the angle $\theta$ enclosed by the metal at the metal-substrate-air corner is assumed to be $90^\circ$ unless otherwise noted. The + sign in the metal-air column data is for a 3\,nm by 3\,nm area (c) at the intersection of the metal-air (ma), metal-substrate (ms), and substrate-air (sa) interfaces, and represents an entry that could be split among the three interface types. It is placed in the metal-air column since there it gives the greatest proportional uncertainty and because $p_\text{ma}+p_\text{c}$ and not $p_\text{ma}$ alone is comparable between the area and surface models. We assume $\epsilon_\text{s}=10$ and surface dielectrics with $\epsilon = 10$, thickness 3\,nm, and loss tangent 0.002.}
\begin{tabular}{llccccccc}
\hline
\hline
type \ \ \ \ \ \ \ \ \ & dimensions\ & capacitance &
\ \ metal-air \ & metal-sub. \ & \ sub.-air \ &
\ \ loss metal-air \ & \ loss metal-sub. \ & \ loss sub.-air \ \\
& ($\mu$m) & pF/m & $p_\text{ma}$ (ppm) & $p_\text{ms}$ (ppm) & $p_\text{sa}$ (ppm) & $\times 10^6$ & $\times 10^6$ & $\times 10^6$ \\
\hline
\hline
coplanar & $w, h, g, d$ \\
\hline
c1 [a] & 5, 0.1, 2, 0.01 & 162 & 119+167 & 2200 & 2541 & 0.24+0.33 & 4.40 & 5.08 \\
\hline
c2 [s] & 5, 0.1, 2, 0.01 & 162 & 56+196 & 2322 & 2624 & 0.11+0.39 & 4.64 & 5.25 \\
\hline
c3 [a] & 5, 0.1, 2, 0 & 163 & 290+387 & 2234 & 2286 & 0.58+0.77 & 4.47 & 4.57 \\
\hline
c4 [s] & 5, 0.1, 2, 0 & 163 & 52+662 & 3065 & 2011 & 0.10+1.32 & 6.13 & 4.02 \\
\hline
c5 [a\cite{Wang2009}] & 5, 0.1, 2, 0 & & 600 & & 2000 & 1.2 & & 4.0 \\
\hline
c6 [s] & 5, 0.1, 2, 2 & 104 & 44+6 & 2690 & 1032 & 0.09+0.01 & 5.38 & 2.06 \\
\hline
c7 [s] & 5, 0.025, 2, 0.01 & 161 & 55+209 & 2376 & 2735 & 0.11+0.42 & 4.75 & 5.47 \\
\hline
c8 [s] & 2, 0.1, 20, 0.01 & 68 & 33+111 & 1394 & 1594 & 0.07+0.22 & 2.79 & 3.19 \\
\hline
c9 [s] & 5, 0.1, 20, 0.01 & 85 & 18+60 & 847 & 928 & 0.04+0.12 & 1.69 & 1.85 \\
\hline
c10 [s] & 5, 0.1, 20, 0 & 85 & 17+207 & 1091 & 764 & 0.03+0.41 & 2.18 & 1.53 \\
\hline
c11 [s] & 5, 0.1, 20, 0,$\theta=45^\circ$ & 169 & 32+1414 & 3727 & 2267 & 0.06+2.83 & 7.45 & 4.53 \\
\hline
c12 [s] & 5, 0.1, 20, 0,$\theta=135^\circ$ & 158 & 104+695 & 2841 & 1963 & 0.21+1.39 & 5.68 & 3.93 \\
\hline
\hline
microstrip & $w, h, s, d$ \\
\hline
m1 [s] & 20, 0.2, 2, 0.01 & 985 & 10+45 & 3155 & 526 & 0.02+0.09 & 6.31 & 1.05 \\
\hline
m2 [s] & 20, 0.2, 0.2, 0.01 & 8964 & 7+55 & 29942 & 409 & 0.01+0.11 & 59.9 & 0.82 \\
\hline
m3 [s] & 10, 0.2, 2, 0.01 & 539 & 19+82 & 3301 & 964 & 0.04+0.16 & 6.60 & 1.93 \\
\hline
m4 [s] & 20, 0.02, 2, 0.01 & 983 & 10+49 & 3185 & 557 & 0.02+0.10 & 6.37 & 1.11 \\
\hline
m5 [s] & 20, 0.2, 2, 0 & 987 & 9.3+189 & 3301 & 397 & 0.02+0.38 & 6.60 & 0.79 \\
\hline
m6 [s] & 20, 0.2, 2, 2 & 914 & 4.6+1.9 & 2924 & 291 & 0.009+0.004 & 5.85 & 0.58 \\
\hline
m7 [s] & 20, 0.2, 2, -2 & 1006 & 1.5+3.2 & 3192 & 241 & 0.003+0.006 & 6.38 & 0.48 \\
\hline
\hline
\end{tabular}
\end{table*}
To determine the participation ratios, we initially treated the interfaces as 3\,nm thick dielectrics with dielectric constant $\epsilon=10$. However, this area approach is computationally expensive since it requires meshing on the nanometer scale over distances of hundreds of microns. As such, we primarily calculated the participation ratios by computing the electric field on all boundary interfaces with the interface dielectrics excluded from the model and then applying Eqs.\,(\ref{SuppEqMA})-(\ref{SuppEqSA}). This is less computationally expensive as there is no thin dielectric layer explicitly included at the interfaces which needs to be carefully meshed. As indicated in Fig.\,\ref{FigAreaSurface} and Table \ref{tab:loss}, for two different pairs of simulations, $p_\text{ms}$, $p_\text{sa}$, and $p_\text{ma}+p_\text{c}$ as calculated by these two approaches typically agree to within 15\%, although $p_\text{ma}$ alone differs by a factor of at least two. This means that the total metal-air interface includes an indeterminate significant fraction of the corners.
We also assumed all surfaces were smooth. Simulations indicate that incorporating smooth bumps on the order of the interface thickness increase the participation ratios and thus loss by a factor of order unity. The value of this factor depends on the interface thickness and on the defect density.
\section{Results for Different Geometries}
Numerical results for a variety of coplanar and microstrip resonator geometries are presented in Table \ref{tab:loss}. We have calculated the participation ratio $p_i$ and loss $p_i\tan\delta$ for a dielectric with thickness 3\,nm, dielectric constant $\epsilon = 10$, and loss tangent $\tan\delta=0.002$, typical values for metal or silicon oxides \cite{Wang2009}. Since the participation ratio is proportional to thickness, these values can easily be scaled for other parameters. Assuming these parameters, the loss from the metal-air interfaces is typically below $10^{-6}$. The second quantity in the sum for the metal-air columns arises from the 3\,nm by 3\,nm corner at the metal/substrate/air interface. As this is a small area, it shows the sensitivity of the loss to this inside corner and indicates the uncertainty in the metal-air prediction.
\begin{figure}
\begin{center}
\includegraphics[width=3.25in]{AllDimsV4.eps}
\end{center}
\caption{Coplanar and microstrip dimensions. (\textbf{a}) Dimensions for coplanar waveguide simulations. We assume $\theta=90^\circ$ unless otherwise specified. (\textbf{b}) Dimensions for microstrip simulations.}
\label{FigDims}
\end{figure}
For coplanar resonators, a significant effect on total loss comes from etching into the substrate within the coplanar gap. This is important because the etching reduces the divergence of the fields at the corner, as shown in Fig.\,2 of the main paper, and because the etching reduces the fields parallel to the metal-substrate interface due to the increased distance from the metal traces. Between the pair of cases c1 and c2 and the pair c3 and c4, it is apparent that $p_\text{ma}$ is reduced by a factor of 2-3. Further deep etching (2\,$\mu$m, case c6) reduces $p_\text{ma}$ by an additional factor of 5 and $p_\text{sa}$ by a factor of 2 while leaving unchanged $p_\text{ms}$. This change has been experimentally tested for Si substrates \cite{Barends2010}, where a feature in the loss versus power saturation curve was identified as substrate-air loss. After etching the substrate caused the feature to disappear and resulted in half the loss, which is consistent with the halving of $p_\text{sa}$ in case c6. This result is consistent with the discussion in the main paper indicating that the substrate-air interface is a dominant loss mechanism.
As experimentally seen by the Delft group \cite{Barends2010}, different geometries of coplanar resonators result in somewhat different losses. In comparing cases c1 and c2, there is minimal difference between the area and surface models except for some change in $p_\text{ma}$ from the inside corner, and comparing cases c4 and c5 indicate that these data are similar to previous work from our group \cite{Wang2009}. The metal thickness is shown to have little effect in case c7. Case c8 shows that decreasing the width makes the loss increase. However, in case c9, increasing the gap from 2 to 20\,$\mu$m gave roughly a factor of 3 reduction in all losses.
Sloped sidewalls are also seen to give different losses by comparing cases c11, c4, and c12, where the metal angle $\theta$ at the substrate-corner corner (Fig.\,\ref{FigDims}) was varied. All interfaces except the metal-air interface had the greatest loss in case c11, where the sidewall slope was $\theta=45^\circ$, and the least loss in case c12, where the overetched sidewalls had $\theta=135^\circ$. This is expected since the electric field is predicted to scale with the distance $r$ from corners as $r^{-3/7}$ for $\theta=45^\circ$, $r^{-1/3}$ for $\theta=90^\circ$, and $r^{-1/5}$ for $\theta=135^\circ$ \cite{Jackson}, thus giving the least field divergence, and thus the lowest loss, at the overetched corner. The metal-air interface exhibits the opposite trend, which is consistent with the same argument for the top corner.
Microstrip resonators show significantly higher capacitance per length, which for the same interface energy results in lower loss. In the base case m1, $p_\text{sa}$ and $p_\text{ma}+p_\text{c}$ are both much less than the corresponding values for coplanar resonators. However, microstrip resonators also have a larger $p_\text{ms}$ contribution, approximately equal to the distance ratio $2t/s$ for oxide thickness $t$. In fact, the thin dielectric of case m2 compared to the base case m1 has a capacitance and $p_\text{ms}$ approximately 10 times that in case m1 and similar $p_\text{sa}$ and $p_\text{ma}$. Another geometric parameter that is important is the width $w$, with which $p_\text{ma}$ and $p_\text{sa}$ scale inversely, as shown by case m3. However, as indicated in case m4, changing the metal height has minimal effect on the participation ratios.
As with coplanar resonators, changing the depth of etching of the exposed substrate also has a significant effect on loss. The lack of dielectric etching in case m5 results in an increase in $p_\text{ma}$ from the inside corner of the metal, which is mostly compensated for by a decrease in $p_\text{sa}$. Case m6 shows that a deeper etch significantly reduces both of these terms. If, instead of etching the exposed dielectric, the dielectric extends up the sidewall of the metal (as in case m7), the losses are similar to that of a deep etch. Overall, the improvements with etching the exposed dielectric for microstrip resonators are primarily at the metal-air interface, but some effect (factor of 2) is seen for the substrate-air interface. However, the most important concern for a microstrip geometry is a low loss metal-substrate interface, as this loss was dominant in all cases.
\section{Derivation of Power Dependence of Participation Ratios}
One potential way to determine which interface dominates the loss is by measuring the power dependence of the loss \cite{Barends2010}. When the saturation of surface two-level states (TLSs) at the field $E_s$ is considered, the surface participation ratio $p$ is given by
\begin{equation}
\frac{p}{t\epsilon} = \int_{r_c}^{r_0} \frac{E^2\,dr}{\sqrt{1+E^2/E_s^2}},
\label{SuppEqSurfaceDef}
\end{equation}
where the interface has thickness $t$ and dielectric constant $\epsilon$. The electric field is assumed to be dominated by a feature such as a corner with length coordinate $r$ from this feature, characteristic length $r_0$, and cutoff length $r_c<r_0$.
For a square corner, the field scales \cite{Jackson} as $E=E_0(r/r_0)^{-1/3}$, so substituting this into Eq.\,(\ref{SuppEqSurfaceDef}) gives a surface participation ratio of
\begin{eqnarray}
\frac{p}{t\epsilon}
&=& E_0^2 \int_{r_c}^{r_0} \frac{(r_0/r)^{2/3}\,dr}{\sqrt{1+(E_0^2/E_s^2)(r_0/r)^{2/3}}} \nonumber \\
&=& 3E_0^2r_0 \left[\sqrt{1+\frac{E_0^2}{E_s^2}} - \sqrt{\left(\frac{r_c}{r_0}\right)^{2/3}+\frac{E_0^2}{E_s^2}}\right].
\end{eqnarray}
For a thin edge at distances much greater than the film thickness, the field scales \cite{Jackson} as $E=E_0(r/r_0)^{-1/2}$, so substituting this into Eq.\,(\ref{SuppEqSurfaceDef}) gives a surface participation ratio of
\begin{eqnarray}
\frac{p}{t\epsilon}
&=& E_0^2 \int_{r_c}^{r_0} \frac{(r_0/r)\,dr}{\sqrt{1+(E_0^2/E_s^2)(r_0/r)}} \nonumber \\
&=& 2E_0^2r_0 \log \frac {1+\sqrt{1+\frac{E_0^2}{E_s^2}}} {\sqrt{\frac{r_c}{r_0}}+\sqrt{\frac{r_c}{r_0}+\frac{E_0^2}{E_s^2}}}.
\end{eqnarray}
|
1,116,691,498,451 | arxiv | \section{Introduction}
Consider the linear regression model
\[
y_i = \beta_0 + x_i^{\top}\beta + \varepsilon_i
\]
where $x_i$ is a $p$-dimensional vector of covariates,
$(\beta_0, \beta)$ are regression coefficients, and $\varepsilon_i$ is the random error. We are interested in the high dimensional case where $p \gg n$ and the model is sparse in the sense that only a small proportion of the coefficients are nonzero. In such a scenario, a key task is identifying and estimating the nonzero coefficients. A popular approach is the penalized regression
\begin{equation}\label{gen_loss}
\min_{\beta_{0},\beta}\frac{1}{n}\sum_{i}\ell(y_{i}-\beta_{0}-x_{i}^{\top}\beta)+ \lambda P(\beta),
\end{equation}
where $\ell$ is a generic loss function and $p$ is a penalty function with a tuning parameter $\lambda \ge 0$.
We consider the elastic-net penalty \citep{zou2005regularization}
\[
P(\beta) \equiv P_{\alpha}(\beta) = \alpha\|\beta\|_{1}+(1-\alpha)\frac{1}{2}\|\beta\|_{2}^{2}, 0 \le \alpha \le 1,
\]
which is a convex combination of the lasso \citep{tibshirani1996regression} ($\alpha = 1$) and the ridge penalty ($\alpha = 0$).
A common choice for $\ell$ is the squared loss $\ell(t) = t^2/2$, corresponding to the least squares regression in classical regression literature. Although the squared loss is analytically simple, it is not suitable for data in the presence of outliers or heterogeneity. Instead, we could consider two widely used robust alternatives, the Huber loss \citep{huber1973robust} and the quantile loss \citep{koenker1978regression}.
The Huber loss is
\begin{equation}\label{huberfunc}
\ell(t) \equiv h_{\gamma}(t)=\begin{cases}
\frac{t^{2}}{2\gamma}, & \text{if } |t|\leq\gamma, \\
|t|-\frac{\gamma}{2}, & \text{if } |t|>\gamma,
\end{cases}
\end{equation}
where $\gamma >0$ is a given constant.
This function is quadratic for $|t|\leq \gamma$ and linear for $|t|>\gamma$. In addition, it is convex and first-order differentiable. These features allow it to combine analytical tractability of the squared loss for the least squares and outlier-robustness of the absolute loss for the LAD regression.
The quantile loss is
\begin{equation}\label{quantfunc}
\ell(t) \equiv \rho_{\tau}(t)=t(\tau-I(t<0)), t \in \mathbb{R},
\end{equation}
where $0 < \tau < 1$.
This is a generalization of the absolute loss with $\tau=1/2$.
Rather than the conditional mean of the response given the covariates, quantile regression models conditional quantiles. For heterogeneous data, the functional relationship between the response and the covariates may vary in different segments of its conditional distribution. By choosing different $\tau$, quantile regression provides a powerful technique for exploring data heterogeneity in addition to outlier-robustness.
The theoretical properties of these two regression models have been systematically studied, yet there are relatively few researches on the algorithmic aspect, especially the penalized versions for high-dimensional data.
\cite{holland1977robust} proposed an iteratively re-weighted least squares algorithm for the unpenalized Huber loss regression. However, this algorithm does not have a natural extension to the penalized version. For unpenalized quantile regression, \cite{portnoy1997gaussian} formulated its dual form as a linear programming problem and proposed an interior point method to solve it. The lasso penalized version can be shown to have a similar dual form, except that it becomes $(n+p)$-dimensional with $p$ extra constraints due to the penalty. Thus it can be solved using the same algorithm and this extension was implemented in
the R package \texttt{quantreg} (\url{http://cloud.r-project.org/package=quantreg}). However, it is not clear if this
approach is scalable to high-dimensional problems. \cite{osborne2011homotopy} proposed a homotopy algorithm for computing solution paths of lasso penalized quantile regression, where the lasso penalty was formulated as a constraint $\sum_{j=1}^p |\beta_j| \leq \kappa$, which is not directly comparable with the unconstrained formulation considered here.
In recent years coordinate descent algorithms have proven to be very effective for pathwise optimization of penalized regression models, see for example,
\cite{friedman2007pathwise} for lasso and fused lasso penalized least squares, \cite{friedman2010regularization} for elastic-net penalized GLM, and
\cite{breheny2011coordinate} for nonconvex penalized least squares and logistic regression.
The loss functions considered by these authors are either quadratic, or twice differentiable which can be approximated quadratically via Taylor expansion. Hence the coordinate descent iterations have close-form solutions. However, the Huber loss is only first-order differentiable and the quantile loss is nondifferentiable, hence the above approach does not work. \cite{wu2008coordinate} proposed a coordinate descent algorithm for lasso penalized LAD regression that amounts to computing a weighted median at each iteration, but did not provide any guarantee for convergence. Recently \cite{peng2015iterative} proposed a QICD algorithm for nonconvex penalized quantile regression that majorizes the penalty functions by weighted lasso and then solves the problem with coordinate descent. The authors proved convergence of QICD to a stationary point, for which the majorization step plays a critical role. But when the lasso penalty is used, which does not need to be majorized, the algorithm becomes the same as the one in \cite{wu2008coordinate}. In addition, it appears that neither algorithm can be easily generalized to the elastic-net penalty with $0< \alpha < 1$.
In this paper, we propose a novel semismooth Newton coordinate descent (SNCD) algorithm for computing solution paths of the elastic-net penalized Huber loss regression and quantile regression. This algorithm combines the coordinate descent algorithm with the semismooth Newton algorithm (SNA) for solving nonsmooth equations. It is highly efficient and scalable in high-dimensional settings. Unlike a typical coordinate descent method which only updates the primal variable $\beta$, the SNCD utilizes both the primal and the dual information (via subgradient) in its iterations. In addition, an adaptive version of the strong rule \citep{tibshirani2012strong} for screening predictors is incorporated to gain extra efficiency. We also provide an implementation of SNCD through a publicly available R package \texttt{hqreg} (\url{https://cran.r-project.org/web/packages/hqreg/index.html}) which currently supports the Huber loss, the quantile loss and the squared loss.
This algorithm can be generalized to other problems with nonsmooth loss functions, like the linear support vector machine with the hinge loss.
The rest of this paper is organized as follows.
In section \ref{section_huber} we introduce SNCD for the penalized Huber loss regression and establish its convergence.
In section \ref{section_quantile} we extend SNCD to penalized quantile regression. Section \ref{section_asr} describes the adaptive strong rule. In section \ref{section_numerical} we investigate the performance of \texttt{hqreg}, our implementation of SNCD, through simulation studies and real datasets.
\section{SNCD for Penalized Huber Loss Regression}
\label{section_huber}
\subsection{Background on Newton Derivatives and SNA}
Based on the concepts of generalized Jacobian \citep{clarke1983optimization} and semismoothness \citep{mifflin1977semismooth}, \cite{qi1993nonsmooth} established superlinear convergence of a Newton-type method for solving finite-dimensional nonsmooth equations, hence the name Semismooth Newton Algorithm (SNA).
The Newton differentiability was introduced later for more general problems including infinite-dimensional cases \citep{chen2000smoothing, ito2008lagrange}. It has a simpler
formulation and is actually a milder condition than semismoothness. Newton derivatives can be calculated
via basic algebra and chain rules as indicated in Lemmas \ref{chain}, \ref{basic} and \ref{piecewise} in Appendix A.
\begin{definition}\label{slant}
A function $F:\mathbb{R}^{m}\rightarrow\mathbb{R}^{l}$ is said to be \emph{Newton differentiable} at $z\in\mathbb{R}^{m}$ if there exists an open neighborhood $\mathcal{N}(z)$ and a mapping $H:\mathcal{N}(z)\rightarrow\mathbb{R}^{l\times m}$ such that $\{H(z+h):z+h\in \mathcal{N}(z), h\neq 0\}$ is uniformly bounded in spectral norm induced by the Euclidean norm and
\[
\|F(z+h)-F(z)-H(z+h)h\|_{2}=o(\|h\|_{2})\quad\text{as}\; h\rightarrow 0.
\]
Here $H$ is called a \emph{Newton derivative} for $F$ at $z$. The set of all Newton derivatives at $z$ is denoted as $\nabla_N F(z)$.
\end{definition}
A function $F:\mathbb{R}^{m}\rightarrow\mathbb{R}^{l}$ is said to be \emph{locally Lipschitz continuous} at $z$ if there exists $L(z)>0$ such that for all sufficiently small $h$,
\[
\|F(z+h)-F(z)\|_{2}\leq L\|h\|_{2}.
\]
Then $F$
is Newton differentiable at $z$ if and only if $F$ is locally Lipschitz continuous at $z$ \citep{chen2000smoothing}.
This gives a simple characterization of the Newton differentiability.
The following result due to \cite{chen2000smoothing} establishes the superlinear convergence of SNA under the Newton differentiability.
\begin{theorem}\label{superlinear}
Suppose that $F:\mathbb{R}^{m}\rightarrow\mathbb{R}^{m}$ is Newton differentiable at a solution $z^*$ of $F(z)=\mathbf{0}$. Let $H$ be a Newton derivative for $F$ at $z^*$.
Suppose there exists a neighborhood $\mathcal{N}(z^*)$ and $M>0$ such that $H(z)$ is nonsingular and $\|H(z)^{-1}\|\leq M$ for all $z \in \mathcal{N}(z^*)$, then the Newton-type iteration
\[
z^{k+1}=z^{k}-H(z^{k})^{-1}F(z^{k}),\ k=0, 1, \ldots
\]
converges superlinearly to $z^*$ provided that $\|z^0 - z^*\|_2$ is sufficiently small,
where $z^0$ is the initial value.
\end{theorem}
\subsection{Algorithm}
\subsubsection{Description}
Consider the Huber loss $\ell = h_\gamma$, then \eqref{gen_loss} becomes
\begin{equation}\label{huber}
\min_{\beta_{0},\beta} f_H(\beta_0, \beta) = \frac{1}{n}\sum_{i}h_\gamma(y_{i}-\beta_{0}-x_{i}^{\top}\beta)+ \lambda P_{\alpha}(\beta).
\end{equation}
Fix $\lambda$ and $\alpha$, and denote the optimizer by $(\widehat{\beta}_0, \widehat{\beta})$. Since the objective function in \eqref{huber} is convex, $(\widehat{\beta}_0, \widehat{\beta})$
satisfies the necessary and sufficient Karush-Kuhn-Tucker (KKT) conditions.
Let $\partial|t|$ denote the set of subgradients of the absolute value
function $|\cdot|$ at $t$, then it can be shown that
\begin{equation}
s \in \partial |t| \text{ if and only if } t = S(t+s),
\end{equation}
where $S$ is the soft-thresholding operator with threshold 1, i.e. $S(z)=\text{sgn}(z)(|z|-1)_{+}$.
As shown in Appendix A, combining this fact with some other convex analysis concepts \citep{rockafellar1970convex, combettes2005signal},
the KKT conditions of \eqref{huber} can be written as
\begin{equation}
\label{KKT1_reformulated}
\begin{cases}
-\frac{1}{n}\sum_{i}h^\prime_{\gamma}(y_{i}-\widehat{\beta}_{0}-x_{i}^{\top}\widehat{\beta}) = 0, \\
-\frac{1}{n}\sum_{i}h^\prime_{\gamma}(y_{i}-\widehat{\beta}_{0}-x_{i}^{\top}\widehat{\beta})x_{ij}+\lambda\alpha \widehat{s}_{j}+\lambda(1-\alpha)\widehat{\beta}_{j} = 0, \\
\widehat{\beta}_{j}-S(\widehat{\beta}_{j}+\widehat{s}_{j})=0, \quad j=1,\ldots, p,
\end{cases}
\end{equation}
where $\widehat{s}_j \in \partial |\widehat{\beta}_j|$ and $h^\prime_{\gamma}(\cdot)$, the derivative of $h_\gamma(\cdot)$, is given by
\[
h^\prime_{\gamma}(t)=\begin{cases}
\frac{t}{\gamma}, & \text{if } |t|\leq\gamma,\\
\text{sgn}(t), & \text{if } |t|>\gamma.
\end{cases}
\]
In this way the optimization problem \eqref{huber} is transformed into a root finding problem for a system of nonsmooth equations \eqref{KKT1_reformulated}. A straightforward approach is applying
SNA to the entire system of equations. As discussed later in section \ref{subsection_compare}, this approach contains many matrix operations that cause $O(np^2)$ computational cost per iteration, which severely limits its scalability.
For better efficiency and scalability, we propose a new algorithm, Semismooth Newton Coordinate Descent (SNCD), that combines SNA with cyclic coordinate descent in solving these equations. Similar to the Gauss-Seidel method for linear equations, SNCD solves the equations of \eqref{KKT1_reformulated} in a cyclic fashion to avoid cumbersome matrix operations. We cycle through $(\beta_0, \beta, s)$ in a pairwise fashion: at each step, a pair $(\beta_j, s_j)$ (and $\beta_0$ by itself) is updated by solving the corresponding part of \eqref{KKT1_reformulated}, while the other variables are fixed at their current values $\tilde{\beta}_k, \tilde{s}_k, k \neq j$. Specifically, we solve the following equations at each step:
\begin{itemize}
\item For $(\beta_j, s_j)$:
\begin{equation}\label{KKT_bj}
\begin{cases}
-\frac{1}{n}\sum_{i}h^\prime_{\gamma}(\tilde{r}_i + x_{ij}\tilde{\beta}_j- x_{ij}\beta_j)x_{ij}+\lambda\alpha s_{j}+\lambda(1-\alpha)\beta_{j} = 0, \\
\beta_j - S(\beta_j+s_j) = 0,
\end{cases}
\end{equation}
\item For $\beta_0$:
\begin{equation}\label{KKT_b0}
-\frac{1}{n}\sum_{i}h^\prime_{\gamma}(\tilde{r}_i + \tilde{\beta}_0- \beta_0) = 0,
\end{equation}
\end{itemize}
where $\tilde{r}_i = y_i - \tilde{\beta}_0-x_j^\top\tilde{\beta}, \quad i = 1,\ldots, n$.
Note that \eqref{KKT_bj} is exactly the KKT conditions of
\[
\min_{\beta_j}f_H(\ldots, \tilde{\beta}_{j-1}, \beta_j, \tilde{\beta}_{j+1}, \ldots),
\]
and \eqref{KKT_b0} the KKT condition of
\[
\min_{\beta_0} f_H(\beta_0, \tilde{\beta}_{1}, \ldots).
\]
Hence SNCD can be seen as a special type of coordinate descent.
Denote
\begin{equation}\label{psi}
\psi_{\gamma}(t)=\frac{1}{\gamma}I(|t|\leq\gamma),
\end{equation}
then $\psi_\gamma \in \nabla_N h_\gamma^\prime(t), \forall t\in \mathbb{R}$.
The SNCD iterations proceed as follows:
\begin{enumerate}[(i)]
\item
Updating $\beta_0$. Let
\[
F_0(z;\tilde{\beta}) = -\frac{1}{n}\sum_{i}h^\prime_{\gamma}(\tilde{r}_i + \tilde{\beta}_0- z).
\]
Since
\[
H_0(z) = \frac{1}{n}\sum_{i}\psi_\gamma(\tilde{r}_{i}+\tilde{\beta}_{0}-z)\in\nabla_N(F_0(z)),
\]
we update $\beta_0$ by solving \eqref{KKT_b0} via SNA
\[
\beta_{0}\leftarrow \tilde{\beta}_0 - H_0(\tilde{\beta}_0)^{-1}F_0(\tilde{\beta}_0) =
\tilde{\beta}_{0}+
\frac{\sum_{i}h^\prime(\tilde{r}_{i})}{\sum_{i}\psi_\gamma(\tilde{r}_{i})}.
\]
\item
Updating $(\beta_j, s_j)$. Let
\[
F_j(z;\tilde{\beta})=\left[\begin{array}{c}
-\frac{1}{n}\sum_{i} h_\gamma^\prime(\tilde{r}_{i}+x_{ij}\tilde{\beta}_{j}-x_{ij}z_1)x_{ij}+\lambda\alpha z_2+\lambda(1-\alpha)z_1\\
z_1-S(z_1+z_2)
\end{array}\right],
\]
where $z=(z_1, z_2)^{\top}$.
Since
\begin{equation}\label{soft-thresholding decompose}
z_1 - S(z_1 + z_2) =
\begin{cases}
-z_2 + \text{sgn}(z_1+z_2) & \text{if } |z_1 + z_2|>1,\\
z_1 & \text{if } |z_1+z_2|\leq 1,
\end{cases}
\end{equation}
we solve for $(\beta_j, s_j)$ from \eqref{KKT_bj} via SNA in two types of updates:
\begin{enumerate}
\item
$|\tilde{\beta}_{j}+\tilde{s}_{j}|>1$. For $z$ with $|z_1 + z_2|>1$, a Newton derivative of $F_j$ at $z$ is
\begin{equation}
\label{ND1}
H_{j}(z)=\left[\begin{array}{cc}
\frac{1}{n}
\sum_i\psi_\gamma(\tilde{r}_{i}+
x_{ij}\tilde{\beta}_{j}-x_{ij}z_{1})x_{ij}^{2}+\lambda(1-\alpha) & \lambda\alpha\\
0 & -1
\end{array}\right]\in \nabla_N F_{j}(z).
\end{equation}
Hence the update is
\[
\left[\begin{array}{c} \beta_{j} \\ s_{j} \end{array}\right]
\leftarrow
\left[\begin{array}{c} \tilde{\beta}_{j} \\ \tilde{s}_{j} \end{array}\right]
- H_j(\tilde{\beta}_j, \tilde{s}_j)^{-1} F_j(\tilde{\beta}_j, \tilde{s}_j) =
\left[\begin{array}{c}
\tilde{\beta}_{j}+
\frac{\frac{1}{n}\sum_{i}h_\gamma^\prime(\tilde{r}_{i})x_{ij}-\lambda\alpha \text{sgn}(\tilde{\beta}_{j}+\tilde{s}_{j})-\lambda(1-\alpha)\tilde{\beta}_{j}}
{\frac{1}{n}\sum_{i}\psi_\gamma(\tilde{r}_i)x_{ij}^{2}+\lambda(1-\alpha)} \\
\text{sgn}(\tilde{\beta}_{j}+\tilde{s}_{j})
\end{array}
\right].
\]
\item
$|\tilde{\beta}_{j}+\tilde{s}_{j}|\leq 1$.
For $z$ with $|z_1 + z_2|\leq1$, a Newton derivative of $F_j$ at $z$ is
\begin{equation}
\label{ND2}
H_{j}(z)=\left[\begin{array}{cc}
\frac{1}{n}
\sum_i\psi_\gamma(\tilde{r}_{i}+x_{ij}\tilde{\beta}_{j}-x_{ij}z_{1})x_{ij}^{2}+
\lambda(1-\alpha) & \lambda\alpha, \\
1 & 0
\end{array}\right]\in \nabla_N F_{j}(z).
\end{equation}
Hence the update is
\[
\left[\begin{array}{c} \beta_{j} \\ s_{j} \end{array}\right]
\leftarrow
\left[\begin{array}{c} \tilde{\beta}_{j} \\ \tilde{s}_{j} \end{array}\right]
- H_j(\tilde{\beta}_j, \tilde{s}_j)^{-1} F_j(\tilde{\beta}_j, \tilde{s}_j) =
\left[\begin{array}{c}
0 \\
\frac{
\frac{1}{n}\sum_ih^\prime_\gamma(\tilde{r}_{i})x_{ij}
+\tilde{\beta}_{j}\cdot\frac{1}{n}
\sum_{i}\psi_\gamma(\tilde{r}_{i})x_{ij}^{2}}{\lambda\alpha}
\end{array}
\right].
\]
\end{enumerate}
\end{enumerate}
\subsubsection{Convergence}
Since SNCD fits in the general coordinate descent framework, its convergence property follows
from the convergence results for coordinate descent
\citep{tseng2001convergence}.
To apply the results, we first show that the optimization problem is of the form
\[
\min\; f(z_{1},\ldots,z_{m})=f_{0}(z_{1},\ldots,z_{m})+\sum_{j=1}^m f_{j}(z_{j}),
\]
where $f_0, f_1, \ldots, f_m$ are convex, $f_0$ is first-order differentiable and the level set $\{z: f(z)\leq f(z^0)\}$ is bounded given any initial point $z^0$. A key fact to notice about this formulation is that the nondifferentiable part $\sum_j f_j(z_j)$ must be separable. The penalized Huber loss regression model in \eqref{huber} clearly satisfies these conditions.
At each coordinate update, SNA is applied to solve the equations, which requires nonsingularity of the Newton derivative and the uniform boundedness of its inverse. When updating $\beta_0$, these requirements are met if $|\sum_{i}\psi_\gamma(y_{i}-\beta_{0}-x_{i}^{\top}\beta)|$ is bounded away from 0. This is true as long as there is at least one observation with $|y_i-\beta_0 - x_i^\top \beta|\leq \gamma$. When updating $\beta_j, s_j$, it can be shown via some algebra that a sufficient condition is $0<\alpha<1$ and $\psi_\gamma$ is bounded. The latter always holds since $\psi_\gamma(t) \in \{0,1/\gamma\}$ for any $t$.
In order for this local SNA strategy to work well, we also need the starting point and the optimal point in each coordinate update to be sufficiently close. Denote the globally initial $f_H$ value by $f_H^0$. Since $f_H$ decreases along SNCD iterations and the level set $\{(\beta_0, \beta): f_H(\beta_0, \beta) \leq f_H^0 \}$ is bounded, the closeness requirement is satisfied if the diameter of the set is sufficiently small.
The above discussions are summarized in the following result.
\begin{theorem}\label{SNCD_huber}
For problem \eqref{huber}, let $\lambda > 0$, $\alpha \in (0,1)$ and the initial $f_H$ value be $f_H^0$. Assume for every point $(\beta_0, \beta)$ in the level set $\mathcal{L} = \{(\beta_0, \beta): f_H(\beta_0, \beta) \leq f_H^0 \}$ there exists $i\in\{1,\ldots,n\}$ such that $|y_{i}-\beta_{0}-x_{i}^{\top}\beta|\leq\gamma$. Then SNCD iterations converge to a global minimizer provided that the diameter of $\mathcal{L}$ is sufficiently small.
\end{theorem}
\subsubsection{Pathwise Optimization}\label{path}
To actually implement the algorithm, we still need to consider an important issue: its convergence relies on a good initial point, which is usually not guaranteed in practice. For low-dimensional problems we can use line search to ensure global convergence with an arbitrary initial point, but since line search methods involve considerable amounts of function and gradient evaluations, they are not well-suited for high-dimensional cases.
The strategy of pathwise optimization with warm start can help globalize the convergence of the algorithm. With a decreasing sequence of $\lambda$ values, this strategy sequentially solves the optimization problem at each $\lambda_k$ using the optimizer at the previous $\lambda_{k-1}$ as the initial value.
When $\lambda_{k-1}$, $\lambda_k$ are reasonably close, the initial point $(\widehat{\beta}_{0}(\lambda_{k-1}),\widehat{\beta}(\lambda_{k-1}))$ will be near the optimizer $(\widehat{\beta}_{0}(\lambda_{k}),\widehat{\beta}(\lambda_k))$ as well.
Hence each optimization problem along the path is warm-started with a good initial point, and fast convergence can be achieved. This strategy generates a solution path, which in turn will be useful for tuning parameter selection.
\subsection{Comparison with SNA and Existing Coordinate Descent Type Algorithms}\label{subsection_compare}
\subsubsection{SNA for Penalized Huber Loss Regression and Its Computational Bottleneck}
\label{subsection SNA}
Denote $\mathcal{S}(z)
=(S(z_{1}),\ldots,S(z_{p}))^\top$ and
$d(\beta_0,\beta)=(h^\prime_{\gamma}(y_{1}-\beta_{0}-x_{1}^{\top}\beta),\ldots,h^\prime_{\gamma}(y_{n}-\beta_{0}-x_{n}^{\top}\beta))^\top$, then the KKT conditions \eqref{KKT1_reformulated} can be written compactly as
\begin{equation} \label{KKT_vector}
F(\beta_0,\beta,s) = \left[
\begin{array}{l}
-\frac{1}{n}1^{\top}d(\beta_0,\beta)\\
-\frac{1}{n}X^{\top}d(\beta_0,\beta)+\lambda\alpha s+\lambda(1-\alpha)\beta\\
\beta-\mathcal{S}(\beta+s)
\end{array}\right]
= \mathbf{0}.
\end{equation}
It is easy to verify $F$ is Newton differentiable, then SNA can be directly applied here for solving $F(\beta_0,\beta,s)=\mathbf{0}$. See Appendix B for details.
In terms of computational cost, the first concern is about matrix inversion, since the Newton derivative of $F$ is a $(1+2p)\times(1+2p)$ matrix, for which inversion becomes intractable when $p$ is large. However, the decomposition \eqref{soft-thresholding decompose} leads to an ``active set strategy'' that helps reduce the dimension. Given the $k$th iteration $(\beta_0^k, \beta^k, s^k)$, define the active set $A_k$ and its complement $B_k$ by
\begin{equation}\label{AB_SNA}
A_k= \{ j:|\beta_{j}^k+s_{j}^k|>1\} \text{ and } B_k= \{j:|\beta_{j}^k+s_{j}^k|\leq 1\}.
\end{equation}
Then the Newton-type iteration of SNA is decomposed into two parts $A_k$ and $B_k$ and only the computation of $\beta_0^{k+1}, \beta_{A_k}^{k+1}$ requires inverting a matrix, the dimension of which is only $(1+|A_k|)\times (1+|A_k|)$. In general, $|A_k|$ can be as large as $p$. But since pathwise optimization is implemented, the algorithm is warm-started at each $\lambda$ value. Hence $A_k$ is usually not too much different from the support of the optimizer, which tends to be a sparse subset of $\{1,\ldots,p\}$.
The real bottleneck is in matrix multiplication. Let $\psi_\gamma$ be as in \eqref{psi}. Let $X^{*}=(\mathbf{1}_{n}\; X)$ and $\Psi_k = \frac{1}{n}\text{diag}(\psi_\gamma(~y_1 -\beta_0^k-x_1^\top\beta^k),\ldots,\psi_\gamma(y_n-\beta_0^k-x_n^\top\beta^k))$.
Then as shown in Appendix B, each iteration includes re-computing and re-partitioning $X^{* \top} \Psi_k X^*$, which involves $O(np^2)$ arithmetic operations that become formidable for large $p$. The diagonality of $\Psi_k$ and the symmetry of $X^{* \top} \Psi_k X^*$ could be utilized to reduce computation, but the magnitude remains $O(np^2)$. Since $X^{*\top}\Psi_{k}X^{*}=
\frac{1}{n}\sum_i\psi_\gamma(y-\beta_{0}^{k}-x_i^{\top}\beta^{k})x_{i}^{*}x_{i}^{*\top}$, caching all the $(1+p)\times (1+p)$ matrices $x_i^{*} x_i^{*\top}$ would also speed up the
computation, but since there are $n$ such matrices, such an implementation would be memory-inefficient.
\subsubsection{SNCD vs. SNA}
The two algorithms mainly differ in the following aspects:
\begin{itemize}
\item
Consider a full update on $(\beta_0, \beta, s)$ as one iteration. The computational cost per iteration of SNCD is $O(np)$, compared with $O(np^2)$ for SNA.
\item
The SNCD iterations consist of univariate and bivariate updates only while SNA involves
matrix inversions.
\item
While SNA has locally superlinear convergence rate in theory, SNCD is at most linear. It is a worthwhile compromise, however, considering that SNCD reduces the computational cost per iteration from $O(np^2)$ to $O(np)$ and that warm-starting due to pathwise optimization strategy allows SNCD to converge quickly.
\item
In practice, SNCD is much faster; and SNCD always converges while SNA diverges in some high-dimensional cases even when
pathwise optimization is used.
\end{itemize}
\subsubsection{SNCD vs. Existing Coordinate Descent Type Algorithms}
SNCD also differs from the existing coordinate descent algorithms for penalized regression (\citealp{friedman2007pathwise}; \citealp{friedman2010regularization}; \citealp{simon2011regularization};
\citealp{breheny2011coordinate})
in the following aspects:
\begin{itemize}
\item
It generalizes coordinate descent to work on a wider class of models where the loss functions, like the Huber loss, only need to be first-order differentiable. As shown in the next section, it is also extended to a case with a nondifferentiable loss, i.e. the quantile loss, via smoothing approximation.
\item
It is directly motivated from the KKT conditions as a root-finding method, where the subgradients $s_j$'s are treated as independent variables that are connected with $\beta_j$'s through the equation $\beta_j - S(\beta_j+s_j) = 0$.
\item
Each pair of $(\beta_j, s_j)$ is updated simultaneously with different formulas for two situations $|\tilde{\beta}_{j}+\tilde{s}_{j}|>1$ and $|\tilde{\beta}_{j}+\tilde{s}_{j}|\leq 1$.
This is quite different from the coordinate descent algorithms mentioned above that only update the coefficients $\beta_j$'s.
\end{itemize}
\section{SNCD for Penalized Quantile Regression}
\label{section_quantile}
\subsection{Description}
For the quantile loss function $\ell = \rho_\tau$, \eqref{gen_loss} becomes
\begin{equation}\label{quantile_enet}
\min_{\beta_{0},\beta} f_Q(\beta_0, \beta) = \frac{1}{n}\sum_{i}\rho_\tau(y_{i}-\beta_{0}-x_{i}^{\top}\beta)+ \lambda P_{\alpha}(\beta).
\end{equation}
SNCD cannot be directly applied to this problem since it requires the first-order derivatives of the loss function,
but $\rho_\tau$ is not differentiable. However, note that
\[
\rho_{\tau}(t) = (1-\tau)t_-+\tau t_+
= \frac{1}{2}\left\{ |t|+(2\tau-1)t\right\}.
\]
Since $h_\gamma(t) \rightarrow |t|$ as $\gamma \rightarrow 0^+$, $\rho_\tau(t) \approx \frac{1}{2}\left\{ h_\gamma(t)+(2\tau-1)t\right\}$ for small $\gamma$ and the solutions to penalized quantile regression can be approximated by
\begin{equation}\label{RA_enet}
\min_{\beta_{0},\beta}\; f_{HA}(\beta_{0},\beta)=\frac{1}{2n}\sum\left\{ h_{\gamma}(y_{i}-\beta_{0}-x_{i}^{\top}\beta)+(2\tau-1)(y_{i}-\beta_{0}-x_{i}^{\top}\beta)
\right\} + \lambda P_{\alpha}(\beta),
\end{equation}
where ``HA" stands for Huber approximation. This problem is easier to handle since its loss function is first-order differentiable.
The following result provides theoretical support for this smoothing approximation.
\begin{theorem}\label{convergence_quantile}
Given any $\lambda \geq 0 $, $0 <\tau < 1$ and $\{\gamma_k\}$ converging to 0, let $(\beta_{0k},\beta_k)$ be a minimizer of $f_{HA}(\beta_0, \beta;\lambda, \tau, \gamma_k)$. Then every cluster point of sequence $\{(\beta_{0k},\beta_k)\}$ is a minimizer of $f_Q(\beta_0, \beta; \lambda, \tau)$.
\end{theorem}
Now we can derive the KKT conditions and apply SNCD to solve \eqref{RA_enet}. Due to its similarity to the penalized Huber loss regression, we omit the details. At each iteration, with the current estimates denoted by $(\tilde{\beta}_0, \tilde{\beta}, \tilde{s})$ and residuals by $\tilde{r}_i$, the SNCD updates are
\begin{enumerate}[(i)]
\item
For $\beta_0$:
\[
\beta_{0}\leftarrow\tilde{\beta}_{0}+\frac{\sum_{i}\left\{ h^\prime_{\gamma}(\tilde{r}_{i})+2\tau-1\right\} }{\sum_{i}\psi_{\gamma}(\tilde{r}_{i})}.
\]
\item
For $(\beta_j, s_j)$:
\begin{enumerate}[(a)]
\item
If $|\tilde{\beta}_{j}+\tilde{s}_{j}|>1$, then
\begin{eqnarray*}
\beta_{j} & \leftarrow & \tilde{\beta}_{j}+\frac{\frac{1}{2n}\sum_{i}\left\{ h^\prime_{\gamma}(\tilde{r}_{i})+2\tau-1\right\} x_{ij}-\lambda\alpha \text{sgn}(\tilde{\beta}_{j}+\tilde{s}_{j})-\lambda(1-\alpha)\tilde{\beta}_{j}}
{\frac{1}{2n}\sum_{i}\psi_{\gamma}(\tilde{r}_i)x_{ij}^{2}+\lambda(1-\alpha)}, \\
s_{j} & \leftarrow & \text{sgn}(\tilde{\beta}_{j}+\tilde{s}_{j}).
\end{eqnarray*}
\item
If $|\tilde{\beta}_{j}+\tilde{s}_{j}|\leq1$, then
\begin{eqnarray*}
\beta_{j} & \leftarrow & 0,\\
s_{j} & \leftarrow & \frac{\frac{1}{2n}\sum_{i}\left\{ h^\prime_{\gamma}(\tilde{r}_{i})+2\tau-1\right\} x_{ij}+\tilde{\beta}_{j}\cdot\frac{1}{2n}
\sum_{i}\psi_{\gamma}(\tilde{r}_{i})x_{ij}^{2}}{\lambda\alpha}.
\end{eqnarray*}
\end{enumerate}
\end{enumerate}
The previous discussions on convergence and pathwise optimization also apply here. And similar to Theorem \ref{SNCD_huber}, we have the following result.
\begin{theorem}\label{SNCD_quant}
For problem \eqref{quantile_enet}, let $\lambda > 0$, $\alpha \in (0,1)$ and the initial $f_{HA}$ value be $f_{HA}^0$. Assume for every point $(\beta_0, \beta)$ in the level set $\mathcal{L} = \{(\beta_0, \beta): f_{HA}(\beta_0, \beta) \leq f_{HA}^0 \}$ there exists $i\in\{1,\ldots,n\}$ such that $|y_{i}-\beta_{0}-x_{i}^{\top}\beta|\leq\gamma$ . Then SNCD iterations
converge to a global minimizer provided that the diameter of $\mathcal{L}$ is sufficiently small.
\end{theorem}
\subsection{The Choice of $\gamma$ Values}
For the approximation to work well, we need to use a sufficiently small $\gamma$; but when $\gamma$ gets too close to 0, the algorithm becomes ill-conditioned. Therefore we designed a data-dependent heuristic method for picking appropriate $\gamma$ values. At each $\lambda_k$, we determine $\gamma_k$ depending on the residuals $\tilde{r}_i$'s given by the previous optimizer $(\widehat{\beta}_0(\lambda_{k-1}), \widehat{\beta}(\lambda_{k-1}))$
as follows.
\begin{enumerate}[i.]
\item Initialize residuals $\tilde{r}_i \leftarrow y_i$;
\item For each $\lambda_k$:
\begin{enumerate}[(a)]
\item
$\gamma_k \leftarrow \mbox{10-th percentile of } \{|\tilde{r}_i|\}$;
\item
$\gamma_k \leftarrow \min\{\gamma_k, \gamma_{k-1}\}$;
\item
$\gamma_k \leftarrow \max\{\gamma_k, 0.001\}$;
\item
solve the problem with $\gamma_k, \lambda_k$ and update $\tilde{r}_i$'s at each iteration.
\end{enumerate}
\end{enumerate}
In step (a) we pick a value smaller than the magnitudes of 90\% of all residuals for which the loss function is the same as the quantile loss so the approximation should work well. This also keeps $\gamma_k$ above the magnitudes of 10\% of the residuals, which ensures the numerical stability of the algorithm. Bracketing in (b) and (c) are additional safeguards for stability.
\subsection{Related Convergence Results}
The key to the smoothing approximation is the fact that $h_\gamma(t)$ converges to $|t|$ as $\gamma$ tends to 0. In fact, it is also easy to see that with $\gamma$ as a scaling factor, $\gamma h_\gamma(t)$ converges to the squared loss $\frac{t^2}{2}$ when $\gamma$ goes to infinity. Hence, in the same spirit of Theorem \ref{convergence_quantile}, we also show the connections between the penalized Huber loss regression and two important regression models with respectively the absolute loss and the squared loss, i.e. the Least Absolute Deviations (LAD) and the Least Squares (LS).
To simplify the notation, let $\theta = (\beta_0, \beta)$ and $P(\cdot)$ be a general penalty function.
Denote
\[
\begin{array}{lll}
\underset{\theta}{\min}\;f_{H}(\theta;\lambda,\gamma) & = & \frac{1}{n}\sum_{i}h_{\gamma}(y_{i}-\beta_{0}-x_{i}^{\top}\beta)+\lambda P(\beta),\\
\\
\underset{\theta}{\min}\;f_{A}(\theta;\lambda) & = & \frac{1}{n}\sum_{i}|y_{i}-\beta_{0}-x_{i}^{\top}\beta|+\lambda P(\beta),\\
\\
\underset{\theta}{\min}\;f_{S}(\theta;\lambda) & = & \frac{1}{2n}\sum_{i}(y_{i}-\beta_{0}-x_{i}^{\top}\beta)^{2}+\lambda P(\beta).
\end{array}
\]
Then we have $f_{H}(\theta;\lambda,\gamma) \rightarrow f_{A}(\theta;\lambda)$ as $\gamma \rightarrow 0$; $\gamma f_{H}(\theta;\lambda/\gamma,\gamma) \rightarrow f_{S}(\theta;\lambda)$ as $\gamma \rightarrow \infty$.
And the following results establish the convergence between their optimizers.
\begin{theorem}\label{convergence_lad}
Given any $\lambda \geq 0$ and $\{\gamma_k\}$ converging to 0, let $\theta_k$ be a minimizer of $f_{H}(\theta;\lambda, \gamma_k)$. Then every cluster point of sequence $\{\theta_k\}$ is a minimizer of $f_A(\theta; \lambda)$.
\end{theorem}
\begin{theorem}\label{convergence_ls}
Given any $\lambda \geq 0$ and $\{\gamma_k\}$ converging to $\infty$, let $\theta_k$ be a minimizer of $f_{H}(\theta;\lambda/\gamma_k,\gamma_k)$. Then every cluster point of sequence $\{\theta_k\}$ is a minimizer of $f_S(\theta; \lambda)$.
\end{theorem}
Therefore, the penalized Huber loss regression bridges the gap between LAD and LS regression as $\gamma$ varies from $0$ to $\infty$.
The solutions of the penalized Huber loss regression constitute a rich spectrum from the solution of LAD regression to that of LS regression. This property gives us more flexibility in fitting high-dimensional regression models.
\section{Adaptive Strong Rule for Screening Predictors}
\label{section_asr}
\cite{tibshirani2012strong} proposed the (sequential) strong rule for screening out predictors in pathwise optimization of penalized regression models for computational efficiency.
However, when applied to the penalized Huber loss regression and quantile regression, we discover that the strong rule suffers from the issue of ``violations" that is explained below. To deal with this issue and enhance algorithmic stability, we develop an adaptive version of the strong rule.
We first describe the strong rule. Consider a general elastic-net penalized regression
\[
\underset{\beta_0, \beta}{\min} \; \frac{1}{n}\sum_{i=1}^n \ell(y_i - \beta_{0} - x_i^\top\beta)+\lambda P_\alpha(\beta).
\]
where $\ell$ is convex and differentiable. Then the optimizer $(\widehat{\beta}_{0}(\lambda),\widehat{\beta}(\lambda))$ satisfies the KKT conditions
\[
\begin{cases}
-\frac{1}{n}\sum_i \ell^\prime(y_i - \widehat{\beta}_{0} - x_i^\top\widehat{\beta}) = 0,\\
-\frac{1}{n}\sum_i \ell^\prime(y_i - \widehat{\beta}_{0} - x_i^\top\widehat{\beta})x_{ij} +\lambda\alpha \widehat{s}_{j}+\lambda(1-\alpha)\widehat{\beta}_{j} = 0,\\
\widehat{s}_{j}\in\partial|\widehat{\beta}_{j}|, \quad j=1,\ldots, p.
\end{cases}
\]
The unpenalized intercept $\beta_0$ is always in the model, so there is no screening rule for it. For $\beta_j$, let $c_{j}(\lambda)=-\frac{1}{n}\sum_i \ell^\prime(y_i - \widehat{\beta}_{0} - x_i^\top\widehat{\beta})x_{ij}$. Assume each $c_j$ is $\alpha$-Lipschitz continuous,
\begin{equation}\label{lip}
|c_{j}(\lambda)-c_{j}(\lambda^{\prime})|\leq\alpha|\lambda-\lambda^{\prime}|,
\ \text{ for every } \ \lambda, \lambda^\prime >0.
\end{equation}
Then at each new $\lambda_k$ in the solution path, given the previous optimizer $(\widehat{\beta}_{0}(\lambda_{k-1}),\widehat{\beta}(\lambda_{k-1}))$ and the corresponding $c_j (\lambda_{k-1})$'s, the strong rule discards predictor $j$ if
\begin{equation}\label{strong}
|c_{j}(\lambda_{k-1})|<\alpha(2\lambda_{k}-\lambda_{k-1}).
\end{equation}
The reasoning is as follows. Assume \eqref{lip} and \eqref{strong} hold, since $\lambda_{k-1} > \lambda_k$, we have
\begin{eqnarray*}
|c_{j}(\lambda_{k})| & \leq & |c_{j}(\lambda_{k})-c_{j}(\lambda_{k-1})|+|c_{j}(\lambda_{k-1})|\\
& < & \alpha(\lambda_{k-1}-\lambda_{k})+\alpha(2\lambda_{k}-\lambda_{k-1})\\
& = & \alpha\lambda_{k}.
\end{eqnarray*}
It follows that $\widehat{\beta}_{j}(\lambda_{k})=0$, since by contradiction $\ensuremath{\widehat{\beta}_{j}(\lambda_{k})\neq0}$ implies $ \widehat{s}_{j}(\lambda_k)=\text{sgn}(\widehat{\beta}_{j}(\lambda_{k}))$ thus $|c_{j}(\lambda_{k})|=\lambda_{k}\alpha+\lambda_{k}(1-\alpha)|\widehat{\beta}_{j}(\lambda_{k})|\geq\lambda_{k}\alpha$.
The effectiveness of the strong rule relies on the assumption \eqref{lip}, which does not necessarily hold. So
application of the rule should always be accompanied with a check of the KKT conditions. A pathwise optimization algorithm incorporating the strong rule proceeds as follows.
For each $\lambda_k$,
\begin{enumerate}[(a)]
\item
Compute the eligible set $E=\{j:\;|c_{j}(\lambda_{k-1})|\geq\alpha(2\lambda_{k}-\lambda_{k-1})\}$;
\item
Solve the problem using only the predictors in $E$;
\item
Check KKT conditions on the solution: $|c_{j}(\lambda_{k})|\leq\alpha\lambda_{k}$ for $j\in E^c$. We are done if there are no violations; otherwise, add violating indices to $E$ and repeat (b) and (c).
\end{enumerate}
For the penalized least squares and logistic regression we have not encountered any violation, but it a different story for the penalized Huber loss regression and quantile regression. Using the strong rule for these two models, we often encounter a large number of violations, indicating that the rule may have been too restrictive.
Since the algorithm is re-run each time violations are found, the overall efficiency is affected. Thus reducing the number of violations can enhance the algorithmic stability and lead to potential speedup.
A simple approach is to use a multiplier $M>1$ and relax the assumption \eqref{lip} to the following: $\forall \lambda, \lambda^\prime >0$,
\[
|c_{j}(\lambda)-c_{j}(\lambda^{\prime})|\leq\alpha M|\lambda-\lambda^{\prime}|.
\]
Accordingly, we will need to change \eqref{strong} to
\[
|c_{j}(\lambda_{k-1})|<\alpha\left(\lambda_{k}+M(\lambda_{k}-\lambda_{k-1})\right).
\]
However, this strategy does not work well in practice, since it is difficult to pre-determine an appropriate value of $M$ that suits all values of $\lambda$ in the solution path.
Hence we propose an ``adaptive" version that allows $M$ to vary with $\lambda$. This rule automatically estimates a localized $M(\lambda)$ that varies and adapts to the trends of the solution paths, which reduces the number of violations by a large margin without sacrificing speed. The idea is as follows.
Let $M(\lambda_0)=1$. Then at each $\lambda_k$,
\begin{enumerate}[(a)]
\item
use $M(\lambda_{k-1})$ to construct the eligible set, i.e. let
\[E=\{j:\;|c_{j}(\lambda_{k-1})|\geq \alpha\left(\lambda_{k}+M(\lambda_{k-1})(\lambda_{k}-\lambda_{k-1})\right)\};
\]
\item
solve the problem using only the predictors in $E$, and check KKT conditions as before; update $E$ and repeat step (b) if violations occur;
\item
compute $M(\lambda_k)$ based on the local trend of $c_j$'s:
\[
M(\lambda_{k})=\frac{\underset{1\leq j\leq p}{\max}|c_{j}(\lambda_{k-1})-c_{j}(\lambda_{k})|}{\alpha(\lambda_{k-1}-\lambda_{k})}.
\]
\end{enumerate}
\section{Numerical results}
\label{section_numerical}
\subsection{Optimization Performance for Penalized Quantile Regression}\label{subsection_opt}
As mentioned in the introduction, \texttt{quantreg} is another publicly available
R package that supports lasso penalized quantile regression.
Since our implementation employs an approximation model, it does not give ``exact" solutions. Hence we want to compare its solutions with the ones computed by \texttt{quantreg} in terms of optimality.
Unlike \texttt{hqreg} that computes a solution path, \texttt{quantreg} computes a single solution for a given $\lambda$ value, and it does not support the general elastic-net penalty with $0 <\alpha < 1$. For comparison, we only consider lasso ($\alpha = 1$). We first computed a solution path along 100 $\lambda$ values using \texttt{hqreg} and then ran \texttt{quantreg} for each $\lambda$ value. Note that \texttt{quantreg} actually uses the formulation
\[
\min_{\beta_{0},\beta}\sum_{i=1}^{n}\rho_{\tau}(y_{i}-\beta_{0}-x_{i}^{\top}\beta)+\lambda\cdot \frac{1}{2}\sum_{j=1}^p|\beta_{j}|
\]
which does not have a $1/n$ scaling factor for the loss part and instead contains a $1/2$ factor for the penalty. This is intended to treat the penalty terms as if median regression were performed on them ($\frac{1}{2}\lambda|\beta_j| = \rho_{0.5}(\lambda\beta_j)$). Due to this difference, for each $\lambda$ value used with \texttt{hqreg}, we equivalently supplied \texttt{quantreg} with $2n\lambda$. Also, while \texttt{hqreg} supports data preprocessing via the argument ``preprocess" with 3 options ``standardize", ``rescale" and ``none", \texttt{quantreg} does not provide such an option.
So we standardized the data beforehand for all the real datasets involved in this section and used the standardized ones for comparison. Consequently, we set \texttt{preprocess = "none"} when calling \texttt{hqreg}. For \texttt{quantreg}, the latest version 5.24 was used.
Let $f_Q(\cdot;\lambda)$ denote the objective function as in \eqref{quantile_enet}, and let $\widehat{\beta}_{\text{hqreg}}$ and $\widehat{\beta}_{\text{quantreg}}$ be the solutions given by the two packages, respectively. For $\alpha = 1$ the model is not strictly convex, so in general it does not have a unique optimizer. Hence the values of the two solutions may not be very close. Instead, a reasonable approach is to compare the values of the objective functions $f_Q(\widehat{\beta}_{\text{hqreg}})$ and $f_Q(\widehat{\beta}_{\text{quantreg}})$. Specifically, we made the comparisons based on the relative difference,
\begin{equation}
\label{Ddef}
D(\lambda) = \frac{f_{Q}(\widehat{\beta}_{\text{hqreg}};\lambda)-f_{Q}(\widehat{\beta}_{\text{quantreg}};\lambda)}{f_{Q}(\widehat{\beta}_{\text{quantreg}};\lambda)}.
\end{equation}
Two datasets were considered:
\begin{itemize}
\item
GDP \citep{koenker1999goodness}: consists of 161 observations on national GDP growth rates, recorded as ``Annual Change Per Capita GDP", and 13 covariates. The first 71 observations are from the period 1965-1975, and the rest from the period 1975-1985. This dataset is available in \texttt{quantreg} via \texttt{data(barro)}.
\item
Riboflavin \citep{buhlmann2014high}: gene-expression data for predicting log transformed riboflavin (vitamin B2) production rate in Bacillus subtilis. It contains 71 observations and 4088 features (gene expressions). This dataset is available in R package \texttt{hdi} via \texttt{data(riboflavin)}. For this task only 1000 features with the largest variances were used.
\end{itemize}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{qr_objective_comparison.png}
\vspace{-7mm}
\caption{Values of objective functions with $\tau=0.5$ along the solution path for GDP and riboflavin datasets. Solid line: \texttt{quantreg}, dashed line: \texttt{hqreg}.}
\label{plot_objective_quant}
\end{center}
\end{figure}
\begin{table}[htbp]
\spacingset{1}
\begin{center}
\begin{tabular}{c|c|rr}
\hline
Dataset & $\tau$ & $\min D(\lambda_i)$
& $\max D(\lambda_i)$ \\
\hline
\multirow{3}{*}{GDP} &0.25 & -2.1e-9 & 1.5e-3 \\
&0.50 & -1.3e-10 & 9.6e-4 \\
&0.75 & -3.0e-10 & 1.7e-3 \\
\hline
\multirow{3}{*}{Riboflavin} &0.25 & 6.5e-5 & 2.6e-2 \\
&0.50 & -3.6e-10 & 2.0e-2 \\
&0.75 & 8.8e-5 & 2.1e-2 \\
\hline
\end{tabular}
\caption
The range of the relative differences $D(\lambda_i), 1 \le i \le 100,$ between \texttt{hqreg} and \texttt{quantreg}.}
\label{compare_optimality_quant}
\end{center}
\end{table}
\begin{table}[htbp]
\spacingset{1}
\begin{center}
\begin{tabular}{c|c|c|ccc}
\hline
\multirow{2}{*}{Dataset} & \multirow{2}{*}{$\tau$} & \multirow{2}{*}{\texttt{hqreg}} & \multicolumn{3}{c}{\texttt{quantreg}} \\
& & & total & $\lambda_1$ & $\lambda_{100}$ \\
\hline
\multirow{3}{*}{GDP} &0.25 & 0.018 & 0.235 & 0.007 & 0.002\\
&0.50 & 0.018 & 0.223 & 0.002 & 0.003\\
&0.75 & 0.027 & 0.240 & 0.003 & 0.002\\
\hline
\multirow{3}{*}{Riboflavin} &0.25 & 2.501 & 538.2 & 3.630 & 4.958 \\
&0.50 & 3.026 & 531.6 & 4.591 & 4.984\\
&0.75 & 2.922 & 588.8 & 7.119 & 5.791\\
\hline
\end{tabular}
\caption{Running time (in seconds) for computing the solution paths}
\label{compare_time_quant}
\end{center}
\end{table}
Figure \ref{plot_objective_quant} displays the computed values of objective functions $f_Q(\widehat{\beta}_{\text{hqreg}})$ and $f_Q(\widehat{\beta}_{\text{quantreg}})$ for $\tau = 0.5$ along the solution path for both datasets. There is no visually detectable discrepancy between the two lines. Hence we also computed the range of $D(\lambda)$ in each case and the results are listed in Table \ref{compare_optimality_quant}. In each case, the range of $D(\lambda_i)$'s is extremely narrow and all values are very close to zero. This indicates the two packages indeed have similar performances.
We also report the running time in Table \ref{compare_time_quant}. The time for \texttt{hqreg} is for one call that fits the entire solution path, and the time for \texttt{quantreg} is the total of time recorded separately for each $\lambda$. For all these cases \texttt{hqreg} is significantly faster than \texttt{quantreg}, although it may not be quite fair for \texttt{quantreg} since it does not rely on warm-start. The timings taken for \texttt{quantreg} on $\lambda_1$ and $\lambda_{100}$ are also listed, which appear to be roughly the same. In the case of riboflavin data, the running time of \texttt{quantreg} on single $\lambda$ values is in fact longer than the time used by \texttt{hqreg} to compute the whole path.
To further investigate their performances in various other scenarios, we ran a large set of experiments on 10000 datasets, each generated with the following settings:
\begin{itemize}
\item
the number of observations $n$ and the number of features $p$ are randomly selected from the set $\{20,100,200,500,1000,2000,5000\}$.
\item
the number of nonzero coefficients is $q = \theta \min(n, p)$ where $\theta$ is uniformly sampled from $\{5\%, 10\%, 20\%, 30\%\}$ and the coefficients values are randomly selected from $\{\pm 1, \ldots, \pm 10 \}$.
\item
each feature vector $x_i$ is generated via $x_{ij} = z_{ij} + 0.5 u_i, \; 1\leq j \leq p$, where $z_{ij}, u_i$'s are i.i.d. standard gaussian, so that each pair of features has the same correlation 0.25.
\item
the outcome $y_i$'s are generated by $y_i = 10 + x_i^\top\beta+\varepsilon_i$, where
$\varepsilon_i$'s are iid sampled from Student's t distribution with $df = 4$.
\end{itemize}
For each dataset and each $\tau \in \{0.25, 0.5, 0.75\}$, we applied \texttt{hqreg} to compute an entire solution path and randomly selected an index $k$ out of $\{10, 20, \ldots, 100\}$, then ran \texttt{quantreg} on $\lambda_k$, the k-th $\lambda$ value for the solution path computed by \texttt{hqreg}. These experiments were performed in parallel via grid computing on a high performance cluster at the University of Iowa.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{relative_diff_10000.png}
\vspace{-5mm}
\caption{Boxplots of the relative difference $D$ on 10000 simulated datasets}
\label{box_objective}
\end{center}
\end{figure}
We calculate the relative difference $D(\lambda)$ for each pair of the solutions and summarize the results in three boxplots plotted on the logarithmic scale shown in Figure \ref{box_objective}. We observe that the values of $D$ have a narrow range between 1e-7 and 1e-1 with the majority falling below 1e-3, and are slightly smaller for $\tau = 0.5$. Besides, the distribution of $D$ appears roughly symmetric on the logarithmic scale in each case.
\subsection{Timing Performance}
\label{subsection_timings}
In addition to the Huber loss and the quantile loss, \texttt{hqreg} also supports the squared loss for the least squares which is not discussed in this paper, but its SNCD iterations can be derived in a similar way as the other two models. Here we consider their running time performances.
We generated Gaussian data with $n$ observations and $p$ features, where each pair of features have an identical correlation $\rho$. To simplify settings and highlight the timing comparison based on the key parameter $\gamma$ and $\tau$, we set $\rho = 0.25$ and $\alpha = 0.9$ for all cases. The responses were generated by
\[
Y=\sum_j X_{j}\beta_{j}+k\cdot E
\]
where $\beta_{j}=(-1)^{j}\exp(-(j-1)/10)$, $E\sim T(df=4)$ and $k$ is determined so that the signal-to-noise ratio is 3.
\subsubsection{Huber Loss Regression and Least Squares}
In this part, we compare the running time of competing methods for the elastic-net penalized Huber loss regression and least squares. For the Huber loss, since there is no other algorithms, we consider only \texttt{hqreg} for SNCD with no variable screening (\texttt{hqreg}-NVS), SNCD with the adaptive strong rule (\texttt{hqreg}-ASR),
and our implementation of pure SNA.
In the experiments we considered 5 values of $\gamma$ ranging from 0.01 to 100. On the other hand, for the least squares we compared {\texttt{hqreg}} with a state-of-the-art coordinate descent algorithm implemented by R package \texttt{glmnet}. For \texttt{glmnet}, the latest version 2.0-5 was used which employs the strong rule for variable screening. All methods considered here are R functions. \texttt{glmnet} does all its computation in Fortran, \texttt{hqreg} does the computation in C, and the SNA implementation is also programmed in C with matrix operations performed via BLAS and LAPACK.
\begin{table}[htbp]
\spacingset{1}
\begin{center}
\begin{tabular}{lrrrrr|c}
& \multicolumn{5}{c|}{Huber} & Least Squares \\
& \multicolumn{5}{c|}{$\gamma$} & \\
& 0.01 & 0.1 & 1 & 10 & 100 & \\
\cline{2-7}\\[0.01ex]
& \multicolumn{6}{c}{$n = 1000, p = 100$} \\
\cline{2-7}
hqreg-NVS &0.61 &0.09 &0.05 &0.04 &0.04 & 0.03\\
hqreg-ASR &0.33 &0.06 &0.03 &0.02 &0.02 & 0.02\\
glmnet & --- &--- &--- &--- &--- & 0.02 \\
\cline{2-7}\\[0.01ex]
& \multicolumn{6}{c}{$n = 5000, p = 100$} \\
\cline{2-7}
hqreg-NVS &2.46 &0.48 &0.24 &0.20 &0.21 & 0.14 \\
hqreg-ASR &1.32 &0.30 &0.15 &0.12 & 0.11 &0.07 \\
glmnet & --- &--- &--- &--- &--- & 0.02 \\
\cline{2-7}\\[0.01ex]
& \multicolumn{6}{c}{$n = 100, p = 1000$} \\
\cline{2-7}
hqreg-NVS & 1.89 &0.39 &0.09 &0.08 &0.08 &0.05 \\
hqreg-ASR &0.52 &0.11 &0.03 &0.02 &0.02 & 0.02 \\
glmnet & --- &--- &--- &--- &--- & 0.02 \\
\cline{2-7}\\[0.01ex]
& \multicolumn{6}{c}{$n = 100, p = 5000$} \\
\cline{2-7}
hqreg-NVS &8.70 &2.09 &0.46 &0.38 &0.38 & 0.29\\
hqreg-ASR &0.85 &0.23 &0.09 &0.11 &0.08 & 0.07 \\
glmnet & --- &--- &--- &--- &--- & 0.08 \\
\cline{2-7}\\[0.01ex]
& \multicolumn{6}{c}{$n = 100, p = 20000$} \\
\cline{2-7}
hqreg-NVS & 30.27 & 8.88 &2.43 & 2.45 & 2.40 & 1.23\\
hqreg-ASR &1.60 &0.54 &0.43 &0.31 &0.30 & 0.32 \\
glmnet & --- &--- &--- &--- &--- & 0.30 \\
\cline{2-7}\\[0.01ex]
& \multicolumn{6}{c}{$n = 100, p = 100000$} \\
\cline{2-7}
hqreg-NVS &175.81 &45.33 & 11.23 & 11.49 & 11.41 & 5.94 \\
hqreg-ASR &4.50 &2.12 &1.69 &1.58 & 1.53 & 1.57 \\
glmnet & --- &--- &--- &--- &--- & 1.39 \\
\cline{2-7}
\end{tabular}
\caption{Running time (in seconds) for computing regularization paths for the elastic-net penalized Huber loss regression and least squares regression. Total time for \texttt{100} $\lambda$ values, averaged over 3 runs.
}
\label{timings}
\end{center}
\end{table}
We have found in practice that convergence of SNA has much higher reliance on initial points than SNCD, and it can fail if the $\lambda$ sequence is not dense enough. Hence we divided the experiments into two parts. In the first part for the Huber loss and the least squares together, we left out SNA and computed each solution path with the usual number of 100 $\lambda$ values. In the second part, we compared only SNA and SNCD(\texttt{hqreg}-NVS) for the Huber loss on dense lambda sequences each consisting of 10000 values.
Table \ref{timings} shows average CPU timings for the first part. First compare the timings for the Huber loss. Across different values of $\gamma$, we observe that for both versions the timings increase when $\gamma$ is nearing 0, and stay almost the same for $\gamma \geq 1$. And clearly \texttt{hqreg}-ASR that employs the adaptive strong rule is much faster and more scalable than \texttt{hqreg}-NVS that has no variable screening. For the least squares, \texttt{hqreg}-ASR and \texttt{glmnet} have similar performances except the case with $n = 5000$. Besides, we discover that the timings for the Huber loss regression with $\gamma \geq 1$ are very close to those of the least squares. Considering that the Huber loss is more difficult to handle than the simple squared loss, the performance of \texttt{hqreg} is very impressive.
Table \ref{timings_small} shows average CPU timings for the second part.
We observe that while SNCD converges in every case, SNA fails in the cases with large $p$ and $\gamma = 0.1$. When $p$ is small, SNCD does not have much advantage. But when $p$ increases, SNCD becomes considerably faster with an increasing speedup relative to SNA. These results show that SNCD is more stable and scalable than SNA.
\begin{table}[htbp]
\spacingset{1}
\begin{center}
\begin{tabular}{lrrr}
& \multicolumn{3}{c}{$\gamma$} \\
& 0.1 & 1 & 10 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 1000, p = 100$} \\
\cline{2-4}
SNA & 3.98 & 5.16 & 5.44 \\
SNCD & 1.89 & 1.27 & 1.19 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 5000, p = 100$} \\
\cline{2-4}
SNA & 17.98 & 24.42 & 26.33 \\
SNCD & 12.38 & 6.84 & 6.31 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 100, p = 1000$} \\
\cline{2-4}
SNA & $\times$ & 11.70 & 10.47 \\
SNCD & 2.24 & 1.62 & 1.51 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n= 100, p = 5000$} \\
\cline{2-4}
SNA & $\times$ & 98.66 & 100.76 \\
SNCD & 9.87 & 8.19 & 8.96 \\
\cline{2-4}\\[0.01ex]
\end{tabular}
\vspace{-5mm}
\caption{Running time (in seconds) for comparing SNCD(\texttt{hqreg}-NVS) and SNA on the penalized Huber loss regression. ``$\times$" represents early exit due to divergence at some $\lambda$ value. Total time for \texttt{10000} $\lambda$ values. }
\label{timings_small}
\end{center}
\end{table}
\subsubsection{Quantile Regression}
\texttt{hqreg} is faster than \texttt{quantreg} for the examples in section \ref{subsection_opt}. However, \texttt{quantreg} does not implement pathwise optimization and rely on warm-start like \texttt{hqreg} does. Instead, for each supplied $\lambda$ value it has to solve the corresponding problem individually ``from scratch". So it is not quite reasonable to compare \texttt{quantreg} with \texttt{hqreg} for computing the whole solution path. For this part, we compare only \texttt{hqreg}-NVS and \texttt{hqreg}-ASR. As shown in Table \ref{timings_quantile}, \texttt{hqreg}-ASR is similar to \texttt{hqreg}-NVS in cases with $p=100$ but considerably faster when $p$ gets larger. \texttt{hqreg}-ASR also shows much better scalability with the dimension $p$.
\begin{table}[htb]
\spacingset{1}
\begin{center}
\begin{tabular}{lrrr}
& \multicolumn{3}{c}{$\tau$} \\
& 0.25 & 0.50 & 0.75 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 1000, p = 100$} \\
\cline{2-4}
hqreg-NVS & 0.21 & 0.18 & 0.19 \\
hqreg-ASR & 0.13 & 0.10 & 0.11 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 5000, p = 100$} \\
\cline{2-4}
hqreg-NVS & 0.56 & 0.58 & 0.54 \\
hqreg-ASR & 0.38 & 0.42 & 0.33 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 100, p = 1000$} \\
\cline{2-4}
hqreg-NVS & 10.77 & 7.37 & 11.90 \\
hqreg-ASR & 2.98 & 1.94 & 2.92 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 100, p = 5000$} \\
\cline{2-4}
hqreg-NVS & 47.08 & 41.46 & 58.97 \\
hqreg-ASR & 3.33 & 2.92 & 4.23 \\
\cline{2-4}\\[0.01ex]
& \multicolumn{3}{c}{$n = 100, p = 100000$} \\
\cline{2-4}
hqreg-ASR & 19.28 & 12.43 & 22.98 \\
\cline{2-4}\\[0.01ex]
\end{tabular}
\vspace{-5mm}
\caption{Running time (in seconds) for computing regularization paths for penalized quantile regression. Total time for \texttt{100} $\lambda$ values, averaged over 3 runs.}
\label{timings_quantile}
\end{center}
\end{table}
\subsection{Real Data Example}
We now compare the modelling performance of penalized Huber loss regression, quantile regression and least squares via an empirical analysis on a real dataset. It is a breast cancer gene expressions dataset that comes from the Cancer Genome Atlas (2012) project (\url{http://cancergenome.nih.gov/}), obtained using Agilent mRNA expression microarrays. It contains expression measurements of 17814 genes on 536 patients, including BRCA1, the first gene identified to be associated with increasing risk of early onset breast cancer. Hence we regress the key gene BRCA1 on the other genes to detect potential interconnections. Before fitting the models, we carried out the following two screening steps: remove any gene for which the range of the expression among all patients is less than 2, and remove any gene for which the sample correlation with BRCA1 is less than 0.05. After the screening, there are 11562 genes left.
Then we consider 7 elastic-net penalized linear regression models using these genes as predictors: the least squares (LS-Enet); 3 Huber loss regression models with values of $\gamma$ being $\text{IQR}(y)$, $\text{IQR}(y)/2$, $\text{IQR}(y)/10$ respectively where $\text{IQR}(y) = 0.93$, denoted as H-Enet($\gamma = \text{IQR}(y)$), H-Enet($\gamma = \text{IQR}(y)/2$), and (H-Enet($\gamma = \text{IQR}(y)/10$); 3 quantile regression models with $\tau = 0.25, 0.50, 0.75$, denoted as Q-Enet($\tau = 0.25$), Q-Enet($\tau = 0.50$), and Q-Enet($\tau = 0.75$). $\alpha = 0.9$ is applied to the elastic-net penalty for all models.
We conduct 50 random partitions. For each partition, we randomly select 300 patients as the training data and the other 236 as the testing data. A five-fold cross validation is applied to the training data to select the tuning parameter $\lambda$. For prediction on the testing set, we consider two error measures. The first one is the commonly used mean absolute prediction error (MAPE). Since MAPE is not sensitive to heterogeneity and may not provide accurate assessment for Q-Enet($\tau = 0.25$) and Q-Enet($\tau = 0.75$) which use asymmetric losses, we also consider using the quantile loss $\rho_\tau$ to measure prediction performance as suggested in \cite{wang2012quantile}. With $\rho_\tau$ for corresponding quantile regression models and $\rho_{0.5}$ for the least squares and the Huber loss regression models, we define quantile-based prediction error (QPE) as $\sum_{i} \rho_\tau(y_i - \widehat{y}_i)/n$.
\begin{table}[htbp]
\spacingset{1}
\begin{center}
\begin{tabular}{lrrr}
\multicolumn{1}{c}{Model} & \multicolumn{1}{c}{Ave \# nonzero} & \multicolumn{1}{c}{Ave MAPE} & \multicolumn{1}{c}{Ave QPE} \\
\hline
LS-Enet & 114.30 (36.99) & 0.335 (0.018) & 0.167 (0.009) \\
H-Enet($\gamma = \text{IQR}(y)$) & 100.14 (44.70) & 0.331 (0.018) & 0.166 (0.009) \\
H-Enet($\gamma = \text{IQR}(y)/2$) & 82.06 (30.40) &
0.310 (0.020) & 0.155 (0.010) \\
H-Enet($\gamma = \text{IQR}(y)/10$) & 114.08 (30.73) & 0.293 (0.021) & 0.146 (0.010) \\
Q-Enet($\tau = 0.25$) & 94.58 (41.60) & 0.373 (0.026) & 0.151 (0.010) \\
Q-Enet($\tau = 0.50$) & 152.90 (51.96) & 0.294 (0.021) & 0.147 (0.012) \\
Q-Enet($\tau = 0.75$) & 104.90 (27.96) & 0.317 (0.027) & 0.109 (0.007) \\
\hline
\end{tabular}
\caption{Analysis of the microarray dataset}
\label{bcdata_comparison}
\end{center}
\end{table}
In Table \ref{bcdata_comparison} we report the average number of nonzero regression coefficients, the average MAPE and the average QPE, where numbers in the parentheses are the corresponding standard errors across the 50 partitions. The standard errors of the estimated numbers of nonzero coefficients are large relative to the averages, showing that all models are affected by noise to some extent. However, the standard errors for MAPE and QPE are relatively small, which indicates the prediction performances are stable. Among all models, H-Enet($\gamma = \text{IQR}(y)/10$) and Q-Enet($\tau = 0.50$) have the best performances in terms of MAPE, and Q-Enet($\tau = 0.75$) dominates QPE, while LS-Enet performs poorly under both criteria. Q-Enet($\tau = 0.75$) seems the best overall and it also tends to select sparser models compared to the aforementioned H-Enet($\gamma = \text{IQR}(y)/10$), Q-Enet($\tau = 0.50$) or LS-Enet.
For each model, different partitions may lead to different selection results.
We select LS-Enet, H-Enet($\gamma = \text{IQR}(y)/10$) and Q-Enet($\tau = 0.75$) to represent their own classes, and report the names and the frequencies of top genes selected (over 40 times) in Table \ref{freq_bcdata} where the genes are ordered alphabetically. We observe that some genes such as DTL, NBR2, PSME3, RPL27 have high frequencies with all three models, while genes such as KHDRBS1 do not. Overall, H-Enet($\gamma = \text{IQR}(y)/10$) and Q-Enet($\tau = 0.75$) select more genes with high frequencies than LS-Enet while their model sizes are smaller on average, especially Q-Enet($\tau = 0.75$). It indicates these two models more consistently capture the important genes.
\begin{table}[htbp]
\spacingset{1}
\begin{center}
\begin{tabular}{lclclc}
\multicolumn{2}{c}{LS-Enet} & \multicolumn{2}{c}{H-Enet($\gamma = \text{IQR}(y)/10$)} & \multicolumn{2}{c}{Q-Enet($\tau = 0.75$)} \\
\hline
Gene & \multicolumn{1}{c}{Frequency} &
Gene & \multicolumn{1}{c}{Frequency} &
Gene & \multicolumn{1}{c}{Frequency} \\
\hline
DTL & 45 & C17orf53 & 46 & C17orf53 & 48 \\
KHDRBS1 & 41 & CENPQ & 42 & CENPM & 45 \\
NBR2 & 50 & DTL & 46 & DTL & 44 \\
PSME3 & 45 & MCM6 & 50 & GCN5L2 & 44 \\
RPL27 & 45 & NBR1 & 47 & KIAA0101 & 40 \\
VPS25 & 43 & NBR2 & 50 & MCM6 & 42 \\
& & NMT1 & 41 & NBR1 & 49 \\
& & PSME3 & 50 & NBR2 & 50 \\
& & RPL27 & 41 & PSME3 & 50 \\
& & & & RPL27 & 50 \\
& & & & SUZ12 & 40 \\
& & & & SYNGR4 & 41 \\
& & & & XRCC2 & 41 \\
\hline
\end{tabular}
\caption{Genes selected with high frequency for the microarray dataset}
\label{freq_bcdata}
\end{center}
\end{table}
\section{Discussions}
\label{section_discuss}
The Huber loss regression and the quantile regression have important applications in many fields.
However, there is a lack of efficient algorithms and publicly available software that can fit these models in high-dimensional settings. In this paper, we develop an efficient and scalable algorithm for computing the solution paths for these models
with the elastic-net penalty. We also provide an implementation via the R package \texttt{hqreg} publicly available on CRAN (\url{http://cloud.r-project.org/package=hqreg}).
|
1,116,691,498,452 | arxiv | \section{Introduction}
Let $T$ be a $\field{C}^*$-torus of rank $n$, i.e., $T=(\field{C}^*)^n$.
A toric variety $X$ of complex dimension $n$ is a normal complex algebraic
variety with an action of $T$ having an open dense orbit.
A fundamental result in the theory of toric varieties says that there
is a one-to-one correspondence between toric varieties and fans, and
among toric varieties,
compact smooth toric varieties, which we call {\em toric manifolds},
are well studied, see \cite{fult93}, \cite{oda88}.
Suppose two toric manifolds $X$ and $X'$ are isomorphic as varieties.
Then they are not necessarily equivariantly isomorphic as varieties, but
\emph{weakly equivariantly} isomorphic as varieties, i.e.
there is a variety isomorphism $\phi\colon X\to X'$ together with
an automorphism $\gamma$ of $T$ such that $\phi(tx)=\gamma(t)
\phi(x)$ for any $t\in T$ and $x\in X$.
This is well-known and follows from the fact that
the automorphism group of a toric manifold is a linear algebraic group
with the acting group $T$ as a maximal algebraic torus
(\cite[Section 3.4]{oda88}). Therefore, classifying toric manifolds
up to variety isomorphism is same as that up to weakly equivariant variety
isomorphism.
The equivariant cohomology of a toric variety $X$ is by definition
\[
H^*_T(X):=H^*(ET\times_T X)
\]
where $ET$ is the total space of the universal principal $T$-bundle and
$ET\times_T X$ is the oribit space of $ET\times X$ by the diagonal $T$-action.
$H^*_T(X)$ contains a lot of geometrical
information on $X$, but its ring structure does not reflect enough
geometrical information on $X$.
In fact, when $X$ is a toric manifold, $H^*_T(X)$ as a ring
is the face ring of the underlying simplicial complex $\Sigma$
of the fan of $X$ and determined by $\Sigma$.
There are toric manifolds which are not isomorphic as varieties but
have the same underlying simplicial complex, so equivariant
cohomology as a ring does not distinguish toric manifolds.
However, $H^*_T(X)$ is not only a ring but also
an algebra over $H^*(BT)$ through the projection map from
$ET\times_T X$ onto $ET/T=BT$. This algebra structure contains more
geometrical information on $X$.
If two toric manifolds $X$ and $X'$ are isomorphic as varieties, then they
are weakly equivariantly isomorphic as varieties as remarked above, so that
$H^*_T(X)$ and $H^*_T(X')$ are \emph{weakly isomorphic} as
algebras over $H^*(BT)$, i.e., there is a ring isomorphism $\Phi\colon
H^*_T(X')\to H^*_T(X)$ together with an automorphism $\gamma$ of $T$ such that
$\Phi(u\omega)=\gamma^*(u)\Phi(\omega)$ for any $u\in H^*(BT)$ and $\omega
\in H^*_T(X')$ where $\gamma^*$ denotes the automorphism of $H^*(BT)$ induced
by $\gamma$. Our main result asserts that the converse holds.
\begin{theo} \label{main}
Two toric manifolds are (weakly equivariantly) isomorphic as varieties if
and only if their equivariant cohomology algebras
are weakly isomorphic.
\end{theo}
The theorem above leads us to ask how much information ordinary
cohomology contains for toric manifolds, in particular we may ask
whether two toric manifolds are homeomorphic (or diffeomorphic)
if their ordinary cohomology rings are isomorphic.
The question is affirmatively solved in some cases
(\cite{ma-pa06}, \cite{ch-ma-su07}) and the author does not know any
counterexample.
This paper is organized as follows. In Section 2 we review how the
equivariant cohomology of a toric manifold $X$ is related to the fan of $X$,
and prove Theorem~\ref{main} in Section 3. In Section 4 we observe
that our argument also works with some modification for quasitoric
manifolds which are a topological counterpart to toric manifolds.
\medskip
\noindent
{\bf Acknowledgment.}
The first version of this paper was written while the author was visiting
Fudan University in fall
2006. He would like to thank Fudan University and Zhi L\"u
for the invitation and providing excellent circumstances to do research.
He also would like to thank Taras Panov, Hiroshi Sato and Dong Youp Suh
for useful discussions.
\bigskip
\section{Equivariant cohomology and fan} \label{eqfan}
Throughout this and next sections, $X$ will denote a toric manifold of
complex dimension $n$ unless otherwise stated.
In this section we shall review how the equivariant cohomology
of $X$ is related to the fan of $X$.
The reader will find that most of the arguments in this and next sections
work with a compact torus $(S^1)^n$ instead of $T=(\field{C}^*)^n$.
There are only finitely many $T$-invariant divisors in $X$, which we denote
by $X_1,\dots,X_m$. Each $X_i$ is a complex codimension-one invariant
closed submanifold of $X$ and fixed pointwise by some $\field{C}^*$-subgroup
of $T$. Since $X$ and $X_i$ are complex manifolds,
they have canonical orientations. Let $\tau_i\in H^2_T(X)$ be the
Poincar\'e dual of $X_i$ viewed as an equivariant cycle in $X$,
in other words, $\tau_i$ is the image of the unit $1\in H^0_T(X_i)$
by the equivariant Gysin homomorphism $\colon H^0_T(X_i)\to H^2_T(X)$ induced
from the inclusion map $\colon X_i\to X$.
We call $\tau_i$ the {\em Thom class} of $X_i$.
We abbreviate a set $\{1,\dots,m\}$ as $[m]$.
The invariant divisors $X_i$ intersect transversally, so a cup product
$\prod_{i\in I}\tau_i$ for a subset $I$ of $[m]$ is the Poincar\'e
dual of the intersection $\cap_{i\in I}X_i$.
In particular, $\prod_{i\in I}\tau_i=0$ if $\cap_{i\in I}X_i=\emptyset$.
Since $H^*(X)$ is generated by elements in $H^2(X)$ as a ring
(see \cite[section 3.3]{oda88}),
we see that $H^*_T(X)$ is generated by $\tau_i$'s as a ring and there is
no more relation among $\tau_i$'s than those mentioned above,
see {\cite[Proposition 3.4]{masu99}} for example. Namely we have
\begin{prop}\label{Tring}
\[
H^*_T(X)=\field{Z}[\tau_1,\dots,\tau_m]/(\prod_{i\in I}\tau_i \mid \bigcap_{i\in I}X_i
=\emptyset) \quad \text{as ring}
\]
where $I$ runs all subsets of $[m]$ such that $\bigcap_{i\in I}X_i
=\emptyset$.
\end{prop}
We set
\[
\Sigma:=\{ I\subset [m]\mid \bigcap_{i\in I}X_i\not=\emptyset\}.
\]
This is an abstract simplicial complex of dimension $n-1$ and the proposition
above says that $H^*_T(X)$ is the face ring (or Stanley-Reisner ring) of the
simplicial complex $\Sigma$.
Let $\pi\colon ET\times_T X\to ET/T=BT$ be the projection on the first factor.
Through $\pi^*\colon H^*(BT)\to H^*_T(X)$, one can regard $H^*_T(X)$ as
an algebra over $H^*(BT)$. Since $T$ is a torus of rank $n$,
$H^*(BT)$ is a polynomial ring in $n$ variables of degree two, in particular,
it is generated by elements of degree two as a ring. Therefore, one can
find the algebra structure of $H^*_T(X)$ over $H^*(BT)$ if one knows
how elements in $H^2(BT)$ map to $H^2_T(X)$ by $\pi^*$.
\begin{prop}
\label{Talge}
To each $i\in [m]$, there is a unique element $v_i\in H_2(BT)$
such that
\begin{equation} \label{keyeq}
\pi^*(u)=\sum_{i=1}^m\langle u,v_i\rangle \tau_i\quad \text{for any $u\in
H^2(BT)$}
\end{equation}
where $\langle\ ,\ \rangle$ is the natural pairing between cohomology and
homology.
\end{prop}
\begin{rema} The identity (\ref{keyeq}) corresponds to the identity
in \cite[Lemma in p.61]{fult93} in algebraic geometry,
which describes a principal divisor
as a linear combination of the $T$-invariant divisors $X_i$.
\end{rema}
\begin{proof} The proposition is proved in
\cite[Lemma 9.3]{ha-ma03} and \cite[Lemma 1.5]{masu99}.
But for the reader's convenience we shall reproduce the proof given
in \cite[Lemma 9.3]{ha-ma03}.
By Proposition~\ref{Tring} $H^2_T(X)$ is freely generated by $\tau_1,
\dots,\tau_m$ over $\field{Z}$. Therefore, for each $u\in H^2(BT)$, one can
uniquely express $\pi^*(u)\in H^2_T(X)$ as
\[
\pi^*(u)=\sum_{i=1}^mv_i(u)\tau_i
\]
with integers $v_i(u)$ depending on $u$. We view $v_i(u)$ as a function
of $u$. Since $\pi^*$ is a homomorphism,
the function $v_i(u)$ is linear; so there
is a unique $v_i\in H_2(BT)$ such that $v_i(u)=\langle u,v_i\rangle$.
\end{proof}
The vectors $v_i$ have a nice geometrical meaning, which we shall explain.
The group $\Hom(\C^*,T)$ of homomorphisms
from $\C^*$ to $T$ can be identified with $H_2(BT)$ as follows.
An element $\rho$ of $\Hom(\C^*,T)$ induces a continuous map
$\bar\rho\colon B\C^*\to BT$ between classifying spaces
and $H_2(B\C^*)$ is isomorphic to $\field{Z}$; so
once we choose and fix a generator, say $\alpha$, of $H_2(B\C^*)$, we get
an element $\bar\rho_*(\alpha)\in H_2(BT)$. A correspondence $\colon
\rho \to \bar\rho_*(\alpha)$ gives an isomorphism from
$\Hom(\C^*,T)$ to $H_2(BT)$ and we denote by $\lambda_v$ the element of
$\Hom(\C^*,T)$ corresponding to $v\in H_2(BT)$.
It turns out that $\lambda_{v_i}(\C^*)$
is the $\field{C}^*$-subgroup of $T$ fixing $X_i$ pointwise,
see \cite[Lemma 1.10]{masu99} for example.
We have obtained two data from $X$, one is the abstract simplicial complex
$\Sigma$ and the other is the set of vectors $v_1,\dots,v_m$ in $H_2(BT)$.
To each $I\in \Sigma$ we form a cone in $H_2(BT)\otimes\field{R}=H_2(BT;\field{R})$
spanned by $v_i$'s $(i\in I)$. Then the collection of these cones is the fan
of $X$. Precisely speaking, we need to add the $0$-dimensional cone consisting
of the origin to this collection to satisfy
the conditions required in the definition of fan, see \cite{fult93} or
\cite{oda88}.
The $0$-dimensional cone corresponds to the empty subset of $[m]$.
Although we formed cones using the data $\Sigma$ and $\{v_i\}$ to define
the fan of $X$, we may think of a pair $(\Sigma,\{v_i\})$ as the fan of $X$.
As is well known $X$ can be recovered
from the fan of $X$. There are at least three ways (gluing affine spaces,
taking quotient by a $\field{C}^*$-torus or symplectic reduction) to recover $X$
from the fan of $X$.
We shall recall the quotient construction.
For $x=(x_1,\dots,x_m)\in \field{C}^m$ we define $I(x)=\{ i\mid x_i=0\}$.
We note that $(\field{C}^*)^m$ acts on $\field{C}^m$ via coordinatewise scalar
multiplication.
\begin{prop}[see \cite{cox95}] \label{recov}
Let $X$ be a toric manifold and $(\Sigma,\{v_i\})$ be the fan of $X$.
We consider
\[
Y:=\{ x\in \field{C}^m\mid I(x)\in \Sigma\cup\{\emptyset\}\}
\]
and a homomorphism
\[
\mathcal V\colon (\field{C}^*)^m\to (\field{C}^*)^n=T
\]
defined by
$$\mathcal V(g_1,\dots,g_m)=\prod_{i=1}^m \lambda_{v_i}(g_i).$$
Then $Y$ is invariant under the $(\field{C}^*)^m$-action,
the kernel $\ker \mathcal V$ of $\mathcal V$ acts on $Y$ freely and
the quotient $Y/\ker \mathcal V$ with the induced $T$-action
is a toric manifold equivariantly isomorphic to $X$.
\end{prop}
\bigskip
\section{Poof of Theorem~\ref{main}}
We continue to use the notation in Section 2.
Let $X^T$ denote the set of $T$-fixed points in $X$. As is well known,
it consists of finitely many points.
For $\xi\in H^2_T(X)$, we denote its restriction to $p\in X^T$ by $\xi|p$
and define
\[
Z(\xi):=\{ p\in X^T\mid \xi|p=0\}.
\]
\begin{lemm} \label{length}
Express $\xi=\sum_{i=1}^m a_i\tau_i$ with integers
$a_i$. If $a_i\not=0$ for some $i$, then $Z(\xi)\subset Z(\tau_i)$. Moreover,
if $a_i\not=0$ and $a_j\not=0$ for some different $i$ and $j$,
then $Z(\xi)\subsetneq Z(\tau_i)$.
\end{lemm}
\begin{proof}
Let $p\in X^T$.
Since $\tau_i$ is the Poincar\'e dual of $X_i$ viewed as an equivariant cycle
in $X$, $\tau_i|p=0$ if $p\notin X_i$.
Moreover, if $p\in X_i$, then $\tau_i|p\in H^2_T(p)=H^2(BT)$ is the
equivariant Euler class of
the complex one-dimensional normal $T$-representation at $p$ to $X_i$.
This implies that
\begin{equation} \label{taup}
\tau_i|p=0 \quad\text{if and only if}\quad p\notin X_i
\end{equation}
and that there are exactly $n$ number of $X_i$'s
containing $p$ and $\{\tau_i|p\mid p\in X_i\}$ forms a basis of $H^2(BT)$.
Suppose $p\in Z(\xi)$. Then $0=\xi|p=\sum_{i=1}^m a_i\tau_i|p$ and
it follows from the observation above that
$\tau_i|p=0$ if $a_i\not=0$. This proves the former statement in the lemma.
If both $a_i$ and $a_j$ are non-zero, then $Z(\xi)\subset Z(\tau_i)\cap
Z(\tau_j)$ by the former statement in the lemma.
Therefore, it suffices to prove
that $Z(\tau_i)\cap Z(\tau_j)$ is properly contained in $Z(\tau_i)$.
Suppose that
$Z(\tau_i)\cap Z(\tau_j)=Z(\tau_i)$. Then $Z(\tau_j)\supset Z(\tau_i)$,
so $X_j^T\subset X_i^T$ by (\ref{taup}).
This implies that $X_j=X_i$, a contradiction.
\end{proof}
Let $S=H^*(BT)\backslash \{0\}$ and let $S^{-1}H^*_T(X)$ denote the localized
ring of $H^*_T(X)$ by $S$.
Since $H^{odd}(X)=0$, $H^*_T(X)$ is free as a module over $H^*(BT)$.
Hence the natural map
\[
H^*_T(X)\to S^{-1}H^*_T(X)\cong S^{-1}H^*_T(X^T)=
\bigoplus_{p\in X^T}S^{-1}H^*_T(p)
\]
is injective, where the isomorphism above is induced from the
inclusion map from $X^T$ to $X$ and is a consequence of the Localization
Theorem in equivariant cohomology (\cite[p.40]{hsia75}). The annihilator
$$\Ann(\xi):=\{\eta \in S^{-1}H^*_T(X)\mid \eta\xi=0\}$$
of $\xi$ in $S^{-1}H^*_T(X)$ is nothing but sum of $S^{-1}H^*_T(p)$
over $p$ with $\xi|p=0$. Therefore it is a free $S^{-1}H^*(BT)$
module of rank $|Z(\xi)|$.
Since $\Ann(\xi)$ is defined using the algebra structure of $H^*_T(X)$,
$|Z(\xi)|$ is an invariant of $\xi$
depending only on the algebra structure of $H^*_T(X)$.
We note that $|Z(\xi)|$ is invariant under an algebra isomorphism.
We call $|Z(\xi)|$ the {\em zero-length} of $\xi$.
\begin{lemm} \label{Thom}
Let $X'$ be another toric manifold ($X'$ might be same as $X$).
If $f\colon H^*_T(X)\to H^*_T(X')$ is an algebra isomorphism,
then $f$ maps the Thom classes in $H^2_T(X)$ to the Thom classes in
$H^2_T(X')$ bijectively up to sign.
\end{lemm}
\begin{proof}
We classify the Thom classes according to their zero-length.
Let $\mathcal T_1$ be the set of Thom classes in $H^2_T(X)$ with largest
zero-length, and let $\mathcal T_2$ be the set of Thom classes in $H^2_T(X)$ with
second largest zero-length, and so on. Similarly we define $\mathcal T_1', \mathcal T_2'$
and so on for the Thom classes in $H^2_T(X')$.
Let $m_k$ (resp. $m_k'$) be the zero-length of elements in $\mathcal T_k$ (resp.
$\mathcal T_k'$).
Since both $f$ and $f^{-1}$ preserve zero-length and are isomorphisms,
$m_1=m_1'$ and $f$ maps $\mathcal T_1$ to $\mathcal T_1'$ bijectively up to sign by
Lemma~\ref{length}.
Take an element $\tau_i$ from $\mathcal T_2$. Since $\mathcal T_1$ and
$\mathcal T_1'$ are preserved under $f$ and $f^{-1}$, $f(\tau_{i})$ is
not a linear combination of elements in $\mathcal T_1'$ .
This together with Lemma~\ref{length} means
that $m_2\leq m_2'$. The same argument for $f^{-1}$ instead of $f$ shows
that $m_2'\leq m_2$, so that $m_2=m_2'$. Again, this together with
Lemma~\ref{length} implies that $f$ maps $\mathcal T_2$ to $\mathcal T_2'$ bijectively
up to sign. The lemma follows by repeating this argument.
\end{proof}
Now we shall complete the proof of Theorem~\ref{main}.
Let $X$ and $X'$ be two toric manifolds whose equivariant cohomology algebras
over $H^*(BT)$ are weakly isomorphic. We note that changing the action of $T$
on $X$ through an automorphism of $T$, we may assume that $H^*_T(X)$ and
$H^*_T(X')$ are isomorphic as algebras over $H^*(BT)$.
We put a prime for notation for $X'$ corresponding to
the Thom classes $\tau_i$, the abstract simplicial complex $\Sigma$ and
the vectors $v_i$ etc. for $X$.
Let $f\colon H^*_T(X)\to H^*_T(X')$ be an isomorphism of algebras over
$H^*(BT)$.
By Lemma~\ref{Thom}, the number of the Thom classes in $H^2_T(X)$ is same as
that in $H^2_T(X')$ and there is a permutation $\bar f$ on $[m]$ such that
\begin{equation} \label{ftau}
f(\tau_i)=\epsilon_i\tau'_{\bar f(i)} \quad\text{with $\epsilon_i=\pm 1$.}
\end{equation}
If $I\subset [m]$ is an element of $\Sigma$, then
$\prod_{i\in I}\tau_i$ is non-zero by Proposition~\ref{Tring} and hence so is
$f(\prod_{i\in I}\tau_i)=\prod_{i\in I}\epsilon_i\tau'_{\bar f(i)}$.
Therefore a subset $\{ \bar f(i)\mid i\in I\}$ of $[m]$ is
a simplex in $\Sigma'$ whenever $I$ is a simplex in $\Sigma$, which means
that $\bar f$ induces a simplicial map from $\Sigma$ to $\Sigma'$.
Applying the same argument to the inverse of $f$, we see that the induced
simplicial map has an inverse, so that it is an isomorphism.
Since $f$ is an algebra map over $H^*(BT)$,
${\pi'}^*=f\circ\pi^*$. Therefore,
sending the identity (\ref{keyeq}) by $f$ and using (\ref{ftau}), we have
\[
{\pi'}^*(u)=f(\pi^*(u))=\sum_{i=1}^m \langle u,v_i\rangle f(\tau_i)
=\sum_{i=1}^m \langle u,v_i\rangle \epsilon_i\tau'_{\bar f(i)}.
\]
Comparing this with the identity (\ref{keyeq}) for $X'$ and
noting that $\bar f$ is a permutation on $[m]$, we conclude that
\begin{equation} \label{fv}
\epsilon_iv_i=v'_{\bar f(i)}\quad \text{for each $i$.}
\end{equation}
We identify $\Sigma$ with $\Sigma'$ through the isomorphism induced by
$\bar f$,
so that we may think of $\bar f$ as the identity map and then the identity
(\ref{fv}) turns into
\[
\epsilon_i v_i=v_i'.
\]
By Proposition~\ref{recov} we may assume $X=Y/\ker\mathcal V$ and $X'=Y'/\ker\mathcal V'$.
Since $\Sigma'$ is identified with $\Sigma$, we have $Y=Y'$.
Therefore it suffices to check that $\ker \mathcal V=\ker \mathcal V'$.
Since $\lambda_{-v}(g)=\lambda_{v}(g)^{-1}=\lambda_{v}(g^{-1})$ for
$v\in H_2(BT)$ and $g\in
\C^*$, an automorphism $\rho$ of $(\field{C}^*)^m$ defined by
$$\rho(g_1,\dots,g_m)=(g_1^{\epsilon_1},\dots,g_m^{\epsilon_m})$$
satisfies $\mathcal V\circ \rho=\mathcal V'$. This implies $\ker \mathcal V=\ker\mathcal V'$ and
completes the proof of Theorem~\ref{main}.
\bigskip
\section{Quasitoric manifolds}
Davis-Januszkiewicz
\cite{da-ja91} introduced the notion of what is now called a
\emph{quasitoric manifold}, see \cite{bu-pa02}.
A quasi-toric manifold is a closed smooth
manifold of even dimension, say $2n$, with a smooth action of a compact torus
group $(S^1)^n$ of dimension $n$ such that the action is locally isomorphic
to a faithful $(S^1)^n$-representation of real dimension $2n$ and that
the orbit
space is combinatorially a simple convex polytope. A toric manifold with the
action restricted to the maximal compact toral subgroup of $T$
often provides an example of a quasitoric manifold, e.g. this is the
case when $X$ is projective. However, there are many quasitoric manifolds
which do not arise from a toric manifold. For instance, $\field{C} P^2\#\field{C} P^2$
with an appropriate action of $(S^1)^2$ is a quasitoric manifold but
does not arise from a toric manifold because $\field{C} P^2\#\field{C} P^2$
does not allow a complex (even almost complex) structure.
We note that the equivariant cohomology of a quasitoric manifold of dimension
$2n$ is an algebra over $H^*(B(S^1)^n$) similarly to the toric case.
The purpose of this section is to prove the following.
\begin{theo} \label{quasi}
Two quasitoric manifolds are equivariantly homeomorphic if
their equivariant cohomology algebras are isomorphic.
\end{theo}
\begin{proof}
When $X$ is a quasitoric manifold, we take $X_i$ to be a connected real
codimension-two closed submanifold of $X$ fixed pointwise by some circle
subgroup of $(S^1)^n$. Then
the proof for Theorem~\ref{main} almost works if we replace $\field{C}^*$ by $S^1$
(and hence $T=(\field{C}^*)^n$ by $(S^1)^n$). The only problem is that we do
not have Proposition~\ref{recov} for quasitoric manifolds, so that the
last paragraph in the previous section needs to be modified.
In the sequel, it suffices to prove that the existence of an isomorphism
$\bar f\colon \Sigma\to \Sigma'$ satisfying (\ref{fv}) implies that the two
quasitoric manifolds $X$ and $X'$ are equivariantly homeomorphic.
Let $P$ be the orbit space of $X$ by the action of
$(S^1)^n$ and let $q\colon X\to P$ be the quotient map. The orbit space
$P$ is a simple convex polytope by the definition of quasitoric manifold.
Then $P_i:=q(X_i)$ is a facet (i.e., a codimension-one face) of $P$.
The dual polytope $P^*$ of $P$ is a simplicial polytope
and its boundary complex agrees with $\Sigma$. The vertices of $\Sigma$
bijectively correspond to the facets of $P$ so that $v_i$ is assigned
to $P_i$. The vectors $v_i$ form a characteristic function on $P$
introduced in \cite{da-ja91}. Any (proper) face of $P$ is obtained as an
intersection $P_I:=\cap_{i\in I}P_i$ for some $I\in\Sigma$.
We define $P_\emptyset$ to be $P$ itself.
For $I\in \Sigma$ we denote by $S_I$ a subgroup of $(S^1)^n$
generated by circle subgroups $\lambda_{v_i}(S^1)$ for $i\in I$.
We define $S_\emptyset$ to be the unit group.
Associated with a pair $(P,\{v_i\})$ we form a quotient space
\[
X(P,\{v_i\}):=P\times (S^1)^n/\sim.
\]
Here $(p_1,g_1)\sim (p_2,g_2)$ if and only if $p_1=p_2$ and $g_1^{-1}g_2
\in S_I$ where $I\in \Sigma\cup\{\emptyset\}$ is determined by the condition
that $p_1=p_2$ is contained in the interior of $P_I$. The natural action
of $(S^1)^n$ on the product $P\times (S^1)^n$ descends to an action on
$X(P,\{v_i\})$ and $X$ is equivariantly homeomorphic to
$X(P,\{v_i\})$ (see \cite[Proposition 1.8]{da-ja91}).
As before, we put a prime to denote elements for $X'$ corresponding to
$P, v_i$ and $\Sigma$.
The isomorphism $\bar f\colon \Sigma\to \Sigma'$ induces an isomorphism
from $P^*$ to ${P'}^*$ and then a face-preserving homeomorphism
from $P$ to $P'$ which we
denote by $\varphi$. A map $\varphi\times id \colon P\times (S^1)^n\to
P'\times (S^1)^n$ descends to a map from $X(P,\{v_i\})$ to $X(P',\{v_i'\})$
by virtue of (\ref{fv}) and the resulting map is an
equivariant homeomorphism, so $X$ is equivariantly homeomorphic to $X'$.
\end{proof}
Similarly to the toric case,
it would be interesting to ask whether two quasitoric manifolds are
homeomorphic (or diffeomorphic) if their ordinary cohomology rings are
isomorphic, see \cite{ma-pa06} and \cite{ch-ma-su07} for some
partial affirmative solutions.
\begin{rema}
Davis-Januszkiewicz \cite{da-ja91} also introduced the notion of a real
version of quasitoric manifold, which they call a {\em small cover}.
A small cover is a closed smooth manifold of dimension, say $n$, with a smooth
action of a rank $n$ mod two torus group $(\field{Z}_2)^n$ such that
the action is locally isomorphic to a faithful $(\field{Z}_2)^n$-representation of
real dimension $n$ and that the orbit space is combinatorially a simple
convex polytope. Our argument also works for small covers with $\field{Z}_2$
coefficient, so that
small covers are equivariantly homeomorphic if their equivariant cohomology
algebras with $\field{Z}_2$ coefficient are isomorphic.
\end{rema}
\bigskip
|
1,116,691,498,453 | arxiv | \section{Numerical details}
In this appendix we give some numerical details regarding the construction of both the lumpy black holes and black rings. We first test numerical convergence. Since we are using spectral collocation methods, we expect to find exponential convergence as the number of points is varied. This is exactly what we see in Fig.~\ref{fig:converge}. On the left panel of Fig.~\ref{fig:converge}, we consider some lumpy BHs and show how the square of the Weyl tensor $\mathcal{C}^2 = C^{abcd}C_{abcd}$ varies as the number of points is changed.
Another quantity we can use to test convergence is the norm of the deTurck vector $\xi^2$. In the DeTurck method \cite{Headrick:2009pv,Figueras:2011va} this is the vector $\xi^\mu=g^{\alpha\beta}\left(\Gamma^{\mu}_{\alpha\beta}+\bar\Gamma^{\mu}_{\alpha\beta}\right)$, where $\bar\Gamma^{\mu}_{\alpha\beta}$ is the Levi-Civita connection for a chosen reference metric $\bar g$. For the boundary value problems considered here, the solutions found are necessarily solutions of the vacuum Einstein equations in the gauge $\xi^\mu=0$, so the norm of the deTurck vector is a measure of how well the gauge condition is satisfied. On the right panel of Fig.~\ref{fig:converge}, we take some black ring solutions and we plot the norm of the DeTurck vector as a function of grid points. We again see exponential convergence.
\begin{figure}[ht]
\centering
\includegraphics[width=.45\textwidth]{convergence.pdf}\qquad
\includegraphics[width=.45\textwidth]{convergencering.pdf}
\caption{{\bf Left:} convergence test for the Weyl tensor for lumpy black holes in $d=7$. We plot $1-|\mathcal{C}^2(N)|_\infty/|\mathcal{C}^2(N+1)|_\infty$, as a function of the number of grid points $N$. From bottom to top, we have $j = 1.097,\,1.117$ and $1.136$. {\bf Right:} convergence test of the norm of the DeTurck vector $|\xi^2|_\infty$ for rings in $d=7$ as a function of the grid points $(N+N)\times N$. From top to bottom, $j=1.05$ (fat branch), $j=1.01$ (thin branch), $j=1.64$ (thin branch). The $j=1.64$ plot plateaus after reaching the limits of machine precision.}\label{fig:converge}
\label{fig:convex}
\end{figure}
There is another test of numerical accuracy for the energy and angular momentum extracted from our solutions. Since our solutions are asymptotically flat, the expansion of the metric off spatial infinity is controlled by perturbations of flat space in a given gauge. We can then take advantage of the fact that flat space can be written as a warped product of a two-dimensional space (spanned by $t,R$, say) and a round $S^{d-2}$ sphere. We can thus use the expansion of \cite{Kodama:2003jz}, to catalog the possible boundary behaviours of our functions in terms of spherical harmonics on the $S^{d-2}$ sphere that preserve the $SO(2)\times SO(d-3)$ symmetry of the line element (\ref{geometry}). If we fix the metric at infinity to have the following form:
\begin{equation}
ds^2 = -dt^2+dR^2+R^2(d\theta^2+\cos^2\theta d\Omega^2_{d-4}+\sin^2\theta d\psi^2)\,,
\end{equation}
then the asymptotic behavior of $g_{tt}$ and $g_{t\psi}$ take the following simple form
\begin{align}
&g_{tt}= -1+E_0\frac{\mathbb{Y}^{0}(\theta)}{R^{d-3}}+\mathcal{O}(R^{-(d-1)})
\\
&g_{t\psi}= J_0\frac{\mathbb{Y}_{\psi}^{1}(\theta)}{R^{d-3}}+\mathcal{O}(R^{-(d-1)})\,,
\end{align}
where $\mathbb{Y}^{\ell_s}(\theta)$ and $\mathbb{Y}^{\ell_v}_a(\theta)$ are scalar and vector harmonics on $S^{d-2}$ with quantum numbers $\ell_s\geq0$ and $\ell_v\geq1$, respectively, and both $E_0$ and $J_0$ are \emph{constants}. Note that these metric functions are gauge invariant for spacetimes with the symmetries detailed above. Both the energy and angular momentum can solely be written as linear functions of these constants.
In $d=7$, our radial coordinate is asymptotically defined as $y \propto R^{-1/2}$, which means we can extract $E_0$ by taking two numerical derivatives of $g_{tt}$. In $d=6$, our radial coordinate is asymptotically $y\propto 1/R$, so we require three derivatives to extract $E_0$. In any dimension, we defined our metric functions such that $J_0$ could be extracted by taking a single numerical derivative. A good test of accuracy is to measure if these two coefficients are constants. This can be best done by performing a $\chi^2$ fit, and extracting the standard error. For all of our solutions, we find that the error in $E_0$ is smaller than $10^{-3}\%$, except for the last few points close to the merger, in which case the error increases to $0.1\%$ for the lumpy black holes, and $1\%$ for the rings. The errors in $J_0$ are much smaller ($\sim10^{-5}\%$).
A final test can be extracted from the the Smarr law. Since we can independently compute the energy, angular momentum, angular velocity, entropy and temperature, we can test whether the Smarr law is satisfied. Again, for the solutions in $d=7$, we find that to be true within $10^{-3}\%$, except for the last couple of points close to the merger where it increases to $0.1\%$. We find that the $d=6$ rings satisfy the Smarr law to within $10^{-3}\%$ except for the points close to the merger, where the error increases to $5\%$.
We emphasise that the error in the Smarr law and in the energy is a generous overestimation of the error in our plots. In our plots for the rings, the energy is obtained directly from the Smarr law after first obtaining the angular momentum, angular velocity, entropy, and temperature. This is more accurate since it only involves taking a single derivative for the angular momentum.
|
1,116,691,498,454 | arxiv | \section{Introduction}
Scalar-valued extrapolation, using the theory of Muckenhoupt weights, has proven to be an essential tool in harmonic analysis. The classical extrapolation result (see \cite{Ru82} and \cite[Chapter IV]{GR85}) says that if a (sub)linear operator $T$ satisfies for a fixed $p_0\in (1,\infty)$ and all weights $w$ in the \emph{Muckenhoupt} class $A_{p_0}$ the norm inequality
\begin{equation}\label{eq:extrapinput}
\|Tf\|_{L^{p_0}(w)}\leq C\,\|f\|_{L^{p_0}(w)}
\end{equation}
for all $f \in L^{p_0}(w)$, then we have for all $p\in (1,\infty)$ and all weights $w\in A_p$
\begin{equation}\label{eq:extrapout}
\|Tf\|_{L^p(w)}\leq C\,\|f\|_{L^p(w)}
\end{equation}
for all $f \in L^{p}(w)$. Numerous generalizations of this result have appeared, see for example \cite{AM07, CMP04, CM17, GM04, HMS88}. We mention several of them.
It was shown by Grafakos and Martell \cite{GM04} that extrapolation extends to the \emph{multilinear} setting. Indeed, they showed that given fixed exponents $p_1,\ldots, p_m\in (1,\infty)$, if for an $m$-(sub)linear operator $T$ and all weights $w_j^{p_j}\in A_{p_j}$ we have
\[
\|T(f_1,\ldots,f_m)\|_{L^{p}(w^p)}\leq C\, \prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j})},
\]
where $w= \prod_{j=1}^m w_j$ and $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}$, then the same estimate holds for all $p_j\in (1,\infty)$, weights $w_j^{p_j}\in A_{p_j}$ and $w$ and $p$ as before.
Considering a different kind of generalization, it was shown by Auscher and Martell \cite{AM07} that a \textit{limited range} version of the extrapolation result holds: if there are exponents $0<p_-<p_+\leq\infty$ such that the estimate \eqref{eq:extrapinput} is valid for a fixed $p_0\in (p_-,p_+)$ and all weights $w$ in the Muckenhoupt and \emph{Reverse H\"older} class $A_{p_0/p_-}\cap\RH_{(p_+/p_0)'}$, then \eqref{eq:extrapout} is valid for all $p\in (p_-,p_+)$ and all weights $w \in A_{p/p_-}\cap\RH_{(p_+/p)'}$.
Vector-valued extensions of the extrapolation theory have also been considered. Through an argument using Fubini's Theorem, the initial estimate \eqref{eq:extrapinput} immediately implies not only the estimate \eqref{eq:extrapout} for all $p \in (1,\infty)$, but also for extensions of the operator $T$ to functions taking values in the sequence spaces $\ell^s$ or more generally Lebesgue spaces $L^s$ for $s \in (1,\infty)$. Moreover, Rubio de Francia showed in \cite[Theorem 5]{Ru85} that one can take this even further. Indeed, this result states that assuming \eqref{eq:extrapinput} holds for some $p_0\in (1,\infty)$ and for all weights $w\in A_{p_0}$, then for each Banach function space $X$ with the $\UMD$ property, $T$ extends to an operator $\widetilde{T}$ on the Bochner space $L^p(X)$ which satisfies
\[
\nrmb{\widetilde{T}f}_{L^p(X)}\leq C \,\|f\|_{L^p(X)}
\]
for all $p\in(1,\infty)$ and all $f \in L^p(X)$ . Recently, it was shown by Amenta, Veraar, and the first author in \cite{ALV17} that given $p_-\in(0,\infty)$, if \eqref{eq:extrapinput} holds for $p_0\in (p_-,\infty)$ and all weights $w\in A_{p_0/p_-}$, then for each Banach function space $X$ such that $X^{p_-}$ has the $\UMD$ property, $T$ extends to an operator $\widetilde{T}$ on the Bochner space $L^p(w;X)$ and satisfies
\[
\nrmb{\widetilde{T}f}_{L^p(w;X)}\leq C \, \|f\|_{L^p(w;X)}
\]
for all $p\in(p_-,\infty)$, all weights $w \in A_{p/p_-}$ and all $f \in L^p(w;X)$. Here $X^{p_-}$ is the $p_-$-concavification of $X$, see Section \ref{sec:preliminaries} for the definition.
Vector-valued estimates in harmonic analysis have been actively developed in the past decades. Important for the mentioned vector-valued extrapolation are the equivalence of the boundedness of the vector-valued Hilbert transform on $L^p(X)$ and the $\UMD$ property of $X$ for a Banach space $X$ (see \cite{Bo83,Bu83}) and the fact that for a Banach function space $X$ the $\UMD$ property implies the boundedness of the lattice Hardy--Littlewood maximal operator on $L^p(X)$ (see \cite{Bo84,Ru86}). For recent results in vector-valued harmonic analysis in $\UMD$ Banach function spaces, see for example \cite{BFR12,DK17, Ho14, HV15,Xu15}.
\bigskip
In the recent work \cite{CM17} by Cruz-Uribe and Martell both the limited range and the multilinear extrapolation result were combined, yielding a unified multilinear limited range version of the extrapolation result in the scalar-valued case. This result also covers vector-valued extensions to $\ell^s$ for certain $s \in (0,\infty)$. This opened the question whether a unified multilinear limited range extrapolation theorem also holds for more general Banach function spaces. In this work, we give a positive answer to this question.
We now state our main result, in which we denote $X \in \UMD_{p_-,p_+}$ for the technical assumption that $\hab{(X^{p_-})^*}^{(p_+/p_-)'}$ has the $\UMD$ property, see Section \ref{sec:UMDp-p+} for a thorough discussion of this assumption. A more general version of this theorem can be found in Theorem \ref{thm:multi-limited-range-extrap} below.
\begin{theorem}\label{thm:maincor} Let $m\in\N$ and fix $0< p_j^-<p_j^+\leq\infty$ for $j\in\{1,\ldots,m\}$. Let $T$ be an operator defined on $m$-tuples of functions and suppose there exist $p_j\in (p_j^-,p_j^+)$ such that for all weights $w_j^{p_j} \in A_{{p_j/p_j^-}}\cap RH_{(p_j^+/p_j)'}$ and $f_j\in L^{p_j}(w_j^{p_j})$ we have
\begin{equation*}
\nrmb{T(f_1,\ldots,f_m)}_{L^p(w^p)} \leq C\,\prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j})},
\end{equation*}
with $w=\prod_{j=1}^mw_j$, $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}$, and where $C>0$ depends only on the characteristic constants of the weights. Moreover, assume that $T$ satisfies one of the following conditions:
\begin{enumerate}[(i)]
\item \label{main:1} $T$ is $m$-linear.
\item \label{main:2} $T$ is $m$-sublinear and positive valued.
\end{enumerate}
Let $X_1,\ldots, X_m$ be quasi-Banach function spaces over a $\sigma$-finite measure space $(S,\mu)$ and define $X=X_1\cdots X_m$. Assume that for all simple functions $f_j:\R^d \to X_j$ the function $\widetilde{T}f:\R^d \to X$ given by
\begin{equation*}
\widetilde{T}(f_1,\ldots,f_m)(x,s) := T(f_1(\cdot, s),\ldots,f_m(\cdot,s))(x), \qquad x\in \R^d, \quad s\in S
\end{equation*}
is well-defined and strongly measurable. If $X_j\in \UMD_{p_j^-,p_j^+}$, then for all $p_j\in (p_j^-,p_j^+)$ and weights $w_j^{p_j} \in A_{{p_j/p_j^-}}\cap RH_{(p_j^+/p_j)'}$, $\widetilde{T}$ extends to a bounded operator on $L^{p_1}(w_1^{p_1};X_1)\times\cdots\times L^{p_m}(w_m^{p_m};X_m)$ with
\begin{equation*}
\nrmb{\widetilde{T}(f_1,\ldots,f_m)}_{L^p(w^p;X)} \leq C'\,\prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j};X_j)},
\end{equation*}
for all $f_j\in L^{p_j}(w_j^{p_j};X_j)$, with $w$ and $p$ are as before, and where $C'>0$ depends on the $p_j$, $p_j^-$, $p_j^+$, the characteristic constants of the weights, and the spaces $X_j$.
\end{theorem}
\begin{remark}\label{rem:mainthm}~
\begin{itemize}
\item If $T$ is a linear operator as in Theorem \ref{thm:maincor}, we have for $f_j \otimes \xi_j \in L^{p_j}(w_j^{p_j}) \otimes X_j$ that
\begin{equation*}
\widetilde{T}(f_1\otimes \xi_1, \cdots, f_m \otimes \xi_m) = T(f_1,\cdots,f_m) \otimes \xi_1\cdots\xi_m \in L^{p}(w^{p}) \otimes X.
\end{equation*}
So in this case $\widetilde{T}$ is automatically well-defined and strongly measurable for all simple functions $f_j:\R^d \to X$.
\item Although we state Theorem \ref{thm:maincor} for Banach function spaces, it extends to spaces isomorphic to a closed subspace of a Banach function space and by standard representation techniques also to certain Banach lattices, see \cite{LT79, MN91} for the details.
\item In \cite{CM17} scalar-valued multilinear limited range extrapolation is proven through off-diagonal extrapolation. Relying on this result, in this paper we prove the vector-valued multilinear limited range result. Our method does not directly generalize to the off-diagonal setting, which leaves vector-valued off-diagonal extrapolation as an open problem.
\item In Theorem \ref{thm:maincor} one could allow for $p_j^- = 0$. In this case one would have to interpret $X_j \in \UMD_{0,p_j^+}$ as $X_j \in \UMD_{p,p_j^+}$ for some $p\in (0,p_j^+)$.
\end{itemize}
\end{remark}
Even in the linear case $m=1$ our result is new in the sense that it extends the main result of \cite{ALV17} to allow for finite $p_j^+$, which yields many new applications. We are now able to consider, for example, Riesz transforms associated to elliptic operators through the weighted estimates obtained in \cite{AM07}. Many more examples of such operators can also be considered through recent advances in the theory of sparse dominations. Indeed, for example for certain Fourier multipliers such as Bochner-Riesz multipliers as well as for spherical maximal operators, sparse bounds have been found. Sparse bounds naturally imply weighted norm estimates which, through our result, yield bounded vector-valued extensions for such operators. For a more elaborate discussion as well as for references we refer the reader to Section \ref{sec:applications}.
Our result is also new in the full range multilinear case, i.e., if $p_j^-=1$, $p_j^+=\infty$ for all $j\in\{1,\ldots, m\}$. This can, for example, be applied to multilinear Calder\'on-Zygmund operators, as these satisfy the appropriate weighted bounds to apply our result. We elaborate on this in Section \ref{sec:applications}.
Finally, for the case $m=2$ our result yields new results for boundedness of the vector-valued bilinear Hilbert transform $\widetilde{\BHT}$, due to known scalar-valued weighted bounds as were first established by Culiuc, di Plinio, and Ou \cite{CDPO16}.
Bounds for the vector-valued bilinear Hilbert transform $\widetilde{\BHT}$ have useful applications in PDEs, see \cite{BM16} and references therein. The precise result we obtain can be found in Theorem \ref{thm:bilhilbo}.
\begin{remark} \label{rem:intro}
In the recent work \cite{LMO18} of Li, Martell and Ombrosi, and the recent work \cite{N18} of the second author, scalar-valued extrapolation results were obtained using the multilinear weight classes from \cite{LOPTT09}, which were made public after this paper first appeared. Rather than considering a condition for each weight individually, these weight classes allow for an interaction between the various weights, making them more appropriately adapted to the multilinear setting. This gives rise to the problem of extending these results to the vector-valued case. To facilitate this, it seems that an appropriate multilinear $\UMD$ condition on tuples of Banach function spaces is required. We leave this as a basis for future research.
\end{remark}
\bigskip
This article is organized as follows:
\begin{itemize}
\item In Section \ref{sec:preliminaries} we summarize the preliminaries on Muckenhoupt weights, product quasi-Banach function spaces and the $\UMD$ property.
\item In Section \ref{sec:UMDp-p+} we discus the $\UMD_{p_-,p_+}$ property and give examples of quasi-Banach function spaces satisfying the $\UMD_{p_-,p_+}$ property.
\item In Section \ref{sec:main} we prove our main result in terms of $(m+1)$-tuples of functions, proving Theorem \ref{thm:maincor} as a corollary.
\item In Section \ref{sec:applications} we prove new vector-valued bounds for various operators.
\end{itemize}
\textbf{Acknowledgement.} The authors thank Mark Veraar and Dorothee Frey for their helpful comments on the draft. Moreover the authors thank Alex Amenta for his suggestion to consider the multilinear setting. Finally the authors would like to thank the anonymous referees for their suggestions.
\section{Preliminaries}\label{sec:preliminaries}
\subsection{Muckenhoupt weights}
A locally integrable function $w:\R^d\to (0,\infty)$ is called a \textit{weight}. For $p \in [1,\infty)$ and a weight $w$ the space $L^p(w)$ is the subspace of all measurable functions $f: \R^d \to \C$, which we denote by $f \in L^0(\R^d)$, such that
\begin{equation*}
\nrm{f}_{L^p(w)}:= \has{\int_{\R^d}\abs{f(x)}^pw(x)\dd x}^{1/p}<\infty.
\end{equation*}
By a cube $Q\subseteq\R^d$ we will mean a half-open cube whose sides are parallel to the coordinate axes and for a locally integrable function $f\in L^0(\R^d)$ we will write $\ip{f}_Q:= \frac{1}{\abs{Q}}\int_Q f \dd x$.
For $p\in[1,\infty)$ we will say that a weight $w$ lies in the \emph{Muckenhoupt class $A_p$} and write $w\in A_p$ if it satisfies
\[
[w]_{A_p}:=\sup_{Q}\ip{w}_{Q}\langle w^{1-p'}\rangle_{Q}^{p-1}<\infty,
\]
where the supremum is taken over all cubes $Q\subseteq\R^d$ and the second factor is replaced by $(\essinf_Q w)^{-1}$ if $p=1$. We define $A_\infty:=\bigcup_{p\in[1,\infty)}A_p$.
\begin{lemma}\label{lem:muckenhoupt}Let $p \in [1,\infty)$ and $w \in A_p$.
\begin{enumerate}[(i)]
\item \label{it:mw2} $w \in A_q$ for all $q \in [p,\infty)$ with $[w]_{A_q} \leq [w]_{A_p}$.
\item \label{it:mw1} If $p>1$, $w^{1-p'} \in A_{p'}$ with $[w]^{\frac{1}{p}}_{A_p} = [w^{1-p'}]_{A_{p'}}^{\frac{1}{p'}}$.
\item \label{it:mw3} If $p>1$, there exists an $\varepsilon>0$ such that $w \in A_{p-\varepsilon}$ and $[w]_{A_{p-\varepsilon}}\leq C_{p} \, [w]_{A_p}$.
\end{enumerate}
\end{lemma}
The first two properties of Lemma \ref{lem:muckenhoupt} are immediate from the definition. For the third see \cite[Corollary 7.2.6]{Gr14a}. The linear estimate of $[w]_{A_{p-\varepsilon}}$ in terms of $[w]_{A_p}$ can be found in \cite[Theorem 1.2]{HPR12}. Note that self-improvement properties for $A_p$ weights are classical. We opt to use this quantitative version of the result for clarity in the proof of our main theorem.
For $s\in[1,\infty)$ we say that $w \in A_\infty$ satisfies a \textit{reverse H\"{o}lder} property and write $w\in\RH_s$ if
\[
[w]_{\RH_s}:=\sup_{Q}\langle w^s\rangle^{\frac1s}_{Q}\langle w\rangle_{Q}^{-1}<\infty.
\]
We will require the following properties of the reverse H\"{o}lder classes, see \cite{JN91}.
\begin{lemma}\label{lem:reverseholder}
Let $r \in (1,\infty)$, $s \in [1,\infty)$ and define $p = s(r-1)+1$. For $w \in A_\infty$ the following are equivalent
\begin{enumerate}[(i)]
\item \label{it:weightsequiv1} $w \in A_r \cap \RH_s$.
\item \label{it:weightsequiv2} $w^s \in A_p$.
\item \label{it:weightsequiv3} $w^{1-r'} \in A_{p'}$.
\end{enumerate}
Moreover we have
\begin{align*}
\max\cbraceb{[w]_{\RH_s}^s, [w]_{A_r}^s} &\leq [w^s]_{A_p} \leq \hab{[w]_{A_r}[w]_{\RH_s}}^s.
\end{align*}
\end{lemma}
For $n\in\N$ we will write $\phi_{a,b,\cdots}$ for a non-decreasing function $[1,\infty)^n \to [1,\infty)$, depending on the parameters $a,b,\cdots$ and the dimension $d$. This function may change from line to line. We need non-decreasing dependence on the Muckenhoupt characteristics in our proofs. In \cite[Appendix A]{ALV17} it is shown how to deduce non-decreasing dependence from a more general estimate in terms of the Muckenhoupt characteristics.
\subsection{Banach function spaces}
Let $(S,\mu)$ be a $\sigma$-finite measure space. A subspace $X$ of $L^0(S)$ equipped with a quasi-norm $\nrm{\, \cdot \,}_X$ is called a \emph{quasi-Banach function space} if it satisfies the following properties:
\begin{enumerate}[(i)]
\item If $\xi\in L^0(S)$ and $\eta\in X$ with $\abs{\xi} \leq \abs{\eta}$, then $\xi\in X$ and $\nrm{\xi}_X \leq \nrm{\eta}_X$.
\item There is an $\xi\in X$ with $\xi>0$.
\item If $0
\leq \xi_n \uparrow \xi$ with $(\xi_n)_{n=1}^\infty$ a sequence in $X$, $\xi \in L^0(S)$ and $\sup_{n \in \N}\nrm{\xi_n}_X < \infty$, then $\xi\in X$ and $\nrm{\xi}_X = \sup_{n\in \N}\nrm{\xi_n}_X$.
\end{enumerate}
It is called a \emph{Banach function space} if $\nrm{\, \cdot \,}_X$ is a norm. A Banach function space $X$ is called \textit{order continuous} if for any sequence $0 \leq \xi_n \uparrow \xi \in X$ it holds that $\nrm{\xi_n -\xi}_X \to 0$. Order continuity of a Banach function space $X$ ensures that its dual $X^*$ is again a Banach function space (see \cite[Section 1.b]{LT79}), and that the Bochner space $L^p(S';X)$ is a Banach function space over $(S \times S',\mu\times\mu')$ for any $\sigma$-finite measure space $(S',\mu')$. As an example we note that any reflexive Banach function space is order-continuous.
A quasi-Banach function space $X$ is said to be \emph{$p$-convex} for $p \in (0,\infty]$ if
\begin{equation*}
\nrms{\has{\sumkn \abs{\xi_k}^p}^\frac{1}{p}}_X \leq \has{\sumkn\nrm{\xi_k}_X^p}^\frac{1}{p}
\end{equation*}
for all $\xi_1,\cdots,\xi_n \in X$ with the usual modification when $p = \infty$. It is said to be \emph{$p$-concave} when the reverse inequality holds. Usually the defining inequalities for $p$-convexity and $p$-concavity include a constant depending on $p$ and $X$, but as shown in \cite[Theorem 1.d.8]{LT79}, $X$ can be renormed equivalently such that these constants equal 1. See \cite[Section 1.d]{LT79} for a thorough introduction of $p$-convexity and concavity in Banach function spaces and see \cite{Ka84} for the quasi-Banach function space case.
We define the \emph{$p$-concavification} of a quasi-Banach function space $X$ for $p \in (0,\infty)$ by
\begin{equation*}
X^p := \cbrace{\xi \in L^0(S): \abs{\xi}^\frac{1}{p} \in X} = \cbrace{\abs{\xi}^p \sgn{\xi}: \xi \in X},
\end{equation*}
equipped with the quasi-norm $\nrm{\xi}_{X^p} := \nrmb{\abs{\xi}^{\frac{1}{p}}}^p_X$. Note that $X^p$ is a Banach function space if and only if $X$ is $p$-convex. In particular, $X$ is a Banach function space if and only if it is $1$-convex.
For two quasi-Banach function spaces $X_0,X_1$ over the same measure space $(S,\mu)$ we define the vector space $X_0 \cdot X_1$ as
\begin{equation*}
X_0 \cdot X_1 := \cbrace{\xi_0\cdot \xi_1:\xi_0 \in X_0, \xi_1 \in X_1}
\end{equation*}
and for $\xi\in X_0 \cdot X_1$ we define
\begin{equation*}
\nrm{\xi}_{X_0 \cdot X_1}:= \inf \cbraceb{\nrm{\xi_0}_{X_0}\nrm{\xi_1}_{X_1}:\abs{\xi} = \xi_0\cdot \xi_1, 0 \leq \xi_0 \in X_0, 0\leq \xi_1 \in X_1}
\end{equation*}
We call $X_0 \cdot X_1$ a \emph{product quasi-Banach function space} if $\nrm{\, \cdot \,}_{X_0 \cdot X_1}$ defines a complete quasi-norm on $X_0 \cdot X_1$. We will mostly be working with so called \emph{Calder\'on-Lozanovskii} products. These are product quasi-Banach function spaces of the form $X_0^{1-\theta}\cdot X_1^\theta$ for some $\theta \in (0,1)$, see \cite{Ca64,Lo69}. Of course the definition of product quasi-Banach function spaces and Calder\'on-Lozanovskii products can be canonically extended to $m$ quasi-Banach function spaces over the same measure space for any $m \in \N$. We give a few examples of product Banach function spaces, see also \cite{Bu87}.
\begin{example}\label{ex:exproducts}
Fix $m \in \N$ and let $(S,\mu)$ be an atomless or atomic $\sigma$-finite measure space.
\begin{enumerate}[(i)]
\item \label{it:exproduct1} Lebesgue spaces: $L^p(S) = L^{p_1}(S) \cdots L^{p_m}(S)$ for $p_j \in (0,\infty)$ and $\frac{1}{p} = \sum_{j=1}^m\frac{1}{p_j}$.
\item \label{it:exproduct2} Lorentz spaces: $L^{p,q}(S) = L^{p_1,q_1}(S) \cdots L^{p_m,q_m}(S)$ for $p_j,q_j \in (0,\infty)$, $\frac{1}{p} = \sum_{j=1}^m\frac{1}{p_j}$ and $\frac{1}{q} = \sum_{j=1}^m\frac{1}{q_j}$.
\item \label{it:exproduct3} Orlicz spaces: $L^{\Phi}(S) = L^{\Phi_1}(S)\cdots L^{\Phi_m}(S)$ for Young functions $\Phi_j$ and $\Phi^{-1} = \Phi_1^{-1} \cdots \Phi_m^{-1}$.
\end{enumerate}
\end{example}
We will use the following properties of product Banach function spaces:
\begin{proposition}\label{prop:pBFS} Let $X,X_0,X_1$ be Banach function spaces over a $\sigma$-finite measure space $(S,\mu)$ and let $\theta \in (0,1)$.
\begin{enumerate}[(i)]
\item \label{it:pBFScomplex} If $X_0$ or $X_1$ is reflexive, then $X_0^{1-\theta}\cdot X_1^\theta=[X_0,X_1]_\theta$.
\item \label{it:pBFSrefl} If $X_0$ or $X_1$ is reflexive, then $X_0^{1-\theta}\cdot X_1^\theta$ is reflexive.
\item \label{it:pBFSdual} $\hab{X_0^{1-\theta}\cdot X_1^\theta}^* = \hab{X_0^*}^{1-\theta}\cdot \hab{X_1^*}^\theta$.
\item \label{it:pBFSdualconcave} $(X^{\theta})^* = (X^*)^{\theta} \cdot L^{1/(1-\theta)}(S)$.
\item \label{it:pBFSUMD} If $X_0$ and $X_1$ have the $\UMD$ property, then $X_0^{1-\theta}\cdot X_1^\theta$ has the $\UMD$ property.
\end{enumerate}
\end{proposition}
Part \ref{it:pBFScomplex} follows from \cite{Ca64}, it has been extended to the product quasi-Banach function space setting in \cite{KMM07,KM98}. Part \ref{it:pBFSrefl} is proven in \cite[Theorem 3]{Lo69}. It also follows from \cite{Ca64} through complex interpolation. Part \ref{it:pBFSdual} is proven in \cite[Theorem 2]{Lo69} and for \ref{it:pBFSdualconcave} see \cite[Theorem 2.9]{Sc10}. Finally part \ref{it:pBFSUMD} follows from part \ref{it:pBFScomplex} and \cite[Proposition 4.2.17]{HNVW16}, see also the next section on the $\UMD$ property.
\subsection{The \texorpdfstring{$\UMD$}{UMD} property}
We say that a Banach space $X$ has the $\UMD$ property if the martingale difference sequence of any finite martingale in $L^p(\Omega;X)$ is unconditional for some (equivalently all) $p \in (1,\infty)$. The $\UMD$ property implies reflexivity and if $X$ has the $\UMD$ property, then $X^*$ has the $\UMD$ property as well. Standard examples of Banach spaces with the $\UMD$ property include reflexive $L^p$-spaces, Lorentz spaces, Orlicz spaces, Sobolev spaces, Besov spaces and Schatten classes. For a thorough introduction to the theory of $\UMD$ spaces we refer the reader to \cite{Bu01,HNVW16}.
Throughout this paper we will consider Banach function spaces with the $\UMD$ property.
In this case we have a characterisation of the $\UMD$ property in terms of the lattice Hardy-Littlewood maximal operator, which for simple $f:\R^d \to X$ is defined by
\begin{equation*}
\widetilde{M}f(x) := \sup_{Q} \ip{\abs{f}}_{Q} \ind_{Q}(x),
\end{equation*}
where the supremum is taken over all cubes $Q \subseteq\R^d$ (see \cite{GMT93} for the details). The boundedness of $\widetilde{M}$ on both $L^p(\R^d;X)$ and $L^p(\R^d;X^*)$ for some (equivalently all) $p \in (1,\infty)$ is equivalent to $X$ having the $\UMD$ property by a result of Bourgain \cite{Bo84} and Rubio de Francia \cite{Ru86}. Moreover, if $X$ has the $\UMD$ property we have the following weighted bound for all $p \in (1,\infty)$, $w \in A_p$ and $f \in L^p(w;X)$
\begin{equation}\label{eq:maximalbdd}
\nrmb{\widetilde{M}f}_{L^p(w;X)} \leq \inc_{X,p}([w]_{A_p}) \nrmb{f}_{L^p(w;X)},
\end{equation}
see \cite{GMT93}. A more precise dependence on the weight characteristic can be found in \cite{HL17}.
The $\UMD$ property of a Banach function space $X$ implies that certain $q$-concavifications of $X$ also have the $\UMD$ property, see \cite[Theorem 4]{Ru86}.
\begin{proposition}[Rubio de Francia]\label{prop:UMDopenRubio}
Let $X$ be a Banach function space over a $\sigma$-finite measure space $(S,\mu)$ such that $X$ has the $\UMD$ property. Then there exists an $\varepsilon>0$ such that such that $X^q$ has the $\UMD$ property for all $0<q<1+\varepsilon$.
\end{proposition}
Note that the difficult part of Proposition \ref{prop:UMDopenRubio} is the claim that $X^q$ has the $\UMD$ property for $1<q<1+\varepsilon$.
\section{The \texorpdfstring{$\UMD_{p_-,p_+}$}{UMDp+p-} property of quasi-Banach function spaces}\label{sec:UMDp-p+}
For our main result we need an extension of the $\UMD$ property, as we will often consider quasi-Banach function spaces of which a concavification has the $\UMD$ property. In particular, we will use the following notion:
\begin{definition}\label{def:UMDp-p+}
Let $X$ be a quasi-Banach function space and let $0<p_-<p_+\leq\infty$. Then we say $X$ has the $\UMD_{p_-,p_+}$ property if and only if $X$ is $p_-$-convex, $p_+$-concave and $\hab{(X^{p_-})^*}^{(p_+/p_-)'}$ has the $\UMD$ property. We denote this by $X \in \UMD_{p_-,p_+}$.
\end{definition}
Note that $X$ is a Banach function space with the $\UMD$ property if and only if $X \in \UMD_{1,\infty}$ and we denote this by $X \in \UMD$.
\begin{remark}~\label{rem:UMDp-p+}
\begin{itemize}
\item The $p_-$-convexity in Definition \ref{def:UMDp-p+} implies that $X^{p_-}$ is a Banach function space, so its dual $(X^{p_-})^*$ is non-trivial. Moreover $(X^{p_-})^*$ is a Banach function space, since it has the $\UMD$ property by Proposition \ref{prop:UMDopenRubio} and is therefore reflexive, which implies that $X^{p_-}$ is order-continuous.
\item The $p_+$-concavity assumption in Definition \ref{def:UMDp-p+} is not restrictive, as any quasi-Banach function space with the $\UMD$ property is actually isomorphic to a Banach function space (see \cite{CCV18}), which implies that $(X^{p_-})^*$ is $(p_+/p_-)'$-convex and thus that $X$ is $p_+$-concave by \cite[Section 1.d]{LT79}
\end{itemize}
\end{remark}
We first show some basic results for the $\UMD_{p_-,p_+}$ property.
\begin{proposition} Fix $0<p_-<p_+ \leq \infty$ and let $X$ be a quasi-Banach function space over a $\sigma$-finite measure space $(S,\mu)$ such that $X \in \UMD_{p_-,p_+}$.
\begin{enumerate}[(i)]\label{prop:UMDp+p-}
\item \label{it:UMDrescale}$X^{p_-} \in \UMD_{1,p_+/p_-}$.
\item \label{it:UMDdual}$X^* \in \UMD_{p_+',p_-'}$ if $p_- \geq 1$.
\item \label{it:UMDinterval}$X \in \UMD_{\tilde{p}_-,\tilde{p}_+}$ for all $\tilde{p}_- \in (0,p_-]$ and $\tilde{p}_+ \in [p_+,\infty]$.
\item \label{it:UMDinterp} If $1<p_-<p_+<\infty$, then $X = [Y,L^2(S)]_{\theta}$ for a Banach function space $Y \in \UMD$ and $\theta = 2/\max\cbrace{p_-',p_+}$.
\item \label{it:UMDLp}$L^p(S';X) \in \UMD_{p_-,p_+}$ for all $p \in (p_-,p_+)$ and any $\sigma$-finite measure space $(S',\mu')$.
\end{enumerate}
\end{proposition}
\begin{proof}
Part \ref{it:UMDrescale} follows directly from the definition. For part \ref{it:UMDdual} the $p_+'$-convexity and $p_-'$-concavity follow from \cite[Section 1.d]{LT79}. If $p_- =1$, the claim is that $\hab{(X^*)^{p_+'}}^* \in \UMD$, which is clear. Assuming $p_->1$, we have by Proposition \ref{prop:pBFS}
\begin{align*}
\hab{\hab{(X^*)^{p_+'}}^*}^{(p'_-/p_+')'} &= \has{\has{\hab{(X^{p_-})^*}^{p_+'/p_-}\cdot L^{p_-'/p_+'}(S)}^*}^{(p'_-/p_+')'} \\
&= \has{\has{\hab{(X^{p_-})^*}^{(p_+/p_-)'\theta}\cdot L^1(S)^{1-\theta}}^*}^{1/\theta}\\
&= \has{\hab{(X^{p_-})^*}^{(p_+/p_-)'}}^*\cdot \hab{L^\infty(S)}^{(1-\theta)/\theta}\\
&= \has{\hab{(X^{p_-})^*}^{(p_+/p_-)'}}^*
\end{align*}
with $\theta := \frac{1}{(p'_-/p_+')'}<1$, since taking a product with $L^\infty(S)^{(1-\theta)/\theta}=L^\infty(S)$ has no effect on the space. Thus we conclude that $X^* \in \UMD_{p_+',p_-'}$.
For part \ref{it:UMDinterval} the $p_+'$-convexity and $p_-'$-concavity follow from \cite[Theorem 4.2]{Ma04}. First assume that $p_-=1$ and let $\tilde{p}_- \in (0,1)$.
By Proposition \ref{prop:pBFS}\ref{it:pBFSdualconcave} we have
\begin{align}\label{eq:UMDp-p+interval}
\hab{(X^{\tilde{p}_-})^*}^{(p_+/\tilde{p}_-)'} = \ha{X^*}^{\tilde{p}_-(p_+/\tilde{p}_-)' } \cdot L^{\frac{p_+-\tilde{p}_-}{(1-\tilde{p}_-)p_+}}(S) = (X^*)^{p_+'\theta} \cdot L^{1}(S)^{1-\theta}
\end{align}
with
\begin{equation*}
\theta := \frac{\tilde{p}_-(p_+-1)}{p_+-\tilde{p}_-} <1.
\end{equation*}
By assumption $(X^*)^{p_+'} \in \UMD$, so
\begin{equation*}
\bracb{(X^*)^{p_+},L^1(S)}^*_\theta = \bracb{\hab{(X^*)^{p_+'}}^*,L^\infty(S)}_\theta = \has{\hab{(X^*)^{p_+'}}^*}^\theta \in \UMD
\end{equation*}
by Proposition \ref{prop:pBFS} and Proposition \ref{prop:UMDopenRubio}. Using Proposition \ref{prop:pBFS}\ref{it:pBFScomplex}, we obtain from \eqref{eq:UMDp-p+interval} that $X \in \UMD_{\tilde{p}_-,p_+}$.
For arbitrary $0<p_-<p_+ \leq \infty$ we know that $X \in \UMD_{\tilde{p}_-,p_+}$ for all $\tilde{p}_- \in (0,p_-]$ by \ref{it:UMDrescale} and Proposition \ref{prop:UMDopenRubio} yields that $X \in \UMD_{\tilde{p}_-,\tilde{p}_+}$ for all $\tilde{p}_+ \in [p_+,\infty]$.
For part \ref{it:UMDinterp} note that $X \in \UMD_{p',p}$ with $p = \max\cbrace{p_-',p_+}$ by part \ref{it:UMDinterval}. Therefore
\begin{equation*}
Y := \has{\hab{(X^{p'})^*}^{(p/p')'}}^* \in \UMD.
\end{equation*}
Then using Proposition \ref{prop:pBFS} we have
\begin{equation*}
X = \has{\hab{\ha{X^{p'}}^*}^{1/{p'}}\cdot L^{p}(S)}^* = \bracb{ \hab{\ha{X^{p'}}^*}^{(p/p')'}, L^2(S)}_{2/p}^* = \brac{ Y , H}_{\theta}.
\end{equation*}
Finally part \ref{it:UMDLp} follows from \cite[Proposition 4.2.15]{HNVW16} as
\begin{equation*}
\hab{\hab{L^p(S';X)^{p_-}}^*}^{(p_+/p_-)'} = L^{\frac{(p_+-p_-)p}{p_+(p-p_-)}}\hab{S';\hab{(X^{p_-})^*}^{(p_+/p_-)'}}. \qedhere
\end{equation*}
\end{proof}
Next we note how product quasi-Banach function spaces work under the $\UMD_{p_-,p_+}$ property. In particular the following result describes some properties of the space $X$ in our main theorem, Theorem \ref{thm:maincor}.
\begin{proposition}
Let $X_1,\cdots,X_m$ be quasi-Banach function spaces. For $j=1,\cdots,m$ let $0<p_j^-<p_j^+\leq \infty$ and assume that $X_j \in \UMD_{p_j^-,p_j^+}$. Let $X = X_1\cdots X_m$, then $X \in \UMD_{p_-,p_+}$, where $\frac{1}{p_-}:= \sum_{j=1}^m \frac{1}{p_j^-}$ and $\frac{1}{p_+}:= \sum_{j=1}^m \frac{1}{p_j^+}$.
\end{proposition}
\begin{proof}
We will prove the proposition for $m=2$. The general case can be proven by induction, cf. the proof of Lemma \ref{lemma:pBFSbochner}. First note that $X^{p_-} = X_1^{p_1^-(p_-/p_1^-)}\cdot X_2^{p_2^-(p_-/p_2^-)}$ is a Banach function space by assumption, so $X$ is $p_-$-convex. By Proposition \ref{prop:pBFS} we have
\begin{align*}
\hab{(X^{p_-})^*}^{(p_+/p_-)'} &= \hab{(X_1^{p_-} \cdot X_2^{p_-})^*}^{(p_+/p_-)'}\\ &= \hab{(X_1^{p^-_1})^*}^{(p_1^+/p_1^-)'(1-\theta)} \cdot \hab{(X_2^{p^-_2})^*}^{(p_2^+/p_2^-)'\theta}\\
\end{align*}
with
\begin{equation*}
\theta = \frac{\frac{1}{p_2^-}-\frac{1}{p_2^+}}{\frac{1}{p_-}-\frac{1}{p_+}}.
\end{equation*}
Thus by Proposition \ref{prop:pBFS}\ref{it:pBFSUMD} and Remark \ref{rem:UMDp-p+} we know that $X \in \UMD_{p_-,p_+}$.
\end{proof}
The $\UMD_{p_-,p_+}$ property of a quasi-Banach function space $X$ looks quite technical. However, as we will see in the next example, this abstract assumption is quite natural for concrete examples of Banach function spaces.
\begin{example}\label{ex:concavification}
Let $0 < p_- <p_+\leq \infty$ and let $X$ be a quasi-Banach function space over an atomless or atomic $\sigma$-finite measure space $(S,\mu)$. Then $X \in \UMD_{p_-,p_+}$ in each of the following cases:
\begin{enumerate}[(i)]
\item \label{it:exconcavification1} The Lebesgue spaces $X=L^p(S)$ for $p \in (p_-,p_+)$.
\item \label{it:exconcavification2} The Lorentz spaces $X=L^{p,q}(S)$ with $p,q \in (p_-,p_+)$.
\item \label{it:exconcavification3} The Orlicz spaces $X=L^\Phi(S)$ for which $t \mapsto \Phi(t^{1/p})$ is a convex function and $t\mapsto \Phi(t^{1/q})$ is a concave function with $p,q \in (p_-,p_+)$.
\end{enumerate}
Note that Theorem \ref{thm:maincor} for the Lebesgue spaces described in Example \ref{ex:concavification}\ref{it:exconcavification1} follows directly from scalar-valued limited range extrapolation using Fubini's theorem, see also \cite{CMP11}.
\end{example}
\begin{proof}
Note that \ref{it:exconcavification1} is a special case of \ref{it:exconcavification2}. For \ref{it:exconcavification2} the $p_-$-convexity and $p_+$-concavity follow from \cite[Theorem 4.4 and Theorem 5.1]{Ma04}. Furthermore by the definition of $L^{p,q}(S)$ and the duality of Lorentz spaces (see \cite{Hu66}) we have that
\begin{equation*}
\bigl((X^{p_-})^*\bigr)^{(p_+/p_-)'} = \bigl(\hab{L^{p/{p_-},q/p_-}(S)}^*\bigr)^{(p_+/p_-)'} = L^{\frac{(p_+-p_-)p}{p_+(p-p_-)},\frac{(p_+-p_-)q}{p_+(q-p_-)}}(S).
\end{equation*}
Since $L^{r,s}(S) \in \UMD$ for $r,s \in (1,\infty)$ (see \cite{HNVW16}), this proves \ref{it:exconcavification2}.
For \ref{it:exconcavification3} note that $L^\Phi(S)$ is $p$-convex and $q$-concave by \cite{Ka98}. So $Y := \bigl((X^{p_-})^*\bigr)^{(p_+/p_-)'}$ is $\frac{(p_+-p_-)p}{p_+(p-p_-)}$-convex and $\frac{(p_+-p_-)q}{p_+(q-p_-)}$-concave. By \cite[Theorem 1.f.1]{LT79} this implies that both $Y$ and $Y^*$ are uniformly convex. Note that $Y$ is an Orlicz space with Young function $\Psi(t) = \varphi^*(t^{(p_+-p_-)/p_-})$, where $\varphi(t) = \Phi(t^{1/p_-})$, and $Y^*$ is an Orlicz space with Young function $\Psi^*$. Therefore we know by \cite[Proposition 1]{Ka98} that both $\Psi$ and its conjugate function $\Psi^*$ satisfy the $\Delta_2$-condition. Thus it follows from \cite[Theorem 6.2]{FG91} that $Y \in \UMD$.
\end{proof}
We end our discussion of the $\UMD_{p_-,p_+}$ property by extending the result of Rubio de Francia for the $\UMD$ property of Banach function spaces in Proposition \ref{prop:UMDopenRubio} to the $\UMD_{p_-,p_+}$ property of quasi-Banach function spaces.
\begin{theorem}\label{thm:UMDopen}
Let $0<p_-<p_+ \leq \infty$ and let $X$ be a quasi-Banach function space over a $\sigma$-finite measure space $(S,\mu)$ such that $X \in \UMD_{p_-,p_+}$. Then there exists an $\varepsilon>0$ such that such that $X \in \UMD_{p_-q_-,p_+/q_+}$ for all $0<q_-,q_+<1+\varepsilon$.
\end{theorem}
\begin{proof}
By Proposition \ref{prop:UMDp+p-}\ref{it:UMDrescale} we may assume $p_-=1$ without loss of generality. Note that the case $p_+ = \infty$ was already included in Proposition \ref{prop:UMDopenRubio}, so we restrict our attention to $p_+< \infty$.
Applying Proposition \ref{prop:UMDopenRubio} to $(X^*)^{p'}$ yields an $r_1>1$ such that $(X^*)^{p_+'r_1} \in \UMD$. Furthermore since $p_+'>1$ we know that $X^* \in \UMD$ and thus also $X \in \UMD$. So by Proposition \ref{prop:UMDopenRubio} applied to $X$ there exists an $r_2>1$ such that $X^{r_2} \in \UMD$. Define $r = \min\cbrace{r_1,r_2, 1+\frac{1}{p_+'}}$.
Let $\theta = \frac{r'}{p_++r'} \in (0,1)$ and define the complex interpolation space
\begin{equation*}
Y := \brac*{(X^r)^*,(X^*)^{p_+'r}}_\theta.
\end{equation*}
Note that since $(X^r)^*, (X^*)^{p_+'r} \in \UMD$, we know by Proposition \ref{prop:pBFS}\ref{it:pBFSUMD} that $Y \in \UMD$ as well. Moreover using Proposition \ref{prop:pBFS} we have
\begin{align*}
Y &= \bigl((X^r)^*\bigr)^{1-\theta} \cdot \bigl((X^*)^{p_+'r}\bigr)^\theta\\
&= \bigl((X^r)^*\bigr)^{1-\theta} \cdot \has{\bigl((X^r)^{1/r}\bigr)^*}^{p_+'r \theta}\\
&= \hab{(X^r)^*}^{\frac{p_+}{p_++r'}} \cdot \has{\bigl((X^r)^*\bigr)^{1/r}\cdot L^{r'}(S)}^{\frac{p_+'rr'}{p_++r'}}\\
&= \hab{(X^r)^*}^{\frac{p_++p_+'r'}{p_++r'}} \cdot L^{\frac{p_++r'}{p_+'r}}(S)
\end{align*}
Define
\[
\alpha := \frac{p_++p_+'r'}{p_++r'},\qquad\beta := \frac{p_+'r}{p_++r'} < p_+' (r-1)<1.
\]
Again by Proposition \ref{prop:pBFS} we have
\begin{align*}
Y &= \bigl((X^r)^*\bigr)^\alpha \cdot L^{\frac{1}{\beta}}(S) = \has{\bigl((X^r)^*\bigr)^{\frac{\alpha}{\alpha+\beta}}\cdot L^{\frac{\alpha+\beta}{\beta}}(S)}^{\alpha+\beta} = \bigl((X^{\frac{r\alpha}{\alpha+\beta}})^*\bigr)^{\alpha+\beta}.
\end{align*}
Take $q_- = \frac{r\alpha}{\alpha+\beta}$. Then we have
\begin{align}\label{eq:q-equality}
q_- = \frac{r \alpha}{\alpha+\beta} = \frac{r(p_++p_+'r')}{p_++p_+'r'+p_+'r} = \frac{p_+r+p_+'r'+p_+'r}{p_++p_+'r'+p_+'r} >1.
\end{align}
Moreover
\begin{equation*}
\alpha+\beta - \frac{r\alpha}{p_+} = \frac{p_++p_+'r'+p_+'r-r - (p_+'-1)(r+r')}{p_++r'} =1
\end{equation*}
and therefore
\begin{align*}
(p_+/q_-)'&= \frac{\alpha+\beta}{\alpha+\beta - \frac{r\alpha}{p_+}} = \alpha+\beta.
\end{align*}
So $Y = \hab{(X^{q_-})^*}^{(p_+/q_-)'}$ and since $Y \in \UMD$, this implies that $X \in \UMD_{q_-,p_+}$. By applying Proposition \ref{prop:UMDopenRubio} once more, we can find a $q_+>1$ such that $X \in \UMD_{q_-,p_+/q_+}$. By Proposition \ref{prop:UMDp+p-}\ref{it:UMDinterval}, this completes the proof with $\varepsilon = \min\cbrace{q_--1,q_+-1}>0$.
\end{proof}
\section{Proof of the main result}\label{sec:main}
In this section we will prove our main result, Theorem \ref{thm:maincor}. The proof of Theorem \ref{thm:maincor} consists of following ingredients:
\begin{itemize}
\item The extension of Rubio de Francia's result for the $\UMD$ property to the setting of the $\UMD_{p_-,p_+}$ property, proven in Theorem \ref{thm:UMDopen}.
\item A vector-valued Rubio de Francia iteration algorithm, see Lemma \ref{lem:rubioalgoritme}.
\item A result for the product of weighted Bochner spaces, proven below in Lemma \ref{lemma:pBFSbochner}.
\end{itemize}
We start with the Rubio de Francia iteration algorithm lemma. We remark that Rubio de Francia iteration algorithms also play a key role in scalar-valued extrapolation, see for example \cite{CMP11}. Recall that we write $\phi_{a,b,\cdots}$ for a non-decreasing function $[1,\infty)^{2} \to [1,\infty)$, depending on the parameters $a,b,\cdots$ and the dimension $d$.
\begin{lemma}\label{lem:rubioalgoritme}
Fix $1<r<r_+\leq \infty$ and let $Y$ be a Banach function space over a $\sigma$-finite measure space $(S,\mu)$ with $Y \in \UMD_{r_+',\infty}$. For all $w \in A_r \cap RH_{(r_+/r)'}$ and nonnegative $u \in L^{r'}(w;Y)$ there is a nonnegative $v \in L^{r'}(w;Y)$ such that $u \leq v$, $\nrm{v}_{L^{r'}(w;Y)} \leq 2\nrm{u}_{L^{r'}(w;Y)}$, and $v(\cdot,s)w \in A_1\cap\RH_{r_+'}$ with
\begin{equation*}
\max\cbraceb{[v(\cdot,s)w]_{A_1},[v(\cdot,s)w]_{\RH_{r_+'}} } \leq \inc_{Y,r,r_+}\hab{[w]_{A_r}, [w]_{RH_{(r_+/r)'}}}
\end{equation*}
for $\mu$-a.e $s \in S$.
\end{lemma}
\begin{proof}
Fix $w \in A_r \cap RH_{(r_+/r)'}$ and $u \in L^{r'}(w;Y)$. Define
\begin{align*}
u_w &:= (uw)^{r_+' } \qquad \text{and} \qquad
X := L^{r'/r_+'}\hab{w^{1-r'};Y^{r_+'}}.
\end{align*}
Then $u_w \in X$. By Lemma \ref{lem:reverseholder}\ref{it:weightsequiv3} we know that for $p:=\ha*{{r_+/r}}'(r-1)+1$ we have $w^{1-r'} \in A_{p'}$ with
\begin{equation*}
[w^{1-r'}]_{A_{p'}}^{\frac{1}{p'}}=\bracb{w^{(r_+/r)'}}_{A_p}^{\frac{1}{p}} \leq \hab{[w]_{A_r}[w]_{\RH_{(r_+/r)'}}}^{(r_+/r)'}
\end{equation*}
So since
\begin{align*}
p' = 1+\frac{1}{\ha*{{r_+/r}}'(r-1)} =
\frac{r'}{r_+'}
\end{align*}
we know that $\widetilde{M}$ is bounded on $X$ by \eqref{eq:maximalbdd} with
\begin{equation}\label{eq:maximalbddweight}
\bigl\|\widetilde{M}\bigr\|_{X \to X} \leq \inc_{Y,r,r_+}\bigl([w]_{A_r} ,[w]_{\RH_{(r_+/r)'}}\bigr).
\end{equation}
Define
\begin{equation*}
v := w^{-1}\cdot\has{\sum_{n=0}^\infty \frac{\widetilde{M}^nu_w}{\hab{2\bigl\|\widetilde{M}\bigr\|_{X \to X}}^n}}^{1/r_+'}
\end{equation*}
where $\widetilde{M}^n$ is given by $n$ iterations of $\widetilde{M}$. As $\widetilde{M}^n u_w$ is nonnegative we know that $u \leq v$. Furthermore $v \in L^{r^\prime}(w;Y)$ with
\begin{align*}
\nrm{v}_{L^{r'}(w;Y)} = \nrms{\sum_{n=0}^\infty \frac{\widetilde{M}^nu_w}{\hab{2\bigl\|\widetilde{M}\bigr\|_{X \to X}}^n}}_X^{1/r_+'} \leq 2\nrm{u_w}_X^{1/r_+'} = 2\nrm{u}_{L^{r'}(w;Y)}.
\end{align*}
Moreover, since
\begin{equation*}
\widetilde{M}\bigl((vw)^{r_+'}\bigr)(\cdot,s) \leq 2 \bigl\|\widetilde{M}\bigr\|_{X \to X} (v(\cdot,s)w)^{r_+'},
\end{equation*}
we know that $(v(\cdot,s)w)^{r_+'} \in A_1$ for $\mu$-a.e $s \in S$. Thus it follows from \eqref{eq:maximalbddweight} and Lemma \ref{lem:reverseholder} that $v(\cdot,s)w \in A_1\cap\RH_{r_+'}$ with
\begin{equation*}
\max\cbraceb{[v(\cdot,s)w]_{A_1},[v(\cdot,s)w]_{\RH_{r_+'}} } \leq \inc_{Y,r,r_+}\bigl([w]_{A_r} ,[w]_{\RH_{(r_+/r)'}}\bigr)
\end{equation*}
for $\mu$-a.e $s \in S$
\end{proof}
Next we prove the result for the product of weighted Bochner spaces, which follows from the properties of product Banach function spaces in Proposition \ref{prop:pBFS} and complex interpolation of weighted Bochner spaces.
\begin{lemma}\label{lemma:pBFSbochner}
Fix $m\in \N$ and $r \in (1,\infty)$. Let $Y_1,\cdots,Y_m$ be reflexive Banach function spaces over a $\sigma$-finite measure space $(S,\mu)$, let $w_1,\cdots,w_m$ be weights and take $\theta_1,\cdots,\theta_m \in (0,1)$ such that $\sum_{j=1}^m \theta_j = 1$. Define $Y = Y_1^{\theta_1}\cdots Y_m^{\theta_m}$ and $w = \prod_{j=1}^m w_j^{\theta_j}$. Then we have
\begin{equation*}
L^r(w;Y)=L^{r}\hab{w_1;Y_1}^{\theta_1} \cdots L^{r}\hab{w_m;Y_m}^{\theta_m}
\end{equation*}
\end{lemma}
\begin{proof}
We will prove the lemma by induction. For $m=1$ the result is trivial. Now assume that the statement holds for $m=k-1$ for some $k \in \N$. We will show the statement for $m=k$.
Let $\tilde{\theta}_j = \frac{\theta_j}{1-\theta_k}$ for $j=1,\cdots,k-1$ and define
\begin{equation*}
X = Y_1^{\tilde{\theta}_1}\cdots Y_{k-1}^{\tilde{\theta}_{k-1}},\qquad v = \prod_{j=1}^m w_j^{\tilde{\theta}_j}.
\end{equation*}
Using Proposition \ref{prop:pBFS}\ref{it:pBFScomplex} twice and complex interpolation of weighted Bochner spaces (see
\cite[Theorem 1.18.5]{Tr78} and \cite{Bu87}) we get
\begin{align*}
L^r(w;Y) &= L^r\hab{w;[X,Y_k]_{\theta_k}} = \bracb{L^r(v;X),L^r(w_k;Y_k)}_{\theta_k} \\ &= L^r\hab{v;X}^{1-\theta_k} \cdot L^{r}\hab{w_k;Y_k}^{\theta_k} \\
&= \has{L^{r}\hab{w_1;Y_1}^{\tilde{\theta}_1} \cdots L^{r}\hab{w_{k-1};Y_{k-1}}^{\tilde{\theta}_{k-1}}}^{1-\theta_k} \cdot \hab{L^{r}\hab{w_k;Y_k}}^{\theta_k} \\
&= L^{r}\hab{w_1;Y_1}^{\theta_1} \cdots L^{r}\hab{w_k;Y_k}^{\theta_k},
\end{align*}
which proves the lemma.
\end{proof}
With these preparatory lemmata we are now ready to prove our main theorem. We first state and prove the result in terms of $(m+1)$-tuples of functions. Afterwards, we present the main result, Theorem \ref{thm:maincor}, as a corollary. We write $\inc^{j=1,\ldots,m}_{a_j,b_j,\cdots}$ for a non-decreasing function $[1,\infty)^{2m} \to [1,\infty)$ depending on the parameters $a_j,b_j,\cdots$ for $j=1,\cdots,m$ and the dimension $d$.
\begin{theorem}[Multilinear limited range extrapolation for vector-valued functions]\label{thm:multi-limited-range-extrap}
Fix $m\in\N$, let $X_1,\ldots,X_m$ be quasi-Banach function spaces over a $\sigma$-finite measure space $(S,\mu)$ and define $X=X_1\cdots X_m$. Let
\[
\mc{F} \subseteq L^0_+(\R^d;X)\times L^0_+(\R^d;X_1)\times\cdots\times L^0_+(\R^d;X_m).
\]
For $j=1,\cdots,m$ fix $0<p^-_j<p^+_j\leq\infty$ and assume that $X_j\in\UMD_{p_j^-,p_j^+}$. Moreover assume that for all $p_j\in(p_j^-,p_j^+)$, weights $w_j^{p_j}\in A_{p_j/p_j^-}\cap\RH_{(p^+_j/p_j)'}$ and $(f,f_1,\ldots,f_m)\in\mc{F}$, we have
\begin{equation}\label{eqn:extrap-assn2}
\nrm{f(\cdot,s)}_{L^p(w^p)} \leq \inc^{j=1,\ldots,m}_{p_j,p_j^-,p_j^+} \bigl([w_j^{p_j}]_{A_{p_j/p_j^-}},[w_j^{p_j}]_{\RH_{(p^+_j/p_j)'}}\bigr) \prod_{j=1}^m\nrm{f_j(\cdot,s)}_{L^{p_j}(w_j^{p_j})}
\end{equation}
for $\mu$-a.e. $s \in S$, where $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}$ and $w=\prod_{j=1}^m w_j$.
Then for all $p_j\in(p_j^-,p_j^+)$, weights $w_j^{p_j}\in A_{p_j/p_j^-}\cap\RH_{(p^+_j/p_j)'}$, and $(f,f_1,\ldots,f_m)\in\mc{F}$, we have
\begin{equation}\label{eqn:extrap-goal2}
\nrm{f}_{L^p(w^p;X)} \leq \inc^{j=1,\ldots,m}_{X_j,p_j,p_j^-,p_j^+} \bigl([w_j^{p_j}]_{A_{p_j/p_j^-}},[w_j^{p_j}]_{\RH_{(p^+_j/p_j)'}}\bigr) \prod_{j=1}^m\nrm{f_j}_{L^{p_j}(w_j^{p_j};X_j)},
\end{equation}
with $w$ and $p$ as before.
\end{theorem}
\begin{proof}
We split the proof in two steps. In the first step we show that the conclusion of the theorem holds for specific choices of $p_j\in(p_j^-,p_j^+)$. In the second step we conclude that the result holds for all $p_j\in(p_j^-,p_j^+)$ through scalar-valued extrapolation.
\textbf{Step 1:} Let $1 < \beta < \min_j \limits \frac{p_j^+}{p_j^-}$. We will first prove the theorem for $p_j := \beta \cdot p_j^-$. Let $(f,f_1,\ldots,f_m) \in \mc{F}$ and take weights $w_j^{p_j}\in A_{p_j/p_j^-}\cap\RH_{(p^+_j/p_j)'}$ for $j=1,\cdots,m$. From Theorem \ref{thm:UMDopen} and Lemma \ref{lem:muckenhoupt}\ref{it:mw3} it follows that there exists an $1<\alpha < \beta$ such that
\begin{equation}\label{eqn:qprop}
X_j\in \UMD_{\alpha p_j^-,p_j^+} \qquad \text{and} \qquad w_j^{p_j}\in A_{p_j/(\alpha p_j^-)}\cap\RH_{\ha*{{p_j^+/p_j}}'}
\end{equation}
with $[w_j^{p_j}]_{A_{p_j/(\alpha p_j^-)}}\leq C_{p_j,p_j^-}\,[w_j^{p_j}]_{A_{p_j/p_j^-}}$ for all $j\in\{1,\ldots, m\}$. We define
\begin{align*}
q_j:=\alpha p_j^- \qquad &\text{and} \qquad q:=\frac{\alpha}{\beta} p .
\intertext{Note that}
\frac{p}{q} = \frac{\beta}{\alpha} = \frac{p_j}{q_j} \qquad &\text{and} \qquad \frac{1}{q} = \frac{\beta}{\alpha} \sum_{j=1}^m \frac{1}{p_j} = \sum_{j=1}^m \frac{1}{q_j}.
\end{align*}
Let $u_j\in L^{(p_j/q_j)'}(w_j^{p_j};(X_j^{q_j})^\ast)$. By Proposition \ref{prop:UMDp+p-} and \eqref{eqn:qprop} we may apply Lemma \ref{lem:rubioalgoritme} for $j=1,\cdots,m$ with
\begin{equation*}
r=p_j/q_j, \qquad r_+=p_j^+/q_j, \qquad Y=(X_j^{q_j})^\ast,
\end{equation*}
and weight $w_j^{p_j}$ to find nonnegative $v_j \in L^{(p_j/q_j)'}(w_j^{p_j};(X^{q_j})^\ast)$ such that
\begin{itemize}
\item $u_j\leq v_j$.
\item $\nrm{v_j}_{L^{(p_j/q_j)'}(w_j^{p_j};(X_j^{q_j})^\ast)} \leq 2$.
\item $v_j(\cdot,s)w_j^{p_j} \in A_{q_j/p_j^-} \cap RH_{(p_j^+/q_j)'}$ with for $\mu$-a.e. $s \in S$
\begin{equation*}
\max\cbraceb{[v_j(\cdot,s)w_j^{p_j}]_{A_{q_j/p_j^-}},[v_j(\cdot,s)w_j^{p_j}]_{\RH_{(p_j^+/q_j)'}} } \leq \inc_{X_j,p_j,p_j^-,p_j^+}\hab{[w_j^{p_j}]_{A_{p_j/p_j^-}}, [w_j^{p_j}]_{RH_{(p_j^+/p_j)'}}}.
\end{equation*}
\end{itemize}
We set $v=\prod_{j=1}^m v_j^{1/q_j}w_j^{p_j/q_j}$ so that
\begin{equation*}
\has{\prod_{j=1}^m u_j^{p/p_j}}w^p\leq\prod_{j=1}^m v_j^{q/q_j}w_j^{qp_j/q_j}=v^q.
\end{equation*}
Let $(f,f_1,\ldots,f_m)\in\mc{F}$. By Fubini's theorem, H\"older's inequality, the assumption \eqref{eqn:extrap-assn2}, and the properties of the $v_j$ we have
\begin{equation}\label{eqn:dcompest}
\begin{split}
\int_{\R^d}&\int_S f^q \prod_{j=1}^m u_j^{p/p_j} \dd\mu \, w^p\dd x \leq\int_S \int_{\R^d} f^q v^q \dd x \, \dd \mu\\
&\leq \int_S \inc^{j=1,\ldots,m}_{q_j,p_j^-,p_j^+} \bigl([v_j(\cdot,s)w_j^{p_j}]_{A_{q_j/p_j^-}},[v_j(\cdot,s)w_j^{p_j}]_{\RH_{(p_j^+/q_j)'}}\bigr)\prod_{j=1}^m\|f_j(\cdot,s)\|^q_{L^{q_j}(v_j w_j^{p_j})} \dd \mu(s) \\
&\leq\inc^{j=1,\ldots,m}_{X_j,p_j,p_j^-,p_j^+}\hab{[w_j^{p_j}]_{A_{p_j/p_j^-}}, [w_j^{p_j}]_{RH_{(p_j^+/p_j)'}}}\prod_{j=1}^m\left(\int_S\int_{\R^d} f_j^{q_j}v_j w_j^{p_j}\,\dd x\,\dd\mu\right)^{q/q_j}\\
&\leq\inc^{j=1,\ldots,m}_{X_j,p_j,p_j^-,p_j^+}\hab{[w_j^{p_j}]_{A_{p_j/p_j^-}}, [w_j^{p_j}]_{RH_{(p_j^+/p_j)'}}}\prod_{j=1}^m\|f_j^{q_j}\|^{q/q_j}_{L^{p_j/q_j}(w_j^{p_j};X_j^{q_j})}\|v_j\|^{q/q_j}_{L^{(p_j/q_j)'}(w_j^{p_j};(X_j^{q_j})^\ast)}\\
&\leq\inc^{j=1,\ldots,m}_{X_j,p_j,p_j^-,p_j^+}\hab{[w_j^{p_j}]_{A_{p_j/p_j^-}}, [w_j^{p_j}]_{RH_{(p_j^+/p_j)'}}} \has{\prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j};X_j)}\|u_j\|^{1/q_j}_{L^{(p_j/q_j)'}(w_j^{p_j};(X_j^{q_j})^\ast)}}^q.
\end{split}
\end{equation}
Now by Lemma \ref{lemma:pBFSbochner} with
\begin{equation*}
r= p/q, \qquad Y_j = X_j^{q_j}, \qquad \theta_j = q / q_j
\end{equation*}
and weights $w_j^{p_j}$, Proposition \ref{prop:pBFS}\ref{it:pBFSdual} and the duality of Bochner spaces (see \cite[Corollary 1.3.22]{HNVW16}), we have
\begin{equation*}
L^{p/q}(w^p;X^q)^* = L^{(p_1/q_1)'}\hab{w_1^{p_1};(X_1^{q_1})^*}^{q/q_1} \cdots L^{(p_m/q_m)'}\hab{w_m^{p_m};(X_m^{q_m})^*}^{q/q_m}.
\end{equation*}
Thus, picking $u\in L^{p/q}(w^p;X^q)^*$ of norm $1$, by taking an infimum over all decompositions $u=\prod_{j=1}^m u_j^{p/p_j}$ with $u_j\in L^{(p_j/q_j)'}\hab{w_j^{p_j};(X_j^{q_j})^\ast}$, we may conclude from \eqref{eqn:dcompest} that
\[
\int_{\R^d}\int_S f^q u \dd\mu \, w^p\dd x\leq\inc^{j=1,\ldots,m}_{X_j,p_j,p_j^-,p_j^+}\hab{[w_j^{p_j}]_{A_{p_j/p_j^-}}, [w_j^{p_j}]_{RH_{(p_j^+/p_j)'}}} \has{\prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j};X_j)}}^q.
\]
Thus, the result for these specific $p_j$'s follows from
\begin{align*}
\|f\|^q_{L^p(w^p;X)} = \|f^q\|_{L^{p/q}(w^p;X^q)}= \sup_{ \|u\|_{L^{p/q}(w^p;X^q)^*}=1} \int_{\R^d}\int_S f^qu\dd\mu\,w^p\dd x.
\end{align*}
\bigskip
{\bf Step 2:} We may finish the proof for general $p_j$'s by appealing to the scalar-valued limited range multilinear extrapolation result by Cruz-Uribe and Martell \cite{CM17}. Indeed, we define a new family
\[
\widetilde{\mc{F}}:=\cbraceb{\hab{\|f\|_{X},\|f_1\|_{X_1},\ldots,\|f_m\|_{X_m}} : (f,f_1,\ldots,f_m) \in \mc{F}}.
\]
Then $\widetilde{\mc{F}} \subset L^0_+(\R^d)^{m+1}$ and by Step 1 we have
\[
\nrmb{\tilde{f}}_{L^p(w^p)}\leq\inc^{j=1,\ldots,m}_{X,p_j^-,p_j^+}\hab{[w_j^{p_j}]_{A_{p_j/p_j^-}}, [w_j^{p_j}]_{RH_{(p_j^+/p_j)'}}} \prod_{j=1}^m\nrmb{\tilde{f}_j}_{L^{p_j}(w_j^{p_j})}
\]
for certain $p_j\in (p_j^-,p_j^+)$, all $(\tilde{f},\tilde{f}_1,\ldots,\tilde{f}_m)\in\widetilde{\mc{F}}$, and all weights $w_j^{p_j}\in A_{p_j/p_j^-}\cap\RH_{(p_j^+/p_j)'}$. The result for general $p_j\in (p_j^-,p_j^+)$ then follows directly from \cite[Theorem 1.3 and Corollary 1.11]{CM17}, proving the assertion.
\end{proof}
Finally, we will prove the main result from the introduction, which is a direct corollary of Theorem \ref{thm:multi-limited-range-extrap}.
\begin{proof}[Proof of Theorem \ref{thm:maincor}]
We wish to apply Theorem \ref{thm:multi-limited-range-extrap} to the collection
\begin{equation*}
\mc{F} = \cbraceb{(\abs{\widetilde{T}(f_1,\ldots,f_m)},\abs{f_1},\ldots,\abs{f_m}): \,\,f_j\colon\R^d \to X_j \text{ simple}}.
\end{equation*}
Our assumption implies that there are $p_j\in(p_j^-,p_j^+)$ so that for all weights $w_j^{p_j}\in A_{p_j/p_j^-}\cap\RH_{(p_j^+/p_j)'}$ the a priori estimate \eqref{eqn:extrap-assn2} in Theorem \ref{thm:multi-limited-range-extrap} holds. By appealing to the scalar-valued limited range multilinear extrapolation result \cite{CM17} we may conclude that \eqref{eqn:extrap-assn2} in fact holds for all $p_j\in(p_j^-,p_j^+)$ and weights $w_j^{p_j}\in A_{p_j/p_j^-}\cap\RH_{(p_j^+/p_j)'}$. Thus, Theorem \ref{thm:multi-limited-range-extrap} implies that
\begin{equation}\label{eq:extend}
\nrmb{\widetilde{T}(f_1,\ldots,f_m)}_{L^p(w^p;X)} \leq C\,\prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j};X_j)}
\end{equation}
for all simple functions $f_j\colon\R^d \to X_j$, where $C$ depends only on the $X_j$, $p_j$, and the characteristic constants of the weights. If $T$ is $m$-linear, then \eqref{eq:extend} extends directly to all $f_j\in L^{p_j}(w_j^{p_j};X_j)$ by density. If $T$ is $m$-sublinear and positive valued, then we fix simple functions $f_j\colon:\R^d \to X_j$ for $j\in\{2,\ldots,m\}$. For any pair of simple functions $f_1,g_1:\R^d\to X_1$ we have
\begin{equation*}
\widetilde{T}(f_1,\ldots, f_m) = \widetilde{T}(f_1-g_1+g_1,f_2\ldots, f_m) \leq \widetilde{T}(f_1-g_1,f_2\ldots, f_m)+\widetilde{T}(g_1,f_2\ldots,f_m)
\end{equation*}
so that
\begin{align*}
\nrmb{\widetilde{T}(f_1,\ldots, f_m)-\widetilde{T}(g_1,f_2\ldots,f_m)}_{L^p(w^p;X)} &\leq \nrmb{\widetilde{T}(f_1-g_1,f_2\ldots, f_m)}_{L^p(w^p;X)}\\
&\leq C\, \|f_1-g_1\|_{L^{p_1}(w_1^{p_1};X_1)}\prod_{j=2}^m\|f_j\|_{L^{p_j}(w_j^{p_j};X_j)}.
\end{align*}
Thus, \eqref{eq:extend} extends to arbitrary $f_1 \in L^{p_1}(w_1^{p_1};X_1)$ by density. Iterating this argument for $j=2,\ldots m$ proves the result.
\end{proof}
\section{Applications}\label{sec:applications}
In this section we apply our main result to various operators, for which we obtain new vector-valued bounds.
\subsection{The bilinear Hilbert transform}
For $d=1$, The bilinear Hilbert transform $\BHT$ is defined by
\[
\BHT(f,g)(x)=\pv\int_{\R}\!f(x-t)g(x+t)\frac{\dd t}{t}.
\]
After its initial introduction by Calder\'on, it took thirty years until $L^p$ estimates were established by Lacey and Thiele \cite{LT99}. They showed that for $p_1,p_2\in (1,\infty]$ with $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}<\frac{3}{2}$ one has
\begin{equation}\label{eqn:bihilisbounded}
\|\BHT(f,g)\|_{L^p}\leq C\|f\|_{L^{p_1}}\|g\|_{L^{p_2}}.
\end{equation}
As for weighted bounds, the first results were obtained by Culiuc, di Plinio, and Ou \cite{CDPO16}, and through the extrapolation result of Cruz-Uribe and Martell the range of exponents was increased \cite{CM17}, in particular recovering the full range of exponents for the unweighted result \eqref{eqn:bihilisbounded}. It was already shown in \cite{CM17} that this result implies corresponding vector-valued bounds for $\BHT$ for certain $\ell^s$-spaces. Moreover, vector-valued bounds for $\BHT$ have also been considered by Benea and Muscalu \cite{BM16}. In particular, they consider functions taking values in iterated $L^s$-spaces, see \cite[Theorem 8]{BM16} including the case $s=\infty$.
Through our main result we are able to obtain a new bounded vector-valued extension of the bilinear Hilbert transform. By combining the weighted estimates in \cite[Theorem 1.18]{CM17} with Theorem \ref{thm:maincor}, we get:
\begin{theorem}\label{thm:bilhilbo}
Let $q_1,q_2\in(1,\infty)$ so that $\frac{1}{q_1}+\frac{1}{q_2}<1$. For $j\in\{1,2\}$, define
\[
p_j^-:=\frac{2q_j}{1+q_j},\qquad p_j^+:=2q_j.
\]
Let $X=X_1\cdot X_2$, where $X_1$, $X_2$ are quasi-Banach function spaces over a $\sigma$-finite measure space $(S,\mu)$ satisfying $X_j\in\UMD_{p_j^-,p_j^+}$.
Then for all $p_1$, $p_2$ with $p_j\in(p_j^-,p_j^+)$ and all weights $w_1$, $w_2$ satisfying $w_j^{p_j}\in A_{p_j/p_j^-}\cap\RH_{(p_j^+/p_j)'}$ we have
\[
\big\|\widetilde{\BHT}(f,g)\big\|_{L^p(w^p;X)}\leq C'
\|f\|_{L^{p_1}(w_1^{p_1};X_1)}\|g\|_{L^{p_2}(w_2^{p_2};X_2)}
\]
for all $f\in L^{p_1}(w_1^{p_1};X_1)$, $g\in L^{p_2}(w_2^{p_2};X_2)$, where $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, $w=w_1w_2$, and where $C'>0$ depends only on the $X_j$, $p_j$, $q_j$, and the characteristic constants of the weights.
\end{theorem}
By Example \ref{ex:exproducts} we have $\ell^s = \ell^{s_1} \cdot \ell^{s_2}$ for $s_1, s_2\in (0,\infty)$ and $\frac{1}{s}=\frac{1}{s_1}+\frac{1}{s_2}$. Thus, we recover \cite[Theorem 1.29]{CM17} by Example \ref{ex:concavification}. It is implicit from the arguments in \cite{CDPO16} that there are more general weighted estimates for $\BHT$ leading to a wider range of vector-valued extensions. For a technical discussion on this, we refer the reader to \cite[Section 5]{CM17}.
Furthermore by Proposition \ref{prop:UMDp+p-}\ref{it:UMDLp} we can also handle iterated $L^s$-spaces as considered by Benea and Muscalu \cite{BM16}, but our results do not overlap as we do not obtain bounds involving $L^\infty$-spaces. Such spaces might be in the scope of a generalized version of our main theorem using multilinear weight classes combined with a multilinear $\UMD$ condition, see also Remark \ref{rem:intro} and \cite{N18}.
Finally, we mention the vector-valued bounds obtained by Hyt\"onen, Lacey, and Parissis \cite{HLP13} for the related bilinear quartile operator (the Fourier-Walsh model of $\BHT$). They consider estimates involving triples of more general $\UMD$ Banach spaces with so called \emph{quartile type} $q$. It is unknown whether these estimates hold for $\BHT$ itself. Note that a Banach function space $X \in \UMD_{p_-,p_+}$ has quartile type $\max\cbrace{p_-',p_+}$ by Proposition \ref{prop:UMDp+p-}\ref{it:UMDinterp} and \cite[Proposition 4.1]{HLP13}.
\subsection{Multilinear Calder\'on-Zygmund operators}
Let $T$ be an $m$-linear operator, initially defined for $m$-tuples $f_1,\ldots,f_m\in C_c^\infty(\R^d)$, that satisfies
\[
T(f_1,\ldots, f_m)(x)=\int_{(\R^d)^m}\!K(x,y_1,\ldots,y_m)\prod_{j=1}^m f_j(y_j)\dd y,
\]
whenever $x\notin\cap_{j=1}^m\supp f_j$, where $K$ is a kernel defined in $(\R^d)^{m+1}$ outside of the diagonal $y_0=y_1=\cdots=y_m$. If $K$ satisfies the estimate
\[
|\partial_{y_0}^{\alpha_0}\cdots\partial_{y_m}^{\alpha_m} K(y_0,\ldots,y_m)|\leq\frac{C}{\left(\sum_{j,k=0}^m|y_j-y_k|\right)^{md+|\alpha_1|+\cdots+|\alpha_m|}}
\]
for all multi-indices $\alpha_j$ so that $\sum_{j=1}^m|\alpha_j|\leq 1$ and if there exist $p_1,\ldots, p_m$ so that $T$ extends to a bounded operator $L^{p_1}\times\cdots L^{p_m}\to L^p$ with $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}$, then $T$ is called an $m$-linear Calder\'on-Zygmund operator.
Multilinear Calder\'on-Zygmund operators first appeared in the work \cite{CM75} by Coifman and Meyer. Weighted estimates for these operators have been considered for example by Grafakos and Torres in \cite{GT02} and subsequently by Grafakos and Martell in \cite{GM04}, where it was shown that for all $p_j\in(1,\infty)$, all weights $w_j^{p_j}\in A_{p_j}$, and all $f_j\in L^{p_j}(w_j^{p_j})$ we have
\begin{equation*}
\nrmb{T(f_1,\ldots,f_m)}_{L^p(w^p)} \leq C\,\prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j})},
\end{equation*}
where $w=\prod_{j=1}^m w_j$ and $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}$, and where $C$ depends only on the characteristic constants of the weights. Thus, by Theorem \ref{thm:maincor} we obtain the following result:
\begin{theorem}
Let $T$ be an $m$-linear Calder\'on-Zygmund operator and suppose $X_1,\ldots,X_m\in\UMD$. Then for all $p_j\in(1,\infty)$, all weights $w_j^{p_j}\in A_{p_j}$, and all $f_j\in L^{p_j}(w_j^{p_j};X_j)$ we have
\begin{equation*}
\nrmb{\widetilde{T}(f_1,\ldots,f_m)}_{L^p(w^p;X)} \leq C'\,\prod_{j=1}^m\|f_j\|_{L^{p_j}(w_j^{p_j};X_j)},
\end{equation*}
where $X=X_1\cdots X_m$, $w=\prod_{j=1}^m w_j$, $\frac{1}{p}=\sum_{j=1}^m\frac{1}{p_j}$, and where $C'$ depends only on the $X_j$, $p_j$, and the characteristic constants of the weights.
\end{theorem}
This result is new, as previously only $\ell^s$-valued extensions had been considered in \cite{GM04}.
We wish to point out that, using a more appropriate multilinear weight condition, more general weighted bounds for multilinear Calder\'on-Zygmund operators have been found in \cite{LOPTT09}.
\subsection{Limited range extrapolation}
The remaining examples are for the linear case $m=1$.
\begin{example}[Fourier multipliers I]
For $a<b$, $q\in[1,\infty)$, and a function $m:[a,b]\to\C$ we define the $q$-variation norm
\[
\nrm{m}_{V^q([a,b])}:=\|m\|_{L^\infty([a,b])}+\sup\left(\sum_{j=0}^{n-1}|m(t_{j+1})-m(t_j)|^q\right)^{\frac{1}{q}},
\]
where the supremum is taken over all partitions $a=t_0<\ldots<t_n=b$ of the interval $[a,b]$. Let $\ms{D}:=\big\{\pm(2^k,2^{k+1}]:k\in\Z\big\}$ be the dyadic decomposition of $\R$. Then we define a class of multipliers
\[
V^q(\ms{D}):=\big\{m:\R\to\C : \sup_{I\in\ms{D}}\nrm{m|_{I}}_{V^q(I)}<\infty\big\}.
\]
For $q>2$ and $p_+:=2\left(\frac{q}{2}\right)'$
it was shown by Kr\'{o}l \cite[Theorem A(ii)]{Kr14} that for all $p\in[2,p_+)$, $w\in A_{p/2}\cap\RH_{(p_+/p)'}$ and $m\in V^q(\ms{D})$ the Fourier multiplier $T_m$ defined by $\ms{F}(T_mf)=m\ms{F}f$ satisfies
\[
\|T_m\|_{L^p(w)\to L^p(w)}<\infty.
\]
Therefore one may readily apply Theorem \ref{thm:maincor} with $p_-=2$ to the linear operator $T_m$. So for any Banach function space $X$ such that $X\in\UMD_{2,p_+}$ we find for all $p\in (2,p_+)$, all $w\in A_{p/2}\cap\RH_{(p_+/p)'}$ and $m\in V^q(\ms{D})$ that
\[
\nrmb{\widetilde{T}_m}_{L^p(w;X)\to L^p(w;X)}<\infty.
\]
Note that \cite[Theorem A(i)]{Kr14} was already extrapolated to the vector-valued setting by Amenta, Veraar, and the first author in \cite{ALV17}, proving that for
$m\in V^q(\ms{D})$ with $q\in[1,2]$ the Fourier multiplier $T_m$ has a bounded vector-valued extension for Banach function spaces $X\in\UMD_{q,\infty}$. Furthermore extensions of \cite[Theorem A]{Kr14} for operator-valued Fourier multipliers have been obtained in \cite{ALV18}.
\end{example}
\begin{example}[Riesz transforms associated with elliptic operators]
Let $A\in L^\infty(\R^d;\C^{d\times d})$ satisfy an ellipticity condition $\re(A(x)\xi\cdot\overline{\xi})\geq\lambda|\xi|^2$ for a.e. $x\in\R^d$, and all $\xi\in\C^d$. Then we may consider a second order divergence form operator
\[
L:=-\divv(A\nabla f),
\]
defined on $L^2$, which due to the ellipticity condition on $A$ generates an analytic semigroup $(e^{-t L})_{t>0}$ in $L^2$. Let $1\leq p_-<p_+\leq \infty$. If both the semigroup and the gradient family $(\sqrt{t}\nabla e^{-t L})_{t>0}$ satisfy $L^{p_-}$--$L^{p_+}$ off-diagonal estimates, then the Riesz transform $R:=\nabla L^{-1/2}$ is a bounded operator in $L^p(w)$ for all $p\in(p_-,p_+)$ and all weights $w\in A_{p/p_-}\cap\RH_{(p_+/p)'}$, see \cite{AM07, BFP16}. The values of $p_-$ and $p_+$ for which such off-diagonal estimates hold depend on the dimension $d$ and on the matrix-valued function $A$ and are studied in detail in \cite{Au07}. The result we obtain is that if a Banach function space $X$ satisfies $X\in\UMD_{p_-,p_+}$, then for all $p\in (p_-,p_+)$ and all weights $w\in A_{p/p_-}\cap\RH_{(p_+/p)'}$ we have
\[
\nrmb{\widetilde{R}}_{L^p(w;X)\to L^p(w;X)}<\infty.
\]
This result is new in the sense that previously such bounds were previously only known for $X=\ell^s$ through the limited range extrapolation result in \cite{AM07}.
\end{example}
Next, we consider a class of operators satisfying a certain sparse domination property. A collection $\mathcal{S}$ of cubes in $\R^d$ is called \textit{sparse} if there is a pairwise disjoint collection of sets $(E_Q)_{Q\in\mathcal{S}}$ so that for each $Q\in\mathcal{S}$ we have $E_Q\subseteq Q$ and $|Q|\leq 2|E_Q|$. We say that a (sub)linear operator $T$ satisfies the sparse domination property with parameters $1\leq p_-<p_+\leq\infty$ if there is a $C>0$ so that for all compactly supported smooth functions $f,g:\R^d\to\C$ we have
\begin{equation}\label{eq:sparsebounds}
|\langle Tf,g\rangle|\leq C\sup_{\mathcal{S}\text{ sparse}}\sum_{Q\in\mathcal{S}}\langle |f|^{p_-}\rangle_{Q}^{\frac{1}{p_-}}\langle |g|^{p_+}\rangle_{Q}^{\frac{1}{p_+}}|Q|,
\end{equation}
where the supremum runs over all sparse collections of cubes $\mc{S}$. For an operator $T$ we denote the optimal constant $C$ appearing in \eqref{eq:sparsebounds} by $\|T\|_{S(p_-,p_+)}$. Estimates in the form \eqref{eq:sparsebounds} were first considered in \cite{BFP16} where it was shown that
\[
\|T\|_{S(p_-,p_+)}<\infty\,\Rightarrow\,\|T\|_{L^p(w)\to L^p(w)}<\infty
\]
for $p \in (p_-,p_+)$ and $w\in A_{p/p_-}\cap\RH_{(p_+/p)'}$ by giving a quantitative estimate in terms of the characteristic constants of the weight. Thus, we may readily apply Theorem \ref{thm:maincor} with $m=1$ to any linear or positive-valued sublinear $T$ such that $\|T\|_{S(p_-,p_+)}<\infty$. This yields the following result:
\begin{theorem}\label{thm:sparsevecext}
Let $T$ be a linear or a positive-valued sublinear operator and let $X$ be a Banach function space over a $\sigma$-finite measure space $(S,\mu)$. Assume that for all simple functions $f:\R^d\to X$ the function $\widetilde{T}f(x,s):=T(f(\cdot,s))(x)$ is well-defined and strongly measurable.
If there are $1\leq p_-<p_+\leq\infty$ such that
\[
X\in\UMD_{p_-,p_+},\qquad \|T\|_{S(p_-,p_+)}<\infty,
\]
then for all $p\in(p_-,p_+)$, all weights $w\in A_{p/p_-}\cap\RH_{(p_+/p)'}$, and all $f\in L^p(w;X)$, we have
\begin{equation}\label{eqn:sparseconc}
\nrmb{\widetilde{T}f}_{L^p(w;X)}\leq C\|f\|_{L^p(w;X)},
\end{equation}
where $C$ depends only on $X$, $p$, $p_-$, $p_+$, and the characteristic constants of $w$.
\end{theorem}
We emphasize again that if $T$ is linear, then $\widetilde{T}f$ is automatically well-defined and strongly measurable for any simple function $f:\R^d\to X$, see also Remark \ref{rem:mainthm}. We conclude this section by giving several examples of operators satisfying sparse bounds.
\begin{example}[Fourier multipliers II]
For each $\delta\geq 0$, the Bochner-Riesz multiplier $B_\delta$ is defined as the Fourier multiplier $\ms{F}(B_\delta f)=(1-|\xi|^2)_+^\delta\ms{F}f$, where $t_+=\max(t,0)$. For $\delta\geq (d-1)/2$, $B_\delta$ satisfies weighted bounds $\|B_\delta\|_{L^p(w)\to L^p(w)}<\infty$ for any $p\in (1,\infty)$ and any $w\in A_p$, see \cite{Bu93, DR86, SS92}.
The situation is more complicated when $0<\delta<(d-1)/2$ and weighted bounds for such $\delta$ have, for example, been considered in \cite{CDL12, Ch85, DMOS08}. The idea to quantify weighted bounds for $B_\delta$ for $0<\delta<(d-1)/2$ through sparse domination was initiated by Benea, Bernicot, and Luque \cite{BBL17}. It was shown by Lacey, Mena, and Reguera that for this range of $\delta$ there are explicit subsets $R_{\delta,d}$ of the plane so that
\[
\\|B_\delta\|_{S(p_-,p_+)}<\infty
\]
for $(p_-,p_+)\in R_{\delta,d}$, see \cite{LMR17}. We also refer the reader to the recent work by Kesler and Lacey \cite{KL17} containing certain sparse endpoint bounds in dimension $d=2$.
As far as we know, the only vector-valued estimates that have been shown for $B_\delta$ have been for $X=\ell^s$, see \cite{BBL17}. For any $p_-$, $p_+$ and $\delta$ for which $\|B_\delta\|_{S(p_-,p_+)}<\infty$, we obtain by Theorem \ref{thm:sparsevecext} that inequality \eqref{eqn:sparseconc} with $\widetilde{T} = \widetilde{B}_\delta$ holds for any Banach function space $X$ satisfying $X\in\UMD_{p_-,p_+}$, yielding new vector-valued estimates.
\end{example}
\begin{example}[Spherical maximal operators]
Let $(S^{d-1},\sigma)$ denote the unit sphere in $\R^d$ equipped with its normalized Euclidean surface measure $\sigma$. For a smooth function $f$ on $\R^d$ we denote by $A_r f(x)$ the average of $f$ over the sphere centered at $x$ of radius $r>0$, i.e.,
\[
A_rf(x):=\int_{S^{d-1}}f(x-r\omega)\dd\sigma(\omega).
\]
We respectively define the lacunary spherical maximal operator and the full spherical maximal operator by
\[
M_{\lac}f:=\sup_{k\in\Z}|A_{2^k}f|,\quad M_{\full}f:=\sup_{r>0}|A_r f|,
\]
the latter having been introduced by Stein \cite{St76} and the former having been studied by Calder\'{o}n \cite{Ca79}. It was shown by Lacey \cite{La17} that for explicit subsets $L_d$, $F_d$ of the plane we have
\begin{align*}
\|M_{\lac}\|_{S(p_-,p_+)}<\infty & ,\quad\text{for $(p_-,p_+)\in L_d$,}\\
\|M_{\full}\|_{S(p_-,p_+)}<\infty & ,\quad\text{for $(p_-,p_+)\in F_d$.}
\end{align*}
These results recover the previous known $L^p$-bounds for these operators and yield weighted bounds.
To apply Theorem \ref{thm:sparsevecext} to $M_{\lac}$ and $M_{\full}$, one needs to check that these operators have well-defined and strongly measurable extensions to $X$-valued simple functions with $X$ a Banach function space over $(S,\mu)$. This can be checked as in \cite[Lemma 3.1]{HL17}. Therefore it follows from Theorem \ref{thm:sparsevecext} that if $(p_-,p_+)\in L_d$ or $(p_-,p_+)\in F_d$, then for any Banach function space $X\in\UMD_{p_-,p_+}$ we obtain the bound \eqref{eqn:sparseconc} for $\widetilde{T}=\widetilde{M}_{\lac}$ or $\widetilde{T}=\widetilde{M}_{\full}$ respectively. As far as we know, this is the first instance that vector-valued extensions have been considered for these operators.
\end{example}
\bibliographystyle{plain}
|
1,116,691,498,455 | arxiv | \section{INTRODUCTION}
A significant hurdle of $\Lambda$CDM-based cosmological simulations has been the so-called ``missing satellite problem''. Simply put, early $\Lambda$CDM-based simulations \citep{Kauffmann+93, Navarro+96} found an order of magnitude more satellite galaxies than observed orbiting the Milky Way \citep{Klypin+99, Moore+99, Bullock+00}. The degree of this problem has since diminished due to a number of new findings which include: models with more detailed physics \citep[e.g.][]{Guo+11, Wetzel+16}, a more complete understanding of the satellite mass function \citep[e.g.][]{Bullock+10, Guo+15}, and the finding that observed satellites are less massive than they were originally due to tidal stripping \citep{Kravtsov+04}. Also, more low-mass galaxies have recently been discovered near the Milky Way \citep[e.g.][]{Belokurov+06, Zucker+06, Martin+07, Koposov+15}, the Andromeda galaxy \citep[e.g.][]{Zucker+07, Irwin+08}, and beyond \citep[e.g.][]{Javanmardi+16, Munoz+15} since the ``missing satellite problem'' was originally posited.
An alternative test of $\Lambda$CDM-based simulations is the comparison of measured to predicted stellar halo masses. Stellar halos are diffuse spherical distributions of stars which are thought to surround most galaxies larger than dwarfs. These halos are predominantly made up of stars which have been stripped via tidal forces from satellite galaxies (which are often destroyed in this process) and deposited onto a larger host galaxy. The number and size of satellite galaxies available to be tidally disrupted is determined by cosmological parameters. It is therefore possible to use cosmological simulations to determine the predicted distribution of stellar halo masses, and then test these predictions with observations.
The relation between stellar halo mass and total stellar mass has been simulated by a few teams thus far \citep{Johnston+08, Purcell+11, Cooper+13, Rodriguez-Gomez+15,Rodriguez-Gomez+16, Elias+18}, who find that the stellar halo mass fraction increases with total stellar mass, but with a high amount of scatter. This scatter is likely due to the finding that the majority of stellar halo mass is accreted from a single, large, satellite \citep{Bullock+05, DSouza+Bell18}. Observational confirmation of these results has proven to be difficult. Stellar halos are amongst the dimmest stellar components of a galaxy at 28~AB~mag~arcsec$^{-2}$ \citep{Cooper+13}, where slight deviations from perfect flat-fielding of images, the extended wings of point spread functions (PSFs) \citep{Sandin14, Sandin15}, and the time required to reach these depths, hinders detections. Nevertheless, several stellar halo masses have been measured in individual galaxies. These include the Milky Way \citep{Carollo+10}, NGC253 \citep{Bailin+11}, M31 \citep{Courteau+11}, M101 \citep{vanDokkum+14}, NGC3115 \citep{Peacock+15}, M63 \citep{Staudaher+15}, and UGC00180 \citep{Trujillo+16}. Stellar halos have also been found within small targeted surveys including GHOSTS \citep{Harmsen+17} and Dragonfly \citep{Merritt+16}. The consensus from these results is that the simulations find more massive stellar halos than observed. However, the methodologies used to calculate stellar halo masses vary greatly: from counting stars in the Milky Way \citep{Carollo+10}, to optical colors \citep{vanDokkum+14, Trujillo+16}, to near-infrared {\it Spitzer Space Telescope} 3.6~${\rm \mu}$m\ imaging \citep{Courteau+11, Staudaher+15}, to {\it Hubble Space Telescope} color-magnitude diagrams which resolve red giant stars \citep{Bailin+11, Peacock+15, Harmsen+17}. A large sample size is also important due to the aformentioned stochastic accretion process of stellar halo populations. To observationally test the predicted halo mass relation requires a large robust sample with consistent measurement techniques.
\begin{figure*}
\includegraphics[width=140mm]{figs/EDGES_sample_hists.pdf}
\caption{The distribution of optical morphology (top left), apparent $B$-band magnitude (top right), Galactic latitude, $|b|$ (bottom left), and distance (bottom right).}
\label{fig:sample_hist}
\end{figure*}
The first large scale observational examination \citep{DSouza+14} of stellar halos was done by grouping $>$45,000 Sloan Digital Sky Survey (SDSS) galaxies by mass bins, and then stacking every image in a given bin to produce an extremely deep image ($\mu_r\sim32$ mag~arcsec$^{-2}$) of an average galaxy. They found that the average stellar halo for galaxies from 10$^{10}$-10$^{11.4}$~M$_{\odot}$ (split into 6 mass bins) are in general agreement with simulations. The Dragonfly group uses a robotic array of refractive telescopes which helps to minimize scattered light and thus leads to improved flat-fielding \citep{Abraham+14}. They are in the process of conducting a survey of stellar halos \citep{Merritt+16} and have successfully measured the stellar halo mass fraction for five galaxies, with three stellar halos remaining undetected down to $\mu_g\sim31$~mag~arcsec$^{-2}$. The Dragonfly results systematically find stellar halo mass fractions lower than predicted by simulations. A promising survey, MADCASH (Magellanic Analog Dwarf Companions And Stellar Halos) has discovered a faint dwarf galaxy \citep{Carlin+16} and will examine resolved stars within stellar halos for several nearby LMC analogs. The first large scale survey of individually resolved stellar halos was completed with data from the Hyper Suprime-Cam-based Subaru Strategic Program \citep{Aihara+17} in \cite{Huang+17}. They studied $\sim$3,000 galaxies between $0.3~<~z~<0.5$ using high quality data that reaches $i~>~28.5$ mag~arcsec$^{-2}$ allowing measurements of stellar halos to 100~kpc. They find an astounding agreement with the Illustris-based simulations of \cite{Rodriguez-Gomez+16}. The majority of their galaxies fit within the 1-$\sigma$ envelope of the predicted relation between the fraction of stellar halo mass and total stellar mass from $11-12$ log($M_{\star}/M_{\odot}$).
We pursue an observational approach to quantifying the stellar halos in nearby galaxies, one that is based on extremely deep and spatially wide near-infrared imaging. The Extended Disc Galaxy Exploration Science (EDGES) program is a Warm Spitzer survey of 92 galaxies designed to measure the dimmest stellar structures in nearby galaxies. In this paper we first discuss the sample selection and explain our general observational strategy. We detail the processing techniques which generate our mosaics and surface brightness profiles. We then show how the surface brightness profiles are decomposed into individual components and how the mass of these components is measured. From these measurements the statistics of up-bending, and down-bending breaks are examined, and these findings are compared to predictions from $\Lambda$CDM-based simulations.
\section{SAMPLE}
\label{sec:sample}
EDGES consists of 92 nearby galaxies spanning a wide range of physical properties and possible formation histories (as shown in \cite{Dale+16} and Dale (2019, in prep.)). Due to the exploratory nature of the project, several galaxies of each morphological type, luminosity, and inclination angle are included in order to explore the relationships between the extended stellar distribution and global properties of galaxies in general. The selected galaxies are drawn from a parent sample of all galaxies listed in the NASA/IPAC Extragalactic Database (NED) with velocities less than 3000 km~s$^{-1}$ and $|b|~>20^{\circ}$. The primary selection criteria include both a minimum and maximum distance cut, a minimum and maximum angular size cut ($2\arcmin~<D_{25}~<13\arcmin)$, an apparent magnitude limit ($m_B~<~16$), and a very strict Galactic latitude constraint ($|b|~>~60^{\circ}$). The latter criterion is imposed based on our experience with data from our pilot projects in Cycle 6 (PIDs 60094 and 60116, see \mbox{\cite{Barnes+14}}), where even moderate galactic latitude fields have significant contamination by foreground stars at the depths of these images. In addition, to minimise confusion with background structure, we exclude all potential targets within 20$^{\circ}$ of the center of the Virgo cluster. Note that the combination of the distance cut and apparent magnitude limit results in a nearly complete parent sample for galaxies brighter than $M_B$ of $-15$; thus, in order to maintain a nearly complete parent sample while still including a representative sample of low mass galaxies, we further limit the sample to those galaxies brighter than $M_B$ of $-14$. This final selection criterion ensures that the majority of galaxies in this sample are representative of systems that encompass the majority of baryonic mass in this volume element.
The volume element for this survey was selected to optimise our ability to identify and trace extended stellar components within the observational constraints of Spitzer and other telescopes. Specifically, we excluded galaxies within 2~Mpc of the Milky Way since both the primary target and any associated stellar streams are too large to map efficiently with IRAC, with angular sizes up to tens of degrees. We also imposed a maximum redshift constraint ($v < 1100$ km s$^{-1}$) to facilitate targeted follow-up observations of stellar features identified in the survey; this outer distance limit enables ancillary observations designed to investigate the resolved stellar populations of selected small fields to further explore their origin and structure. The redshift limit allows such observations to probe the tip of the red giant branch of these targeted fields with reasonable integration times using existing instrumentation available on ground-based telescopes and the Hubble Space Telescope.
Minimum and maximum angular size constraints were imposed to best match our observational mapping strategy to the areal coverage required to trace the extended stellar population ($5\times R_{25}$). We excluded small angular extent targets from the sample in order to best match the minimum primary field-of-view (FOV) from our mapping strategy ($10 \arcmin \times 10 \arcmin$ for an $8 \times 8$ grid) with the known optical extent. While this criterion preferentially excluded low mass galaxies from the final sample, our science goals only require a representative sample, with well defined selection criteria, which can then be compared to a sample selected in a similar manner from $\Lambda$CDM simulations; thus, a slight over-representation of massive galaxies (relative to the parent sample) is acceptable. The maximum size limit was imposed so that the maximum FOV to be mapped could be accomplished within a single Spitzer Astronomical Observing Request (AOR). While it was nominally possible to split observations into multiple AORs to map the large FOVs required for the 7 extremely large nearby galaxies in this survey volume, the potential systematic errors associated with matching sky levels for data taken at different times would have posed significant challenges to meeting our science goals.
The above constraints resulted in a potential sample of 122 galaxies. We examined each field visually to eliminate galaxies that would require masking a significant fraction of the FOV due to either numerous conspicuous foreground or background source contamination (Milky Way stars, galaxies, and galaxy clusters). We also examined each potential field with the Spitzer planning software package SPOT to verify that each target has low infrared sky levels at both 3.6~${\rm \mu}$m\ and 4.5~${\rm \mu}$m. As expected given the location of these galaxies, the typical infrared sky levels for this sample are less than 0.1~MJy~sr$^{-1}$ at 3.6~${\rm \mu}$m.
The final galaxy sample is a well defined and statistically representative sample of normal galaxies within this volume (see Figure~\ref{fig:sample_hist} for the general properties of this sample and Table~\ref{table:general} for the properties of individual galaxies). Our deep observations of a large FOV (at least $5~R_{25}$) around this representative sample of 92 galaxies provides an unprecedented view of the faint stellar populations associated with nearby galaxies.
\section{OBSERVATIONAL STRATEGY AND DATA PROCESSING}
\subsection{Mapping strategy}
Our observations are designed to map a field of view that corresponds to at least a factor of 5 beyond a galaxy's bright optical component. The need for wide field mapping observations is driven by the estimated sizes of dark matter halos relative to their bright baryonic components and by the observed extent of the largest gaseous discs currently known. Kinematic tracers indicate that the dark matter halos extend at least a factor of 5 beyond the high surface brightness component for both early- \citep[e.g][]{Rhode+07} and late-type galaxies \citep[e.g.][]{Zaritsky+93, Christlein+Zaritsky08}. Further, extremely large atomic gas discs (HI-to-optical size ratios of 7-10) have been identified in several low mass dwarf irregular galaxies \citep[e.g. NGC~3741][]{Begum+08}, indicating that the baryonic component may extend well beyond the high surface brightness regime even in isolated galaxies. Finally, \cite{Regan+06} find smoothly varying stellar surface brightnesses at 3.6~${\rm \mu}$m\ out to at least 2~$R_{25}$, the limit of the SINGS and S$^4$G data \citep{Kennicutt+03, Sheth+10}, indicating that deeper observations would reveal a more extended stellar component in most spiral galaxies. Thus, while our observations are exploratory in nature, our observing strategy samples the faint extended stellar distributions for every galaxy.
Astronomical Observing Requests (AORs) were constructed using the successful strategy employed for our Cycle~6 pilot studies (see \cite{Barnes+14}). Mosaics are built upon a grid of 100$\arcsec$ spacings ($\sim$one-third the IRAC FOV). Two sets of maps are obtained for each source to enable asteroid removal and to enhance map sensitivity and redundancy. At any given location within the map cores there are a total of 18 100~s frames resulting in a net integration per sky position of 1800~s (along with a 1200~s, 100$\arcsec$-wide ``inner periphery'' and a 600~s, 100$\arcsec$-wide ``outer periphery''). The mosaics for 3.6~${\rm \mu}$m\ observations are centered on the target galaxies, but even our smallest maps have sufficient sky coverage that the FOV of the corresponding 4.5~${\rm \mu}$m\ mosaic includes the galaxy as well. The smallest maps are $8 \times 8$ mosaics, providing 10$\arcmin$ map cores at the deepest 1800~s effective integration. For highly inclined galaxies, the mosaic pattern was modified to generate long rectangular strips for 31 fields (such restrictions were not required for the face-on galaxies for which there is no preferred direction for the map).
\subsection{Pre-processing and mosaicking}
The EDGES observations took place from 2011-06-20 to 2013-03-13 and are currently freely available at the Spitzer Heritage Archive under PID:80025, and we plan to make the fully processed mosaics available via NED. The raw data of EDGES are a massive trove of 33,434 individual pointings totaling over a month of exposure time (1005.3~hours). In order to analyse these data we have created mosaics using the MOPEX pipeline with the BCD images including standard pre- and post-processing fixes for Warm Mission data. The pre-processing steps include the removal of a frame-wide bias caused by ``stuck'' pixels, the so-called ``column-pulldown effect'', the ``first-frame effect'', and a dependence of the bias on frame number. A more detailed description of this process is found in \cite{Staudaher+15}, and see also \cite{Krick+11} for an analysis of the effects removed in pre-processing.
\subsection{Post-processing}
Post-processing of the mosaics begins with the removal of the 3.6~${\rm \mu}$m\ sky by subtracting a fitted plane gradient to a suite of individually defined rectangular sky regions with the IDL function {\tt MPFITFUN}. The subset of galaxies with saturated pixels at their centers are corrected with shallower archival data. Archival mosaics are generated from BCD-level data using our pipeline with our pixel mapping solution (the Fiducial Image Frame), and saturated pixels in EDGES mosaics are replaced with the unsaturated values from the newly generated archival mosaics. Also, as a further check, the pixel values within the archival mosaics are compared to the pixel values in the EDGES mosaics.
Foreground stars and background galaxies are the largest contaminants in the EDGES mosaics due to the extremely low background noise level of $\sim2.5$~kJy~sr$^{-1}$. To remove these contaminants a SExtractor \citep{Bertin+96} catalog of sources is generated for every mosaic. The SExtractor algorithm correctly identifies stars and background galaxies beyond the extent of EDGES galaxies. However, within the diffuse emission of our targets SExtractor overestimates the extent of point sources and misidentifies spiral structure as independent galaxies. To remedy these issues the extent of point sources within extended structure are adjusted by hand and misidentified spiral structures are removed. Once the catalogs have been properly edited they are used to generate masks for each mosaic.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/smooth_test.pdf}
\caption{The dependence of the standard deviation within sky apertures on smoothing kernel size.}
\label{fig:smooth_test}
\end{figure}
After masking, the mosaics are smoothed to reach the 3.6~${\rm \mu}$m\ Spitzer warm-mission limiting surface brightness of $\sim0.5$~kJy~sr$^{-1}$, (1~$\sigma$ per pixel; see \cite{Krick+11} for examples and discussion). The smoothing is accomplished using the python function ${\tt astropy.convolve}$ \citep{astropy+13} with a 2D-Gaussian kernel. The size of this kernel was determined by convolving every mosaic with a variety of filters of different sizes and measuring the standard deviation of the sky (using the same manually defined sky regions used in the measurement of the sky gradient) for the entire sample (see Figure~\ref{fig:smooth_test}). The 4 pixel ($3\arcsec$) wide kernel is the first to reach the Warm Mission's limiting surface brightness and is chosen for this study. We also note that the convolution algorithm uses the smoothing kernel to interpolate over missing values in the image. Normally this is not ideal as the interpolated data lack the noise characteristics of the original image, the structure beneath the interpolated-over source may not necessarily be smooth, and this interpolation may recover sources which have not been fully masked. However, missing data causes the surface brightness profile fitting (described below) to fail prematurely, and thus we use astropy's interpolative convolution method while noting the above issues. See Figure~\ref{fig:mosaics} for a suite of example mosaics.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=75mm]{figs/images/NGC3922.pdf} &
\includegraphics[width=75mm]{figs/images/NGC3998.pdf} \\
(a) NGC 3922 & (b) NGC 3998 \\[6pt]
\includegraphics[width=75mm]{figs/images/NGC4151.pdf} &
\includegraphics[width=75mm]{figs/images/NGC5005.pdf} \\
(c) NGC 4151 & (d) NGC 5005 \\[6pt]
\end{tabular}
\caption{A sampling of 3.6~${\rm \mu}$m\ mosaics. Black ellipses are $R_{25}$, red ellipses are Type-II breaks, and blue ellipses are Type-III breaks.}
\label{fig:mosaics}
\end{figure*}
\renewcommand{\thefigure}{\arabic{figure} (Cont.)}
\addtocounter{figure}{-1}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=75mm]{figs/images/NGC3726.pdf} &
\includegraphics[width=75mm]{figs/images/NGC3917.pdf} \\
(e) NGC 3726 & (f) NGC 3917 \\[6pt]
\includegraphics[width=75mm]{figs/images/UGC06161.pdf} &
\includegraphics[width=75mm]{figs/images/IC3687.pdf} \\
(g) UGC 06161 & (h) IC 3687 \\[6pt]
\end{tabular}
\caption{A sampling of 3.6~${\rm \mu}$m\ mosaics.}
\end{figure*}
\renewcommand{\thefigure}{\arabic{figure}}
\section{RESULTS}
\subsection{Surface brightness profiles}
Once background and foreground sources have been removed from the mosaics and the mosaics have been smoothed, the IRAF task {\tt ELLIPSE} is run to generate surface brightness profiles. The ellipticity, central position, and position angle are left as free parameters to minimise uncertainties caused by differences of orientation and position between the outer and inner isophotes. The cause of these differences may be from interactions with nearby galaxies, the presence of distinct thick and thin discs or a bar, or extreme radial migration. The initial galaxy centers are defined by their RC3 values \citep{DeVaucouleurs+91} with the Spitzer-calibrated world coordinate system. In cases where ELLIPSE fitting fails to find a solution using the RC3 values the centers are defined manually. An iterative 4$\sigma$ rejection is used to minimise the effect of cosmic rays and unidentified point sources, and the surface brightness profile is sampled linearly with an annular width of 4~pixels ($3\arcsec$) per bin. Also, a pixel list file is generated to properly mask any missing data. Once the surface brightness is measured the profiles are converted into AB-based magnitudes with the calibration from \cite{Reach+05}. Also, the stellar mass density is calculated using a $M/L$ ratio of 0.5$\pm0.1$. This simple relation has been found to be consistent at 3.6~${\rm \mu}$m\ with a variety of techniques: optically calibrated stellar population synthesis models \citep{Oh+08, Eskew+12, McGaugh+14}, rotation curve decomposition \citep{Barnes+14}, stellar population synthesis calibrated with [3.6]-[4.5]~${\rm \mu}$m\ colors \cite{Meidt+14}, and with the Tully-Fisher relation \citep{McGaugh+15}. Examples which span the range of morphological types of EDGES is found in Figure~\ref{fig:profiles}, and surface brightness profiles for the entire sample are available within the supplementary online-only version of this paper.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=90mm]{figs/profiles/NGC3922.pdf} &
\includegraphics[width=90mm]{figs/profiles/NGC3998.pdf} \\
(a) NGC 3922 & (b) NGC 3998 \\[6pt]
\includegraphics[width=90mm]{figs/profiles/NGC4151.pdf} &
\includegraphics[width=90mm]{figs/profiles/NGC5005.pdf} \\
(c) NGC 4151 & (d) NGC 5005 \\[6pt]
\end{tabular}
\caption{Surface brightness profiles. $R_{25}$ is denoted as the solid and labeled line, the dashed line denotes the extent of the spiral structure, and the colored lines mark the locations of the breaks (blue for an up-bending break, red for a down-bending break). The blue line is the model fit convolved with the PSF. Also included is the local scale length, $h_{\rm local}$ (middle), and the asymmetry parameters $a$ (bottom).}
\label{fig:profiles}
\end{figure*}
\renewcommand{\thefigure}{\arabic{figure} (Cont.)}
\addtocounter{figure}{-1}
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=90mm]{figs/profiles/NGC3726.pdf} &
\includegraphics[width=90mm]{figs/profiles/NGC3917.pdf} \\
(e) NGC 3726 & (f) NGC 3917 \\[6pt]
\includegraphics[width=90mm]{figs/profiles/UGC06161.pdf} &
\includegraphics[width=90mm]{figs/profiles/IC3687.pdf} \\
(g) UGC 06161 & (h) IC 3687 \\[6pt]
\end{tabular}
\caption{Surface brightness profiles.}
\end{figure*}
\renewcommand{\thefigure}{\arabic{figure}}
\subsection{Component fitting and modeling}
Variations within the surface brightness profiles are caused from a multitude of sources, including the PSF, transitions from one galactic component to another, (e.g. bulges, bars, discs, and stellar halos), spiral structure, asymmetries in a component's two-dimensional profile, and radial migration. To study these phenomena individually, we model the surface brightness profiles as a number of overlapping components. These models include a bulge component which is based upon the 3.6~${\rm \mu}$m\ PSF, and disc components based upon S{\'e}rsic disc profiles \cite{Sersic68}.
\subsubsection{The bulge model}
While traditionally bulges have been modeled as S{\'e}rsic profiles with $n>2$, we have found that the bulge profile is dominated by the PSF in the EDGES surface brightness profiles after smoothing the PSF to $3\arcsec$. To measure the bulge as a point source we have created a PSF specifically for EDGES with the aid of the Spitzer Warm Mission extended Point Response Function (PRF) from the NASA/IPAC Infrared Science Archive \footnote{http://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/calibrationfiles/psfprf/}. The PRF is transformed to the EDGES platescale with the IDL task ${\tt CONGRID}$, then smoothed using a Gaussian with a 4~pixel standard deviation with the python function ${\tt astropy.convolve}$ \citep{astropy+13}, the same procedure used to smooth the mosaics. This result is then normalised to sum to unity, resulting in the EDGES PSF. We note that the PSF differs between mosaics since the set of images used to construct the mosaics were split into subsets taken on different days, resulting in subsets of images taken with different telescope orientation angles. The diffraction spikes do not overlap with different orientation angles, creating differences in the final PSF for each mosaic. However, with only a single spatial dimension, the surface brightness profile's PSF does not suffer from this orientation issue, and is the same for each surface brightness profile. The contribution from the central point source to the surface brightness profile is the following ratio:
\begin{equation}
S_{\rm bulge} = \frac{S_0}{\rm PSF_{cen}},
\end{equation}
where $S_{\rm bulge}$ is the total surface brightness within the bulge, $S_0$ is the surface brightness at $R=0$ on the surface brightness profile, and ${\rm PSF_{cen}}$ is the value of the central pixel in the normalised PSF. We note that in many cases the galaxy in question will not have a significant contribution from a bulge or any other central point source. In these cases the modeled central surface brightness of the disc (described below) is compared to the measured central surface brightness. If they are within 2$\sigma$ of one another the contribution from the central point source is set to zero.
\subsubsection{disc models}
\label{sec:final_model}
To model the disc we assume S{\'e}rsic disc profiles (i.e. $n=1$), of the general form:
\begin{equation}
S(R) = S_0 {\rm exp}[-(R/h)], \\
\end{equation}
where $S$ is the surface brightness at a semi-major axis, $R$, $S_0$ is the surface brightness of the disc at $R=0$, and $h$ is the scalelength. To include breaks in the model, where the slope of the surface brightness profile abruptly changes, we allow for multiple S{\'e}rsic disc profiles defined for specific radial regimes. Our final model takes the form:
\begin{equation}
\label{eq:final_model}
S(R) = \begin{cases}
S_{\text{bulge}}, & R=0, \\
S_{0_1} {\rm exp}[-(R/h_1)], & 0 < R \leqslant R_{b_1}, \\
S_{0_2} {\rm exp}[-(R/h_2)], & R_{b_1} < R \leqslant R_{b_2}, \\
&... \\
S_{0_n} {\rm exp}[-(R/h_n)], & R_{b_{n-1}} < R \leqslant {\rm 200~kpc}
\end{cases}
\end{equation}
where the numerical subscripts denote with which S{\'e}rsic disc profile a parameter is associated, $n$ denotes the total number of these profiles, and $R_b$ is a break radius.
To locate the break radii a first pass is performed with the ``mark-the-disc'' method from \cite{Pohlen+Trujillo06}. The first step is to mark the extent of the disc-like section of the surface brightness profile where the S{\'e}rsic index is one. The inner mark denotes the extent of the bulge, or as is the case with the EDGES data, the extent of the contribution from the central point source. For galaxies with a prominent bulge this is determined by eye, and for galaxies without a prominent bulge (described above) the inner mark is set to zero. The outer mark is determined visually by examining the mosaics and drawing an aperture over the extent of each galaxy.
Once the disc has been marked, the local scalelength $h_{\rm local}$ profile is measured along the disc. This is done by calculating the slope of the surface brightness profile within eleven neighboring points with scipy's ${\tt curve\_fit}$, a Levenburg-Marquardt least squares fitting algorithm. The weighted mean of the local scalelength between the inner and outer marks is calculated and the points where the local scalelength crosses the weighted mean are used as the initial break radii. These values are then used to fit the initial parameters of the model (see above). For galaxies with no significant break in their 3.6~${\rm \mu}$m\ surface brightness profile many false breaks can be identified with this approach due to slight variations in the local scalelength caused by spiral structure, or from random scatter in $h_{\rm local}$ due to the uncertainties in the surface brightness profile. This method may also miss breaks in galaxies with multiple breaks, where a break may occur entirely above or below the weighted mean. For galaxies with either no breaks or multiple breaks the radii are determined by eye, then the model is fit (see Equation~\ref{eq:final_model}) and the breaks are adjusted iteratively until the model correctly reproduces the observed surface brightness profile. See Figure~\ref{fig:profiles} for example surface brightness profiles with the local scalelength profile, break locations, the fits to the surface brightness profile between each break, and the complete model.
\subsection{Extended PSF}
\label{sec:exten_psf}
A serious concern for any study of extended surface brightness profiles is the influence of the PSF on the extreme outskirts of the surface brightness profile. In some cases, the extended wings of PSFs cause a break to appear. Large ground-based reflectors in particular suffer from this problem \citep[e.g.][]{Sandin14, Trujillo+16}, and this is especially true for red optical observations with thinned CCDs, where even shallow images are affected \citep{Michard02}. The Dragonfly group foregoes large reflectors entirely and has achieved incredibly deep images ($\mu_B\sim$32~${\rm \mu~(mag/\square \arcsec)}$) with a fully automated array of commercially manufactured refractors (48 lenses for an equivalent aperture of 99~cm at the time of this writing). These Canon-branded instruments were originally designed for sports or birding photography \citep{Abraham+14}, have an effective aperture of only $\sim$400~mm, and do not suffer from extended wings in their PSFs since they are refractors and because of a proprietary chemical coating on the lens designed to limit diffraction spikes from bright stadium lights.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/NGC4151_PSF_model.pdf}
\caption{NGC~4151 modeled without the up-bending break at 1.8\arcmin. Observed data are black points with included uncertainties. The model is represented as a blue dashed line and contains only the bulge and first disc component. The PSF is scaled to appear on this plot and is the orange dashed-dotted line. The model convolved with the PSF is the green solid line. Note that a false up-bending break is produced but appears at 29~AB~mag~arcsec$^{-2}$, below the EDGES detection limit.}
\label{fig:N4151PSF}
\end{figure}
We investigate the effect of the Spitzer 3.6~${\rm \mu}$m\ PSF on the EDGES data set by constructing a model with the method used in \cite{Sandin14, Sandin15}. This method is based upon the supposition that if a model bulge and disc are convolved with the PSF and an up-bending break is observed, then observed up-bending breaks may be due to the wings of the PSF and not from a distinct component within the galaxy. For our model the bulge and disc are based upon measurements from NGC~4151, an extreme case as it has the brightest 3.6~${\rm \mu}$m\ central pixel in the entire EDGES sample due to its status as a Seyfert galaxy. The bulge is modeled as a point source at $R=0$ with $S_0=1340$~MJy/sr, and a disc with S{\'e}rsic disc profile with $S_0=0.85$~MJy/sr and $h=1.44$\arcmin\ (see Equation~\ref{eq:final_model}). The inner section of the PSF is based upon the surface brightness profile of the model PSF used to measure the contribution from the bulge (see \S~\ref{sec:final_model}). The extent of the model PSF is 2\farcm2, but we extend it by fitting an inverse square law to the the model PSF beyond 1\arcmin\ (which most PSFs tend to follow in this radial regime). The PSF and model image are then convolved to produce Figure~\ref{fig:N4151PSF}. This convolution correctly reproduces the bulge and disc section of the galaxy but there is no evidence of a stellar halo caused by the wings of the PSF. A false up-bending break is produced, but this break does not match the location and brightness of the observed up-bending break. This same procedure is used to test any up-bending break in the entire sample but we find no evidence of false up-bending breaks due to the extended PSF in EDGES data.
\subsection{Break classification}
\label{sec:break_class}
The discs of each galaxy in EDGES are decomposed into their individual components according to the breaks in the surface brightness profile (see above). This is accomplished by measuring the central surface brightness and scalelength of the profile between each break with Equation~\ref{eq:final_model} and the Levenburg-Marquardt least squares fitting algorithm ${\tt curve\_fit}$. Each break is then classified according to its type defined according to the following commonly used metric \citep{Erwin+08, Pohlen+Trujillo06, Gutierrez+11, Laine+14, Laine+16}, where {\bf Type-I} signifies no break, {\bf Type-II} signifies a down-bending break, or a decrease in scalelength, and {\bf Type-III} signifies an up-bending break, or an increase in scalelength. The Type-II breaks are subdivided into {\bf Type-II.OLR} (for Outer Linblad Resonance), and {\bf Type-II.CT} (for Classical Truncations). Outer Linblad Resonances cause down-bending breaks via interactions with the bar and occur at less than 2 times the bar radius; thus any Type-II break with a radius less than 2 times the bar radius (measured by examining the mosaics by eye) is classified as Type-II.OLR, and any other down-bending break is classified as Type-II.CT. Type-III breaks which occur within the spiral structure of the galaxy are classified as {\bf Type-III.S} (for spiral) and Type-III breaks which occur beyond the extent of the spiral structure are defined as {\bf Type-III.O} (for outer). The extent of the spiral structure was found by creating a difference image between the mosaic, and a smooth, homogeneous model created by the ${\tt IRAF}$ task ${\tt bmodel}$. This difference image removes the power-law profile from the images and leaves spiral structure. The extent of the structure is measured with a circular aperture. See Figure~\ref{fig:M63Sub} for an illustration of this process.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/M63_disc.png}
\caption{The subtracted {\tt bmodel} image of Messier~063, highlighting the grand-design spiral structure.}
\label{fig:M63Sub}
\end{figure}
\subsection{Mass measurement}
\label{sec:mass_measure}
To examine the physical properties of the EDGES galaxies we measure the mass for each component within each galaxy. The total flux density between two points for the S{\'e}rsic disc components $(n=1)$ is:
\begin{equation}
f(R_i < R < R_o)= 2 \pi S_0 h^2
\left\{
\gamma \left( 2, \frac{R_o}{h} \right)
- \gamma \left( 2, \frac{R_i}{h} \right)
\right\}, \\
\end{equation}
where $f$ is the total flux density between the inner point, $R_i$, and the outer point, $R_o$, $S_0$ is the surface brightness at $R=0$, $h$ is the scalelength, and $\gamma$ is the incomplete gamma function. For components between breaks, $R_i$ and $R_o$ span the distance between the breaks, and we use $R_o=200$~kpc for the final component. To compute the mass we use:
\begin{equation}
\label{eq:mass_derivation}
M[M_{\odot}] = \Upsilon_{\star}^{3.6} \frac{f[Jy]}{c_m^{3.6}[Jy]} \times 10^{0.4(\mu + M_{\odot}^{3.6})},
\end{equation}
where $M$ is the mass, $\Upsilon_{\star}^{3.6}=0.5$ is the stellar mass-to-light ratio at 3.6~${\rm \mu}$m\ \citep{Oh+08, Eskew+12, Barnes+14, Meidt+14}, $c_m^{3.6}$ is the 3.6~${\rm \mu}$m\ calibration factor \citep{Reach+05}, $\mu$ is the distance modulus taken from the Extragalactic Distance Database \citep{Tully+09}, and $M_{\odot}^{3.6}=6.02$ is the absolute AB-based magnitude of the Sun at 3.6~${\rm \mu}$m\ \citep{Oh+08}. The masses of the individual galaxies, along with their respective model components, are found in Table~\ref{table:mass}.
\section{ANALYSIS AND DISCUSSION}
\subsection{Break statistics}
The entire sample contains 121 breaks; $6\%$ are Type-II.OLR, $31\%$ are Type-CT, $20\%$ are Type-III.S, and $43\%$ are Type-III.O. Seven galaxies have three breaks (8$\%$), twenty-nine galaxies have two breaks (32$\%$), forty-two galaxies have one break (46$\%$), and fourteen galaxies have no breaks (15$\%$). This is compared to only 7/90 (8$\%$) galaxies with two breaks in \cite{Pohlen+Trujillo06}, 5/47 (11$\%$) in \cite{Gutierrez+11}, and 4/328 (1$\%$) from \cite{Laine+14}. Note that these differences are sensitive to the wavelength of the surface brightness profiles under study \citep{Laine+16}, the method in which the breaks are determined, and the depth of the imaging. These details are further expanded upon below.
The mean break occurs at 6.6~kpc, the median at 5.2~kpc, and the furthest break occurs at 29.6~kpc. The distribution of break radii is found in Figure~\ref{fig:break_hist}. These break-radii are similar to a number of optical-based studies of disc-breaks \citep[i.e.][]{Pohlen+Trujillo06, Erwin+08, Gutierrez+11}, with a slightly higher fraction of Type-III breaks which occur further away. The differences from the optical studies \citep{Pohlen+Trujillo06, Erwin+08, Gutierrez+11} are likely due to differences between near-infrared and optical data (such as dust lanes and the greater effect of metallicity on optical data). In some cases these differences between optical and infrared data are enough to change the statistics of the break parameters, as seen in the near-infrared based S$^4$G and NIRS0S \citep{Laine+14, Laine+16}. Despite also being in the near-infrared, EDGES also differs from S$^4$G, with less Type-I galaxies and more Type-III galaxies. Galaxies labeled Type-III in EDGES would be labeled Type-I in S$^4$G when the break occurs beyond the surface brightness limit of S$^4$G (the unsmoothed noise level of EDGES is 2.5~kJy~sr$^{-1}$ versus 7.2~kJy~sr$^{-1}$ for S$^4$G \citep{Sheth+10}, a difference of 1.15~magnitudes). Of the five Type-I galaxies in \citep{Laine+14} which are also in EDGES, we classify four as Type-III and one as Type-II.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/bend_fraction_vs_t.pdf}
\caption{The percentage of galaxies with a certain T-type. Type-II and Type-III galaxies are slightly offset to emphasize their differences.}
\label{fig:t_type}
\end{figure}
Figure~\ref{fig:t_type} shows break type by morphological $T$-type. Note that in this plot galaxies with multiple breaks are considered to be the type of their first break, which allows a comparison to studies which did not find many galaxies with multiple breaks. The bins are defined to mimic \cite{Pohlen+Trujillo06} with three bins added to represent ellipticals, Sa, and irregular/dwarf galaxies. The bins are thusly defined as: $T \leq 0.5$, $0.5 < T \leq 2.5$, $2.5 < T \leq 4.5$, $4.5 < T \leq 6.5$, $6.5 < T \leq 8.5$, $T > 8.5$ which roughly maps to ellipticals, Sa, Sb, Sc, Sd, and irregular/dwarfs respectively. EDGES has a relatively small sample of galaxies (92) and cannot fill every bin. With that caveat, our data fall within the uncertainties of Figure~5 in \cite{Laine+14}. In general, Type-I profiles are rare in all galaxies except irregulars and dwarfs, Type-II profiles dominate in disc galaxies, and Type-III profiles are common in all galaxies types.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/break_plot.pdf}
\caption{The relation between the ratio of the inner break radius to the inner scale-length, $R_{break}/h_i$ with the log of the ratio of the outer scale-length to the inner scale-length, log$_{10}(h_0/h_i)$ for each EDGES galaxy. The data are classified according to the break-type of the first break.}
\label{fig:break}
\end{figure}
To further investigate the break statistics in EDGES, Figure~\ref{fig:break} presents the break radius, $R_{\rm break}$ normalized by the scalelength interior to the break $h_{\rm i}$, versus the log of the ratio between the interior scalelength and the outer scalelength, $h_{\rm o}$. The data-points are identified by their break type, either Type-II.CT, Type-II.OLR, Type-III.S, or Type-III.O as discussed in $\S~\ref{sec:break_class}$. The x-axis represents the strength and type of the break; large values for strong breaks, small values for weak breaks, and positive values for Type-III breaks, and negative values for Type-II breaks. The y-axis is analogous to the location of the break, normalised by the physical size of the galaxy (this normalisation is important because discs with high-valued scalelengths tend to be larger as they reach the surface brightness limit more gradually). These data are similar to the results in \cite{Laine+14} with a few key differences. Due to the greater depth of EDGES a larger number of Type-III breaks are detected farther out in the disc, and with a slightly larger difference between interior and outer scalelengths. This causes the overall break statistics to be more heavily weighted towards Type-III breaks (in $45\%$ of EDGES galaxies the first break is Type-III versus $38\%$ in \cite{Gutierrez+11} and $21\%$ in \cite{Laine+14}), and our results are a continuation of the general trend in Figure~\ref{fig:break}.
\subsection{Mass analysis}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/lvl_edges_log_comparison.pdf}
\caption{A comparison of LVL derived masses versus EDGES derived masses.}
\label{fig:lvl_comparison}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/total_mass_hist.pdf}
\caption{The distribution of total mass for every galaxy in EDGES (solid line) and in the LVL survey (dashed line).}
\label{fig:total_mass_hist}
\end{figure}
The total stellar mass within each galaxy is found by summing the components of each model from Equation~\ref{eq:final_model}. To verify these results, we compare the total stellar mass for the galaxies which appear in EDGES and also in LVL \citep{Dale+09} in Figure~\ref{fig:lvl_comparison}. The LVL masses are found with Equation~\ref{eq:mass_derivation} along with the integrated flux from \citep{Dale+09}, and distances from the Extragalactic Distance Database \citep{Tully+09}. We find that, on average, the total stellar mass found in EDGES is larger than the total stellar mass in the LVL sample. This offset is due to the differences in integration techniques, and the depth of the data; EDGES integrates a model to 200~kpc, whereas in LVL the total flux was calculated using aperture photometry which encompasses the shallower LVL emission (which averages to 1.13~$R_{25}$). In addition, we have computed the stellar mass for the entirety of LVL to showcase the differences between EDGES and a volume-limited sample in Figure~\ref{fig:total_mass_hist}. In general, EDGES galaxies are more massive, and have a flatter distribution than the galaxies found within LVL. This is due to the sampling strategy (see \S~\ref{sec:sample}), and subsequent lack of dwarf-irregular galaxies in EDGES in comparison to LVL.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/mass_type_hist.pdf}
\caption{The distribution of total mass for the EDGES sample, subdivided by galaxies with a single Type-I break (top-left), a single Type-II break (top-right), a single Type-III break (bottom-left), and for galaxies with multiple breaks (bottom-right).}
\label{fig:mass_type_hist}
\end{figure}
In Figure~\ref{fig:mass_type_hist} we plot the distribution of total stellar mass according to the galaxy break-type. In this case, galaxies with no breaks are considered Type-I, galaxies with a single down-bending break are Type-II, galaxies with a single up-bending break are Type-III, and galaxies with more than one break are classified as ``Multiple Breaks''. In general, Type-I galaxies have the least stellar mass, followed by Type-II galaxies, Type-III galaxies, and galaxies with multiple breaks have the greatest stellar mass.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/mu0_vs_mass.pdf}
\caption{The dependence of the central surface brightness, $\mu_0$, of the first S{\'e}rsic disc on total stellar mass for each EDGES galaxy. The data are classified according to the break-type of the first break.}
\label{fig:mu_mass}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/h_i_vs_mass.pdf}
\caption{The dependence of the inner scale-length, $h_i$, on total stellar mass for each EDGES galaxy. The data are classified according to the break-type of the first break.}
\label{fig:h_mass}
\end{figure}
These differences are coupled with the relation of galaxy break type with central surface brightness and scale-length of the disc components. In Figure~\ref{fig:mu_mass} we plot the relation between the central surface brightness of the inner-most disc and total stellar mass. EDGES galaxies with high stellar mass tend to have the brightest disc central surface brightness. In Figure~\ref{fig:h_mass} we plot the relation between the scale-length of the inner-most disc, and total stellar mass. This relation is relatively weak, but positive, where galaxies with high total stellar mass have longer scale-lengths. Both of these relations are found in the optically-based SDSS \citep{Gadotti+09}, as well as in the near-infrared based S$^4$G survey \citep{Laine+16}. In addition, we differentiate each point by the type of galaxy as defined above. Type-I galaxies tend to have low stellar mass, with dim cores, and short scale lengths. Type-II galaxies have slightly higher mass, brighter cores, and longer scale-lengths. Type-III galaxies have even higher mass, brighter cores, and longer scale-lengths. Galaxies with multiple breaks have the highest mass with the brightest cores and the longest scale-lengths.
\subsection{Stellar halos}
\label{sec:stellar_halos}
The Type-III, or up-bending, breaks in the EDGES sample may be caused by a disc which is distinct from the inner disc, a stellar halo, substructure in dwarf galaxies, the extended wings of the EDGES PSF, or by variance as the data approach the noise-floor. We address the extended wings possibility in \S~\ref{sec:exten_psf}, and while the noise-floor may cause up-bending breaks, these breaks are not statistically significant, and do not occur often in our analysis after the disc has been correctly marked (see \S~\ref{sec:final_model}). This leaves distinct discs, stellar halos, and substructure from dwarf galaxies. While distinct discs and substructure within dwarfs have implications for the dynamic processes which form and shape galaxies, we focus on stellar halos as they have a direct link to our broader, cosmological, understanding of the Universe. As discussed in the introduction, the ratio of the mass in the stellar halo to the total stellar mass of a galaxy follows a relation predicted by the latest cosmological models of galaxy evolution \citep{Purcell+07, Cooper+13, Rodriguez-Gomez+15}. The prohibitive depth required to detect stellar halos ($\mu$>28~AB~mag~arcsec$^{-2}$) makes verification of the results from simulations difficult. With the depth of EDGES, some percent of the Type-III breaks are likely associated with stellar halos. In this section we consider if a Type-III break is the result of a stellar halo, or if the break is associated with another cause.
A method to determine where discs end and stellar halos begin is with resolved stellar populations. Stellar halos are populated with stars born from multiple galaxies while disc stars are generally born within the parent galaxy, causing a myriad of differences between the stellar populations of disc and halo stars. The GHOSTS survey has used this strategy of differentiating between discs and halos with stellar populations to investigate stellar halos within their sample \citep{Bailin+11, Harmsen+17}. This method has also found success in detecting ultra-faint dwarfs \citep{Smercina+17}. Differing stellar populations are also reflected within radial optical color differences \citep{Chonis+11,Dale+16}. Lacking resolved stellar populations and optical colors, we compare our solely mass-based observations with predictions from simulations \citep{Cooper+13, Rodriguez-Gomez+15} and observations of aggregated stellar halos in SDSS \citep{DSouza+14}.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/r_break_kpc_hist.pdf}
\caption{The distributions of break radii, $R_{\rm break}$. These distributions are grouped by break-type, with a red solid line representing Type-II.CT breaks, a red dashed line representing Type-II.OLR, a blue solid line representing Type-III.S, and a blue dashed line representing Type-III.O}
\label{fig:break_hist}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figs/Type_III_Frac.pdf}
\caption{The dependence of stellar mass fraction of disc components which occur after a final Type-III break on total stellar mass. These data are subdivided according to the radius at which the Type-III break occurs, with filled circles being Type-III breaks found beyond 15~kpc, grey circles are found between 6 and 15~kpc, and white circles are found under 6~kpc.}
\label{fig:type_III_frac}
\end{figure}
Type-III breaks associated with stellar halos are beyond the disc. Therefore, we only consider breaks beyond the observed spiral structure, Type-III.O. Also, for galaxies with multiple breaks, we only consider the final breaks (those which occur at the largest radii). In Figure~\ref{fig:type_III_frac} we plot component mass fraction (the mass integrated from the final Type-III.O component over total stellar mass) versus total stellar mass. These data are subdivided according to the distribution of Type-III break radii (Figure~\ref{fig:break_hist}). Type-III breaks with radii less than the peak of the distribution at $\sim6$~kpc are shown in white, galaxies above 6~kpc but below the tail at 15~kpc are shown in gray, and any break in the tail of the distribution, at 15~kpc, is shown as black. The white points are generally associated with the dwarf galaxies within the EDGES sample, with total stellar masses ranging from $\sim10^8-2\times10^9~M_{\odot}$. These breaks are most likely caused by substructures within these small, often disturbed, galaxies and are not necessarily caused by stellar halos. The twenty six grey points fall nicely within the locus of the simulations of \citep{Cooper+13}. While it is possible that these breaks are associated with stellar halos, they appear much closer to the center of their galaxies than simulations predict \citep{Purcell+07, Cooper+13, Rodriguez-Gomez+16}. A survey of the stellar populations found within these galaxies would determine if these are due to stellar halos or some other influence. The seven black points, the galaxies with final Type-III.O breaks beyond 15~kpc are retained as candidate galaxies with detected stellar halos. These galaxies are NGC~3675, NGC~3953, M~109, NGC~4013, NGC~4100, NGC~5005, and M~63.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/Halo_Frac.pdf}
\caption{The dependence of stellar halo mass fraction on total stellar mass. The blue polygon represents the model from \protect\cite{Cooper+13}, where the middle line is the median, nd the lower and upper limits are a single standard deviation above and below the median. The green polygon is similarly defined and represents the model from \protect\cite{Rodriguez-Gomez+15}. The literature data-points (see \S~\ref{sec:stellar_halos}) are represented by blue squares. The aggregated stellar halo measurements from \protect\cite{DSouza+14} are purple triangles, and the 7 galaxies with stellar halo candidates from this work are represented by filled circles.
}
\label{fig:halo_frac}
\end{figure}
The ratio of the mass due to final Type-III.O breaks to the total stellar mass of the galaxy versus the total stellar mass is shown in Figure~\ref{fig:halo_frac}. We include the predictions from the simulations of \cite{Cooper+13} and \cite{Rodriguez-Gomez+15}, the average stellar halo in SDSS from \cite{DSouza+14}, and a sampling of observed stellar halos found within the literature: M~101 \citep{vanDokkum+14}, the Milky Way \citep{Carollo+10}, M~31 \citep{Courteau+11}, M~63 \citep{Staudaher+15}, NGC~0253 \citep{Bailin+11}, M~33 \citep{McConnachie+10}, NGC~2403 \citep{Barker+12}, NGC~3115 \citep{Peacock+15}, and UGC~00180 \citep{Trujillo+16}. In general, the literature points appear below the median values of the simulations. However, there is significant scatter amid the few literature points, and given the stochastic nature of merger events \citep{Bullock+05, DSouza+Bell18} a more robust sample is needed. Our seven candidate stellar halos are also included in this figure as black points. The EDGES data fall within a single standard deviation of the predictions from the models.
\begin{figure}
\includegraphics[width=\columnwidth]{figs/NGC3953.pdf}
\caption{The full 3.6~${\rm \mu}$m\ mosaic of NGC~3953. Note the looping tidal stream south of the disc.}
\label{fig:N3953 Stream}
\end{figure}
Another distinctive characteristic of stellar halos is highly structured filaments; tidal streams caused by the stochastic accretion of satellite galaxies \citep{Cooper+13, Rogriguez-Gomez+16} due to stochastic accretion of a handful of high mass systems \citep{Bullock+05}. We find similarly distinctive structures in the seven candidate galaxies, especially in M~63, NGC~4013, and NGC~3953, which either have known tidal streams in the cases of M~63 \citep{Chonis+11, Staudaher+15} and NGC~4013 \citep{Martinez-Delgado+09}, or their stellar streams are newly discovered in the EDGES data (for NGC~3953, see Figure~\ref{fig:N3953 Stream}). Barring new data, the Type-III.O breaks beyond 15~kpc within these seven galaxies are consistent with stellar halos.
\section{SUMMARY}
In this paper we describe the observational strategy for the EDGES survey, along with the data processing techniques used to create deep 3.6~${\rm \mu}$m\ mosaics for each of the 92 galaxies within the sample. The surface brightness profiles of these mosaics reach a depth of 28~AB~mag~arcsec$^{-2}$. We model these profiles as a combination of distinct S{\'e}rsic discs, along with a bulge component.
\begin{enumerate}
\item We find many more galaxies with multiple breaks (where the slope of the profile increases or decreases) than previous, shallower, studies.
\item The type of the break depends upon the radius at which the break occurs, and also on whether the break occurred first, second, or third. In general, Type-II breaks occur closer to the cores of galaxies, and Type-III breaks occur further out.
\item Galaxies without breaks are preferentially lower mass than galaxies with breaks. In addition, galaxies with a single Type-II break are less massive than galaxies with a single Type-III break. Galaxies with multiple breaks tend to be the most massive galaxies in the EDGES sample.
\item Type-III breaks are associated with stellar halos for at least seven galaxies in our sample, with the possibility of more. In general, the fraction of the mass within these halos agrees with simulations, but many galaxies in EDGES show no evidence for stellar halos despite the depth of the survey.
\item A new tidal stream is discovered near NGC~3953.
\end{enumerate}
\section*{Acknowledgments}
This work is based [in part] on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Support for this work was provided by NASA through an award issued by JPL/Caltech. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr).
|
1,116,691,498,456 | arxiv | \section{Perturbation theory for non-Hermitian systems}
\label{sec:PerturbSols}
In this section, we show in more details how one can use the Magnus expansion to find approximate solutions
of the dynamial system (Eq.~\eqref{eq:EigFrameDynSysI} of the main text)
\begin{equation}
\dot{\Phi}_\mathrm{I} (t) =
D_\mathrm{I} (t) \Phi_\mathrm{I} (t) =
\dot{\theta} (t) \left(e^{2 i \Lambda (t)} \sigma_+ - e^{-2 i \Lambda (t)} \sigma_- \right) \Phi_\mathrm{I} (t).
\label{eq:DIsup}
\end{equation}
Using the Magnus expansion, we can formally write the exact solution as
\begin{equation}
\Phi_\mathrm{I} (t) = \exp\left[ \sum_{k=1}^\infty \epsilon^k \Omega_k (t) \right] = \mathbbm{1} + \sum_{j=1}^\infty \frac{1}{j!} \left[
\sum_{k=1}^\infty \epsilon^k \Omega_k (t) \right]^j,
\label{eq:MagnusSol}
\end{equation}
where the second equality follows from expanding the exponential function with a Taylor series (Eq.~\eqref{eq:PhiIApprox} of the
main text) and we use the parameter $\epsilon$ for bookkeeping.
Approximate solutions are found by truncating the series at a desired order in $\epsilon$. Keeping at most terms that are fourth
order in $\epsilon$, we find
\begin{equation}
\begin{aligned}
\Phi_\mathrm{I} (t) &= \mathbbm{1} + \epsilon \Omega_1 (t) + \epsilon^2\left[ \frac{1}{2} \Omega_1^2 (t) + \Omega_2 (t) \right] +
\epsilon^3 \left[ \frac{1}{3!} \Omega_1^3 (t) + \frac{1}{2}\left\{\Omega_1 (t),\Omega_2 (t)\right\} + \Omega_3 (t)
\right] \\
%
&\phantom{={}}
+ \epsilon^4\left[ \frac{1}{4!} \Omega_1^4 (t) + \frac{1}{2} \Omega_2^2 (t) + \frac{1}{2}\left\{\Omega_1
(t),\Omega_3 (t)\right\} + \frac{1}{3!} \left\{\Omega_1 (t)^2,\Omega_2 (t)\right\} + \frac{1}{3!} \Omega_1 (t)
\Omega_2 (t) \Omega_1 (t) + \Omega_4 (t) \right] \\
%
&\phantom{={}}
+\mathcal{O}\left( \epsilon^5 \right),
\end{aligned}
\label{eq:MagnusDysonSol}
\end{equation}
where $\{A_1,A_2\} = A_1 A_2 + A_2 A_1$ denotes the anticommutator of the matrices $A_1$ and $A_2$ and the Magnus elements
$\Omega_k (t)$ with $k\in \{1,4\}$ are given by
\begin{equation}
\begin{aligned}
\Omega_1 (t) &= \int_0^t \di{t_1} D_\mathrm{I} (t_1) = f^{(1)}_+ (t) \sigma_+ - f^{(1)}_- (t) \sigma_-,\\
%
\Omega_2 (t) &= \frac{1}{2} \int_0^t \di{t_1} \left[ D_\mathrm{I} (t_1), \Omega_1 (t_1) \right] = \frac{1}{2} f^{(2)}_z
(t) \sigma_z, \\
%
\Omega_3 (t) &= \int_0^t \di{t_1}\left\{ \frac{1}{2} \left[ D_\mathrm{I} (t_1), \Omega_2 (t_1) \right]
+ \frac{1}{12} \left[ \Omega_1 (t_1), \left[ \Omega_1 (t_1),D_\mathrm{I} (t_1) \right] \right] \right\}
= f^{(3)}_+ (t) \sigma_+ + f^{(3)}_- (t) \sigma_-, \\
%
\Omega_4 (t) &= \int_0^t \di{t_1}\left\{ \frac{1}{2} \left[ D_\mathrm{I} (t_1), \Omega_3 (t_1) \right]
+ \frac{1}{12} \left[ \Omega_2 (t_1), \left[ \Omega_1 (t_1),D_\mathrm{I} (t_1) \right] \right]
+ \frac{1}{12} \left[ \Omega_1 (t_1), \left[ \Omega_2 (t_1),D_\mathrm{I} (t_1) \right] \right] \right\} \\
&= \frac{1}{2} f^{(4)}_z (t) \sigma_z,
\end{aligned}
\label{eq:MagnusEls4th}
\end{equation}
where $[A_1,A_2] = A_1 A_2 - A_2 A_1$ denotes the commutator of the matrices $A_1$ and $A_2$. We have decomposed the Magnus
elements in the basis of Pauli matrices and we have introduced $f_\pm^{(2k-1)} (t)$ and $f_z^{(2k)} (t)$ with $k\in \{1,2\}$ to
denote the time-dependent coefficients of the decomposition.
Substituting Eq.~\eqref{eq:MagnusEls4th} into Eq.~\eqref{eq:MagnusDysonSol} and setting $\epsilon =1$, we find
\begin{equation}
\begin{aligned}
\Phi_\mathrm{I}^{(4)} (t) &=
\left\{ 1 +
\frac{1}{2}\left( f_+^{(1)} (t) f_-^{(3)} (t) - f_+^{(1)} (t) f_-^{(1)} (t) - f_-^{(1)} (t)f_+^{(3)} (t)\right)
+ \frac{1}{4!} \left[ f_+^{(1)} (t) f_-^{(1)} (t) \right]^2 + \frac{1}{8} \left[ f_z^{(2)} (t)
\right]^2\right\} \mathbbm{1} \\
%
&\phantom{={}}
+ \frac{1}{2}\left[ f_z^{(2)} (t) + f_z^{(4)} (t) - \frac{1}{3!} f_-^{(1)} (t) f_+^{(1)} (t) f_z^{(2)} (t)\right]
\sigma_z
+ \left[f_+^{(1)} (t) + f_+^{(3)} (t) - \frac{1}{3!} \left[ f_+^{(1)} (t) \right]^2 f_-^{(1)} (t)
\right] \sigma_+ \\
%
&\phantom{={}}
- \left[f_-^{(1)} (t) - f_-^{(3)} (t) - \frac{1}{3!} \left[ f_-^{(1)} (t) \right]^2 f_+^{(1)} (t) \right]
\sigma_-,
\end{aligned}
\label{eq:PhiIApprox4th}
\end{equation}
where we used the notation $\Phi_\mathrm{I}^{(n)} (t)$ introduced in the main text and which is defined via $\Phi_\mathrm{I} (t) =
\Phi_\mathrm{I}^{(n)} (t) + \mathcal{O} (\epsilon^{n+1})$.
Exact closed-form expressions for the coefficients $f_\pm^{(2k-1)} (t)$ and $f_z^{(2k)} (t)$ [see Eq.~\eqref{eq:MagnusEls4th}] are
difficult to obtain. We can, however, find series representations in powers of $1/(\Gamma t_\mathrm{f})$ by iteratively integrating by
parts Eq.~\eqref{eq:MagnusEls4th}. The general strategy is reminiscent of the standard procedure used when trying to
approximate the integral of a fast oscillating function multiplied by a slow varying envelope function, but here we need to take into
account that the frequency of the fast oscillating function is explicitly time-dependent. As an example, we show below the first
iteration for the functions $f_\pm^{(1)} (t)$. We have
\begin{equation}
\begin{aligned}
f_\pm^{(1)} (t) = \int_0^t \di{t_1} e^{\pm 2 i \Lambda (t_1)} \dot{\theta} (t_1)
%
&= \int_0^t \di{t_1} \left\{\drv{}{t_1}\left[\mp \frac{i}{2}\frac{1}{\lambda (t_1)} e^{\pm 2 i \Lambda (t_1)}\right]
\mp \frac{i}{2}\frac{\dot{\lambda} (t_1)}{\lambda^2 (t_1)} e^{\pm 2 i \Lambda (t_1)} \right\} \dot{\theta} (t_1) \\
%
&= \mp \frac{i}{2} \left[ \frac{\dot{\theta} (t)}{\lambda (t)} e^{\pm 2 i \Lambda (t)} - \frac{\dot{\theta} (0)}{\lambda
(0)}\right] \pm \frac{i}{2} \int_0^t \di{t_1} e^{\pm 2 i \Lambda (t_1)} \left[ \frac{\ddot{\theta} (t_1)}{\lambda (t_1)} -
\frac{\dot{\lambda} (t1)}{\lambda (t_1)}\frac{\dot{\theta} (t_1)}{\lambda (t_1)}\right].
\end{aligned}
\label{eq:fpmExact}
\end{equation}
By truncating the series representations at fourth order in $1/(\Gamma t_\mathrm{f})$, we find
\begin{equation}
\begin{aligned}
f_{\pm}^{(1)} (t) &= \mp \frac{i}{2} \left[ \frac{\dot{\theta} (t)}{\lambda (t)} e^{\pm 2 i \Lambda (t)}
- \frac{\dot{\theta} (0)}{\lambda(0)}\right]
%
+\frac{1}{4}\left[ \left(\frac{\ddot{\theta} (t)}{\lambda^2 (t)}
-\frac{\dot{\theta}(t)}{\lambda (t)} \frac{\dot{\lambda}(t)}{\lambda^2 (t)}\right)e^{\pm 2 i \Lambda (t)}
- \left(\frac{\ddot{\theta} (0)}{\lambda^2 (0)}
-\frac{\dot{\theta}(0)}{\lambda (0)} \frac{\dot{\lambda} (0)}{\lambda^2 (0)}\right) \right]\\
%
&\phantom{={}}
\pm \frac{i}{8} \left\{\left[ \frac{\theta^{(3)}(t)}{\lambda^3 (t)}
-\frac{3\ddot{\theta}(t) \dot{\lambda} (t)}{\lambda^4 (t)}
+ \frac{\dot{\theta}(t)}{\lambda (t)}\left(\frac{3 \dot{\lambda}^2(t)}{\lambda^4 (t)} -
\frac{\ddot{\lambda}(t)}{\lambda^3 (t)} \right)\right] e^{\pm 2 i \Lambda (t)}
- \left[\frac{\theta^{(3)}(0)}{\lambda^3 (0)}
-\frac{3\ddot{\theta}(0) \dot{\lambda} (0)}{\lambda^4 (0)}
+ \frac{\dot{\theta}(0)}{\lambda (0)}\left( \frac{3\dot{\lambda}^2(0)}{\lambda^4 (0)} -
\frac{\ddot{\lambda}(0)}{\lambda^3 (0)} \right)\right]
\right\} \\
%
&\phantom{={}}
-\frac{1}{16} \left\{\left[\frac{\theta^{(4)}(t)}{\lambda^4 (t)}
-\frac{6\theta^{(3)}(t) \dot{\lambda}(t)}{\lambda^5 (t)}
+ \frac{\ddot{\theta}(t)}{\lambda^2 (t)}\left( \frac{15 \dot{\lambda}^2 (t)}{\lambda^4 (t)} -
\frac{4\ddot{\lambda}(t)}{\lambda^3 (t)}\right)
+ \frac{\dot{\theta}(t)}{\lambda (t)} \left( \frac{10 \dot{\lambda} (t) \ddot{\lambda}(t)}{\lambda^5 (t)}
- \frac{\lambda^{(3)}(t)}{\lambda^4 (t)} - \frac{15\dot{\lambda}^3 (t)}{\lambda^6 (t)}\right)
\right]e^{\pm 2 i \Lambda (t)} \right. \\
%
&\phantom{={}-\frac{1}{16}[} \left.
-\left[\frac{\theta^{(4)}(0)}{\lambda^4 (0)}
-\frac{6\theta^{(3)}(0) \dot{\lambda}(0)}{\lambda^5 (0)}
+ \frac{\ddot{\theta}(0)}{\lambda^2 (0)}\left( \frac{15 \dot{\lambda}^2 (0)}{\lambda^4 (0)} -
\frac{4\ddot{\lambda}(0)}{\lambda^3 (0)}\right)
+ \frac{\dot{\theta}(0)}{\lambda (0)} \left( \frac{10 \dot{\lambda} (0) \ddot{\lambda}(0)}{\lambda^5 (0)}
- \frac{\lambda^{(3)}(0)}{\lambda^4 (0)} - \frac{15\dot{\lambda}^3 (0)}{\lambda^6 (0)}\right)
\right] \right\} \\
&\phantom{={}}
+\mathcal{O}\left[ \frac{1}{\left( \Gamma t_\mathrm{f} \right)^5} \right],
\end{aligned}
\label{eq:fpm1Approx}
\end{equation}
where we have defined $h^{(n)} (t) = \drv{^n h (t)}{t^n}$ for $n>2$. Proceeding similarly, we find
\begin{equation}
\begin{aligned}
f_z^{(2)} (t) &= -i \int_0^t \di{t_1} \left ( \frac{\dot{\theta}^2 (t_1)}{\lambda (t_1)} \right)
%
+ \frac{1}{4}\frac{\dot{\theta}(0)}{\lambda(0)}\frac{\dot{\theta}(t)}{\lambda(t)}\left(e^{2i \Lambda(t)} - e^{-2i \Lambda(t)}\right)
%
+ \frac{i}{4} \int_0^t \di{t_1} \left[\frac{\dot{\theta} (t_1)}{\lambda (t_1)}\drv{}{t_1}\left( \frac{\ddot{\theta}
(t_1)}{\lambda^2 (t_1)} - \frac{\dot{\theta} (t_1) \dot{\lambda} (t_1)}{\lambda^3 (t_1)} \right) \right]\\
%
&\phantom{={}}
%
+ \frac{i}{8}\left[
\left(\frac{\dot{\theta} (0) \dot{\lambda} (0)}{\lambda^3 (0)}-\frac{\ddot{\theta}
(0)}{\lambda^2 (0)}\right) \frac{\dot{\theta} (t) }{\lambda (t)} \left(e^{2 i \Lambda(t)} + e^{-2 i \Lambda(t)}\right)
%
- \frac{\dot{\theta} (0) }{\lambda (0)} \left(\frac{\dot{\theta} (t) \dot{\lambda} (t)}{\lambda^3(t)}
- \frac{\ddot{\theta}(t)}{\lambda^2 (t)}\right) \left(e^{2 i \Lambda(t)} + e^{-2 i \Lambda(t)}\right)
\right] \\
%
&\phantom{={}}
%
-\frac{1}{16}\left\{
\left[
\frac{\theta^{(3)}(0)}{\lambda^3 (0)}
-\frac{3 \ddot{\theta}(0) \dot{\lambda}(0)}{\lambda^4 (0)} +\frac{\dot{\theta} (0)}{\lambda (0)}
\left(\frac{3 \dot{\lambda}^2(0)}{\lambda^4 (0)} - \frac{\ddot{\lambda}(0)}{\lambda^3 (0)}\right)
\right]
\frac{\dot{\theta} (t)}{\lambda (t)}
%
+ 3\frac{\dot{\theta} (0)}{\lambda (0)} \left(\frac{\ddot{\theta}(t) \dot{\lambda}(t)}{\lambda^4 (t)}
-\frac{\dot{\theta} (t) \dot{\lambda}^2 (t)}{\lambda^5 (t)}\right) \right .\\
%
&\phantom{={}-\frac{1}{16}\{} \left .
%
+ \left(\frac{\ddot{\theta} (0)}{\lambda^2 (0)}-\frac{\dot{\theta} (0) \dot{\lambda} (0)}{\lambda^3 (0)}\right)
\left(\frac{\ddot{\theta} (t)}{\lambda^2 (t)}-\frac{\dot{\theta} (t) \dot{\lambda} (t)}{\lambda^3 (t)}\right)
%
+ \frac{\dot{\theta} (0)}{\lambda (0)} \left(\frac{\dot{\theta}(t) \ddot{\lambda} (t)}{\lambda^4 (t)}
-\frac{\theta^{(3)}(t)}{\lambda^3 (t)}\right) \right\} \left(e^{2 i \Lambda (t)}-e^{-2 i \Lambda (t)}\right) \\
%
&\phantom{={}}
%
+\mathcal{O}\left[ \frac{1}{\left( \Gamma t_\mathrm{f} \right)^5} \right],
\end{aligned}
\label{eq:fz2Approx}
\end{equation}
\begin{equation}
\begin{aligned}
f_\pm^{(3)} (t) &= \pm \frac{1}{4} \left( \frac{\dot{\theta} (t)}{\lambda (t)} e^{\pm 2 i \Lambda (t)} +
\frac{\dot{\theta} (0)}{\lambda (0)}\right) \int_0^t \di{t_1} \left(\frac{\dot{\theta}^2 (t_1)}{\lambda (t_1)}
\right) \\
%
&\phantom{={}}
%
+\frac{i}{2}\left\{ -\frac{1}{12}\frac{\dot{\theta} (0)}{\lambda(0)} \int_0^t \di{t_1} \left(
\frac{\dot{\theta}(t_1)}{\lambda(t_1)} \drv{}{t_1}\frac{\dot{\theta} (t_1)}{\lambda (t_1)}\right)
%
+\frac{1}{24} \frac{\dot{\theta}^2 (0) \dot{\theta} (t)}{\lambda^2 (0)\lambda (t)}
\left(e^{\pm 2 i \Lambda(t)}-e^{\mp 2 i \Lambda(t)}\right) \right .\\
%
&\phantom{={} + \frac{i}{2} \Big[}
%
+\frac{1}{3} \left( \frac{\dot{\theta}^3 (t)}{\lambda^3 (t)}e^{\pm 2 i \Lambda(t)} -\frac{\dot{\theta}^3(0)}{\lambda^3 (0)}\right)
%
+\frac{1}{24} \frac{\dot{\theta} (0)}{\lambda (0)}\left(\frac{\dot{\theta}^2 (t)}{\lambda^2 (t)} e^{\pm 4
i\Lambda(t)} - \frac{\dot{\theta}^2(0)}{\lambda^2 (0)} \right)\\
%
&\phantom{={} + \frac{i}{2} \Big[} \left .
%
+\frac{1}{4} \left[
\left(\frac{\ddot{\theta} (t)}{\lambda^2 (t)}-\frac{\dot{\theta} (t) \dot{\lambda} (t)}{\lambda^3 (t)} \right)
e^{\pm 2 i \Lambda (t)}
+ \left(\frac{\ddot{\theta} (0)}{\lambda^2 (0)}-\frac{\dot{\theta} (0) \dot{\lambda}(0)}{\lambda^3 (0)}\right) \right]
\int_0^t \di{t_1} \left( \frac{\dot{\theta}^2 (t_1)}{\lambda (t_1)} \right) \right\} \\
%
&\phantom{={}} \pm \frac{1}{16}\left\{
%
\int_0^t \di{t_1} \left[\frac{\dot{\theta} (0) \dot{\theta} (t_1)}{\lambda (0) \lambda (t_1)} \left(3
\frac{\dot{\lambda} (t_1)}{\lambda^2 (t_1)}\drv{}{t_1} \frac{\dot{\theta} (t_1)}{\lambda (t_1)} +
\frac{\dot{\theta} (t_1) \ddot{\lambda} (t_1)}{\lambda^3 (t_1)} - \frac{\theta^{(3)} (t_1)}{\lambda^2 (t_1)}
\right) \right. \right .\\
%
&\phantom{= \pm \frac{1}{16} \{\int_0^t \di{t_1}\Big[+}
%
+ \frac{\dot{\theta}^2 (t_1)}{\lambda (0) \lambda (t_1)} \left( 3 \frac{\dot{\lambda} (0)}{\lambda^2 (0)}
\left[\drv{}{t_1} \frac{\dot{\theta} (t_1)}{\lambda (t_1)}\right]_{t_1 = 0} + \frac{\dot{\theta} (0)
\ddot{\lambda} (0)}{\lambda^3 (0)} - \frac{\theta^{(3)} (0)}{\lambda^2 (0)} \right) \\
%
&\phantom{= \pm \frac{1}{16} \{\int_0^t \di{t_1}\Big[+}
%
\left .
+ \frac{1}{3} \left( \frac{\ddot{\theta} (0) \dot{\theta} (t_1)}{\lambda^2 (0) \lambda (t_1)} -
\frac{\dot{\theta} (0) \dot{\lambda} (0) \dot{\theta} (t_1)}{\lambda^3 (0) \lambda (t_1)}\right)\drv{}{t_1}
\frac{\dot{\theta} (t_1)}{\lambda (t_1)}\right] \\
%
&\phantom{=\pm \frac{1}{16}[ }
%
- \int_0^t \di{t_1} \left[\frac{\dot{\theta} (t_1)}{\dot{\lambda} (t_1)}
\left( -3 \frac{\dot{\lambda}(t_1)}{\lambda^2 (t_1)} \drv{}{t_1}\frac{\dot{\theta}(t_1)}{\lambda (t_1)}
- \frac{\dot{\theta} (t_1) \ddot{\lambda}(t_1)}{\lambda^3 (t_1)} + \frac{\theta^{(3)} (t_1)}{\lambda^2 (t_1)} \right) \right]
\frac{\dot{\theta} (t)}{\lambda (t)} e^{\pm 2 i \Lambda(t)} \\
%
&\phantom{=\pm \frac{1}{16}[ }
%
+\frac{1}{3} \left[ \frac{\ddot{\theta} (0) \dot{\theta} (t)}{\lambda^2 (0)} - \frac{\dot{\theta} (0)}{\lambda (0)}
\left(\frac{\dot{\lambda} (0) \dot{\theta} (t)}{\lambda^2 (0)} + \frac{1}{2} \drv{}{t} \frac{\dot{\theta} (t)}{\lambda (t)}\right)
\right] \frac{\dot{\theta} (0)}{\lambda (0) \lambda (t)} e^{\mp 2 i \Lambda(t)} \\
%
&\phantom{=\pm \frac{1}{16}[ }
%
+\frac{1}{3} \left[ \frac{\ddot{\theta} (0) \dot{\theta} (t)}{2 \lambda^2 (0)} - \frac{\dot{\theta} (0)}{\lambda(0)}
\left(\frac{\dot{\lambda} (0) \dot{\theta} (t)}{2 \lambda^2 (0)} + \drv{}{t} \frac{\dot{\theta} (t)}{\lambda (t)}\right) \right]
\frac{\dot{\theta} (t)}{\lambda^2 (t)}e^{\pm 4 i \Lambda(t)} \\
%
&\phantom{=\pm \frac{1}{16}[ }
%
- \left[\left(\frac{\dot{\theta}^2 (0)}{6 \lambda^2 (0) \lambda (t)} + \frac{16}{3} \frac{\dot{\theta}^2
(t)}{\lambda^3 (t)}\right) \drv{}{t} \frac{\dot{\theta} (t)}{\lambda (t)} - \int_0^t \di{t_1} \frac{\dot{\theta}^2 (t_1)}{\lambda (t_1)}
\left(3 \frac{\dot{\lambda} (t)}{\lambda^3 (t)} \drv{}{t} \frac{\dot{\theta} (t)}{\lambda (t)}
+ \frac{\dot{\theta} (t)\ddot{\lambda} (t)}{\lambda^4 (t)} - \frac{\theta^{(3)} (t)}{\lambda^3 (t)} \right)\right]
e^{\pm 2 i \Lambda(t)} \\
%
&\phantom{=\pm \frac{1}{16}[ } \left .
%
+\frac{11}{2} \frac{\dot{\theta}^2 (0)}{\lambda^3 (0)}\left[\drv{}{t} \frac{\dot{\theta} (t)}{\lambda (t)}\right]_{t=0} \right\}
%
+ \mathcal{O}\left[ \frac{1}{\left( \Gamma t_\mathrm{f} \right)^5} \right],
\end{aligned}
\label{eq:fpm3Approx}
\end{equation}
where $[\mathrm{d} f(t)/\mathrm{d} t]_{t=0} = \dot{f} (0)$ denotes that we evaluate the derivative at $t=0$. Finally, we have
\begin{equation}
\begin{aligned}
f_z^{(4)} (t) &= \frac{i}{3}\left[ \int_0^t \di{t_1} \frac{\dot{\theta} (t_1)}{\lambda (t_1)} \left(
\frac{\dot{\theta}^2 (0) \dot{\theta} (t_1) }{4 \lambda^2 (0)} + \frac{\dot{\theta}^3 (t_1)}{\lambda^2 (t_1)}
+ \frac{1}{2} \left(\drv{}{t_1} \frac{\dot{\theta} (t_1)}{\lambda(t_1)}\right)\int_0^{t_1} \di{t_2}
\frac{\dot{\theta}^2 (t_2)}{\lambda (t_2)} \right) \right. \\
%
&\phantom{=\frac{i}{3} [} \left.
%
+ \frac{1}{2} \frac{\dot{\theta} (0) \dot{\theta} (t)}{\lambda (0) \lambda (t)} \int_0^t \di{t_1}
\left(\frac{\dot{\theta}^2 (t_1)}{\lambda (t_1)}\right) \left( e^{2 i \Lambda(t)} + e^{-2 i
\Lambda(t)}\right)\right] \\
%
&\phantom{={}}
%
+\frac{1}{4}\left\{
\frac{1}{4} \frac{\dot{\theta} (0) \dot{\theta} (t)}{\lambda (0) \lambda (t)}\left(- \frac{1}{3} \int_0^t \di{t_1}
\left( \frac{\dot{\theta} (t_1)}{\lambda (t_1)} \drv{}{t_1} \frac{\dot{\theta} (t_1)}{\lambda (t_1)} \right) +
\frac{\dot{\theta}^2 (t)}{6 \lambda^2 (t)} - \frac{3 \dot{\theta}^2 (0)}{2 \lambda^2 (0)}\right)
\right. \\
%
&\phantom{={} +\frac{1}{4}\Big[}
%
+ \frac{\dot{\theta} (t)}{3 \lambda (t) \lambda (0)} \left[\drv{}{t}\frac{\dot{\theta} (t)}{\lambda (t)}
\right]_{t=0} \int_0^t \di{t_1} \left( \frac{\dot{\theta}^2 (t_1)}{\lambda (t_1)} \right) \\
%
&\phantom{={} +\frac{1}{4}\Big[} \left .
%
- \frac{\dot{\theta} (0)}{3 \lambda (0)} \left[ \frac{\dot{\theta}^3 (t)}{\lambda^3 (t)} + \frac{1}{\lambda (t)}
\left(\drv{}{t}\frac{\dot{\theta} (t)}{\lambda (t)}\right) \int_0^t \di{t_1} \left( \frac{\dot{\theta}^2 (t_1)}{\lambda (t_1)}
\right) \right]\right\} \left( e^{2 i \Lambda(t)} - e^{-2 i \Lambda(t)}\right)\\
%
&\phantom{={}}
%
-\frac{\dot{\theta}^2 (0) \dot{\theta}^2 (t)}{96 \lambda^2 (0) \lambda^2 (t)} \left( e^{4 i \Lambda(t)} - e^{-4 i \Lambda(t)}\right)
%
+ \mathcal{O}\left[ \frac{1}{\left( \Gamma t_\mathrm{f} \right)^5} \right].
\end{aligned}
\label{eq:fz4Approx}
\end{equation}
Substituting the truncated series representations of $f^{(2k-1)}_\pm (t)$ and $f^{(2k)}_z (t)$ $[k\in\{1,2\}]$ in
Eq.~\eqref{eq:PhiIApprox4th}, we can evaluate the matrix elements $\mbox{\boldmath$c$}_i^\mathsf{T} \Phi (t) \mbox{\boldmath$c$}_j$ with $i,\,j\in{\mathrm{G},\mathrm{L}}$ and
their modulus squared (not shown here due to the length of the expression).
The asymptotic expression shown in the main text [see Eq.~\eqref{eq:ApproxSolEls}] is obtained by keeping solely the exponential
large terms that governed the dynamics in the long-time regime, i.e., $\Gamma t \gg 1$.
\section{Magnus-based strategy for control}
\label{sec:Magnus}
In this section, we show in more detail how we obtained the linear system of equations determining the Fourier coefficients of the
control fields $\Delta_\mathrm{c} (t)$ and $g_\mathrm{c} (t)$.
In the interaction picture defined by $\Phi_0 (t)$ (see Eq.~\eqref{eq:Phi0} of the main text), the control matrix $W (t)$ (see
Eq.~\eqref{eq:W} of the main text) takes the form
\begin{equation}
\begin{aligned}
W_\mathrm{I} (t) &= \sum_n \tilde{w}_z^{(n)} (t) \sigma_z + \tilde{w}_x^{(n)} (t) \sigma_x + \tilde{w}_y^{(n)} (t) \sigma_y \\
%
&=\sum_n \left[-\left( \frac{i \frac{\Gamma}{2} + \Delta (t)}{\lambda (t)} \Delta_\mathrm{c}^{(n)} (t)
- \frac{g (t)}{\lambda (t)} g_\mathrm{c}^{(n)} (t) \right) \sigma_z \right.
%
- \cos\left[ \Lambda (t) \right] \left( \frac{g (t)}{\lambda (t)} \Delta_\mathrm{c}^{(n)} (t)
+ \frac{i \frac{\Gamma}{2} + \Delta (t)}{\lambda (t)} g_\mathrm{c}^{(n)} (t) \right) \sigma_x \\
%
&\phantom{={}} \left.
%
+ \sin\left[ \Lambda (t) \right] \left(\frac{g (t)}{\lambda (t)} \Delta_\mathrm{c}^{(n)} (t)
+ \frac{i \frac{\Gamma}{2} + \Delta (t)}{\lambda (t)} g_\mathrm{c}^{(n)} (t) \right)\sigma_y \right].
\end{aligned}
\label{eq:WILin}
\end{equation}
The first order correction is found by solving Eq.~\eqref{eq:EqWI} of the main text for $n=1$. Since we want $W(t)$ to cancel the
effects of $V_\mm{bad} (t)$ independently of the orientation of the control loop, we must solve the system of equations
\begin{equation}
\begin{aligned}
\int_0^{t_\mathrm{f}} \di{t} W_\mathrm{I}^{(1)} (t) &= -i \int_0^{t_\mathrm{f}} \di{t} V_{\mm{bad},\mathrm{I}}^{\circlearrowright} (t), \\
\int_0^{t_\mathrm{f}} \di{t} W_\mathrm{I}^{(1)} (t) &= -i \int_0^{t_\mathrm{f}} \di{t} V_{\mm{bad},\mathrm{I}}^{\circlearrowleft} (t).
\end{aligned}
\label{eq:LinSysStep1}
\end{equation}
where we have defined
\begin{equation}
\begin{aligned}
V_{\mm{bad},\mathrm{I}}^{\circlearrowright} (t) &= \tilde{v}_z^\circlearrowright (t) \sigma_z +
\tilde{v}_x^\circlearrowright (t) \sigma_x + \tilde{v}_y^\circlearrowright (t) \sigma_y = e^{2 i \Lambda (t)]}
\dot{\theta} (t) \left( \sigma_x + i \sigma_y \right), \\
%
V_{\mm{bad},\mathrm{I}}^{\circlearrowleft} (t) &= \tilde{v}_z^\circlearrowleft (t) \sigma_z +
\tilde{v}_x^\circlearrowleft (t) \sigma_x + \tilde{v}_y^\circlearrowleft (t) \sigma_y = -e^{-2 i \Lambda (t)]}
\dot{\theta} (t) \left( \sigma_x - i \sigma_y \right).
\end{aligned}
\label{eq:VIPauli}
\end{equation}
Using the decomposition of $W_\mathrm{I} (t)$ [see Eq.~\eqref{eq:WILin}] and $V_{\mm{bad},\mathrm{I}}^s (t)$ [see Eq.~\eqref{eq:VIPauli}] into
the basis of Pauli matrices and taking into account that the coefficients of the decomposition are complex,
Eq.~\eqref{eq:LinSysStep1} can be written as
\begin{equation}
\begin{aligned}
\mm{Re}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_z^{(1)}\right] &= \mm{Re}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_z^\circlearrowright (t)\right], \\
\mm{Im}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_z^{(1)}\right] &= \mm{Im}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_z^\circlearrowright (t)\right], \\
\mm{Re}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_x^{(1)}\right] &= \mm{Re}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_x^\circlearrowright (t)\right], \\
\mm{Im}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_x^{(1)}\right] &= \mm{Im}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_x^\circlearrowright (t)\right], \\
\mm{Re}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_y^{(1)}\right] &= \mm{Re}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_y^\circlearrowright (t)\right], \\
\mm{Im}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_y^{(1)}\right] &= \mm{Im}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_y^\circlearrowright (t)\right], \\
\mm{Re}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_z^{(1)}\right] &= \mm{Re}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_z^\circlearrowleft (t)\right], \\
\mm{Im}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_z^{(1)}\right] &= \mm{Im}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_z^\circlearrowleft (t)\right], \\
\mm{Re}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_x^{(1)}\right] &= \mm{Re}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_x^\circlearrowleft (t)\right], \\
\mm{Im}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_x^{(1)}\right] &= \mm{Im}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_x^\circlearrowleft (t)\right], \\
\mm{Re}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_y^{(1)}\right] &= \mm{Re}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_y^\circlearrowleft (t)\right], \\
\mm{Im}\left[\int_0^{t_\mathrm{f}} \di{t} \tilde{w}_y^{(1)}\right] &= \mm{Im}\left[-i \int_0^{t_\mathrm{f}} \di{t} \tilde{v}_y^\circlearrowleft (t)\right],
\end{aligned}
\label{eq:LinSysStep2}
\end{equation}
Substituting Eq.~\eqref{eq:DeltaAndgCorr} of the main text into Eq.~\eqref{eq:LinSysStep2}, we can carry out the time integration
and we are left with a linear system of $12$ equations for the unknown Fourier coefficients. As shown in Ref.~\cite{roque2021}, the
system of equations can be written in matrix form as
\begin{equation}
M \mbox{\boldmath$x$}^{(1)} = \mbox{\boldmath$y$}^{(1)}
\label{eq:LinSysMatrix}
\end{equation}
with $M$ a known $12 \times N_\mm{coeffs}$ matrix characterizing the evolution of the system under the flow $\Phi_0 (t)$,
$\mbox{\boldmath$y$}^{(1)}$ is the known vector of length $12$ that encodes the spurious elements, and $\mbox{\boldmath$x$}^{(1)}$ is the unknown vector of Fourier
coefficients of length $N_\mm{coeffs}$. Here, $N_\mm{coeffs}$ is the total number of Fourier coefficients that one is free to
choose. As noted in Ref.~\cite{roque2021}, for $N_\mm{coeffs} \neq 12$ the system of equations can be solved using the
Moore-Penrose pseudo-inverse.
Higher-order coefficients are found by solving the linear system of equations
\begin{equation}
M \mbox{\boldmath$x$}^{(n)} = \mbox{\boldmath$y$}^{(n)},
\label{eq:LinSysMatrix2}
\end{equation}
where $M$ is the same matrix as in Eq.~\eqref{eq:LinSysMatrix} and the vector of spurious elements $\mbox{\boldmath$y$}^{(n)}$ is determined using
Eq.~\eqref{eq:EqWI} of the main text.
\end{appendix}
\end{document}
|
1,116,691,498,457 | arxiv | \section{Introduction}
Stars form as a result of the gravitational collapse of clouds of gas and dust. This process can take place
in isolated Bok globules \citep[e.g., B335;][]{keene1983,stutz2008,stutz2010} or within fragmented giant molecular
clouds \citep[e.g. Orion;][]{johnstone1999, motte2001, polychroni2013}.
The earliest recognizable phase of the star formation process is the Class 0
phase \citep{andre1993}, the beginning of which is marked by the formation of a hydrostatically
supported protostar within an infalling cloud of gas and dust.
The robust identification of the youngest sources is imperative for characterizing the initial conditions
at the time of protostar formation, before feedback from the formation process significantly alters local
physical conditions \citep[e.g.,][]{arce2006,offner2014} and probe the earliest stages
of the collapse of the gas onto nascent protostars \citep[e.g.,][]{foster1993}. The
density profile, overall mass, and angular momentum of the
initially collapsing envelope will determine the potential for fragmentation,
how quickly the protostar may accumulate mass, and the growth of the circumstellar disk.
At the start of protostellar collapse, just prior to
protostar formation, there is a theoretical prediction of
a short-lived ($\sim$1000 yr) first
hydrostatic core (FHSC), a phase just before or at the start of the Class 0
phase \citep[e.g.,][]{larson1969,commercon2012}. Several candidate
FHSCs have been identified \citep{enoch2010,chen2010,pineda2011,schnee2012}, but
their identification as true FHSCs remains uncertain, given that these objects could simply be
very low luminosity protostars \citep[e.g., VeLLOs;][]{bourke2006}.
Since the FHSC phase is thought to be short,
it is unlikely that many of the candidates in the nearest star forming regions
(i.e. Perseus, Taurus, Ophiuchus) are true FHSCs due to their relatively small populations of protostars.
Above all else, it is uncertain if there is truly a FHSC phase and if it can be uniquely distinguished
from Class 0 protostars. Nonetheless, detecting and characterizing the youngest
protostellar sources are key steps towards understanding the
star formation process. To capture short-lived
phenomena, like the early Class 0 protostars and FHSCs, it is advantageous
to look toward more populous regions of star formation.
The Orion molecular clouds are the nearest regions of active star formation. The \textit{Spitzer} Orion
survey by \citet{megeath2012} found 488 protostellar candidates
amongst a total of $\sim$3000 young stellar objects.
A subset of 329 protostars from this sample were selected for
observations in the far-infrared as part of the \textit{Herschel} Orion
Protostar Survey (HOPS) \citep[e.g.,][]{fischer2010,stanke2010,ali2010,manoj2013}.
Within the fields observed by HOPS, \citet{stutz2013}, hereafter ST13,
serendipitously identified 11 protostars with bright 70 \micron\ and 160 \micron\ emission
that were not part of the original \textit{Spitzer}-selected sample.
At 24 \micron, these sources were either non-detections (8 sources) or so faint that they were
flagged as potential extragalactic contamination in the \textit{Spitzer} surveys \citep{megeath2012,kryukova2012}. Moreover, within the original HOPS sample,
7 protostars had $[24\,\mu{\rm m}]-[70\,\mu{\rm m}]$ colors
(in log ($\lambda$F$_{\lambda}$) space) redder than 1.65, consistent with
the \textit{Herschel}-detected sources. ST13 refers to the
18 protostars satisfying the extremely red color criterion as the PACS Bright Red Sources (PBRS).
Analysis of the PBRS spectral energy distributions (SEDs) by ST13, which were augmented by APEX 350 \micron\ and
870 \micron\ mapping, found that these PBRS sources have
very cold bolometric temperatures \citep[T$_{\rm bol}$;][]{myers1993} (29 K to 45 K)
and high ratios of submillimeter luminosity to bolometric luminosity (L$_{\rm bol}$)
(0.6\% to 6.1\%). Most PBRS are not
detected shortward of 24 \micron, but some display faint
features in the \textit{Spitzer} IRAC 4.5 \micron\
band, possibly indicative of shocked H$_{2}$ emission associated with outflows.
Despite their deeply embedded nature, sources
emitting at 70 \micron\ must be self-luminous. For example, starless core models
show that the emission would otherwise be too faint to detect at
70 \micron\ \citep[][ST13]{ragan2012}. It is important to point
out that the PBRS sources are \textit{not} low-luminosity objects like the VeLLOs \citep[e.g.,][]{bourke2006} since
they have L$_{\rm bol}$\ ranging between 0.65 L$_{\sun}$
and 30.6 L$_{\sun}$, with the median being $\sim$3 L$_{\sun}$. These luminosities
are large enough such that they are not dominated by external heating \citep{dunham2008}.
The characteristics of the PBRS indicate that these protostars could be very young
Class 0 sources with very dense envelopes (ST13). There is, however, a degeneracy in the
interpretation of protostar SEDs between envelope density and
inclination due to bipolar cavities being evacuated by the outflows. The envelope properties were
also difficult to study with only the APEX submillimeter (submm) data available in ST13, due to the low resolution
and blending of the envelope emission with extended cloud structure. Furthermore, the lack
emission shortward of 10 \micron\ toward most PBRS, made the inclinations impossible to constrain
from SED modeling. The only way to derive more
detailed envelope properties, independent of inclination, is to observe these sources with an interferometer.
We have obtained observations of a subset of the PBRS sample with the Combined Array for Research
in Millimeter-wave Astronomy (CARMA). There are a total of
19 PBRS, 18 of which were presented in ST13 and 1 additional
PBRS will be described in this paper. We
have observed
14 PBRS, focusing on the new, \textit{Herschel}-detected subset of
PBRS. We focused on this subset because they had the least amount of complementary data and
non-detections at most wavelengths shorter than 70 \micron.
We observed the PBRS in both the dust
continuum and spectral line emission to examine both the envelope and outflow
properties of these sources; the outflow results will be presented in a future paper.
We discuss
the observations in Section 2, our results for the dust continuum emission and
model comparison are given in Section 3, we discuss the results within the
broader context of star formation in Section 4,
and summarize our main conclusions in Section 5.
\section{CARMA Observations and Data Reduction}
CARMA is a heterogeneous interferometer array comprised of 23 antennas (6 - 10.4 m,
9 - 6.1 m, and 8 - 3.5 m) located in the Inyo mountains of California. Our
observations were carried out with the main, 15-element CARMA array using
the 10.4 m and 6.1 m antennas in two configurations.
We observed a subset of the PBRS identified in ST13
with CARMA in the D configuration during late 2012 and early 2014.
We also followed-up a subset of the sources observed in
D-configuration with higher-resolution observations in C-configuration in early 2014.
The angular resolutions
in D and C configurations were $\sim$5\arcsec\ and $\sim$2\arcsec\ respectively.
The central frequency was 107.77 GHz and four spectral windows were configured
for 500 MHz bandwidth to observe the dust continuum; two windows
were configured for 8 MHz bandwidth to observe para-NH$_2$D ($J=1_{11}\rightarrow1_{01}$) and
C$^{18}$O ($J=1\rightarrow~0$); and the two remaining windows had 31 MHz
bandwidth to observe $^{13}$CO ($J=1\rightarrow0$) and $^{12}$CO ($J=1\rightarrow0$).
Two or three sources were observed per track, with further details given in Table 1.
The C-configuration observations did not observe para-NH$_2$D and another
500 MHz continuum band was allocated. Generally,
we only detect $^{12}$CO ($J=1\rightarrow0$) and the 2.9 mm continuum;
there were a few weak detections in the other lines which will not
be discussed further. Our root-mean-squared (RMS)
sensitivity is typically 0.2 Jy beam$^{-1}$ channel $^{-1}$
for the CO ($J=1\rightarrow0$) and 1 mJy beam$^{-1}$ for the continuum data.
The data were reduced, edited, and imaged using standard procedures within the MIRIAD
software package \citep{sault1995}. The uncertainty in the absolute flux
is estimated to be $\sim$20\%.
We will only present the continuum results in this paper, the CO outflow data will be
presented in an upcoming paper.
\section{Results}
The observations of $\lambda$ = 2.9 mm continuum emission
enable us to probe the properties of the protostellar envelopes, in
terms of mass and density profiles. We will discuss the overall flux densities, visibility
amplitudes, and comparison of the visibility amplitudes to radiative transfer models.
We also report the data observed toward additional sources located within our field of view,
but primarily discuss the PBRS in the main text; the PBRS and non-PBRS are denoted in Table 2.
\subsection{Integrated 2.9 mm Dust Continuum Emission}
We detect all the observed PBRS sources in the 2.9 mm continuum and deconvolved images
using natural weighting are shown in Figure 1.
Our observations are sensitive to spatial scales between $\sim$1000 AU and $\sim$10000 AU.
On these scales, most sources have some resolved structure, in terms of extended envelope emission
and in the case of HOPS 373, there is a binary source separated by $\sim$4\arcsec.
Two sources (082012 and 061012) also have companions $\sim$20\arcsec\ (9400 AU) away.
Images of the sources made with Robust weighting factor of -1 \citep{briggs1995}
did not reveal significant structure on smaller-scales.
The flux densities measured from the deconvolved images
are presented in Table 2.
The 2.9 mm flux densities of the sample exhibit a relatively
large amount of heterogeneity given the extremely red colors selection of the sample.
The brightest PBRS is 082012 at 155.6 mJy and the faintest is 119019 at 10.2 mJy. Indeed, 12 of 14 PBRS
have flux densities $>$ 30 mJy and their values of L$_{\rm bol}$\ also span an order of magnitude.
The combined D- and C-configuration images agree with D-configuration-only flux
densities within the statistical uncertainties.
Note that we also present an additional
PBRS, 135003, that did not appear in ST13. This source was left out from the sample
due to the 70 \micron\ FWHM being extended more than the cutoff value 7\farcs8. More details of this source
and its infrared and submillimeter imaging are given in the Appendix; its inclusion
raises the number of PBRS in the Orion clouds to 19.
The strength of dust continuum emission from the PBRS sources prompted us to
collect $\lambda$ $\sim$ 3 mm flux densities from the literature of other
Class 0 or Class I protostellar sources observed with interferometers
for comparison \citep[Table 3; ][]{looney2000,arce2006,tobin2011}).
These observations had comparable resolution and sampling of the uv-plane.
To match the 2.9 mm flux densities better, we have scaled the flux densities of the comparison sources.
We have done the scaling by assuming that the relative flux densities only depend on the the dust
opacity spectral index ($\beta$) and the
function F$_{\lambda}$ $\propto$ $\lambda^{-(2 + \beta)}$. This assumption is reasonable
given the similar wavelengths of the samples. With the further assumption that $\beta$ $\sim$ 1,
the scaling factors for the 2.7 mm flux densities and 3.4 mm flux densities
are 0.8 and 1.6 respectively.
We have converted all the flux densities to 2.9 mm luminosities (L$_{2.9mm}$) using
the distances provided in Table 3
and assuming a bandwidth of 0.11 mm (4 GHz).
We plot L$_{2.9mm}$ versus L$_{\rm bol}$\ and
the ratio of L$_{2.9mm}$ to L$_{\rm bol}$\ versus L$_{\rm bol}$\ for all the data in Figure \ref{mmflux}.
The comparison sources span the range of observed L$_{\rm bol}$\ for the PBRS, but
most PBRS have lower L$_{\rm bol}$\ values, comparable to those in \citet{tobin2011}.
They, however, have L$_{2.9mm}$ values
that are comparable to the \citet{looney2000}
sources, which are among the brightest nearby protostars a millimeter wavelengths
(e.g. NGC 1333 IRAS 4A, NGC 1333 IRAS2A, IRAS 16293-2422) and are more
luminous than most PBRS. Furthermore, the PBRS
have among the largest values of L$_{2.9mm}$ and L$_{2.9mm}$/L$_{\rm bol}$\ ratios.
This behavior is true at all luminosities, but especially evident at L$_{\rm bol}$\ $\sim$ 1 L$_{\sun}$.
The non-PBRS sources in our observations generally have higher L$_{\rm bol}$, higher T$_{\rm bol}$, and lower L$_{2.9mm}$/L$_{\rm bol}$\ ratios;
however, the source HOPS 68 does intermingle with the PBRS in Figure \ref{mmflux}. Note that the results do not significantly change
whether or not scaling is applied to the literature data.
Also evident in Figure \ref{mmflux} is the lack of a clear
relationship between L$_{\rm bol}$\ and L$_{2.9mm}$. This indicates that the millimeter
emission is decoupled from the central source
properties (central source refers to both the protostar and accretion processes generating
luminosity). The PBRS are tracing a new region of parameter space with their
large amounts of circumstellar material traced by the 2.9 mm flux densities and lower values
of L$_{\rm bol}$.
\subsection{Circumstellar Masses}
The integrated flux densities of the protostars enable us to directly probe
the circumstellar mass associated with the protostars, without significant
contributions from the surrounding molecular cloud. In this case, the interferometer
filtering works to our advantage by separating the envelope emission from
the surrounding background cloud. To convert a flux density into a mass, we assume that the emission is optically thin
and isothermal, and apply the equation
\begin{equation}
M = \frac{D^2 F_{\lambda} }{ \kappa_{2.9mm} B_{\lambda}(T_{dust}) };
\end{equation}
where $B_{\lambda}$ is the Planck function. We have assumed that $T_{dust}$ = 20 K,
$\kappa_{2.9mm}$ = 0.00215 cm$^{2}$~g$^{-1}$ using \citet{ossenkopf1994}
(Table 1, Column 5) extrapolated to 2.9 mm, and $D$ = 420 pc.
The extrapolation to 2.9 mm uses the dust opacity spectral index ($\beta$) of the \citet{ossenkopf1994}
dust model between 700 \micron\ to 1.3 mm which has $\beta$ = 1.78. The opacity given
is the dust+gas opacity, assuming a gas-to-dust ratio of 100.
The calculated masses are given in Table 2, the uncertainties given are statistical only (not
including the uncertainty in absolute flux calibration)
and the masses themselves are likely only valid at the order of magnitude
level given the assumptions. There may be optically thick regions of the envelope, but those are on scales of
order a few hundred AU and will make only a small contribution to the overall mass.
With these assumptions,
all the PBRS sources (except 119019) have more
than 1 $M_{\sun}$ of surrounding material, with the largest
being almost 10 $M_{\sun}$. These masses are reflected in the high integrated flux densities observed toward these protostars.
We also note that the masses are systematically larger than those calculated by ST13.
The masses calculated in ST13, however, are from modified blackbody fits to the emission from 70 \micron\ to 870 \micron\ and
the ST13 modified blackbody fits
systematically underpredict the 870 \micron\ flux densities. The underprediction of the 870 \micron\ flux densities likely
results from fitting a single temperature to data that reflect a superposition of temperatures and
span an order of magnitude in wavelength. If masses were calculated
directly using the 870 \micron\ flux, closer agreement is expected. Several
non-PBRS have masses listed in Table 2 that are comparable to the mass
of the PBRS. Many of these sources, however, have higher L$_{\rm bol}$\ and T$_{\rm bol}$,
suggesting that the dust temperatures could be larger and by extension the masses
are overestimated. We assumed T$_{dust}$ = 20 K,
and the actual masses will be different by the ratio T$_{dust}$/20~K.
The estimated masses will also change if we assume
a different dust opacity law; \citet{ossenkopf1994} has $\beta$ = 1.78 at millimeter
wavelengths (Table 1, Column 5). If we instead assume $\beta$ = 1 and use the normalization of
0.1($\nu$/1200 GHz)$^{\beta}$ from \citet{beckwith1990},
the masses would be a factor of 4.5 lower.
\subsection{Visibility Amplitudes of 2.9 mm Continuum}
The integrated flux densities are only one aspect of the continuum data,
the visibility amplitudes as a function of uv-distance/baseline length can reveal
more about the source structure than the deconvolved images alone. We show the visibility amplitudes
for all detected sources in Figure \ref{uvamps}.
How slowly (or rapidly) the amplitude decreases with increasing uv-distance reveals
how concentrated the emission is toward a particular source, in addition to structural changes in the emitting
material.
Similar to the order of magnitude span in 2.9 mm flux density, the
visibility amplitudes profiles themselves are quite varied but generally fall within two groups. About half of
the observed PBRS have amplitudes that drop quickly with
increasing uv-distance (HOPS 373, 082012, 302002, 119019, HOPS 372, 019003A, 061012),
meaning that there is more emission on larger spatial scales relative to small spatial scales.
The other half of the sample have amplitudes that are flat or slowly decreasing
with increasing uv-distance (093005, 090003, 091016, 097002), indicative
of most emission arising from compact, unresolved structure.
The visibility amplitudes of 082005 and 091015
are most consistent with the flat visibility amplitude sources,
but decrease more rapidly than the others.
We note that 135003 was only observed in D-configuration and is located in a more complex region,
so it is uncertain if its flat visibilities extend toward larger uv-distances.
To examine the scales at which most flux is being emitted, we plot the ratio of the visibility amplitudes
at 5 k$\lambda$ ($\sim$41\arcsec, 17300 AU) F(5k$\lambda$) to those at
30 k$\lambda$ ($\sim$7\arcsec, 3000 AU) against F(30k$\lambda$) and L$_{\rm bol}$\
in Figure \ref{uvratio}. We see that the brightest PBRS sources at 30 k$\lambda$ also have the lowest ratios, meaning that
most of their flux is emitted from scales smaller than 3000 AU; 082012 is an outlier from this trend.
When plotted against L$_{\rm bol}$, the F(5k$\lambda$)/F(30k$\lambda$) ratio tends to be $<$ 2 for sources
with luminosities of $\sim$1 L$_{\rm bol}$, while
higher luminosity sources and non-PBRS tend to have ratios $>$ 2. We note,
however, that a source composed of just a circumstellar disk would
appear to have a ratio of 1 on these plots and the non-PBRS sources
that have F(5k$\lambda$)/F(30k$\lambda$) ratios $\sim$ 2 are likely
more-evolved protostars whose millimeter emission is likely to be dominated by a disk on
small spatial scales.
\subsection{Comparison to Protostellar Envelope Models}
The visibility amplitude profiles can also indicate the density profiles of the envelope.
We ran a small grid of radiative transfer models to obtain qualitative results for the
interpretation of the visibility amplitude data. The goal is to determine what
density profiles are consistent with the data and if a compact, unresolved
source is a necessary component for the models to fit the data.
We use the Hyperion code \citep{robitaille2011} to perform the radiative equilibrium calculations
and produce ray-traced images of 2.9 mm continuum emission, with a 5.0 $M_{\sun}$
envelope, 1 L$_{\sun}$ central protostar, 10000 AU outer radius, and radial density profiles
of $\rho$ $\propto$ R$^{-1.5,-2.0,-2.5}$, and a 50 AU radius
embedded disk with $M_{disk}$ = 0.0 $M_{\sun}$, 0.01 $M_{\sun}$, and 0.1 $M_{\sun}$. We also ran envelope models using
the density structure for a rotationally-flattened, infalling envelope
\citep[CMU envelope;][]{ulrich1976, cassen1981}.
For the CMU models, we explored the same disk masses,
but we used four centrifugal radii ($R_C$ = 50 AU, 100 AU, 300 AU, and 500 AU) and
assumed that the disk radius was equal to $R_C$; $R_C$ is the radius at which infalling material can
be rotationally-supported due to conservation of angular momentum.
The overall envelope masses of the CMU
models were the same as those of the power-law envelopes. The inclination of the system only
has a minor effect on the visibility amplitudes and, we assume an inclination
angle of 60\degr\ for simplicity.
We use the dust opacities calculated by \citet{ormel2011} for icy silicate grains and
bare graphite grains grown for a period of 3$\times$10$^5$ yr. These dust opacities
are similar to those of \citet{ossenkopf1994} (Table 1, Column 5), but are calculated
down to $\lambda$ $\sim$0.1 \micron\ and include scattering properties. The dust opacity spectral
index ($\beta$) of the \citet{ormel2011} models, however, is $\sim$2 at submillimeter and millimeter wavelengths,
consistent with ISM-sized dust grains. This
steep $\beta$,
however, results in a very
low dust opacity at 2.9 mm and very faint envelope emission. Therefore,
we have altered the dust opacity model and at wavelengths greater
than 90 \micron\ we transition to the \citet[][Table 1 Column 5]{ossenkopf1994} dust opacity
model. This model has $\beta$ $\sim$ 1.78, yielding $\kappa_{2.9mm}$ = 0.00215, producing
2.9 mm millimeter fluxes more consistent with our observations.
We could have simply increased the envelope masses such that the flux densities were consistent with our data.
The \citet{ormel2011} dust opacities, however, are a factor of 2.35 lower
than \citet[][Table 1 Column 5]{ossenkopf1994}. Thus, it would
have been necessary to increase envelope masses to $\sim$12 $M_{\sun}$, which
may be unrealistically large for many of our sources. Furthermore,
the larger masses would increase the envelope opacity at shorter wavelengths
and make the overall dust temperatures lower. There is evidence for
millimeter-sized dust grains in protostellar envelopes which would cause a
shallower $\beta$ of $\sim$1 \citep{sadavoy2013,kwon2009,schnee2014} and we
therefore feel justified in adopting a hybrid dust opacity model.
We generated 30000 AU $\times$
30000 AU model images with 15 AU resolution, corresponding to 2048$\times$2048 pixel images for emission
between 2.8 mm and 3.0 mm.
Such high resolution was necessary to ensure that we did not introduce false structure
when Fourier transforming the images to compare with the observed visibility data. We used the
MIRIAD task \textit{fft} to calculate the Fourier transform of each model image and
we azimuthally averaged the Fourier transformed image to construct a 1-dimensional visibility amplitude
profile. To facilitate model comparisons, we normalized the flux densities of the envelopes and data
in our comparisons at uv-distances of
22.5 k$\lambda$. This normalization is reasonable because most
emission should be optically thin at $\lambda$ $\sim$ 3 mm on spatial scales
larger than our best resolution ($\sim$2000 AU) and we are only interested
in comparing the density profiles, not fitting model envelopes in detail.
We perform a simple model comparison for the sources with the best
signal-to-noise ratios at the uv-distances probed by our observations:
082005, 093005, 090003, 082012, and 097002, see Figures \ref{uvcomps} and \ref{uvcomps-cmu}.
This sample is representative of the range in visibility amplitudes profiles observed, e.g., from
very flat (090003, 097002, and 093005), intermediate (082005), and
rapidly declining (082012).
The flat visibility amplitude sources (090003, 097002, and 093005) are
consistent with a power-law envelope with $\rho$ $\propto$ R$^{-2.5}$
envelope if there is no unresolved component.
When a 0.01 $M_{\sun}$ unresolved component is included in a power-law envelope,
the flat visibility sources are still most consistent with a $\rho$
$\propto$ R$^{-2.5}$ envelope. We note that the observed visibility amplitude profiles of
090003, 097002, and 093005 are systematically elevated with respect to
the $\rho$ $\propto$ R$^{-2.5}$ envelope both with and without the inclusion of a 0.01 $M_{\sun}$ disk.
When comparing the flat visibility sources to models with a
0.1 $M_{\sun}$ unresolved component, all three power-law envelope
density models provide a reasonable match to the visibility data since the compact
source dominates the emission at uv-distances $>$ 20 k$\lambda$, although the curvature of visibility
curve of the R$^{-1.5}$ model is opposite to that apparent in the data. In the case of
the CMU envelope, the flat visibility amplitude sources
are \textit{inconsistent} with no disk component and are \textit{consistent} with a 0.1 $M_{\sun}$ disk
with $R_C$ $\le$ 100 AU, but the curvature of the model visibility amplitude profile
is in the opposite sense as the data.
The intermediate source between the extremes of flat visibilities and rapidly declining visibilities
(082005) is consistent with a density profile between $\rho$ $\propto$ R$^{-2.0}$ and $\rho$ $\propto$ R$^{-2.5}$
if no unresolved component is included. It is consistent with $\rho$ $\propto$ R$^{-2.0}$ when a
0.01 $M_{\sun}$ unresolved component is included, but is marginally inconsistent with a power-law envelope
and a 0.1 $M_{\sun}$ unresolved component. Assuming a CMU envelope, it is most consistent with
$R_C$ = 300 AU and a 0.1 $M_{\sun}$ disk.
The rapidly declining visibility amplitude source 082012 is well-matched
by the power-law envelope models with no unresolved component and a density profile between
$\rho$ $\propto$ R$^{-2.0}$ and $\rho$ $\propto$ R$^{-1.5}$. The data are also consistent
with $\rho$ $\propto$ R$^{-1.5}$ when a 0.01 $M_{\sun}$
disk is included. If we assume a CMU envelope structure, then
082012 is also consistent with a CMU envelope with $R_C$ = 100 - 300 AU, containing
a 0.01 $M_{\sun}$ disk. Thus, sources with rapidly declining
visibility amplitudes (082012 and others in the sample)
are \textit{inconsistent} with both density profiles steeper than $\rho$ $\propto$ R$^{-2}$ and
disk components more massive than 0.01 $M_{\sun}$.
In addition to comparing the visibility amplitude profiles directly, we show the visibility
amplitude ratios from the envelope-only models in Figure \ref{uvratio}. These ratios can be thought of as
limiting cases in the absence of a massive protostellar disk. The visibility amplitude ratio
for the $\rho$ $\propto$ R$^{-2.5}$ envelope is the smallest, but it is still in excess of observed sources with the smallest
ratios. Most observed sources have visibility amplitude ratios in between the values
found for the $\rho$ $\propto$ R$^{-2.5}$ and $\rho$ $\propto$ R$^{-2.0}$ models. The addition of an unresolved
component to any of the models in Figure \ref{uvratio} would decrease the ratios, making the models more consistent
with the observations, but with a shallower density profile.
The qualitative model comparison shows that multiple physical structures can
be invoked to explain both the flat and rapidly declining visibility amplitude
sources. The flat visibility sources can be explained with having most
flux in the unresolved component or a very steep ($\rho$ $\propto$ R$^{-2.5}$) density profile.
Thus, a steep density profile
is essentially indistinguishable from a compact source with our current data.
The rapidly declining visibility sources, on the other hand,
are inconsistent with a massive unresolved
component (0.1 $M_{\sun}$) within a power-law or CMU envelope.
There may, however, be some additional dependence on the disk density structure
that we do not explore here.
Higher resolution data will be necessary to break these degeneracies between
power-law envelopes with no or an unresolved component and rotationally-flattened
envelopes with a large, massive disk component.
\section{Discussion}
The PBRS represent an intriguing piece to the puzzle of low-mass
star formation. Their 2.9 mm luminosities are quite large relative to their
bolometric luminosities, and their millimeter luminosities are comparable to the
brightest millimeter sources known in the nearby star forming regions. At the same time
4 out of the 14 observed PBRS have some of the flattest 2.9 mm
visibility amplitudes observed toward any protostellar source; indeed, the most comparably flat source
is NGC 1333 IRAS 4B \citep{looney2003}. In the following subsections,
we compare and contrast the PBRS to known protostars in nearby star forming regions and
theoretical models to examine their significance in the star formation process as a whole.
\subsection{Envelope Density Profiles}
The favored interpretation of the the very red 24 \micron\ to 70 \micron\ colors exhibited by
the PBRS is very high envelope densities (ST13). Most of the observed PBRS
envelopes appear to be quite massive as measured from their 2.9 mm flux densities (Table 2).
Furthermore, the low 5 k$\lambda$ to 30 k$\lambda$ flux ratios
indicate that there is a significant amount of unresolved emission
at spatial scales less than 3000 AU (in diameter)
for 6 of 14 sources. Two possible explanations
for the flat visibility amplitudes are either steep envelope density profiles or massive--compact structures with
densities in excess of a smooth power-law density profile.
Analytic protostellar collapse models predict several radial
density profiles that we could expect to observe.
The Larson-Penston solution \citep{larson1969} (and the numerical solution)
predicts that the free-fall collapse of a constant
density cloud would result in a $\rho$ $\propto$ R$^{-2}$ density profile.
Bonnor-Ebert spheres \citep{bonnor1956} on the other hand, have a high,
constant density region, with a surrounding envelope with a
$\rho$ $\propto$ R$^{-2}$ density profile. As a Bonnor-Ebert
sphere collapses, the entire density profile approaches $\rho$ $\propto$ R$^{-2}$.
A Bonnor-Ebert sphere with a small, flat inner region cannot account for the observed density
structures since the density of the inner region joins smoothly with the outer power-law envelope
and our observations would require a jump to higher density. Moreover, since all the sources
we observe are protostellar, we do not expect there to be a flat density region at small
spatial scales.
A singular isothermal sphere (SIS) also has a $\rho$ $\propto$ R$^{-2}$ density profile and
this is the initial condition of the \citep{shu1977} protostellar
collapse model. The free-fall collapse of a SIS is inside-out, meaning there is an
outwardly propagating rarefaction wave that bounds the infalling region of the envelope;
the infalling region has a $\rho$ $\propto$ R$^{-1.5}$ density profile.
This model was extended to include rotation by \citet{tsc1984} and
the region inside the centrifugally supported radius has a density profile
of $\rho$ $\propto$ R$^{-0.5}$. Within the context of these models, the envelope emission
of the youngest sources is expected to be dominated by the $\rho$ $\propto$ R$^{-2}$ region
since both the infall and rotationally supported regions are small at early times.
For an initial sound speed (c$_s$) of 0.2 \mbox{km s$^{-1}$}\ (assuming T = 10 K),
the infalling region would
extend to a radius of $\sim$1050 AU (2\farcs5) 25 kyr after collapse begins (r = $c_s$ $\times$ t). Thus,
for extremely young sources, one would expect that data with a maximum resolution of $\sim$2\arcsec\
to be dominated by the $\rho$ $\propto$ R$^{-2}$ component of the envelope (within the context of this model).
If some of the sources, however, are more evolved, with larger collapsing regions, the shallower density
profiles and possibly the rotationally-flattened portion
of the density structure would be apparent in the visibility amplitudes (see Figure \ref{uvcomps-cmu}).
The envelope models that we compare to the observed visibility amplitudes
in Figure \ref{uvcomps} have density profiles that encompass those
expected from the
theories of protostellar collapse. Considering \textit{only} the envelope
contribution to the visibility amplitudes \textit{without} additional unresolved
source emission, however, the sources with
flat visibility amplitudes (090003, 093005, 097002, 135003
and 091016) models are most consistent with the envelope density profiles
that are as steep as $\rho$ $\propto$ R$^{-2.5}$.
The sources with more rapidly declining
visibility amplitudes (082012 and 302002) are consistent with
the $\rho$ $\propto$ R$^{-1.5}$ density profile.
The sources in between (082005 and 091015) are consistent with $\rho$ $\propto$ R$^{-2}$
density profiles. Thus, if we consider only envelopes with smooth radial density
profiles, the flat visibility amplitude sources appear to be
inconsistent with the analytic protostellar collapse theories, while the sources with intermediate and rapidly
declining visibility amplitudes fall within theoretical expectations.
Such steep density profiles
have been obtained previously toward more nearby protostars \citep{looney2003,chiang2008,kwon2009,chiang2012}.
The sources with flat visibility amplitudes appear similar to
NGC 1333 IRAS 4B, a source with very flat visibility
amplitudes out to $\sim$80 k$\lambda$ \citep{looney2003,chiang2008};
this is equivalent to flat visibility amplitudes out to
$\sim$150 k$\lambda$ at the distance to Orion.
The other sources with more rapidly declining visibility amplitudes
appear more similar to NGC 1333 IRAS 4A, or IRAS 2A.
Our results further confirm that the dust continuum structure of some protostellar envelopes
indicate radial density profiles steeper than expected from analytic
models of collapse, consistent with previous findings by \citet{looney2003} and \citet{kwon2009}.
The density profiles steeper than the analytic models could result from having
asymmetric envelope structures on these scales. \citet{tobin2010,tobin2012} showed that
envelopes are often filamentary and asymmetric on $>$1000 AU scales and that infall might
come through a filamentary envelope rather than a spherical envelope. The \textit{Spitzer}
IRAC and the APEX 350 \micron\ and 870 \micron\ images in ST13 often show
filamentary structure on larger scales that may persist on
smaller scales (e.g., Figures 8 and 13b of ST13).
The 2.9 mm continuum images described here, however, do not have features suggestive of
strong asymmetry in most cases, but \citet{tobin2010} argued
that asymmetry can be difficult to observe in the dust continuum
due to the emission resulting from a combination of density and temperature.
If additional, unresolved components to the dust emission are considered (i.e., added to the
envelope density profile), the flat visibility amplitude sources may be
consistent with shallower density profiles. With an unresolved component
of 0.1 $M_{\sun}$, then the $\rho$ $\propto$ R$^{-2.0}$
and R$^{-1.5}$ density profiles (in addition to $\rho$ $\propto$ R$^{-2.5}$)
are able to reproduce the flat visibility amplitudes observed for some sources
(Figure \ref{uvcomps}). Even with the unresolved component, however, the curvature in the visibility amplitudes
of the $\rho$ $\propto$ R$^{-1.5}$ density profile is inconsistent with the data. The steeper density
profiles also do not fully capture the curvature observed in the data, but the effect
is less dramatic than for $\rho$ $\propto$ R$^{-1.5}$.
A circumstellar disk is a natural compact structure that is expected to develop during the
star formation process \citep[e.g.,][and references therein]{williams2011} and are
likely to have a variety of sizes \citep{maury2010,tobin2012}. Therefore, a comparison to the CMU models
with and without disk components is also of interest, see Figure \ref{uvcomps-cmu}). The
rapidly declining and intermediate sources could be consistent with having a large $R_C$
region (and a large protostellar disk), depending on the disk mass. The
sources with flat visibility amplitudes can only be consistent with small $R_C$ and a massive disk.
Again, however, the curvature of the model visibility amplitude curves with small (R = 50 AU, 100 AU), 0.1 $M_{\sun}$ disks
are dissimilar to that of the data. Thus, the models show that disk components are possible in the intermediate and rapidly
declining cases, but the exact parameters are not well-constrained. The flat visibility amplitude sources on the
other hand are grossly consistent with a disk component within the context of the CMU model,
but, as stated previously, the curvature of the visibility amplitude profiles of the models
relative to the observations are different.
Rather than a massive disk component, it is also possible that the envelope
density profile itself is not a smooth power-law.
Simulations of protostellar collapse including
magnetic fields have been shown to
create density enhancements of infalling material during collapse
that depart from a smooth power-law density profile
\citep{tassis2005}. Such structures can
cause flattening of the visibility amplitudes due to the mass build-up
at small spatial scales. The robustness of this model, however,
is unclear.
Regardless of the exact nature of the structure, we can conclude that the envelopes with
flat visibility amplitudes are
inconsistent with the often assumed $\rho$ $\propto$ R$^{-1.5}$ density profile for
protostellar envelopes. The flat visibility amplitude profiles are most consistent with
either a
steep density profile ($\rho$ $\propto$ R$^{-2.5}$)
or the visibility amplitudes are dominated by dense, unresolved structure. The unresolved structure
could be a disk, or it could be departures from a power-law density profile. Higher
resolution data that resolve down to the expected scales of a disk are necessary
to distinguish between these two (radically different) scenarios.
\subsection{Nature of the PBRS}
The defining characteristic of the PBRS from the \textit{Herschel}
study by ST13 is the very red color of the PBRS sample as a whole,
having $[24\,\mu{\rm m}]-[70\,\mu{\rm m}]$ colors
(in log ($\lambda$F$_{\lambda}$) space) redder than 1.65. Furthermore,
the T$_{\rm bol}$\ measurements of the PBRS are in a narrow range of 20 - 45 K, consistent
with little observed emission shortward of
24 \micron\ for most PBRS. Thus, while the
coldness (and redness) of the SEDs of the PBRS is consistent throughout the sample, they
are very heterogeneous in terms of their ratio of L$_{submm}$ to L$_{\rm bol}$\
(0.6\% to 6.1\%), L$_{\rm bol}$\ itself, and 2.9 mm luminosity.
Nevertheless, the PBRS are characterized by higher L$_{2.9mm}$
to L$_{\rm bol}$\ ratios than previously identified in protostellar samples (see Figure \ref{mmflux}).
The expected evolutionary trend for protostars is that they become more luminous as they accrete mass due to
increased photospheric luminosity and greater accretion luminosity \citep{young2005, dunham2010}.
T$_{\rm bol}$\ is also expected to also increase with decreasing envelope density and
hence optical depth. At the same time, clearing of the envelope
is likely driven by the influence of the protostellar outflow \citep{arce2006,offner2014}.
Thus, it is expected that very young Class 0 protostars will have larger millimeter
flux densities (or larger envelope mass), with rather low luminosities; in other words they will have
large fractions of millimeter luminosity relative to L$_{\rm bol}$\ \citep[e.g.,][]{andre1993}.
These changes, however, may not be due solely to evolution: both initial conditions and evolution
play roles in the observed T$_{\rm bol}$\ and
L$_{submm}$/L$_{\rm bol}$\ ratios observed toward particular protostars \citep[e.g.][]{young2005}.
Further complicating matters, T$_{\rm bol}$\ can be strongly influenced by the
orientation of a given source in the plane of the sky, such that a more
evolved source viewed edge-on can appear younger
\citep[ST13; ][]{jorgensen2009,launhardt2013,dunham2014}.
Within the sample of PBRS, T$_{\rm bol}$\ is confined
to a narrow range, but there are a few sources that have
low luminosities and low 2.9 mm flux densities (061012 and 119019).
Meanwhile, other PBRS have both relatively high luminosities and high millimeter flux densities
(082012 and HOPS 373). Such variations in the observed properties of the sample
indicate that, despite the stringent color selection (ST13), the PBRS as a
whole may not be characterized by a single evolutionary state.
Furthermore, the fact that the PBRS present such a narrow range
in T$_{\rm bol}$\ but exhibit a broad range in other properties poses
possible problems for a T$_{\rm bol}$-based classification of protostars.
If even at the lowest values of T$_{\rm bol}$\ and for a uniformly selected
sample we see clear variations, then even larger variations may be seen across the T$_{\rm bol}$\
range encompassing the Class 0 and Class I protostellar phases. Thus, not all
protostars of equivalent T$_{\rm bol}$\ are equal.
With the interferometry data, we are able
to determine the spatial scales from which we are detecting emission due to the
high-resolution and analysis of visibility amplitude profiles.
The visibility amplitude profiles of the PBRS have many features, but
they can be broadly described as flat or rapidly declining. The visibility amplitude
ratios from 5 k$\lambda$ to 30 k$\lambda$ (flux at $\sim$17,000 AU to 3000 AU scales) plotted versus
30 k$\lambda$ flux density and L$_{\rm bol}$\ (see Figure \ref{uvratio}), further enable the sample
to be examined as a whole. The expected evolutionary trends
for visibility amplitude ratios and profiles are uncertain due to the unknown
contribution of the disk at a given time. Nevertheless, we can use the analytic models
for protostellar envelopes as limiting cases.
If we consider the collapse of a
singular isothermal sphere with a density profile $\propto$ R$^{-2}$, the visibility amplitude
ratios would be smaller initially and then increase as material falls in from
larger radii. The density profile of the infalling region
will be proportional to R$^{-1.5}$ and this region grows with time.
So, in the absence of a massive disk, the visibility amplitude ratio for such a model
will be between the R$^{-2}$ and R$^{-1.5}$ values (see ratios taken from the
models in Figure \ref{uvratio}). Then as a disk grows and
the envelope dissipates, the 5 k$\lambda$ to 30 k$\lambda$ visibility amplitude ratios
will decrease (trending toward 1 on the scales examined)
as the disk begins to dominate the flux density of the system at millimeter wavelengths. Thus, to summarize
the visibility amplitude ratios will start small, then increase, and then decrease.
The exact evolution will depend on how quickly a large disk forms and
how massive such a disk is.
We note that if a singular isothermal sphere formed from a Bonnor-Ebert sphere, a Bonnor-Ebert sphere
would initially have a large visibility amplitude ratio, depending on the size of the flat density region \citep[e.g.,][]{schnee2012}.
The visibility amplitude ratios would then decrease as the Bonnor-Ebert sphere evolved toward a singular isothermal sphere.
Two trends are evident in the visibility amplitude ratio plots shown in Figure \ref{uvratio}.
Most of the sources with the highest flux densities at 30 k$\lambda$ also have the lowest 5 k$\lambda$ to
30 k$\lambda$ ratios (i.e., spatially compact flux density). Then
looking at the 5 k$\lambda$ to 30 k$\lambda$ ratio versus L$_{\rm bol}$\, we see that 6 out of 14
of the lower luminosity PBRS sources (L$_{\rm bol}$\ $<$ 3 L$_{\sun}$)
have ratios between 1 and 1.5. The non-PBRS and higher luminosity PBRS tend to have higher ratios, except for the
sources that are the most evolved (i.e., HOPS 223 and HOPS 59). We therefore suggest that the visibility amplitude
ratios enable us to divide the PBRS sample into two groups. The PBRS with the
smallest visibility amplitude ratios (flattest profiles) are the youngest of the PBRS and the sources with larger
ratios (rapidly declining amplitudes) are likely more evolved, though still Class 0 protostars.
The youngest sources would then have the most compact, dense envelopes initially
that may then be accreted rapidly due to the short
free-fall time. The flat visibility amplitudes indicate that there is likely
large amounts of mass within only a few thousand AU, implying high average densities.
A source with 2 $M_{\sun}$ of envelope material
a radius of 1500 AU has an average density of 3$\times$ 10$^7$ cm$^{-3}$,
corresponding to a local free-fall time of $\sim$10 kyr. Thus, a substantial amount
of the final protostellar mass could be accumulated in this short period of time, much less
than the expected lifetime of the Class 0 phase ($\sim$150 kyr) \citep{dunham2014}.
Therefore, the free-fall times suggest that the Class 0 phase may begin with a
short period of rapid infall that may only last $\sim$10\% of the Class 0 phase.
Based on the number of detected sources, ST13 suggested that if the PBRS represented a
phase of protostellar evolution distinct from the Class 0 phase, it may only
last $\sim$ 25 kyr. Rather than necessarily being a distinct phase, we believe that the
PBRS with flat visibility amplitudes (093005, 090003, 091015, 091016, 082005, and 097002)
are among the youngest Class 0 protostars,
and the large amount of mass on small spatial scales could indicate that they
are in a brief period of high-infall/accretion.
The PBRS that do not have flat visibility amplitudes are still young, but may be more
comparable to typical Class 0 sources. We suggest that the sources with bright
2.9 mm flux densities, but rapidly declining visibility
amplitudes (302002, 082012, HOPS 373) are slightly more evolved that the PBRS with flat
visibility amplitudes. At least a fraction of their
inner envelopes is likely to have been accreted onto the disk and/or protostar. The remaining sources with
declining visibility amplitudes and low flux densities (119012, 061012, HOPS 372, 135003, 019003)
are still consistent with being young Class 0 sources.
Their cold T$_{\rm bol}$\ values and extremely red 24 \micron\ to 70 \micron, however,
colors could result from high density envelopes, but with less overall mass. Alternatively,
they could be edge-on sources; we will further explore the properties of these sources in relation to
their outflows in an upcoming paper (Tobin et al. in prep.).
If the proposed scenario is true, then the Class 0 phase might be a two-phase process, with a
short, rapid accretion phase \citep[like a Bonnor-Ebert collapse; see ][]{foster1993}), lasting $\sim$10 - 25 kyr. This
phase is then followed by a period of slower mass assembly, for the remainder of the Class 0
phase ($\sim$100 kyr - 150 kyr), assuming a Class 0 lifetime of $\sim$ 160 kyr \citep{dunham2014}.
This idea is consistent with the models of \citet{offner2014} that show most protostellar mass
being accreted during the Class 0 phase, before the outflow destroys the envelope.
\subsection{Comparison to VeLLOs and candidate FSHCs}
The infrared and millimeter properties of the PBRS
distinguish them from typical Class 0 protostars and indicate
that at least some of the PBRS may be very young Class 0 objects in a period of
high infall. Two other sub-classes of protostars identified by \textit{Spitzer} and submm/millimeter
observations are the VeLLOs and candidate FHSCs
and it is important to distinguish the PBRS from these sources
based on their millimeter properties.
First, many of the VeLLOs and candidate FHSCs are very faint at 2.9 mm.
For instance, the brightest VeLLO/candidate FHSC at 2.9 mm is
Per-Bolo 58 with a flux density of
13 mJy at d $\sim$ 230 pc \citep{schnee2010, enoch2010}.
If this source was at the distance to Orion,
it would have a flux density of only $\sim$3.9 mJy and only appear
as a $\sim$4$\sigma$ detection in our data.
This flux density is less than half that
of the faintest source in the PBRS sample (119019). None of the other VeLLOs or FHSC
candidates would be detectable at the distance to Orion with the sensitivity of our
CARMA observations. The PBRS 061012 may be the most similar to a VeLLO,
having the lowest luminosity, but it
has a higher 2.9 mm flux density than other VeLLOs.
A comparison to the visibility amplitudes of the VeLLOs is less straightforward. Most
VeLLOs/candidate FHSCs are not bright enough to enable analysis of their visibility amplitude profiles.
Per-Bolo-58 is found to have rapidly declining visibility amplitudes and
The candidate FHSC L1451-MMS \citep{pineda2011}, however, has
flat visibility amplitudes at 1.3 mm, similar to some PBRS sources.
The visibility amplitudes are 30 mJy
out to $\sim$200 k$\lambda$ or 230 AU scales. At 2.9 mm, the visibility amplitudes
of this source would be 2.5 mJy, assuming $\beta$ = 1. At the distance to Orion, however,
the visibility amplitudes would be below our detection limits at 0.8 mJy.
While the overall emitting mass of this source is much lower than the PBRS, it does have
a similar 5 k$\lambda$ to 30 k$\lambda$ flux ratio, meaning that the envelope density profile
might be similar to the most concentrated PBRS.
However, L1451-MMS is undetected at 70 \micron\ and 100 \micron, unlike the PBRS.
In summary, the millimeter properties of the PBRS, combined with the far-infrared constraints from
ST13, distinguish the PBRS from the VeLLOs and candidate FHSCs.
\section{Summary and Conclusions}
We have presented CARMA 2.9 mm dust continuum observations toward 14 PBRS \citep{stutz2013} in the Orion
A and B star forming regions, twelve of these protostars were first identified by \textit{Herschel}
observations. This sample of 14 PBRS also includes 135003, a new PBRS that
discovered by \textit{Herschel} and was not included in the \citet{stutz2013} sample due to
their stringent FWHM cut-off. The inclusion
of this source increases the total number of PBRS in Orion to 19. The PBRS classification in \citet{stutz2013} required
$[24\,\mu{\rm m}]-[70\,\mu{\rm m}]$ colors or limits (in log $\lambda$F$_{\lambda}$ space)
in excess of 1.65. In addition, we also report the continuum properties of 4 protostars and
1 apparent starless/pre-stellar core within the fields observed toward the PBRS.
All 14 PBRS are detected in dust continuum emission. Twelve out of 14
have flux densities $>$ 30 mJy and three have flux densities $\ge$~90
mJy. The 8 PBRS with L$_{\rm bol}$\ $\sim$ 1 L$_{\sun}$ exhibit
higher 2.9~mm luminosities than other known protostars with similar
L$_{\rm bol}$\ values, and therefore have characteristics not previously
identified. The PBRS with L$_{\rm bol}$\ $>$ 2.7 L$_{\sun}$ have comparable
2.9~mm luminosities yet lower L$_{\rm bol}$\ values than the the brightest
sources in the more nearby regions of Perseus and Ophiuchus. Furthermore, the 2.9~mm luminosity does not
strongly correlate with L$_{\rm bol}$. This lack of correlation indicates
that the 2.9~mm luminosity is not strongly dependent on either the the
central protostellar luminosity or the accretion luminosity.
Six PBRS sources (097002, 090003, 093005, 082005, 091015, and 091016)
have flat 2.9~mm visibility amplitudes (and 5~k$\lambda$ to
30~k$\lambda$ visibility amplitude ratios of less than 2). As a
consequence, more than $\sim\,$50\% of the total flux density arises
from scales that are smaller than 7\arcsec\ ($\sim\,$3000~AU) in
diameter. This behavior indicates either steep envelope density
profiles or the presence of significant mass contained within a
compact, unresolved structure. We suggest that these particular PBRS
are the youngest of the sample and may be in a brief period of high
infall rate. Indeed, the average density on scales $<$~3000~AU implies
local free-fall times of $\sim\,$10 kyr, in agreement with independent
life-time estimates based on the ratio of PBRS to protostars
\citep{stutz2013}. The PBRS with large 2.9~mm flux densities but
rapidly declining visibility amplitudes (302002, 082012, HOPS 372) are
still considered to be young Class~0 protostars, but may be more
evolved than the PBRS with flat visibility amplitudes. The sources
with lower 2.9~mm flux densities and declining visibility amplitudes
(119012, 061012, HOPS 372, 135003, 019003) are also still consistent
with being Class~0 sources, but may have edge-on orientations and/or
lower envelope masses.
To better characterize the density profiles of the sample, we compare
the observed visibility amplitudes of the sources to Hyperion
radiative transfer models of axisymmetric envelopes with varying
radial density profiles and unresolved components (represented by a
disk component). We also compare with rotating collapse models with
various centrifugal radii and disk masses. We find that without an
unresolved component to the emission, the flat visibility amplitude
sources are most consistent with a $\rho$ $\propto$ R$^{-2.5}$ radial
density profile. If a compact structure is massive enough, however,
then all three envelope density profiles tested here ($\rho$ $\propto$
R$^{-1.5,-2.0,-2.5}$) are able to provide a
reasonable match to the data. Thus, with the current data we cannot
distinguish between these two scenarios and higher resolution data are
required to understand the nature of these sources. Furthermore,
sources with more rapidly decreasing visibility amplitudes may be
consistent with shallower density profiles and are inconsistent with
having a massive unresolved component.
While the PBRS occupy a narrow range to T$_{\rm bol}$, their 2.9~mm flux
densities and visibility amplitude profiles show a large amount of
heterogeneity suggesting that they are not all in exactly the same
evolutionary stage. We suggest an evolutionary trend in which the
sources with flat visibility amplitude profiles are the youngest and
perhaps have a dense inner envelope that may be rapidly accreted. The
sources large 2.9~mm flux densities and rapidly declining visibility
amplitudes may be slightly more evolved than those with flat
visibility amplitudes. Moreover, the sources with flat visibility
amplitude profiles also tend to have lower L$_{\rm bol}$\ values than those
with rapidly declining visibility amplitudes, consistent with the
expected evolutionary trend of increasing L$_{\rm bol}$. The PBRS also draw a
sharp contrast with candidate FHSCs and VeLLOs. The PBRS have higher
L$_{\rm bol}$\ and larger 2.9 mm luminosities than all of the candidate FHSCs
and VeLLOs; at the distance to Orion, none of the known candidate
FHSCs or VeLLOs would have been confidently detected.
In summary, the PBRS have properties that are consistent with placing
them among the youngest Class~0 protostars. Their millimeter and
infrared properties distinguish them from typical Class~0 protostars,
as well as from candidate FHSCs and VeLLOs. While the data presented
here have enabled us to postulate a tentative evolutionary scenario,
further characterization at higher resolutions and in molecular lines
will be necessary to more firmly establish their place in the context
of the star formation process.
The authors wish to thank the anonymous referee for constructive comments
that improved quality and clarity of the manuscript.
J.T. acknowledges support provided by NASA through Hubble Fellowship
grant \#HST-HF-51300.01-A awarded by the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy,
Inc., for NASA, under contract NAS 5-26555. J.T. also acknowledges funding from
\textit{Herschel} OT2 JPL grant \#1458263.
The work of A.M.S. and S.E.R. was supported by the Deutsche
Forschungsgemeinschaft priority program 1573 ('Physics of the
Interstellar Medium'). Support for STM and WJF was provided by
NASA through awards issued by JPL/Caltech. Support for CARMA construction
was derived from the states of Illinois, California, and Maryland, the
James S. McDonnell Foundation, the Gordon and Betty Moore Foundation, the Kenneth
T. and Eileen L. Norris Foundation, the University of Chicago, the Associates of the
California Institute of Technology, and the National Science Foundation. Ongoing
CARMA development and operations are supported by the National Science Foundation
under a cooperative agreement, and by the CARMA partner universities.
The National Radio Astronomy
Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc.
|
1,116,691,498,458 | arxiv | \section{Introduction}\label{sec:intro}
Virtual personal assistants, such as Siri, Cortana, Google Now, and Alexa, have made commercial use of interactive spoken language technology. However, commercial exploitation of advanced spoken dialogue technology requires new methods for cost-effective development and efficient adaptation to new domains. In this paper, we argue that this problem can be addressed by taking a \emph{multi\hyp dimensional\ }approach.
Current systems focus almost exclusively on the primary task underlying the conversation, for example travel booking or seeking tourist information. The behaviour resulting from such an approach is quite different from natural human dialogue, where several other aspects besides the task itself are addressed as well, such as giving and eliciting feedback, following social conventions, and managing turn-taking and timing. Humans frequently perform \emph{multi-functional} utterances, where several of these aspects, or \emph{dimensions}, are addressed simultaneously \cite{Bunt:2011et}. Consider the following example interaction (annotated with different functions for each turn):
\begin{center}
\small
\begin{tabular}{@{}l@{\;}l@{}}
\toprule
\textbf{Usr}: & \textsl{\textbf{Hello, I am looking for a \underline{cheap} \underline{Indian} restaurant}} \\[1mm]
\multicolumn{2}{l}{\hspace{3mm}\textsc{Social:Greet; Task:Inform; Turn:Release}} \\[2mm]
\midrule
\textbf{Sys}: & \textsl{\textbf{Okay, let me see, \dots}} \\
\multicolumn{2}{l}{\hspace{3mm}\textsc{AutoPositive; Time:Pausing; Turn:Keep}} \\[2mm]
\textbf{Sys}: & \textsl{\textbf{The Rice Boat is an \underline{Indian} restaurant}} \\[1mm]
& \hspace{3mm}\textsl{\textbf{in the \underline{cheap} pricerange}} \\[2mm]
\multicolumn{2}{l}{\hspace{3mm}\textsc{Auto-feedback:Inform; Task:Inform}} \\
\bottomrule
\end{tabular}
\end{center}
The user both greets the system and tells the system they want a cheap Indian restaurant, before releasing the turn; the system then takes the turn with positive feedback and indicates that it needs more time to retrieve the requested information; in the second part the system both provides this information and gives feedback about understanding the user's question (underlined).
Following the notion of multi-dimensionality of dialogue as described by Bunt~\shortcite{Bunt:2011et} and early exploratory work on multi\hyp dimensional\ dialogue management by Keizer and Bunt~\shortcite{Keizer:2006vn,Keizer:2007ve}, we present a new framework for statistical dialogue management which explicitly accounts for these different dimensions of communication. By separating out domain-independent dimensions, our approach has the potential to learn a set of transferable conversational skills, enabling more efficient cross-domain adaptation.
In \cref{sec:md-dial} we discuss the theoretical background of our approach, followed in \cref{sec:md-pomdp-dial} by its embedding into a statistical dialogue system framework. In \cref{sec:sds-impl} we present the first implementation of our multi\hyp dimensional\ statistical dialogue manager, including components for state monitoring and action selection, and the user simulator used for testing, training and evaluation. We then present preliminary experiments in \cref{sec:sim-exp}, demonstrating the potential of our method for cross-domain transfer. We conclude the paper in \cref{sec:concl}.
\section{Multi\hyp dimensional\ Dialogue Modelling}\label{sec:md-dial}
In Bunt's account of multi-dimensionality in dialogue, utterances are represented as combinations of dialogue acts from a multi\hyp dimensional\ dialogue act taxonomy, thus accounting for their multifunctional nature \cite{Bunt:2011et}. This taxonomy, which is part of the ISO standard for dialogue act annotation \cite{ISO-SemAnnot}, includes the following 9 core dimensions: \textsl{Task/Activity}, \textsl{Auto-}, and \textsl{AlloFeedback}, \textsl{Turn-}, and \textsl{TimeManagement}, \textsl{Partner-} and \textsl{Own Processing Management}, \textsl{Discourse Structuring}, and \textsl{Social Obligations Management}. In producing utterances, dialogue partners select one or more dialogue acts, at most one from each dimension. The second system utterance in the example of \cref{sec:intro} is the result of the system selecting an answer act in the Task dimension and an inform act in the AutoFeedback dimension, which are then combined and realised as a single multi-functional utterance. However, some combinations of dialogue acts can only be realised sequentially in a natural language, such as the greeting and the question in the user utterance of the example in \cref{sec:intro}, even though these acts were selected simultaneously by the agent.
A key feature of the dialogue act taxonomy we aim to exploit, is that all dimensions except Task are domain-independent. A dialogue agent uses the same dialogue acts for managing the turn-taking process, or for following social conventions such as greeting and thanking, regardless of the underlying task or activity. Moreover, we believe that to some extent, the strategies for selecting these domain-independent dialogue acts can be largely transferred across tasks/activities. When changing from one task to the other, a dialogue participant does not need to learn from scratch how to achieve mutual understanding through feedback dialogue acts; they merely need to adapt their strategy to the new circumstances. Turn management for example, will depend on the communicative settings of the dialogue, i.e., whether the dialogue is a telephone conversation (speech only) or face-to-face (speech and gestures). In safety critical domains, giving and eliciting feedback will be more explicit. In more informal settings, or domains where for example empathy is important, the strategy for handling social conventions will be more elaborate, or at least different.
Our dialogue system follows the ISO standard in the same way as Keizer and Bunt~\shortcite{Keizer:2006vn,Keizer:2007ve}, featuring multiple dialogue act agents, each dedicated to selecting dialogue acts from one of the dimensions, and a process of evaluating combinations of dialogue act candidates. However, in our framework this is incorporated into a statistical dialogue manager, where the action selection policies are jointly optimised using multi-agent reinforcement learning. This combination of multi-dimensional modelling and machine learning opens up the opportunity to use transfer learning methods for more efficient cross-domain adaptation of dialogue systems.
\section{Multi\hyp dimensional\ POMDP-based dialogue management}\label{sec:md-pomdp-dial}
Recent advances in statistical dialogue systems have investigated Reinforcement Learning (RL) to optimise dialogue policies \cite{rl:springer11,young-etal-ieee-review}. The underlying problem is modelled as a Partially Observable Markov Decision Process (POMDP) to account for uncertainty introduced by automatic speech recognition and spoken language understanding (ASR \& SLU). A conventional POMDP-based spoken dialogue system typically consists of a pipeline of components for speech recognition and understanding (ASR \& SLU), dialogue management (DM), and natural language generation and speech synthesis (NLG \& TTS), see \cref{fig:sys_diag}, where the DM consists of \emph{belief monitoring} (updating the \emph{belief state} $b(s)$, i.e., a distribution over dialogue state hypotheses, based on an N-best list of user act hypotheses $\tilde{a}_u^i$), and \emph{action selection} (deciding which system act $a_m$ to generate, given the current belief state). By combining probabilistic belief monitoring with reinforcement learning of dialogue policies, these systems have been demonstrated to be more robust to speech processing errors and more scalable to larger application domains.
\begin{figure}[htb]
\centering
\includegraphics[width=.95\columnwidth]{system_diagram}
\caption{Typical dialogue system architecture, contrasting a conventional statistical dialogue manager with a multi\hyp dimensional\ version.}
\label{fig:sys_diag}
\end{figure}
A major limitation of data-driven approaches to spoken dialogue systems is their reliance on substantial amounts of (annotated) data in the target domain. As the number of application domains is growing every day, accelerated by the emergence of the Internet of Things (IoT) in particular, new methods for cost-effective development of conversational interfaces for these domains are needed. More recently, researchers have started to address this issue by looking at \emph{transfer learning} techniques \cite{Taylor:2009ur,Pan:2010dm,Lazaric:2012}, with the aim to speed up learning dialogue models and policies for a target domain by leveraging data and/or knowledge from a source domain. Recent domain adaptation work, however, has primarily focused on identifying and exploiting similarities between domain ontologies in slot-filling task domains. Ga\v{s}i\'{c} et al \shortcite{gasic-EtAl:2013:SIGDIAL} used Gaussian Process Reinforcement Learning (GPRL) to adapt a dialogue policy to a new slot being added to the domain. Since their approach relies on correlations between belief states rather than the belief states themselves, such adaptation is feasible, as long as the correlations are sufficiently similar between the domains. Using the GPRL framework extended with a Bayesian committee machine, they have also demonstrated successful transfer in a multi-domain setting, where the domains have different, but overlapping sets of slots \cite{gasic-EtAl:2015:ASRU}. In similar multi-domain settings, transfer learning methods have been developed for state tracking \cite{Mrksic_ea-2015} and natural language generation \cite{Wen_ea-2016}.
Rather than focusing on the domain ontology and the task, our proposed multi-dimensional framework distinguishes domain-independent dimensions such as social obligations management and time management, which can be transferred directly between domains. These transferable skills are trained jointly in one domain, and can be re-used and adapted in a new domain. Task/domain oriented approaches as used in the domain extension and multi-domain settings discussed above, might be used within our framework as well. In that case, we not only transfer domain-independent policies, but also the domain-specific policy associated with the task dimension.
Instead of selecting one dialogue act $a_m$ out of a single set of possible acts in each turn (the `conventional' setting A in \cref{fig:sys_diag}), our proposed DM selects responses that consist of combinations of dialogue acts $a_m^i$ (the `multidimensional' setting B in \cref{fig:sys_diag}). The multi-dimensional POMDP model can also be represented with the graphical model shown in \cref{fig:sds_pomdp_fa}, which incorporates multiple action nodes, each associated with actions in one dimension, and affecting different sets of state variables. A naive alternative to this factorisation of the action space would be to collapse the dimensions of dialogue acts back into a single set of actions, and then follow the conventional approach. However, under this architecture the state-action space would grow exponentially and therefore unlikely to tractably accommodate the proposed richness of interaction.
\begin{figure}[htb]
\centering
\includegraphics[width=.9\columnwidth]{sds_pomdp_fa}
\caption{Graphical model of a {POMDP} based SDS with factored action space.}
\label{fig:sds_pomdp_fa}
\end{figure}
\section{System implementation}\label{sec:sds-impl}
We have created a generic statistical dialogue manager for slot-filling domains, adopting many design features of the POMDP systems described in \cite{Young:2010vy,Thomson:2010dg}). The dialogue manager consists of a probabilistic state monitoring model and an MDP-based action selection model. To test, train and evaluate the dialogue manager, we have built an agenda-based user simulator based on \cite{Schatzmann:2007uc} and a basic error model, based on \cite{Thomson:2012wn}. The user simulator generates dialogue acts in response to the dialogue manager, following a randomly selected user goal. The error model then generates from this `true' user act an n-best list of user act hypotheses with confidence scores, to be passed to the dialogue manager. Our simulated experiments have been carried out for the restaurant information domain, containing 4 `informable' slots ({\tt foodtype, pricerange, area, near}), 5 `requestable' slots ({\tt name, phonenumber, address, price, postcode}), and a database of 149 restaurants in Cambridge (UK).
\subsection{State Monitoring}
The dialogue state representation follows directly from the domain ontology and consists of user goal belief states for each of the informable slots (multinomial distributions over the slot values), beliefs about whether a requestable slot is indeed requested by the user (Bernoulli distributions), and other relevant information such as the dialogue history (previous dialogue acts), a list of database entities matching the user goal top hypothesis, and the database entity under discussion (if any).
The user goal beliefs $b(s,v)$ are updated as follows:
\begin{equation}\small
b'(s,v) = \left\{
\begin{array}{ll}
c(s,v) & \mbox{\small\; if evidence seen} \\
& \mbox{\;\; for the first time} \\[2mm]
c(s,v) \cdot b(s,v) & \mbox{\; otherwise}
\end{array}
\right.
\end{equation}
where $(s,v)$ is a slot-value pair, $c(s,v)$ is a confidence score on evidence about a slot-value pair in the input n-best list of user act hypotheses. This relatively simple belief tracker supports accumulation of evidence for slot values across multiple turns, where the slots are treated as independent.
Orthogonal to belief tracking, we also track grounding states (such as {\tt user\_informed} and {\tt system\_confirmed}) of user goal item hypotheses, which are updated according to a finite state machine similar to the model used in \cite{Young:2010vy}, based originally on \cite{Traum:Phd1994}.
\subsection{Action Selection}
Based on the updated dialogue state, the dialogue manager selects response dialogue acts using one or more MDPs, each of which uses state features extracted from the full dialogue state and selects a summary action (e.g., `recommend a venue' or `ask slot preference') using a trainable policy, to be mapped back to a full dialogue act using information from the dialogue state (e.g., which venue to recommend or which slot to ask about). The MDPs are trained using Monte Carlo Control reinforcement learning with linear value function approximation. The reward signal is provided by the user simulator, assigning a score of -1 for each turn and a score of +30 when the user's goal is satisfied.
The MDPs consist of states $s \in S$, actions $a \in A$, and a policy $\pi:S \rightarrow A$ which maps states to actions. The policy is based on the state action value function $Q:S \times A \rightarrow \mathbb{R}$ which approximates the long term cumulative reward when taking action $a$ in state $s$ and following the policy onwards. During training we use $\epsilon$-greedy action selection. The Q-values are approximated by a linear function of the state features $\phi_i(s)$:
\begin{equation}
Q(s,a) = \sum_i \theta_{i,a} \cdot \phi_i(s)
\end{equation}
After each dialogue/episode, the weight vectors $\mathbf{\theta}_a$ for each action $a$ are updated using gradient descent, minimising the squared difference between the current value estimates $Q(s,a)$ and the cumulative discounted rewards $R_t=\sum_{k=t}^{T-1} \gamma^{k-t} \cdot r_k$ for each visited state-action pair $(s_t,a_t)$ in the episode ($t = 0,\dots,T-1$), where $r_k$ are the immediate rewards received after each visited state action pair, and $\gamma=0.95$ is the discount factor.
\section{Preliminary experiments in simulation}\label{sec:sim-exp}
As a first proof-of-concept experiment, we have created a one-dimensional and a multi-dimensional version of our dialogue manager, which generate dialogue acts from the same action set. Using the simulated user, we have carried out extensive policy optimisation experiments and compared the two systems.
\subsection{Experimental setup}
The two versions of the dialogue manager were created as follows. The one-dimensional version uses a single MDP model, using an action set of 7 possible summary actions. The multi-dimensional version uses three MDP models, corresponding to the dimensions \emph{Task} (5 actions, including asking for user preferences, making recommendations, presenting restaurant information), \emph{AutoFeedback} (3 actions, including asking clarification questions), and \emph{SocialOblMan} (2 actions, including goodbye acts). The selected summary actions are combined into single system dialogue acts in a rule-based manner \cite{Keizer:2007ve}, ensuring the same range of output dialogue acts as the one-dimensional version. For example, negative feedback acts cancel task acts, and goodbye acts are kept only if no candidate acts in the task dimension were selected (`null' actions). In this restricted setting, the multi-dimensional version is expected to be more challenging to train, given the larger action space: $5\!\times\!3\!\times\!2=30$ action combinations versus $7$ actions. \Cref{tab:mdp-action-stats} shows a description of the 7 actions in the one-dimensional system, the dimension of the resulting dialogue act, and the number of action combinations in the multi-dimensional system that map to this system act. Since all dialogue act candidates are cancelled in the presence of negative feedback, all $5 \times 2 = 10$ combinations of the negative feedback act with Task and SocialOblMan acts are mapped to a negative feedback output act (see action index 0 in \cref{tab:mdp-action-stats}). On the other hand, a returnGoodbye act is only allowed in combination with a `null' act from the task agent and if no negative feedback act is generated, leaving only 2 combinations (see action index 5 in \cref{tab:mdp-action-stats}).
\begin{table*}[tb]
\centering
\setlength{\tabcolsep}{3pt}
\begin{tabular}{ l c c }
\toprule
{\bf Action index \& description} & {\centering\bf Dimension} & \parbox[c]{23mm}{\centering\bf \# Action combinations} \tabularnewline
\midrule
0 -- negative feedback (``could you repeat that please?'') & AutoFeedback & 10 \tabularnewline
1 -- propositional question feedback (``did you say/mean \dots?'') & AutoFeedback & 1 \tabularnewline
2 -- answer to setQuestion w.r.t. task (``The address is \dots'') & Task & 4 \tabularnewline
3 -- answer to propQuestion w.r.t. task (``Yes'', or ``No, it serves \dots'') & Task & 4 \tabularnewline
4 -- venue recommendation (``\dots is a nice place in the city centre'') & Task & 4 \tabularnewline
5 -- returnGoodbye act (``goodbye!'') --- closes the dialogue & SocialOblMan & 2 \tabularnewline
6 -- setQuestion w.r.t task (``What kind of food do you like?'') & Task & 4 \tabularnewline
\bottomrule
\end{tabular}
\caption{Specification and quantitative comparison between one- and multi-dimensional MDP action sets.}\label{tab:mdp-action-stats}
\end{table*}
\subsection{Policy optimisation}
In all our policy optimisation experiments, 10 independent training runs have been carried out, and the evaluation results are averages over the 10 corresponding policy evaluations. The one-dimensional system was trained over 40k dialogues with an exploration rate linearly decaying from $\epsilon=0.4$ to $\epsilon=0$ and a fixed learning rate of $\alpha=0.001$. The multi-dimensional system was trained using the same settings, but now running the three MDP models simultaneously and updating their policies based on the same reward function. This training process involves implicit coordination between the policies, within the restrictions of the combination rules. For example, the task policy learns to stop making recommendations when the user is satisfied and says goodbye, whereas the social policy learns to respond to the user saying goodbye act and thus end the dialogue, but not before the task is completed.
The learning curves in \cref{fig:curves} show the performance of trained policies at different training stages, where each data point represents the average reward over 3000 evaluation dialogues. As expected, the one-dimensional system (purple, with triangular markers) achieves higher rewards than the multi-dimensional system (red, with square markers), in particular in the early stages of training. However, after around 25k training dialogues, they have converged to similar performance levels (average reward 17--18; average dialogue length 11; average success rate 94--97\%).
\begin{figure}[htb]
\centering
\includegraphics[width=.45\textwidth]{learn_curves_avgRew}
\caption{Policy evaluation results of the one- and multi-dimensional systems in terms of average success rate at different training stages (20\% error rate was used throughout).}
\label{fig:curves}
\end{figure}
After jointly optimising the three MDP policies, two domain-independent policies have been obtained that have the potential to be re-used in a new domain. To demonstrate this potential in a first preliminary test without actually creating a new domain, we re-trained the dialogue manager in the same domain by retaining the trained auto-feedback and social obligations management policies (as if they were trained in a different source domain) and training the task policy from scratch (for the `new' target domain). This domain transfer exercise was carried out in two settings: 1)~\textsl{multi-dim transfer}: only updating the task policy, i.e., keeping the trained domain-independent policies fixed, and 2)~\textsl{multi-dim transfer+adapt}: updating all three polices during training, i.e., adapting the trained domain-independent policies to the `new' domain. The effectiveness of domain transfer is demonstrated by the corresponding learning curves in \cref{fig:curves}, which show improved performance levels at the earlier stages of training in comparison to the non-transferred multi-dimensional system. Setting 1 (blue, with circular markers) shows clear and consistent improvement, whereas the improvement in setting 2 (green, with diamond markers) is more modest and training seems less stable (see the dips in performance at the 5k and 15k stages). At the very early training stages, we even see improvements in comparison to the one-dimensional system.
\subsection{Discussion}\label{sec:disc}
Although the results of our initial experiments are encouraging, the next step of course is to extend our multi-dimensional system by refining the MDP models and allowing for system responses containing multiple dialogue acts. For example, combinations of task and auto-feedback acts will be considered, as in the example dialogue exchange in \cref{sec:intro}, for which the auto-feedback MDP model will be extended to include decisions about which user-provided information the agent should give feedback about. Our hypothesis is that a one-dimensional solution for generating such dialogue act combinations is less scalable.
Although some coordination between the dimensions will still be required, the MDP agents are also planned to be more independent. To accommodate this, the reward function will also be decomposed into dimension-specific components. For example, the SocialOblMan MDP agent can be extended with actions for apologies and responses to thanking acts, and trained based on a combination of an overall reward signal shared between all agents and a reward signal related specifically to social conventions and which the other agents will not receive or use.
In the above transfer experiment, adaptation of the domain-general policies was not necessary, since the target domain was identical. For new domains, however, adaptation will be needed, for example safety critical domains where more explicit feedback is required, or informal domains where social interaction is more appropriate.
In the one-dimensional version, the system outputs are restricted to single dialogue acts by definition of the action space. Each MDP action leads to a single dialogue act and the actions are mutually exclusive by definition. In the multi-dimensional version, this restriction is still in place through the current combination rules, but we now have a more flexible mechanism in which such restrictions can be lifted. For encoding logical conflicts between dialogue act candidates from different dimensions, for example answers in the task dimension and negative feedback \cite{Keizer:2007ve}, some combination rules can be retained. For dealing with strategic and stylistic issues when evaluating dialogue act candidates, an additional MDP agent could be introduced and optimised jointly with the dimension-specific MDP agents.
To support the coordination process during training and thus make training more stable and efficient, each of the MDP models could be extended with information about the actions selected by the other MDPs. The more dependent the dimensions turn out to be, the more explicit coordination might be required for learning, which in the most extreme case would lead to a model that is equivalent to the one-dimensional model. However, the design of the used dialogue act taxonomy is such that a high level of independence between the MDPs can be expected, and therefore only modest explicit coordination might be needed.
\section{Conclusion and Future Work}\label{sec:concl}
We have argued for a multi\hyp dimensional\ approach to spoken dialogue system development, in order to enable more efficient cross-domain adaptation. As a proof-of-concept, we have presented a first implementation of our multi-dimensional statistical dialogue manager and illustrated our approach with initial experiments in simulation, demonstrating the feasibility of training transferable conversational skills using multi-agent reinforcement learning and using these to speed up training in a new domain.
In future work, we will extend our dialogue manager and user simulator to support a wider range of dialogue act combinations, without the restrictions used in the initial experiments. This will require further investigation into training settings for the multi-agent reinforcement learning framework, including dimension-specific reward functions and explicit coordination between the agents. As we expand the action sets of the MDP agents, their state spaces will also need to be expanded, and value function approximation for policy optimisation will need to be upgraded from linear models to for example deep neural networks \cite{Mrksic_ea-2015,Zhao:2016us}. We are also building an end-to-end system for the restaurant and smart home domains, in order to demonstrate our results on real data and across domains.
\paragraph*{Acknowledgments}
The MaDrIgAL project is funded by the EPSRC (EP/N017536/1).
\bibliographystyle{eacl2017}
|
1,116,691,498,459 | arxiv | \section{Introduction}
Let $H$ be a real or complex Hilbert space, $\mathcal{B}(H)$ the space of bounded operators on $H$, and $\mathfrak{A}$ a linear subspace of $\mathcal{B}(H)$. For each $x\in H$ write
\[\mathfrak{A}x\equiv \left\{ Ax:A\in \mathfrak{A}\right\} , \]
and, \emph{if it exists}, denote the projection of $H$ onto the closure $\overline{\mathfrak{A}x}$ of $\mathfrak{A}x$ by $\left[ \mathfrak{A}x\right]$. Projections of this type play a very big part in the classical theory of operator algebras, in which context $\mathfrak{A}$ is normally a subalgebra of $\mathcal{B}(H)$; see, for example, \cite{Dixmier,KR,Sakai,Topping}. However, in the constructive\footnote{Our \emph{constructive setting} is that of Bishop \cite{Bishop,BB,BV}, in which the mathematics is developed with intuitionistic, not classical, logic, in a suitable set- or type-theoretic framework \cite{Aczel,ML} and with dependent choice permitted.} setting---the one of this paper---we cannot even guarantee that $\left[ \mathfrak{A}x\right] $ exists. Our aim is to give sufficient conditions on $\mathfrak{A}$ and $x$ under which $\left[\mathfrak{A}x\right] $ exists, or, equivalently, the set $\mathfrak{A}x$ is located, in the sense that
\[\rho \left( v,\mathfrak{A}x\right) \equiv \inf \left\{ \left\Vert v-Ax\right\Vert :A\in \mathfrak{A}\right\} \]
exists for each $v\in H$.
We require some background on operator topologies. Specifically, in addition to the standard uniform topology on $\mathcal{B}(H)$, we need
\begin{itemize}
\item[$\vartriangleright $] the \emph{\textbf{strong operator topology:}} the weakest topology on $\mathcal{B}(H)$ with respect to which the mapping $T\rightsquigarrow Tx$ is continuous for all $x\in H$;
\item[$\vartriangleright $] the \textbf{\emph{weak operator topology:}} the weakest topology on $\mathcal{B}(H)$ with respect to which the mapping $T\rightsquigarrow \left\langle Tx,y\right\rangle $ is continuous for all $x,y\in H$.
\end{itemize}
These topologies are induced, respectively, by the seminorms of the form $T\rightsquigarrow \left\Vert Tx\right\Vert $ with $x\in H$, and $T\rightsquigarrow \left\vert \left\langle Tx,y\right\rangle \right\vert$ with $x,y\in H$. The unit ball\footnote{Note that it is not constructively provable that every element $T$ of $\mathcal{B}(H)$ is normed, in the sense that the usual operator norm of $T$ exists. Nevertheless, when we write `$\left\Vert T\right\Vert \leqslant 1$', we are using a shorthand for `$\left\Vert Tx\right\Vert \leqslant \left\Vert x\right\Vert$ for each $x\in H$'. Likewise, `$\left\Vert T\right\Vert <1$' means that there exists $c<1$ such that $\left\Vert Tx\right\Vert \leqslant c\left\Vert x\right\Vert $ for each $x\in H$; and `$\left\Vert T\right\Vert>1$' means that there exists $x\in H$ such that $\left\Vert Tx\right\Vert
>\left\Vert x\right\Vert $.}
\[\mathcal{B}_{1}(H)\equiv \left\{ T\in \mathcal{B}(H):\left\Vert T\right\Vert\leqslant 1\right\}\]
of $\mathcal{B}(H)$ is classically weak-operator compact, but constructively the most we can say is that it is weak-operator totally bounded (see \cite{BVwo}). The evidence so far suggests that in order to make progress when dealing constructively with a subspace or subalgebra $\mathfrak{A}$ of $ \mathcal{B}(H)$, it makes sense to add the weak-operator total boundedness of
\[\mathfrak{A}_{1}\equiv \mathfrak{A}\cap \mathcal{B}_{1}(H)\]
to whatever other hypothesis we are making; in particular, it is known that $\mathfrak{A}_{1}$ is located in the strong operator topology---and hence $\mathfrak{A}_{1}x$ is located for each $x\in H$---if and only if it is weak-operator totally bounded \cite{BHV,Spitters}.
Recall that the \emph{\textbf{metric complement}} of a subset $S$ of a metric space $X$ is the set $-S$ of those elements of $X$ that are bounded away from $X$. When $Y$ is a subspace of $X$, $y\in Y$, and $S\subset Y$, we define
\[\rho _{Y}\left( y,-S\right) \equiv \inf \left\{ \rho \left( y,z\right) :z\in Y\cap -S\right\}\]
if that infimum exists.
We now state our main result.
\begin{thm}\label{2202a}
Let $\mathfrak{A}$ be a uniformly closed subspace of $\mathcal{B}(H)$ such that $\mathfrak{A}_{1}$ is weak-operator totally bounded, and let $x$ be a point of $H$ such that $\mathfrak{A}x$ is closed and\thinspace $\rho _{\mathfrak{A}x}\left( 0,-\mathfrak{A}_{1}x\right) $ exists. Then the projection $\left[ \mathfrak{A}x\right] $ exists.
\end{thm}
Before proving this theorem, we discuss, in Section \ref{sec:2}, some general results about the locatedness of sets like $\mathfrak{A}x$, and we derive, in Section 3, a generalisation of the open mapping theorem that leads to the proof of Theorem \ref{2202a}. Finally, we show, by means of a Brouwerian example, that the existence of $\rho _{\mathfrak{A}x}\left( 0,-\mathfrak{A}_{1}x\right) $ cannot be dropped from the hypotheses of our main theorem.
\section{Some general locatedness results for $\mathfrak{A}x$}\label{sec:2}
We now prove an elementary, but helpful, result on locatedness in a Hilbert space.
\begin{prop}\label{2202b}
Let $\left( S_{n}\right) _{n\geqslant 1}$ be a sequence of located, convex subsets of a Hilbert space $H$ such that $S_{1}\subset S_{2}\subset \cdots $ , let $S_{\infty }=\mathop{\displaystyle \bigcup }\limits_{n\geqslant 1}S_{n}$, and let $x\in H$. For each $n$, let $x_{n}\in S_{n}$ satisfy $\left\Vert x-x_{n}\right\Vert <\rho \left( x,S_{n}\right) +2^{-n}$. Then
\begin{equation}
\rho \left( x,S_{\infty }\right) =\inf_{n\geqslant 1}\rho(x,S_{n})=\lim_{n\rightarrow \infty }\rho \left( x,S_{n}\right),\label{6}
\end{equation}
in the sense that if any of these three numbers exists, then all three do and they are equal. Moreover, $\rho \left( x,S_{\infty }\right) $ exists if and only if $\left( x_{n}\right) _{n\geqslant 1}$ converges to a limit $x_{\infty }\in H$; in that case, $\rho \left( x,S_{\infty }\right)=\left\Vert x-x_{\infty }\right\Vert $, and $\left\Vert x-y\right\Vert>\left\Vert x-x_{\infty }\right\Vert $ for all $y\in S_{\infty }$ with $y\neq x_{\infty }$.
\end{prop}
\begin{proof}
Suppose that $\rho \left( x,S_{\infty }\right) $ exists. Then $\rho \left(x,S_{\infty }\right) \leqslant \rho \left( x,S_{n}\right) $ for each $n$. On the other hand, given $\varepsilon >0$ we can find $z\in S_{\infty }$ such that $\left\Vert x-z\right\Vert <\rho \left( x,S_{\infty }\right)+\varepsilon $. Pick $N$ such that $z\in S_{N}$. Then for all $n\geqslant N$,
\[\rho \left( x,S_{\infty }\right) \leqslant \rho \left( x,S_{n}\right)\leqslant \rho \left( x,S_{N}\right) \leqslant \left\Vert x-z\right\Vert<\rho \left( x,S_{\infty }\right) +\varepsilon.\]
The desired conclusion (\ref{6}) now follows.
Next, observe that (by the parallelogram law in $H$) if $m\geqslant n$, then
\begin{eqnarray*}
\left\Vert x_{m}-x_{n}\right\Vert ^{2} &\leqslant &\left\Vert \left(x-x_{m}\right) -\left( x-x_{n}\right) \right\Vert ^{2} \\
&=&2\left\Vert x-x_{m}\right\Vert ^{2}+2\left\Vert x-x_{n}\right\Vert^{2}-4\left\Vert x-\frac{1}{2}\left( x_{m}+x_{n}\right) \right\Vert ^{2} \\
&\leqslant &2\left( \rho \left( x,S_{m}\right) +2^{-m}\right) ^{2}+2\left(\rho \left( x,S_{n}\right) +2^{-n}\right) ^{2}-4\rho \left( x,S_{m}\right)^{2},
\end{eqnarray*}
since $\frac{1}{2}\left( x_{m}+x_{n}\right) \in S_{m}$. Thus
\begin{eqnarray}
\left\Vert x_{m}-x_{n}\right\Vert ^{2} &\leqslant &2\left( \left( \rho\left( x,S_{m}\right) +2^{-m}\right) ^{2}-\rho \left( x,S_{m}\right)^{2}\right) \nonumber \\
&&+2\left( \left( \rho \left( x,S_{n}\right) +2^{-n}\right) ^{2}-\rho \left(x,S_{m}\right) ^{2}\right) . \label{5}
\end{eqnarray}
If $\rho (x,S_{\infty })$ exists, then, by the first part of the proof, $\rho \left( x,S_{n}\right) \rightarrow \rho \left( x,S_{\infty }\right) $ as $n\rightarrow \infty $. It follows from this and (\ref{5}) that $\left\Vert x_{m}-x_{n}\right\Vert ^{2}\rightarrow 0$ as $m,n\rightarrow \infty $; whence $\left( x_{n}\right) _{n\geqslant 1}$ is a Cauchy sequence in $H$ and therefore converges to a limit $x_{\infty }\in \overline{S_{\infty }}$. Then
\begin{eqnarray*}
\rho \left( x,S_{\infty }\right) &=&\rho \left( x,\overline{S_{\infty }}\right) \leqslant \left\Vert x-x_{\infty }\right\Vert \\
&=&\lim_{n\rightarrow \infty }\left\Vert x-x_{n}\right\Vert \\
&\leqslant &\lim_{n\rightarrow \infty }\left( \rho \left( x,S_{n}\right)+2^{-n}\right) =\rho \left( x,S_{\infty }\right).
\end{eqnarray*}
Thus $\rho \left( x,S_{\infty }\right) =\left\Vert x-x_{\infty }\right\Vert$.
Conversely, suppose that $x_{\infty }=\lim_{n\rightarrow \infty }x_{n}$ exists. Let $0<\alpha <\beta $ and $\varepsilon =\frac{1}{3}\left( \beta-\alpha \right) $. Pick $N$ such that $2^{-N}<\varepsilon $ and $\left\Vert x_{\infty }-x_{n}\right\Vert <\varepsilon $ for all $n\geqslant N$. Either $\left\Vert x-x_{\infty }\right\Vert >\alpha +2\varepsilon $ or $\left\Vert x-x_{\infty }\right\Vert <\beta $. In the first case, for all $n\geqslant N$,
\begin{eqnarray*}
\rho \left( x,S_{n}\right) &>&\left\Vert x-x_{n}\right\Vert -2^{-n} \\
&\geqslant &\left\Vert x-x_{\infty }\right\Vert -\left\Vert x_{\infty}-x_{n}\right\Vert -\varepsilon \\
&>&\left( \alpha +2\varepsilon \right) -\varepsilon -\varepsilon =\alpha .
\end{eqnarray*}
In the other case, there exists $\nu >N$ such that $\left\Vert x-x_{\nu}\right\Vert <\beta $; we then have
\[\rho \left( x,S_{\nu }\right) \leqslant \left\Vert x-x_{\nu }\right\Vert<\beta.\]
It follows from this and the constructive least-upper-bound principle (\cite{BV}, Theorem 2.1.18) that
\[\inf \left\{ \rho \left( x,S_{n}\right) :n\geqslant 1\right\}\]
exists; whence, by (\ref{6}), $d\equiv \rho \left( x,S_{\infty }\right)$ exists.
Finally, suppose that $x_{\infty }$ exists, and consider any $y\in S_{\infty}$ with $y\neq x_{\infty }$. We have
\begin{eqnarray*}
0 &<&\left\Vert y-x_{\infty }\right\Vert ^{2}=\left\Vert y-x-\left(x_{\infty }-x\right) \right\Vert ^{2} \\
&=&2\left\Vert y-x\right\Vert ^{2}+2\left\Vert x_{\infty }-x\right\Vert^{2}-4\left\Vert \frac{y+x_{\infty }}{2}-x\right\Vert ^{2} \\
&=&2\left( \left\Vert y-x\right\Vert ^{2}-d^{2}\right) +2\left( \left\Vert x_{\infty }-x\right\Vert ^{2}-d^{2}\right) =2\left( \left\Vert y-x\right\Vert ^{2}-d^{2}\right),
\end{eqnarray*}
so $\left\Vert x-y\right\Vert >d$.
\end{proof}
For each positive integer $n$ we write
\[\mathfrak{A}_{n}\equiv n\mathfrak{A}_{1}=\left\{ nA:A\in \mathfrak{A}_{1}\right\}.\]
If $\mathfrak{A}_{1}$ is weak-operator totally bounded and hence strong-operator located, then $\mathfrak{A}_{n}$ has those two properties as well.
Our interest in Proposition \ref{2202b} stems from this:
\begin{cor}\label{2202c}
Let $\mathfrak{A}$ be a linear subspace of $\mathcal{B}(H)$ with $\mathfrak{A}_{1}$ weak-operator totally bounded, and let $x,y\in H$. For each $n$, let $y_{n}\in \mathfrak{A}_{n}$ satisfy $\left\Vert y-y_{n}\right\Vert <\rho \left( x,\mathfrak{A}_{n}x\right) +2^{-n}$. Then
\[\rho \left( y,\mathfrak{A}x\right) =\inf_{n\geqslant 1}\rho (y,\mathfrak{A}_{n}x)=\lim_{n\rightarrow \infty }\rho \left( y,\mathfrak{A}_{n}x\right).\]
Moreover, $\rho \left( y,\mathfrak{A}x\right) $ exists if and only if $\left( y_{n}\right) _{n\geqslant 1}$ converges to a limit $y_{\infty }\in H$; in which case, $\rho \left( y,\mathfrak{A}x\right) =\left\Vert y-y_{\infty}\right\Vert $, and $\left\Vert y-Ax\right\Vert >\left\Vert y-y_{\infty}\right\Vert $ for each $A\in \mathfrak{A}$ such that $Ax\neq y_{\infty}$.
\end{cor}
One case of this corollary arises when the sequence $\left( \rho \left( y,\mathfrak{A}_{n}x\right) \right) _{n\geqslant 1}$ stabilises:
\begin{prop}\label{2202d}
Let $\mathfrak{A}$ be a linear subspace of $\mathcal{B}(H)$ such that $\mathfrak{A}_{1}$ is weak-operator totally bounded. Let $x,y\in H$, and suppose that for some positive integer $N$, $\rho \left( y,\mathfrak{A}_{N}x\right) =\rho \left( y,\mathfrak{A}_{N+1}x\right) $. Then $\rho \left(y,\mathfrak{A}x\right) $ exists and equals $\rho \left( y,\mathfrak{A}_{N}x\right) $.
\end{prop}
\begin{proof}
By Theorem 4.3.1 of \cite{BV}, there exists a unique $z\in \overline{\mathfrak{A}_{N}x}$ such that $\rho \left( y,\mathfrak{A}_{N}x\right)=\left\Vert y-z\right\Vert $. We prove that $y-z$ is orthogonal to $\mathfrak{A}x$. Let $A\in \mathfrak{A}$, and consider $\lambda \in \mathbf{C}$ so small that $\lambda A\in \mathfrak{A}_{1}$. Since,
\[z-\lambda Ax\in \overline{\mathfrak{A}_{N+1}x},\]
we have
\begin{eqnarray*}
\left\langle y-z-\lambda Ax,y-z-\lambda Ax\right\rangle &\geqslant &\rho\left( y,\mathfrak{A}_{N+1}x\right) ^{2} \\
&=&\rho \left( y,\mathfrak{A}_{N}x\right) ^{2}=\left\langle y-z,y-z\right\rangle .
\end{eqnarray*}
This yields
\[\left\vert \lambda \right\vert ^{2}\left\Vert Ax\right\Vert ^{2}+2\func{Re}\left( \lambda \left\langle y-z,Ax\right\rangle \right) \geqslant 0.\]
Suppose that $\func{Re}\left\langle y-z,Ax\right\rangle \neq 0$. Then by taking a sufficiently small real $\lambda$ with
\[\lambda \func{Re}\left\langle y-z,Ax\right\rangle <0,\]
we obtain a contradiction. Hence $\func{Re}\left\langle y-z,Ax\right\rangle =0$. Likewise, $\func{Im}\left\langle y-z,Ax\right\rangle =0$. Thus $\left\langle y-z,Ax\right\rangle =0$. Since $A\in \mathfrak{A}$ is arbitrary, we conclude that $y-z$ is orthogonal to $\mathfrak{A}x$ and hence to $\overline{\mathfrak{A}x}$. It is well known that this implies that $z$ is the unique closest point to $y$ in the closed linear subspace $\overline{\mathfrak{A}x}$. Since $\mathfrak{A}x$ is dense in $\overline{\mathfrak{A}x}$, it readily follows that $\rho \left( y,\mathfrak{A}x\right) =\rho \left( y,\overline{\mathfrak{A}x}\right) =\left\Vert y-z\right\Vert $.
\end{proof}
The final result in this section will be used in the proof of our main theorem.
\begin{prop}
\label{2301a2}Let $\mathfrak{A}$ be a linear subspace of $\mathcal{B}(H)$ with weak-operator totally bounded unit ball, and let $x\in H$. Suppose that there exists $r>0$ such that
\[\mathfrak{A}_{1}x\supset B_{\mathfrak{A}x}(0,r)\equiv \mathfrak{A}x\cap B(0,r).\]
Then $\mathfrak{A}x$ is located in $H$; in fact, for each $y\in H$, there exists a positive integer $N$ such that $\rho \left( y,\mathfrak{A}x\right)=\rho \left( y,\mathfrak{A}_{N}x\right) $.
\end{prop}
\begin{proof}
Fixing $y\in H$, compute a positive integer $N>2\left\Vert y\right\Vert /r$. Let $A\in \mathfrak{A}$, and suppose that
\[\left\Vert y-Ax\right\Vert <\rho \left( y,\mathfrak{A}_{N}x\right).\]
We have either $\left\Vert Ax\right\Vert <Nr$ or $\left\Vert Ax\right\Vert>2\left\Vert y\right\Vert $. In the first case, $N^{-1}Ax\in B_{\mathfrak{A}x}(0,r)$, so there exists $B\in \mathfrak{A}_{1}$ with $N^{-1}Ax=Bx$ and therefore $Ax=NBx$. But $NB\in \mathfrak{A}_{N}$, so
\[\left\Vert y-Ax\right\Vert =\left\Vert y-NBx\right\Vert \geqslant \rho\left( y,\mathfrak{A}_{N}x\right),\]
a contradiction. In the case $\left\Vert Ax\right\Vert \geqslant Nr>2\left\Vert y\right\Vert $, we have
\[\left\Vert y-Ax\right\Vert \geqslant \left\Vert Ax\right\Vert -\left\Vert y\right\Vert >\left\Vert y\right\Vert \geqslant \rho \left( y,\mathfrak{A}_{N}x\right),\]
another contradiction. We conclude that $\left\Vert y-Ax\right\Vert \geqslant \rho \left( y,\mathfrak{A}_{N}x\right) $ for each $A\in \mathfrak{A}$. On the other hand, given $\varepsilon >0$, we can find $A\in \mathfrak{A}_{N}$ such that $\left\Vert y-Ax\right\Vert <\rho \left( y,\mathfrak{A}_{N}x\right) +\varepsilon $. It now follows that $\rho \left( y,\mathfrak{A}x\right) $ exists and equals $\rho \left( y,\mathfrak{A}_{N}x\right) $.
\end{proof}
\section{Generalising the open mapping theorem}\label{sec:3}
The key to our main result on the existence of projections of the form $\left[ \mathfrak{A}x\right] $ is a generalisation of the open mapping theorem from functional analysis (\cite{BV}, Theorem 6.6.4). Before giving that generalisation, we note a proposition and a lemma.
\begin{prop}
\label{2802a0}If $C$ is a balanced, convex subset of a normed space $X$, then $V\equiv \mathop{\displaystyle \bigcup }\limits_{n\geqslant 1}nC$ is a linear subspace of $X$.
\end{prop}
\begin{proof}
Let $x\in V$ and $\alpha \in \mathbf{C}$. Pick a positive integer $n$ and an element $c$ of $C$ such that $x=nc$. If $\alpha \neq 0$, then since $C$ is balanced, $\left\vert \alpha \right\vert ^{-1}\alpha c\in C$, so
\[\alpha x=\alpha nc=\left\vert \alpha \right\vert n\left\vert \alpha\right\vert ^{-1}\alpha c\in \left\vert \alpha \right\vert nC\subset \left(1+\left\vert \alpha \right\vert \right) nC.\]
In the general case, we can apply what we have just proved to show that
\[\left( 1+\alpha \right) x\in \left( 1+\left\vert 1+\alpha \right\vert\right) nC\subset \left( 2+\left\vert \alpha \right\vert \right) nC\text{.}\]
Now, since $C$ is balanced,
\[-x=n\left( -c\right) \in nC\subset (2+\left\vert \alpha \right\vert )nC.\]
Hence, by the convexity of $(2+\left\vert \alpha \right\vert )nC$,
\[\alpha x=2\frac{(1+\alpha )x-x}{2}\in 2(2+\left\vert \alpha \right\vert )nC.\]
Taking $N$ as any integer $>2(2+\left\vert \alpha \right\vert )n$, we now see that $\alpha x\in NC\subset V$. In view of the foregoing and the fact that $\left( nC\right) _{n\geqslant 1}$ is an ascending sequence of sets, if $x^{\prime }$ also belongs to $V$ we can take $N$ large enough to ensure that $\alpha x$ and $x^{\prime }$ both belong to $NC$. Picking $c,c^{\prime}\in C$ such that $\alpha x=Nc$ and $x^{\prime }=Nc^{\prime }$, we obtain
\[\alpha x+x^{\prime }=2N\left( \frac{c+c^{\prime }}{2}\right) \in 2NC,\]
so $\alpha x+x^{\prime }\in V$.
\end{proof}
We call a bounded subset $C$ of a Banach space $X$ \textbf{\emph{superconvex}} if for each sequence $\left( x_{n}\right) _{n\geqslant 1}$ in $C$ and each sequence $\left( \lambda _{n}\right) _{n\geqslant 1}$ of nonnegative numbers such that $\sum_{n=1}^{\infty }\lambda _{n}$ converges to $1$ and the series $\sum_{n=1}^{\infty }\lambda _{n}x_{n}$ converges, we have $\sum_{n=1}^{\infty }\lambda _{n}x_{n}\in C$. In that case, $C$ is clearly convex.
\begin{lem}
\label{2802a2}Let $C$ be a located, bounded, balanced, and superconvex
subset of a Banach space $X$, such that $X=\mathop{\displaystyle \bigcup }\limits_{n\geqslant 1}nC$%
. Let $y\in X$ and $r>\left\Vert y\right\Vert $. Then there exists $\xi \in
2C$ such that if $y\neq \xi $, then $\rho \left( z,C\right) >0$ for some $z$
with $\left\Vert z\right\Vert <r$.
\end{lem}
\begin{proof}
Either $\rho \left( y,C\right) >0$ and we take $z=y$, or else, as we suppose, $\rho \left( y,C\right) <r/2$. Choosing $x_{1}\in 2C$ such that $\left\Vert y-\frac{1}{2}x_{1}\right\Vert <r/2$ and therefore $\left\Vert 2y-x_{1}\right\Vert <r$, set $\lambda _{1}=0$. Then either $\rho \left(2y-x_{1},C\right) >0$ or $\rho \left( 2y-x_{1},C\right) <r/2$. In the first case, set $\lambda _{k}=1$ and $x_{k}=0$ for all $k\geqslant 2$. In the second case, pick $x_{2}\in 2C$ such that $\left\Vert 2y-x_{1}-\frac{1}{2}x_{2}\right\Vert <r/2$ and therefore $\left\Vert 2^{2}y-2x_{1}-x_{2}\right\Vert <r$, and set $\lambda _{2}=0$. Carrying on in this way, we construct a sequence $\left( x_{n}\right) _{n\geqslant 1}$ in $2C$, and an increasing binary sequence $\left( \lambda _{n}\right)_{n\geqslant 1}$ with the following properties.
\begin{itemize}
\item If $\lambda _{n}=0$, then
\[\rho \left( 2^{n-1}y-\sum_{i=1}^{n}2^{n-i-1}x_{i},C\right) <\frac{r}{2}\]
and
\[\left\Vert 2^{n}y-\sum_{i=1}^{n}2^{n-i}x_{i}\right\Vert <r.\]
\item If $\lambda _{n}=1-\lambda _{n-1}$, then
\[\rho \left( 2^{n-1}y-\sum_{i=1}^{n}2^{n-i-1}x_{i},C\right) >0\]
and $x_{k}=0$ for all $k\geqslant n$.
\end{itemize}
Compute $\alpha >0$ such that $\left\Vert x\right\Vert <\alpha $ for all $x\in 2C$. Then the series $\sum_{i=1}^{\infty }2^{-i}x_{i}$ converges, by comparison with $\left\vert \alpha \right\vert \sum_{i=1}^{\infty }2^{-i}$, to a sum $\xi $ in the Banach space $X$. Since $\sum_{i=1}^{\infty }2^{-i}=1$ and $C$ is superconvex, we see that
\[\sum_{i=1}^{\infty }2^{-i}x_{i}=2\sum_{i=1}^{\infty }2^{-i}\left( \frac{1}{2}x_{i}\right) \in 2C.\]
If $y\neq \xi $, then there exists $N$ such that
\[\left\Vert y-\sum_{i=1}^{N}2^{-i}x_{i}\right\Vert >2^{-N}r\]
and therefore
\[\left\Vert 2^{N}y-\sum_{i=1}^{N}2^{N-i}x_{i}\right\Vert >r.\]
It follows that we cannot have $\lambda _{N}=0$, so $\lambda _{N}=1$ and therefore there exists $\nu \leqslant N$ such that $\lambda _{\nu}=1-\lambda _{\nu -1}$. Setting
\[z\equiv 2^{\nu -1}y-\sum_{i=1}^{\nu -1}2^{\nu -i-1}x_{i},\]
we see that $\rho (z,C)>0$ and $\left\Vert z\right\Vert <r$, as required.
\end{proof}
We now prove our generalisation of the open mapping theorem.
\begin{thm}
\label{2802a3}Let $X$ be a Banach space,and $C$ a located, bounded, balanced, and superconvex subset of $X$ such that $\rho \left( 0,-C\right)$ exists and $X=\mathop{\displaystyle \bigcup }\limits_{n\geqslant 1}nC$. Then there exists $r>0$ such that $B\left( 0,r\right) \subset C$.
\end{thm}
\begin{proof}
Consider the identity
\[X=\mathop{\displaystyle \bigcup }\limits_{n\geqslant 1}\overline{nC}.\]
By Theorem 6.6.1 of \cite{BV} (see also \cite{BHV1}), there exists $N$ such that the interior of $\overline{NC}$ is inhabited. Thus there exist $y_{0}\in NC$ and $R>0$ such that $B\left( y_{0},R\right) \subset \overline{NC}$. Writing $y_{1}=N^{-1}y_{0}$ and $r=\left( 2N\right) ^{-1}R$, we obtain $B\left( y_{1},2r\right) \subset \overline{C}.$It follows from Lemma 6.6.3 of \cite{BV} that $B\left( 0,2r\right) \subset \overline{C}$. Now consider any $y\in B\left( 0,2r\right) $. By Lemma \ref{2802a2}, there exists $\xi \in 2C$ such that if $y\neq \xi $, then there exists $z\in B(0,2r)$ with $\rho\left( z,C\right) >0$. Since $B\left( 0,2r\right) \subset \overline{C}$, this is absurd. Hence $y=\xi \in 2C$. It follows that $B\left( 0,2r\right)\subset 2C$ and hence that $B\left( 0,r\right) \subset C$.
\end{proof}
Note that in Lemma \ref{2802a2} and Theorem \ref{2802a3} we can replace the superconvexity of $C$ by these two properties: $C$ is convex, and for each sequence $\left( x_{n}\right) _{n\geqslant 1}$ in $C$, if $\sum_{n=1}^{\infty }2^{-n}x_{n}$ converges in $H$, then its sum belongs to $C$.
We now derive two corollaries of Theorem \ref{2802a3}.
\begin{cor}[\textbf{The open mapping theorem} (\cite{BV},
Theorem 6.6.4)\label{2802a4}\footnote{This is but one version of the open mapping
theorem; for another, see \cite{BI}.}]
Let $X,Y$ be Banach spaces, and $T$ a sequentially continuous linear mapping of $X$ onto $Y$ such that $T\left( \overline{B(0,1)}\right) $ is located and $\rho \left( 0,-T\left( \overline{B(0,1)}\right) \right) $ exists. Then there exists $r>0$ such that $B\left(0,r\right) \subset T\left( \overline{B\left( 0,1\right) }\right) $.
\end{cor}
\begin{proof}
In view of Theorem \ref{2802a3}, it will suffice to prove that $C\equiv T\left( \overline{B\left( 0,1\right) }\right)$ is superconvex. But if $\left( x_{n}\right) _{n\geqslant 1}$ is a sequence in $\overline{B\left(0,1\right) }$ and $\left( \lambda _{n}\right) _{n\geqslant 1}$ is a sequence of nonnegative numbers such that $\sum_{n=1}^{\infty }\lambda _{n}=1$, then $\left\Vert \lambda _{n}x_{n}\right\Vert \leqslant \lambda _{n}$ for each $n$, so $\sum_{n=1}^{\infty }\lambda _{n}x_{n}$ converges in $X$; moreover,
\[\left\Vert \sum_{n=1}^{\infty }\lambda _{n}x_{n}\right\Vert \leqslant\sum_{n=1}^{\infty }\lambda _{n}=1,\]
so, by the sequential continuity of $T$,
\[T\left( \sum_{n=1}^{\infty }\lambda _{n}x_{n}\right) \in C.\]
Thus $C$ is superconvex.
\end{proof}
Theorem \ref{2802a3} also leads to the \emph{\textbf{proof of Theorem \ref{2202a}:}}
\begin{proof}
Taking $C\equiv \mathfrak{A}_{1}x$, we know that $C$ is located (since $\mathfrak{A}_{1}$ is weak-operator totally bounded and hence, by \cite{BHV,Spitters}, strong-operator located), as well as bounded and balanced. To prove that $C$ is superconvex, consider a sequence$~\left( A_{n}\right)_{n\geqslant 1}$ in $\mathfrak{A}_{1}$, and a sequence $\left( \lambda_{n}\right) _{n\geqslant 1}$ of nonnegative numbers such that $\sum_{n=1}^{\infty }\lambda _{n}$ converges to $1$. For $k\geqslant j$ we have
\[\left\Vert \sum_{n=j}^{k}\lambda _{n}A_{n}\right\Vert \leqslant\sum_{n=j}^{k}\lambda _{n},\]
so $\sum_{n=1}^{\infty }\lambda _{n}A_{n}$ converges uniformly to an element $A$ of $\mathcal{B}_{1}(H)$. Since $\mathfrak{A}$ is uniformly closed, $A\in \mathfrak{A}_{1}$, so $\sum_{n=1}^{\infty }\lambda _{n}A_{n}x=Ax\in \mathfrak{A}_{1}x$. Thus $C$ is superconvex. We can now apply Theorem \ref{2802a3}, to produce $r>0$ such that $B_{\mathfrak{A}x}\left( 0,r\right)\subset C$. The locatedness of $\mathfrak{A}x$, and the consequent existence of the projection $\left[ \mathfrak{A}x\right] $, now follow from Proposition \ref{2301a2}.\label{ere}
\end{proof}
We now discuss further the requirement, in Theorem \ref{2202a}, that $\rho_{\mathfrak{A}x}\left( 0,-\mathfrak{A}_{1}x\right) $ exist, where $\mathfrak{A}_{1}$ is weak-operator totally bounded. We begin by giving conditions under which that requirement is satisfied.
If $\mathfrak{A}x$ has positive, finite dimension---in which case it is both closed and located in $H$---then $\mathfrak{A}x-\mathfrak{A}_{1}x$ is inhabited, so Proposition (1.5) of \cite{LNM873} can be applied to show that $\mathfrak{A}x-\mathfrak{A}_{1}x$ is located in $\mathfrak{A}x$. In particular, $\rho _{\mathfrak{A}x}\left( 0,-\mathfrak{A}_{1}x\right)$ exists. On the other hand, if $P$ is a projection in $\mathcal{B}(H)$ and
\[\mathfrak{A}\equiv \left\{ PTP:T\in \mathcal{B}(H)\right\},\]
then $\mathfrak{A}$ can be identified with $\mathcal{B}(P(H))$, so $\mathfrak{A}_{1}$ is weak-operator totally bounded. Moreover, if $x\neq 0$, then $\mathfrak{A}x=P(H)$ and so is both closed and located, $\mathfrak{A}_{1}x=\overline{B}(0,\left\Vert Px\right\Vert )\cap P(H)$, and $\rho _{\mathfrak{A}x}(0,-\mathfrak{A}_{1}x)=\left\Vert Px\right\Vert$.
We end with a Brouwerian example showing that we cannot drop the existence of $\rho_{\mathfrak{A}x}\left( 0,-\mathfrak{A}_{1}x\right)$ from the hypotheses of Theorem \ref{2202a}. Consider the case where $H=\mathbf{R}\times \mathbf{R}$, and let $\mathfrak{A}$ be the linear subspace (actually an algebra) of $\mathcal{B}(H)$ comprising all matrices of the form
\[T_{a,b}\equiv \left(\begin{array}{cc}a & 0 \\ 0 & b\end{array}\right)\]
with $a,b\in \mathbf{R}$. It is easy to show that $\mathfrak{A}$ is uniformly closed: if $\left( a_{n}\right) ,\left( b_{n}\right) $ are sequences in $\mathbf{R}$ such that $\left( T_{a_{n},b_{n}}\right)_{n\geqslant 1}$ converges uniformly to an element $T\equiv \left(\begin{array}{cc}a_{\infty } & p \\ q & b_{\infty }\end{array}\right)$, then
\[a_{n}=T_{a_{n},b_{n}}\left(\begin{array}{c}1 \\ 0\end{array}\right) \rightarrow T\left(\begin{array}{c}1 \\ 0\end{array}\right) =a_{\infty },\]
Likewise, $b_{n}\rightarrow b_{\infty }$, $p=0$, and $q=0$. Hence $T=T_{a_{\infty },b_{\infty }}\in \mathfrak{A}$.
Now, if $\left( x,y\right) $ is in the unit ball of $H$, then
\begin{eqnarray*}
\left\Vert T_{a,b}\left(\begin{array}{c}x \\ y\end{array}\right) \right\Vert ^{2} &=&\left\Vert \left(\begin{array}{c}ax \\ by\end{array}\right) \right\Vert ^{2}=a^{2}x^{2}+b^{2}y^{2} \\
&=&a^{2}\left( x^{2}+y^{2}\right) +\left( b^{2}-a^{2}\right) y^{2} \\
&=&a^{2}+\left( b^{2}-a^{2}\right) y^{2}\text{.}
\end{eqnarray*}
We see from this that if $a^{2}\geqslant b^{2}$, then $\left\Vert T_{a,b}\right\Vert ^{2}\leqslant a^{2}$; moreover, $T_{a,b}\left( 1,0\right)=a$, so $\left\Vert T_{a,b}\right\Vert ^{2}=a^{2}$. If $a^{2}<b^{2}$, then a similar argument shows that $\left\Vert T_{a,b}\right\Vert ^{2}=b^{2}$. It now follows that $\left\Vert T_{a,b}\right\Vert $ exists and equals $\max\left\{ \left\vert a\right\vert ,\left\vert b\right\vert \right\} $. Also, since, relative to the uniform topology on $\mathcal{B}(H)$, $\mathfrak{A}_{1}$ is homeomorphic to the totally bounded subset
\[\left\{ \left( a,b\right) :\max \left\{ \left\vert a\right\vert ,\left\vert b\right\vert \right\} \leqslant 1\right\}\]
of $\mathbf{R}^{2}$, it is uniformly, and hence weak-operator, totally bounded.
Consider the vector $\xi \equiv \left( 1,c\right) $, where $c\in \mathbf{R}$. If $c=0$, then $\mathfrak{A}\xi =\mathbf{R}\times \left\{ 0\right\} $, the projection of $H$ on $\mathfrak{A}\xi $ is just the projection on the $x$-axis, and $\rho \left( \left( 0,1\right) ,\mathfrak{A}\xi \right) =1$. If $c\neq 0$, then
\[\mathfrak{A\xi }=\left\{ \left( a,cb\right) :a,b\in \mathbf{R}\right\} =\mathbf{R}\times \mathbf{R},\]
the projection of $H$ on $\mathfrak{A\xi }$ is just the identity projection $I$, and $\rho \left( \left( 0,1\right) ,\mathfrak{A\xi }\right) =0$. Suppose, then, that the projection $P$ of $H$ on $\mathfrak{A\xi }$ exists. Then either $\rho \left( \left( 0,1\right) ,\mathfrak{A\xi }\right) >0$ or $\rho \left( \left( 0,1\right) ,\mathfrak{A\xi }\right) <1$. In the first case, $c=0$; in the second, $c\neq 0$. Thus if $\left[ \mathfrak{A}x\right]$ exists for each $x\in H$, then we can prove that
\[\forall _{x\in \mathbf{R}}\left( x=0\vee x\neq 0\right),\]
a statement constructively equivalent to the essentially nonconstructive omniscience principle \textbf{LPO}:
\begin{quote}
For each binary sequence $\left( a_{n}\right) _{n\geqslant 1}$, either $a_{n}=0$ for all $n$ or else there exists $n$ such that $a_{n}=1$.
\end{quote}
It follows from this and our Theorem \ref{2202a} that if $\rho _{\mathfrak{A}x}\left( 0,-\mathfrak{A}_{1}x\right)$ exists for each $x\in H$, then we can derive \textbf{LPO}.
\section*{Acknowledgement}
This research was partially done when the author was a visiting fellow at the Isaac Newton Institute for the Mathematical Sciences, in the programme \emph{Semantics \& Syntax: A Legacy of Alan Turing}. The author thanks the referees for helpful comments that improved the presentation of the paper.
|
1,116,691,498,460 | arxiv | \section{Introduction}
Lorentz-violating theories have been extensively studied and used as an
effective probe to test the limits of Lorentz covariance. Nowadays, these
theories are encompassed in the framework of the Extended Standard Model
(SME), \cite{Colladay},\ as a possible extension of the minimal Standard
Model of the fundamental interactions. Such kind of idea has driven much
attention mainly after some authors argued the possibility of \ Lorentz and
CPT spontaneous breaking in the context of string theory \cite{Samuel}. The
SME is the suitable framework to investigate properties of Lorentz violation
on physical systems involving photons \cite{photons1}, \cite{photons2},
radiative corrections \cite{Radiative}, fermions \cite{fermions}, neutrinos
\cite{neutrinos}, topological defects \cite{Defects}, topological phases
\cite{Phases}, cosmic rays \cite{CosmicRay}, supersymmetry \cite{Susy},
particle decays \cite{Iltan}, and other relevant aspects \cite{Lehnert1},
\cite{General}. The SME has also been used as a framework to propose Lorentz
symmetry violation \cite{Tests} and CPT\ \cite{CPT} probing experiments,
which have amounted to the imposition of stringent bounds on the
Lorentz-symmetry violating (LV) coefficients.
To take into account how this violation is implemented, in the fermion
sector of the SME, for example, there are two CPT-odd terms, $v_{\mu }%
\overline{\psi }\gamma ^{\mu }\psi ,b_{\mu }\overline{\psi }\gamma
_{5}\gamma ^{\mu }\psi $, where $v_{\mu },b_{\mu }$ are the LV backgrounds.
The modified Dirac theory has already been examined in literature \cite%
{Hamilton}, and its non-relativistic limit, with special attention to the
hydrogen spectrum \cite{Manojr} is realized. A similar study has also been
developed for the case of a non-minimal coupling with the background, with
new outcomes \cite{Nonmini}.\textbf{\ }Atomic and optical physics are other
areas in which Lorentz symmetry violation has been intensively studied.
Indeed, there are several works examining Lorentz violation in
electromagnetic cavities and optical systems \cite{Cavity}, \cite{Masers},
which contributed to establish upper bounds on the LV coefficients.
Works by Belinfante, Case, Fronsdal and Schwinger in the fifties \cite{fifth}%
, and Yang and Lee in the early sixties, \cite{Yang}, came to the general
result that charged truly-elementary particles, coupled to the
electromagnetic field, exhibit a gyromagnetic ratio given by the inverse of
its corresponding spin. For spin-half particles, described by the Dirac
field, this is a well-celebrated result. However, for higher spins, this
general result is not correct. Indeed, theoretical evidences, based on the
high-energy behavior of amplitudes and unitarity bounds \cite{Wein} and on
the dynamics of higher-spin particles propagating in electromagnetic
backgrounds as dictated by string theories, indicate that, for charged
genuinely elementary particles, the gyromagnetic ratio is always 2, no
matter what the spin of the particle is. For charged spin-1 vector bosons,
like the W-particles of the electroweak interactions, the value 2 is
reconciled by means of the Yang-Mills interactions that yield a non-minimal
(but renormalisable) coupling between the electromagnetic field-strength and
the potentials associated to the charged bosons \cite{ferrara}. Charged
spin-1 matter fields with self-interactions and topological terms have been
studied to provide a possible microscopic description for the origin of
Lorentz symmetry violation\cite{cpw}.
In our contribution, we reassess this issue in an environment dominated by a
background vector that parametrises a tiny violation of Lorentz symmetry. We
come to the conclusion that, also independently of the spin of the particle,
the Lorentz-symmetry violating background vector may yield the same
contribution to the magnetic moment and the Aharonov-Casher phase of the
particle, even if it is electrically neutral; whenever a particular
non-minimal coupling of the particle to the electromagnetic field and the
Lorentz-breaking vector is considered. Everything goes as if
Lorentz-symmetry violating background endows each elementary particle, even
those spinless and electrically neutral, with a universal contribution to
its magnetic moment and, consequently, to its Aharonov-Casher phase. In this
context, we come back to the interesting question that concerns the magnetic
properties of neutrinos\cite{jwv}. Incidentally, from Neutrino Physics, more
specifically, from the observation of non-zero neutrino masses, there
emerges a striking evidence in favour of a Physics Beyond the Standard Model%
\cite{troden}. More recently, theoretical bounds for the neutrino mass and
magnetic moment have been calculated that could be tested in the new
experiments \cite{cal}. In our considerations, we propose that magnetic
moment contributions to neutral and Majorana fermions can be obtained
already at the tree-level approximation by means of non-minimal couplings.
Besides the cases of the charged and neutral massive spin-1 particles and
the Majorana fermions themselves, another contribution we shall present in
this work refers to the way the $\left( k_{F}\right) _{\mu \nu \kappa
\lambda }$-parameter \cite{Colladay} may contribute to the magnetic dipole
moment of neutral vector bosons. This result shall be used to give us a
possible experimental bound on the magnetic moment of a neutral massive
spin-1 particle.
One of our motivations to consider the magnetic moment contributions from
Lorentz-symmetry breaking for massive neutral particles is partly based on
the fact that this issue is always discussed in connection with (loop)
radiative corretions in field-theoretic models. Our focus is to set up a
discussion, at the level of Quantum Mechanics which, in field theory, would
correspond to the generation of magnetic moment contributions (for neutral
particles) at the tree-level. We understand that, in a scenario where the
Lorentz covariance is violated, the symmetry breaking parameters may be
responsible for the appearance of a magnetic moment for neutral particles at
the tree-level. So, we adpot this scenario to discuss magnetic properties of
neutral particles at the quantum-mechanical level. We stress that this is
one of the main goals of our work. Indeed, the discussion on magnetic dipole
moments for neutral particles is a question of relevance in connection with
results \ coming from some Physics Beyond the Standard Model. Finally, we
would like to point out that the paper of ref.\cite{gamma} reports an
interesting \ calculation of the photon magnetic moment in connection with
(external) strong magnetic fields.
The organization of our paper is given as follows: in Section II, we briefly
report on the attainment of the value 2 for the gyromagnetic ratio for
massive charged spin-1 bosons non-minimally coupled to an electromagnetic
field. In Section III, we introduce the Lorentz symmetry violation term and
we discuss the magnetic moment of a neutral massive spin-1 boson
non-minimally coupled to the background associated to the breaking of
Lorentz symmetry and an external electromagnetic field. A $\left(
k_{F}\right) -$contribution to the magnetic moment of spin-$1$ bosons is
also reported in this Section.The discussion involving Majorana fermions is
carried out in Section IV. Finally, we cast our Concluding Remarks in
Section V.
\section{Massive and Charged Spin-1 Field.}
We start off from the Lagrangian that describes a massive charged vector
matter field, $W_{\mu }$, minimally coupled to an external electromagnetic
field as below:
\begin{equation}
\mathcal{L}=-\frac{1}{2}(W^{\mu \nu })^{\ast }W_{\mu \nu }+m^{2}(W^{\mu
})^{\ast }W_{\mu },
\end{equation}%
where $W^{\mu \nu }=D^{\mu }W^{\nu }-D^{\nu }W^{\mu },$ $D_{\mu }=\partial
_{\mu }+ieA_{\mu }$. The minimal electromagnetic coupling yields a wrong $%
(g=1)$ gyromagnetic ratio: this may be found by considering the Pauli - type
equation for the charged vector boson in the presence of a magnetic field,
namely,
\begin{equation}
\frac{1}{2m}\left( \vec{p}-e\vec{A}\right) ^{2}W_{i}-\frac{e}{2m}\vec{B}%
\cdot \text{ }\vec{S}_{ij}W_{j}=E_{nr}W_{i},
\end{equation}%
where $E_{nr}$ means the non-relativistic energy, and $\vec{S}_{ij}$ is the
spin matrix:
\begin{equation}
\vec{\mu}=\frac{e}{2m}\vec{S}\;,
\end{equation}%
$(S_{k})_{ij}=-i\varepsilon _{kij}$. To by-pass the conflict of the
gyromagnetic ratio, we have to introduce a renormalisable non-minimal
electromagnetic coupling \cite{ferrara}. The motivation for this new
interaction term comes from the Electroweak Theory: its $SU(2)\times U(1)$
gauge symmetry dictates the coupling between $A_{\mu }$ and the charged
gauge bosons as given below, after the spontaneous breaking of SU(2) takes
place:
\begin{equation}
ieF_{\mu \nu }W^{\mu \ast }W^{\nu }.
\end{equation}%
This indeed cancels high-energy divergences and corrects the gyromagnetic
factor to the right value $g=2,$ as it should be. So, taking into account
this new interaction, one may consider the vector field equation that
follows:
\begin{equation}
D_{\mu }W^{\mu \nu }+m^{2}W^{\nu }+ieW_{\mu }F^{\mu \nu }=0.
\end{equation}%
It directly implies the subsidiary condition $D_{\mu }W^{\mu }=0$. In the
non-relativistic limit, this condition yields:
\begin{equation}
W^{0}\cong \frac{\vec{p}}{m}\cdot \vec{W}-\frac{e}{m}\vec{A}\cdot \vec{W},
\end{equation}%
which shows that the time component of the $W$-field is of \ the order $%
\displaystyle{\left( \frac{v}{c}\right) }$ of its space components. It is
worthy to remark that, by considering the time component of the field
equation above, one exactly arrives at the same relation that follows from
the subsidiary condition: we then go straight to consider the space
components of the field equation for $W^{\mu }$. By properly \ carrying out
the non-relativistic approximation, after some algebraic steps, we show that
the gyromagnetic ratio comes out with its correct value equal to $2$:
\begin{equation}
\frac{1}{2m}\left( \vec{p}-e\vec{A}\right) ^{2}W_{i}-\frac{e}{m}\vec{S}%
_{ij}\cdot \vec{B}W_{j}=EW_{i}.\text{\ }
\end{equation}%
In the course of these calculations, it becomes clear that the net effect of
the non-minimal coupling, inherited from the non-Abelian $SU(2)$ symmetry of
the Electroweak Theory, is to add up the piece which was missing to yield
the right value for $g$. We now turn into the discussion of the gyromagnetic
ratio in situations where there occurs violation of Lorentz symmetry. This
is motivated by the fact that one may use magnetic moment measurements of
higher spin particles to get new bounds on the Lorentz symmetry violation
parameter.
\section{A Lorentz-symmetry violating background parametrised by a 4-vector,
$v_{\protect\mu }$}
Assuming that the background vector couples to the electromagnetic field,
the covariant derivative operator can be modified to introduce a non-minimal
coupling according to the expression below:
\begin{equation}
D_{\mu }=\partial _{\mu }+ieA_{\mu }+igv^{\alpha }\widetilde{F}_{\mu \alpha
},\text{ }
\end{equation}%
where $\widetilde{F}_{\mu \alpha }$ stands for the dual of the
electromagnetic field strength. It is worthy to mention that
Lorentz-symmetry violation does not conflict with gauge invariance. Gauge
symmetry is not violated by the action term $\varepsilon _{\mu \nu \kappa
\lambda }v^{\mu }A^{\nu }F^{\kappa \lambda }$\ introduced by Carrol, Field
and Jackiw\cite{photons1}. So, if we are to consider the dynamics of charged
particles under the action of the electromagnetic field, a gauge covariant
derivative must be adopted. In our proposal, we go a step further: we
extended the usual covariant derivative by adding up a term that implements
a (non-minimal) coupling of the particle to the external $A^{\mu }-$field
and, contemporarily, to the background vector, $v^{\mu }$. The term $%
v^{\alpha }\widetilde{F}_{\mu \alpha }$ is clearly gauge invariant, so it
does not harm the status of $D_{\mu }$ as a covariant derivative. This means
that a local phase transformation, $e^{i\alpha }$, performed on the charged
matter fields acts upon $D_{\mu }$\ according to the usual gauge
transformation of a genuine covariant derivative: $D_{\mu }^{^{\prime
}}=e^{i\alpha }D_{\mu }e^{-i\alpha }.$
In $\left( 1+2\right) D$, due to the fact that the Levi-Civita tensor is a
rank-3 tensor, the dual of the field strength is a vetor; so, we can define
a covariant derivative\cite{khare} as below:%
\begin{equation}
{\mathcal{D}}_{\mu }=\partial _{\mu }+ieA_{\mu }+ig\widetilde{F}_{\mu }.
\end{equation}%
A direct consequence of the non-minimal coupling introduced in $D_{\mu }$\
is that scalar particles display a non-trivial magnetic moment. Another
contribution of this covariant $\left( 1+2\right) D$\ derivative is the
geneneration of electrically charged vortices in the Abelian Higgs Model\cite%
{khare}.
Based on the result referred to above, and considering that the $v^{\mu }-$\
background may, in some special case, lead to an effective $\left(
1+2\right) D$\ model $\left( v^{\mu }=\left( v^{0},v^{1},v^{2},0\right)
\right) $, we then introduce the term $igv^{\mu }\widetilde{F}_{\mu \nu }$\
as the $4-$dimensional counterpart (whenever there is Lorentz-symmetry
breaking) of the non-minimal term studied in\cite{khare}. As a consequence,
we may investigate electrically charged vortices in the $4D$\ Abelian Higgs\
model \cite{General} and the anomalous magnetic moment generation of spin-$%
\frac{1}{2}$\ particles also in $4$-dimensional space-times\cite{Phases}.
\textbf{\bigskip }The effect of the non-minimal interaction term above on a
charged vector field, as considered in the previous Section (but, now, with
the covariant derivative modified as above), is to endow the particle
associated to the $W$-field with a universal magnetic moment given by
\begin{equation}
\vec{\mu}=\frac{1}{2}g\vec{v}, \label{mu}
\end{equation}%
as it is the case for the scalar and the spin-$\frac{1}{2}$, according to
the results reported in the work of reference \cite{Phases}. This is a very
peculiar outcome. Everything happens as if the presence of the background
modifies the structure of the particle and endows it with the universal
magnetic moment given above. According to the previous studies carried out
by Colladay and Kosteleck\'{y}, in the works of reference \cite{Colladay},
different particle species may have different independent Lorentz-breaking
parameters. Our non-minimal coupling present in $D_{\mu }$ is taken the same
for all charged particle species, for it accompanies the minimal coupling
term in the covariant derivative defined above. In the same way the minimal
coupling is universal, our term $gv^{\alpha }\widetilde{F}_{\mu \alpha }$
follows the same pattern. What is highlighted here is that its net effect,
no matter which spin the particle possesses, is to yield the same value for $%
\overrightarrow{\mu }$, as given above, in eq. \ref{mu}.
This is also the situation in the case of neutral vector particles, once the
non-minimal coupling above is switched on. Indeed, to explicitly see this
result, we take the simpler case where a non-charged ($e=0$) massive spin-$1$
particle is non-minimally coupled to the background and to the
electromagnetic field as described in the wave equation given below:
\begin{equation}
\left( \partial _{\mu }+igv^{\kappa }\widetilde{F}_{\mu \kappa }\right)
Z^{\mu \nu }+m^{2}Z^{\nu }=0, \label{cb1}
\end{equation}%
where $Z_{\mu }$ is the wave function of the spin-1 particle. The subsidiary
condition in this case takes the form:
\begin{equation}
\partial _{\mu }Z^{\mu }+igv^{\kappa }\widetilde{F}_{\mu \kappa }Z^{\mu }=0,
\label{dmuzmu}
\end{equation}%
where we can notice that, differently from the Proca case, it sets up
non-trivial relations among the $Z$-field components. (Incidentally, by
introducing an external electric field, we can get how the Aharonov-Casher
(AC) phase looks like.) In the non-relativistic limit, the subsidiary
condition yields:
\begin{equation}
Z^{0}\cong \frac{\vec{p}}{m}\cdot \vec{Z}-\frac{1}{m}(g\vec{v}\times \vec{E}%
)\cdot \vec{Z},
\end{equation}%
where we point out the presence of a sort of AC term. By replacing the
expression above for $Z^{0}$ in the space components of the $Z^{\mu }$%
-equations, and by properly keeping he terms that survive the
non-relativistic approximation, we get:
\begin{equation}
\frac{1}{2m}\left[ \vec{p}+\frac{g}{2}\left( \vec{v}\times \vec{E}\right) %
\right] ^{2}Z_{i}=E_{nr}Z_{i}.
\end{equation}%
This result suggests \ that the quantity $\frac{1}{2}g\vec{v}$ could be
interpreted as the magnetic moment acquired by the neutral particle due to
the presence of the background vector, $\vec{v}$. We can observe that the
immediate consequence is the appearance of a universal AC phase for
different spins, by virtue of the breaking of Lorentz symmetry under the
particle point of view. Moreover, in the presence of an external electric
field, $\vec{E}$, the wave function of every particle, charged or neutral,
with or without spin, acquires a non-trivial phase given by $\frac{g}{2}%
\left( \vec{v}\times \vec{E}\right) $. To show that the quantity $\frac{1}{2}%
g\vec{v}$ is actually the magnetic moment, we consider our neutral vector
particle under the action of an external magnetic field, $\vec{B}$. To do
that, we take Eq. \eqref{cb1} and we switch on a constant magnetic field
given by
\begin{equation}
F_{ij}=-\varepsilon _{ijk}B_{k}.
\end{equation}%
From the subsidiary condition, we get that
\begin{equation}
Z^{0}\cong \frac{\vec{p}}{m}\cdot \vec{Z}-\frac{gv^{0}}{m}\vec{B}\cdot \vec{Z%
},
\end{equation}%
and, by means of this relation and the space components of Eq. \eqref{cb1}
taken in the non-relativistic limit, the resulting wave equation for the
space components, $Z_{i},$ reads as below:
\begin{equation}
\frac{{\vec{p}}^{\,2}}{2m}Z_{i}+\frac{1}{2}g\vec{v}\cdot \vec{B}%
Z_{i}=E_{nr}Z_{i},
\end{equation}%
where it is easy to recognise the gyromagnetic ratio and to see that it gets
the same expression as in the scalar and spin-$\frac{1}{2}$ cases, as we had
already mentioned.
Before closing this Section and getting to the discussion on the Majorana
fermions, we belive it is worthwhile to mention another result valid for the
case of the neutral spin-1 bosons, namely, the contribution of the $k_{F}-$%
parameter\cite{Colladay} to the magnetic dipole moment of this sort of
particle, previously described by $Z^{\mu }$(present section).
The $k_{F}-$violationg term modifies the $Z^{\mu }-$field equations as given
below:%
\begin{equation}
D_{\mu }Z^{\mu \nu }-\frac{1}{2}k_{F}^{\nu \kappa \lambda \rho }D_{\kappa
}Z_{\lambda \rho }+m^{2}Z^{\nu }=0, \label{betha}
\end{equation}%
where
\begin{equation}
{\mathcal{D}}_{\mu }=\partial _{\mu }+igv^{\alpha }\widetilde{F}_{\mu \alpha
}.
\end{equation}
Following along the same steps as we have shown previously (Section II), we
place the spin-$1$\ particle in an external magnetostatic field and consider
the non-relativistic regime of the corresponding field equation to read off
its corresponding magnetic moment contribution. We calculate the subsidiary
condition out of the equation above and, by considering the space components
of these field equations, where we insert the expression for $Z^{0}$ coming
from the subsidiary condition, we get, after some algebraic manipulations
and the use of the conditions for the non-relativistic regime, that the $%
k_{F}-$parameter induces the correction given by%
\begin{equation}
\overrightarrow{B}_{\cdot }\overrightarrow{\mu }_{ij}Z_{j},
\end{equation}%
where the n-th component of $\overrightarrow{\mu }_{ij}$ is given by%
\begin{equation}
\left( \mu _{n}\right) _{ij}=\frac{1}{2}gv^{0}\left( k_{F}\right) _{nij0}.
\end{equation}
With this result, in our Concluding Remarks, we shall be able to present a
bound on the $Z^{0}$ `s magnectic moment. For that, we propose a discussion
on the magnetic moment of Majorana-type neutrinos in the sequel, from which
we will be able to get information on the product of parameters $gv^{0}$. As
for $k_{F}$, we shall be adopting a result presented in the work of ref.
\cite{klink}, so that an estimation of the magnetic moment given in the
expression above can be obtained.
\section{The case of Majorana fermions}
Neutrino magnetic dipole moments in the Standard Model are calculated as
radiative corrections and the tiny values obtained from loop calculations
may be used as good precision tests. In our case, we adopt the same
procedure followed to study the case of (massive) neutral vector bosons: we
assume a tiny deviation from the situation of Lorentz symmetry and we
non-minimally couple the (neutral) Majorana fermions to an external
electromagnetic field and the background vector that parametrizes the
breaking of the relativistic symmetry. To implement this scenario, we set up
the Dirac equation below for a Majorana spinor:
\begin{equation}
i\gamma ^{\mu }(\partial _{\mu }+igv^{\nu }\widetilde{F}_{\mu \nu }\gamma
_{5})\Psi -m\Psi =0. \label{Dirac1}
\end{equation}%
The introduction of the chirality matrix in the non-minimal electromagnetic
coupling is dictated by the Majorana character of the fermion we consider.
We adopt to work with the Majorana fermion by writing its wave function, $%
\Psi $, in the Majorana representation for the $\gamma $-matrices:
\begin{eqnarray}
&\gamma ^{0}=\left(
\begin{array}{cc}
\sigma _{y} & 0 \\
0 & -\sigma _{y}%
\end{array}%
\right) ,\gamma ^{1}=\left(
\begin{array}{cc}
i\sigma _{x} & 0 \\
0 & -i\sigma _{x}%
\end{array}%
\right) ,\gamma ^{2}=\left(
\begin{array}{cc}
i\sigma _{z} & 0 \\
0 & -i\sigma _{z}%
\end{array}%
\right) ,\gamma ^{3}=\left(
\begin{array}{cc}
0 & i1 \\
i1 & 0%
\end{array}%
\right) &, \notag \\
&\gamma ^{5}=i\gamma ^{0}\gamma ^{1}\gamma ^{2}\gamma ^{3}=\left(
\begin{array}{cc}
0 & -i1 \\
i1 & 0%
\end{array}%
\right) .&
\end{eqnarray}%
The charge conjugation matrix is $C=-\gamma ^{0}$; thus, in this
representation, a Majorana spinor $\left( \Psi ^{c}=C\bar{\Psi}^{t}=\Psi
\right) $ exhibits 4 real components. We should remark that parity may be
preserved despite the appearance of the chirality matrix in the action term $%
\bar{\Psi}\gamma ^{\mu }\gamma _{5}\Psi v^{\nu }\widetilde{F}_{\mu \nu }$ ;
the particular property of $v^{\mu }$\ under parity (vector or a
pseudo-vector) may be suitably chosen so as to make this term
parity-preserving.
As first step, we must probe the neutral particle by subjecting it to an
external magnetic field, $\widetilde{F}_{0i}=\vec{B}$. This shall reveal the
(eventual) contributions of the Lorentz-symmetry violating parameters, v0
and $\vec{v}$, to the magnetic moment of the neutral fermion. We start of
from the coupled Dirac's Eq. \eqref{Dirac1} ,
\begin{equation}
\left( \gamma ^{0}E-\vec{\gamma}\vec{p}-m-g\vec{v}\cdot \vec{B}\gamma
^{0}\gamma _{5}+gv^{0}\vec{B}\cdot \vec{\gamma}\gamma _{5}\right) \Psi =0
\end{equation}
The coupled Dirac%
\'{}%
s equation equation above splits up into 2 equations for the 2-component
spinors, $\xi $ and$\ \chi $. They read as follows:
\begin{eqnarray}
M\xi +N\chi &=&0, \notag \\
P\xi -Q\chi &=&0
\end{eqnarray}
where
\begin{eqnarray}
&M\equiv m-E\sigma _{y}+ip_{x}\sigma _{x}+ip_{y}\sigma
_{z}+gv^{0}B_{z};\;\;N\equiv ip_{z}-ig\vec{v}\cdot \vec{B}\sigma
_{y}-gv^{0}B_{x}\sigma _{x}-gv^{0}B_{y}\sigma _{z},& \notag \\
&P\equiv -ip_{z}+ig\vec{v}\cdot \vec{B}\sigma _{y}+gv^{0}B_{x}\sigma
_{x}+gv^{0}B_{y}\sigma _{z};\;\;Q\equiv m-E\sigma _{y}+ip_{x}\sigma
_{x}+ip_{y}\sigma _{z}+gv^{0}B_{z};&
\end{eqnarray}
\begin{eqnarray}
\chi &=&Q^{-1}P\xi ; \notag \\
\left( M+NQ^{-1}P\right) \xi &=&0.
\end{eqnarray}
Using the quaternionic unities \cite{qua}, we cast the operators $M,N,P$ an $%
Q$ in the form below:
\begin{eqnarray}
M &\equiv &\left( m+gv^{0}B_{z}\right) +p_{x}I+iEJ+p_{y}K; \notag \\
N &\equiv &ip_{z}+igv^{0}B_{x}I-g\vec{v}\cdot \vec{B}J+igv^{0}B_{y}K, \notag
\\
P &\equiv &-ip_{z}-igv^{0}B_{x}I+g\vec{v}\cdot \vec{B}J-igv^{0}B_{y}K;
\notag \\
Q &=&\left( m+gv^{0}B_{z}\right) +p_{x}I-iEJ+p_{y}K.
\end{eqnarray}%
The $(M+NQ^{-1}P)$-operator, that yields a Pauli-type equation, is worked
out, but, by analyzing its explicit form, the magnetic moment cannot be
properly identified. This shows us that the Majorana representation is not
suitable for the sake of taking the non-relativistic approximation. We
better go over into the (usual) Dirac 's representation and we also propose
a more general situation, where we try to compare the competitive effect of
two non-minimal couplings that may be contemporarily present in the Dirac%
\'{}%
s equation for a Majorana fermion; both the couplings are collected in the
expression below:%
\begin{equation}
i\gamma ^{\mu }(\partial _{\mu }+igv^{\nu }\widetilde{F}_{\mu \nu }\gamma
_{5})\Psi +\widetilde{g}F_{\mu \nu }\Sigma ^{\mu \nu }\gamma _{5}\Psi -m\Psi
=0.
\end{equation}%
So, from this complete equation, we follow the necessary steps to work out
the non-relativistic approximation and to arrive at a Pauli-type equation
for the $\xi $-component. By properly treating the terms that dominate in
the non-relativistic limit and taking care of the relations imposed by the
Clifford algebra, we find out the non-relativistic equation for $\xi $,
which turns out to be:
\begin{equation}
i\hbar \frac{\partial }{\partial t}\xi =\left\{ \frac{1}{2m}\left( \vec{p}+2~%
\widetilde{g}~\vec{\sigma}\times \vec{B}\right) ^{2}+~g~v^{0}\vec{\sigma}%
\cdot \vec{B}-~g~\vec{E}\cdot \left( \vec{v}\times \vec{\sigma}\right) -~%
\tilde{g}~\vec{\sigma}\cdot \vec{E}\right\} \xi .
\end{equation}%
The expression above opens up a number of interesting remarks on the
(non-minimal) electromagnetic effects of the spin of a neutral
self-conjugated fermion. We identify the interaction term that leads to the
magnetic dipole moment as being given by $g~v^{0}\vec{\sigma}\cdot \vec{B}$;
this then shows that, instead of the space component, $\vec{v}$, it is now
the time component, $v^{0}$, the responsible for the magnetic moment
generation, and the Pauli-type coupling (the one given by $\widetilde{g}$)
does not contribute to the magnetic interaction as in the ordinary case.
Instead, it induces a coupling to the electric field and a new type of phase
($\Phi $) in the fermion wave function, given by
\begin{equation}
\Phi =\frac{2~\widetilde{g}}{~g~v^{0}}\int \dot{d}\vec{l}\cdot (\vec{\mu}%
\times \vec{B}).
\end{equation}
\section{Concluding Remarks}
The main effort in our work has been to show how, in an environment where
Lorentz symmetry is violated, truly elementary neutral particles may show up
magnetic properties only due to their spin, once non-minimal couplings to
the electromagnetic field are allowed. The background vector responsible for
the Lorentz-symmetry violation couples to both the electromagnetic field and
the particle itself and then the electromagnetic properties of the spin are
revealed through Aharonov-Casher and Pauli-type couplings of the magnetic
dipole moment of the particle. For charged spin-0, spin-$\frac{1}{2}$,
spin-1 particles and neutral vector bosons, we have seen that there appears
a universal magnetic dipole moment for each particle, $\vec{\mu}=\frac{1}{2}g%
\vec{v}$, as a result of the presence of the background vector.
Nevertheless, other contributions for the magnetic moment may be derived
which depend on the type of the particle, as discussed in the treatment of
the Majorana fermion, for which the non-minimal coupling with the presence
of the chirality matrix produced a new sort of phase which involves the
magnetic field. This means that, for neutral particles like the neutrinos,
the tiny magnetic dipole moments they have (less than $10^{-10}$ Bohr
magnetons), which in the framework of the Electroweak Theory are understood
and computed as an effect of the radiative corrections, may also be
attributed to possible effects of an eventual violation of Lorentz symmetry.
What we conclude is that purely electromagnetic effects of the spin may
emerge if neutral particles interact with an external electromagnetic field
via a background that realizes the tiny breaking of Lorentz symmetry. In the
present paper, this has been done for a background vector; however, from our
results, we can safely state that the same conclusions hold through if the
Lorentz-symmetry breaking background is of a tensor nature.\
With the result on the magnetic moment for Majorana-type fermions presented
in the previous Section, and the experimental bounds on the neutrino
magnetic moment\cite{pdg}, we can set a bound on the product $gv^{0}$,
namely, $gv^{0}<0.9\times 10^{-10}\mu _{B}$. By considering the results of
the work of ref. \cite{rei}, and a recent paper by Klinkhamer and Shereck
\cite{klink}, it is reasonable to take the bound $\left\vert \left(
k_{F}\right) _{nij0}\right\vert <2\times 10^{-7}$, so that we can get an
estimation on the magnetic moment for $Z^{0}$: $\mu (Z^{0})<1,65\times
10^{-14}\mu _{N}$, where $\mu _{N}$\ stands for the nuclear magneton. In
possess of the results presented in this work, we are now concentrating our
efforts to systematically get bounds on Lorentz-symmetry breaking parameters
from our investigation of their influence on the calculation of gyromagnetic
ratios and magnetic moments for different particle species, with special
interest on the sector of neutral fundamental fermions and vector bosons.
These results shall be soon reported in a forthcoming paper.
\acknowledgements
J. Moraes and R. Turcati are kindly acknowledged for long discussion. HB,
TCS, JAHN and MDTO acknowledge CNPq for the invaluable financial help.
|
1,116,691,498,461 | arxiv | \section*{Introduction}
Breast cancer is the most common cancer in women worldwide, after skin cancers and about 42,170 women will die from breast cancer in the United States for 2020, according to The American Cancer Society's estimate~\cite{breastcancer}. Screening mammography, a low-dose X-ray examination, is typically used for early detection of breast cancer. The United States Preventive Services Task Force suggests women urdergo such exams every two years if they are 50 to 74 years old and are at average risk for breast cancer~\cite{jama2019}. Although multiple studies have demonstrated that screening mammography reduces breast cancer mortality~\cite{lees2010theoretical, 10.1001/jama.2015.12783, jama2019, Monticciolo}, performance benchmarks demonstrate that 10\% of the performed exams are recalled for additional imaging, and approximately 80\% of biopsies subsequently performed are benign~\cite{lehman2017national}. The yearly national cost of breast-care caused by the false positive mammograms is estimated to be several billion of dollars~\cite{vlahiotis2018analysis, ong2015national} and for women with a false positive diagnosis, their mean cost of breast-care is even higher than the cost of breast-care services for women with cancer~\cite{chubak2010cost}. It is therefore an important task to reduce the recall and biopsy rates so that to decrease patients’ anxiety and reduce healthcare costs while still maintaining optimal cancer detection rates, according to relevant guidelines~\cite{10.1001/jama.2015.12783}.
Traditional computer-aided detection (CAD) tools for mammography neither detected more breast cancers nor decreased the recall rates for additional imaging~\cite{lehman2015diagnostic, fenton2007influence}. Early studies used deep neural networks (DNNs) to assist radiologists interpreting screening mammograms by making predictions for cancer of each breast~\cite{zhu2017deep, kyono2018mammo, aboutalib2018deep, kim2018applying, wu2019deep, shen2020interpretable, mckinney2020international}. This task is frequently considered in literature. It can be viewed as breast-level classification, and models developed accordingly have shown comparable performance to radiologists~\cite{wu2019deep, shen2020interpretable, mckinney2020international}. However, these models suffer from performance degradation when evaluated on a population only containing exams which lead to biopsies, without healthy breasts as negative cases~\cite{wu2019deep}. Meanwhile, models built for the breast-level classification task cannot provide independent risk estimations for multiple areas of interests appearing in the same breast. It is common to encounter cases with multiple findings~\cite{cohen2020multiple}. For example, multiple bilateral circumscribed breast masses are detected in approximately 1.7\% of routine screening mammograms~\cite{leung2000multiple}. In the NYU Breast Cancer Screening Dataset~\cite{wu2019nyu}, a representative sample of screening mammograms from 2010 to 2017, there are 7.45\% images with more than one annotated lesions, and 25.75\% of these images have lesions of different categories. Some examples are shown in Figure~\ref{fig:multi_lesion}. In light of this, the previously proposed models for breast-level classification are difficult to use for the goal of reducing unnecessary biopsies.
\begin{figure}[h!]
\centering
\begin{tabular}{ll}
\includegraphics[width=0.25\textwidth, clip, trim=0mm 0mm 0mm 0mm]{figures/figure0_1.png} & \includegraphics[width=0.25\textwidth, clip, trim=0mm 10mm 0mm 10mm]{figures/figure0_2.png} \\
\textbf{a} & \textbf{b}
\end{tabular}
\caption{Images with both malignant lesion and benign tissue. \textbf{a}. An image of the left breast from mediolateral oblique view (L-MLO). The breast has two lesions confirmed by biopsy, one as malignant (annotated with red), and the other as benign (annotated with green). \textbf{b}. An L-MLO mammogram image from another patient. There are two lesions on the image, one as malignant (annotated with red), and the other as high-risk benign (annotated with yellow).}
\label{fig:multi_lesion}
\end{figure}
Besides breast-level malignancy classification, deep learning methods have also been used to identify high-risk lesions~\cite{ribli2018detecting, liu2020cross, samala2016mass, agarwal2019automatic}. Some of these works can provide risk estimation across regions of the breast, but usually only consider information in a small local region~\cite{samala2016mass, agarwal2019automatic}. The majority of the existing works often utilize object detection models such as Mask-RCNN~\cite{he2017mask}, which neither explicitly utilize fine details nor consider global context. In contrast, radiologists often consider global context factors to make their diagnoses~\cite{wei2011association,pereira2009spatial}. These global context factors include the mammographic breast density, i.e. the global amount of fibroglandular tissue, and the associated parenchymal and nodular patterns of the breasts~\cite{pereira2009spatial}. Dense fibroglandular breast tissue is a known risk factor for breast cancer~\cite{10.1001/jamaoncol.2018.7078}. Other global context factors include the distribution of microcalcifications in the tissue adjacent to an index lesion, or throughout the breast. These global findings often affect radiologists' level of suspicion for a particular lesion. Within deep learning methodology, these scenarios could be viewed as utilizing global image context for classifying a patch of an image. This motivates investigating whether global context is as important for neural networks as it is for human experts.
In this study, we consider lesion-level classification, and design models to directly distinguish biopsy-confirmed lesions as being either benign or malignant. With this strategy, we enable the models to make accurate lesion-wise predictions. To show that deep learning approaches can benefit from utilizing global image context in classifying local findings on mammograms, we first train DNNs with cropped image patches to enable the learning of fine details from a specific region, then integrate the extracted local information with the global context. The global context is provided in the form of saliency maps (Figure~\ref{fig:saliency}) extracted by a model classifying the entire image. Here we use Globally-Aware Multiple Instance Classifier~\cite{shen2020interpretable} as the model to provide such saliency maps. In addition, we evaluate the models' performance on a challenging population which consists only of cases that are difficult to diagnose and the radiologist requested a biopsy for. This further differentiates our work from previous works~\cite{wu2019deep, shen2019globally, shen2020interpretable, kyono2018mammo, mckinney2020international} and makes our results not directly comparable to theirs. This is because these methods were developed and evaluated for the screening population, which contains a lot of negative cases not requiring biopsy, which can inflate their evaluation metrics~\cite{wu2019deep}.
\begin{figure}[h!]
\centering
\begin{tabular}{ccc}
\includegraphics[width=0.7\textwidth, clip, trim=0mm 0mm 0mm 0mm]{figures/saliency.pdf}
\end{tabular}
\caption{An example of saliency maps. From left to right: a mammogram image of a right breast from craniocaudal view view (R-CC) with an annotated malignant lesion, a saliency map indicating suspicious regions for benign tissue, a saliency map indicating suspicious regions for malignant lesions.}
\label{fig:saliency}
\end{figure}
Our results show that DNNs trained with image patches can effectively decrease the number of unnecessary biopsies, and that this ability can be further improved by utilizing global image context. Our best model is able to distinguish between benign and malignant findings on a test set of 600 lesions, achieving an area under the receiver operating characteristic curve (AUC) of 0.799$\pm$0.002. If the model is utilized to assist in reading mammograms, about 2\% unnecessary biopsies could be avoided while catching all malignancies, in addition to cases that radiologists can easily exclude as benign or normal and not needing additional imaging. It reduces 15\% more unnecessary biopsies than the model using only local information, when under the same level of missed malignancies (2\%). It is worth noting that these performance measurements are computed on the population from which we exclude benign cases that radiologists can discount confidently by reading mammograms or other imaging exams. Overall, our results strongly suggest that the proposed strategy can be considered as a viable and valuable enhancer for deep learning methods in reducing unnecessary biopsies based on screening mammography.
\section*{Materials and Methods}
This retrospective study was approved by our IRB and is compliant with the Health Insurance Portability and Accountability Act. Informed consent was waived.
\subsection*{Data}
We utilize a dataset consisting of 229,426 digital screening mammogramphy exams (1,001,093 images) from 141,473 unique patients screened between 2010 and 2017~\cite{wu2019nyu}. Each exam has four standard views and the resolution of images is approximately 2000${\times}$3000 pixels. We asked fellowship-trained breast imagers to annotate both benign tissues (e.g. cyst, fibroadenoma, fibrocystic change) and malignant lesions (e.g. IDC, ILC, DCIS), on the pixel-level. In the entire dataset, there are 8842 lesions from 8080 images with diagnosis confirmed by biopsy, which reveals the fact that a single breast can contain multiple lesions of differing types. The dataset is divided into disjoint training (80\%), validation (10\%) and test (10\%) sets. Detailed statistics of training, validation and test sets are in Table~\ref{tab:data_overall_stats}.
\begin{table}[ht]
\centering
\caption{Number of biopsy-confirmed lesions and number of mammogram images presenting no lesions, benign tissues, and malignant lesions in the training, validation and test set.}
\begin{tabular}{@{}lcccccc @{}}
\toprule
& \multicolumn{3}{c}{\textbf{images}} &\phantom{a}&\multicolumn{2}{c}{\textbf{lesions/tissues}} \\
\cmidrule{2-4} \cmidrule{6-7}
& \textbf{negative} & \textbf{benign} & \textbf{malignant} &&\textbf{benign} & \textbf{malignant} \\
\midrule
\textbf{training} & 808,730 & 5,188 &1,648 && 5,602 & 1,790\\
\textbf{validation} & 123,130& 687 & 110 && 722 & 128\\
\textbf{test} & 60,959& 432& 116& &464 & 136\\
\midrule
\textbf{overall} & 992,819 & 6,307& 1,874&&6,788&2,054 \\
\bottomrule \\
\end{tabular}
\label{tab:data_overall_stats}
\end{table}
\subsection*{The proposed method}
Lesions in mammograms vary in size and shape, so if we crop these regions entirely and resize them to the same size to use them as inputs to standard deep neural networks, we will introduce information distortion and lose the fine details of the lesions. Therefore, we start by learning features of a number of image patches that are cropped from regions overlapping with the lesion, and then aggregate information from all patches to render a prediction for that lesion.
To extract information from image patches, we train a deep convolutional neural network (DCNN) to classify image patches of 256${\times}$256 pixels as one of the four classes: ``malignant'', ``benign'', ``outside'' and ``negative.'' Malignant and benign patches are cropped from windows that overlap with the segmentation of a malignant lesion or benign tissue. Besides cropping image patches that overlap with the annotations, we sample patches that have no overlap with any lesion (``outside''), as well as patches from breasts without records of biopsy (``negative''). The inclusion of these additional data is intended to regularize the model similarly to data augmentation. Examples of patches from each class are shown in Figure~\ref{fig:patches}.
\begin{figure}[htb!]
\centering
\begin{tabular}{cc}
\multicolumn{2}{c}{\includegraphics[width=0.85\textwidth]{figures/patch_top.pdf}}\\
\hspace{2mm}\textbf{a} & \textbf{b} \\
\multicolumn{2}{c}{\includegraphics[width=0.85\textwidth]{figures/patch_down.pdf}}\\
\hspace{2mm}\textbf{c} & \textbf{d}
\end{tabular}
\caption{Examples of image patches along with the mammogram images from which they come. \textbf{a}. ``malignant'' patches, which overlap only with malignant findings (marked with red); \textbf{b}. ``benign'' patches, which overlap only with benign findings (marked with yellow or green); \textbf{c}. ``outside'' patches, which are from regions outside the annotated lesions; \textbf{d}. ``negative'' patches, which are from images without any biopsied findings. }
\label{fig:patches}
\end{figure}
This DCNN is used to produce representations of local information. It is denoted as $f_{loc}$ and is shown in Figure~\ref{fig:model}a. We use DenseNet-161~\cite{huang2017densely} as its architecture. We add an additional fully-connected layer with 32 neurons between the global average pooling layer and the classification layer to obtain concise representations of the patch. The additional layer results in the feature vector $\mathbf{h} \in \mathbb{R}^{32}$ for the patch, which we use as the representation for the local information extracted by $f_{loc}$.
To further incorporate global image context and curate the local information extracted by the DCNN, we train an ``aggregation network'' with inputs formed by aggregating maps containing information relative to the patch and the image it is cropped from, as illustrated in Figure~\ref{fig:model}b. This aggregation network is a shallow convolutional network, denoted as $f_{agg}$. It consists of two convolutional layers, each with 32 $3{\times}3$ convolutional filters, a global average pooling layer and, finally, a classification layer. We apply batch normalization and the ReLU activation function prior to each convolutional layer. This network is trained for the same patch classification task. The maps formed as inputs to the aggregation network are described in the following paragraphs.
The first type of maps are saliency maps which represent global context. We generate the saliency maps by training a network on full-resolution mammography images to predict the presence of benign and malignant lesions in the breast. We refer to this network as the ``context network.'' We use Globally-Aware Multiple Instance Classifier~\cite{shen2019globally, shen2020interpretable} as the context network, which is explicitly designed to provide interpretability by highlighting the most informative regions of the input images. To be more precise, the feature maps obtained after the last residual block of the context network are transformed by a $1{\times}1$ convolutional layer with sigmoid activation into two saliency maps, denoted as $\mathbf{S}_m$ $\in [0,1]^{46,30}$ and $\mathbf{S}_b$ $\in [0,1]^{46,30}$. Each pixel in the saliency map corresponds to a region in the full image, and its element denotes a score indicating the contribution of this region towards classifying the input image as containing malignant lesions or benign tissues. A pair of saliency maps for an image is shown in Figure~\ref{fig:saliency}.
Another type of maps are location indicator maps. Given a patch, this map indicates its cropping window's location on the mammogram, but is downscaled to be the same size as the saliency maps. The location indicator map is denoted as $\mathbf{I} \in [0,1]^{46,30}$. Same with the saliency maps, each pixel on the location indicator map corresponds to a region in the full image and the value of this pixel reflects how much the region is covered by the patch.
The last type of maps, called embedding maps, are formed by utilizing the representation $\mathbf{h}$ generated by $f_{loc}$. To construct this map, for each $\mathbf{h}_k \in \mathbf{h}$, we take a copy of the location indicator map of the patch $\mathbf{I}$ and replace its nonzero elements by $\mathbf{h}_k$. We denote the obtained map by $\mathbf{E}_k$. We concatenate $\mathbf{E}_k$'s to form the full embedding map $\mathbf{E} \in \mathbb{R}^{46, 30, 32}$. The embedding maps contain information learned by $f_{loc}$ specific to the fine details in the image patch.
These maps can be concatenated along the last dimension and served as inputs to the aggregation network, denoted as $\mathbf{X} \in \mathbb{R}^{46, 30, M}$ where $M$ is the number of maps. For example, when using both embedding maps and saliency maps as inputs, $M$ would be 34.
\begin{figure}[htb!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.26\textwidth]{figures/figure1_b.pdf} &\includegraphics[width=0.4\textwidth]{figures/figure1_c.pdf}
\end{tabular}
\caption{Illustration of the proposed method. Left: a deep convolutional neural network which takes image patches of 256$\times$256 pixels as inputs, denoted as $f_{loc}$. Right: the aggregation network, $f_{agg}$, that takes the concatenation of three types of maps as inputs: 1) location indicator map, $\mathbf{I}$, in gray, generated by downscaling the binary mask indicating the cropping window's location, 2) saliency maps, $\mathbf{S}_m$ and $\mathbf{S}_b$, in red and green, generated by the context network, based on Globally-Aware Multiple Instance Classifier~\cite{shen2020interpretable}, 3) embedding map, $\mathbf{E}$, in blue, formed by the representation $\mathbf{h}$, produced by $f_{loc}$.}
\label{fig:model}
\end{figure}
\subsection*{Model training}
We first train $f_{loc}$ that takes image patches as inputs, followed by the aggregation network $f_{agg}$, which takes the concatenated maps, $\mathbf{X}$, as inputs. We use 20, 35, 5000, 4945 patches for malignant, benign, outside and negative patch classes in each training epoch. For data augmentation, we use random rotations (-30 to 30 degrees), and random sizes (128${\times}$128 to 384${\times}$384 pixels) when setting the cropping window to obtain the patch.
In order to address the extreme class imbalance, we use weighted cross-entropy as the training loss. The class weight for each patch class is set as inverse to the ratio of patches from this class among all patches used in each epoch. Therefore, losses on incorrect predictions of ``malignant'' and ``benign'' patches are appropriately up-weighted. For both $f_{loc}$ and $f_{agg}$, we adopt the same configuration while using $256{\times}256$ image patches as inputs for $f_{loc}$, and the concatenated maps, $\mathbf{X}$, as inputs for $f_{agg}$, in which the embedding maps is produced by the best performing $f_{loc}$ and saved to be used.
We minimize the training loss with the Adam optimizer~\cite{kingma2014adam}, setting the batch size to 25 for training $f_{loc}$, and 100 for $f_{agg}$. We initialize weights in $f_{loc}$ of all layers except the last two fully connected layers with weights from DenseNet-161~\cite{huang2017densely} pretrained on the ImageNet ILSVRC-2012 dataset~\cite{russakovsky2015imagenet}, then fine-tune the entire network. We randomly initialized the weights of $f_{agg}$. We optimize the hyperparameters using random search~\cite{bergstra2012random}. Specifically, we search for the learning rate on the logarithmic scale in $[10^{-6}, 10^{-4}]$ for $f_{loc}$, and in $[10^{-5}, 10^{-3}]$ for $f_{agg}$. Early stopping is performed if we observed that the AUC on the validation set has not increased for ten epochs. We implement the models in PyTorch~\cite{paszke2019pytorch}, and use NVIDIA Tesla V100 GPUs for model training and inference.
\subsection*{Model evaluation}
During training, we consider patches from mammograms with and without lesions, and perform multi-class classification over four classes: malignant, benign, outside and negative. In the validation and test phases, we only consider patches from images with lesions, and transform the patch-level predictions into a malignancy prediction for each lesion in the images. To get a prediction for a lesion, as shown in Figure~\ref{fig:evaluation}, we crop 100 patches that overlap with the segmentation of the lesion. The size of the cropping window varies from 128${\times}$128 to 384${\times}$384 pixels, which is the same range we used for data augmentation. After cropping, each patch is resized to 256${\times}$256 pixels, and we use it as input to $f_{loc}$ to produce a feature vector. Then, we apply $f_{agg}$ on the concatenated maps, including the embedding maps transformed by the feature vector and we get its prediction, each as four scores for the four patch classes. For each patch, we normalize the scores for malignant and benign patch classes so that they sum to one. Finally, we average the 100 normalized scores of the 100 sampled patches to obtain a prediction for the lesion. Based on these estimated probabilities, we compute the AUC that the model achieves in classifying the lesions as malignant or benign. We use the AUC computed on the 850 lesions from the validation set for model selection, and report the AUC computed for the 600 lesions from the test set.
\begin{figure}[htb!]
\centering
\begin{tabular}{cc}
\multicolumn{2}{c}{\includegraphics[width=0.85\textwidth]{figures/evaluation.pdf} }\\
\end{tabular}
\caption{Multiple patches are sampled to obtain a prediction for one lesion. In each black box, we present a biopsied finding from the test set and ten out of 100 patches used by the model to make a prediction for the finding. Lesions are marked in red (malignant) or green (benign) and the cropping windows of the patches are marked by blue boxes on the image. Cropped patches are shown on the right.}
\label{fig:evaluation}
\end{figure}
\section*{Results}
\subsection*{Model performance}
We report the model's performance on the test set consisting of 600 lesions which are present on 534 images from 260 patients. There are 44 images containing more than one lesion, and 14 images have both benign and malignant lesions. We emphasize that if we were to use deep learning methods that provide only breast-level risk estimation, it would be impossible to tell which lesion is the one with higher risk of malignancy.
Both model components, $f_{loc}$, which takes patches as inputs, and the aggregation network $f_{agg}$, are selected according to their performance on the validation set. The best performing aggregation network using embedding maps and saliency maps achieved an AUC of 0.799$\pm$0.002 on the test set. In Table~\ref{tab:fnr}, we include more results on the portion of unnecessary biopsies that can be avoided while missing a given portion of malignancies when using the model’s prediction as a second reader to assist radiologists. It can help to further reduce 1.7\% unnecessary biopsies in addition to cases that radiologists can easily exclude as benign or normal and not needing additional imaging, while catching all malignancies. According to the estimated yearly cost related with unnecessary biopsies~\cite{vlahiotis2018analysis, ong2015national}, it can be translated to saving more than a million dollars each year in the US for breast-care. If reducing biopsies is prioritized further, up to 13.5\% could be avoided while only missing 1\% of malignancies and up to 23.1\% could be avoided while only missing 2\% of malignancies.
\begin{table}[htb!]
\centering
\caption{True negative rate (TNR) achieved by our model when its false negative rate (FNR) are 0.01, 0.02, 0.03 and 0.05 as we vary prediction threshold for assigning observations to a positive class indicating malignancy. The 95\% confidence interval of the estimated TNR and the clinical meaning are also provided.}
\begin{tabular}{@{}cccc@{}}
\toprule
clinical meaning & FNR & TNR & 95\% CI of TNR \\
\midrule
\begin{tabular}{c}
1.7\% unnecessary biopsies we can help to avoid \\
while missing no malignancies
\end{tabular} & 0.00 & 0.017 & $[-0.005, 0.040]$\\
\begin{tabular}{c}
13.5\% unnecessary biopsies we can help to avoid \\
while missing 1\% malignancies
\end{tabular} & 0.01 & 0.135 & $[0.039, 0.232]$\\
\begin{tabular}{c}
23.1\% unnecessary biopsies we can help to avoid \\
while missing 2\% malignancies
\end{tabular} & 0.02 & 0.231 & $[0.190, 0.273]$\\
\begin{tabular}{c}
27.6\% unnecessary biopsies we can help to avoid \\
while missing 3\% malignancies
\end{tabular} & 0.03 & 0.276 & $[0.240, 0.312]$\\
\begin{tabular}{c}
43.5\% unnecessary biopsies we can help to avoid \\
while missing 5\% malignancies
\end{tabular} & 0.05 & 0.435 & $[0.407, 0.562]$\\
\bottomrule
\end{tabular}
\label{tab:fnr}
\end{table}
\subsection*{Ablation experiments}
We conduct the following experiments to justify the choice of model architecture, to verify impact of transfer learning, and to elaborate the importance of utilizing both local fine details and global image context in identifying malignant lesions.
\paragraph{Architecture search.}
To choose the architecture for $f_{loc}$, we considered a number of ResNet~\cite{he2016deep} and DenseNet~\cite{huang2017densely} variants. These architectures use skip connections, which improve information flow between layers, and allow for effective training of very deep networks. They both achieved strong results across a wide range of image classification tasks~\cite{guan2020multi, wang2017chestx, hannun2019cardiologist}. The specific ResNet and DenseNet variants we experimented with are: ResNetV2-50, ResNetV2-101, ResNetV2-152, DenseNet-121, DenseNet-161, and DenseNet-169. We compared the performance of $f_{loc}$ when being parameterized as the above architectures. The results are shown in Figure~\ref{fig:perform}. In this experiment, we did not consider global context and used the predictions made by $f_{loc}$ for each lesion.
\paragraph{Transfer learning.}
Transfer learning by pretraining the network on a different task is widely adopted to improve neural networks' performance. We experiment with initializing $f_{loc}$ with weights from networks pretrained on the ImageNet ILSVRC-2012 dataset~\cite{russakovsky2015imagenet}, and compare it to initializing the weights randomly using He initialization~\cite{he2015delving}. Since images from ImageNet are RGB while mammograms are grayscale, we duplicated each patch across the three channels. The AUCs achieved by $f_{loc}$ with or without using transfer learning are presented in Figure~\ref{fig:perform}. Without transfer learning, ResNetV2-50 achieved the highest AUC of 0.762$\pm$0.015, while DenseNet-121 achieved the lowest AUC of 0.748$\pm$0.098. When we applied transfer learning, the performance is improved for most of the architectures except ResNetV2-152, and DenseNet-169 becomes the best performer with 0.782$\pm$0.014 AUC. We conclude from these results that transfer learning from the ImageNet dataset~\cite{russakovsky2015imagenet} clearly improves our results, even though the natural image domain and the mammography image domain are so different.
\begin{figure}[htb!]
\centering
\includegraphics[width=0.75\textwidth]{figures/performan_table_1.pdf}
\caption{Test performance in classifying biopsied findings, achieved by $f_{loc}$, when using different architectures and weights initialization strategies. }
\label{fig:perform}
\end{figure}
\paragraph{Importance of global context.}
To assess the importance of global context in classifying lesions localized to small regions of the image, we performed further ablation experiments. We trained networks using different combinations of saliency maps, location indicator map and embedding maps as inputs. Selected maps are concatenated along the last dimension and used by the aggregation network. Table~\ref{tab:combin} presents the results when using all possible combinations. Since ImageNet-pretrained DenseNet-161 as $f_{loc}$ achieved the highest AUC on the validation set, we used it in this set of experiments.
As expected, for the task of classifying biopsy-confirmed lesions, most of the predictive power comes from local features: the aggregation network trained with only embedding maps achieved an AUC of 0.778$\pm$0.002. In comparison, the network trained only with saliency maps achieved an AUC of 0.695$\pm$0.003, indicating that global context alone was not highly predictive. When we introduced location indicator maps together with saliency maps into the network, the AUC increased to 0.721$\pm$0.011. We observed that using location indicator maps and embedding maps together does not improve performance. This is unsurprising since embedding maps contain the same location information conveyed by the location indicator maps. Finally, networks using the combination of embedding maps and saliency maps achieved an AUC of 0.799$\pm$0.002, which is the highest among all combinations. The fact that combining local features with global context outperformed each of them in isolation confirms the importance of the global image context in classifying lesions localized to small regions of the mammogram.
To further evaluate the impact of the global context in our task, we investigate the relation between the true negative rate (indicating biopsies that can help to avoid) and the false negative rate (indicating missed malignancies). Specifically, we compare the aggregation network using only the ensemble maps and the aggregation network using both the ensemble maps and the saliency maps. Figure~\ref{fig:fnr} visualizes this relationship. We considered scenarios when there are less than 5\% malignancies missed. For all considered false negative rates utilizing the saliency maps resulted in lower true negative rate.
\begin{table}[htb!]
\centering
\caption{Test performance of the aggregation network when using different information combinations as inputs. Models utilizing both local and global information achieved better performance than the counterparts using single type of maps.}
\begin{tabular}{@{}lc@{}}
\toprule
& AUC \\
\midrule
location indicator maps & 0.474 $\pm$ 0.031\\
embedding maps & 0.778 $\pm$ 0.002\\
saliency maps & 0.695 $\pm$ 0.003\\
location indicator maps + embedding maps & 0.777 $\pm$ 0.002\\
location indicator maps + saliency maps & 0.721 $\pm$ 0.011\\
embedding maps + saliency maps& \textbf{0.799 $\pm$ 0.002}\\
location indicator maps + saliency maps + embedding maps & 0.797 $\pm$ 0.001\\
\bottomrule
\end{tabular}
\label{tab:combin}
\end{table}
\begin{figure}[htb!]
\centering
\includegraphics[width=0.75\textwidth]{figures/performan_curve.pdf}
\caption{True negative rate (TNR) and false negative rate (FNR) achieved by the aggregation network using ensemble maps or using both ensemble maps and saliency maps as we vary the prediction threshold for assigning observations to a given class.}
\label{fig:fnr}
\end{figure}
\section*{Discussion}
Regular screening mammography is widely acknowledged to be the best way to detect breast cancer early. However, mammogram-based diagnosis performed by radiologists suffers from a high false positive rate, resulting in both unnecessary imaging and tissue biopsies. Developing deep learning technologies to assist breast cancer screening is promising, but previous works in the literature rarely focused on reducing unnecessary biopsies. Besides achieving radiologist-level performance at detecting breast cancer in mammograms, deep learning models are expected to play a more important role in distinguishing whether a given lesion is malignant or benign. This distinction is highly beneficial for the case of suspicious-appearing but ultimately benign findings that result in unnecessary biopsies by the radiologist.
In this study, we presented a method to combine local features in small image patches with global context in high resolution mammogram images. We showed that it is necessary to consider both fine details in a small region and the global image context to improve deep learning models' performance when classifying localized lesions on the high resolution images, while previous works usually consider only image patches or downscaled mammogram images~\cite{xi2018abnormality, spanhol2016breast, aboutalib2018deep}. Our resulting deep learning model achieved an AUC of $0.799\pm0.002$ in classifying biopsy-confirmed lesions as being malignant or benign. It can help to further reduce over 23\% of unnecessary biopsies while missing only 2\% of cancer as the second reader on regions that radiologists have low confidence on. Compared with works performing breast-level classification~\cite{aboutalib2018deep, wu2019deep, elter2007prediction, xi2018abnormality, shen2019globally, shen2020interpretable, mckinney2020international}, our model can provide prediction for each individual suspicious lesion, and therefore present precise guidance for followup procedure including biopsy and surgery.
We acknowledge some limitations of this work. For instance, we did not capture the levels of difficulty related to different types of cancer, which is clinically valuable. We leave this for future work. In addition, the context network we considered in the study did not perform cross-view reasoning, and we expect that networks utilizing all four standard views in a mammogram exam can introduce more complete information and result in more reliable cancer detection.
\section*{Conclusion}
Besides performing breast-level classification, deep learning methods can help further reduce unnecessary biopsies by classifying suspicious small regions as being benign or malignant. Furthermore, incorporating global image context can improve the network's ability to distinguish between localized benign and malignant lesions on high resolution images. Future research on techniques for combining local information with global context may be promising for breast cancer screening.
\bibliographystyle{custom}
|
1,116,691,498,462 | arxiv | \section{Introduction}}
\label{sec:intro}
\IEEEPARstart{U}supervised node embedding is an exciting field, in which a significant amount of progress has been made in recent years \cite{survey_tkde}. The task consists of mapping each node of a graph to a vector in a low-dimensional Euclidean space. The main goal is to \emph{extract features} that can be utilized downstream in order to perform a variety of unsupervised or (semi-)supervised learning tasks, such as node classification, link prediction, or clustering \cite{survey2}. Ideally, it is desired for the embedded nodal vectors to convey at least as much information as the original graph. Nevertheless, an appropriate embedding can boost the performance of certain learning tasks because they allow one to work with the more ``friendly'' and intuitive Euclidean representation, and deploy mature and widely implemented feature-based algorithms such as (kernel) support vector machines (SVMs), logistic regression, and K-means.
Early embedding works mostly focused on a structure-preserving dimensionality reduction of feature vectors (instead of nodes); see for instance \cite{mds, isomap, lle, lpp, partially}. In this context, graphs are constructed from pairwise feature vector relations and are treated as representations of the manifold that data lie on; embedded vectors are then generated so that they preserve the corresponding pair-wise proximities on the manifold. More recently, nodal vector embedding of a graph has attracted considerable attention in different fields, and is often posed as the factorization of a properly defined node similarity matrix
\cite{large_scale_fact,text_fact,mf,hope,netmf,grarep,factor2,factor3}. Efforts in this direction mostly focus on designing meaningful similarity metrics to factorize. While some methods (e.g. \cite{large_scale_fact,hope}) maintain scalability by factorizing similarity matrices in an implicit manner (without explicitly forming them), others such as \cite{netmf,grarep} form and/or factorize dense similarity matrices that scale poorly to large graphs. Another line of work opts to gradually fit pairs of embedded vectors to existing edges using stochastic optimization tools \cite{LINE,PTE}. Such approaches are naturally scalable and entail a high degree of locality. Recently, stochastic edge-fitting has been generalized to implicitly accommodate long-range node similarities \cite{VERSE}. Meanwhile, other works have approached node embeddings using random-walk-based tools and concepts originating from natural language processing \cite{deepwalk,node2vec,attention}; see also related works on embedding of knowledge graphs \cite{know1,know2,know3}. Methods that rely on graph convolutional neural networks and autoencoders have also been proposed for node embedding \cite{neural1,neural2,neural3}. Moreover, a gamut of related embedding tasks are gaining traction, such as embedding based on structural roles of nodes \cite{struct2vec,graphwave}, supervised embeddings for classification \cite{planetoid}, and inductive embedding methods that utilize multiple graphs \cite{inductive}
\\
We identify the following \emph{challenges} that need to be addressed in order to design embedding methods that are applicable in practice:
\begin{itemize}
\item \textbf{Diversity}. Since graphs that arise from different domains are generally characterized by a diverse set of properties, there may not be a ``one-size-fits-all'' node embedding approach.
\item \textbf{No supervision}. At the same time, node embedding may need to be performed in a \emph{fully unsupervised} manner, that is, without extra information (node attributes, labels, or groundtruth communities) to guide the parameter tuning process with cross-validation.
\item \textbf{Scalability}. While some real-world networks are of moderate size, others may contain massive numbers of nodes and edges. Specifically, graphs encountered with social networks, transportation networks, knowledge graphs and others, typically scale to millions of nodes and tens of millions of edges. Thus, strict computational constraints must be accounted by the design of node embedding methods.
\end{itemize}
In response to these challenges, we propose a scalable node embedding framework that is based on factorizing an adaptive node similarity matrix. The first challenge is addressed by utilizing a large family of node similarity metrics, parametrized by placing different weights on node proximities of different orders; see also our precursor work \cite{adadif}. Experiments indicate that the proposed model for similarity metrics is expressive enough to describe real-world graphs from diverse domains and with different structures. To address the second challenge (lack of supervision), we put forth a self-supervised parameter learning scheme based on predicting randomly removed edges. Finally, we accommodate scalability by constraining the parametrization of similarity matrices such that the proximity order parameters carry over to the embedded vectors in a smooth manner. This allows for learning proximity order parameters directly on the feature vectors. Consequently, dense similarity matrices do not need to be explicitly formed and factorized, thus endowing the proposed method with the desired level of scalability.
The rest of the paper is organized as follows. Section 2 introduces the problem and the proposed similarity model. Section 3 presents a numerical study on model properties, while Section 4 deals with learning the model parameters in an unsupervised manner. Finally, Section 5 discusses related methods, and Section 6 contains experiments on real graphs, comparisons with competing alternatives, and interpretation of the results. While notation is defined wherever it is introduced, we also summarize the most important symbols that appear throughout the paper in Table 1. \\
\begin{table}[t]\caption{Important Notation}
\centering
\begin{tabular}{r c p{6cm} }
\toprule
$\mathcal{V}$ & $\triangleq$ & Set of nodes\\
$\mathcal{E}$ & $\triangleq$ & Set of edges\\
$\mathbf{A}$ & $\triangleq$ & $N\times N$ adjacency matrix\\
$\mathbf{D}$ & $\triangleq$ & $\mathrm{diag}(\mathbf{1}^T\mathbf{A})$ diagonal degree matrix \\
$\mathbf{E}$ & $\triangleq$ & $N\times d$ matrix of embeddings\\
$\mathbf{e}_i$ & $\triangleq$ & Embedding vector of node $v_i$ \\
$s_\mathcal{G}(\cdot,\cdot)$ & $\triangleq$ & Node -- to -- node similarity \\
$s_k (\cdot,\cdot)$ & $\triangleq$ & $k-$hop node -- to -- node similarity \\
$s_\mathcal{E}(\cdot,\cdot)$ & $\triangleq$ & Embedding -- to -- embedding similarity \\
$\ell(\cdot,\cdot)$ & $\triangleq$ & Distance (loss) between similarities \\
$\mathbf{S}_{\mathcal{G}}$& $\triangleq$ & Final node similarity matrix \\
$\mathbf{S}$ & $\triangleq$ & Basic sparse (single-hop) and symmetric node similarity matrix \\
$\theta_k$& $\triangleq$ & Coefficient of $k$-hop paths \\
$\boldsymbol{\theta}$& $\triangleq$ & $[\theta_1, \ldots, \theta_K]^T$ vector of coefficients \\
$\mathcal{S}^K$ & $\triangleq$ & $K-$dimensional probability simplex \\
$\mathcal{S}^+$ & $\triangleq$ & Set of sampled positive edges\\
$\mathcal{S}^-$ & $\triangleq$ & Set of all sampled negative edges\\
$\mathcal{S}$ & $\triangleq$ & $\mathcal{S}^+\cup\mathcal{S}^-$ all sampled edges\\
$N_s$ & $\triangleq$ & Number of sampled edges\\
$\boldsymbol{\theta}^\ast_\mathcal{S}$ & $\triangleq$ & Optimal coefficients that fit sample $\mathcal{S}$\\
$T_s$ & $\triangleq$ & Number of different edge samples\\
\bottomrule
\end{tabular}
\label{tab:TableOfNotationForMyResearch}
\end{table}
\section{Problem Statement and Modeling}\label{sec:problem}
Given an undirected graph $\mathcal{G}:=\{ \mathcal{V},\mathcal{E}\}$, where $\mathcal{V}$ is the set of $N$ nodes, and $\mathcal{E} \subseteq \mathcal{V}\times \mathcal{V}$ is the set of edges, the task of node embedding boils down to determining $f(\cdot):\mathcal{V}\rightarrow\mathbb{R}^d$, where $d\ll N$. In other works, a function is sought to map every node of $\mathcal{G}$ to a vector in the $d-$dimensional Euclidean space. Typically, the embedding is low dimensional with $d$ much smaller than the number of nodes. Given $f(\cdot)$, the low-dimensional vector representation of each node $v_i$ is
\begin{equation*}
\mathbf{e}_i =f(v_i) ~~\forall v_i \in \mathcal{V}\;.
\end{equation*}
Since the number of nodes is finite, instead of finding a general $f(\cdot)$ (induction), one may pose the embedding task in its most general form as the following minimization problem over the embedded vectors
\begin{equation} \label{most_general}
\{\mathbf{e}^\ast_i\}_{i=1}^N = \arg\underset{\{\mathbf{e}_i\}_{i=1}^N}{\min} \sum_{v_i,v_j\in\mathcal{V}} \ell\left(s_{\mathcal{G}}(v_i,v_j), s_{\mathcal{E}}(\mathbf{e}_i,\mathbf{e}_j)\right)
\end{equation}
where $\ell(\cdot,\cdot): \mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ is a loss function; $s_{\mathcal{G}}(\cdot,\cdot): \mathcal{V}\times\mathcal{V}\rightarrow\mathbb{R}$ is a similarity metric over pairs of graph \emph{nodes}; and $s_{\mathcal{E}}(\cdot,\cdot): \mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}$ a similarity metric over pairs of \emph{vectors} in the $d-$dimensional Euclidean space.
In par with \eqref{most_general}, node embedding can be viewed as the design of nodal vectors $\{\mathbf{e}_i\}_{i=1}^N$ that successfully ``encode'' a certain notion of pairwise similarities among graph nodes.
\subsection{Embedding as matrix factorization}
Starting from the generalized framework in \eqref{most_general}, one may arrive at concrete approaches by specifying choices of $s_{\mathcal{G}}(\cdot,\cdot)$, $s_{\mathcal{E}}(\cdot,\cdot)$, and $\ell(\cdot,\cdot)$. To start, suppose that the node similarity metric is symmetric; that is, $s_{\mathcal{G}}(v_i,v_j)=s_{\mathcal{G}}(v_j,v_i)~\forall v_i,v_j\in\mathcal{V}$. Furthermore, let the loss function be quadratic
\begin{equation*}
\ell(x,x^\prime) = \left( x - x^\prime \right)^2
\end{equation*}
and the nodal vector similarity be the inner product
\begin{equation*}
s_{\mathcal{E}}(\mathbf{e}_i,\mathbf{e}_j) = \mathbf{e}_i^\top\mathbf{e}_j.
\end{equation*}
Using these specifications, \eqref{most_general} reduces to the following symmetric matrix factorization problem
\begin{equation} \label{general}
\mathbf{E}^\ast = \arg\underset{\mathbf{E}\in\mathbb{R}^{N\times d}}{\min}
\|\mathbf{S}_{\mathcal{G}}-\mathbf{E}\mathbf{E}^\top\|_F^2
\end{equation}
where $\mathbf{S}_{\mathcal{G}}\in\mathbb{R}^{N\times N}$ is the symmetric similarity matrix with $\left[\mathbf{S}_{\mathcal{G}}\right]_{i,j}=\left[\mathbf{S}_{\mathcal{G}}\right]_{j,i}=s_{\mathcal{G}}(v_i,v_j)$, and matrix $\mathbf{E} := \left[\mathbf{e}_1 \ldots \mathbf{e}_N\right]^\top$ concatenates all node embeddings as rows.
A well-known analytical solution to \eqref{general} relies on the singular value decomposition (SVD) of the similarity matrix, that is $\mathbf{S}_{\mathcal{G}} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^T$, where $\mathbf{U}$ and $\mathbf{V}$ are the $N\times N$ unitary matrices formed by the left and right singular vectors, and $\boldsymbol{\Sigma}$ is diagonal with non-negative singular values sorted in decreasing order; in our case, $\mathbf{U}=\mathbf{V}$ since $\mathbf{S}_{\mathcal{G}}$ is symmetric. Given the SVD of $\mathbf{S}_{\mathcal{G}}$, the low-rank ($d\ll N$) solver in \eqref{general} is
$\mathbf{E}^\ast = \mathbf{U}_d \boldsymbol{\Sigma}_d^{1/2}$, where $\boldsymbol{\Sigma}_d$ contains the $d$ largest singular values, and $\mathbf{U}_d$ the corresponding singular vectors~\cite{svd}. Matrices $\mathbf{U}_d$ and $\boldsymbol{\Sigma}_d$ can be obtained directly using the reduced-complexity scheme known as \emph{truncated} SVD.
If in addition $\mathbf{S}_{\mathcal{G}}$ is \emph{sparse}, \eqref{general} can be solved even more efficiently, with complexity that scales with the number of edges. One such example with sparse similarities is when $\mathbf{S}_{\mathcal{G}}=\mathbf{A}$, where $\mathbf{A}$ is the graph
adjacency matrix. Embeddings generally gain scalability by avoiding the explicit construction of a \emph{dense} $\mathbf{S}_{\mathcal{G}}$. In fact, simply storing $\mathbf{S}_{\mathcal{G}}$ in the working memory becomes prohibitive even for graphs of moderate sizes (say $N > 10^5$).
In the ensuing section, we will design a family of dense similarity matrices that (among other properties) can be decomposed implicitly, at the cost of input sparsity.
\subsection{Multihop graph node similarities}
Having reduced the node embedding problem to the one in \eqref{general}, it remains to specify the graph similarity metric that gives rise to $\mathbf{S}_{\mathcal{G}}$. Towards this end, and in order to maintain expressibility, we will design a parametric model for $\mathbf{S}_{\mathcal{G}}$, with each pairwise node similarity metric expressed as
\begin{equation} \label{multipath_similarity}
s_{\mathcal{G}}(v_i,v_j;\boldsymbol{\theta}) = \sum_{k=1}^K \theta_k s_k(v_i,v_j), ~~~\mathrm{s.t.}~~~ \boldsymbol{\theta} \in \mathcal{S}^K
\end{equation}
where $\mathcal{S}^K:=\{ \boldsymbol{\theta} \in \mathbb{R}^K : \boldsymbol{\theta}\geq \mathbf{0}, \boldsymbol{\theta}^\top \mathbf{1}=1 \}$ is the $K$-dimensional probability simplex, and $s_k (v_i,v_j)$ is a similarity metric that depends on all $k$-hop paths of possibly repeated nodes that start from $v_i$ and end at $v_j$ (or vice-versa). Thus, $s_{\mathcal{G}}(\cdot,\cdot;\boldsymbol{\theta})$ contains all $k$-hop interactions between two nodes, each weighted by a non-negative importance score $\theta_k$ with $k=1,\ldots,K$.
Let $\mathbf{S}$ be any similarity matrix that is characterized by the same sparsity pattern as the adjacency matrix, that is
\begin{align} \label{sparse_pattern}
S_{i,j}=\left\{ \begin{array}{cc}
s_{i,j},~&~ (i,j) \in \mathcal{E} \\
0,~&~(i,j) \notin \mathcal{E}
\end{array} \right.,
\end{align}
where $\{s_{i,j}\}$s denote the generic non-negative values of entries that correspond to edges of $\mathcal{G}$. Maintaining the same sparsity pattern as $\mathbf{A}$ allows for the $(i,j)$ entry of $\mathbf{S}^k$ to be interpreted as a measure of influence between $v_i$ and $v_j$ that depends on all $k$-hop paths that connect them; that is, $\left[\mathbf{S}^k\right]_{i,j}=s_k(v_i,v_j)$. For instance, selecting $\mathbf{S}=\mathbf{A}$ is equivalent to using the $k$-step similarity
$s_k(v_i,v_j) = | \{k-\mathrm{length~paths~connecting}~ v_i~ \mathrm{to}~ v_j \} |$ \cite{arope}. Likewise, if $\mathbf{S}=\mathbf{A}\mathbf{D}^{-1}$ where $\mathbf{D} = \mathrm{diag}(\mathbf{1}^T\mathbf{A})$, then $s_k(v_i,v_j)$ can be interpreted as the probability that a random walk starting from $v_j$ lands on $v_i$ after exactly $k$ steps, e.g., \cite{grarep}. Thus, for a properly selected $\mathbf{S}$ with entries as in \eqref{sparse_pattern}, tunable multihop similarity metrics in \eqref{multipath_similarity} can be collected as entries of the power series matrix
\begin{equation} \label{our_similarity}
\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta}) = \sum_{k=1}^K \theta_k \mathbf{S}^k, ~~~\mathrm{s.t.}~~~ \boldsymbol{\theta} \in \mathcal{S}^K\;.
\end{equation}
Upon substituting \eqref{our_similarity} into \eqref{general} yields the tunable embeddings $\mathbf{E}^\ast(\boldsymbol{\theta})$ that depend on the choice of parameters $\boldsymbol{\theta}$. From the eigen-decomposition $\mathbf{S} = \mathbf{U}\boldsymbol{\Sigma} \mathbf{U}^\top $, and given that $\mathbf{U}^\top\mathbf{U}=\mathbf{I}$, we readily arrive at
\begin{equation} \label{power}
\mathbf{S}^k = \mathbf{U}\boldsymbol{\Sigma}^k \mathbf{U}^\top
\end{equation}
and after plugging \eqref{power} into \eqref{our_similarity}, we obtain
\begin{equation} \label{nice_form}
\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta}) = \mathbf{U} \left( \sum_{k=1}^K \theta_k \boldsymbol{\Sigma}^k \right) \mathbf{U}^\top , ~~~\mathrm{s.t.}~~~ \boldsymbol{\theta} \in \mathcal{S}^K\:.
\end{equation}
Furthermore, the truncated singular pairs of $\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta})$ conveniently follow from those of $\mathbf{S}$, and they have to be computed once. Specifically, the truncated singular vectors and singular values are $\mathbf{U}_d(\boldsymbol{\theta})=\mathbf{U}_d$ and $\boldsymbol{\Sigma}_d(\boldsymbol{\theta})=\sum_{k=1}^K \theta_k \boldsymbol{\Sigma}_d^k$, respectively. Thus, if $\mathbf{S}\in \mathrm{Sym}_N$ the solution to \eqref{general} with $\mathbf{S}_{\mathcal{G}}$ parametrized by $\boldsymbol{\theta}$ is simply given as
\begin{equation}\label{solution}
\mathbf{E}^\ast(\boldsymbol{\theta}) = \mathbf{U}_d \sqrt{\boldsymbol{\Sigma}_d(\boldsymbol{\theta})}\;.
\end{equation}
Note that this holds only for non-negative parameters $\theta_k\geq 0~\forall~k$. If $\theta_k<0$ for at least one $k\in \{1,\ldots,K\}$, then the diagonal entries of $\boldsymbol{\Sigma}_d(\boldsymbol{\theta})$ cannot be guaranteed to be non-negative and sorted in decreasing order, which would cause $\left(\mathbf{U}_d(\boldsymbol{\theta}), \boldsymbol{\Sigma}_d(\boldsymbol{\theta})\right)$ to \emph{not} be a valid SVD pair.
Having narrowed down $\mathbf{S}_{\mathcal{G}}$ to belong to the parametrized family in \eqref{our_similarity}, we proceed to select an appropriate sparsity-preserving $\mathbf{S}$ in order to obtain a solid model.
\subsection{Spectral multihop embeddings }
While any symmetric $\mathbf{S}$ that obeys \eqref{sparse_pattern} can be used for constructing multihop similarities (cf. \eqref{our_similarity}), judicious designs of
$\mathbf{S}$ can effect certain desirable properties. Bearing this in mind, consider the following identity
\begin{equation}\label{equiv}
\mathbf{S}\in \mathcal{P}^+_N ~\iff~ \mathbf{S} = \mathbf{U}\boldsymbol{\Sigma} \mathbf{U}^\top = \mathbf{U}\boldsymbol{\Lambda} \mathbf{U}^\top
\end{equation}
where $\mathcal{P}^+_N$ denotes the space of $N\times N$ symmetric positive definite (SPD) matrices, and $\boldsymbol{\Lambda}$ is the diagonal matrix that contains the eigenvalues of $\mathbf{S}$ sorted in decreasing order. For SPD matrices as in \eqref{equiv}, the SVD is identical to the eigenvalue decomposition (EVD). Thus, if $\mathbf{S}\in \mathcal{P}^+_N$, the solution to \eqref{general} is also given as (cf. \eqref{solution})
\begin{equation}\label{solution2}
\mathbf{E}^\ast(\boldsymbol{\theta}) = \mathbf{U}_d \sqrt{\boldsymbol{\Lambda}_d(\boldsymbol{\theta})}
\end{equation}
where $\mathbf{U}_d$ are also the first $d$ \emph{eigenvectors} of $\mathbf{S}$, and $\boldsymbol{\Lambda}_d(\boldsymbol{\theta})=\sum_{k=1}^K \theta_k \boldsymbol{\Lambda}_d^k$ is the $K$th order polynomial of its eigenvalues defined by $\boldsymbol{\theta}$.
Consider now specifying $\mathbf{S}$ as
\begin{equation} \label{my_matrix}
\mathbf{S} = \frac{1}{2}\left( \mathbf{I} + \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2} \right).
\end{equation}
Recalling that $\lambda_i\left( \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2} \right)\in [-1,1]~\forall~i$, and after using the identity shifting and scaling, we deduce that $\lambda_i(\mathbf{S})\in [0,1]~\forall~i$; hence, matrix $\mathbf{S}$ in \eqref{my_matrix} is SPD. It can also be readily verified that the first $d$ eigenvectors of $\mathbf{S}$ coincide with the eigenvectors corresponding to the $d$ smallest eigenvalues of the symmetric normalized Laplacian matrix
\begin{equation}\label{lsym}
\mathbf{L}_{\mathrm{sym}}:= \mathbf{I} - \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}.
\end{equation}
These smallest eigenvalues are known to contain useful information on cluster structures of different resolution levels, a key property that has been successfully employed by spectral clustering \cite{spectral}. Intuitively, assigning weight $\boldsymbol{\theta}_k$ to $k$-hop paths in the node similarity of \eqref{our_similarity}, is equivalent to shrinking the $d$-dimensional spectral node embeddings (rows of $\mathbf{U}_d$) coordinates according to $\boldsymbol{\Lambda}_d(\boldsymbol{\theta})$. Interestingly, assigning large weights to longer paths ($K\gg 1$) is equivalent to fast shrinking the coordinates that correspond to small eigenvalues and capture the fine-grained structures and local relations, what leads to a coarse, high-level cluster description of the graph.
\subsection{Relation to random walks}
Apart from the spectral embedding interpretation discussed in the last subsection, using powers of $\eqref{my_matrix}$ to capture multihop similarities also admits an interesting random walk interpretation. We begin by expressing the $k$th power of $\mathbf{S}$ as
\begin{align}\nonumber
\mathbf{S}^k &= \frac{1}{2^k}\left( \mathbf{I} + \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2} \right)^k \\ \label{expand1}
&=\sum_{\tau=0}^k \alpha_{\tau}(k) \left(\mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}\right)^\tau
\end{align}
where the sequence
\begin{equation} \label{binom}
\alpha_{\tau}(k) :=\left\{ \begin{array}{cc}
\frac{1}{2^k}\binom{k}{\tau},~&~ 0 \leq \tau \leq k \\
0,~&~ \mathrm{else}
\end{array} \right.
\end{equation}
can be interpreted as nonzero weights that $\mathbf{S}^k$ assigns to all paths with the number of hops \emph{up to} $k$ (see Fig. \ref{fig:Sk}).
Using \eqref{expand1} and \eqref{binom}, the multihop similarity in \eqref{our_similarity} becomes
\begin{align} \nonumber
\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta}) &= \sum_{\tau=0}^K c_\tau(\boldsymbol{\theta}) \left(\mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2}\right)^\tau \\ \label{expand2}
& = \mathbf{D}^{-1/2} \left( \sum_{\tau=0}^K c_\tau(\boldsymbol{\theta}) \mathbf{P}^\tau \right) \mathbf{D}^{1/2}
\end{align}
where
\begin{equation}\label{c}
c_\tau(\boldsymbol{\theta}) := \sum_{k=1}^K \theta_k \alpha_{\tau}(k)
\end{equation}
and $\mathbf{P} = \mathbf{AD}^{-1}$ is the probability transition matrix of a simple random walk defined over $\mathcal{G}$; that is, $P_{i,j}$ is the probabiity that a random walker positioned on node (state) $j$ transitions to node $i$ in one step. Thus, the $k$-hop similarity function defined in \eqref{multipath_similarity} is expressed as
\begin{equation}
s_{\mathcal{G}}(v_i,v_j,\boldsymbol{\theta}) = \sqrt{\frac{d_j}{d_i}} \sum_{\tau=0}^K c_\tau(\boldsymbol{\theta}) \Pr \{ X_\tau = v_i | X_0 = v_j\}
\end{equation}
where $\Pr \{ X_\tau = v_i | X_0 = v_j\}:=\left[\mathbf{P}^\tau\right]_{ij}$ is the probability that a random walk starting from $v_j$ lands on $v_i$ after $\tau$ steps.
Interestingly, $\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta})$ does not weigh landing probabilities of different lengths independently. Instead, it accumulates the latter as weighted combinations (cf. \eqref{c}) in a basis of ``wavelet''-type functions of different resolution (see Fig. 1).
Having established links to spectral clustering and random walks, our novel $\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta})$ is well motivated as a family of node similarity matrices. Nevertheless, before devising an algorithm for learning $\boldsymbol{\theta}$ and testing it on real graphs, we will evaluate how well the basis $\{ \mathbf{S}^k\}_{k=1}^K$, on which $\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta})$ is built, can capture underlying node similarities.
\setlength\figW{0.99\columnwidth}
\setlength\figH{0.53\columnwidth}
\begin{figure}[t!]
\hspace{0.25in}\input{figs/a_tau_k_save.tex}
\vspace{-0.15in}
\caption{Matrix $\mathbf{S}^k$ is equivalent to applying ``wavelet''-type weights $\alpha_\tau(k)$ over walks with hops $\leq k$.}
\label{fig:Sk}
\end{figure}
\section{Model expressiveness}
\begin{figure*}[t!]
\centering
\begin{subfigure}[b]{0.28\textwidth}
\centering
{True SBM similarities ($\mathbf{S}^\ast$)}
\vspace{-0.3cm}
\includegraphics[width=\textwidth]{figs/A_ideal-eps-converted-to.pdf}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.28\textwidth}
\centering
{PageRank ($\hat{\mathbf{S}}_{PPR}$)}
\vspace{-0.25cm}
\includegraphics[width=\textwidth]{figs/A_pgr-eps-converted-to.pdf}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.28\textwidth}
\centering
{Adamic-Adar ($\hat{\mathbf{S}}_{AA}$)}
\vspace{-0.25cm}
\includegraphics[width=\textwidth]{figs/A_adam-eps-converted-to.pdf}
\label{fig:gull}
\end{subfigure}
\begin{subfigure}[b]{0.28\textwidth}
\centering
{Proposed ($\mathbf{S}^1$)}
\vspace{-0.25cm}
\includegraphics[width=\textwidth]{figs/A1-eps-converted-to.pdf}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.28\textwidth}
\centering
{Proposed ($\mathbf{S}^6$)}
\vspace{-0.25cm}
\includegraphics[width=\textwidth]{figs/A6-eps-converted-to.pdf}
\label{fig:gull}
\end{subfigure}
~
\begin{subfigure}[b]{0.28\textwidth}
\centering
{Proposed ($\mathbf{S}^{15}$)}
\vspace{-0.25cm}
\includegraphics[width=\textwidth]{figs/A15-eps-converted-to.pdf}
\label{fig:gull}
\end{subfigure}
\vspace{-0.2in}
\caption{ Depiction of groundtruth and estimated similarity matrices, as yielded from an instance of the numerical experiments described in Section 3.1.} \label{matplot}
\vspace{-0.2in}
\end{figure*}
\setlength\figW{0.99\columnwidth}
\setlength\figH{0.6\columnwidth}
\begin{figure*}[t!]
\vspace{0.2in}
\centering
\input{figs/p_3_q_1_new.tex}
\input{figs/p_015_q_05_new.tex}
\input{figs/p_10_q_10_new.tex}
\phantom{ppppp}\ref{named2}
\caption{ Quality of match between true SBM similarity and various estimates, as yielded from experiments of Section 3.1.} \label{curves}
\end{figure*}
This section introduces a performance metric that quantifies how well a node similarity matrix derived from the graph itself matches the ``true'' underlying similarity structure between nodes. The discussion is followed by numerical evaluation of the performance of different similarity matrices (including the one in \eqref{expand1}) on graphs that are generated according to the stochastic block model \cite{sbm}.
To begin, suppose that for a given set of nodes, an adjacency matrix $\mathbf{A}$ is generated as
\begin{equation*}
\mathbf{A} \sim f_A(\mathbf{A})
\end{equation*}
where $f_A(\mathbf{A})$ is a probability density function defined over the space of all possible adjacency matrices. Let the ``true'' underlying similarity between nodes $v_i$ and $v_j$ be
\begin{equation*}
s^\ast(v_i,v_j) := \Pr\{ (i,j) \in \mathcal{E} \} = \mathbb{E}_{f_A}\left[ A_{i,j} \right]
\end{equation*}
which is the probability that the two nodes are connected. The ``true'' similarity matrix is thus given as the expected adjacency matrix
\begin{equation*}
\mathbf{S}^\ast := \mathbb{E}_{f_A}\left[ \mathbf{A} \right].
\end{equation*}
We define the quality-of-match (QoM) between the underlying $\mathbf{S}^\ast$ and any similarity $\hat{\mathbf{S}} = F(\mathbf{A})$ estimated from the adjacency matrix as
\begin{equation}\label{quality}
\mathrm{QoM} := \mathbb{E}_{f_A}\left[\mathrm{PC}\left(\mathbf{S}^\ast,F(\mathbf{A})\right)\right]
\end{equation}
where
\begin{equation}\label{pearson}
\mathrm{PC}\left(\mathbf{X}_1,\mathbf{X}_2\right):= \frac{\left(\mathrm{vec}\left(\mathbf{X}_1\right)\right)^\top \mathrm{vec}\left(\mathbf{X}_2\right)}{ \|\mathbf{X}_1\|_F \|\mathbf{X}_2\|_F }
\end{equation}
is the Pearson correlation between two matrices $\mathbf{X}_1$ and $\mathbf{X}_2$, with $\mathrm{vec}\left(\mathbf{X}\right)$ denoting matrix vectorization. The latter is used for appropriate rescaling of the ``true'' similarity matrix in order for the comparison with $\mathbf{S}_{\mathcal{G}}$ to be meaningful. Intuitively, \eqref{quality} measures how well the estimated node similarities in $\hat{\mathbf{S}}$ are expected to match the pattern of true underlying similarities in $\mathbf{S}^\ast$, when edges are generated according to the known $f_A(\cdot)$.
\subsection{Numerical experiments and observations}
We numerically evaluate the QoM achieved by different similarity matrices, on a set of $N$ nodes whose interconnections are generated according to a stochastic block model (SBM). For this set of experiments, we divided the nodes into three clusters of equal size
\begin{equation*}\label{clusters}
\mathcal{C}_l = \{ i: (l-1)N/3 \leq i \leq lN/3 \},~l\in \{ 1,2,3 \}
\end{equation*}
with inter- and intra-connection probabilities
\begin{equation}\label{probs}
\Pr\{ (i,j) \in \mathcal{E} \} = \left \{ \begin{array}{cc}
p~&~, (i,j) \mathrm{~in~the~same~}\mathcal{C}_l \\
cq~&~, i\in\mathcal{C}_1\mathrm{~and~}j\in\mathcal{C}_3 \\
q~&~\mathrm{else}
\end{array} \right.
\end{equation}
where $p$ is the probability of connection when two nodes belong to the same cluster, and $c<1$ introduces asymmetry and a hierarchical clustering organization (see Fig. 2-top left), by making two of the clusters less likely to connect; we have related Python scripts available.\footnote{https://github.com/DimBer/ASE-project/tree/master/sim\textunderscore tests} The SBM probability matrix \cite{sbm} is given as
\begin{equation} \label{sbm_mat}
\mathbf{W}_\mathrm{sbm} =\left[ \begin{array}{ccc}
p~&~ q~&~ c q \\
q~&~p~&~q \\
c q~&~q~&~p
\end{array} \right]
\end{equation}
and the underlying similarity can be expressed as
\begin{equation}\label{true_similarity}
\mathbf{S}^\ast = \mathbb{E}\left[\mathbf{A}\right] = \mathbf{W}_\mathrm{sbm} \otimes \left( \mathbf{1}_{N/3}\mathbf{1}_{N/3}^T \right) - \mathrm{diag}(p\mathbf{1}_N)
\end{equation}
where $\otimes$ denotes the Kronecker product.
For each experiment, we set $N=150$ and generated a graph according to
\eqref{probs}. We then compared the QoM between \eqref{true_similarity} and the $k$th power of the proposed \eqref{my_matrix}, the $k$th power of the adjacency ($\mathbf{A}^k$), as well as each of the following well known similarity metrics:
\begin{itemize}
\item $\hat{\mathbf{S}}_{PPR}:= (1-\alpha)(\mathbf{I} - \alpha \mathbf{AD}^{-1})^{-1}$: the steady state probability that a random walk restarting at $v_j$ with probability $1-\alpha$ at every step is located at $v_i$. Essentially a personalized PageRank (PPR) computed for every node of the graph, inheriting the properties of the celebrated centrality measure \cite{brin2012reprint,GleichBeyond,kloumann2017block}.
\item $\hat{\mathbf{S}}_{KATZ}:= (1-\beta)(\mathbf{I} - \beta \mathbf{A})^{-1}\mathbf{A}$ : the Katz index \cite{arope}, an exponentially weighted summation over paths of all possible hops between
two nodes.
\item $\hat{\mathbf{S}}_{NEIGH}:=\mathbf{A}^2$: the number of common neighbors that every pair of nodes shares.
\item $\hat{\mathbf{S}}_{AA}:=\mathbf{A}\mathbf{D}^{-1}\mathbf{A}$: Adamic-Adar \cite{adamic} is a variant of common neighbors
where each set of neighbors is weighted inversely proportional to its cardinality.
\end{itemize}
The resulting QoM was averaged over 200 experiments. Parameters $\alpha$ in $\hat{\mathbf{S}}_{PPR}$ and $\beta$ in $\hat{\mathbf{S}}_{KATZ}$ were tuned to maximize the performance of the metrics. Figure 3 depicts QoM as a function of $k$, for three different scenarios.
In the first scenario (Fig. 3-a), with graphs being dense and clustered ($p=0.3$, $q=0.1$), the proposed $\mathbf{S}^k$ improves sharply in the first few steps, reaching maximum QoM after 4 or 5 steps, and gradually decreases as $k$ continues to increase. The $k$th order proximities that are given as entries of $\mathbf{A}^k$ follow a similar trend, however their QoM peaks shortly after 2 or 3 steps and declines fast for larger $k$. The matrix plots of a randomly selected experiment depicted in Fig. 2 can aid in understanding the underlying mechanism that gives rise to this highly step-dependent behavior. Specifically, $\mathbf{S}^1$ (bottom left) that has the same sparsity pattern as the adjacency is a poor match to the dense block-structure of $\mathbf{S}^\ast$. On the other side of the spectrum, $\mathbf{S}^{15}$ (bottom right) is too ``flat'' and also a poor similarity metric. Meanwhile, taking $k=6$ promotes enough mixing without ``dissipating.'' As a result, $\mathbf{S}^{6}$ (bottom center) visibly matches the structure of $\mathbf{S}^\ast$.
Interestingly, for $k\in[4,10]$ the proposed $\mathbf{S}^k$ surpasses in QoM all other similarity metrics that were tested. Nevertheless, the simple $2-$hop Adamic-adar, common-neighbors similarities perform reasonably well by exploiting the relatively dense structure of the graphs.
Results were markedly different in the second scenario shown in Fig. 3-b. Here, graphs were generated with the same clustering structure but significantly sparser, with edge probability parameters $p=0.15$ and $q=0.05$. For sparser graphs, $\mathbf{A}^k$ and $\mathbf{S}^k$ require more steps to reach peak QoM (4 and 9 respectively). Similarly, PPR which relies on long paths performs much better than the short-reaching Adamic-Adar. This behavior is intuitively reasonable because the sparser a graph is, the longer become the paths that need to be explored around each node, in order for the latter to ``gauge'' its position on the graph.
Finally, a third scenario (Fig. 3-c) was examined, where each graph was generated without a clustering structure ($p=q=0.1$ and $c=1$); essentially an Erdos-Renyi graph. For this degenerate case that is of no real practical interest, all pairs of nodes are equally similar; this type of similarity requires infinitely long paths to be described.
In a nutshell, the presented numerical study hints at the two following facts. First, $\mathbf{S}^k$ can successfully model similarities that are based on grouping nodes in arbitrary and multilevel sets with variable degrees of homophily and heterophily. The second fact, is that the performance of $\mathbf{S}^k$ varies significantly with $k$. Moreover, the way that $k$ affects performance may also vary from graph to graph, depending on the underlying properties -- what suggests viewing this way as a graph ``signature'' that is also validated by the real graphs in Section 6. Thus, a principled means of specifying $\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta})$ by learning the parameters that match this graph ``signature'' in an unsupervised mode, is highly motivated.
\section{ Unsupervised similarity learning}\label{sec:main}
We have arrived at the point where for a given graph, it is prudent to select a specific $\boldsymbol{\theta}\in\mathcal{S}^K$ without supervision. Following the discussion in Section 3, it would be ideal to fit $\mathbf{S}_{\mathcal{G}}(\boldsymbol{\theta})$ to a true $\mathbf{S}^\ast$
by minimizing an expected cost
\begin{equation}\label{ideal}
\boldsymbol{\theta}^\ast = \arg\min_{\boldsymbol{\theta}\in \mathcal{S}^K}\mathbb{E}_{f_A}\left[\ell\left(\mathbf{S}^\ast,\mathbf{S}_\mathcal{G}(\mathbf{A};\boldsymbol{\theta})\right)\right] \:.
\end{equation}
Unfortunately, we only have one realization $\mathbf{A}$ of $f_A(\cdot)$, which means that without prior knowledge, the best approximation of $\mathbf{S}^\ast$ that we can obtain is the adjacency matrix itself, that is $\mathbf{S}^\ast \approx \mathbf{A}$. Using this approximation yields
\begin{equation}\label{wrong}
\min_{\boldsymbol{\theta}\in \mathcal{S}^K}\ell\left(\mathbf{A},\mathbf{S}_\mathcal{G}(\mathbf{A};\boldsymbol{\theta})\right).
\end{equation}
While straightforward, \eqref{wrong} yields embeddings with limited generalization capability. Simply put, regardless of the choice of $\ell(\cdot)$, solving \eqref{wrong} amounts to predicting a set of edges by tuning a similarity metric that is generated by the \emph{same} set of edges.
To mitigate overfitting but also promote generalization of the similarity metric and of the resulting embeddings, we explore the following idea. Suppose we are given a pair $\mathbf{A}_1,\mathbf{A}_2$ of adjacency matrices both drawn independently from $f_A(\cdot)$. In this case, we would be able to use one as approximation of $\mathbf{S}^\ast \approx \mathbf{A}_1$, and the other to form the multihop similarity matrix $\mathbf{S}_\mathcal{G}(\mathbf{A}_2;\boldsymbol{\theta})$; parameters $\boldsymbol{\theta}$ can then be learned by solving
\begin{equation}\label{ok1}
\min_{\boldsymbol{\theta}\in \mathcal{S}^K}\ell\left(\mathbf{A}_1,\mathbf{S}_\mathcal{G}(\mathbf{A}_2;\boldsymbol{\theta})\right).
\end{equation}
Since separate samples are not available, we approximate the aforementioned process by randomly extracting part of $\mathbf{A}$ and approaching \eqref{ok1} as
\begin{equation}\label{ok2}
\min_{\boldsymbol{\theta}\in \mathcal{S}^K}\ell_\mathcal{S}\left(\mathbf{A},\mathbf{S}_\mathcal{G}(\mathbf{A}\ast\mathbf{S}^c;\boldsymbol{\theta})\right)
\end{equation}
where $\mathcal{S} \in \{ 1, \ldots, N \}^2$ is a subset of all possible pairs of nodes with $|\mathcal{S}| = N_s$, and $\mathbf{S}^c$ is an $N\times N$ binary section matrix with $S^c_{i,j} =0$, if $\{i,j\}\in \mathcal{S}$, and $S^c_{i,j} =1$, otherwise. Furthermore, $\ell_\mathcal{S}(\cdot,\cdot)$ in \eqref{ok2} denotes cost $\ell(\cdot,\cdot)$ applied selectively only to entries of the matrix variables that belong to $\mathcal{S}$. Here, such that $ \mathcal{S} = \mathcal{S^+} \cup \mathcal{S^-}$, with $\mathcal{S^+}\in\mathcal{E}$ being as subset of the edges and $\mathcal{S^-}\in \{1, \ldots, N\}^2\setminus\mathcal{E}$ a subset of node index tuples that are not connected (non-edges). To balance the influence of existing and non-existing edges, we use subsets of equal cardinality, that is $|\mathcal{S}^+| =|\mathcal{S}^-| = N_s/2$.
To arrive from the unsupervised similarity learning framework \eqref{ok2} to a practical method, it remains to specify two modular sub-systems: one responsible for sampling edges, and one specifying $\ell(\cdot,\cdot)$ to find $\boldsymbol{\theta}^\ast$ by solving \eqref{ok2}.
\subsection{Edge sampling }
The choice of the sampling scheme for $\mathcal{S}$ plays an important role in the overall performance of the proposed adaptive embedding framework. Ideally, edge sampling should take into account the following criteria.
\begin{itemize}
\item Sample $\mathcal{S}^+$ should be representative of the graph;
\item Edge removal should inflict minimal perturbation;
\item Edge removal should avoid isolating nodes; and
\item Sampling scheme should be simple and scalable.
\end{itemize}
Aiming at a `sweet spot' of these objectives, we populate $\mathcal{S}^+$ by sampling edges according to the following procedure: first, a node $v_1$ is sampled uniformly at random from $\mathcal{V}$; then, a second node $v_2$ is sampled uniformly from the neighborhood set $\mathcal{N}_\mathcal{G}(v_1)$ of $v_1$. The selected edge is removed only if both adjacent nodes have degree greater than one. Non-edges $\mathcal{S}^-$ are obtained by uniform sampling without replacement over $\{1, \ldots, N\}^2\setminus\mathcal{E}$. The overall procedure is summarized in Algorithm \ref{alg:ES}. For $N_s\ll N$, sampling probabilities remain approximately unchanged despite the removals, since the probability of selecting the same node is relatively small. Thus, one may approximate $\Pr\{e_t = (i,j)\}\approx\Pr\{e_0 = (i,j)\}$, and assuming for simplicity that $d_i>1\forall i$, it follows that
\begin{align}\nonumber
\Pr\{e_0 = (i,j)\} &= \Pr\{v_1 = i, v_2 = j\} + \Pr\{v_1 = j, v_2 = i\}\\\nonumber
&=\Pr\{v_2 = i|v_1 = j\}\Pr\{v_1 = j\}\\\nonumber
&~+ \Pr\{v_2 = j|v_1 = i\}\Pr\{v_1 = i\}\\ \label{probability}
&= \frac{1}{d_j}\frac{1}{N} + \frac{1}{d_i}\frac{1}{N} \propto \frac{d_i + d_j}{d_i d_j},
\end{align}
meaning that edge $e=(i,j)$ is removed with probability that is proportional to the harmonic mean of the degrees of the nodes that it connects. As shown in \cite{pertubation}, the perturbation that the removal of edge $e=(i,j)$ inflicts on the spectrum of an undirected graph is proportional to $d_id_j$; that is, removing edges that connect high-degree nodes leads to higher perturbation. Thus, Algorithm \ref{alg:ES} tends to inflict minimal perturbation by sampling with probability that is inversely proportional to $d_id_j$ for $d_i,~d_j\gg 1$; this is because the denominator of \eqref{probability} dominates its numerator for large degrees. On the other hand, for smaller $d_i$ and $d_j$, the numerator ensures relatively high probabilities for moderate-degree nodes. The combination of the two effects yields edge samples that are fairly representative of the graph, while inflicting low perturbation when removed.
\subsection{Parameter training}
Subsequently, for a given sample $\mathcal{S}$, we can obtain the corresponding optimal parameters as (cf. \eqref{ok2})
\begin{equation}\label{good}
\boldsymbol{\theta}^\ast_\mathcal{S} = \arg\min_{\boldsymbol{\theta}\in \mathcal{S}^K}\sum_{i,j\in \mathcal{S}}\ell\left( A_{i,j}, s_{\mathcal{G}^-}(v_i,v_j;\boldsymbol{\theta}) \right)
\end{equation}
where $\mathcal{G}^- := \left( \mathcal{V}, \mathcal{E}\setminus\mathcal{S^+} \right)$ is the original graph with the randomly sampled subset $\mathcal{S^+}$ of edges removed.
Interestingly, one way that \eqref{good} could be solved is by explicitly computing the entries of $\mathbf{S}_\mathcal{G}(\boldsymbol{\theta})$ that are in $\mathcal{S}$. This would require performing $K$ sparse matrix-vector products to obtain every column of $\mathbf{S}^k$ for $k\in \{1,\ldots,K\}$, for all the columns that contain sampled entries. In the worst case, if all nodes in the tuples of $\mathcal{S}$ correspond to different columns of $\mathbf{S}_\mathcal{G}(\boldsymbol{\theta})$, two random walks are required for every tuple, for a total of $2N_s$ random walks. This requires $\mathcal{O}\left(N_sK|\mathcal{E}|\right)$ computations, and $\mathcal{O}\left(N_sN\right)$ memory if they are to be performed concurrently or in matrix form. Since $K$ will typically be in the order of tens, these requirements will be affordable, if $N_s$ is relatively small. Nevertheless, they quickly become cumbersome for $N_s\gg K$, which may be necessary to estimate the $K$-dimensional $\boldsymbol{\theta}$.
\begin{algorithm}[h!]
\caption{\textsc{Adaptive Similarity Embedding}}
\label{alg:ASE}
\begin{algorithmic}
\State \textbf{Input:} $\mathcal{G}$~~\textbf{Output:} $\mathbf{E}$
\State
\vspace{-0.1in}
\State // Training phase
\State $\boldsymbol{\Theta} =\emptyset$
\While { $|\boldsymbol{\Theta}|<T_s$ }
\State $\mathcal{G}^-$, $\mathcal{S}^+$, $\mathcal{S}^-~=$ \textsc{Sample Edges}( $\mathcal{G}$ )
\State $\boldsymbol{\theta}^\ast_\mathcal{S}~=$ \textsc{Train Parameters}( $\mathcal{G}^-,\mathcal{S}^+, \mathcal{S}^-$)
\State $\boldsymbol{\Theta} = \boldsymbol{\Theta} \cup \boldsymbol{\theta}^\ast_\mathcal{S}$
\EndWhile
\State $\boldsymbol{\theta}^\ast = T_s^{-1}\sum_{\boldsymbol{\theta}\in\boldsymbol{\Theta}}\boldsymbol{\theta}$
\State
\vspace{-0.1in}
\State // Embedding phase
\State $\mathbf{S} = \frac{1}{2}\left( \mathbf{I} + \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2} \right)$
\vspace{0.03in}
\State $\mathbf{S} = \mathbf{U}_d\boldsymbol{\Sigma}_d\mathbf{U}_d^T$
\vspace{0.03in}
\State $\boldsymbol{\Sigma}_d(\boldsymbol{\theta}^\ast) = \sum_{k=1}^K \theta_k^\ast \boldsymbol{\Sigma}_d^k$
\State
\vspace{-0.1in}
\State \Return $\mathbf{E} = \mathbf{U}_d \sqrt{\boldsymbol{\Sigma}_d(\boldsymbol{\theta}^\ast)}$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h!]
\caption{\textsc{Sample Edges}}
\label{alg:ES}
\begin{algorithmic}
\State \textbf{Input:} $\mathcal{G}$~~
\textbf{Output:} $\mathcal{G}^-,\mathcal{S}^+, \mathcal{S}^-$
\State
\vspace{-0.1in}
\State // Sample edges
\State $\mathcal{S}^+=\emptyset$, $\mathcal{G}^- = \mathcal{G}$
\While { $|\mathcal{S}^+|<N_s/2$ }
\State Sample $v_1\sim\mathrm{Unif}\left(\mathcal{V}\right)$
\If { $|\mathcal{N}_{\mathcal{G}^-}(v_1)|>1$ }
\State Sample $v_2\sim\mathrm{Unif}\left(\mathcal{N}_{\mathcal{G}^-}(v_1)\right)$
\If { $|\mathcal{N}_{\mathcal{G}^-}(v_2)|>1$ }
\State $\mathcal{S}^+=\mathcal{S}^+ \cup (v_1,v_2)$
\State $\mathcal{G^-} = \mathcal{G^-} \setminus (v_1,v_2)$
\EndIf
\EndIf
\EndWhile
\State
\vspace{-0.1in}
\State // Sample non-edges
\State $\mathcal{S}^-=\emptyset$
\While { $|\mathcal{S}^-|<N_s/2$ }
\State Sample $(v_1,v_2)\sim\mathrm{Unif}\left(\mathcal{V}\times\mathcal{V}\right)$
\If {$(v_1,v_2)\notin \mathcal{E}$}
\State $\mathcal{S}^-=\mathcal{S}^- \cup (v_1,v_2)$
\EndIf
\EndWhile
\State \Return $\mathcal{G^-}$, $\mathcal{S}^+$, $\mathcal{S}^-$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h!]
\caption{\textsc{Train Parameters}}
\label{alg:TRAIN}
\begin{algorithmic}
\State \textbf{Input:} $\mathcal{G}$, $\mathcal{S}^+$, $\mathcal{S}^-$ \textbf{Output:}
\vspace{0.03in} $\boldsymbol{\theta}^\ast_\mathcal{S}$
\State $\mathbf{S} = \frac{1}{2}\left( \mathbf{I} + \mathbf{D}^{-1/2} \mathbf{A} \mathbf{D}^{-1/2} \right)$
\vspace{0.03in}
\State $\mathbf{S} = \mathbf{U}_d\boldsymbol{\Sigma}_d\mathbf{U}_d^T$
\vspace{0.03in}
\State $\mathcal{S} = \mathcal{S}^+\cup \mathcal{S}^-$
\State Form $\mathcal{X_S}=\{ \mathbf{x}_{(i,j)}\}_{(i,j)\in\mathcal{S}}$ as in \eqref{feats}
\vspace{0.03in}
\vspace{0.03in}
\State \Return $\boldsymbol{\theta}^\ast_\mathcal{S}=~$\textsc{SimplexSVM}( $\mathcal{X_S},\mathcal{S}^+, \mathcal{S}^-$)
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[h!]
\caption{\textsc{SimplexSVM}}
\label{alg:SVMs On Simplex}
\begin{algorithmic}
\State \textbf{Input:} $\mathcal{X},\mathcal{S}^+,\mathcal{S}^-$
\textbf{Output:} $\boldsymbol{\theta}^\ast$
\vspace{0.03in}
\State $\boldsymbol{\theta}_0= \frac{1}{K} \mathbf{1},~t=1$
\vspace{0.03in}
\While { $\|\boldsymbol{\theta}_t - \boldsymbol{\theta}_{t-1}\|_\infty \geq \mathrm{tol}$ }
\State $t=t+1,$ $\eta_t=a/\sqrt{t}$
\vspace{0.03in}
\State $\mathcal{S}^+_a = \{e\in \mathcal{S}^+|~ \mathbf{x}_e^T\boldsymbol{\theta}_{t-1} \leq \epsilon \}$
\vspace{0.03in}
\State $\mathcal{S}^-_a = \{e\in \mathcal{S}^-|~ \mathbf{x}_e^T\boldsymbol{\theta}_{t-1} \geq -\epsilon \}$
\vspace{0.03in}
\State $\mathbf{g}_t = \sum_{e\in\mathcal{S}^-_a}\mathbf{x}_e - \sum_{e\in\mathcal{S}^+_a}\mathbf{x}_e$
\vspace{0.03in}
\State $\mathbf{z}_t=(1-2\eta_t\lambda)\boldsymbol{\theta}_{t-1}-\frac{\eta_t}{N_s}\mathbf{g}_t$
\vspace{0.03in}
\vspace{0.03in}
\State $\boldsymbol{\theta}_{t}=$\textsc{SimplexProj}( $\mathbf{z}_t$ )
\EndWhile
\State \Return $\boldsymbol{\theta}_{t}$
\end{algorithmic}
\end{algorithm}
Instead, we will rely on the fact that the proposed embeddings are smooth and differentiable wrt to $\boldsymbol{\theta}$ (cf. \eqref{solution2}), to develop a solution that allows for selecting arbitrarily large $N_s$, using the approximation
\begin{align}\nonumber
s_{\mathcal{G}^-}(v_i,v_j;\boldsymbol{\theta}) & \approx
s_{\mathcal{E}}(\mathbf{e}_i^-(\boldsymbol{\theta},\mathbf{e}_j^-(\boldsymbol{\theta})) \\ \nonumber
& = \left(\mathbf{e}_i^-(\boldsymbol{\theta})\right)^\top \mathbf{e}_j^-(\boldsymbol{\theta}) \\ \nonumber
& = \left(\sqrt{\boldsymbol{\Sigma}^-_d(\boldsymbol{\theta})}~\mathbf{u}_i^-\right)^\top \sqrt{\boldsymbol{\Sigma}^-_d(\boldsymbol{\theta})}~\mathbf{u}_j^-\\ \nonumber
& = \left( \mathbf{u}_i^-\right)^\top \boldsymbol{\Sigma}^-_d(\boldsymbol{\theta})\mathbf{u}_j^- \\ \label{approx}
& = \mathbf{x}_{i,j}^\top ~\boldsymbol{\theta}
\end{align}
where
\begin{equation}\label{feats}
\mathbf{x}_{i,j} = \left( \mathbf{u}_i^- \ast \mathbf{u}_j^- \right)^\top \boldsymbol{\Sigma}_d^K,
\end{equation}
and
\begin{equation*}
\boldsymbol{\Sigma}_d^K = \left[ \begin{array}{cccc}
\sigma_1~&~\sigma_1^2~&~\cdots~&~~\sigma_1^K \\
\vdots~&~\vdots~&~\ddots~&~\vdots \\
~~~\sigma_{d-1}~&~~~~\sigma_{d-1}^2~&~\cdots~&~~~\sigma_{d-1}^K\\
\sigma_d~&~\sigma_d^2~&~\cdots~&~\sigma_d^K
\end{array} \right].
\end{equation*}
Conveniently, $\{\mathbf{x}_{i,j}\}$s act as features over every possible pair of nodes, which when linearly combined with weights $\boldsymbol{\theta}$ to produce similarities, allow us to approach \eqref{good} using well-understood learning and optimization tools. Among the various loss functions one may fit the removed edges\footnote{In our implementation, we also provide learning mechanisms based on least-squares, logistic regression, as well as finding the best single $k$. Due to space constrains though we only present and report results of the SVM-based approach.} using the hinge loss
\begin{equation}
\ell(y,f) := \max(0,\epsilon - yf)
\end{equation}
which is suitable for real-world graphs thanks to its robustness properties \cite{svm}; note that target variables here are defined as $y_{i,j} = 2A_{i,j}-1$ so that $y_{i,j}\in \{-1,1\}$. We can then equivalently express \eqref{good} as
\begin{equation}\label{final}
\boldsymbol{\theta}^\ast_\mathcal{S} = \arg\min_{\boldsymbol{\theta}\in \mathcal{S}^K}\sum_{i,j\in \mathcal{S}}\max(0,\epsilon - y_{i,j}\mathbf{x}_{i,j}^\top ~\boldsymbol{\theta}) + \lambda \|\boldsymbol{\theta}\|_2^2
\end{equation}
where $\lambda\geq0$ is the regularization parameter of the $\ell_2$ regularization typically used to improve the robustness and generalization capability of SVMs \cite{svm}. To solve our variant of simplex-constrained SVMs (cf. \eqref{final}), we employ the projected-gradient descent approach \cite{bertsekas} that we describe in Algorithm 4, where \textsc{SimplexProj}( $\cdot$ ) is a subroutine that implements projections onto $\mathcal{S}^K$; the latter can be performed with $\mathcal{O}(K\log K)$ complexity as described in \cite{simplex_proj}. The overall parameter learning procedure for a given sample is summarized in Algorithm 3.
In general, if runtime or computational resources allow, the sampling and training process described in the last two sections can be repeated $T_s$ times to obtain different $\{\boldsymbol{\theta}^\ast_\mathcal{S}\}$s, which can then be averaged in order to reduce their variance. In practice, this may not be necessary if $N_s$ is large enough, which will yield a near-deterministic $\boldsymbol{\theta}$. The overall proposed adaptive-similarity embedding (ASE) framework is summarized in Algorithm 1.
\subsection{Complexity} The computational complexity of ASE is dominated by the cost of performing the truncated SVD of $\mathbf{S}$ in the training as well as testing phases of Algorithm 1. Relying on the sparsity ($|\mathcal{E}|\ll N^2$) and symmetry of $\mathbf{S}$, the Lanczos algorithm followed by EVD of a tridiagonal matrix yield the truncated SVD in a very efficient manner. Provided that $d \ll N$, the decomposition can be achieved in $\mathcal{O}(|\mathcal{E}|d)$ time and using $\mathcal{O}(Nd)$ memory. Therefore, for the $T_s\geq 1$ training rounds and a single embedding round of Algorithm 1, the overall complexity is $\mathcal{O}((T_s + 1)|\mathcal{E}|d)$.
\section{Related work}\label{sec:remarks}
Two recent embedding methods also pursue similarity matrices that combine walks of different lengths \cite{arope,attention}. Most relevant to the proposed ASE is the ``Arbitrary-Order Proximity Preserved Network Embedding'' \cite{arope} approach, where a method is proposed for obtaining the SVD of a polynomial of the adjacency matrix without having to recompute the singular vectors.
Compared to \cite{arope}, we put forth the following contributions. First, we introduce a family of multihop similarities whose decomposition leads to embeddings that inherit the rich information contained in the spectral embeddings (cf. Section 2.3). An equally important contribution in terms of modeling is that our embeddings can be differentiated with respect to (wrt) weights $\boldsymbol{\theta}$ (cf. \eqref{approx}-\eqref{final}), whereas the embeddings in \cite{arope} are non-differentiable wrt the weights. Hence, \cite{arope} can only proceed in a ``forward'' fashion given some order proximity weights $\boldsymbol{\theta}$, whereas our approach allows for ``navigating'' the space of possible similarity functions $s(v_i,v_j;\boldsymbol{\theta})$ in a smooth fashion, meaning that $\boldsymbol{\theta}$ can be learned with simple optimization on well-defined fitting models such as logistic regression or SVMs (cf. \eqref{final}). This leads to the third main contribution, which is a means of learning ``personalized'' $\boldsymbol{\theta}$ (cf. Section 4) in an unsupervised fashion, meaning without downstream information such as node or edge labels/attributes that can guide cross-validation in high-dimensional discretized parameter grids.
The second related embedding method presented in \cite{attention} builds on the concept of graph attention mechanisms to place weights on lengths of truncated random walks. These mechanisms are used to build a similarity matrix containing co-occurrence probabilities. The matrix is jointly decomposed by maximizing a graph-likelihood function. The model in \cite{attention} is a generalization of the ones implicitly adopted by \cite{deepwalk} and \cite{node2vec}, building on similar tools and concepts that emerge from natural language processing. Different from \cite{deepwalk, node2vec} and the proposed ASE, \cite{attention} explicitly constructs and factorizes a dense $N\times N$ similarity matrix. The detailed procedure incurs complexity that is \emph{cubic} wrt $N$, and becomes at best \emph{quadratic} after model approximations, meaning that \cite{attention} scales rather poorly beyond small graphs.
\section{Experimental Evaluation}
\label{sec:experiments}
The present section reports extensive experimental results on a variety of real-world networks. The aim of the presented tests is twofold. First, to determine and quantify the quality of the proposed ASE embeddings for different downstream learning tasks. Second, to analyze and interpret the resulting embedding parameters for different networks.
\noindent \textbf{Datasets.} In our experiments, we used the following real-world networks (see also Table 2).
\begin{itemize}
\item \textbf{\texttt{ca-AstroPh}}. The Astro Physics collaboration network is from the e-print arXiv and covers scientific collaborations between co-authored papers submitted to Astro Physics category~\cite{snap}. If an author $i$ co-authored a paper with author $j$, the graph contains a undirected edge from $i$ to $j$. If the paper is co-authored by $k$ authors, this generates a completely connected (sub)graph on $k$ nodes.
\item \textbf{\texttt{ca-CondMat}}. Condense Matter Physics collaboration network from ArXiv \cite{snap}.
\item \textbf{\texttt{CoCit}}. A co-citation network of papers citing other papers extracted by \cite{VERSE}; labels represent conferences in which papers were published.
\item \textbf{\texttt{com-DBLP}}. Computer science research bibliography collaboration network \cite{snap}.
\item \textbf{\texttt{com-Amazon}}. Network collected by crawling Amazon website \cite{snap}. It is based on ``Customers Who Bought This Item Also Bought'' feature of the Amazon website. If a product $i$ is frequently co-purchased with product $j$, the graph contains an undirected edge from $i$ to $j$.
\item \textbf{\texttt{vk2016-17}}. VK is a Russian all-encompassing social network. In \cite{VERSE}, two snapshots of the network were extracted in November 2016 and May 2017, to obtain information about link appearance.
\item \textbf{\texttt{email-Enron}}. Enron email communication network covering all the email communication within a dataset of around half a million emails \cite{snap}.
\item \textbf{\texttt{PPI (H.Sapiens)}}. Subgraph of the protein-protein interaction network for Homo Sapiens. The subgraph corresponds to the graph induced by nodes for which labels (representing biological states) were obtained from the hallmark gene sets \cite{node2vec}.
\item \textbf{\texttt{Wikipedia}}. This is a co-occurrence network of words
appearing in the first million bytes of the Wikipedia dump. The labels represent the Part-of-Speech (POS) tags inferred using the Stanford POS-Tagger \cite{node2vec}.
\item \textbf{\texttt{BlogCatalog}}. A network of social relationships of the bloggers listed on the BlogCatalog website. The labels represent blogger interests inferred through the meta-data provided by the bloggers.
\end{itemize}
\begin{table}[t]
\centering
\caption{ Network Characteristics }
\rowcolors{2}{}{gray!7}
\begin{tabular} {ccccc}
\toprule
Graph & $|\mathcal{V}|$ & $|\mathcal{E}|$ & $|\mathcal{Y}|$ & Density \\
\midrule
\texttt{PPI (H. Sapiens)} & 3,890 & 76,584 & 50 & $10^{-2}$ \\
\texttt{Wikipedia} & 4,733 & 184,182 & 40 & $1.6\times10^{-2}$ \\
\texttt{BlogCatalog} & 10,312 & 333,983 & 39 & $6.2\times10^{-3}$ \\
\texttt{ca-CondMat} & 23,133 & 93,497 & - & $3.5\times10^{-4}$ \\
\texttt{ca-AstroPh} &18,772 & 198,110 & - & $1.1\times10^{-3}$ \\
\texttt{email-Enron} &36,692 & 183,831 & - & $2.7\times10^{-4}$ \\
\texttt{CoCit} & 44,312 & 195,362 & 15 & $2\times10^{-4}$ \\
\texttt{vk2016-17} & 78,593 & 2,680,542 & - & $8.7\times10^{-4}$ \\
\texttt{com-Amazon} & 334,863 & 925,872 & - & $1.7\times10^{-5}$ \\
\texttt{com-DBLP} &317,080 & 1,049,866 & - & $2.1\times10^{-5}$ \\
\bottomrule
\end{tabular}\label{tab:graphs}
\end{table}
\begin{table*}[t]
\centering
\caption{ Inferred parameters and interpretation }
\rowcolors{2}{}{gray!7}
\begin{tabular} {ccccccccccccc}
\toprule
Graph & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ & $\theta_5$ & $\theta_6$ & $\theta_7$ & $\theta_8$ & $\theta_9$ & $\theta_{10}$ & range & strength \\
\midrule
\texttt{PPI (H. Sapiens)} & 0.00 & \textbf{0.14} & \textbf{0.31} & \textbf{0.29} & \textbf{0.21} & \textbf{0.04} & 0.00 & 0.00 & 0.00 & 0.00 & medium& medium \\
\texttt{Wikipedia} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & \textbf{0.01} & \textbf{0.37} & \textbf{0.62} & long & strong \\
\texttt{BlogCatalog} & \textbf{1.00} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & short & very strong \\
\texttt{ca-CondMat} & \textbf{0.55} & \textbf{0.33} & \textbf{0.12} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & short & strong \\
\texttt{ca-AstroPh} & \textbf{0.76} & \textbf{0.24} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & short & strong \\
\texttt{email-Enron} & \textbf{0.24} & \textbf{0.25} & \textbf{0.18} & \textbf{0.14} & \textbf{0.1} & \textbf{0.06} & \textbf{0.02} & 0.00 & 0.00 & 0.00 & medium & weak \\
\texttt{CoCit} & \textbf{0.61} & \textbf{0.33} & \textbf{0.06} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & short & strong \\
\texttt{vk2016-17} & \textbf{0.71} & \textbf{0.29} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & short & strong\\
\texttt{com-Amazon} & \textbf{0.10} & \textbf{0.10} & \textbf{0.10} & \textbf{0.10} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & short & very weak \\
\texttt{com-DBLP} & \textbf{0.11} & \textbf{0.10} & \textbf{0.10} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & \textbf{0.09} & \textbf{0.08} & short & very weak \\
\bottomrule
\end{tabular}\label{tab:thetas}
\end{table*}
\begin{figure*}[t!]
\centering
\input{figs/homo_single.tex}
\input{figs/blog_single.tex}
\input{figs/wiki_single_temp.tex}
\input{figs/cocit_single.tex}
\phantom{ppppp}\ref{named5}
\caption{ Micro and Macro $F_1$ scores for the four labeled graphs, when the ``pure'' $k-$order $\mathbf{S}^k$ is used for embedding, given as a function of $k$. Red shade denotes the corresponding $k$'s where ASE assigned non-zero $\boldsymbol{\theta}_k$'s; see also Table 2. } \label{fig:single}
\end{figure*}
\noindent \textbf{Methods.} Experiments were run using the following \emph{unsupervised} and \emph{scalable} embedding methods.
\begin{itemize}
\item \textbf{ASE}. Our proposed adaptive similarity embedding. Based on observations made in Sections 3, and to retain optimization stability, we set the maximum number of steps to $K=10$. We also use the default SVM regularizer ($\lambda=1$). To have a single learning round with learned parameters having small enough variance, we sampled with $N_s/2 = 1,000$. We made our implementation of ASE freely available~\footnote{https://github.com/DimBer/ASE-project}.
\item \textbf{VERSE} \cite{VERSE}. This is a scalable framework for generating node embeddings according to a similarity function by minimizing a KL-divergence-objective via stochastic optimization. We used the default version with similarity (PPR with $\alpha=0.85$), as suggested and implemented by the authors.\footnote{https://github.com/xgfs/verse}
\item \textbf{Deepwalk} \cite{deepwalk}. This approach learns an embedding by sampling random walks from each node, and applying word2vec-based learning on those walks. We use the default parameters proposed in \cite{deepwalk}, i.e., walk length $t= 80$, number of walks per node $\gamma= 80$, window size $w= 10$, and the scalable C++ implementation\footnote{https://github.com/xgfs/deepwalk-c} provided in \cite{VERSE}.
\item \textbf{HOPE} \cite{hope}. This SVD-based approach approximates high-order proximities and leverages directed edges. We report the results obtained with the default parameters, i.e, Katz similarity as the similarity measure with $\beta$ inversely proportional to the spectral radius.
\item \textbf{AROPE} \cite{arope}. An approach for fast computation of thin SVD of different polynomials of $\mathbf{A}$. We used the official Python implementation \footnote{\url{https://github.com/ZW-ZHANG/AROPE}} to produce the embeddings. We selected the polynomial (hyper) parameters of AROPE using a set of validation edges that was sampled similarily to ASE (Algorithm 2). We consider proximity orders in the range $[1,10]$, and perform grid search over the different proximity weights as suggested in \cite{arope}.
\item \textbf{LINE} \cite{LINE}. This approach learns a $d$-dimensional embedding
in two steps, both using adjacency similarity. First, it learns $d/2$ dimensions using first-order proximity; then, it learns another $d/2$ features using second-order proximity. Last, the two halves are normalized and concatenated. We obtained a copy of the code\footnote{https://github.com/tangjianpku/LINE}, and
run experiments with $T= 10^{10}$ samples (although $T= 10^9$ yielded the same accuracy for smaller graphs), and $s = 5$ negative samples, as described in the paper.
\item \textbf{Spectral}. This approach relies on the first $d$ eigenvectors of $\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}$. The baseline was developed for clustering \cite{spectral}, and has also been run as a benchmark for node embeddings \cite{node2vec}. In our case, spectral embedding is of particular interest since it can be obtained by column-wise normalization of the embeddings generated by the proposed method.
\end{itemize}
We excluded comparisons with Node2vec \cite{node2vec} because they use cross-validation on node labels for hyper-parameter selection. Thus comparing Node2vec to methods such as LINE, Deepwalk, HOPE, VERSE, and EMB that all operate with \emph{fixed} hyperparameters in a fully \emph{unsupervised} manner would be unfair. We also excluded comparisons with GraRep \cite{grarep} and M-NMF \cite{netmf} due to their limited scalability ($\mathcal{O}(N^2d)$ computational and $\mathcal{O}(N^2)$ memory complexity). \\
\noindent \textbf{Evaluation methodology}. Our experiment setting follows the one in \cite{VERSE}. All methods are set to embed nodes to dimension $d=100$. Using the resulting embeddings as feature vectors, we evaluated their performance in terms of node classification and link prediction accuracy, and clustering quality. All experiments were repeated 10 times and reported are the averaged results. \\
\begin{figure*}[t!]
\vspace{0.2in}
\centering
\input{figs/HomoSapiens_micro.tex
\input{figs/BlogCatalog_micro.tex
\input{figs/wiki_micro.tex
\input{figs/cocit_micro.tex}
\input{figs/HomoSapiens_macro.tex}
\input{figs/BlogCatalog_macro.tex}
\hspace{-0.1cm}\input{figs/wiki_macro.tex}
\hspace{-0.2cm}\input{figs/cocit_macro.tex}
\phantom{ppppp}\ref{named3}
\caption{Micro (upper row) and Macro (lower row) $F_1$ scores that different embeddings + logistic regression yield on labeled graphs, as a function of the labeling rated (percentage of training data)} \label{fig:class}
\vspace{-0.35cm}
\end{figure*}
\noindent \textbf{Interpretation of results}. One interesting aspect of the proposed ASE method, is that the inferred parameters $\boldsymbol{\theta}^\ast$ from the first phase of Algorithm 1 can be used to characterise the underlying similarity structure of the graph, and the way nodes ``interact'' over different path lengths (short, medium, and long range). The ``strength'' of interactions is inferred by how uniform the coefficients of $\boldsymbol{\theta}^\ast$ are, and depend on the value of $\lambda$. Since the default value was $\lambda=1$ for all graphs, the results can be interpreted as relative interaction strengths between them. The resulting $\{\boldsymbol{\theta}^\ast\}$s for all graphs are listed in Table 3.
It can be immediately observed that the type of node interactions varies significantly across different graphs, with similar behavior for graphs that belong to the same domain. Specifically, \texttt{ca-CondMat, ca-AstroPh}, and \texttt{CoCit} that belong to the citation/co-authorship domain all show relatively strong interactions of short range. \texttt{BlogCatalog} shows very strong short-range similarities of only one-hop neighborhood interactions among bloggers. On the other hand, the \texttt{Wikipedia} word co-occurrence network shows a strong tendency for long-range interactions; while other graphs, such as the \texttt{PPI} protein interaction network stay on the medium range.\\
\noindent \textbf{Node classification}.
Graphs with labeled nodes are frequently used to measure the ability of embedding methods to produce features suitable for classification. For each experiment, nodes were randomly split to a training set and a test set. Similar to other works, and to cope with multi-label targets, we fed the training features and labels into the one-vs-the-rest configuration of logistic regression classifier provided by the \texttt{sklearn} Python library. In the testing phase, we sorted the predicted class probabilities for each node in decreasing order, and extracted the top-$k_i$ ranking labels, were $k_i$ is the true number of labels of node $v_i$. We then computed the Micro- and Macro-averaged $F_1$ scores \cite{manning2008ir} of the predicted labels.
Apart from comparisons with alternative embedding methods, node classification can reveal whether available node labels (metadata) are distributed in a manner that matches the node relations/interactions that are inferred by ASE. To reveal this information, we obtain embeddings for every $k\in \{1,\ldots,10\}$ by ignoring the training phase and ``forcing'' $\boldsymbol{\theta}^\ast=\mathbf{e}_k$ (i.e., 1 at the $k$-th entry and $0$ elsewhere) in Algorithm \ref{alg:ASE}, and then using each embedding for classification with $10\%$ labeling rate. Figure \ref{fig:single} plots Micro and Macro $F_1$ for all labeled graphs as a function of $k$, while red shade is placed on the hops where the \emph{unsupervised} ASE parameters $\boldsymbol{\theta}^\ast$ are non-zero (cf. Table 1). As seen in Fig. \ref{fig:single}, the accuracy on the four labeled graphs evolves with $k$ in a markedly different manner. Nevertheless, ASE identifies the trends and tends to assign non-zero weights to hops that yield a desirable trade-off between Micro and Macro $F_1$. Bearing in mind that ASE does \emph{not} use labels for training or validation, this is rather remarkable considering the fact that $\boldsymbol{\theta}^\ast$ depends only on the graph.
\begin{figure*}[t!]
\centering
\input{figs/cluster_astro.tex}
\input{figs/cluster_cond.tex}
\input{figs/cluster_enron.tex}
\input{figs/cluster_blog.tex}
\input{figs/cluster_amazon.tex}
\input{figs/cluster_dblp.tex}
\phantom{ppppp}\ref{named4}
\vspace{-0.2cm}
\caption{Average conductance of different embeddings used by kmeans for clustering, as a function of number of clusters. } \label{fig:cluster}
\vspace{-0.3cm}
\end{figure*}
We also compared the classification accuracy of ASE embeddings with those of the alternative embedding approaches, with results plotted in Fig. \ref{fig:class}. The plots for some method-graph pairs are not discernible when values are too low. While the relative performance of any given method varies from graph to graph, ASE adapts to each graph and yields consistently reliable embeddings, with accuracy that in most cases reaches or surpasses that of state-of-the-art methods, especially in terms of Macro $F_1$. The two exceptions are the Macro $F_1$ in \texttt{CoCit}, and Micro $F_1$ in \texttt{Wikipedia}, where VERSE and HOPE are correspondingly more accurate. Interestingly, HOPE achieving high Micro $F_1$ and low Macro $F_1$ in \texttt{Wikipedia} is in agreement with the findings in Fig. \ref{fig:single}, combined with the fact that HOPE focuses on longer paths.\\
\begin{table}[h]
\centering
\caption{ Link Prediction Accuracy on \texttt{vk2016-17} }
\rowcolors{2}{}{gray!7}
\begin{tabular} { c c c c c c c }
\toprule
VERSE & ASE & LINE & Deepwalk & AROPE & HOPE & Spectral \\
\midrule
0.79 & 0.75 & 0.74 & 0.69 & 0.65 & 0.62 & 0.60 \\
\bottomrule
\end{tabular}\label{tab:link_pred}
\end{table}
\noindent \textbf{Link prediction}. Link prediction is the task of estimating the probability that a link
between two unconnected nodes will appear in the future. We repeat the experiment performed in \cite{VERSE} on the \texttt{vk2016-17} social network. For every possible edge, we build a feature vector as the Hadamard product between the embedded vectors of its two adjacent nodes. Using the two time instances of \texttt{vk2016-17}, we predict whether a new friendship link appears between November
2016 and May 2017, using $50\%$ of the new links for training and $50\%$
for testing. To train the binary logistic regression classifier, we also randomly sample non-existing
edges as negative examples. The link prediction accuracy for different embeddings is reported in Table 3. While for this experiment ASE does not reach the accuracy of VERSE, it provides the second most accurate link prediction, far surpassing the also SVD-based HOPE and spectral embeddings.
\\
\begin{figure*}[t!]
\centering
\input{figs/lambda_sense.tex}
\vspace{0.3cm}
\input{figs/S_sense.tex}
\vspace{0.3cm}
\input{figs/k_sense.tex}
\vspace{0.3cm}
\input{figs/d_sense.tex}
\vspace{-0.9 cm}
\caption{ Sensitivity (\textcolor{blue}{F-1 Micro} on left axes, and \textcolor{orange}{Runtime} on right axes) of ASE on \texttt{PPI} graphs wrt various parameters. } \label{fig:sense}
\vspace{-0.3 cm}
\end{figure*}
\begin{figure*}[t!]
\centering
\input{figs/runtime.tex}
\caption{ Runtime of various embedding methods across different graphs } \label{fig:runtime}
\vspace{-0.4 cm}
\end{figure*}
\noindent \textbf{Node clustering}. Finally, the embedded vectors were used to cluster the nodes into different communities, using the \texttt{sklearn} library K-means with the default K-means++ initialization \cite{kmeans++}. We evaluate the quality of node clustering with conductance, a well-known metric for measuring the goodness of a community \cite{leskovec2009community}; conductance is minimized for large, well connected communities that are also well separated from the rest of the graph. Each plot in Fig. \ref{fig:cluster} gives the average conductance across communities, as a function of the total number of clusters. Results indicate that the proposed ASE as well as the spectral clustering benchmark yield much lower conductance compared to other embeddings. Apparently, since ASE builds on the same basis of eigenvectors used by normalized spectral clustering, it inherits the property of the latter to approximately minimize the normalized-cut metric \cite{spectral}, which is very similar to conductance. A closer look at the resulting clusters, reveals that clustering beased on VERSE, Deepwalk, LINE, and HOPE splits graphs into very large communities of roughly equal size, cutting a large number of edges in the process. This is an indication that these methods are subject to a \emph{resolution limit}, which is the inability to detect well-separated communities that are below a certain size \cite{resolution}. On the other hand, Spectral and the proposed ASE separate the graph into a large-core component, and many smaller well-separated communities, a structure that many large-scale information networks have been observed to have \cite{leskovec2009community}. Indeed, the conductance gap is smaller for \texttt{BlogCatalog}, which is relatively small and with less pronounced communities.\\
\noindent \textbf{Parameter sensitivity}. We also present results after varying ASE parameters and measured embedding runtime for \texttt{PPI} as well as classification Micro $F_1$ accuracy with $10\%$ labeling rate. The aim is to assess the sensitivity of ASE wrt its basic parameters. The plot on the left shows how increasing $\lambda$ (cf. \eqref{final}) may decrease accuracy by forcing the entries of $\boldsymbol{\theta}^\ast$ to be close to uniform, thus losing the benefits of graph-specific adaptation. Regarding the number of sampled edges $N_s$, results (middle plot) indicate relative robustness of ASE embeddings, given a minimum number of samples. As expected, sampling a large number of edges may cause noticeable perturbation on the graph (even using the minimally-perturbing Algorithm 2); this may be causing a slight decrease in accuracy. Sensitivity is also measured wrt $K$ (i.e., the maximum walk length considered in the optimization). As expected, the accuracy increases sharply with $K$ for the first few steps, and then plateaus as higher order coefficients of \texttt{PPI} take zero values (c.f., Table 3) and do not affect the results. Finally, the plot on the left depicts accuracy across a range of embedding dimensions $d$. \\
\noindent \textbf{Runtime}. Finally, we compared different embedding methods in terms of runtime. Results for all graphs are reported in Fig. \ref{fig:runtime}. All experiments were run on a personal workstation with a quad-core i5 processor, and 16 GB of RAM. For our proposed ASE, we provide a light-weight yet highly portable implementation~\footnote{https://github.com/DimBer/ASE-project/tree/master/portable} that uses the SVDLIBC library \cite{svdlibc} for sparse SVD. We also developed a more scalable implementation~\footnote{https://github.com/DimBer/ASE-project/tree/master/slepc\textunderscore based} that relies on (and requires installation of) the SLEPc package \cite{slepc}; this scalable version can perform large-scale sparse SVD on multiple processes and distributed memory environments using the message-passing interface (MPI) \cite{mpi}. We used the high-performance implementation for the five larger graphs, and the portable one for the five smaller ones. Evidently, ASE and HOPE that are SVD-based are orders of magnitudes faster than VERSE, Deepwalk, and LINE. The main factor that slows the latter down seems to be the large number of stochastic optimization iterations that these methods must perform to reach accurate embeddings. Nevertheless, it should be noted that sampling based methods enjoy nearly-full parallelization and could thus benefit more from highly
multi-threaded environments. On the other hand, methods that rely on SVD (and EVD) can greatly benefit from decades of research on how to efficiently perform these decompositions, and a suite of stable and highly optimized software tools.
\section{Conclusions and Future work}
\label{sec:conclusions}
We presented a scalable node embedding framework that is based on factorizing an adaptive node similarity matrix. The model is carefully studied, interpreted, and numerically evaluated using stochastic block models, with an algorithmic scheme proposed for training the model parameters efficiently and without supervision.
The novel framework opens up several interesting future research directions. For instance, one can explore larger families of node similarity metrics that can be learned using the graph. Furthermore, it would be interesting to assess the performance of different randomized edge sampling methods, and generalize the notion of adaptive-similarity to heterogeneous and multi-layered graph embedding, as well as to edge embedding.
|
1,116,691,498,463 | arxiv | \section{Introduction} \label{sec:intro}
It is known that most stars form in clusters \citep{ladalada2003} and that a substantial fraction of solar-type stars in the field are binaries \citep[e.g.,][]{DM1991,raghavan2010}. Thus it is expected that most solar-type stars must form in binaries \citep[e.g.,][]{lada2006} and that the binary fraction in star-forming clusters should be at least as high as that observed in the field.
\par
At the same time, the binary fraction is not expected to be a static quantity. Wide binaries in particular are not expected to last long in dense stellar regions due to the larger number of interactions between systems, as well as the smaller binding energy of the binary. Indeed, \citet{ghez1993} used speckle imaging to suggest that the occurrence of solar-type binaries with separations of hundreds of AU declines by a factor of $\sim$3.5 from the pre--main-sequence (PMS) to the main sequence. Tight binaries ($a < 20~AU$) are expected to be able to last longer due to their higher binding energy. \cite{mason1998} studied the binary fraction among Gyr-aged populations and found, using chromospheric activity, that the binary fraction was decreasing with age on Gyr timescales.
To properly understand the binary fraction and therefore how stars form in populations, we must understand where this dynamical processing of the binary fraction begins to take place, for both wide and tighter (spectroscopic) binary systems.
Estimates for the spectroscopic binary fraction at field ages are typically $\sim$10\% \citep[e.g.,][]{DM1991}.
It is necessary to also characterize the spectroscopic binary fraction of star forming regions to establish the possible evolution of the spectroscopic binary fraction during the PMS phase.
Finally, the spectroscopic binary fraction especially at young ages is important for understanding planet formation in binary systems, including circumbinary planets around tight binaries and the dynamical evolution of planets around individual stars in wide binaries \citep[see, e.g.,][]{kraus2015, kraus2016}.
\par
Observing such young stellar systems at ages of a few Myr can be challenging, since not all of these systems will have cleared the gas and dust around them. This gas and dust obscures the stars in the optical. For example, in the Orion Nebula Cluster the visual extinction, $A_V$, can range from a few tenths of a magnitude to tens of magnitudes, depending on the sightline and distance into the cloud \citep[see, e.g.,][]{hillenbrand2007, dario2014}.Infrared observations are able to penetrate through the gas and dust so that spectroscopic measurements of radial velocity variations may be performed to identify spectroscopic binaries \cite[see, e.g.,][]{prato2002a, prato2002b}.
\par
The Sloan 2.5m telescope \citep{gunn2006} feeds light to the Apache Point Observatory Galactic Evolution Experiment(APOGEE) spectrograph \citep{majewski2017}. The APOGEE spectrograph primarily observes in the H-band (1.51$\mu$m-1.7$\mu$m) of the near-infrared spectrum. The spectrograph can observe up to 300 source targets per plate, with a 2 arcsecond fiber size, and a nominal spectral resolution of 22,500. The APOGEE survey looked at 100,000 stars within the Milky Way galaxy, focusing on red giants. The Infrared Spectroscopy of Young Nebulous Clouds (IN-SYNC) survey was an ancillary project during SDSS-III that used the APOGEE spectrograph to carry out high volume, high precision observations of pre-main sequence stellar populations. The IN-SYNC survey provides a very powerful and unique opportunity to study binary systems and the binary fractions of young star forming populations as well as the kinematics of the youngest star forming regions in multiple types of star forming environments having carried out observations within the galactic bulge, halo, and disk.
In this paper, we measure and compare the spectroscopic binary fraction of five PMS clusters and the MS Pleiades cluster using the multi-epoch IN-SYNC data. Binary candidates are identified via RV-variability. Straightforward comparison of the raw cluster binary fractions would be misleading due to significant cluster-to-cluster differences in observational cadence and target sampling. After careful accounting of these observational differences and uncertainties within a probabilistic model of the binary fraction within each cluster, we find that the spectroscopic binary fraction for the main sequence Pleiades cluster is a factor of $\sim$3--4 less than the average spectroscopic binary fraction for the PMS clusters. We attribute this to the dissolution of relatively wide (orbital periods 10$^2$--10$^4$~d) spectroscopic binaries, not well studied previously among PMS clusters, that are probed by our sample.
We discuss the data we use in this paper in Section 2, go over the processes employed to characterize the candidates with an unseen companion in Section 3, discuss the Bayesian Inference framework we used to correct for observational differences in Section 4, discuss the derived cluster binary fraction distributions in Section 5, frame our results within a broader scientific context in Section 6, and state our conclusions in Section 7.
\section{Data and Sample}\label{sec:data}
\subsection{The IN-SYNC Survey and APOGEE} \label{ssec:data_overview}
The IN-SYNC survey was an ancillary project to the SDSS III APOGEE program \citep{IN-SYNC_paper_1}. Using the APOGEE spectrograph, the IN-SYNC survey obtained multi-epoch high-resolution (R=22500) spectroscopy of dust obscured young star forming regions. Multiple regions were observed during the IN-SYNC survey, consisting of IC348, and NGC1333 in the Perseus Cloud, NGC2264, and the Orion A molecular cloud complex (see Figure~\ref{time_baseline_nepochs}). The IN-SYNC team has already performed numerous analyses into these regions. \cite{IN-SYNC_paper_2} looked at the kinematics of the embedded pre-main sequence population of NGC1333. \cite{IN-SYNC_paper_3} looked at the dynamical state of IC348. \cite{IN-SYNC_paper_4} looked the Orion A Molecular Cloud Complex and compared derived stellar parameters to previous literature catalogs, and most recently \cite{IN-SYNC_paper_5} looked at the kinematics and dynamical state of the Orion A population.
\par
The IN-SYNC survey obtained multi-epoch spectra for over 3000 pre-main sequence stars. The IN-SYNC survey derived stellar parameters for the Pleiades which were observed with the APOGEE spectrograph \citep{ahn2014}, and whose spectra were processed using the same IN-SYNC data reduction pipeline. The IN-SYNC data reduction pipeline uniformly derived effective temperatures, surface gravities, radial velocities, rotational velocities, H-band veiling, and corresponding stellar parameter errors. For full details of the IN-SYNC pipeline reduction process, please see \cite{IN-SYNC_paper_1} (Section 3.1.1 -- 3.1.3).
\begin{figure*}
\epsscale{1.2}
\plotone{Fig1.pdf}
\parbox{18cm}{\caption{ Left Panel: Empirical distribution functions of the observational time baselines in $log_{10}$ days for the 5 IN-SYNC clusters and the Pleiades. Right Panel: Empirical distribution functions of the number of epochs per observation for the 5 IN-SYNC clusters and the Pleiades. The values within both these panels are for the final vetted sample of stars from Section \ref{ssec:stellar_systematics} }}
\label{time_baseline_nepochs}
\end{figure*}
\subsubsection{Cluster membership of study sample}
\label{ssec:membership}
The raw IN-SYNC pipeline catalog contains derived stellar parameters for 12945 individual observations of 4771 stars. Not all of these observed stars necessarily belonged to any of the five observed regions and so cross-matching of 2MASS ids were used between the observed targets and literature catalogs to separate true members of IC348, NGC1333, NGC2264, Orion A, and the Pleiades from observed field stars. \kojdel{We initially demand that every observation in our raw dataset have a signal to noise $\ge$ 20, thus removing 131 stars, with 1412 observations.}
For NGC1333 we used the \cite{rebull2015} catalog of known NGC1333 members and cross-matched their locations within the \cite{cutri2003} 2MASS point source catalog to extract member 2MASS IDs. \kojdel{Within this set of 854 ids we then found 88 stars that were science targets observed for IN-SYNC with derived stellar parameters for 291 observations.} The NGC1333 cluster has been estimated to have an age of $\sim$1 Myr \citep{gutermuth2008}, which we employ here.
\par
For NGC2264 we used the final cross-matched set of 2MASS ids composed from the catalogs of \cite{lamm2005}, \cite{makidon2004}, \cite{sung2008}, and \cite{sung2009} by the IN-SYNC team. \kojdel{and found 114 stars to be members of NGC2264 with derived stellar parameters for 636 observations.} The age of the NGC2264 cluster has been cited to be around $\sim$3 Myr \citep{venuti2014}.
For the Orion A molecular cloud cross matching was performed using a provided catalog (Dr. Nicola Da Rio, Private Communication) of membership lists composed from Optical Spectra (\cite{hillenbrand1997}, \cite{fang2009}, \cite{fang2013}, \cite{hsu2012}, \cite{hsu2013}, \cite{dario2012}), Infrared excess (\cite{getman2005}, \cite{pillitteri2013}), and X-ray (\cite{megeath2012}). \kojdel{This catalog provided us with 2691 stars, from which we found 3594 stars that were observed science targets for IN-SYNC with derived stellar parameters for 3580 observations.} It is known that the Orion A molecular cloud is home to numerous different stellar populations. From \cite{IN-SYNC_paper_4}, the argument is made that given the approximate size of the entire Orion A cloud of 40pc, as well as its relatively young age, it is not possible for dynamical interactions to homogenize the ages of the young stellar populations throughout the filament. We therefore cannot look at the cloud as one cluster and decide to separate it two smaller, sub-clusters.
\par
Using \ensuremath{\log g} \ -- \ensuremath{T_\mathrm{eff}} \ isochrones, \cite{IN-SYNC_paper_4} finds that there is distinct separation in the ages of the Orion A region around $\delta \sim 6^{\circ}$. \cite{IN-SYNC_paper_5} also finds this distinction between populations occurring at $\delta \sim -6^{\circ}$ when looking at the position-position-velocity space of the population. We therefore decide to split the Orion A region into two sub-clusters which we study separately. We delineate the northern sub-cluster as all systems with a declination of $\delta \ge -6^{\circ}$. This region, which we call Orion A(N), is dominated by the Orion Nebula Cluster. \kojdel{, with 1360 stars and 2309 observations.} The southern sub-cluster is designated as all systems with a declination of $\delta < -6^{\circ}$, which we call Orion A(S). \kojdel{, with 719 stars and 1285 observations.} Following \cite{IN-SYNC_paper_4}, we use the \ensuremath{\log g} \ -- \ensuremath{T_\mathrm{eff}} \ estimated ages of $\sim$1.5 Myr and $\sim$2.5 Myr for Orion A(N), and Orion A(S), respectively.
\par
For IC348, we cross matched identified members within the catalogs of \cite{luhman1999}, \cite{luhman2003}, \cite{lada_et_al2006}, and \cite{muench2007} to develop a set of member 2MASS IDs. \kojdel{From this set we find 198 stars within IC348 that were observed science targets within IN-SYNC with derived stellar parameters for 808 observations.} \cite{bell2013b} recently found the IC348 region to be $\sim$6 Myr, which we use for our population sample.
\par
Finally within the Pleiades we used the online catalog of identified cluster members from \cite{IN-SYNC_paper_1}. \kojdel{to find 1425 Pleiades cluster members with 2MASS ids. From this set we find that there are 68 stars that were observed IN-SYNC science targets with derived stellar parameters for 202 observations.}
While the age of the Pleiades has been well understood to be $\sim$100~Myr\citep{meynet1993}, the exact age has been difficult to constrain. Using Lithium depletion and K-band photometry, \cite{martin2001} found the age of the Pleiades to be $\sim$115 Myr. \cite{stauffer1998} found the age of the region to be $\sim$125 Myr using Keck-II spectroscopic data. We decide to employ the median age of 115 Myr for this population sample from \cite{martin2001}
\begin{figure*}
\epsscale{1.2}
\plotone{Fig2.pdf}
\parbox{18cm}{\caption{Empirical distribution functions of \ensuremath{T_\mathrm{eff}} (left panel), \ensuremath{\mathrm{\it{v} \sin \it{i}}} (center panel), and \ensuremath{\log g} (right panel) for the final vetted sample of the five IN-SYNC clusters and the Pleiades. The difference in the EDFs of the surface gravity within the Pleiades compared to the 5 IN-SYNC clusters is due to the inherent age difference between the Pre-main sequence clusters and the Main-sequence clusters.}}
\label{stellar_parameters_edfs}
\end{figure*}
\subsubsection{Data Quality Assurance}
\label{ssec:parameters}
Having identified all stars that are cluster members, we proceed to ensure that the derived stellar parameters are accurate. We are motivated to drop all individual observations with a signal to noise below 20 since these observations were too noisy to provide accurate information on stellar parameters (see \cite{IN-SYNC_paper_3}, Section 2.3). Stars with only one observation are also removed since they offer no information for looking at radial velocity variability.
Over 1600 stars have had repeat observations during the IN-SYNC survey. Spectral parameters of interest-- \ensuremath{\log g}, \ensuremath{T_\mathrm{eff}}, \ensuremath{\mathrm{\it{v} \sin \it{i}}}, RV, and SN -- were determined for each individual observation and reported as per epoch parameters within the IN-SYNC catalog. We identify candidate binary systems by measuring the variation in radial velocity associate with each star (see Section~\ref{sec:sb_search}). Following \citet{IN-SYNC_paper_1} and \citet{IN-SYNC_paper_4}, we looked at the per epoch variation of \ensuremath{\mathrm{\it{v} \sin \it{i}}}, \ensuremath{\log g}, \ensuremath{T_\mathrm{eff}}\ and found that highly variable per epoch \ensuremath{T_\mathrm{eff}}\ measurements for any single system were the most significant indicator of problems with the data extraction and thus of spurious, non-binary radial velocity variability within the IN-SYNC observations. Therefore, we looked to the per epoch measurements of effective temperature for each star and remove those epochs with \ensuremath{T_\mathrm{eff}}\ measurements that deviate significantly beyond what would be expected from standard gaussian error.
In about 75\% of the stars observed within our data, we have fewer than 4 epochs of measurements. Therefore we calculate the Median Absolute Deviation (MAD) for each set of measurements. The MAD is a robust statistic, compared to the standard deviation. To use the MAD as an estimator for outlier rejection similar to the standard deviation, we employ a constant scale factor, which changes depending on the underlying distribution.
For a gaussian distribution, this scale factor is 1.4286, and 1.4286$\times$MAD can be used to designate the interquartile range of values assuming a gaussian distribution \citep[see][]{huber_ronchetti2009}. We calculate the MAD value for the set of \ensuremath{T_\mathrm{eff}} measurements from stars with more than 2 epochs of observation and remove epochs with parameter deviations exceeding 3 times the value of 1.4286$\times$MAD, which is equivalent to demanding that the stellar parameters lie within 3$\sigma$ of their distribution.
For stars with only 2 epochs of observation we cannot rely on the MAD statistic, as both measurements of \ensuremath{T_\mathrm{eff}}\ are equidistant from the calculated median of the set and would not be removed as outliers using the MAD, even if the two values of \ensuremath{T_\mathrm{eff}}\ were highly variable. Instead we calculate the $\chi^{2}$ of the set of two measurements of \ensuremath{T_\mathrm{eff}}, and find the probability of that $\chi^{2}$ value within a $\chi^{2}$ distribution with 1 degree of freedom. We reject stars with a probability less than $10^{-3}$ as this implies both measurements came from separate distributions.
We also perform \ensuremath{T_\mathrm{eff}}\ and \ensuremath{\mathrm{\it{v} \sin \it{i}}}\ cuts to ensure that the radial velocity variability within the IN-SYNC sample is due to an unseen companion and not the result of poor spectral fitting. We drop stars whose median \ensuremath{T_\mathrm{eff}}\ and/or \ensuremath{\mathrm{\it{v} \sin \it{i}}}\ measurements do not fall in the range of $ 2500K \le \ensuremath{T_\mathrm{eff}}\ \le 6000K$, and $\ensuremath{\mathrm{\it{v} \sin \it{i}}} \le 100\ensuremath{\mathrm{km\:s}^{-1}}$, respectively. We implemented these cuts following \cite{IN-SYNC_paper_3} and summarize their reasoning for these cuts as follows:
\begin{itemize}
\item The IN-SYNC survey did not go deep enough to observe cool stars ($<2500K$), and such low \ensuremath{T_\mathrm{eff}}\ fits indicate noisy fits within the pipeline.
\item IN-SYNC stars with \ensuremath{T_\mathrm{eff}}\ $>6000K$ and \ensuremath{\mathrm{\it{v} \sin \it{i}}}\ $>100\ensuremath{\mathrm{km\:s}^{-1}}$ have hydrogen lines in their spectra that affected radial velocity derivation, making them too noisy and unreliable.
\end{itemize}
From an initial raw sample of 12945 observations of 4771 stars, we removed non-cluster members using 2MASS ids, implemented a signal to noise cut on individual observations, ensured that sets of observations did not deviate from standard gaussian behavior within \ensuremath{T_\mathrm{eff}}\, and removed stars that were too cool/too hot and too rapidly rotating.
Our final vetted catalog of data contains 4642 measurements for 1418 stars. Table~\ref{tab:bin_frac} contains a tally of the number of stars in each cluster included in the analysis that follows. The empirical distribution functions (EDFs), which gives the fraction of a sample for a variable that are at or below any value of the measured variable, of the time baselines, number of observations, and stellar parameters for the final measurements that result from the steps described above are summarized in Figures~\ref{time_baseline_nepochs} and \ref{stellar_parameters_edfs}.
\section{Identifying Binary Systems in IN-SYNC } \label{sec:sb_search}
The IN-SYNC survey achieved radial velocity precision down to 0.3~\ensuremath{\mathrm{km\:s}^{-1}}, which in principle make these data well suited to study multiplicity within these young star forming regions.
Temporal evolution of a star's radial velocity is a clear indication of an orbiting companion \citep{iben_tutukov1996}. Most studies use RV variation to both identify stars with unseen companions and reconstruct the orbits of the binary system. Orbit-reconstruction requires several to many measurement epochs to ensure adequate phase coverage. With multiple epoch measurements, binary candidates can be identified by the RMS scatter of radial velocities, or any other variance measure \citep[{\rm e.g.},][]{troup2016}. However since many of the stars observed within IN-SYNC are pre-main sequence stars, with high temperatures, and high rotational velocities, we expect their radial velocity RMS scatter to be primarily driven by \ensuremath{\mathrm{\it{v} \sin \it{i}}}\, which has been seen before amongst young F spectral type stars \citep[{\rm e.g.},][]{galland2005}. Indeed checking the radial velocity RMS of the IN-SYNC stars against \ensuremath{\mathrm{\it{v} \sin \it{i}}}, we find a Kendall's Rank Correlation Coefficient of 0.315, with p-value $<<$ 0.05. We therefore cannot rely on the radial velocity RMS as an effective measure of binarity.
\par
\citet{fernandez2017} have recently presented orbit solutions for a small number of spectroscopic binaries in the APOGEE IN-SYNC data for which a relatively large number of epochs were available. We do utilize this sample to check that our metric for binary candidate identification (see below) is reliable. However, here we are not concerned with orbit-reconstruction, only binary candidate identification, and we also cannot rely on radial velocity scatter to categorize a system as a spectroscopic binary. We therefore employ a simple, robust measure of RV variability between any pair of epochs which we develop below. Given a set of $N\geq2$ RV measurements $\{v_n\}$ of a star, the maximum variation in radial velocity is \ensuremath{\Delta\mathrm{RV}}$=\ensuremath{v_\mathrm{max}}\ - \ensuremath{v_\mathrm{min}}$, where \ensuremath{v_\mathrm{max}}$= max(\{v_n\})$ and \ensuremath{v_\mathrm{min}}$= min(\{v_n\})$. Each radial velocity measurement $v$ has an associated measurement error $\sigma_v$. We can determine the statistical significance of \ensuremath{\Delta\mathrm{RV}}\ via comparison with its uncertainty $\sigma_{\ensuremath{\Delta\mathrm{RV}}}$, propagated from the individual measurement errors $\sigma_{\ensuremath{v_\mathrm{max}}}$ and $\sigma_{\ensuremath{v_\mathrm{min}}}$ of \ensuremath{v_\mathrm{max}}\ and \ensuremath{v_\mathrm{min}}, respectively. We thus define the Normalized Delta RV (\ensuremath{\mathrm{NDRV}}) as the RV variation, \ensuremath{\Delta\mathrm{RV}}, normalized by the error $\sigma_{\ensuremath{\Delta\mathrm{RV}}}$,
\begin{equation}
NDRV = \frac{\ensuremath{\Delta\mathrm{RV}}}{\sigma_{\ensuremath{\Delta\mathrm{RV}}}} = \frac{ \ensuremath{v_\mathrm{max}} - \ensuremath{v_\mathrm{min}}}{\sqrt{\sigma_{\ensuremath{v_\mathrm{max}}}^{2} + \sigma_{\ensuremath{v_\mathrm{min}}}^{2}}}.
\label{eq:ndrv}
\end{equation}
\ensuremath{\mathrm{NDRV}}\ is our primary statistic for identifying spectroscopic binary candidates.
\subsection{Investigation of NDRV behavior with SN}
\label{ssec:ndrv_sn_correction}
\par
The \ensuremath{\mathrm{NDRV}}\ statistic is a ratio of the maximum radial velocity variation within a set of measurements (\ensuremath{\Delta\mathrm{RV}}), weighted by the quadrature error of both measurements on radial velocity(\ensuremath{\sigma_{\Delta RV}}). It is thus important to ensure that there are no inconsistencies between the behavior of either \ensuremath{\Delta\mathrm{RV}}\ or \ensuremath{\sigma_{\Delta RV}}. We checked the \ensuremath{\mathrm{NDRV}}\ values for our final vetted sample against the derived stellar parameters, \ensuremath{T_\mathrm{eff}}\ , \ensuremath{\mathrm{\it{v} \sin \it{i}}}\ , \ensuremath{\log g}\ , as well as SN. We find no inconsistent behavior between \ensuremath{\Delta\mathrm{RV}}\ or \ensuremath{\sigma_{\Delta RV}}\ when comparing the two to the derived stellar parameters. We do find that the \ensuremath{\mathrm{NDRV}}\ value went down as we approached smaller SN values. We then checked both \ensuremath{\Delta\mathrm{RV}}\ and \ensuremath{\sigma_{\Delta RV}}, each scaled by the median value of \ensuremath{\Delta\mathrm{RV}}\ and \ensuremath{\sigma_{\Delta RV}}, respectively, against SN. We plot both these comparisons in Figure~\ref{sn_comp}.
Looking at the bottom panel of Figure~\ref{sn_comp}, we find that there is a relatively flat correlation between the scaled \ensuremath{\Delta\mathrm{RV}}\ and SN, which means any trend to be found between \ensuremath{\mathrm{NDRV}}\ and SN originates from \ensuremath{\sigma_{\Delta RV}}. From \cite{IN-SYNC_paper_1}, we know that the IN-SYNC pipeline produced stellar parameters errors using signal to noise (see Eqs 3,4, and 5 therein), calibrated by MCMC machinery to ensure that the uncertainty properly matched the epoch-to-epoch variability within stellar parameters. From the top panel in Figure~\ref{sn_comp}, we see that the scaled \ensuremath{\sigma_{\Delta RV}}\ values are indeed correlated with SN at the low values of SN. This suggests that the IN-SYNC pipeline was likely more conservative than warranted at low SN for \ensuremath{\sigma_{RV}}. Checking the scaled per epoch radial velocity errors, we find a similar correlation with signal to noise.
We proceeded to fit a univariate spline to the scaled per epoch radial velocity errors and signal to noise. Using the resulting function from this fit, we corrected the individual \ensuremath{\sigma_{RV}}\ values using their corresponding signal to noise, and recalculated the \ensuremath{\mathrm{NDRV}}\ values to ensure that both \ensuremath{\Delta\mathrm{RV}}, and \ensuremath{\sigma_{\Delta RV}}\ behave consistently flat across the signal to noise range of our final vetted sample.
\kojdel{\textbf{We do note that as the fitted spline corrects the \ensuremath{\sigma_{\Delta RV}}\ values across the entire range of SN, at the higher SN we find an increase in \ensuremath{\sigma_{\Delta RV}}. Though the increase in \ensuremath{\sigma_{\Delta RV}}\ is minor, which could affect our results, we find it a necessary step in order to have consistent \ensuremath{\mathrm{NDRV}}\ behavior across the entire SN range.}}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[scale=.7]{Fig3.pdf}
\caption{Top Panel: Scatter plot of the \ensuremath{\sigma_{\Delta RV}}\ for our final vetted sample, scaled by the median \ensuremath{\sigma_{\Delta RV}}\, against signal to noise. The blue line is the median overlapping trend line with 50 objects/bin. The horizontal black dashed line demarcates unity with the median \ensuremath{\sigma_{\Delta RV}}\ value. The inset in the upper right shows a density plot of the overall sample space. Bottom Panel: Scatter plot of the \ensuremath{\Delta\mathrm{RV}}\ for our final vetted sample, scaled by the median \ensuremath{\Delta\mathrm{RV}}\, against signal to noise. The blue line is the median overlapping trend line with 50 objects/bin. The horizontal black dashed line demarcates unity with the median \ensuremath{\Delta\mathrm{RV}}\ value. The inset in the upper right shows a density plot of the overall sample space.}
\label{sn_comp}
\end{center}
\end{figure*}
\subsection{Robustness of NDRV Statistic Against Stellar Parameters}
\label{ssec:stellar_systematics}
\par
Having accounted for drivers of radial velocity variability resulting from poor data quality, as well as removing stars that have physical characteristics that produce spurious radial velocity variability, we are confident in the capability of the \ensuremath{\mathrm{NDRV}}\ statistic to detect radial velocity variability arising solely from an unseen companion. As a final argument, we check the \ensuremath{\mathrm{NDRV}}\ values of our final vetted sample against the derived stellar parameters.
We looked at the behavior of the \ensuremath{\mathrm{NDRV}}\ statistic as a function of other stellar parameters that should not, in principle, be related to intrinsic radial-velocity variations, in order to see if any biases still existed in the \ensuremath{\mathrm{NDRV}}\ as a function of the parameters. We calculate the median \ensuremath{\mathrm{NDRV}}\ values within equal number, half-overlapping bins of the rotational velocity, surface gravity, and effective temperature in order to investigate any potential systematic trends, and plot the \ensuremath{\mathrm{NDRV}}\ against the aforementioned parameters in Figure~\ref{ndrv_scatter_all}. We find that the \ensuremath{\mathrm{NDRV}}\ statistic does not significantly correlate with any of the three stellar parameters.
\kojdel{Figure~\ref{vrad_rms_vsini} shows that indeed the \ensuremath{\mathrm{NDRV}}\ exhibits significant correlation with the \ensuremath{\mathrm{\it{v} \sin \it{i}}} \ above $\sim$100~\ensuremath{\mathrm{km\:s}^{-1}}, and with the \ensuremath{T_\mathrm{eff}} \ above $\sim$6000~K. We recognize that these stars are F-type stars, which are known for their fast rotation during their pre-main sequence stage.
Thus in our analyses that follow, we have opted to eliminate all stars with a median \ensuremath{\mathrm{\it{v} \sin \it{i}}} \ and/or \ensuremath{T_\mathrm{eff}} \ measurement above these limits. We remove 39 stars with 275 observations from our final parent sample with these two constraints.
Recall that we have already accounted above for the expected effect of rapid rotation (generally among hot stars) on the radial velocity measurements by excluding stars with \ensuremath{\mathrm{\it{v} \sin \it{i}}} \ $ > 100$ ~\ensuremath{\mathrm{km\:s}^{-1}} and/or \ensuremath{T_\mathrm{eff}} $ > 6000$~K.}
\begin{figure*}[!ht]
\epsscale{1.1}
\plotone{Fig4.pdf}
\parbox{18cm}{\caption{Scatter plot of the Normalized Delta RV$_{max}$ values against the rotational velocity (top panel), effective temperature (middle panel), and surface gravity (bottom panel) values for each star observed in the 5 IN-SYNC clusters and the Pleiades, colored by either \ensuremath{T_\mathrm{eff}}\ or \ensuremath{\log g}. The diamonds are those stars flagged as binary detections using the functional \ensuremath{\mathrm{NDRV}}\ value, while the circles are stars that were flagged as non-binaries. The gray shaded area is the variable region of $3\sigma$ NDRV values. The lower and upper bounds of the gray shaded area are the $3\sigma$ NDRV values at 2epochs of observation and at 13epochs of observation, respectively, as described in Section~\ref{ssec:ndrv_behavior}. The black trend line in the top, middle, and bottom panel shows the median \ensuremath{\mathrm{NDRV}} value as a function of half over lapping bins of \ensuremath{\mathrm{\it{v} \sin \it{i}}}, \ensuremath{T_\mathrm{eff}}, and \ensuremath{\log g}\ with 50 objects per bin, respectively. The white triangles with blue outlines are sources within our final vetted that were found to be SB2s within \cite{fernandez2017}.}}
\label{ndrv_scatter_all}
\end{figure*}
\clearpage
\subsection{Establishing an NDRV Threshold for Detecting Spectroscopic Binaries}
\label{ssec:ndrv_behavior}
\kojdel{We select spectroscopic binary candidates as those with \ensuremath{\mathrm{NDRV}}\ $\geq$ 3.0, i.e., $\Delta RV_{max}$ is at least $3$ times greater than the propagated velocity error.}
We wish to select binaries as $3\sigma$ detections using the \ensuremath{\mathrm{NDRV}}\ statistic. We cannot assume a static threshold for all the stars within the IN-SYNC final vetted sample due to the different number of epochs of observation carried out during the IN-SYNC survey (see Figure~\ref{time_baseline_nepochs}). This is due to the $\Delta$RV term (equation~\ref{eq:ndrv}), which measures the greatest variation between any pair of radial velocity measurements within the per epoch data for any one star, increasing as the total number of observations on a star increases as well. The correlation between the number of RV measurements and the measured $\Delta$RV was previously described in \cite{moaz_badenes2012}.
To establish the $3\sigma$ \ensuremath{\mathrm{NDRV}}\ threshold, we measure the false positive rate by randomly sampling the RV distribution of a constant RV star with some measurement uncertainty. This distribution is a simple Gaussian with which we generate $10^7$ realizations of randomly sampled pairs, representing two epochs, of RV measurements and calculate their NDRV according to equation~\ref{eq:ndrv}. We then find the \ensuremath{\mathrm{NDRV}}\ value corresponding to the $3\sigma$ percentile of the distribution. We repeat this process for the the range of the number of epochs ($2$ to $13$) observed in our final vetted sample. We find that the $3\sigma$ threshold monotonically increases with the number of epochs observed. This is as expected; the increased number of samples per realization will increase the expectation value of $\Delta$RV.
We find that the \ensuremath{\mathrm{NDRV}}\ $3\sigma$ threshold ranges from $\sim$3.00 for 2 epochs of observation up to $\sim$4.11 for 13 epochs of observation. We list the complete list of \ensuremath{\mathrm{NDRV}}\ $3\sigma$ threshold values in Table~\ref{tab:ndrv_limits}. To characterize binary systems within the IN-SYNC final vetted sample we employ a functional form of the NDRV statistic by comparing any star's \ensuremath{\mathrm{NDRV}}\ value against the 3$\sigma$ threshold value given how many observations that star had in total within the final vetted sample. Candidate binary systems must have an \ensuremath{\mathrm{NDRV}}\ value greater than the \ensuremath{\mathrm{NDRV}}\ value at which there is at least 3$\sigma$ significance.
\jcbdel{We generate distributions of NDRV values from a single star with a gaussian radial velocity distribution by performing 1e7 samplings with the number of values sampled increasing from 2 up to 13, and then find the NDRV value for each distribution that is equivalent to a 3$\sigma$ value.}
\kojdel{Though a significant fraction of all
stars are likely in binary systems (Section~\ref{sec:intro}), relatively few stars ($\approx7\%$) exceed the NDRV$>3$ threshold due to RV resolution limits and observational properties, both of which we later take into account (Section~\ref{sec:binary_fraction_prob_curve}). Still, the NDRV distribution qualitatively suggests NDRV$=3$ lies between two major populations: a large group of stars concentrated at low NDRV, presumably single stars or undetected binaries, and a tail of binary systems extending towards high NDRV values (Figure ~\ref{ndrv_distribution_hist}).}
\begin{table}[H]
\begin{center}
\begin{tabular}{| c | c | c |}
\hline
$ N_{stars}$ & $N_{epochs}$ & $ 3\sigma\ \ensuremath{\mathrm{NDRV}}\ Threshold $ \\ \hline
792 & 2 & 3.00 \\
310 & 3 & 3.31\\%790281718 \\
74 & 4 & 3.49\\%037345275 \\
60 & 5 & 3.62\\%871974106 \\
72 & 6 & 3.72\\%069935786 \\
27 & 7 & 3.81\\%588840047 \\
6 & 8 & 3.88\\%507831385 \\
13 & 9 & 3.94\\%772245054 \\
12 & 10 & 4.04\\%881493 \\
18 & 11 & 4.03\\%104011547 \\
14 & 12 & 4.08\\%52577357 \\
20 & 13 & 4.11\\%022132646\\
\hline
\end{tabular}
\caption{\ensuremath{\mathrm{NDRV}}\ threshold values at $3\sigma$ significance based on the number of epochs from which the \ensuremath{\mathrm{NDRV}}\ value was calculated. Values listed in second column calculated from Monte Carlo simulations. \label{tab:ndrv_limits}}
\end{center}
\end{table}
The \ensuremath{\mathrm{NDRV}}\ statistic depends on accurate reporting of the RV measurement uncertainty. We can use our characterization of the false positive rate to assess the accuracy of the reported RV measurement errors. As above, random measurement errors will lead to non-zero NDRV values due to fluctuating radial velocity measurements. Since NDRV is normalized by the measurement errors, the width of the NDRV distribution for non-RV variable objects will show if the RV errors are under or over estimated.
\kojdel{Since NDRV is normalized by the measurement error, the distribution of NDRV due to random error should repeat measurements of single stars with non-zero radial velocity errors will produce non-zero NDRV values}
\kojdel{The distribution of NDRV qualitatively suggests NDRV$=3$ lies between two major components: a large group of stars concentrated at low NDRV, presumably single stars or undetected binaries, and a tail of binary systems extending towards high NDRV values (Figure ~\ref{ndrv_distribution_hist}). We can validate our choice of NDRV threshold and assess the accuracy of the reported RV measurement errors via the detailed shape of the NDRV distribution.}
In Figure~\ref{ndrv_distribution_hist}, we show the overall distribution of \ensuremath{\mathrm{NDRV}}\ values for all stars with only 2 epochs within the IN-SYNC sample. Unless the RV uncertainties are vastly overestimated, stars with \ensuremath{\mathrm{NDRV}}$< 3$ can safely be assumed to be non-binaries. For \ensuremath{\mathrm{NDRV}}\ $\lesssim 3$, the distribution closely follows a folded normal distribution with $\mu=0$, and $\sigma=1$. The same folded normal distribution is an excellent fit to the distribution of $10^7$ \ensuremath{\mathrm{NDRV}}\ values calculated to determine the false positive rate using two measurement epochs. About $\sim55\%$ of the stars in our final vetted sample have just two epochs. The strong similarity of the \ensuremath{\mathrm{NDRV}}\ distribution for non-binaries and the folded normal distribution with $\sigma=1$ (solid blue line in Figure~\ref{ndrv_distribution_hist}) indicates that \ensuremath{\mathrm{NDRV}}\ values for our sample have been calculated with properly estimated errors. If the radial velocity errors were underestimated by a factor of two, the \ensuremath{\mathrm{NDRV}}\ distribution would broaden by a factor of $2$ (dashed purple line); likewise, if the radial velocity errors were overestimated by a factor of two, the \ensuremath{\mathrm{NDRV}}\ distribution would broaden by a factor of $1/2$ (dotted-dashed orange line). We conclude that the reported RV uncertainties are accurate. In addition, there are more stars with \ensuremath{\mathrm{NDRV}}$ \gtrsim 3$ than expected if all stars in our sample were solitary; as demonstrated, these stars are likely to be in binary systems.
\begin{figure*}[!ht]
\plotone{Fig5.pdf}
\parbox{18cm}{\caption{Cumulative distribution of the \ensuremath{\mathrm{NDRV}} values for the final vetted sample of stars with only two observations within the 6 IN-SYNC clusters, denoted by the black solid line. The solid blue line is a folded normal distribution with $\mu$=0 and $\sigma$=1. The purple dashed line is a folded normal distribution with $\mu$=0 and $\sigma$=2. The orange dotted-dashed line is a folded normal distribution with $\mu$=0 and $\sigma$=.5.}}
\label{ndrv_distribution_hist}
\end{figure*}
\subsection{Vetting the NDRV Statistic Against Known Spectroscopic Binary Samples}\label{ssec:ndrv_vetting}
The IN-SYNC derived radial velocity measurements assume that each spectrum eminantes from a single star. It is unclear how this assumption and the NDRV statistic itself affect detection of double lined spectroscopic binaries (SB2s). We now crossmatch our catalog with established spectroscopic binary samples to both characterize how SB2s appear within our analysis framework and validate the recovery of single lined systems (SB1s) using the NDRV statistic.
\cite{fernandez2017} used the APOGEE spectral cross-correlation function to identify 104 SB2 systems within the IN-SYNC survey footprint. Our final vetted sample contains 34 of these systems; the remaining 70 candidate SB2s do not appear in our sample because they either have only one measurement epoch or do not meet are data quality criteria (see Sec.~\ref{ssec:parameters}). Despite using the same the APOGEE / IN-SYNC spectra as \cite{fernandez2017}, we detect only 6 of the 34 crossmatched systems as RV variable using \ensuremath{\mathrm{NDRV}}. This low SB2 detection rate is perhaps unsurprising given that the IN-SYNC pipeline was not designed to extract parameters from SB2 systems.
Still, it does imply that our inferred spectroscopic binary fractions for each cluster (Section~\ref{sec:binary_fraction_prob_curve}) underestimates the true spectroscopic binary fraction. However, the absolute binary fraction is not critical to our analysis.
We are more concerned with the relative cluster-to-cluster binary fractions. Given the distribution of stellar parameters in each cluster (Figure.~\ref{stellar_parameters_edfs}), a correlation between SB2 detection with photospheric parameters could influence even relative measurements. The full 34 known SB2 systems do not exhibit qualitatively different spectral parameters from the rest of the sample (white triangle with blue outlines denote SB2 systems in Fig.~\ref{ndrv_scatter_all}). Splitting the SB2 systems into NDRV-detected and non-detected subgroups, we find no statistic difference in their stellar parameter distributions. We calculate a 2-sided KS statistic of $\sim0.3$ with two-tailed p-values of 0.49, 0.45, and 0.65 for \ensuremath{T_\mathrm{eff}}, \ensuremath{\mathrm{\it{v} \sin \it{i}}}, and \ensuremath{\log g}, respectively. Since there is no significant statistical difference in the properties of SB2 systems that we do and do not detect, we conclude that we can fairly compare the cluster \ensuremath{\mathrm{NDRV}}\ derived SB fractions.
We proceed to validate the fidelity of our method to recover previously identified SB1s, the assumed binary type in our analysis. \cite{kounkel2016} (hereafter K16) measured radial velocities for 2057 stars in the ONC and NGC 2264, identifying 130 sources as RV variable. Of these 130 RV variable sources, 17 stars appear in our final vetted sample.
However, none of these stars were detected as RV variable according to the \ensuremath{\mathrm{NDRV}}\ statistic using the IN-SYNC observations.
Starting with our sample of RV variable stars, we identify 11 stars that also appear in the K16 catalog. All 11 of these systems were designated as single stars in K16.
The disagreement in RV variable classification between our two studies is not a failure of our method to recover SB1 systems, but likely the result of differences in the observations of the two surveys. For the 17 RV variable systems only detected in K16, both the median number of epochs (5 in K16; 2 in IN-SYNC) and time baselines ($\sim1100\ \mathrm{d}$ in K16; $\sim11\ \mathrm{d}$ in IN-SYNC) strongly favored RV variability detection in K16. Conversely, the IN-SYNC data favored variability detection (median number of epochs 4.5 in IN-SYNC; 3 in K16) for the 11 systems that only we designate as RV variable.
As the observational properties of each survey strongly affect detection probability of an individual system, we also test whether the \ensuremath{\mathrm{NDRV}}\ statistic can re-identify the K16 RV variable stars using the K16 data.
We first remove 28 SB2 stars from the K16 RV variable sample and calculate the \ensuremath{\mathrm{NDRV}}\ of the remaining 102 SB1s using the RV measurements and errors reported in K16. We recover $94\%$ of the K16 variable stars; 96 of the 102 systems had NDRV values exceeding our variability threshold. This successful comparison strongly suggests that the \ensuremath{\mathrm{NDRV}}\ statistic is a robust method for SB1 detection and gives us confidence to apply the NDRV method to make fair comparisons of the cluster SB fractions using IN-SYNC observations.
\section{\jcbins{Developing Binary Fraction Probability Functions}} \label{sec:binary_fraction_prob_curve}
\subsection{Raw Binary Fractions}
Comparing the \ensuremath{\mathrm{NDRV}}\ of each star with their corresponding threshold \ensuremath{NDRV}\ values, we determine the number of spectroscopic binary (SB) counts for each cluster region, with 10 SB systems out of a total of 88 systems in NGC1333, 19 SB systems out of a total of 455 systems in the Orion A(N) cluster, 11 SB systems out of a total of 312 systems in the Orion A(S) cluster, 7 SB systems out of a total of 110 systems in NGC2264, 22 SB systems out of a total of 237 systems in IC348, and 3 SB system out of a total of 216 systems in the Pleiades. These values are listed in Table \ref{tab:bin_frac}. We also list the 2MASS IDs, \ensuremath{\mathrm{NDRV}}\ values, and positions of the spectroscopic binary candidates in Table \ref{tab:candidates}.
\subsection{Bayesian Inference Approach} \label{ssec:bayes_infer}
In reality, the raw spectroscopic binary fractions can not be directly compared because of the differing IN-SYNC observational cadences, time baselines, and visitation coverage of the different clusters (Figure~\ref{time_baseline_nepochs}). We have already demonstrated that observational differences between surveys affect RV variability detection (Section~\ref{ssec:ndrv_vetting}). Cluster to cluster differences in observations must be accounted for to ensure a fair comparison of spectroscopic binary fractions across the IN-SYNC survey.
We therefore develop a probabilistic model of the binary fraction in each cluster to account for the differences in observational data. In addition, we wish to know how the samples we have observed in the IN-SYNC survey relate to the intrinsic populations that exist in these clusters. Our probabilistic approach also accounts for this sampling uncertainty. To constrain the cluster binary fractions, we must calculate the posterior probability $P( X_{f} | O , I )$, where $X_f$ is the binary fraction, $O$ represent the detected binary systems, and $I$ the relevant background information for the data used to determine $O$. Using Bayes' theorem, the posterior probability distribution is proportional to the likelihood of our detected binary systems and any prior on the binary fraction itself
\begin{equation}\label{equation3}
P( X_{f} | O , I ) \propto P( O | X_{f} , I) \times P( X_{f} | I)
\end{equation}
.
\cite{clark2012} (Hereafter C12) successfully employed a similar framework to construct the binary fraction probability distribution of close period M-type Dwarfs from SDSS data.
We now define the likelihood $P(O| X_f, I)$. Each cluster has $N$ confirmed members of which the subset $O$ are defined as RV variable at the $3\sigma$ level using the \ensuremath{\mathrm{NDRV}}\ statistic (Equation ~\ref{eq:ndrv} and Section~\ref{ssec:ndrv_behavior}). Since the \ensuremath{NDRV}\ measurement of any star is independent of all other stars, \ensuremath{P (O | X_f, I)}\ is the probability of detecting RV variability for each star in $O$ while finding no RV variability for all other cluster stars. The likelihood of assigning the stars to one of these two mutual exclusive groups given the data and a binary fraction $X_f$ is the joint probability of each star's velocity measurements $\{v\}_i$ producing an NDRV value indicative of RV variability ($i \in O$) or not ($i \notin O$):
\begin{equation}
P( O | X_{f}, I ) = \prod_{i \in O}{P( \{v\}_i | X_f, I)} \times \prod_{i \notin O}{P( \{v\}_{i} | X_f, I)}
\end{equation}
where the products are done over all stars in each cluster classified as spectroscopic binaries ($i \in O$) and classified as not spectroscopic binaries ($i \notin O$) using \ensuremath{\mathrm{NDRV}}.
We are confident that the RV variability quantified by the \ensuremath{NDRV}\ statistic is due to the interaction of the primary star with an unseen companion. While this is already a common assertion in the literature \cite[{\rm e.g.},][C12]{maxted_jeffries2005}, we have demonstrated that neither spectroscopic parameters (Section ~\ref{ssec:stellar_systematics}) nor RV measurement uncertainties (Section~\ref{ssec:ndrv_behavior}) drive the \ensuremath{NDRV}\ statistic .
Therefore, the probability of any system $i$ being detected as a binary is the sum of detection probability given the true binary fraction and the false-positive probability
\begin{equation}
P(\{v\}_i | X_f, I) = X_f \times \ensuremath{p_{\mathrm{detect},i}} + (1 - X_f) \times 10^{-2.57}
\label{eq:pvi}
\end{equation}
where the factor $10^{-2.57}$ is the false positive rate of a $3\sigma$ detection of a binary candidate using \ensuremath{\mathrm{NDRV}}. The \ensuremath{p_{\mathrm{detect},i}}\ term is the probability of detecting a star $i$ as a binary (\ensuremath{\mathrm{NDRV}}$ > $ \ensuremath{NDRV} at $3\sigma$) assuming it is part of a binary system. In the next section, we employ simulated observations of each system to calculate \ensuremath{p_{\mathrm{detect},i}}. Since each star is classified as either a binary ($\in O$) or not ($\notin O$), it follows that the probability of non-detection is $1 - P(\{v\}_i | X_f, I)$.
The final term of the binary fraction posterior probability is the prior distribution, $P(X_{f}|I)$. This is simply the probability of the model we have chosen given the information we have on hand. Like C12, we also follow \cite{allen2007} in assuming an uninformative prior. We treat the binary fraction as a scale parameter since we are concerned \textbf{only} with relative differences in the binary fraction between clusters. With this information and following C12 we can use Jeffrey's Prior from \cite{sivia_skilling2006} to compose our prior distribution such that $P(X_{f}|I) \propto \frac{1}{X_{f}}$.
\subsection{Calculating P$_{detect}$ with Monte Carlo Simulations}
\label{ssec:monte_carlo_sims}
We still must characterize the probability of falsely classifying a star as non-binary; conversely, the probability of recovering stars in real binary systems as RV-variable binary candidates (\ensuremath{p_{\mathrm{detect},i}}). Stars in binary systems trace out periodic velocity curves due to Keplerian motion about the center of mass. The measured instantaneous radial velocity of the star depends upon the orbital phase at which the measurement took place and the intrinsic properties of the binary system. From \cite{lovis2010} we employ the projected velocity vector equation used to find unseen exo-planets orbiting an observed star. Assuming a circular orbit, on-edge binary system, the equation for the expected $RV$ of the primary simplifies to
\begin{equation}
RV(\theta) = RV_{amplitude} \times \sin(\theta),
\label{eq:rv_theta}
\end{equation}
where $\theta$ is the orbital phase of the primary mass relative to our line of sight. Using Kepler's equations of motion, the amplitude of the radial velocity curve under our orbital assumptions is
\begin{equation}
RV_{amplitude} =\sqrt{ \frac{G M_{2}^2}{a(M_1 + M_2)}},
\label{eq:rv_amp}
\end{equation}
where G is the gravitational constant, $M_1$ ($M_2$) is the mass of the primary (secondary), and $a$ is the semi-major axis of the relative binary orbit.
To calculate \ensuremath{p_{\mathrm{detect},i}}, we perform mock observations of the primary star motion predicted by equations ~\ref{eq:rv_theta} and ~\ref{eq:rv_amp} within a suite of simulated binary systems. The cadence of the observations and synthetic measurement errors follow directly from the observational properties of our sample (discussed below). The simulated binary systems themselves result from Monte Carlo sampling of the parameter distributions that affect the radial velocity curve -- the binary separation $a$ and the mass ratio $q$.
The mass ratio $q\equiv M_2/M_1$ for binary stars is relatively well constrained. Following \cite{allen2007}, we model the probability distribution function of $q$ as a power-law $P(q) \propto q^{1.8}$ for $0.02 < q < 0.5$; $0$ otherwise. We do not test other parameterizations of the mass ratio distribution; C12 found no difference in the inferred short-period ($P < 10~d$) binary fraction of SDSS field stars when varying the index of the \cite{allen2007} power-law.
We perform first order mass derivations for the PMS sample and the MS using both Baraffe PMS isochrones \cite{baraffe2015} and Padova PMS isochrones \cite{bressan2012} in order to set an appropriate primary mass for our simulations that would realistically reflect the populations IN-SYNC observed. Choosing Orion A(N) and the Pleiades for our PMS and MS sample, repsectively, we find a median primary mass in both samples of $\sim0.5 \ensuremath{M_{\odot}}$ which we use to set our primary mass limit on our simulated binaries. The secondary mass follows directly from the mass ratio as the primary mass is kept constant.
Kepler's Third Law determines the binary separation for stars of known mass given the orbital period of the secondary ($P$). We therefore sample $P$ from an established literature distribution and calculate $a$ for each system via $a^3 = G P^2 (M_1 + M_2) / 4\pi^2$. The distribution of orbital period in our simulated systems is log-normal with mean $\overline{P}= 10^{4.8}$ days and dispersion $\sigma_{P}=10^{2.3}$ days, following \cite{DM1991}. The empirical distribution functions of companion mass and the period distributions that were sampled are plotted in Figure \ref{comb_period_comp_mass}.
\begin{figure*}
\plotone{Fig6.pdf}
\parbox{18cm}{\caption{The green dashed line is the empirical distribution function of the periods distribution of binary system simulations used to generate $p_{detect}$ for systems observed in IN-SYNC clusters. Sampled from \cite{DM1991} with a mean of Log10(days) = 4.8, this distribution has been constrained to be below Log10(days) $\le$ 4.5 from tests to calculate the effective period limit at detecting binaries with the IN-SYNC observational parameters. The blue dotted-dashed line is the empirical distribution function of the masses of companions in binary systems simulations to generate recoverability fraction. Sampled from a power law distribution with $\alpha$=1.8 and ranging from 0.02M$_{\odot}$ to 0.5M$_{\odot}$ }}
\label{comb_period_comp_mass}
\end{figure*}
We impose both minimum and maximum period limits on our simulated binary systems. Our minimum period comes from the minimum separation necessary to have a detached, circular 0.5\ensuremath{M_{\odot}} - 0.5\ensuremath{M_{\odot}} \ binary system. For each system composed from randomly sampled companion mass and period, we use the equation for the Roche-Lobe, R$_{l}$, from \cite{eggleton1983}, given below, to ensure that the system is detached,
\begin{equation}
\frac{R_{l}}{a} = \frac{0.49\ q^{-2/3}}{0.6\ q^{-2/3} + \ln(1+q^{-1/3})}.
\end{equation}
We stipulate that the Roche lobe of the primary be less than half of the orbital separation, {\rm i.e.}} \newcommand{\etc}{{\rm etc.}, $R_l/a < 2$.
For the maximum orbital period, we recall that the typical RV error is 1.5~\ensuremath{\mathrm{km\:s}^{-1}}, and so even the most advantageous IN-SYNC observational cadence and measurement errors are likely to be unable to recover RV variability at 3$\sigma$ significance for orbital amplitudes $\lesssim 4.5$~\ensuremath{\mathrm{km\:s}^{-1}}. For a 0.5\ensuremath{M_{\odot}}\ binary twin system, this gives a maximum period length of $10^{4.5}$ d.
To corroborate this choice of maximum period, we calculate the fraction of simulated binary systems recovered using \ensuremath{NDRV}\ out of a group of 1e5 simulated orbits for each of our 6 clusters (see Figure \ref{ndrv_recovery}). In each cluster we find that the maximum period recoverable is around $10^{4.1}$~d. We thus set our maximum period cutoff in the simulated orbits to $10^{4.5}$~d, to conservatively assess the recoverability capability of simulated binaries with the IN-SYNC survey. We plot the empirical distribution functions of the probability of detections for each of the 6 clusters in Figure \ref{p_detect_dist}.
It is crucial that the mock observations have the same observational properties as the data. Recall that each star in our final sample has $n = 1 .. N$ ($N \geq 2$) RV measurements with associated errors $\{\sigma_{RV}\}$ collected on $N$ epochs represented by Julian Date and denoted $\{\mathrm{JD}\}$. To observe a simulated system with period $P$, we determine the relative phase $\theta_n$ of our observations, $\theta_n = (\mathrm{JD}_n - \mathrm{JD}_{n-1})/ P$ for $n=1 .. N$; $\theta_0 =0$. The trial radial velocity curve has $N$ RV measurements, computed as $RV_n = RV_{amplitude} \times \sin (\Theta + 2\pi\theta_n)$, where $\Theta$ is a randomly chosen initial orbital phase angle. We then calculate the
\ensuremath{\mathrm{NDRV}}\ value of the set of $RV_n$ and the corresponding $\sigma_{RV_n}$ from the actual IN-SYNC observations. For each star and set of observations in our parent sample, we generate $10^5$ binary systems and measure $10^5$ \ensuremath{\mathrm{NDRV}}\ values. The fraction of these systems with \ensuremath{\mathrm{NDRV}} $>$ the $3\sigma$ \ensuremath{\mathrm{NDRV}}\ threshold corresponding to the total number of observations,i.e., tagged as RV variable, is $p_{\mathrm{detect}}$.
\begin{figure*}
\begin{center}
\includegraphics[scale=.8]{Fig7.pdf}
\parbox{18cm}{\caption{The fractional recovery of simulated systems from Monte Carlo simulations using the observational parameters of the 6 clusters. Each data point represents $\ge$ 1000 simulated binary systems grouped by their period on the x-axis. For each line, there are 1e5 * N$_{cluster}$ simulated systems in total. The y-axis shows the fractional percentage of these systems that produced a $3\sigma$ level \ensuremath{\mathrm{NDRV}}\ value. As it can be seen here, most of the 6 clusters do not recover any simulated binary systems with the \ensuremath{\mathrm{NDRV}} statistic with a period of $Log_{10}(days) \ge 4.0$}}
\label{ndrv_recovery}
\includegraphics[scale=1.]{Fig8.pdf}
\parbox{18cm}{\caption{Empirical distribution functions of the $P_{detect}$ values produced from observations of simulated binary star systems produced from Monte Carlo sampling for the 5 IN-SYNC clusters and the Pleiades. }}
\label{p_detect_dist}
\end{center}
\end{figure*}
\subsection{The Binary Fraction Posterior Distribution}
\label{ssec:posteriors}
\par
With full expressions for both the likelihood distribution and the prior distribution, the posterior distribution for the binary fraction of a cluster can then be written as follows:
\begin{multline}
p( X_{f} | O , I ) \propto p(O | X_f, I) p(X_f | I) \\
\propto \prod_{i \in O}{p(\{v\}_i | X_f, I)} \times \prod_{i \notin O}{p(\{v\}_i| X_f, I)} \centerdot p(X_f | I) \\
= \bigg[ \prod_{i \in O}{\Big[ X_f \times \ensuremath{p_{\mathrm{detect},i}} + (1 - X_f) \times 10^{-2.57}\Big]} \centerdot \\
\prod_{i \notin O}{\Big[1 - \big(X_f \times \ensuremath{p_{\mathrm{detect},i}} + (1 - X_f) \times 10^{-2.57}\big) \Big]}\bigg] \frac{1}{X_{f}}
\end{multline}
\kojdel{[KARL: OLD equation below
\begin{multline}
p( X_{f} | O , I ) \propto p(O | X_f, I) p(X_f | I) \\
\propto \prod_{i \in O}{p(\{v\}_i | X_f, I)} \times \prod_{i \notin O}{p(\{v\}_i| X_f, I)} \centerdot p(X_f | I) \\
= \bigg[ \prod_{i \in O}{\Big[ X_f \times \ensuremath{p_{\mathrm{detect},i}} + (1 - X_f) \times 10^{-2.657}\Big]} \centerdot \\
\prod_{i \notin O}{\Big[1 - \big(X_f \times \ensuremath{p_{\mathrm{detect},i}} + (1 - X_f) \times 10^{-2.89}\big) \Big]}\bigg] \frac{1}{X_{f}}
\end{multline}
]}
where the products are taken over each star $i$ in the cluster classified as a binary ($i \in O$) or single ($i \notin O$) system.
Finally, we normalize the posterior probability distribution by requiring $\int_{0}^{1} P( X_{f} | O , I ) dX_{f} = 1$.
\section{results}
\label{sec:results}
Applying the methodology laid out in Section~\ref{sec:binary_fraction_prob_curve} to the IN-SYNC observations described in Section~\ref{sec:data}, we obtain the primary results of this study: the posterior distributions of spectroscopic binary frequency for each of the six clusters in our study.
We calculate $p(X_f| O, I)$ for each cluster for $X_f$ in the range of 0 to 1. We also report the median binary fraction within each posterior probability distribution as well as the difference between the median and the 16th and 84th percentiles. These values are also in Table \ref{tab:bin_frac} along with the number of stars flagged as binary systems using the \ensuremath{\mathrm{NDRV}}\ statistic, the total number of stars observed within each cluster, the total number of observations within our final vetted data for each cluster, and the age of each cluster from the literature.
\begin{deluxetable*}{ccccccccc}
\tablecaption{In the first 3 columns above we have for each cluster: the 16th percentile of the posterior distribution, the 50th percentile of the posterior distribution, and the 84th percentile of the posterior distribution. In the last four columns we have the number of real systems flagged as binaries, the total number of systems observed for each cluster, the total number of observations within each cluster, the raw spectroscopic binary fraction of each cluster, and the adopted literature age for the cluster. $^{\dagger}$ Age is in Myr. \label{tab:bin_frac}}
\tablehead{\colhead{Cluster} & \colhead{$X_{f,16}$} & \colhead{$X_{f,50}$} & \colhead{$X_{f,84}$} & \colhead{$N_{SB}$} & \colhead{$N_{tot}$} & \colhead{$N_{obs}$} & \colhead{$f_{SB}$} & \colhead{$Age^{\dagger}$} }
\startdata
Orion A(N) & 0.104 & 0.134 & 0.168 & 19 & 455 & 1026 & 0.042 & 1.5\\
Orion A(S) & 0.087 & 0.126 & 0.173 & 11 & 312 & 697 & 0.035 & 2.5\\
IC 348 &0.157 & 0.196 & 0.239 & 22 & 237 & 1375 & 0.093 & 6.0\\
NGC 1333 & 0.132 & 0.184 & 0.246 & 10 & 88 & 423 & 0.114 &1.0\\
NGC 2264 & 0.079 & 0.121 & 0.174 & 7 & 110 & 555 & 0.064 &3.0 \\
Pre Main-Seq & 0.139 & 0.158 & 0.178 & 69 & 1202 & 4076 & & $\approx$~5 \\
Pleiades & 0.0086 & 0.0329 & 0.0698 & 3 & 216 & 566 & 0.014 &115\\
\enddata
\end{deluxetable*}
Figure \ref{prob_bf} summarizes these key results, where we have plotted the calculated binary fraction posterior distributions for our set of 6 clusters. The inset shows the relationship between the median binary fraction for the 6 clusters and their ages (as discussed in Section~\ref{ssec:data_overview}).
We see that there is a decrease of the binary fraction from the pre-main sequence ($\sim$1--10~Myr) to the main sequence ($\sim$100~Myr) by a factor of $\sim$ 3--4.
The shape of the binary fraction posterior distributions of each cluster reflect how the bayesian framework takes into account the number of systems observed overall, the number of systems detected as spectroscopic binaries using the \ensuremath{\mathrm{NDRV}}\ statistic, and the probability of a binary detection for any star within a cluster. The 5 pre-main sequence clusters have large enough sample sizes, and high enough $p_{detect}$, that the binary fraction posterior distribution can be effectively reconstructed. The Pleiades has the smallest raw binary fraction and the smallest achieved $p_{detect}$ values, which manifests into an asymptotic probability distribution toward the lower binary fraction values. From Figure~\ref{prob_bf} it can be seen that the Pleiades distribution has a turnover occur at the median, after which it becomes asymptotic as it approaches a binary fraction of 0.0. This is likely a result of using Jeffrey's prior, which weights lower binary fraction values as more probabilistically possible given our framework.
The EDFs of the $p_{detects}$ from Figure~\ref{p_detect_dist} reflect how the monte carlo simulations convolve the different time baselines, number of observations (see Figure~\ref{stellar_parameters_edfs}), and real radial velocity error in order to develop the probability of detection to be used within the bayesian framework.
Looking only at the PMS cluster probability distributions, Orion A(N) has the highest peaked cluster. This is most likely due to the cluster having the largest number of systems observed, as well as high values of $p_{detect}$. Comparing with the posterior distribution of IC348, and NGC1333, we find that the the time baselines carried out in each cluster do not play as large a role in constructing the binary fraction distributions as the other factors accounted for. Considering that IC348, and NGC1333 have some of the highest time baselines, as well as the highest $p_{detect}$ values out of the 6 clusters, their resulting binary fraction distribution are more spread out and not as peaked as the two Orion A sub-clusters, suggesting that increasing the number of stars observed will sharpen the resulting distribution, as the intrinsic population will be more comprehensively sampled.
The shape of the Pleiades distribution arises not only from there being only three detected binary within the final 216 stellar systems, but also from the smaller range of detection probabilities that is achieved for the Pleiades cluster. \kojins{For the 5 pre-main sequence clusters, the probability of detection ranges from $\sim$10\% -- 25\% at the lower end up to $\sim$50\% -- 75\% at the upper end. The Pleiades cluster, however, only achieves probabilities of detection ranging from $\sim$15\% to $\sim$35\%.} The shape of the EDFs of the probability of detection is also reflected within the shape of the fraction of detected simulated systems for each cluster. We see that smoothly varying $p_{detect}$ distributions result in smoothly varying behavior in the fractional recovery of simulated binary systems.
The longer right tail of the Pleiades is most likely due to the small number of \ensuremath{\mathrm{NDRV}}\ binaries found within the cluster sample, and the values of probability of detection achieved. Since the $p_{detect}$ values were the smallest out of the 6 clusters observed by IN-SYNC, there is a higher possibility that some of the stars designated by the \ensuremath{\mathrm{NDRV}}\ statistic as non-binaries were indeed false-negatives, and suggests that the true binary fraction of the Pleiades could be higher.
We can use these probability distributions to assess the statistical significance of the our finding of a declining binary fraction. Considering for example just the pairwise comparison of the Orion A North region versus the Pleiades, and using a simple $\chi^2$ null hypothesis test, we conclude that the difference is modestly significant at $\lesssim$2$\sigma$. However, the same difference in both sense and magnitude remains for each of the clusters, strengthening confidence in the result.
We can also consider the joint probability distribution of all five pre--main-sequence clusters as one. Having a total number of 1202 stars, as well as combining some of the higher distributions of probability of detection, it is the most sharply defined and most highly peaked distribution. This is shown in Table \ref{tab:bin_frac} and Figure~\ref{prob_bf} (gray solid line). Taking the pair wise comparison of the joint pre-main sequence distribution and the pleiades, the difference from the pre--main-sequence to the main sequence becomes more highly significant at the 3--4$\sigma$ level.
\begin{figure*}
\includegraphics[scale=1.2 ]{Fig9.pdf}
\parbox{18cm}{\caption{The normalized posterior distributions of the binary fraction for each cluster. The 5 IN-SYNC clusters (plotted in the dashed lines), and the Pleiades (red solid line). The gray solid line is the joint distribution of the 5 pre-main sequence IN-SYNC clusters. The inset box in the top right has the ages of the 6 clusters plotted against their median binary fraction, along with their $16^{th}$ and $84^{th}$ percentile values. The joint distribution of the pre-main sequence clusters is also in the inset plot, marked by the gray circle, with $16^{th}$ and $84^{th}$ percentile values. The 'age' of the joint distribution is chosen to be $\sim$5Myr.}}
\label{prob_bf}
\end{figure*}
\section{discussion}
\label{sec:discussion}
In this work, we have used simulated observations within a Bayesian framework to infer the spectroscopic binary fractions of young star clusters, based on observations of the pre-main sequence aged IN-SYNC clusters to the main-sequence aged Pleiades cluster. From our simulations of detached, edge-on, circular spectroscopic binaries, using the \ensuremath{\mathrm{NDRV}}\ statistic and accounting for the IN-SYNC time-baselines, number of observations, and radial velocity errors, we find that the IN-SYNC survey has a sensitivity to spectroscopic binaries with orbital periods ranging from $10^{2}$~d -- $10^{4.1}$~d, depending on the cluster observed. The observed samples are found to all represent a similar mass regime, with a median estimated mass of $\sim$0.5~\ensuremath{M_{\odot}}. Thus, our results apply to low-mass binaries of short- and intermediate-periods.
Our results suggests that the fraction of low-mass binaries with orbital periods up to $10^{4.1}$~d appears to be decreasing by a factor of $\sim$3-4 from $\sim$1~Myr to $\sim$100~Myr.
In the following subsections, we discuss the relevant observational literature to place our findings in context. We then specifically discuss prior theoretical findings regarding the role of dynamics in the evolution of binaries. Finally, we consider potentially important effects that we have not yet included in our analysis.
\subsection{Comparison with previous observational results}\label{ssec:}
We find that there is general agreement between our inferred spectroscopic binary fractions for the IN-SYNC pre-sequence clusters and previous observational studies of pre-main sequence spectroscopic binaries.
For example, looking at the Ophiuchus region in the H-band, \cite{prato2007} found a spectroscopic binary fraction of $12^{+8}_{-3.5}$. Their sample consisted of 33 T-Tauri stars observed over 3 years with 10m Keck-II Telescope, mostly consisting of M-type stars. Our derived spectroscopic binary fractions for all 5 of our pre-main-sequence regions fall within this observed range.
Similarly, \cite{tobin2009, tobin2013} looked at the Orion Nebula Cluster's spectroscopic binary fraction, gathering multi-epoch data for a larger sample of 727 stars, achieving sensitivity to spectroscopic binaries out to $4\times10^{3}$ days. With this multi-epoch data they find 89 binaries \kgsdelete{within their sample}, giving a spectroscopic binary fraction of 11.5\% for the ONC, which is slightly lower than but comparable to our spectroscopic binary fraction of $\sim$~13.4\%.
\citet{kounkel2016} did a reanalysis of the \citet{tobin2009} Orion data and also performed an analysis of NGC2264. They found their binary fractions within separations of 10~AU to be $\sim$~5.3\% and $\sim$~5.8\%, respectively. These observed binary fractions are roughly half of the values we have inferred using our Bayesian framework of $\sim$~13.4\% and $\sim$~12.1\% for Orion A(N) and NGC~2264, respectively.
However, the IN-SYNC survey
has a sensitivity out to 20~AU and so we would expect a higher number of binaries to be detected as the parameter space of detection is increased.
\cite{mathieu1989} also found a relatively low pre-main sequence spectroscopic binary fraction. They found 6 young spectroscopic binaries within the naked T-Tauri star populations of Taurus-Auriga, Scorpius-Ophiuchus, and the Corona-Australis star forming regions for short periods (p$<$100 days), corresponding to a spectroscopic binary fraction of $9\pm 4$\%. This again is not necessarily inconsistent with the higher binary fractions found by IN-SYNC and the other studies noted above that probe a much larger range of binary orbital periods.
Next we consider how our inferred spectroscopy binary fraction for the Pleiades compares to previous findings for main-sequence clusters. Our results are most directly comparable to \cite{abt1987} who derived a short period main-sequence binary fraction of 12\%. This is a factor of $\sim$3 higher than our measurement, however
\cite{abt1987} considered stars roughly 2--3 times more massive than those observed by IN-SYNC, and the short-period binary fraction has been found to increase linearly with primary mass \citep[][]{clark2012}.
In contrast, using the CORAVEL radial velocity survey, \cite{raboud_mermilliod1998} (hereafter RM1998) found the Pleiades overall binary fraction to be a higher $\sim$20\% within a mass range from 0.5\ensuremath{M_{\odot}} -- 1.0\ensuremath{M_{\odot}}.
As with \cite{abt1987}, the sample observed by RM1998 covered a higher mass range (see their Figure 10)
In addition \kgsdelete{to this}, the time baseline of the \kgsdelete{spectrocopic CORAVEL data used in} RM1998 observations spanned 17 years, much longer than the $\sim$1.3 year maximum time baseline of the IN-SYNC survey.
Thus, the binary fraction determined by RM1998 represents both a larger period range and a higher mass range than the IN-SYNC sample, which as noted above is more directly comparable to the \cite{abt1987} results.
\subsection{Dynamical evolution of short-period versus intermediate-period binaries} \label{ssec:dynamics}
Observationally, there has been evidence for a decreasing overall binary fraction on long timescales among solar-type stars \citep[$\gtrsim$1~Gyr][]{} \citep{mason1998}. During the pre-main-sequence phase, observational results have been sparse, but generally have also suggested a overall decreasing binary fraction from pre-main-sequence ages to field ages \citep[e.g.,][]{ghez1993, ghez1997}. However, these results have typically been based on longer period binary populations, much wider than those accessible to the IN-SYNC observations.
The fact that our results pertain to shorter period spectroscopic binaries could on its face be surprising, considering that simulations have shown short-period binaries in fact become more tightly bound as they evolve within a cluster \kgsdelete{a behavior commonly known as {\it Heggie's Law}} \citep[see][and references therein]{heggie1975}. This may imply that it is specifically the intermediate-period binaries (periods of $\sim$10$^2$--10$^4$~d), which have not been well probed in previous observational studies of pre-main sequence populations, that are undergoing dynamical disruption as they evolve toward Pleiades age.
It is expected that dynamical processing via interactions with other stars in the cluster will modify the number of binaries, and the distributions of their properties, over time. Current theory relies largely on N-body simulations to reproduce the observed states of star clusters.
For example, \cite{kroupa_2001_binaries} finds that the overall number of binaries with period range $10^0$--$10^9$~d of a simulated cluster decreases after just $\sim$2.5~Myr, comparable to the typical age of the IN-SYNC cluster sample, as binary disruption occurs among the widest binaries, unbinding the smaller mass secondary star. \cite{goodwin_kroupa2005} further finds that the disruption is most pronounced for lower-mass M-dwarf binaries due to the lower binding energy.
Thus, the decrease in the spectroscopic binary fraction that we observe between the pre-main sequence clusters and the Pleiades, could be a manifestation of these effects, particularly given the low masses that characterize our observational samples and the sensitivity of our observations to relatively long-period binaries.
At the same time, the role of cluster environment is a potentially confounding factor. \cite{duchene1999} did not find clear evidence for a decrease in the binary fraction on the basis of comparing their IC~348 sample with the Trapezium \citep{prosser1994, petr1998}, the Pleaides \citep{bouvier1997}, and solar-type stars in the field \citep{DM1991}. However, they do postulate that there may be an anti-correlation between binary fraction and cluster density; i.e., that the densest star-forming environments (e.g., Orion Trapezium region) have a lower binary fraction than low-density environments (e.g., Taurus-Auriga). That result would be consistent with the literature on binary fractions in older globular clusters \citep[][but see \citet{milone2012}]{sollima2007}.
More generally, the argument of cluster density driving the the observed differences of the binary fraction among different clusters would complicate the apparent evolution of the binary fraction in our samples if the Pleiades cluster had started as a pre-main sequence cluster with higher stellar density than the IN-SYNC clusters.
\cite{converse_stahler2009} used simulations in an attempt to recreate the initial dynamical state of the Pleiades. Their results suggest that the Pleiades cluster already had a high degree of mass segregation, and expanded rapidly, leaving behind only a dense core remnant.
\cite{fuente_marcos1997}, \cite{fuente_marcos1998}, and \cite{adams2000} corroborate the scenario that dense older clusters like the Pleiades are the dense bound remnant core of a larger open cluster that has already evaporated away.
However, \cite{kroupa_2001_sc} suggests that the Pleiades open core remnant observed today could have arisen from a high stellar density OB association such as the ONC. This cluster nucleus would be all that was left after the high mass O stars expelled the gas cluster gas, causing the majority of the cluster to expand outward on shorter relaxation time scales, while the core became more bound due to two-body relaxation avoiding the cluster dissolution. Simulations performed by \cite{kah2001} support this scenario, finding a cluster simulation of the ONC leading to a Pleiades-like remnant core. They find that the binary fraction within this dense OB association would also decrease over the evolution of the cluster to a dense core remnant.
It is clear that more work is needed to solidify the conclusion that the Pleiades represents the later-stage evolution of pre-main sequence clusters similar in density to those in our IN-SYNC sample. However, assuming that the IN-SYNC clusters and the Pleiades may indeed be fairly compared, we suggest that the most relevant consideration for the present work is that the parameter spaces of previous observational studies had a dearth of observational information on pre-main-sequence binaries with periods ranging from $\sim10^{2}$days -- $10^{5}$days. Considering that the IN-SYNC survey achieved sensitivity over the period range of $10^2$--$10^4$~d, we speculate that the binary fraction decrease found in our results stems from these relatively shorter period binaries which are still wide enough to be disrupted by dynamical interactions over the 100~Myr timescale we consider here.
\subsection{Caveats and issues not addressed in this work }
We have not attempted to treat the influence of triples and higher-order hierarchal systems. \cite{tokovinin2006}, looking at a sample of 165 solar-type spectroscopic binaries, found that as many as $\sim$80\% possess a tertiary component when the period of the inner binary pair is less than 7~days, decreasing to $\sim$30\% for inner-binary period $\gtrsim$20~days. While our observed and simulated parameter space spans a very much larger period range, the apparent decrease in binary fraction that we observe could in principle result if the IN-SYNC clusters possess a higher fraction of triples than the Pleiades, thus artificially inflating the incidence of (easier to detect) short-period spectroscopic binaries in our pre-main sequence samples.
The Bayesian framework that we have developed should be useful for future efforts to infer binary populations from inhomogeneous datasets. Even so, there are also directions for improving our framework. With regard to our simulations, we have ignored the effects of eccentricity and inclination on generating radial velocities to be simulated using the observational time baselines and observational cadences of our 6 clusters. Thus all of the radial velocity curves for our simulated binaries before being `observed' are all symmetric and sinusoidal.
Fortunately, however, for the purposes of our work, these effects will largely be systematic in nature, but the relative binary fractions should be largely preserved. For example, inclination will systematically decrease the amplitude of the radial velocity curves of the simulated binaries, and thus will lower the $p_{detect}$ values for all 6 clusters, thus underestimating the inferred binary fractions.
The effect of eccentricity is somewhat more complex. On the one hand, circularization has been found observationally to occur for binaries with a period $\le$10 days \citep{meibom2005, raghavan2010}, and so our assumption of circular orbits in our simulations should be robust for the shortest period spectroscopic binaries. For longer period binaries, there is a competing effect of eccentric binaries being easier to detect spectroscopically near periastron but more difficult to detect near apastron. We expect that the net effect will not be large, but this is an avenue for future extensions of the simulations we have presented here.
In addition, we have not attempted to model the effects of the increased magnetic activity of pre-main-sequence stars is known to drive spurious RV jitter. \cite{hillenbrand2015} calculated the lower limit of mass for planetary companions around stars younger than our Sun and found RV jitter is correlated with chromospheric activity. They find the highest levels of chromospheric activity produce RV jitter of $\sim$200~\ensuremath{\mathrm{m \cdot s}^{-1}}. \cite{stassun2004} finds that sunspot activity can cause spurious RV jitter because of contrast between the sunspot and the stellar photosphere. This RV jitter can be of order 1~\ensuremath{\mathrm{km\:s}^{-1}}\ in the optical. We tested what effect this would have upon our results and introduced a sunspot RV jitter error correction into the denominator of our \ensuremath{\mathrm{NDRV}}\ statistic (see equation \ref{eq:ndrv}) of $\sigma_{sunspot}= 0.25~\ensuremath{\mathrm{km\:s}^{-1}}$. We adopted a value of 0.25~\ensuremath{\mathrm{km\:s}^{-1}}\ for sunspot activity as \cite{marchwinski2015} found that RV jitter in the near IR is up to 4x less than in the optical \cite[see also][]{mahmud2011}. If we assume that this error applies to all of the stars in our pre-main sequence samples, then our finding of a decreasing binary fraction from the pre-main sequence to the Pleiades declines to $\sim2\sigma$ significance.
More generally, our work relies on the measured RVs and RV errors from the IN-SYNC pipeline. In any analysis, the robustness of the measurements is crucial to the robustness of the results derived from them. Throughout this work we have attempted to characterize and account for systematics present within the data we use---across stellar parameter space, within a given cluster, across the various clusters---as well as systematics that may arise from our analysis machinery itself. As with any work, any deeper or hidden systematic effects that we have not considered will of course remain unaccounted for. In this section we have attempted to identify a number of these potential additional systematic effects. We cannot know the extent to which they actually impact our data or analysis, but we may speculate that our estimates above suggest that even if present these effects do not negate the findings of this work.
\section{summary}
\label{sec:summary}
Using the high precision ($\sigma_{RV}\sim .3\ensuremath{\mathrm{km\:s}^{-1}}$), high volume ($>10^{3}$ stars) H-Band spectroscopic data from the IN-SYNC survey, we studied the derived stellar properties of five IN-SYNC clusters to find signals of possible unseen sub-stellar companions. We used the \ensuremath{NDRV}\ statistic to find candidates that could have unseen stellar companions within the IN-SYNC data. The advantage of the \ensuremath{NDRV}\ is its ability to capture the maximum $\Delta$RV within a set of radial velocity measurements and weighting it by the relative error of the two measurements that produce the highest $\Delta$RV with as little as two observations of a star.
We performed Monte Carlo simulations within a Bayesian framework in order to account for the differing observational cadences and other observational biases in the observations from one cluster to the next.
Due to the small number of systems within each region observed by the IN-SYNC survey as well as the Pleiades, it was not enough to simply bootstrap our data in order to characterize the error on the raw spectroscopic binary fraction. We could not assume that the sample of systems observed in the IN-SYNC survey is representative of the population of star systems within these 5 clusters. In order to characterize the spectroscopic binary fraction of these clusters, we developed a Bayesian framework in order to infer the underlying distributions for each cluster while taking into account the calculated effectiveness of IN-SYNC observations at capturing radial velocity variations from unseen companions using simulated observations of constructed binaries. The framework that we have presented should itself be useful for future studies that seek to make inferences about binary populations from inhomogeneous datasets.
We find the spectroscopic binary fraction for the five pre--main-sequence clusters, with ages in the range $\approx$1--10~Myr, to be in the range $\approx$20--30\%.
Performing the same \ensuremath{NDRV}\ analysis on similarly reduced, APOGEE derived, spectroscopic data from the Pleiades, we find a smaller spectroscopic binary fraction of 5--10\%. Even with all of the biases and sampling effects folded into the analysis, this decline in binary fraction is modestly significant when comparing any one of the pre--main-sequence clusters to the Pleiades. That the same decline in sense and in magnitude emerges for each of the pre--main-sequence clusters, bolsters this conclusion, and becomes significant at the 3--4$\sigma$ level when all of the pre--main-sequence clusters are considered together.
There are additional effects, in particular random line-of-sight inclinations and eccentric orbits for long-period binaries, that we have not included here but that would allow our framework to produce inferred binary fractions that are closer to the absolute binary fraction. In addition to this, in this work we considered the effect of RV jitter from pre-main-sequence magnetic activity in a simplistic fashion by assuming that it could negatively affect every star in the sample, which in turn would decrease the statistical significance of our result. This effect could be more fully modeled or treated more probabilistically in future work.
In the meantime, we have focused here on relative differences across the ages of the clusters in our study.
Importantly, the APOGEE observations considered in this work have a sufficient cadence and span a sufficient time baseline to allow us to probe low-mass spectroscopic binaries with orbital periods as long as $\sim$10$^4$~d. Most previous observational studies of binary populations in pre-main sequence clusters have been sensitive to either much tighter spectroscopic binaries and/or to much wider visual binaries. Binaries in the period range $\sim$10$^2$--10$^4$~d have not been well studied by spectroscopic observations heretofore. Our results suggest that these intermediate-period binaries are tight enough to be detectable in the APOGEE radial-velocity measurements yet soft enough to be susceptible to dynamical sculpting in their birth clusters prior to arrival on the main sequence.
\acknowledgements
We would like to thank the referee whose comments greatly helped to improve the quality and scope of this paper. We would also like to thank Dr. Kaitlin Kratter and Dr. Moe Maxwell for fruitful discussions that helped clarify the impact of the results in this paper.
This work was funded by an NSF LSAMP Bridge to the Doctorate grant. K.G.S.\ acknowledges partial support from NSF PAARE grant AST-1358862. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III web site is http://www.sdss3.org/.
SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.
\vspace*{1.5cm}
|
1,116,691,498,464 | arxiv | \section{\label{intro}Introduction}
\begin{bf}
\noindent
I. INTRODUCTION
\end{bf}
\vglue .3cm
Recently, we have calculated the complete set of massive one-loop master integrals
\cite{KMR}
needed in the calculation of the next-to-next-to-leading order (NNLO) parton model
corrections to the hadroproduction of heavy flavors \cite{KMR05}. We used Feynman
parametrization to evaluate the one-loop master integrals in $n=4-2\varepsilon$ dimensions.
We obtained the coefficients of the Laurent
series expansion of the relevant scalar integrals in terms of the parameter $\varepsilon$
up to ${\cal O}(\varepsilon^2)$ as needed for the NNLO calculation. We found that the
real parts of some of the $\varepsilon^2$ coefficients contain
a new class of functions which can be written in terms of one--dimensional integral
representations involving products of log and dilog functions. These so--called
single and triple index $L$ functions cannot be expressed in terms of
classical
polylogaritms but can be seen to belong to a generalization of the classical
polylogarithms which are called multiple polylogarithms.
Functions analogous to the triple index functions $L_{\sigma_1\sigma_2\sigma_3}$
also arise in the approach of \cite{Andrei} when one analytically
continues their ${\cal O}(\varepsilon^2)$ integral representation for a general vertex
function. Methods differing from ours have been used for the
derivation of master $N$-point integrals such as
the differential equations method \cite{diffeqs} or the nested sum method
\cite{nested}. Depending on the number of scales involved,
the results include multiple polylogarithms \cite{Multilogs} and/or harmonic
\cite{Remiddi1} or two-dimensional harmonic \cite{Remiddi2}
polylogarithms. The latter function all are subsets of multiple polylogarithms.
Presenting our results in terms of multiple polylogarithms will facilitate
a comparison with the results of possible rederivations of the scalar one--loop
integrals using other methods. It is very likely that future results of
multiloop calculations will be presented in terms of multiple polylogarithms
or their subclasses.
Alongside with this the necessary tools will be developed to deal with multiple
polylogarithms, be it analytically or numerically. In fact, recently a computer code
has been written for the
numerical evaluation of the multiple polylogarithms \cite{Vollinga}. It is
therefore timely that we express the results of \cite{KMR} also in terms of
multiple polylogarithms.
It is a purpose of this paper to show that the single and triple index
$L$ functions
introduced in \cite{KMR} can all be related to multiple polylogarithms. This is
done in explicit form. We are thus able to present our results for the scalar
massive one--loop master integrals in terms of multiple polylogarithms and
classical
polylogarithms \cite{Lewin}. In Sec.~II we recapitulate material on the definition
of the single and triple index $L$ functions as they arise in the approach of
\cite{KMR}. Simple symmetry relations allow one to restrict the discussion to
the triple index $L$ functions $L_{-++}$ and $L_{+++}$, and to the single index
$L$ function $L_{+}$. In Sec.~II we also recapitulate the definition of
multiple polylogarithms. In the subsequent sections we will
write down the formulas needed to transform the
$L$ functions to multiple polylogarithms for general arguments. The general
formulas are not always applicable when the arguments take special values as they
do in the massive one-loop calculation. For these special values one must
carefully
discuss the limiting behavior of the general formulas. In Sec.~III A
we derive the
general formula which relates the $L_{-++}$ functions to the set of multiple
polylogarithms.
Section~III B considers special cases of the general relation.
Similarly,
Sec.~IV A gives general relations which allow one to express the $L_{+++}$ functions
in terms of multiple polylogarithms.
In Sec.~IV B we discuss special cases for the arguments of the
$L_{+++}$ functions.
Sections~V A and~V B repeat the discussion for the single
index $L_{+}$ functions.
Finally, Sec.~VI presents our conclusions.
As remarked on before, the $L$ functions appear only in the real parts of some
of the ${\cal O}(\varepsilon^2)$ coefficient functions of the masive one--loop
integrals. In the notation of \cite{KMR}
these are the three--point coefficient functions ${\rm Re}\,C_1^{(2)}$,
${\rm Re}\,C_2^{(2)}$ and ${\rm Re}\,C_5^{(2)}$, and the four--point
coefficient functions ${\rm Re}\,D_1^{(2)}$, ${\rm Re}\,D_2^{(2)}$ and
${\rm Re}\,D_3^{(2)}$. For the sake of brevity we have decided to present
multiple polylogarithm results in this paper only for the four--point coefficient
function ${\rm Re}\,D_1^{(2)}$. This result is listed in the Appendix.
The corresponding results for the other five coefficient functions are readily
available in electronic form \cite{epaps}.
\vglue 1.0cm
\begin{bf}
\noindent
II. BASIC FEATURES
\end{bf}
\vglue .3cm
In order to make the paper self--contained, we write down a number of basic
definitions for the $L$ functions and the multiple polylogarithms in this section,
as well as some symmetry properties and domains of
definitions for the single and triple index $L$ functions. These will be of help
when presenting the
subsequent material.
The definition for the $L$ functions is as follows \cite{KMR}:
\begin{equation}
\label{Lfunction}
L_{\sigma_1\sigma_2\sigma_3}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=
\int_0^1 dy \frac{\ln (\alpha_1+\sigma_1 y) \ln (\alpha_2+\sigma_2 y)
\ln (\alpha_3+\sigma_3 y)}{\alpha_4+y}
\end{equation}
and
\begin{equation}
\label{Lpfunction}
L_{\sigma_1}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=
\int_0^1 dy \frac{\ln (\alpha_1+\sigma_1 y) {\rm Li}_2(\alpha_2+\alpha_3 y)
}{\alpha_4+y}.
\end{equation}
Here the $\sigma_i\,\, (i=1,2,3)$ take the values $\pm 1$ and the $\alpha_j$'s
are
either integers $\{1,0,-1\}$ or else
kinematical variables. We want to emphasize that the numerical evaluation of
the $L$ functions is straightforward.
The $L$ functions possess simple symmetry properties as follows.
One notices that
a change of the integration variable $y\rightarrow 1-y$ results
in the identity
\begin{equation}
L_{\sigma_1} (\alpha_1, \alpha_2, \alpha_3, \alpha_4) = -
L_{-\sigma_1} (\alpha_1 + \sigma_1, \alpha_2 + \alpha_3, -\alpha_3,
-\alpha_4
- 1)
\end{equation}
which implies that $L_{-}$ can always be related to $L_{+}$, and
vice versa. We have thus written our results for the three-point and
four-point functions in \cite{KMR} only in terms of the $L_{+}$
functions.
Turning to the triple index $L$ function one notices that $L_{\sigma_1
\sigma_2 \sigma_3} (\alpha_1, \alpha_2, \alpha_3, \alpha_4)$
is symmetric under permutations of any
two pairs of indices and arguments $\{\sigma_i, \alpha_i\}$ and $\{\sigma_j,
\alpha_j\}$ for $(i\ne j)$. The same change of variables as above
$y\rightarrow
1-y$ results in
\begin{equation}
\label{symmetry2}
L_{\sigma_1 \sigma_2 \sigma_3} (\alpha_1, \alpha_2, \alpha_3, \alpha_4) = -
L_{-\sigma_1 -\sigma_2 -\sigma_3} (\alpha_1 + \sigma_1, \alpha_2 + \sigma_2,
\alpha_3 + \sigma_3, -\alpha_4 - 1).
\end{equation}
Therefore, from the eight functions $L_{---}$, $L_{--+}$, $L_{-+-}$,
$L_{+--}$, $L_{-++}$, $L_{+-+}$, $L_{++-}$, and $L_{+++}$ only two are
independent. We have chosen to write our results in terms of $L_{-++}$ and
$L_{+++}$.
The domains of definition of the functions $L_{+++}, L_{-++}$, and $L_{+}$ that
follow from the requirement that these functions take real values can be
read off from the defining relations Eqs.~(\ref{Lfunction}) and
(\ref{Lpfunction})
considering the arguments of the log and dilog functions in the integrands, as well
as from ensuring that the denominator of Eqs.~(\ref{Lfunction}) and
(\ref{Lpfunction}) does
not change sign on the integration path. One has
\begin{eqnarray}
\label{domain}
\begin{array}{r@{\quad:\quad}l}
L_{+++}(\alpha_{1},\alpha_2, \alpha_{3}, \alpha_{4}) & \alpha_{1}>0
, \alpha_2 > 0, \alpha_{3} > 0, \alpha_{4}<-1 \quad {\rm or}\quad
\alpha_{4}>0; \\
L_{-++}(\alpha_{1},\alpha_2, \alpha_{3}, \alpha_{4}) & \alpha_{1} >
1, \alpha_2 > 0, \alpha_{3}> 0, \alpha_{4}<-1\quad {\rm or}\quad
\alpha_{4}>0; \\
L_{+}(\alpha_{1},\alpha_2, \alpha_{3}, \alpha_{4}) & \alpha_{1} >0
, \alpha_2 \leq 1,\alpha_{2}+ \alpha_{3}\leq 1, \alpha_{3}\neq 0,
\alpha_{4}<-1\quad {\rm or}\quad \alpha_{4}>0.
\end{array}
\end{eqnarray}
Looking at the definition of the triple index $L$ function in
(\ref{Lfunction}) one concludes that the boundary points $\alpha_1=0$
and/or $\alpha_2=0$ and/or $\alpha_3=0$ can be included in the domain of
the definition
for $L_{+++}$. The same holds true for $\alpha_1=1$ and/or $\alpha_2=0$
and/or $\alpha_3=0$ for $L_{-++}$. Also, from the definition of the
single index
function $L_+$ in (\ref{Lpfunction}) one concludes that
the boundary point $\alpha_1=0$ can be added to its domain of definition.
The points $\alpha_{4}=\{-1, 0\}$ can also be included in the domain if the
values taken by the other parameters $\alpha_{i}$ guarantee the convergence of
the integral.
We mention that for all of our purposes the conditions (\ref{domain}),
with the boundary points included, are
satisfied, e.g., our results for the integrals are real.
Nevertheless, it is of course always possible to analytically continue the
parameters to the complex plane.
There are some further relations for the $L$ functions which result from applying
integration-by-parts identities. They are not listed here but can be found in
Appendix~C of \cite{KMR}. They have been used to reduce the set of
$L$ functions occurring in the master integrals to a subset of $L$ functions
having real values in physical phase space \cite{KMR}.
Multiple polylogarithms are defined as a limit of Z sums \cite{Multilogs}, e.g.,
\begin{eqnarray}
\label{zsum}
Li_{m_{k},...,m_{1}}(x_{k},...,x_{1})=\lim_{n_1 \rightarrow \infty } \sum_{
n_{1} > n_{2}...> n_{k} > 0}
\frac{x_{1}^{n_{1}}x_{2}^{n_{2}}...x_{k}^{n_{k}}
}{n_{1}^{m_{1}}n_{2}^{m_{2}}...n_{k}^{m_{k}} }.
\end{eqnarray}
The number $w=m_{1}+...+m_{k}$ is called the weight and $k$ is called
the depth of the multiple polylogarithm. The
power series (\ref{zsum}) is convergent for $|x_{i}|<1$, and can be
analytically
continued via the iterated integral representation:
\begin{eqnarray}
\label{intrepr}
Li_{m_{k},...,m_{1}}(x_{k},...,x_{1})=\int \limits_{0}^{x_{1}x_{2}...x_{k}}
\left(
\frac{dt}{t} \circ \right)^{m_{1}-1}
\frac{dt}{x_{2}x_{3}...x_{k}-t} \circ
\nonumber \\ \left( \frac{dt}{t} \circ \right)^{m_{2}-1}
\frac{dt}{x_{3}...x_{k}-t} \circ ... \circ \left( \frac{dt}{t} \circ
\right)^{m_{k}-1} \frac{dt}{1-t},
\end{eqnarray}
where the following notation is used for the iterated integrals:
\begin{eqnarray}
\int \limits_{0}^{\lambda} \frac{dt}{a_{n}-t}\circ ...\circ
\frac{dt}{a_{1}-t} = \int \limits_{0}^{\lambda} \frac{dt_{n}}{a_{n}-t_{n}}
\int \limits_{0}^{t_{n}} \frac{dt_{n-1}}{a_{n-1}-t_{n-1}} \times...\times
\int
\limits_{0}^{t_{2}}\frac{dt_1}{a_{1}-t_{1}}.
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
III. TRANSFORMATION OF $L_{-++}$ TO MULTIPLE POLYLOGARITHMS
\end{bf}
\vglue .3cm
In this section we will show that all our $L_{-++}$ functions can be
expressed in terms of multiple polylogarithms.
\vglue 1.0cm
\begin{bf}
\noindent
A. General case for the $L_{-++}$ function
\end{bf}
\vglue .3cm
We begin with the $L_{-++}$ function Eq.~(\ref{Lfunction}),
\begin{equation}
\label{l3gen1}
L_{-++}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=
\int \limits_0^1 dy \frac{\ln (\alpha_1- y) \ln (\alpha_2+ y)
\ln (\alpha_3 + y)} {\alpha_4 + y} .
\end{equation}
After changing the integration variable $y=\alpha_{1} t$ one gets
\begin{eqnarray}
\label{getB27General}
\int \limits_{0}^{1/\alpha_1} dt
\frac{\ln(\alpha_1-\alpha_{1} t) \ln(\alpha_{2}+\alpha_{1}
t)\ln(\alpha_3+\alpha_{1} t)}{\frac{\alpha_{4}}{\alpha_{1}}+t} =
\int \limits_{0}^{1/\alpha_1} dt
\frac{\ln\alpha_1 \ln(\alpha_{2}+\alpha_{1} t)\ln(\alpha_3+\alpha_{1}
t)}{\frac{\alpha_{4}}{\alpha_{1}}+t}\nonumber\\
+\int \limits_{0}^{1/\alpha_1} dt
\frac{\ln(1-t) [ \ln\alpha_1+\ln(\frac{\alpha_2}{\alpha_1}+t)] [
\ln\alpha_1+\ln(\frac{\alpha_3}{\alpha_1}+ t) ]
}{\frac{\alpha_4}{\alpha_1}+t}=\nonumber\\
\ln\alpha_{1} \int \limits_0^1 dy \frac{ \ln (\alpha_2+ y)\ln (\alpha_3 +
y)}
{\alpha_4 + y}+\ln^{2}\alpha_1
\int \limits_{0}^{1/\alpha_1} dt
\frac{\ln(1-t)}{\frac{\alpha_4}{\alpha_1}+t}\nonumber\\
+\ln\alpha_1
\int \limits_{0}^{1/\alpha_1} dt
\frac{\ln(1-t)\ln(\frac{\alpha_2}{\alpha_1}+
t)}{\frac{\alpha_4}{\alpha_1}+t}
+\ln\alpha_1
\int \limits_{0}^{1/\alpha_1} dt
\frac{\ln(1-t)\ln(\frac{\alpha_3}{\alpha_1}+
t)}{\frac{\alpha_4}{\alpha_1}+t}~~~\\
+\int \limits_{0}^{1/\alpha_1} dt
\frac{\ln(1-t)\ln(\frac{\alpha_2}{\alpha_1}+ t)
\ln(\frac{\alpha_3}{\alpha_1}+ t) }{\frac{\alpha_4}{\alpha_1}+t}\,
. \nonumber
\end{eqnarray}
With the help of (\ref{intrepr}) the integral in the
second term of the last equation of (\ref{getB27General}) can be written as \begin{eqnarray}
\label{secondForB27} \int \limits_{0}^{1/\alpha_1} dt
\frac{\ln(1-t)}{\frac{\alpha_4}{\alpha_1}+t}= \int
\limits_{0}^{1/\alpha_1}
\frac{dt_1}{-\frac{\alpha_4}{\alpha_1}-t_{1}} \int \limits_{0}^{t_1}
\frac{dt_2}{1-t_{2}}=Li_{1,1}(-\frac{\alpha_4}{\alpha_1},
-\frac{1}{\alpha_4} ).
\end{eqnarray}
The third and fourth terms of the last equation in (\ref{getB27General}) contain
integrals of the form
\begin{eqnarray} \label{defB23} \int \limits_{0}^{t_m} dt
\frac{ \ln(1-t) \ln(\beta_1 + t)} {\beta_2 + t}\, . \end{eqnarray}
To express
such integrals in terms of multiple polylogarithms one proceeds as
follows:
\begin{eqnarray} \label{getB23} -Li_{1,1,1} \left(
-\beta_2,\frac{\beta_1}{\beta_2},\frac{-t_m}{\beta_1} \right)=
\int \limits_{0}^{t_m} \frac{dt_{2}}{\beta_1 + t_{2}} \int
\limits_{0}^{t_{2}}
dt_{1} \frac{\ln(1-t_{1})} {\beta_2 + t_{1}}=
\int \limits_{0}^{t_m} dt_{1} \frac{\ln(1-t_{1})} {\beta_2 + t_{1}} \int
\limits_{t_{1}}^{t_m} \frac{dt_{2}}{\beta_1 + t_{2}}= \nonumber \\
\ln(\beta_1 + t_m) \int \limits_{0}^{t_m} dt_{1} \frac{ \ln(1-t_{1})}
{\beta_2
+ t_{1}}
- \int \limits_{0}^{t_m} dt_{1} \frac{ \ln(1-t_{1}) \ln(\beta_1 + t_{1})}
{\beta_2 + t_{1}} = ~~~ \\
\nonumber
\ln(\beta_1 + t_m) Li_{1,1} \left( -\beta_2, \frac{-t_m}{\beta_2} \right)
- \int \limits_{0}^{t_m} dt_{1} \frac{ \ln(1-t_{1})\ln(\beta_1 + t_{1})}
{\beta_2 + t_{1}}\, .
\end{eqnarray}
In the first line of (\ref{getB23}) we have changed the order of integration in the
two-dimensional integral.
We shall frequently use this trick further on.
From Eq.~(\ref{getB23}) one immediately concludes that
\begin{equation}
\label{B23}
\int \limits_{0}^{t_m} dt \frac{ \ln(1-t)
\ln(\beta_1 + t)} {\beta_2 + t}=
Li_{1,1,1} \left( -\beta_2,\frac{\beta_1}{\beta_2},\frac{-t_m}{\beta_1}
\right) +
\ln\left(\beta_1 + t_m\right) Li_{1,1} \left( -\beta_2, \frac{-t_m}{\beta_2}
\right).
\end{equation}
Let us now turn to the more involved integral [first term of
Eq.~(\ref{getB27General})]:
\begin{eqnarray}
\label{getB23A}
\int \limits_0^1 dy \frac{ \ln (\alpha_2+ y)\ln (\alpha_3 + y)} {\alpha_4 +
y}\,
\stackrel{y\rightarrow-\alpha_2 t }{\, =\, } \,
-\int \limits_0^{-1/\alpha_2} dt \frac{[\ln\alpha_2+ \ln (1- t)] \ln
(\alpha_3
-\alpha_2 t)} {\frac{\alpha_4}{\alpha_2} -t}=\nonumber\\
-\ln\alpha_2\int \limits_0^{-1/\alpha_2} dt \frac{\ln (\alpha_3 -\alpha_2
t)}
{\frac{\alpha_4}{\alpha_2} -t}
-\int \limits_0^{-1/\alpha_2} dt \frac{\ln (1- t)[\ln\alpha_2 + \ln
(\frac{\alpha_3}{\alpha_2} -t)]} {\frac{\alpha_4}{\alpha_2} -t}=\\
+\ln\alpha_2\int \limits_0^{1} dy \frac{\ln (\alpha_3 +y)} {\alpha_4 +y}
-\ln\alpha_2 \int \limits_0^{-1/\alpha_2} dt \frac{\ln (1- t)}
{\frac{\alpha_4}{\alpha_2} -t}
-\int \limits_0^{-1/\alpha_2} dt \frac{\ln (1- t)\ln
(\frac{\alpha_3}{\alpha_2}
-t)} {\frac{\alpha_4}{\alpha_2} -t}\nonumber \, .
\end{eqnarray}
The integral in the first term can be expressed as
\begin{eqnarray}
\label{B28}
\int \limits_0^{1} dy \frac{\ln (\alpha_3 +y)} {\alpha_4 +y}\,
\stackrel{y\rightarrow-\alpha_3 t }{\, =\, } \,
-\int \limits_0^{-1/\alpha_3} dt \frac{[\ln\alpha_3 +\ln (1 -t)]}
{\frac{\alpha_4}{\alpha_3} -t}=\nonumber\\
\ln\alpha_3\ln\left(\frac{\alpha_{4}+1}{\alpha_{4}}\right)+Li_{1,1}
\left(\frac{\alpha_4}{ \alpha_3 },-\frac{1}{ \alpha_3 } \right )\, .
\end{eqnarray}
The integral in the second term can be written as
\begin{eqnarray}
\label{B28A}
\int \limits_0^{-1/\alpha_2} dt \frac{\ln (1- t)} {\frac{\alpha_4}{\alpha_2}
-t}=-Li_{1,1}\left( \frac{\alpha_4}{ \alpha_2},-\frac{1}{ \alpha_2 }
\right)\, .
\end{eqnarray}
The third term from the last line of Eq.~(\ref{getB23A}) has a form which is
an
analogue of the integral (\ref{defB23})
and can be calculated in a similar way,
\begin{eqnarray}
\label{B23C}
\int \limits_{0}^{t_m} dt \frac{ \ln(1-t)
\ln(\beta_1 - t)} {\beta_2 - t}= Li_{1,1,1} \left(
\beta_2,\frac{\beta_1}{\beta_2},\frac{t_m}{\beta_1}
\right) +
\ln\left(\beta_1 - t_m\right) Li_{1,1} \left( \beta_2, \frac{t_m}{\beta_2}
\right)\, .
\end{eqnarray}
Combining the Eqs.~(\ref{B28}),~(\ref{B28A}) and~(\ref{B23C}) we arrive at
the
result for Eq.~(\ref{getB23A}),
\begin{eqnarray}
\label{B23A}
\int \limits_0^1 dy \frac{ \ln (\alpha_2+ y)\ln (\alpha_3 + y)} {\alpha_4 +
y}=
Li_{1,1,1}\left(\frac {\alpha_4}{\alpha_2},
\frac{\alpha_{3}}{\alpha_{4}},-\frac{1}{\alpha_{3}}\right)
+\ln\alpha_{2}Li_{1,1}\left(
\frac{\alpha_{4}}{\alpha_{3}},-\frac{1}{\alpha_{4}}
\right )\nonumber \\
+\ln(1+\alpha_{3})Li_{1,1}\left(
\frac{\alpha_{4}}{\alpha_{2}},-\frac{1}{\alpha_{4}} \right )
+\ln\alpha_{2}\ln\alpha_{3}\ln\left(\frac{\alpha_{4}+1}{\alpha_{4}}\right).
\end{eqnarray}
Because the initial integrand is symmetric under the exchange of the
parameters
$\alpha_{2}$ and $\alpha_{3}$,
the rhs of (\ref{B23A}) can be rewritten in a symmetric form if desired.
We are now left with the fifth term in (\ref{getB27General}). The fifth term
is an integral of the type
\begin{eqnarray}
\label{typeB27}
\int \limits_{0}^{t_m} dt \frac{ \ln(1-t) \ln(\gamma_1 + t)
\ln(\gamma_2 + t)} {\gamma_3 +t}\, .
\end{eqnarray}
In order to express such integrals in terms of multiple polylogarithms one can
perform the
following chain of transformations resulting in a multiple polylogarithm of weight four:
\begin{eqnarray}
\label{l3gen4}
- Li_{1,1,1,1}\left(-\gamma_3, \frac{\gamma_2}{\gamma_3},
\frac{\gamma_1}{\gamma_2}, \frac{-t_m}{\gamma_1} \right) =
\int \limits_{0}^{t_m} \frac{dt_{4}}{\gamma_1 +t_{4}} \int
\limits_{0}^{t_{4}}
\frac{dt_{3}}{\gamma_2 +t_{3}}
\int \limits_{0}^{t_{3}} \frac{dt_{2}}{\gamma_3 +t_{2}} \int
\limits_{0}^{t_{2}}
\frac{dt_{1}}{1-t_{1}} = \\
\nonumber
- \int \limits_{0}^{t_m} \frac{dt_{4}}{\gamma_1 +t_{4}} \int
\limits_{0}^{t_{4}}
\frac{dt_{3}}{\gamma_2 +t_{3}}
\int \limits_{0}^{t_{3}} d t_{2} \frac{\ln(1-t_{2})}{\gamma_3 +t_{2}} =
- \int \limits_{0}^{t_m} \frac{dt_{4}}{\gamma_1 +t_{4}} \int
\limits_{0}^{t_{4}}
d t_{2} \frac{\ln(1-t_{2})}{\gamma_3 +t_{2}}
\int \limits_{t_{2}}^{t_{4}} \frac{dt_{3}}{\gamma_2 +t_{3}}= \\
\nonumber
- \int \limits_{0}^{t_m} dt_{4} \frac{\ln(\gamma_2 +t_{4})}{\gamma_1 +t_{4}}
\int
\limits_{0}^{t_{4}} d t_{2} \frac{ \ln(1-t_{2})}{\gamma_3 + t_{2}} +
\int \limits_{0}^{t_m} \frac{dt_{4}}{\gamma_1 + t_{4}} \int
\limits_{0}^{t_{4}}
dt_{2} \frac{\ln(1-t_{2}) \ln(\gamma_2 + t_{2})}{\gamma_3
+t_{2}}=\label{big} \\
\nonumber
-I'(t_m) + \int \limits_{0}^{t_m} dt_{2} \frac{ \ln(1-t_{2}) \ln(\gamma_2 +
t_{2})} {\gamma_3 +t_{2}} \int \limits_{t_{2}}^{t_m}
\frac{dt_{4}}{\gamma_1
+ t_{4}}= \\
\nonumber
-I'(t_m) + I''(t_m)
- \int \limits_{0}^{t_m} dt_{2} \frac{ \ln(\gamma_1 + t_{2})
\ln(\gamma_2 + t_{2})\ln(1-t_{2})} {\gamma_3 +t_{2}},
\end{eqnarray}
where we have introduced the notation
\begin{eqnarray}
\label{iprimes}
I'(t_m) = \int \limits_{0}^{t_m} dt_{4} \frac{\ln(\gamma_2
+t_{4})}{\gamma_1
+t_{4}} \int \limits_{0}^{t_{4}} d t_{2} \frac{ \ln(1-t_{2})}{\gamma_3 +
t_{2}},
\\ \nonumber
I''(t_m) = \ln(\gamma_1 + t_m) \int \limits_{0}^{t_m} dt_{2} \frac{
\ln(1-t_{2})
\ln(\gamma_2 + t_{2})} {\gamma_3 + t_{2}}.
\end{eqnarray}
The third term on the last line of (\ref{l3gen4}) is exactly the integral of
the required type Eq.~(\ref{typeB27}).
The integral in $I''(t_m)$ has the form of (\ref{B23}). For the integral
$I'(t_m) $ we write
\begin{equation}
I'(t_m) = \int \limits_{0}^{t_m} dt_{4} \frac{\ln(\gamma_2
+t_{4})}{\gamma_1
+t_{4}} \int \limits_{0}^{t_{4}} d t_{2} \frac{ \ln(1-t_{2})}{\gamma_3 +
t_{2}} =
\int \limits_{0}^{t_m} dt_{4} \frac{\ln(\gamma_2 + t_{4})} {\gamma_1 +
t_{4}}
Li_{1,1}\left(-\gamma_3,\frac{-t_{4}}{\gamma_3}\right) .
\end{equation}
On the other, hand one has
\begin{eqnarray}
Li_{1,1,1,1} \left( -\gamma_3, \frac{\gamma_1}{\gamma_3}, \frac{\gamma_2}
{\gamma_1}, \frac{-t_m}{\gamma_2} \right) =
\int \limits_{0}^{t_m} \frac{ dt_{2}}{\gamma_2 + t_{2}}
\int \limits_{0}^{t_{2}} \frac{dt_{1}} {\gamma_1 + t_{1}}
Li_{1,1} \left( -\gamma_3, \frac{-t_{1}}{\gamma_3} \right)= \\
\nonumber
\int \limits_{0}^{t_m} \frac{dt_{1}}{\gamma_1 + t_{1}}
Li_{1,1} \left( -\gamma_3, \frac{-t_{1}}{\gamma_3} \right)
\int \limits_{t_{1}}^{t_m} \frac{ dt_{2}}{\gamma_2
+ t_{2}}= \\
\nonumber
\ln(\gamma_2 + t_m) \int \limits_{0}^{t_m} \frac{dt_{1}} {\gamma_1 + t_{1}}
Li_{1,1} \left( -\gamma_3, \frac{-t_{1}}{\gamma_3} \right)
- \int \limits_{0}^{t_m} dt_{1} \frac{ \ln(\gamma_2 + t_{1})}{\gamma_1 +
t_{1}}
Li_{1,1}\left( -\gamma_3, \frac{-t_{1}}{\gamma_3} \right) = \\
\nonumber
- \ln(\gamma_2 + t_m) Li_{1,1,1}\left( -\gamma_3, \frac{\gamma_1}{\gamma_3},
\frac{-t_m}{\gamma_1} \right) - I'(t_m).
\end{eqnarray}
One then concludes that
\begin{equation}
\label{sub2}
I'(t_m) = - Li_{1,1,1,1} \left( -\gamma_3, \frac{\gamma_1}{\gamma_3},
\frac{\gamma_2} {\gamma_1}, \frac{-t_m}{\gamma_2} \right) - \ln(\gamma_2 +
t_m)
Li_{1,1,1} \left( -\gamma_3, \frac{\gamma_1}{\gamma_3},
\frac{-t_m}{\gamma_1}
\right).
\end{equation}
Finally, substituting $I'(t_m) $ and $I''(t_m)$ into Eq.~(\ref{l3gen4})
we write down the result for the integral of the required type
Eq.~(\ref{typeB27}),
\begin{eqnarray}
\label{B27}
\int \limits_{0}^{t_m} dt \frac{ \ln(1 - t) \ln(\gamma_1 + t)
\ln(\gamma_2 + t)}
{\gamma_3 + t} =
\ln(\gamma_1 + t_m) \ln(\gamma_2 + t_m)
Li_{1,1}\left( -\gamma_3, \frac{-t_m} {\gamma_3} \right) + \nonumber
\\
\ln(\gamma_2 + t_m) Li_{1,1,1}\left( -\gamma_3, \frac{\gamma_1}
{\gamma_3},
\frac{-t_m}{\gamma_1} \right)
+ \ln(\gamma_1 + t_m)
Li_{1,1,1}\left( -\gamma_3, \frac{\gamma_2} {\gamma_3}, \frac{-t_m}
{\gamma_2} \right) \\
+ Li_{1,1,1,1}\left( -\gamma_3, \frac{\gamma_2} {\gamma_3}, \frac{\gamma_1}
{\gamma_2}, \frac{-t_m}{\gamma_1} \right)
+ Li_{1,1,1,1} \left( -\gamma_3, \frac{\gamma_1} {\gamma_3}, \frac{\gamma_2}
{\gamma_1}, \frac{-t_m}{\gamma_2} \right). \nonumber
\end{eqnarray}
We are now in the position to collect all required contributions to express
the $L_{-++}$ function in terms of multiple polylogarithms.
Taking into account Eqs.~(\ref{secondForB27}), (\ref{B23}), (\ref{B23A}), and
(\ref{B27}) we obtain
\begin{eqnarray}
\label{B27General}
L_{-++}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=
Li_{1,1,1,1}\left(-\frac{\alpha_4}{\alpha_1}, \frac{\alpha_2}{\alpha_4},
\frac{\alpha_3}{\alpha_2},- \frac{1}{\alpha_3}\right)\nonumber\\
+Li_{1,1,1,1}\left(-\frac{\alpha_4}{\alpha_1}, \frac{\alpha_3}{\alpha_4},
\frac{\alpha_2}{\alpha_3}, - \frac{1}{\alpha_2}\right)
+\ln\alpha_{1} Li_{1,1,1}\left(\frac{\alpha_4}{\alpha_2},
\frac{\alpha_3}{\alpha_4}, - \frac{1}{\alpha_3}\right)\nonumber\\
+\ln(1+\alpha_{2}) Li_{1,1,1}\left(-\frac{\alpha_4}{\alpha_1},
\frac{\alpha_3}{\alpha_4}, - \frac{1}{\alpha_3}\right)
+\ln(1+\alpha_{3}) Li_{1,1,1}\left(-\frac{\alpha_4}{\alpha_1},
\frac{\alpha_2}{\alpha_4}, - \frac{1}{\alpha_2}\right)\nonumber\\
+\ln\alpha_{1}\ln\alpha_{2} Li_{1,1}\left(\frac{\alpha_4}{\alpha_3}, -
\frac{1}{\alpha_4}\right)
+\ln\alpha_{1}\ln(1+\alpha_{3}) Li_{1,1}\left(\frac{\alpha_4}{\alpha_2}, -
\frac{1}{\alpha_4}\right) \\
+\ln(1+\alpha_{2})\ln(1+\alpha_{3})
Li_{1,1}\left(-\frac{\alpha_4}{\alpha_1}, -
\frac{1}{\alpha_4}\right)
+\ln\alpha_{1}\ln\alpha_{2}\ln\alpha_{3}
\ln\left(\frac{\alpha_{4}+1}{\alpha_{4}}\right).\nonumber
\end{eqnarray}
Some remarks are in order at this place. The final formula~(\ref{B27General})
contains multiple polylogarithms up to weight four.
All multiple polylogarithms up to weight three can be expressed in terms of
logarithms and clasical polylogarithms $Li_2$ and $Li_3$ .
This fact is used by us when we reexpress our results for the massive
scalar integrals in terms of multiple polylogarithms, i.e., our final results
will contain only multiple polylogarithms of weight four.
For the variables $\alpha_{i}$ the conditions~(\ref{domain}) are assumed.
But in the results for the massive scalar integrals there are also cases
when $\alpha_{1}=1$ and/or $ \alpha_{2}=0$ and/or $ \alpha_{3}=0$ and/or
$\alpha_{4}=\{-1,0\}$.
In such cases the general formula ~(\ref{B27General}) is no longer valid and
these cases must be studied separately.
\vglue 1.0cm
\begin{bf}
\noindent
B. Special cases for the $L_{-++}$ function
\end{bf}
\vglue .3cm
In the Laurent series expansion of the massive scalar one-loop integrals one
encounters special values of the arguments $\alpha_{i}$ for which the general
formula Eq.~(\ref{B27General}) no longer applies. This is quite obvious from
the list of special cases discussed in the following.
\vglue 1.0cm
\begin{bf}
\noindent
{\em 1. ${\bf \alpha_{1}=1, \,\, \alpha_{4}=0}$ }
\end{bf}
\vglue .3cm
In such case one can make use of Eq.~(\ref{B27}).
One should find the limit of the expression on the right-hand side for $t_{m}=1,
\gamma_{3}\rightarrow 0$. One obtains
\begin{eqnarray}
\label{getChangeLmpp}
\int \limits_{0}^{1} dt \frac{ \ln(1 - t) \ln(\gamma_1 + t) \ln(\gamma_2 + t)}
{t} =\hspace{9.8cm} \nonumber\\
\lim_{\gamma_{3} \rightarrow 0} \Big\{ \ln(\gamma_1 + 1) \ln(\gamma_2 + 1)
\int \limits_{0}^{1} \frac{dt_{2}}{-\gamma_3 -t_{2}} \int
\limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}}
+
\ln(\gamma_2 + 1)
\int \limits_{0}^{1} \frac{dt_{3}}{-\gamma_1 -t_{3}}\int \limits_{0}^{t_{3}}
\frac{dt_{2}}{-\gamma_3 -t_{2}} \int \limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}} \nonumber\\
+ \ln(\gamma_1 + 1)
\int\limits_{0}^{1} \frac{dt_{3}}{-\gamma_2 -t_{3}}\int \limits_{0}^{t_{3}}
\frac{dt_{2}}{-\gamma_3 -t_{2}} \int \limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}}
+ \int \limits_{0}^{1} \frac{dt_{4}}{-\gamma_1 -t_{3}}\int \limits_{0}^{t_{4}}
\frac{dt_{3}}{-\gamma_2 -t_{3}}
\int \limits_{0}^{t_{3}} \frac{dt_{2}}{-\gamma_3 -t_{2}} \int
\limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}} \nonumber \\
+ \int \limits_{0}^{1} \frac{dt_{4}}{-\gamma_2 -t_{3}}\int \limits_{0}^{t_{4}}
\frac{dt_{3}}{-\gamma_1 -t_{3}}
\int \limits_{0}^{t_{3}} \frac{dt_{2}}{-\gamma_3 -t_{2}} \int
\limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}}
\Big \}=
- \ln(\gamma_1 + 1) \ln(\gamma_2 + 1)
\int \limits_{0}^{1} \frac{dt_{2}}{t_{2}} \int \limits_{0}^{t_{2}}\frac{dt_{1}}{1
-t_{1}}\nonumber\\
-\ln(\gamma_2 + 1)
\int \limits_{0}^{1} \frac{dt_{3}}{-\gamma_1 -t_{3}}\int \limits_{0}^{t_{3}}
\frac{dt_{2}}{t_{2}} \int \limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}}
- \ln(\gamma_1 + 1)
\int\limits_{0}^{1} \frac{dt_{3}}{-\gamma_2 -t_{3}}\int \limits_{0}^{t_{3}}
\frac{dt_{2}}{t_{2}} \int \limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}} \nonumber\\
-\int \limits_{0}^{1} \frac{dt_{4}}{-\gamma_1 -t_{3}}\int \limits_{0}^{t_{4}}
\frac{dt_{3}}{-\gamma_2 -t_{3}}
\int \limits_{0}^{t_{3}} \frac{dt_{2}}{t_{2}} \int
\limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}}
- \int \limits_{0}^{1} \frac{dt_{4}}{-\gamma_2 -t_{3}}\int \limits_{0}^{t_{4}}
\frac{dt_{3}}{-\gamma_1 -t_{3}}
\int \limits_{0}^{t_{3}} \frac{dt_{2}}{t_{2}} \int
\limits_{0}^{t_{2}}\frac{dt_{1}}{1 -t_{1}}\, .\nonumber\\
\end{eqnarray}
In order to get the expression under the sign of the limit in
Eq.~(\ref{getChangeLmpp})
one applies the definition (\ref{intrepr}) for the multiple polylogarithms in
Eq.~(\ref{B27}).
Using the same definition for the final multidimensional integrals in
(\ref{getChangeLmpp})
and making the change $\gamma_{1}\rightarrow\alpha_{2},
\gamma_{2}\rightarrow\alpha_{3} $ one finally arrives at the result
for the case $\alpha_{1}=1$ and $\alpha_{4}=0$,
\begin{eqnarray}
\label{ChangeLmpp}
L_{-++}\left(1,\alpha_{2},\alpha_{3},0 \right)=
-Li_{2,1,1}\left(-\alpha_{3},\frac{\alpha_{2}}{\alpha_{3}}, -\frac{1}{\alpha_{2}}
\right)
-Li_{2,1,1}\left(-\alpha_{2},\frac{\alpha_{3}}{\alpha_{2}}, -\frac{1}{\alpha_{3}}
\right) .\nonumber\\
-\ln(\alpha_{3} + 1)Li_{2,1}\left(-\alpha_{2},-\frac{1}{\alpha_{2}} \right)
-\ln(\alpha_{2} + 1)Li_{2,1}\left(-\alpha_{3},-\frac{1}{\alpha_{3}} \right)\nonumber\\
-\ln(\alpha_{2} + 1) \ln(\alpha_{3} + 1)\zeta(2) .
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 2. ${\bf \alpha_{1}=1, \,\, \alpha_{2}=\alpha_{3}=0}$ }
\end{bf}
\vglue .3cm
For these values of the parameters $\alpha_{i}$ one has an integral of the very
simple form
\begin{eqnarray}
L_{-++}(1,0,0,\alpha_{4})=\int \limits_0^1 dy \frac{\ln (1- y) \ln^{2}y} {\alpha_4 +
y}\, . \nonumber
\end{eqnarray}
After a change of variable $y\rightarrow 1-t$ one gets
\begin{eqnarray}
\int \limits_0^1 dt \frac{\ln t \ln^{2}(1-t)} {\alpha_4+1 - t}=- \int \limits_0^1
dt_{1} \frac{ \ln^{2}(1-t_{1})} {\alpha_4+1 - t_{1}}
\int\limits_{t_{1}}^{1} \frac{dt_{2}}{t_{2}}=\nonumber\\
- \int \limits_0^1 \frac{dt_{2}}{t_{2}} \int\limits_{0}^{t_{2}} dt_{1}\frac{
\ln^{2}(1-t_{1})} {\alpha_4+1 - t_{1}}=
-2 \int \limits_0^1 \frac{dt_{2}}{t_{2}} \int\limits_{0}^{t_{2}} \frac{dt_{1}}
{\alpha_4+1 - t_{1}} \int \limits_0^{t_{1}} \frac{dt_{3}} {1 - t_{3}}
\int \limits_0^{t_{3}} \frac{dt_{4}} {1 - t_{4}}\, .
\end{eqnarray}
Applying the definition~(\ref{intrepr}) one obtains
\begin{eqnarray}
L_{-++}(1,0,0,\alpha_{4})=-2 Li_{1,1,2}\left(1,\alpha_{4}+1,\frac{1}{\alpha_{4}+1}
\right).
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 3. ${\bf \alpha_{1}=1, \,\, \alpha_{2}=0}$ (and ${\bf \alpha_{4}=-1}$)
}
\end{bf}
\vglue .3cm
We shall again find the limit of the rhs of (\ref{B27}) for $t_{m}=1$ and $
\gamma_{1}\rightarrow 0$.
The first and the third terms are equal to 0 because of the limit $
\lim_{\gamma_1\rightarrow 0 } \ln(\gamma_{1}+1)=0$. The other terms
transform into
\begin{eqnarray}
\lim_{\gamma_1\rightarrow 0 } Li_{1,1,1}\left( -\gamma_3, \frac{\gamma_1} {\gamma_3},
\frac{-1}{\gamma_1} \right) = -Li_{1,2}\left(-\gamma_{3},-\frac{1} {\gamma_3}
\right), \nonumber \\
\lim_{\gamma_{1}\rightarrow 0 }
Li_{1,1,1,1}\left( -\gamma_3, \frac{\gamma_2} {\gamma_3}, \frac{\gamma_1}{\gamma_2},
\frac{-t_m}{\gamma_1} \right) =
-Li_{1,1,2}\left( -\gamma_3, \frac{\gamma_2} {\gamma_3}, \frac{-1}{\gamma_2}
\right),\nonumber\\
\lim_{\gamma_{1}\rightarrow 0 }
Li_{1,1,1,1} \left( -\gamma_3, \frac{\gamma_1} {\gamma_3}, \frac{\gamma_2}
{\gamma_1}, \frac{-t_m}{\gamma_2} \right)=-Li_{1,2,1}\left( -\gamma_3,
\frac{\gamma_2}{\gamma_3}, \frac{-1}{\gamma_2} \right). \nonumber
\end{eqnarray}
Finally we write
\begin{eqnarray}
L_{-++}\left(1, 0, \alpha_{3}, \alpha_{4} \right)=-Li_{1,1,2}\left( -\alpha_4,
\frac{\alpha_3} {\alpha_4}, \frac{-1}{\alpha_3} \right)
-Li_{1,2,1}\left( -\alpha_4, \frac{\alpha_3} {\alpha_4}, \frac{-1}{\alpha_3}
\right)\nonumber\\
-\ln(\alpha_{3}+1)Li_{1,2}\left(-\alpha_{4},-\frac{1} {\alpha_4} \right).
\end{eqnarray}
For the special case $\alpha_{4}=-1$ one gets
\begin{eqnarray}
L_{-++}\left(1, 0, \alpha_{3}, -1 \right)=-Li_{1,1,2}\left( 1, -\alpha_3,
\frac{-1}{\alpha_3} \right)
-Li_{1,2,1}\left( 1,- \alpha_3, \frac{-1}{\alpha_3} \right)\nonumber\\
-\ln(\alpha_{3}+1)\, \zeta(3).
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 4. ${\bf \alpha_{2}=\alpha_{3}=0}$ \,\, (and ${\bf \alpha_{4}=-1}$) }
\end{bf}
\vglue .3cm
In this case one proceeds along the following lines:
\begin{eqnarray}
L_{-++}(\alpha_1, 0, 0, \alpha_4)=
\int \limits_0^1 dy \frac{\ln (\alpha_1- y) \ln^{2} y} {\alpha_4 + y} \,
\stackrel{y\rightarrow1-t }{\, =\, } \,
\int \limits_0^1 dt \frac{\ln (\alpha_1- 1+t) \ln^{2}(1-t)} {\alpha_4 +1- t}=\nonumber\\
-\int \limits_0^1 dt_{1} \frac{\ln^{2}(1-t_{1})} {-\alpha_4 -1+ t_{1}}\int
\limits_{-\alpha_{1}+2}^{t_{1}}\frac{dt_{2}} {\alpha_1 -1+ t_{2}}=
-\int \limits_0^1 dt_{1} \frac{\ln^{2}(1-t_{1})} {-\alpha_4 -1+ t_{1}}
\left \{ \int \limits_{1}^{t_{1}} + \int \limits_{-\alpha_{1}+2}^{1} \right \}
\frac{dt_{2}} {\alpha_1 -1+ t_{2}}=\nonumber\\
\int \limits_0^1 \frac{dt_{2}} {\alpha_1 -1+ t_{2}} \int \limits_0^{t_{2}}dt_{1}
\frac{\ln^{2}(1-t_{1})} {-\alpha_4 -1+ t_{1}}
-\ln\alpha_{1} \int \limits_0^1 dt_{1} \frac{\ln^{2}(1-t_{1})} {-\alpha_4 -1+
t_{1}}= \nonumber\\
2 \int \limits_0^1 \frac{dt_{2}} {\alpha_1 -1+ t_{2}} \int \limits_0^{t_{2}}
\frac{dt_{1}} {-\alpha_4 -1+ t_{1}}
\int \limits_0^{t_{1}}\frac{dt_{3}}{1-t_{3}}\int
\limits_0^{t_{3}}\frac{dt_{4}}{1-t_{4}}
-2\ln\alpha_{1} {\rm Li}_{3}\left(-\frac{1}{\alpha_{4}} \right) .\nonumber\\
\end{eqnarray}
Using the definition (\ref{intrepr}) we arrive at the result
\begin{eqnarray}
L_{-++}(\alpha_1, 0, 0, \alpha_4)
=2 Li_{1,1,1,1}\left(1, \alpha_{4}+1, \frac{1-\alpha_{1}}{\alpha_{4}+1},
\frac{1}{1-\alpha_{1}} \right)-2\ln\alpha_{1} {\rm
Li}_{3}\left(-\frac{1}{\alpha_{4}} \right)\, .
\end{eqnarray}
For the case $\alpha_{4}=-1$ one obtains
\begin{eqnarray}
L_{-++}(\alpha_1, 0, 0, -1)=-2 Li_{1,2,1}\left(1, 1- \alpha_{1},
\frac{1}{1-\alpha_{1}} \right)-2\ln\alpha_{1}\, \zeta(3)\, .
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 5. ${\bf \alpha_{2}=0}$ (and ${\bf \alpha_{4}=-1}$) }
\end{bf}
\vglue .3cm
For this integral we change the integration variable $y \rightarrow 1-t $,
\begin{eqnarray}
\label{changeB27A}
\int \limits_0^1 dy \frac{\ln (\alpha_1- y) \ln y \ln(\alpha_{3}+y)} {\alpha_4 + y}=
\int \limits_0^1 dt \frac{\ln (1- t) \ln(\alpha_{1}-1+t) \ln(\alpha_{3}+1-t)}
{\alpha_4+1 - t}= \nonumber \\
\int \limits_0^1 dt \frac{\ln (1- t) \ln(\gamma_{1}+t) \ln(\gamma_{2}-t)}
{\gamma_{3} - t} \, .
\end{eqnarray}
One notes that the last integral is an analogue of the integral in Eq.~(\ref{B27}).
The calculation proceeds in a similar way,
\begin{eqnarray}
\label{getB27A}
- Li_{1,1,1,1}\left(\gamma_3, \frac{\gamma_2}{\gamma_3},
-\frac{\gamma_1}{\gamma_2},- \frac{1}{\gamma_1} \right) =
\int \limits_{0}^{1} \frac{dt_{4}}{\gamma_1 +t_{4}} \int \limits_{0}^{t_{4}}
\frac{dt_{3}}{\gamma_2 -t_{3}}
\int \limits_{0}^{t_{3}} \frac{dt_{2}}{\gamma_3 -t_{2}} \int \limits_{0}^{t_{2}}
\frac{dt_{1}}{1-t_{1}} = \nonumber \\
\nonumber
- \int \limits_{0}^{t_m} \frac{dt_{4}}{\gamma_1 +t_{4}} \int \limits_{0}^{t_{4}}
\frac{dt_{3}}{\gamma_2 -t_{3}}
\int \limits_{0}^{t_{3}} d t_{2} \frac{\ln(1-t_{2})}{\gamma_3 -t_{2}} =
- \int \limits_{0}^{t_m} \frac{dt_{4}}{\gamma_1 +t_{4}} \int \limits_{0}^{t_{4}}
d t_{2} \frac{\ln(1-t_{2})}{\gamma_3 -t_{2}}
\int \limits_{t_{2}}^{t_{4}} \frac{dt_{3}}{\gamma_2 -t_{3}}= \\
\nonumber
\int \limits_{0}^{1} dt_{4} \frac{\ln(\gamma_2 -t_{4})}{\gamma_1 +t_{4}} \int
\limits_{0}^{t_{4}} d t_{2} \frac{ \ln(1-t_{2})}{\gamma_3 -t_{2}} -
\int \limits_{0}^{1} \frac{dt_{4}}{\gamma_1 + t_{4}} \int \limits_{0}^{t_{4}}
dt_{2} \frac{\ln(1-t_{2}) \ln(\gamma_2 - t_{2})}{\gamma_3 - t_{2}}= \\
\nonumber
Y'(1) - \int \limits_{0}^{1} dt_{2} \frac{ \ln(1-t_{2}) \ln(\gamma_2 -
t_{2})} {\gamma_3 -t_{2}} \int \limits_{t_{2}}^{1} \frac{dt_{4}}{\gamma_1
+ t_{4}}= \nonumber\\
Y'(1)
-\ln(\gamma_{1}+1) \int \limits_{0}^{1} dt_{2} \frac{ \ln(1-t_{2}) \ln(\gamma_2
-t_{2})} {\gamma_3 -t_{2}}+
\int \limits_{0}^{1} dt_{2} \frac{ \ln(1-t_{2}) \ln(\gamma_1 + t_{2})
\ln(\gamma_2 - t_{2})} {\gamma_3 -t_{2}}=\nonumber\\
=Y'(1)
-Y''(1)+
\int \limits_{0}^{1} dt_{2} \frac{ \ln(1-t_{2}) \ln(\gamma_1 + t_{2})
\ln(\gamma_2 - t_{2})} {\gamma_3 -t_{2}}\, , \nonumber\\
\end{eqnarray}
where we have introduced the notation
\begin{eqnarray}
\label{yprimes}
Y'(t_m) = \int \limits_{0}^{t_m} dt_{4} \frac{\ln(\gamma_2 - t_{4})}{\gamma_1
+t_{4}} \int \limits_{0}^{t_{4}} d t_{2} \frac{ \ln(1-t_{2})}{\gamma_3 - t_{2}},\nonumber
\\
Y''(t_m) = \ln(\gamma_1 + t_m) \int \limits_{0}^{t_m} dt_{2} \frac{ \ln(1-t_{2})
\ln(\gamma_2 - t_{2})} {\gamma_3 - t_{2}}.
\end{eqnarray}
The last term in (\ref{getB27A}) is the required integral.
The expansion of the integral $Y'(t_{m})$ in terms of multiple polylogarithms is
similar to the evaluation of $I'(t_{m})$ in Eq.~(\ref{iprimes}).
The result of the calculation is
\begin{eqnarray}
\label{B26Cnew}
Y'(t_m) =Li_{1,1,1,1}\left( \gamma_{3},
-\frac{\gamma_{1}}{\gamma_{3}},-\frac{\gamma_{2}}{\gamma_{1}},
\frac{t_{m}}{\gamma_{2}} \right)
+\ln(\gamma_{2}-t_{m}) Li_{1,1,1}\left( \gamma_{3},
-\frac{\gamma_{1}}{\gamma_{3}},-\frac{t_{m}}{\gamma_{1}} \right)\, .
\end{eqnarray}
For the calculation of $Y''(t_m)$ one can make use of (\ref{B23C}). Finally using
Eqs.~(\ref{getB27A}) and (\ref{B26Cnew}) one arrives at the result
\begin{eqnarray}
\label{B27A}
\int \limits_{0}^{1} dt \frac{ \ln(1-t) \ln(\gamma_1 + t)
\ln(\gamma_2 - t)} {\gamma_3 -t}=
-Li_{1,1,1,1}\left(\gamma_{3}, -\frac{\gamma_{1}}{\gamma_{3}}, -
\frac{\gamma_{2}}{\gamma_{1}}, \frac{1}{\gamma_{2}} \right)\nonumber\\
-Li_{1,1,1,1}\left(\gamma_{3}, \frac{\gamma_{2}}{\gamma_{3}}, -
\frac{\gamma_{1}}{\gamma_{2}}, - \frac{1}{\gamma_{1}} \right)
-\ln(\gamma_{1}+1)Li_{1,1,1}\left(\gamma_{3}, \frac{\gamma_{2}}{\gamma_{3}},
\frac{1}{\gamma_{2}} \right)\\
-\ln(\gamma_{2}-1)Li_{1,1,1}\left(\gamma_{3}, -\frac{\gamma_{1}}{\gamma_{3}}, -
\frac{1}{\gamma_{1}} \right)
-\ln(\gamma_{1}+1)\ln(\gamma_{2}-1)Li_{1,1}\left(\gamma_{3}, \frac{1}{\gamma_{3}}
\right).\nonumber
\end{eqnarray}
To obtain the formula for the $L$ function with $\alpha_{2}=0$ we must only change
$\gamma_{1}, \gamma_{2}$, and $\gamma_{3}$ to
$\alpha_{1}-1\, , \alpha_{3}+1$, and $\alpha_{4}+1$ according to
Eq.~(\ref{changeB27A}):
\begin{eqnarray}
\label{fromB27}
L_{-++}(\alpha_{1}, 0, \alpha_{3}, \alpha_{4})=\int \limits_0^1 dy \frac{\ln
(\alpha_1- y) \ln y \ln(\alpha_{3}+y)} {\alpha_4 + y}=\nonumber\\
-Li_{1,1,1,1}\left(1+\alpha_{4}, \frac{1-\alpha_{1}}{1+\alpha_{4}},
\frac{1+\alpha_{3}}{1-\alpha_{1}}, \frac{1}{1+\alpha_{3}} \right)
-Li_{1,1,1,1}\left(1+\alpha_{4} ,
\frac{1+\alpha_{3}}{1+\alpha_{4}},\frac{1-\alpha_{1}}{1+\alpha_{3}},
\frac{1}{1-\alpha_{1}} \right)\nonumber\\
-\ln\alpha_{1}Li_{1,1,1}\left(1+\alpha_{4}, \frac{1+\alpha_{3}}{1+\alpha_{4}},
\frac{1}{1+\alpha_{3}} \right)
-\ln\alpha_{3} Li_{1,1,1}\left(1+\alpha_{4}, \frac{1-\alpha_{1}}{1+\alpha_{4}},
\frac{1}{1-\alpha_{1}} \right)\nonumber\\
-\ln\alpha_{1}\ln\alpha_{3} Li_{1,1}\left(1+\alpha_{4}, \frac{1}{1+\alpha_{4}}
\right)\,.\,\,\,\,\,\,\,\,
\end{eqnarray}
For the case $\alpha_{4}=-1$ we calculate the limit of the rhs of
(\ref{fromB27}) for $\alpha_4 \rightarrow -1$ and obtain
\begin{eqnarray}
L_{-++}(\alpha_{1}, 0, \alpha_{3}, -1)=
Li_{2,1,1}\left(1-\alpha_{1} , \frac{1+\alpha_{3}}{1-\alpha_{1}},
\frac{1}{1+\alpha_{3}} \right)\nonumber\\
+Li_{2,1,1}\left(1+\alpha_{3},\frac{1-\alpha_{1}}{1+\alpha_{3}},
\frac{1}{1-\alpha_{1}} \right)
+\ln\alpha_{1}Li_{2,1}\left(1+\alpha_{3}, \frac{1}{1+\alpha_{3}} \right)\nonumber\\
+\ln\alpha_{3} Li_{2,1}\left(1-\alpha_{1}, \frac{1}{1-\alpha_{1}} \right)
+\ln\alpha_{1}\ln\alpha_{3}\, \zeta(2) \,.
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
IV. TRANSFORMATION OF $L_{+++}$ TO MULTIPLE POLYLOGARITHMS
\end{bf}
\vglue .3cm
In this section we will show that all our $L_{+++}$ functions can be
expressed in terms of multiple polylogarithms.
\vglue 1.0cm
\begin{bf}
\noindent
A. General case for the $L_{+++}$ function
\end{bf}
\vglue .3cm
We now proceed with the transformation of the triple index function $L_{+++}$,
\begin{equation} \label{getLpppGen}
L_{+++}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)= \int \limits_0^1 dy
\frac{\ln (\alpha_1+y)\ln (\alpha_2+ y)\ln (\alpha_3 + y)}
{\alpha_4 + y}\, .
\end{equation}
After changing the
integration variable $y=-\alpha_{1} t$ we obtain
\begin{eqnarray}
\label{getLpppGeneral}
-\int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln(\alpha_1-\alpha_{1} t) \ln(\alpha_{2}-\alpha_{1}
t)\ln(\alpha_3-\alpha_{1} t)}{\frac{\alpha_{4}}{\alpha_{1}}-t} =
-\int \limits_{0}^{-1/\alpha_1} dt \frac{\ln\alpha_1
\ln(\alpha_{2}-\alpha_{1} t)\ln(\alpha_3+\alpha_{1} t)}
{\frac{\alpha_{4}}{\alpha_{1}}-t}\nonumber\\
-\int \limits_{0}^{-1/\alpha_1} dt \frac{\ln(1-t) [
\ln\alpha_1+\ln(\frac{\alpha_2}{\alpha_1}-t)] [
\ln\alpha_1+\ln(\frac{\alpha_3}{\alpha_1}- t) ] }
{\frac{\alpha_4}{\alpha_1}-t}=\nonumber\\
\ln\alpha_{1} \int \limits_0^1 dy \frac{ \ln (\alpha_2+ y)\ln
(\alpha_3 + y)} {\alpha_4 + y}-\ln^{2}\alpha_1 \int
\limits_{0}^{-1/\alpha_1} dt
\frac{\ln(1-t)}{\frac{\alpha_4}{\alpha_1}-t}\nonumber\\
-\ln\alpha_1 \int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln(1-t)\ln(\frac{\alpha_2}{\alpha_1}-
t)}{\frac{\alpha_4}{\alpha_1}-t}
-\ln\alpha_1 \int
\limits_{0}^{-1/\alpha_1} dt
\frac{\ln(1-t)\ln(\frac{\alpha_3}{\alpha_1}- t)}{\frac{\alpha_4}{\alpha_1}-t}\nonumber\\
-\int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln(1-t)\ln(\frac{\alpha_2}{\alpha_1}- t)
\ln(\frac{\alpha_3}{\alpha_1}- t)
}{\frac{\alpha_4}{\alpha_1}-t}\,.
\hspace{2cm}
\end{eqnarray}
The first integral on the rhs of (\ref{getLpppGeneral}) has been calculated in
Eq.~(\ref{B23A}). For the
second integral one makes use of the formula (\ref{B28A}) (the
only change is $\alpha_2\rightarrow \alpha_1$). For the
evaluation of the third and fourth integrals one uses Eq.
(\ref{B23C}). We are left with the most complicated fifth
integral. Let us consider an integral of the type
\begin{eqnarray}
\int \limits_{0}^{t_m} dt \frac{\ln(1-t)\ln(\gamma_1- t)
\ln(\gamma_2- t) }{\gamma_3-t}\,.
\end{eqnarray}
This integral is an analogue of the integral in Eq.~(\ref{B27}). The calculation
proceeds in a similar way. One obtains the result
\begin{eqnarray}
\int \limits_{0}^{t_m} dt \frac{\ln(1-t)\ln(\gamma_1- t)
\ln(\gamma_2- t) }{\gamma_3-t}=-\ln(\gamma_1 -t_m) \ln(\gamma_2 -t_m)
Li_{1,1}\left( \gamma_3, \frac{t_m} {\gamma_3} \right) \nonumber
\\
-\ln(\gamma_2 -t_m) Li_{1,1,1}\left( \gamma_3, \frac{\gamma_1} {\gamma_3},
\frac{t_m}{\gamma_1} \right)
- \ln(\gamma_1- t_m)
Li_{1,1,1}\left( \gamma_3, \frac{\gamma_2} {\gamma_3}, \frac{t_m}
{\gamma_2} \right) \\
- Li_{1,1,1,1}\left( \gamma_3, \frac{\gamma_2} {\gamma_3}, \frac{\gamma_1}
{\gamma_2}, \frac{t_m}{\gamma_1} \right)
- Li_{1,1,1,1} \left( \gamma_3, \frac{\gamma_1} {\gamma_3}, \frac{\gamma_2}
{\gamma_1}, \frac{t_m}{\gamma_2} \right)\, . \nonumber
\label{B27F}
\end{eqnarray}
Taking into account everything mentioned above for Eq.~(\ref{getLpppGeneral}) we
arrive at the final result for the $L_{+++}$ function,
\begin{eqnarray}
\label{LpppGeneral}
L_{+++}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=
Li_{1,1,1,1}\left(\frac{\alpha_4}{\alpha_1}, \frac{\alpha_2}{\alpha_4},
\frac{\alpha_3}{\alpha_2},- \frac{1}{\alpha_3}\right)\nonumber\\
+Li_{1,1,1,1}\left(\frac{\alpha_4}{\alpha_1}, \frac{\alpha_3}{\alpha_4},
\frac{\alpha_2}{\alpha_3}, - \frac{1}{\alpha_2}\right)
+\ln\alpha_{1} Li_{1,1,1}\left(\frac{\alpha_4}{\alpha_2},
\frac{\alpha_3}{\alpha_4}, - \frac{1}{\alpha_3}\right)\nonumber\\
+\ln(1+\alpha_{2}) Li_{1,1,1}\left(\frac{\alpha_4}{\alpha_1},
\frac{\alpha_3}{\alpha_4}, - \frac{1}{\alpha_3}\right)
+\ln(1+\alpha_{3}) Li_{1,1,1}\left(\frac{\alpha_4}{\alpha_1},
\frac{\alpha_2}{\alpha_4}, - \frac{1}{\alpha_2}\right)\nonumber\\
+\ln\alpha_{1}\ln\alpha_{2} Li_{1,1}\left(\frac{\alpha_4}{\alpha_3}, -
\frac{1}{\alpha_4}\right)
+\ln\alpha_{1}\ln(1+\alpha_{3}) Li_{1,1}\left(\frac{\alpha_4}{\alpha_2}, -
\frac{1}{\alpha_4}\right) \\
+\ln(1+\alpha_{2})\ln(1+\alpha_{3}) Li_{1,1}\left(\frac{\alpha_4}{\alpha_1}, -
\frac{1}{\alpha_4}\right)
+\ln\alpha_{1}\ln\alpha_{2}\ln\alpha_{3}\ln\left(\frac{\alpha_{4}+1}
{\alpha_{4}}\right) \, .\nonumber
\end{eqnarray}
For this equation the conditions \eqr{domain} are assumed. We emphasize that the
arguments of $L_{+++}$ functions occuring in the actual calculation of the massive
scalar one--loop integrals are not of the most general type as assumed in
the derivation of (\ref{LpppGeneral}). We have nevertheless
included a discussion of the general
case because Eq.~(\ref{LpppGeneral}) may be useful in other applications.
In the results for the massive scalar integrals one has only the special
cases where
$\alpha_{1}=\alpha_{2}$ or $\alpha_{1}=\alpha_{3}$
as well as the cases $\alpha_{1}=0$ and/or $\alpha_{2}=0$ and/or $
\alpha_{3}=0$ and/or $\alpha_{4}=\{-1,0\}$.
If some $\alpha$'s coincide with each other Eq.~(\ref{LpppGeneral}) becomes
simpler. In this case one can also make use of symmetry properties to obtain
simpler relations between the $L_{+++}$ functions and multiple polylogarithms.
For the cases $\alpha_{1}=0$ and/or $ \alpha_{2}=0$ and/or $
\alpha_{3}=0$
and/or $\alpha_{4}=\{-1,0\}$ the general formula (\ref{LpppGeneral}) is no longer
valid and these cases must be studied separately.
\vglue 1.0cm
\begin{bf}
\noindent
B. Special cases for the $L_{+++}$ function
\end{bf}
\vglue .3cm
In the Laurent series expansion of the massive scalar one-loop integrals the
following special cases for the $\alpha_{i}$ are present.
\vglue 1.0cm
\begin{bf}
\noindent
{\em 1. ${\bf \alpha_{1}=\alpha_{2}}$ or ${\bf \alpha_{1}=\alpha_{3}}$ }
\end{bf}
\vglue .3cm
As it was stated in Sec.~II the $L_{+++}$ function is symmetric under the
permutations $\alpha_{i} \leftrightarrow \alpha_{j}$.
Therefore, it suffices to consider the case $\alpha_{1}=\alpha_{2}$.
We must evaluate the integral
\begin{eqnarray}
L_{+++}(\alpha_1,\alpha_1,\alpha_3,\alpha_4)=\int \limits_0^1 dy
\frac{\ln^{2} (\alpha_1+y)\ln (\alpha_3 + y)}
{\alpha_4 + y}\, .
\end{eqnarray}
This integral can be expressed in different ways. First of all one can directly use
Eq.~(\ref{LpppGeneral})
replacing $\alpha_{2}$ by $\alpha_{1}$ . The second possibility
is to use symmetry properties. One takes into account the rhs of
Eq.~(\ref{LpppGeneral}) and
notes that the part with multiple polylogarithms of weight four is symmetric under
the exchange $\alpha_{2}\leftrightarrow\alpha_{3}$. It allows one to reduce the
number of the multiple polylogarithms from two to one.
First we apply Eq.~(\ref{LpppGeneral}) for the case $\alpha_{2} =\alpha_{3}$
replacing $\alpha_{3}$ by $\alpha_{2}$. Second we change
$\alpha_{1}\rightarrow\alpha_{3}$ and $\alpha_{2} \rightarrow \alpha_{1}$. After
these transformations one obtains the following result:
\begin{eqnarray}
\label{secversion}
L_{+++}(\alpha_1,\alpha_1,\alpha_3,\alpha_4)=\int \limits_0^1 dy
\frac{\ln^{2} (\alpha_1+y)\ln (\alpha_3 + y)}
{\alpha_4 + y}= \nonumber\\
+2 Li_{1,1,1,1}\left(\frac{\alpha_4}{\alpha_3}, \frac{\alpha_1}{\alpha_4},1,-
\frac{1}{\alpha_1}\right)
+\ln\alpha_3\, Li_{1,1,1}\left(\frac{\alpha_4}{\alpha_1},
\frac{\alpha_1}{\alpha_4}, - \frac{1}{\alpha_1}\right)\nonumber\\
+2 \ln(1+\alpha_1) Li_{1,1,1}\left(\frac{\alpha_4}{\alpha_3},
\frac{\alpha_1}{\alpha_4}, - \frac{1}{\alpha_1}\right)
+\ln\alpha_3 \left[\ln(\alpha_1+1)+\ln\alpha_1 \right]
Li_{1,1}\left(\frac{\alpha_4}{\alpha_1}, - \frac{1}{\alpha_4}\right) \\
+\ln^{2}(1+\alpha_1) Li_{1,1}\left(\frac{\alpha_4}{\alpha_3}, -
\frac{1}{\alpha_4}\right)
+\ln^2\alpha_1 \ln\alpha_3 \ln\left(\frac{\alpha_{4}+1}{\alpha_{4}}\right)\, .\nonumber
\end{eqnarray}
There is also the third possibility to express
$L_{+++}(\alpha_1,\alpha_1,\alpha_3,\alpha_4) $ in terms of multiple polylogarithms:
\begin{eqnarray}
\label{getthirdversion}
\int \limits_0^1 dy
\frac{\ln^{2} (\alpha_1+y)\ln (\alpha_3 + y)}
{\alpha_4 + y}\, \stackrel{y\rightarrow-\alpha_1 t }{\, =\, } \,
-\int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln^{2}(\alpha_1-\alpha_{1} t) \ln(\alpha_3-\alpha_{1}
t)}{\frac{\alpha_{4}}{\alpha_{1}}-t}=\nonumber\\
-\int \limits_{0}^{-1/\alpha_1} dt \frac{\left[\ln^2\alpha_1+2\ln\alpha_{1}\ln(1-t)
+\ln^{2}(1-t)\right]\left[ \ln\alpha_{1}+ \ln(\frac{\alpha_3}{\alpha_{1}}-t)\right]}
{\frac{\alpha_{4}}{\alpha_{1}}-t}=\nonumber\\
+\ln^{3} \alpha_1 \int \limits_0^1
\frac{dy}
{\alpha_4 + y}
+\ln^{2} \alpha_1 \int \limits_0^1 dy
\frac{\ln (\alpha_3 + y)}
{\alpha_4 + y}
-2 \ln^{2}\alpha_{1}\int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln(1-t)}{\frac{\alpha_{4}}{\alpha_{1}}-t}\nonumber\\
- \ln\alpha_{1}\int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln^{2}(1-t)}{\frac{\alpha_{4}}{\alpha_{1}}-t}
-2\ln\alpha_{1}\int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln(1-t)\ln(\frac{\alpha_3}{\alpha_{1}}-t)}
{\frac{\alpha_{4}}{\alpha_{1}}-t}\nonumber\\
-\int \limits_{0}^{-1/\alpha_1} dt \frac{\ln^{2}(1-t)\ln(\frac{\alpha_3}{\alpha_{1}}-t)}
{\frac{\alpha_{4}}{\alpha_{1}}-t}\, .
\end{eqnarray}
The first term can be integrated immediately. For the second and third term one uses
Eq.~(\ref{B28}) and Eq.~(\ref{B28A}), respectively.
The integral of the fourth term can be rewritten as
\begin{eqnarray}
\int \limits_{0}^{-1/\alpha_1} dt
\frac{\ln^{2}(1-t)}{\frac{\alpha_4}{\alpha_1}-t}= 2\int
\limits_{0}^{-1/\alpha_1}\frac{dt_{1}}{\frac{\alpha_4}{\alpha_1}-t_{1}}
\int \limits_{0}^{t_1}\frac{dt_{2}}{1-t_{2}}
\int
\limits_{0}^{t_2}\frac{dt_{3}}{1-t_{3}}=
2Li_{1,1,1}\left(1,\frac{\alpha_{4}}{\alpha_{1}},\frac{-1}{\alpha_{4}} \right)\, .
\end{eqnarray}
The fifth term is calculable with Eq.~(\ref{B23}). To integrate the last term one
first evaluates the following integral:
\begin{eqnarray}
\label{B27E}
\int \limits_{0}^{t_{m}} dt \frac{\ln^{2}(1-t)\ln(\beta_{1}-t)}
{\beta_{2}-t}=-\int \limits_{0}^{t_{m}} dt_{1} \frac{\ln^{2}(1-t_{1})}
{\beta_{2}-t_{1}}
\left \{ \int \limits_{t_{m}}^{t_{1}} + \int \limits_{\beta_{1}-1}^{t_{m}} \right
\} \frac{dt_{2}}{\beta_1-t_{2}}=\nonumber\\
\int \limits_{0}^{t_{m}}\frac{dt_{2}}{\beta_1-t_{2}}\int \limits_{0}^{t_2}
dt_{1}\frac{\ln^{2}(1-t_{1})}
{\beta_{2}-t_{1}} +\ln(\beta_{1}-t_{m})\int \limits_{0}^{t_{m}} dt \frac{\ln^{2}(1-t)}
{\beta_{2}-t}=\\
2 Li_{1,1,1,1}\left(1,\beta_{2},\frac{\beta_{1}}{\beta_{2}},\frac{t_{m}}{\beta_{1}}
\right)+2\ln(\beta_{1}-t_{m})Li_{1,1,1}\left(1,\beta_{2},\frac{t_{m}}{\beta_{2}}\right)\,
.\nonumber
\end{eqnarray}
Then to calculate the last term of Eq.~(\ref{getthirdversion})
one only has to change $\beta_{1}, \beta_{2}$, and $t_{m}$ by the corresponding
combinations of $\alpha_{i}$.
Finally we arrive at the result for the
$L_{+++}(\alpha_1,\alpha_1,\alpha_3,\alpha_4) $ function,
\begin{eqnarray}
\label{thirdversion}
L_{+++}(\alpha_1,\alpha_1,\alpha_3,\alpha_4)=
-2Li_{1,1,1,1}\left(1,\frac{\alpha_{4}}{\alpha_{1}},\frac{\alpha_{3}}{\alpha_{4}}, -
\frac{1}{\alpha_{3}}\right)
-2\ln(\alpha_{3}+1) Li_{1,1,1}\left(1,\frac{\alpha_{4}}{\alpha_{1}}, -
\frac{1}{\alpha_{4}}\right)\nonumber\\
+2\ln \alpha_{1}
Li_{1,1,1}\left(\frac{\alpha_{4}}{\alpha_{1}},\frac{\alpha_{3}}{\alpha_{4}} , -
\frac{1}{\alpha_{3}}\right)
+2\ln\alpha_{1}\ln(\alpha_{3}+1) Li_{1,1}\left(\frac{\alpha_{4}}{\alpha_{1}}, -
\frac{1}{\alpha_{4}}\right)\nonumber \\
+\ln^{2}\alpha_{1} Li_{1,1}\left(\frac{\alpha_{4}}{\alpha_{3}}, -
\frac{1}{\alpha_{4}}\right)
+ \ln^{2}\alpha_1\ln\alpha_{3}\ln\left(\frac{\alpha_{4}+1}{\alpha_{4}} \right)\,
.~~~~~
\end{eqnarray}
This is the third possibility to express
$L_{+++}(\alpha_1,\alpha_1,\alpha_3,\alpha_4) $ function in terms of multiple
polylogaritms.
Each of the Eqs.~(\ref{secversion}) and (\ref{thirdversion}) contains only one
multiple polylogarithm
of weight four and they are both equally acceptable from this point of view.
One has a free choice to apply any of these equations for the
required $L$ functions.
The situation with the $L_{+++}(\alpha_1,\alpha_1,\alpha_3,\alpha_4) $ function is
an example of the statement that the expansion of the $L$ functions in terms
of multiple polylogarithms is not unique.
\vglue 1.0cm
\begin{bf}
\noindent
{\em 2. ${\bf \alpha_{1}=0}$ (or ${\bf \alpha_{2}=0}$ or ${\bf \alpha_{3}=0}$)
}
\end{bf}
\vglue .3cm
For this integral we change the integration variable $y \rightarrow 1-t $,
\begin{eqnarray}
L_{+++}(0,\alpha_2,\alpha_3,\alpha_4)= \int \limits_0^1 dy
\frac{\ln y\ln (\alpha_2+ y)\ln (\alpha_3 + y)}
{\alpha_4 + y}=\nonumber\\
\int \limits_0^1 dt
\frac{\ln(1-t) \ln (\alpha_2+ 1-t)\ln (\alpha_3 + 1-t)}{\alpha_4 + 1-t}
\end{eqnarray}
and using Eq.~(\ref{B27F}) we arrive at the result
\begin{eqnarray}
\label{fromB27F}
L_{+++}(0,\alpha_2,\alpha_3,\alpha_4)=
-Li_{1,1,1,1}\left(1+\alpha_{4}, \frac{1+\alpha_{2}}{1+\alpha_{4}},
\frac{1+\alpha_{3}}{1+\alpha_{2}}, \frac{1}{1+\alpha_{3}} \right)\nonumber\\
-Li_{1,1,1,1}\left(1+\alpha_{4},
\frac{1+\alpha_{3}}{1+\alpha_{4}},\frac{1+\alpha_{2}}{1+\alpha_{3}},
\frac{1}{1+\alpha_{2}} \right)
-\ln\alpha_{2}Li_{1,1,1}\left(1+\alpha_{4}, \frac{1+\alpha_{3}}{1+\alpha_{4}},
\frac{1}{1+\alpha_{3}} \right)\nonumber\\
-\ln\alpha_{3} Li_{1,1,1}\left(1+\alpha_{4}, \frac{1+\alpha_{2}}{1+\alpha_{4}},
\frac{1}{1+\alpha_{2}} \right)
-\ln\alpha_{2}\ln\alpha_{3} Li_{1,1}\left(1+\alpha_{4}, \frac{1}{1+\alpha_{4}}
\right).\,\,
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 3. ${\bf \alpha_{1}=\alpha_{2}=0}$ }
\end{bf}
\vglue .3cm
To calculate this integral we again change the integration variable $y
\rightarrow 1-t $,
\begin{eqnarray}
L_{+++}(0,0,\alpha_3,\alpha_4)= \int \limits_0^1 dy
\frac{\ln^{2} y \ln (\alpha_3 + y)}
{\alpha_4 + y}=\nonumber\\
\int \limits_0^1 dt
\frac{\ln^{2}(1-t) \ln (\alpha_3 + 1-t)}{\alpha_4 + 1-t} .
\end{eqnarray}
For the last integral we use Eq.~(\ref{B27E}). An additional simplification can be
done if one notes that
\begin{eqnarray}
Li_{1,1,1}\left(1,\alpha_{4}+1,\frac{1}{\alpha_{4}+1}\right)=-{\rm
Li}_{3}\left(-\frac{1}{\alpha_4} \right) .
\end{eqnarray}
Finally one has
\begin{eqnarray}
\label{fromB27E}
L_{+++}(0,0,\alpha_3,\alpha_4)=
2Li_{1,1,1,1}\left(1,\alpha_{4}+1,\frac{\alpha_{3}+1}
{\alpha_{4}+1},\frac{1}{\alpha_{3}+1} \right)
-2\ln\alpha_{3}{\rm Li}_3 \left(-\frac{1}{\alpha_{4}}\right).
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 4. ${\bf \alpha_{1}=\alpha_{2}=0, \,\, \alpha_4=-1 }$ (or
${\bf \alpha_2=\alpha_{3}=0, \,\, \alpha_{4}=-1 }$) }
\end{bf}
\vglue .3cm
In this case one should calculate the limit of the rhs of (\ref{fromB27E}) for
$t_{m}=1$ and $ \alpha_{4}\rightarrow -1$.
After this procedure one obtains
\begin{eqnarray}
L_{+++}(0,0,\alpha_3,-1)=-2 Li_{1,2,1}\left(1,\alpha_{3}+1,\frac{1}{\alpha_{3}+1} \right)
-2\ln\alpha_{3}\zeta(3).
\end{eqnarray}
For the case $\alpha_2=\alpha_{3}=0$ and $\alpha_{4}=-1$ one can use the same formula.
The only change is $\alpha_{3}\rightarrow\alpha_{1}$.
\vglue 1.0cm
\begin{bf}
\noindent
{\em 5. ${\bf \alpha_{1}=0, \,\, \alpha_{4}=-1}$ }
\end{bf}
\vglue .3cm
To obtain the solution for these values of the $\alpha_{i}$ we must find the
limit of the rhs of (\ref{fromB27F}) for $\alpha_{4}\rightarrow -1$.
After taking the limit one arrives at the result
\begin{eqnarray}
L_{+++}(0,\alpha_2,\alpha_3,-1)=
+Li_{2,1,1}\left(1+\alpha_{2}, \frac{1+\alpha_{3}}{1+\alpha_{2}},
\frac{1}{1+\alpha_{3}} \right)\nonumber\\
+Li_{2,1,1}\left(1+\alpha_{3}, \frac{1+\alpha_{2}}{1+\alpha_{3}},
\frac{1}{1+\alpha_{2}} \right)
+\ln\alpha_{2}Li_{2,1}\left(1+\alpha_{3}, \frac{1}{1+\alpha_{3}} \right)\nonumber\\
+\ln\alpha_{3} Li_{2,1}\left(1+\alpha_{2}, \frac{1}{1+\alpha_{2}} \right)
+\ln\alpha_{2}\ln\alpha_{3} \zeta(2) .
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
V. TRANSFORMATION OF $L_{+}$ TO MULTIPLE POLYLOGARITHMS
\end{bf}
\vglue .3cm
In this section we will show that all our $L_{+}$ functions can be
expressed in terms of multiple polylogarithms.
\vglue 1.0cm
\begin{bf}
\noindent
A. General case for the $L_{+}$ function
\end{bf}
\vglue .3cm
Here we derive the general formula for the single index $L_{+}$ function
Eq.~(\ref{Lpfunction}),
\begin{equation}
L_{+}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=
\int\limits_0^1 dy\, \frac{\ln (\alpha_1+y)} {\alpha_4+y} {\rm
Li}_2(\alpha_2+\alpha_3 y).
\end{equation}
After changing the integration variable $y\rightarrow(t-\alpha_2)/\alpha_3$ one gets
\begin{equation}
L_{+}=\int \limits_{\alpha_2}^{\alpha_2+\alpha_3} \frac{dt}{\alpha_3}
\frac{\ln (\alpha_1+ \frac{t-\alpha_2}{\alpha_3})}
{\alpha_4+\frac{t-\alpha_2}{\alpha_3}} {\rm Li}_2(t) =
\int \limits_{\alpha_2}^{\alpha_2+\alpha_3}
dt \frac{-\ln\alpha_3 + \ln (\alpha_1
\alpha_3 - \alpha_2 + t)}{\alpha_3 \alpha_4 - \alpha_2 + t} {\rm
Li}_2(t).
\end{equation}
The integration interval can be split into two pieces, $[\alpha_2, 0]$ and
$[0, \alpha_2 + \alpha_3]$. One can then write $L_{+}$ as a sum of four
terms,
\begin{equation}
\label{sum4}
L_{+} =- \ln \alpha_3 \left\{ \int
\limits_{0}^{\alpha_2+\alpha_3} - \int \limits_{0}^{\alpha_2} \,\, \right\}
\frac{dt} {\gamma+t} {\rm Li}_{2}(t)
+ \left\{ \int \limits_{0}^{\alpha_2+\alpha_3}
- \int \limits_{0}^{\alpha_2} \,\,\right\}
dt \frac{\ln(\alpha+t)} {\gamma+t} {\rm Li}_{2}(t),
\end{equation}
where we have introduced the notation
\begin{equation}
\label{algama}
\alpha = \alpha_1 \alpha_3 - \alpha_2, {\rm \hspace{0.4in}}
\gamma = \alpha_3 \alpha_4 - \alpha_2.
\end{equation}
Looking at Eq.~(\ref{sum4}) it is clear that there are only two different
types of integrals to be dealt with,
\begin{equation}
\label{types}
\int \limits_{0}^{t_m} \frac{dt} {\gamma+t} {\rm Li}_{2}(t)
{\rm \hspace{0.4in} and} {\rm \hspace{0.4in}}
\int \limits_{0}^{t_m} dt \frac{\ln(\alpha+t)} {\gamma+t} {\rm Li}_{2}(t).
\end{equation}
The upper limits are $t_m = \alpha_2 + \alpha_3$ or $t_m = \alpha_2$.
The first integral can be evaluated analytically in terms
of standard logarithms and classical polylogarithms up to ${\rm Li}_3$. However,
the same integral can also be expressed in terms of multiple polylogarithms via
the integral representation (\ref{intrepr}), e.g.,
\begin{equation}
\label{multi1}
\int \limits_{0}^{t_m}\frac{dt}{\gamma+t}{\rm Li}_{2}(t)=
\int \limits_{0}^{t_m}\frac{dt_{1}}{\gamma+t_{1}}
\int \limits_{0}^{t_1} \frac{dt_{2}}{t_2}
\int \limits_{0}^{t_2} \frac{dt_3}{1-t_3} =
-Li_{2,1}\left(-\gamma,\frac{-t_m}{\gamma}\right).
\end{equation}
We now deal with
the second integral in (\ref{types}). Consider the following multiple
polylogarithm of weight four:
\begin{eqnarray}
Li_{2,1,1} \left( -\gamma, \frac{\alpha}{\gamma},\frac{t_m}{-\alpha} \right)=
\int \limits_{0}^{t_m} \frac{dt_{2}}{-\alpha-t_{2}} \int \limits_{0}^{t_{2}}
\frac{dt_{1}}{-\gamma-t_{1}}{\rm Li}_{2}(t_{1}) =
\int \limits_{0}^{t_m}\frac{dt_{1}}{\gamma+t_{1}}{\rm Li}_{2}(t_{1}) \int
\limits_{t_{1}}^{t_m} \frac{dt_{2}}{\alpha+t_{2}}= \nonumber \\
\int \limits_{0}^{t_m}\frac{dt_{1}}{\gamma+t_{1}}{\rm Li}_{2}(t_{1})
\ln(\alpha+t_m) - \int
\limits_{0}^{t_m}\frac{dt_{1}}{\gamma+t_{1}}{\rm Li}_{2}(t_{1}) \ln(\alpha+t_{1}).~~~
\end{eqnarray}
In the first step we have used the usual trick to change the order of
integration.
As already noted before [see Eq.~(\ref{multi1})] the first term on the second line
can be expressed through a multiple polylogarithm of weight three. Thus one has
\begin{equation}
\label{multi2}
\int \limits_{0}^{t_m} dt \frac{\ln(\alpha+t)} {\gamma+t} {\rm Li}_{2}(t)
=
- Li_{2,1,1}\left(-\gamma, \frac{\alpha}{\gamma},\frac{t_m}{-\alpha}\right) -
Li_{2,1}\left(-\gamma,\frac{-t_m}{\gamma}\right) \ln (\alpha+t_m).
\end{equation}
Finally, substituting Eqs.~(\ref{multi1}) and (\ref{multi2}) into
Eq.~(\ref{sum4}) we arrive at the desired relation
\begin{eqnarray}
\label{LpGeneral}
L_{+}(\alpha_1,\alpha_2,\alpha_3,\alpha_4)=
Li_{2,1,1}\left(\alpha_{2}-\alpha_{3}\alpha_{4},
\frac{\alpha_{2}-\alpha_{1}\alpha_{3} }{\alpha_{2}-\alpha_{3}\alpha_{4}},
\frac{\alpha_2}{\alpha_{2} -\alpha_{1}\alpha_{3} }\right) \nonumber\\
-Li_{2,1,1}\left(\alpha_{2}-\alpha_{3}\alpha_{4},
\frac{\alpha_{2}-\alpha_{1}\alpha_{3} }{\alpha_{2}-\alpha_{3}\alpha_{4}},
\frac{\alpha_2+\alpha_{3}}{\alpha_{2} -\alpha_{1}\alpha_{3} }\right)
+\ln\alpha_{1} Li_{2,1} \left( \alpha_{2}-\alpha_{3}\alpha_{4} ,
\frac{\alpha_2}{\alpha_{2} -\alpha_{3}\alpha_4} \right) \\
-\ln(\alpha_{1}+1)
Li_{2,1} \left( \alpha_{2}-\alpha_{3}\alpha_{4} ,
\frac{\alpha_2+\alpha_{3}}{\alpha_{2} -\alpha_{3}\alpha_{4}} \right) .\nonumber
\end{eqnarray}
We should note that, similar to Eq.~(\ref{B27General}), the conditions~(\ref{domain})
are assumed for the variables $\alpha_{i}$.
Also, one cannot directly use Eq.~(\ref{LpGeneral}) if
$\alpha_{2}-\alpha_{3}\alpha_{4}=0$ or $\alpha_{2}-\alpha_{1}\alpha_{3}=0$.
However, in the results for the massive scalar integrals precisely these special cases
appear, as well as the cases
where $\alpha_{1}=0$ and/or $\alpha_{2}=0$ and/or $ \alpha_{3}=0$ and/or
$\alpha_{4}=\{-1,0\}$.
In such cases the general formula~(\ref{LpGeneral}) is no longer valid and these
cases must be studied separately.
\vglue 1.0cm
\begin{bf}
\noindent
B. Special cases for the $L_{+}$ function
\end{bf}
\vglue .3cm
In the Laurent series expansion of the massive scalar one-loop integrals the
following special cases appear
for the arguments of the $L_{+}$ functions:
\vglue 1.0cm
\begin{bf}
\noindent
{\em 1. ${\bf \alpha_{2}-\alpha_{3}\alpha_{4}=0 }$
(or ${\bf \alpha_{2}-\alpha_{1}\alpha_{3}=0 }$) }
\end{bf}
\vglue .3cm
In this case one must find the limit of the rhs of Eq.~(\ref{LpGeneral}) for
$\alpha_{2}\rightarrow \alpha_{3}\alpha_{4} $.
First we rewrite the rhs of Eq.~(\ref{LpGeneral}) in terms of multidimensional
integrals via the definition~(\ref{intrepr}). Second we replace
$\alpha_{2}$ by $\alpha_{3}\alpha_{4}$. We finally again use the
definition~(\ref{intrepr}) to obtain the result
\begin{eqnarray}
L_{+}(\alpha_1, \alpha_3 \alpha_4 ,\alpha_3,\alpha_4)=
-Li_{3,1}\left(\alpha_{3}(\alpha_{4}-\alpha_{1}),\frac{\alpha_4}{\alpha_{4}
-\alpha_{1} }\right)\nonumber \\
+ Li_{3,1}\left(\alpha_{3}(\alpha_{4}-\alpha_{1}),\frac{\alpha_4+1}{\alpha_{4}
-\alpha_{1} }\right)
-\ln\alpha_{1}{\rm Li}_{3}(\alpha_{3}\alpha_{4})+\ln(\alpha_{1}+1){\rm
Li_3}\left(\alpha_3(\alpha_{4}+1)\right) .
\end{eqnarray}
When $\alpha_{2}-\alpha_{1}\alpha_{3}=0$ one must find the limit of the rhs
of Eq.~(\ref{LpGeneral}) for
$\alpha_{2}\rightarrow \alpha_{1}\alpha_{3} $. We again rewrite the rhs of
Eq.~(\ref{LpGeneral})
in terms of multidimensional
integrals. We then replace $\alpha_{2}$ by $\alpha_{1}\alpha_{3}$ and use the
definition~(\ref{intrepr}). We arrive at the result
\begin{eqnarray}
L_{+}(\alpha_1, \alpha_1 \alpha_3 ,\alpha_3,\alpha_4)=
- Li_{2,2}\left(\alpha_{3}(\alpha_{1}-\alpha_{4}),\frac{\alpha_1}{\alpha_{1}
-\alpha_{4} }\right)\nonumber \\
+Li_{2,2}\left(\alpha_{3}(\alpha_{1}-\alpha_{4}),\frac{\alpha_1+1}{\alpha_{1}
-\alpha_{4} }\right)
+\ln\alpha_1 Li_{2,1}\left(\alpha_{3}(\alpha_{1}-\alpha_{4}),\frac{\alpha_1}
{\alpha_{1}-\alpha_{4} }\right) \\
-\ln(\alpha_{1}+1)
Li_{2,1}\left(\alpha_{3}(\alpha_{1}-\alpha_{4}),\frac{\alpha_1+1}
{\alpha_{1}-\alpha_{4} }\right) \nonumber .
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 2. ${\bf \alpha_1=0 }$ }
\end{bf}
\vglue .3cm
Unfortunately in this case one cannot use Eq.~(\ref{LpGeneral}) for $\alpha_1=0$
because one is immediately faced with the problem of a logarithmic
infinity. One must find another algorithm to express the $L_{+}(0,
\alpha_2,\alpha_3,\alpha_4)$ function in terms of multiple polylogarithms.
After changing the integration variable $y\rightarrow 1-t$ one gets
\begin{eqnarray}
\label{togetChangeLp1}
\int\limits_0^1 dy\, \frac{\ln y} {\alpha_4+y}
{\rm Li_2}(\alpha_2+\alpha_3 y)= \int\limits_0^1 dt\, \frac{\ln (1-t)}
{\alpha_4+1-t} {\rm Li}_2(\alpha_2+\alpha_3 -\alpha_{3}t)=\nonumber\\
\int \limits_0^1 dt_{1} \frac{\ln (1-t_{1})} {\alpha_4+1-t_{1}}
\int \limits_{\alpha_2/ \alpha_3+1}^{t_{1}} dt_{2}
\frac{\ln(1-\alpha_{2}-\alpha_{3}+\alpha_{3} t_{2})
}{\frac{\alpha_{2}}{\alpha_{3}}+1-t_{2}}=\nonumber\\
\nonumber\\
\int \limits_0^1 dt_{1} \frac{\ln (1-t_{1})} {\alpha_4+1-t_{1}} \left\{\int
\limits_{1}^{t_{1}}+\int \limits_{\alpha_2/ \alpha_3+1}^{1} \right\}
dt_{2} \frac{\ln(1-\alpha_{2}-\alpha_{3}+\alpha_{3} t_{2})
}{\frac{\alpha_{2}}{\alpha_{3}}+1-t_{2}}
=\\
\nonumber\\
-\int \limits_{0}^{1} dt_{2} \frac{\ln(1-\alpha_{2}-\alpha_{3}+\alpha_{3} t_{2})
}{\frac{\alpha_{2}}{\alpha_{3}}+1-t_{2}}
\int \limits_0^{t_{2}} dt_{1} \frac{\ln (1-t_{1})} {\alpha_4+1-t_{1}}
-{\rm Li}_{2}(\alpha_{2}) Li_{1,1}\left(\alpha_{4}+1,\frac{1}{\alpha_{4}+1 }
\right) .\nonumber
\end{eqnarray}
The last integral is an analogue of $I'(t_{m})$ in Eq.~(\ref{iprimes}). First one
notes that
\begin{eqnarray}
\label{smallhelp}
\int \limits_0^{t_{2}} dt_{1} \frac{\ln (1-t_{1})} {\alpha_4+1-t_{1}}=
-Li_{1,1}\left(\alpha_{4}+1,\frac{t_{1}}{\alpha_{4}+1 }\right) .
\end{eqnarray}
Then one considers the following chain of transformations:
\begin{eqnarray}
\label{getB26F}
\int \limits_{0}^{1} \frac{dt_{2}}{1-\alpha_{2}-\alpha_{3}+\alpha_{3} t_{2}}
\int \limits_{0}^{t_{2}}\frac{dt_{1}}{\frac{\alpha_{2}}{\alpha_{3}}+1-t_{1}}
Li_{1,1}\left(\alpha_{4}+1,\frac{t_{1}}{\alpha_{4}+1 }\right)=\nonumber\\
\int \limits_{0}^{1} \frac{dt_{1}}{\frac{\alpha_{2}}{\alpha_{3}}+1-t_{1}}
Li_{1,1}\left(\alpha_{4}+1,\frac{t_{1}}{\alpha_{4}+1 }\right)
\int \limits_{t_{1}}^{1} \frac{dt_{2}}{1-\alpha_{2}-\alpha_{3}+\alpha_{3} t_{2}}=\nonumber\\
\frac{1}{\alpha_{3}}\ln(1-\alpha_{2})
\int \limits_{0}^{1} \frac{dt_{1}}{\frac{\alpha_{2}}{\alpha_{3}}+1-t_{1}}
Li_{1,1}\left(\alpha_{4}+1,\frac{t_{1}}{\alpha_{4}+1 }\right)\\
-\frac{1}{\alpha_{3}}
\int \limits_{0}^{1}dt_{1} \frac{\ln(1-\alpha_{2}-\alpha_{3}+\alpha_{3} t_{1})
}{\frac{\alpha_{2}}{\alpha_{3}}+1-t_{1}}
Li_{1,1}\left(\alpha_{4}+1,\frac{t_{1}}{\alpha_{4}+1 }\right) \nonumber
\end{eqnarray}
Using Eq.~(\ref{smallhelp}) we see that the last integral is exactly the integral
required in Eq.~(\ref{togetChangeLp1}).
The initial integral of Eq.~(\ref{getB26F}) and the first integral of the rhs of
Eq.~(\ref{getB26F}) can be expressed in terms of
multiple polylogarithms due to the definition (\ref{intrepr}). Finally for the
$L_{+}(0, \alpha_2,\alpha_3,\alpha_4)$ function we obtain
\begin{eqnarray}
\label{changeLp1}
L_{+}(0, \alpha_2,\alpha_3,\alpha_4)=
Li_{1,1,1,1}\left(\alpha_{4}+1,\frac{\alpha_2+\alpha_3}{\alpha_{3}(\alpha_4 +1) },
\frac{\alpha_2+\alpha_3-1}{ \alpha_2+\alpha_3 }, \frac{\alpha_3}{
\alpha_2+\alpha_3-1 } \right) ~~~\\
+\ln(1-\alpha_{2})Li_{1,1,1}\left(\alpha_{4}+1,\frac{\alpha_2+\alpha_3}
{\alpha_{3}(\alpha_4 +1) }, \frac{\alpha_3}{ \alpha_2+\alpha_3 } \right)
-{\rm Li}_{2}(\alpha_{2})Li_{1,1}\left(\alpha_{4}+1,\frac{1}{\alpha_4 +1 } \right) . \nonumber
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 3. ${\bf \alpha_1=0, \,\, \alpha_{4}=-1 }$ }
\end{bf}
\vglue .3cm
For these values of the $\alpha_{i}$ one uses Eq.~(\ref{changeLp1}) to calculate
the limit of the rhs for $\alpha_{4}\rightarrow -1$.
One arrives at the result
\begin{eqnarray}
L_{+}(0, \alpha_2,\alpha_3,-1)=-Li_{2,1,1}\left(\frac{\alpha_2+\alpha_3}{\alpha_{3} },
\frac{\alpha_2+\alpha_3-1}{ \alpha_2+\alpha_3 }, \frac{\alpha_3}{
\alpha_2+\alpha_3-1 } \right)\\
-\ln(1-\alpha_{2})Li_{2,1}\left(\frac{\alpha_2+\alpha_3}{\alpha_{3} },
\frac{\alpha_3}{ \alpha_2+\alpha_3 } \right)
+ {\rm Li}_{2}(\alpha_{2}) \zeta(2) . \nonumber
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 4. ${\bf \alpha_1=0, \,\, \alpha_{2}+\alpha_{3}=1}$ (and
${\bf \alpha_{4}=-1}$) }
\end{bf}
\vglue .3cm
If one takes a look at Eq.~(\ref{changeLp1}) one realizes that there is a problem if
$\alpha_{2}+\alpha_{4}=1$.
To express the $L_{+}$ function for this configuration of the $\alpha_{i}$ the limit
of the rhs of ~(\ref{changeLp1})
for $\alpha_{2}\rightarrow 1 - \alpha_{3}$ must be found.
The result is
\begin{eqnarray}
L_{+}(0,1-\alpha_{3},\alpha_{3} ,\alpha_{4})=
-Li_{1,1,2}\left(\alpha_{4}+1,\frac{1}{\alpha_{3}(\alpha_{4}+1) }, \alpha_{3} \right)\\
+\ln\alpha_{3}Li_{1,1,1}\left(\alpha_{4}+1,\frac{1}{\alpha_{3}(\alpha_{4}+1) },
\alpha_{3} \right)
-{\rm Li}_{2}(1-\alpha_{3})Li_{1,1}\left(\alpha_{4}+1,\frac{1}{\alpha_{4}+1 }
\right) . \nonumber
\end{eqnarray}
For the case $\alpha_1=0$, $\alpha_{2}+\alpha_{3}=1$, and $\alpha_{4}=-1$ one must
find in addition the limit for $\alpha_{4}\rightarrow-1$.
One arrives at the result
\begin{eqnarray}
L_{+}(0,1-\alpha_{3},\alpha_{3} ,-1)=
Li_{2,2}\left(\frac{1}{\alpha_{3} }, \alpha_{3} \right)
-\ln\alpha_{3}Li_{2,1}\left(\frac{1}{\alpha_{3} }, \alpha_{3} \right)
+\zeta(2) {\rm Li_{2}}(1-\alpha_{3}) .
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 5. ${\bf \alpha_1=0, \,\, \alpha_{2}=-\alpha_{3} }$ }
\end{bf}
\vglue .3cm
To obtain the result for this case one must calculate the limit of the rhs
of~(\ref{changeLp1}) for
$\alpha_{3}\rightarrow-\alpha_{2}$.
After taking the limit one has
\begin{eqnarray}
L_{+}(0,\alpha_{2},-\alpha_{2} ,\alpha_{4})=
-Li_{1,1,2}\left(\frac{\alpha_{2}}{\alpha_{2}-1 },- \alpha_{4},
-\frac{1}{\alpha_{4}} \right)\nonumber\\
+\ln(1-\alpha_{2})Li_{1,2}\left(- \alpha_{4}, -\frac{1}{\alpha_{4}} \right)
+{\rm Li}_{2}(\alpha_{2}) {\rm Li}_{2}\left(-\frac{1}{\alpha_{4}}\right).
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 6. ${\bf \alpha_1=0, \,\, \alpha_{2}=0}$ }
\end{bf}
\vglue .3cm
For this case one can directly use Eq.~(\ref{changeLp1}),
\begin{eqnarray}
\label{LpChange7V2}
L_{+}(0, 0,\alpha_3,\alpha_4)=Li_{1,1,1,1}\left(\alpha_{4}+1,\frac{1}{\alpha_4 +1 },
\frac{\alpha_3-1}{\alpha_3 }, \frac{\alpha_3}{\alpha_3-1 } \right).
\end{eqnarray}
But there is also another very simple possibility. We first change the integration
variable $y\rightarrow t/\alpha_{3}$,
\begin{eqnarray}
\int\limits_0^1 dy\, \frac{\ln y} {\alpha_4+y} {\rm Li_2}(\alpha_3 y)=
\int\limits_0^{\alpha_{3}} dt \frac{\ln(t / \alpha_{3}) } {\alpha_{3}\alpha_4+t}
{\rm Li_2}(t)=
\int\limits_0^{\alpha_{3}} \frac{dt_{1} } {\alpha_{3}\alpha_4+t_{1}} {\rm Li_2}(t_{1})
\int\limits_{\alpha_3}^{t_{1}} \frac{dt_{2} } {t_{2}}=\nonumber\\
-\int\limits_{0}^{\alpha_{3}} \frac{dt_{2} } {t_{2}}
\int\limits_0^{t_{2}} \frac{dt_{1} } {\alpha_{3}\alpha_4+t_{1}} {\rm Li_2}(t_{1}) =
\int\limits_{0}^{\alpha_{3}} \frac{dt_{2} } {t_{2}}
\int\limits_0^{t_{2}} \frac{dt_{1} } {-\alpha_{3}\alpha_4+t_{1}}
\int\limits_{0}^{t_{1}} \frac{dt_{3} } {t_{3}} \int\limits_{0}^{t_{3}}
\frac{dt_{4} } {1-t_{4}}.
\end{eqnarray}
Now using the definition~(\ref{intrepr}) we obtain the result
\begin{eqnarray}
\label{LpChange7V1}
L_{+}(0,
0,\alpha_3,\alpha_4)=Li_{2,2}\left(-\alpha_{3}\alpha_{4},\frac{-1}{\alpha_4}\right).
\end{eqnarray}
The reader has a free choice to use either formula (\ref{LpChange7V2}) or
(\ref{LpChange7V1}). Both equations contain
multiple polylogarithm of weight four. The depth of the multiple polylogarithm in
Eq.~(\ref{LpChange7V1}) is two against
four in Eq.~(\ref{LpChange7V2}). For $\alpha_{4}=-1$ Eq.~(\ref{LpChange7V1}) can be
directly used. However, in the case of Eq.~(\ref{LpChange7V2})
one must first calculate the limit for $\alpha_{4}\rightarrow -1$.
\vglue 1.0cm
\begin{bf}
\noindent
{\em 7. ${\bf \alpha_1=0, \,\, \alpha_{2}=1 }$ }
\end{bf}
\vglue .3cm
Unfortunately, in this case one cannot use Eq.~(\ref{changeLp1}) because of the term
$\ln(1-\alpha_{2})$.
To express this $L_{+}$ function in terms of multiple polylogarithms we first make use of
a standard relation between dilogs with arguments $x$ and $1-x$
for the function ${\rm Li}_{2}$ under the sign of the integral:
\begin{eqnarray}
\label{getChangeLp8}
\int\limits_0^1 dy \frac{\ln y} {\alpha_4+y} {\rm Li_2}(1+\alpha_3 y)=
\int\limits_0^1 dy \frac{\ln y} {\alpha_4+y} [\zeta(2)
-\ln(-\alpha_{3}y)\ln(1+\alpha_{3}y)-{\rm Li_2}(-\alpha_3 y)]=\nonumber\\
\zeta(2)\int\limits_0^1 dy\, \frac{\ln y} {\alpha_4+y}
-\int\limits_0^1 dy \frac{\ln y} {\alpha_4+y}{\rm Li_2}(-\alpha_3 y)
-\int\limits_0^1 dy \frac{\ln y [ \ln(-\alpha_{3})+\ln y ] \ln(1+\alpha_{3}y)
} {\alpha_4+y} =\nonumber\\
\zeta(2){\rm Li}_{2}\left(-\frac{1}{\alpha_{4}} \right)
-Li_{1,1,1,1}\left(\alpha_{4}+1,\frac{1}{\alpha_4 +1 },
\frac{\alpha_3+1}{\alpha_3 }, \frac{\alpha_3}{\alpha_3+1 } \right)~~~ \\
-\ln(-\alpha_{3}) \int\limits_0^1 dy\, \frac{\ln y \ln(1+\alpha_{3}y) } {\alpha_4+y}
-\int\limits_0^1 dy \frac{\ln^{2} y \ln(1+\alpha_{3}y) } {\alpha_4+y}\, ,~~~\nonumber
\end{eqnarray}
where the $Li_{1,1,1,1}$ function was obtained with the help of
Eq.~(\ref{LpChange7V2}).
To obtain the last integral in Eq.~(\ref{getChangeLp8}) one proceeds as follows:
\begin{eqnarray}
\label{helpLp8a}
\int\limits_0^1 dy \frac{\ln^{2} y \ln(1+\alpha_{3}y) } {\alpha_4+y} \,
\stackrel{y\rightarrow1-t }{\, =\, } \,
\int\limits_0^1 dt \frac{\ln^{2} (1-t) \ln(1+\alpha_{3}-\alpha_{3}t) }
{\alpha_4+1-t}=\nonumber\\
\int\limits_0^1 dt_{1} \frac{\ln^{2} (1-t_{1}) } {\alpha_4+1-t_{1}}
\int\limits_1^{t_{1}} \frac{-\alpha_{3} dt_{2}}{1+\alpha_{3}-\alpha_{3}t_{2} }=
\int\limits_0^{1} \frac{ dt_{2}}{\frac{1}{\alpha_{3}}+1-t_{2}
}\int\limits_0^{t_{2}} dt_{1} \frac{\ln^{2} (1-t_{1}) } {\alpha_4+1-t_{1}}=\nonumber\\
2 \int\limits_0^{1} \frac{ dt_{2}}{\frac{1}{\alpha_{3}}+1-t_{2}
}\int\limits_0^{t_{2}}\frac{dt_{1} } {\alpha_4+1-t_{1}}
\int\limits_0^{t_{1}}\frac{dt_{3}}{1-t_{3}}
\int\limits_0^{t_{3}}\frac{dt_{4}}{1-t_{4}}= \\
2 Li_{1,1,1,1}\left(1,\alpha_{4}+1,\frac{\alpha_3+1}{\alpha_3(\alpha_{4}+1 ) },
\frac{\alpha_3}{\alpha_3+1 } \right). \nonumber
\end{eqnarray}
Similarly one can evaluate the remaining integral
\begin{eqnarray}
\label{helpLp8b}
\int\limits_0^1 dy \frac{\ln y \ln(1+\alpha_{3}y) } {\alpha_4+y}=
-Li_{1,1,1}\left(\alpha_{4}+1,\frac{\alpha_3+1}{\alpha_3(\alpha_{4}+1 ) },
\frac{\alpha_3}{\alpha_3+1 } \right).
\end{eqnarray}
Now combining Eqs.~(\ref{getChangeLp8}), (\ref{helpLp8a}), and (\ref{helpLp8b}) one
arrives at the result
\begin{eqnarray}
\label{ChangeLp8}
L_{+}(0,1,\alpha_{3},\alpha_{4})=
-2 Li_{1,1,1,1}\left(1,\alpha_{4}+1,\frac{\alpha_3+1}{\alpha_3(\alpha_{4}+1 ) },
\frac{\alpha_3}{\alpha_3+1 } \right)\nonumber\\
-Li_{1,1,1,1}\left(\alpha_{4}+1,\frac{1}{\alpha_4 +1 },
\frac{\alpha_3+1}{\alpha_3 }, \frac{\alpha_3}{\alpha_3+1 } \right)\\
+\ln(-\alpha_{3})
Li_{1,1,1}\left(\alpha_{4}+1,\frac{\alpha_3+1}{\alpha_3(\alpha_{4}+1 ) },
\frac{\alpha_3}{\alpha_3+1 } \right)
+\zeta(2){\rm Li}_{2}\left(-\frac{1}{\alpha_{4}} \right). \nonumber
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
{\em 8. ${\bf \alpha_1=0, \,\, \alpha_{2}=-\alpha_{3}=1 }$ }
\end{bf}
\vglue .3cm
For these values of the $\alpha_{i}$ we must find the limit of the rhs of
Eq.~(\ref{ChangeLp8}) for $\alpha_{3}\rightarrow -1$. After taking the limit
we obtain
\begin{eqnarray}
L_{+}(0,1,-1,\alpha_{4})=
Li_{1,1,2}\left(\alpha_{4}+1,\frac{1}{\alpha_{4}+1 }, 1 \right)\nonumber\\
+2 Li_{1,1,2}\left(1,\alpha_{4}+1,\frac{1}{\alpha_{4}+1 } \right)
+\zeta(2){\rm Li}_{2}\left(-\frac{1}{\alpha_{4}} \right).
\end{eqnarray}
\vglue 1.0cm
\begin{bf}
\noindent
VI. CONCLUSIONS
\end{bf}
\vglue .3cm
We have presented all the necessary relations to transform the
$L$ functions [as defined in Eqs.~\eqr{Lfunction} and \eqr{Lpfunction}] that occur in
our ${\cal O}(\varepsilon^2)$ results \cite{KMR} for the Laurent series expansion
of massive scalar one-loop integrals to multiple polylogarithms.
We have used these relations to transform our results on massive one-loop
integrals involving $L$ functions to corresponding results involving
multiple polylogarithms.
The multiple polylogarithms results are readily available in electronic form
\cite{epaps}.
Despite of the fact that the relations between the $L$ functions and the
multiple polylogarithms have been derived having the massive scalar one-loop
integrals in mind they can also be used in a more general setting. In fact, any
definite integral given by
\begin{eqnarray}
\int\limits_A^B \frac{ \ln (a_{1}+b_{1} x ) \ln (a_{2}+b_{2} x) \ln
(a_{3}+b_{3}x)
dx } { a_{4}+b_{4}x }~ {\rm or} ~
\int\limits_A^B \frac{ \ln (a_{1}+b_{1} x ) {\rm Li}_{2}(a_{2}+b_{2} x
)dx } {
a_{3}+b_{3}x }\nonumber
\end{eqnarray}
can be written in terms of multiple polylogarithms with the help of the relations
presented in this paper.
It is worthwhile to mention that all the equations presented in the present
paper have been also checked numerically.
We have found several examples where the representation of the $L$ functions in
terms of multiple polylogarithms is not unique. This reflects the fact that
multiple polylogarithms obey
quasishuffle and shuffle Hopf algebras and hence satisfy numerous identities
as is the case for the classical polylogarithms.
More information about identities between multiple polylogarithms can be found,
e.g., in \cite{nested} and \cite{Vollinga}, and references therein.
For future parton model applications of our results numerical efficiency is
an important issue. We are presently writing numerical
$C$\raisebox{0.05cm}{\scriptsize++}
codes to
compare the numerical efficiency of the two representations in terms of
$L$ functions and multiple polylogarithms.
\vglue 1.0cm
\begin{bf}
\noindent
ACKNOWLEDGMENTS
\end{bf}
\vglue .3cm
One of the authors (Z.~M.) would like to thank the Particle Theory Group of the
Institut f{\"u}r Physik, Universit{\"a}t Mainz, for hospitality.
The work of one of the authors (Z.~M.) was supported by a DFG (Germany) grant under contract No.
436 GEO 17/6/05. One of the authors (M.~R.) was
supported by the DFG through the Graduiertenkolleg ``Eichtheorien''
at the University of Mainz and by the Helmholtz Gemeinschaft
under contract No. VH-NG-105.
\setcounter{equation}{0}
\renewcommand{\theequation}{A\arabic{equation}}
\vglue 1.0cm
\begin{bf}
\noindent
APPENDIX
\end{bf}
\vglue .3cm%
In this Appendix we consider as an example the real part of the
${\cal O}(\varepsilon^2)$ coefficient ${\rm Re}\,D_1^{(2)}$ of the Laurent series
expansion of the massive box $D_1$ with three massive
propagators. Using the rules written down in the main text of this paper we have
expressed the corresponding results of \cite{KMR} involving $L$ functions
in terms of multiple polylogarithms. The $L$ function structure of
${\rm Re}\,D_1^{(2)}$ in \cite{KMR} is sufficiently rich to provide an
illustration
of the corresponding complexity in terms of multiple polylogarithms when
transforming to the latter representation. We mention that all multiple
polylogarithms up to weight three
have been reexpressed in terms of classical polylogarithms. We then used
automatic program codes to simplify the classical polylogarithms as much
as possible, as was also done in \cite{KMR}.
We use the notation and the conventions of \cite{KMR}. In brief,
we use the Mandelstam-type variables
\begin{equation}
\label{s-t-u}
s\equiv (p_1+p_2)^2, \quad t\equiv T-m^2
\equiv (p_1-p_3)^2-m^2,
\quad u\equiv U-m^2\equiv (p_2-p_3)^2-m^2
\end{equation}
for the $2\to 2$ partonic process
${a}(p_1)+{b}(p_2)\to {Q}(p_3)+{\overline Q}(p_4)$ with
$p_1^2=p_2^2=0$ and $p_3^2=p_4^2=m^2$. We also introduce the abbreviations
$\Big(\beta=\sqrt{1-4m^2/s}\Big)$
\begin{eqnarray}
\label{notations}
\nonumber &&
z_3 \equiv (s + 2 t + s\beta)/2, {\rm \hspace{.4in}}
z_4 \equiv (s + 2 t - s\beta)/2, \\
&& z_5 \equiv (2 m^2 + t + t \beta)/2, {\rm \hspace{.4in}}
z_6 \equiv (2 m^2 + t - t \beta)/2, \\
\nonumber &&
l_s \equiv \ln \frac{s}{m^2}, {\rm \hspace{.2in}}
l_t \equiv \ln \frac{-t}{m^2}, {\rm \hspace{.2in}}
l_T \equiv \ln \frac{-T}{m^2}, {\rm \hspace{.2in}}
l_x \equiv \ln x, {\rm \hspace{.2in}} \\
\nonumber &&
l_{\beta} \equiv \ln \beta, {\rm \hspace{.2in}}
l_{z3} \equiv \ln \frac{z_3}{m^2}, {\rm \hspace{.2in}}
l_{z4} \equiv \ln \frac{-z_4}{m^2} \, .
\end{eqnarray}
One finds
\begin{eqnarray}
{\rm Re}\,D_1^{(2)}&=& \frac{1}{st\beta} \left[ \frac{1}{192} \left\{ - 109 l_s^4
+ 240 l_t^4 + 32 l_T^3 l_x + 264 l_T^2 l_x^2 - 200 l_T l_x^3 -
177 l_x^4 -
\right.\right.
\\ \nonumber &&
96 l_T^2 l_x l_{z3} + 192 l_T l_x^2 l_{z3} + 12 l_x^3 l_{z3} +
96 l_T l_x l_{z3}^2 - 32 l_x l_{z3}^3 - 480 l_T^2 l_x l_{z4} +
\\ \nonumber &&
24 l_T l_x^2 l_{z4} + 180 l_x^3 l_{z4} -
144 l_x^2 l_{z3} l_{z4} + 480 l_T l_x l_{z4}^2 -
336 l_x^2 l_{z4}^2 - 96 l_x l_{z3} l_{z4}^2 +
\\ \nonumber &&
320 l_x l_{z4}^3 - 168 l_{z4}^4 + 480 l_T^2 l_x l_{\beta} -
480 l_T l_x^2 l_{\beta} -
40 l_x^3 l_{\beta} - 192 l_T l_x l_{z3} l_{\beta} +
\\ \nonumber &&
336 l_x^2 l_{z3} l_{\beta} + 96 l_x l_{z3}^2 l_{\beta} -
384 l_T l_x l_{z4} l_{\beta} +
336 l_x^2 l_{z4} l_{\beta} + 192 l_x l_{z3} l_{z4} l_{\beta} +
\\ \nonumber &&
96 l_x l_{z4}^2 l_{\beta} + 192 l_{z4}^3 l_{\beta} +
96 l_T l_x l_{\beta}^2 + 24 l_x^2 l_{\beta}^2 -
96 l_x l_{z3} l_{\beta}^2 - 96 l_{z4}^2 l_{\beta}^2 +
32 l_x l_{\beta}^3 +
\\ \nonumber &&
32 l_{\beta}^4 - 32 l_t^3
( 9 l_T + 20 l_x + 25 l_{z3} + l_{z4} + 13 l_{\beta} ) -
4 l_s^3 ( 8 l_T - 36 l_x - 9 l_{z3} -
\\ \nonumber &&
43 l_{z4} + 94 l_{\beta} ) -
6 l_s^2 \left( 52 l_t^2 - 28 l_T^2 + 15 l_x^2 + 26 l_x l_{z3} +
46 l_x l_{z4} + 24 l_{z3} l_{z4} + \right.
\\ \nonumber &&
32 l_{z4}^2 - 4 l_T
( 9 l_x + 8 l_{z3} - 15 l_{z4} - 4 l_{\beta} )
- 60 l_x l_{\beta} - 24 l_{z3} l_{\beta} -
56 l_{z4} l_{\beta} + 76 l_{\beta}^2 +
\\ \nonumber && \left.
l_t ( -44 l_T + 60 l_x - 8 l_{z3} - 60 l_{z4} +
8 l_{\beta}^{} ) \right) - 24 l_t^2
\left( 4 l_T^2 + 15 l_x^2 - 8 l_T ( 2 l_x + \right.
\\ \nonumber &&
2 l_{z3} + 4 l_{z4} - 3 l_{\beta} )
+ 4 l_x ( 3 l_{z3} - 8 l_{z4} + 13 l_{\beta} ) +
2 ( -4 l_{z3}^2 + l_{z4}^2 + 12 l_{z3} l_{z4} -
\\ \nonumber && \left.
12 l_{z3} l_{\beta} - 8 l_{z4} l_{\beta} + 10 l_{\beta}^2 ) \right) +
8 l_t \left( 8 l_T^3 - 31 l_x^3 + l_x^2 ( -6 l_{z3} + 33 l_{z4}
+ 6 l_{\beta} ) + \right.
\\ \nonumber &&
6 l_x ( 3 l_{z3}^2 -
22 l_{z4}^2 - 2 l_{z3} ( 5 l_{z4} -
9 l_{\beta} ) + 20 l_{z4} l_{\beta} - 10 l_{\beta}^2 ) +
4 ( -2 l_{z3}^3 + 6 l_{z3}^2 l_{z4} +
\\ \nonumber &&
16 l_{z4}^3 + 9 l_{z3}
( l_{z4} - l_{\beta} )^2 -
18 l_{z4}^2 l_{\beta} + 9 l_{z4} l_{\beta}^2 + 2 l_{\beta}^3) +
6 l_T^2 ( 3 l_x - 4 l_{z3} + 4 l_{z4} -
\\ \nonumber && \left.
4 l_{\beta} ) - 3 l_T (5 l_x^2 - 4 l_x ( 5 l_{z3} - 4 l_{z4} ) -
8 ( l_{z3}^2 - 2 l_{z3} l_{z4} -
3 l_{z4}^2 + 4 l_{z4} l_{\beta} + l_{\beta}^2 )) \right) -
\\ \nonumber &&
4 l_s \left( 80 l_t^3 + 8 l_T^3 + 18 l_x^3 -
27 l_x^2 l_{z3} - 8 l_{z3}^3 -
165 l_x^2 l_{z4} - 72 l_x l_{z3} l_{z4} + 48 l_x
l_{z4}^2 - \right.
\\ \nonumber &&
24 l_{z3} l_{z4}^2 -
64 l_{z4}^3 + 18 l_x^2 l_{\beta} +
120 l_x l_{z3} l_{\beta} + 24 l_{z3}^2 l_{\beta} +
72 l_x l_{z4} l_{\beta} + 48 l_{z3} l_{z4} l_{\beta} +
\\ \nonumber &&
72 l_{z4}^2
l_{\beta} - 84 l_x l_{\beta}^2 -
24 l_{z3} l_{\beta}^2 - 48 l_{z4}
l_{\beta}^2 + 40 l_{\beta}^3 -
12 l_T^2 (13 l_x + 2 l_{z3} -
6 l_{z4} +
\\ \nonumber &&
6 l_{\beta} ) + 12 l_t^2 ( 8 l_T + 5 l_x -
26 l_{z3} - 10 l_{z4} + 12 l_{\beta} )
- 12 l_t
( 5 l_T^2 + 6 l_x^2 - 7 l_{z3}^2 -
\\ \nonumber &&
10 l_{z3} l_{z4} - 16 l_{z4}^2 +
l_x ( 14 l_{z3} + 27 l_{z4} - 4 l_{\beta} )
+ 10 l_{z3} l_{\beta} + 16 l_{z4} l_{\beta} +
2 l_{\beta}^2 +
\\ \nonumber &&
l_T ( -5 l_x - 2 l_{z3} + 4 l_{z4} + 8 l_{\beta} ))
+ 12 l_T
( 11 l_x^2 + l_x ( 8 l_{z3} +
7 l_{z4} - 4 l_{\beta} ) +
2 l_{z3}^2 -
\\ \nonumber && \left.\left.
6 l_{z4}^2 - 4 l_{z3} l_{\beta} +
8 l_{z4} l_{\beta} + 2 l_{\beta}^2 )\right)\right\} +
\left( 3 l_s^2/4 - 7 l_t^2/2 + l_T l_x + 3 l_x^2/4 + 5 l_x l_{z3} + \right.
\\ \nonumber &&
5 l_x l_{z4} + l_{z4}^2/2 +
l_t ( 2 l_T - l_x + 10 l_{z3} +
2 l_{z4} - 15 l_{\beta}) -
8 l_x l_{\beta} + 2 l_{z4} l_{\beta} - 11 l_{\beta}^2 -
\\ \nonumber && \left.
l_s ( 7 l_t + l_T +
8 l_x + 5 l_{z3} + l_{z4} +
6 l_{\beta}^{} )\right) \zeta(2) +
( -3 l_s + 2 l_x) \zeta(3) - 35 \zeta(4)/4 -
\\ \nonumber &&
2 {\rm Li}_2^2\left(\frac{m^2}{z_5}\right) +
2 {\rm Li}_2^2\left(\frac{-t (1 - \beta)}{2 m^2}\right) +
\frac{1}{8} {\rm Li}_2\left(\frac{m^2}{z_5}\right)
\left( -11 l_s^2 - 4 l_t^2 + 25 l_x^2 + \right.
\\ \nonumber &&
8 l_x l_{z4} - 24 l_{z4}^2 + l_s ( 4 l_t +
26 l_x + 24 l_{z4} - 8 l_{\beta} ) +
24 l_x l_{\beta} + 16 l_{z4} l_{\beta} - 8 l_{\beta}^2 -
\\ \nonumber && \left.
4 l_t ( 9 l_x - 4 l_{z4} +
4 l_{\beta}^{})\right) + {\rm Li}_2\left(\frac{m^2 x}{-T}\right)
\left( 11 l_s^2/8 - l_t^2 +
2 l_t ( l_x + l_{z4} - 2 l_{\beta}) + \right.
\\ \nonumber && \left.
l_s ( -l_t +
7 l_x/4 - l_{z4} + 2 l_{\beta}) +
l_x ( 19 l_x - 24 l_{z4} +
16 l_{\beta}^{} )/8 \right) + \frac{1}{8} {\rm Li}_2(-x) \times
\\ \nonumber &&
\left( -15 l_s^2 + 4 l_t^2 + l_x ( -39 l_x + 32 l_{z4} -
16 l_{\beta} ) + 2 l_s ( 10 l_t + 5 l_x - 8 l_{\beta} ) + \right.
\\ \nonumber && \left.
l_t ( -44 l_x + 32 l_{\beta}^{} )\right) +
{\rm Li}_2\left(\frac{z_3}{z_4}\right)
\left( -2 {\rm Li}_2\left(\frac{m^2}{z_5}\right)
+ \frac{1}{4} \left( -5 l_s^2 + 20 l_t^2 - 7 l_x^2 -
\right.\right.
\\ \nonumber &&
4 l_x l_{z3} + 12 l_x l_{z4} - 4 l_{z4}^2 +
l_s ( -2 l_t + 4 ( 2 l_x + l_{z3} + l_{z4} - l_{\beta} )) -
4 l_x l_{\beta} +
\\ \nonumber && \left.
l_t ( -22 l_x - 8 l_{z3} + 8 l_{\beta}^{} ) -
8 \zeta(2) \right) \bigg) +
{\rm Li}_2(x)\left( 2 {\rm Li}_2\left(\frac{m^2}{z_5}\right) +
\right.
\\ \nonumber &&
2 {\rm Li}_2\left(\frac{-t (1 - \beta)}{2 m^2}\right) -
7 l_s^2/4 - 3 l_t^2/2 -
11 l_x^2/4 + l_x l_{z3} +
3 l_x l_{z4} + l_{z4}^2 -
\\ \nonumber &&
2 l_x l_{\beta} + l_t
( -4 l_x + 2 l_{z3} + 4 l_{\beta} ) +
l_s ( 4 l_t + l_x/2 - l_{z3} - l_{z4} - 2 l_{\beta} )
- 6 \zeta(2) \bigg) +
\\ \nonumber &&
{\rm Li}_2\left(\frac{T}{m^2}\right)
\left( -4 {\rm Li}_2(x) - 4 {\rm Li}_2\left(\frac{z_3}{z_4}\right) -
4 {\rm Li}_2\left(\frac{m^2}{z_5}\right) +
4 {\rm Li}_2\left(\frac{-t (1 - \beta)}{2 m^2}\right) + \right.
\\ \nonumber &&
l_s^2/8 - 3 l_t^2/2 + l_x^2/8 +
2 l_x l_{z3} - 4 l_{z4}^2 + l_t ( 3 l_x/2 +
4 l_{z3} + 4 l_{z4} - 4 l_{\beta} ) -
\\ \nonumber && \left.
2 l_x l_{\beta} + 4 l_{z4} l_{\beta} - 2 l_{\beta}^2 -
\frac{l_s}{4} ( 18 l_t + 7 l_x +
8 l_{z3} - 16 l_{z4} + 8 l_{\beta} ) -
4 \zeta(2) \right) +
\\ \nonumber &&
{\rm Li}_2\left(\frac{T}{z_3}\right) \left(-\frac{l_s^2}{8} - 2 l_t^2
+ l_s (\frac{9}{4} l_x - l_{z4}) - 2 l_t (l_x - l_{z4}) -
l_x (\frac{17}{8} l_x - l_{z4}) \right) +
\\ \nonumber &&
{\rm Li}_2\left(\frac{-t (1 - \beta)}{2 m^2}\right)
\left( -2 {\rm Li}_2\left(\frac{z_3}{z_4}\right) - 7 l_t^2 - l_x^2/2 +
2 l_x l_{z3} - l_x l_{z4} - l_{z4}^2 + \right.
\\ \nonumber &&
l_s ( l_t +
3 l_x/2 - 2 l_{z3} + l_{z4} -
l_{\beta} ) + 3 l_x l_{\beta} +
2 l_{z4} l_{\beta} - l_{\beta}^2 -
l_t ( l_x - 4 l_{z3} -
2 l_{z4} +
\\ \nonumber &&
2 l_{\beta}) + 12 \zeta(2) \bigg) +
{\rm Li}_3\left(\frac{-1 + \beta}{2\beta}\right) (4 l_s - 7 l_t) +
5 {\rm Li}_3\left(\frac{z_5}{t\beta}\right) l_t +
{\rm Li}_3\left(\frac{m^2}{z_5}\right) \times
\\ \nonumber &&
(5 l_s - 6 l_t - 11 l_x)/2 -
{\rm Li}_3\left(\frac{z_3}{t}\right) (4 l_t + 6 l_x) +
{\rm Li}_3\left(\frac{z_6}{m^2}\right) (3 l_s/2 - 5 l_t -
\\ \nonumber &&
7 l_x/2 ) + 4 {\rm Li}_3\left(\frac{z_4}{t}\right) (l_s -
l_t - 2 l_x) + {\rm Li}_3\left(-\frac{m^2 x z_3}{sT\beta}\right)
(l_s - 2 l_t - l_x) +
\\ \nonumber &&
{\rm Li}_3\left(\frac{-x^2}{1 - x^2}\right) (l_s - l_t -
l_x) + {\rm Li}_3\left(\frac{z_3}{s\beta}\right) (l_s +
5 l_t - l_x) + 2 {\rm Li}_3\left(\frac{m^2}{-t}\right) l_x +
\\ \nonumber &&
{\rm Li}_3\left(\frac{z_5}{T}\right)
\left( -\frac{5}{2} l_s - 3 l_t - \frac{5}{2} l_x + 4 l_{z4} - 4
l_{\beta}\right) +
{\rm Li}_3\left(\frac{-t (1 - \beta)}{2 z_5}\right) \times
\\ \nonumber &&
\left( -\frac{5}{2} l_s - l_t - \frac{3}{2} l_x + 4 l_{z4} - 4
l_{\beta}\right) +
{\rm Li}_3\left(\frac{-2 z_6}{t(1 + \beta)}\right)
\left( -\frac{3}{2} l_s - 2 l_t + 2 l_{z4} - \right.
\\ \nonumber && \left.
\frac{l_x}{2} - 2 l_{\beta}\right) +
{\rm Li}_3\left(\frac{z_3}{z_4}\right) \left( \frac{l_s}{2} - l_t -
\frac{l_x}{2} + 2 l_{z4} - 2 l_{\beta}\right) +
2 {\rm Li}_3(-x) (3 l_s +
\\ \nonumber &&
2 l_t + 2 l_x - 4 l_{z4} +
4 l_{\beta}) + {\rm Li}_3\left(\frac{z_6}{z_5}\right)
\left( \frac{l_s}{2} + 4 l_t - \frac{l_x}{2} - 2 l_{z4} + 2 l_{\beta}
\right) +
\\ \nonumber &&
{\rm Li}_3\left(\frac{2 z_6}{m^2 (1 + \beta)}\right)
(\frac{3}{2} l_s + 2 l_t + \frac{l_x}{2} - 2 l_{z4} + 2 l_{\beta}) +
{\rm Li}_3\left(\frac{z_4}{T}\right) (3 l_s + 3 l_t +
\\ \nonumber &&
3 l_x - 4 l_{z4} + 4 l_{\beta}) +
{\rm Li}_3\left(\frac{T}{z_3}\right) \left( \frac{l_s}{2} +
l_t + \frac{7}{2} l_x - 2 l_{z4} + 2 l_{\beta} \right) +
\\ \nonumber &&
{\rm Li}_3\left(\frac{m^2 (1 - \beta)}{2 z_5}\right)
\left( \frac{3}{2} l_s + 3 l_t + \frac{5}{2} l_x - 4 l_{z4} +
4 l_{\beta} \right) +
\\ \nonumber &&
{\rm Li}_3(x) \left( \frac{15}{2} l_s - 2 l_t +
\frac{5}{2} l_x - 6 l_{z4} + 6 l_{\beta} \right) +
{\rm Li}_3\left(\frac{T}{z_6}\right) ( - 2 l_s +
\\ \nonumber &&
l_t - 2 l_x + 2 l_{z4} - 2 l_{\beta}) +
2 {\rm Li}_4(x) - 4 {\rm Li}_4\left(\frac{z_3}{t}\right) +
4 {\rm Li}_4\left(\frac{z_4}{t}\right) -
{\rm Li}_4\left(\frac{z_4}{T}\right) -
\\ \nonumber &&
{\rm Li}_4\left(\frac{T z_4}{D}\right) +
{\rm Li}_4\left(\frac{z_5}{T}\right) +
2 {\rm Li}_4\left(\frac{s(1 - \beta)}{-2t}\right) +
3 {\rm Li}_4\left(\frac{s (1 - \beta)}{2 z_4}\right) +
\\ \nonumber &&
{\rm Li}_4\left(\frac{T}{z_6}\right) +
4 {\rm Li}_4\left(\frac{-1 + \beta}{2\beta}\right) +
2 {\rm Li}_4\left( \frac{-2t}{s(1 + \beta)}\right) +
4 {\rm Li}_4\left(\frac{2\beta}{1 + \beta}\right) +
\\ \nonumber &&
3 {\rm Li}_4\left(\frac{2 z_3}{s(1 + \beta)}\right) +
2 Li_{3, 1} \left(-\frac{m^2 x z_3}{s T \beta}, \frac{-T}{m^2 x}\right) -
\\ \nonumber &&
2 Li_{3, 1} \left(-\frac{m^2 x z_3}{s T \beta},
\frac{2T}{t(1-\beta)}\right) -
6 Li_{1, 2, 1} \left(1, \frac{s (1 - \beta)}{2 z_4},
\frac{z_5}{m^2}\right) +
\\ \nonumber &&
6 Li_{1, 2, 1} \left(1, \frac{s(1 + \beta)}{2 z_3},
\frac{z_6}{m^2}\right) -
2 Li_{2, 1, 1} \left(1, \frac{z_4}{z_3}, \frac{z_3}{t}\right) -
\\ \nonumber &&
2 Li_{2, 1, 1} \left(\frac{m^2}{T}, \frac{T}{z_5},
\frac{z_5}{m^2}\right) +
2 Li_{2, 1, 1} \left(\frac{m^2}{T}, \frac{T}{z_6},
\frac{z_6}{m^2}\right) -
\\ \nonumber &&
2 Li_{2, 1, 1} \left(\frac{z_3}{z_4}, \frac{z_4}{z_3},
\frac{z_3}{t}\right) -
2 Li_{2, 1, 1} \left(\frac{m^2}{z_5}, 1, \frac{z_5}{m^2}\right) -
2 Li_{2, 1, 1} \left(\frac{m^2}{z_5}, \frac{z_5}{T},
\frac{T}{m^2}\right) +
\\ \nonumber &&
2 Li_{2, 1, 1} \left(\frac{s(1 - \beta)}{2 z_4}, \frac{z_5}{z_6},
\frac{z_6}{m^2}\right) +
2 Li_{2, 1, 1} \left(-\frac{m^2 x z_3}{sT\beta}, -\frac{sT\beta}{m^2xz_3},
\frac{z_3}{s\beta}\right) -
\\ \nonumber &&
2 Li_{2, 1, 1} \left(-\frac{m^2xz_3}{sT\beta},
-\frac{sT\beta}{m^2xz_3}, -\frac{z_6}{t\beta}\right) +
2 Li_{2, 1, 1} \left(\frac{s(1 + \beta)}{2z_3}, 1,
\frac{z_6}{m^2}\right) +
\\ \nonumber &&
2 Li_{2, 1, 1} \left(\frac{s(1 + \beta)}{2z_3}, \frac{z_6}{T},
\frac{T}{m^2}\right) -
2 Li_{2, 1, 1} \left(\frac{s(1 + \beta)}{2z_3}, \frac{z_6}{z_5},
\frac{z_5}{m^2}\right) +
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(1, \frac{T}{z_6}, \frac{z_6}{z_5},
\frac{z_5}{m^2}\right) -
Li_{1, 1, 1, 1} \left(1, \frac{s(1 - \beta)}{2z_4}, \frac{z_5}{T},
\frac{T}{m^2}\right) +
\\ \nonumber &&
3 Li_{1, 1, 1, 1} \left(1, \frac{s(1 - \beta)}{2z_4}, \frac{z_5}{z_6},
\frac{z_6}{m^2}\right) +
Li_{1, 1, 1, 1} \left(1, \frac{s(1 + \beta)}{2z_3}, \frac{z_6}{T},
\frac{T}{m^2}\right) -
\\ \nonumber &&
3 Li_{1, 1, 1, 1} \left(1, \frac{s(1 + \beta)}{2z_3}, \frac{z_6}{z_5},
\frac{z_5}{m^2}\right) -
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, 1, \frac{T}{z_3},
\frac{z_3}{t}\right) +
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, 1, \frac{T}{z_4},
\frac{z_4}{t}\right) +
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, \frac{T}{z_3}, 1,
\frac{z_3}{t}\right) -
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, \frac{T}{z_3}, \frac{z_3}{T},
\frac{T}{t}\right) -
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, \frac{T}{z_3}, \frac{z_3}{z_4},
\frac{z_4}{t}\right) -
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, \frac{T}{z_4}, 1,
\frac{z_4}{t}\right) +
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, \frac{T}{z_4}, \frac{z_4}{T},
\frac{T}{t}\right) +
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{T}, \frac{T}{z_4}, \frac{z_4}{z_3},
\frac{z_3}{t}\right) +
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_3}, 1, \frac{z_3}{T},
\frac{T}{t}\right) -
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_3}, \frac{z_3}{T}, 1,
\frac{T}{t}\right) +
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_3}, \frac{z_3}{T}, \frac{T}{z_3},
\frac{z_3}{t}\right) -
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_3}, \frac{z_3}{T}, \frac{T}{z_4},
\frac{z_4}{t}\right) -
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_3}, \frac{z_3}{z_4}, \frac{z_4}{T},
\frac{T}{t}\right) -
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_4}, 1, \frac{z_4}{T},
\frac{T}{t}\right) +
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_4}, \frac{z_4}{T}, 1,
\frac{T}{t}\right) +
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_4}, \frac{z_4}{T}, \frac{T}{z_3},
\frac{z_3}{t}\right) -
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_4}, \frac{z_4}{T}, \frac{T}{z_4},
\frac{z_4}{t}\right) +
\\ \nonumber &&
2 Li_{1, 1, 1, 1} \left(\frac{t}{z_4}, \frac{z_4}{z_3}, \frac{z_3}{T},
\frac{T}{t}\right) -
2 Li_{1, 1, 1, 1} \left(\frac{T}{z_4}, \frac{z_4}{z_3}, 1,
\frac{z_3}{t}\right) +
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(\frac{z_4}{T}, \frac{T}{z_3}, 1,
\frac{z_3}{t}\right) -
Li_{1, 1, 1, 1} \left(\frac{m^2}{z_5}, 1, \frac{z_5}{T},
\frac{T}{m^2}\right) +
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(\frac{m^2}{z_5}, \frac{z_5}{T}, 1,
\frac{T}{m^2}\right) -
Li_{1, 1, 1, 1} \left(\frac{m^2}{z_5}, \frac{z_5}{T}, \frac{T}{z_5},
\frac{z_5}{m^2}\right) -
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(\frac{m^2}{z_6}, \frac{z_6}{T}, 1,
\frac{T}{m^2}\right) -
Li_{1, 1, 1, 1} \left(\frac{m^2}{z_6}, \frac{z_6}{T}, \frac{T}{z_5},
\frac{z_5}{m^2}\right) -
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(\frac{m^2}{z_6}, \frac{z_6}{z_5}, 1,
\frac{z_5}{m^2}\right) -
Li_{1, 1, 1, 1} \left(\frac{m^2}{z_6}, \frac{z_6}{z_5}, \frac{z_5}{T},
\frac{T}{m^2}\right) -
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(\frac{z_6}{T}, \frac{T}{z_5}, 1,
\frac{z_5}{m^2}\right) +
Li_{1, 1, 1, 1} \left(\frac{z_6}{z_5}, \frac{z_5}{T}, 1,
\frac{T}{m^2}\right) -
\\ \nonumber &&
3 Li_{1, 1, 1, 1} \left(\frac{s(1 - \beta)}{2z_4}, 1, \frac{z_5}{z_6},
\frac{z_6}{m^2}\right) +
Li_{1, 1, 1, 1} \left(\frac{s(1 - \beta)}{2z_4}, \frac{z_5}{T},
\frac{T}{z_6}, \frac{z_6}{m^2}\right) -
\\ \nonumber &&
3 Li_{1, 1, 1, 1} \left(\frac{s(1 - \beta)}{2z_4}, \frac{z_5}{z_6}, 1,
\frac{z_6}{m^2}\right) +
Li_{1, 1, 1, 1} \left(\frac{s(1 - \beta)}{2z_4}, \frac{z_5}{z_6},
\frac{z_6}{T}, \frac{T}{m^2}\right) -
\\ \nonumber &&
3 Li_{1, 1, 1, 1} \left(\frac{s(1 - \beta)}{2z_4}, \frac{z_5}{z_6},
\frac{z_6}{z_5}, \frac{z_5}{m^2}\right) +
Li_{1, 1, 1, 1} \left(\frac{s(1 + \beta)}{2z_3}, 1, \frac{z_6}{T},
\frac{T}{m^2}\right) -
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(\frac{s(1 + \beta)}{2z_3}, 1, \frac{z_6}{z_5},
\frac{z_5}{m^2}\right) +
Li_{1, 1, 1, 1} \left(\frac{s(1 + \beta)}{2z_3}, \frac{z_6}{T},
\frac{T}{z_6}, \frac{z_6}{m^2}\right) -
\\ \nonumber &&
Li_{1, 1, 1, 1} \left(\frac{s(1 + \beta)}{2z_3}, \frac{z_6}{z_5},
\frac{z_5}{z_6}, \frac{z_6}{m^2}\right) -
i \pi (l_s - 2 l_t - l_x)
\left( \frac{l_s^2}{2} + l_s l_T + l_s l_x + \right.
\\ \nonumber &&
\frac{l_x^2}{2} - l_T l_{z3} + \frac{l_{z3}^2}{2} -
l_s l_{z4} + \frac{l_{z4}^2}{2} + l_s l_{\beta} + l_T
l_{\beta} + l_x l_{\beta} - l_{z4} l_{\beta} + \frac{l_{\beta}^2}{2} +
\\ \nonumber && \left.\left.
2 {\rm Li}_2(-x) + 2 {\rm Li}_2(x) -
{\rm Li}_2\left(\frac{m^2 x}{-T}\right)
+ {\rm Li}_2\left(\frac{T}{z_3}\right) +
{\rm Li}_2\left(\frac{z_3}{z_4}\right) + \zeta(2) \right)
\right].
\end{eqnarray}
At the very end of the expression one finds an explicit imaginary part. Since
the whole expression must be real this clearly indicates that the same imaginary
contribution with opposite sign must be contained in multiple polylogarithms,
e.g., some of them are sitting on branch cuts. This is in fact true for the
multiple polylogarithms
\begin{eqnarray} \nonumber &&
Li_{3, 1} \left(-\frac{m^2 x z_3}{s T \beta}, \frac{-T}{m^2 x}\right),
\qquad
Li_{3, 1} \left(-\frac{m^2 x z_3}{s T \beta}, \frac{2T}{t(1-\beta)}\right),
\\ \nonumber &&
Li_{2, 1, 1} \left(-\frac{m^2 x z_3}{sT\beta}, -\frac{sT\beta}{m^2xz_3},
\frac{z_3}{s\beta}\right), \qquad
Li_{2, 1, 1} \left(-\frac{m^2xz_3}{sT\beta}, -\frac{sT\beta}{m^2xz_3},
-\frac{z_6}{t\beta}\right).
\end{eqnarray}
Indeed, one finds that the imaginary contributions cancel out when one
numerically evaluates the result.
As regards the length the representations of ${\rm Re}\,D_1^{(2)}$ in terms of
$L$ functions in \cite{KMR} and in terms of multiple polylogarithms are of
similar size. The representation in terms of $L$ functions contains 43 different
$L$ function expressions against 59 different multiple polylogarithm
expressions.
|
1,116,691,498,465 | arxiv | \section*{Introduction}
The precise equivalence between the $4$-spinor and $2$-spinor settings
for electrodynamics was exposed by Jadczyk and myself
in~\cite{CJ97a,CJ97b,C98,C00b}.
In summary one sees that, from an algebraic point of view,
the only notion of a complex $2$-dimensional vector space $\S$ yields,
naturally and without any further assumptions,
all the needed algebraic structures through functorial constructions;
conversely in a $4$-spinor setting, provided one makes the minimum assumptions
which are needed in order to formulate the standard physical theory,
the $4$-spinor space naturally splits (Weyl decomposition)
into the direct sum of two 2-dimensional subspaces
which are anti-dual to each other.
In a sense, which setting one regards as fundamental is then
mainly a matter of taste.
The $4$-spinor setting is closer to standard notations,
and some formulas can be written in a more compact way,
while the relations among the various objects are somewhat more involved.
The $2$-spinor setting
turns out to give a much more direct formulation,
in which all the basic objects and the relations among them
naturally set into their places;
just from $\S$ one authomatically gets
\emph{exactly} the needed algebraic structure, nothing more, nothing less:
4-spinor space ${\boldsymbol{W}}$ with the `Dirac adjoint' anti-isomorphism,
Minkowski space $\H$ and Dirac map $\gamma:\H\to\operatorname{End}({\boldsymbol{W}})$
with the required properties.
Further objects which are commonly considered
depend on the choice of a gauge of some sort,
whose nature is precisely described.
When we consider a vector bundle $\S\to{\boldsymbol{M}}$,
where now the fibres are complex 2-dimensional
and ${\boldsymbol{M}}$ is a real 4-dimensional manifold,
then we don't have to assign any further background structure
in order to formulate a full Einstein-Cartan-Maxwell-Dirac theory.
In fact we naturally get a vector bundle $\H\to{\boldsymbol{M}}$
whose fibres are Minkowski spaces, a 4-spinor bundle ${\boldsymbol{W}}\to{\boldsymbol{M}}$ and so on.
Any object which is not determined by geometric construction
from the unique geometric datum $\S\to{\boldsymbol{M}}$ is a \emph{field} of the theory,
namely we consider: the tetrad $\Theta:\mathrm{T}{\boldsymbol{M}}\to{\mathbb{L}}{\,\otimes\,}\H$,
the 2-spinor connection $\Cs$, the electromagnetic and Dirac fields.
(Even coupling factors naturally arise as covariantly constant sections
of the real line bundle ${\mathbb{L}}$ of \emph{length units},
which is geometrically constructed from $\S$.)
The gravitational field is described by the tetrad
(which can be seen as a `square root' of spacetime metric)
and by the connection induced by $\Cs$ on $\H$,
while the remaining part of the spinor connection
can be viewed as the electromagnetic potential.
A natural Lagrangian density for all these fields is then introduced;
the relations between metric and connection
and between e.m.\ potential and e.m.\ field follow from the
(Euler-Lagrange) field equations.
All considered, this setting has some original aspects
but is not in contrast to the (by now classical) Penrose formalism~\cite{PR84}.
In~\Sec\ref{S:Dirac algebra in two-spinor terms}
and~\Sec\ref{S:Clifford group and its subgroups}
I'll show how the above said algebraic setting,
and in particular the natural splitting of the 4-spinor space
into the direct sum of its Weyl subspaces,
enables us to examine the structures of the Dirac algebra, the Clifford group
and its subgroups from a different perspective.
In~\Sec\ref{S:Spinors and particle momenta}
I'll show the strict relation existing between the two-spinor setting
and the geometry of particle momenta,
in particular the bundle structure of ${\boldsymbol{W}}$ over the space of momenta.
These results
are a preparation to a 2-spinor formulation
of quantum electrodynamics along le lines of a previous paper~\cite{C05},
in which the classical structure underlying electron states
is a 2-fibred bundle over spacetime.
\section{Two-spinor geometry}
\label{S:Two-spinor geometry}
In this section we'll see how all the fundamental geometric structures
needed for Dirac theory naturally arise through functorial constructions
from a two-dimensional complex vector space,
with no further assumptions.
\subsection{Complex conjugated spaces}\label{s:Complex conjugated spaces}
If ${\boldsymbol{A}}$ is a set and $f:{\boldsymbol{A}}\to{\mathbb{C}}$ is any map,
then $\bar f:{\boldsymbol{A}}\to{\mathbb{C}}:a\mapsto\overline{f(a)}$ is the conjugated map.
Let ${\boldsymbol{V}}$ be a complex vector space of finite-dimension $n$\,;
its \emph{dual} space $\V{}^\lin$ and \emph{antidual} space ${\boldsymbol{V}}^{\overline{\scriptscriptstyle\bigstar}}$
are defined to be the $n$-dimensional complex vector spaces
of all maps ${\boldsymbol{V}}\to{\mathbb{C}}$ which are respectively linear and antilinear.
One then has the distinguished anti-isomorphism $\V{}^\lin\to{\boldsymbol{V}}^{\overline{\scriptscriptstyle\bigstar}}:\l\mapsto{\bar\lambda}$\,.
Set now $\cj{\V}:={\boldsymbol{V}}^{{\scriptscriptstyle\bigstar}{\overline{\scriptscriptstyle\bigstar}}}$, and call this the \emph{conjugate space} of ${\boldsymbol{V}}$.
One has the natural isomorphisms
$${\boldsymbol{V}}\cong{\boldsymbol{V}}^{{\scriptscriptstyle\bigstar}\lin}\cong{\boldsymbol{V}}^{{\overline{\scriptscriptstyle\bigstar}}\,{\overline{\scriptscriptstyle\bigstar}}}~,\quad
\cj{\V}:={\boldsymbol{V}}^{{\scriptscriptstyle\bigstar}{\overline{\scriptscriptstyle\bigstar}}}\cong{\boldsymbol{V}}^{{\overline{\scriptscriptstyle\bigstar}}{\scriptscriptstyle\bigstar}}~.$$
Summarizing, one one gets the four distinct spaces
$${\boldsymbol{V}}\leftrightarrow\cj{\V}~,\quad \V{}^\lin\leftrightarrow\V{}^\alin~,$$
where the arrows indicate the conjugation anti-isomorphisms.
Accordingly, coordinate expressions have four types of indices.
Let $({\mathsf{b}}_{\scriptscriptstyle A})$, $1\leq A\leq n$\,, be a basis of ${\boldsymbol{V}}$
and $({\mathsf{b}}^{{\scriptscriptstyle A}})$ its dual basis of $\V{}^\lin$.
The corresponding indices in the conjugate spaces are distinguished by a dot,
namely one writes
$$\bar{\mathsf{b}}_{{\sA\.}}:=\overline{{\mathsf{b}}_{\scriptscriptstyle A}}~,\quad \bar{\mathsf{b}}^{{\sA\.}}
:=\overline{{\mathsf{b}}^{{\scriptscriptstyle A}}}~,$$
so that $\{\bar{\mathsf{b}}_{{\sA\.}}\}$ is a basis of $\cj{\V}$ and
$\{\bar{\mathsf{b}}^{{\sA\.}}\}$ its dual basis of $\cj{\V}{}^\lin$.
For $v\in{\boldsymbol{V}}$ and $\l\in\V{}^\lin$ one has
\begin{align*}&
v=v^{\scriptscriptstyle A}\,{\mathsf{b}}_{\scriptscriptstyle A}~,\quad {\bar v}={\bar v}^{\sA\.}\,\bar{\mathsf{b}}_{\sA\.}~,\\&
\l=\l_{\scriptscriptstyle A}\,{\mathsf{b}}^{\scriptscriptstyle A}~,\quad {\bar\lambda}={\bar\lambda}_{\sA\.}\,\bar{\mathsf{b}}^{{\sA\.}}~,
\end{align*}
where ${\bar v}^{\sA\.}=\overline{v^{\scriptscriptstyle A}}$, ${\bar\lambda}_{\sA\.}:=\overline{\l_{\scriptscriptstyle A}}$
and Einstein summation convention is used.
The conjugation morphism can be extended to tensors of any rank and type;
if $\t$ is a tensor then all indices of $\bar\t$
are of reversed (dotted/non-dotted) type;
observe that dotted indices cannot be contracted with non-dotted indices.
In particular if $K\in\operatorname{Aut}({\boldsymbol{V}})\subset{\boldsymbol{V}}{\,\otimes\,}\V{}^\lin$
then $\bar K\in\operatorname{Aut}(\cj{\V})\subset\cj{\V}{\,\otimes\,}\cj{\V}{}^\lin$ is the induced
conjugated transformation
(under a basis transformation,
dotted indices transform with the conjugate matrix).
\subsection{Hermitian tensors}\label{s:Hermitian tensors}
The space ${\boldsymbol{V}}{\,\otimes\,}\cj{\V}$ has a natural real linear (complex anti-linear) involution
$w\mapsto w^\dag$,
which on decomposable tensors reads
$$(u{\,\otimes\,}{\bar v})^\dag=v{\,\otimes\,}{\bar u}~.$$
Hence one has the natural decomposition of ${\boldsymbol{V}}{\,\otimes\,}\cj{\V}$ into the direct sum
of the \emph{real} eigenspaces of the involution with eigenvalues $\pm1$,
respectively called the \emph{Hermitian} and \emph{anti-Hermitian} subspaces,
namely
$${\boldsymbol{V}}{\,\otimes\,}\cj{\V}=({\boldsymbol{V}}{\,\bar\vee\,}\cj{\V})\oplus \mathrm{i}\,({\boldsymbol{V}}{\,\bar\vee\,}\cj{\V})~.$$
In other terms, the Hermitian subspace ${\boldsymbol{V}}{\,\bar\vee\,}\cj{\V}$ is constituted by
all $w\in{\boldsymbol{V}}{\,\otimes\,}\cj{\V}$ such that $w^\dag=w$,
while an arbitrary $w$ is uniquely decomposed into the sum of an Hermitian
and an anti-Hermitian tensor as
$$w=\tfrac{1}{2}(w+w^\dag)+\tfrac{1}{2}(w-w^\dag)~.$$
In terms of components in any basis,
$w=w^{{\scriptscriptstyle A}{\sB\.}}{\mathsf{b}}_{\scriptscriptstyle A}{\,\otimes\,}\bar{\mathsf{b}}_{\sB\.}$ is Hermitian (anti-Hermitian)
iff the matrix $(w^{{\scriptscriptstyle A}{\sB\.}}\,)$ of its components is of the same type,
namely ${\bar w}^{{\sB\.}{\scriptscriptstyle A}}=\pm w^{{\scriptscriptstyle A}{\sB\.}}$.
Obviously $\V{}^\lin{\,\otimes\,}\cj{\V}{}^\lin$ decomposes in the same way,
and one has the natural isomorphisms
$$({\boldsymbol{V}}{\,\bar\vee\,}\cj{\V})^*\cong\V{}^\lin{\,\bar\vee\,}\cj{\V}{}^\lin~~,\quad (\mathrm{i}\,{\boldsymbol{V}}{\,\bar\vee\,}\cj{\V})^*\cong\mathrm{i}\,\V{}^\lin{\,\bar\vee\,}\cj{\V}{}^\lin~,$$
where ${}^*$ denotes the \emph{real} dual.
A Hermitian $2$-form is defined to be a Hermitian tensor $h\in\cj{\V}{}^\lin{\,\bar\vee\,}\V{}^\lin$.
The associated quadratic form $v\mapsto h(v,v)$ is real-valued.
The notions of signature and non-degeneracy of Hermitian $2$-forms
are introduced similarly to the case of real bilinear forms.
If $h$ is non-degenerate then it yields the isomorphism
$h^\flat:\cj{\V}\to\V{}^\lin:{\bar v}\mapsto h({\bar v},\_)$;
its conjugate map is an anti-isomorphism $\cj{\V}\to\cj{\V}{}^\lin$ which,
via composition with the canonical conjugation, can be seen as the isomorphism
${\bar h}^\flat:{\boldsymbol{V}}\to\cj{\V}{}^\lin:v\mapsto h(\_,v)$.
The inverse isomorphisms $h^\#$ and ${\bar h}^\#$ are similarly related
to a Hermitian tensor $h^{-1}\in\cj{\V}{\,\bar\vee\,}{\boldsymbol{V}}$.
One has the coordinate expressions
\begin{align*}
&(h^\flat({\bar v}))_{\scriptscriptstyle B}=h_{{\sA\.}{\scriptscriptstyle B}}{\bar v}^{\sA\.}~~,\quad
&({\bar h}^\flat(v))_{\sA\.}=h_{{\sA\.}{\scriptscriptstyle B}}v^{\scriptscriptstyle B}={\bar h}_{{\scriptscriptstyle B}{\sA\.}}\,v^{\scriptscriptstyle B}~~,\\
&(h^\#({\bar\lambda}))^{\scriptscriptstyle B}=h^{{\sA\.}{\scriptscriptstyle B}}{\bar\lambda}_{\sA\.}~~,\quad
&({\bar h}^\#(\l))^{\sA\.}=h^{{\sA\.}{\scriptscriptstyle B}}\l_{\scriptscriptstyle B}={\bar h}^{{\scriptscriptstyle B}{\sA\.}}\l_{\scriptscriptstyle B}~~,
\end{align*}
where $h^{{\sC\.}{\scriptscriptstyle A}}h_{{\sC\.}{\scriptscriptstyle B}}=\d\Ii{{\scriptscriptstyle A}}{{\scriptscriptstyle B}}$\,,
$h^{{\sA\.}{\scriptscriptstyle C}}h_{{\sB\.}{\scriptscriptstyle C}}=\d\Ii{{\sA\.}}{{\sB\.}}$\,.
\subsection{Two-spinor space}\label{s:Two spinor space}
Let $\S$ be a $2$-dimensional complex vector space.
Then $\weu{2}\S$ is a $1$-dimensional complex vector space;
its dual space $(\weu{2}\S)^{\scriptscriptstyle\bigstar}$ will be identified with
$\weu{2}\S{}^\lin$ via the rule\footnote{
Here, $s{{\,\wedge\,}}s':=\tfrac{1}{2}(s{\otimes}s'{-}s'{\otimes}s)$\,.
This contraction, defined in such a way to respect
usual conventions in two-spinor literature,
corresponds to $1/4$ standard exterior-algebra contraction.}
$${\omega}(s{{\,\wedge\,}}s'):=\tfrac{1}{2}{\omega}(s,s')~, \quad\forall~{\omega}\in\weu{2}\S{}^\lin,~s,s'\in\S~.$$
Any ${\omega}\in\weu2\S{}^\lin\setminus\{0\}$
(a `symplectic' form on $\S$)
has a unique `inverse' or `dual' element ${\omega}^{-1}$\,.
Denoting by ${\omega}^\flat:\S\to\S{}^\lin$ the linear map defined by
$\bang{{\omega}^\flat(s),t}:={\omega}(s,t)$
and by ${\omega}^\#:\S{}^\lin\to\S$ the linear map defined by
$\bang{\mu,{\omega}^\#(\l)}:={\omega}^{-1}(\l,\mu)$\,,
one has $${\omega}^\#=-({\omega}^\flat)^{-1}~.$$
The Hermitian subspace of $(\weu{2}\S){\,\otimes\,}(\weu{2}\cj{\S})$
is a 1-dimensional real vector space with a distinguished orientation,
whose positively oriented semispace
$${\mathbb{L}}^2:=[(\weu{2}\S){\,\bar\vee\,}(\weu{2}\cj{\S})]^{+}:=\{w{\,\otimes\,}{\bar w},~w\in\weu{2}\S\}$$
has the square root semi-space ${\mathbb{L}}$,
called the space of \emph{length units}.\footnote{
A \emph{unit space} is defined to be a 1-dimensional real semi-space,
namely a positive semi-field associated with the semi-ring ${\mathbb{R}}^+$
(see~\cite{CJM,CJ97a} for details).
The \emph{square root} ${\mathbb{U}}^{1/2}$ of a unit space ${\mathbb{U}}$,
is defined by the condition that ${\mathbb{U}}^{1/2}{\,\otimes\,}{\mathbb{U}}^{1/2}$ be isomorphic to ${\mathbb{U}}$.
More generally, any \emph{rational power} of a unit space
is defined up to isomorphism
(negative powers correspond to dual spaces).
In this article we only use the unit space ${\mathbb{L}}$ of lengths and its powers;
essentially, this means that we take $\hbar=c=1$\,.}
Next, consider the complex $2$-dimensional space
$${\boldsymbol{U}}:={\mathbb{L}}^{-1/2}{\,\otimes\,}\S~.$$
This is our \emph{$2$-spinor space}.
Observe that the $1$-dimensional space
$${\boldsymbol{Q}}:=\weu{2}{\boldsymbol{U}}={\mathbb{L}}^{-1}{\,\otimes\,}\weu{2}\S$$
has a distinguished Hermitian metric,
defined as the unity element in
$$\cj{\Q}{}^\lin{\,\bar\vee\,}\Q{}^\lin\equiv(\weu{2}\cj{\U}{}^\lin){\,\bar\vee\,}(\weu{2}\U{}^\lin)
={\mathbb{L}}^{-2}{\,\otimes\,}(\weu{2}\S{}^\lin){\,\bar\vee\,}(\weu{2}\S{}^\lin)\cong{\mathbb{R}}~.$$
Hence there is the distinguished set of normalized symplectic forms on ${\boldsymbol{U}}$,
any two of them differing by a phase factor.\footnote{
One says that
elements of ${\boldsymbol{U}}$ and of its tensor algebra are `conformally invariant',
while tensorializing by ${\mathbb{L}}^r$ one obtains `conformal densities'
of weight $r$.}
Consider an arbitrary basis
$(\xi_{\scriptscriptstyle A})$ of $\S$ and its dual basis $({\mathsf{x}}^{\scriptscriptstyle A})$ of $\S{}^\lin$.
This determines the mutually dual bases
$${\mathsf{w}}:=\varepsilon^{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,\xi_{\scriptscriptstyle A}{\,\wedge\,}\xi_{\scriptscriptstyle B}~,\quad
{\mathsf{w}}^{-1}:=\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,{\mathsf{x}}^{\scriptscriptstyle A}{\,\wedge\,}{\mathsf{x}}^{\scriptscriptstyle B}~,$$
respectively of $\weu{2}\S$ and $\weu{2}\S{}^\lin$
(here $\varepsilon^{{\scriptscriptstyle A}{\scriptscriptstyle B}}$ and $\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}$ both denote
the antisymmetric Ricci matrix),
and the basis
$$l:=\sqrt{{\mathsf{w}}{\,\otimes\,}\bar{\mathsf{w}}} \quad\text{of}\quad {\mathbb{L}}~.$$
Then one also has the induced mutually dual, {\em normalized\/} bases
$$({\zeta_\sA}):=(l^{-1/2}{\,\otimes\,}\xi_{\scriptscriptstyle A})~,\quad ({\mathsf{z}}^\sA):=(l^{1/2}{\,\otimes\,}{\mathsf{x}}^{\scriptscriptstyle A})$$
of ${\boldsymbol{U}}$ and $\U{}^\lin$, and also
\begin{align*}
&\varepsilon:=l{\,\otimes\,}{\mathsf{w}}^{-1}=\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,{\mathsf{z}}^\sA{\,\wedge\,}{\mathsf{z}}^\sB\in\Q{}^\lin\equiv\weu{2}\U{}^\lin~, \\[6pt]
&\varepsilon^{-1}\equiv l^{-1}{\,\otimes\,}{\mathsf{w}}=\varepsilon^{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,{\zeta_\sA}{\,\wedge\,}{\zeta_\sB}\in{\boldsymbol{Q}}\equiv\weu{2}{\boldsymbol{U}}~.
\end{align*}
\smallbreak\noindent{\bf Remark.}~
In contrast to the usual $2$-spinor formalism,
no symplectic form is fixed.
The $2$-form $\varepsilon$ is unique up to a phase factor
which depends on the chosen 2-spinor basis,
and determines isomorphisms
\begin{align*}
& \varepsilon^\flat:{\boldsymbol{U}}\to\U{}^\lin:u\mapsto u^\flat~,~~
\bang{u^\flat,v}:=\varepsilon(u,v){\quad\Rightarrow\quad} (u^\flat)_{\scriptscriptstyle B}=\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,v^{\scriptscriptstyle A}~,\\[6pt]
& \varepsilon^\#:\U{}^\lin\to{\boldsymbol{U}}:\l\mapsto\l^\#~,~~
\bang{\mu,\l^\#}:=\varepsilon^{-1}(\l,\mu){\quad\Rightarrow\quad} (\l^\#)^{\scriptscriptstyle B}=\varepsilon^{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,\l_{\scriptscriptstyle A}~.
\end{align*}
If no confusion arises, we'll make the identification $\varepsilon^\#\equiv\varepsilon^{-1}$.
\smallbreak
\subsection{2-spinors and Minkowski space}
\label{s:2-spinors and Minkowski space}
Though a normalized element $\varepsilon\in\Q{}^\lin$ is unique only up to a phase factor,
certain objects which can be expressed through it
are natural geometric objects.
The first example is the unity element in $\Q{}^\lin{\,\otimes\,}\cj{\Q}{}^\lin$,
which can be written as $\varepsilon{\,\otimes\,}{\bar\varepsilon}$\,;
it can also be seen as a bilinear form $g$ on ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$,
given for decomposable elements by
$$g(p{\,\otimes\,}{\bar q},r{\,\otimes\,}{\bar s})=\varepsilon(p,r)\,{\bar\varepsilon}({\bar q},{\bar s})~.$$
The fact that any $\varepsilon$ is non-degenerate implies that $g$
is non-degenerate too.
In a normalized 2-spinor basis $({\zeta_\sA})$ one writes
$w=w^{{\sA\cA}}\,{\zeta_\sA}{\,\otimes\,}{\bze_\cA}\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U}$,
$g_{{\sA\cA}\,{\sB\cB}}=\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,{\bar\varepsilon}_{{\sA\.}{\sB\.}}$ and\footnote{
Note how $\det w\equiv\det\bigl(w^{{\sA\cA}}\:\bigr)$ is intrinsically defined
through $\varepsilon$\,, even if $w$ is not an endomorphism.}
$$g(w,w)=\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,{\bar\varepsilon}_{{\sA\.}{\sB\.}}\,w^{{\sA\cA}}\,w^{{\sB\cB}}=2\,\det w~.$$
Next, consider the Hermitian subspace
$$\H:={\boldsymbol{U}}{\,\bar\vee\,}\cj{\U}\subset{\boldsymbol{U}}{\,\otimes\,}\cj{\U}~.$$
This is a $4$-dimensional \emph{real} vector space;
for any given normalized basis $({\zeta_\sA})$ of ${\boldsymbol{U}}$ consider, in particular,
the \emph{Pauli basis} $(\t_\l)$ of $\H$ associated with $({\zeta_\sA})$,
namely
$$\t_\l\equiv\t\iI{\l}{{\sA\cA}}\,{\zeta_\sA}{\,\otimes\,}{\bze_\cA}
\equiv\tfrac{1}{\surd2}\,\si{\l}{{\sA\cA}}\,{\zeta_\sA}{\,\otimes\,}{\bze_\cA}~,\quad \l=0,1,2,3~,$$
where $(\si{\l}{{\sA\cA}})$ denotes the $\l$-th Pauli matrix.\footnote{
$\quad\sigma_0:=\left(\begin{smallmatrix}~1&~\hm0~\\[5pt]
~0&~\hm1~\end{smallmatrix}\right)~,\quad
\sigma_1:=\left(\begin{smallmatrix}~0&~\hm1~\\[5pt]
~1&~\hm0~\end{smallmatrix}\right)~,\quad
\sigma_2:=\left(\begin{smallmatrix}~0&~-\mathrm{i}~\\[5pt]
~\mathrm{i}&~\hm0~\end{smallmatrix}\right)~,\quad
\sigma_3:=\left(\begin{smallmatrix}~1&~\hm0~\\[5pt]
~0&~-1~\end{smallmatrix}\right)~.$ }
The restriction of $g$ to the Hermitian subspace $\H$
turns out to be a Lorentz metric with signature $(+,-,-,-)$\,.
Actually, a Pauli basis is readily seen to be orthonormal, namely
$g_{\l\mu}:=g(\t_\l\,,\t_\mu)=\eta_{\l\mu}:=2\,\d^0_\l\d^0_\mu-\d_{\l\mu}$\,.
It's not difficult to prove:
\begin{proposition}
An element $w\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U}={\mathbb{C}}{\,\otimes\,}\H$ is null, that is $g(w,w)=0$\,,
iff it is a decomposable tensor: $w=u{\,\otimes\,}{\bar s}$, $u,s\in{\boldsymbol{U}}$\,.
\end{proposition}
A null element in ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$ is also in $\H$
iff it is of the form $\pm u{\,\otimes\,}{\bar u}$.
Hence the \emph{null cone} ${\boldsymbol{N}}\subset\H$
is constituted exactly by such elements.
Note how this fact yields a way of distinguish between time orientations:
by convention, one chooses the \emph{future} and \emph{past}
null-cones in $\H$ to be, respectively,
$${\boldsymbol{N}}^+:=\{u{\,\otimes\,}{\bar u},~u\in{\boldsymbol{U}}\}~,\quad {\boldsymbol{N}}^-:=\{-u{\,\otimes\,}{\bar u},~u\in{\boldsymbol{U}}\}~.$$
\begin{proposition}\label{p:existence2spinorPaulibases}
For each $g$-orthonormal positively oriented basis $({\mathsf{e}}_\l)$ of $\H$,
such that ${\mathsf{e}}_0$ is timelike and future-oriented,
there exists a normalized 2-spinor basis $({\zeta_\sA})$ whose associated
Pauli basis $(\t_\l)$ coincides with $({\mathsf{e}}_\l)$\,.
\end{proposition}
\smallbreak\noindent{\bf Remark.}~From the above proposition it follows
that any future-pointing timelike vector can be written as
\hbox{$u{\,\otimes\,}{\bar u}+v{\,\otimes\,}{\bar v}$}\,,
for suitable $u,v\in{\boldsymbol{U}}$\,.
\smallbreak
\subsection{From 2-spinors to 4-spinors}\label{s:From 2-spinors to 4-spinors}
Next observe that an element of ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$
can be seen as a linear map $\cj{\U}{}^\lin\to{\boldsymbol{U}}$,
while an element of $\cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin$ can be seen as a linear map ${\boldsymbol{U}}\to\cj{\U}{}^\lin$.
Then one defines the linear map
\begin{align*}
&\gamma:{\boldsymbol{U}}{\,\otimes\,}\cj{\U}\to\operatorname{End}({\boldsymbol{U}}\oplus\cj{\U}{}^\lin):y\mapsto
\gamma(y):=\sqrt2\,\bigl(y,y^{\flat{\scriptscriptstyle\bigstar}}\bigr)~,\phantom{\text{i.e$.$}\quad}\\[6pt]
\text{i.e$.$}\quad
&\gamma(y)(u,\chi)=\sqrt2\bigl(y\mathord{\rfloor}\chi\,,u\mathord{\rfloor} y^\flat \bigr)~,
\end{align*}
where $y^\flat:=g^\flat(y)\in\U{}^\lin{\,\otimes\,}\cj{\U}{}^\lin$ and $y^{\flat{\scriptscriptstyle\bigstar}}\in\cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin$
is the transposed tensor.
In particular for a decomposable $y=p{\,\otimes\,}{\bar q}$ one has
$$\tilde\gamma(p{\,\otimes\,}{\bar q})(u,\chi)
=\sqrt2\bigl(\bang{\chi,{\bar q}}\,p\,,\bang{p^\flat,u}\,{\bar q}^\flat\,\bigr)~.$$
\begin{proposition}
For all $y,y'\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U}$ one has
$$\gamma(y)\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}\gamma(y')+\gamma(y')\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}\gamma(y)=2\,g(y,y')\,{1\!\!1}~.$$
\end{proposition}
{\sc proof:~}
It is sufficient to check the statement's formula for any couple of null
i.e$.$\ decomposable elements in ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$.
Using the identity
$$\varepsilon(p,q)\,r^\flat+\varepsilon(q,r)\,p^\flat+\varepsilon(r,p)\,q^\flat=0~,\quad p,q,r\in{\boldsymbol{U}}~,$$
which is in turn easily checked, a straightforward calculation gives
\begin{multline*}
[\gamma(p{\,\otimes\,}{\bar q})\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}\gamma(r{\,\otimes\,}{\bar s})
+\gamma(r{\,\otimes\,}{\bar s})\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}\gamma(p{\,\otimes\,}{\bar q})](u+\chi)=\\[6pt]
=2\,\varepsilon(p,r)\,{\bar\varepsilon}({\bar q},{\bar s})\,(u,\chi)=
2\,g(p{\,\otimes\,} {\bar q},r{\,\otimes\,} {\bar s})\,(u,\chi)~.
\end{multline*}
\EndBox{\square}
Now one sees that $\gamma$ is a \emph{Clifford map} relatively to $g$
(see also \Sec\ref{s:Dirac algebra});
thus one is led to regard $${\boldsymbol{W}}:={\boldsymbol{U}}\oplus\cj{\U}{}^\lin$$ as the space of Dirac spinors,
decomposed into its Weyl subspaces.
Actually, the restriction of $\gamma$ to the Minkowski space $\H$
turns out to be a Dirac map.
The 4-dimensional complex vector space ${\boldsymbol{W}}$ is naturally endowed
with a further structure:
the obvious anti-isomorphism
$${\boldsymbol{W}}\to\W{}^\lin:(u,\chi)\mapsto({\bar\chi},{\bar u})~.$$
Namely, if $\psi=(u,\chi)\in{\boldsymbol{W}}$ then $\bar\psi=({\bar u},{\bar\chi})\in\cj{\W}$
can be identified with $({\bar\chi},{\bar u})\in\W{}^\lin$\,;
this is the so-called `Dirac adjoint' of $\psi$\,.
This operation can be seen as the ``index lowering anti-isomorphism''
related to the Hermitian product
$$\mathrm{k}:{\boldsymbol{W}}\times{\boldsymbol{W}}\to{\mathbb{C}}:\Bigl((u,\chi),(u',\chi')\Bigr)
\mapsto\bang{{\bar\chi},u'}+\bang{\chi',{\bar u}}~,$$
which is obviously non-degenerate;
its signature turns out to be $(+\,+\,-\,-)$,
as it can be seen in a ``Dirac basis'' (below).
Let $({\zeta_\sA})$ be a normalized basis of ${\boldsymbol{U}}$\,;
the \emph{Weyl basis} of ${\boldsymbol{W}}$
is defined to be the basis $({\zeta}_\a)$, $\a=1,2,3,4$, given by
$$({\zeta}_1\,,{\zeta}_2\,,{\zeta}_3,{\zeta}_4):=({\zeta}_1\,,{\zeta}_2\,,-{\bar{\mathsf{z}}}^1,-{\bar{\mathsf{z}}}^2)~.$$
Above, ${\zeta}_1$ is a simplified notation for $({\zeta}_1\,,0)$, and the like.
Another important basis is the \emph{Dirac basis} $({\zeta}'_\a)$, $\a=1,2,3,4$,
where
\begin{align*}
& {\zeta}'_1=\tfrac{1}{\surd2}({\zeta}_1\,,{\bar{\mathsf{z}}}^1)\equiv\tfrac{1}{\surd2}({\zeta}_1-{\zeta}_3)~,
&& {\zeta}'_2=\tfrac{1}{\surd2}({\zeta}_2\,,{\bar{\mathsf{z}}}^2)\equiv({\zeta}_2-{\zeta}_4)~,\\[6pt]
& {\zeta}'_3=\tfrac{1}{\surd2}({\zeta}_1\,,-{\bar{\mathsf{z}}}^1)\equiv({\zeta}_1+{\zeta}_3)~,
&& {\zeta}'_4=\tfrac{1}{\surd2}({\zeta}_2\,,-{\bar{\mathsf{z}}}^2)\equiv({\zeta}_2+{\zeta}_4)~.
\end{align*}
Setting
$$\gamma_\l:=\gamma(\t_\l)\in\operatorname{End}({\boldsymbol{W}})$$
one recovers the usual Weyl and Dirac representations as the matrices
$\bigl(\gamma_\l\bigr)$\,, $\l=0,1,2,3$\,,
in the Weyl and Dirac bases respectively.
\subsection{Further structures}\label{s:Further structures}
Some other operations on 4-spinor space, commonly used in the literature,
actually depend on particular choices or conventions.
Similarly to the choice of a basis or of a gauge
they are useful in certain arguments or calculations,
but don't need to be fixed in the theory's foundations.
I'll describe the cases of a Hermitian form on ${\boldsymbol{U}}$,
of \emph{charge conjugation}, \emph{parity} and \emph{time reversal};
I'll show the relations among these objects
and how they are related to the notion of \emph{observer}.
A Hermitian 2-form $h$ on ${\boldsymbol{U}}$ is an element in $\cj{\U}{}^\lin{\,\bar\vee\,}\U{}^\lin$\,,
hence it can be seen as an element in $\H^*$\,;
more precisely, ${\bar h}\in\H^*$\,.
One says that $h$ is \emph{normalized} if it is non-degenerate, positive
and $g^\#(h)=h^{-1}$;
the latter condition is equivalent to $g(h,h)=2$\,.
If $h$ is normalized then it is necessarily
a future-pointing timelike element in $\H^*$\,.
For example, consider the Pauli basis $(\t_\l)$
determined by a normalized 2-spinor basis $({\zeta_\sA})$\,,
and let $(\tt^\l)$ be the dual basis;
then $\sqrt2\,\bar\tt^0={\bar{\mathsf{z}}}^1{\,\otimes\,}{\mathsf{z}}^1+{\bar{\mathsf{z}}}^2{\,\otimes\,}{\mathsf{z}}^2$ is normalized;
conversely, every positive-definite normalized Hermitian metric $h$
can be expressed in the above form for some suitable
normalized 2-spinor bases.\footnote{
Similarly, negative-definite Hermitian metrics correspond to
past-pointing timelike covectors.
Hermitian metrics of mixed signature $(1,-1)$ correspond to
spacelike covectors;
actually, such metrics can always be written as proportional
to $\sqrt2\,\bar\tt^3={\bar{\mathsf{z}}}^1{\,\otimes\,}{\mathsf{z}}^1-{\bar{\mathsf{z}}}^2{\,\otimes\,}{\mathsf{z}}^2$\,,
in appropriate normalized 2-spinor bases.}
The basic observation resulting from the above discussion is that
the assignments of an `observer' in $\H$ and of a positive-definite
Hermitian metric on ${\boldsymbol{U}}$ are equivalent;
actually, the two objects are nearly the same thing.
In 4-spinor terms, the above equivalence is only slightly less obvious.
If $h$ is assigned, then it extends naturally
to a Hermitian metric $h$ on ${\boldsymbol{W}}$,
which can be characterized by\footnote{
In the traditional notation,
$\gamma_\l^\dag$ indicates the $h$-adjoint of $\gamma_\l$\,,
and then depends on the chosen observer.}
$$h(\psi,\phi)=\mathrm{k}(\gamma_0\psi,\phi)~.$$
Charge conjugation depends on the choice of a normalized $2$-form
${\omega}=\mathrm{e}^{\mathrm{i} t}\,\varepsilon\in\weu2\U{}^\lin$, and is defined as the anti-isomorphism
$${\mathcal{C}}_{\omega}:{\boldsymbol{W}}\to{\boldsymbol{W}}:\psi\mapsto{\mathcal{C}}_{\omega}(\psi)\equiv{\mathcal{C}}(u,\chi)
=\bigl({\omega}^\#({\bar\chi}),-\bar{\omega}^\flat({\bar u})\bigr)
=\mathrm{e}^{-\mathrm{i} t}\,\bigl(\varepsilon^\#({\bar\chi}),-{\bar\varepsilon}^\flat({\bar u})\bigr)~.$$
Thus ${\mathcal{C}}_{\omega}=\mathrm{e}^{-\mathrm{i} t}\,{\mathcal{C}}_\varepsilon$\,.
One also gets
\begin{align*}&
{\mathcal{C}}_{\omega}\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}{\mathcal{C}}_{\omega}=\Id{\scriptscriptstyle{\boldsymbol{W}}}~,\\&
\gamma_y\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}{\mathcal{C}}_{\omega}+{\mathcal{C}}_{\omega}\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}\gamma_y=0 \quad\Leftrightarrow \quad
{\mathcal{C}}_{\omega}\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}\gamma_y\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}{\mathcal{C}}_{\omega}=-\gamma_y~,\quad y\in\H\,.
\end{align*}
Finally, parity is an isomorphism of ${\boldsymbol{W}}$
dependent on the choice of an observer,
while time-reversal is an anti-isomorphism dependent on the choice
of an observer \emph{and} of a normalized $2$-form;
they are defined by
$${\mathcal{P}}:=\gamma_0\equiv\gamma(\t_0)~,\qquad {\mathcal{T}}_{\omega}:=\gamma_\eta\gamma_0{\mathcal{C}}_{\omega}~,$$
where the chosen observer is expressed as $\t_0$ in a suitable Pauli basis,
and $\gamma_\eta$ is the canonical element of the Dirac algebra
corresponding to the $g$-normalized volume form of $\H$,
and expressed in a Pauli basis as $\gamma_\eta=\gamma_0\gamma_1\gamma_2\gamma_3$
(see~\Sec\ref{s:Dirac algebra}).
\smallbreak\noindent{\bf Remark.}~An observer, seen as a Hermitian metric on ${\boldsymbol{U}}$,
also determines an isomorphism ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}\to{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin\equiv\operatorname{End}({\boldsymbol{U}})$\,.
Through it, one can view `world spinors' as endomorphisms,
thus recovering the algebraic structure for the
Galileian treatment of spin~\cite{CJM}.
\smallbreak
\subsection{2-spinor groups}\label{s:2-spinor groups}
The group $\operatorname{Aut}(\S)\cong\operatorname{Aut}({\boldsymbol{U}})\subset{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin$ has the natural subgroups
\begin{align*}
& \mathrm{Sl}({\boldsymbol{U}}):=\{K\in\operatorname{Aut}({\boldsymbol{U}}):\det K=1\}~, &&\dim_{\scriptscriptstyle{\mathbb{C}}}\mathrm{Sl}({\boldsymbol{U}})=3~,
\\[6pt]
& \mathrm{Sl}^c({\boldsymbol{U}}):=\{K\in\operatorname{Aut}({\boldsymbol{U}}):|\det K|=1\}~,&&\dim_{\scriptscriptstyle{\mathbb{R}}}\mathrm{Sl}^c({\boldsymbol{U}})=7~.
\end{align*}
The former is the group of all automorphisms of $\S$ (of ${\boldsymbol{U}}$)
which leave any complex volume form invariant;
the latter is the group of all automorphisms
which leave any complex volume form invariant up to a phase factor,
and thus it can be seen as the group which preserves
the two-spinor structure.
One has the Lie algebras
\begin{align*}
& \mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})\cong\{A\in\operatorname{End}({\boldsymbol{U}}):\operatorname{Tr} A=0\}~,\\[6pt]
& \mathfrak{L}\mathrm{Sl}^c({\boldsymbol{U}})\cong\{A\in\operatorname{End}({\boldsymbol{U}}):\Re\operatorname{Tr} A=0\}=\mathrm{i}\,{\mathbb{R}}\oplus\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})~.
\end{align*}
If $h\in\U{}^\lin{\,\bar\vee\,}\cj{\U}{}^\lin$ is a positive Hermitian metric then one sets
\begin{align*}
&\mathrm{U}({\boldsymbol{U}},h):=\{K\in\operatorname{Aut}({\boldsymbol{U}}):K^\dag=K^{-1}\}\subset\mathrm{Sl}^c({\boldsymbol{U}})~,\\[6pt]
&\mathrm{SU}({\boldsymbol{U}},h):=\{K\in\operatorname{Aut}({\boldsymbol{U}}):K^\dag=K^{-1}\,,~\det K=1\}\subset\mathrm{Sl}({\boldsymbol{U}})~,
\end{align*}
where $K^\dag$ denotes the $h$-adjoint of $K$\,.
One gets the Lie algebras
\begin{align*}
& \mathfrak{L}\mathrm{U}({\boldsymbol{U}},h)=\{A\in\operatorname{End}({\boldsymbol{U}}):A+A^\dag=0\}=\mathrm{i}\,{\mathbb{R}}\oplus\mathfrak{L}\mathrm{SU}({\boldsymbol{U}},h)~,
\\[6pt]
& \mathfrak{L}\mathrm{SU}({\boldsymbol{U}},h)=\{A\in\operatorname{End}({\boldsymbol{U}}):A+A^\dag=0\,,~\operatorname{Tr} A=0\}~.
\end{align*}
Now observe that $\operatorname{End}({\boldsymbol{U}})$ can be decomposed into the direct sum
of the subspaces of all $h$-Hermitian and anti-Hermitian endomorphisms;
the restriction of this decomposition to $\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})$ gives then
$$\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})=\mathfrak{L}\mathrm{SU}({\boldsymbol{U}},h)\oplus\mathrm{i}\,\mathfrak{L}\mathrm{SU}({\boldsymbol{U}},h)~.$$
When a 2-spinor basis is fixed, then one gets group isomorphisms
$\mathrm{Sl}({\boldsymbol{U}})\to\mathrm{Sl}(2,{\mathbb{C}})$\,, $\mathrm{SU}({\boldsymbol{U}},h)\to\mathrm{SU}(2)$ and the like.
\subsection{2-spinor groups and Lorentz group}
\label{s:2-spinor groups and Lorentz group}
Up to an obvious transposition we can make the identification
$$\operatorname{End}({\boldsymbol{U}}){\,\otimes\,}\operatorname{End}(\cj{\U})\cong\operatorname{End}({\boldsymbol{U}}{\,\otimes\,}\cj{\U})~.$$
We then write\footnote{
The elements of the dual Pauli basis can be written as
$\tt^\l=\t\Ii\l{\sA\cA}\,{\mathsf{z}}^\sA{\,\otimes\,}\bzz^\cA$ with
$\t\Ii\l{\sA\cA}=g^{\l\mu}\,\varepsilon^{{\scriptscriptstyle A}{\scriptscriptstyle B}}\,{\bar\varepsilon}^{{\sA\.}{\sB\.}}\,\t\iI\mu{\sB\cB}$\,.
}
\begin{align*}
&(K{\,\otimes\,}\bar H)\Ii{{\sA\cA}}{{\sB\cB}}=K\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}\,\bar H\Ii{\sA\.}{\sB\.}~,
\quad K\in\operatorname{End}({\boldsymbol{U}})~,\\[6pt]
&(K{\,\otimes\,}\bar H)\Ii\l\mu=K\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}\,\bar H\Ii{\sA\.}{\sB\.}\,\t\Ii\l{\sA\cA}\,\t\iI\mu{\sB\cB}~.
\end{align*}
The group $\operatorname{Aut}({\boldsymbol{U}})\times\operatorname{Aut}(\cj{\U})$ can be identified
with the subgroup of $\operatorname{Aut}({\boldsymbol{U}}{\,\otimes\,}\cj{\U})$
constituted of all elements of the type $K{\,\otimes\,}\bar H$ with $K,H\in\operatorname{Aut}{\boldsymbol{U}}$\,.
This subgroup is sometimes written as $\operatorname{Aut}({\boldsymbol{U}}){\,\otimes\,}\operatorname{Aut}(\cj{\U})$\,,
which of course must not be intended as a true tensor product.
It has the proper subgroup $\operatorname{Aut}({\boldsymbol{U}}){\,\bar\vee\,}\operatorname{Aut}(\cj{\U})$\,,
constituted of all automorphisms of the type $K{\,\otimes\,}\bar K$\,, $K\in\operatorname{Aut}({\boldsymbol{U}})$\,.
\begin{proposition}
$\operatorname{Aut}({\boldsymbol{U}}){\,\bar\vee\,}\operatorname{Aut}(\cj{\U})$ preserves the splitting ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}=\H\oplus\mathrm{i}\,\H$
and the causal structure of $\H$\,.
\end{proposition}
{\sc proof:~}
There exist bases of $\H$ composed of isotropic elements;
these are also complex bases of isotropic elements of ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$.
Then $A\in\operatorname{Aut}({\boldsymbol{U}}{\,\otimes\,}\cj{\U})$ preserves the splitting and the causal structure
iff it sends any element of the form $u{\,\otimes\,}{\bar u}$
in an element of the form $v{\,\otimes\,}{\bar v}$\,.\EndBox{\square}
Accordingly, on sets
$$\mathrm{Sl}^c({\boldsymbol{U}}){\,\bar\vee\,}\mathrm{Sl}^c(\cj{\U})=\mathrm{Sl}({\boldsymbol{U}}){\,\bar\vee\,}\mathrm{Sl}(\cj{\U})
:=\{K{\,\otimes\,}\bar K:K\in\mathrm{Sl}({\boldsymbol{U}})\,\}~.$$
Since $K$ preserves $\varepsilon$ up to a phase factor,
$K{\,\otimes\,}\bar K$ preserves $\varepsilon{\,\otimes\,}{\bar\varepsilon}\equiv g$\,;
moreover it is immediate to check that any Pauli basis is transformed
to another Pauli basis.
From proposition~\ref{p:existence2spinorPaulibases} it then follows
that $\mathrm{Sl}({\boldsymbol{U}}){\,\bar\vee\,}\mathrm{Sl}(\cj{\U})$ restricted to $\H$ coincides
with the special ortochronous Lorentz group $\mathrm{Lor}_+^{\scriptscriptstyle\uparrow}(\H,g)$\,.
Actually, the epimorphism $\mathrm{Sl}({\boldsymbol{U}})\to\mathrm{Lor}_+^{\scriptscriptstyle\uparrow}(\H,g)$
turns out to be \hbox{2-to-1}\,.
The Lie algebra of $\mathrm{Sl}({\boldsymbol{U}}){\,\bar\vee\,}\mathrm{Sl}(\cj{\U})$ is the Lie subalgebra of
$\operatorname{End}({\boldsymbol{U}}){\,\otimes\,}\operatorname{End}(\cj{\U})$ constituted by all elements
which can be written in the form
$$A{\,\otimes\,}\Id{\cj{\U}}+\Id{{\boldsymbol{U}}}{\,\otimes\,}\bar A~,\quad A\in\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})~.$$
One easily checks that these restrict to endomorphisms of $\H$,
actually they constitute the vector space of all $g$-antisymmetric
endomorphisms of $\H$ namely the Lie algebra $\mathfrak{L}\mathrm{Lor}(\H,g)$\,.
Let a normalized 2-spinor basis be fixed;
then the isomorphism $\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})\leftrightarrow\mathfrak{L}\mathrm{Lor}(\H,g)$\,,
taking into account the isomorphism $\mathfrak{L}\mathrm{Lor}(\H,g)\leftrightarrow\weu2\H^*$
induced by the Lorentz metric $g$\,,
associates the basis $(\nu_i\,;\check\nu_i)$
with the basis $(\r_i\,;\check\r_i)$\,, $i=1,2,3$\,,
where\footnote{
Here again $(\sigma\iIi i{\scriptscriptstyle A}{\scriptscriptstyle B})$ denotes the $i$-th Pauli matrix.
$(\tt^\l)$ is the dual Pauli basis.
Also note that the Hodge isomorphism restricts to a complex structure
on $\weu2\H^*$.}
\begin{align*}
& \nu_i:=-\mathrm{i}\,\check\nu_i
&&\check\nu_i:=\tfrac{1}{2}\,\sigma_i\equiv\tfrac{1}{2}\,\sigma\iIi i{\scriptscriptstyle A}{\scriptscriptstyle B}\,{\zeta_\sA}{\,\otimes\,}{\mathsf{z}}^\sB~,
~,\\[6pt]
&\r_i:=-{*}\check\r_i~, &&\check\r_i:=2\,\tt^0{\,\wedge\,}\tt^i~.
\end{align*}
A Hermitian metric $h$ on ${\boldsymbol{U}}$, besides the above said
(\Sec\ref{s:2-spinor groups})
splitting of $\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})$\,,
also determines an ``observer'' $\t_0:=\tfrac{1}{\surd2}\,{\bar h}^\#$\,,
hence also the splitting of $\mathfrak{L}\mathrm{Lor}(\H,g)$ into ``infinitesimal rotations''
and ``infinitesimal boosts'' as
$$\mathfrak{L}\mathrm{Lor}(\H,g)=\mathfrak{L}\mathrm{Lor}_{\scriptscriptstyle\mathrm{R}}(\H,g,\t_0)
\oplus\mathfrak{L}\mathrm{Lor}_{\scriptscriptstyle\mathrm{B}}(\H,g,\t_0)~.$$
If one chooses a normalized 2-spinor basis
such that the element $\t_0$ of the corresponding Pauli basis of $\H$
coincides with the given observer,
then the bases $(\nu_i\,;\check\nu_i)$ and $(\r_i\,;\check\r_i)$
turn out to be adapted to the respective splittings.
\smallbreak\noindent{\bf Remark.}~On $\mathfrak{L}\mathrm{Lor}(\H,g)$ one has the pseudo-metric induced by $g$\,;
moreover, consider the real symmetric 2-form
$$\mathrm{K}_{\scriptscriptstyle\mathfrak{L}\mathrm{Sl}}:\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})\times\mathfrak{L}\mathrm{Sl}({\boldsymbol{U}})\to{\mathbb{R}}:
(A,B)\mapsto 2\,\Re\operatorname{Tr}(A\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}} B)~.$$
Then it turns out that the bases
$(\nu_i\,;\check\nu_i)$ and $(\r_i\,;\check\r_i)$
are orthonormal,
and that the signature of both metrics is
$(-\,,\,-\,,\,-\,,\,+\,,\,+\,,\,+)$\,.
So, the splittings of the two algebras
determined by the choice of an ``observer''
can't be into arbitrary subspaces:
the two components must be mutually orthogonal subspaces
of opposite signature.
\section{Two-spinor bundles}
\subsection{Two-spinor connections}\label{s:Two-spinor connections}
Consider any real manifold ${\boldsymbol{M}}$ and a vector bundle
$\S\to{\boldsymbol{M}}$ with complex $2$-dimensional fibres.
Denote base manifold coordinates as $({\mathsf{x}}^a)$;
choose a local frame $(\xi_{\scriptscriptstyle A})$ of $\S$,
determining linear fibre coordinates $({\mathsf{x}}^{\scriptscriptstyle A})$.
According to the constructions of the previous sections,
one now has the bundles ${\boldsymbol{Q}}$, ${\mathbb{L}}$, ${\boldsymbol{U}}$, $\H$ over ${\boldsymbol{M}}$,
with smooth natural structures;
the frame $(\xi_{\scriptscriptstyle A})$ yields the frames $\varepsilon$, $l$, $({\zeta_\sA})$ and $(\t_\l)$\,,
respectively.
Moreover for any rational number $r\in{\mathbb{Q}}$
one has the semi-vector bundle ${\mathbb{L}}^r$\,.
Consider an arbitrary ${\mathbb{C}}$-linear connection $\Cs$ on $\S\to{\boldsymbol{M}}$,
called a \emph{$2$-spinor connection}.
In the fibred coordinates $({\mathsf{x}}^a,{\mathsf{x}}^{\scriptscriptstyle A})$\, $\Cs$ is expressed
by the coefficients $\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle B}}:{\boldsymbol{M}}\to{\mathbb{C}}$\,,
namely the covariant derivative of a section $s:{\boldsymbol{M}}\to\S$ is expressed as
$$\nabla s=(\partial_a s^{\scriptscriptstyle A}-\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle B}}s^{\scriptscriptstyle B})\,\dO{\mathsf{x}}^a{\,\otimes\,}\xi_{\scriptscriptstyle A}~.$$
The rule
$\nabla{\bar s}=\overline{\nabla s}$
yields a connection $\bar\Cs$ on $\cj{\S}\to{\boldsymbol{M}}$,
whose coefficients are given by
$$\bar\Cs\iIi{a}{{\sA\.}}{{\sB\.}}=\overline{\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle B}}}~.$$
Actually, $\Cs$ determines linear connections on each of the above said
induced vector bundles over ${\boldsymbol{M}}$
(in particular, it is easy to see that any ${\mathbb{C}}$-linear connection
on a complex vector bundle determines a ${\mathbb{R}}$-linear connection
on the induced Hermitian tensor bundle).
Denote by $2\,G$ and $2\,Y$ the connections induced on ${\mathbb{L}}$ and ${\boldsymbol{Q}}$
(this notation makes sense because the fibres are 1-dimensional), namely
\begin{gather*}
\nabla l=-2\,G_a\,\dO{\mathsf{x}}^a{\,\otimes\,} l~,\quad \nabla\varepsilon=2\,\mathrm{i}\,Y_a\,\dO{\mathsf{x}}^a{\,\otimes\,}\varepsilon~,\\
\nabla{\mathsf{w}}^{-1}\equiv\nabla(l^{-1}{\,\otimes\,}\varepsilon)=2(G_a+\mathrm{i}\,Y_a)\,\dO{\mathsf{x}}^a{\,\otimes\,} l^{-1}{\,\otimes\,}\varepsilon
\end{gather*}
and the like. By direct calculation we find
\begin{align*}
G_a&=\Re(\tfrac{1}{2}\,\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle A}})=
\tfrac{1}{4}(\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle A}}+\bar\Cs\iIi{a}{{\sA\.}}{{\sA\.}})~,
\\[8pt]
Y_a&=\Im(\tfrac{1}{2}\,\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle A}})=
\tfrac{1}{4\mathrm{i}}(\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle A}}-\bar\Cs\iIi{a}{{\sA\.}}{{\sA\.}})~.
\end{align*}
Note that since $Y_a$ are real the induced linear connection on ${\boldsymbol{Q}}$
is Hermitian (preserves its natural Hermitian structure).
The coefficients of the connection $\tilde\Cs$ induced on ${\boldsymbol{U}}$ are given by
$$\tilde\Cs\iIi a{\scriptscriptstyle A}{\scriptscriptstyle B}=\Cs\iIi a{\scriptscriptstyle A}{\scriptscriptstyle B}-G_a\,\d\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}~.$$
Let $\tilde\Gamma$ be the connection induced on ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$,
and $\Gamma'$ the connection induced on $\S{\,\otimes\,}\cj{\S}$. Then
\begin{align*}
\Gamma'\iIi{a}{{\sA\cA}}{{\sB\cB}}&=\Cs\iIi a{\scriptscriptstyle A}{\scriptscriptstyle B}\,\d\Ii{\sA\.}{\sB\.}
+\d\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}\,\bar\Cs\iIi a{\sA\.}{\sB\.}~,
\\[8pt]
\tilde\Gamma\iIi{a}{{\sA\cA}}{{\sB\cB}}&=
\Cs\iIi a{\scriptscriptstyle A}{\scriptscriptstyle B}\,\d\Ii{\sA\.}{\sB\.}+\d\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}\,\bar\Cs\iIi a{\sA\.}{\sB\.}
-2\,G_a\,\d\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}\,\d\Ii{\sA\.}{\sB\.}~.
\end{align*}
Since the above coefficients are real,
$\Gamma'$ and $\tilde\Gamma$ turn out to be reducible to real connections on
$\S{\,\bar\vee\,}\cj{\S}$ and $\H\equiv{\boldsymbol{U}}{\,\bar\vee\,}\cj{\U}$, respectively.
Moreover
\begin{proposition}
The connection $\tilde\Gamma$ induced on $\H$ by any $2$-spinor connection is metric,
namely $\nabla[\tilde\Gamma]g=0$\,.
\end{proposition}
{\sc proof:~}
The Lorentz metric $g$ of $\H$
can be identified with the identity of the bundle ${\mathbb{L}}^{-2}$,
namely it is the canonical section
$1\equiv\varepsilon^{-1}{\,\otimes\,}\varepsilon:{\boldsymbol{M}}\to{\mathbb{L}}^{-2}{\,\otimes\,}{\mathbb{L}}^2\equiv{\boldsymbol{M}}\times{\mathbb{R}}^+$\,,
which obviously has vanishing covariant derivative.
\EndBox{\square}
Because of metricity the coefficients $\tilde\Gamma\iIi a\l\mu$ of $\tilde\Gamma$
in the frame $(\t_\l)$ are antisymmetric and traceless, namely
$$\tilde\Gamma\iI{a}{\l\mu}+\tilde\Gamma\iI{a}{\mu\l}=0~,\quad\tilde\Gamma\iIi{a}{\l}{\l}=0$$
(the second formula says $\nabla\eta=0$\,,
where $\eta$ is the $g$-normalized volume form of $\H$).
The above relations between $\Cs$ and the induced connections can be inverted
as follows:
\begin{proposition} One has
$$\Cs\iIi a{\scriptscriptstyle A}{\scriptscriptstyle B}=(-G_a+\mathrm{i}\,Y_a)\,\d\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}+\tfrac{1}{2}\,\Gamma'\iIi a{{\sA\cA}}{{\scriptscriptstyle B}{\sA\.}}
=(G_a+\mathrm{i}\,Y_a)\,\d\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}+\tfrac{1}{2}\,\tilde\Gamma\iIi a{{\sA\cA}}{{\scriptscriptstyle B}{\sA\.}}~.$$
\end{proposition}
\smallbreak
In $4$-spinor formalism the above relation reads
$$\Cs\iIi{a}{\a}{\b}=
(G_a+\mathrm{i}\,Y_a)\,\d\Ii{\a}{\b}
+\tfrac{1}{4}\,\tilde\Gamma\iI{a}{\l\mu}(\gamma_\l\,\gamma_\mu)\Ii{\a}{\b}~,$$
where now $\Cs\iIi{a}{\a}{\b}$ stands for the coefficients of the
naturally induced connection $(\Cs,\bar\Cs^{\scriptscriptstyle\bigstar})$ on ${\boldsymbol{W}}\equiv{\boldsymbol{U}}\dir{{\boldsymbol{M}}}\cj{\U}{}^\lin$
in any 4-spinor frame, $\a,\b=1,..,4$.
A similar relation holds among the curvature tensors, namely
\begin{align*}
R\iIi{ab}{{\scriptscriptstyle A}}{{\scriptscriptstyle B}}&=
2\,(\mathrm{d} G-\mathrm{i}\,\mathrm{d} Y)_{ab}\,\d\Ii{{\scriptscriptstyle A}}{{\scriptscriptstyle B}}+\tfrac{1}{2}\,R'\iIi{ab}{{\sA\cA}}{{\scriptscriptstyle B}{\sA\.}}=
\\[8pt]
&
=-2\,(\mathrm{d} G+\mathrm{i}\,\mathrm{d} Y)_{ab}\,\d\Ii{{\scriptscriptstyle A}}{{\scriptscriptstyle B}}+\tfrac{1}{2}\,\tilde R\iIi{ab}{{\sA\cA}}{{\scriptscriptstyle B}{\sA\.}}~,
\end{align*}
where $R$, $R'$ and $\tilde R$ are the curvature tensors of
$\Cs$, $\Gamma'$ and $\tilde\Gamma$, respectively.
\smallbreak\noindent{\bf Remark.}~Under a local gauge transformation ${\mathsf{K}}:{\boldsymbol{M}}\to\mathrm{Gl}(2,{\mathbb{C}})$
the above coefficients transform as
\begin{align*}
&\Cs\iIi{a}{{\scriptscriptstyle A}}{{\scriptscriptstyle B}} \mapsto
({\mathsf{K}}^{-1})^{\scriptscriptstyle A}_{\scriptscriptstyle C}\,{\mathsf{K}}^{\scriptscriptstyle D}_{\scriptscriptstyle B}\,\Cs\iIi{a}{{\scriptscriptstyle C}}{{\scriptscriptstyle D}}
-({\mathsf{K}}^{-1})^{\scriptscriptstyle A}_{\scriptscriptstyle C}\,\partial_a{\mathsf{K}}^{\scriptscriptstyle C}_{\scriptscriptstyle B}~,
\\[6pt]
&G_a \mapsto G_a-\tfrac{1}{2}\,\partial_a\log\left|\det {\mathsf{K}}\right|~,\quad
Y_a \mapsto Y_a-\tfrac{1}{2}\,\partial_a\arg\det {\mathsf{K}}~,
\\[6pt]
& \tilde\Gamma\iIi{a}{\l}{\mu} \mapsto
(\tilde {\mathsf{K}}^{-1})^\l_\nu\,\tilde {\mathsf{K}}^\r_\mu\,\tilde\Gamma\iIi{a}{\nu}{\r}
-(\tilde {\mathsf{K}}^{-1})^\l_\nu\,\partial_a\tilde {\mathsf{K}}^\nu_\mu~.
\end{align*}
\smallbreak
\subsection{Two-spinor tetrad}\label{s:Two-spinor tetrad}
Henceforth I'll assume that ${\boldsymbol{M}}$ is a real $4$-dimensional manifold.
Consider a linear morphism
$$\Theta:\mathrm{T}{\boldsymbol{M}}\to\S{\,\otimes\,}\cj{\S}={\mathbb{C}}{\,\otimes\,}{\mathbb{L}}{\,\otimes\,}\H~,$$
namely a section
$$\Theta:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}{\,\otimes\,}\H{\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}$$
(all tensor products are over ${\boldsymbol{M}}$).
Its coordinate expression is
$$ \Theta=\Theta_a^\l\,\t_\l{\,\otimes\,}\dO{\mathsf{x}}^a=\Theta_a^{\sA\cA}\,{\zeta_\sA}{\,\otimes\,}{\bze_\cA}{\,\otimes\,}\dO{\mathsf{x}}^a~,
\qquad \Theta_a^\l,\Theta_a^{\sA\cA}:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}~.$$
We'll assume that $\Theta$ is non-degenerate
and valued in the Hermitian subspace ${\mathbb{L}}{\,\otimes\,}\H\subset\S{\,\otimes\,}\cj{\S}$\,;
then $\Theta$ can be viewed as a `scaled' \emph{tetrad}
(or \emph{soldering form}, or \emph{vierbein});
the coefficients $\Theta_a^\l$ are real (i.e$.$\ valued in ${\mathbb{R}}{\,\otimes\,}{\mathbb{L}}$)
while the coefficients $\Theta_a^{\sA\cA}$ are Hermitian,
i.e$.$\ $\bar\Theta_a^{{\sA\.}{\scriptscriptstyle A}}=\Theta_a^{\sA\cA}$.
\smallbreak\noindent{\bf Remark.}~Most of what follows actually still holds
in the case of a degenerate tetrad.
The inverse $\Theta^{-1}$ is not used.
This will give rise to a more natural theory,
in which all field equations are of the first order.
Possible degeneracy might also have a physical meaning,
as discussed in~\cite{C98}.
\smallbreak
Through a tetrad, the geometric structure of the fibres of $\H$
is carried to a similar, scaled structure on the fibres of $\mathrm{T}{\boldsymbol{M}}$.
It will then be convenient, from now on, to distinguish by a tilda
the objects defined on $\H$,
so I'll denote by $\tilde g$\,, $\tilde\eta$ and $\tilde\gamma$
the Lorentz metric,
the $\tilde g$-normalized volume form and the Dirac map of $\H$\,, and set
\begin{align*}
g&:=\Theta^*\tilde g:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^2{\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}{\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}~,
\\[6pt]
\eta&:=\Theta^*\tilde\eta:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^4{\,\otimes\,}\weu{4}\TO^{*}\!{\boldsymbol{M}}~,
\\[6pt]
\gamma&:=\tilde\gamma\mathbin{\raisebox{1pt}{$\scriptstyle\circ$}}\Theta:\mathrm{T}{\boldsymbol{M}}\to{\mathbb{L}}{\,\otimes\,}\operatorname{End}({\boldsymbol{W}})~,
\end{align*}
which have the coordinate expressions
\begin{align*}
g&=\eta_{\l\mu}\,\Theta_a^\l\,\Theta_b^\mu\,\dO{\mathsf{x}}^a{\,\otimes\,}\dO{\mathsf{x}}^b
=\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}\varepsilon_{{\sA\.}{\sB\.}}\,\Theta_a^{\sA\cA}\,\Theta_b^{\sB\cB}\,\dO{\mathsf{x}}^a{\,\otimes\,}\dO{\mathsf{x}}^b~,
\\[6pt]
\eta&=\det(\Theta)\,\dO{\mathsf{x}}^0{\,\wedge\,}\dO{\mathsf{x}}^1{\,\wedge\,}\dO{\mathsf{x}}^2{\,\wedge\,}\dO{\mathsf{x}}^3~,
\\[6pt]
\gamma&=\sqrt2\,\Theta_a^{\sA\cA}\,
({\zeta_\sA}{\,\otimes\,}{\bze_\cA}+\varepsilon_{{\scriptscriptstyle A}{\scriptscriptstyle B}}\varepsilon_{{\sA\.}{\sB\.}}\,\bzz^\cB{\,\otimes\,}{\mathsf{z}}^\sB){\,\otimes\,}\dO{\mathsf{x}}^a~.
\end{align*}
The above objects turn out to be a Lorentz metric,
the corresponding volume form and a Clifford map.
Moreover
$$ \Theta_\mu^b:=\Theta_a^\l\,\eta_{\l\mu}\,g^{ab}=(\Theta^{-1})_\mu^b:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^{-1}~,
\quad
g^{ab}:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^{-2}~.$$
A non-degenerate tetrad, together with a two-spinor frame,
yields mutually dual orthonormal frames
$(\Theta_\l)$ of ${\mathbb{L}}^{-1}{\,\otimes\,}\mathrm{T}{\boldsymbol{M}}$
and $(\ost\Theta{}^\l)$ of ${\mathbb{L}}{\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}$\,, given by
$$\Theta_\l:=\Theta^{-1}(\t_\l)=\Theta_\l^a\,\partial{\mathsf{x}}_a~,\quad
\ost\Theta{}^\l:=\Theta^*(\tt^\l)=\Theta_a^\l\,\dO{\mathsf{x}}^a~.$$
We also write
\begin{align*}&
\gamma=\gamma_\l{\,\otimes\,}\ost\Theta{}^\l=\gamma_a{\,\otimes\,}\dO{\mathsf{x}}^a~,\quad
\gamma_\l:=\gamma(\Theta_\l):{\boldsymbol{M}}\to\operatorname{End}({\boldsymbol{W}})~,\\&
\gamma_a:=\gamma(\partial{\mathsf{x}}_a)=\Theta_a^\l\,\gamma_\l:{\boldsymbol{M}}\to{\mathbb{L}}{\,\otimes\,}\operatorname{End}({\boldsymbol{W}})~.
\end{align*}
\subsection{Cotetrad}
One defines a natural `exterior' product of elements in
the fibres of $\H\ten{{\boldsymbol{M}}}\TO^{*}\!{\boldsymbol{M}}$
by requiring that, for decomposable tensors, it is given by
$$(y_1{\,\otimes\,}\a_1){\,\wedge\,}(y_2{\,\otimes\,}\a_2)=(y_1{\,\wedge\,} y_2){\,\otimes\,}(\a_1{\,\wedge\,}\a_2)~,
\quad \a_1\,,\a_2\in\TO^{*}\!{\boldsymbol{M}}\,,~u_1\,,u_2\in\H~.$$
We'll consider the exterior products
$$\weu{q}\Theta:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^q{\,\otimes\,}\weu{q}\H{\,\otimes\,}\weu{q}\TO^{*}\!{\boldsymbol{M}}~,
\quad q=1,2,3,4~.$$
In particular, one has $\weu2\Theta\equiv\Theta{\,\wedge\,}\Theta$\,, that is
$$\weu{2}\Theta(u{\,\wedge\,} v)=\Theta(u){\,\wedge\,}\Theta(v) {\quad\Rightarrow\quad}
\weu2\Theta=\Theta_a^\l\Theta_b^\mu\,(\t_\l{\,\wedge\,}\t_\mu){\,\otimes\,}(\dO{\mathsf{x}}^a{\,\wedge\,}\dO{\mathsf{x}}^b)~.$$
Next, consider the linear map over ${\boldsymbol{M}}$
$${\breve\Theta}:(\S{\,\otimes\,}\cj{\S}){\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^4{\,\otimes\,}\weu{4}\TO^{*}\!{\boldsymbol{M}}$$
defined by
$${\breve\Theta}(\xi):=\tfrac{1}{3!}\,\tilde\eta\mid(\xi{\,\wedge\,}\Theta{\,\wedge\,}\Theta{\,\wedge\,}\Theta)=
\tfrac{1}{3!}\,\tilde\eta\mid[\xi{\,\wedge\,}(\weu{3}\Theta)]~.$$
Its coordinate expression is
\begin{align*}
& {\breve\Theta}(\xi)={\breve\Theta}_\l^a\,\xi_a^\l\,\mathrm{d}^4{\mathsf{x}}:=
\tfrac{1}{3!}\,\varepsilon^{abcd}\,\varepsilon_{\l\mu\nu\r}\,
\Theta_b^\mu\Theta_c^\nu\Theta_d^\r\,\xi_a^\l\,\mathrm{d}^4{\mathsf{x}}~,
\\[6pt]
& \xi=\xi_a^\l\,\t_\l{\,\otimes\,}\dO{\mathsf{x}}^a~,\quad \xi_a^\l:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}~.
\end{align*}
Now ${\breve\Theta}$ can be seen as a bilinear map
$(\S{\,\otimes\,}\cj{\S})\times\TO^{*}\!{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^4{\,\otimes\,}\weu{4}\TO^{*}\!{\boldsymbol{M}}$
over ${\boldsymbol{M}}$, or also as a linear map
$$\S{\,\otimes\,}\cj{\S}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^4{\,\otimes\,}\mathrm{T}{\boldsymbol{M}}{\,\otimes\,}\weu{4}\TO^{*}\!{\boldsymbol{M}}$$
over ${\boldsymbol{M}}$.
Using the latter point of view, if $\Theta$ is non-degenerate then one has
$${\breve\Theta}=\Theta^{-1}{\,\otimes\,}\eta~.$$
Namely, in general one may regard ${\breve\Theta}$\,,
which is called the \emph{co-tetrad},
as a kind of `pseudo-inverse' of $\Theta$\,,
defined even if $\Theta$ is degenerate.
The above construction can be easily generalized, for $p=0,1,2,3,4$,
to a map
$${\breve\Theta}^{(p)}:
\weu{p}(\S{\,\otimes\,}\cj{\S}){\,\otimes\,}(\weu{p}\TO^{*}\!{\boldsymbol{M}})\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^4{\,\otimes\,}\weu{4}\TO^{*}\!{\boldsymbol{M}}~.$$
We'll be concerned with ${\breve\Theta}^{(1)}={\breve\Theta}$ and ${\breve\Theta}^{(2)}$.
Note that ${\breve\Theta}^{(0)}=\eta$.
\subsection{Tetrad and connections}\label{s:Tetrad and connections}
If $\Cs$ is a complex-linear connection on $\S$,
and $G$ and $\tilde\Gamma$ are the induced connections on ${\mathbb{L}}$ and $\H$,
then a non-degenerate tetrad $\Theta:\mathrm{T}{\boldsymbol{M}}\to{\mathbb{L}}{\,\otimes\,}\H$ yields a unique
connection $\Gamma$ on $\mathrm{T}{\boldsymbol{M}}$, characterized by the condition
$$\nabla[\Gamma{\,\otimes\,}\tilde\Gamma]\Theta=0~.$$
Moreover $\Gamma$ is metric, i.e$.$\ $\nabla[\Gamma]g=0$.
Denoting by $\Gamma\iIi{a}{\l}{\mu}$ the coefficients of $\Gamma$
in the frame $\Theta_\l'\equiv\Theta^{-1}(l{\,\otimes\,}\t_\l)$ one obtains
$$\Gamma\iIi{a}{\l}{\mu}=\tilde\Gamma\iIi{a}{\l}{\mu}+2\,G_a\,\d\Ii{\l}{\mu}~.$$
The curvature tensors of $\Gamma$ and $\tilde\Gamma$ are related by
$R\iIi{ab}{\l}{\mu}=\tilde R\iIi{ab}{\l}{\mu}$\,, or
$$R\iIi{ab}{c}{d}=\tilde R\iIi{ab}{\l}{\mu}\,\Theta_\l^c\,\Theta_d^\mu~.$$
Hence the Ricci tensor and the scalar curvature are given by
\begin{align*}
R_{ad}&=R\iIi{ab}{b}{d}=\tilde R\iIi{ab}{\l}{\mu}\,\Theta_\l^b\,\Theta_d^\mu~,\\
R\iI{a}{a}&=\tilde R\iI{ab}{\l\mu}\,\Theta_\l^b\,\Theta_\mu^a~.
\end{align*}
In general,
the connection $\Gamma$ will have non-vanishing torsion,\footnote{
This is the tensor field $T:{\boldsymbol{M}}\to\mathrm{T}{\boldsymbol{M}}{\,\otimes\,}\weu2\TO^{*}\!{\boldsymbol{M}}$ defined by
$T(u,v)=\nabla\!_uv-\nabla\!_vu-[u,v]$\,,
where $u,v:{\boldsymbol{M}}\to\mathrm{T}{\boldsymbol{M}}$ are any two vector fields,
and has the coordinate expression
$T\Ii c{ab}=-\Gamma\iIi acb+\Gamma\iIi bca$\,.}
which can be expressed\footnote{
Taking into account
$0=\nabla\!_a\Theta_b^\l=\partial_a\Theta_b^\l-\Gamma\iIi a\l\mu\,\Theta_b^\mu+\Gamma\iIi acb\,\Theta_c^\l$\,.}
as
$$\Theta_c^\l\,T\Ii{c}{ab}=\partial_{[a}^{\phantom{a}}\Theta_{b]}^\l
+\Theta_{[a}^\mu\,\tilde\Gamma\iIi{b]}{\l}{\mu}
+2\,\Theta_{[a}^\l\,G^{\phantom{a}}_{b]}~.$$
\smallbreak\noindent{\bf Remark.}~The torsion can be seen as the Fr\"olicher-Nijenhuis bracket
$$\tilde T:=T\mathord{\rfloor}\Theta=[\Gamma',\Theta]:{\boldsymbol{M}}\to\weu2\TO^{*}\!{\boldsymbol{M}}\ten{{\boldsymbol{M}}}H'~,$$
where $H'={\mathbb{L}}{\,\otimes\,}\H$,
$\Gamma':\H'\to\TO^{*}\!{\boldsymbol{M}}\ten{\H'}\mathrm{T}\H'$
is the induced connection on $\H'\to{\boldsymbol{M}}$,
and $\Theta$ is seen as a vertical-valued form
$\Theta:\H'\to\TO^{*}\!{\boldsymbol{M}}\ten{\H'}\mathrm{V}\H'$\,.
\smallbreak
\subsection{The Dirac operator}\label{s:The Dirac operator}
Given a tetrad and a two-spinor connection,
one introduces the Dirac operator acting on sections
$\psi:{\boldsymbol{M}}\to{\mathbb{L}}^{-3/2}{\,\otimes\,}{\boldsymbol{W}}$.
Writing
$\tilde\gamma^\#:{\boldsymbol{M}}\to\H{\,\otimes\,}\operatorname{End}({\boldsymbol{W}})$\,,
$\nabla\psi:{\boldsymbol{M}}\to{\mathbb{L}}^{-3/2}{\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}\ten{{\boldsymbol{M}}}{\boldsymbol{W}}$\,,
one has
$$\tilde\gamma^\#\nabla\psi:{\boldsymbol{M}}\to{\mathbb{L}}^{-3/2}{\,\otimes\,}\H{\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}{\,\otimes\,}{\boldsymbol{W}}~,$$
where contraction in ${\boldsymbol{W}}$ is understood.
Next, one contracts the factors $\H$ and $\TO^{*}\!{\boldsymbol{M}}$ above via
$${\breve\Theta}:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^3{\,\otimes\,}\H^*{\,\otimes\,}\mathrm{T}{\boldsymbol{M}}{\,\otimes\,}\weu4\TO^{*}\!{\boldsymbol{M}}~,$$
obtaining
$$\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi:=\bang{{\breve\Theta},\tilde\gamma^\#\nabla\psi}:
{\boldsymbol{M}}\to{\mathbb{L}}^{3/2}{\,\otimes\,}{\boldsymbol{W}}{\,\otimes\,}\weu4\TO^{*}\!{\boldsymbol{M}}~,$$
which has the coordinate expression
$$\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi={\breve\Theta}^a_\l\,\left(
\sigma^{\l{\sA\cA}}\,\nabla\!_a\chi_{\sA\.}\,{\zeta_\sA}\,,\,\sigma\Ii{\l}{{\sA\cA}}\nabla\!_a u^{\scriptscriptstyle A}\,\bzz^\cA\,
\right){\,\otimes\,}\mathrm{d}^4{\mathsf{x}}~.$$
This definition works even if $\Theta$ were degenerate;
in the non-degenerate case one simply has $\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi={\rlap{\raise1pt\hbox{\,/}}\nabla}\psi{\,\otimes\,}\eta$\,.
\section{Two-spinors and field theories}
\label{S:Two-spinors and field theories}
\subsection{The fields}\label{s:The fields}
In this section I'll present a ``minimal geometric data'' field theory:
actually, the unique ``geometric datum'' is a vector bundle $\S\to{\boldsymbol{M}}$
with complex 2-dimensional fibres and real 4-dimensional base manifold.
All other bundles and fixed geometric objects are determined just
by this datum through functorial constructions,
as we saw in the previous sections;
no further background structure is assumed.
Any considered bundle section which is not functorially fixed
by our geometric datum is a field.
In this way one obtains a field theory
which turns out to be essentially equivalent to a classical theory
of Einstein-Cartan-Maxwell-Dirac fields.
The fields are taken to be the tetrad $\Theta$\,,
the $2$-spinor connection $\Cs$,
the electromagnetic field $F$ and the electron field $\psi$\,.
The gravitational field is represented by $\Theta$
(which can be viewed as a `square root' of the metric)
and the traceless part of $\Cs$, namely $\tilde\Gamma$,
seen as the gravitational part of the connection.
If $\Theta$ is non-degenerate one obtains,
as in the standard metric-affine approach~\cite{GrHe,HCMN,Re,FK82},
essentially the Einstein equation and the equation for torsion;
the metricity of the spacetime connection is a further consequence.
But note that the theory is non-singular also in the degenerate case.
The connection $G$ induced on ${\mathbb{L}}$ will be assumed to have vanishing curvature,
$\mathrm{d} G=0$, so that one can always find local charts such that $G_a=0$;
this amounts to gauging away the conformal (`dilaton') symmetry.
Coupling constants will arise as covariantly constant sections of ${\mathbb{L}}$,
which now becomes just a vector space.
The Dirac field is a section
$$\psi:{\boldsymbol{M}}\to{\mathbb{L}}^{-3/2}{\,\otimes\,}{\boldsymbol{W}}:={\mathbb{L}}^{-3/2}{\,\otimes\,}({\boldsymbol{U}}\oplus\cj{\U}{}^\lin)~,$$
assumed to represent a semiclassical particle with one-half spin,
mass $m\in{\mathbb{L}}^{-1}$ and charge $q\in{\mathbb{R}}$\,.
The electromagnetic potential can be thought of as
the Hermitian connection $Y$ on $\weu2{\boldsymbol{U}}$ determined by $\Cs$\,,
whose coefficients are indicated as $\mathrm{i}\,Y_a$\,;
locally one writes
$$Y_a\equiv q\,A_a~,$$
where $A:{\boldsymbol{M}}\to\TO^{*}\!{\boldsymbol{M}}$ is a local 1-form.
The electromagnetic field is represented by a spinor field
$$\tilde F:{\boldsymbol{M}}\to{\mathbb{L}}^{-2}{\,\otimes\,}\weu2\H^*$$
which, via $\Theta$\,, determines the 2-form $F:=\Theta^*\tilde F:{\boldsymbol{M}}\to\weu2\TO^{*}\!{\boldsymbol{M}}$\,.
The relation between $Y$ and $F$ will follow as one of the field equations;
note how this setting allows a first-order linear Lagrangian and
non-singularity in the degenerate case also for the electromagnetic sector.
The total Lagrangian and the Euler-Lagrange operator will be the sum of
a gravitational, an electromagnetic and a Dirac term
$${\mathcal{L}}={\mathcal{L}}{}_{\mathrm{g}}+{\mathcal{L}}{}_{\mathrm{em}}+{\mathcal{L}}{}_{\scriptscriptstyle{\mathrm{D}}}~,\quad
{\mathcal{E}}={\mathcal{E}}\!{}_{\mathrm{g}}+{\mathcal{E}}\!{}_{\mathrm{em}}+{\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}}~.$$
Observe that all Lagrangian $4$-forms are defined
in terms of the cotetrad ${\breve\Theta}$,
while a direct translation of the standard formulation in terms of our fields
would force one to use $\Theta^{-1}$,
resulting in a less simple and natural theory.
\subsection{Gravitational Lagrangian}\label{s:Gravitational Lagrangian}
The tetrad $\Theta$ and the curvature tensor $\tilde R$ of $\tilde\Gamma$ can be assembled
into a $4$-form ${\mathcal{L}}{}_{\mathrm{g}}$ which, in the non-degenerate case,
turns out to be the usual gravitational Lagrangian density:
$${\mathcal{L}}{}_{\mathrm{g}}:=\frac{1}{4\,\Bbbk}\,{\breve\Theta}^{(2)}(\tilde R^\#)=
\frac{1}{8\,\Bbbk}\,\tilde\eta\mid(\tilde R^\#{\,\wedge\,}\Theta{\,\wedge\,}\Theta):{\boldsymbol{M}}\to\weu4\mathrm{T}^*{\boldsymbol{M}}~,$$
where $\tilde R^\#:{\boldsymbol{M}}\to\weu{2}\mathrm{T}^*{\boldsymbol{M}}{\,\otimes\,}\weu{2}\H$ is the curvature tensor
of $\tilde\Gamma$ with one index raised via $\tilde g$\,,
and $\Bbbk\in{\mathbb{L}}^2$ is Newton's gravitational constant.
Note how this is necessary in order to obtain a true (non-scaled)
$4$-form on ${\boldsymbol{M}}$ and the correct coupling with the spinor field.
One has the coordinate expression
${\mathcal{L}}{}_{\mathrm{g}}=\ell{}_{\mathrm{g}}\,\mathrm{d}^4{\mathsf{x}}$ with
$$\ell{}_{\mathrm{g}}=\frac{1}{8\,\Bbbk}\,
\varepsilon_{\l\mu\nu\r}\,\varepsilon^{abcd}\,\tilde R\iI{ab}{\l\mu}\,\Theta_c^\nu\,\Theta_d^\r
=\frac{1}{2\,\Bbbk}\,R\,\det\Theta~,$$
where $R$ is the scalar curvature
and the last equality holds if $\Theta$ is non-degenerate.
A calculation gives the $\Theta$- and $\tilde\Gamma$-components of the gravitational part ${\mathcal{E}}\!{}_{\mathrm{g}}$ of the Euler-Lagrange operator:
\begin{align*}
({\mathcal{E}}\!{}_{\mathrm{g}})_\nu^c&=
\frac1{4\,\Bbbk}\,\varepsilon_{\l\mu\nu\r}\,\varepsilon^{abcd}\,R\iI{ab}{\l\mu}\,\th_d^\r~,
\\[6pt]
({\mathcal{E}}\!{}_{\mathrm{g}})\Ii{a}{\l\mu}&=
\frac1{2\,\Bbbk}\,\varepsilon_{\l\mu\nu\r}\,\varepsilon^{abcd}\,
(\partial_b\Theta_c^\nu+\Theta_b^\sigma\,\tilde\Gamma\iIi{c}{\nu}{\sigma}\,)\,\Theta_d^\r~.
\end{align*}
In the non-degenerate case these are essentially the Einstein tensor
and the torsion of the spacetime connection, respectively.
The first, in particular, can be written
$$({\mathcal{E}}\!{}_{\mathrm{g}})_\nu^c=
\frac1{4\,\Bbbk}\,\Theta_{[\l}^a\,\Theta_\mu^b\,\Theta_{\nu]}^c\,\det\Theta
=\frac1\Bbbk\,(R\iI{ab}{bc}-\tfrac{1}{2}\,R\iI{db}{bd}\,\d_a^c)\Theta_\nu^a\,\det\Theta~.$$
The $\tilde\Gamma$-component of ${\mathcal{E}}\!{}_{\mathrm{g}}$ can be expressed
in terms of the torsion as
$$({\mathcal{E}}\!{}_{\mathrm{g}})\Ii{a}{\l\mu}=\frac1{4\,\Bbbk}\,
\varepsilon_{\l\mu\nu\r}\,\varepsilon^{abcd}\,T\Ii{e}{bc}\,\Theta_e^\nu\,\Theta_d^\r~.$$
\subsection{Electromagnetic Lagrangian}\label{s:Electromagnetic Lagrangian}
The electromagnetic potential and the
Maxwell field will be considered independent fields.
The former is represented by a local section
$A:{\boldsymbol{M}}\to\TO^{*}\!{\boldsymbol{M}}$\,,
related to the connection $Y$ induced by $\Cs$ on $\weu{2}{\boldsymbol{U}}$
by the relation $Y=q\,A~.$
The Maxwell field is a section
$\tilde F:{\boldsymbol{M}}\to{\mathbb{L}}^{-2}{\,\otimes\,}\weu2\H^*$,
written in coordinates as
$\tilde F=\tilde F_{\l\mu}\,\tt^\l{\,\otimes\,}\tt^\mu$\,.
The e.m.\ Lagrangian density is defined to be
$${\mathcal{L}}{}_{\mathrm{em}}=\ell{}_{\mathrm{em}}\,\mathrm{d}^4{\mathsf{x}}=\Bigl[
-\tfrac{1}{2}\,\Theta^{(2)}(\mathrm{d} A{\,\otimes\,}\tilde F)+\tfrac{1}{4}\,(\tilde F{\cdot}\tilde F)\Bigr]\,\eta~,$$
with coordinate expression
$$\ell{}_{\mathrm{em}}=
-\tfrac{1}{4}\,\varepsilon^{abcd}\,\varepsilon_{\l\mu\nu\r}\,\partial_aA_b\,\tilde F^{\l\mu}\,\Theta_c^\nu\Theta_d^\r
+\tfrac{1}{4}\,\tilde F^{\a\b}\tilde F_{\a\b}\,\det\Theta~.$$
In the non-degenerate case,
this turns out to be essentially the Lagrangian used in the ADM formalism.
Since $\tilde F$ does not appear in the other terms of the total Lagrangian,
the $\tilde F$-component of the field equations is immediately seen to yield
$$-\tfrac{1}{2}\,\varepsilon^{abcd}\,\varepsilon_{\l\mu\nu\r}\,\partial_aA_b\,\Theta_c^\nu\,\Theta_d^\r
+\tilde F_{\l\mu}\,\det\Theta=0~,$$
which in the non-degenerate case gives
$$F:=\Theta^*\tilde F=2\,\mathrm{d} A
\quad\Rightarrow\quad
{\mathcal{L}}{}_{\mathrm{em}}=-\tfrac{1}{4}\,F^2\,\eta~.$$
The $A$-component of the Euler-Lagrange operator is
\begin{equation*}\begin{split}
({\mathcal{E}}\!{}_{\mathrm{em}})^a&=\tfrac{1}{2}\,\varepsilon^{abcd}\,\varepsilon_{\l\mu\nu\r}\,
(\partial_b\tilde F^{\l\mu}\,\Theta_c^\nu\,\Theta_d^\r+
2\,\tilde F^{\l\mu}\,\partial_b\Theta_c^\nu\,\Theta_d^\r)=\\
&=\tfrac{1}{2}\,\varepsilon^{abcd}\,(\mathrm{d}{*}F)_{bcd}~.
\end{split}\end{equation*}
The $\Theta$-component is
$$({\mathcal{E}}\!{}_{\mathrm{em}})_\nu^c=
-\tfrac{1}{2}\,\varepsilon^{abcd}\,\varepsilon_{\l\mu\nu\r}\,\partial_aA_b\,\tilde F^{\l\mu}\Theta_d^\r
+\tfrac{1}{4}\,\tilde F^2\,{\breve\Theta}_\nu^c~,$$
which in the non-degenerate case becomes
essentially the usual Maxwell stress-energy tensor
$$({\mathcal{E}}\!{}_{\mathrm{em}})_\nu^c=(F_{ab}F^{ac}-\tfrac{1}{4}\,F^2\,\d_b^c){\breve\Theta}_\nu^b~.$$
\subsection{Dirac Lagrangian}\label{s:Dirac Lagrangian}
The Dirac spinor field and its `Dirac adjoint' are sections
\begin{align*}
\psi&=(u,\chi):{\boldsymbol{M}}\to{\mathbb{L}}^{-3/2}{\,\otimes\,}{\boldsymbol{W}}={\mathbb{L}}^{-3/2}{\,\otimes\,}({\boldsymbol{U}}\oplus\cj{\U}{}^\lin)~,\\
\bar\psi&=({\bar\chi},{\bar u}):{\boldsymbol{M}}\to{\mathbb{L}}^{-3/2}{\,\otimes\,}(\U{}^\lin\oplus\cj{\U})={\mathbb{L}}^{-3/2}{\,\otimes\,}\W{}^\lin~.
\end{align*}
In coordinates:
\begin{align*}
& u=u^{\scriptscriptstyle A}\,{\zeta_\sA}~,\quad \chi=\chi_{\sA\.}\,\bzz^\cA~,\quad
u^{\scriptscriptstyle A}\,,\chi_{\sA\.}\,:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^{-3/2}
\\[6pt]
& \bang{\bar\psi,\psi}=({\bar u}^{\sA\.}\,\chi_{\sA\.}+{\bar\chi}_{\scriptscriptstyle A}\,u^{\scriptscriptstyle A}):
{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}{\mathbb{L}}^{-3}~.
\end{align*}
The Dirac operator (\Sec\ref{s:The Dirac operator}) yields a section
$$\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi:{\boldsymbol{M}}\to{\mathbb{L}}^{3/2}{\,\otimes\,}{\boldsymbol{W}}{\,\otimes\,}\weu4\mathrm{T}^*{\boldsymbol{M}}~,$$
so that
$$\bang{\bar\psi,\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi}:{\boldsymbol{M}}\to{\mathbb{C}}{\,\otimes\,}\weu4\mathrm{T}^*{\boldsymbol{M}}~.$$
Now we introduce the scalar density
$${\mathcal{L}}{}_{\scriptscriptstyle{\mathrm{D}}}=
\tfrac{\mathrm{i}}{2}\,\bigl(\bang{\bar\psi,\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi}-\bang{\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\bar\psi,\psi}\bigr)
-m\,\bang{\bar\psi\,,\psi}\,\eta
:{\boldsymbol{M}}\to\weu4\mathrm{T}^*{\boldsymbol{M}}~,$$
where $\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\bar\psi:=\overline{\breve{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi}$\,,
and $m\in{\mathbb{L}}^{-1}$ is the described particle's mass.
This is a version of the Dirac Lagrangian which remains non-singular
when $\Theta$ is degenerate.
In the non-degenerate case one also has
$${\mathcal{L}}{}_{\scriptscriptstyle{\mathrm{D}}}=\bigl[
\tfrac{\mathrm{i}}{2}\,\left(\bang{\bar\psi,{\rlap{\raise1pt\hbox{\,/}}\nabla}\psi}-\bang{{\rlap{\raise1pt\hbox{\,/}}\nabla}\bar\psi,\psi}\right)
-m\,\bang{\bar\psi\,,\psi}\bigr]\,\eta~;$$
in $2$-spinor terms this reads
$${\mathcal{L}}{}_{\scriptscriptstyle{\mathrm{D}}}=\tfrac{\mathrm{i}}{\surd2}{\breve\Theta}\Bigl(
\nabla\! u{\,\otimes\,}{\bar u}-u{\,\otimes\,}\nabla\!{\bar u}+\tilde g^\#({\bar\chi}{\,\otimes\,}\nabla\!\chi-\nabla\!{\bar\chi}{\,\otimes\,}\chi)\Bigr)
-m\,\Bigl(\bang{\chi,{\bar u}}+\bang{{\bar\chi},u}\Bigr)\,\eta~,$$
with the coordinate expression
\begin{multline*}
\ell{}_{\scriptscriptstyle{\mathrm{D}}}=\tfrac{\mathrm{i}}{\surd2}\,{\breve\Theta}^a_{{\sA\cA}}\,\Bigl(\nabla\!_au^{\scriptscriptstyle A}\,{\bar u}^{\sA\.}-u^{\scriptscriptstyle A}\,\nabla\!_a{\bar u}^{\sA\.}
+\varepsilon^{{\scriptscriptstyle A}{\scriptscriptstyle B}}{\bar\varepsilon}^{{\sA\.}{\sB\.}}({\bar\chi}_{\scriptscriptstyle B}\,\nabla\!_a\chi_{\sB\.}-\nabla\!_a{\bar\chi}_{\scriptscriptstyle B}\,\chi_{\sB\.}\,)
\Bigr)
\\
-m\,({\bar\chi}_{\scriptscriptstyle A} u^{\scriptscriptstyle A}+\chi_{\sA\.}\,{\bar u}^{\sA\.}\,)\,\det\Theta~.
\end{multline*}
Next we compute the Euler-Lagrange operator ${\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}}$\,.
The ${\bar u}$-component is
$$({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})_{\sA\.}=\sqrt2\,\mathrm{i}\,{\breve\Theta}^a_{{\sA\cA}}\,\nabla\!_au^{\scriptscriptstyle A}
-m\,\chi_{\sA\.}\,\det\Theta+\tfrac{\mathrm{i}}{\surd2}\,T_{\sA\cA}\,u^{\scriptscriptstyle A}~,$$
where
$T_{\sA\cA}:={\breve\Theta}^a_{\sA\cA}\,T\Ii{b}{ab}$
is used for replacing the term with $\partial_a\Theta_b^\mu$
(see \Sec\ref{s:Tetrad and connections}).
The ${\bar\chi}$-component is
$$({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})^{\scriptscriptstyle A}=
\sqrt2\,\mathrm{i}\,{\breve\Theta}^{a{\sA\cA}}\,\nabla\!_a\chi_{\sA\.}-m\,u^{\scriptscriptstyle A}\,\det\Theta
+\tfrac{\mathrm{i}}{\surd2}\,T^{\sA\cA}\,\chi_{\sA\.}~,$$
with ${\breve\Theta}^{a{\sA\cA}}:={\breve\Theta}^a_{{\sB\cB}}\,\varepsilon^{{\scriptscriptstyle B}{\scriptscriptstyle A}}{\bar\varepsilon}^{{\sB\.}{\sA\.}}$ and
$T^{\sA\cA}:=\varepsilon^{{\scriptscriptstyle B}{\scriptscriptstyle A}}{\bar\varepsilon}^{{\sB\.}{\sA\.}}\,T_{\sB\cB}$\,.
The $\tilde\Gamma$-component is
\begin{multline*}
({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})^a_{\l\mu}=
\tfrac{\mathrm{i}}{4\,\surd2}\,[
({\breve\Theta}^a_{{\scriptscriptstyle A}{\sC\.}}\,\ti{[\l}{{\scriptscriptstyle D}{\sC\.}}\t_{\mu]{\scriptscriptstyle D}{\sA\.}}^{\phantom{a}}
-{\breve\Theta}^a_{{\scriptscriptstyle C}{\sA\.}}\,\ti{[\l}{{\scriptscriptstyle C}{\sD\.}}\t_{\mu]{\scriptscriptstyle A}{\sD\.}}^{\phantom{a}})u^{\scriptscriptstyle A}{\bar u}^{\sA\.}
\\
+({\breve\Theta}^{a{\scriptscriptstyle B}{\sC\.}}\ti{[\l}{{\scriptscriptstyle D}{\sB\.}}\t_{\mu]{\scriptscriptstyle D}{\sC\.}}^{\phantom{a}}
-{\breve\Theta}^{a{\scriptscriptstyle C}{\sB\.}}\ti{[\l}{{\scriptscriptstyle B}{\sD\.}}\t_{\mu]{\scriptscriptstyle C}{\sD\.}}^{\phantom{a}})
{\bar\chi}_{\scriptscriptstyle B}\chi_{\sB\.}\,]~.
\end{multline*}
The $\Theta$-component is
\begin{align*}
({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})^c_\nu&=
\varepsilon^{abcd}\,\varepsilon_{\l\mu\nu\r}\,\Theta_b^\mu\,\Theta_d^\r \Bigl[
\\ &\qquad
\tfrac{\mathrm{i}}{2\,\surd2}\,\Bigl(
\nabla\!_au^{\scriptscriptstyle A}\,{\bar u}^{\sA\.}-u^{\scriptscriptstyle A}\,\nabla\!_a{\bar u}^{\sA\.}
+\varepsilon^{{\scriptscriptstyle B}{\scriptscriptstyle A}}{\bar\varepsilon}^{{\sB\.}{\sA\.}}({\bar\chi}_{\scriptscriptstyle B}\nabla\!_a\chi_{\sB\.}-\nabla\!_a{\bar\chi}_{\scriptscriptstyle B}\chi_{\sB\.}\,)\Bigr)
\t\Ii{\l}{{\sA\cA}}
\\ & \hspace{5.5cm}
-\tfrac{1}{3!}\,m\,({\bar\chi}_{\scriptscriptstyle A}\,u^{\scriptscriptstyle A}+\chi_{\sA\.}\,{\bar u}^{\sA\.}\,)\,
\Theta_a^\l\Bigr]=
\\[8pt]
&=
\tfrac{\mathrm{i}}{4}\,\varepsilon^{abcd}\,\varepsilon_{\l\mu\nu\r}\,\Theta_b^\mu\,\Theta_d^\r
\Bigl(\bar\psi\tilde\gamma^\l\nabla\!_a\psi-\Bar{\tilde\gamma}^\l\nabla\!_a\bar\psi\,\psi\Bigr)
-m\,\bar\psi\psi\,{\breve\Theta}_\nu^c~.
\end{align*}
The $A$-component is simply
$$({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})^a=
\sqrt2\,q\,{\breve\Theta}_{{\sA\cA}}^a\,\Bigl(
u^{\scriptscriptstyle A}\,{\bar u}^{\sA\.}+\varepsilon^{{\scriptscriptstyle B}{\scriptscriptstyle A}}\,{\bar\varepsilon}^{{\sB\.}{\sA\.}}\,{\bar\chi}_{\scriptscriptstyle B}\,\chi_{\sB\.}\Bigr)=
q\,{\breve\Theta}_\l^a\,(\bar\psi\tilde\gamma^\l\psi)~.$$
\subsection{Field equations}\label{s:Field equations}
Having calculated the various pieces of
${\mathcal{E}}={\mathcal{E}}\!{}_{\mathrm{g}}+{\mathcal{E}}\!{}_{\mathrm{em}}+{\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}}$,
writing down the field equations ${\mathcal{E}}=0$ is a simple matter.
These equations are non-singular also when $\Theta$ is degenerate;
in the non-degenerate case one expects this approach to reproduce
essentially the usual Einstein-Cartan-Maxwell-Dirac field equations.
The $\Theta$-component
$$({\mathcal{E}}\!{}_{\mathrm{g}})^c_\nu=-({\mathcal{E}}\!{}_{\mathrm{em}}+{\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})^c_\nu~,$$
corresponds to the Einstein equation;
actually, as already discussed, in the non-degenerate case
the left-hand side is essentially the Einstein tensor,
while the right-hand side can be viewed as the sum of the energy-momentum
tensors of the electromagnetic field and of the Dirac field.
The $\tilde\Gamma$-component gives the equation for torsion
$$({\mathcal{E}}\!{}_{\mathrm{g}})\Ii{a}{\l\mu}=-({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})\Ii{a}{\l\mu}~.$$
From this one sees that the spinor field is a source for torsion,
and that in this context one cannot formulate a torsion-free theory.
It was already seen (\Sec\ref{s:Electromagnetic Lagrangian})
that the $\tilde F$-component reads $F=2\,\mathrm{d} A$ in the non-degenerate case,
and of course this yields the first Maxwell equation $\mathrm{d} F=0$.
The $A$-component is
$$-\tfrac{1}{2}\,\,\varepsilon^{abcd}(\mathrm{d}{*}F)_{bcd}
+q\,{\breve\Theta}_\l^a\,(\bar\psi\tilde\gamma^\l\psi)=0
\quad\text{i.e$.$}\quad
\tfrac{1}{2}\,c\,\varepsilon^{abcd}(\mathrm{d}{*}F)_{bcd}=q\,{\breve\Theta}_\l^a\,(\bar\psi\tilde\gamma^\l\psi)~.$$
In the non-degenerate case this gives the second Maxwell equation
$$\tfrac{1}{2}\,{*}\mathrm{d}{*}F=j~,$$
where $j:{\boldsymbol{M}}\to{\,\otimes\,}\TO^{*}\!{\boldsymbol{M}}$ is the \emph{Dirac current},
with coordinate expression
$$j:=\frac qc\,\Theta_a^\l\,(\bar\psi\tilde\gamma_\l\psi)\,\dO{\mathsf{x}}^a~.$$
The ${\bar u}$- and ${\bar\chi}$-components
$({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})_{\scriptscriptstyle A}=0$ and $({\mathcal{E}}\!{}_{\scriptscriptstyle{\mathrm{D}}})^{\sB\.}=0$
give the following generalized form of the standard \emph{Dirac equation}:
\begin{equation*}\begin{cases}
\sqrt2\,\mathrm{i}\,{\breve\Theta}^a_{{\sA\cA}}\,\nabla\!_au^{\scriptscriptstyle A}-m\,\chi_{\sA\.}\,\det\Theta
+\tfrac{\mathrm{i}}{\surd2}\,T_{\sA\cA}\,u^{\scriptscriptstyle A}=0
\\[8pt]
\sqrt2\,\mathrm{i}\,{\breve\Theta}^{a{\sA\cA}}\,\nabla\!_a\chi_{\sA\.}-m\,u^{\scriptscriptstyle A}\,\det\Theta
+\tfrac{\mathrm{i}}{\surd2}\,T^{\sA\cA}\,\chi_{\sA\.}=0
\end{cases}\quad.\end{equation*}
Denoting by $\breve T$ the $1$-form obtained from the torsion by contraction,
with coordinate expression $\breve T_a\equiv T_a=T\Ii{b}{ab}$\,,
the above equation can be written in coordinate-free form as
$$\left(\mathrm{i}\,{\rlap{\raise1pt\hbox{\,/}}\nabla}-m+\tfrac{\mathrm{i}}{2}\,\gamma^\#(\breve T)\right)\psi=0~.$$
\section{Dirac algebra in two-spinor terms}
\label{S:Dirac algebra in two-spinor terms}
\subsection{Dirac algebra}\label{s:Dirac algebra}
If ${\boldsymbol{V}}$ is a finite-dimensional real vector space
endowed with a non-degenerate scalar product,
then its \emph{Clifford algebra} ${\boldsymbol{C}}({\boldsymbol{V}})$
is the associative algebra generated by ${\boldsymbol{V}}$
where the product of any $u,v\in{\boldsymbol{V}}$ is subjected to the condition
$$u\,v+v\,u=2\,u\cdot v~, \quad u,v\in{\boldsymbol{V}}~.$$
The Clifford algebra fulfills the following \emph{universal property}:
if ${\boldsymbol{A}}$ is an associative algebra with unity and $\gamma:{\boldsymbol{V}}\to{\boldsymbol{A}}$
is a linear map such that
$\gamma(v)\,\gamma(v)=v\cdot v~\forall\:v\in{\boldsymbol{V}}$\,,
then $\gamma$ extends to a unique homomorphism $\hat\gamma:{\boldsymbol{C}}({\boldsymbol{V}})\to{\boldsymbol{A}}$\,.
It turns out that ${\boldsymbol{C}}({\boldsymbol{V}})$ is isomorphic, as a vector space,
to the vector space underlying the exterior algebra $\wedge{\boldsymbol{V}}$\,;
through this isomorphism one identifies $v_1{\,\wedge\,} \dots{\,\wedge\,} v_p$
with the antisymmetrized Clifford product
$$\frac1{p!}\,\bigl(
v_1 v_2{\cdot}{\cdot}v_p-v_2 v_1{\cdot}{\cdot}v_p+\,\cdots~\bigr)$$
where the sum is extended to all permutations of the set $\{1,\dots,p\}$\,,
with the appropriate signs.
In other terms,
one has two distinct algebras on the same underlying vector space:
any element of ${\boldsymbol{C}}({\boldsymbol{V}})$ can be uniquely expressed as a sum of terms,
each of well-defined exterior degree.
For example, one has $u\,v=u{\,\wedge\,} v+u\cdot v$\,;
from this one sees that the Clifford algebra product does not preserve
the exterior algebra degree, but only its parity:
${\boldsymbol{C}}({\boldsymbol{V}})$ is ${\mathbb{Z}}_2$-graded.
If $\phi\in\weu{r}{\boldsymbol{V}}$, $\th\in\weu{s}{\boldsymbol{V}}$,
then the Clifford product $\phi\,\th$ turns out to be a sum of terms of
exterior degree $r{+}s,~r{+}s{-}2~,\dots,|r{-}s|$.
\bigbreak
The Clifford algebra ${\boldsymbol{D}}:={\boldsymbol{C}}(\H)$ of Minkowski space $\H$
(\Sec\ref{s:2-spinors and Minkowski space})
is called the \emph{Dirac algebra}.
The Dirac map $\gamma:\H\to\operatorname{End}({\boldsymbol{W}})$ is a Clifford map,
hence by virtue of the above said universal property
one can see the Dirac algebra
as a real vector subspace ${\boldsymbol{D}}\subset\operatorname{End}({\boldsymbol{W}})$ of dimension $2^4=16$\,.
Since this coincides with the \emph{complex} dimension
of $\operatorname{End}({\boldsymbol{W}})\equiv{\boldsymbol{W}}{\,\otimes\,}\W{}^\lin$, one gets $\operatorname{End}({\boldsymbol{W}})={\mathbb{C}}{\,\otimes\,}{\boldsymbol{D}}$\,.
The Dirac algebra ${\boldsymbol{D}}$ is multiplicatively generated
by $\gamma(\H)\subset\operatorname{End}({\boldsymbol{W}})$\,, simply identified with $\H$.
One has the natural decompositions
$${\boldsymbol{D}}={\boldsymbol{D}}^{(\!{+}\!)}\oplus{\boldsymbol{D}}^{(\!{-}\!)}=
\bigl({\mathbb{R}}\oplus\weu2\H\oplus\weu4\H\bigr)\oplus\bigl(\H\oplus\weu3\H\bigr)~,$$
where ${\boldsymbol{D}}^{(\!{+}\!)}$ and ${\boldsymbol{D}}^{(\!{-}\!)}$
denote the even-degree and odd-degree subspaces, respectively
(the former is a subalgebra).
Also, one has the distinguished elements
$$1\equiv\Id{{\boldsymbol{W}}}\subset{\mathbb{R}}\subset{\boldsymbol{D}}^{(\!{+}\!)}~,\quad
\eta^\#\subset\weu4\H\subset{\boldsymbol{D}}^{(\!{+}\!)}~,$$
where $\eta^\#\equiv g^\#(\eta)$ is the contravariant tensor corresponding
to the unimodular volume form $\eta$\,.
One gets
$$\eta^\#\,\eta^\#=-1~,\quad
\vartheta\,\eta^\#={*}\vartheta~~\forall\vartheta\in\wedge\H~,$$
where ${*}$ is the Hodge isomorphism.
\subsection{Decomposition of $\operatorname{End}{\boldsymbol{W}}$ and $\varepsilon$-transposition}
\label{s:Decomposition of EndW and e-transposition}
One has the natural decomposition
$$\operatorname{End}({\boldsymbol{W}})\equiv\operatorname{End}({\boldsymbol{U}}\oplus\cj{\U}{}^\lin)=
({\boldsymbol{U}}{\,\otimes\,}\U{}^\lin)\oplus({\boldsymbol{U}}{\,\otimes\,}\cj{\U})\oplus(\cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin)\oplus(\cj{\U}{}^\lin{\,\otimes\,}\cj{\U})~.$$
Accordingly, any $\Phi\in\operatorname{End}({\boldsymbol{W}})$ is a 4-uple of tensors,
which will be conveniently written in matricial form as
$$\Phi=\begin{pmatrix}K&P\{\boldsymbol{Q}}&J\end{pmatrix}~,\qquad
K\in{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin\,,~P\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U}\,,~Q\in\cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin\,,~J\in\cj{\U}{}^\lin{\,\otimes\,}\cj{\U}\,.$$
We now introduce an operation which acts on each
of the above 4 types of tensors in a similar way.
This operation, called \emph{$\varepsilon$-transposition},
is actually independent of the particular normalized $\varepsilon\in\weu2\U{}^\lin$ chosen;
it is defined by
\begin{align*}
& {\boldsymbol{U}}{\,\otimes\,}\U{}^\lin\to\U{}^\lin{\,\otimes\,}{\boldsymbol{U}}:K\mapsto\tilde K:=\bang{\varepsilon^\flat{\,\otimes\,}\varepsilon^\#, K}=
\varepsilon_{{\scriptscriptstyle C}{\scriptscriptstyle A}}\,K\Ii{\scriptscriptstyle C}{\scriptscriptstyle D}\,\varepsilon^{{\scriptscriptstyle D}{\scriptscriptstyle B}}\,{\mathsf{z}}^\sA{\,\otimes\,}{\zeta_\sB}~,
\\[6pt]
& {\boldsymbol{U}}{\,\otimes\,}\cj{\U}\to\U{}^\lin{\,\otimes\,}\cj{\U}{}^\lin:P\mapsto\tilde P:=\bang{\varepsilon^\flat{\,\otimes\,}{\bar\varepsilon}^\flat, P}=
\varepsilon_{{\scriptscriptstyle C}{\scriptscriptstyle A}}\,P^{{\scriptscriptstyle C}{\sD\.}}\,{\bar\varepsilon}_{{\sD\.}{\sB\.}}\,{\mathsf{z}}^\sA{\,\otimes\,}\bzz^\cB~,
\\[6pt]
& \cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin\to\cj{\U}{\,\otimes\,}{\boldsymbol{U}}:Q\mapsto\tilde Q:=\bang{{\bar\varepsilon}^\#{\,\otimes\,}\varepsilon^\#, Q}=
{\bar\varepsilon}^{{\sC\.}{\sA\.}}\,Q_{{\sC\.}{\scriptscriptstyle D}}\,\varepsilon^{{\scriptscriptstyle D}{\scriptscriptstyle B}}\,{\zeta_\sA}{\,\otimes\,}{\bze_\cB}~,
\\[6pt]
& \cj{\U}{}^\lin{\,\otimes\,}\cj{\U}\to\cj{\U}{\,\otimes\,}\cj{\U}{}^\lin:J\mapsto\tilde J:=\bang{{\bar\varepsilon}^\#{\,\otimes\,}{\bar\varepsilon}^\flat, J}=
{\bar\varepsilon}^{{\sC\.}{\sA\.}}\,J\iI{\sC\.}{\sD\.}\,{\bar\varepsilon}_{{\sD\.}{\sB\.}}\,{\bze_\cA}{\,\otimes\,}\bzz^\cB~.
\end{align*}
Namely, \emph{$\varepsilon$-transposition} changes the position (either high or low)
of both indices of the tensor it acts on.
For elements in ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$ or $\cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin$ it essentially amounts
to index lowering (resp.\ raising) by the Lorentz metric $g$
in complexified Minkowski space;
for invertible elements
in ${\boldsymbol{U}}{\,\otimes\,}\U{}^\lin\equiv\operatorname{End}({\boldsymbol{U}})$ or $\cj{\U}{}^\lin{\,\otimes\,}\cj{\U}\equiv\operatorname{End}(\cj{\U}{}^\lin)$,
$\varepsilon$-transposition amounts to
$$\tilde X=(\det X)\,(X^{-1})^{\scriptscriptstyle\bigstar}~,$$
where the superscript ${\scriptscriptstyle\bigstar}$ denotes standard transposition.
It is clear that $\varepsilon$-transposition can be similarly defined\footnote{
One could introduce $\varepsilon$-transposition on further spaces
such as ${\boldsymbol{U}}{\,\otimes\,}{\boldsymbol{U}}$, ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}{}^\lin$ and so on.
These extensions however would depend from the chosen normalized $\varepsilon$\,;
phase factors cancel out only in the considered cases.}
on $\U{}^\lin{\,\otimes\,}{\boldsymbol{U}}$, $\U{}^\lin{\,\otimes\,}\cj{\U}{}^\lin$, $\cj{\U}{\,\otimes\,}{\boldsymbol{U}}$ and $\cj{\U}{\,\otimes\,}\cj{\U}{}^\lin$,
and in all cases one gets
$$\Tilde{\Tilde X}=X~,\quad (\tilde X)^{\scriptscriptstyle\bigstar}=(X^{\scriptscriptstyle\bigstar})^{\scriptscriptstyle\sim}~,\quad
\tilde X\,X^{\scriptscriptstyle\bigstar}=X^{\scriptscriptstyle\bigstar} \tilde X=(\det X)\,{1\!\!1}~,\quad \det X=\det\tilde X~.$$
\begin{remark}
The determinant is uniquely defined, via any $\varepsilon$\,,
also for elements in ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$, $\U{}^\lin{\,\otimes\,}\cj{\U}{}^\lin$, $\cj{\U}{\,\otimes\,}{\boldsymbol{U}}$ and $\cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin$.
In these cases, the determinant of a tensor
equals one-half its Lorentz pseudo-norm.
\end{remark}\smallbreak
Moreover, whenever the composition of tensors $X$ and $Y$ is defined, one has
$$(X\,Y)^{\scriptscriptstyle\sim}=\tilde X\,\tilde Y~,\quad
\operatorname{Tr}(\tilde X\,\tilde Y)=\operatorname{Tr}(X\,Y)~.$$
Whenever $A$ and $B$ are tensors of the same type, one has
$$\det(A+B)=\det(A)+\det(B)+\operatorname{Tr}(A^{\scriptscriptstyle\bigstar}\tilde B)~,$$
where the \emph{scalar product} $(A,B)\mapsto\operatorname{Tr}(A^{\scriptscriptstyle\bigstar}\tilde B)$
is \emph{symmetric}.\footnote{
On ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$ and $\cj{\U}{\,\otimes\,}{\boldsymbol{U}}$
(resp.\ $\U{}^\lin{\,\otimes\,}\cj{\U}{}^\lin$ and $\cj{\U}{}^\lin{\,\otimes\,}\U{}^\lin$)
this coincides with $2\,g$ (resp.\ $2\,g^\#$).}
\begin{proposition}\label{p:inverseofPhi}
Let $\Phi=\begin{pmatrix}K&P\\ Q&J\end{pmatrix}\in{\boldsymbol{W}}{\,\otimes\,}\W{}^\lin$ be non-singular.
Then
\begin{align*}
& \det\Phi=(\det K)\,(\det J)+(\det P)\,(\det Q)
-\operatorname{Tr}(K^{\scriptscriptstyle\bigstar}\,\tilde P\,J^{\scriptscriptstyle\bigstar}\,\tilde Q)~,
\\[8pt]
& (\det\Phi)\,\Phi^{-1}=\begin{pmatrix}
(\det J)\,\tilde K{}^{\scriptscriptstyle\bigstar}-\tilde Q{}^{\scriptscriptstyle\bigstar}\,J\,\tilde P{}^{\scriptscriptstyle\bigstar} &&
(\det P)\,\tilde Q{}^{\scriptscriptstyle\bigstar}-\tilde K{}^{\scriptscriptstyle\bigstar}\,P\,\tilde J{}^{\scriptscriptstyle\bigstar} \\[6pt]
(\det Q)\,\tilde P{}^{\scriptscriptstyle\bigstar}-\tilde J{}^{\scriptscriptstyle\bigstar}\,Q\,\tilde K{}^{\scriptscriptstyle\bigstar} &&
(\det K)\,\tilde J{}^{\scriptscriptstyle\bigstar}-\tilde P{}^{\scriptscriptstyle\bigstar}\,K\,\tilde Q{}^{\scriptscriptstyle\bigstar}
\end{pmatrix}~.
\end{align*}
\end{proposition}
{\sc proof:~} It can be checked by a direct calculation,
taking into account the above identities.\EndBox{\square}
\subsection{$\varepsilon$-adjoint and characterization of ${\boldsymbol{D}}$}
\label{s:e-adjoint and characterization of D}
If $X$ is a tensor of any of the above types, then its \emph{$\varepsilon$-adjoint}
is the tensor
$$X^\ddag:=\tb X~.$$
Using this operation one defines the real involution
$$\ddag:{\boldsymbol{W}}{\,\otimes\,}\W{}^\lin\to{\boldsymbol{W}}{\,\otimes\,}\W{}^\lin:
\begin{pmatrix}K & P \\ Q & J\end{pmatrix}\mapsto
\begin{pmatrix}J^\ddag & Q^\ddag \\ P^\ddag & K^\ddag \end{pmatrix}~.$$
\begin{proposition}
${\boldsymbol{D}}$ and $\mathrm{i}\,{\boldsymbol{D}}$ are the eigenspaces of $\ddag$
corresponding to eigenvalues $+1$ and $-1$\,, respectively.
Namely, ${\boldsymbol{D}}$ is the real subspace of ${\boldsymbol{W}}{\,\otimes\,}\W{}^\lin$ constituted by
all endomorphisms which can be written in the form
$$\begin{pmatrix}K & P \\ P^\ddag & K^\ddag \end{pmatrix}~,\qquad
K\in{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin~,~~P\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U}~.$$
Moreover one has the following characterisations
\begin{align*}
&{\boldsymbol{D}}^0\equiv{\mathbb{R}}=\Bigl\{r\begin{pmatrix}\Id{{\boldsymbol{U}}} & 0 \\ 0& \Id{\cj{\U}{}^\lin}\end{pmatrix}~,
\quad r\in{\mathbb{R}}\Bigr\}~,
\displaybreak[2]\\[8pt]
&{\boldsymbol{D}}^1\equiv\H=\Bigl\{\begin{pmatrix}0 & P \\ P^\ddag & 0\end{pmatrix}~,
\quad P\in\H\Bigr\}~,
\displaybreak[2]\\[8pt]
&{\boldsymbol{D}}^2\equiv\weu2\H=\Bigl\{\begin{pmatrix}K & 0 \\ 0 & K^\ddag\end{pmatrix}~,
\quad K\in{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin\,,~\operatorname{Tr} K=0\Bigr\}~,
\displaybreak[2]\\[8pt]
&{\boldsymbol{D}}^3\equiv\weu3\H=\Bigl\{\begin{pmatrix}0 & P \\ P^\ddag & 0\end{pmatrix}~,
\quad P\in\mathrm{i}\,\H\Bigr\}~,
\displaybreak[2]\\[8pt]
&{\boldsymbol{D}}^4\equiv\weu4\H=
\Bigl\{\mathrm{i}\,r\begin{pmatrix}\Id{{\boldsymbol{U}}} & 0 \\ 0& -\Id{\cj{\U}{}^\lin}\end{pmatrix}~,
\quad r\in{\mathbb{R}}\Bigr\}~,
\displaybreak[2]\\[8pt]
&{\boldsymbol{D}}^{(\!{+}\!)}={\boldsymbol{D}}^0\oplus{\boldsymbol{D}}^2\oplus{\boldsymbol{D}}^4=
\Bigl\{\begin{pmatrix}K & 0 \\ 0 & K^\ddag\end{pmatrix}~,
\quad K\in{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin\Bigr\}~,
\displaybreak[2]\\[8pt]
&{\boldsymbol{D}}^{(\!{-}\!)}={\boldsymbol{D}}^1\oplus{\boldsymbol{D}}^3=
\Bigl\{\begin{pmatrix}0 & P \\ P^\ddag & 0\end{pmatrix}~,
\quad P\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U} \Bigr\}~.
\end{align*}
\end{proposition}
{\sc proof:~}
The Dirac map $\gamma:\H\to\operatorname{End}{\boldsymbol{W}}$ can be written as
$$\gamma:v\mapsto\begin{pmatrix}0&\sqrt2\,v \\ \sqrt2\,v^\ddag & 0\end{pmatrix}~,$$
whence the characterization of ${\boldsymbol{D}}^1$.
It immediately follows that
${\boldsymbol{D}}^{(\!{+}\!)}$ is constituted by diagonal-block elements,
while ${\boldsymbol{D}}^{(\!{-}\!)}$ is constituted by off-diagonal-block elements.
The other characterizations can be checked by matrix calculations.\EndBox{\square}
\section{Clifford group and its subgroups}
\label{S:Clifford group and its subgroups}
\subsection{Clifford group}\label{s:Clifford group}
Let ${\boldsymbol{D}}^{\bullet}:={\boldsymbol{D}}\cap\operatorname{Aut}{\boldsymbol{W}}$ be the group of all invertible
elements in ${\boldsymbol{D}}$.
The Clifford group $\mathrm{Cl}\equiv\mathrm{Cl}({\boldsymbol{W}})$ is defined to be~\cite{Cr,Gr}
the subgroup of ${\boldsymbol{D}}^{\bullet}$ under whose adjoint action $\H$ is stable.
In other terms, $\Phi\in{\boldsymbol{D}}^\bullet$ is an element of $\mathrm{Cl}$ iff
$$\operatorname{Ad}[\Phi]v\equiv\Phi\,\gamma(v)\,\Phi^{-1}\in\gamma(\H)~,\quad\forall\:v\in\H~.$$
Using proposition~\ref{p:inverseofPhi} we write the adjoint action as
\begin{align*}
(\det\Phi)\,\operatorname{Ad}[\Phi]v &=
\begin{pmatrix}K && P \\[6pt] P^\ddag && K^\ddag \end{pmatrix}
\begin{pmatrix}0 && V \\[6pt] V^\ddag && 0\end{pmatrix}
\begin{pmatrix}X && Y \\[6pt] Y^\ddag && X^\ddag \end{pmatrix}=
\\[12pt]
&=\begin{pmatrix}
P\,V^\ddag\,X+K\,V\,Y^\ddag && P\,V^\ddag\,Y+K\,V\,X^\ddag
\\[8pt]
K^\ddag\,V^\ddag\,X+P^\ddag\,V\,Y^\ddag &&
K^\ddag\,V^\ddag\,Y+P^\ddag\,V\,X^\ddag
\end{pmatrix}~,
\end{align*}
where $V\equiv\sqrt2\,v$ and
\begin{align*}
& X\equiv (\det{\bar K})\,\tilde K{}^{\scriptscriptstyle\bigstar}-\bar P{}^{\scriptscriptstyle\bigstar}\,\tb K\,\tilde P{}^{\scriptscriptstyle\bigstar}~,
&& Y\equiv (\det P)\,\bar P{}^{\scriptscriptstyle\bigstar}-\tilde K{}^{\scriptscriptstyle\bigstar}\,P\,\bar K{}^{\scriptscriptstyle\bigstar}~,
\\[8pt]
& X^\ddag=(\det K)\,\bar K{}^{\scriptscriptstyle\bigstar}-\tilde P{}^{\scriptscriptstyle\bigstar}\,K\,\bar P{}^{\scriptscriptstyle\bigstar}~,
&& Y^\ddag=(\det\bar P)\,\tilde P{}^{\scriptscriptstyle\bigstar}-\bar K{}^{\scriptscriptstyle\bigstar}\,\tb P\,\tilde K{}^{\scriptscriptstyle\bigstar}~.
\end{align*}
\begin{lemma}\label{l:Cleitheroddoreven}
An element of ${\boldsymbol{D}}^\bullet$ which belongs to the Clifford group
is necessarily either odd or even,
so that the Clifford group is the disjoint union $\mathrm{Cl}=\mathrm{Cl}^{(\!{+}\!)}\cup\mathrm{Cl}^{(\!{-}\!)}$
where $\mathrm{Cl}^{(\!{+}\!)}\equiv\mathrm{Cl}\cap{\boldsymbol{D}}^{(\!{+}\!)}$\,,
$\mathrm{Cl} ^{(\!{-}\!)}\equiv\mathrm{Cl}\cap{\boldsymbol{D}}^{(\!{-}\!)}$\,.
\end{lemma}
{\sc proof:~}
If $\Phi$ is in $\mathrm{Cl}$ then the ${\boldsymbol{U}}{\,\otimes\,}\U{}^\lin$-component of $\operatorname{Ad}[\Phi]v$
vanishes for all $v\in\H$, namely
$$K\,V\,\tb Y=-P\,\tb V\,X~,\quad \forall\:V\in\H~.$$
Composing both sides with $\tilde V{}^{\scriptscriptstyle\bigstar}\,\tilde K{}^{\scriptscriptstyle\bigstar}$ on the left
and with $\tilde X{}^{\scriptscriptstyle\bigstar}$ on the right one finds
$$(\det{K})\,(\det{V})\,\tb Y\,\tilde X{}^{\scriptscriptstyle\bigstar}=
-(\det\Phi)(\det\bar K)\,\tilde V{}^{\scriptscriptstyle\bigstar}\,\tilde K{}^{\scriptscriptstyle\bigstar}\,P\,\tb V~.$$
Now the above equality is certainly fulfilled in the particular case
when $\det K=0$\,.
Suppose $\det K\neq0$ for the moment
(the other case will be considered later).
The left-hand side vanishes for all null elements $V\in\H$,
thus also $\tilde V{}^{\scriptscriptstyle\bigstar}\,\tilde K{}^{\scriptscriptstyle\bigstar}\,P\,\tb V$ vanishes
for all null vectors $V$\,;
it's not difficult to see that this implies $\tilde K{}^{\scriptscriptstyle\bigstar}\,P=0$\,,
which on turn implies $P=0$\,.
Summarizing, if $\Phi\in\mathrm{Cl}$ and $\det K\neq0$ then $P=0$\,.
By a similar argument,
composing the equation $K\,V\,\tb Y=-P\,\tb V\,X$
on the left by $\bar V{}^{\scriptscriptstyle\bigstar}\,\tilde P{}^{\scriptscriptstyle\bigstar}$
and on the right by $\bar Y{}^{\scriptscriptstyle\bigstar}$,
one finds that if $\Phi\in\mathrm{Cl}$ and $\det P\neq0$ then $K=0$\,.
The case which remains to be considered is that when $\det K=\det P=0$\,.
Since $\det P=\tfrac{1}{2}\,g(P,P)$\,, $P$ is an isotropic element of ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$,
and as such it is decomposable.
Similarly, $K$ is decomposable. Namely one can write
$$K=k{\,\otimes\,}\l~,~~P=p{\,\otimes\,}{\bar q}~,~~V=s{\,\otimes\,}{\bar s}~,\qquad k,p,q,s\in{\boldsymbol{U}},~\l\in\U{}^\lin~.$$
A little two-spinor algebra then yields
\begin{align*}
& P\,\tb V\,X+K\,V\,\tb Y={\bar\varepsilon}(\bar k,\bar p)\,\Bigl[\,
\bang{\l,q}\,|\bang{\l,s}|^2\,k{\,\otimes\,} k^\flat
-\bang{{\bar\lambda},{\bar q}}\,|\varepsilon(s,q)|^2\,p{\,\otimes\,} p^\flat\,\Bigr]~,
\\[6pt]
& \det\Phi=-\operatorname{Tr}(K\,\bar P{}^{\scriptscriptstyle\bigstar}\,\tb K\,\tilde P{}^{\scriptscriptstyle\bigstar}\,)=
|\varepsilon(k,p)|^2\,|\bang{\l,q}|^2~.
\end{align*}
Now one sees that in order that $\det\Phi\neq0$ one must have
$\bang{\l,q}\neq0$ and $\varepsilon(k,p)\neq0$\,.
Thus $k{\,\otimes\,} k^\flat$ and $p{\,\otimes\,} p^\flat$
are linearly independent elements of ${\boldsymbol{U}}{\,\otimes\,}\U{}^\lin$
and, in order that $P\,\tb V\,X+K\,V\,\tb Y$ vanishes for all $V$,
one must have $\bang{\l,s}=\varepsilon(q,s)$ for all $s\in{\boldsymbol{U}}$\,,
which implies $\l=0$ and $q=0$ that is $K=0$ and $P=0$\,,
a contradiction.
Thus the case $\det K=\det P=0$ cannot yield an element $\Phi\in\mathrm{Cl}$\,.
\EndBox{\square}
\begin{proposition}~\\
$a)$~$\mathrm{Cl}^{(\!{+}\!)}$ is the $7$-dimensional real submanifold of ${\boldsymbol{D}}^{(\!{+}\!)}$
constituted of all elements in ${\boldsymbol{W}}{\,\otimes\,}\W{}^\lin$ which are of the type
$$\sKM~,\quad K\in{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin\,,~\det K\in{\mathbb{R}}\setminus\{0\}~.$$
$b)$~$\mathrm{Cl}^{(\!{-}\!)}$ is the $7$-dimensional real submanifold of ${\boldsymbol{D}}^{(\!{-}\!)}$
constituted of all elements in ${\boldsymbol{W}}{\,\otimes\,}\W{}^\lin$ which are of the type
$$\sPM~,\quad P\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U}\,,~\det P\in{\mathbb{R}}\setminus\{0\}~.$$
\end{proposition}
{\sc proof:~}\\
$a)$~Let $\Phi=\sKM$\,, $K\in{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin$, $\det K\neq0$\,.
Then
$$(\det\Phi)\,\operatorname{Ad}[\Phi]v=
\begin{pmatrix}0 && (\det K)\,K\,V\,\bar K{}^{\scriptscriptstyle\bigstar} \\[8pt]
(\det\bar K)\tb K\,\tb V\,\,\tilde K{}^{\scriptscriptstyle\bigstar} && 0 \end{pmatrix}~,\quad
V\equiv\sqrt2\,v\in\H~.$$
For $\operatorname{Ad}[\Phi]v$ to be in $\H$,
the two non-zero entries of the above matrix must be in $\H\equiv{\boldsymbol{U}}{\,\bar\vee\,}\cj{\U}$
and in $\cj{\U}{}^\lin{\,\bar\vee\,}\U{}^\lin$, respectively.
Consider the ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$-entry.
Since $\bar V=V^{\scriptscriptstyle\bigstar}$ because $V$ is Hermitian, one finds
$$[(\det K)\,K\,V\,\bar K{}^{\scriptscriptstyle\bigstar}]^{\overline{\scriptscriptstyle\bigstar}}=
(\det\bar K)\,K\,V\,\bar K{}^{\scriptscriptstyle\bigstar}~,$$
and $(\det K)\,K\,V\,\bar K{}^{\scriptscriptstyle\bigstar}$ is Hermitian
for all $V\in\H$ iff $\det K=\det\bar K$
(this argument gives the same result for the other non-zero entry).
\\[6pt]
$b)$~Let $\Phi=\sPM$\,, $P\in{\boldsymbol{U}}{\,\otimes\,}\cj{\U}$, $\det P=\tfrac{1}{2}\,g(P,P)\neq0$\,. Then
$$(\det\Phi)\,\operatorname{Ad}[\Phi]v=
\begin{pmatrix}0 && (\det P)\,P\,\tb V\,\bar P{}^{\scriptscriptstyle\bigstar} \\[8pt]
(\det\bar P)\tb P\,V\,\,\tilde P{}^{\scriptscriptstyle\bigstar} && 0 \end{pmatrix}~.$$
By the same argument as before, $\Phi\in\mathrm{Cl}$ iff $\det P=\det\bar P$\,.\EndBox{\square}
Now it is not difficult to show that any complex $2\times2$-matrix
with real determinant can be written as a product of Hermitian matrices.
Using this, one recovers a well-known result:
\begin{proposition}\label{p:Clmgenerated}
$\mathrm{Cl}$ is multiplicatively generated by $\H^\bullet\subset\H$,
the subset of all elements in $\H$ with non-vanishing Lorentz pseudo-norm.
\end{proposition}
Namely any element of $\mathrm{Cl}$ can be written as
$$\Phi=v_1\,v_2\,\dots\,v_n~,\quad v_j\in\H,~g(v_j,v_j)\neq0~;$$
its inverse is
$$\Phi^{-1}=\frac1{\nu(\Phi)}\,v_n\,\dots\,v_2\,v_1~,\quad
\nu(\Phi):=g(v_1,v_1)\,g(v_2,v_2)\,\dots\,g(v_n,v_n)~.$$
Setting now $V_i\equiv\sqrt2\,v_i$ one has $\det V_i=\det\tb V_i=g(v_i,v_i)$\,,
hence
$$\nu(\Phi)=\det\bigl(V_1\,\tb V_2\,V_3\,\tb V_4\,\dots\bigr)
=\mathop{\Pi}\limits_{i=1}^n \det(V_i)~.$$
Namely, if $\Phi=\sKM\in\mathrm{Cl}^{(\!{+}\!)}$
then $\nu(\Phi)=\det K=\det K^\ddag$\,;
if $\Phi=\sPM\in\mathrm{Cl}^{(\!{-}\!)}$
then $\nu(\Phi)=\det P=\det P^\ddag$\,.
\begin{remark}
Actually, it can be seen that any complex $2\times2$-matrix
with real determinant can be written as a product
of just \emph{three} Hermitian matrices
(but not, in general, of two matrices).
This implies that an element in $\mathrm{Cl}^{(\!{-}\!)}$ can be written as $\sPM$
with $P=V_1\,V_2^\ddag\,V_3$\,,
and an element in $\mathrm{Cl}^{(\!{+}\!)}$ can be written as $\sKM$
with $K=V_1\,V_2^\ddag\,V_3\,V_4^\ddag$\,,
$V_i\in\H^\bullet$\,.
\end{remark}\bigbreak
The adjoint action of any $w\in\H$ on $\H$ is easily checked to be
the negative of the reflection
through the hyperplane orthogonal to $w$\,.
It follows that $\mathrm{Cl}^{(\!{+}\!)}$ is the subgroup of all elements in $\mathrm{Cl}$
whose adjoint action preserves the orientation of $\H$.
Moreover, the subgroup
$$\mathrm{Cl}^{\scriptscriptstyle\uparrow}:=\{\Phi\in\mathrm{Cl}:\nu(\Phi)>0\,\}$$
is constituted of all elements of $\mathrm{Cl}$
whose adjoint action preserves the time-orientation of $\H$.
Its representation as $\Phi=v_1\,v_2\,\dots\,v_n$
has an even number of spacelike factors
and any number of timelike factors.
\bigbreak
The unit element of $\mathrm{Cl}$ is ${1\!\!1}\in{\boldsymbol{D}}^{(\!{+}\!)}\subset{\boldsymbol{D}}$.
Thus the Lie algebra of $\mathrm{Cl}$ is a 7-dimensional vector subspace
$$\mathfrak{L}\mathrm{Cl}\subset{\boldsymbol{D}}^{(\!{+}\!)}={\mathbb{R}}\oplus\weu2\H\oplus\weu4\H
\equiv {\mathbb{R}}\,{1\!\!1}\oplus\hat\gamma(\weu2\H)\oplus\hat\gamma(\weu4\H)~.$$
Now observe that $\weu4\H$ is not contained in $\mathfrak{L}\mathrm{Cl}$ since
$$t\in{\mathbb{R}}{\quad\Rightarrow\quad}\exp(t\,\eta^\#)=
\exp\begin{pmatrix} -\mathrm{i}\,t\,\Id{{\boldsymbol{U}}} & 0 \\
0 & \mathrm{i}\,t\,\Id{\cj{\U}{}^\lin} \end{pmatrix}
=\begin{pmatrix}\mathrm{e}^{-\mathrm{i}\,t}\,\Id{{\boldsymbol{U}}} & 0 \\
0 & \mathrm{e}^{\mathrm{i}\,t}\,\Id{\cj{\U}{}^\lin} \end{pmatrix}$$
is not in $\mathrm{Cl}$ because the two component endomorphsims
$\mathrm{e}^{-\mathrm{i}\,t}\,\Id{{\boldsymbol{U}}}\in{\boldsymbol{U}}{\,\otimes\,}\U{}^\lin$ and $\mathrm{e}^{\mathrm{i}\,t}\,\Id{\cj{\U}{}^\lin}\in\cj{\U}{}^\lin{\,\otimes\,}\cj{\U}$
have non-real determinant.
Hence, just by a dimension argument, one finds
$$\mathfrak{L}\mathrm{Cl}={\mathbb{R}}\oplus\weu2\H~.$$
\subsection{$\mathrm{Pin}$ and $\mathrm{Spin}$}\label{s:Pin and Spin}
If $\Phi\in\mathrm{Cl}$ and $a\in{\mathbb{R}}\setminus\{0\}$
then $\operatorname{Ad}[a\,\Phi]=\operatorname{Ad}[\Phi]:\H\to\H$.
It is then natural to consider the subgroup
$$\mathrm{Pin}:=\{\Phi\in\mathrm{Cl}:\nu(\Phi)=\pm1\}~,$$
which is multiplicatively generated by all elements in $\H$
whose Lorentz pseudo-norm is $\pm1$\,.
It has the subgroups
\begin{align*}
& \mathrm{Spin}:=\mathrm{Pin}^{(\!{+}\!)}\equiv\mathrm{Pin}\cap\mathrm{Cl}^{(\!{+}\!)}=\{\Phi\in\mathrm{Cl}^{(\!{+}\!)}:\nu(\Phi)=\pm1\}~,
\\[6pt]
& \mathrm{Pin}^{\scriptscriptstyle\uparrow}:=\mathrm{Pin}\cap\mathrm{Cl}^{\scriptscriptstyle\uparrow}=\{\Phi\in\mathrm{Cl}:\nu(\Phi)=1\}~,
\\[6pt]
& \mathrm{Spin}^{\scriptscriptstyle\uparrow}:=\mathrm{Spin}\cap\mathrm{Cl}^{\scriptscriptstyle\uparrow}=\{\Phi\in\mathrm{Cl}^{(\!{+}\!)}:\nu(\Phi)=1\}~.
\end{align*}
These share the same Lie algebra
$$\weu2\H=\mathfrak{L}\mathrm{Pin}=\mathfrak{L}\mathrm{Spin}=\mathfrak{L}\mathrm{Pin}^{\scriptscriptstyle\uparrow}=\mathfrak{L}\mathrm{Spin}^{\scriptscriptstyle\uparrow}~.$$
\bigbreak
The automorphisms of ${\boldsymbol{U}}$ which have unit determinant constitute the group
$\mathrm{Sl}\equiv\mathrm{Sl}({\boldsymbol{U}})$\,;
thus
\begin{align*}
& \mathrm{Cl}^{{(\!{+}\!)}{\scriptscriptstyle\uparrow}}\equiv\mathrm{Cl}^{(\!{+}\!)}\cap\mathrm{Cl}^{\scriptscriptstyle\uparrow}
=\left\{\, \sKM\in\operatorname{End}{\boldsymbol{W}} : K\in{\mathbb{R}}^+\times\mathrm{Sl} \,\right\}~,
\\[6pt]
& \mathrm{Spin}^{\scriptscriptstyle\uparrow}=\left\{\, \sKM\in\operatorname{End}{\boldsymbol{W}} : K\in\mathrm{Sl} \,\right\}~.
\end{align*}
In particular, one has the isomorphism
$$\mathrm{Spin}^{\scriptscriptstyle\uparrow}\leftrightarrow\mathrm{Sl}:\sKM\leftrightarrow K~.$$
Now remember that
\begin{align*}
& \hat\gamma(\weu2\H)=
\left\{\,\cliffordplusmatrix{A}\in\operatorname{End}{\boldsymbol{W}} : \operatorname{Tr} A=0\,\right\}~,
\\[8pt]
& \hat\gamma({\mathbb{R}}\oplus\weu2\H)=
\left\{\,\cliffordplusmatrix{A}\in\operatorname{End}{\boldsymbol{W}} : \Im\operatorname{Tr} A=0\,\right\}~;
\end{align*}
moreover $\operatorname{End}{\boldsymbol{U}}$ can be decomposed into the direct sum
of the subspace of all traceless endomorphism,
which is just $\mathfrak{L}\mathrm{Sl}$\,,
and the subspace ${\mathbb{C}}\,{1\!\!1}$ generated by the identity.
Then one has the Lie algebra isomorphisms
\begin{align*}
& \mathfrak{L}\mathrm{Cl}=\mathfrak{L}\mathrm{Cl}^{{(\!{+}\!)}{\scriptscriptstyle\uparrow}}={\mathbb{R}}\oplus\weu2\H ~\longrightarrow~
({\mathbb{R}}\,{1\!\!1})\oplus\mathfrak{L}\mathrm{Sl}~,
\\[6pt]
& \mathfrak{L}\mathrm{Pin}=\mathfrak{L}\mathrm{Spin}^{\scriptscriptstyle\uparrow}=\weu2\H ~\longrightarrow~ \mathfrak{L}\mathrm{Sl}~.
\end{align*}
\bigbreak
\begin{proposition}
Let
$$\Phi=\sKM\in\mathrm{Spin}~,~~v\in\H~,~~
\gamma(v)=\cliffordplusmatrix{V}\equiv
\left(\begin{smallmatrix}\sqrt2\,v&0\\
0&\sqrt2\,v^\ddag\end{smallmatrix}\right)~.$$
Then
$$\operatorname{Ad}[\Phi]\gamma(v)=\pm
\begin{pmatrix}0 && [K{\otimes}\bar K](V) \\[6pt]
\bigl([K{\otimes}\bar K](V)\bigr)^\ddag && 0\end{pmatrix}~,$$
where the $+$ sign holds iff $\Phi\in\mathrm{Spin}^{\scriptscriptstyle\uparrow}$.
\end{proposition}
{\sc proof:~}
Remembering the previous results one finds
$$\operatorname{Ad}[\Phi]\gamma(v)=
\frac1{\det K}\begin{pmatrix}
0 && K\,V\,\bar K{}^{\scriptscriptstyle\bigstar} \\[6pt]
\bigl(K\,V\,\bar K{}^{\scriptscriptstyle\bigstar}\bigr)^\ddag && 0
\end{pmatrix}~.$$
Moreover
$$(K\,V\,\bar K{}^{\scriptscriptstyle\bigstar})^{\sA\cA}
=K\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}\,V^{\sB\cB}\,(\bar K{}^{\scriptscriptstyle\bigstar})\iI{\sB\.}{\sA\.}
=K\Ii{\scriptscriptstyle A}{\scriptscriptstyle B}\,V^{\sB\cB}\,\bar K\Ii{\sA\.}{\sB\.}
=(K{\,\otimes\,}\bar K)\Ii{\sA\cA}{\sB\cB}\,V^{\sB\cB}~.$$
\EndBox{\square}
Now remember (\Sec\ref{s:2-spinor groups and Lorentz group}) that the group
$\{K{\,\otimes\,}\bar K:K\in\operatorname{Aut}({\boldsymbol{U}})\,\}$
is constituted of automorphisms of ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}$
which preserve the splitting ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}=\H\oplus\mathrm{i}\,\H$
and the causal structure of $\H$.
Its subgroup
$\{K{\,\otimes\,}\bar K:K\in\mathrm{Sl}({\boldsymbol{U}})\,\}$
coincides with $\mathrm{Lor}_+^{\scriptscriptstyle\uparrow}(\H)$\,.
Thus one sees that the group isomorphism $\mathrm{Sl}\to\mathrm{Spin}^{\scriptscriptstyle\uparrow}$
determines the \hbox{2-to-1} epimorphism $\mathrm{Spin}^{\scriptscriptstyle\uparrow}\to\mathrm{Lor}_+^{\scriptscriptstyle\uparrow}$\,.
\bigbreak
One also finds that $\mathrm{Spin}^{\scriptscriptstyle\uparrow}$ is the subgroup of $\operatorname{End}{\boldsymbol{W}}$
preserving $(\gamma,\mathrm{k},g,\eta,\varepsilon)$ as well as time-orientation.
Let's review these properties in terms of two-spinors.
\bigbreak\noindent
$\bullet$~Obviously, $\mathrm{Spin}^{\scriptscriptstyle\uparrow}$ preserves the splitting ${\boldsymbol{W}}={\boldsymbol{U}}\oplus\cj{\U}{}^\lin$.
If $\Phi=\sKM$\,, $K\in\mathrm{Sl}({\boldsymbol{U}})$\,, then $\tilde K=K^{-1}$\,,
so for $\psi\equiv(u,\chi),\psi'\equiv(u',\chi')\in{\boldsymbol{W}}$ one gets
\begin{align*}
\mathrm{k}(\Phi\psi,\Phi\psi')&=
\mathrm{k}\bigl((K\,u,\chi\,\bar K{}^{-1}),(K\,u',\chi'\,\bar K{}^{-1})=
\bang{{\bar\chi}\,K^{-1},K\,u'}+\bang{\chi'\,\bar K{}^{-1},\bar K\,{\bar u}}=\\[6pt]
&=\bang{{\bar\chi},u'}+\bang{\chi',{\bar u}}=\mathrm{k}(\psi,\psi')~.
\end{align*}
\smallbreak\noindent
$\bullet$~Since $K{\,\otimes\,}\bar K:{\boldsymbol{U}}{\,\otimes\,}\cj{\U}\to{\boldsymbol{U}}{\,\otimes\,}\cj{\U}$
sends Hermitian tensors to Hermitian tensors
and anti-Hermitian tensors to anti-Hermitian tensors,
it preserves the splitting ${\boldsymbol{U}}{\,\otimes\,}\cj{\U}=\H\oplus\mathrm{i}\,\H$.
Also, remember that $K{\,\otimes\,}\bar K=\operatorname{Ad}[\Phi]$\,.
\smallbreak\noindent
$\bullet$~$K{\,\otimes\,}\bar K=\operatorname{Ad}[\Phi]\in\mathrm{Lor}_+^{\scriptscriptstyle\uparrow}(\H)$\,,
the subgroup of the Lorentz group which preserves orientation
and time-orientation.
\smallbreak\noindent
$\bullet$~$\Phi$ preserves the Dirac map $\gamma$\,. In fact if $y\in\H$ then
\begin{align*}
&\gamma[y]=\begin{pmatrix}0 && \sqrt2\,y \\ \sqrt2\,y^\ddag && 0\end{pmatrix}~,
\quad y^\ddag\equiv\tb y=\tilde y{}^{\scriptscriptstyle\bigstar}~,
\\[8pt]
& \operatorname{Ad}[\Phi]\gamma[y]=
\begin{pmatrix}0 && \sqrt2\,[K{\,\otimes\,}\bar K]\,y \\
\sqrt2\,([K{\,\otimes\,}\bar K]\,y)^\ddag && 0\end{pmatrix}
=\gamma\bigl[[K{\,\otimes\,}\bar K]\,y\bigr]~.
\end{align*}
\smallbreak\noindent
$\bullet$~If $K\in\mathrm{Sl}$ then $K$ preserves any simplectic form $\varepsilon\in\weu2\U{}^\lin$.
Hence $\Phi\equiv\sKM\in\mathrm{Spin}^{\scriptscriptstyle\uparrow}$ preserves the corresponding
simplectic form $(\varepsilon,{\bar\varepsilon}^\#)\in\weu2\W{}^\lin$ and charge conjugation.
\section{Spinors and particle momenta}
\label{S:Spinors and particle momenta}
\subsection{Particle momentum in two-spinor terms}
\label{s:Particle momentum in two-spinor terms}
It has already been observed (\Sec\ref{s:2-spinors and Minkowski space})
that any future-pointing non-spacelike element in $\H$
can be written in the form
$$u{\,\otimes\,}{\bar u}+v{\,\otimes\,}{\bar v}~,\quad u,v\in{\boldsymbol{U}}~.$$
If $u$ and $v$ are not proportional to each other, that is $\varepsilon(u,v)\neq0$\,,
then the above expression is a timelike future-pointing vector;
if $\varepsilon(u,v)\neq0$\,, then it is a null vector.
Future-pointing elements in $\H$ are a contravariant,
``conformally invariant'' version of \emph{classical particle momenta}
(translation to a scaled and/or covariant version, when needed,
will be effortless).
Let ${\boldsymbol{K}}$ and ${\boldsymbol{N}}$ be the subsets of $\H$ constituted
of all future-pointing timelike vectors
and of all future-pointing null vectors, respectively;
moreover, set ${\boldsymbol{J}}:={\boldsymbol{K}}\cup{\boldsymbol{N}}$
(all these sets do not contain the zero element).
Consider now the real quadratic maps
\begin{align*}
&\tilde{\mathrm{p}}:{\boldsymbol{U}}\times{\boldsymbol{U}}\to{\boldsymbol{J}}:(u,v)\mapsto\tfrac{1}{\surd2}\,(u{\,\otimes\,}{\bar u}+v{\,\otimes\,}{\bar v})~,
\\[6pt]
&\mathrm{p}:{\boldsymbol{W}}\cong{\boldsymbol{U}}\times\cj{\U}{}^\lin\to{\boldsymbol{J}}:
(u,\chi)\mapsto\tfrac{1}{\surd2}\,(u{\,\otimes\,}{\bar u}+{\bar\chi}^\#{\,\otimes\,}\chi^\#)~.
\end{align*}
When a normalized symplectic form $\varepsilon\in\weu2\U{}^\lin$ is \emph{fixed},
$\tilde{\mathrm{p}}$ and $\mathrm{p}$ are essentially the same objects,
as one can represent a given element
$\tfrac{1}{\surd2}\,(u{\,\otimes\,}{\bar u}+v{\,\otimes\,}{\bar v})$ of ${\boldsymbol{J}}$
by writing $v{\,\otimes\,}{\bar v}$ as $({\bar\chi}{\,\otimes\,}\chi)^\#$\,;
here, $u,v\in{\boldsymbol{U}}$, $\chi\in\cj{\U}{}^\lin$.
In such case I'll set
\begin{align*}
& v:=-{\bar\chi}^\# \quad\iff\quad \chi={\bar v}^\flat~,\\[6pt]
\Rightarrow\quad
&\bang{{\bar\chi},u}=\bang{v^\flat,u}=\varepsilon(v,u)~,\quad
\bang{\chi,{\bar u}}=\bang{{\bar v}^\flat,{\bar u}}={\bar\varepsilon}({\bar v},{\bar u})~.
\end{align*}
If $p=\mathrm{p}(u,\chi)\equiv\tilde{\mathrm{p}}(u,v)$ then we'll use the shorthands
\begin{align*}
& \mu^2:=g(p,p)=2\,|\varepsilon(u,v)|^2=2\,|\bang{{\bar\chi},u}|^2~,\\[6pt]
& h:=\frac{\sqrt2}\mu\,\bar p^\flat
=\frac1{|\bang{{\bar\chi},u}|}\,({\bar u}^\flat{\,\otimes\,} u^\flat+\chi{\,\otimes\,}{\bar\chi})~.
\end{align*}
Then, $h$ can be seen as an $\varepsilon$-normalized Hermitian metric on ${\boldsymbol{U}}$.
\begin{proposition}\label{p:pgeneratingcouple}
Let $(u,\chi)\equiv(u,{\bar v}^\flat)\in{\boldsymbol{W}}$\,, $\bang{{\bar\chi},u}\neq0$\,;
let $p\in{\boldsymbol{K}}$\,.
Then, the following conditions are equivalent:
\begin{enumerate}
\item[{\bf i)}]
$p=u{\,\otimes\,}{\bar u}+({\bar\chi}{\,\otimes\,}\chi)^\#$\,,
\item[{\bf ii)}]
$\gamma[p](u,\chi)=\mu\,\bigl(\mathrm{e}^{-\mathrm{i}\th}u,\mathrm{e}^{\mathrm{i}\th}\,\chi\bigr)$\,,~
$\th\in{\mathbb{R}}$\,,
\item[{\bf iii)}]
${\bar h}^\flat(u)=\mathrm{e}^{\mathrm{i}\th}\,\chi$\,,
\item[{\bf iv)}]
$h^\#(\chi)=\mathrm{e}^{-\mathrm{i}\th}\,u$\,,
\item[{\bf v)}]
$h({\bar u},v)=0$ and $|\bang{{\bar\chi},u}|=h({\bar u},u)$\,,
\item[{\bf v')}]
$h({\bar u},v)=0$ and $|\bang{{\bar\chi},u}|=h({\bar v},v)$\,,
\end{enumerate}
where $\mu$ and $h$ are defined in terms of $(u,\chi)$ as above.
\end{proposition}
{\sc proof:~}
By straightorwaed calculations one sees that condition {\bf i}
implies conditions {\bf ii}, {\bf iii}, {\bf iv}, {\bf v} and {\bf v'}.
Moreover:
\smallbreak\noindent
{\bf (\:ii}~$\Leftrightarrow$~{\bf iii\,)\,:}~%
It follows from
$\gamma[\t](u,\chi)=\tfrac{1}{\surd2}\,\gamma[{\bar h}^\#](u,\chi)
=\bigl(h^\#(\chi),{\bar h}^\flat(u)\bigr)$\,.
\smallbreak\noindent
{\bf (\:iii}~$\Leftrightarrow$~{\bf iv\,)\,:}~%
If ${\bar h}^\flat(u)=\mathrm{e}^{\mathrm{i}\th}\,\chi$ then
$u=h^\#({\bar h}^\flat(u))=h^\#(\mathrm{e}^{\mathrm{i}\th}\,\chi)=\mathrm{e}^{\mathrm{i}\th}\,h^\#(\chi)$\,.\\
Similarly, if $h^\#(\chi)=\mathrm{e}^{-\mathrm{i}\th}\,u$ then
$\chi={\bar h}^\flat(h^\#(\chi))={\bar h}^\flat(\mathrm{e}^{-\mathrm{i}\th}\,u)
=\mathrm{e}^{-\mathrm{i}\th}\,{\bar h}^\flat(u)$\,.
\smallbreak\noindent
{\bf (\:iv}~$\Rightarrow$~{\bf v\,)\,:}~%
$h({\bar u},v)=\bang{h^\flat({\bar u}),-{\bar\chi}^\#}=-\bang{\mathrm{e}^{-\mathrm{i}\th}\,{\bar\chi},{\bar\chi}^\#}
=\mathrm{e}^{-\mathrm{i}\th}\,\varepsilon^\#({\bar\chi},{\bar\chi})=0$\,.\\
Moreover
$h({\bar u},u)=\bang{{\bar h}^\flat(u),{\bar u}}=\bang{\mathrm{e}^{\mathrm{i}\th}\,\chi,{\bar u}}
=\bang{{\bar\chi},u}\,\bang{\chi,{\bar u}}/|\bang{{\bar\chi},u}|=|\bang{{\bar\chi},u}|$\,.
\smallbreak\noindent
{\bf (\:v}~$\Rightarrow$~{\bf iv\,)\,:}~%
From $0=h({\bar u},v)=\bang{h^\flat({\bar u}),-{\bar\chi}^\#}=-\varepsilon^\#({\bar\chi},h^\flat({\bar u}))$
one has ${\bar\chi}=c\,h^\flat({\bar u})$\,, $c\in{\mathbb{C}}$\,.
Then $\bang{{\bar\chi},u}=c\,h({\bar u},u)=c\,|\bang{{\bar\chi},u}|{\quad\Rightarrow\quad} c=\mathrm{e}^{\mathrm{i}\th}$\,.
\smallbreak\noindent
{\bf (\:v}~$\Rightarrow$~{\bf v'\,)\,:}~%
From {\bf iv} (equivalent to {\bf v}) one has
$h({\bar v},v)=\bang{h,\chi^\#{\,\otimes\,}{\bar\chi}^\#}=\bang{h^\#,\chi{\,\otimes\,}{\bar\chi}}
=\bang{h^\#(\chi),{\bar\chi}}=\mathrm{e}^{-\mathrm{i}\th}\,\bang{{\bar\chi},u}=|\bang{{\bar\chi},u}|$\,,
hence also $h({\bar v},v)=|\bang{{\bar\chi},u}|$\,.
\smallbreak\noindent
{\bf (\:v'}~$\Rightarrow$~{\bf iv\,)\,:}~%
As in {\bf v}~$\Rightarrow$~{\bf iv} one has ${\bar\chi}=c\,h^\flat(u)$\,, $c\in{\mathbb{C}}$\,,
or $u=\frac1{\bar c}\,h^\#(\chi)$\,.
Then, from
$\bang{{\bar\chi},u}=\bang{{\bar\chi},\frac1{\bar c}\,h^\#(\chi)}
=\frac1{\bar c}\,h^\#(\chi,{\bar\chi})=\frac1{\bar c}\,h({\bar v},v)$
one has $\bar c=\mathrm{e}^{-\mathrm{i}\th}$ i.e$.$\ $c=\mathrm{e}^{\mathrm{i}\th}$\,.
\smallbreak\noindent
{\bf (\:v}~$\Rightarrow$~{\bf i\,)\,:}~%
Using also {\bf v'} (already seen to be equivalent to {\bf v})
one sees that the couple $({\zeta}_u\,,{\zeta}_v)\equiv(u,v)/\sqrt{|\bang{{\bar\chi},u}|}$
is an $h$-orthonormal basis of ${\boldsymbol{U}}$\,;
hence
$h^\#={\bar\zeta}_u{\,\otimes\,}{\zeta}_u+{\bar\zeta}_v{\,\otimes\,}{\zeta}_v
=\frac1{|\bang{{\bar\chi},u}|}\,\bigl({\bar u}{\,\otimes\,} u+{\bar v}{\,\otimes\,} v\bigr)~.$
Condition {\bf i} then follows.
\EndBox{\square}
\subsection{Bundle structure of 4-spinor space over momentum space}
\label{s:Bundle structure of 4-spinor space over momentum space}
The previous results show that the restriction
$\mathrm{p}:{\boldsymbol{W}}\setminus\{0\}~\longrightarrow~{\boldsymbol{J}}$
is surjective.
Since the Lorentz ``length'' of $\mathrm{p}(u,\chi)$ is $\sqrt2\,|\bang{{\bar\chi},u}|$
one sees that the subset of all elements in ${\boldsymbol{W}}$
which project onto ${\boldsymbol{N}}$ is the 6-dimensional real submanifold
$${\boldsymbol{W}}^0:=\mathrm{p}^{-1}({\boldsymbol{N}})=
\bigl\{(u,\chi)\in{\boldsymbol{W}}{\setminus}\{0\}:\bang{{\bar\chi},u}=0\bigr\}
\subset{\boldsymbol{W}}~.$$
The subset of all elements in ${\boldsymbol{W}}$ which project onto ${\boldsymbol{K}}$ is
the open submanifold
$${\boldsymbol{W}}^\backprime:=\mathrm{p}^{-1}({\boldsymbol{K}})=
\bigl\{(u,\chi)\in{\boldsymbol{W}}:\bang{{\bar\chi},u}\neq0\bigr\}~,$$
and one has
$${\boldsymbol{W}}{\setminus}\{0\}={\boldsymbol{W}}^0\cup{\boldsymbol{W}}^\backprime~.$$
Moreover, consider the subsets ${\boldsymbol{W}}^+,{\boldsymbol{W}}^-\subset{\boldsymbol{W}}^\backprime$ defined to be
$${\boldsymbol{W}}^\pm:=\bigl\{(u,\chi)\in{\boldsymbol{W}}:\bang{{\bar\chi},u}\in{\mathbb{R}}^\pm\bigr\}~.$$
Recalling condition {\bf ii} of proposition~\ref{p:pgeneratingcouple} one has
$$\gamma[\mathrm{p}\psi]\psi=\mu\,(\mathrm{e}^{-\mathrm{i}\th}u,\mathrm{e}^{\mathrm{i}\th}\,\chi)~,$$
which holds for every $\psi\equiv(u,\chi)\in{\boldsymbol{W}}$
(if $\psi\in{\boldsymbol{W}}^0$ then $\mu=0$).
In particular
$${\boldsymbol{W}}^\pm=\bigl\{\: \psi\equiv(u,\chi)\in{\boldsymbol{W}}{\setminus}\{0\} :
\gamma[\mathrm{p}\psi]\psi=\pm\mu\,\psi~,\mu\equiv|\bang{{\bar\chi},u}|\:\bigr\}~.$$
Next, consider the subset
$$\tilde{\boldsymbol{W}}{}^\backprime:=\{(u,v):\varepsilon(u,v)\neq0\}\subset{\boldsymbol{U}}\times{\boldsymbol{U}}~,$$
and note that when a normalized symplectic form $\varepsilon\in\weu2\U{}^\lin$ is fixed,
$\tilde{\boldsymbol{W}}{}^\backprime$ can be identified with ${\boldsymbol{W}}^\backprime$ via the correspondence
${\bar v}^\flat\leftrightarrow\chi$\,.
$\tilde{\boldsymbol{W}}{}^\backprime$ is a fibred set over ${\boldsymbol{K}}$\,;
for each $p\in{\boldsymbol{K}}$, the fibre of $\tilde{\boldsymbol{W}}{}^\backprime$ over $p$ is the subset
$$\tilde{\boldsymbol{W}}_{\!\!p}^\backprime:=\tilde{\mathrm{p}}{}^{\scriptscriptstyle-1}(p)=\bigl\{
(u,v)\in\tilde{\boldsymbol{W}}{}^\backprime:\tfrac{1}{\surd2}\,(u{\,\otimes\,}{\bar u}+v{\,\otimes\,}{\bar v})=p \bigr\}~.$$
\begin{proposition}\label{p:uvU2}
$\tilde{\mathrm{p}}:\tilde{\boldsymbol{W}}{}^\backprime\to{\boldsymbol{K}}$
is a trivializable principal bundle with structure group $\mathrm{U}(2)$\,.
\end{proposition}
{\sc proof:~}
Let $p=\tilde{\mathrm{p}}(u,v)=\tilde{\mathrm{p}}(u',v')$\,.
From proposition~\ref{p:pgeneratingcouple} one then sees that
$(u,v)$ and $(u',v')$ are orthonormal bases of ${\boldsymbol{U}}$
relatively to the Hermitian metric
$h\equiv\sqrt2\,\bar p^\flat/\mu$.
Then there exists a unique transformation $K\in\mathrm{U}({\boldsymbol{U}},h)$ such that
$$u'=K(u)~,\quad v'=K(v)~;$$
hence,
$\tilde{\boldsymbol{W}}_{\!\!p}^\backprime$ is a group-affine space, with derived group $\mathrm{U}(2)$\,.
Let now $({\zeta_\sA})$ be an $\varepsilon$-normalized basis of ${\boldsymbol{U}}$ and $(\t_\l)$
the associated Pauli frame.
For each $p\in{\boldsymbol{K}}$ let $L_p\in\mathrm{Lor}_+^{\scriptscriptstyle\uparrow}(\H)$ be the boost
such that $L_p\t_0=p/\mu$\,, where $\mu^2\equiv g(p,p)$\,;
up to sign there is a unique $B_p\in\mathrm{Sl}({\boldsymbol{U}})$ such that
$L_p=B_p{\,\otimes\,}\bar B_p$\,,
and a consistent smooth way of choosing one such $B_p$ for each $p$
can be fixed.
It turns out that the basis $\bigl(\sqrt\mu\,B_p{\zeta_\sA}\bigr)$
is orthonormal relatively to $\sqrt2\,\bar p^\flat/\mu$
seen as a Hermitian metric on ${\boldsymbol{U}}$,
hence $\tilde{\mathrm{p}}(\sqrt\mu\,B_p{\zeta}_1\,,\sqrt\mu\,B_p{\zeta}_2)=p$\,.
In this way one selects an ``origin'' element in each fibre of $\tilde{\mathrm{p}}$\,,
so getting a trivialization $\tilde{\boldsymbol{W}}{}^\backprime\to{\boldsymbol{K}}\times\mathrm{U}(2)$\,.
\EndBox{\square}
Using a little two-spinor algebra it is not difficult to prove:
\begin{proposition}\label{p:Kuchiexpr}
Let $\psi,\psi'\in{\boldsymbol{W}}^\backprime$,
$\psi\equiv(u,\chi)$\,, $\psi'\equiv(u',\chi')$\,;
let $K\in\operatorname{Aut}{\boldsymbol{U}}$ be the unique automorphism of ${\boldsymbol{U}}$ such that
$$K\,u=u~,\quad K\,{\bar\chi}^\#={\bar\chi}'{}^\#~.$$
Then
$$K=\frac1{\bang{{\bar\chi},u}^2}\,\bigl[
\bang{{\bar\chi},u'}\,u{\,\otimes\,}{\bar\chi}-\varepsilon^\#({\bar\chi},{\bar\chi}')\,u{\,\otimes\,} u^\flat
+\varepsilon(u,u')\,{\bar\chi}^\#{\,\otimes\,}{\bar\chi}+\bang{{\bar\chi}',u}\,{\bar\chi}^\#{\,\otimes\,} u^\flat \bigr]~.$$
Moreover, one has
$$\chi'=K^\ddag\,\chi~.$$
Conversely,
the conditions $u'=Ku$ and $\chi'=K^\ddag\chi$ determine $K$ uniquely.
\end{proposition}
\bigbreak
The above expression for $K$ is invariant
relatively to the transformation $\varepsilon\mapsto\mathrm{e}^{\mathrm{i}\,\th}\varepsilon$\,;
hence, $K$ is independent of the particular normalized
symplectic form $\varepsilon$ chosen.
When a normalized $\varepsilon\in\weu2\U{}^\lin$ is given,
one has the real vector bundle isomorphism
${\boldsymbol{W}}^\backprime\leftrightarrow\tilde{\boldsymbol{W}}{}^\backprime:(u,v)\leftrightarrow(u,{\bar v}^\flat)$.
Through this correspondence,
${\boldsymbol{W}}^\backprime\to{\boldsymbol{K}}$ turns out to be a trivializable principal bundle
with structure group $\mathrm{U}(2)$\,.
If $\psi,\psi'\in{\boldsymbol{W}}_{\!\!p}^\backprime$\,, let
$$(K)=c \begin{pmatrix}\phantom{-} a & \bar b~ \\ -b & \bar a\end{pmatrix}
\in\mathrm{U}(2)~,\quad
a,b,c\in{\mathbb{C}}:|a|^2+|b|^2=|c|^2=1~,$$
be the matrix
of $K\in\operatorname{Aut}{\boldsymbol{U}}$ sending $\psi$ to $\psi'$
(according to proposition~\ref{p:Kuchiexpr})
relatively to the basis $(u,v)$\,.
Then
$$\begin{cases}
u'=c\,(a\,u-b\,v)~,\\[6pt]
v'=c\,(\bar b\,u+\bar a\,v)~,
\end{cases}
\qquad\quad\Longleftrightarrow\quad\qquad
\begin{cases}
u'=c\,(a\,u+b\,{\bar\chi}^\#)~,\\[6pt]
\chi'=\bar c\,(a\,\chi+b\,{\bar u}^\flat)~.
\end{cases}$$
If you take a different normalized symplectic form $\varepsilon\to\mathrm{e}^{\mathrm{i}\,\th}\varepsilon$\,,
then $K$ does not change,
while the corresponding matrix $(K)\in\mathrm{U}(2)$ changes
according to
\hbox{$c\to c$}\,, \hbox{$a\to a$}\,, \hbox{$b\to\mathrm{e}^{\mathrm{i}\,\th}b$}\,.
\bigbreak
The above $\mathrm{U}(2)$-action does not preserve ${\boldsymbol{W}}^\pm\subset{\boldsymbol{W}}^\backprime$.
In fact it's straightforward to prove:
\begin{proposition}
Let $\psi,\psi'\in{\boldsymbol{W}}_{\!\!p}^+$ (resp.\ $\psi,\psi'\in{\boldsymbol{W}}_{\!\!p}^-$),
$\psi\equiv(u,\chi)$\,, $\psi'\equiv(u',\chi')$\,;
let $K$ be the unique automorphism of ${\boldsymbol{U}}$ such that
$Ku=u$\,, $K^\ddag\chi=\chi'$\,.
Then $K\in\mathrm{SU}({\boldsymbol{U}},h)$\,, where $h\equiv\sqrt2\,\bar p^\flat/\mu$\,.
\end{proposition}
Hence, ${\boldsymbol{W}}^+\to{\boldsymbol{K}}$ and ${\boldsymbol{W}}^-\to{\boldsymbol{K}}$
turn out to be trivializable principal bundles,
with structure group $\mathrm{SU}(2)$\,.
\vfill\newpage
|
1,116,691,498,466 | arxiv | \section{Introduction}\label{sec:introduction}
\indent An application in image processing led to the search for orthogonal matrices, all of whose elements have modulus $\leq 1$ and which have maximal or high determinant.
$Cretan$ matrices were first discussed during a conference in Crete in 2014 by N. A. Balonin, M. B. Sergeev and colleagues of the Saint Petersburg State University of Aerospace Instrumentation but were well known using \textit{Heritage names}, \cite{BMS02,BM06,BMS01,BMS03}. This paper follows closely the joint work of N. A. Balonin, Jennifer Seberry and M. B. Sergeev \cite{BNSJ14,BNSJ14a,BNSJSM14}.
In this and future papers we use some \textit{names, definitions, notations}
differently to how they have been have in the past \cite{BM06}. This we hope, will cause less confusion, bring our nomenclature closer to common usage, conform for mathematical purists and clarify the similarities and differences between some matrices. We have chosen to use the word level, instead of value for the entries of a Cretan matrix, to conform to earlier writings \cite{BM06, BMS01, BMS03}.
We know of no references where the \textbf{Co-Existence of Hadamard Matrices and Cretan$(4t-1,2)$-Mersenne Matrices Theorem} is stated.
\subsection{Preliminary Definitions}
The absolute value of the determinant of any matrix is not altered by 1) interchanging any two rows, 2) interchanging any two columns, and/or 3) multiplying any row/or column by $-1$. These equivalence operations are called \textit{Hadamard equivalence operations}. So the absolute value of the determinant of any matrix is not altered by the use of Hadamard equivalence operation.
Although it is not the definition used by purists we use orthogonal matrix as below. Write $I_n$ for the identity matrix of order $n$ and let $\omega $ be a constant.
When a matrix $S$ is written in the following form
\newcommand{\BigFig}[1]{\parbox{16pt}{\Huge #1}}
\newcommand{\BigFig{B}}{\BigFig{B}}
\[S = \begin{bmatrix}
x & \sigma & \dots & \sigma \\
\sigma \\
\vdots & & \BigFig{B}\\
\sigma
\end{bmatrix}\]
$B$ is said to be the \textit{core of } $S$ and the $\sigma $'s are the \textit{borders} of $B$ in $S$. The variable $x$ is set as zero for conference matrices, $\sigma$ for Hadamard matrices and $a$ for Fermat matrices.
\begin{definition} [\textbf{Orthogonal Matrix, Hadamard Matrices, Cretan Matrix}] \label{had-orthog-cretan}
An \textit{orthogonal matrix}, $S= (s_{ij})$ of order $n$, is square and has the modulus of all its entries $\leq 1$, and satisfies $SS^{\top} = \omega I_n$ for $n = 1, 2, 4t$.
An \textit{Hadamard matrix} of order has entries $\pm 1$ and satisfies
$HH^{\top} = nI_n$ for $n$ = 1, 2, $4t$, $t > 0$ an integer. Any Hadamard matrix can be put into \textit{normalized form}, that is having the first row and column all plus 1s using Hadamard equivalence operations: that is it can be written with a core.
A \textit{Cretan matrix}, $S$, of order $v$ has entries with modulus $\leq 1$ and at least one element in each row and column must be 1, and which satisfies $SS^{\top} = \omega I_v$. A $Cretan(n; \tau; \omega )$ matrix, or $CM(n; \tau ; \omega)$ has $\tau $ levels or values for its entries.
\end{definition}
In this work we will only use orthogonal to refer to matrices comprising real elements with modulus $\leq 1 $, where at least one entry in each row and column must be one.
Hadamard matrices and weighing matrices are the best known of these orthogonal matrices. We refer to \cite{BNSJ14a,SY92,JH1893,JSW72} for more definitions. We recall Barba \cite{Barba33} showed that for matrices whose entries have modulus $\leq 1$, $B$, of order $n$
\begin{equation}\label{eq:B}
\det{B} \leq \sqrt{2n-1}(n-1)^{\frac{n-1}{2}} \textnormal{ or asymptotically } \approx 0.858(n)^{\frac{n}{2}}\,.
\end{equation}
Wotjas \cite{Wojtas64} showed that for matrices whose entries have modulus $\leq 1$, $B$, of order $n \equiv 2 \pmod{4}$ we have
\begin{equation*}\label{eq:C}
\det{B} \leq 2(n-1)(n-2)^{\frac{n-2}{2}} \textnormal{ or asymptotically } \approx 0.736(n)^{\frac{n}{2}}\,.
\end{equation*}
More details of Cretan matrices can be found in \cite{NJ15a}.
When Hadamard introduced his famous inequality \cite{JH1893}, for matrices with moduli $\leq 1$ it was noticed that Hadamard matrices, which are orthogonal and with entries $\pm 1$ satisfied the equality of Hadamard's inequality.
However the inequality applies to other matrices and orders with entries on the unit disk, these have been named ``Fermat", Mersenne and Euler matrices for different congruence classes $\pmod{4}$, \cite{BNSJ14a,NJ15a}.
We emphasize: in general a Cretan($n,\tau $) or $CM(n,\tau)$ or $S$, has $\tau$ levels, they are made from \textit{$\tau $-variable orthogonal matrices} by replacing the variables by appropriate real numbers with moduli $\leq 1$, where at least one entry in each row and column is 1. After this stage, $\tau $-variable orthogonal matrices and Cretan($n,\tau $), are used, \textit{loosely} to denote one-the-other.
For 2-level Cretan matrices we will denote the levels/values by $x ,y$ where $0 \leq |y| \leq x = 1$. We also use the notations \textit{Cretan(v)}, \textit{Cretan(v)-SBIBD} and \textit{Cretan-SBIBD} for Cretan matrices of 2-levels and order $v$ constructed using $SBIBD$s.
\begin{definition}[\textbf{SBIBD and Incidence Matrix}]\label{def:incidence-matrix-SBIBD}
For the purposes of this paper we will consider an $SBIBD(v, k, \lambda)$, $B$, to be a $v \times v$ matrix, with entries $0$ and $1$, $k$ ones per row and column, and the inner product of distinct pairs of rows and/or columns to be $\lambda$. This is called the \textit{incidence matrix} of the SBIBD. For these matrices $\lambda(v-1) = k(k-1)$.
\end{definition}
For every $SBIBD(v, k, \lambda)$ there is a complementary $SBIBD(v, v-k, v-2k + \lambda)$. One can be made from the other by interchanging the $0$'s of one with the $1$'s of the other. The usual use $SBIBD$ convention that $v >2k$ and $k >2\lambda$ is followed.
A combinatorial trick allows us to say that any matrix of order $n$ satisfying $AA^{\top} =aI +bJ$ has determinant $\left(\sqrt{a + nb}\right)a^{\frac{n-1}{2}}$.
Writing $A$ for the incidence matrix of the $SBIBD(v, k, \lambda)$, which has entries 0 and 1, and $B=2A -J$, $J$ the matrix of all ones, for the $\pm 1$ it forms we have:
\begin{multline}\label{eq:A}
AA^{\top} = (k-\lambda)I + \lambda J \hspace{0.5cm} AJ=kJ \textnormal{~and~} \\BB^{\top} =2(k - \lambda )I + (v- 2(k-\lambda ))J \hspace{0.5cm} BJ=(2k-v)J.
\end{multline}
Thus
\[det(A) = k(k-\lambda)^{\frac{v-1}{2}} \quad \textnormal{and} \quad det(B) = \sqrt{k^2 + (v-k)^2)}(2(v-k)g^{\frac{v-1}{2}}\,.\]
We now define our important concepts the \textit{orthogonality equation}, the \textit{radius equation(s)}, the \textit{characteristic equation(s)} and the \textit{weight} of our matrices.
\begin{definition}[\textbf{Orthogonality equation, radius equation(s), characteristic equation(s), weight}]\label{def:or-rad-char}
Consider the matrix $S= (s_{ij})$ comprising the variables $x$ and $y$.
The \textit{matrix orthogonality equation}
\[ S^{\top }S = SS^{\top} = \omega I_n \]
yields two types of equations: the $n$ equations which arise from taking the inner product of each row/column with itself (which leads to the diagonal elements of $\omega I_n$ being $\omega$) are called \textit{radius equation(s)}, $g(x,y)=\omega$, and the $n^2 -n$ equations, $f(x,y)=0$, which arise from taking inner products of distinct rows of $S$ (which leads to the zero off diagonal elements of $\omega I_n$) are called \textit{characteristic equation(s)}. The \textit{orthogonality equation} is $\sum_{j=1}^n s_{ij}^2 = \omega$. $\omega$ is called the \textit{weight} of $S$. \qed
\end{definition}
\begin{example}\label{eg:4a-b}
We consider the 2-variable $S$ matrix given by
\[ S = \begin{bmatrix}
x & y & y & y & y \\
y & x & y & y & y \\
y & y & x & y & y \\
y & y & y & x & y \\
y & y & y & y & x
\end{bmatrix}\,.
\]
By definition, in order to become an orthogonal matrix, it must satisfy the orthogonality equation, $SS^{\top}= \omega I$, the radius and the characteristic equations, so we have
\[ x^2 + 4y^2 = \omega, \qquad 2xy +3 y^2 = 0\,.\]
To make a Cretan(5;2;$\frac{10}{3}$) we force $x =1$, (since we require that at least one entry per row/column is 1), and the characteristic equation gives $y=-\frac{2}{3}$. Hence $\omega = 3{\frac{1}{3}}$. The determinant is $(\frac{10}{3})^{\frac{5}{2}}$ = 20.286. Thus we now have an $S = Cretan(5;2;{\frac{10}{3}};20.286).$ \qed
\end{example}
\subsection{Notation Transitions}\label{notation-transition}
In transiting from one mother tongue to another (Russian to English and English to Russian) and from previous to newer usage, some words reoccur: we need a shorthand. To simplify references we note:
\begin{table}[h]
\begin{center}
\begin{tabular}{ll|l}
\textbf{Heritage Usage}& Cretan Matrix & References \\
\hline
Fermat & Cretan(4t+1) & \cite{BM06,BNSJ14a,NJ15,BMS01} \\
Hadamard & Cretan(4t) & \cite{BNSJ14,BMS03,BNSJ14a,JH1893,SY92}\\
Mersenne & Cretan(4t-1) & \cite{BMS02,BM06,BMS03,BNSJ14a,Sorrento,Singapore}\\
Euler & Cretan(4t-2) & \cite{BMS02,BMS01,BMS03}.
\end{tabular}
\caption{Cretan and Heritage Names}
\label{table:cretan-and-heritage-names}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\begin{tabular}{ll|l}
\textbf{Usage}& Dual Usage & Heritage name \\
\hline
CM(4t+1) & Core of $CM(4t+2)$ & ``Fermat" matrices\\
CM(4t) & Core of $CM(4t+1)$ & Hadamard matrix \\
CM(4t-1) & Core of $CM(4t)$ & Mersenne matrix \\
CM(4t-2) & Core of $CM(4t-1)$ & Euler matrix \\
CM(4t-3) & Core of $CM(4t-2)$ & ``Fermat" matrices\\
\hline
\end{tabular}
\caption{Cretan and Core Names}
\label{table:cretan-and-core-names}
\end{center}
\end{table}
\section{Preliminary Results and Hadamard Mathematical Foundations}\label{sec:Had}
As an historical note we point out that J. A. Todd's \cite{Todd33} article showing the relationship of $SBIBD(4t-1,2t-1,t-1)$ and Hadamard matrices of order $4t$ appeared in the same issue of the \textit{Journal of Mathematics and Physics} as the famous paper by R. E. A. C. Paley \cite{Paley33} using Legendre symbols to construct orthogonal matrices.
\begin{example} \label{example:b}
An $SBIBD(7,4,2) = B$ and its complementary $SBIBD(7,3,1)$ can be written with incidence matrices: they are still complementary if permutations of rows/columns are applied to one and other permutations of rows/columns to the other. This is because if $P$ and $Q$ are permutation matrices $PBQ$ is equivalent to the $SBIBD(7,4,2)$.
\begin{multline*}
SBIBD(7,4,2) = \begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 0 & 0\\
0 & 1 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 1 & 1 & 1 & 0 & 1 \\
1 & 0 & 0 & 1 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 1 & 1 \\
1 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 1
\end{bmatrix}\\
SBIBD(7,3,1) = \begin{bmatrix}
0 & 1 & 1 & 0 & 1 & 0 & 0\\
0 & 0 & 1 & 1 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 1 & 0 & 1 \\
1 & 0 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 0 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 0 & 0 & 0 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 0
\end{bmatrix}
\end{multline*}
The second incidence matrix is still a complement of the incidence matrix of the first $SBIBD$ even after permutations of its rows and/or columns have been performed.
To prepare to make Cretan matrices from $SBIBD(4t-1,2t-1,t-1)$ we first put the two $SBIBD$ in variable forms: or $SBIBD(7,4,2)$ \text{circ}($x,x,x,y,x,y,y$) and for the complementary $SBIBD(7,3,1)$, circ($y,x,x,y,x,y,y$), for
\allowdisplaybreaks
\begin{multline*}
SBIBD(7,4,2) = \begin{bmatrix}
x & x & x & y & x & y & y\\
y & x & x & x & y & x & y \\
y & y & x & x & x & y & x \\
x & y & y & x & x & x & y \\
y & x & y & y & x & x & x \\
x & y & x & y & y & x & x \\
x & x & y & x & y & y & x
\end{bmatrix}\\
SBIBD(7,3,1) = \begin{bmatrix}
y & x & x & y & x & y & y\\
y & y & x & x & y & x & y \\
y & y & y & x & x & y & x \\
x & y & y & y & x & x & y \\
y & x & y & y & y & x & x \\
x & y & x & y & y & y & x \\
x & x & y & x & y & y & y
\end{bmatrix}
\end{multline*}
We could just have exchanged $x$ and $y$ in the definition \text{circ}($x,x,x,y,x,y,y$) to get the second $SBIBD$, but we wanted to illustrate that there are many other possibilities for the first row of the second $SBIBD$.
\begin{figure}[h]
\centering
\subfloat[][the principal solution]{\includegraphics[width=0.4\textwidth]{S7}} \qquad \qquad
\subfloat[][the complementary solution]{\includegraphics[width=0.4\textwidth]{S7_2}}\\
\caption{Orthogonal matrices for order $7$: Balonin-Sergeev Family}
\label{fig:S7}
\end{figure}
Consider the 2-variable $SBIBD(7,4,2)$: it has characteristic equation $2x^{2} + 4xy+ y^{2} = 0$, and radius equation $\omega = 4x^2 + 3y^2$. $\det(S) = \omega^{\frac{7}{2}}$. This is an $CM(7,2)$. To make a Cretan matrix we now set $x=1$ and solve the characteristic equation to find $y$ in terms of $x$. This value/level is then used to give the Cretan$(7;2;5.0294)$. Thus the principal solution has
\[x = 1,\hspace{1cm} y = - 2 + \sqrt{2},\; \hspace{1cm} \omega = 4x^{2} + 3y^{2} = 5.0294\;.\]
The $SBIBD(7,3,1)$ with characteristic equation $x^{2} + 4xy+ 2y^{2} = 0$, and radius equation $\omega = 3x^2 + 4y^2$, $\det(S) = \omega^{\frac{7}{2}}$, (smaller than above values replacing $y$), has a feasible solution for the characteristic equation. Proceeding as before we obtain a second solution
\[x = 1,\hspace{1cm} y = \frac{-2+\sqrt{2}}{2}\;, \hspace{1cm} \omega = 3x^{2} + 4y^{2} = 3.3431\;.\]
Thus it gives a Cretan$(7;2;3.3431)$ matrix.
The two determinants are $285.31$ and $69.319$ respectively.
Loosely we write $Cretan(7;2:5.2904)-SBIBD$, (2-variable orthogonal $SBIBD$ $(7,4,2)$), and obtain the following 2-level Cretan-Mersenne matrix:
\[ \begin{bmatrix*}[c]
1 & 1 & 1 &-2+\sqrt{2} & 1 &-2+\sqrt{2} &-2+\sqrt{2}\\
-2+\sqrt{2} & 1 & 1 & 1 &-2+\sqrt{2} & 1 &-2+\sqrt{2} \\
-2+\sqrt{2} &-2+\sqrt{2} & 1 & 1 & 1 &-2+\sqrt{2} & 1 \\
1 &-2+\sqrt{2} &-2+\sqrt{2} & 1 & 1 & 1 &-2+\sqrt{2} \\
-2+\sqrt{2} & 1 &-2+\sqrt{2} &-2+\sqrt{2} & 1 & 1 & 1 \\
1 &-2+\sqrt{2} & 1 &-2+\sqrt{2} &-2+\sqrt{2} & 1 & 1 \\
1 & 1 &-2+\sqrt{2} & 1 &-2+\sqrt{2} &-2+\sqrt{2} & 1
\end{bmatrix*}
\]\qed
\end{example}
\section{Cretan-SBIBD(v,2) Theorem: The Cretan 2-level Matrices from SBIBD Theorem: }\label{sec:main SBIBD-CRETAN}
Although our main theorem appeared as a corollary in \cite{NJ15a} it is in fact worthy of being a theorem in its own right. The paper \cite{NJ15a} gives many relevant details. We write
\begin{theorem}[\textbf{The Cretan-SBIBD(v,2) Theorem: The Cretan 2-level Matrices from SBIBD Theorem}] \label{principal-2-var-theorem}
Whenever there exists an $SBIBD(v,k,\lambda)$ there exists a 2-level Cretan-SBIBD$(v,2)$, or $S$ or $CM$, as follows,
\begin{itemize}
\item Cretan$\left(v;2;kx^2 + (v-k)y^2; \textnormal{ y,x;determinant} \right);$
\item Cretan$\left(v;2;(v-k)x^2 +ky^2; \textnormal{ y,x;determinant}\right)$.
\end{itemize}
We have used the notation Cretan(order;$\tau ;\omega;$ \textnormal{ y,x;determinant}).
\qed
\end{theorem}
In all these $Cretan-Hadamard$ cases (but not in all cases) the Balonin-Sergeev-Cretan$(4t-1,2)$ matrix with higher determinant comes from $SBIBD(4t-1,2t,t)$ while the $SBIBD(4t-1,2t-1,t-1)$ gives a Cretan$(4t-1,2)$ matrix with smaller determinant. These examples have been given as they may give circulant SBIBD when other matrices do not necessarily do so.
\section{Main Equivalence Theorem}\label{sec:main-equiv-theorem}
\begin{lemma}[\textbf{Hadamard to SBIBD Lemma}] \label{lem:H toSBIBD}
Whenever there exists an Hadamard matrix of order $4t$ there exists an $SBIBD(4t-1,2t-1,t-1)$.
\end{lemma}
\begin{proof}
Let $H$ be the Hadamard matrix of order $4t$. We use the Hadamard equivalence operations to make the normalized Hadamard matrix $G$ with the first row and column all $+1$'s.
It now has the form
\[G = \begin{bmatrix}
1 & 1 & \dots & 1 \\
1 \\
\vdots & & \BigFig{B}\\
1
\end{bmatrix}\]
where $B$ is a $\pm 1$ matrix of order $4t-1$ containing $2t$ $-1$'s and $2t-1$ 1's per row and column. $B$ satisfies $BB^{\top} = 4tI - J$. We form $A$ by replacing the $-1$ elements of $B$ by zero, that is $A =\frac{1}{2}(J+B)$. Then $A$ satisfies $AA^{\top} = tI + (t-1)J$ and is the incidence matrix of an $SBIBD(4t-1,2t-1,t-1)$ as required.
\end{proof}
\begin{lemma}[\textbf{SBIBD to Cretan$(4t-1,2)$ Lemma}] \label{lem:SBIBD to Cretan(4t-1)}
Whenever there exists an $SBIBD(4t-1,2t-1,t-1)$, there exists an Cretan-Mersenne 2-level matrix.
\end{lemma}
\begin{proof}
We take the $SBIBD$ and replace the zeros by $x$ and the ones by $y$. Call this matrix $C$.
We choose $x=1$ and $y = \frac{-t + \sqrt{t}}{t-1}$.
We first show this satisfies the orthogonality equation. The inner product of any row with itself is $2t + (2t-1)\left(\frac{-t + \sqrt{t}}{t-1}\right)^2=\omega$ which is a constant given $t$. Thus we have the radius equation. Using Hadamard equivalence operations we permute the columns of $C$ until we have
\begin{gather*}
\overbrace{y,y,\dots,y,y,~~y,y,\dots, \;y,\;y}^{2t-1} \qquad \overbrace{x,x,\dots,x,x,~~x,x,\dots,x,x}^{2t}\\
\underbrace{y,y,\dots,y,y}_{t-1} ~~\underbrace{x,x,\dots,x,x}_{t} \qquad \underbrace{y,y,\dots,y,y}_{t} ~~ \underbrace{x,x,\dots,x,x}_{t}
\end{gather*}
Then the inner product of rows $i$ and $j$ is
\[(t-1)y^2+2txy+tx^2= (t-1)\left (\frac{-t + \sqrt{t}}{t-1}\right )^2 + 2t\left(\frac{-t + \sqrt{t}}{t-1}\right) + t =0\,.\]
Hence $CC^{\top} =\omega I$ and we have the matrix orthogonality equation. Thus we have formed the required 2-level Cretan-Mersenne matrix.
\end{proof}
\begin{lemma} [\textbf{Cretan$(4t-1,2)$ to Hadamard Lemma}] \label{lem:Cretan(4t-1) to Had}
Whenever there exists an $SBIBD(4t-1,2t-1,t-1)$, there exists an Hadamard matrix of order $4t$ .
\end{lemma}
\begin{proof}
We take the Cretan$(4t-1,2)$ matrix and replace the entries which are not $1$ by zero.
This gives the incidence matrix $A$ of the $SBIBD(4t-1,2t-1,t-1)$. By definition of $SBIBD$ the inner product of any pair of distinct columns is $\lambda = t-1$ so it satisfies $AA^{\top} = tI + (t-1)J$, $J$ the matrix of all $1$'s. We note that it satisfies $AA^{\top} = tI + (t-1)J$. It has a total of $4t-1$ entries, $2t-1$ 1's and $2t$ zeros in each row and column. We now form $B = 2A - J$, of order $4t-1$ and note $BB^{\top} = 4tI -J$. Bordering $B$ with a row and column of $1$'s gives the required Hadamard matrix.
\end{proof}
\begin{theorem} [\textbf{Co-Existence of Hadamard Matrices and Cretan$(4t-1,2)$\\-Mersenne Matrices Theorem}] \label{th:equivalence}
The existence of an Hadamard matrix of order $4t$ is equivalent to the existence of a Cretan$(4t-1,2)$-Mersenne matrix.
\end{theorem}
\begin{proof}
We have shown that and Hadamard matrix of order $4t$ can be used to form an $SBIBD(4t-1,2t-1,t-1)$ using Lemma \ref{lem:H toSBIBD} as its core.
Lemma \ref{lem:SBIBD to Cretan(4t-1)} was used to show that whenever there exists an $SBIBD(4t-1,2t-1,t-1)$, there exists an Cretan-Mersenne 2-level matrix.
This shows that an Hadamard matrix can always be used to obtain a Cretan-Mersenne 2-level matrix.
To prove the converse we use the Cretan$(4t-1,2)$ to Hadamard Lemma \ref{lem:Cretan(4t-1) to Had} to show how a 2-level Cretan$(4t-1,2)$ matrix can always be used to form an Hadamard matrix.
Whenever there exists an $SBIBD(4t-1,2t-1,t-1)$, there exists an Hadamard matrix of order $4t$ .
This shows that a Cretan-Mersenne 2-level matrix can always be used to obtain a Cretan-Mersenne 2-level matrix.
Thus we have shown the existence of one gives the existence of the other. This completes the proof.
\end{proof}
{\bf Conjecture:} Since Hadamard matrices are conjectured to exist for all orders $4t$, $t>0$ an integer, 2-level Cretan$(4t-1,2)$-Mersenne matrices are conjectured to exist for all orders $4t-1$, $t>0$ an integer.
\section{Conclusions}
Cretan matrices are a very new area of study. They have many research lines open: what is the minimum number of variables that can be used; what are the determinants that can be found for Cretan($n;\tau$) matrices; why do the congruence classes of the orders make such a difference to the proliferation of Cretan matrices for a given order; find the Cretan matrix with maximum and minimum determinant for a given order; can one be found with fewer levels?
We conjecture that $\omega \approxeq v $ will give unusual conditions.\qed
\section{Acknowledgements}
The authors also wish to sincerely thank Mr Max Norden, BBMgt(C.S.U.), for his work preparing the content and LaTeX version of this article.
We acknowledge use of \url{http://www.mathscinet.ru} and \url{ http://www.wolframalpha.com} sites for the number and symbol calculations in this paper.
\bibliographystyle{elsarticle-num}
|
1,116,691,498,467 | arxiv | \section{Introduction}
Crystalline materials doped with impurities such as rare-earth-ions (REI), or diamond silicon-vancancy (SiV) and nitrogen-vacancy (NV) centers, have found many applications in fields as diverse as quantum information processing \cite{Riedmatten2015,heshami_quantum_2016,Sipahigil2016,Hensen2015}, quantum memories \cite{Hedges2010,saglamyurekbroadband2011,Maurer1283}, sensing \cite{hong2013}, lasers \cite{Powell98}, and phosphors \cite{kenyon_recent_2002,Justel98}. Nanometer-sized structures fabricated from these materials have begun to be investigated for on-chip implementations of these applications. In addition, small-sized nano-phosphors are desired for high-quality window materials used in lamps as well as for state-of-the-art displays \cite{Wang10,Downing1185}. Finally, nano-powders have also been proposed for optical refrigeration applications where their modified phonon spectrum and particle morphology could enhance the cooling efficiency \cite{Ruan2006}.
Nano-materials can be obtained through different routes: nano-structures can be milled or etched from high-quality bulk materials, and nanocrystals can be obtained through chemical synthesis, mechanical crushing, or ablation techniques. The transition to nano-sized structures generally introduces detrimental effects such as poor crystal quality, surface effects, and amorphous behavior that can severely restrict practical applications \cite{coprec,Lutz2017}. While some of these effects, such as the increasing surface-to-volume ratio, are fundamental, others can be minimized by optimizing the fabrication process \cite{coprec}. Indeed, in some cases, both chemical synthesis as well as fabrication methods starting with bulk materials produced high-quality materials \cite{Zhong2015,Ferrier2013}. However, none of those structures have allowed studying the effects of decreasing dimensions on phonon-mediated population dynamics. Furthermore, a general procedure for achieving consistently high-quality nano-materials is still unknown and many open questions remain regarding the transition to smaller sizes, requiring further systematic studies.
During the transition from a bulk material to nano-structures, many material properties change. One interesting effect is the predicted modification of the vibrational density of states (VDOS) in small structures. Whereas a bulk crystal has a Debye VDOS (a continuous function that increases with the square of the vibrational frequency), the distribution becomes discrete in small crystallites, exhibiting gaps and even a cutoff below which no phonons are supported. Furthermore, phononic crystals---nano-machined, periodic structures---can feature engineered frequency band-gaps where vibrations are forbidden \cite{Lutz2016}. These approaches to phonon engineering could potentially benefit applications in the field of quantum information, in particular quantum memories, since the absence of phonons could enhance spin-population lifetimes as well as optical coherence times. Modifications of population dynamics in REI-doped nanocrystals have been previously reported by Meltzer \textit{et\,al.}\ \cite{Yang1999,Yang1999a}, Liu \textit{et\,al.}\ \cite{Liu2002} and Mercier \textit{et\,al.}\ \cite{Mercier2006}, and it was suggested that the changes were due to phonon suppression in the nanocrystals. However, the particles employed in some of those studies were not sufficiently small to suppress phonons at the desired frequencies, and locally elevated temperatures caused by the optical excitation of the powder materials might explain some of the observed effects. Thus, unambiguous confirmation of the suppression of phonon-mediated relaxation in optical centers through VDOS-engineering remains an open challenge.
In this manuscript we examine the effect of the transition from bulk crystals to $\le$ 40 nm particles (see Fig.~\ref{fig:tb_yag_levels}) on the population dynamics between excited-state crystal-field levels in Tb$^{3+}$-doped Y$_3$Al$_5$O$_{12}$ (YAG). Specifically, we study the influence of size restriction on relaxation dynamics and equilibrium population distribution between the crystal-field levels, i.e. thermalization. We observe that the population dynamics are strongly modified for smaller particles, which can be explained by a modified density of states. However, we also find that the thermal population distribution exhibited by the nano-powders is the same as in the bulk material. As described in detail in the discussion in later sections, the observation of rapid thermalization suggests that in addition to possible phonon suppression, other non-phononic processes---e.g. related to surface defects or energy transfer \cite{Forster}---are introduced, enabling rapid thermalization of population between the closely-spaced energy levels.
\section{Experimental Details}
We chose Tb$^{3+}$ doped Y$_3$Al$_5$O$_{12}$ (YAG) since the combination of its energy level structure and its high acoustic velocity is well suited to investigate the effects of size on the direct phonon process. More precisely, the small excited-state splitting $\Delta_e=35$~GHz between the Tb$^{3+}$ crystal-field levels $^5$D$_4 \, a/b$ (see Fig.~\ref{fig:tb_yag_levels}), together with the acoustic velocity of 6400 m/s \cite{mezeix_comparison_2006}, results in an expected suppression of the direct phonon process for relatively large particles of $\sim100$ nm diameter. In addition, in the bulk crystal, the inhomogeneous broadening of about 20~GHz allows one to selectively address each of the $^5$D$_4 \, a/b$ crystal-field levels. Furthermore, the ground state splitting $\Delta_g=83$~GHz is large enough that resonant phonons are not expected to be inhibited in the $> 40$~nm diameter nanocrystals, and we can therefore directly measure the internal sample temperature through the ratio of population in the two levels $^7$F$_6\, a/b$.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\columnwidth]{TbYAG_levels}
\caption{Relevant energy levels of Tb$^{3+}$:Y$_3$Al$_5$O$_{12}$ (vertical axis not to scale) for the measurement of population relaxation between the first two crystal-field levels within the $^5$D$_4$ excited-state manifold. A pulsed laser excites the ions from the $^7$F$_6 \, a$ ground state to the $^5$D$_4 \, f$ excited state, from where they decay rapidly into $^5$D$_4 \, a$ and $b$. The resulting fluorescence due to the four $^5$D$_4 \, a/b \rightarrow ^7$F$_5 \, a/b$ transitions are collected, spectrally resolved, and then analyzed.}
\label{fig:tb_yag_levels}
\end{figure}
We created our powders using a sol-gel synthesis (method 1 \cite{kaithwas}) and a modified sol-gel synthesis with a freeze drying step under vacuum and at temperatures below -20~C to restrict agglomeration (method 2 \cite{freeze_dry}). Each method leads to slightly different particle morphologies and size distributions. Additional size control can be achieved by changing the annealing duration and temperature. In this way we were able to vary the nanocrystal diameter $d$ between 40 and 500 nm, and 40 and 70 nm using method 1 and 2, respectively. We evaluated the crystal quality of our powders using a scanning electron microscope, a transmission electron microscope, x-ray diffraction, and optical spectroscopy methods (APPENDIX A and B). From these measurements we conclude that the bulk crystal quality can be maintained for crystallites with diameters down to $\sim 80$~nm. For smaller sizes, we observe a slight decrease in crystal quality that manifests itself in an increase of the inhomogeneous broadening. Measurements of the radiative lifetime as a function of particle size (APPENDIX A) confirm that we can approximately treat the powders as individual particles rather than large agglomerates, with method 2 producing less agglomeration than method 1.
All powders were mounted in an unsealed glass cuvette within an Oxford Instruments cryostat. The samples were held at temperatures from $\sim$ 1 K up to 10 K; for temperatures below 2.17 K, the samples were immersed in superfluid liquid helium, whereas for higher temperatures, the samples were cooled by a continuous flow of helium vapor. A pulsed H\"ansch-style nitrogen-laser-pumped dye laser \cite{Haensch1972} with peak powers of up to 10 kW, a pulse duration of 6 ns and a repetition rate of 6 Hz was used with Coumarin 481 dye to provide excitation light at 485 nm. As shown in Fig. \ref{fig:tb_yag_levels}, we resonantly excited Tb ions from the ground state $^7$F$_6\, a$ to the $^5$D$_4 \, f$ \ level, a transition that provides strong absorption. From the $^5$D$_4\, f$ \ level, the population rapidly decays ( $<$ ns) non-radiatively by emission of high-frequency acoustic phonons into the two levels $^5$D$_4\, a$ \ and $b$. Using a SPEX 1401 monochromator ($<$ 3 GHz resolution), we selectively collected fluorescence from each of the four $^5$D$_4\, a/b$ $\rightarrow$ $^7$F$_5\, a/b$ transitions. The collection was at an angle of 90$\degree$ relative to the excitation laser,
and the fluorescence was measured using a photomultiplier tube (Hamamatsu R928) terminated with a variable resistance that allowed for time resolutions as fast as 100 ns.
For all powders and experimental configurations described below, we directly measured the local temperature through the relative population in the two levels $^7$F$_6\, a$ \ and $b$, as detailed in APPENDIX C. We found no local heating, thus confirming that observed changes in relaxation dynamics were not due to elevated sample temperatures.
\section{Time-Resolved Fluorescence Measurements}
Before studying the population dynamics of the $^5$D$_4 \, a/b$ \ levels, we first ensured
that we could selectively collect fluorescence from the two excited levels $^5$D$_4 \, a/b$ for each of our samples. We recorded fluorescence spectra by scanning the monochromator over the four lines connecting $^5$D$_4 \, a/b \rightarrow ^7$F$_5 \, a/b$, with typical fluorescence spectra shown in Fig.~\ref{fig:fluospect}. We observed that the smallest nanocrystals feature an increased inhomogeneous broadening compared to the bulk (for details see APPENDIX B). As a consequence, some fraction of the detected emission originates from the neighboring transition. Since this can lead to observations that could wrongfully be interpreted as modifications in relaxation dynamics, it must be considered in the analysis of any obtained data.
\begin{figure}[]
\centering
\includegraphics[width=1\columnwidth]{fluo_spectra.pdf}
\caption{Fluorescence spectra at 5 K of (a) a 40 nm diameter powder and (b) a 500 nm diameter powder (synthesized via method 1). Each spectrum (black dots) is fit with the sum of four identical Lorentzian lines with the same pairwise energy splitting $\Delta$ ($\Delta^{\textrm{'}}$). For $d=40$ nm the width is 53 GHz and $\Delta$ is 40 GHz, and for $d=500$ nm the width is 13 GHz and $\Delta^{\textrm{'}}$ is 39 GHz.}
\label{fig:fluospect}
\end{figure}
Following this initial characterization, we recorded fluorescence decays from the two crystal-field levels $^5{\rm D}_4 \, a/b$ at a temperature of 1.5~K. The individual decays were collected by sequentially tuning the monochromator on resonance with each of the four transitions $^5{\rm D}_4\ \, a/b \rightarrow \, ^7{\rm F}_5\, a/b$. The specific frequencies of these transitions were obtained from the fluorescence spectra measured for each sample as described above.
After excitation of the $^5{\rm D}_4 \, f\;$ level, 2 THz above $^5{\rm D}_4 \, a$, the population first decays rapidly into both $^5{\rm D}_4 \, a/b$ levels and from there into $^7{\rm F}_5 \, a/b$. The dynamics are captured by the following rate equations:
\begin{figure}[t!]
\centering
\includegraphics[width=1\columnwidth]{decays.pdf}
\caption{Fluorescence decays $^5{\rm D}_4 \, a$ $\rightarrow$ $^7{\rm F}_5 \, a$ (red dots) and $^5{\rm D}_4 \, b$ $\rightarrow$ $^7{\rm F}_5 \, a$ (blue dots) at 1.5 K in large crystallites ($d=500$~nm). The experimental points are fit with Eq.~\ref{eq:2expdecay} (solid lines), resulting in $T_\alpha=$ 235 $\mu$s and $T_\beta = $ 4 ms. Inset: relevant level structure and rates associated with the fluorescence decays (see definitions in the main text).}
\label{fig:decays}
\end{figure}
\begin{align}
\dot{n}_a(t) &= \gamma_{ba} \left[ n_b(t) - n_a(t) \right] - \gamma_a n_a(t) \\
\dot{n}_b(t) &= - \, \gamma_{ba} \left[ n_b(t) - n_a(t) \right] - \gamma_b n_b(t)
\label{rateequations}
\end{align}
where $n_{a/b}(t)$ denote the populations in the levels $^5{\rm D}_4 \, a/b$ at a time $t$ after excitation, $\gamma_{a/b}$ are the rates of the radiative decay from each level into the $^7{\rm F}_5$ multiplet, and $\gamma_{ba}$ is the rate of the non-radiative process coupling the levels $^5{\rm D}_4 \, a/b$. Note that relaxation into the ground state multiplet is ignored (experimentally and in Eq. \ref{rateequations}) due to the transitions' comparably small rates.
For all four transitions (see Fig.~\ref{fig:decays} for two examples), we observed fluorescence decays composed of two components. The first component corresponds to the non-radiative decay of population from $^5{\rm D}_4 \, b$ to $^5{\rm D}_4 \, a$ and manifests in the fluorescence decays from $^5{\rm D}_4 \, a$ to $^7{\rm F}_5 \, a/b$ as a fill-in, i.e. an increase of the fluorescence intensity with time, and in the fluorescence from $^5{\rm D}_4 \, b$ to $^7{\rm F}_5 \, a/b$ as an initial, fast decay, both with the same characteristic time of about 0.2~ms.
The second, long component corresponds to the radiative decay. Thus, we fit the recorded, time-dependent fluorescence intensities $I_{a,b}(t)$ from the levels $^5{\rm D}_4 \, a,b$ using
\begin{equation}
I_{a,b}(t) = \tilde{\alpha}_{a,b} e^{-t/T_{\alpha}} + \beta_{a,b} e^{-t/T_{\beta}} ,
\label{eq:2expdecay}
\end{equation}
where $T_\alpha$ and $T_\beta$ are the time constants of the fast (non-radiative) and slow (radiative) components with corresponding amplitudes $\tilde{\alpha}_{a,b}$ and $\beta_{a,b}$. As discussed before, due to increased inhomogeneous broadening in the small powders, the fluorescence signals collected from transitions starting in either $^5{\rm D}_4 \, a$ or $b$ \ contain a certain amount of emission from the other transition. Fitting our fluorescence spectra with four Lorentzians allows us to compute the percentage $P_{a,b}$ of the collected signal that originates from the neighboring transition. This affects the recorded signal since the fill-in and the fast decay originating from $^5{\rm D}_4 \, a$ and $b$, respectively, compensate each other to a certain degree. However, the time constants and the amplitudes of the slow decays are not affected. After obtaining $P_{a,b}$ from our fits, we subsequently compute the corrected amplitudes $\alpha_{a,b}=\tilde{\alpha}_{a,b}/(1-2P_{a,b})$. Since we obtained similar results from decays to the levels $^7{\rm F}_5 \, a$ and $b$, the following observations correspond to averages over all four transitions unless otherwise stated.
\begin{figure}[t!]
\centering
\includegraphics[width=1\columnwidth]{Anr_Tb_vs_size.pdf}
\caption{Amplitude ratio $\alpha / \beta$ obtained from an average of the fits of Eq.~\ref{eq:2expdecay} to the four decays $^5{\rm D}_4 \, a,b$ $\rightarrow$ $^7{\rm F}_5 \, a,b$ as a function of average nanocrystal diameter $d$ for powder 1 (red circles) and powder 2 (blue squares) at 1.5 K. The horizontal error bars correspond to one standard deviation of the size distribution of the respective sample, obtained through SEM analysis. Simulations are depicted using dashed lines. As described in the main text, the difference between the two simulations is the width of the phonon modes that can be different due to the different morphologies of the two synthesis methods. Note that the amplitude of the fast decay has been corrected for the partially overlapping lines as described in the main text. Inset: Characteristic time of the fast decay component as a function of nanocrystal diameter $d$}
\label{fig:Anr_Tb_vs_size}
\end{figure}
By reducing the size of the particles, we expect to introduce a frequency cutoff in the VDOS below which phonons are not allowed. Consequently, we expect to observe a change in the non-radiative relaxation between the two closely spaced crystal-field levels $^5{\rm D}_4 \, a$ and $b$, described by the first term in Eq.~\ref{eq:2expdecay}. The cutoff frequency is given by
\begin{equation}
\nu_{\rm min}= \eta \, \frac{c}{\pi d} ,
\label{eq:cutoff}
\end{equation}
where $c$ is the sound velocity, $d$ is the diameter of the crystal, and $\eta$ is a numerical constant that equals 2.05 for one spherical particle \cite{Lamb1881} and phonon modes with negligible broadening. According to this formula, YAG particles (where $c=6400$~m/s \cite{mezeix_comparison_2006}) with diameters below 100~nm should not be able to support phonons at $\Delta=35$~GHz, corrresponding to the splitting between the two crystal-field levels $^5{\rm D}_4 \, a/b$. Because our powder samples inevitably exhibit some distribution of particle sizes (the horizontal error bars in Fig. \ref{fig:Anr_Tb_vs_size} correspond to one standard deviation in the respective size distribution), the non-radiative relaxation should be suppressed in more and more particles as the mean of the size distribution is reduced below $\sim$~100~nm. This should lead to a gradual decrease in the amplitude of the fast component $\alpha$ of the fluorescence decays. Note that increased inhomogeneous broadening causing partial overlap of the levels $^5{\rm D}_4 \, a$ and $b$ could also lead to similar observations. In our case, we account for this effect by computing the actual amplitude $\alpha$ from the measured one $\tilde{\alpha}$ as described above. As shown in Fig.~\ref{fig:Anr_Tb_vs_size}, we find that for particles created using method 1, the decay ratio begins to decrease at around 130~nm diameter, whereas the onset for particles created via method 2 starts at sizes of $\sim$~70 nm.
We modeled this effect by calculating the VDOS \cite{Lutz2016}, assuming that each mode contributes a Lorentzian with a width $\Delta\nu$ to the VDOS, for $10^5$ particles with different particle diameter following a Gaussian distribution. We obtained the mean diameter and the standard deviation of this distribution from the SEM images of our samples. We assumed that ions in a nanocrystal with diameter $d$ have a fast decay only if the VDOS of the nanocrystal at $\Delta = 35$~GHz is greater than zero.
For each of the $10^5$ particles, we simulate the fluorescence decay, either a sum of two exponential functions in case the fast decay is allowed or only a single exponential decay with the radiative decay time in case the fast decay is forbidden. We then average the individual decays to obtain the overall fluorescence decay for the ensemble of $10^5$ particles. Finally, we fit it with a sum of two exponential functions which results in the amplitudes $\alpha$ and $\beta$ of the fast and slow decay, respectively.
The only free parameter in the simulations is the width $\Delta\nu$ of the individual vibrational modes, which we obtain through a fit to the experimental data. Note, the mode width is forced to be the same for all powders fabricated using a specific method. The results of the simulations are presented in Fig.~\ref{fig:Anr_Tb_vs_size}.
For large crystal diameters $d$, the simulated amplitude ratios (for crystals created via either method) are around 2, which corresponds to the bulk value at 1.5~K. When we reduce $d$, the amplitude ratios start decreasing at two distinct diameters (130 nm for method 1 and 70 nm for method 2 crystals). This can be explained by different mode widths ($\Delta\nu=0.5$ and $6$~GHz for powders from methods 1 and 2, respectively) obtained from the simulations. If phonon modes are broad, as in powders from method 2, it is more likely that they overlap with the transition between the $^5{\rm D}_4 \, a/b$ levels, even for small crystals. However, for sharp modes (powders from method 1), overlap becomes significant only for larger particles in which more modes exist. Note that the width of the phonon modes is related to the powder quality. In particular, crystallites with reduced surface roughness should feature narrower phonon modes. This leads us to conclude that powder 1 should be of higher quality, which, however, cannot be verified given the insufficient resolution of our SEM pictures. Note that the optical inhomogeneous linewidths suggest that method 2 produces powders with less internal strain; however, it is not known if there is a relationship between particle surface morphology and internal strain.
Overall, the simulated values are in good agreement with the experimental data for powders produced by either method, consistent with suppression of phonon-induced relaxation in sufficiently small powders. In particular, we observed the complete transition from large particles, where the relevant phonon processes are fully allowed, to the smallest particles, where we could not measure any contribution of the phonon-induced component to population relaxation.
In addition to a change in the amplitude of the fast decay, we also expect a change in its characteristic decay time $T_\alpha$. For the fraction of particles with $\nu_{\rm min} \simeq \Delta_e$, the phonon density of states at $\Delta_e$ should deviate from the bulk value. With small enough particles we expect a decrease in VDOS as the lowest phonon mode moves towards higher frequencies. Thus, because the rate of the non-radiative relaxation is proportional to the VDOS, the phonon-induced decay rate for these nanocrystals should be slower than the one for the bulk, i.e. $T_\alpha$ should increase with decreasing particle size. Note that in some cases, an enhanced VDOS can occur due to phonon confinement, which would lead to faster decay rates.
Experimentally (see inset Fig.~\ref{fig:Anr_Tb_vs_size}), we observe a decrease in $T_\alpha$ for some particles but we do not observe the expected increase for particles from method 1. This is consistent with the conjecture of having sharp phonon modes for particles from method 1 (resulting from the fit of the decay ratios in Fig.~\ref{fig:Anr_Tb_vs_size}), in which case the phonon cutoff occurs abruptly as the size is reduced so that the particles either experience a phonon rate equal or larger than the bulk or no phonon decay at all.
For powders from method 2, the fit of the decay ratios predicts broader phonon modes, and we thus expect a smoother transition (rather than an abrupt cutoff as for particles from method 1)in the VDOS from the bulk value to zero. Therefore, more particles should experience decay times longer than the bulk before the decay is suppressed completely. Indeed, except for the smallest powder created using method 2, we see signatures of such an increase in $T_\alpha$, consistent with the predictions of our model. At this stage, the scatter of our experimental data unfortunately does not allow for a more in-depth analysis and interpretation.
\section{Measurement of temperature dependent population re-distribution}
Another indication of the restriction of non-radiative transition processes, including phonon modes, is the inhibition of thermalization (population re-distribution) between the two crystal-field levels $^5{\rm D}_4 \, a/b$ after their initial population through decay from $^5{\rm D}_4 \, f$. The ratio $n_b/n_a$ will subsequently change and approach thermal equilibrium due to any non-radiative transitions between $^5{\rm D}_4 \, a/b$. For $t >T_\alpha$, the population ratio is described by
\begin{equation}
n_b/n_a =(1-N_0) \, e^{-\Delta/k_B T}+N_0
\label{eq:boltzmann}
\end{equation}
i.e. the Boltzmann distribution that assumes thermalization through non-radiative processes, with an additional offset $N_0$ that accounts for the overlap of the studied transitions, as described below. The population ratio $n_b/n_a$ for $t>T_\alpha$ is directly related to the ratio of the amplitudes $\beta_{b}$ and $\beta_{a}$ of the long fluorescence decay components (see Eq.~\ref{eq:2expdecay}): $\beta_{b}/\beta_{a}\propto~n_b(t)/n_a(t)$. This allows us to extract the actual population distribution for different temperatures.
\begin{figure}[t!]
\centering
\includegraphics[width=1\columnwidth]{thermalization.pdf}
\caption{Ratio of population $n_b/n_a$ as a function of temperature in the bulk crystal (black squares) and in nanocrystals from method 1 of average diameter $d=72$~nm (blue triangles) and 40 nm (red dots). Solid lines are best fits using Eq.~\ref{eq:boltzmann}, and shaded areas represent uncertainties.}
\label{fig:thermalization}
\end{figure}
The measured temperature dependence of $n_b/n_a$, calculated by averaging $\beta_{a}$ (and $\beta_{b}$) over both transitions starting in $^5$D$_4\, a$ (and $^5$D$_4\, b$), is shown in Fig.~\ref{fig:thermalization} for the bulk crystal as well as for two nanocrystal samples ($d=40$ and 72~nm) created via method 1. We fit the experimental points using Eq.~\ref{eq:boltzmann} after fixing the energy separation $\Delta_e$ to the bulk value of 35 GHz and leaving $N_0$ as a free variable. For both powders we observe a non-zero offset $N_0$. The fit gave $N_0$=0.21$\pm$0.1 and 0.14$\pm$0.07 for the two powders with $d=72$~nm and 40 nm, respectively, and for the bulk crystal it resulted in $N_0$=0, suggesting that there is the expected difference in thermalization in the powders. However, this offset can be fully explained by the amount of emission $P_{a,b}$ originating from the neighboring transition due to the increased inhomogeneous broadening as discussed above. Indeed, calculating $N_0$ by taking only line overlap into account, we find $N_0=$ 0.3$\pm$0.02 and 0.26$\pm$0.02 for $d=72$~nm and 40 nm, respectively. This seems to be at odds with the observed reduction of the nanocrystal's fast decay amplitude (Fig. \ref{fig:Anr_Tb_vs_size}), which is also supported by VDOS simulations. Thus, the observation of thermalization indicates that other processes, happening on time scales smaller than the 100 ns resolution of our detector, are responsible for population re-distribution in our smallest nanocrystals.
Such fast relaxation could be caused by coupling of the Tb$^{3+}$ ions to tunneling modes characteristic of amorphous materials \cite{anderson_anomalous_1972,phillips_tunneling_1972} (note that the increase of amorphous character as the particle size is reduced is supported by the observation of larger inhomogeneous broadening). Other explanations are relaxations driven by energy transfer \cite{Forster} or interactions between ions and surface states.
\section{Conclusion}
In conclusion, we observed modifications in relaxation dynamics between crystal field levels of Tb$^{3+}$:Y$_3$Al$_5$O$_{12}$\ crystals as the particle size is varied from bulk to 40 nm, and confirmed via absorption measurements that this effect is not due to local heating. One possible explanation is a modification of the VDOS in the nanocrystals, which restricts phonon processes between the two first crystal-field levels in the $^5{\rm D}_4$ excited state that are separated by 35 GHz.
However, other measurements suggest a different explanation: population redistribution is still observed within the two closely-spaced levels, meaning that other, fast, non-radiative processes must enable this transition. These processes may arise from a partially amorphous character of the nanocrystals, even though significant effort was dedicated to achieving good crystal quality by exploring various synthesis methods and modifying different important parameters in each of them, such as the addition of surfactants or the annealing temperature. We note that the case of YAG is particularly difficult because of the high annealing temperature---which favors particle growth--- required to crystallize the particles.
Improving the nanocrystal quality by optimizing fabrication methods, as well as switching to a different material with a lower annealing temperature, such as fluoride crystals, may enable one to observe the full phonon restriction. However, there may exist a fundamental limit to how small a particle can become while still preserving the spectroscopic properties of a large crystal -- this limit is frequently estimated to be around 10 nm \cite{Bunzli}. Our results suggest that, for YAG, it may be around 100 nm. Measurements of crystal structure may shine more light on this important question.\\
\section{Acknowledgements}
The authors acknowledge support from Alberta Innovates Technology Futures (ATIF), the National Engineering and Research Council of Canada (NSERC), and the National Science Foundation of the USA (NSF) under award nos. PHY-1415628 and CHE-1416454, and the Montana Research and Economic Development Initiative. W. T. is a Senior Fellow of the Canadian Institute for Advanced Research (CIFAR).
\section{APPENDIX A: Powder characterization}
For all investigations of population dynamics presented in the main text, information about the morphology and size distribution of the various powders is needed to interpret the results. We obtained this information for powders from method 1 and 2 using a scanning electron microscope (SEM), with example images shown in Fig. \ref{fig:images}. The images indicate that method 2 produces slightly less agglomerated powders compared to method 1. In addition, we confirmed that we obtained good single-phase crystalline YAG particles using powder x-ray diffraction (XRD) analysis. Figures \ref{fig:images} (e,f) show the perfect overlap between the XRD spectrum of the Tb$^{3+}$:Y$_3$Al$_5$O$_{12}$\ powders produced by method 1 and 2 with the reference spectrum for YAG (JCPDS \# 30-0040). For selected powder samples, we also directly probed the quality of the crystal structure using a transmission electron microscope (TEM), as shown in Fig. \ref{fig:images} (d). The TEM analysis revealed that the crystallite orientations in agglomerated nanocrystals can remain nearly aligned throughout multiple grains when they are fused together. Since it is possible that phonon modes extend across several crystallites in these cases, we considered the effective particle size in agglomerated samples to be equal to the larger size of the approximately aligned agglomerations rather than the individual grain size. The effect of agglomeration on phonon propagation dynamics cannot be quantified at this stage.\\
\begin{figure}[]
\centering
\includegraphics[width=1\columnwidth]{images}
\caption{Microscope images of 1\% Tb$^{3+}$:Y$_3$Al$_5$O$_{12}$ \ created using method 1 and annealed at 1400 C (a), or 900 C (b,c), and from method 2 annealed at 900 C (d). Panels (a,b,d) are SEM images, showing the size distribution of the nanocrystals. Panels (e) and (f) show XRD spectra of powders produced by method 1 and 2, respectively (solid blue lines), and the corresponding reference spectrum (JCPDS \# 30-0040; red circles) for YAG. Panel (c) is a high-resolution TEM image showing the crystalline structure (narrow white lines), which can extend over several particles if they are agglomerated.}
\label{fig:images}
\end{figure}
In addition to using the TEM measurements, we also investigated the degree to which the crystallites within the powder act as isolated particles versus being part of a larger agglomerated mass by observing the increase in radiative lifetime as the particle sizes are reduced. As known from relations such as the Strickler-Berg equation \cite{Strickler62}, the radiative lifetime $T_{\rm rad}$ of an electric dipole transition depends on the average index of refraction, $n$, of the material.
Because of this, when the size of a fluorescing particle becomes comparable to the wavelength of the emitted light, the index of the medium surrounding the particle can have a significant effect on the radiative lifetime\cite{Yang1999,Schniepp02,Aubret16}. Consequently, we expect to observe an increase in the fluorescence lifetime as the average particle size in our powders is reduced. Note that strongly agglomerated particles would effectively act as a single, larger particle in this case.\\
We employ a simple analytical model to estimate the size dependence of the fluorescence lifetime for perfectly isolated particles. More precisely, we used the form of the Lorentz local field, sometimes referred-to as the virtual cavity model \cite{Scheel1999}, where the radiative lifetime in the medium $T_{\rm rad}$ is related to the lifetime in vacuum $T_0$ according to $1/T_{\rm rad} = (1/T_0) \, n_{\rm eff}(n_{\rm eff}^2+2)^2 / 9$, with $n_{\rm eff}$ being an effective index of refraction averaged over the surrounding medium within a distance on the order of the wavelength of light from the ion.
For particles smaller than the wavelength of light, the electric field extends beyond the particle. To evaluate $n_{\rm eff}$ for such particles, we assumed that the electric field of the emitted light experiences the bulk crystal dielectric constant within the particle, and the vacuum dielectric constant outside the particle. We furthermore assumed that the electric field outside the particle decays as $E(r)=E_0 e^{-\frac{r}{l}}$ over the decay length $l$, which is equal to the evanescent field decay length outside of a bulk dielectric given by $l=\lambda_0/2 \pi\sqrt{n^2-1}$ \cite{Fornel00}. Here $n$ is the refractive index of the bulk material and $\lambda_0$ the wavelength of the transition. The value of $n_{\rm eff}$ experienced by the emitting ion, and the resulting change in radiative lifetime, was then estimated by calculating the field-strength-weighted average dielectric constant over the area of non-zero electric field.\\
By using this simple model, we estimated the change in lifetime with particle size using only the known bulk crystal index, transition wavelength, and lifetime with no free parameters, resulting in the solid line in Fig.~\ref{fig:trad}. We find that the measured lifetimes in our samples agree reasonably well with the calculated dependence (Fig.~\ref{fig:trad}). For method 1, the radiative lifetime increases up to 7 ms, as the crystallite size decreases, but powders smaller than 50 nm show lifetimes similar to the bulk crystal, indicating that some degree of agglomeration is present. For method 2, the radiative lifetime increases up to 13 ms, indicating that the particles in powders synthesized with this method indeed behave as individual particles with sizes approximately equal to the values estimated from the SEM and TEM analysis.
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{tradvssize}
\caption{Particle size ($d$) dependence of the radiative lifetime of the $^5$D$_4\, a$ level for powders from method 1 (red circles) and powders from method 2 (blue squares). The solid line shows the expected dependence.}
\label{fig:trad}
\end{figure}
\section{APPENDIX B: Spectroscopic investigations of powder quality}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{ing_vs_size.pdf}
\caption{Particle-size-dependence of the inhomogeneous linewidth for powders from method 1 (red circles) and powders from method 2 (blue squares). Inset: Splitting $\Delta$ between the $^5$D$_4 \, a/b $ levels versus particle size. The solid lines are guides to the eye.
\label{fig:inh}
\end{figure}
To ensure that we can selectively collect fluorescence from the two excited levels $^5$D$_4 \, a/b$ for each of our samples, we recorded fluorescence spectra by scanning the monochromator over the four lines connecting $^5$D$_4 \, a/b \rightarrow \, ^7$F$_5 \, a/b$. As shown in Fig.~\ref{fig:inh}, we observed that for both fabrication methods, the smallest nanocrystals feature an increased inhomogeneous broadening compared to the bulk. This increase, which was expected due to an increased amount of strain, was not observable in the XRD spectra due to the limited resolution of our XRD diffractometer (Rigaku Multiflex). The observation of increased inhomogeneous broadening is consistent with the emergence of relaxation that is facilitated by amorphous phases and surface defects (see main text). However, as shown in the inset of Fig.~\ref{fig:inh}, the splitting $\Delta$ between the $^5$D$_4 \, a/b$ levels does not change with particle size, which indicates that the ions' crystal field splittings and local lattice symmetry are not measurably different in the small powders.
\section{APPENDIX C: Local temperature measurement}
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{tempcal.pdf}
\caption{Ratio $n_2/n_1$ of populations in the ground manifold $^7$F$_6 \; b/a$ levels as a function of the temperature $T_{\rm set}$ read by the cryostat sensor for a bulk crystal (black dots) and nanocrystalline powder samples with diameters $d=124\pm 30$~nm and produced by method 1 (blue squares) and $d=63 \pm 11$~nm, produced by method 2 (red triangles). The solid line is the fit to a Boltzmann distribution with $\Delta_g = 83.5$~GHz. Note that the large error bar for 3.2 K and $d = 124$~nm is caused by a large uncertainty of the fit to that particular absorption spectrum.}
\label{fig:tempcal}
\end{figure}
In past measurements, the thermal conductivity of small powders in a gas environment was observed to decrease with particle size due to two effects. First, the phonon scattering length is reduced in small powders, and second, heat flow is hindered by the surface resistance of the small particles contained in the powder \cite{Garrett1974,Rettelbach1995,Brodie1965}. These effects could lead to a locally elevated powder temperature, especially when the powder is probed using a high power laser. An elevated sample temperature in turn would produce significant changes in population relaxation and thermalization that could potentially be misinterpreted as arising from other effects. To ensure that the laser excitation did not induce localized heating, a direct temperature measurement that enables the true internal temperature of the particles to be monitored is required.
We measured the internal temperature for each sample by recording the absorption spectrum $^7$F$_6 \, a,b$ $\rightarrow$ $^5$D$_4 \, f$ and comparing the populations $n_1$ and $n_2$ in the first and second crystal field level $^7$F$_6 \, a$ and $b$ of the ground state multiplet, which are separated by $\Delta_g = 83$~GHz. This allowed us to determine the effective internal sample temperature $T$ via the Boltzmann law $n_2/n_1 = A \exp\{-\Delta_g /k_B T\}$ with $k_B$ the Boltzmann constant. We obtained the coefficient $A$, which depends on the relative oscillator strengths of the transitions, from a calibration measurement using a 1\% Tb:YAG bulk crystal, that we assumed to thermalize quickly to the temperature $T_{\rm set}$ that was measured by the cryostat sensor. The temperature-dependent population ratio $n_2/n_1$ is shown in Fig.~\ref{fig:tempcal} for a selected set of nanocrystals. The close overlap between the bulk and powder results confirms that all powders thermalize as well as the bulk when immersed in liquid helium or vapor and that the laser is not measurably heating the nanocrystals, even for the smallest sizes. Since these measurements are conducted using the same laser power and focusing parameters as those used for all other measurements described in the main text, they confirm that the changes in dynamics that we observe in the powders are not caused by local heating. Also, with this method, we ensure that the ions that are studied in the relaxation dynamics measurements (as in the main text) are also the same ions that are used to measure the temperature.
|
1,116,691,498,468 | arxiv | \section*{Introduction}
In quantum cosmology the whole universe is treated
quantum-mechanically and is described by a wave function rather than
by a classical space-time. This quantum approach to cosmology may
help us avoid the cosmological singularity problem and
understand what determined the initial state of the universe.
The wave function of the universe $\psi$ satisfies the Wheeler-DeWitt
equation
\begin{equation}
{\cal H}\psi=0
\end{equation}
which is analogous to the Schrodinger equation in ordinary quantum
mechanics. To solve this equation, one has to specify some boundary
conditions for $\psi$. In quantum mechanics, the boundary conditions
are determined by the physical setup external to the system. But
since there is nothing external to the universe, it appears that
boundary conditions for the wave function of the universe should be
postulated as an independent physical law. The possible form of this law has
been debated for some 15 years and is one of the central points of our
debate here.
There are at least three proposals on the table:
the Hartle-Hawking wave function \cite{HH}, the Linde wave function
\cite{L1}, and the tunneling wave function \cite{V84,V86}. In
this talk I will first review the tunneling wave function. Then,
since this is supposed to be a debate, I will attack my opponents.
And finally, I will comment on the recent work on quantum creation of
open universes.
\section*{The tunneling wave function}
To introduce the tunneling wave function, let us consider a very
simple model of a closed Robertson-Walker universe filled with a
vacuum of constant energy density $\rho_v$ and some radiation. The
total energy density of the universe is given by
\begin{equation}
\rho=\rho_v+\epsilon/a^4,
\label{1}
\end{equation}
where $a$ is the scale factor and $\epsilon$ is a constant
characterizing the amount of radiation.
The evolution equation for $a$ can be
written as
\begin{equation}
p^2+a^2-a^4/a_0^2=\epsilon.
\label{2}
\end{equation}
Here, $p=-a{\dot a}$ is the momentum conjugate to $a$ and
$a_0=(3/4)\rho_v^{-1/2}$.
\begin{figure}[b!]
\centerline{\epsfig{file=cosm3.eps,height=3.5in,width=3.5in}}
\vspace{10pt}
\caption{The potential for the scale factor in Eq.(\ref{2}). Instead
of recollapsing, the universe can tunnel through the potential
barrier to the regime of unbounded expansion.}
\label{fig1}
\end{figure}
Eq.(\ref{2}) is identical to that for a ``particle'' of energy
$\epsilon$ moving in a potential $U(a)=a^2-a^4/a_0^2$. For
sufficiently small $\epsilon$, there are two types of classical trajectories.
The universe can start at $a=0$, expand to a maximum radius $a_1$ and
then recollapse. Alternatively, it can contract from
infinite size, bounce at a minimum radius $a_2$ and then re-expand
(see Fig. 1). But in quantum cosmology there is yet another
possibility. Instead of recollapsing, the universe can
tunnel through the potential barrier to the regime of unbounded
expansion. The semiclassical tunneling probability can be estimated
as
\begin{equation}
{\cal P}\sim\exp\left(-2\int_{a_1}^{a_2} |p(a)|da\right).
\end{equation}
It is interesting that this probability does not vanish in the limit
of $\epsilon\to 0$, when there is no radiation and
the size of the initial universe shrinks to
zero. We then have tunneling from {\it nothing} to a closed universe
of a finite radius $a_0$; the corresponding probability is
\begin{equation}
{\cal P}\sim\exp\left(-2\int_0^{a_0} |p(a)|da\right)
=\exp\left(-{3\over{8\rho_v}}\right).
\end{equation}
The tunneling approach to quantum cosmology assumes that
our universe originated in a tunneling event of this kind. Once it
nucleates, the universe immediately begins a de Sitter inflationary
expansion.
The Wheeler-DeWitt equation for our simple model can be obtained by
replacing the momentum $p$ in (\ref{2}) by a differential operator,
$p\to -id/d a$,
\begin{equation}
\left({d^2\over{da^2}}-a^2+{a^4\over{a_0^2}}\right)\psi(a)=0.
\label{WDW}
\end{equation}
This equation has outgoing and ingoing wave solutions corresponding to
expanding and contracting universes in the classically allowed range
$a>a_0$ and exponentially growing and decaying solutions in the
classically forbidden range $0<a<a_0$.
The boundary condition that selects the tunneling wave function
requires that $\psi$ should include only an outgouing wave at
$a\to\infty$ (see Fig.2). The under-barrier wave function is then a linear
combination of the growing and decaying solutions. The two solutions
have comparable magnitudes near the classical turning point, $a=a_0$,
but the decaying solution dominates in the rest of the under-barrier
region.
\begin{figure}[b!]
\centerline{\epsfig{file=cosm2.eps,height=3.5in,width=3.5in}}
\vspace{10pt}
\caption{The tunneling wave function for the simple model (\ref{WDW}).}
\label{fig2}
\end{figure}
In a more realistic model, the constant vacuum energy density $\rho_v$
is replaced by the potential $V(\phi)$ of some scalar field $\phi$.
If $V(\phi)$ is a sufficiently slowly-varying function of $\phi$, one
finds the same result as before, with the replacement $\rho_v\to
V(\phi)$,
\begin{equation}
{\cal P}\sim\exp\left(-{3\over{8V(\phi)}}\right).
\label{prob}
\end{equation}
Eq.(\ref{prob}) can be interpreted as the probability distribution for the
initial values of $\phi$ in the ensemble of nucleated universes.
The highest probability is obtained
for the largest values of $V(\phi)$ (and smallest initial size
$a_0$). Thus, the tunneling wave function `predicts' that the
universe is most likely to nucleate with the largest possible vacuum
energy. This is just the right initial condition for inflation.
In the general case, the wave function of the universe is defined on
superspace, which is the space of all 3-dimensional geometries and
matter field configurations,
$\psi [g_{ij}({\bf x}), \phi ({\bf x})]$,
where $g_{ij}$ is the 3-metric, and matter fields are represented by a
single field $\phi$. The tunneling boundary condition can be extended
to full superspace by requiring that $\psi$ should include only
outgoing waves at the boundary of superspace, except the part of the
boundary corresponding to vanishing 3-geometries.
Alternatively, the tunneling wave function can be defined as a path
integral
\begin{equation}
\psi_T(g,\phi)=\int_\emptyset^{(g,\phi)}e^{iS},
\label{psiT}
\end{equation}
where the integration is over paths interpolating between a vanishing
3-geometry $\emptyset$ (`nothing') and $(g,\phi)$. In other words,
the integration is over compact Lorentzian geometries
bounded by the 3-geometry $g$ with the field configuration $\phi$.
At present these general definitions of the tunneling wave function
remain largely formal since we do not know how to solve the
Wheeler-DeWitt equation or how to calculate the path integral
(\ref{psiT}), except for simple models and small perturbations about them.
\section*{The Hartle-Hawking wave function}
The Hartle-Hawking wave function is expressed as a path integral over
compact Euclidean grometries
bounded by a given 3-geometry $g$,
\begin{equation}
\psi_{HH}(g,\phi)=\int^{(g,\phi)}e^{-S_E}.
\label{HH}
\end{equation}
The Euclidean rotation of the time axis,
$t\to i\tau$,
is often used in quantum field theory because it improves the
convergence of the path integrals. However, in quantum gravity the
situation is the opposite. The gravitational part of the Euclidean
action $S_E$ is unbounded from below, and the integral (\ref{HH}) is
badly divergent. One can hope that the problem will somehow be fixed
in the future theory of quantum gravity, but at present we cannot
meaningfully define an integral such as (\ref{HH}).
In practice, one assumes that the dominant contribution to the path
integral is given by the stationary points of the action (the
instantons) and evaluates $\psi_{HH}$ simply as
$\psi_{HH}\sim e^{-S_E}$.
For our simple model, $S_E\approx -3/8V(\phi)$ and
\begin{equation}
{\cal P}\sim\exp\left(+{3\over{8V(\phi)}}\right).
\label{probHH}
\end{equation}
The wave function $\psi_{HH}(a)$ for this model is shown in Fig.3.
It has only the growing solution under the barrier and a superposition
of ingoing and outgoing waves with equal amplitudes in the classically
allowed region. This wave function appears to describe a contracting
and re-expanding universe.
\begin{figure}[b!]
\centerline{\epsfig{file=cosm1.eps,height=3.5in,width=3.5in}}
\vspace{10pt}
\caption{Hartle-Hawking (top) and Linde (bottom) wave functions for
the model (\ref{WDW}).}
\label{fig1}
\end{figure}
The distribution (\ref{probHH}) is similar to Eq.(\ref{prob}) for the
tunneling wave function, but
there is a crucial difference in sign. The distribution (\ref{probHH}) is
peaked at the smallest values of $V(\phi)$, and thus the
Hartle-Hawking wave function tends to predict initial conditions that
disfavor inflation.
In fairness, one has to admit that this objection is not fatal.
Without inflation, a tiny nucleated universe will never reach a
macroscopic size and will not evolve any living creatures, so there
will be nobody there to observe it. If universes are predominantly of
this kind, then most of them will never be observed. The right
question to ask, then, is not what a typical universe looks like, but
what a typical observer will see \cite{M}.
The larger the universe is, the
more stars it contains, the more civilizations are likely to develop.
If we are a typical civilization, then we can expect
to live in a large and populous universe characterized by a large
amount of inflation \cite{M,LGB,Page}.
Moreover, in many models inflation is eternal to the future, provided
that the initial value of $V(\phi)$ is sufficiently large \cite{V83,L2}.
An eternally inflating universe produces an infinite number of
observers, and thus we expect to find ourselves in such a universe
with a 100\% probability, even if the probability of its nucleation is
very low.
\section*{The Linde wave function}
Linde suggested that the wave function of the universe is given by a
Euclidean path integral like (\ref{HH}), but with the Euclidean time rotation
performed in the opposite sense,
$t\to +i\tau$,
yielding
\begin{equation}
\psi_L=\int^{(g,\phi)}e^{+S_E}.
\end{equation}
For our simple model, this wave function gives the same nucleation
probability (\ref{prob}) as the tunneling wave function.
The problem with this proposal is that the Euclidean
action is also unbounded from above and, once again,
the path integral is divergent.
This divergence is even more disastrous than in the Hartle-Hawking
case, because now all integrations over matter fields and over
inhomogeneous modes of the metric are divergent.
It is not clear how Linde's proposal can be extended beyond the
simplest model. This problem of Linde's wave function makes it an
easy target, and I suspect it is for this reason that Stephen likes to
confuse $\psi_L$ and $\psi_T$ and refer to both of them as ``the
tunneling wave function''. In fact, the two wave functions are quite
different, even in the simple model (\ref{WDW}) \cite{discord}.
The Linde wave
function includes only the decaying solution under the barrier and a
superposition of ingoing and outgoing modes with equal amplitudes
outside the barrier (see Fig. 3).
To summarize, the Hartle-Hawking and Linde wave functions have serious
problems with divergent integrals. In addition, the Hartle-Hawking
wave function has a potential problem with inflation. The tunneling
wave function appears to do reasonably well on both accounts.
\section*{Quantum creation of open universes}
\begin{figure}[b!]
\centerline{\epsfig{file=cosm5.eps,height=2.33in,width=3.5in}}
\vspace{10pt}
\caption{The Hawking-Turok instanton.}
\label{fig4}
\end{figure}
Hawking and Turok (HT) have recently argued \cite{19} that open
universes can be spontaneously created from nothing and suggested an
instanton to describe this process. They considered a model with a
very simple potential,
\begin{equation}
V(\phi) = {1\over{2}}m^2\phi^2.
\label{V}
\end{equation}
This model is known not to have regular
instanton solutions, and indeed, the HT instanton is
singular. Geometrically, it is like a sphere with a thorn, the tip of
the thorn being the singularity where the curvature and the scalar
field are infinite (see Fig. 4). HT point out, however, that the singularity is
integrable and the instanton action is finite. Analytic continuation
of this instanton gives a closed, singular spacetime. A part of this
spacetime, is isometric to an open Robertson-Walker universe. The
singularity has the form of an expanding singular bubble.
However, it never hits an observer in the Robertson-Walker
part of the universe, and HT argue that the singularity is therefore
not a problem \cite{24}.
HT instantons have a free parameter corresponding to the strength of
the singularity. As this parameter is varied, the density parameter
$\Omega$ of the open universe also changes, and HT use an anthropic
approach to find the most probable value of
$\Omega$.
I think there are serious problems with HT approach.
In a singular instanton, the field equations are not
satisfied at the singularity, and such an instanton is not, therefore,
a stationary point of the action. It is not clear why such instantons
should dominate the path integral. Moreover, if HT instanton is
allowed, we will then have to admit a host of other instantons with
integrable singularities. I am going to give some examples of such
instantons and argue that they lead to unacceptable consequences.
First, I will construct a singular
instanton for nucleation of open universes which has a lower action than
the Hawking-Turok one. Take two copies of HT instanton, cut off their
thorns and match what remains of the instantons across the cut, as in Fig.5.
\begin{figure}[b!]
\centerline{\epsfig{file=cosm6.eps,height=1.75in,width=3.5in}}
\vspace{10pt}
\caption{This singular instanton is made out of two HT instantons with
their thorns cut off. It has a lower action than the HT instanton.}
\label{fig5}
\end{figure}
The resulting instanton
has an integrable domain-wall-type singularity at the matching
surface.\footnote{Note that the field $\phi$ should be continuous
across the matching surface in order for the action to be finite.}
Its action is about twice as negative as that of the HT instanton. One can
go further and use HT-type instantons with thorns on both
sides\footnote{Instantons with two singular thorns
have been discussed by
R. Bousso and A. Linde \cite{21}} to
construct singular instantons of arbitrarily large negative action
(see Fig.6). In the HT approach, such instantons should completely
dominate the path integral.
\begin{figure}[b!]
\centerline{\epsfig{file=cosm4.eps,height=1.75in,width=5in}}
\vspace{10pt}
\caption{In this way one can construct singular instantons of
arbitrarily large negative action.}
\label{fig6}
\end{figure}
As a second example, for the same model (\ref{V}) I have constructed an
asymptotically-flat singular instanton \cite{23}. Geometrically, it
looks like a flat space with a thorn. The behavior of the fields near
the singularity is identical to that in the HT instanton and the
action is finite, so there is absolutely no reason to reject my
instanton if HT instanton is legitimate. The analytic continuation of
my instanton gives a flat space with a singular sphere which expands
at a speed close to the speed of light. If this were indeed a
legitimate instanton, then we would have to conclude that flat space
is unstable with respect to nucleation of singular bubbles. The
nucleation probability can be made very high by adjusting the strength
of the singularity, and since this strength is a free parameter, the
universe in this picture would have already been overrun by expanding
singular bubbles. Since this is in a glaring contradiction with
observations, we have to conclude that HT instanton, as it stands,
cannot be used to describe the creation of open universes.
Let me conclude by pointing out one thing that I think is good about
the HT instanton. It has invigorated the debate about the basic
issues of quantum cosmology and will hopefully lead to a better
understanding of some of these issues.
|
1,116,691,498,469 | arxiv | \section{Introduction}
The Zadko Telescope \citep{cow10} is a 1.0 metre f/4 Ritchey-Chrétien telescope located in Western Australia, in the shire of Gingin (see Figure \ref{fig:observatory}). Constructed by DFM Engineering Inc. in Longmont Colorado USA and donated to the University of Western Australia by Resource Company Claire Energy CEO, Jim Zadko. The telescope has been in operation since 1st April 2009. Specifications of the Zadko Observatory are listed in Table \ref{table1}.
\begin{figure}[!b]\centering
\vspace{0pt}
\includegraphics[width=7.5cm]{obs.jpg}
\caption{External view of the Zadko Observatory.}
\label{fig:observatory}
\end{figure}
\section{Evolution of the Observatory}
\label{sec:intro}
Due to operational issues the original 6.7 metre fibreglass domed observatory which the Zadko Telescope \citep{cow10,cow17} was originally housed was replaced in 2011 with a state of the art purpose built robotic rolling roof autonomous observatory with a dedicated 21 m$^2$ constant temperature climate controlled operations room. In addition to this a climate controlled telescope service room to mirror the operation room was also added. This had the benefit of allowing for possible future conversion into a second control room when other telescopes were added to the increased 63 m$^2$ telescope viewing area. Robotic control of the rolling roof is performed using a PLC-Burgess system designed and manufactured by the electronics workshop at Observatoire de Haute-Provence. A brief system overview is explained here: The PLC is connected to the PC-Zosma via an Ethernet connection. The PLC receives commands from the PC-Zosma and sends status to the PC-Zosma through a socket connection. The weather station and the cloud sensor are connected to the serial ports of PC-Zosma. The PC processes this data and sends commands to the PLC in order to control the roof in a safe condition. All the safety devices are controlled by the PLC. In the event of a communication failure with the PC, the PLC closes the roof. Refer to Figure \ref{fig:widefig1} for floor plan of the Zadko Observatory.
\begin{figure*}[!t]\centering
\vspace{0pt}
\includegraphics[scale=0.4]{florr_plan.pdf}
\hspace*{\columnsep}
\caption{Zadko Observatory mechanical drawings, showing floor plans for top and side views, including the position of the 1-m Zadko Telescope.}
\label{fig:widefig1}
\end{figure*}
\begin{table}[!bht]\centering
\setlength{\tabnotewidth}{\columnwidth}
\tablecols{2}
\caption{Zadko Observatory physical parameters. } \label{table1}
\begin{tabular}{lr}
\toprule
Property & Value \\
\midrule
Observatory code & D20 \\
Longitude & 115$^o$ 42' 49'' E \\
Latitude & 31$^o$ 21' 24'' S \\
Altitude & 50 meters \\
Seeing & Variable (1-4\arcsec) \\
magnitude & R$\sim$21.5 (180s exposure) \\
Zadko Camera & QHY 160M \\
Zadko Filters & Sloan g', r', i', Clear \\
Commercial clients & 4 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Weather station}
A critical component of all robotic observatories is the ability to monitor site conditions using a variety of instruments. The Zadko Observatories instruments feed back to the PLC-Burgess and the TCS to signal when it is safe to open the roof and commence viewing. Alternately if a detrimental weather event is detected the roof will automatically close thereby protecting the telescopes from water, dust or smoke ingress.
The Saia Burgess Control PCD3 module that controls the roof enables bypass of any supporting computer allowing an immediate response to close the roof. The PCD3 module also acts as a failsafe in the event that any detector reports incoherent values or is lost the observatory will go into safe mode by closing the roof and forbidding it to re-open without human intervention.
Instruments that form part of the Zadko Observatories control are:
\begin{itemize}
\item A {\bf Hydreon RG-11 Rain Gauge}\footnote{https://rainsensors.com/} senses water hitting its outside surface using beams of infrared light. It uses the same sensing principle used in millions of automotive rain sensing windshield wiper controls. The RG-11 is optical - not mechanical, chemical, or conductive. Consequently, it is far more rugged, sensitive and reliable than any other technology. The sensor is extremely sensitive, and virtually immune to false trips. Yet, it is completely unaffected by jostling and motion. There are no exposed conductors to corrode, and no openings for bugs to crawl into. There is no place for leaves or other debris to collect. It is also inexpensive compared to other similar instruments on the market. The RG-11 detector has been in operation for over twelve months without error.
\item A {\bf Diffraction Limited Boltwood II Cloud Sensor}\footnote{http://diffractionlimited.com/product/boltwood-cloud-sensor-ii/} that measures the amount of cloud cover by comparing the temperature of the sky to the ambient ground level temperature. The sky temperature is determined by measuring the amount of radiation in the 8 to 14 micron infrared band. A large difference indicates clear skies, whereas a small difference indicates dense, low-level clouds. This allows the sensor to continuously monitor the {\bf clarity} of the skies, and to trigger appropriate alerts on your computer. The device also includes a {\bf moisture sensor} which detects rain and snow. To prevent false alarms due to frost or dew, a heater keeps the sensor slightly above ambient temperature. When rain or snow falls, the sensor is automatically heated to 70 degrees Celsius. This clears the sensor quickly when the precipitation ends, and ensures that the sky-measuring thermophile has a clear view. The Boltwood Cloud Sensor II measures {\bf wind speed}, using a specially-designed anemometer with no moving parts. The sensor will warn you when winds speeds are too high for safe operation of the observatory. Plus the sensor detects {\bf daylight} and can be set to automatically close the roof to prevent any possibility of sunlight entering the telescope. The sensor also measures {\bf humidity}, and provides a continuous readout of temperature, humidity, and dew point.
\item A Vaisala weather control unit\footnote{http://www.vaisala.com} providing humidity, temperature, and wind data.
\item A GPS antenna for precision (microsecond) timing
\item An external webcam allowing a clear view of the observatories roof status and sky to the horizon for remote manual inspection.
\end{itemize}
All of these instruments are located on a six metre high mast external to the observatory at approximately three metre distance from the control room. A flashing amber light is installed at the top to signal roof movement as a safety measure.
\begin{figure}[!h!]\centering
\includegraphics[width=7.5cm]{weather.pdf}
\caption{Mast housing weather station sensors for cloud, rain, temperature, humidity, wind speed/direction, internet camera and GPS antenna.}
\label{fig:weather}
\end{figure}
\subsection{Mirror re-coating}
Due to the Zadko telescope important science capability it is essential that the primary and secondary astronomical reflecting surfaces (mirrors) are kept in an optimal state of cleanliness to reduce contaminants which can significantly degrade their reflectivity, IR emissivity, and light-scattering properties. In October 2018 the Zadko telescopes Primary and Secondary mirrors were removed and returned to Evaporated Metal Films Corporation in Ithaca NY, USA for specialized treatment including coating with standard enhanced aluminium (Al-SiOx/TiOx) with R average $\ge 93 \%$ over range 450 to 650 nm.
This was an improvement on the mirrors original average reflectivity, which had an R average of 86-88\% or better with standard protected aluminium coating.
As the Zadko Observatory is located in a hostile environment (high pollen and dust) at 50 metres above sea level surrounded by native bush and only 18 kilometres from the coast, the protective aluminium coating has served its purpose well and provided a ten-year useable life before re-coating became necessary. Due to the work required for removal and reinstallation of the mirrors plus the cost involved in the re-coating process, obtaining twice the recommended period for re-coating has been considered a bonus.
Further improvements to the telescope room environment will be investigated over the coming months to improve air quality and enhance the temperature control.
\begin{figure}[!t]\centering
\includegraphics[width=7.5cm]{mirror.pdf}
\caption{Zadko Telescope primary mirror after re-coating.}
\label{fig:mirror}
\end{figure}
\section{Observatory management}
To maintain a fully autonomous observatory requires careful strategic planning including long-term goal setting and financial support plus a dedicated and committed team. Over the past decade the Zadko Observatory has steadily extended its research capability and has become part of several large collaborations, with an emphasis on transient astronomy and gravitational wave follow-up \citep{ant20,and17,abb17}.
\subsection{Commercial clients}
With a view to future expansion the new Zadko Observatory, which originally housed just the Zadko Telescope, was designed to incorporate up to six other telescopes situated on two metre high piers equally spaced around the viewing room. This was taken with the view that to ensure the future viability of the Observatory non-government sources of funding would be required.
We were fortunate through our relationship with the European Space Agency (ESA) to host two ground based optical stations for Ariane used for space surveillance and space traffic management (\url{https://www.ariane.group}).
A USA based company Numerica Corporation have installed a ground based optical telescope for space debris and satellite tracking, plus general surveillance and observations for commercial and research purposes (\url{https://www.numerica.us/}).
As a part of a European Space Agency initiative ESA have requested time on the Zadko Telescope for follow-up observations of hazardous NEOs selected by the ESA NEOCC centre, with the most critical operations being NEO-alerts. This will require obtaining observations within 4-6 hours of a newly discovered NEO which has a non-zero probabilty of impacting the Earth (\url{https://www.esa.int/}).
Two piers house SPIRIT 3 and 4 telescopes, an initiative that incorporates a full life cycle of teacher and student professional learning opportunities and activities delivered via the outreach program at the International Centre for Radio Astronomy Research (ICRAR). SPIRIT 3 is a 35cm Schmidt-Cassegrain telescope manufactured by Celestron Telescopes. Its imaging camera is an Apogee Alta U6, which provides a 20 arc minute square field of view, and a default resolution of 1.2 arc seconds per pixel. SPIRIT 4 is a 32cm Corrected Dall-Kirkham telescope manufactured by Planewave Instruments. Its imaging camera is an SBIG STX-16803, offering a 50 arc minute field of view and a native resolution of 0.7 arc seconds per pixel.
ICRAR have also installed a telescope (Starfox) making use of next generation anti-reflection lens coatings to design complex new astrographs to minimise scattered light in the optical train optimising the instrument for low surface brightness imaging of the universe (\url{https://www.icrar.org/}).
\section{Conclusion}
After several years of upgrades the future is looking positive for the Zadko Observatory. Its unique location on the planet is becoming recognized internationally for time-domain astronomy. This is especially the case for gamma ray burst studies, space situational awareness and near Earth object follow-up. The Observatory now has a solid foundation to continue providing commercial services to international space agencies and companies.
Future projects of the Zadko Telescope include a complete upgrade to the Telescope Control System, installation of a new Guide Aquired Module allowing expanded capability for spectroscopy and the purchase of a new CMOS camera allowing for time domain photometry. These upgrades are expected to attract further commercial interest in the observatory.
\begin{figure}[!t]\centering
\includegraphics[scale = 0.25,angle=-90]{Zadko_Observatory-1.pdf}
\caption{Zadko Telescope (foreground) along with several independent commercial instruments hosted inside the observatory.}
\label{fig:telescope}
\end{figure}
\section{Acknowledgments}
The Zako Telescope receives partial support from the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), project number CE170100004. We thank the Zadko family for providing funding for the Zadko Telescope and recognise the untimely passing of Jim Zadko.
|
1,116,691,498,470 | arxiv | \section{Introduction}
A group $G$ is \emph{discriminated} by another group $\Gamma$ (or is \emph{fully residually $\Gamma$}) if for every finite set of non-trivial elements $\{g_{1},\ldots, g_{n}\}$ of
$G$ there exists a homomorphism $\phi: G\rightarrow \Gamma$ such that $\phi(g_{i})\neq 1$ for $i=1,\ldots, n$. If this condition is only required to hold for $n=1$ we say that $G$ is \emph{separated} by $\Gamma$
(or is \emph{residually $\Gamma$}).
The class of fully residually \emph{free} groups (when $\Gamma$ is a non-abelian free group) has been extensively studied in the last 15 years, particularly in connection with
\emph{Tarski's problems} on the elementary theory of a free group (\cite{KM06}, \cite{Sel06}), and now have a well-developed theory. Further, many algorithmic problems related to
these groups have been solved in recent years.
Generalizing to the case when $\Gamma$ is a hyperbolic group, much of the theory has been developed and is similar, but many algorithmic questions remain open. This paper is motivated by the following problem.
\begin{problem} Is the elementary theory $\mathrm{Th}(G)$ of a (torsion-free) hyperbolic group $G$ decidable? \end{problem}
Notice, that it was proved in \cite{KM05JSJ} that the universal theory of a finitely generated
fully residually free group is decidable and in \cite{Dah09}
that the universal theory of a hyperbolic group is
decidable. We will give another proof of this result (for torsion-free hyperbolic groups) in Corollary~\ref{Cor:UnivTh}.
Fix throughout this paper a torsion-free hyperbolic group $\Gamma = \GammaPresentation$.
One of the characterizations of finitely generated groups $G$ discriminated by $\Gamma$ is that they embed
into the \emph{Lyndon completion} $\Gamma^{\integers[t]}$ of $\Gamma$, or equivalently, into a group obtained from $\Gamma$ by a series of extensions of centralizers \cite{KM09}.
If $G$ is separated by $\Gamma$, there is an embedding into a finite direct product of such groups. The case when $\Gamma$ is a free group was proved by
Kharlampovich and Myasnikov, who also provided an algorithm to construct the embedding (in both the discriminated and separated cases) \cite{KM98b}.
For the general case, however, the embedding described in \cite{KM09} is not effective. We provide an algorithm to
construct the embedding of any residually $\Gamma$ group $G$ into a direct product of groups obtained from $\Gamma$ by extensions of centralizers. When $G$ is
fully residually $\Gamma$, we effectively construct a finite collection of homomorphisms from $G$ into groups obtained from $\Gamma$ by extensions of centralizers, at least
one of which is an embedding (Theorem 3.17).
The first step of our approach is to use \emph{canonical representatives} for certain elements of $\Gamma$, developed by Rips and Sela in their study of equations
over hyperbolic groups \cite{RS95}, to reduce part of the problem to the free group case. As a corollary of this reduction, we are able to effectively construct a finite diagram
that describes the complete set $\mathrm{Hom}(G, \Gamma)$ of homomorphisms from an arbitrary finitely presented group $G$ to $\Gamma$ (Theorem 2.3).
\subsection{Algebraic geometry over groups}
We will use throughout the language of \emph{algebraic geometry over groups} \cite{BMR99}. We recall here some important notions and establish notation.
Let $\Gamma$ be a finite group generated by $A$ (`constants') and $X$ a finite set (`variables') and set $\Gamma[X]=\Gamma\ast F(X)$. Let
$\mathrm{Hom}_{\Gamma}(\Gamma[X], \Gamma)$ denote the set of homomorphisms from $\Gamma[X]$ to $\Gamma$ that are identical on $\Gamma$
(`$\Gamma$-homomorphisms').
To each element $s$ of $\Gamma[X]$ we associate a formal expression `$s=1$' called an
\emph{equation over $\Gamma$}. A \emph{solution} to this equation is a homomorphism $\phi\in \mathrm{Hom}_{\Gamma}(\Gamma[X], \Gamma)$ such that
$s^{\phi}=1$. A subset $S$ of $\Gamma[X]$ corresponds to the \emph{system of equations} `$S=1$' which we also denote `$S(X,A)=1$'. Define
\[
\Gamma_{S} = \Gamma[X] / \ncl{S}
\]
and note that every solution to the system $S$ factors through $\Gamma_{S}$. If $\Gamma$ has the presentation $\GammaPresentation$, then
\[
\Gamma_{S} \simeq \langle X,A \, | \, S, \mathcal{R}\rangle.
\]
Define the \emph{radical} $R_{\Gamma}(S)$ of $S$ over $\Gamma$ by
\[
R_{\Gamma}(S) = \{ t\in \Gamma[X] \; | \; \forall_{\phi\in \mathrm{Hom}_{\Gamma}(\Gamma[X], \Gamma)}\, \forall_{s\in S} \; (s^{\phi}=1 \implies t^{\phi}=1) \}
\]
and define the \emph{coordinate group} of $S$ over $\Gamma$ by
\[
\Gamma_{R_{\Gamma}(S)} = \Gamma[X] / R_{\Gamma}(S).
\]
Every solution to $S$ factors through $\Gamma_{R_{\Gamma}(S)}$.
When $S$ is a subset of $F(X)$ we say that the system is \emph{coefficient-free} and we may consider $F(X)$ in place of $\Gamma\ast F(X)$ and
the ordinary set of homomorphisms $\mathrm{Hom}(F(X), \Gamma)$ in place of $\mathrm{Hom}_{\Gamma}(\Gamma[X], \Gamma)$. In particular, for any
group $G$ presented by $\GPresentation$ we may consider $S$ as a system of equations in variables $Z$. In the general case, when $S\subset \Gamma[X]$, we may
consider $S$ as a system of equations over any group $G$ that has $\Gamma$ as a fixed subgroup (i.e. any $G$ in the `category of $\Gamma$-groups').
\subsection{Notation}
Fix $\Gamma=\GammaPresentation$ a finitely presented torsion-free hyperbolic group, $F$ the free group on $A$,
and $\pi:F\rightarrow \Gamma$ the
canonical epimorphism.
The map $\pi$ induces an epimorphism $F[X]\rightarrow \Gamma[X]$, also denoted $\pi$, by fixing each $x\in X$. For a system of equations
$S\subset F[X]$, we study the corresponding system $S^{\pi}\subset\Gamma[X]$ which we may denote again by $S$, depending on context.
The radical of $S^{\pi}$ over $\Gamma$ may be denoted $R_{\Gamma}(S^{\pi})$, $R_{\Gamma}(S)$, or $R(S^{\pi})$. Likewise,
the coordinate group $\Gamma_{R_{\Gamma}(S^{\pi})}$ may be denoted simply $\Gamma_{R(S)}$.
Notice that the relators $\mathcal{R}$ of $\Gamma$ are in the radical $R_{\Gamma}(S)$ for \emph{every} system of equations $S$, hence
\[
F_{R_{\Gamma}(S)} = \Gamma_{R(S)}.
\]
In denoting a coordinate group $\Gamma_{R(S)}=\Gamma[X] / R(S)$ we always assume that $X$ is precisely the set of variables appearing
in $S$.
\subsection{Toral relatively hyperbolic groups}
A group $G$ that is hyperbolic relative to a collection $\{H_{1},\ldots,H_{k}\}$ of subgroups is called \emph{toral} if $H_{1},\ldots,H_{k}$ are all abelian and $G$ is
torsion-free.
Many algorithmic problems in
(toral) relatively hyperbolic groups are decidable, and in particular we take note of the following for later use.
\begin{lemma}\label{Lem:AlgorithmsRelativelyHyperbolic}
In every toral relatively hyperbolic group $G$, the following hold.
\begin{arabicenumerate}
\item The conjugacy problem in $G$, and hence the word problem, is decidable.
\item If $G$ is non-abelian then we may effectively construct two non-commuting elements of $G$.
\item If $g\in G$ is a hyperbolic element (i.e. not conjugate to any element of any $H_{i}$), then the centralizer $C(g)$ of $g$ is an infinite cyclic group.
Further, a generator for $C(g)$ can be effectively constructed.
\end{arabicenumerate}
\end{lemma}
\begin{proof}
The word problem was solved in \cite{Far98} and the conjugacy problem in \cite{Bum04}.
For the second statement, we need only enumerate pairs $(g,h)\in G\times G$ until we find a pair with $[g,h]\neq 1$.
For the third statement, let $G=\langle A\rangle$ and let $g\in G$ be a hyperbolic element.
Theorem~4.3 of \cite{Osi06IJAC} shows that the subgroup
\[
E(g)= \{ h\in G \sst \exists\qs n\in\naturals :\qs h^{-1}g^{n}h=g^{n}\}
\]
has a cyclic subgroup of finite index. Since $G$ is torsion-free, $E(g)$ must be infinite cyclic (see for example the proof of
Proposition~12 of \cite{MR96}). Clearly $C(g)\leq E(g)$, hence $C(g)$ is infinite cyclic.
To construct a generator for $C(g)$, consider the following results of D. Osin (see the proof of Theorem~5.17 and Lemma~5.16 in \cite{Osi06Memoirs}) :
\begin{romanenumerate}
\item there exists a constant $N$, which depends on $G$ and the word length $\wl{g}$ \label{Osin2}
and can be computed, such that if $g=f^{n}$ for some $f\in G$ and with $n$ positive then $n\leq N$;
\item there is a computable function $\beta:\naturals \rightarrow\naturals$ \label{Osin1}
such that if $f$ is an element of $G$ with $f^{n}=g$ for some positive $n$, then $f$ is conjugate to some element $f_{0}$ satisfying $\wl{f_{0}}\leq \beta(\wl{g})$.
\end{romanenumerate}
We proceed as follows. Let
$\mathcal{F}$ be the set of all $f\in G$ such that $\wl{f}\leq \beta(\wl{g})$ and $h^{-1}f^{n}h = g$ for some $h\in G$ and $1\leq n\leq N$.
It is finite, non-empty (since $g$ is an element), and can be computed (since conjugacy is decidable).
Let $f$ be an element of $\mathcal{F}$ such that the exponent $n$ is maximum amongst elements of $\mathcal{F}$ and find an element $h\in G$ such
that $h^{-1} f^{n} h = g$ (we may find $h$ by enumeration).
We claim that if $\overline{g}$ is a generator of $C(g)$ then either $h^{-1}f h=\overline{g}$ or $h^{-1}f h = \overline{g}^{-1}$. Indeed,
$h^{-1} f h\in C(g)$ since it commutes with $g=(h^{-1} f h)^{n}$, hence $h^{-1} f h=\overline{g}^k$ for some $k$ and so
\begin{equation*
g = (h^{-1}f h)^{n}=\overline{g}^{kn}.
\end{equation*}
Suppose $k>0$. Since $\overline{g}^{kn}=g$, (\ref{Osin1}) implies that $\overline{g}$ is conjugate to some element $g_{0}$ with $\wl{g_{0}}\leq \beta(\wl{g})$.
Then $g_{0}^{kn}$ is conjugate to $g$, so by (\ref{Osin2}) $kn\leq N$ hence $g_{0}\in \mathcal{F}$.
By maximality of the exponent in the choice of $f$, $k$
must be 1 and $h^{-1} f h=\overline{g}$. If $k<0$, a similar argument shows that $h^{-1}f h = \overline{g}^{-1}$.
\end{proof}
\section{Effective description of homomorphisms to $\Gamma$}
In this section, we describe an algorithm that takes as input a system
of equations $S$ over $\Gamma$ and produces a tree diagram $\mathcal{T}$ that encodes the set
$\mathrm{Hom}_{\Gamma}(\Gamma_{R(S)},\Gamma)$. When $S$ is a system without coefficients, we interpret $S$ as relators for a finitely presented group
$G=\GPresentation$ and the diagram $\mathcal{T}$ encodes instead the set $\mathrm{Hom}(G,\Gamma)$.
Though the diagram $\mathcal{T}$ will give a finite description of $\mathrm{Hom}_{\Gamma}(\Gamma_{R(S)},\Gamma)$, it is not a `Makanin-Razborov diagram' in the
sense of \cite{Gro05}. We discuss this further at the end of this section.
There are two ingredients in this construction: first,
the reduction of the system $S$ over $\Gamma$ to finitely many systems of equations over free groups, and second, the construction of Hom-diagrams (Makanin-Razborov diagrams)
for systems of equations over free groups.
\begin{notation}
Let $\overline{\phantom{c}}$ denote the canonical epimorphism $F(Z,A)\rightarrow \Gamma_{R(S)}$.
For a homomorphism $\phi: F(Z,A)\rightarrow K$ we define $\overline{\phi}: \Gamma_{R(S)} \rightarrow K$
by
\[
\overline{\phi}\big(\overline{w}\big) = \phi(w),
\]
where any preimage $w$ of $\overline{w}$ may be used. We will always ensure that $\overline{\phi}$ is a well-defined homomorphism.
\end{notation}
\subsection{Reduction to systems of equations over free groups}
In \cite{RS95}, the problem of deciding whether or not a system of equations $S$ over a torsion-free hyperbolic group $\Gamma$ has a solution was solved by constructing
\emph{canonical representatives} for certain elements of $\Gamma$. This construction reduced the problem to deciding the existence of solutions in finitely many
systems of equations over free groups, which had been previously solved. The reduction may also be used to find all solutions to $S$ over $\Gamma$, as described
below.
\begin{lemma}\label{Lem:RipsSela1}
Let $\Gamma=\GammaPresentation$ be a torsion-free $\delta$-hyperbolic group and $\pi : F(A)\rightarrow \Gamma$ the canonical epimorphism. There
is an algorithm that, given a system $S(Z,A)=1$
of equations over $\Gamma$, produces finitely many systems of
equations
\begin{equation}
S_{1} (X_{1},A)=1,\ldots,S_{n}(X_{n},A)=1
\end{equation}
over $F$ and homomorphisms $\rho_{i}: F(Z,A)\rightarrow F_{R(S_{i})}$ for $i=1,\ldots,n$
such that
\begin{romanenumerate}
\item for every $F$-homomorphism $\phi : F_{R(S_{i})}\rightarrow F$, the map $\overline{\rho_{i}\phi\pi}:\Gamma_{R(S)}\rightarrow \Gamma$ is a $\Gamma$-homomorphism, and
\item for every $\Gamma$-homomorphism $\psi: \Gamma_{R(S)}\rightarrow \Gamma$ there is an integer $i$ and an $F$-homomorphism
$\phi : F_{R(S_{i})}\rightarrow F(A)$ such that $\overline{\rho_{i}\phi\pi}=\psi$.
\end{romanenumerate}
Further, if $S(Z)=1$ is a system without coefficients, the above holds with $G=\GPresentation$ in place of $\Gamma_{R(S)}$ and `homomorphism' in place of
`$\Gamma$-homomorphism'.
\end{lemma}
\begin{proof}
The result is an easy corollary of Theorem~4.5 of \cite{RS95}, but we will provide a few details.
\begin{figure}[htbp]
\begin{center}
\[
\xymatrix{
& F(Z,A) \ar[ld]_{\overline{\phantom{\phi}}} \ar[d]^{\rho_{i}} \\
\Gamma_{R(S_{i})} \ar[ddr]_{\overline{\rho_{i}\phi\pi}} & F_{R_{(S_{i})}} \ar[d]_{\phi} \\
& F(A) \ar[d]^{\pi} \\
& \Gamma
}
\]
\caption{Commutative diagram for Lemma~\ref{Lem:RipsSela1}.}
\label{Figure:CanonicalReps}
\end{center}
\end{figure}
We may assume that the system $S(Z,A)$, in variables $z_{1},\ldots,z_{l}$, consists of $m$ constant equations and $q-m$ triangular equations, i.e.
\[
S(Z,A) = \braced{z_{\sigma(j,1)}z_{\sigma(j,2)}z_{\sigma(j,3)}=1}{j=1,\ldots,q-m}{z_{s} = a_{s}}{s=l-m+1,\ldots,l}
\]
where $\sigma(j,k)\in\{1,\ldots,l\}$ and $a_{i}\in\Gamma$. An algorithm is described in \cite{RS95}
which, for every $m\in\naturals$, assigns to each element $g\in \Gamma$ a word $\theta_{m}(g)\in F$ satisfying
\[
\theta_{m}(g)=g \mbox{ in } \Gamma
\]
called its \emph{canonical representative}. The representatives $\theta_{m}(g)$ are not `global canonical representatives', but do satisfy
useful properties for certain $m$ and certain finite subsets of $\Gamma$, as follows.
Let\footnote{The constant of hyperbolicity $\delta$ may be computed from a presentation of $\Gamma$ using the results of \cite{EH01}.} $L=q\cdot 2^{5050(\delta+1)^{6}(2|A|)^{2\delta}}$.
Suppose $\psi: F(Z,A)\rightarrow \Gamma$ is a solution of $S(Z,A)$ and denote
\[
\psi(z_{\sigma(j,k)})=g_{\sigma(j,k)}.
\]
Then there exist
$h_{k}^{(j)}, c_{k}^{(j)}\in F(A)$ (for $j=1,\ldots,q-m$ and $k=1,2,3$) such that
\begin{romanenumerate}
\item each $c_{k}^{(j)}$ has length less than\footnote{The bound of $L$ here, and below, is extremely loose. Somewhat tighter, and more intuitive, bounds are given in \cite{RS95}.} $L$
(as a word in $F$), \label{RepsCond1}
\item $c_{1}^{(j)}c_{2}^{(j)}c_{3}^{(j)} = 1$ in $\Gamma$, \label{RepsCond2}
\item there exists $m\leq L$\ such that the canonical representatives satisfy the following equations in $F$:\label{RepsCond3}
\begin{eqnarray}
\theta_{m} (g_{\sigma(j,1)}) & = & h_{1}^{(j)} c_{1}^{(j)} \left(h_{2}^{(j)}\right)^{-1} \label{CanonReps1}\\
\theta_{m} (g_{\sigma(j,2)}) & = & h_{2}^{(j)} c_{2}^{(j)} \left(h_{3}^{(j)}\right)^{-1}\\
\theta_{m} (g_{\sigma(j,3)}) & = & h_{3}^{(j)} c_{3}^{(j)} \left(h_{1}^{(j)}\right)^{-1}.\label{CanonReps3}
\end{eqnarray}
\end{romanenumerate}
In particular, when $\sigma(j,k)=\sigma(j',k')$ (which corresponds to two occurrences in $S$ of the variable $z_{\sigma(j,k)}$) we have
\begin{equation}
h_{k}^{(j)} c_{k}^{(j)} \left(h_{k+1}^{(j)}\right)^{-1} = h_{k'}^{(j')} c_{k'}^{(j')} \left(h_{k'+1}^{(j')}\right)^{-1}.\label{Hequality}
\end{equation}
Consequently, we construct the systems $S(X_{i},A)$ as follows.
For every positive integer $m\leq L$ and every choice of $3(q-m)$ elements $c_{1}^{(j)},c_{2}^{(j)},c_{3}^{(j)}\in F$ ($j=1,\ldots,q-m$)
satisfying (i) and (ii)\footnote{The word problem in hyperbolic groups is decidable.}
we build a system $S(X_{i},A)$
consisting of the equations
\begin{eqnarray}
x_{k}^{(j)}c_{k}^{(j)}\left(x_{k+1}^{(j)}\right)^{-1} & = & x_{k'}^{(j')}c_{k'}^{(j')}\left(x_{k'+1}^{(j')}\right)^{-1} \label{Eqn:SC1}\\
x_{k}^{(j)}c_{k}^{(j)}\left(x_{k+1}^{(j)}\right)^{-1} & = & \theta_{m}(a_s) \label{Eqn:SC2}
\end{eqnarray}
where an equation of type (\ref{Eqn:SC1}) is included whenever $\sigma(j,k)=\sigma(j',k')$ and an equation of type (\ref{Eqn:SC2}) is included whenever
$\sigma(j,k)=s\in\{l-m+1,\ldots,l\}$. To define $\rho_{i}$, set
\[
\rho_i (z_s) = \braced{x_{k}^{(j)}c_{k}^{(j)}\left(x_{k+1}^{(j)}\right)^{-1},}{1\leq s \leq l-m \mbox{ and } s=\sigma(j,k)}{ \theta_{m}(a_s),}{l-m+1\leq s \leq l}
\]
where for $1\leq s \leq l-m$ any $j,k$ with $\sigma(j,k)=s$ may be used.
If $\psi:F(Z)\rightarrow \Gamma$ is any solution to $S(Z,A)=1$, there is a system $S(X_{i},A)$ such that $\theta_{m}(g_{\sigma(j,k)})$ satisfy
(\ref{CanonReps1})-(\ref{CanonReps3}). Then the required solution $\phi$ is given by
\[
\phi\big(x_{j}^{(k)}\big) = h_{j}^{(k)}.
\]
Indeed, (iii) implies that $\phi$ is a solution to $S(X_{i},A)=1$. For $s=\sigma(j,k)\in\{1,\ldots,l-m\}$,
\[
z_{s}^{\rho_{i}\phi} = h_{k}^{(j)} c_{k}^{(j)} \left(h_{k+1}^{(j)}\right)^{-1} = \theta_{m}(g_{\sigma(j,k)})
\]
and similarly for $s\in\{l-m+1,\ldots,l\}$, hence $\psi= \rho_{i}\phi\pi$.
Conversely, for any solution $\phi\big(x_{j}^{(k)}\big)= h_{j}^{(k)}$ of $S(X_{i})=1$ one sees that by (\ref{Eqn:SC1}),
\[
z_{\sigma(j,1)}z_{\sigma(j,2)}z_{\sigma(j,3)} \xmapsto{\rho_{i}\phi} h_{1}^{(j)} c_{1}^{(j)}c_{2}^{(j)}c_{3}^{(j)} \big(h_{1}^{(j)}\big)^{-1}
\]
which maps to 1 under $\pi$ by (ii), hence $\rho_{i}\phi\pi$ induces a homomorphism.
\end{proof}
\subsection{Encoding solutions with the tree $\mathcal{T}$}\label{section:HomDiagrams}
An algorithm is described in \S 5.6 of \cite{KM05Implicit} which constructs, for a given system of equations $S(X,A)$ over the free group $F$,
a diagram encoding the set of solutions of $S$. The diagram consists of a directed finite rooted tree $T$ with
the following properties. Let $G=F_{R(S)}$.
\begin{romanenumerate}
\item Each vertex $v$ of $T$ is labelled by a pair $(G_{v},Q_{v})$ where $G_{v}$ is an $F$-quotient of $G$ and $Q_{v}$ a finitely generated subgroup of $\mathrm{Aut}_{F}(G_{v})$.
The root $v_0$ is labelled by $(G,1)$ and every leaf is labelled by $(F(Y)\ast F,1)$ where $Y$ is some finite set (called \emph{free variables}).
Each $G_{v}$, except possibly $G_{v_{0}}$, is fully residually $F$.
\item Every (directed) edge $v\rightarrow v'$ is labelled by a proper surjective $F$-homomorphism $\pi(v,v'):G_{v}\rightarrow G_{v'}$.
\item For every $\phi\in\mathrm{Hom}_{F}(G,F)$ there is a path $p=v_0 v_1 \ldots v_k$ where $v_k$ is a leaf labelled by
$(F(Y)\ast F,1)$, elements $\sigma_{i}\in Q_{v_{i}}$, and a $F$-homomorphism
$\phi_{0}: F(Y)\ast F\rightarrow F$ such that
\begin{equation}
\phi = \pi(v_0,v_1) \sigma_1 \pi(v_1,v_2) \sigma_2 \cdots \pi(v_{k-2},v_{k-1})\sigma_{k-1}\pi(v_{k-1},v_{k})\phi_{0}.
\end{equation}
\end{romanenumerate}
The algorithm gives for each $G_{v}$ a finite presentation $\langle A_{v}|\mathcal{R}_{v}\rangle$, and for each $Q_{v}$ a finite list of
generators in the form of functions $A_{v}\rightarrow (A_{v}\cup A_{v}^{-1})^{*}$. Note that the choices for $\phi_{0}$ are
exactly parametrized by the set of functions from $Y$ to $F$.
Let $S(Z,A)=1$ be a system of equations over $\Gamma$. We will construct a diagram $\mathcal{T}$ to encode the set of solutions of $S(Z,A)=1$, as follows.
Apply Lemma~\ref{Lem:RipsSela1} to construct the systems $S_{1}(X_{1},A),\ldots,S_{n}(X_{n},A)$.
Create a root vertex $v_{0}$ labelled by $F(Z,A)$. For each of the systems $S_{i}(X_{i},A)$, let $T_{i}$
be the tree constructed above. Build an edge from $v_{0}$ to the root of $T_{i}$ labelled by the homomorphism $\rho_{i} \pi_{S_{i}}$, where $\pi_{S_{i}}:F(X_{i},A)\rightarrow F_{R(S_{i})}$
is the canonical projection.
For each leaf $v$ of $T_{i}$, labelled by $F(Y)\ast F$, build a new vertex $w$ labelled by $F(Y)\ast\Gamma$ and an edge $v\rightarrow w$ labelled by the homomorphism
$\pi_{Y}:F(Y)\ast F\rightarrow F(Y)\ast \Gamma$ which is induced from $\pi:F\rightarrow \Gamma$ by acting as the identity on $F(Y)$.
Define a \emph{branch} $b$ of $\mathcal{T}$ to be a path $b=v_{0} v_{1} \ldots v_{k}$ from the root $v_{0}$ to a leaf $v_{k}$.
Let $v_{1}$ be labelled by $F_{R(S_{i})}$ and $v_{k}$ by $F(Y)\ast \Gamma$.
We associate to $b$ the set $\Phi_{b}$ consisting of all homomorphisms $F(Z)\rightarrow \Gamma$ of the form
\begin{equation}
\rho_{i}\pi_{S_{i}}\pi(v_{1},v_{2}) \sigma_{2} \cdots \pi(v_{k-2},v_{k-1})\sigma_{k-1}\pi(v_{k-1},v_{k})\pi_{Y}\phi
\end{equation}
where $\sigma_{j}\in Q_{v_{j}}$ and $\phi\in \mathrm{Hom}_{\Gamma}(F(Y)\ast\Gamma,\Gamma)$.
Since $\mathrm{Hom}_{\Gamma}(F(Y)\ast\Gamma,\Gamma)$ is in bijective correspondence with the set of functions $\Gamma^{Y}$, all elements of $\Phi_{b}$
can be effectively constructed. We have obtained the following theorem.
\begin{theorem} \label{Thm:EffectiveSolutions}
There is an algorithm that, given a system $S(Z,A)=1$ of equations over $\Gamma$, produces a diagram encoding its set of solutions. Specifically,
\[
\mathrm{Hom}(\Gamma_{R(S)},\Gamma) = \{ \overline{\phi} \sst \phi\in\Phi_{b}\cs \mbox{$b$ is a branch of $\mathcal{T}$}\}
\]
where $\mathcal{T}$ is the diagram described above. When the system is coefficient-free, then the diagram encodes $\mathrm{Hom}(G, \Gamma)$ where
$G=\GPresentation$.
\end{theorem}
Note that in the diagram $\mathcal{T}$, the groups $G_{v}$ appearing at vertices are not quotients of coordinate group $\Gamma_{R(S)}$ and
that to obtain a homomorphism from $\Gamma_{R(S)}$ to $\Gamma$ one must compose maps along a complete path ending at a leaf of $\mathcal{T}$. In \cite{Gro05} it
is shown that for any toral relatively hyperbolic group there exist Hom-diagrams with the property that every group $G_{v}$ is a quotient of
$\Gamma_{R(S)}$ and that every edge map $\pi(v,v')$ is a
proper surjective homomorphism. However, we are not aware of a algorithm for constructing these
diagrams.
\section{Embedding into extensions of centralizers}
The proof given in \cite{KM09} that any $\Gamma$-limit group $G$ embeds into extensions of centralizers of $\Gamma$ involves two steps: first, $G$ is shown to
embed into the coordinate groups of an \emph{NTQ system} (see \S \ref{subsection:NTQ}), and second, such groups are shown to embed into extensions of centralizers of $\Gamma$.
The first step of this construction relies on the following theorem (Theorem 1.1 of \cite{Gro05}):
there exists a finite collection $\{ L_{i}\}$ of proper quotients of $G$ such that any homomorphism from
$G$ to $\Gamma$ factors through one of the $L_{i}$ (up to a certain equivalence). Algorithmic construction of the set $\{ L_{i}\}$ is not given, nor are we aware of
an algorithm for constructing it.
Instead, we use canonical representatives and results from \cite{KM98b} regarding equations over free groups to construct a collection of NTQ groups $F_{R(S_{i})}$ and
maps from the generating set of $G$ to each $F_{R(S_{i})}$. Regarding each system of equations $S_{i}$ over $\Gamma$ rather than $F$, at least one of these maps induces
an embedding $G\hookrightarrow \Gamma_{R(S_{i})}$. The coordinate groups $\Gamma_{R(S_{i})}$ can be embedded into extensions of centralizers of $\Gamma$ using
the techniques from \cite{KM09}.
\subsection{Quadratic equations and NTQ systems}\label{subsection:NTQ}
An equation $s\in G[X]$ over a group $G$ is said to be (strictly) \emph{quadratic} if every variable appearing in $s$ appears at most (exactly) twice, and a system
of equations $S(X)\subset G[X]$ is (strictly) quadratic if every variable that appears in $S$ appears at most (exactly) twice. Here we count both $x$ and $x^{-1}$ as
an appearance of $x$.
Constructing NTQ systems involves considerable analysis of quadratic equations, and is aided by considering certain standard forms.
\begin{definition}
A \emph{standard quadratic equation} over a group $G$ is an equation of one of the following forms, where $c_{i}$ and $d$ are all nontrivial
elements of $G$:
\begin{eqnarray}
\prod_{i=1}^{n}[x_{i},y_{i}] & = & 1, \;\;\; n \geq 1; \label{Eqn:st1}\\
\prod_{i=1}^{n}[x_{i},y_{i}] \prod_{i=1}^{m}z_{i}^{-1}c_{i} z_{i} d & = & 1,\;\;\;
n,m\geq 0, n+m \geq 1 ; \label{Eqn:st2}\\
\prod_{i=1}^{n}x_{i}^2 & = & 1, \;\;\; n \geq 1; \label{Eqn:st3}\\
\prod_{i=1}^{n}x_{i}^2 \prod_{i=1}^{m}z_{i}^{-1}c_{i} z_{i} d & = & 1, \;\;\; n,m
\geq 0, n+m \geq 1.\label{Eqn:st4}
\end{eqnarray}
The left-hand sides of the above equations are called the \emph{standard quadratic words}.
\end{definition}
The following result allows us to assume that quadratic equations always appear in standard form.
\begin{lemma}
Let $s(X)\in G[X]$ be a strictly quadratic word over a group $G$. Then there is
a $G$-automorphism $\phi$ such that $s^{\phi}$ is a
standard quadratic word over $G$.
\end{lemma}
\begin{proof}
Follows easily from \S I.7 of \cite{LS77}.
\end{proof}
To each quadratic equation we associate a punctured surface. To (\ref{Eqn:st1}) we associate the orientable surface of genus $n$ and zero punctures, to (\ref{Eqn:st2})
the orientable surface of genus $n$ with $m+1$ punctures, to (\ref{Eqn:st3}) the non-orientable surface of genus $n$, and to (\ref{Eqn:st4}) the non-orientable surface of
genus $n$ with $m+1$ punctures. For a standard quadratic equation $S$, denote by $\chi(S)$ the
Euler characteristic of the corresponding surface.
Quadratic words of the form $ [x,y]$, $x^{2}$, and $z^{-1}cz$
where $c \in G$, are called \emph{atomic quadratic words} or simply \emph{atoms}. An atom $[x,y]$ contributes $-2$ to the Euler characteristic of $S$ while
$x^{2}$ and $z^{-1} c z$ (as well as $d$) each contribute $-1$.
A standard quadratic equation $S = 1$ over $G$ has the form
\[
r_{1} r_{2} \ldots r_{k} d = 1,
\]
where $r_{i}$ are atoms and $d \in G$. We classify solutions to quadratic equations based on the extent to which the images
of the atoms commute, as follows.
\begin{definition} Let $S = 1$ be a standard quadratic equation written in the
atomic form
$r_{1}r_{2}\ldots r_{k}d = 1 $ with $k \geq 2$. A solution $\phi : G_{R(S)}
\rightarrow G$
of $S = 1$ is called
\begin{romanenumerate}
\item \emph{degenerate}, if $r_{i}^{\phi} = 1$ for some $i$, and
\emph{non-degenerate} otherwise;
\item \emph{commutative}, if $[r_{i}^{\phi},r_{i+1}^{\phi}]=1$ for all
$i=1,\ldots ,k- 1,$ and \emph{non-commutative} otherwise;
\item in \emph{general position}, if $[r_{i}^{\phi},r_{i+1}^{\phi}] \neq 1$ for all
$i=1,\ldots ,k-1,$.
\end{romanenumerate}
\end{definition}
When the group $G$ is commutation transitive, a commutative solution satisfies $[r_{i}^{\phi},r_{j}^{\phi}]=1$ for all $i,j$. We will only be
interested in the case when $G$ is toral relatively hyperbolic, hence commutation transitive\footnote{Toral relatively hyperbolic groups are CSA, hence commutation
transitive. See \cite{KM09} or \cite{Gro09}.}. In this case, solutions also
have the following important property.
\begin{lemma}\label{Lem:GenPosOrAllComm}
Let $S\in G[X]$ be a standard quadratic equation over a toral relatively hyperbolic group $G$ such that $S$ has at least two atoms and such that $S=1$ has a solution in $G$. Then either
\begin{arabicenumerate}
\item $S$ has a solution in general position, or
\item every solution of $S$ is commutative.
\end{arabicenumerate}
Further, there is an algorithm that distinguishes the cases.
\end{lemma}
\begin{proof}
The dichotomy is true for all CSA groups, by Proposition~3 of \cite{KM98a}.
For the algorithm, let $S$ have the atomic form $r_{1} r_{2}\ldots r_{k} d$ with variables $x_{1},\ldots,x_{n}$. Consider the sentences
\[
\mathcal{S}_{i} : \exists{x_{1}}\ldots\exists{x_{n}}\; (S=1)\land ([r_{i},r_{i+1}]\neq 1)
\]
for $i=1,\ldots,k-1$.
Then all solutions of $S=1$ are commutative if and only if none of the sentences $\mathcal{S}_{i}$ is true in $G$.
The existential theory of toral relatively hyperbolic groups is decidable (\cite{Dah09}),
hence we can decide whether or not each $\mathcal{S}_{i}$ is true
in $G$.
\end{proof}
Now we define NTQ systems.
Let $G$ be a group generated by $A$ and let $S(X,A)$ be a system of equations. Suppose $S$ can be partitioned into subsystems
\begin{eqnarray*}
S_1(X_1, X_2, \ldots, X_n,A) & = & 1,\\
S_2(X_2, \ldots, X_n,A) & = & 1,\\
& \ldots & \\
S_n(X_n,A) & = & 1
\end{eqnarray*}
where $\{X_{1},X_{2},\ldots,X_{n}\}$ is a partition of $X$. Define groups $G_{i}$ for $i=1,\ldots,n+1$ by
\begin{eqnarray*}
G_{n+1} & = & G \\
G_{i} & = & G_{R(S_{i},\ldots,S_{n})}.
\end{eqnarray*}
We interpret $S_{i}$ as a subset of $G_{i-1}*F(X_{i})$, i.e. letters from $X_{i}$ are considered variables and letters from $X_{i+1}\cup\ldots\cup X_{n}\cup A$ are considered as constants from $G_{i}$.
A system $S(X,A)=1$ is
called \emph{triangular quasi-quadratic} (TQ) if it can be partitioned as above such that for each $i$ one of the following holds:
\begin{Romanenumerate}
\item $S_{i}$ is quadratic in variables $X_{i}$; \label{NTQ1}
\item $S_{i} = \{[x,y]=1\cs [x,u]=1\sst x\cs y\in X_i\cs u\in U_{i} \}$ where $U_{i}$ is a finite subset of $G_{i+1}$
such that $\langle U_{i} \rangle=C_{G_{i+1}}(g)$ for some $g\in G_{i+1}$; \label{NTQ2}
\item $S_i = \{[x,y]=1\sst x\cs y\in X_i \}$; \label{NTQ3}
\item $S_i$ is empty. \label{NTQ4}
\end{Romanenumerate}
The system is called \emph{non-degenerate triangular quasi-quadratic} (NTQ) if for every $i$ the system $S_{i}(X_{i}, \ldots, X_{n},A)$ has a solution in the coordinate group
$G_{R(S_{i+1}, \ldots, S_n)}$.
\begin{definition}
A group $H$ is called a \emph{$G$-NTQ group} if there is a NTQ system $S$ over $G$ such that $H\simeq G_{R(S)}$.
\end{definition}
For any quadratic system $S$ over $G$ one can, by eliminating linear variables,
find a strictly quadratic system $S'$ over $G$ such that every variable occurs in exactly
one equation and $G_{S}\simeq G_{S'}$. Consequently, if $H$ is an NTQ group with $H\simeq G_{R(S)}$ then we may assume that every
system $S_{i}$ of $S$ that has the form (\ref{NTQ1}) consists of a single quadratic equation in standard form.
In order to study NTQ groups by induction on the height $n$ of the NTQ system, we will need the following lemma.
\begin{lemma}
Let $S(X,A)$ and $T(Y,A)$ be systems of equations over a group $G$ with $X\cap Y=\emptyset$ and let $G_{1} = G[X] / R_{G}(S)$. Then
\[
G_{R(S\cup T)} \simeq G_{1}[Y] / R_{G_{1}}(T).
\]
\end{lemma}
\begin{proof}
Let $X=\{x_{1},\ldots,x_{n}\}$, $Y=\{y_{1},\ldots,y_{m}\}$, $u=u(x_{1},\ldots,x_{n},y_{1},\ldots,y_{m})\in G[X\cup Y]$. We will show that the natural map, which sends
$u$ to the element represented by $u$ in $G_{1}[Y] / R_{G_{1}}(T)$, is an isomorphism.
To see that the map is well-defined, suppose $u\in R_{G}(S\cup T)$. It suffices to show that $u\in R_{G_{1}}(T)$. Let $\varphi: Y\rightarrow G_{1}$ be any solution of $T$
over $G_{1}$ and denote $y_{j}^{\varphi} = w_{j}(x_{1},\ldots,x_{n})$. We need to show that $u^{\varphi}=1$ in $G_{1}$, i.e. $u^{\varphi}\in R_{G}(S)$. Let $\psi: X\rightarrow G$ be
any solution of $S$ over $G$, and denote $x_{i}^{\psi} = g_{i}$. Consider the map $\alpha: X\cup Y\rightarrow G$ defined by
\begin{eqnarray*}
x_{i} & \rightarrow & g_{i},\\
y_{j} & \rightarrow & w_{j}(g_{1},\ldots,g_{n}),
\end{eqnarray*}
for $i=1,\ldots,n$ and $ j=1,\ldots,m$. The map $\alpha$ is a solution to $S\cup T$. Indeed, if $s\in S$ then $s^{\alpha} = s^{\psi}$, and $\psi$ is a solution to $S$
so $s^{\psi}=1$.
If $t\in T$ then
\[
t^{\alpha} = t(w_{1}(g_{1},\ldots,g_{n}),\ldots,w_{m}(g_{1},\ldots,g_{n})) = \left(t^{\varphi}\right)^{\psi}.
\]
Since $\varphi$ is a solution to $T$ over $G_{1}$, we have that $t^{\varphi}\in R_{G}(S)$ and since $\psi$ is a solution to $S$ over $G$ we have that
$\left( t^{\varphi}\right)^{\psi} = 1$ in $G$, proving that $\alpha$ is a solution to $S\cup T$. Since $u\in R_{G}(S\cup T)$, $u^{\alpha}=1$ hence
\[
1 = u^{\alpha} = \left( u^{\varphi}\right) ^ {\psi}
\]
so $u^{\varphi}\in R_{G}(S)$ as required.
The fact that the natural map is surjective is trivial, so it remains to prove injectivity. Let $u\in G[X\cup Y]$ with $u\not\in R_{G}(S\cup T)$. We must show
that $u\not\in R_{G_{1}}(T)$. Since $u\not\in R_{G}(S\cup T)$, there exists a solution
$\alpha: X\cup Y\rightarrow G$ of $S\cup T$ such that $u^{\alpha}\neq 1$. The restriction $\alpha\vert_{Y}$ of $\alpha$ to $Y$ is a solution
to $T$ over $G_{1}$. Indeed, if $t\in T$ then variables of $X$ do not occur in $t$, so
\[
t^{\alpha\vert_{Y}} = t^{\alpha} = 1
\]
in $G$, hence $t^{\alpha\vert_{Y}}=1$ in $G_{1}$ as well. Since $\alpha\vert_{X}$ is a solution to $S$ over $G$ and
\[
\left(u^{\alpha\vert_{Y}}\right)^{\alpha\vert_{X}} = u^{\alpha} \neq 1
\]
we conclude that $u^{\alpha\vert_{Y}}$ is non-trivial in $G_{1}$ hence $u$ is not in $R_{G_{1}}(T)$, as required.
\end{proof}
It follows from the lemma that for every $i=1,\ldots,n$,
\begin{equation}
G_{i} \simeq G_{i+1}[X_{i}] / R_{G_{i+1}}(S_{i}).
\end{equation}
Note that this isomorphism holds for any system of equations that can be partitioned in triangular form, not just for NTQ
systems.
It is essential to observe that when $R_{G_{i+1}}(S_{i}) = \nclofin{S_{i}}{G_{i+1}}$, $G_{i}$ admits the presentation
\[
G_{i} = \langle G_{i+1}\cs X_{i} \gst S_{i} \rangle.
\]
In this case, $G_{i}$ has a graph of groups decomposition of one of the following four types, according to the form of $S_{i}$:
\begin{Romanenumerate}
\item as a graph of groups with vertices $v_{1}$, $v_{2}$ where $G_{v_1}=G_{i-1}$ and $G_{v_{2}}$ is a QH-subgroup;
\item as a graph of groups with vertices $v_{1}$, $v_{2}$ where $G_{v_1}=G_{i-1}$, $G_{v_{2}}$ is a free abelian group of rank $m$
and the edge groups generate a maximal abelian subgroup of $G_{v_1}$ (`rank $m$ extension of centralizer');
\item as a free product with a finite rank free abelian group;
\item as a free product with a finitely generated free group.
\end{Romanenumerate}
A frequently used method of proving that $R_{G_{i+1}}(S_{i}) = \nclofin{S_{i}}{G_{i+1}}$ is the following well-known fact.
\begin{lemma}\label{Lem:ResidualNullstellensatz}
Let $S(X)$ be a system of equations over a group $G$. If $G_{S}$ is residually $G$, then $R_{G}(S) = \nclofin{S}{G}$ and hence
$G_{R(S)} = G_{S}$.
\end{lemma}
\begin{proof}
It is always the case that $\nclofin{S}{G}\subset R_{G}(S)$, so assume for contradiction that there exists $w\in R_{G}(S)\setminus \nclofin{S}{G}$.
Then $w\neq 1$ in $G_{S}$, so there exists a homomorphism $\phi: G_{S} \rightarrow G$ such that $w^{\phi}\neq 1$. But $\phi$ is a solution to $S$
and $w\in R_{G}(S)$ so $w^{\phi} =1$, a contradiction.
\end{proof}
For NTQ systems over toral relatively hyperbolic groups, \cite{KM09} has shown that the condition $R_{G_{i+1}}(S_{i}) = \nclofin{S_{i}}{G_{i+1}}$ holds
except in some exceptional cases. We recall the relevant definitions from \cite{KM09}.
\begin{definition}
A standard quadratic equation $S=1$ over a group $G$ is said to be \emph{regular} if either $\chi (S) \leq -2$ and $S$ has a non-commutative solution over $G$,
or $S=1$ is an equation of the form $[x,y]d=1$ or $[x_{1},y_{1}][x_{2},y_{2}]=1$. An NTQ system is called \emph{regular} if every quadratic equation appearing in
case (\ref{NTQ1}) is regular.
\end{definition}
\begin{proposition}[\cite{KM09}]
Let $G$ be a toral relatively hyperbolic group and $S=S_{1}\cup\ldots\cup S_{n}$ a regular NTQ system over $G$. Then for all $i=1,\ldots,n$,
\[
R_{G_{i+1}}(S_{i}) = \nclofin{S_{i}}{G_{i+1}}.
\]
\end{proposition}
The condition $R_{G_{i+1}}(S_{i}) = \nclofin{S_{i}}{G_{i+1}}$ allows us to use the graph of groups decomposition of $G_{i}$ to derive
properties of NTQ groups inductively. In particular, we have the following.
\begin{lemma}\label{Lem:PropertiesOfGammaNTQ}
Let $\Gamma=\GammaPresentation$ be a toral relatively hyperbolic group and
$G$ a $\Gamma$-NTQ group such that $R_{G_{i+1}}(S_{i}) = \nclofin{S_{i}}{G_{i+1}}$ for all $i=1,\ldots,n$.
Then $G$ is toral relatively hyperbolic and fully residually $\Gamma$.
\end{lemma}
\begin{proof}
The second statement is proved in \cite{KM09}. For the first, we proceed by induction on the height of the NTQ system. The base $\Gamma$ is toral relatively hyperbolic.
Now assume that $G_{n-1}$ is toral relatively hyperoblic. We will show that $G_{n}$ is toral relatively hyperbolic by
applying Theorem~0.1 of \cite{Dah03} (`Combination theorem') to the four possible decompositions of
$G_{i}$ described above.
Cases (\ref{NTQ4}) and (\ref{NTQ3}) follow from Theorem~0.1 parts (3) and (2), respectively, by amalgamating over the trivial subgroup. Note that to use Theorem~0.1 (2) we
need the fact that
if $G$ is hyperbolic relative to the collection of subgroups $\mathcal{H}$ then it is also hyperbolic relative to $\mathcal{H}\cup \{1\}$.
Case (\ref{NTQ2}) follows from Theorem~0.1 (2) by amalgamating over $P=\langle U_{i}\rangle$, which is maximal abelian in $G_{i-1}$.
For case (\ref{NTQ1}), consider first the case when the surface corresponding to the quadratic equation has punctures. In this case we form $G_{i}$ by amalgamating $G_{i-1}$
with a free group over a $\integers$ subgroup, followed HNN-extensions over $\integers$ subgroups. It follows from the results of \cite{Osi06Memoirs} that these $\integers$ subgroups
are maximal parabolic subgroups, hence we may apply Theorem~0.1 (3), (3').
\end{proof}
\begin{remark}\label{Rem:ParabolicsAreComputable}
From the Combination Theorem it follows that $G$ has finitely many maximal non-cyclic abelian subgroups up to conjugation, and we can construct, by induction, the list of them
along with a finite generating set for each. In the base group $\Gamma$ this is possible using the results of \cite{Dah08}.
\end{remark}
NTQ groups over free groups played a central role in the solution to Tarski's problems by Kharlampovich-Miasnikov and Sela. In Sela's work, they are called
\emph{$\omega$-residually free towers} \cite{Sel01}.
\subsection{Embedding into extensions of centralizers}
Let $G=\langle Z\gst S\rangle$ be a finitely presented group.
We consider $S$ as
a (coefficient-free) system of equations over $\Gamma$. Let $\Gamma$ be presented by $\GammaPresentation$.
For a system of equations over free groups, Kharlampovich and Miasnikov proved that every solution factors through one of finitely many NTQ groups, which can
be effectively constructed.
\begin{proposition}\cite{KM98b}\label{Prop:FreeNTQ}
There is an algorithm that, given a system of equations $T(X,A)=1$ over a free group, produces
finitely many $F$-NTQ systems
\[
T_{1}(X_{1},A)=1,T_{2}(X_{2},A)=1,\ldots,T_{n}(X_{n},A)=1
\]
and homomorphisms
\[
\mu_{i}: F(X)\rightarrow F_{R(T_{i})}
\]
such that
for every homomorphism $\psi:F_{R(T)}\rightarrow F$ there is an integer $i$ and a
homomorphism $\phi:F_{R(T_{i})}\rightarrow F$ such that
\[
\psi = \mu_{i}\phi.
\]
\end{proposition}
Given this result, we may assume that the systems $S_{1}(X_{1},A),\ldots,S_{n}(X_{n},A)$ constructed in Lemma~\ref{Lem:RipsSela1} are in fact NTQ systems.
For each of these systems $S_{i}$ we consider the system $S_{i}^{\pi}$ over $\Gamma$.
In the following lemma, we construct
homomorphisms from $G$ to the coordinate groups $\Gamma_{R(S_{i})}$, one of which must be an embedding if $G$ is fully residually $\Gamma$.
\comment{
\[
\xymatrix{
& F(Z,A) \ar[ld]_{\overline{\phantom{m}}} \ar[d]^{\rho_{i}} \\
G \ar[ddr]_{\psi} & F_{R(T_{i})} \ar[d]^{\phi} \\
& F(A) \ar[d]^{\pi} \\
& \Gamma
}
\]
}
\begin{lemma}\label{Lem:EmbeddingIntoCoordinateGroups}
There is an algorithm that, given a finitely presented group $G=\langle Z\gst S\rangle$,
produces
\begin{romanenumerate}
\item finitely many $F$-NTQ systems $S_{1}(X_{1},A),\ldots, S_{m}(X_{m},A)$, and
\item homomorphisms $\alpha_{i}: G\rightarrow \Gamma_{R(S_{i})}$
\end{romanenumerate}
such that
\begin{arabicenumerate}
\item if $G$ is fully residually $\Gamma$, then there exists $i\in \{1,\ldots,m\}$ such that $\alpha_{i}$ is injective, and
\item if $G$ is residually $\Gamma$, then for every $g\in G$ there exists $i\in \{1,\ldots,m\}$ such that $g^{\alpha_{i}}\neq 1$.
\end{arabicenumerate}
\end{lemma}
\begin{proof}
Refer to Figure~\ref{Figure:CommDiagram} for a diagram of the maps constructed in this proof.
\begin{figure}[htbp]
\begin{center}
\[
\xymatrix{
& F(Z) \ar[ld]_{\overline{\phantom{\phi}}} \ar[rd]^{\rho_{i}} \\
G \ar[ddrr]_{\overline{\rho_{i}\phi\pi}} \ar[r]^{\alpha_{i}} & \Gamma_{R_{\Gamma}(S_{i})} & F_{R(S_{i})} \ar@{->>}[l]^{\gamma_{i}} \ar[d]^{\phi} &\\
& & F(A) \ar[d]^{\pi} \\
& & \Gamma
}
\]
\caption{Commutative diagram for Lemma~\ref{Lem:EmbeddingIntoCoordinateGroups}.}
\label{Figure:CommDiagram}
\end{center}
\end{figure}
Construct the $F$-NTQ systems $S_{1}(X_{1},A),\ldots,S_{n}(X_{n},A)$ and the homomorphisms $\rho_{i}: F(Z)\rightarrow F_{R(S_{i})}$ from Lemma~\ref{Lem:RipsSela1}.
Let $\gamma_{i}: F_{R(S_{i})}\rightarrow \Gamma_{R(S_{i})}$ be the canonical epimorphism and set $\alpha_{i}=\overline{\rho_{i}\gamma_{i}}$. That is,
for any $\overline{u}\in G$,
\[
\overline{u}^{\alpha_{i}} = u^{\rho_{i}\gamma_{i}}.
\]
Since $\rho_{i}$ is given as a word mapping, so is $\alpha_{i}$.
To check that $\alpha_{i}$ is well-defined, let
$u\in F(Z)$ with $\overline{u}=1$ (in $G$). Since $u\in\nclofin{S}{F(Z)}$, there exist $s_{j}\in S$ and $w_{j}\in F(Z)$ such that
$u = \prod_{j=1}^{n} s_{j}^{w_{j}}$
hence
\[
u^{\rho_{i}\gamma_{i}} = \prod_{j=1}^{m} (s_{j}^{\rho_{i}\gamma_{i}})^{w_{j}^{\rho_{i}\gamma_{i}}}.
\]
Recall from the description of canonical representatives in Lemma~\ref{Lem:RipsSela1} that $s_{j}$ has the form $s_{j}=z_{1}z_{2}z_{3}$ and hence $s_{j}^{\rho_{i}}$ has the form
\[
s_{j}^{\rho_{i}} = (x_{1}c_{1}x_{2}^{-1})(x_{2}c_{2}x_{3}^{-1})(x_{3}c_{3}x_{1}^{-1})
\]
where $c_{1} c_{2} c_{3}=1$ in $\Gamma$ and $x_{1},x_{2},x_{3}\in X_{i}$.
Hence
\[
s_{j}^{\rho_{i}}=(c_{1}c_{2}c_{3})^{x_{1}}.
\]
Since the relators of $\Gamma$ are elements of $R_{\Gamma}(S_{i})$ we have that $s_{j}^{\rho_{i}\gamma_{i}}=1$ in $\Gamma_{R(S_{i})}$
hence $u^{\rho_{i}\gamma_{i}}=1$ and $\alpha_{i}$ is well-defined.
Suppose now that $G$ is fully residually $\Gamma$.
For each
$i\in \{1,\ldots,n\}$ set
\[
\Phi_{i} = \{\overline{\rho_{i}\phi\pi}\sst \phi\in\mathrm{Hom}(F_{R(S_{i})},F) \}.
\]
From Lemma~\ref{Lem:RipsSela1} we know that
\[
\mathrm{Hom} (G,\Gamma) = \bigcup_{i=1}^{n} \Phi_{i}.
\]
Since $G$ is fully residually $\Gamma$, there exists $i$ such that $\Phi_{i}$ is a discriminating family of homomorphisms. Indeed,
if no $\Phi_{i}$ discriminates
$G$, then for each $i$ there is a finite subset $W_{i}\subset G$ such that every $\phi\in\Phi_{i}$ is not injective on $W_{i}$. Then
$\bigcup_{i=1}^{m} W_{i}$ is a finite subset that is not discriminated by $\bigcup_{i=1}^{m}\Phi_{i}$.
Let $i$ be such that $\Phi_{i}$ is a discriminating family and let $\overline{u}$ be any non-trivial element of $G$.
Then there exists
$\overline{\rho_{i}\phi\pi}\in\Phi_{i}$ with $u^{\rho_{i}\phi\pi}\neq 1$.
The homomorphism $\phi\pi: F_{R(S_{i})}\rightarrow \Gamma$ is a solution to $S_{i}$ over $\Gamma$ which does \emph{not} send $u^{\rho_{i}}$ to 1, hence $u^{\rho_{i}}$ is
not in $R_{\Gamma}(S_{i})$. Consequently,
\[
\overline{u}^{\alpha_{i}} = (u^{\rho_{i}})^{\gamma_{i}} \neq 1
\]
so $\alpha_{i}$ is injective.
Now suppose that $G$ is residually $\Gamma$ and let $\overline{u} \in G$. Since $\mathrm{Hom} (G,\Gamma) = \bigcup_{i=1}^{n} \Phi_{i}$, there exists $i$ and
$\overline{\rho_{i}\phi\pi}\in\Phi_{i}$ such that $u^{\rho_{i}\phi\pi}\neq 1$. As above, this implies $\overline{u}^{\alpha_{i}}\neq 1$.
\end{proof}
\begin{remark}
Though at least one of the homomorphisms $\alpha_{i}$ must be injective when $G$ is fully residually $\Gamma$, we are not aware of a method for determining which one
(there may be several).
\end{remark}
Our objective now is to construct an effective embedding of each coordinate group $\Gamma_{R(S_{i})}$ into a group obtained from $\Gamma$ by a series of
extensions of centralizers. We will need the following lemma in order to argue by induction.
\begin{lemma}\label{Lem:CanonicalEmbedding}
Let $H\leq G$ be any torsion-free groups and let $S$ be a system of equations over $H$ such that $S$ has one of the NTQ forms (\ref{NTQ1})--(\ref{NTQ4}). Then
the canonical homomorphism $H_{S} \rightarrow G_{S}$ is an embedding.
\end{lemma}
\begin{proof}
The case when $S$ is a standard quadratic equation is Proposition~2 of \cite{KM98a}, the case when $S$ is an extension of a centralizer follows immediately
from the theory of normal forms for HNN-extensions, and the cases of a free product with a free group or free abelian group are obvious.
\end{proof}
We now prove the main technical lemma.
\begin{lemma}\label{Lem:Embedding}
Let $\Gamma=\GammaPresentation$ a finitely presented torsion-free hyperbolic group.
There exists an algorithm that, given an NTQ system $S(X,A)$ over the free group $F$, constructs
a group $H$ obtained from $\Gamma$ by a series of extensions of centralizers and an embedding
\[
\beta:\Gamma_{R(S)}\hookrightarrow H.
\]
Further, both groups $\Gamma_{R(S)}$ and $H$ are toral relatively hyperbolic and a generating set for any maximal abelian subgroup can be effectively
constructed.
\end{lemma}
\begin{proof}
Let $S(X,A)$ be partitioned as an NTQ system as $S_{1},\ldots,S_{n}$. Consider $S$ as a system of equations over $\Gamma$,
with $G_{n+1} = \Gamma$ and
\[
G_{i} = G_{i+1}[X_{i}] / R_{G_{i+1}}(S_{i}).
\]
Note that $\Gamma_{R(S)} = G_{1}$.
We proceed by induction on $n$. For the base
case $n=0$ there are no equations or variables in $S$ so $\Gamma_{R(S)}=\Gamma$ so we may take $H=\Gamma$ and $\beta$ the identity.
Now assume the theorem holds up to $n-1$. That is, assume we have constructed a group $H'$ obtained by
extensions of centralizers of $\Gamma$ and an embedding $\beta':G_{2}\rightarrow H'$.
We argue based of the form
(\ref{NTQ1})--(\ref{NTQ4}) of the system of equations
$S_{1}(X_{1},A)$. In the following we will frequently use without mention Lemma~\ref{Lem:ResidualNullstellensatz} to obtain a presentation of $G_{1}$, and
Lemma~\ref{Lem:PropertiesOfGammaNTQ} and Remark~\ref{Rem:ParabolicsAreComputable} to show that $G_{1}$ is toral relatively hyperbolic
with a finite collection of maximal abelian subgroups (up to conjugation), generating sets of which can be effectively constructed.
\textbf{Form (\ref{NTQ4}): Free product with a free group}. Suppose $S_{1}$ has the form (\ref{NTQ4}), that is, $S_{1}$ is empty.
We will show that the group $\langle G_{2}, X_{1} \gst -\rangle\simeq G_{2} \ast F(X_{1})$ embeds in a group obtained from $G_{2}$
by extensions of centralizers . It will suffice to consider the case of two variables, $X_{1}=\{x, y\}$.
Let $u, v\in G_{2}$ such that $C(u)\cap C(v) = 1$, and consider the series of extensions of centralizers
\begin{eqnarray*}
G_{2}' & = & \langle G_{2}, t \gst [C(u), t] \rangle, \\
G_{2}'' & = & \langle G_{2}', s \gst [C(v), s] \rangle, \\
G_{2}''' & = & \langle G_{2}'', r \gst [C(ust), r] \rangle.
\end{eqnarray*}
One checks that $C_{G_{2}'}(v) = C_{G_{2}}(v)$, that $t$ and $s$ generate a rank two free subgroup in $G_{2}'''$, that $C_{G_{2}''}(ust) \cap G_{2} = 1$,
and that $C_{G_{2}''}(ust)\cap \langle t, s\rangle = 1$.
Define $\phi: G_{2} \ast F(x, y) \rightarrow G_{2}'''$ by $x^{\phi} = t^{r}$, $y^{\phi} = s^{r}$, and $g^{\phi} = g$ for $g\in G_{2}$. A non-trivial element
$w\in G_{2} \ast F(x, y)$ has reduced form
\[
w = g_{1} w_{1}(x, y) g_{2} w_{2}(x, y) \ldots g_{m} w_{m}(x, y) g_{m+1}
\]
and is sent under $\phi$ to
\[
w^{\phi} = g_{1} r^{-1} w_{1}(t, s) r g_{2} r^{-1} w_{2}(t, s) r \dots g_{m} r^{-1} w_{m}(t, s) r g_{m+1}.
\]
This word has no reduction of the form $r g_{i} r^{-1} \rightarrow g_{i}$, since $C_{G_{2}''}(ust) \cap G_{2} = 1$, and no reduction of the form
$r^{-1} w_{i}(t, s) r \rightarrow w_{i}(t, s)$, since $C_{G_{2}''}(ust)\cap \langle t, s\rangle = 1$ and $\langle t,s\rangle$ is free of rank two.
Hence $w^{\phi}$ is reduced and therefore non-trivial by
Britton's Lemma, so $\phi$ is injective.
We conclude that $\langle G_{2}, X_{1}\gst -\rangle$ is residually $G_{2}$, hence $G_{1} = G_{1} \ast F(x, y)$ and $G_{1}$ is toral relatively
hyperbolic. By Lemma~\ref{Lem:CanonicalEmbedding}, $G_{1}$ embeds canonically in $H' \ast F(x ,y)$. Repeating the construction above with $H'$ in place
of $G_{2}$ we may construct an embedding of $H'\ast F(x, y)$ into a group $H$ obtained by extensions of centralizers from $H'$.
\textbf{Form (\ref{NTQ3}): Free product with a free abelian group}. Suppose $S_{1}$ has the form (\ref{NTQ3}).
First, suppose that $|X_{1}|=2$, and so
$\langle G_{2}, X_{1} \gst S_{1}\rangle\simeq G_{2} \ast \integers^{2}$.
Lemma~16 of \cite{KM98a} shows that $G_{2} \ast \integers^{2}$
embeds in every non-trivial extension extension of a centralizer of $G_{2}$.
Consequently, $G_{2}\ast \integers^{2}$ is residually $G_{2}$ so $G_{1} \simeq G_{2} \ast \integers^{2}$ and is toral relatively hyperbolic.
From Lemma~\ref{Lem:CanonicalEmbedding}, $G_{1}$ embeds canonically in $H'\ast\integers^{2}$. Apply Lemma~16 of \cite{KM98a} again to
embed $H'\ast\integers^{2}$ in an extension of centralizers $H$ of $H'$.
It follows immediately from the proof that the embedding is effective, provided we can
produce two non-commuting elements of $H'$, which we may do by Lemma~\ref{Lem:AlgorithmsRelativelyHyperbolic}.
If $|X_{1}|>2$, we partition $S_{1}$ into
two subsystems
\begin{eqnarray*}
S_{1,a} & = & \{ [x_{i},x_{j}]=1, [x_{i},u]=1, \sst i,j\in \{3,\ldots, m\}, u\in U_{1,a}\}\\
S_{1,b} & = & \{ [x_{1},x_{2}]=1 \}
\end{eqnarray*}
where $X_{1} = \{x_{1},\ldots,x_{n}\}$ and $U_{1,a}=\{x_{1},x_{2}\}$. The system $S_{1,b}$ has the form (\ref{NTQ3}) with two variables, which we have dealt with above,
and $S_{1,a}$ is an extension of the centralizer $C_{G_{1,b}}(x_{1}) = \langle x_{1},x_{2}\rangle $ in $G_{1,b}\simeq G_{2} \ast \langle x_{1},x_{2}\rangle$, which we deal with
in form (\ref{NTQ2}) below.
\textbf{Form (\ref{NTQ2}): Extension of a centralizer}.
Suppose $S_{1}$ has the form (\ref{NTQ2}). If $U_{1}$ generates the trivial subgroup in $G_{2}$, which we may check since the word problem in $G_{2}$ is decidable,
then we have the form (\ref{NTQ3}) and we may argue as above.
Otherwise, let $U'$ be the centralizer of $U_{1}$ in $G_{2}$. In general,
$\langle U_{1}\rangle$ is a proper subgroup of $U'$. We must construct a generating set $u_{1},\ldots,u_{m}$ for $U'$.
By induction, $G_{2}$ has, up to conjugation, finitely many parabolic (i.e. abelian of rank at least two) subgroups $P_{1}, \ldots, P_{l}$
and we have constructed a generating set for each one. The centralizer $U'$ is a maximal abelian subgroup of $G_{2}$, hence is either conjugate to
one of the $P_{i}$ or is cyclic.
It follows from \cite{Bum04} and the fact that conjugacy in the abelian groups $P_{i}$ is decidable (see also Theorem~5.6 of \cite{Osi06Memoirs}), that
for any element $g\in G_{2}$ and parabolic subgroup $P_{i}$ we can decide whether or not $g$ is conjugate to an element of $P_{i}$, and if so
find a conjugating element. Applying this to any non-trivial element $g$ of $U_{1}$, we either identify $U'$ as a conjugate of one of the $P_{i}$ and construct a generating set by
conjugating the generating set of $P_{i}$, or we determine that $U'$ is in fact cyclic and we find a generator using Lemma~\ref{Lem:AlgorithmsRelativelyHyperbolic}.
Now consider the system of equations
\[
S_{1}' = \{ [x,u_{i}], [x,y] \sst x,y\in X_{1}\cs i\in \{1,\ldots,m\} \}
\]
over $G_{2}$.
Since $G_{2}$ is commutation-transitive, we know that if $\phi: X_{1}\rightarrow G_{2}$
is any solution to the system $S_{1}$ then $[x^{\phi},u_{i}]=1$ for all $x\in X_{1}$ and $i=1,\ldots,m$. Consequently, $[x,u_{i}] \in R_{G_{2}}(S_{1})$ for all $x\in X_{1}$
and $i=1,\ldots,m$ so $S_{1}' \subset R_{G_{2}}(S_{1})$. The group $\langle G_{2}, X\gst {S_{1}'}\rangle$ is an extension of a centralizer of $G_{2}$. It follows from
$\S 5$ of \cite{BMR02} and Proposition~1.1 of \cite{KM09} that any extension of a centralizer of a toral relatively hyperbolic group $K$ is (fully) residually $K$, so
$\langle G_{2}, X\gst {S_{1}'}\rangle$ is residually $G_{2}$. Then by Lemma~\ref{Lem:ResidualNullstellensatz},
\[
R_{G_{2}}(S_{1}) = R_{G_{1}}(S_{1}')=\nclofin{S_{1}'}{G_{2}}
\]
hence $G_{1} = \langle G_{2}\cs X_{1} \gst S_{1}'\rangle$ and is toral relatively hyperbolic.
We need to show that $G_{1}$ embeds in an extension of centralizer of $H'$. By induction, we may construct a finite generating set
$v_{1},\ldots,v_{l}$ for the maximal abelian subgroup of $H'$ that contains $U'$. Consider the system of equations
\[
T = \{ [x, v_{i}], [x,y] \sst x,y\in X_{1}\cs i\in \{1,\ldots,l\} \}
\]
and the group $H=\langle H', X_{1} \gst T\rangle$, which is an extension of centralizer of $H'$.
Define the map $\beta: G_{1} \rightarrow H$ by $x^{\beta} = x$ for $x\in X_{1}$ and $g^{\beta} = g^{\beta'}$ for
$g\in G_{2}$. One easily checks that $\beta$ is a (well-defined) homomorphism. To show that $\beta$ is injective, let
$w \in G_{1}$ be non-trivial. Since $G_{1}$ is residually $G_{2}$, there is a function
$\phi: X_{1}\rightarrow G_{2}$ which is a solution to $S_{1}'$ and such that $w^{\phi}$ is a non-trivial element of $G_{2}$. We claim that $\phi\beta': X_{1} \rightarrow H'$
is a solution to $T$. For $x,y\in X_{1}$ we have
\[
[x^{\phi\beta'},y^{\phi\beta'}] = [x^{\phi},y^{\phi}]^{\beta'} = 1^{\beta'} = 1.
\]
For $x\in X$ and any $u_{i}$ we have that
\[
[x^{\phi\beta'},u_{i}^{\beta'}] = [x^{\phi},u_{i}]^{\beta'}=1^{\beta'} = 1
\]
so by commutation-transitivity $[x^{\phi\beta'},v_{j}]=1$ for
all $j$. Hence $\phi\beta'$ is a solution as required, and induces a homomorphism $\phi\beta': G_{1}\rightarrow H'$. The image of $w^{\beta}$ under this homomorphism
is
\[
(w^{\beta})^{\phi\beta'} = w^{\phi\beta'}
\]
and is non-trivial since $w^{\phi}\neq 1$ and $\beta'$ is injective. Consequently, $w^{\beta}\neq 1$ in $H$ as required.
\textbf{Form (\ref{NTQ1}): Quadratic equation}. Suppose that $S_{1}$ is a quadratic equation. Then $S_{1}$ has one of the standard forms
(\ref{Eqn:st1})--(\ref{Eqn:st4}). The words $c_{i}$ and $d$ in the standard form are non-trivial in $F_{R(S_{2}\cup\ldots\cup S_{n})}$, but may be trivial in $G_{2}$.
We can check which are trivial by solving the word problem in $G_{2}$. Form an equation $S_{1,a}$
by
\begin{romanenumerate}
\item erasing from $S_{1}$ each atom $c_{i}^{z_{i}}$ such that $c_{i}=1$ in $G_{2}$, and
\item if $d=1$ in $G_{2}$, by erasing $d$ and replacing the rightmost atom of the form $c_{i}^{z_{i}}$ by $c_{i}$.
\end{romanenumerate}
Let $Z$ be the set of variables of $X_{1}$ not appearing in $S_{1,a}$ (i.e. the $z_{i}$ from the erased atoms, as well as the rightmost $z_{i}$ if $d=1$).
Partition $X$ into $X\setminus Z$ and $Z$.
The system of equations $S_{1}(X_{1},A)$ is equivalent over $G_{2}$ to the union of the systems $S_{1,b}=\emptyset$ in variables $Z$ and
$S_{1,a}$ in variables $X_{1}\setminus Z$, so we replace $S_{1}(X_{1},A)$ with these two systems and apply case (\ref{NTQ4}) to $S_{1,b}$.
The equation $S_{1,a}$ is a quadratic equation in standard form over $G_{2}$. To simplify notation, we rename $S_{1,a}$ to $S_{1}$ and
$X_{1}\setminus Z$ to $X_{1}$. We study cases based on the Euler characteristic $\chi(S_{1})$ of the surface associated with $S_{1}$.
\textbf{Case $\mathbf{\chi(S_{1})\leq -2}$.} Assume that $\chi(S_{1})\leq -2$. First, check using Lemma~\ref{Lem:GenPosOrAllComm} whether
or not $S_{1}$ has a solution in general position over $G_{2}$. If so, then $S_{1}$ is regular.
Whenever $S_{1}$ is regular and $G_{2}$ is toral relatively hyperbolic,
Theorem~4.1 of \cite{KM09} proves that the group $\langle G_{2},X_{1} \gst S_{1}\rangle$ embeds into
a group $H$ obtained from $G_{2}$ by a series of extensions of centralizers. Consequently, this group is residually $G_{2}$ hence
$G_{1} = \langle G_{2},X_{1} \gst S_{1}\rangle$ and $G_{1}$ is toral relatively hyperbolic. Embed $G_{1}$ canonically
into $\langle H', X_{1} \gst S_{1}\rangle$, using Lemma~\ref{Lem:CanonicalEmbedding}.
The equation $S_{1}$ is regular over $H'$, and $H'$ is toral relatively hyperbolic, so again applying Theorem~4.1 of \cite{KM09} we obtain that
$\langle H', X_{1} \gst S_{1}\rangle$ embeds into a group obtained from $H'$ by a series of extensions of centralizers. It suffices to show that
this embedding is effective. The reader may verify that in order to obtain an effective embedding from the proof given in \cite{KM09}, one must be
able to solve the following three problems: (a) solve the word problem in $H'$, (b) decide whether or not an quadratic equation over $H'$ has a non-commutative solution,
and (c) find such
a solution. We can solve (a) by Lemma~\ref{Lem:AlgorithmsRelativelyHyperbolic} since $H'$ is toral relatively hyperbolic,
(b) by Lemma~\ref{Lem:GenPosOrAllComm}, and (c) by enumerating all possible solutions
until we find a non-commutative one.
Now suppose that $S_{1}$ does not have a solution in general position over $G_{2}$. By Lemma~\ref{Lem:GenPosOrAllComm}, all solutions
are commutative. We consider cases based on the from of $S_{1}$.
\emph{Orientable forms}. Suppose $S_{1}$ contains a commutator. If $S_{1}=[x_{1},y_{1}][x_{2},y_{2}]$, then $S_{1}$ is regular by definition
and we may proceed as above. Otherwise, by
Proposition~4.3 of
\cite{KM09}, $S_{1}$ has a solution in general position
in a group $K$ obtained from $G_{2} \ast F$, where $F$ is a finite-rank free group, by a series of centralizer extensions.
Since $K$ is discriminated by $G_{2}$ (see form (IV)), it follows that $S_{1}$ has a solution in general position in $G_{2}$, which contradicts
the fact that all solutions are commutative.
\emph{Genus zero forms}.
Suppose that $S_{1}$ has the form
\[
c_{1}^{z_{1}}\ldots c_{k}^{z_{k}}d.
\]
Although $\chi(S_{1})\leq -2$ implies that $k\geq 3$, we will assume only $k\geq 2$. Since $G_{2}$ has the CSA property, we may apply Corollary~3 of \cite{KM98a}
to obtain that
\[
R_{G_{2}}(S_{1})=\ncl{\left\{\:[a_{i}^{-1}z_{i},C],[a_{i}^{-1}z_{i},a_{j}^{-1}z_{j}]\sst i,j=1\ldots k\right\}}
\]
where $C=C_{G_{2}}(c_{1}^{a_{1}},\ldots,c_{m}^{a_{m}})$ and $z_{j}\rightarrow a_{j}$ is a solution to $S_{1}$.
A solution must exist since $S_{1}$ has a solution over $F_{R(S_{2}\cup\ldots\cup S_{n})}$, and $G_{2}$ is a quotient of $F_{R(S_{2}\cup\ldots\cup S_{n})}$.
We may find such a solution by enumerating all possible solutions.
Since $G_{2}$ is CSA, the group $C$ is precisely the maximal abelian subgroup which is the centralizer of $c_{1}^{a_{1}}$. By assumption, we may compute a
generating set $\{u_{1},\ldots,u_{m}\}$ for $C$. Then
\[
G_{1} \simeq \langle G_{2}, t_{1},\ldots, t_{k} \gst [t_{i},u_{l}]\cs [t_{i},t_{j}]\cs 1\leq i,j\leq k\cs 1\leq l \leq m\rangle
\]
via the isomorphism $t_{i}\rightarrow a_{i}^{-1}z_{i}$. Since this is an extension of a centralizer, we complete the argument by reasoning as in Case~(\ref{NTQ2}).
\emph{Non-orientable forms}.
Suppose that $S_{1}$ corresponds to a non-orientable surface.
Suppose $S_{1}$ has the form
\[
x_{1}^2 \cdots x_{p}^2
\]
where, by assumption, $p\geq 4$. Then any two non-commuting elements $g,h\in G_{2}$ yield the non-commutative solution $x_{1}\rightarrow g$, $x_{2}\rightarrow g^{-1}$,
$x_{3}\rightarrow h$, $x_{4}\rightarrow h^{-1}$, and $x_{i}\rightarrow 1$ for $i\geq 4$. This contradicts the assumption that all solutions of $S_{1}$ are commutative.
Suppose $S_{1}$ has the form
\[
x_{1}^2 \cdots x_{p}^2 d
\]
with $d\neq 1$ and $p\geq 3$. For any commutative solution $x_{i}\rightarrow s_{i}$ and any $g\not\in C_{G_{2}}(s_{1})$,
the function $x_{1}\rightarrow g$, $x_{2}\rightarrow g^{-1}$,
$x_{3}\rightarrow s_{1}\cdots s_{p}$, and $x_{i}\rightarrow 1$ for
$i> 3$ is a non-commutative solution, which is a contradiction.
Suppose $S_{1}$ has the form
\[
x_{1}^2 \ldots x_{p}^{2}c_{1}^{z_{1}}\cdots c_{k}^{z_{k}}d.
\]
with $p\geq 2$. Though $\chi (S_{1})\leq -2$ implies $k\neq 0$, the following argument applies for all $k\geq 0$.
Construct any (commutative) solution $x_{i}\rightarrow s_{i}$, $z_{j}\rightarrow a_{j}$. From transitivity of commutation, it follows that
\[
[c_{i}^{a_{i}}, c_{j}^{a_{j}}] = [c_{i}^{a_{i}}, s_{1}\ldots s_{p}]=1
\]
for all $i,j=1,\ldots,k$. Let $U=C_{G_{2}}(c_{1}^{a_{1}},\ldots,c_{k}^{a_{k}},s_{1}\ldots s_{p})$
and construct a generating set $\{u_{1},\ldots,u_{m}\}$ for $U$.
From the proof of Proposition~8 of \cite{KM98a}, which needs only the fact
that $G_{2}$ is commutation-transitive and torsion-free, we see that $G_{1}$ is isomorphic to the group
\[
\langle G_{2},t_{1},\ldots,t_{p+k-1} \sst [u_{l},t_{j}]\cs [t_{i},t_{j}]\cs 1\leq i,j \leq p+k-1\cs 1\leq l \leq m\rangle \ast \langle x_{p} \rangle
\]
via the isomorphism $t_{i}\rightarrow a_{i}^{-1} z_{i}$ for $i=1,\ldots, k$ and $t_{i} \rightarrow x_{i}$ for $i=k+1,\ldots,k+p-1$. This group
is an extension of a centralizer followed by free product with $\integers$, so we proceed as in Case~(\ref{NTQ2}) and Case~(\ref{NTQ4}).
Finally, suppose $S_{1}$ has the form
\[
x_{1}^2 c_{1}^{z_{1}}\ldots c_{k}^{z_{k}}d.
\]
It is shown in the proof of Proposition~8 of \cite{KM98a} that there exists $s\in G_{2}$ such that every solution of $S_{1}$ sends $x_{1}$ to $s$.
Consequently, $s^{-1}x_{1}$ is in the radical of $S_{1}$ over $G_{2}$, hence
\[
G_{1} \simeq G_{2} [z_{1},\ldots,z_{k}] / R_{G_{2}}( c_{1}^{z_{1}}\ldots c_{k}^{z_{k}}d )
\]
and we may argue as in the genus zero case above. Note that we may find $s$ by finding any solution.
\textbf{Case $\mathbf{\chi(S_{1}) > -2}$.} Assume that $\chi(S_{1}) > -2$. We consider cases based on the form of $S_{1}$.
\emph{Orientable forms}.
There are two possible forms, $[x,y]d$ and $[x,y]$. The form $[x,y]d$ is a regular quadratic equation (by definition),
and the argument for regular equations given at the beginning of the case $\chi(S_{1})\leq -2$ applies.
For the form $[x,y]$, we apply Case~(\ref{NTQ3}).
\emph{Non-orientable forms}.
The possible forms are $x^{2}$, $x^{2}d$, $x^{2} y^{2}$, $x^{2} y^{2} d$, and $x^{2} y^{2} z^{2}$.
For the form $x^2$, $x\rightarrow 1$ is the unique solution since $G_{2}$ is torsion-free. Hence $x\in R_{G_{2}}(S_{1})$ and
$G_{1} \simeq G_{2}$, so there is nothing further to prove.
For the form $x^2 d$, find a solution $x\rightarrow a$. Note that $d=a^{-2}$. Suppose, for contradiction, that there exists a second
solution $x\rightarrow b$.
Then since $[a,a^{-2}]=1$, $[b,b^{-2}]=1$, and $a^{-2}=b^{-2}$ we conclude $[a,b]=1$ by transitivity of commutation. Then
\[
(ab^{-1})^{2} = a^{2} b^{-2} = d^{-1} d = 1
\]
which implies $a=b$ since $G_{2}$ is torsion-free.
Consequently, $x\rightarrow a$ is the unique solution and $xa^{-1}$ is in the radical of $x^{2} d$ over $G_{2}$.
Then $\langle G_{2}, x \gst xa^{-1},x^{2}d\rangle \simeq G_{2}$ hence $R_{G_{2}}(\{x^{2}d\})=\nclofin{xa^{-1}}{G_{2}}$ and $G_{1}\simeq G_{2}$.
For the form $x^{2} y^{2}$, the analysis is similar. First, we may check for the existence of a non-trivial solution using the fact that
the existential theory of toral relatively hyperbolic groups is decidable (\cite{Sel09}, \cite{Dah09}). If all
solutions are trivial, then $G_{1}\simeq G_{2}$. Otherwise, let $x\rightarrow a$, $y\rightarrow b$ be a non-trivial solution. Since $[a,a^{2}]=1$ and
$[b,b^{-2}]=1$ we obtain $[a,b]=1$ by transitivity of commutation. As above, $(ab)^{2}=1$ implies $ab=1$ hence $xy$ is in the radical of $x^{2} y^{2}$. The group
$\langle G_{2},x,y\gst xy,x^{2}y^{2}\rangle \simeq G_{2} * \langle x \rangle$ is fully residually $G_{2}$ hence $R_{G_{2}}(x^{2}y^{2}) = \nclofin{xy}{G_{2}}$ so
\[
G_{1} \simeq G_{2}\ast\integers
\]
and we may argue as in Case~(\ref{NTQ4}).
\comment{
For the form $x^2 y^2 d$, first consider the case in which all solutions are commutative. If $x\rightarrow a$, $y\rightarrow b$ is any (commutative) solution, then
$d^{-1}=(ab)^{2}$ and since $\Gamma_{S,n-1}$ is torsion-free we obtain that $xy=ab$ is in the radical. The group $\langle \Gamma_{S,n-1},x,y\sst xy=ab\rangle $
is isomorphic to $\Gamma_{S,n-1}*\langle x\rangle$ hence is residually $\Gamma_{S,n-1}$ and
\[
\Gamma_{S,n}\simeq \Gamma_{S,n-1}*\integers.
\]
Now suppose that $x\rightarrow a$, $y\rightarrow b$ is a non-commutative solution. Since $[a^2,b^2]\neq 1$ it follows that $[a,b]\neq 1$. Let
$\varphi:\Gamma_{S,n-1}\rightarrow \Gamma$ be a homomorphism such that $\varphi([a,b])\neq 1$. Since $\varphi(a)\neq 1$ and $\varphi(b)\neq 1$, the
subgroup $\langle \varphi(a),\varphi(b)\rangle$ is not cyclic. By \cite{Pap04}, there exists $k\in \integers$ such that $\langle \varphi(a)^{k}, \varphi(b)^{k}\rangle$ is
free of rank two. We claim that $\langle a^{k},b^{k}\rangle$ is also free of rank two. Indeed, a non-trivial relation $w(a^{k},b^{k})=1$ gives a non-trivial
relation $w(\varphi(a)^{k}, \varphi(b)^{k})=1$.
Now define $H$ to be the extension of centralizer
\[
H = \langle \Gamma_{S,n-1}, u\sst [C(d),u] \rangle.
\]
We will show that the map $\psi:\GammaNC{n-1} \rightarrow H$ given by
\[
\psi(x) = (a^{k})^{u}\cs \psi(y) = (b^{k})^{u}
\]
is a monomorphism. It is a homomorphism since
\[
\psi(x^{2} y^{2} d) = u^{-1}
\]
oh wait, no it isn't. :(
}
For the form $x^{2} y^{2} d$, first we determine whether or not all solutions are commutative, using Lemma~\ref{Lem:GenPosOrAllComm}. If all solutions
are commutative, the proof given for the case $\chi(S_{1})\leq -2$ and $S_{1}=x_{1}^{2}\ldots x_{p}^{2} c_{1}^{z_{1}}\ldots c_{k}^{a_{k}} d$ with $p\geq 2$ applies,
since there we allowed $k=0$. Otherwise, find any (non-commutative) solution $x\rightarrow a$, $y\rightarrow b$. Consider the series of extensions of centralizers
\begin{eqnarray*}
G_{2}' & = & \langle G_{2}, t \gst [C(ab), t] \rangle, \\
G_{2}'' & = & \langle G_{2}', s \gst [C(atat), s] \rangle, \\
G_{2}''' & = & \langle G_{2}'', r \gst [C(s^{-1}atst^{-1}b), r] \rangle,
\end{eqnarray*}
and the map $\psi: \langle G_{2}, x, y \gst x^{2} y^{2} d \rangle \rightarrow G_{2}'''$ given by
\begin{eqnarray*}
x & \rightarrow & (at)^{s}r \\
y & \rightarrow & r^{-1} t^{-1} b.
\end{eqnarray*}
Since $(x^{2} y^{2} d)^{\psi} = 1$, hence $\psi$ defines a homomorphism. Using normal forms for elements of HNN-extensions, we can show that
$\psi$ is injective (see for example the proofs in \S 5 of \cite{KM98a}). Consequently, $\langle G_{2}, x, y \gst S_{1}\rangle $ is residually $G_{2}$ hence
$G_{1} = \langle G_{2}, x, y \gst S_{1}\rangle $. By Lemma~\ref{Lem:CanonicalEmbedding}, $G_{1}$ embeds canonically into
$\langle H', x, y \gst S_{1}\rangle$. We then apply the construction above to $\langle H', x, y \gst S_{1}\rangle$ to embed this group into a group $H$ obtained from $H'$,
hence from $\Gamma$, by extensions of centralizers.
For the form $x^{2} c^{z} d$, first
we determine whether or not all solutions are commutative, using Lemma~\ref{Lem:GenPosOrAllComm}. Suppose all solutions are commutative. Find any (commutative)
solution $x\rightarrow a$, $z\rightarrow b$. Let $x\rightarrow a_{1}$, $z\rightarrow b_{1}$ be any other solution. We have that $d=(a^{2} c^{b})^{-1} = (a_{1}^{2} c^{b_{1}})^{-1}$
and $[c^{b}, d] = [c^{b_{1}}, d]=1$ since both solutions are commutative. By transitivity of commutation, $[c^{b}, c^{b_{1}}]=1$, and from the CSA property it follows that
$[b_{1} b^{-1}, c]=1$. This equation may be rewritten as $c^{b} = c^{b_{1}}$, and consequently $a_{1}^{2} = a^{2}$. If $a=1$, then $a_{1}=1$ since $G_{2}$ is torsion-free. If $a\neq 1$,
then by transitivity of commutation $[a_{1}, a]=1$ hence
$(a_{1} a)^{2} =1$ so $a_{1} = a$. In either case, $xa^{-1} \in R_{G_{2}}(S_{1})$. Since
\[
\langle G_{2}, x, z \gst xa^{-1}\cs x^{2} c^{z} d\rangle \simeq \langle G_{2}, z \gst c^{z} d a^{2} \rangle
\]
we may apply the argument for the case $S_{1} = c^{z} d$, given below.
If not all solutions are commutative, find any (non-commutative) solution $x\rightarrow a$, $z\rightarrow b$.
As was done in \cite{KM98a}, consider the sequence of extensions of centralizers
\begin{eqnarray*}
G_{2}' & = & \langle G_{2}, t \gst [C(d), t] \rangle, \\
G_{2}'' & = & \langle G_{2}', s \gst [C(c^{b}), s] \rangle, \\
G_{2}''' & = & \langle G_{2}'', r \gst [C(c^{bt}), r] \rangle,
\end{eqnarray*}
and the map $\psi: \langle G_{2}, x, y \gst x^{2} c^{z} d \rangle \rightarrow G_{2}'''$ given by
\begin{eqnarray*}
x & \rightarrow & a^{t} \\
y & \rightarrow & bstr.
\end{eqnarray*}
As in the previous case, $(x^{2} c^{z} d)^{\psi} = 1$ and we may prove using normal forms that $\psi$ is injective and complete the argument as above.
For the form $x^{2} y^{2} z^{2}$, first we determine whether or not all solutions are commutative, using Lemma~\ref{Lem:GenPosOrAllComm}. Suppose
all solutions are commutative. It follows from commutation-transitivity of $G_{2}$ that $[x,y], [x,z], [y,z]\in R_{G_{2}}(S_{1})$, and then from the fact that $G_{2}$
is torsion-free that $xyz \in R_{G_{2}}(S_{1})$. Let $S_{1}'$ be the system of equations $\{ x^{2}y^{2}z^{2}, [x,y], [x,z], [y,z], xyz \}$. Then
\[
\langle G_{2}, x, y ,z \gst S_{1}' \rangle \simeq G_{2} \ast \integers^{2}.
\]
It follows from Case~(\ref{NTQ3}) that this group is fully residually $G_{2}$ and hence
\[
\nclofin{S_{1}'}{G_{2}} = R_{G_{2}}(S_{1}') = R_{G_{2}}(S_{1}).
\]
Then $G_{1} = G_{2} \ast \integers^{2}$ and we may argue as in Case~(\ref{NTQ3}).
Now find any solution $x\rightarrow a$, $y\rightarrow b$, $z\rightarrow c$ of $S_{1}$ in general position. Consider the series of six
extensions of centralizers
\begin{eqnarray*}
G_{2}^{(1)} & = & \langle G_{2}, s \gst [s, C(ab)] \rangle, \\
G_{2}^{(2)} & = & \langle G_{2}^{(1)}, r \gst [r, C(s^{-1}bc)] \rangle, \\
G_{2}^{(3)} & = & \langle G_{2}^{(2)}, v \gst [v, C(abrs^{-1}bc)] \rangle, \\
G_{2}^{(4)} & = & \langle G_{2}^{(3)}, t \gst [t, C(vasvas)] \rangle, \\
G_{2}^{(5)} & = & \langle G_{2}^{(4)}, u \gst [u, C(s^{-1}brs^{-1}br)] \rangle, \\
G_{2}^{(6)} & = & \langle G_{2}^{(5)}, w \gst [w, C(r^{-1}cv^{-1}r^{-1}cv^{-1})] \rangle,
\end{eqnarray*}
and the map $\psi: \langle G_{2}, x, y, z \gst x^{2} y^{2} z^{2} \rangle \rightarrow G_{2}^{(6)}$ given by
\begin{eqnarray*}
x & \rightarrow & (vas)^{t} \\
y & \rightarrow & (s^{-1}br)^{u} \\
z & \rightarrow & (r^{-1}cv^{-1})^{w}.
\end{eqnarray*}
As in the previous case, $(x^{2} y^{2} z^{2})^{\psi} = 1$ and we may prove, with a lengthy argument using normal forms, that $\psi$ is injective and complete the argument
as before.
\emph{Genus zero forms}. The possible forms are $c^{z}d$ and $c_{1}^{z_{1}}c_{2}^{z_{2}}d$. The form $c_{1}^{z_{1}}c_{2}^{z_{2}}d$ was covered
under genus zero forms for $\chi(S_{1})\leq -2$, since the proof there needed only $k\geq 2$.
For the form $c^{z} d$, find a solution $z\rightarrow a$ and a generating set $\{u_{1},\ldots,u_{m}\}$
for $C_{G_{2}}(c)$. We claim that
$[za^{-1}, u_{i}]$ is in the radical of $c^{z} d$, for all $i$. Indeed, if $z\rightarrow b$ is any solution to $c^{z} d=1$ over $G_{2}$ then
\[
[ba^{-1},c] = ab^{-1}c^{-1}ba^{-1}c=a d a^{-1} c = c^{-1}c=1
\]
and by transitivity of commutation we have $[ba^{-1},u_{i}]=1$, hence $[za^{-1},u_{i}]$ is in the radical. Then
\[
\langle G_{2},z\sst [za^{-1},u_{i}]\cs i=1,\ldots,m\rangle \simeq \langle G_{2},t \sst [t,u_{i}]\cs i=1,\ldots,m\rangle
\]
is an extension of the centralizer of $c$, hence is residually $G_{2}$. Consequently, $G_{1}$ is isomorphic to the extension of centralizer
\[
G_{1} \simeq \langle G_{2}, t \sst [t,u_{i}]\cs i=1,\ldots,m\rangle
\]
and we may argue as in Case~(\ref{NTQ2}).
All possible forms of $S_{1}$ have been covered, so the proof is complete.
\end{proof}
We may now prove the main result of the paper.
\begin{theorem}\label{Thm:FinitelyManyEmbeddings}
Let $\Gamma$ be any torsion-free hyperbolic group.
There is an algorithm that, given a finitely presented group $G$, constructs
\begin{romanenumerate}
\item finitely many groups
$H_{1},\ldots, H_{n}$, each given as a series of extensions of centralizers of $\Gamma$, and
\item homomorphisms $\phi_{i}: G\rightarrow H_{i}$,
\end{romanenumerate}
such that
\begin{arabicenumerate}
\item if $G$ is fully residually $\Gamma$, then at least one of the $\phi_{i}$ is injective, and
\item if $G$ is residually $\Gamma$, the map $\phi_{1}\times\ldots\times \phi_{n}: G\rightarrow H_{1}\times\ldots\times H_{n}$ is injective.
\end{arabicenumerate}
This also holds for $G$ in the category of $\Gamma$-groups.
\end{theorem}
\begin{proof}
For each system of equations $S_{i}$ constructed in Lemma~\ref{Lem:EmbeddingIntoCoordinateGroups}, let $\beta_{i}: \Gamma_{R(S_{i})}\rightarrow H_{i}$ be as
constructed in Lemma~\ref{Lem:Embedding} and set $\phi_{i} = \alpha_{i}\beta_{i}$. The result then follows from Lemma~\ref{Lem:EmbeddingIntoCoordinateGroups} and
the fact the each $\beta_{i}$ is injective.
\end{proof}
As a corollary, we obtain a polynomial-time solution to the word problem in any finitely presented residually $\Gamma$ group.
\begin{corollary}\label{Cor:WPinGammaLimit}
Let $\Gamma$ be a torsion-free hyperbolic group and $G=\GPresentation$ any finitely presented group that is known to be residually $\Gamma$.
There is an algorithm that, given a word $w$ over the alphabet $Z^{\pm}$, decides whether or not $w=1$ in $G$ in time polynomial in $\wl{w}$.
\end{corollary}
\begin{proof}
We compute in advance the embedding $\phi: G \rightarrow H_{1}\times \ldots \times H_{n}$, i.e. we compute $z^{\phi}$ for each $z\in Z$.
Given the input word $w$, we need only
compute $w^{\phi}$ and solve the word problem in $H_{1}\times \ldots \times H_{n}$. There is a fixed constant $L$ such that $\wl{\pi_{H_{i}}(w^{\phi})}\leq L \wl{w}$,
where $\pi_{H_{i}}$ is projection onto $H_{i}$,
so we have a polynomial reduction to $n$ word problems in the groups $H_{1}, \ldots, H_{n}$. It then suffices to show that each $H_{i}$ has a polynomial
time word problem.
Let $H_{i}$ be formed by a sequence of $m$ extensions of centralizers and proceed by induction. If $m=0$, then $H_{i}=\Gamma$ so the word problem in $H_{i}$ is decidable in
polynomial time. Now assume that
\begin{equation}\label{Eqn:HNNinduction}
H_{i} = \langle H_{i}', t \gst [t, C(u)] \rangle
\end{equation}
where $u\in H_{i}'$ and $H_{i}'$ is formed from $\Gamma$ by a sequence of $m-1$ extensions of centralizers and has a polynomial time word problem. Let $w$ be a word
in $H_{i}$. It suffices to produce a reduced form for $w$ as an element of the HNN-extension (\ref{Eqn:HNNinduction}):
if any $t^{\pm 1}$ appears in the reduced form then $w\neq 1$, and if no $t^{\pm 1}$ appears then
$w\in H_{i}'$ and we check whether or not $w=1$ using the word problem algorithm for $H_{i}'$.
We produce a reduced form for $w$ by examining all subwords of the form $t v t^{-1}$ and $t^{-1} v t$ where no $t^{\pm 1}$ appears in $v$, and
making reductions
\[
t v t^{-1} \rightarrow v, \;\;\;\; t^{-1} v t \rightarrow v
\]
whenever $v\in C_{H_{i}'}(u)$. The element $v$ is in $C_{H_{i}'}(u)$ if and only if $[v,u]=1$ in $H_{i}'$,
which is an instance of the word problem in $H_{i}'$ and so may be checked in polynomial time.
It is clear that we need only examine a polynomial number of subwords $t v t^{-1}$ and
$t^{-1} v t$ before reaching a reduced form.
\end{proof}
The result below follows from the proof of Lemma \ref{Lem:Embedding}.
\begin{proposition} Let $\Gamma=\GammaPresentation$ be a torsion-free hyperbolic group.
There exists an algorithm that, given a system of equations $U(X,A)=1$ over $\Gamma$, constructs a finite number of $\Gamma$-NTQ systems $\overline S_i(X,A)=1$ over
$\Gamma$ that correspond to the fundamental sequences of solutions of $U(X,A)=1$ that satisfy the second restriction on fundamental sequences as in Section 7.9 in \cite{KM06}. Namely,
a) edge groups in the decompositions on each level are not mapped along this sequence into trivial elements,
b) images of QH subgroups on each level are non-cyclic,
c) images of rigid subgroups are non-cyclic.
Each homomorphism $\Gamma _{R(U)}\rightarrow \Gamma$ factors through one of these fundamental sequences.
(Such fundamental sequences correspond to strict resolutions in Sela's terminology \cite{Sel09}.)
\end{proposition}
\begin{corollary} \label{Cor:UnivTh}
The universal theory of a torsion-free hyperbolic group is decidable.\end{corollary}
\begin{proof} To show that the universal theory of $\Gamma$ is decidable we have to show that there is an algorithm to decide whether the conjunction
of a system of equations $U(X,A)=1$ and a system of inequalities $V(X,A)\neq 1$ has a solution in $\Gamma.$ The conjunction has a solution if and only if there exists an index $i$ such that
the images of all elements from $V(X,A)$ are non-trivial in $\Gamma _{R(\overline S_i(X,A))}.$ This we can check because the word problem in $\Gamma _{R(\overline S_i(X,A))}$ is solvable.
\end{proof}
\bibliographystyle{alpha}
|
1,116,691,498,471 | arxiv | \section{\label{Introduction}Introduction}
With the rapid development of laser technologies, laser intensity of $10^{22} ~\mathrm{W/cm^2}$ has been demonstrated \cite{Yanovsky}. Extreme laser intensity like $10^{23} ~\mathrm{W/cm^2}$ is available in the next few years, which means the electron dynamics approaching nonlinear quantum electrodynamics (QED) regime \cite{Dipiazza,Mourou}. Such laser intensity will allow studying bright $\gamma$-ray emission, $e^+e^-$ pair production, QED-cascade and particles acceleration in laboratories \cite{Remington,Rufni}. Intense $\gamma$-ray sources are useful for simulating the celestial process and extreme environments \cite{Aharonian}. In the past decades, many researches focused on the $\gamma$-ray emission and pair production \cite{Avetissian,Shen,Chen,Liang,Avetissian2,Shkolnikova,Berezhiani}. At extremely high laser intensity, nonlinear Compton scattering is an important way for $\gamma$-ray emission through colliding relativistic electrons with intense laser pulse \cite{Di Piazza,Ta Phuoc,Sarri,Sakai}. This high energy $\gamma$-photons colliding with lasers enables the laser energy to convert into $e^+e^-$ pairs via multi-photon Breit-Wheeler (BW) process \cite{Breit,Nikishov}.
Several schemes are proposed to generate bright $\gamma$-ray and pair production via nonlinear Compton scattering and BW process. Among them, one way is to enhance the laser intensity by selecting appropriate polarized lasers \cite{Gelfer,Yuan,Marija} or/and focusing and redistributing the lasers energy\cite{Bulanov,Gonoskov,Esirkepov,Kirk,Vranic}. Another way is to change the plasma target configuration, such as one or multiple laser interaction with near-critical-density plasma \cite{JIN-JIN LIU,Huang,Zhu,Brady}, solid Al target \cite{Ridgers1,Luo,Chang} or gas plasma \cite{Liu,Lobet}. Among them, laser wakefield acceleration \cite{Liu J X} and laser ponderomotive acceleration \cite{HAN-ZHEN LI,Hu} are generally used to enhance electron acceleration and constraint. Recently, the radiation pressure acceleration (RPA) of ultra-thin foils is also applied to $\gamma$-ray emission and dense $e^+e^-$ pairs production \cite{Lihanzhen}, as it is capable of obtaining high energy electrons and quasi-monoenergetic ion beams \cite{Henig,Macchi,Lv,Xueren}. However, the laser intensity $5 \times 10^{23} ~\mathrm{W/cm^2}$ is too high to obtain experimentally and on the other hand, the plane target cannot prevent the electrons from transverse escaping.
In this paper, a diamond-like carbon (DLC) circular target is presented as an alternative to prevent the electrons from escaping transversely. It is obvious that when the circular target is used, laser energy conversion efficiency to $\gamma$-photons is enhanced and the $\gamma$-photons number density is about twice higher than that of the plane target. Besides, the circular target allows an interaction with multi-lasers at the same time, the optical trap generated in situ can reduce the electrons escaping more efficiently. Eventually, an ultrabright $\gamma$-ray emission with a high density of $5164 n_c$ is obtained at $14 T_0$ (where $T_0$ is the laser period) under the laser intensity $8 \times 10^{22} ~\mathrm{W/cm^{2}}$ through the nonlinear Compton back-scattering (NCBS) process. Further these $7.5 \times 10^{14}$ photons with the average energy of $16 ~\mathrm{MeV}$ colliding with lasers can produce dense positrons with more than $20 n_c$ density via multi-photon BW process. The total positron yield can be as high as $2.7 \times 10^{11}$, with average energy is about $230 ~\mathrm{MeV}$.
The paper is organized as follows. Section 2 outlines the basic target configurations and simulation parameters. The $\gamma$-ray emission by two circularly polarized (CP) laser-driven target is also discussed in detail. Section 3 examines the ultrabright $\gamma$-ray emission and $e^+e^-$ pair production through RPA by four CP lasers irradiating a circular target. Among them, the optimal target radius and the deviation of lasers mismatching are also taken into account. Lastly, a brief summary is given in Sec. 4.
\section{Ultrabright $\gamma$-ray emission by two lasers irradiating a circular target}
The 2D3V simulation results of ultrabright $\gamma$-ray emission by two laser-driven DLC target is performed via QED-PIC code EPOCH \cite{Ridgers,Duclous}.
The DLC foils are ideal materials for self-supporting targets in experiments due to it's high tensile strength, hardness and heat resistance \cite{Liechtenstein}. In our scheme, a circular DLC target as shown in Fig.\ref{target}(b) is used instead of the plane DLC target as shown in Fig.\ref{target}(a) \cite{Lihanzhen} to get brighter $\gamma$-ray and denser $e^+e^-$ pairs through RPA. The DLC target is a plasma consisting of electrons, protons and full ionized carbon ions with charge state $Z_i=6$ and mass $m_i=12\times 1836 m_e$, where $m_e$ is the electron mass. The density of target is $n_e=200n_c$, mixed with $20 \%$ protons in number density, where $n_c=m_e\omega_0^2/4\pi e^2$ ($\omega_0$ is the frequency and $-e$ is the charge) is the critical density of plasma. As Fig.\ref{target} shows, the simulation box size is $20\lambda \times 20\lambda$ with $2000$ $\times$ $1400$ grid cells. Two identical CP laser pulses are incident from the center of left and right boundary of the box simultaneously. Each laser has a peak intensity of $8\times 10^{22} ~\mathrm{W/cm^2}$ and rises in about 1 $T_0$ and then keeps the maximum amplitude for $9$ $T_0$, where $T_0=\lambda/c$ is the laser period, $\lambda=1 ~\mathrm{\mu m}$ is the wavelength of laser and $c$ is the speed of light. The laser is Gaussian profile in $y$ direction with a spot size of ${4 ~\mathrm{\mu m}}$ [full width at half maximum (FWHM)]. When the laser intensity is $8\times 10^{22} ~\mathrm{W/cm^2}$, the optimal thickness and foil gap of plane target for $\gamma$-ray emission and pair production have been studied in detail \cite{Lihanzhen}. So, both targets in our scheme have a thickness of $L=0.25 ~\mathrm{\mu m}$ and the coordinates of the target centre is $(x, y)=(10 ~\mathrm{\mu m}, 0)$. The foil gap is $G=13.5 ~\mathrm{\mu m}$ for the target and the radius is $R=5 ~\mathrm{\mu m}$ for circular target. Note that, only the $\gamma$-photons whose energy is larger than $1 ~\mathrm{MeV}$ are counted in the following simulations.
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig1.pdf}
\caption{\label{target} (Color online) Simulation box and initial target structure. Initial plasma density of plane target (a) and circular (b). The density is normalized by the critical density $n_c$.}
\label{fig:1}
\end{figure*}
In the initial stage, the electrons, carbon ions and protons are separated and form a big charge separation field resulting in an inefficient acceleration of electrons due to the heavier carbon ions by using of DLC target. As time goes on, most of the laser waves penetrates through the target and begins to collide with the relativistic electrons accelerated by opposite laser from the other side through RPA. At this point, high energy $\gamma$-photons is generated through NCBS. When the high energy $\gamma$-photons collide with the lasers, the $e^+e^-$ pairs is produced via multi-photon BW process. For the plane target, the laser intensity along the axis increases since the laser is further focused in the inner surfaces as the target undergoes significant deformation. So, a large number of electrons escape from the foil, which result in a low density of $\gamma$-ray, as shown in Fig.\ref{twolaser}(a). The circular target we proposed can enhance the $\gamma$-photons density to $800 n_c$ which is about $2$ times the $\gamma$-photons density of plane target. There are two reasons for this enhancement. On the one hand, the circular target structure can slow down the laser pulse focusing and laser intensity increasing which will reduce the electrons escaping. On the other hand, the lasers pull the electrons out of the circular target continually and replenish the electrons source when the foil deforms and $\gamma$-ray emits. Besides, the high density $\gamma$-ray, shown in Fig.\ref{twolaser}(b), can sustain about $5 T_0$ which may become a stable $\gamma$-ray source in the future laboratory.
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig2.pdf}
\caption{\label{twolaser} (Color online) Distributions of photon density of plane target (a) and circular target (b) at $20 T_0$. Laser energy conversion efficiency to $\gamma$-photons (c). The density is normalized by the critical density $n_c$.}
\label{fig:1}
\end{figure*}
The laser energy conversion efficiency to $\gamma$-photons for plane target (the blue triangle curve) and circular target (the red circular curve) is plotted in Fig.\ref{twolaser}(c). The energy conversion efficiency of laser-to-photons for the plane target is about $6 \%$, which is comparable with the 3D simulation result of Ref. \cite{Lihanzhen}. It is evident that the circular target can significantly enhance the energy conversion efficiency of laser-to-photons to about $9 \%$ as time goes on. However, compared with the plane target, the lower laser intensity caused by circular structure also reduces the cutoff energy of electrons and $\gamma$-photons at the same time.
While the lower $\gamma$-photons energy may reduce the possibility of $e^+e^-$ pairs production to some extent, the circular target irradiated by multiple lasers has still an obvious advantage that can be seen in the following study. It not only affords a stable and high density $\gamma$-ray source but also provides a chance to get higher density $\gamma$-photons and more $e^+e^-$ pairs.
\section{$\gamma$-ray emission and $e^+e^-$ pairs production by multi-lasers driven circular target}
In order to demonstrate the enhancement of ultrabright $\gamma$-ray emission and dense $e^+e^-$ pairs production by multi-lasers driven DLC circular target, we perform the 2D3V simulation using QED-PIC code EPOCH. The simulation parameters are the same as presented in section $\expandafter{2}$ except that two additional CP lasers are incident from the center of up and down boundary of the simulation box and these two lasers are Gaussian profile in $x$ direction.
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig3.pdf}
\caption{\label{electrondensity}(Color online) Distributions of transverse electric field $E_y$ (a-c), electron density (d-f) and photon density (g-i) at $10 T_0$, $14 T_0$ and $20 T_0$, respectively. Here the electric field is normalized by $E_0=m_e\omega_0 c/e$ and the density is normalized by the critical density $n_c$.}
\label{fig:3}
\end{figure*}
\subsection{$\gamma$-ray emission}
Figure \ref{electrondensity} presents the transverse electric field (a-c), electrons density (d-f) and photons density (g-i) distribution of circular target at different stage. The probability rate for $\gamma$-ray emission in the QED regime is determined by a quantum invariant $\chi_{e^-} = (1/a_s)\sqrt{(\varepsilon_{e^-} E + P_{e^-} \times B)^2 - (P_{e^-} \cdot E)^2}$, where $a_s = eE_s/m_ec\omega_0 = m_ec^2/\hslash\omega_0$ is the normalized QED critical field, $E_s = m_ec^3/(\hslash e) = 1.32 \times 10^{18} ~\mathrm{Vm^{-1}}$ is the Schwinger field \cite{Schwinger}, $\varepsilon_{e^-} = \gamma_{e^-}m_ec^2$ is the electron energy, $\gamma_{e^-}$ is the Lorentz factor, $P_{e^-}$ is the electron momentum, $E$ and $B$ are the electromagnetic fields. Through analyzing, we know that $\chi_{e^-} \simeq 0$ and almost no high energy $\gamma$-photons is produced if the electrons interact with co-propagating lasers. When the electrons collide with the counter-propagating lasers, the quantum invariant $\chi_{e^-}$ can be $\chi_{e^-} \simeq 2\gamma_{e^-}E/E_s$. Hence, the $\gamma$-ray emission is generated if $\chi_{e^-} \geq 1$, which rely on the electrons energy and electric field intensity of lasers.
At the first stage, the initial circular target is distorted to be the four cone structures by the four lasers. Some electrons are first pulled out from the inner wall of target and then rapidly accelerated to high speed by the laser pressure and form overdense relativistic electron layers, as shown in Fig.\ref{electrondensity}(d). A big charge separation field is formed meanwhile due to the heavy protons and heavier carbon ions of DLC target materials which in turn pull ions forward. The accelerated electrons interact with the reflected laser waves resulting in $\gamma$-ray emission by NCBS, as seen in Fig.\ref{electrondensity}(g).
As shown in Fig.\ref{electrondensity}(b), the lasers in cone top are further focused and the intensity is enhanced when the target is expanding. So, the central residual electrons of the target are pulled off, as shown in Fig.\ref{electrondensity}(e), and the relativistic transparency of the DLC target occurs now, which means the lasers will penetrate through the target and collide with the counter-propagating electrons at about $14 T_0$. Through the NCBS, the $\gamma$-ray emission is enhanced resulting in a ultrabright $\gamma$-ray with a high peak density of $5164 n_c$, see Fig.\ref{electrondensity}(h). At $14 T_0$, for two-lasers driven DLC circular target, the photon peak density is $1914 n_c$. This indicates that the peak density of photons is increased about $2.7$ times when other two lasers are injected on side. In addition, the high density $\gamma$-photons can sustain about $20~\mathrm{fs}$. One reason for these benefits is the escaped electrons with transverse velocity will also interact with side lasers to realize the $\gamma$-ray emission enhancement. Another more important reason is that the optical traps created by multiple lasers prevent the electrons escaping from the region of maximum laser intensity.
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig4.pdf}
\caption{\label{magnetic}(Color online) The distribution of transverse magnetic fields $B_z$ at $10 T_0$ (a), $14 T_0$ (b), $16 T_0$ (c) and $20 T_0$ (d). Here the transverse magnetic field is normalized by $m_e\omega c/e$.}
\label{fig:5}
\end{figure*}
Due to the limitation of our computer sources, here we only present a 2D optical trap with four lasers in Fig.\ref{magnetic}, which is a mimic of a real 3D optical trap formation by six laser beams. Some striking features are kept in our 2D simulations. It was shown that the lattice-like magnetic field $B_z$ is formed and the intensity is enhanced as the lasers began to overlap at $14 T_0$. At $16 T_0$, the lasers are fully overlap and the magnetic field $B_z$ is enhanced from $300 ~\mathrm{MG}$ to about $700 ~\mathrm{MG}$, which is beneficial for trapping electrons and maximising the $\gamma$-ray emission. It is obvious that the different lattice-like optical trap structures at different stage correspond to different lattice structures of density distributions of electrons and $\gamma$-photons, which can be seen by comparing Fig.\ref{electrondensity} and Fig.\ref{magnetic}. This confirms our judgement more efficiently that the optical traps created by multiple lasers are the main reason for $\gamma$-ray emission enhancement.
The lattice-like optical trap structure is diffused and the $B_z$ is reduced as well, as shown in Fig.\ref{magnetic}(d). In this last stage, after the lasers penetrating across the central intersection area, the overlapping area of lasers would be reduced gradually. The lasers push the electrons and $\gamma$-photons away from the center which results in a low number density of electrons and $\gamma$-photons, as shown in Fig.\ref{electrondensity}(i). The growth rate of electrons and $\gamma$-photons number at different times is presented in Fig.\ref{energy}(a). It can be seen from this that the total $\gamma$-photons number is $7.5 \times 10^{14}$, which is enhanced by over an order of magnitude compared to the $\gamma$-ray source in Ref. \cite{Lihanzhen}. Besides, the average energy of obtained $\gamma$-photons can be about $16 ~\mathrm{MeV}$, which can be seen in Fig.\ref{radius}(b).
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig5.pdf}
\caption{\label{energy}(Color online) The numbers of electrons (blue triangle line) and photons (red roundness line) at different times (a), the energy spectrum of electrons (b), of $\gamma$-photons (c) and divergence distribution of photons (d) at $12 T_0$ (yellow line), $16 T_0$ (blue line) and $20 T_0$ (red line), respectively.}
\label{fig:4}
\end{figure*}
Above all, $7.5 \times 10^{14}$ $\gamma$-photons with average energy $16 ~\mathrm{MeV}$ is obtained via multi-lasers driven DLC circular target. The maximum density of $\gamma$-ray can be $5164 n_c$ at $14 T_0$, which can be as an extremely dense and ultrabright $\gamma$-ray source for future application. This high quality photons will also have a significant benefit for pair production in BW process.
\subsection{Dense $e^+e^-$ pairs production}
In the QED region, multi-photon BW process is an very important mechanism for pair production through photon-photon annihilation ($\gamma + n\hslash\omega_l \rightarrow e^- + e^+$). The probability for pair production via multi-photon BW process is determined by another quantum parameter $\chi_\gamma = (1/a_s)\sqrt{(\varepsilon_\gamma E + P_\gamma \times B)^2 - (P_\gamma \cdot E)^2} \simeq (2\hslash\omega_\gamma/m_ec^2)E/E_s$, here, $\varepsilon_\gamma = \hslash\omega_\gamma$, $P_\gamma = \hslash\omega_\gamma / c$ ($\omega_\gamma$ the photon frequency). So, the pair production depend on the photons energy $\hslash\omega_\gamma$ and electric field $E$ in interaction zone.
Figure \ref{energy}(c) illustrates the energy spectrum of $\gamma$-photons at different times. Here, due to Doppler red shift, the reflected laser is weakened so that the maximal value of $\gamma$-photons is only $380 ~\mathrm{MeV}$ at the first stage, such as $12 T_0$, which is not enough to produce $e^+e^-$ pairs. At the second stage, the maximal cutoff energy of $\gamma$-photons can be $850 ~\mathrm{MeV}$ by NCBS while it will decrease since the high energy $\gamma$-photons are continually applied to BW process. The spectrum has a wide distribution and the average energy $\bar{\varepsilon_\gamma}$ can be $16 ~\mathrm{MeV}$ at $17 T_0$. Here the high photon energy can greatly enhance the possibility of pair production. As an example, Fig.\ref{positrons}(a) and \ref{positrons}(b) present the positrons density distribution at $14 T_0$ and $17 T_0$, respectively. At $14 T_0$, when high energy $\gamma$-photons and lasers collide, the positron yield starts being considerable. In second stage, the positron density remains at about $20 n_c$ and the maximum value can be $29 n_c$ at $17 T_0$, as Fig.\ref{positrons}(b) shows. These long-lasting and high-bunching positrons show a good prospect for potential applications in future.
The energy spectrum of positrons at different times has also be plotted in Fig.\ref{positrons}(c). The maximum energy of positrons obtained can be as high as $~\mathrm{GeV}$ at $14 T_0$. However, this higher energy positrons also oscillate in laser field and emit $\gamma$-ray resulting in a decrease of cutoff energy as time goes on. Beyond that, there is a monoenergetic peak at $200 ~\mathrm{MeV}$, which means monoenergetic positrons can be achieved through our scheme. The significant increase of positron number through this process is plotted in Fig.\ref{positrons}(d), which shows that the final number of positrons is $2.7 \times 10^{11}$. Besides the mean positron energy can be $230 ~\mathrm{MeV}$ at $17 T_0$. Note that the third dimensional size is assumed as $4 ~\mathrm{\mu m}$ according to the laser spot size and the positron distribution in order to a reasonable mimic of full 3D reality as much as possible.
To make the entire process more intuitive, we also calculate the time-dependent of laser efficiency to electrons, to $\gamma$-photons and to positrons. The general finding is that the laser energy conversion efficiency to $\gamma$-ray and positrons have a rapid growth in the second stage. As time goes on, the total laser energy conversion efficiency to $\gamma$-photons and positrons are about $27 \%$ and $0.2 \%$, respectively, which is a really high exploitation of laser energy.
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig6.pdf}
\caption{\label{positrons}(Color online) The density distributions [(a) and (b)] and energy spectrum (c) of positrons. The laser energy conversion efficiency to electrons, to $\gamma$-photons
and to positrons as well as the number of positrons at different times (d).}
\label{fig:6}
\end{figure*}
\subsection{The effect of the target radius on $\gamma$-ray emission and $e^+e^-$ pairs production}
In our previous simulations, the circular target radius is chosen as $5 ~\mathrm{\mu m}$. Actually the radius of circular target plays an important role in $\gamma$-ray emission and pair production in real application.
Figure \ref{radius}(a) shows the peak number density of $\gamma$-photons, laser-to-electrons and laser-to-photons energy conversion efficiency at $17 T_0$ with different target radius. On the one hand, the number density of photons is the highest when the radius is $5 ~\mathrm{\mu m}$, which is comparable to laser spot size. When the radius is small, the number and accelerating distance of electrons under target is reduced accordingly, which means a low energy of produced electrons resulting in a low rate for the $\gamma$-ray emission. However, if the radius is large, the transverse Rayleigh-Taylor-like instability develops quickly \cite{Chen1,Wang1}, which will also lower the energy of electrons resulting in a undesirable $\gamma$-ray emission. So, the ultrabright $\gamma$-photons can be achieved when the laser field and circular structure collimate the electrons together. On the other hand, the laser-to-photons energy conversion efficiency is considerable when $R = 5 ~\mathrm{\mu m}$. It means the optimal target radius is $5 ~\mathrm{\mu m}$ for $\gamma$-ray emission when both the number and average energy of $\gamma$-photons are taken into account.
However, the average energy of $\gamma$-photons has a slow decrease as the target radius increases appropriately, as shown in Fig.\ref{radius}(b). The reason is that more high energy $\gamma$-photons are used for pair production by multi-photon BW process, although the electrons accelerating and high energy $\gamma$-ray emission become more significant as radius increases. As shown in Fig.\ref{radius}(b), when $R \ge 4 ~\mathrm{\mu m}$, the average energy of $\gamma$-photons decreases as $R$ increases. However, the number and average energy of positrons have a significant increase as $R$ increases when $R \ge 4 ~\mathrm{\mu m}$. It is obvious that the positron yield and average energy are almost the minimum value when the number density of photons and laser-to-photons energy conversion efficiency are maximum, comparing the Fig.\ref{radius}(b) with Fig.\ref{radius}(a). So, the circular target radius should be increased appropriately if it is designed for pair productions.
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig7.pdf}
\caption{\label{radius}(Color online) The peak number density of photons and laser energy efficiency to electrons and photons at $17 T_0$ with different R (a). The number of positrons, average energy of photons and positrons at $17 T_0$ with different R (b).}
\label{fig:7}
\end{figure*}
\subsection{The effect of incident laser beam mismatching}
In experiments, the deviation of incident lasers becomes a key issue for pair production. In order to check the influence of the mismatching of lasers on the $\gamma$-ray emission and pair production, we assume one laser is incident with a deviation $C$, where $C$ is a transverse offset compared to initial ideal case.
The deviation of lasers will reduce the collision interaction of lasers and high energy $\gamma$-photons, which maybe decrease the probability of pair production. As Fig.\ref{deviation}(a) shows, the laser energy conversion efficiency to electrons, $\gamma$-photons and positrons at $17 T_0$ are diminished as $C$ increases. Besides, the number and peak density of positrons are also reduced as $C$ increases, as shown in Fig.\ref{deviation}(b). Above all, both the ultrabright $\gamma$-ray source and high quality positrons can be obtained if the deviation of lasers is controlled within $1 ~\mathrm{\mu m}$.
\begin{figure*}[htbp]\suppressfloats
\includegraphics[width=17cm, height=10cm]{Fig8.pdf}
\caption{\label{deviation}(Color online) The laser energy conversion efficiency to electrons, $\gamma$-photons and positrons at $17 T_0$ with different deviation C (a). The peak number density and number of positrons at $17 T_0$ with different deviation C (b). Here the number density is normalized by $n_c$.}
\label{fig:8}
\end{figure*}
\section{Summary and conclusion}
In summary, a DLC circular target is proposed to replace the plane target and to enhance the $\gamma$-ray emission and $e^+e^-$ pairs production in present study by using the 2D3V QED-PIC code EPOCH. When two counter-propagating lasers are incident from the center of left and right boundary of the simulation box interact with the target, the circular target can enhance the laser-to-photons energy conversion efficiency by comparing the plane target. The density of $\gamma$-photons is increased about $2$ times of the plane target at $20 T_0$. Moreover, when another two counter-propagating lasers are incident from the center of up and down boundary of the simulation box, the overlap of multi-lasers will enhance the laser intensity and form a stable lattice-like optical trap. This optical trap can prevent the high energy electrons accelerated by RPA escaping from central interaction zone. Eventually, $7.5 \times 10^{14}$ $\gamma$-photons with average energy $16 ~\mathrm{MeV}$ is obtained and through NCBS, which is an order of magnitude higher than the photons yield from the plane target. The maximum density of $\gamma$-photons can be $5164 n_c$ at $14 T_0$, which may be ultrabright $\gamma$-ray source in the future application.
Compared with the two laser-driven circular target, we found the number and density of $\gamma$-photons has a nonlinear growth when another lasers is incident. These high quality photons collide with lasers resulting in above $20 n_c$ dense positrons via multi-photon BW process. As time goes on, the total positrons with average energy $230 ~\mathrm{MeV}$ yield can be $2.7 \times 10^{11}$. Furthermore, the optimal radius of circular target for $\gamma$-ray emission and pair production has also been analyzed and discussed respectively. For $\gamma$-ray emission, the optimal radius of target should be $5 ~\mathrm{\mu m}$. However, the radius should be increased suitably if one need more positrons. Lastly, the deviation of lasers is considered for real application, we found there is almost no effect on $\gamma$-ray emission and pair production if the deviation of lasers is controlled within $1 ~\mathrm{\mu m}$.
\section*{Acknowledgements}
This work was supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11875007, and No. 11305010. Guoxing Xia's work is supported by the STFC Cockcroft Institute core grant. The computation was carried out at the HSCC of the Beijing Normal University. The authors are particularly grateful to CFSA at University of Warwick for allowing us to use the EPOCH.
|
1,116,691,498,472 | arxiv | \section{Introduction}
Human \index{languages} languages dististinguish us from other animals, but also birds
or ants have systems of communication. Also, humans have invented
alphabets and other formalized forms of writings. In principle the
methods to be described here could be applied also to these other forms of
communication, but mostly we are interested here in the presently about $10^4$
different human languages on this planet\cite{grimes}. We leave it to
linguists to distinguish languages from dialects or language families;
when we mention "language" readers may read dialect or family instead.
Everyday language contains thousands of words for different aspects of
life, and with the special words of science, medicine, $\dots$ we get
much more. For the same concept of everyday life, each different
language in general has a different word, and thus the number of
possible languages is enormous and difficult to simulate. Things
become easier if we look only at \index{grammar} grammar; do we order (subject,
object, verb) or (subject, verb, object) or $\dots$? Briscoe
\cite{briscoe} mentioned about 30 independent binary grammatical
choices, which leads to a manageable $2^{30} \simeq 10^9$ possible
languages, which can be symbolized by a string of $\ell = 30$
bits. Thus many of the simulations described here use bit-strings with
$\ell = 8, 16 \dots 64$.
The present situation is not in equilibrium; about every ten days a
human language dies out, and in Brazil already more than half of the
indigenous languages have vanished as a result of the European
conquest. On the other hand, \index{Latin} Latin has split in the last two millennia
into several languages, from Portuguese to Romanian, and many experts
believe that Latin and the other \index{Indo-European} Indo-European languages spoken 600
years ago from Iceland to Bengal (and now also in the Americas,
Australia, Africa) have originated from the people who invented
agriculture in the \index{Konya} Konya plane of Turkey, $10^4$ years ago. Thus
similar to biology, also languages can become extinct or speciate
into several daughter languages.
In contrast to biology, humans do not eat humans of other languages as
regular food, and thus one does not have a complex ecosystem of
predators eating prey as in biology. Instead, languages are meant for
communication, and thus there is a tendency of only one language
dominating in one region, like German in Germany etc. Will
globalisation lead to all of us speaking one language in the distant
future? For physics research, that situation has already arrived many
years ago. If we follow the Bible, then at the beginning Adam and Eve
spoke one language only, and only with the destruction of the Tower of
Babel different languages originated.
Thus in the history mankind we may have had first a rise, and later a
decay, in the number of different languages spoken. In \index{Papua} Papua New Guinea
\cite{novotny} there are now $10^3$ languages, each spoken by about $10^3$
people; can this situation survive if television and
mobile phones become more widespread there?
While we cannot answer these questions, we can at least simulate such
"survival of the fittest" among languages, in a way similar but not
identical to biology. We will not emphasize here the longer history of
computer simulations of how children learn a language \cite{nowak}, see also
\cite{roma},
or how mankind developed the very first language out of ape
sounds\cite{brighton}. Instead we talk about the competition between different
languages for adults. And we will emphasize the \index{agent-based} "agent-based" models
simulating individuals, analogous to \index{Monte Carlo} Monte Carlo and Molecular
Dynamics for spins and molecules
The second section deals with \index{differential equations} differential equations (a method we
regard as outdated), followed by agent-based simulations with few
languages in section 3 and with many languages in section 4. Further
results from the two many-language models are given in the appendix.
\section{Differential equations}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.5]{langchak1a.eps}
\end{center}
\caption{
Development of one dominating language, or lack of such dominance, for
the model of Nowak et al \cite{nowak}, with random matrix
elements. We start from the dominance of another
language. The different symbols correspond to two suitably
selected languages and two slightly different mutation rates $p \simeq 0.3$.
From \cite{newbook}.
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.5]{langchak2.eps}
\end{center}
\caption{Size histogram, ignoring the dominating language, for the
model of Nowak et al \cite{nowak} with random matrix
elements. The number of simulated languages varies from 80 on the
right to its real value 8000 on the left; The straight line has slope
1 in this log-log plot. From \cite{newbook}.
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.5]{langchak3.eps}
\end{center}
\caption{Size histogram for human languages, from \cite{grimes}.
We bin language sizes by factors of two, just
as in Fig.2: Thus the leftmost point corresponds to size = 1, the
second sums sizes 2 and 3, the third sums sizes 4 to 7, etc.
}
\end{figure}
Already Nettle \cite{nettle} suggested a very simple differential
equation to see how the number $L$ of languages changes with time:
$$dL/dt = 70/t - L/20$$
Here the time unit is thousand years. (Actually $L$ is the number of
different language groups, and time is discrete). The second term on
the RHS means a loss of five percent per millennium; the first term
indicates the formation of new languages which became more difficult
when the population became higher since then the higher demand for
communication reduced the chances of new languages to develop. The
aim was to explain why the recently populated Americas have a higher
language diversity than Africa and Eurasia with their older human
population. For long times, this differential equation means that $L$
decays exponentially towards zero.
Nowak et al \cite{nowak} use
$$dx_j/dt = (\sum_i f_iQ_{ij}x_i) - \phi x_j$$
for the fraction $x_j$ of a population speaking language $j=1,2,3
\dots L$. (Actually they apply this equation to the learning of languages
or grammars by children; the interpretation for competition between
adult languages is ours.) Here the \index{fitness} fitness $f_i = \sum_j F_{ij}x_j$ of
language $i$ is determined by the degree $F_{ij}$ to which a speaker
of language $i$ is understood by people speaking language $j$. The
average fitness is $\phi = \sum_i f_ix_i$ and is subtracted to keep
the sum over all fractions $x_j$ independent of time. The
probability that children from $i$-speaking parents later speak
language $j$ is $Q_{ij}$.
For a large number $L$ of languages, there are numerous free
parameters in the matrices $Q_{ij}$ and $F_{ij}$. With most of them the same
one finds a sharp \index{phase transition} phase transition \cite{nowak} as a function of
\index{mutation} mutation rates $Q_{ij}$. If one starts with only one
language, then at low mutation rates most of the people still speak
this language and only a minority has switched to other languages. For
increasing mutation rates, suddenly the system jumps from this \index{dominance}
dominance of one language to a \index{fragmentation} fragmentation state where
all languages are spoken equally often. If, in turn, we start from such a
fragmented state then it stays fragmented at high mutation rates. With
decreasing mutation rates it suddenly jumps to the dominance of one
language (numerically, one then has to give this one language a very
slight advantage). The two jumps do not occur at the same mutation
rate but show hysteresis: Starting with dominance and increasing the
mutation rate allows dominance for higher mutation rates then when we
start with fragmentation and decrease the mutation rate.
Qualitatively these properties remain if the many matrix elements are
selected randomly instead of being the same \cite{newbook} except that
the hysteresis has become very small. Fig.1 shows the case where we
start with dominance and looks similar to the case where we start with
fragmentation. The time development for two of the 30 simulated
languages is shown for two slightly different mutation rates, and we
see how for the lower mutation rate but not for the higher rate one of
the two languages starts to dominate, at the expense of the other.
These 30 languages are more mathematical exercises, but Fig.2 applies
these methods to up to $L =8000$ languages, using two $8000 \times 8000$
random matrices $F$ and $Q$. We show the size distribution of
languages, where the size is the fraction of people speaking this
language. On this log-log plot we see roughly parabolas, shifting to
the left with increasing number $L$ of languages. These parabolas
correspond to log-normal distributions, roughly as observed
empirically in Fig.3.
(Similar to Komarova \cite{nowak} we assume the average $F$ to be 0.3 except
for $F_{ii} = 1$ and the average $Q$ to be $p/(L-1)$ except $Q_{ii}
= 1-p$; the actual values are selected randomly between zero and
twice their average.)
There are two problems in this comparison of Figures 2 and 3: In these
simulations, the (logarithmic) range over which the language sizes vary is
quite small and does not change with increasing $L$. And the
real distribution is unsymmetric, having higher values for small
languages \cite{sutherland} than the log-normal distribution; this
enhancement is missing in the simulation of Fig.2. Finally, we
cheated: Fig.2 was taken in the dominance regime and the dominating
language was ignored in the statistics.
Much more attractive for physicists was the one-page paper of Abrams and
Strogatz \cite{abrams} which was within weeks followed by a poster of
Patriarca and Leppanen \cite{finland}. This pair of papers then triggered
apparently independent research in Spain \cite{spain}, Greece \cite{kosmidis},
Germany \cite{schulze}, Argentina \cite{argentina} and at two different places
in Brazil \cite{schwammle,gomes}, all on language competition.
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.5]{langchak5.eps}
\end{center}
\caption{
Exponential decay for the language with lower status, consisting
initially of half the population. The symbols give Monte Carlo simulations
where each individual is influenced by the whole population $N$, while the
like is the result of the differential equation of Abrams and Strogatz.
$a = 1.31, \; s = 0.4, \; N = 10^3, 10^6, 10^9$.
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.30]{langchak6.eps}
\includegraphics[angle=-90,scale=0.30]{langchak6b.eps}
\end{center}
\caption{
Part a: As for Fig.4 but on a $101 \times 101, \; 301 \times 301, \; 1001 \times 1001$
square lattices with $a = 1.31, \; s = 0.1$. Part b: Three-dimensional lattices,
at the symmetry point $x(t=0) = s = 1/2, \; a = 1$. After a long time, the
concentration moves towards zero or one.
}
\end{figure}
The Abrams-Strogatz differential equation for the competition of a language Y
with higher social status $1-s$ against another language X with lower social
status $s$ is
$$ {\rm d}x/{\rm d}t = (1-x)x^as - x(1-x)^a(1-s) $$
where $a \simeq 1.3$ and $0 < s \le 1/2$. Here $x$ is the fraction in the
population speaking language X with lower social status $s$ while the fraction
$1-x$ speaks language Y. Figure 4 with no status difference, $s = 1/2$, shows
as intended that language to win which is initially in the majority; the other
language dies out. For $x(t=0) < 1/2$ the language Y wins and for $x(t=0) > 1/2$
the language X wins. This is highly plausible: If we would immigrate to Brazil
where in most places most of the people speak Portuguese, then also we would
have to learn Portuguese, not because of status but because of numbers. If
the initial minority language has the higher status, as happened 500 years ago
when Portuguese ships landed in Brazil, then it may win at the end, thanks to
guns, writing, and other status aspects, as is the case in Brazil. Figures
for unequal status are published in \cite{newbook}.
The Finnish group \cite{finland} generalized this simple differential equation
to a square lattice, where it became a partial differential equation including
a La\-placian $\nabla^2 x({\bf r})$. Then having in one part of the lattice a
higher status for X compared to Y, and in the other part the opposite status
relation, they showed that the languages X and Y can coexist besides each
other, with a narrow interface in between. This reminds us of Canal Street
in \index{New Orleans} New Orleans which in earlier times separated the French quarter from
the English speaking part. We will later return to such geographical coexistence
also without status differences, arising merely from an initial separation
into an X part and a Y part \cite{schulze,tibihmo}.
If we set $a=1$ the simple logistic \index{Verhulst} Verhulst equation results \cite{verhulst},
$$ {\rm d}x/{\rm d}t = (2s-1)(1-x)x $$
which was applied to languages \cite{ke} already before Abrams and Strogatz.
This case was generalized to two Verhulst equations \cite{argentina} describing
the two populations of people speaking languages X and Y. Now as in
Lotka-Volterra equations for predators eating prey, both populations can
coexist with each other in some parameter range, which in the usual
Abrams-Strogatz model is possible only for $x(t=0) = s = 1/2$.
The competition of two languages is changed if some people become bilingual
\index{bilingual}, that means they learn to speak the other language which
was not their mother tongue \cite{spain}. This was applied to Gallego versus
Castellano in Spain \index{Spain};
of course, some may regard Castellano spoken in Madrid as the proper Spanish,
and Gallego as its dialect spoken in Galicia. As citizens of the Prussian
occupied Westbank of the Rhine River, we know that publicly going into such
details before liberation may be dangerous. A language is a dialect with an
army and a navy behind it.
Of course, all these differential equations are dangerous approximations, just
as \index{mean field theory} mean field theory for critical phenomena in statistical physics is
dangerous. We know since 80 years that the one-dimensional Ising model has
a positive Curie temperature $T_c$ in the mean-field approximation, while in
reality $T_c = 0$. Thus do the Abrams-Strogatz results remain correct if we
deal with individuals which randomly change from one language to the other,
with probabilities corresponding to the original differential equation?
In general, the answer is yes \cite{mallorca}: As long as not both $s$ and
the initial concentration $x(t=0)$ are 1/2, one language still dies out, and
it does so exponentially. This holds for the case of everybody influencing
everybody, Fig.4, as well as for a square lattice where everybody is influenced
only by its four lattice neighbours, Fig.5. The line in Fig.4 is the solution
of the differential equation and agrees qualitatively with the
\index{Monte Carlo} Monte Carlo
results represented by the separate symbols for various total constant
populations $N$. Only for a completely symmetric start, $s = x(t=0) = 1/2$,
when the differential equation gives an equilibrium (stable for $a < 1$ and
unstable for $a > 1$), the microscopic Monte Carlo simulation gives
one or the other language dying out, while the differential equation then
predicts both to always comprise half of the population each.
More details are given in \cite{mallorca}.
Finally we mention the model of \cite{novotny} which also does not deal with
individuals but avoids differential equations.
\section{Microscopic models}
Here we deal with the more modern methods of language simulations, based on
individuals instead of on overall concentrations. Such methods are applied in
physics since half a century and are called \index{agent-based}
agent-based in some fields outside
physics. First we review two models for only two (or a few) languages, then
in much greater detail the two models for many languages.
\subsection{Few languages}
The model of Kosmidis et al \cite{kosmidis} for mixing two languages X and Y
uses \index{bit-string} bit-strings of length 20; each bit can be 0 (representing a word or
grammatical aspect which is not learned) or 1 (an element which this individual
has learned). If
someone speaks language X perfectly and language Y not at all, the bit-string
for this person is 11111111110000000000 while 00000000001111111111 corresponds
to a perfect Y-speaker. People can become perfectly bilingual, having
all 20 bits at 1, but this is rare. This model is particularly simple to explain
the generation of a mixture language Z out of the two original languages X and
Y. One merely has to take about ten bits equal to one and distribute them
randomly among the 20 bit positions. This may then correspond to the creation
of Shakespeare's \index{English} English out of the Germanic language spoken by the
Anglo-Saxons and the French spoken by the Normannic conquerors of the year 1066.
Biological \index{ageing} ageing was included in the model of Schwammle \cite{schwammle}, using
the well-established Penna model \cite{penna,verhulst,newbook} of mutation
accumulation. Two languages X and Y are modelled. Individuals learn to speak
from father and mother (and thus may become \index{bilingual} bilingual) and move on a square
lattice in search of emptier regions. Bit-strings are used also here, but
only for the ageing part to store genetic diseases; the two languages have
no internal structure here. A bilingual person surrounded by neighbours speaking
mostly language X forgets with some probability the language Y, and vice versa.
The model allows for the coexistence of the two languages, each in a different
region of the lattice, as in \cite{finland} but without giving one language
a higher status than the other.
In his later model \cite{schwammle}, that author allows for up to 16 languages.
Again the structure of languages is ignored.
Only young people can learn languages from others, and sometimes they learn
a new language by themselves. As a function of the "mutation" probability to
learn independently a new language, the model gives \index{dominance} dominance
of one language for small mutation rates, and \index{fragmentation}
fragmentation of the population into many languages for high mutation rates,
with a sharp \index{phase transition} phase transition separating
these two possibilities, e.g. at a mutation rate near 1/4. This phase transition
is similar to that found by \cite{nowak} as reviewed above.
\subsection{Many languages}
To explain the existence of the $10^4$ present human languages, we need
different models \cite{gomes,schulze} which we review now.
\subsubsection{Colonization}
After the first human beings came to the American continent by crossing the
Bering street several ten thousand years ago, presumably they first all
spoke one language. Then they moved southward from Alaska and separated into
different tribes which slowly evolved different languages. This first
\index{colonization}
colonization was modeled by Viviane de Oliveira and collaborators \cite{gomes}
by what we call the Viviane model \index{Viviane model}.
Languages have no internal structure but are labelled by integers 1,2,3 ...
Human population starts at the centre site of a square
lattice \index{lattice} with language
1, and from then on humans move to empty neighbour sites of already populated
areas. Each site can carry a population of up to about $10^2$ people, selected
randomly. The size or fitness of a language is the number of people speaking
it. On every new site, the population selects as its own language that of a
populated neighbour site, with a probability proportional to the fitness of the
neighbouring language. In addition, the language can mutate into a new language
with a probability $\alpha$/size. To prevent this \index{mutation}
mutation rate to become too
small, this denominator is replaced in their later simulations
by some maximum $\simeq 10^3$, if the actual
language size is larger than this cut-off value. The simulation stops when
all lattice sites have been populated. A complete \index{Fortran} Fortran
program is listed in \cite{schulze}(d).
For a mutation coefficient $\alpha = 0.256$ the simulated language sizes in the
Viviane model can reach the thousand millions of Chinese \cite{schulze}(d),
but the shape differs from Fig.3 and corresponds more to two power laws than
to one roughly log-normal parabola. In contrast to other models
\cite{nowak,schwammle,schulze} there is no sharp phase transition between
the dominance of one language and the fragmentation into numerous languages.
\cite{schulze}(d).
This Brazilian group \cite{brazil} earlier had claimed that
the language size distribution follows two power laws, both
indicating a decay of the number of languages with increasing
language size. This fit, however, applies only to the large-size
tail and not for small sizes where the power law would
indicate an unrealistic divergence. Fig.3 in contrast shows there
a very small number, with less than $10^2$ of the $10^4$
languages spoken by only one person \cite{sutherland}. The
cumulative number of languages spoken by at least $s$ people
thus should be quite flat for small $s$ instead of diverging
with a power law for $s \rightarrow 0$, as fitted in
\cite{brazil}. A log-normal distribution gives a much better
overall fit and is for large sizes not necessarily worse than
the two power laws of \cite{brazil}.
Further results from the Viviane model are given in our appendix.
\subsubsection{Bit-string model}
Our own model uses bit-strings as in \cite{kosmidis,schwammle} but for
different purposes. Each different \index{bit-string} bit-string
represents a different
language though one may also define slightly different bit-strings as
representing different dialects of the same language. Lengths $\ell$ of
8 to 64 bits have been simulated, and the results for 16 bits differed little
from those of longer strings, while 8 bits behaved differently.
We used three different probabilities $p,\, q,\, r$ though most properties
can be also obtained form the special cases $q = 0, \; r = 1$. When a
new individual is born its language is mutated with probability $p$ compared
to that of the mother. One of the $\ell$ bits is selected randomly and reverted,
which means a zero bit becomes one and a one bit becomes zero. This $p$
is the mutation probability per bit-string; the probability per bit is
therefore $p/\ell$.
When $q$ is not zero, then the above mutation process is modified. With
probability $1-q$ it happens randomly as above, and with probability $q$
the new value of the bit is obtained not by reverting it but by taking
over the corresponding bit value of a randomly selected individual from
the whole population. This \index{transfer} transfer
probability $q$ thus describes the effect
that one language can learn concepts from other languages. Many words of
higher civilization in the German language came from French, while French
beers sometimes have German names.
Thus far the simulations are similar to biology with vertical ($p$) and
horizontal ($q$) gene transfer. Specific human thinking enters into the third
probability $(1-x^2)r$ (also $(1-x)^2$ instead of $1-x^2$ was used)
to give up the own language and to switch to the
language of another randomly selected person. Here $x$ is the fraction of
people speaking the old language, and thus this probability to abandon the
old language is particularly high for small languages. The new language is
selected by a random process, but since it is that of a randomly selected
person and not a randomly selected language, most likely the new language
is one of the major languages in the population. In this way we simulate
the same trend towards dominating language which was already modelled
by Abrams and Strogatz, as described above in the example of our emigration
to Brazil. This \index{flight} flight
from small to large languages, through the parameter $r$,
distinguishes the language competition from biological competition between
species in an ecosystem, and takes into account human consideration of
the utility of the language.
The population size is kept from going to infinity by a \index{Verhulst}
Verhulst death probability proportional to the actual population size. Thus if
we start with one person, the population will grow until it reaches the carrying
capacity given by the reciprocal proportionality factor. More practical is
an initial population which is already about equal to the final equilibrium
population. With the latter choice one can start with either everybody
talking the same language, or everybody talking a randomly selected language.
A complete \index{Fortran} Fortran
program for the simple case $q=0, \; r=1$ is listed in \cite{newbook}.
Compared to the Viviane model explained above, our model is more complicated
since it has three probabilities $p,q,r$ instead of only one coefficient
$\alpha$. However, one can set $q=0, \; r=1$ in our model and then has
the same number of free parameters. The Viviane model simulates the
flight from small to large languages by a mutation probability inversely
proportional to the size of the languages while we separate the mutations
(independent of language size) from the flight probability $(1 - x^2)r$.
Moreover, we simulate a continuous competition of languages while the
Viviane model simulates the unique historical event of a human population
spreading over a continent where no humans lived before.
The results of our model are reported in \cite{newbook,schulze}. Most important
is the sharp \index{phase transition} phase transition, for increasing mutation
rate $p$ at fixed $q$ and $r$, between \index{dominance} dominance at small and
\index{fragmentation} fragmentation at large $p$. For dominance,
at least three quarters of the population speak one language, and most of the
others speak a variant differing by only one bit from that language. For
fragmentation, on the other hand, the population spreads over all possible
languages. If we start with dominance, the phase transition to fragmentation
was already described in the biblical story of the Tower of Babel. If we
start with fragmentation, we get dominance for long enough times and small
enough mutation rates, if we
use $(1-x)^2$ instead of $1-x^2$ for the flight probability. Fig.6 shows
the phase diagram for $\ell = 8$ and 16 if we start from fragmentation. In
Fig.7, particularly long simulations for $\ell = 64$ and one million people
show how an initial dominance decays into fragmentation.
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.5]{langchak8.eps}
\end{center}
\caption{Phase diagram for dominance in the upper left part and fragmentation
in the lower right part. The higher the mutation rate $p$ and the lower the
transfer rate $q$ is the more fragmented is the population into many
different languages. We start with an equilibrium distribution of 100,000
languages. each speaking a randomly selected language. The curve corresponds
to $\ell = 8$ bits, the nearly straight line to $\ell = 16$; $r = 1$ in both
cases. From \cite{schulze}.
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.5]{langchak9.eps}
\end{center}
\caption{Phase transition from dominance to fragmentation for one million
people and 64 bits, i.e. much more possible languages than people. We show
the size of the most-often spoken language after 300 iterations; it jumps from
$10^6 $ to $10^2$.
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.28]{langchak10a.eps}
\includegraphics[angle=-90,scale=0.28]{langchak10b.eps}
\end{center}
\caption{Difference between languages, as measured by the normalized
Hamming distance = fraction of different bits. We show both the average
distance between all pairs and that between the two largest languages,
for 10,000 people and $q=0$. The top part starts with dominance, the
bottom part with fragmentation. From F.W.S. Lima, priv. comm.
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,width=.72\textwidth]{langchak14.eps}
\end{center}
\caption{
Size distribution far from equilibrium during the phase transiton
from fragmentation to dominance, $\ell = 16, \; q=0$. Additional
smoothening by random multiplications was applied \cite{wichmann}.
}
\end{figure}
Tesileanu and Meyer-Ortmanns \cite{tibihmo} introduced into this model
the \index{Hamming} Hamming
distance as a measure of dissimilarity between languages. This Hamming
distance counts the number of different bits in a position-by-position
comparison of two bit-strings. Thus the $\ell=4$ strings 0101 and 1010 have a
Hamming distance of four. This distance can be normalized to lie
between zero and one, through division by $\ell$. Fig.8 shows this normalized
Hamming distance for both the two largest languages and the average
over all possible pairs. Not much difference is seen except that the
one for the single pair fluctuates much stronger than the average over
all pairs. And for dominance the difference is very small while for
fragmentation is it nearly 1/2. Thus for fragmented populations, the
various languages are nearly uncorrelated, and half their bits agree
accidentally while the other half disagrees. For dominance, the minor
languages are mostly one-bit mutants of the dominating language. Fig.8, like
Fig.7 before, shows a clear first-order \index{phase transition} phase
transition, that means a sharp jump. Thus far we were not able to
modify this model such that it gives a second-order transition where
the fraction of people speaking the largest language goes
continuously to zero at a sharp critical point. Such a modification
might give a more realistic distribution of language sizes.
The time dependence of the size of the largest cluster, if we start
with fragmentation, suggests a complicated nucleation
process. Originally all languages are about equal in size, and then
due to random fluctuations one language happens to be somewhat more
equal than the others. This language then wins over, first slowly,
then rapidly. The time needed for one language to win increases
about logarithmically when the population increases from $10^3$ to
$10^8$. Thus for an infinite population, as simulated by
deterministic differential equations of the Nowak et al style
\cite{nowak}, the emergence of dominance out of a fragmented
population might never happen in our model.
First-order phase transitions like those in Figs. 7 and 8 are usually
accompanied by hysteresis, like when undercooled liquid water is to crystallize
into ice. Thus we should get different positions of the effective
transition (for fixed population size and fixed observation time)
depending on whether we start from dominance or fragmentation. This is
shown in \cite{schulze}(b), using in both cases $(1-x)^2$ for the flight
probability.
The language size distribution shows the desired shape of a
slightly asymmetric parabola on the log-log plot (log-normal distribution)
but the actual language sizes are far too small compared with
reality. This is not due to lack of computer power but comes from the
sharp first-order transition, Figs.7 and 8. Either one language
dominates as if 80 percent of the world speaks Chinese. Or all $2^\ell$
languages are equivalent apart from fluctuations and thus each is
spoken only by a small population. If the first-order transition would
be changed into a second-order one, the results for mutation rates
slightly below the critical point might be better.
An alternative was suggested by linguist Wichmann \cite{wichmann}: The
present language distribution is not in equilibrium. If we assume that
parts of the world are on one side and parts on the other side of the
phase transition from dominance to fragmentation (or from fragmentation
to dominance), then the above equilibrium results are not
good. Instead, we show in Fig. 9 two runs for a \index{non-equilibrium}
non-equilibrium
situation of about 5000 iterations at very low mutation rate, starting
from fragmentation. The results are averaged over the second half of
the simulation with the time adjusted such that the phase transition
of fig.11 happened during that second half. Now the language sizes
vary over five orders of magnitude, much better than before. (If
we start from dominance the size distribution is similar but more
symmetric \cite{wichmann}.)
\section{Conclusion}
The last few years have seen the development of a variety of
different approaches to simulate the competition between existing
languages of adult humans. Each model has its advantages and
disadvantages.
If we follow the tradition of physics, that theories should
explain precise experimental data, then the size histogram
of the $10^4$ human languages, Fig.3, seems to be best candidate.
Empirically it is based on Grimes \cite{grimes} and was
analyzed e.g. by \cite{sutherland,grimes,novotny,brazil}.
In order to simulate this language size distribution, we need
models for $10^4$ different languages, and only two of them
have been published thus far, the Viviane model and our model
\cite{schulze,gomes}.
Future work with these models could look at the similarities and
differences between the languages (bit-strings), as started in
\cite{tibihmo} and Fig.8, or the \index{geography} geography of languages and
their dialects \cite{goebl}, as started in \cite{gomes}.
We thank our coauthors \cite{wichmann,mallorca} for collaboration.
\section{Appendix}
This appendix brings some more results for many languages, first on
the Viviane model \cite{gomes} and then on our model \cite{schulze}.
\subsection{Viviane colonization model}
For the model of \cite{gomes}, one can look at the \index{history} history how
languages split from a mother languages, and later produced more
daughter languages. In contrast to linguistic field research, which looks
only at the
last few thousand years, computer simulations can store and analyze
the whole history since the beginning. \cite{schulze}(d) shows for a small $64
\times 64$ lattice, how one language split into
daughters, etc, very similar to biological speciation trees. For
clarity we omitted numerous languages which had no "children". For
larger lattices we found that even for many thousand languages a few
steps suffice on average to reach from any of the languages in this
\index{tree} tree the oldest ancestor language on the top of the tree. Other tree
simulations were published in \cite{wang}.
Often a conquering population imposes its language to the native
population. Perhaps in Europe, before the arrival of Indo-European
farmers, the Cro Magnon people spoke a language family of which the Basque
language is the only present survivor. Better documented, though not
necessarily more true, is the story of the single Gallic village in
today's France which resisted the Roman conquest two millennia ago, thanks to
the efforts of \index{Asterix} Asterix and Obelix (helped by doping). In the
Viviane model, where people may adopt the language of their
neighbours, such a single resistance center can influence many other
sites during the later spread of languages. indeed a
rather large fraction of the total population is influenced by
Asterix, particularly for large mutation rates \cite{schulze}(d). In
physics, such simulation of the influence of a single "error" are
called \index{damage} "damage spreading".
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.27]{langchak19a.eps}
\includegraphics[angle=-90,scale=0.27]{langchak19c.eps}
\end{center}
\caption{
Domain formation if flight and transfer happen only to/from a language learned
from a
lattice neighbour. We mark the sites where the largest language is spoken, after
240 and 450 iterations. For $t \ge 514$ nearly everybody speaks this
language. $(L = 200,\; p = 0.016, \; q = 0.9, \; r=1, \; \ell=16, \; Q=2,$
periodic boundary conditions).
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,scale=0.3]{langchak20a.eps}
\includegraphics[angle=-90,scale=0.3]{langchak20b.eps}
\end{center}
\caption{
Decay of population originally speaking one particular language for
$\ell = 8$ (top) and 16 (bottom).
}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\includegraphics[angle=-90,width=.72\textwidth]{langchak21.eps}
\end{center}
\caption{
Distribution of the number of languages in a language family,
from sums over 100 or 10 independent simulations at various
population sizes, $p = 0.0064$. The observation time increases slightly with
increasing population size.
From ongoing work with S. Wichmann and F.W.S. Lima.
}
\end{figure}
\subsection{Our bit-string model}
While the Viviane model always happens on a lattice \index{lattice}, for our model the
lattice is optional. If we want to study the geographical coexistence
of two languages in adjacent regions, then of course a lattice is
needed \cite{schulze}(b). Now on every lattice site live many people.
Without any difference in status, as opposed to
\cite{finland}, on one side one language dominates and on the other
side the other language, if initially each region was occupied
only by speakers of its own language. Also in the transition
region the other $2^\ell-2$ languages play no major role. The situation in this
figure may correspond to \index{New Orleans} New Orleans a long time
ago, where Canal Street separated the French quarter from the newer
English settlement. These methods could be applied to dialectometry,
\index{dialectometry} as documented for France by Goebl \cite{goebl}.
Bit-strings allow only $Q=2$ choices per position, but the lattice model was
also generalized to $Q = 3$ and 5 choices. Surprisingly, the phase
transition curve \cite{schulze}(d) between dominance for low and fragmentation
for high mutation rates was independent of this number $Q$ of choices. Only
when $\ell$ was changed, the different transition curves were
obtained.
If we want to apply the lattice model to \index{geography} geography we want
compact
geographical language regions to emerge from a fragmented start. Then not
only the transfer of language elements but also the flight to another
language needs to be restricted to lattice neighbours, i.e. people learn
new elements or a new language only from one of the four nearest
neighbours, randomly selected. Fig.10 shows how one language, accidentally
the largest at intermediate times, grows until
is covers nearly the whole lattice.
One may look, without lattice, on the \index{history} history of people
speaking one randomly selected language in an initially fragmented
population. Because of mutations, after a long enough time everybody
has moved at least once to another language. But since the number
$L = 2^\ell$ of possible languages is finite, some people move back to
their original language, like emigrants whose offspring later return
to their old country. Thus after 50 iterations to
give an equilibrium, we mark all those speaking language zero. Their
offspring carries that mark also, even if they mutate their
language, and we count at each time step the number of marked people
speaking language zero.
Then we see a rapid decrease of that number; to slow down the decay we
modified the flight probability to $0.1(1-x^2)y^2$ where $y$ is the
fraction for the language which the individual from fraction $x$
considers to switch to. Then a slower decay as in Fig.11 results,
faster for higher mutation rates. For $\ell = 8$ bits we see nicely
the random background of less than thousand people (from 50 million)
who returned to the language zero of their ancestors; for $\ell = 16$
both the initial and the final number of zero speakers are much
smaller since the 50 million can now distribute among 65536 instead of
only 256 possible languages.
Human languages can be grouped into families, like the Indo-European family
\index{family} of $10^2$ different languages. To simulate language families
we need a criterion which bit-strings belong into one family.
Thus we worked with $\ell = 64$ bits and assumed, following
Wichmann, that the leading 19 bits determine the family and the
remaining 45 bits the different languages within one family. The
numbers $2^{19}$ and $2^{45}$ of possible families and languages
are so large that our computer simulations with less than a
million people do not notice their finite size. Indeed, the
results in Fig.12 for 500,\, 5000,\, 50,000 and 500,000 people are
roughly independent of population size and show a mostly monotonically
decaying probability distribution function for the number of
languages within one family. Empirical observations were
published e.g. by Wichmann \cite{grimes}.
\def6mm{6mm}
|
1,116,691,498,473 | arxiv | \section{Introduction}
Pretrained word embeddings \citep{DBLP:conf/nips/MikolovSCCD13} are the basis of many other natural language processing and machine learning systems. Word embeddings of a specific language contain rich syntax and semantic information. \citet{DBLP:journals/corr/MikolovLS13} stated that the continuous embedding spaces exhibit similar structures across different languages, and we can exploit the similarity by a linear transformation from source embedding space to target embedding space. This similarity derives the \textbf{Bilingual Lexicon Induction(BLI)} task. The goal of bilingual lexicon induction is to align two languages' embedding space and generates word translation lexicon automatically. This fundamental problem in natural language processing benefits much other research such as sentence translation \citep{rapp1995identifying, fung1995compiling}, unsupervised machine translation \citep{lample2017unsupervised}, cross-lingual information retrieval \citep{lavrenko2002cross}.
Recent endeavors \cite{DBLP:conf/iclr/LampleCRDJ18, DBLP:conf/emnlp/Alvarez-MelisJ18, DBLP:conf/aistats/GraveJB19, artetxe2017learning} have proven that unsupervised BLI's performance is even on par with the supervised methods. A crucial part of these approaches is the \textbf{matching procedure}, i.e., how to generate the translation plan. \citet{DBLP:conf/emnlp/Alvarez-MelisJ18} used Gromov-Wasserstein distance to approximate the matching between languages. \citet{DBLP:conf/aistats/GraveJB19} regarded it as a classic optimal transport problem and used the sinkhorn algorithm \citep{DBLP:conf/nips/Cuturi13} to compute the translation plan.
In this work, we follow the previous iterative framework but use a different matching procedure. Previous iterative algorithms required to compute an approximate 1 to 1 matching every step. This 1 to 1 constraint brings out many redundant matchings. Thus in order to avoid this problem, we relax the constraint and control the relaxation degree by adding two KL divergence regularization terms to the original loss function. This relaxation derives a more precise matching and significantly improves performance. Then we propose a bidirectional optimization framework to optimize the mapping from source to target and from target to source simultaneously. In the section of experiments, we verify the effectiveness of our method, and results show our method outperforms many SOTA methods on the BLI task.
\section{Background}
The early works for the BLI task require a parallel lexicon between languages. Given two embedding matrices $X$ and $Y$ with shape $n \times d$ ($n$:word number, $d$:vector dimension) of two languages and word $x_i$ in $X$ is the translation of word $y_i$ in $Y$, i.e., we get a parallel lexicon $X \rightarrow Y$. \citet{DBLP:journals/corr/MikolovLS13} pointed out that we could exploit the similarities of monolingual embedding spaces by learning a linear transformation $W^{\star}$ such that
\begin{equation}
W^{\star}= \mathop{\arg\min}_{W \in M_{d}(\mathbb{R})}\|XW-Y\|^{2}_{F}
\end{equation}
where $M_{d}(\mathbb{R})$ is the space of $d \times d$ matrices of real numbers. \citet{DBLP:conf/naacl/XingWLL15} stated that enforcing an orthogonal constraint on $W$ would improve performance. There is a closed-form solution to this problem called \textbf{Procrutes}: $W^{\star}=Q=UV^T $ where $USV^T=XY^T$.
Under the unsupervised condition without parallel lexicon, i.e., vectors in $X$ and $Y$ are totally out of order, \citet{DBLP:conf/iclr/LampleCRDJ18} proposed a domain-adversarial approach for learning $W^{\star}$. On account of the ground truth that monolingual embedding spaces of different languages keep similar spatial structures, \citet{DBLP:conf/emnlp/Alvarez-MelisJ18} applied the Gromov-Wasserstein distance based on infrastructure to find the corresponding translation pairings between $X$ and $Y$ and further derived the orthogonal mapping Q. \citet{DBLP:conf/aistats/GraveJB19} formulated the unsupervised BLI task as
\begin{equation}
\setlength\abovedisplayskip{10pt}
\setlength\belowdisplayskip{10pt}
\min_{Q \in \mathcal{O}_d , P \in \mathcal{P}_n} \|XQ - PY\|^{2}_{F}
\label{loss}
\end{equation}
where $\mathcal{O}_{d}$ is the set of orthogonal matrices and $\mathcal{P}_n$ is is the set of permutation matrices.Given $Q$, estimating $P$ in Problem~\eqref{loss} is equivalent to the minimization of the 2-Wasserstein distance between the two sets of points: $XQ$ and $Y$.
\begin{equation}
W^2_2(XQ, Y) = \min_{P \in \mathcal{P}_n} \langle D, P \rangle
\label{W}
\end{equation}
where $D_{ij} = \| x_iQ - y_j\|^2_2$ and $\langle D, P\rangle =\sum_{i, j} P_{ij} D_{ij} $ denotes the matrix inner product. \citet{DBLP:conf/aistats/GraveJB19} proposed a stochastic algorithm to estimate $Q$ and $P$ jointly. Problem~\eqref{W} is the standard optimal transport problem that can be solved by Earth Mover Distance linear program with $ \mathcal{O}(n^3)$ time complexity. Considering the computational cost, \citet{zhang2017earth} and \citet{DBLP:conf/aistats/GraveJB19} used the Sinkhorn algorithm~ \citep{DBLP:conf/nips/Cuturi13} to estimate $P$ by solving the entropy regularized optimal tranpsort problem~\citep{peyre2019computational}.
We also take Problem~\eqref{loss} as our loss function and our model shares a similar alternative framework with \citet{DBLP:conf/aistats/GraveJB19}. However, we argue that the permutation matrix constraint on $P$ is too strong, which leads to many inaccurate and redundant matchings between $X$ and $Y$, so we relax it by unbalanced optimal transport.
\citet{DBLP:journals/corr/abs-1811-01124} extended the line of BLI to the problem of aligning multiple languages to a common space. \citet{DBLP:conf/naacl/ZhouMWN19} estimated Q by a density matching method called normalizing flow. \citet{DBLP:conf/aaai/ArtetxeLA18} proposed a multi-step framework of linear transformations that generalizes a substantial body of previous work. \citet{DBLP:journals/corr/abs-1912-01706} further investigated the robustness of \citet{DBLP:conf/aaai/ArtetxeLA18}'s model by introducing four new languages that are less similar to English than the ones proposed by the original paper. \citet{DBLP:conf/acl/ArtetxeLA19a} proposed an alternative approach to this problem that builds on the recent work on unsupervised machine translation.
\section{Proposed Method}
In this section, we propose a method for the BLI task. As mentioned in the background, we take Problem~\eqref{loss} as our loss function and use a similar optimization framework in \citet{DBLP:conf/aistats/GraveJB19} to estimate $P$ and $Q$ alternatively. Our method focuses on the estimation of $P$ and tries to find a more precise matching $P$ between $XQ$ and $Y$. Estimation of $Q$ is by stochastic gradient descent. We also propose a bidirectional optimization framework in section 3.2.
\subsection{Relaxed Matching Procedure}
Regarding embedding set $X$ and $Y$ as two discrete distributions
$\mu = \sum_{i=1}^I u_i \delta_{x_i}$ and
$\nu = \sum_{j=1}^J v_j \delta_{y_j}$, where $u$ (or $v$) is column vector satisfies $\sum_i u_i = 1, u_i > 0$ ($v$ is similar), $\delta_x$ is the Dirac function supported on point $x$.
Standard optimal transport enforces the optimal transport plan to be the joint distribution $P \in \mathcal{P}_n$. This setting leads to the result that every mass in $\mu$ should be matched to the same mass in $\nu$. Recent application of unbalanced optimal transport ~\citep{DBLP:journals/corr/abs-1904-10294} shows that the relaxation of the marginal condition could lead to more flexible and local matching, which avoids some counterintuitive matchings of source-target mass pairs with high transportation cost.
The formulation of unbalanced optimal transport~\citep{chizat2018interpolating} differs from the balanced optimal transport in two ways. Firstly, the set of transport plans to be optimized is generalized to $\ensuremath{\mathbb{R}}_+^{I\times J}$. Secondly, the marginal conditions of the Problem~\eqref{W} are relaxed by two KL-divergence terms.
\begin{equation}
\setlength \abovedisplayskip{0em}
\begin{split}
\min_{P \in \ensuremath{\mathbb{R}}_+^{I\times J}} \langle D, P\rangle
+ \lambda_1 & \KL{P \mathbbm{1}_J}{u} \\
& + \lambda_2 \KL{P^T \mathbbm{1}_I}{v}
\end{split}
\label{RelaxedW}
\end{equation}
where $\KL{p}{q} = \sum_i p_i\log\left(\frac{p_i}{q_i}\right) - p_i + q_i$ is the KL divergence.
We estimate $P$ by considering the relaxed Problem~\eqref{RelaxedW} instead of the original Problem~\eqref{W} in~\citep{DBLP:conf/aistats/GraveJB19}.
Problem~\eqref{RelaxedW} could also be solved by entropy regularization with the generalized Sinkhorn algorithm~\citep{chizat2018scaling,DBLP:journals/corr/abs-1904-10294,peyre2019computational}.
\begin{algorithm}[t]
\caption{Generalized Sinkhorn Algorithm}
\begin{algorithmic}[1]
\REQUIRE source and target measure $\mu_i\in \ensuremath{\mathbb{R}}^m_+, \nu_j\in \ensuremath{\mathbb{R}}^n$, entropy regularizer $\epsilon$, KL relaxation coefficient $\lambda_1,\lambda_2$ and distance matrix$D_{ij}$.
\ENSURE Transport Plan$P_{ij}$
\STATE Initialize $u \gets 0 \in \ensuremath{\mathbb{R}}^m$, $v \gets 0\in \ensuremath{\mathbb{R}}^n$, $K \gets e^{-D/\gamma} \in \ensuremath{\mathbb{R}}^{m\times n}$
\WHILE{not converge}
\STATE $u \gets \left(\frac{\mu}{Kv}\right)^{\frac{\lambda_1}{\epsilon+\lambda_1}}$
\STATE $v \gets \left(\frac{\nu}{K^\top u}\right)^{\frac{\lambda_2}{\epsilon+\lambda_2}}$
\ENDWHILE
\STATE $P \gets \text{diag}(u) K \text{diag}(v)$
\end{algorithmic}
\end{algorithm}
In short, we already have an algorithm to obtain the minimum of the Problem~\eqref{RelaxedW}. In order to avoid the hubness phenomenon, we replace $\mathcal{l}_2$ distance of embedding with the $rcsls$ distance proposed in \citet{DBLP:conf/emnlp/JoulinBMJG18} formalized as $D_{ij} = rcsls(x_iQ, y_j)$.
$rcsls$ can not provide significantly better results than euclidean distance in our evaluation. However, previous study suggests that RCSLS could be considered as a better metric between words than euclidean distance. So we propose our approach with RCSLS. The "relaxed matching" procedure and the "bi-directional optimization" we proposed bring most of the improvement.
We call this relaxed estimation of $P$ as \textbf{Relaxed Matching Procedure(RMP)}. With RMP only when two points are less than some radius apart from each other, they may be matched together. Thus we can avoid some counterintuitive matchings and obtain a more precise matching $P$. In the section of experiments we will verify the effectiveness of RMP.
\subsection{Bidirectional Optimization}
Previous research solved the mapping $X$ to $Y$ and the mapping $Y$ to $X$ as two independent problems, i.e., they tried to learn two orthogonal matrix $Q_1$ and $Q_2$ to match the $XQ_1$ with $Y$ and $YQ_2$ with $X$, respectively. Intuitively from the aspect of point cloud matching, we consider these two problems in opposite directions are symmetric. Thus we propose an optimization framework to solve only one $Q$ for both directions.
In our approach, we match $XQ$ with $Y$ and $YQ^T$ with $X$ simultaneously. Based on the stochastic optimization framework of \citet{DBLP:conf/aistats/GraveJB19}, we randomly choose one direction to optimize at each iteration.
The entire process of our method is summarized in Algorithm~\ref{alg:1}. At iteration $i$, we start with sampling batches $X_b$, $Y_b$ with shape $\mathbb{R}^{b \times d}$. Then we generate a random integer $rand$ and choose to map $X_bQ$ to $Y_b$ or map $Y_bQ^T$ to $X_b$ by $rand$'s parity. Given the mapping direction, we run the RMP procedure to solve Problem~\eqref{RelaxedW} by sinkhorn and obtain a matching matrix $P^*$ between $X_bQ$ and $Y_b$(or $Y_bQ^T$ and $X$). Finally we use gradient descent and procrutes to update $Q$ by the given $P^*$. The procedure of $Q$'s update is detailed in \citet{DBLP:conf/aistats/GraveJB19}.
\begin{algorithm}[t]
\caption{Bidirectional Optimization with RMP}
\label{alg:1}
\begin{algorithmic}[1]
\REQUIRE word vectors from two languages$X$, $Y$
\ENSURE Transformation $Q$
\FOR{each $e \in [1,E]$}
\FOR{each $i \in [1,I]$}
\STATE Draw $X_b$, $Y_b$ of size $b$ from $X$ and $Y$
\STATE set $rand = random()$
\IF{$rand \mod 2 = 1$}
\STATE $Y_b, X_b, Q \Leftarrow X_b, Y_b, Q^T$
\ENDIF
\STATE Run RMP by solving Problem~\eqref{RelaxedW} and obtain $P^*$\
\STATE Update Q by gradient descent and Procrutes
\IF{$rand \mod 2 = 1$}
\STATE $Q \Leftarrow Q^T$
\ENDIF
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
\vspace{-0.3cm}
\section{Experiments}
In this section, we evaluate our method in two settings. First, We conduct distillation experiments to verify the effectiveness of RMP and bidirectinal optimization.
Then we compare our method consisting of both RMP and bi-directional optimization with various SOTA methods on the BLI task.
\begin{table*}[]
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{DECCCCCCCCCCCC}
\toprule[2pt]
Method & & \multicolumn{2}{c}{EN-ES} & \multicolumn{2}{c}{EN-FR} & \multicolumn{2}{c}{EN-DE} & \multicolumn{2}{c}{EN-RU} & \multicolumn{2}{c}{EN-IT} & Avg.\\ \hline
& Supervision & $\rightarrow$ & $\leftarrow$ & $\rightarrow$ & $\leftarrow$ & $\rightarrow$ & $\leftarrow$ & $\rightarrow$ & $\leftarrow$ & $\rightarrow$ & $\leftarrow$ \\ \hline
Proc. & 5K words & 81.9 & 83.4 & 82.1 & 82.4 & 74.2 & 72.7 & 51.7 & 63.7 & 77.4 & 77.9 &74.7 \\
RCSLS & 5K words & 84.1 &86.3 &83.3 &84.1 &79.1 &76.3 &57.9 &67.2 &&& 77.3 \\ \hline
GW & None & 81.7 & 80.4 & 81.3 & 78.9 & 71.9 & \textbf{78.2} & 45.1 & 43.7 & 78.9 & 75.2 & 71.5\\
Adv. - Refine & None & 81.7 & 83.3 & 82.3 & 82.1 & 74.0 & 72.2 & 44.0 & 59.1 & 77.9 & 77.5 & 73.4\\
W.Proc. - Refine & None & \textbf{82.8} & 84.1 & 82.6 & 82.9 & 75.4 & 73.3 & 43.7 & 59.1 & & &73.0\\
Dema - Refine & None & 82.8 & 84.9 & 82.6 & 82.4 & 75.3 & 74.9 & 46.9 & 62.4 &&& 74.0\\ \hline
Ours - Refine & None & 82.7 & \textbf{85.8 } &
\textbf{83.0 } & \textbf{83.8} & \textbf{76.2} & 74.9 & \textbf{48.1} & \textbf{64.7} & \textbf{79.1} & \textbf{80.3} & \textbf{75.9}\\
\bottomrule[2pt]
\end{tabular}
\end{center}
\caption{Comparison between SOTA methods on BLI task. Methods in Line 1-2 are supervised. Methods in Line 3-8 are unsupervised. Except the GW’ method, other unsupervised methods are refined. In bold, the best among unsupervised approaches. All numbers of others are taken from
their papers. ('EN': English, 'ES': Spanish, 'FR': French, 'DE': German, 'RU': Russian, 'IT': Italian).}
\label{Table1}
\end{table*}
\textbf{DataSets\footnote{https://github.com/facebookresearch/MUSE}} We conduct word translation experiments on 6 pairs of languages and use pretrained word embedding from fasttext. We use the bilingual dictionaries opensourced in the work \cite{DBLP:conf/iclr/LampleCRDJ18} as our evaluate set.We use the CSLS retrieval method for evaluation as \citet{DBLP:conf/iclr/LampleCRDJ18} in both settings. All the translation accuracy reported is the precision at 1 with CSLS criterion. We open the source code on Github\footnote{https://github.com/BestActionNow/bidirectional-RMP}.
\subsection{Main Results}
Through the experimental evaluation, we seek to demonstrate the effectiveness of our method compared to other SOTA methods. The word embeddings are normalized and centered before entering the model. We start with a batch size 500 and 2000 iterations each epoch. We double the batch size and quarter the iteration number after each epoch. First 2.5K words are taken for initialization, and samples are only drawn from the first 20K words in the frequently ranking vocabulary. The coefficients $\lambda_1$ and $\lambda_2$ of the relaxed terms in Problem~\eqref{RelaxedW} are both set to 0.001.
\textbf{Baselines} We take basic Procrutes and RCSLS-Loss of \citet{DBLP:conf/emnlp/JoulinBMJG18} as two supervised baselines.
Five unsupervised methods are also taken into accounts:
the Gromov Wasserstein matching method of \citet{DBLP:conf/emnlp/Alvarez-MelisJ18},
the adversarial training(Adv.-Refine) of \citet{DBLP:conf/iclr/LampleCRDJ18}, the Wasserstein Procrutes method(W.Proc.-Refine) of \citet{DBLP:conf/aistats/GraveJB19}, the density matching method(Dema-Refine) of \citet{DBLP:conf/naacl/ZhouMWN19}.
In Table~\ref{Table1}, it's shown that leading by an average of 2 percentage points, our approach outperforms other unsupervised methods in most instances and is on par with the supervised method on some language pairs. Surprisingly we find that our method achieves significant progress in some tough cases such as English - Russian, English - Italian, which contain lots of noise. Our method guarantees the precision of mapping computed every step which achieves the effect of noise reduction.
However, there still exists an noticeable gap between our method and the supervised RCSLS method, which indicates further research can be conducted to absorb the superiority of this metric to unsupervised methods.
We also compare our method with W.Proc on two non-English pairs including FR-DE and FR-ES to show how bidirectional relaxed matching improves the performance and results are presented in Table~\ref{Table2}. Most of the recent researches didn't report results of non-English pairs, which makes it hard for fair comparison. However from the results in Table~\ref{Table2}, we could find that our method keeps an advantage over W.Proc. Note that the W.Proc. results here are our implementation rather than that are reported in the original paper.
\begin{table}[t]
\begin{center}
\renewcommand{\arraystretch}{1.5}
\renewcommand{\tabcolsep}{3.0pt}
\begin{tabular}{lcccc}
\toprule
& FR-DE & DE-FR & FR-ES & ES-FR\\ \hline
W.Proc.& 65.8 & 73.5 & 82.0 & 84.9\\
Ours-Refine & 67.7 & 74.0 & 83.3 & 84.9\\
\bottomrule
\end{tabular}
\end{center}
\caption{Comparision bewtween W.Proc. and our method on non-English language pairs}
\label{Table2}
\vspace{-0.7cm}
\end{table}
\subsection{Ablation Study}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figures/data.pdf}
\caption{Ablation study of our methods' effectiveness. 'WP' refers to the original Wasserstein Procrutes Method proposed by \citet{DBLP:conf/aistats/GraveJB19}. 'WP-RMP' applies RMP to 'WP'. 'WP-RMP-bidiretion' applies bidirectional optimization framework to 'WP-RMP'. 'WP-RMP-bidirection-refine' applies the refinement procedure to 'WP-RMP-bidirection'.('EN': English, 'ES': Spanish, 'FR': French, 'DE': German, 'RU': Russian, 'IT': Italian).}
\label{fig:fig1}
\vspace{-0.3cm}
\end{figure}
The algorithms for BLI could be roughly divided into three parts: 1. initialization, 2 iterative optimization, and 3. refinement procedure, such as \citet{lample2017unsupervised}. W.Proc.\citep{DBLP:conf/aistats/GraveJB19} only covers the first two parts. Our approaches, i.e. relaxed matching and bi-directional optimization are categorized into the second part. To ensure a fair comparison, W.Proc.-Refine is compared to ours-Refine which is discussed in next section. To verify the effectiveness of RMP and bidirectional optimization directly, we apply them to the method proposed in \citet{DBLP:conf/aistats/GraveJB19} one by one. We take the same implementation and hyperparameters reported in their paper and code
\footnote{https://github.com/facebookresearch/fastText/alignment} but using RMP to solve $P$ instead of ordinary 2-Wasserstein.
On four language pairs, We applied RMP, bidirectional optimization and refinement procedure to original W.Proc. gradually and evaluate the performance change. In Figure \ref{fig:fig1} it's clearly shown that after applying bidirectional RMP, the translation accuracy improves by 3 percentage averagely. The results of 'WP-RMP' are worse than 'WP-RMP-bidirection' but better than original 'WP'. Moreover, we find that by applying RMP, a more precise $P$ not only eliminates many unnecessary matchings but also leads to a faster converge of the optimization procedure. Furthurmore, the effectiveness of refinement procedure is quite significant.
To summarize, we consider the average of scores (from en-es to ru-en). By mitigating the counter-intuitive pairs by polysemies and obscure words, the "relaxed matching" procedure improves the average score about 2 points, the "bi-directional optimization" improves the average score about 0.6 points. From the results we could get some inspiration that our ideas of relaxed matching and bidirectional optimization can also be applied to other frameworks such as adversarial training by \citet{lample2017unsupervised} and Gromov-Wasserstein by \citet{DBLP:conf/emnlp/Alvarez-MelisJ18}.
\section{Conclusion}
This paper focuses on the matching procedure of BLI task. Our key insight is that the relaxed matching mitigates the counter-intuitive pairs by polysemy and obscure words, which is supported by comparing W.Proc.-RMP with W.Proc in Table~\ref{fig:fig1}. The optimal transport constraint considered by W.Proc. is not proper for BLI tasks. Moreover, Our approach also optimizes the translation mapping Q in a bi-directional way, and has been shown better than all other unsupervised SOTA models with the refinement in Table 1.
\section{Acknowledgement}
This work was supported by the National Natural Science Foundation of China (11871297, 91646202), National Key R\&D Program of China(2018YFB1404401, 2018YFB1402701), Tsinghua University Initiative Scientific Research Program.
|
1,116,691,498,474 | arxiv | \section{Introduction}
\label{intro}
A number of meson candidates, dubbed the $XYZ$ mesons, that contain charmed-quark
anticharmed-quark ($c\bar{c}$) pairs but do not match expectations for any of the
unassigned levels of the $[c\bar{c} ]$ charmonium meson spectrum, have been observed in recent
experiments~\cite{Olsen:2017bmm}. In some cases, the distinction between the new
states that are nonstandard hadrons and conventional charmonium mesons remains
controversial.
This is especially the case for the $X(3915)$ that was first observed by
Belle~\cite{Abe:2004zs} and confirmed by BaBar~\cite{Aubert:2007vj,delAmoSanchez:2010jr}
as a near-threshold peak in the
$\omegaJ/\psi$ invariant mass distribution in exclusive $B\rightarrow K\omegaJ/\psi$ decays (see
Fig.~\ref{fig:fig1}a). An
$\omegaJ/\psi$ mass peak with similar mass and width was seen in the two-photon fusion
process $\gamma\gamma\rightarrow\omegaJ/\psi$, again by both Belle~\cite{Uehara:2009tx} and
BaBar~\cite{Lees:2012xs} (see Fig.~\ref{fig:fig1}b); BaBar reported its $J^{PC}$ to be
$0^{++}$. The similar masses and widths of
the peaks seen in the two production modes suggest that these are being produced
a single state (i.e., the $X(3915)$). The Particle Data Group's (PDG) average values for
the mass and width measurements from both production channels are~\cite{Tanabashi:2018oca}:
\begin{eqnarray}
M(X(3915)) &=& 3918.4\pm 1.9~{\rm MeV}~~~{\rm and}~~~\Gamma(X(3915)) = 20.0 \pm 5.0~{\rm MeV},
\label{eqn:x3915-m}
\end{eqnarray}
and the product branching fraction for $X(3915)$ production in $B^+$ meson decays is
\begin{eqnarray}
{\mathcal B}(B^+\rightarrow K^+ X(3915))\times{\mathcal B}(X\rightarrow\omegaJ/\psi)&=&3.0\pm 0.9\times 10^{-5}.
\label{eqn:prodbf}
\end{eqnarray}
The measured $\gamma\gamma\rightarrow\omegaJ/\psi$ production rates are used to extract the
($J^{PC}$-dependent) widths:
\begin{eqnarray}
\Gamma_{\gamma\gamma}(X(3915))\times {\mathcal B}(X\rightarrow\omegaJ/\psi) &=&
54\pm 9\ {\rm eV}~(0^{++})~~{\rm or}~~11.4\pm 2.7~{\rm eV}~(2^{++}).
\label{eqn:gamee-wjpsi}
\end{eqnarray}
\begin{figure}[htb]
\begin{minipage}[t]{70mm} \includegraphics[height=0.75\textwidth,width=0.85\textwidth]{fig1a.pdf}
\end{minipage}
\begin{minipage}[t]{70mm}
\includegraphics[height=0.75\textwidth,width=0.80\textwidth]{fig1b.pdf}
\end{minipage}\hspace{\fill}
\caption{\footnotesize {\bf a)} The $\omegaJ/\psi$ invariant mass spectrum for $B\rightarrow K\omegaJ/\psi$
decays from {\it (top)} Belle~\cite{Abe:2004zs} and {\it (bottom)}
Babar~\cite{delAmoSanchez:2010jr}. The low mass peak in the BaBar data is attributed to
$X(3872)\rightarrow\omegaJ/\psi$ (see inset); the higher mass peak is the
$X(3915)\rightarrow\omegaJ/\psi$ signal. The Belle analysis did not consider the possible presence
of an $X(3872)\rightarrow\omegaJ/\psi$ signal. {\bf b)} The $\omegaJ/\psi$ mass spectrum for
$\gamma\gamma\rightarrow\omegaJ/\psi$ from {\it (top)} Belle~\cite{Uehara:2009tx} and {\it (bottom)}
Babar~\cite{Lees:2012xs}.
}
\label{fig:fig1}
\end{figure}
\section{The \boldmath{$X(3915)$} is not the \boldmath{$\chi_{c0}^{\prime}$} charmonium state?}
The Babar group's $J^{PC}$ determination was based on an analysis of angular
correlations amongst the final-state particles in their
$\gamma\gamma\rightarrow\omegaJ/\psi$ event sample~\cite{Lees:2012xs}. The important angles
for distinguishing $J=2^+$ from $J=0^{\pm}$
are $\theta_{\rm n}^*$, the angle between $\vec{\rm n}$, the normal to the $\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$
decay plane, and the $\gamma\gamma$ axis in the omega rest frame, and $\theta_{\rm ln}$, the angle
between $\vec{\rm n}$ and the direction of the $\ell^+$ from $J/\psi\rightarrow\ell^{+}\ell^{-}$ decay (see
Fig.~\ref{fig:babar-angles}a). Figure~\ref{fig:babar-angles}b shows the BaBar
$\cos\theta_{\rm n}^*$ distribution
together with the expectation for $J=0^{\pm}$ as a solid red line and $J=2^+$ as a dashed
blue curve. There is a strong $\chi^2$ penalty for the near-zero event likelihood near
$\cos\theta_{\rm n}^* = \pm 1$ for the $J=2^+$ hypothesis to fluctuate {\it upward} to the
observed levels of $\sim 8$ and $\sim 9$ events, and this is the main support BaBar's
$J=0$ conclusion. The $J=2$ hypothesis seems to fit the BaBar $\cos\theta_{\rm ln}$
distribution (see Fig.~\ref{fig:babar-angles}c) better than
that for $J=0$. But in this case, the likelihood of $\sim 6$ expected events near
$\cos\theta_{\rm ln}=\pm 1$ to fluctuate {\it downward} to the observed $\simeq 2$~events
is not so improbable. With $J=0$ established, the $0^+$ {\it vs.} $0^-$ discrimination
mostly relies on the angle $\theta_{\rm n}$,
which is the angle between the $\omega$'s flight path and $\vec{\rm n}$ in the $\omegaJ/\psi$
restframe. The BaBar $\cos\theta_{\rm n}$ distribution shown in
Fig.~\ref{fig:babar-angles}d favors $0^+$ over $0^-$, mostly because of the $\simeq 10$
events near $\cos\theta_{\rm n}=+1$, where the $0^-$ expectation is zero.
\begin{figure}[h]
\centering
\includegraphics[height=0.25\textwidth,width=0.95\textwidth]{babar-angles.pdf}
\caption{\footnotesize {\bf a)}~Directions used in the BaBar study of
$\gamma\gamma\rightarrow\omegaJ/\psi$,
where $\omega\rightarrow\pi^{+}\pi^{-}\pi^{0}$ and $J/\psi\rightarrow\ell^{+}\ell^{-}$. {\bf b)}~Comparison of the $\cos\theta_{\rm n}^*$
distribution with $J^{P}=0^{\pm}$ (solid red) and $2^+$ (dashed blue) expectations. {\bf c)}~The
corresponding plot for $\cos\theta_{\rm ln}$.
{\bf d)}~The $\cos\theta_{\rm n}$ distribution with expectations for $0^+$ in solid red
and $0^-$ in dashed blue. (From ref.~\cite{Lees:2012xs}.)
}
\label{fig:babar-angles}
\end{figure}
BaBar's $J^{PC}=0^{++}$ assignment led them to suggest it as a suitable candidate
for the $2^3P_0$ charmonium state, commonly known as the $\chi_{c0}^{\prime}$, and it was listed
as such in the 2014 PDG tables~\cite{Agashe:2014kda}. However, this assignment
had some problems and was challenged for a number of reasons~\cite{Guo:2012tv}:
the partial width for $X(3915)\rightarrow\omegaJ/\psi$, which would be an OZI-suppressed
decay mode for a charmonium state, was too large; the lack of evidence for
$X(3915)\rightarrowD\bar{D}$, which would be the dominant mode for the $\chi_{c0}^{\prime}$; and
the small, $\simeq 9$~MeV, mass splitting between the $\chi_{c0}(2P)$ and the
$X(3915)$, which is an order-of-magnitude lower than the smallest theoretical
estimates for $M_{\chi_{c0}(2P)}-M_{\chi_{c0}^{\prime}}$~\cite{Wang:2014voa,Olsen:2014maa}. This
assignment was finally put to rest in 2017 by Belle, when they reported the
observation of the $X^*(3860)$, a $D\bar{D}$ resonance with mass
$3862^{+47}_{-35}$~MeV in $e^{+}e^{-}\rtJ/\psiD\bar{D}$ annhilations with preferred
spin-parity of $0^{++}$~\cite{Chilikin:2017evr}. These properties, particularly
the strong $D\bar{D}$ decay mode, match well the
expectations for the $\chi_{c0}^{\prime}$, and the $X^*(3862)$ is clearly a much stronger
candidate for this state than the $X(3915)$.
\section{Is it the \boldmath{$\chi_{c0}(2P)$} charmonium state?}
The $\chi_{c0}(2P)$ was first spotted by Belle~\cite{Uehara:2005qd} and
subequently confirmed by BaBar~\cite{Aubert:2010ab} as a
prominent $M(D\bar{D})$ peak in the two-photon fusion process
$\gamma\gamma\rightarrow D\bar{D}$ that has a distinct $\sin^4\theta^*$ production angle
dependence that is characteristic of a $J=2$ state. The
mass and width~\cite{Tanabashi:2018oca}:
\begin{eqnarray}
M(\chi_{c2}^{\prime}) &=& 3927.2\pm 2.6~{\rm MeV}~~~{\rm and}~~~\Gamma(\chi_{c2}^{\prime}) = 24.0 \pm 6.0~{\rm MeV},
\label{eqn:c2p-mass}
\end{eqnarray}
are consistent with charmonium expectations for the $\chi_{c2}^{\prime}$ and there are no
reasons to question this assignment. The Belle (BaBar) $M(D\bar{D})$ and $dN/d|\cos\theta^*|$
distributions are shown in Fig.~\ref{fig:z3930}a (b). Belle and BaBar measurements of its
two-photon production rate are also in good agreement and are characterized by the product
\begin{eqnarray}
\Gamma_{\gamma\gamma}(\chi_{c0}(2P))\times {\mathcal B}(\chi_{c0}(2P)\rightarrow D\bar{D})
&=& 210\pm 40~{\rm eV}.
\label{eqn:c2p-ggwidth}
\end{eqnarray}
\begin{figure}[htb]
\centering
\includegraphics[height=0.2\textwidth,width=0.81\textwidth]{z3930.pdf}
\caption{\footnotesize {\bf a)} {\it left}: The $M(D \bar{D})$ distribution for
$\gamma\gamma\rightarrow D\bar{D}$. The open histogram the $D$ mass-sideband-determined
background. The solid (dashed) curve
shows results of a fit that includes (excludes) a $\chi_{c2}^{\prime}$ signal.
{\it right}:~$dN/d|\cos\theta^*|$ for peak-region events with a
solid (dashed) curve showing $J=2$ ($J=0$) expectations.
The histogram is the non-resonant contribution. (From ref.~\cite{Uehara:2005qd}.)
{\bf b)} Corresponding plots from BaBar~\cite{Aubert:2010ab}.
}
\label{fig:z3930}
\end{figure}
BaBar's $J^{PC}=0^{++}$ assignment for the $X(3915)$
was based on a comparison to a $2^{++}$ scenario that only considered a
helicity-2 component ($h_2$) and ignored the possibility of any helicity-0 contribution.
This assumption of ``helicity-2 dominance'' originate from a theoretical analysis
that found that in two-photon production of tensor mesons, the helicity-0
component $(h_0)$ is zero in the non-relativistic limit~\cite{Krammer:1977an}. The
authors of ref.~\cite{Zhou:2015uva} point out that in the case of charmonium,
the suppression of helicity-0 contributions only applies to mesons that are
100\% $c\bar{c}$, which is generally considered to be unlikely for charmonium
mesons with masses above the $2m_D$ open-charm threshold (see, e.g.,
ref.~\cite{Pennington:2007xr}).
This is important because if the $J^{PC}$ of the $X(3915)$ is $2^{++}$, the
mass peak identified with the $X(3915)$ could be conceivably be due to an
$\omegaJ/\psi$ decay mode of the $\chi_{c2}(2P)$ charmonium state.
The dashed lines in Fig.~\ref{fig:zhou-etal}a show the ref.~\cite{Zhou:2015uva}
comparison of the Belle $M(D\bar{D})$ and $|\cos\theta|$ with an $h_0\simeq 1.5 h_2$
mixture to represent the $X(3915)$. Figure~\ref{fig:zhou-etal}b) shows BaBar's
$\cos\theta_{\rm n}^*$ and $\cos\theta_{\rm ln}$ distributions with expectations
for $0^{++}$, and $2^{++}$ with $h=0$ \& $h=2$. With the inclusion of some $h=0$
contribution, the $\chi^2$ distinction between $0^{++}$ and $2^{++}$ angular distributions
is diminished and the authors conclude that the $X(3915)$ could be a $\chi_{c0}(2P)$ state
that contains a sizable non-$c\bar{c}$ component.
\begin{figure}[htb]
\centering
\includegraphics[height=0.2\textwidth,width=0.81\textwidth]{zhou-etal.pdf}
\caption{\footnotesize {\bf a)} Belle $M(D \bar{D})$ ({\it left}) and
$|\cos\theta^*|$ ({\it right}) distributions for
$\gamma\gamma\rightarrow D\bar{D}$ production. The solid (dashed) curves
show expectations for $h_0=0$ ($h_0=1.5 h_2$). {\bf b)} BaBar $\cos\theta_{\rm n}^*$
distribution ({\it left}) with a solid (dotted) curve showing expectations for $2^{++}$ with
$h=0$ ($h=2$); the dashed curve is for $0^{++}$. ({\it right}) The $\cos\theta_{\rm ln}$
distribution with a solid curve for $2^{++}$ with $h=0\ {\rm or}\ 2$, and a
dashed curve for $0^{++}$. (From ref.~\cite{Zhou:2015uva}.)
}
\label{fig:zhou-etal}
\end{figure}
\subsection{Other aspects of the \boldmath{$X(3915)=\chi_{c0}(2P)$} assignment}
In addition to violating helicity-2 dominance, which ref.~\cite{Zhou:2015uva}
claims may not be a problem, there are other concerns with the $X(3915)=\chi_{c0}(2P)$ assignment.
These are briefly discussed here.
\subsubsection{Mass and width differences}
Belle and BaBar measurements of the $\gamma\gamma\rightarrow\omegaJ/\psi$ mass peak,
$3915\pm 4$ and $3919\pm 3$~MeV, respectively, are both lower, by $\simeq 2\sigma$,
than their respective $\chi_{c0}(2P)\rightarrow D\bar{D}$ mass peak measurements, $3929\pm 5$ and
$3927\pm 3$~MeV. Since the measurements reference
well known masses -- $\omega$ and $J/\psi$ for the $X(3915)$ and
$D$-meson for the $\chi_{c0}(2P)$-- systematic effects
are small.
On the other hand, a recent LHCb report on the $M(D\bar{D})$ distribution for
inclusive $D$-meson pair production in high energy proton-proton collisions
included observation of a distinct peak in the $\chi_{c0}(2P)$ mass region, shown in
Fig.~\ref{fig:B2kchic2}a, with mass $M=3921.9\pm 0.6\pm 0.2$~MeV, $2\sigma$
below the $\chi_{c0}(2P)$ value listed in eqn.\ \ref{eqn:c2p-mass}~\cite{Aaij:2019evc}.
The reported width, $\Gamma=36.6\pm 1.9\pm 0.9$~MeV, is $2\sigma$ higher than the
eqn.\ \ref{eqn:c2p-mass} value. The LHCb group attributes this peak to the $\chi_{c0}(2P)$.
\begin{figure}[htb]
\centering
\includegraphics[height=0.225\textwidth,width=0.81\textwidth]{B2Kchic2.pdf}
\caption{\footnotesize {\bf a)} The $M(D^+D^-)$ distribution for inclusive
$D$-meson pair production at the LHCb. The peak at $3842$~MeV is the first
observation of the $\psi_3$, the $1^3D_3$ charmonium level. The broader
peak near $3920$~MeV is attributed by the LHCb group to the
$\chi_{c0}(2P)$~\cite{Aaij:2019evc}.
{\bf b)} The $M(\omegaJ/\psi)$ distribution for $e^{+}e^{-}\rightarrow Y(4220)\rightarrow\omegaJ/\psi$
events from BESIII. An $X(3872)\rightarrow\omegaJ/\psi$ signal is evident. Additional
peaks near 3925~MeV and 3960~MeV each have about $3\sigma$
significance~\cite{Ablikim:2019zio}.
{\bf c)} $B^+\rightarrow K^+\chi_{c1}$ and $K^+\chi_{c2}$ signals from the full Belle
data set~\cite{Bhardwaj:2011dj}.
}
\label{fig:B2kchic2}
\end{figure}
Figure~\ref{fig:B2kchic2}b shows recent BESIII $M(\omegaJ/\psi)$ results for
$e^{+}e^{-}\rightarrow Y(4220)\rightarrow\gamma\omegaJ/\psi$, where there is a
strong $X(3872)\rightarrow\omegaJ/\psi$ signal and $3\sigma$ ``evidence''
for two higher mass peaks~\cite{Ablikim:2019zio}. The fitted mass
of the middle peak is $M=3926.4\pm 2.5$~MeV, near the Belle and BaBar
results for $\chi_{c0}(2P)\rightarrow D\bar{D}$. Thus, the current situation with mass measurements
is inconclusive.
\subsubsection{A large OZI-violating {$\omegaJ/\psi$} decay width for a $[c\bar{c} ]$ meson}
With the $\Gamma_{\gamma\gamma}\times {\mathcal B}$
values listed in eqns.~\ref{eqn:gamee-wjpsi} and \ref{eqn:c2p-ggwidth},
the $\chi_{c0}(2P)$ assignment implies that
\begin{eqnarray}
\frac{{\mathcal B}(\chi_{c0}(2P)\rightarrow \omegaJ/\psi)}{{\mathcal B}( \chi_{c0}(2P)\rightarrow D\bar{D})}
&=& 0.05\pm 0.02,
\label{eqn:wjpsibf}
\end{eqnarray}
which is large for an
OZI-rule-violating decay of an above-open-charm-threshold charmonium state,
and more than an order-of-magnitude higher than the measured corresponding
ratio for $\psi''\rightarrow\pipiJ/\psi$ and $D\bar{D}$. If $\chi_{c0}(2P)\rightarrow D\bar{D}$ and
$D\bar{D}^*$ are the dominant decay modes and
$\Gamma_{\chi_{c0}(2P)}(D\bar{D}^*)\approx\Gamma_{\chi_{c0}(2P)}(D\bar{D})$ (as predicted
in ref.~\cite{Barnes:2005pb}), then
$\Gamma_{\chi_{c0}(2P)}(\omegaJ/\psi)>200$~keV (at the $\sim$90\% CL), and
much larger than any measured OZI-violating width for a charmonium state.
\subsubsection{${\mathcal B}(B\rightarrow K\chi_{c0}(2P))>>{\mathcal B}(B\rightarrow K\chi_{c2})$ ?}
In 2011,
with their full event sample accumulated over ten years, Belle
reported $\sim 3\sigma$ evidence for $B^+\rightarrow K^+\chi_{c2}$ based on the
$33\pm 11$~event signal shown in
Fig.~\ref{fig:B2kchic2}c~\cite{Bhardwaj:2011dj}.
The inferred branching fraction,
${\mathcal B}(B^+\rightarrow K^+\chi_{c2})=1.1\pm 0.4\times 10^{-5}$, is smaller that the {\it product}
branching fraction for $X(3915)\rightarrow\omegaJ/\psi$ production in $B^+$ meson decays
(eqn.\ \ref{eqn:prodbf}).
Since ${\mathcal B}(\chi_{c0}(2P)\rightarrow D\bar{D})$ cannot exceed unity, eqn.\ \ref{eqn:wjpsibf}
implies ${\mathcal B}(\chi_{c0}(2P)\rightarrow \omegaJ/\psi)<0.08$ (90\% CL). Thus, if the $X(3915)$
produced in $B\rightarrow K\omegaJ/\psi$ is the $\chi_{c0}(2P)$, the $B$-meson decay width to $K^+\chi_{c0}(2P)$
would be more than an order of magnitude larger than that to $ K^+\chi_{c2}$. This contradicts
theoretical expectations that $B\rightarrow K[c\bar{c} ]$ decay widths decrease with increasing
radial $[c\bar{c} ]$ quantum numbers~\cite{Bodwin:1992qr}.
Suppression of $B\rightarrow K\chi_{c2}^{(')}$ is not unexpected. The primary mechanism for
$B$-meson ($\bar{b}q$) decays to $K[c\bar{c}]$ final states is $\bar{b}\rightarrow \bar{c}$
plus a virtual $W^+$ that, in turn, materializes as $c\bar{s}$. The final-state
$c$- and $\bar{c}$-quark form the $[c\bar{c} ]$ state and the $\bar{s}$- and
``spectator'' $q$-quark form the $K$. This process is only allowed for
$J^{PC}=0^{-+}, 1^{--}\ {\rm and}\ 1^{++}$ $[c\bar{c} ]$ states, decays to $[c\bar{c}]$
states with other $J^{PC}$ values are higher-order and
expected to be ``factorization suppressed''~\cite{Beneke:1999br}. The Belle
results on $B\rightarrow K\chi_{c2}$ shown in Fig.~\ref{fig:B2kchic2}c demonstrate that for
$J^{PC}=2^{++}$ $[c\bar{c} ]$ states, factorization suppression is very effective:
${\mathcal B}(B\rightarrow K\chi_{c2} )< 0.04\times{\mathcal B}(B\rightarrow K\chi_{c1})$ (90\% CL).
\section{Summary and conclusions}
Despite its observation by different experiments in a variety of production
channels, the nature of the $X(3915)$ remains a mystery. If it is a nonstandard
$XYZ$ meson, it cannot be easily interpreted by any of the proposed models for these states.
For example: its mass is too low for a QCD-hybrid~\cite{Liu:2012ze}, and not near
an appropriate threshold for a molecular state or a cusp effect~\cite{Olsen:2018ikz};
the lack of evidence for a $\eta\eta_c$ decay mode~\cite{Vinokurova:2015txd} is
problematic for a diquark-diantiquark assignmment~\cite{Lebed:2016yvr}. Thus, if it is
an $XYZ$ meson, it is a very interesting one.
The sum total of existing
data on $\omegaJ/\psi$ and $D\bar{D}$ production in
the $\sim 3925$~MeV mass region {\it cannot} be explained as
being simply due to the $\chi_{c0}(2P)$ charmonium state. While a (tenuous)
case could be made that the near-3925~MeV mass peaks seen by the LHCb in
$pp\rightarrow D\bar{D} X$, Belle and BaBar in $\gamma\gamma\rightarrow\omegaJ/\psi$ \& $D\bar{D}$
and BESIII in $Y(4220)\rightarrow\gamma\omegaJ/\psi$ are all due to decays of the $\chi_{c0}(2P)$,
the existing evidence is not conclusive. Moreover, a very strong case can be made
{\it against} a $\chi_{c0}(2P)$ interpretation of the $\omegaJ/\psi$ peak seen in
$B\rightarrow K\omegaJ/\psi$ decays.
More refined mass and width measurements are needed, and reliable, separate $J^{PC}$
determinations for the $\omegaJ/\psi$ peaks produced via $\gamma\gamma$ fusion, radiative
$Y(4220)$ transitions, and $B$-meson decays that eschew the
helicity-2 dominance constraint are essential. The LHCb group has demonstrated that they can
isolate clean $B^+\rightarrow K^+\omegaJ/\psi$ signals with good efficiency~\cite{Andreassia:2014phd} and
I look forward to high-statistics results from them in the near future.
\section{Acknowledgements}
I congratulate Phi-to-Psi-2018 organizers for an interesting and provocative meeting. This work
is supported by the CAS President’s International Fellowship Initiative.
|
1,116,691,498,475 | arxiv | \section{Introduction}
\label{sec:intro}
The stellar initial mass function (IMF) is one of the most fundamental distribution functions of astrophysics. The number of stars that form in one event with stellar masses in the interval $m,m+dm$ is $dN=\xi(m)dm$, where $\xi(m)$ is the IMF. Star-formation theory predicts systematic variations of the IMF with changes of the physical conditions \citep[e.g.][]{ml96,l98,e04,tp05,blz07}. Nevertheless, the IMF has been found to be essentially invariant \citep{man98,maw01,k01,k02}, which is still a challenge for theoreticians.
\citet*[hereafter DMPP]{dmpp07} reported a surprising relation (see Fig. \ref{fig:DMPP}) between the slope $\alpha$ of the low-mass stellar mass-function $\left(dN/dm=m^{-\alpha}\right)$ of globular clusters (GCs) and their concentration parameter $c=\log\left(r_t/r_c\right)$, i.e. the logarithmic ratio of tidal- and core-radius. All high concentration clusters in their sample had a steep mass-function (MF), while low concentration clusters tended to have a flatter MF (smaller $\alpha$). The canonical IMF has a slope of $\alpha=+1.3$ and $\alpha=+2.3$ in the mass range $0.08\msun-0.5\msun$ and $0.5\msun-150\msun$ \citep*{k01,wk04}, respectively\footnote{\textit{Note:} DMPP defined their MF as $dN/dm=m^{+\alpha}$.}. The mechanism usually believed to be responsible for high cluster concentration is core-collapse driven by two-body relaxation and the same process causes low-mass stars to move to the outer cluster parts where they are removed by the external tidal field. So one would naively expect the relation to be the other way round, and this is indeed confirmed by the direct N-body simulations of initially non-mass-segregated clusters of \citet{bm03}.
\begin{figure}
\centering
\includegraphics[width=8.3cm]{fig1.eps}
\caption{Observed trend between MF index $\alpha$ and the concentration parameter $c$ (taken from fig.1 in DMPP). The dashed line is a fit to the distribution (Section \ref{sec:compobs}). The slope $\alpha$ is measured in the mass-range $0.3-0.8\msun$ and the canonical IMF has $\alpha\approx1.72$ over this mass-range.}
\label{fig:DMPP}
\end{figure}
Among other approaches to a solution discussed in DMPP, a variation of the IMF, e.g. depending on cluster density, could possibly solve this problem, if the present day concentration is primordial. This would for the first time be a strong observational evidence for a variable IMF. However, before adopting such an exciting solution, it is important to check for alternative solutions.
Many observations show very young clusters to be (partially) mass-segregated \citep[e.g. recent papers by][]{sabbi07,cdgz07,dib07}. The reaction of initially mass-segregated clusters to gas expulsion and the influence which this has for the stellar MF in a cluster has never been investigated but may be important for understanding the structure and stellar content of present day globular clusters. In this paper we study for the first time the influence of the gas expulsion process on the MFs of initially mass-segregated globular clusters (GCs) with an \textit{in}variable IMF using the recent results by \citet*[see Section \ref{sec:models}]{bk07} for the dynamical response of star clusters to gas expulsion.
The idea behind this is as follows: Star clusters form from massive, dense cores in giant molecular clouds. Stars form in these cores with a given star formation efficiency (SFE) reproducing the stellar IMF. The radiation of the newly born stars will immediately begin to disrupt their surrounding gas. Especially the winds of high-mass (O- and B-type) stars and the first supernovae will lead to ejection and disruption of the entire cloud. The stars have to react to the change in the potential and depending on the gas removal time-scale, the stars will or will not have time to react to this change, so by gas expulsion the final properties of the cluster will be affected. The cluster expands and thereby preferentially stars from the cluster periphery are lost. If mass-segregation is a primordial feature of star clusters, mostly the low-mass stars are lost due to gas expulsion. Thus, the low-mass stellar MF changes strongly, while the high-mass part is left unaffected. Also the concentration is affected by cluster formation parameters such as the SFE and the gas expulsion time. Intuitively we expect the most-disruptive cases of gas expulsion to lead to remnant clusters that have the lowest density and largest depletion of low-mass stars.
The paper is divided in three parts: In Section \ref{sec:models} we explain the models used and the procedure of our analysis. This is followed by a presentation of the results in Section \ref{sec:result} and a discussion \& conclusion in Section \ref{sec:concl}.
\section{The Models}
\label{sec:models}
\citet*[hereafter BK07]{bk07} carried out a large set of $N$-body integrations studying the effect of residual-gas expulsion on the survival rate and final properties of star clusters \citep*[cf.][]{lmd84,kah01}. They placed star clusters on a circular orbit around a spherical galactic potential and varied $(i)$ the star formation efficiency
\begin{equation}
\epsilon=\frac{M_{\rm ecl}}{M_{\rm ecl}+M_{\rm gas}}\,,
\end{equation}
where $M_{\rm ecl}$ is the stellar mass of the embedded cluster and $M_{\rm gas}$ is the mass of the residual gas within the cluster volume, $(ii)$ the ratio of the gas expulsion time to the crossing time of a cluster, $\tau_M/t_{\rm cross}$, and $(iii)$ the strength of the external tidal field in terms of the initial half-mass radius in units of the tidal-radius $r_h/r_t$. In their models, they assumed that the SFE does not depend on the position inside the cluster, so gas and stars followed the same density distribution initially, which was given by a Plummer model. The influence of the gas on the stars was modelled as a modification to the equation of motion of stars. Gas expulsion was assumed to start at a certain time $t_D$, which was set equal to one $N$-body time unit \citep{hm85}, equivalent to $1/\sqrt{8}$ of a crossing time at the clusters virial (= gravitational) radius \citep{bt87}. After the delay time $t_D$, the gas density was decreased exponentially on a characteristic time-scale $\tau_M$, the gas expulsion time-scale, so the total gas left at later times was given by \citep{kah01}
\begin{equation}
M_{gas}(t) = M_{gas}(0) \; e^{-(t-t_D)/\tau_M} \; \; .
\end{equation}
All calculations were performed with the collisional $N$-body code NBODY4 \citep{a99} on the GRAPE6 computers \citep{mfkn03} at the Argelander Institute. All clusters contained $20000$ equal-mass stars initially, distributed according to a Plummer sphere. On the one hand, the restriction to equal-mass stars is necessary as otherwise a grid of cluster mass would need to be computed for each of the above parameter combinations. This is presently not feasible. On the other hand, $20000$ stars give good statistics. The integration proceeded for $1000$ initial $N$-body times (equivalent to about $300$ initial crossing times).
The above procedure captures the essence of the physics and their results can be summarised briefly as follows: Both the star formation efficiency and the speed with which the gas is removed have a strong influence on the evolution of star clusters. In the case of instantaneous gas removal ($\tau_M\ll t_{\rm cross}$), clusters have to form with SFEs $\ge 33$ per cent in order to survive gas expulsion \citep{bk03a,bk03b}. This limit is significantly lowered for gas removal on longer time-scales and clusters with SFEs as low as $10$ per cent can survive gas expulsion in the adiabatic limit ($\tau_M\gg t_{\rm cross}$) if the external tidal field is weak. External tidal fields have a significant influence on the cluster evolution only if the ratio of $r_h/r_t$ is larger than about 0.05. Below this value, star clusters behave nearly as if they are isolated.
\subsection{Procedure \& Assumptions}
\label{sec:assum}
For each of the 20000 stars in the models of BK07 we selected a mass from the canonical IMF using a C-routine described in the appendix of \citet*{pk06}. This reproduces the steep MF of the high concentration clusters, which have values around $\alpha\approx1.5$ in the mass range $0.3\msun-0.8\msun$ (hereafter referred to as the low-mass part of the MF) considered by DMPP, if MFs of these clusters still resemble their IMFs\footnote{A power-law fit to the canonical IMF gives $\alpha\approx1.72$ over this mass-range.}. We further omitted the brown dwarf (BD) part of the IMF, i.e. masses below $0.08\msun$, since BDs have a negligible effect on the overall evolution of the cluster and they can't be detected in GCs.
We distributed stellar masses among the 20000 stars according to their initial total energy: The star with the smallest specific energy, i.e. the star that is strongest bound, gets the largest mass and the one with the largest energy is assigned the smallest mass. However, a segregation of 100 per cent as for this mass-assignment is most likeley exaggerated and at least the outer regions of a cluster wouldn't be expected to be totally mass-segregated, so our assumption probes one extreme. We also performed our analysis with initially \textit{not} segregated clusters, where the masses have been assigned randomly to the stars to probe the other extreme. Note again, that the models of BK07 have equal-mass stars, but as the mass in gas in most cases is much larger than the total mass in stars, the dynamical evolution is dominated by the gas expulsion process for the time-span over which the models were computed. Since BK07 included the effect of an external tidal field in the so-called 'near field approximation' \citep*{a85}, the specific energy of a star is given by
\begin{equation}
E_{\rm tot}=\frac{1}{2}\dot{\textbf{\textit{r}}}^2-\Phi\left(\textbf{\textit{r}}\right)-\frac{1}{2}\omega^2\left(3x^2\textbf{\textit{e}}_x+z^2\textbf{\textit{e}}_z\right)\;,
\end{equation}
where the first term is the specific kinetic energy of a star, $\Phi$ is the potential due to the residual-gas and the other stars in the cluster and the last term is a combination of centrifugal and tidal energy. The vector $\textbf{\textit{r}}=\left(x,y,z\right)$ is the position-vector of each individual star measured from the cluster centre, the $\textbf{\textit{e}}_i\;(i=x,\,z)$ are unit vectors and the angular velocity of the cluster around the spherical galactic potential,
\begin{equation}
\omega=\sqrt{\frac{GM_G}{R_G^3}}\;,
\end{equation}
is determined by the mass of the galaxy, $M_G$, and the galactocentric distance, $R_G$.
Recently analytic techniques became available to create mass-segregation in modeled clusters \citep*{gg08,skb08,bkdm08}. \citet{gg08} created a radius dependent 'segregated MF' for the high-mass stars while the low-mass part is radius independent. While our procedure has all massive stars but no low-mass stars in the centre, their formulation of mass-segregation mixes both low and high mass stars in the inner part. Both techniques have no high-mass stars in the cluster outskirts.
In order to determine the final concentration of the modelled clusters, the tidal- and core-radii are needed. In order to determine these, we first calculated the position of the cluster centre (projected centre in the case of the core-radius) using the method by \citet*{ch85}.
The tidal radius of the cluster was then determined iteratively, by first assuming that all stars still in the calculation are bound and calculating the tidal radius,
\begin{equation}
r_t=\sqrt[3]{\frac{G\,M_{\rm ecl}}{3M_G}}R_G\;,
\label{eq:tidalradius}
\end{equation}
at the end of the the $N$-body runs, when the residual-gas is expelled. In a second step we computed the mass of all stars inside $r_t$ and used it to obtain a new estimate for the tidal radius from equation (\ref{eq:tidalradius}). This was repeated until a stable solution was found.
To identify the core-radius, we placed radial annuli around the projected cluster-centre and searched for the radius at which the projected number-density of stars dropped to half its central value. This method is of course sensitive to the choice of the width of the rings. A too small width leads to a very noisy surface-density profile, too large widths result in inaccurate values for the core-radius. Also the lower the number of stars at the end of the computations the noisier the surface-density-profile gets. So we decided to increase the ring-width for clusters with less than 5000 (25 per cent) and 2000 (10 per cent) stars at the end of the $N$-body integrations, respectively, in order to compensate for that effect. This hardly changes the measured core-radii for smooth density-profiles, but improves them significantly for those with a not so stable profile. The central-density was determined by averaging over the surface-densities in the first two to five annuli (depending on the ring-width) to reduce the influence of outliers. We took the mean value of the measured core-radii from three different projections as the best approximation of the true core-radius in the models. Their standard-deviation was taken to be the error of $r_c$. Models which survived gas expulsion but kept less than five per cent of their stars have been left out of consideration because radii-determinations were too difficult.
With the tidal- and core-radius, the concentration and its error follow from
\begin{equation}
c=\log_{10}\left(\frac{r_t}{r_c}\right)
\end{equation}
and error propagation.
\begin{figure*}
\begin{center}
\includegraphics[width=17cm]{fig2.eps}
\end{center}
\caption{Example mass-spectra for our analysis. The plots show the number of stars, $N(m)$, per mass-interval normalised to the bin-width, $\Delta m$, per bin versus the mass $m$, both on logarithmic scales. The left and right panels, respectively, show the same models (see title), one without (upper) and one with unresolved binaries (lower). The fraction of stars remaining in the cluster is $f_{\rm st}$. Crosses indicate the IMF, open circles the resulting MF. The solid vertical lines limit the mass range $0.3-0.8\msun$. The dashed line is the fit to the final MF in this region. In mass-segregated clusters only the low mass-part of their MF is affected: The slope of the MF shifts to lower values for $\alpha$ if binaries are included in our analysis. Also note that the MFs with unresolved binaries are flatter for larger binary-fractions \citep*[cf.][]{kgt91}.}
\label{fig:massspectra}
\end{figure*}
Furthermore, at the end of each integration, we plotted the MF of the stars that lie within the final tidal radius following the binning-method described by \citet*[their experiment 3]{ma05} assuming a power-law behaviour in the interesting mass-range: We chose to distribute the masses in $2\times N^{2/5}$ bins \citep*[second recommendation of][]{ds86} with an equal number of stars in each bin\footnote{Binning was performed over the whole mass-range $0.08\ldots150\msun$.}. Afterwards we determined the slope $\alpha$ of the resulting MF in the mass range $0.3\msun-0.8\msun$ by applying a least-squares fit. Although the slope may change over this range (see upper right panel of Fig. \ref{fig:massspectra}), we assigned just one $\alpha$-value for each model. This may not be completely representative for the actual mass-distribution for some models, but the uncertainties of the best-fitting line give an idea of how strong the data deviates from an ideal power law. The errors (indicated in Fig. \ref{fig:comparison}) can be seen as the uncertainty in the determination of the slope.
\section{Results}
\label{sec:result}
\subsection{The effect of gas expulsion on the stellar MF}
\label{sec:MFgas}
\begin{figure*}
\begin{center}
\includegraphics[width=17cm]{fig3col.eps}
\end{center}
\caption{The slope $\alpha$ of the MF at the end of the $N$-body integrations in the mass-range $0.3\msun-0.8\msun$ as a function of the cluster concentration $c$. The upper panels and the lower left show the same data, but coded for a different parameter used in the set of models of BK07: \textit{top left:} SFE $\epsilon$; \textit{top right:} strength of the tidal field measured by the ratio of $r_h/r_t$; \textit{bottom left:} for the characteristic gas expulsion-time-scale, $\tau_M$, measured in crossing-times, $t_{\rm cross}$, of the initial cluster. The horizontal dashed line gives the slope of the canonical IMF in the mass-range $0.3\msun-0.8\msun$; all initial models lie on this line and have $1\lesssim c_{\rm init}\lesssim 2.5$. The lower right plot compares the position of the points in the c-alpha-plane with (open symbols and crosses) and without (filled dots) binaries in dependence of the chosen binary-fraction. For the sake of clarity, error bars are omitted in these plots. Uncertainties are indicated in Fig. \ref{fig:comparison}}
\label{fig:calpha}
\end{figure*}
Fig. \ref{fig:calpha} depicts the effect of gas expulsion on the slope of the MF in the low-mass part in dependence of the cluster final concentration. As can be seen, the MFs of most high-concentration clusters still resemble their IMFs and don't show a strong change of the slope in the low-mass part, while for clusters with low-$c$ several models do show a strong change in the slope. The exact position of a point in the diagram depends on the set of parameters for the corresponding model.
In the upper part of the diagrams (above $\alpha\approx1$) one finds the models which kept most of their stars. These are preferentially models with a gas expulsion time-scale $\gtrsim t_{\rm cross}$. Except for the lowest concentration clusters, most of them still lie on their IMF slope. Those models with a smaller $\tau_M$ therefore have an extremely high SFE (above 40 per cent) and experience a weak or intermediate tidal field only. Just three models with a gas expulsion time-scale larger or equal to the cluster's initial crossing time are found to be shallower than $\alpha\approx-1$ which is mainly due to their low SFE of $\lesssim15$ per cent. Below a concentration of $c\sim0.8$ the strong tidal fields in these models lift even the large-$\tau_M$ clusters with a large SFE away from the IMF slope.
In the left part of the diagrams ($c\lesssim1.4$) several models show a strongly flattened MF (below $\alpha\approx1$), which are predominantly the models with fast gas removal ($\tau_M\ll t_{\rm cross}$). These clusters experience intermediate or strong tidal-fields and have large SFEs ($\gtrsim33$ per cent), which is responsible for cluster survival.
The upper right panel reveals the dependency of the final concentration on the the initial tidal field strength. Starting almost exclusively with weak tidal fields at large concentrations, stronger tidal fields become successively more common as the concentration decreases. This is easy to understand: Due to general cluster expansion the tidal boundary of the cluster shrinks and the core radius grows, the final concentration can be expected to be lower for models in which the two radii are lying closer together initially, i.e. for the clusters in a stronger tidal field. Of course the final concentration is also affected by the gas expulsion time and the SFE, since they determine how large cluster expansion exactly is. That is, the models with different tidal field strength overlap slightly, but the overall $c$-value is established by the tidal field. Strong tidal fields at large galactocentric distances may have occurred for those GCs forming in the immediate vicinity of pre-Milky Way gaseous building blocks.
\begin{figure*}
\begin{center}
\includegraphics[width=17cm]{fig4col.eps}
\end{center}
\caption{Dependence of the slope $\alpha$ of the low-mass stellar MF (as in Fig. \ref{fig:calpha}) on the gas expulsion time $\tau_M$ for a fixed value of the SFE $\epsilon$ (as indicated in the plots). The horizontal dashed line shows the IMF slope.}
\label{fig:alphatgas}
\end{figure*}
The lower left panel in Fig. \ref{fig:calpha} suggests a systematic behaviour of the MF slope with the concentration for a given gas expulsion time-scale. This can explicitly be seen for $\tau_M\geq t_{\rm cross}$. The panels of Fig. \ref{fig:alphatgas} emphasize the strong dependence on this parameter for fixed SFEs. As the gas expulsion time decreases the MF becomes shallower. Furthermore, a stronger tidal-field leads to enhanced mass-loss and thus to a lower MF-slope. So the models with the strongest tidal-fields are the ones with the lowest value of $\alpha$ for a fixed gas expulsion time. The lower the SFE gets, the stronger is the effect of the gas expulsion time. For a high SFE, very strong deviations from the IMF slope start for the lowest gas expulsion times and the strongest tidal-fields first, migrating to larger $\tau_M$ and lower $r_h/r_t$ as the SFE decreases.
The influence of the exact set of parameters is so strong, that for gas expulsion times $\lesssim0.33\,t_{\rm cross}$ we even observe four strongly affected MFs in the high-concentration regime.
\subsection{Unresolved binaries}
\label{sec:unrbin}
An error occurring in any observational determination of MF slopes is due to unresolved binaries. \citet*{kgt91} and \citet{k01} showed that this leads to an additional apparent flattening of the MF. We tried to account for this fact in our analysis as follows: We randomly picked stars from the stellar mass grid and randomly chose a secondary mass for them from the canonical IMF (random pairing). A 'centre-of-mass' star is then assigned the sum of the two selected masses. These 'binaries' are still treated as single stars, i.e. they are 'unresolved', but they now have the combined mass of their components. So the companions will stay together throughout the simulation and stay bound to or leave the cluster as a stellar system, i.e. possible hardening effects or dissolution of binaries is not directly being considered. But it is included indirectly by assuming a variable binary-fraction in dependence of $c$.
The binary-fraction of a cluster is defined as
\begin{equation}
f_{\rm bin}=\frac{\rm number\;of\;binary\;stars}{\rm number\;of\;single\;stars+binary\;stars}\;.
\end{equation}
In order to account for a change of the binary-fraction of a cluster due to dynamical evolution, we vary $f_{\rm bin}$ for different models. In denser clusters close encounters between stars are more common than in less dense models \citep*{k95}. So dissolution of binary- and multiple-systems is more likeley in denser environments. Assuming that star clusters form with similar binary-fractions and that concentration is a measure of cluster density\footnote{Depending on the actual value of $r_t$ and $r_c$: in principle a cluster with larger $c$ can be less dense than a cluster with lower $c$.}, clusters with a high concentration after gas expulsion are expected to retain a smaller number of binaries than low-concentration clusters after sufficient time for dynamical evolution. As has been shown in Section \ref{sec:MFgas}, the concentration at the end of our $N$-body integrations approximately follows the strength of the tidal field. So we chose to distribute binary-fractions of $f_{\rm bin}=10\%,\;20\%,\;$\ldots$60\%$ according to models with tidal-field strength $r_h/r_t=0.01,\;0.033,\;\ldots0.2$, respectively. Thus, clusters experiencing a weaker tidal field are given a smaller binary-fraction and vice-versa.
\begin{figure*}
\begin{center}
\includegraphics[width=17cm]{fig5.eps}
\end{center}
\caption{\textit{Left panel:} The $c-\alpha$ plane for initially non-mass-segregated clusters with unresolved binaries. The dashed line is the relation proposed by DMPP (see Section \ref{sec:compobs}). Just a slight decrease of the slope can be seen due to enhanced binary-fractions in low-concentration clusters. \textit{Right panel:} resulting mass-spectrum (as in Fig. \ref{fig:massspectra}) for no mass-segregation assuming unresolved binaries for the strongly affected cluster in the mass-segregated case with parameters $\epsilon=50\%,\;\tau_M=0.33t_{\rm cross},\;r_h=0.15r_t$ (compare with lower right panel of Fig. \ref{fig:massspectra}). The shape of the MF stays more or less the same, i.e. all stars are equally affected by gas expulsion.}
\label{fig:masssegr}
\end{figure*}
A choice of $f_{\rm bin}$ independent from the concentration is not suitable to enhance the trend, because the strength of the shift depends on the binary-fraction. In principle one could explain the relation with unresolved binaries alone, if the true MFs of all clusters still resemble their IMF. But since GCs are old objects and should therefore be at least partially dynamically evolved, this is not expected for all of them (Section \ref{sec:intro}). For our modelling, gas expulsion is in any case needed to initiate the trend and to span the whole range in concentration.
After selecting the binaries and assigning masses we distributed these stellar masses to the stars as described in Section \ref{sec:assum} and re-ran our analysis.
As expected, unresolved binaries cause an additional shift of the MF slope. The effect can be seen in the mass-spectra of Fig. \ref{fig:massspectra} (comparison of upper and lower panels) and its influence on the $\alpha-c$-plane is shown in the lower right panel of Fig. \ref{fig:calpha}. The slope of the MF changes according to the chosen binary-fraction towards lower values for $\alpha$ when compared with the mass-spectra or the location of the points in the $\alpha-c$-plots, that consider gas expulsion only. The effect becomes stronger the larger the binary-fraction $f_{\rm bin}$ is. By our assumption, the strongest change in the slope of the MF is seen in the least concentrated final models, because these have been assigned the largest binary-fractions. All models with binaries are flatter than their IMF. The binary MF slopes differ from the non-binary MF slopes by $\Delta\alpha=\left|\alpha_{\rm bin}-\alpha_{\rm no-bin}\right|=0.005$ (one of the $f_{\rm bin}=10$ per cent models) to $\Delta\alpha=2.8$ ($f_{\rm bin}=60$ per cent) with an average change of $\Delta\alpha=0.5$.
\subsection{Not-mass-segregated models}
\label{sec:masssegr}
Fig. \ref{fig:masssegr} (left panel) shows that primordial non-mass-segregated clusters do not show a strong trend of the MF slope with cluster concentration. Just a slight decrease of $\alpha$ is seen as $c$ becomes smaller, which is due to the larger binary-fractions assigned for low $c$-values.
The reason for this behaviour can nicely be seen in the mass-spectrum shown in the same figure (right panel). The diagram displays the same model parameters as Fig. \ref{fig:massspectra} (lower right panel), but for the case of no mass-segregation. Here, the MF is just shifted along the y-axis while the slope hardly changes. Hence, stars of all stellar masses are equally removed from a non-mass-segregated cluster during expansion following gas expulsion leaving $\alpha$ unchanged.
\subsection{Comparison with observations}
\label{sec:compobs}
\begin{figure*}
\begin{center}
\includegraphics[width=17cm]{fig6col.eps}
\end{center}
\caption{Comparison of our results with the observational data by DMPP. The left panel shows the result considering gas expulsion only, the right one includes primordial binaries. The large open circles are the observed clusters, the small filled dots (with error bars) are from the $N$-body integrations. The solid line is an eye-ball fit from DMPP (eq. (\ref{eq:eyeball})); we added the dashed lines as limits to their observations (see text). The agreement is better for the models including the effect of unresolved binaries.}
\label{fig:comparison}
\end{figure*}
The panels of Fig. \ref{fig:comparison} compare the influence of gas expulsion on the MFs of mass-segregated clusters with and without unresolved binaries, with the DMPP data. Fitting by eye, DMPP proposed a relation of the form (solid line)
\begin{equation}
\alpha\left(c\right)=-\frac{2.3}{c}+2.5\,.
\label{eq:eyeball}
\end{equation}
In this Figure we added upper and lower limits (dashed lines) to their observational data
\begin{equation}
\alpha_u\left(c\right)=-\frac{1.2}{c}+2.5\;,\\
\alpha_l\left(c\right)=-\frac{3.5}{c}+2.5\,.
\end{equation}
Both, the models with and without unresolved binaries, agree with the general trend that less concentrated clusters show a shallower MF, but the agreement is much better for the models including unresolved binaries. Even high-$c$ clusters with a flattened MF are within or near the observational limits. The not mass-segregated models, on the other hand, aren't able to reproduce the observed trend (Fig. \ref{fig:masssegr}). Unfortunately we do not reach values around $c\sim2.5$ with our models, but we wouldn't expect significant deviations from the IMF there. Nevertheless, our data points suggest a slightly steeper run than that proposed by DMPP, which can be understood, because DMPP had reliable observational data for only 20 GCs and the relation was an eye-ball fit. In addition, the N-body models of BK07 did not include the effect of dynamical evolution of the clusters beyond the 300 initial crossing times (Section \ref{sec:models}), which will also change the MF slope because of low-mass star evaporation \citep*{bm03}. Dynamical evolution, however, also affects the concentration and the core-collapse process may produce concentrations as large as in the observed clusters with $c\sim2.5$. The difference between the models and observed clusters may also be explained by the exact number of unresolved binaries. How clusters with different initial stellar content and concentration evolve exactly after gas expulsion is still an open issue.
The low-concentration model clusters in the upper left corner of Fig. \ref{fig:comparison} with only one observed cluster, both with and without binaries, are expected to shift to lower slopes during dynamical evolution. Such low-$c$ clusters can indeed survive for a Hubble time and wander downwards in the $c-\alpha$-plane to slightly larger concentrations and join the strongly affected clusters before they dissolve, if these clusters start mass-segregated \citep*[in prep.]{bkdm08}. The low-$c$ clusters with a short gas expulsion time-scale, which expanded a lot and lost many low-mass stars, may be left in a collision-less state such that dynamical evolution doesn't affect the properties of these clusters much.
Although we find several models (including binaries) with MF slopes $-1\gtrsim\alpha\gtrsim-2$ in the lowest concentration regime, which follow the DMPP relation, they do not observe any clusters there. This may be understood because most of the computed clusters in this domain experience the strongest tidal fields and these models are unlikely to survive for a Hubble-time as they expand by an additional $\sim30$ per cent due to stellar evolution and will probably dissolve rather quickly \citep*{fh95,bm03}.
The four model clusters at $c\sim1.7$, which are more strongly depleted in low-mass stars than the other models in that part of the diagram are also further away from the observational data, although three of them are still within the observational limits set by us. This may be a selection effect, because we find just four out of 31 clusters above $c=1.4$ with a strong effect on the MF slope and DMPP observed just six clusters in total in that regime.
Note that we measured global MFs (GMF) from our integrations while in observations only local MF measurements are available, which are subsequently transformed to GMFs by assuming that the MF at the half-mass radius approaches closely the GMF \citep*{dmpp00,bm03}. We checked this in our N-body runs by looking at the local MF (LMF) of the stars near the half-mass radius, i.e. between the 40 and 60 per cent Lagrange radii, and found that we can reproduce the trend as well. The mean difference between LMFs and GMFs in our analysis is $\Delta\alpha=\left|\alpha_{\rm GMF}-\alpha_{\rm LMF}\right|\approx0.25$. This difference might be removed by dynamical evolution.
\section{Summary \& Conclusions}
\label{sec:concl}
We have for the first time studied the effect of gas expulsion with and without unresolved binaries on the stellar MF of initially mass-segregated clusters with an invariable (canonical) IMF.
Due to gas expulsion the cluster expands and stars in the cluster halo are more easily lost than stars near the centre. In a mass-segregated cluster this leads to the loss of the less massive stars causing a change of the low-mass MF. Our computations showed a flattening of the MFs in the low-mass part for less concentrated final clusters, but not for strongly concentrated clusters. This can explain the \citet*{dmpp07} data, while initially non-mass-segregated models don't produce the observed trend. Depending on the parameters of the models, the final concentration and the slope of the MF are determined. Primordial binaries induce an additional apparent shift in the MF slopes (compare panels in Fig. \ref{fig:comparison}), which lead to an even better agreement with the DMPP data. If the difference between theory and observation is due to unresolved binaries, adjusting the exact value of $f_{\rm bin}$ of the model clusters to best reproduce the observational data could lead to predictions of binary-fractions for the observed clusters.
The models can approximately be divided into two different parts: The models with a high concentration of which most still have their IMF, and clusters with a low concentration, where a few systems lost a large portion of their (preferentially) low-mass stars and thus show a significant deviation from their IMF slope.
Our models constitute a possible solution to the DMPP problem without invoking a variable IMF. This contribution emphasizes the importance of choosing physically realistic initial models for understanding cluster evolution: The ability of our models to reproduce the observed $c-\alpha$ trend strongly suggests that initial mass-segregation and gas expulsion play a critical role in star-cluster evolution and that we can't exclude the possibility of a universal IMF for all globular clusters.
However, the MF slope and concentration in our calculations are measured shortly after the residual-gas has been expelled from the cluster once the surviving clusters have re-virialized. Later dynamical evolution of initially mass-segregated clusters will additionally affect these two parameters. This topic is investigated in \citet*[in prep.]{bkdm08}.
If gas expulsion is the determining factor for the observed trend between $c$ and $\alpha$ then, by direct measurement of these parameters, observers may be able to constrain starting conditions of the clusters such as the SFE and the gas expulsion time-scale from Fig. \ref{fig:calpha}, unless the $c$-$\alpha$-plane is degenerate. From our results it is not clear if the slope of the MF is uniquely determined by just one set of parameters for a given concentration. A different combination of the three parameters may yield a similar MF slope (but also the concentration is affected by them). This is especially true for clusters with less affected MFs, i.e. large $\tau_M$ and SFE, but also in other parts of the diagram some data points accumulate. In the case of ambiguous results, at least another parameter would be needed to lift this degeneracy.
\bibliographystyle{aa}
|
1,116,691,498,476 | arxiv | \section{Introduction}
\label{sec:introduction}
It remains a great challenge to construct meta-stable de Sitter (dS) solutions in string theory. A useful framework is provided by $\mathcal{N}=1$ supergravity specified through the choice of a real K{\"a}hler potential $K$ and a holomorphic superpotential $W$. This is phenomenologically motivated, sufficiently restricted to provide non-trivial constraints, and general enough to allow for interesting solutions.
To obtain a dS solution, supersymmetry needs to be broken, either through explicit breaking, \emph{e.g.} by adding an anti-brane, or through spontaneous supersymmetry breaking when F-terms are turned on.
Many of the most studied models in the literature, including those inspired by string theory, include exponential contributions to the superpotential motivated by non-perturbative effects \cite{Kachru:2003aw}. These terms play a crucial role through their contribution to the stability of the models. This is the case in \cite{Kallosh:2014oja, Marsh:2014nla}, where attempts were made to find stable dS in a constructive way through spontaneous breaking of supersymmetry. The starting point for the constructions is either no-scale or supersymmetric Minkowski (Mkw).
In no-scale theories supersymmetry is partially broken through an F-term, and the vacuum energy guaranteed to be zero thanks to a translational invariance of the superpotential. The theory is then deformed so that the two or three flat directions at the Mkw point become massive, while the vacuum energy is lifted to a positive value. These theories have a natural separation between the scale of SUSY-breaking (as given by the gravitino mass and the value of the superpotential), and the possibly small lifting to the positive cosmological constant of the dS.
A possible disadvantage of this scenario is that the same separation of scales also occurs between the gravitino mass and the much smaller uplifted masses of the no-scale directions. A way to remedy this is to turn to supersymmetric Mkw. There, the gravitino mass is small from the start, and the challenge is to break SUSY in such a way that all scalar masses become larger than the gravitino mass, while the cosmological constant remains small.
The generic supersymmetric Mkw is fully stable and lacks flat directions. Unfortunately, as proven in \cite{Kallosh:2014oja}, any small deformation of the superpotential (or the K{\"a}hler potental) just yields another supersymmetric vacuum -- Mkw or AdS. Hence, one can never get a dS in this way. The proposed solution of \cite{Kallosh:2014oja} is to add a new field, referred to as a Polonyi field, to generate an additional, flat direction. The Mkw minimum can then be uplifted to a stable dS. Through appropriate choices of parameters, in \cite{Kallosh:2014oja} it is argued that all scalar masses can be made larger than the gravitino mass, and that the cosmological constant can be set at a still much smaller scale.
A drawback of the non-perturbative models is that the exact form of the corrections is not known in general, and it may become difficult to blindly trust the results. One is therefore motivated to try to do without such non-perturbative effects and generate dS solutions perturbatively. This has turned out to be remarkably complicated in simple string motivated examples, and various no-go results have been obtained in the context of geometric compactifications of type II theories \cite{Maldacena:2000mw,Hertzberg:2007wc,Caviezel:2008tf}.
The natural models to study are the so called STU-models obtained through compactifying type IIB, type IIA (or M-theory) on twisted tori with orientifolds \cite{Derendinger:2004jn,Dall'Agata:2005fm,Danielsson:2009ff,Dall'Agata:2009gv,Dibitetto:2011gm,Danielsson:2011au,Danielsson:2012et}. These models have an isotropic sector with three complex moduli ($S$, $T$ and $U$), while the full non-isotropic theory has $7$ complex moduli. Different theories are specified through various kinds of fluxes due to the metric and gauge fields living on the manifold. In order to find stable dS solution it has turned out to be crucial to turn on non-geometric fluxes, and explicit, successful examples can be found in refs~\cite{deCarlos:2009fq,Danielsson:2012by,Blaback:2013ht}.
The stable dS in \cite{deCarlos:2009fq,Danielsson:2012by,Blaback:2013ht} were all found to be close to Mkw vacua with no special properties. The aim of our work is to investigate the existence of examples close to either no-scale or fully supersymmetric Mkw. In this paper we will see that this is indeed the case.
In fact, we will be able to show that all vacua generated through the addition of non-perturbative terms can be captured through the presence of non-geometric fluxes. The key to this is the fact that all the relevant properties of the vacua, including the properties of the mass matrix only depend on up to third derivatives of the superpotential. As we will see, the number of real degrees of freedom in the complex superpotential up to third order derivatives is $32$, which coincides with the number of real fluxes in the most general duality invariant superpotential. From this point of view the non-perturbative terms do not introduce anything new beyond the polynomial form.
In case of no-scale, we will give examples of non-perturbative potentials admitting stable dS solutions approaching Mkw points with two as well as three massless directions. In both cases we provide the equivalent polynomial superpotential that gives the same values for the cosmological constant and the masses. We also provide the first example where one of the squared masses go through zero and changes sign at the no-scale point. The existence of such a possibility was argued for in \cite{Marsh:2014nla} but no explicit example was provided. It is crucial that our example is of a more general type than those accessible through the exponential.
In case of supersymmetric Mkw, our analysis more or less automatically selects the restricted class of points that have flat directions. Hence, we can find deformations that take us to dS vacua, and with some care these can be made stable. Our construction does not need the introduction of ad hoc Polonyi fields.
The paper is organised as follows. In section 2 we review STU-models arising from type IIB compactifications, and the duality-covariant form of their superpotential induced by generalised fluxes. In section 3 we outline a systematic way of solving the equations of motion, while keeping track of the stability of the theory close to Mkw or no-scale points. In section 4 we have collected all our explicit examples. Section 5 contains our conclusions and some outlook. We also provide two appendices that classify all supersymmetric Mkw points, as well as no-scale points, for the full polynomial superpotential.
\section{Type IIB compactifications with O$3$/O$7$-planes}
\label{sec:Z2xZ2}
Type IIB compactifications on $T^{6}/\left(\mathbb{Z}_{2}\,\times\,\mathbb{Z}_{2}\right)$ with O$3$/O$7$-planes and D$3$/D$7$-branes as well as all dual backgrounds, can be incorporated within a class of $\mathcal{N}=1$
supergravity theories \emph{a.k.a.} STU-models. Such 4D effective descriptions enjoy, in their isotropic incarnation, an $\textrm{SL}(2)^{3}$ global symmetry which can be interpreted as string duality
relating dual ten-dimensional backgrounds to each other.
Isotropic STU-models of this type arise from the coupling between the gravity multiplet and three chiral multiplets. These theories are then free of vectors and possess a scalar sector containing three
complex scalars denoted by $\Psi^{{\alpha}}\,\equiv\,\left(S,\,T,\,U\right)$ spanning the $\left(\frac{\textrm{SL}(2)}{\textrm{SO}(2)}\right)^{3}$ coset. Adopting the type IIB language, the $S$ modulus is the one
that contains the ten-dimensional dilaton, whereas the $T$ and $U$ moduli are interpreted as K\"ahler and complex structure moduli respectively.
The kinetic Lagrangian
\begin{equation}
\mathcal{L}_{\textrm{kin}} = \frac{\partial S\partial \overline{S}}{\left(-i(S-\overline{S})\right)^2} \,+ \, 3\,\frac{\partial T\partial \overline{T}̣}{\left(-i(T-\overline{T})\right)^2}\, + \,
3\,\frac{\partial U\partial \overline{U}}{\left(-i(U-\overline{U})\right)^2} \ ,
\end{equation}
can be derived from the following K\"ahler potential
\begin{equation}
\label{Kaehler_STU}
K\,=\,-\log\left(-i\,(S-\overline{S})\right)\,-\,3\,\log\left(-i\,(T-\overline{T})\right)\,-\,3\,\log\left(-i\,(U-\overline{U})\right)\ .
\end{equation}
Note that we are adopting a set of conventions where the imaginary parts of the complex fields are those ones carrying geometric meaning and hence they appear in the above K\"ahler potential and
need to be strictly positive. The real parts, on the contrary, just represent axionic scalars and have no sign restriction. As a consequence, our choice for the \emph{origin} of moduli space is given by
\begin{equation}
\label{origin}
S_{0} \ = \ T_{0} \ = \ U_{0} \ = \ i \ .
\end{equation}
In this paper we will analyse different mechanisms giving rise to scalar potentials for the $\left(S,\,T,\,U\right)$ moduli. Since there are no vector fields available, a potential cannot be induced by
means of a gauging procedure. However, some other massive deformations turn out to be consistent with minimal supersymmetry.
Such deformations are controlled by an arbitrary holomorphic function $W(S,T,U)$ called \emph{superpotential}, which induces a scalar potential through
\begin{equation}
\label{V_N=1}
V\,=\,e^{K}\left(-3\,|W|^{2}\,+\,K^{{\alpha}\bar{{\beta}}}\,D_{{\alpha}}W\,D_{\bar{{\beta}}}\overline{W}\right)\ ,
\end{equation}
where $K^{{\alpha}\bar{{\beta}}}$ is the inverse K\"ahler metric and $D$ denotes the K\"ahler-covariant derivative.
The value of $|W|$ at a given point in moduli space sets the scale of the gravitino mass, whereas the size of the F-terms $|DW|$, which are responsible for supersymmetry breaking sets the mass scale of the
spin-$\tfrac{1}{2}$ fermions.
There are mainly two distinct ways of inducing a holomorphic superpotential in the context of type II string compactifications: a \emph{perturbative} one and a \emph{non-perturbative} one.
The former includes the possibility of adding \emph{e.g.} gauge fluxes and metric flux. Within the latter class instead, there appear phenomena such as gaugino condensation in the open-string sector or
D-brane instanton effects.
In the context of isotropic STU-models, sets of stable dS critical points have been found both with pertubatively
\cite{deCarlos:2009fq,Danielsson:2012by,Blaback:2013ht} and non-pertubatively \cite{Danielsson:2013rza,Blaback:2013qza} induced superpotentials. These critical points always turn out to be organised into thin regions in parameter space attached to a line of Mkw solutions.
Following the strategy of searching for stable dS solutions around Mkw points, it becomes natural to study the possibility of
deforming special Mkw solutions, \emph{e.g.} of SUSY or no-scale type, into a stable dS vacuum.
This work is inspired by the line of investigation opened up in refs~\cite{Kallosh:2014oja,Marsh:2014nla}. There, the importance of approximate no-scale dS vacua in particular is discussed and a technical machinery is developed for constructing simple analytical families of such solutions. In the construction, the key ingredient seems to be the presence of a non-zero \emph{third} derivative of the superpotential w.r.t. the K\"ahler moduli. However, all explicit realisations shown there, seem to rely on the presence of
non-perturbative effects, whereas the possibility of finding examples within perturbatively-induced superpotentials of polynomial type
still remains to be verified.
The main goal of this paper will be that of testing STU-models with polynomial superpotentials induced by generalised fluxes when it comes to searching for dS solutions allowing for full analytical treatment. We will follow the approach of \cite{Kallosh:2014oja} in order to construct simple examples of stable dS vacua both close to SUSY Mkw and close to no-scale points. All of this will be achieved
within a string-inspired STU-model with a duality-covariant superpotential.
\subsection*{Perturbatively-induced superpotentials}
In the type IIB duality frame of our interest, gauge fluxes of $F_{(3)}$ \& $H_{(3)}$ type turn out to be allowed by a combination of the orbifold and orientifold involution, other RR gauge fluxes and metric flux in
completely projected out. This is what identifies the complete set of type IIB geometric fluxes, \emph{i.e.} deformations admitting a 10D understanding.
In the following subsections we will see that superpotentials which are purely induced by gauge fluxes are unable to generate a dependence on the K\"ahler moduli.
In order to obtain such a dependence in this context, one can add generalised fluxes obtained by acting on the geometric ones with a duality chain.
\subsubsection*{Models with only gauge fluxes}
Type IIB compactifications with only $F_{(3)}$ \& $H_{(3)}$ fluxes were originally studied in ref.~\cite{Giddings:2001yu}. These backgrounds are supported by the presence of O$3$-planes and D$3$-branes in
flat geometry. The corresponding flux-induced superpotential reads \cite{Gukov:1999ya}
\begin{equation}
\label{W_GKP}
W_{\textrm{GKP}}\,=\underbrace{\,a_{0}\,-\,3a_{1}\,U\,+\,3a_{2}U^{2}\,-\,a_{3}\,U^{3}\,}_{F_{(3)} \textrm{ flux}}\,-\,\underbrace{\,S\,\left(b_{0}\,-\,3b_{1}\,U\,+\,3b_{2}U^{2}\,-\,b_{3}\,U^{3}\right)\,}_{H_{(3)} \textrm{ flux}}\ ,
\end{equation}
where all superpotential couplings are consistently chosen to be real thanks to our conventions \eqref{origin} for the origin of moduli space.
The $\mathcal{N}=1$ supergravity defined by the above superpotential has a so-called \emph{no-scale} symmetry due to the absence of the K\"ahler modulus $T$. This implies that its real and imaginary parts respectively
appear as a completely flat and a \emph{run-away} direction in the scalar potential. The latter in turn implies that the only maximally symmetric solutions that these models can have must be Minkowski.
\subsubsection*{Models with generalised fluxes}
Starting from a geometric STU-model, one can start acting with $\textrm{SL}(2)^{3}$ transformations to obtain dual models. In this way, it becomes natural to conjecture the existence of a completely
duality-covariant superpotential \cite{Shelton:2005cf} containing all possible STU-terms up to linear in $S$ and up to cubic in $T$ \& $U$.
We will now summarise here the correspondence between generalised isotropic fluxes and superpotential couplings appearing in the $\mathcal{N}=1$ effective 4D description.
The complete generalised flux-induced superpotential can be written as
\begin{equation}
\label{W_all_fluxes}
W_{\textrm{pert.}} = (P_{F} - P_{H} \, S ) + 3 \, T \, (P_{Q} - P_{P} \, S ) + 3 \, T^2 \, (P_{Q'} - P_{P'} \, S ) + T^3 \, (P_{F'} - P_{H'} \, S ) \ ,
\end{equation}
where\footnote{\label{c1_tilde}Please note that, in principle, the truncation to the isotropic sector gives rise to $32+8=40$ fluxes, where all the fluxes transforming in the mixed symmetry representations of $\textrm{GL}(6)$ (\emph{i.e.} $Q$, $P$ and their primed counterparts) have in fact two fluxes $(c_{1},\tilde{c}_{1})$ etc. giving rise to one single coupling $(2c_{1}-\tilde{c}_{1})$ etc., so without loss of generality, we set throughout the text $\tilde{c}_{1}=c_{1}$ etc..}
the couplings in
\begin{equation}
\begin{array}{lcll}
\label{Poly_unprim}
P_{F} = a_0 - 3 \, a_1 \, U + 3 \, a_2 \, U^2 - a_3 \, U^3 & \hspace{5mm},\hspace{5mm} & P_{H} = b_0 - 3 \, b_1 \, U + 3 \, b_2 \, U^2 - b_3 \, U^3 & , \\[2mm]
P_{Q} = c_0 + c_{1} \, U - c_{2} \, U^2 - c_3 \, U^3 & \hspace{5mm},\hspace{5mm} & P_{P} = d_0 + d_{1} \, U - d_{2} \, U^2 - d_3 \, U^3 & ,
\end{array}
\end{equation}
are introduced and explained in table~\ref{table:unprimed_fluxes}, whereas the details of the couplings in
\begin{equation}
\begin{array}{lcll}
\label{Poly_prim}
P_{F'} = a_3' + 3 \, a_2' \, U + 3 \, a_1' \, U^2 + a_0' \, U^3 & \hspace{3mm},\hspace{3mm} & P_{H'} = b_3' + 3 \, b_2' \, U + 3 \, b_1' \, U^2 + b_0' \, U^3 & , \\[2mm]
P_{Q'} = -c_3' + c'_{2} \, U + c'_{1} \, U^2 - c_0' \, U^3 & \hspace{3mm},\hspace{3mm} & P_{P'} = -d_3' + d'_{2} \, U + d'_{1} \, U^2 - d_0' \, U^3 & ,
\end{array}
\end{equation}
are given in table~\ref{table:primed_fluxes}.
The first half of the terms (see table~\ref{table:unprimed_fluxes}) are characterised by lower powers in $T$, \emph{i.e.} up to linear, and represent fluxes which admit a locally geometric interpretation in type IIB (unprimed fluxes).
The remaining ones (primed fluxes) instead, appear with quadratic and cubic behaviour in $T$ and represent additional generalised fluxes which do not even admit a locally geometric description. These were first formally introduced in ref.~\cite{Aldazabal:2006up} as dual counterparts of the unprimed fluxes.
\begin{table}[h!]
\renewcommand{\arraystretch}{1.25}
\begin{center}
\scalebox{0.92}[0.92]{
\begin{tabular}{ | c || c | c |}
\hline
couplings & Type IIB & fluxes \\
\hline
\hline
$1 $& $ {F}_{ ijk} $& $ a_0 $ \\
\hline
$U $& ${F}_{ ij c} $& $ a_1 $ \\
\hline
$U^2 $& ${F}_{i b c} $& $ a_2 $\\
\hline
$U^3 $& ${F}_{a b c} $& $ a_3 $ \\
\hline
\hline
$S $& $ {H}_{ijk} $& $ -b_0$ \\
\hline
$S \, U $& ${H}_{ij c} $& $ -b_1 $ \\
\hline
$S \, U^2 $& ${H}_{ i b c}$ & $ -b_2 $ \\
\hline
$S \, U^3 $& $ {H}_{a b c} $& $ -b_3 $ \\
\hline
\hline
$T $& $ {Q_k}^{a b} $& $ c_0 $ \\
\hline
$T \, U $& $ {Q_k} ^{a j} = {Q_k}^{i b} \,\,\,,\,\,\, {Q_a}^{b c} $& $c_1 \,\,\,,\,\,\, \tilde {c}_1 $ \\
\hline
$T \, U^2 $& $ {Q_c}^{ib} = {Q_c}^{a j} \,\,\,,\,\,\, {Q_k}^{ij} $ & $c_2 \,\,\,,\,\,\,\tilde{c}_2 $ \\
\hline
$T \, U^3 $& $ {Q_{c}}^{ij} $& $c_3 $ \\
\hline
\hline
$S \, T $& $ {P_k}^{a b}$ & $ -d_0 $ \\
\hline
$S \, T \, U $& $ {P_k}^{a j} = {P_k}^{i b} \,\,\,,\,\,\, {P_a}^{b c} $& $-d_1 \,\,\,,\,\,\, -\tilde{d}_1 $ \\
\hline
$S \, T \, U^2 $& $ {P_c}^{ib}= {P_c}^{a j} \,\,\,,\,\,\, {P_k}^{ij} $& $-d_2 \,\,\,,\,\,\,-\tilde{d}_2 $\\
\hline
$S \, T \, U^3 $& $ {P_{c}}^{ij} $& $-d_3 $ \\
\hline
\end{tabular}
}
\end{center}
\caption{{\it Mapping between unprimed fluxes and couplings in the superpotential in type IIB with O3 and O7. The six internal directions are split into $\,``-"$ labelled by $i=1,3,5$ and $\,``\,|\,"$ labelled by $a=2,4,6$.}}
\label{table:unprimed_fluxes}
\end{table}
\begin{table}[h!]
\renewcommand{\arraystretch}{1.25}
\begin{center}
\scalebox{0.92}[0.92]{
\begin{tabular}{ | c || c | c |}
\hline
couplings & Type IIB & fluxes \\
\hline
\hline
$T^3 \, U^3 $& $ {F'}^{ijk} $& $ a_0' $ \\
\hline
$T^3 \, U^2 $& ${F'}^{ ij c} $& $ a_1' $ \\
\hline
$T^3 \, U $& ${F'}^{i b c} $& $ a_2' $ \\
\hline
$ T^3 $& ${F'}^{a b c} $& $ a_3' $ \\
\hline
\hline
$S \, T^3 \, U^3 $& $ {H'}^{ ijk} $& $ -b_0'$ \\
\hline
$S \, T^3 \, U^2 $& $ {H'}^{i jc} $& $ - b_1' $ \\
\hline
$S \, T^3 \, U $& $ {H'}^{ i b c} $& $ -b_2' $ \\
\hline
$S \, T^3 $& $ {H'}^{a b c} $& $ -b_3' $ \\
\hline
\hline
$T^2 \, U^3 $& $ {{Q'}_{a b}}^k $& $ c_0' $\\
\hline
$T^2 \, U^2 $& $ {{Q'}_{a j}}^k = {{Q'}_{i b}}^k \,\,\,,\,\,\, {{Q'}_{b c}}^a $& $c_1' \,\,\,,\,\,\, \tilde{c}_1' $\\
\hline
$T^2 \, U $& $ {{Q'}_{ib}}^c = {{Q'}_{a j}}^c \,\,\,,\,\,\, {{Q'}_{ij}}^k $& $c_2' \,\,\,,\,\,\,\tilde{c}_2' $ \\
\hline
$T^2 $& $ {{Q'}_{ij}}^{c} $ &$c_3' $ \\
\hline
\hline
$S \, T^2 \, U^3$& $ {{P'}_{a b}}^k $ &$ -d_0' $ \\
\hline
$S \, T^2 \, U^2 $& $ {{P'}_{a j}}^k = {{P'}_{i b}}^k \,\,\,,\,\,\, {{P'}_{b c}}^a $ & $-d_1' \,\,\,,\,\,\, -\tilde{d}_1' $ \\
\hline
$S \, T^2 \, U $& $ {{P'}_{ib}}^c = {{P'}_{a j}}^c \,\,\,,\,\,\, {{P'}_{ij}}^k $& $-d_2' \,\,\,,\,\,\,-\tilde{d}_2' $ \\
\hline
$S \, T^2 $& $ {{P'}_{ij}}^{c} $& $-d_3' $\\
\hline
\end{tabular}
}
\end{center}
\caption{{\it Mapping between primed fluxes and couplings in the superpotential. The conventions are the same as in table~\protect\ref{table:unprimed_fluxes}.}}
\label{table:primed_fluxes}
\end{table}
\subsection*{Non-perturbatively-induced superpotentials}
Starting again from the geometric superpotential \eqref{W_GKP} induced by gauge fluxes, one could instead introduce non-perturbative effects in order to generate a $T$-dependence in the superpotential,
which is typically exponential. Following the idea of \cite{Saltman:2004sn}, these could be used in order to further fix $T$ in perturbatively-constructed dS solutions where it appears as a \emph{run-away} direction.
Such non-perturbative effects can thus be seen as an alternative way of breaking the no-scale symmetry of \eqref{W_GKP} w.r.t. generalised fluxes.
As mentioned earlier, amongst possible mechanisms to generate this type of superpotentials, we find the phenomenon of gaugino condensation within the gauge theory sector living on D$7$-branes or D-brane
instanton effects.
In particular, the line of including gaugino condensation \cite{Font:1990nt} has been considered in the literature as a possible mechanism to further stabilise the K\"ahler moduli (see \emph{e.g.}
refs~\cite{Witten:1996bn, Achucarro:2006zf}).
A fairly generic prototype of non-perturbatively-induced superpotential can be written as
\begin{equation}
\label{W_nonpert}
W_{\textrm{non-pert.}} = \underbrace{\,(P_{F} - P_{H} \, S )\,}_{W_{\textrm{GKP}}} \, + \, P_{Z} \, e^{i\,{\alpha} T} \ ,
\end{equation}
where $P_{Z}$ is a priori an arbitrary holomorphic function of $S$ \& $U$ and ${\alpha}>0$ is some characteristic constant that depends on the physics of the explicit non-perturbative phenomenon in question
\footnote{In the case of gaugino condensation, a rough estimation for ${\alpha}$ is $\frac{2\pi}{N}$, $N$ being the rank of the corresponding $\textrm{SU}(N)$ gauge group.}.
Unfortunately, not much is known about the explicit form of the function $P_{Z}(S,U)$ since it would be extremely difficult to compute from stringy principles. In the construction of
ref.~\cite{Kachru:2003aw}, an argument was presented that would allow one to ignore such an $(S,U)$-dependence in $P_{Z}$, thus reducing the prefactor in front of the exponential to a constant, at least
in the large volume regime.
\section{Solving the field equations systematically}
\label{sec:method}
In order to solve the field equations for the six real scalars in the STU-model of our interest and find maximally symmetric solutions, we combine two crucial observations that transform the extremality
conditions for the scalar potential \eqref{V_N=1} into an algebraic system of linear equations in the superpotential couplings.
The first fact that we use is that a general non-compact $\textrm{SL}(2)^{3}$ duality transformation takes any point in moduli space to the origin \eqref{origin}. This statement holds for any supergravity
model in which the scalars span a homogenous space, just as \emph{e.g.}, $\left(\frac{\textrm{SL}(2)}{\textrm{SO}(2)}\right)^{3}$. Due to the general form of the scalar potential, the formulation of its
extremality conditions in the origin is just given by a set of six algebraic quadratic equations in the superpotential couplings.
Not only does this simplify the problem significantly, but such a restricted search for critical points can even turn out to be exhaustive if one moreover includes a set of superpotential terms which
happens to be closed under non-compact duality transformations \cite{Dibitetto:2011gm}. In our case, for a pertubatively-induced superpotential, this is in particular true if one keeps the complete
duality-invariant superpotential given in \eqref{W_all_fluxes}.
It may be worth mentioning that, for the case of non-pertubatively induced superpotentials of the type in \eqref{W_nonpert}, such a search for solutions will generically imply a loss of generality.
This is specifically related to the fact that the $\textrm{SL}(2)_{T}$ symmetry appears to be broken by the presence of the exponential term \eqref{W_nonpert}, unless a non-trivial compensating transformation
under $\textrm{SL}(2)_{T}$ is allowed for the constant ${\alpha}$ appearing there.
The second prescription that we make use of, is that of treating the derivatives of the superpotential w.r.t. the three complex fields
as independent quantities. This line of investigation was first pursued in ref.~\cite{Danielsson:2012by} where the so-called \emph{SUSY-breaking parameters} were introduced. By looking at the form of the F-terms in the origin of moduli space,
\begin{equation}
F_{{\alpha}} \ \equiv \ \left. D_{{\alpha}}W \right|_{\textrm{origin}} \ = \ \textrm{constants} \ ,
\end{equation}
one can reexpress six of the fluxes as functions of the above constants. Upon doing so, one can use the remaining fluxes to easily solve the field equations, since they can appear there at most linearly.
However, it was later noted in ref.~\cite{Kallosh:2014oja} that such a method is nothing but an overconstrained case of the more general situation where all derivatives of the superpotential evaluated in the origin up to third order take part in a various-step-procedure. Specifically, if one starts out with the following arbitrary cubic superpotential\footnote{Such a form of the superpotential might even be supported by geometric arguments, like \emph{e.g.} those in ref.~\cite{Catino:2013ppa}, where a cubic term of this form was proposed in order to reproduce truncations of some particular gaugings in maximal supergravity describing relevant M-theory compactifications including the $7$-sphere.}
\begin{equation}
\label{W_Phi}
W\left(\Phi^{{\alpha}}\right) \ = \ W_{0} \ + \ W_{{\alpha}} \, \Phi^{{\alpha}} \ + \ \frac{1}{2!} \, W_{{\alpha}{\beta}} \, \Phi^{{\alpha}}\Phi^{{\beta}} \ + \ \frac{1}{3!} \, W_{{\alpha}{\beta}{\gamma}} \, \Phi^{{\alpha}}\Phi^{{\beta}}\Phi^{{\gamma}} \ ,
\end{equation}
where $\Phi^{{\alpha}} \ \equiv \ \left(S-i,\,T-i,\,U-i\right)$ and all $W$ derivatives appear as arbitrary complex numbers, one can:
\begin{itemize}
\item Choose $W_{0}$ in order to fix the gravitino mass scale,
\item Choose $W_{{\alpha}}$ in order to fix the SUSY-breaking scale,
\item Fix part of the $W_{{\alpha}{\beta}}$'s in order to solve the field equations (where they only appear \emph{linearly}), while suitably choosing the remaining ones,
\item Fix $W_{{\alpha}{\beta}{\gamma}}$ such that the mass matrix (where, as well, they only appear \emph{linearly}) be positive definite.
\end{itemize}
The mass matrix for the six real moduli fields at a critical point has the following general form
\begin{equation}
{\left(m^{2}\right)^{I}}_{J} \ = \ \left(
\begin{array}{cc}
K^{{\alpha}\bar{{\gamma}}}\,D_{\bar{{\gamma}}}D_{{\beta}}V & K^{{\alpha}\bar{{\gamma}}}\,D_{\bar{{\gamma}}}D_{\bar{{\beta}}}V \\
K^{\bar{{\alpha}}{\gamma}}\,D_{{\gamma}}D_{{\beta}}V & K^{\bar{{\alpha}}{\gamma}}\,D_{{\gamma}}D_{\bar{{\beta}}}V
\end{array}\right) \ = \ \left(
\begin{array}{cc}
K^{{\alpha}\bar{{\gamma}}}\,V_{\bar{{\gamma}}{\beta}} & K^{{\alpha}\bar{{\gamma}}}\,V_{\bar{{\gamma}}\bar{{\beta}}} \\
K^{\bar{{\alpha}}{\gamma}}\,V_{{\gamma}{\beta}} & K^{\bar{{\alpha}}{\gamma}}\,V_{{\gamma}\bar{{\beta}}}
\end{array}\right) \ ,
\end{equation}
where $V_{{\alpha}{\beta}}$ etc. just denote ordinary derivatives of $V$ w.r.t. the complex fields $\Psi^{{\alpha}}$. In order to significantly simplify the problem of getting all positive eigenvalues and hence construct
proper minima of the potential, it might be very useful in some cases to observe \cite{Kallosh:2014oja} that the mass matrix becomes block-diagonal upon imposing $V_{{\alpha}{\beta}}\,=\,0$. Such a condition, which
can be easily solved in terms of the $W_{{\alpha}{\beta}{\gamma}}$, also turns out to enforce a pairwise organisation of the mass spectrum, which will in such cases only possess three distinct eigenvalues.
The positivity of the cosmological constant, instead, is here controlled by the sign of the following quadratic combination
\begin{equation}
V_{0} \ \propto \ -3\,\left|W_{0}\right|^2 \ + \ \left|W_{{\alpha}} \, + \, K_{{\alpha}}W_{0}\right|^2 \ ,
\end{equation}
which means that such a procedure can systematically produce analytical stable dS solutions provided that all the different complex coefficients in the \eqref{W_Phi} can be fixed independently.
Due to the form of the K\"ahler potential \eqref{Kaehler_STU}, we should restrict to those complex coefficients that do not generate
superpotential terms with degree higher than one w.r.t. $S$, \emph{i.e.} $W_{SS}\,=\,W_{SSS}\,=\,W_{SST}\,=\,W_{SSU}\overset{!}{=}\,0$.
This implies that the most general parametrisation of the form \eqref{W_Phi} compatible with the class of STU-models presented in section~\ref{sec:Z2xZ2} counts $16$ complex parameters, of which $1$ is given by $W_{0}$, $3$ by $W_{{\alpha}}$, $5$ by $W_{{\alpha}{\beta}}$ and the remaining $7$ by $W_{{\alpha}{\beta}{\gamma}}$.
The final crucial observation is now that such a parameter space is in \emph{one-to-one} correspondence with the complete set of $32$ real superpotential couplings collected in tables~\ref{table:unprimed_fluxes} and \ref{table:primed_fluxes} describing the most general duality-invariant superpotential for our STU-model. The mapping relating generalised fluxes to complex superpotential derivatives is, in fact, linear and invertible. This in particular implies that, whenever a stable dS solution is found for a certain superpotential derivative configuration, this will always admit an STU-realisation in terms of $32$ generalised perturbative fluxes.
\subsection*{Stable dS vacua close to SUSY Mkw points}
Let us now apply the above prescription in order to find interesting dS vacua obtained by starting from a SUSY Mkw solution and subsequently breaking SUSY by small amounts. The details concerning the most general supersymmetric Mkw solution within our STU-model can be found in appendix~\ref{app:SUSY_Mkw}.
In order to apply the above formalism, we first need to understand our Mkw points in the language of $W$ derivatives. These are identified by the following choice:
\begin{equation} \label{NoScalePointConditions}
\begin{array}{lcclclclcc}
W_{0} \, = \, 0 & , & & \textrm{and} & & W_{{\alpha}} \, = \, 0 & , & {\alpha}\,=\,S,\,T,\,U & & ,
\end{array}
\end{equation}
whereas all the other $12$ complex higher derivatives stay completely arbitrary. One can check that the above choice already automatically implies the field equations and guarantees the
(semi-)positiveness of the mass matrix.
Now we need to construct a consistent \emph{Ansatz} to break SUSY by small amounts and hence move away from the SUSY Mkw critical point. To this end, we introduce a small parameter $\epsilon$, \emph{i.e.}
such that $|\epsilon|\,\ll\,1$, that can be taken to zero whenever one wants to go back to the supersymmetric situation. Here we can make use of the no-go theorem in ref.~\cite{Kallosh:2014oja}, which guarantees that our construction automatically selects points close to a SUSY Mkw with two real massless directions. Without flat directions the theorem only allows for deformations into other SUSY points, and any dS critical point is excluded.
Such a deformation \emph{Ansatz} reads
\begin{equation}
\label{Close2SUSY}
\begin{array}{lcclclcl}
W_{0} \, = \, \kappa_{0} \, \epsilon & , & & \textrm{and} & & W_{{\alpha}} \, = \, \kappa_{{\alpha}} \, \epsilon & ,
\end{array}
\end{equation}
where $\left(\kappa_{0},\,\kappa_{S},\,\kappa_{T},\,\kappa_{U}\right)$ are arbitrary complex numbers of $\mathcal{O}(1)$. By plugging the \eqref{Close2SUSY} into the field equations, the second
derivatives of $W$ can only appear there linearly. Hence, one can use six out of the ten real independent parameters in $W_{{\alpha}{\beta}}$ in order to find extrema of the potential. The other four of them can
be arbitrarily chosen.
As far as the mass matrix is concerned, the remarkable feature of these dS points close to a SUSY Mkw with two flat directions is that the light directions always coincide with the two real sGoldstini
\begin{equation}
g_{{\alpha}} \ \equiv \ \frac{F_{{\alpha}}}{\sqrt{F_{{\beta}}\,\overline{F}^{{\beta}}}} \ ,
\end{equation}
\emph{i.e.} the superpartners of the Goldstone fermions responsible for the SUSY-breaking mechanism. This implies that these are the only directions that can turn into tachyons in the small $\epsilon$
limit.
At this point it is useful to impose the pairwise degeneracy condition for the mass spectrum, \emph{i.e.} $V_{{\alpha}{\beta}}\,=\,0$, in order for the two sGoldstini directions to become degenerate. Such a situation turns out to be very
special since it produces two sGoldstini directions with the same mass given by their \emph{average}, which happens to be subject to a universal geometric stability bound \cite{Covi:2008cn,Borghese:2012yu}
\begin{equation}
\label{sG_bound}
\eta_{sG} \ \equiv \ \frac{1}{V} \, g^{\bar{{\alpha}}}\bar{g}^{{\beta}} \, D_{\bar{{\alpha}}}D_{{\beta}}V \ = \ \frac{2}{3{\gamma}} \ - \ \frac{1+{\gamma}}{{\gamma}} \, \tilde{\mathcal{R}} \ \overset{!}{\geq} \ 0 \ ,
\end{equation}
where ${\gamma} \, \equiv \, \frac{V}{3\,e^{K}\,|W|^{2}}$ and $\tilde{\mathcal{R}} \, \equiv \, \mathcal{R}_{\bar{{\alpha}}{\beta}\bar{{\gamma}}{\delta}} \, g^{\bar{{\alpha}}}\bar{g}^{{\beta}}g^{\bar{{\gamma}}}\bar{g}^{{\delta}}$
denotes the sectional curvature of the K\"ahler manifold along the $g_{{\alpha}}$ plane.
As already noted in ref.~\cite{Danielsson:2012by}, the sectional curvature can be rewritten as $\tilde{\mathcal{R}}\,=\,\frac{2}{n_{\textrm{eff}}}$, where
\begin{equation}
n_{\textrm{eff}} \ = \ \frac{\left(F_{{\alpha}}\overline{F}^{{\alpha}}\right)^{2}}{\left(F_{S}\overline{F}^{S}\right)^{2}\,+\,\frac{1}{3} \, \left(F_{T}\overline{F}^{T}\right)^{2}
\,+\,\frac{1}{3} \, \left(F_{U}\overline{F}^{U}\right)^{2}} \ ,
\end{equation}
effectively represents the number of complex fields\footnote{The quantity $n_{\textrm{eff}}$ is only defined when SUSY is broken and it ranges from $1$ to $7$.
The field $S$ has weight one, whereas $T$ \& $U$ have weight three due to the origin of this STU-model as the isotropic limit of the $\textrm{SL}(2)^{7}$ model.} taking part in the SUSY-breaking
mechanism. As an interesting consequence, the geometric bound in \eqref{sG_bound} only depends on information encoded in the zero-th \& first derivatives of the superpotential, \emph{i.e.} $W_{0}$ \& $W_{{\alpha}}$,
which are parametrised by $\left(\kappa_{0},\,\kappa_{S},\,\kappa_{T},\,\kappa_{U}\right)$ as in \eqref{Close2SUSY} close to a SUSY Mkw point.
Summarising, one can adopt the following combination of a system of equalities and a system of inequalities
\begin{equation} \label{SUSYIneqs}
\begin{array}{ccccc}
\left\{\begin{array}{lc}
V_{{\alpha}} \ \overset{!}{=} \ 0 & , \\
V_{{\alpha}{\beta}} \ \overset{!}{=} \ 0 & ,
\end{array}\right.
& & \textrm{and} & & \left\{\begin{array}{lc}
{\gamma} \ \overset{!}{>} \ 0 & , \\
n_{\textrm{eff}} \ \overset{!}{>} \ 3\,(1\,+\,{\gamma}) & ,
\end{array}\right.
\end{array}
\end{equation}
as a constructive procedure in order to analytically produce simple examples of stable dS vacua close to a SUSY Mkw critical point. We remind the reader that the two sets of equations on the left can be
both linearly solved by fixing six real parameters within $W_{{\alpha}{\beta}}$ and ten real parameters in $W_{{\alpha}{\beta}{\gamma}}$, respectively. The inequalities on the right, can instead be satisfied by suitably choosing
$\kappa_{0}$ \& $\kappa_{{\alpha}}$'s. It is worth mentioning that the whole construction still leaves $4+4=8$ real free parameters within the second and third $W$ derivatives.
In section~\ref{sec:examples}, we will present an explicit example of dS vacuum close to a SUSY Mkw point within our class of STU-models inspired by type IIB compactifications with generalised fluxes, which was obtained by following the above procedure.
\subsection*{Stable dS vacua close to no-scale Mkw points}
Another simple way of constructing Mkw points stable up to flat directions, is to consider the so-called
no-scale models, where SUSY is broken by large amounts but only in a single direction (usually $T$ in our notation) in such a way that
the corresponding F-term contribution to the potential exactly cancels the negative contribution from $-3|W|^2$.
A realisation of a no-scale supergravity model is given by the GKP superpotential \eqref{W_GKP} arising from type IIB compactifications
with NS-NS and R-R 3-form gauge fluxes. The possibility of obtaining stable dS solutions in this context was first considered in
ref.~\cite{Covi:2008ea} and later extensively studied in refs~\cite{Kallosh:2014oja,Marsh:2014nla}, where explicit examples of analytical stable dS solutions close to no-scale Mkw points are constructed by means of non-perturbative superpotentials of the type
\eqref{W_nonpert}.
These examples making use of non-perturbative effects are special realisations of superpotentials within the class given in
\eqref{W_Phi} with non-trivial zero-th, first, second and third derivatives. Due to the correspondence shown in the beginning of this
section, they all have a realisation within perturbative (polynomial) superpotentials with generalised fluxes.
In the next section, by making use of this correspondence, we will be able to present stable dS examples close to no-scale points with both two and three massless directions, in the context of our STU-model with $32$ generalised fluxes.
In order to go beyond the pre-existing examples borrowed from non-perturbative models and construct new solutions close to no-scale Mkw
points from scratch, in analogy with the SUSY case, we need to understand no-scale points in the language of $W$ derivatives.
No-scale Mkw points are identified by
\begin{equation}
\label{Close2NoScale}
\begin{array}{lc}
\begin{array}{lcccl}
W_{0} \, = \, \textrm{arbitrary} & & , & &
\left\{\begin{array}{l}
W_{S} \, = \, - K_{S} \, W_{0} \\
W_{U} \, = \, - K_{U} \, W_{0}
\end{array}\right. \end{array} & , \\[6mm]
\left\{\begin{array}{l}
W_{T} \, = \, 0 \\
W_{ST} \, = \, W_{TU} \, = \, W_{TT} \, = \, 0 \\
W_{STU} \, = \,W_{TUU} \, = \, W_{STT} \, = \, W_{TTU} \, = \, W_{TTT} \, = \, 0
\end{array}\right.
& , \\[3mm]
\end{array}
\end{equation}
whereas all the other complex derivatives stay completely aribitrary. One can check that the above choice already automatically implies the field equations and guarantees the semi-positiveness of the mass matrix. Generically these Mkw solutions will have two
flat directions, but upon imposing a degeneracy condition on the remaining free parameters, one will hit special no-scale points with an extra massless mode.
By translating the conditions in \eqref{Close2NoScale} into the language of generalised fluxes, one gets easily convinced of the existence of \emph{generalised no-scale points}, \emph{i.e.} for which the explicit flux-induced superpotential may in general depend
on $T$, but in such a way that all its $T$ derivatives up to third order vanish. An exhaustive classification of such models is presented in appendix~\ref{app:no-scale_Mkw}.
Now we can move away from the generalised no-scale points defined by the conditions in \eqref{Close2NoScale} by deforming them into
the following \emph{Ansatz} for the first $W$ derivatives
\begin{equation} \label{Close2NSWs}
\left\{\begin{array}{lc}
W_{S} \, = \, - K_{S} \, W_{0} \ + \ \kappa_{S} \, \epsilon & ,\\
W_{T} \, = \, \kappa_{T} \, \epsilon & ,\\
W_{U} \, = \, - K_{U} \, W_{0} \ + \ \kappa_{U} \, \epsilon & ,
\end{array}\right.
\end{equation}
where $\epsilon$ is again a small real parameter and $\left(\kappa_{S},\,\kappa_{T},\,\kappa_{U}\right)$ are arbitrary
$\mathcal{O}(1)$ complex parameters. By fixing some of the $W_{{\alpha}{\beta}}$'s in order to solve the field equations lineraly, one correctly
approaches a generalised no-scale point with two light directions that are generically lifted at quadratic order in $\epsilon$, just
like in those solutions which are obtained from adding non-perturbative effects. In this situation, fixing some $W_{{\alpha}{\beta}{\gamma}}$'s to solve
the condition $V_{{\alpha}{\beta}} \ \overset{!}{=} \ 0$ can be very helpful in finding stable dS critical points.
On the other hand, in order to approach a generalised no-scale Mkw with three flat directions, it turns out that one needs to violate
such a condition since having an odd number of massless modes is not compatible with the pairwise organisation of the mass spectrum
that it causes. In the next section we will show an example of a dS solution exhibiting three light modes, two of which are lifted
at $\mathcal{O}(\epsilon^2)$ while the extra one is lifted at $\mathcal{O}(\epsilon)$, thus providing the first realisation of the
general situation discussed in ref.~\cite{Marsh:2014nla}.
\section{Relevant Examples}
\label{sec:examples}
In this section we will present explicit analytical examples of (stable) dS solutions obtained by making use of the technical machinery presented in section~\ref{sec:method}.
We will first start by giving a generalised-flux realisation of stable dS close to a SUSY Mkw; in this case we will have two light modes lifted at $\mathcal{O}(\epsilon^{2})$.
Secondly, we will move to discussing dS solutions close to no-scale Mkw points. There we will present both generalised-flux realisations of solutions borrowed from non-perturbative superpotentials
and new examples which cannot be interpreted as coming from such exponential superpotentials. Only within these new cases will we be able to find the first realisation of an approximate no-scale
dS solution with three light modes, one of which is lifted at linear level, while the other two get lifted at quadratic level. However, such a solution will turn out to be unstable.
\subsection*{Stable dS close to a SUSY Mkw point}
This is a simple example of the application of the constraints in (\ref{SUSYIneqs}). A particularly simple choice that reduces the number of involved real parameters from 6 to 1 is
\begin{equation} \label{SUSYExFs}
F_{{\alpha}} \, = \, \big( \, (1+i) \, \epsilon \, , \, (\lambda_T +i) \, \epsilon \, , \, (1+i) \, \epsilon \, \big) ,
\end{equation}
where $\lambda_T$ can then be seen as a relative scaling between $\mathrm{Re}[F_{T}]$ and $\mathrm{Re}[F_{S}]$ or $\mathrm{Re}[F_{U}]$. In the language of the prescription given in (\ref{Close2SUSY}), we may additionally pick $\mathcal{O}(\epsilon)$ terms for $W_0$ and write
\begin{equation}
W_{0} \, = \, \left(\frac{8}{3}+\frac{2 i}{3}\right) \epsilon \ \ \ , \ \ \ W_{S} \, = \, \left(\frac{4}{3}-\frac{i}{3}\right) \epsilon \ \ \ , \ \ \ W_{T} \, = \, \left( \lambda_T+1-3 i\right) \epsilon \ \ \ , \ \ \ W_{U} \, = \, (2-3 i) \epsilon \, .
\end{equation}
Hence the cosmological constant takes the form
\begin{equation}
V \, = \, \frac{1}{96} \left(\lambda_T^2-8\right) \, \epsilon^2 \,.
\end{equation}
This two-parameter set of solutions can then be run through the inequalities in (\ref{SUSYIneqs}) which at $\mathcal{O}(\epsilon^0)$ provide the simple bounds
\begin{equation} \label{SUSYbounds}
-\sqrt{\frac{1}{2} \left(15+\sqrt{385}\right)}< \lambda_T<-2 \sqrt{2} \,\,\,\,\,\, \mathrm{or}\,\,\,\,\,\, 2 \sqrt{2}< \lambda_T<\sqrt{\frac{1}{2} \left(15+\sqrt{385}\right)}
\end{equation}
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\scalebox{0.92}[0.92]{
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values\\
\hline
\hline
$ a_0 $ & $ \frac{347}{39304}-\frac{12859}{235824} \epsilon $ \\
\hline
$ a_1 $ & $ -\frac{231}{9826}+\frac{25591}{235824}\epsilon $ \\
\hline
$ a_2 $ & $ \frac{407}{29478}+\frac{33905}{235824} \epsilon $ \\
\hline
$ a_3 $ & $ -\frac{623}{39304}-\frac{18281}{235824}\epsilon $ \\
\hline
\hline
$ b_0 $ & $ -\frac{2029}{39304}-\frac{19429}{78608}\epsilon $ \\
\hline
$ b_1 $ & $ \frac{13945}{353736}+\frac{5311}{235824}\epsilon $ \\
\hline
$ b_2 $ & $ -\frac{2455}{176868}-\frac{18209}{78608}\epsilon $ \\
\hline
$ b_3 $ & $ \frac{1349}{9826}+\frac{10481}{78608}\epsilon $ \\
\hline
\hline
$ c_0 $ & $ -\frac{3749}{58956} -\frac{37007}{78608} \epsilon $ \\
\hline
$ c_1 $ & $ -\frac{9661}{117912} -\frac{158563}{235824}\epsilon $ \\
\hline
$ c_2 $ & $ \frac{4241}{353736}-\frac{12115}{78608}\epsilon $ \\
\hline
$ c_3 $ & $ \frac{2569}{58956}-\frac{143}{235824} \epsilon $ \\
\hline
\hline
$ d_0 $ & $ \frac{15107}{176868} +\frac{45635}{235824}\epsilon $ \\
\hline
$ d_1 $ & $ -\frac{1625}{58956} +\frac{3083}{78608}\epsilon $ \\
\hline
$ d_2 $ & $ -\frac{20293}{117912} -\frac{10159}{78608}\epsilon$ \\
\hline
$ d_3 $ & $ \frac{11795}{353736} +\frac{11821}{235824}\epsilon $ \\
\hline
\end{tabular} \quad
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values\\
\hline
\hline
$ a_0' $ & $ \frac{1333}{39304}+\frac{20613}{78608}\epsilon $ \\
\hline
$ a_1' $ & $ \frac{6943}{176868}+\frac{78155}{235824}\epsilon $ \\
\hline
$ a_2' $ & $ -\frac{4307}{176868} +\frac{15851}{235824}\epsilon $ \\
\hline
$ a_3' $ & $ \frac{23869}{117912} +\frac{253991}{235824}\epsilon $ \\
\hline
\hline
$ b_0' $ & $ \frac{133}{19652}+\frac{6347}{235824}\epsilon $ \\
\hline
$ b_1' $ & $ \frac{110}{14739}-\frac{10311}{78608}\epsilon $ \\
\hline
$ b_2' $ & $ -\frac{8225}{353736} -\frac{32561}{235824}\epsilon $ \\
\hline
$ b_3' $ & $ \frac{3441}{39304}+\frac{119707}{235824}\epsilon $ \\
\hline
\hline
$ c_0' $ & $ \frac{2681}{88434}+\frac{7843}{235824}\epsilon $ \\
\hline
$ c_1' $ & $ \frac{43471}{353736}+\frac{95315}{235824}\epsilon $ \\
\hline
$ c_2' $ & $ -\frac{32513}{353736} -\frac{46999}{78608}\epsilon $ \\
\hline
$ c_3' $ & $ -\frac{1691}{14739} -\frac{27649}{235824}\epsilon $ \\
\hline
\hline
$ d_0' $ & $ -\frac{1075}{117912} -\frac{9989}{235824}\epsilon $ \\
\hline
$ d_1' $ & $ \frac{32869}{353736}-\frac{12817}{235824}\epsilon $ \\
\hline
$ d_2' $ & $ -\frac{1370}{14739} -\frac{66721}{235824}\epsilon $ \\
\hline
$ d_3' $ & $ -\frac{2578}{44217} -\frac{23815}{78608}\epsilon $ \\
\hline
\end{tabular}
}
\end{center}
\caption{{\it An explicit example of a generalised flux realisation of a stable dS solution which analytically approaches a SUSY Mkw point as $\epsilon\rightarrow 0$. It may be checked explicitly that the 0-th order values of the fluxes define a SUSY Mkw solution with two flat directions.}}
\label{table:SUSYExFluxes}
\end{table}
The equations in (\ref{SUSYIneqs}) can be solved to fix $W_{ST}$, $W_{TT}$ and $W_{TU}$ (with $V_{\alpha} = 0$) as well as $W_{STT}$, $W_{STU}$, $W_{TTT}$, $W_{TTU}$ and $W_{TUU}$ (with $V_{\alpha \beta} = 0$). Then it only remains to choose values for the 8 remaining free real parameters. Here we will present one particular realization of fluxes obtained with
\begin{equation}
W_{SU} \, = \, \frac{1}{3}\left(1+i\right) \ \ \ , \ \ \ W_{UU} \, = \, \frac{1}{3}\left(1+i\right) \ \ \ , \ \ \ W_{SUU} \, = \, W_{UUU} = 0 \, .
\end{equation}
\begin{figure}
\centering
\includegraphics[bb=0 0 7in 6.32in,keepaspectratio,viewport= 0 0 7in 6.32in,clip,scale=0.5]{SUSY}
\caption{{\it Stable (red) and dS (blue) points for solutions close to a SUSY Mkw point (in the origin). The axes correspond to $\mathrm{Re}[F_{S}] \, = \, \epsilon$ and $\mathrm{Re}[F_{T}] \, = \, \lambda_T \, \epsilon$.}}
\label{SUSYPlot}
\end{figure}
If $\lambda_T$ is left arbitrary, then these solutions become a set parameterised solely by $\epsilon$ and $\lambda_T$. To see their behaviour, we study the values of the cosmological constant and stability regions in a plot of $\mathrm{Re}[F_{T}]$ vs $\mathrm{Re}[F_{S}]$ (see figure~\ref{SUSYPlot}). This plot can be seen as a polar plot, in which the origin corresponds to a SUSY Mkw point. From (\ref{SUSYExFs}) we see that in the plot $\arctan (\lambda_T)$ is the corresponding polar angle and any point lays at a radial distance of $\sqrt{\mathrm{Re}[F_{S}] ^2 \, + \, \mathrm{Re}[F_{T}] ^2} = \sqrt{1 \, + \, \lambda_T^2 }\,\epsilon$ from a SUSY point.
The superposition of the stable regions (red) and dS regions (blue) is then clear and this overlap can be understood quite naturally as merely a manifestation of (\ref{SUSYbounds}). It should be pointed out that the origin of this plot represents a different SUSY Mkw solution for each direction in which one approaches it, i.e. the Mkw point depends on $\lambda_T$.
In order to be more concise, we present the explicit solution with $\lambda_T = 4$. The fluxes resulting fluxes are given in table~\ref{table:SUSYExFluxes}. Due to its linear dependence, the limit $\epsilon \rightarrow 0$ can be read quite easily. As it was discussed before, the eigenvalues of ${\left(m^{2}\right)^{I}}_{J}$ are organised by pairs. For $\lambda_T = 4$ they take the values
\begin{equation}
\left\{\frac{1}{150} \, \epsilon^2 \,+ \, \mathcal{O}(\epsilon^{4}) \,\,\, , \,\,\, \frac{3511-31 \sqrt{6061}}{187272} \, + \, \mathcal{O}(\epsilon^{1}) \,\,\, , \,\,\, \frac{3511+31 \sqrt{6061}}{187272} \, + \, \mathcal{O}(\epsilon^{1}) \,\right\} \ ,
\end{equation}
and the cosmological constant is then simply
\begin{equation}
V \, = \, \frac{1}{12} \, \epsilon^2 \ .
\end{equation}
\subsection*{Stable dS close to no-scale}
Two examples will be discussed of stable dS close to no-scale points. We will use a special kind of solutions connected to (\ref{Close2NSWs}), which were originally motivated by non-perturbative superpotentials of the kind of (\ref{W_nonpert}). Due to the simple mapping between 16 complex values of the $W$ derivatives and a 32 fluxes superpotential, it was possible to construct non-geometric polinomial superpotentials that share the same fundamental physical properties. We will provide explicit solutions making emphasis on this equivalence.
For these examples it is sufficient to consider superpotentials with the functional form of (\ref{W_nonpert})
\begin{equation} \label{WPertExamps}
W_{\textrm{non-pert.}} = \,(P_{F} - P_{H} \, S )\, + \, P_{Z} \, e^{i\,{\alpha} T} \ ,
\end{equation}
with
\begin{equation}
P_Z \, = \, \left( \alpha_0 \, + \, i \, \alpha_1 \right) + \left( \alpha_2 \, + \, i \, \alpha_3 \right) \, U \ ,
\end{equation}
with $\alpha_0$, $\alpha_1$, $\alpha_2$ and $\alpha_3$ being real parameters.
As it was mentioned, conditions (\ref{NoScalePointConditions}) define a no-scale point. We may also start, as we did in the previous examples, by establishing a particular behaviour for the supersymmetry braking parameters. By doing so, it can readily be seen that the $\mathcal{O}(\epsilon^0)$ terms in $\mathrm{Re}[F_T]$ characterise the $\mathcal{O}(\epsilon^0)$ values of $W_0$ via $W_T = 0$ and, via equations of motion or (\ref{NoScalePointConditions}), the $\mathcal{O}(\epsilon^0)$ values of $W_S$ and $W_U$ as well. Hence, with this approach much of the freedom left is in the higher order terms in $\epsilon$.
In the present examples we will \emph{not} enforce $V_{\alpha \beta} = 0$. This is partly motivated by the fact that we want to show some explicit realizations of no-scale points with 3 massless modes. But most importantly, in the superpotential (\ref{WPertExamps}) we have 12 real fluxes which naturally fit with a description in terms of 6 real supersymmetry breaking parameters and 6 real constraints placed by the equations of motion (see \cite{Danielsson:2012by}). Hence, once picked $F_\alpha$ with the required $\epsilon \rightarrow 0$ limit, exact and well defined solutions can be found, with $\epsilon$ parameterizing a continuous trajectory from the chosen Mkw point.
In this case it will be sufficient to break supersymmetry as
\begin{equation} \label{NScaleExFs}
F_{{\alpha}} \, = \, \big( \, \epsilon \, , \, 1 \, , \, \lambda_U \, \epsilon \, \big)\ .
\end{equation}
Solutions corresponding to this choice are then parameterised in terms of $\epsilon$, $\lambda_U$ and the exponent factor in the non-perturbative term $\alpha$. The $\mathcal{O}(\epsilon^0)$ behaviour will be naturally of the no-scale type. This is all the background we need to find examples of stable dS close to no-scale points with 2 or 3 massless modes. In analogy with the SUSY case, $\lambda_U$ will help us parameterise the direction in which we approach no-scale points. In particular, it will determine whether we land in a 2 or 3 massless Mkw. In addition, these no-scale points will be of the GKP type. We will present simultaneously their corresponding manifestations as 32 fluxes models, in which, interestingly, only 16 will end up being different from 0. Hence, through this map, we will find for free non-geometric polynomial superpotentials with 16 fluxes that show dS close to a no-scale point.
For instance, if one picks $\alpha = \tfrac{2}{5}$, very interesting physics occurs. The corresponding solution in terms of the 12 fluxes of the non-perturbative superpotential (\ref{WPertExamps}) is given in the left part of table~\ref{table:NoSc2MExFluxesNONPERT} for $\lambda_U = 2$. As explained, we may compute the corresponding $W$ derivatives and then map them into the 32 fluxes non-geometric superpotential. The resulting solution is given in table~\ref{table:NoSc2MExFluxesNONGEO}. The fact that only 16 are non-zero is related to the fact that the $W$ derivatives have a very particular parity: $W_0$, $W_{\alpha \beta}$ are all imaginary while $W_\alpha$ and $W_{\alpha \beta \gamma}$ are all real (notice that the chosen supersymmetry braking parameters are all real).
\begin{figure}
\centering
\includegraphics[bb=0 0 8.50in 7.74in,keepaspectratio,viewport= 0 0 8.50in 7.74in,clip,scale=0.32]{NoScaleSmall}
\ \ \ \ \ \
\includegraphics[bb=0 0 8.50in 8.31in,keepaspectratio,viewport= 0 0 8.50in 8.31in,clip,scale=0.3]{NoScaleBig}
\caption{{\it Stable (red) and dS (blue) points for solutions close to a no-scale point (in the origin). The axes correspond to $\mathrm{Re}[F_{S}] \, = \, \epsilon$ and $\mathrm{Re}[F_{U}] \, = \, \lambda_U \, \epsilon$. Each plot shows a different scale. The regions of overlap are connected no-scale points with 2 massless modes.}}
\label{NoScale2MPlots}
\end{figure}
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\scalebox{0.92}[0.92]{
\begin{tabular}{ | c | c |}
\hline
labels & values\\
\hline
\hline
$ a_0 $ & $ 0 $ \\
\hline
$ a_1 $ & $ \frac{38425 \epsilon ^3+7045 \epsilon ^2+990 \epsilon +24}{7800 \epsilon ^2+8880 \epsilon +576} $ \\
\hline
$ a_2 $ & $ 0 $ \\
\hline
$ a_3 $ & $ \frac{375 \epsilon ^3+4215 \epsilon ^2-566 \epsilon -120}{24 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
\hline
$ b_0 $ & $ \frac{-2175 \epsilon ^3-600 \epsilon ^2+877 \epsilon +60}{3900 \epsilon ^2+4440 \epsilon +288} $ \\
\hline
$ b_1 $ & $ 0 $ \\
\hline
$ b_2 $ & $ -\frac{1525 \epsilon ^3-740 \epsilon ^2+105 \epsilon +12}{3900 \epsilon ^2+4440 \epsilon +288} $ \\
\hline
$ b_3 $ & $ 0 $ \\
\hline
\hline
$\alpha_0 $ & $ 0 $ \\
\hline
$ \alpha_1 $ & $ -\frac{25 e^{2/5} \epsilon \left(105 \epsilon ^2+19 \epsilon -12\right)}{4 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ \alpha_2 $ & $ \frac{75 e^{2/5} \epsilon \left(245 \epsilon ^2+17 \epsilon -4\right)}{4 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ \alpha_3 $ & $ 0 $ \\
\hline
\end{tabular} \qquad
\begin{tabular}{ | c | c |}
\hline
labels & values \\
\hline
\hline
$ a_0 $ & $ 0 $ \\
\hline
$ a_1 $ & $ \frac{\epsilon (1315 \epsilon +506)}{33 (20 \epsilon +31)} $ \\
\hline
$ a_2 $ & $ 0 $ \\
\hline
$ a_3 $ & $ \frac{75 \epsilon ^2+25 \epsilon -31}{60 \epsilon +93} $ \\
\hline
\hline
$ b_0 $ & $ \frac{-75 \epsilon ^2-25 \epsilon +31}{60 \epsilon +93} $ \\
\hline
$ b_1 $ & $ 0 $ \\
\hline
$ b_2 $ & $ \frac{(16-25 \epsilon ) \epsilon }{60 \epsilon +93} $ \\
\hline
$ b_3 $ & $ 0 $ \\
\hline
\hline
$\alpha_0 $ & $ 0 $ \\
\hline
$ \alpha_1 $ & $ 0 $ \\
\hline
$ \alpha_2 $ & $ \frac{600 e^{11/10} \epsilon ^2}{220 \epsilon +341} $ \\
\hline
$ \alpha_3 $ & $ 0 $ \\
\hline
\end{tabular}
}
\end{center}
\caption{{\it Two examples of stable dS solutions analytically connected to a 2-massless (left) and 3-massless (right) no-scale Mkw point realised within a non-perturbative superpotential of the type in \eqref{W_nonpert}.}}
\label{table:NoSc2MExFluxesNONPERT}
\end{table}
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\scalebox{0.92}[0.92]{
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values \\
\hline
\hline
$ a_0 $ & $ 0 $ \\
\hline
$ a_1 $ & $ \frac{-201065 \epsilon ^3+193449 \epsilon ^2+39660 \epsilon +960}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ a_2 $ & $ 0 $ \\
\hline
$ a_3 $ & $\frac{229515 \epsilon ^3+189541 \epsilon ^2-31460 \epsilon -4800}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
\hline
$ b_0 $ & $ \frac{-45165 \epsilon ^3-6811 \epsilon ^2+39740 \epsilon +4800}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ b_1 $ & $ 0 $ \\
\hline
$ b_2 $ & $ -\frac{10385 \epsilon ^3-73001 \epsilon ^2+15540 \epsilon +960}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ b_3 $ & $ 0 $ \\
\hline
\hline
$ c_0 $ & $ -\frac{\epsilon \left(114205 \epsilon ^2+14707 \epsilon -7820\right)}{320 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ c_1 $ & $ 0 $ \\
\hline
$ c_2 $ & $ -\frac{\epsilon \left(65765 \epsilon ^2-2149 \epsilon +4820\right)}{320 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ c_3 $ & $ 0 $ \\
\hline
\hline
$ d_0 $ & $ 0 $ \\
\hline
$ d_1 $ & $ -\frac{\epsilon \left(31465 \epsilon ^2-4529 \epsilon +5380\right)}{320 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ d_2 $ & $ 0 $ \\
\hline
$ d_3 $ & $ -\frac{\epsilon \left(111615 \epsilon ^2+13801 \epsilon -7140\right)}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
\end{tabular} \quad
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values \\
\hline
\hline
$ a_0' $ & $ 0 $ \\
\hline
$ a_1' $ & $ -\frac{\epsilon \left(8715 \epsilon ^2+6661 \epsilon -5460\right)}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ a_2' $ & $ 0 $ \\
\hline
$ a_3' $ & $ -\frac{\epsilon \left(191835 \epsilon ^2+18149 \epsilon -7380\right)}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
\hline
$ b_0' $ & $ \frac{\epsilon \left(128835 \epsilon ^2+41189 \epsilon -30420\right)}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ b_1' $ & $ 0 $ \\
\hline
$ b_2' $ & $ \frac{\epsilon \left(111615 \epsilon ^2+13801 \epsilon -7140\right)}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ b_3' $ & $ 0 $ \\
\hline
\hline
$ c_0' $ & $ \frac{\epsilon \left(945 \epsilon ^2+2303 \epsilon -1980\right)}{320 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ c_1' $ & $ 0 $ \\
\hline
$ c_2' $ & $ \frac{\epsilon \left(20685 \epsilon ^2-4621 \epsilon +4980\right)}{320 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
$ c_3' $ & $ 0 $ \\
\hline
\hline
$ d_0' $ & $ 0 $ \\
\hline
$ d_1' $ & $ -\frac{\epsilon \left(31465 \epsilon ^2-4529 \epsilon +5380\right)}{320 \left(325 \epsilon ^2+370 \epsilon +24\right)}$ \\
\hline
$ d_2' $ & $ 0 $ \\
\hline
$ d_3' $ & $ -\frac{\epsilon \left(111615 \epsilon ^2+13801 \epsilon -7140\right)}{960 \left(325 \epsilon ^2+370 \epsilon +24\right)} $ \\
\hline
\end{tabular}
}
\end{center}
\caption{{\it An example of a non-geometric superpotential which analytically connects a 2-massless no-scale Mkw point with stable dS. This is the result of mapping the first solution given in table~\ref{table:NoSc2MExFluxesNONPERT} into the 32 fluxes model.}}
\label{table:NoSc2MExFluxesNONGEO}
\end{table}
With $\epsilon$ and $\lambda_U$ arbitrary, we may make plots looking for stability and dS points around the no-scale point. In figure~\ref{NoScale2MPlots} we show $Re[F_S]$ vs $Re[F_U]$. As before, the origin corresponds to a Mkw point while $\arctan (\lambda_U)$ corresponds to the polar angle. The origin is a distinct no-scale point depending on the value of $\lambda_U$. In particular, if $\lambda_U$ takes any of the values $-3$, $-1$ , $1$ or $3$, the no-scale point has 3 massless modes. Any other direction lands in a no-scale with 2 massless modes.
For this value of $\alpha$, it can be seen that none of the 3-massless directions lands in stable dS (i.e. they land in stable Ads, unstable dS or unstable AdS). The superposition of the stable (red) and dS (blue) regions is clear and a particular realization of stable dS close to a no-scale with 2 massless modes is found when $\lambda_U = 2$. Notice also that, once we zoom out from the no-scale point, interesting phenomena are also manifest around other Mkw points. The cosmological constant with $\alpha = \tfrac{4}{10}$ and $\lambda_U = 2$ is
\begin{eqnarray}
V &=& -\frac{\epsilon ^2 \left(870625 \epsilon ^4-203875 \epsilon ^3+78200 \epsilon ^2+5240 \epsilon -96\right)}{48 \left(325 \epsilon ^2+370 \epsilon +24\right)^2}
\\
&=& \frac{1}{288} \epsilon ^2 -\frac{1025}{3456} \, \epsilon ^3 \, + \,\mathcal{O}(\epsilon^4).
\end{eqnarray}
while the eigenvalues of ${\left(m^{2}\right)^{I}}_{J}$ take the values
\begin{eqnarray}
&& \frac{79}{3240} \, \epsilon^2 \,+ \, \mathcal{O}(\epsilon^{3}) \,\,\, , \,\,\,
\frac{23}{2700} \, \epsilon^2 \,+ \, \mathcal{O}(\epsilon^{3}) \,\,\, , \,\,\, \frac{1}{1152} \, + \, \mathcal{O}(\epsilon^{1}) \,\,\, , \,\,\,
\\
&& \frac{1}{1152} \, + \, \mathcal{O}(\epsilon^{1}) \,\,\, , \,\,\,\,\,\,\,\,
\frac{25}{1152} \, + \, \mathcal{O}(\epsilon^{1}) \,\,\,\, \mathrm{and} \,\,\,\, \frac{1}{128} \, + \, \mathcal{O}(\epsilon^{1}) \, ,
\end{eqnarray}
The superpotential
\begin{equation}
W \, = \frac{1}{24} \, \left(\,3 \, S U^2 \,- \,5 \, S\,+\,5 \, U^3\,-\,3 \, U\,\right)+ \mathcal{O}(\epsilon^{1}) \,,
\end{equation}
is clearly of the GKP type at $\mathcal{O}(\epsilon^0)$.
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\scalebox{0.92}[0.92]{
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values \\
\hline
\hline
$ a_0 $ & $ 0 $ \\
\hline
$ a_1 $ & $ \frac{23 (160-31 \epsilon ) \epsilon }{240 (20 \epsilon +31)} $ \\
\hline
$ a_2 $ & $ 0 $ \\
\hline
$ a_3 $ & $ \frac{8253 \epsilon ^2+2000 \epsilon -2480}{240 (20 \epsilon +31)} $ \\
\hline
\hline
$ b_0 $ & $ \frac{-8253 \epsilon ^2-2000 \epsilon +2480}{4800 \epsilon +7440} $ \\
\hline
$ b_1 $ & $ 0 $ \\
\hline
$ b_2 $ & $ \frac{(1280-1007 \epsilon ) \epsilon }{240 (20 \epsilon +31)} $ \\
\hline
$ b_3 $ & $ 0 $ \\
\hline
\hline
$ c_0 $ & $ -\frac{751 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ c_1 $ & $ 0 $ \\
\hline
$ c_2 $ & $ -\frac{1833 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ c_3 $ & $ 0 $ \\
\hline
\hline
$ d_0 $ & $ 0 $ \\
\hline
$ d_1 $ & $ -\frac{1413 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ d_2 $ & $ 0 $ \\
\hline
$ d_3 $ & $ -\frac{331 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
\end{tabular} \quad
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values \\
\hline
\hline
$ a_0' $ & $ 0 $ \\
\hline
$ a_1' $ & $ \frac{89 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ a_2' $ & $ 0 $ \\
\hline
$ a_3' $ & $ -\frac{509 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
\hline
$ b_0' $ & $ -\frac{751 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ b_1' $ & $ 0 $ \\
\hline
$ b_2' $ & $ \frac{331 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ b_3' $ & $ 0 $ \\
\hline
\hline
$ c_0' $ & $ -\frac{331 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ c_1' $ & $ 0 $ \\
\hline
$ c_2' $ & $ \frac{227 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ c_3' $ & $ 0 $ \\
\hline
\hline
$ d_0' $ & $ 0 $ \\
\hline
$ d_1' $ & $ -\frac{1413 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
$ d_2' $ & $ 0 $ \\
\hline
$ d_3' $ & $ -\frac{331 \epsilon ^2}{80 (20 \epsilon +31)} $ \\
\hline
\end{tabular}
}
\end{center}
\caption{{\it An example of a non-geometric superpotential which analytically connects a 3-massless no-scale Mkw point with stable dS. This is the result of mapping the second solution given in table~\ref{table:NoSc2MExFluxesNONPERT} into the 32 fluxes model.}}
\label{table:NoSc3MExFluxesNONGEO}
\end{table}
To see an example of stable dS close to a no-scale wth 3-massless modes one can simple take $\alpha = \tfrac{11}{10}$ and $\lambda_U = 1$. The solutions in terms of the 12 parameters in the non-perturbative superpotential can be seen in the right part of table~\ref{table:NoSc2MExFluxesNONPERT} while its non-geometric counterpart is given in table~\ref{table:NoSc3MExFluxesNONGEO}. It shares many of the features we discussed previously. The corresponding potential is then
\begin{eqnarray}
V &=&\frac{\epsilon ^2 \left(-500 \epsilon ^2+640 \epsilon +31\right)}{24 (20 \epsilon +31)^2}
\\
&=& \frac{1}{744} \epsilon ^2 +\frac{25}{961} \, \epsilon ^3 \, + \,\mathcal{O}(\epsilon^4).
\end{eqnarray}
exhibits, as expected, a simple GKP form.
and the scaling behaviour of the eigenvalues of ${\left(m^{2}\right)^{I}}_{J}$ can be observed in figure~\ref{NoScale3MPlot}. Notice that in this case all the massless modes grow quadratically. We will discuss an example with a mass growing linearly with $\epsilon$ in the next subsection. The $\mathcal{O}(\epsilon^0)$ superpotential
\begin{equation}
W \, = \frac{1}{3}\, \left(\,U^3\,-\,S\,\right) + \mathcal{O}(\epsilon^{1}) \ ,
\end{equation}
exhibits, as expected, a simple GKP form.
\begin{figure}
\centering
\includegraphics[bb=0 0 11.97in 5.32in,keepaspectratio,viewport= 1.2in 0 11.97in 5.32in,clip,scale=0.33]{3masslessmasses}
\caption{{\it Eigenvalues of ${\left(m^{2}\right)^{I}}_{J}$ (red) as a function of $\epsilon$, shown in a loglog plot for a stable dS solution close to a no-scale point with 3 massless states. For reference, linear (blue), quadratic (yellow) and cubic (green) dependences are shown. Two of the massive states share a similar value (around $10^{-2}$).}}
\label{NoScale3MPlot}
\end{figure}
It is worth mentioning that there exist plenty of approximately no-scale stable dS solutions which can be constructed analytically without the aid of explicit non-perturbative terms by
using the procedure sketched in section~\ref{sec:method} which may be combined with solving the pairwise degeneracy constraint for the mass matrix $V_{{\alpha}{\beta}}\,=\,0$. These solutions generically approach no-scale Mkw points with two flat directions, but of a generalised type (see appendix~\ref{app:no-scale_Mkw}).
\subsection*{Massless Mode lifted at linear order}
This particular example shows a solution that moves continuously from a no-scale Mkw point with 3 massless modes to an unstable dS. One of the massless modes is lifted at $\mathcal{O}(\epsilon^1)$ and becomes positive for positive $\epsilon$. Nevertheless, there is a massless mode that becomes tachyonic with a negative $\mathcal{O}(\epsilon^2)$ term. This specific solution is given in table~\ref{table:LinearMass}.
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\scalebox{0.92}[0.92]{
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values \\
\hline
\hline
$ a_0 $ & $ \frac{21 \epsilon ^3}{8}-\frac{21 \epsilon ^2}{4}+\frac{17 \epsilon }{8}-\frac{13}{48} $ \\
\hline
$ a_1 $ & $ \frac{7 \epsilon ^3}{4}-\frac{11 \epsilon ^2}{6}+\frac{3 \epsilon }{8}+\frac{1}{8} $ \\
\hline
$ a_2 $ & $ -\frac{7 \epsilon ^3}{8}-\frac{\epsilon ^2}{4}+\frac{3 \epsilon }{8}-\frac{7}{48} $ \\
\hline
$ a_3 $ & $ -\epsilon ^2+\frac{3 \epsilon }{8}-\frac{1}{24} $ \\
\hline
\hline
$ b_0 $ & $ \frac{7 \epsilon ^3}{4}-\frac{9 \epsilon ^2}{4}+\frac{\epsilon }{8}+\frac{1}{8} $ \\
\hline
$ b_1 $ & $ -\frac{7 \epsilon ^3}{8}+\frac{3 \epsilon }{8}-\frac{3}{16} $ \\
\hline
$ b_2 $ & $ -\frac{11 \epsilon ^2}{12}+\frac{7 \epsilon }{24}-\frac{1}{24} $ \\
\hline
$ b_3 $ & $ -\frac{7 \epsilon ^3}{8}+\frac{\epsilon ^2}{2}+\frac{\epsilon }{8}-\frac{1}{16} $ \\
\hline
\hline
$ c_0 $ & $ \frac{7 \epsilon ^3}{6}-\frac{\epsilon ^2}{4}-\frac{17 \epsilon }{24}+\frac{5}{24} $ \\
\hline
$ c_1 $ & $ \frac{21 \epsilon ^3}{8}-\frac{11 \epsilon ^2}{6}+\frac{\epsilon }{12}-\frac{3}{16} $ \\
\hline
$ c_2 $ & $ \frac{7 \epsilon ^3}{4}-\frac{19 \epsilon ^2}{12}+\frac{11 \epsilon }{24}-\frac{1}{8} $ \\
\hline
$ c_3 $ & $ \frac{7 \epsilon ^3}{24}-\frac{\epsilon }{12}+\frac{1}{48} $ \\
\hline
\hline
$ d_0 $ & $ -\frac{7 \epsilon ^3}{8}+\frac{7 \epsilon ^2}{12}-\frac{\epsilon }{4}+\frac{1}{48} $ \\
\hline
$ d_1 $ & $ \frac{7 \epsilon ^3}{4}-\frac{5 \epsilon ^2}{3}+\frac{13 \epsilon }{24}-\frac{1}{8} $ \\
\hline
$ d_2 $ & $ -\frac{7 \epsilon ^3}{8}+\frac{\epsilon ^2}{4}+\frac{\epsilon }{4}-\frac{3}{16} $\\
\hline
$ d_3 $ & $ \frac{5 \epsilon ^2}{6}-\frac{7 \epsilon }{24}+\frac{1}{24} $ \\
\hline
\end{tabular} \quad
\begin{tabular}{ | c | c |}
\hline
flux labels & flux values \\
\hline
\hline
$ a_0' $ & $ \frac{7 \epsilon ^3}{8}-\epsilon ^2+\frac{1}{48} $ \\
\hline
$ a_1' $ & $ \frac{7 \epsilon ^2}{12}-\frac{5 \epsilon }{24}+\frac{1}{24} $ \\
\hline
$ a_2' $ & $ \frac{7 \epsilon ^3}{8}-\frac{5 \epsilon ^2}{6}+\frac{1}{16} $ \\
\hline
$ a_3' $ & $ \frac{7 \epsilon ^3}{4}+\frac{\epsilon ^2}{4}-\frac{3 \epsilon }{8}+\frac{1}{24} $ \\
\hline
\hline
$ b_0' $ & $ \frac{7 \epsilon ^3}{4}-\frac{7 \epsilon ^2}{2}+\frac{9 \epsilon }{8}-\frac{5}{24} $ \\
\hline
$ b_1' $ & $ -\frac{7 \epsilon ^3}{8}+\frac{3 \epsilon ^2}{4}-\frac{1}{16} $ \\
\hline
$ b_2' $ & $ -\frac{2 \epsilon ^2}{3}+\frac{7 \epsilon }{24}-\frac{1}{24} $ \\
\hline
$ b_3' $ & $ -\frac{7 \epsilon ^3}{8}+\frac{3 \epsilon ^2}{4}-\frac{1}{48} $ \\
\hline
\hline
$ c_0' $ & $ \frac{7 \epsilon ^3}{12}-\frac{2 \epsilon ^2}{3}+\frac{5 \epsilon }{24}-\frac{1}{24} $ \\
\hline
$ c_1' $ & $ \frac{7 \epsilon ^3}{8}-\frac{5 \epsilon ^2}{4}+\frac{\epsilon }{8}-\frac{1}{16} $ \\
\hline
$ c_2' $ & $ \frac{5 \epsilon ^2}{6}-\frac{11 \epsilon }{24}+\frac{1}{8} $ \\
\hline
$ c_3' $ & $ -\frac{7 \epsilon ^3}{24}+\frac{17 \epsilon ^2}{12}-\frac{\epsilon }{8}-\frac{5}{48} $ \\
\hline
\hline
$ d_0' $ & $ -\frac{7 \epsilon ^3}{8}+\frac{\epsilon ^2}{2}+\frac{\epsilon }{8}-\frac{5}{48} $ \\
\hline
$ d_1' $ & $ \frac{7 \epsilon ^3}{4}-\frac{23 \epsilon ^2}{12}+\frac{13 \epsilon }{24}-\frac{1}{8} $ \\
\hline
$ d_2' $ & $ -\frac{7 \epsilon ^3}{8}+\epsilon ^2-\frac{\epsilon }{8}-\frac{1}{16} $ \\
\hline
$ d_3' $ & $ \frac{5 \epsilon ^2}{12}-\frac{7 \epsilon }{24}+\frac{1}{24} $ \\
\hline
\end{tabular}
}
\end{center}
\caption{{\it An example of a no-scale point with a massless mode growing linearly as it moves towards unstable dS.}}
\label{table:LinearMass}
\end{table}
While the cosmological constant takes the simple form
\begin{equation}
V \, = \, \frac{\epsilon^2 }{12} \ ,
\end{equation}
the described scaling behaviour of the masses can be observed in figure~\ref{LinearMass}.
\begin{figure}
\centering
\includegraphics[bb=0 0 11.6in 5.39in,keepaspectratio,viewport=1.0in 0 11.6in 5.39in,clip,scale=0.3]{LinearMass1}
\ \ \
\includegraphics[bb=0 0 11.6in 5.32in,keepaspectratio,viewport= 1.2in 0 11.6in 5.32in,clip,scale=0.3]{LinearMass2}
\caption{{\it Eigenvalues of ${\left(m^{2}\right)^{I}}_{J}$ (red) as a function of $\epsilon$, shown in a loglog plot for an unstable dS solution close to a no-scale point with 3 massless states. In the plot on the left we see the 5 positive masses (notice that one is linear). Two of the massive states share a similar value (around 0.003). The sixth mass is negative and in the plot on the right we plot its absolute value. For reference, linear (blue), and quadratic (yellow) dependences are shown.}}
\label{LinearMass}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
In this paper we have shown how dS vacua can be constructed close to no-scale as well as supersymmetric Mkw points using a polynomial superpotential with non-geometric fluxes. As far as the existence of critical points and the features of the corresponding mass spectra are concerned, the superpotential with generalised fluxes covers all possibilities, and explicit non-perturbative terms will not bring in anything new.
The case of supersymmetric Mkw is particularly interesting. In contrast to previous constructions, such as \cite{Kallosh:2014oja}, we did not need to add a Polonyi field to be able to break supersymmetry. Instead, we managed to arrange for flat directions already within the STU-model itself. However, whether it is actually possible to tune parameters to obtain the desired hierarchy with $V \ll m^2_{\rm gravitino} \ll m^2_{\rm scalars}$ still remains to be seen. The large mass of the Polonyi field of \cite{Kallosh:2014oja} was arranged through tuning of its K{\"a}hler potential. Since we have a large number of fluxes available we expect to be able to perform similar tunings in our case. It would be interesting to try to construct a phenomenologically viable example and we hope to come back to this issue in the future.
In contrast to the non-perturbative terms the generalised fluxes are fully determined by string dualities and the number of parameters limited to $32$. Furthermore, a picture is emerging where at least some combinations of the non-geometric fluxes can be given a fully geometric interpretation through compactifications of string/M-theory on other topologies than twisted tori \cite{Danielsson:2015tsa}. Examples includes M-theory on $S^7$ and on $S^4 \times T^3$.
It will be interesting to see whether any of the stable dS vacua that can be constructed using the techniques of this paper can be given a fully geometric interpretation. Which features of the topology will be crucial for success? The advantage of such a solution is that it will be under firm controll from within supergravity and not dependent upon non-perturbative and string-inspired corrections under limited control.
\section*{Acknowledgments}
We would like to thank D.~Marsh, B.~Vercnocke and T.~Wrase for stimulating and valuable discussions. JB is supported by the John Templeton Foundation Grant 48222 and the CEA Eurotalents program. The work of UD, GD and SV is supported by the Swedish Research Council (VR).
\newpage
|
1,116,691,498,477 | arxiv | \@startsection{section}{1}{\z@{\@startsection{section}{1}{\z@}
{3.5ex plus 1.0ex minus 0.2ex}{2.3ex plus .2ex}{\normalsize}}
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\begin{document}
\begin{titlepage}
\begin{center}
Meson Decay Constants from the Valence Approximation\\
to Lattice QCD
\end{center}
\vspace{\baselineskip}
\vspace{\baselineskip}
\begin{center}
F. Butler, H. Chen,
J. Sexton\footnote{permanent address: Department of Mathematics,
Trinity College, Dublin 2,
Republic of Ireland},
A. Vaccarino,\\
and D. Weingarten \\
IBM Research \\
P.O. Box 218, Yorktown Heights, NY 10598
\end{center}
\vspace{\baselineskip}
\vspace{\baselineskip}
\begin{center}
ABSTRACT
\end{center}
\begin{quotation}
We evaluate $f_{\pi}/ m_{\rho}$, $f_K/ m_{\rho}$, $1/f_{\rho}$, and $
m_{\phi}/(f_{\phi} m_{\rho})$, extrapolated to physical quark mass, zero
lattice spacing and infinite volume, for lattice QCD with Wilson quarks
in the valence (quenched) approximation. The predicted ratios differ
from experiment by amounts ranging from 12\% to 17\% equivalent to
between 0.9 and 2.8 times the corresponding statistical uncertainties.
\end{quotation}
\vspace{\baselineskip}
\vspace{\baselineskip}
\end{titlepage}
\@startsection{section}{1}{\z@{INTRODUCTION}
In a recent paper~\cite{Butler93} we presented lattice QCD predictions
for the masses of eight low-lying hardrons, extrapolated to physical
quark mass, zero lattice spacing, and infinite volume, using Wilson
quarks in the valence (quenched) approximation. The masses we found were
within 6\% of experiment, and all differences between prediction and
experiment were consistent with the calculation's statistical
uncertainty. We argued that this result could be interpreted as
quantitative confirmation of the low-lying mass predictions both of QCD
and of the valence approximation. It appeared to us unlikely that eight
different valence approximation masses would agree with experiment yet
differ significantly from QCD's predictions including the full effect of
quark-antiquark vacuum polarization.
We have now evaluated the infinite volume, zero lattice spacing,
physical quark mass limit of $f_{\pi}/ m_{\rho}$, $f_K/ m_{\rho}$,
$1/f_{\rho}$, and $m_{\phi}/(f_{\phi} m_{\rho})$. To our knowledge
there have been no previous systematic calculations of this physical
limit of lattice meson decay constants. A review of earlier work is
given in Ref. \cite{Toussaint}. The predicted $m_{\phi}/(f_{\phi}
m_{\rho})$ lies above its observed value by about its statistical
uncertainty of approximately 15\%. The predicted $f_{\pi}/ m_{\rho}$
and $1/f_{\rho}$ are close to 12\% below experiment, equivalent to about
0.9 times the corresponding statistical uncertainties, and $f_K/
m_{\rho}$ is below experiment by 17\%, equal to 2.8 times its
statistical uncertainty.
Although overall the predicted values could be considered in fair
agreement with experiment, the result that three of four decay constants
range from 0.9 to 2.8 standard deviations below experiment suggests the
possiblity that the true, underlying values, if determined without
statistical uncertainties, would fall somewhat below experiment. One
possible source of such disagreement is that we have assumed exact
isospin symmetry and ignored electromagnetic effects. In particular,
although our prediction for $f_{\pi}$ lies below measured values
determined from charged pion decays, reported numbers for the neutral
pion decay constant \cite{Cello} are quite close to our result. The
significance of the closer agreement between our predicted $f_{\pi}$ and
the neutral pion decay constant is unclear to us, however, as a
consequence of the systematic uncertainties arising in the experimental
determination of the neutral pion decay constant.
Another possible source of disagreement between our numbers and
experiment may be the valence approximation itself. A simple physical
argument tends to support of this alternative
\cite{Fermilab}. The valence approximation may be viewed as replacing
the momentum and frequency dependent color dielectric constant arising
from quark-antiquark vacuum polarization with its low momentum limit
\cite{Weingar81}. At low momentum, then, the effective quark charge
appearing in the valence approximation will agree with the low momentum
effective charge of the full theory. The valence approximation might
thus be expected to be fairly reliable for low-lying baryon and meson
masses, which are determined largely by the long distance behavior of
the chromoelectric field. The valence approximation's effective quark
charge at higher momentum can be obtained from the low momentum charge
by the Callan-Symanzik equation. As a consequence of the absence of
dynamical quark-antiquark vacuum polarization, the quark charge in the
valence approximation will fall faster with momentum than it does in the
full theory, at short distance the attractive quark-antiquark potential
in the valence approximation will be weaker than in the full theory, and
meson wave functions in the valence approximation will be pulled into
the origin less than in the full theory. Since decay constants are
proportional to the square of wave functions at the origin, decay
constants in the valence approximation could be expected to be smaller
than in the full theory.
The calculations described here were done on the GF11 parallel computer
at IBM Research \cite{Weingar90} and use the same collection of gauge
configurations and quark propagators generated for the mass calculations
of Ref. \cite{Butler93}. The full set of mass and decay constant
calculations took approximately one year to complete. GF11 was used in
configurations ranging from 384 to 480 processors, with sustained speeds
ranging from 5 Gflops to 7 Gflops. With the present set of improved
algorithms and 480 processors, these calculations could be repeated in
less than four months.
\@startsection{section}{1}{\z@{DEFINITIONS}\label{sect:defs}
The normalization we adopt for pseudoscalar and vector decay constants
in continuum QCD is
\begin{eqnarray}
<\! 0 | J^{\mu}_j(0) | V(p, \epsilon, j)\! > \; & = & \epsilon^{\mu} m_j
F_j, \nonumber \\
<\! 0 | J^{5 \mu}_j(0) | P(p, j)\! > & = & p^{\mu} f_j, \nonumber
\end{eqnarray}
for vector and pseudoscalar states, $| V(p, \epsilon, j)\! >$ and $|
P(p, j)\! >$, respectively, normalized by
\begin{eqnarray}
< \!p | q \!> \; = (2 \pi)^3 p^0 \delta( \vec{p} - \vec{q}). \nonumber
\end{eqnarray}
Here $j$ is a flavor-SU(3) octet index and vector and axial vector
flavor-SU(3) currents $J^{\mu}_j(x)$ and $J^{5 \mu}_j(x)$ are related to
cannonical continuum quark and antiquark fields, $\overline{\psi}_c$
and ${\psi}_c$, and an orthonormal set of
flavor-SU(3) matrices $\lambda_j$ by
\begin{eqnarray}
J^{\mu}_j(x) & = & \overline{\psi}_c(x) \gamma^{\mu} \lambda_j
\psi_c(x), \nonumber \\ J^{5 \mu}_j(x) & = & \overline{\psi}_c(x)
\gamma^5 \gamma^{\mu} \lambda_j
\psi_c(x), \nonumber \\
tr( \lambda_j \lambda_j) & = & \frac{1}{2}. \nonumber
\end{eqnarray}
Assuming exact isospin symmetry, we have
\begin{center}
\begin{tabular}{ll}
$f_i = f_{\pi}$ & $i = 1, \ldots 3,$ \\
$f_i = f_K$ & $i = 4, \ldots 7,$ \\
$F_i = \frac{m_{\rho}}{ f_{\rho}}$ & $i = 1, \ldots 3.$
\end{tabular}
\end{center}
In the valence approximation we have, in addition,
\begin{eqnarray}
F_8 = \frac{3 m_{\phi}}{\sqrt{2} f_{\phi}} \nonumber.
\end{eqnarray}
For simplicity, we will also
use the names $F_{\rho}$ and $F_{\phi}$ for $F_1$ and $F_8$,
respectively. Our normalization gives $f_{\pi}$ the value $(93.15 \pm
0.11)$ MeV.
The hadron mass calculation of Ref. \cite{Butler93} was done using
gaussian smeared lattice quark and antiquark fields defined in Coulomb gauge.
Smeared fields have, therefore, also been adopted for the present
calculation. The smeared field $\phi_r(\vec{x},t)$ is related by
the lattice antiquark field $\psi_{\ell}(\vec{x},t)$ by
\begin{eqnarray}
\phi_r(\vec{x},t) & = & \sum_{\vec{y}} G_r(\vec{x} - \vec{y})
\psi_{\ell}(\vec{y},t), \nonumber \\
G_r(\vec{z}) & = & (\sqrt{\pi}r)^{-3} exp( - \frac{|\vec{z}|^2}{ r^2}).
\nonumber
\end{eqnarray}
The field $\overline{\phi}_r(\vec{x},t)$ is defined correspondingly
from $\overline{\psi}_{\ell}(\vec{x},t)$. We
take the smeared fields $\phi_0(x)$ and $\overline{\phi}_0(x)$ to be
$\psi(x)$ and $\overline{\psi}(x)$, respectively. From these fields,
define smeared currents by
\begin{eqnarray}
J^5_{j r}(x) & = & \overline{\phi}_r(x) \gamma^5 \lambda_j \phi_r(x), \nonumber \\
J^{\mu}_{j r}(x) & = & \overline{\phi}_r(x) \gamma^{\mu} \lambda_j \phi_r(x) ,
\nonumber \\
J^{5 \mu}_{j r}(x) & = & \overline{\phi}_r(x) \gamma^5 \gamma^{\mu} \lambda_j
\phi_r(x) . \nonumber
\end{eqnarray}
Define the correlation functions
$C^P_{j r' r}(t)$, $C^V_{j r' r}(t)$ and
$C^A_{j r' r}(t)$ to be
\begin{eqnarray}
C^P_{j r' r}(t) & = &
\sum_{\vec{x}} <\! [J^5_{ j r'}(\vec{x},t)]^{\dagger} J^5_{j
r}(0,0)\!>, \\
C^V_{j r' r}(t) & = &
\sum_{\vec{x}} <\![J^i_{j r'}(\vec{x},t)]^{\dagger} J^i_{ j r}(0,0)\!>, \\
C^A_{j r' r}(t) & = &
\sum_{\vec{x}} <\![J^{5 0}_{ j r'}(\vec{x},t)]^{\dagger} J^{5 0}_{j r}(0,0)\!>.
\end{eqnarray}
Then for a set of constants $Z^P_{j r' r}$, $Z^V_{j r' r}$ and
$Z^A_{j r' r}$, these correlation functions
have the asymptotic forms, for a large time separation
$t$ and lattice time direction period $T$,
\begin{eqnarray}
\label{asymP}
C^P_{j r' r}(t)
& \rightarrow & Z^P_{j r' r} \{ exp( -m^P_j t) + exp[ -m^P_j (T - t)]\}, \\
\label{asymV}
C^V_{j r' r}(t)
& \rightarrow & Z^V_{j r' r}\{ exp( -m^V_j t) + exp[ -m^V_j (T -
t)]\} \\
\label{asymA}
C^A_{j r' r}(t)
& \rightarrow & Z^A_{j r' r} \{ exp( -m^P_j t) + exp[ -m^P_j (T - t)]\}.
\end{eqnarray}
Here $m^P_j$ and $m^V_j$ are pseudoscalar and
vector masses, respectively. Eqs. (\ref{asymP}) and (\ref{asymA}) imply, in addition,
the asymptotic form
\begin{eqnarray}
\label{asymAoverP}
\frac{C^A_{j r' r''}(t)}{C^P_{j r' r}(t)} \rightarrow
\frac{Z^A_{j r' r''}}{Z^P_{j r' r}}.
\end{eqnarray}
Measured in units of the lattice spacing $a$ the decay constants $f_j a$
and $F_j a$ are then given, for any choice of smearing size $r$, by
\begin{eqnarray}
\label{deffj}
(f_j a)^2 & = &
\frac{ 2 (z^A_j Z^{AP}_{j 0 r})^2}
{m_j a Z^P_{jrr}} \\
\label{defFj}
(F_j a)^2 & = &
\frac{ 2 (z^V_j Z^V_{j 0 r})^2}
{m_j a Z^V_{jrr}}.
\end{eqnarray}
The coefficients
$z^A_j$ and $z^V_j$ are finite renormalizations chosen so that the
lattice currents $z^A_j a^3 J^{5 \mu}_{j 0}$ and $z^V_j a^3 J^{\mu}_{j 0}$
approach the continuum currents $J^{5 \mu}_j$ and $J^{\mu}_j$,
respectively, as the lattice spacing approaches zero.
These constants are often given the ``naive'' values
\begin{eqnarray}
\label{defnaive}
z^{A N}_j & = & 2 k^P_j, \nonumber \\
z^{V N}_j & = & 2 k^V_j.
\end{eqnarray}
where $k^P_j$ and $k^V_j$ are the hopping constants corresponding to the
mass of the quark and antiquark for a pseudoscalar or vector meson,
respectively, with flavor $j$, assumed here to have $m_q =
m_{\overline{q}}$. A standard derivation of the naive finite
renormalization constants, however, which relates the quark terms in the lattice
action to those in the continuum action is actually not correct \cite{Lepage}.
Altough the naive finite renormalization constants do lead to the correct continuum
limit, naive renormalization contributes to decay constant
an additional extraneous
dependence on lattice spacing.
Numerical evidence for this behavior will be given in Section \ref{sect:contlim}.
A consistent
calculation of finite renormalizations \cite{Lepage}, correct to zeroth
in a mean-field theory improved perturbation expansion, gives
\begin{eqnarray}
\label{defz0}
z^{A 0}_j & = & 1 - \frac{3 k^P_j}{4 k_c}, \nonumber \\
z^{V 0}_j & = & 1 - \frac{3 k^V_j}{4 k_c},
\end{eqnarray}
where $k_c$ is
the critical hopping constant at which the pion's mass becomes zero.
To first order in improved perturbation theory, the finite
renormalizations become $z^{A 1}_j z^{A 0}$ and
$z^{V 1}_j z^{V 0}$ where
\begin{eqnarray}
\label{defz1}
z^{A 1} & = & 1 - 0.31 \alpha_{\overline{ms}}( 1 / a),
\nonumber \\
z^{V 1} & = & 1 - 0.82 \alpha_{\overline{ms}}( 1 / a).
\end{eqnarray}
Decay constants for mesons with $m_q \neq m_{\overline{q}}$ will be
discussed below.
For equal values of the up and down quark masses,
$m_u$ and $m_d$, we adopt the convention that a missing flavor index
j always has the value 1. Thus
\begin{eqnarray}
C^P_{r' r}(t) & = & C^P_{1 r' r}(t), \\ \nonumber
Z^P_{r' r} & = & Z^P_{1 r' r}, \\ \nonumber
C^V_{r' r}(t) & = & C^V_{1 r' r}(t), \\ \nonumber
Z^V_{r' r} & = & Z^V_{1 r' r}, \\ \nonumber
C^A_{r' r}(t) & = & C^A_{1 r' r}(t), \\ \nonumber
Z^A_{r' r} & = & Z^A_{1 r' r}. \\ \nonumber
\end{eqnarray}
\@startsection{section}{1}{\z@{PROPAGATORS}\label{sect:props}
Table~\ref{tab:lattices} lists the lattice sizes, parameter values,
sweeps skipped between gauge configurations, and number of
configurations used in the ensembles from which decay constants were
calculated. Gauge configurations were generated with the
Cabbibo-Marinari-Okawa algorithm. A variety of different test support
the expectation that the large number of sweeps run between successive
configurations were more than sufficient to produce statistically
independent values for all of the quantities required by the
present calculations. A discussion of the algorithms by which quark
propagators were found is given in Ref. \cite{Butler93}.
Vector and pseudoscalar masses and the coefficients $Z^P_{r' r}$,
$Z^V_{r' r}$, and $Z^A_{r' r}$ were then determined by fitting
$C^P_{r' r}(t)$, $C^V_{r' r}(t)$, and $C^A_{r' r}(t)$ to
the asymptotic forms of Eqs. (\ref{asymP}) - (\ref{asymA}). As an
alternative means of determining $Z^A_{r' r}$ given a value of
$Z^P_{r' r}$, fits of $C^A_{r' r}(t) / C^P_{r' r}(t)$
were also made to the asymptotic form of Eqs. (\ref{asymAoverP}).
For the lattice $8^3 \times 32$ at $\beta$ of 5.70 we calculated
propagators only for source $r$ and sink $r'$ of size 0. In all other
cases we calculated propagators only for source size $r$ of 2. To
determine decay constants, according to Eqs. (\ref{deffj}) and
(\ref{defFj}), fits for the lattice $8^3 \times 32$ to propagators for
the single sink of size 0 are sufficient, while for all other lattices
fits are needed for sink sizes of both 0 and 2. To determine the range
of time separations to be used in fitting for each $\beta$ and $k$
value, we evaluated effective masses $m^P(t)$, $m^V(t)$ and $m^{P'}(t)$
by fitting $C^P_{ r' r}(t)$, $C^V_{ r' r}(t)$, and $C^A_{ r'
r}(t)$, respectively, to Eqs. (\ref{asymP}) - (\ref{asymA}) at time
separations $t$ and $t+1$. The largest interval at large $t$ showing an
approximate plateau in an effective mass graph we chose as the initial
trial range on which to fit each propatator to the corresponding
asymptotic form of Eqs. (\ref{asymP}) - (\ref{asymA}). Similarly,
the largest interval at large $t$ showing an approximate plateau in a
graph of $C^A_{ r' r''}(t) / C^P_{ r' r}(t)$ we chose as the
initial trial range for a fit to Eq. (\ref{asymAoverP}).
Figures (\ref{fig:570mP}) - (\ref{fig:617mV})
show the plateaus at large $t$ in the
effective masses $m^P(t)$, $m^{P'}(t)$,
the plateau at large $t$ in
the ratio $C^A_{ 0 2}(t) / C^P_{ 0 2}(t)$, and the effective mass
$m^V(t)$ for propagators
with source $r$ of 2 and sink $r'$ of 0.
Figures (\ref{fig:570mP}) - (\ref{fig:570mV}) show results for the
lattice $16^3 \times 32$ at $\beta$ of 5.70 and the largest
corresponding $k$ value, 0.1675. Figures (\ref{fig:593mP}) -
(\ref{fig:593mV}) show results for the lattice $24^3 \times 36$ at
$\beta$ of 5.93 and the largest corresponding $k$, 0.1581. Figures
(\ref{fig:617mP}) - (\ref{fig:617mV}) show results for $30 \times
32^2 \times 40$ at $\beta$ of 6.17 and the largest corresponding $k$, 0.1532.
Fits to data for a range of $t$ were done by minimizing the fit's
$\chi^2$ determined from the full correlation matrix for the data being
fit. An automatic fitting program repeatedly carried out fits on every
connected interval of four or more points within the initial trial
fitting range. For fits to Eq. (\ref{asymAoverP}), which require only a
single fitting parameter, we looked at intervals of three or more
points. The final fitting range was chosen by the program to be the
interval with the smallest value of $\chi^2$ per degree of freedom.
Altough a variety of other criteria could be used to determine the final
fitting range, an advantage of the method we adopted is that it can be
implemented automatically thereby reducing the potential for biases.
The reliability of our in our final extrapolated results depends
to some degree on adopting a consistent choice of fitting ranges at
different parameter values.
The horizontal lines in Figures (\ref{fig:570mP}) - (\ref{fig:617mV})
show the fitted values of masses and $Z^A_{0 2} / Z^P_{0 2}$, and the
pair of vertical lines in each figure indicates the interval of $t$
values in the final fitting range chosen. It is perhaps useful to
mention that since the effective mass at each $t$ depends on data both
at $t$ and $t+1$, the effective mass shown at the highest $t$ within
each final fitting range depends on data outside the fitting range.
Thus the fitted lines tend to approximate the average of the effective
masses within the fitting range but with the effective mass at highest
$t$ omitted. In all but one case, the data shows clear plateaus at
large $t$ extending over more than four time values. These plateaus
appear to be fitted fairly reliably. For the rho propagator,
$C^V_{02}(t)$, on the lattice $24^3 \times 36$, the plateau is more
ambiguous. The fit is made over the four $t$ values from 9 to 12, for
which the three corresponding effective masses fall slowly. The
effective mases at $t$ of 12 to 15 then fall about one standard
deviation below the fit. At $t$ of 16 and 17, the effective masses then
return to the original fitted value. This behavior is consistent with a
statistical fluctuation although it makes the identication of the
plateau at which to fit more ambiguous. Comparable ambiguities do not
occur elsewhere in our data. The interval chosen is approximately a
rescaling of the rho fitting intervals of 5 to 8, at $\beta$ of 5.70 and
13 to 16, at $\beta$ of 6.17. Also, the rho decay constant obtained
from this fit in Section \ref{sect:decays} is interpolates smoothly
values obtained from less ambiguous fits at other values of quark mass
and lattice spacing. If this point were simply elimated from our
extrapolations of $f_{\rho}$ in quark mass and in lattice spacing, our
final continuum predictions would be nearly unchanged.
Value of $\chi^2$ for the fits in Figures (\ref{fig:570mP}) -
(\ref{fig:617mV}) are shown in Table
\ref{tab:propchi2}. Our fits for sinks with size $r'$ of 2, fits at smaller
$k$, and fits on the lattice $24^3 \times 32$ are of comparable quality to
those shown and give comparable $\chi^2$.
Statistical uncertainties of parameters obtained from fits and of any
function of these parameters were determined by the bootstrap method
\cite{Efron}. From each ensemble of N gauge configurations, 100
bootstrap ensembles were generated. Each bootstrap ensemble consists of
a set of N gauge configurations randomly selected from the underlying N
member ensemble allowing repeats. For each bootstrap ensemble the entire
fit was repeated, including a possibly new choice of the final fitting
interval. The collection of 100 bootstrap ensembles thus yields a
collection of 100 values of any fitted parameter or any function of any
fitted parameter. The statistical uncertainty of any parameter is taken
to be half the difference between a value which is higher than all but
15.9\% of the bootstrap values and a value which is lower than all but
15.9\% of the bootstrap values. In the limit of large N the collection
of bootstrap values of a parameter $p$ approaches a gaussian
distribution and the definition we use for statistical uncertainty
approaches the dispersion, $d$, given by $\sqrt{< p^2 > - < p >^2}$.
In the absence of some independent method for determing the predictions
of QCD, it appears inevitable that the choice of $t$ interval on which
to fit data to a large $t$ asymptotic form must be made by some
procedure which depends on the Monte Carlo data itself. Thus the
statistical uncertainties in the data lead to a corresponding
uncertainty in the choice of fitting interval which, in turn, leads to
some additional uncertainty in the fitted result. Another advantage of our
procedure for choosing the fitting interval combined with bootstrap
evaluation of statistical uncertainties is that the values we obtain for
statistical uncertainties include the uncertainty arising from the
choice of fitting interval. A comparison of the error bars found for
our final fits with the error bars found using the same fitting range
held fixed across the bootstrap ensemble shows that typically about 10\%
of the final statistical uncertainty comes from fluctuations over the
bootstrap ensemble of the fitting range itself.
\@startsection{section}{1}{\z@{DECAY CONSTANTS}\label{sect:decays}
To construct decay constants using the finite renomalizations of
Eqs. (\ref{defz0}) or (\ref{defz1}), we require values of the critical
hopping constant $k_c$ and of the strong coupling constant
$\alpha_{\overline{ms}}( 1 / a)$. For $k_c$ we used the values
determined in Ref. \cite{Butler93}. These are listed in Table (\ref{tab:kcrit}).
To determine $\alpha_{\overline{ms}}( 1 / a)$ we first calculated
$\alpha_{\overline{ms}}( \pi / a)$ using the mean-field improved
perturbation theory relation \cite{Lepage,Fermilab}
\begin{eqnarray}
\label{meanfield}
\frac{1}{g^2_{\overline{MS}}( \pi/a)} = \frac{< Tr U / 3>}{g^2_{lat}} + 0.025,
\end{eqnarray}
where $< Tr U>$ is the expectation value of the trace of a plaquette and
$g^2_{lat}$ is $6/\beta$.
We then found $\alpha_{\overline{ms}}( \pi / a)$
given by $ 4 \pi / g^2_{\overline{MS}}( \pi/a) $ and used the two-loop
Callan-Symanzik equation to determine $\alpha_{\overline{ms}}( 1 / a)$.
The corresponding values of $z^{A 1}$ and $z^{V 1}$ are shown
in Table (\ref{tab:zAandzV}).
Values of the decay constant in lattice units $f_{\pi} a$ for the
various lattices, $\beta$ and $k$ shown in Table (\ref{tab:lattices})
are listed in Tables (\ref{tab:f8}) - (\ref{tab:f32}). As a measure of
the lattice spacing in each case, Table (\ref{tab:mrho}) gives the rho
mass in lattice units $m_{\rho}(m_n) a$, extrapolated to the ``normal''
quark mass $m_n$ which produces the physical value of
$m_{\pi}(m_n)/m_{\rho}(m_n)$ \cite{Butler93}. We list also values of
$m_n$ itself.
These
parameters are not given for the lattice $8^3 \times 32$ since in this
case we were not able to calculate propagators at small enough quark
mass to perform the required extrapolation reliably. The finite
renormalizations for the decay constants in Tables (\ref{tab:f8}) -
(\ref{tab:f32}) all include both the leading term $z^{A 0}$ of Eqs.
(\ref{defz0}) and the first order mean-field improved perturbative
correction $z^{A 1}$ of Eqs. (\ref{defz1}). The second column in each
table gives values of $f_{\pi} a$ found from $Z^A_{ r' r}$
determined from a direct fit of $C^A_{ r' r}( t)$ to Eq.
(\ref{asymA}). The third column gives $f_{\pi} a$ found from $Z^A_{
r' r}$ determined by fitting the ratio $C^A_{ r' r}( t) / C^P_{
r' r}( t)$ to Eq. (\ref{asymAoverP}) and then using the value of
$Z^P_{ r' r}$ found from a fit of $C^P_{ r' r}( t)$ to Eq.
(\ref{asymP}). The two sets of data in all cases are statistically
consistent and in all cases, except for the lattice $8^3 \times 32$ for
$k$ below 0.1500, $f_{\pi}$ determined from ratio fits has a smaller statistical error
$f_{\pi}$ determined from direct fits.
The ratio method tends to give less statistical
noise, in effect, because it uses $C^P_{ r' r}( t)$, which is
relatively less noisey, to determine $m^P$ and then extracts only
$Z^A_{ r' r}$ from the more noisey propagator $C^A_{ r' r}( t)$.
The direct method extracts both $Z^A_{ r' r}$ and $m^P$ from
$C^A_{ r' r}( t)$ yielding an $m^P$ with greater noise, which is
then multiplied by a possibly large $t$ and exponentiated, feeding
additional noise back into the value of $Z^A_{ r' r}$. In the
remainder of this paper we use only values of $f_{\pi}$ determined from
ratio fits.
Values of the decay constant in lattice units $F_{\rho} a$ for the
lattices shown in Table (\ref{tab:lattices}) are listed in the fourth
column of Tables (\ref{tab:f8}) - (\ref{tab:f32}). The finite
renormalizations for $F_{\rho}$ shown in these tables all include both
the leading term $z^{V 0}$ of Eqs. (\ref{defz0}) and the first order
mean-field improved perturbative correction $z^{V 1}$ of Eqs.
(\ref{defz1}). The values of $Z^V_{ r' r}$ used to determine
$F_{\rho}$ were all extracted from direct fits of $C^V_{ r' r}( t)$
to Eq. (\ref{asymV}).
\@startsection{section}{1}{\z@{VOLUME DEPENDENCE}\label{sect:vol}
Percentage changes in decay constants going from $8^3 \times 32$ and
$16^3 \times 32$ to $24^3 \times 32$, at $\beta$ of 5.70, are given in
Table~\ref{tab:voldep}. These changes are the same for all choices of
finite renormalization. All of the differences appear to be of marginal
statistical significance and may therefore best be viewed as upper
bounds on the volume dependence of our results. A variety of different
arguments suggest that, for the range of $k$, $\beta$, and lattice
volume we have examined, the errors in valence approximation decay
constants due to calculation in a finite volume $L^3$ are bounded by an
expression of the form $C e^{- L/R}$. A simple non-relativistic
potential model gives this expression with the radius of a hadron's wave
function for $R$. A more elaborate field theory argument gives for $R$
the Compton wave length of a pair of pions, which is the lightest state
that can be exchanged between a pair of identical pseudoscalar or vector
mesons. At $\beta$ of 5.70, $R$ is thus very likely to be between 3 and
5 lattice units. Since the changes in decay constants shown in
Table~\ref{tab:voldep} between $16^3$ and $24^3$ are all less than 5\%
for k $\ge 0.1650$, it follows that the differences between these values
in $24^3$ and in infinite volume should be less than 1.3\%. For
extrapolations to physical quark masses, we use only decay constants
with $k \ge 0.1650$.
\@startsection{section}{1}{\z@{QUARK MASS EXTRAPOLATION}\label{sect:mextrap}
At the largest $k$ on each lattice, the ratio $m_{\pi} / m_{\rho}$ is
significantly larger than its experimentally observed value of 0.179.
Thus to produce decay constants for hadrons containing only light
quarks, our data has to be extrapolated to larger $k$ or, equivalently,
to smaller quark mass. We did not calculate directly at larger $k$ both
because the algorithms we used to find quark propagators became too slow
and because the statistical errors we found in trial calculations became
too large.
Define the
quark mass in lattice units $m_q a$ to be
\begin{eqnarray}
\label{defmq}
m_q a = \frac{1}{2 k} - \frac{1}{2 k_c},
\end{eqnarray}
where $k_c$ is the critical hopping constant at which $m_{\pi}$ becomes zero.
We found $f_{\pi} a$ and $F_{\rho} a$, for all three possible choices of
finite renormalization in Eqs. (\ref{defnaive}) - (\ref{defz1}) to be
nearly linear functions of $m_q a$ over the entire range of $k$
considered on each lattice.
Figure~\ref{fig:mextrap} shows $f_{\pi}$ and $F_{\rho}$, with finite
renormalizations including the first order mean-field improved
perturbative correction, as functions of $m_q$. Data is shown from the
three lattices of Table~\ref{tab:lattices} which we use to evaluate
continuum limits, $16^3 \times 32$, $24^3 \times 36$ and $30 \times 32^2
\times 40$. For convenience, decay constants at each $\beta$ are shown
in units of the central value of the rho mass $m_{\rho}(m_n)$, at the same
$\beta$, extrapolated to the the ``normal'' quark mass $m_n$. Quark
masses $m_q$ for each $\beta$ are shown in units of the central value of
the strange quark mass $m_s$ at the same $\beta$. The data
Figure~\ref{fig:mextrap} has been scaled by the central values of $m_{\rho}(m_n)$
and $m_s$ taken as arbitrary external parameters, and the error bars
shown do not include the effect of statistical fluctuations in
$m_{\rho}(m_n)$ or $m_s$.
Table
(\ref{tab:mrho}) gives values of $m_s$ found in Ref. \cite{Butler93} by
requiring $m_{\pi}[ (m_n + m_s)/2] / m_{\rho}(m_n)$ to be equal to the
physical value of $m_K/m_{\rho}$. The lines in Figure~\ref{fig:mextrap}
are fits of
decay constants measured in lattice units, $f_{\pi} a$ and $F_{\rho} a$,
to linear functions of the quark mass in lattice units, $m_q a$, at the
three smallest quark masses in the data set at each $\beta$. These fits
were obtained by minimizing $\chi^2$ obtained form the full correlation
matrix amoung the data points. The correlation matrix was calculated by
taking averages of data values and products of data values over
bootstrap ensembles generated as described in Section~(\ref{sect:props}). The
$\chi^2$ for these fits, and corresponding fits on the lattice $24^3
\times 32$ at $\beta$ of 5.70, are given in Table \ref{tab:mqchi2}. The
fits in Figure~\ref{fig:mextrap} appear to be quite good and provide, we
believe, a reliable method for extrapolating decay constants down to
light quark masses.
With naive finite renormalization, Eq.
(\ref{defnaive}), or zero order mean-field finite renormalization, Eq.
(\ref{defz0}), $f_{\pi} a$ and $F_{\rho} a$ fit straight lines in $m_q a$
about as well as the first order perturbatively renormalized data of
Figure~\ref{fig:mextrap}.
The linear fits of Figure~\ref{fig:mextrap} permit the determination of
$f_K$ and $F_{\phi}$ in addition to $f_{\pi}$ and $F_{\rho}$. For a
pion composed of a quark and antiquark with mass $m_q \neq
m_{\overline{q}}$, Figure~\ref{fig:mextrap} suggests
\begin{eqnarray}
\label{fpiofmq}
f_{\pi} = \alpha_q m_q + \alpha_{\overline{q}} m_{\overline{q}} + \beta.
\end{eqnarray}
Charge conjugation invariance then gives $\alpha_q =
\alpha_{\overline{q}}$. It follows that the kaon, which is a pion with,
say, $m_q = m_s$ and $m_{\overline{q}} = m_n$, will have the same decay
constant as a pion composed of a single type of quark and antiquark with
$m_q = m_{\overline{q}} = (m_s + m_n)/2$. On the other hand, the linear
fits of Figure~\ref{fig:mextrap} permit $F_{\rho}$ to be extrapolated to
the point $m_q = m_{\overline{q}} = m_s$ which, in the valence
approximation, gives $F_{\phi}$. Tables \ref{tab:fPofa} and
\ref{tab:fVofa} give the value
of $f_{\pi} a$, $f_K
a$, $F_{\rho} a$ and $F_{\phi} a$
obtained from the fits in Figure~\ref{fig:mextrap}.
The statistical uncertainties in
these quantities were obtained by a further application of the bootstrap
method of Section~(\ref{sect:props}). Bootstrap ensembles of the
underlying gauge configurations were generated, and on each bootstrap
ensemble the extrapolated decay constants were recalculated.
The uncertainty in each decay constant was obtained from the resulting
distribution. The correlation matrices used to fit bootstrap data to
linear functions of $m_q a$ were taken to be the same as the correlation
matrices for the full ensemble. To
recalulate correlation matrices separately on each bootstrap ensemble by a further
bootstrap would have been too time consuming.
\@startsection{section}{1}{\z@{CONTINUUM LIMIT}\label{sect:contlim}
The ratios $f_{\pi} / m_{\rho}$, $f_K / m_{\rho}$, $F_{\rho} / m_{\rho}$
and $F_{\phi} / m_{\rho}$ for physical quark masses we then extrapolated
to zero lattice spacing. For Wilson fermions the leading asymptotic
lattice spacing dependence in these decay ratios is expected to be
linear in $a$. On the other hand, as shown in Ref.~\cite{Butler93},
$m_{\rho} a$ follows the two-loop Callan-Symanzik scaling prediction in
$\alpha_{\overline{ms}}$ quite well for the range of $\beta$ considered
here. Thus assuming the asymptotic form of the lattice spacing
dependence of decay ratios appears to be reasonable.
Figure~\ref{fig:aextrap} shows decay constants with first order perturbative
renormalization along with fits to linear functions of $m_{\rho} a$.
The quantity $m_{\rho} a$ may be viewed as the lattice spacing $a$
measured in units of the physical rho Compton wavelength, $1/m_{\rho}$.
The vertical bars at $m_{\rho} a$ of 0, offset slightly for visibility,
are the extrapolated predictions' uncertainties.
The horizontal lines at $m_{\rho} a$ of 0 lying within or slightly above
the range of each prediction are the corresponding experimental values.
The data points in Figure~\ref{fig:aextrap} are from the lattices $16^3
\times 32$, $24^3 \times 36$ and $30 \times 32^2
\times 40$. The values of $\beta$ for these lattices were chosen so
that the physical volume in each case is nearly the same. For lattice
period L, the quantity $m_{\rho} L$ is respectively, 9.08 $\pm$ 0.13,
9.24 $\pm$ 0.19 and, averaged over three directions, 8.67 $\pm$ 0.12
\cite{Butler93}.
The fits shown in Figure~\ref{fig:aextrap} were found by minimizing
$\chi^2$ obtained from the full correlation matrix among the fitted
data. Since both the x and y coordinates of each of the three fitted
points on each line have statistical uncertainties, we evaluated
$\chi^2$ among all six pieces of data and chose as fitting parameters
the slope and intercept of the line along with the x coordinate of each
point. The required correlation matrices were found by the bootstrap
method as were the statistical uncertainties of the extrapolated
predictions. The correlation matrices used in fits for each bootstrap
ensemble were again taken from the full ensemble and not recalculated on
each bootstrap ensemble independently. The $\chi^2$ per degree of
freedom for the fits in Figure~\ref{fig:aextrap}, are given in
Table~\ref{tab:slopechi}.
For the lattice spacing dependence of decay ratios found using zeroth
order mean-field finite renormalization and using naive finite
renormalization we also made fits to linear functions of $m_{\rho} a$.
The slopes with respect to $m_{\rho a}$ of the ratios $f_{\pi} /
m_{\rho}$, $f_K / m_{\rho}$, $F_{\rho} / m_{\rho}$ and $F_{\phi} /
m_{\rho}$ along with $\chi^2$ for each fit are also given in
Table~\ref{tab:slopechi}. It is clear from these results, as
menitioned in Section \ref{sect:defs}, that naive renormalization leads
to decay ratios with significantly stronger lattice spacing dependence
than found for either zeroth or first order improved perturbative
renormalization. There also appears to be some tendency for first order
perturbative renormalization to lead to weaker lattice spacing
dependence than zeroth order. It follows that the extrapolations we have
done to zero lattice spacing are likely to be most reliable for first
order perturbative renormalzation and least reliable for naive
renormalization.
\@startsection{section}{1}{\z@{INFINITE VOLUME LIMIT}
The continuum ratios we found in finite volume were then corrected to
infinite volume by an adaptation of the method used in Ref.
\cite{Butler93} to correct finite volume continuum mass ratios to
infinite volume. From $f_{\pi} / m_{\rho}$, with first order
perturbative renormalization,
as a function of lattice spacing $a$ and lattice period $L$, both
measured in physical units, define the finite volume correction term
$\Delta( a, L)$ to be
\begin{eqnarray}
\label{defdelta}
\Delta( a, L) = \frac{f_{\pi}}{m_{\rho}}( a, \infty) -
\frac{f_{\pi}}{m_{\rho}}( a, L).
\end{eqnarray}
The quantity which we would like to determine
is $\Delta( 0, 9/m_{\rho})$.
Now the ratio $f_{\pi} / m_{\rho}$,
for $L$ of $9/m_{\rho}$,
undergoes a relative change of a bit less than 20\% as $a$ goes from its value
$a_{5.7}$ at
$\beta$ of 5.70 to 0.
Thus we would expect an error of about 20\% of $\Delta( 0, 9/m_{\rho})$
for the approximation
\begin{eqnarray}
\label{approxina}
\Delta( 0, \frac{9}{m_{\rho}}) \approx \Delta( a_{5.7}, \frac{9}{m_{\rho}}).
\end{eqnarray}
Moreover, from our earlier discussion
of the exponential approach of decay ratios to their infinite volume limits,
it follows that
with an additional
error of about 20\% of $\Delta( 0, 9/m_{\rho})$ we have
\begin{eqnarray}
\label{approxinL}
\Delta( a_{5.7}, \frac{9}{m_{\rho}}) \approx
\frac{f_{\pi}}{m_{\rho}}( a_{5.7}, \frac{13.5}{m_{\rho}}) -
\frac{f_{\pi}}{m_{\rho}}( a_{5.7}, \frac{9}{m_{\rho}}),
\end{eqnarray}
where $L$ of $13.5 / m_{\rho}$ corresponds to the lattice
$24^3 \times 32$ at $\beta$ of 5.70.
Finally, a direct evaluation of the
right side of Eq.~(\ref{approxinL}) shows it is
quite small, with statistical errors of
about 6\% of $f_{\pi}/m_{\rho}$.
Combining Eqs. (\ref{defdelta}) - (\ref{approxinL}), we obtain
\begin{eqnarray}
\label{approxfinal}
\frac{f_{\pi}}{m_{\rho}}( 0, \infty) \approx
\frac{f_{\pi}}{m_{\rho}}( 0, \frac{9}{m_{\rho}}) +
\frac{f_{\pi}}{m_{\rho}}( a_{5.7}, \frac{13.5}{m_{\rho}}) -
\frac{f_{\pi}}{m_{\rho}}( a_{5.7}, \frac{9}{m_{\rho}}).
\end{eqnarray}
The error in this approximation should be less than about 40\% of 6\% of
$f_{\pi}/m_{\rho}$, which is 2.4\% of $f_{\pi}/m_{\rho}$. Table
\ref{tab:syserrs} lists estimates of the systematic uncertainties in
equations corresponding to Eq. (\ref{approxfinal}) for other decay
constants and other choices of finite renormalization. First order
perturbative renormalization consistently gives the smallest systematic
error in volume correction largely because the lattice spacing
dependence of these decay ratios is smallest.
The ratios $f_{\pi} / m_{\rho}$, $f_K / m_{\rho}$, $F_{\rho} / m_{\rho}$
and $F_{\phi} / m_{\rho}$, for all three different choices of
finite renormalization, extrapolated to zero lattice spacing with
$m_{\rho} L$ fixed at 9, and then corrected to infinite volume are shown
in Table~\ref{tab:res}. For first order perturbative renormalization
we also give, in Table~\ref{tab:resoverfk}, finite and infinite volume
values of the ratios $f_{\pi} / f_K$, $F_{\rho} / f_K$
and $F_{\phi} / f_K$.
The errors shown for infinite volume ratios
are statistical only and do not include the estimates we have just given
for the systematic error in our procedure for making infinite volume
corrections.
For first order perturbative finite
renormalization, a comparision of Tables \ref{tab:syserrs} and
\ref{tab:res} shows, however, that the systematic errors arising from our
method of obtaining infinite volume results are
much smaller than the statistical errors. For the range of
$\beta$ used in our extrapolation to zero lattice spacing, first order
mean-field theory improved perturbation expansions have been
shown~\cite{Lepage} to work quite well for a wide variety of different
quantities. In addition, as we mentioned earlier, the extrapolation to zero lattice
spacing should be most reliable for this renormalization scheme.
Thus we believe that the numbers in Table~\ref{tab:res}
obtained using first order mean-field perturbative renormalization are
significantly more reliable than those found using the other two
renormalization methods. Zeroth order perturbative renormalization
has been included,
however, to provided some measure of the degree to which our results may
be sensitive to the choice of renormalization. Half of the difference
between first order and zeroth order mean-field perturbative
renormalization appears to us to be a conservative estimate of the
systematic uncertainty in the first order results arising from the
missing second and higher order perturbative renormalization contributions.
In all cases this uncertainty is significantly less than the statistical
errors.
Predictions obtained with naive renormalization have been included in
Table \ref{tab:res} largely as a curiosity. It is
interesting to notice, however, that the difference between
the final, infinite volume results found with naive
renormalization and those found with first order perturbative
renormalization is still less than 1.5 times the naive renormalization
statistical errors.
The predicted infinite volume ratios in Table~\ref{tab:res} are all
statistically consistent with the corresponding finite volume ratios.
The main consequence of the correction to infinite volume is an increase
in the size of the statistical uncertainty in each prediction.
The experimental numbers shown in Table~\ref{tab:res} for $f_{\pi}$ and
$f_K$ are from charged particle decays and for $f_{\rho}$ from neutral
decays. In all cases the uncertainties in the experimental values are
0.001 or less. As mentioned in the introduction, an experimental value
for the neutral pion decay \cite{Cello} gives $f_{\pi} / m_{\rho}$ of
$0.110 \pm 0.005$, which is quite close to our prediction. The
systematic uncertainties in this experimental number, however, are
larger than those for the charged pion decay. As a result the
significance of the improved agreement of our prediction with the
observed neutral pion decay constant is unclear to us.
We would like to thank Paul Mackenzie for discussions, and Mike Cassera,
Molly Elliott, Dave George, Chi Chai Huang and Ed Nowicki for their work
on GF11. We are particularly grateful to Chris Sachrajda for calling our
attention to an error in an earlier version of this paper.
|
1,116,691,498,478 | arxiv | \section{Introduction}
Photoproduction of mesons off nuclei involves in general many different final
states of the meson-nucleus system and can contribute to a wide range of topics
(see \cite{Krusche_11} for an overview). Very interesting for many questions
are two limiting cases. In `quasi-free' processes, the reaction involves one
specific nucleon, called `participant', which is kicked out of the nucleus,
and the rest of the nucleus can be regarded as a `spectator' system that only
compensates the momentum of the bound participant. For light nuclei,
this process is a powerful tool for the study of reactions off quasi-free
neutrons \cite{Krusche_11}; for heavy nuclei it can be used as a testing
ground for meson - nucleus interactions and hadron in-medium properties
\cite{Krusche_04}.
In `coherent' reactions, ideally the meson is produced via a superposition of
the reaction amplitudes from all nucleons and, in the final state, the nucleus
remains in its ground state. A similar process in which no nucleon is removed
from the nucleus but the nucleus is excited to a higher lying nuclear state, is
sometimes called `incoherent' production.
The advantage of the coherent process is the simplicity of the
final state; the ground-state properties of nuclei are well under control.
This reaction is well suited for the study of the in-medium properties of
mesons and nucleon resonances. The undisturbed final state can be easily
constructed from the plane-wave impulse approximation and any deviations may be
attributed to nuclear effects like meson-nucleus final-state interactions or
in-medium modifications of hadron properties. Such programs have been pursued
in particular for the study of medium effects on the production and propagation
of the $\Delta$-resonance in medium via the coherent $\gamma A\rightarrow A\pi^0$
reaction (see e.g. \cite{Drechsel_99,Krusche_02}). The same reaction was also
exploited for the study of nuclear properties such as nuclear mass form factors
\cite{Krusche_05}, and in incoherent production, for nuclear transition form
factors \cite{Tabert_08}. Nuclear form factors in the region of helium and
lithium isotopes have gained much new interest in connection with the study
of halo nuclei (see e.g. \cite{Tomaselli_00,Sick_11}). Coherent pion photoproduction
allows the direct study of the nuclear mass distribution because production of
$\pi^0$ mesons in the $\Delta$-resonance region couples identically to protons
and neutrons.
Until now, coherent photoproduction of heavier mesons off nuclei has almost not been
investigated since such measurements are very demanding. For mesons like the
$\eta$, large momenta are transfered to the nucleus, which suppresses the production
cross section due to the nuclear form factors. Background from breakup reactions,
where the participating nucleon is removed from the nucleus, dominates the
production process. This background must be suppressed either by detection of the
recoil nucleons or by conditions on the reaction kinematics, demanding detector
systems with large solid-angle coverage, large detection efficiency, and excellent
energy and angular resolution.
Recently, coherent photoproduction of $\eta$-mesons from light nuclei
has attracted interest as a tool for the search of so-called $\eta$-mesic nuclei
\cite{Pfeiffer_04,Pheron_12}.
The question is whether the strong interaction allows the formation of quasi-bound
meson-nucleus states, which would be the ideal system for the study of meson-nucleus
interactions. The interaction of low-energy pions with nuclei is too weak for
quasi-bound states but the situation is much different for $\eta$-mesons.
Production of $\eta$-mesons in the threshold region is dominated by the excitation
of the s-wave S$_{11}$(1535) resonance \cite{Krusche_95,Krusche_97}, which couples
strongly (branching ratio $\approx$ 50\% \cite{PDG}) to $N\eta$. As a consequence,
the interaction of $\eta$-mesons with nuclear matter is important also for very
small momenta of the mesons. Typical absorption cross sections are around 30 mb and
are over a wide range of kinetic energy ($T\approx$ 1~MeV - 1~GeV) almost independent
of $T$ \cite{Roebig_96,Mertens_08}. First evidence for an attractive s-wave $\eta N$
interaction, which might lead to the formation of quasi-bound states,
was reported from coupled channel analyses of pion-induced $\eta$-production
reactions \cite{Bhalerao_85,Liu_86} in the 1980s. However, it is still
controversially discussed whether the interaction is strong enough to form such
states. The original prediction was for nuclei with mass numbers $A$ in the range
slightly above 10. However, refined values for the $\eta N$-scattering length
extracted from more precise recent $\eta$-production data extended the discussion
to very light nuclei like hydrogen and helium isotopes. (See \cite{Pheron_12} and
refs. therein for a summary of recent results.)
A much explored experimental approach to identify $\eta$-mesic states is the study
of the threshold behavior of $\eta$-production reactions. Quasi-bound states in the
vicinity of the production threshold should give rise to an enhancement of the
respective cross section over phase-space behavior. Many hadron-induced reactions
(see refs. in \cite{Pheron_12}) have been studied for this purpose. Interesting
threshold effects have been observed for many of them. Particularly strong
enhancements were found for the $pd\rightarrow\eta ^3\mbox{He}$ \cite{Mayer_96} and
$dp\rightarrow\eta ^3\mbox{He}$ reactions \cite{Smyrski_07,Mersmann_07,Rausmann_09},
implying a large $\eta^3$He scattering length. If such effects
are due to a resonance in the $\eta$-nucleus system, they should exist independently
on the initial state of the reaction.
Electromagnetic induced reactions, like photoproduction of mesons, offer a very
clean way to study the $\eta$-nucleus final state, but have small production cross
sections, in particular for the coherent process. Photoproduction of $\eta$-mesons
in the threshold region has been studied for several hydrogen and helium isotopes
\cite{Pfeiffer_04,Krusche_95,Krusche_95a,Hoffmann_97,Weiss_01,Weiss_03,Hejny_99,Hejny_02}
and these results allowed the characterization of the spin and isospin
structure of the relevant transition amplitudes \cite{Krusche_03}. The reaction is
dominated by the excitation of the S$_{11}$(1535) resonance via the
$E_{0+}$-multipole, which involves a spin-flip of the participating nucleon.
This means, that coherent $\eta$-production is practically forbidden for nuclei
with spin $J=0$ ground states. Also for nuclei with non-zero ground-state spins,
depending on the nuclear structure, only a fraction of the nucleons (those which
can participate in spin-flip transitions) may contribute. Furthermore, the
electromagnetic excitation of the S$_{11}$(1535) resonance is mainly isovector
($A_{1/2}^{IS}/A_{1/2}^p \approx 0.1$, where $A_{1/2}^p$ is the helicity coupling for
the proton and $A^{IS}_{1/2}$ is the isoscalar part of the helicity coupling)
\cite{Krusche_03}, so that contributions from protons and neutrons will cancel to
a large extent in coherent $\eta$-production. Together with the large momentum
transfers involved, these features lead to very small reaction cross sections.
\begin{figure}[thb]
\centerline{
\resizebox{0.45\textwidth}{!}
\includegraphics{fig_01.eps}
}}
\caption{Total cross section for the $\gamma ^3\mbox{He}\rightarrow \eta^3\mbox{He}$
reaction \cite{Pheron_12} compared to plane-wave impulse approximation.
Vertical dashed lines indicate coherent and breakup threshold for $\eta$-production.
Insert: ratio of data and impulse approximation.
}
\label{fig:helium}
\end{figure}
Only nuclei with ground-state spin $J$ and isospin $I$ different from zero are
promising candidates for the observation of the coherent process. Previous
experimental results are consistent with this picture. The cross section for
coherent production off the deuteron ($J$=1, $I$=0) is small \cite{Weiss_01},
(typical values for $d\sigma/d\Omega$ are on the order of 10 nb/sr). Only upper limits
have been extracted for the $J=I=0$ nucleus $^4$He \cite{Hejny_99}. The most
interesting case studied so far is the $J=I=$1/2 nucleus $^3$He
\cite{Pfeiffer_04,Pheron_12}. The coherent process was clearly identified.
The energy dependence of the total cross section shown in Fig.~\ref{fig:helium}
\cite{Pheron_12} is different from the expectation for reaction phase-space.
A strong threshold enhancement relative to the plane-wave impulse approximation (PWIA),
similar to the results from hadron-induced reactions
\cite{Smyrski_07,Mersmann_07,Rausmann_09}, is observed. The angular distributions
close to threshold are more isotropic than expected from the shape of the
nuclear form factor \cite{Pheron_12}. Both observations together have been taken
as indication for the formation of a resonant-like meson-nucleus state
\cite{Pfeiffer_04,Pheron_12}.
So far, this is the only isolated case where coherent $\eta$-threshold production
off nuclei could be studied. Almost nothing is known experimentally about its
systematics and the validity of the simple plane-wave impulse approximation used
in \cite{Pfeiffer_04,Pheron_12}.
The present work therefore aimed at the measurement of this reaction from a different
light nucleus. Apart from $^3$H, the mirror nucleus of $^3$He, which, however,
is difficult to handle as a target, the lightest stable isotope with nonzero
ground-state spin ($J^{\pi}=$ 3/2$^-$) and isospin ($I=$ 1/2) is $^7$Li. In the
relevant range of momentum transfer its squared form factor \cite{Suelzle_67},
which is expected to be proportional to the cross section, is roughly smaller by
an order of magnitude compared to $^3$He \cite{McCarthy_77}. However, a factor of
$\approx$ 3 in counting statistics may be recovered from the target thickness
(number of nuclei/cm$^2$), making the measurement feasible.
This paper is organized as follows. The assumptions and inputs for the
modelling of coherent $\pi^0$- and $\eta$-pho\-to\-pro\-duc\-tion off $^7$Li in
plane-wave impulse approximation are discussed in Sec.~\ref{sec:pwia}.
The experimental setup is described in Sec. \ref{sec:setup}
and the data analysis, in particular the identification of events from the coherent
process, is discussed in Sec. \ref{sec:ana}. The measured cross sections for coherent
$\pi^0$ and $\eta$-production are summarized in Sec. \ref{sec:results} and compared
to the results of the PWIA modelling.
\section{Plane wave impulse approximation}
\label{sec:pwia}
The PWIA of the coherent meson production follows the work of Drechsel et al.
\cite{Drechsel_99}, taking into account the specific features of the $\pi^0-A$
and $\eta-A$ final states. The main inputs are nuclear form factors and
the amplitudes for the elementary meson production reactions off the free nucleon.
The elastic charge form factor $F_{C}$ of $^7$Li has been measured with electron
scattering over a wide range of momentum transfer $q$
\cite{Suelzle_67,Bumiller_72,Lichtenstadt_89}.
Lichtenstadt et al. \cite{Lichtenstadt_89} also reported results for the inelastic
transition form factor $F_{Cx}$ related to the excitation of the 478-keV state in
$^7$Li. Since the charge and mass rms radii of $^7$Li are similar
\cite{Tomaselli_00,Kajino_88}, we can use the charge form factors as basis. However,
they include the effects from the charge distribution of the proton. For the meson
production reactions we need instead the distribution of point-like nucleons.
Therefore, the measured charge form factors must be divided by the proton dipole
form factor $F_p^2(q^2)$; the ratios are denoted by $F_{C*}$ and $F_{Cx*}$.
Figure \ref{fig:chff} summarizes the charge form factors and their parametrizations
used in the PWIA modelling. For the elastic form factor, the parametrization
of $F_{C*}$ is also shown. The $q$-dependence of the inelastic form factor $F_{Cx}$
for small values of $q$ is approximated by the model results cited in
\cite{Lichtenstadt_89}.
\begin{figure}[htb]
\resizebox{0.49\textwidth}{!}
\includegraphics{fig_02.eps}
}
\caption{Elastic charge form factors from Bumiller et al. \cite{Bumiller_72},
(black) triangles, Suelzle et al.\cite{Suelzle_67}, (blue) stars,
and Lichtenstadt et al. (green) squares.
Dashed line: parametrization of form factor $F_{C}$, Solid line: form factor
$F_{C*}$. (Magenta) dots: inelastic form factor for 478-keV excitation
\cite{Lichtenstadt_89}. Dotted curve: parametrization of inelastic form factor.
}
\label{fig:chff}
\end{figure}
The construction of the transition amplitudes starts from the effective total
energy $W=\sqrt{s^{\rm eff}}$ of the incident photon (four-momentum $P_{\gamma}$,
laboratory energy $E_{\gamma}$) and an off-shell nucleon (four-momentum $P_{N}$)
with three-momentum $\vec{p}_{N}$ from its motion inside the nucleus
\begin{equation}
s_{\rm eff} = (P_{\gamma} + P_{N})^2.
\end{equation}
The nucleon momentum $\vec{p}_N$ is obtained in the factorization approximation
\cite{Drechsel_99} from the momentum transfer $\vec{q}$ to
the nucleus by
\begin{equation}
{\bf{p}}_N = -\frac{A-1}{2A}{\bf{q}} = -\frac{3}{7}{\bf{q}}\; ,
\end{equation}
where $A$ is the nuclear mass number and all momenta are in the laboratory frame
(note that the expressions in \cite{Drechsel_99} refer to the center-of-momentum frame).
The amplitudes of the elementary reactions are then evaluated at
$W(E_{\gamma},\vec{q})$.
The amplitude for meson photoproduction off nuclei is in general given by
\begin{equation}
{\cal{F}} = L +i \sigma K, \;\;\; d\sigma = |L|^2 +|K|^2
\end{equation}
with the spin-independent part $L$ and the spin-dependent part $K$. It is
efficiently evaluated in the CGLN parameterization \cite{Chew_57}, involving the
four invariant amplitudes $F_1$,...,$F_4$.
The simplest case is coherent $\pi^0$-production from spin $J=0$ nuclei in the
$\Delta$-resonance region \cite{Drechsel_99}. The elementary production amplitudes
are identical for protons and neutrons. The dominant contribution to
$\gamma A\rightarrow \pi^0 A$ for spin $J=0$ nuclei thus involves the
spin/isospin-independent part of the production amplitude.
In the CGLN representation a spin-independent piece arises only from the term
with the $F_2$ amplitude. Due to the pseudoscalar nature of the pion and the
overall symmetry of the problem, this term has a $\mbox{sin}(\Theta^{\star}_{\pi})$
factor ($\Theta_{\pi}^{\star}$: pion polar angle in photon-nucleus cm-system) in
the amplitude \cite{Drechsel_99}. Since the dominant excitation of the
$\Delta$-resonance is not isospin dependent, all amplitudes from protons and
neutrons add coherently, which is reflected in a factor $A$ in the amplitude.
The full evaluation of the $L$-piece gives:
\begin{equation}
\frac{d\sigma_0}{d\Omega} = \frac{1}{2} \frac{q_{\pi}^{\star}}{k_{\gamma}^{\star}}
|F_{2}(W)|^2 A^2 \mbox{sin}^2(\Theta_{\pi}^{\star}) F^2_{C*}(q^2)\;,
\label{eq:pi_no}
\end{equation}
where the ratio of pion and photon momenta $q_{\pi}^{\star}$, $k_{\gamma}^{\star}$
gives the phase-space factor for the photon-nucleus system. Numerical values for
the CGLN amplitude $F_{2}$ were taken from the MAID analysis of pion photoproduction
\cite{MAID}.
The $^7$Li case is complicated by the unpaired proton in the $1p_{3/2}$ orbit,
which gives rise to additional contributions involving also spin-flip amplitudes
that may contribute to all four CGLN amplitudes. Apart from elastic reactions, the
$1p_{3/2}$ proton may be excited to the $1p_{1/2}$ orbit, populating the low lying
$1/2^-$ state of $^7$Li with an excitation energy of 478 keV. Incoherent pion
production to this final state cannot be separated experimentally from the coherent
process and is thus included in the measured cross sections. The spin-dependent
contribution must be small compared to the spin-independent contribution because
it is lacking the $A^2$ factor, but it is important for extreme forward or backward
angles (because it has a $\mbox{cos}^2(\Theta_{\pi}^{\star})$ dependence instead of the
$\mbox{sin}^2(\Theta_{\pi}^{\star})$ for the spin-independent part).
These contributions are approximated from the leading $M_{1+}$ multipole.
Evaluation of the multipole expansion of the CGLN amplitudes for the spin-dependent
part of the cross section leads to
\begin{equation}
\frac{d\sigma_{sf}}{d\Omega} \approx \frac{q_{\pi}^{\star}}{k_{\gamma}^{\star}}
|M_{1+}(W)|^2 \mbox{cos}^2(\Theta_{\pi}^{\star}) \left(F^2_{C*}(q^2) +F^2_{Cx*}(q^2)\right)
\label{eq:pi_sf}
\end{equation}
when all multipoles except the leading $M_{1+}$ are neglected. The incoherent
excitation of the nucleus is included, but the contribution turns out to be
negligible (see Sec.~\ref{sec:pires}). The amplitudes $M_{1+}$ are again taken from
the MAID-model \cite{MAID}. For the full PWIA cross section the incoherent sum
\begin{equation}
\frac{d\sigma_{\pi A}}{d\Omega} = \frac{d\sigma_{0}}{d\Omega}
+ \frac{d\sigma_{sf}}{d\Omega}
\label{eq:pi_pwia}
\end{equation}
is used.
The situation is different for $\eta$-production. Since the elementary reaction is
completely dominated by an isovector, spin-flip amplitude, there is no piece
corresponding to Eq.~\ref{eq:pi_no} in pion production. Like in the $^3$He case
\cite{Pheron_12}, the main contribution to coherent production comes from the
S$_{11}$ excitation of the unpaired nucleon via a spin-flip transition. The main
difference is that $^3$He has an unpaired neutron while $^7$Li has an unpaired
proton. We use therefore a similar PWIA as for $^3$He in
\cite{Pheron_12}, including the incoherent excitation via the $F_{Cx*}$-term
\begin{equation}
\frac{d\sigma_{\eta A}}{d\Omega} =
\left(\frac{q_{\eta}^{(A)}}{k_{\gamma}^{(A)}}
\frac{k_{\gamma}^{(N)}}{q_{\eta}^{(N)}}\right)
\frac{d\sigma_{\rm elem}}{d\Omega}
\left(F^2_{C*}(q^2) + F^2_{Cx*}(q^2)\right)
\label{eq:eta}
\end{equation}
with a parameterization of the measured $\gamma p\rightarrow p\eta$ cross section
from \cite{McNicoll_10} for the elementary cross section $d\sigma_{\rm elem}$.
The change of phase space between the different c.m. systems is derived from the
photon and $\eta$ three-momenta in the photon-nucleon
($k_{\gamma}^{(N)}$, $q_{\eta}^{(N)}$),
and photon-nucleus ($k_{\gamma}^{(A)}$, $q_{\eta}^{(A)}$) c.m. systems.
\section{Experimental setup}
\label{sec:setup}
The experimental setup was identical to the one used in \cite{Schumann_10,Zehr_12}, apart
from the target (liquid hydrogen targets for \cite{Schumann_10,Zehr_12}, threshold settings,
and trigger conditions.
\begin{figure}[htb]
\centerline{\resizebox{0.48\textwidth}{!}
\includegraphics{fig_03.eps}
}}
\caption{Setup of the electromagnetic calorimeter combining the
Crystal Ball and TAPS detectors. Detectors for charged particle identification
were mounted in the Crystal Ball (PID and MWPC) and in front of the TAPS forward
wall (TAPS Veto-detector).
}
\label{fig:calo}
\end{figure}
The measurement used the tagged photon beam \cite{Anthony_91,Hall_96} from a primary
883 MeV electron beam of the Mainz MAMI accelerator \cite{Herminghaus_83,Walcher_90}.
The photons irradiated a $^7$Li target (enrichment 99\%) of 5.4 cm length and a
density of 0.534 g/cm$^{3}$, corresponding to a surface density of 0.264
nuclei/barn. The reaction products were detected with an electromagnetic
calorimeter composed of the Crystal Ball (CB) \cite{Starostin_01} and TAPS detectors
\cite{Novotny_91,Gabler_94}. The 672 NaI crystals of the CB covered the full
azimuthal angle for polar angles between 20$^{\circ}$ and 160$^{\circ}$ around the
target, which was mounted in the center of the CB. TAPS covered polar angles between
1$^{\circ}$ and 20$^{\circ}$ as a hexagonal wall of 510 BaF$_{2}$ crystals, mounted
1.75 m downstream from the target. Individual plastic detectors in front of each
crystal were used for charged particle identification. A schematic view of the
setup, which covered $\approx$ 98\% of 4$\pi$, is shown in Fig.~\ref{fig:calo}.
It was complemented by a cylindrical Particle Identification Detector (PID)
\cite{Watts_04}, mounted around the target inside the CB, which covered the same
solid angle as the CB.
The experiment trigger was based on a subdivision of the CB and TAPS into logical
sectors. For TAPS these were eight sectors of 64 modules in a pizza-like geometry, and
for the CB 45 rectangles. The trigger required signals in at least two logical sectors
of the calorimeter above a threshold of 20~MeV and an analog energy sum of the CB
modules above 50 MeV. Once a valid trigger had been generated, thresholds for the
readout of individual modules were 5~MeV in TAPS and 2~MeV in the CB.
\section{Data analysis}
\label{sec:ana}
The different analysis steps for the identification of photons, charged pions, and
recoil nucleons are discussed in more detail in \cite{Schumann_10,Zehr_12}.
The analysis of coherent neutral meson production off nuclei is special in so far
as no charged particles (no charged pions, no recoil protons) may occur in the
final state. Detection of charged particles was only used to veto events, which
simplifies the analysis (there was no need to separate charged pions from protons
or to extract energy information for the charged particles). Accepted were only
events with exactly two photons (from the $\pi^0\rightarrow \gamma\gamma$ or
$\eta\rightarrow \gamma\gamma$ decays) or with exactly six photons
($\eta\rightarrow 3\pi^0\rightarrow 6\gamma$). These are particularly clean
data samples.
The invariant-mass spectrum of photon pairs for incident photon energies below
300~MeV is shown in Fig.~\ref{fig:pi_inv}. It is practically background free.
No other reactions with significant cross section produce two or more photons
in this energy range.
\begin{figure}[thb]
\centerline{\resizebox{0.45\textwidth}{!}
\includegraphics{fig_04.eps}
}}
\caption{Invariant mass spectrum for two-photon events for incident photon energies
below 300 MeV. Statistical uncertainties smaller than symbol sizes.
The solid (red) curve is a Monte Carlo simulation of the detector
response.
}
\label{fig:pi_inv}
\end{figure}
\begin{figure}[htb]
\centerline{\resizebox{0.45\textwidth}{!}
\includegraphics{fig_05.eps}
}}
\caption{Missing energy analysis for single $\pi^0$ production for different
incident photon energies. Black dots: measurement (statistical uncertainties smaller
than symbol size),
solid (red) curves: MC-simulation for coherent events,
dashed (green) curves: MC for breakup events,
dotted (blue): sum of both.
}
\label{fig:pi_misse}
\end{figure}
\clearpage
\begin{figure}[thb]
\resizebox{0.43\textwidth}{!}
\includegraphics{fig_06.eps}
}
\caption{Invariant-mass spectra for two-photon (left-hand side) and six-photon
(right-hand side) events in the energy region of the $\eta$-production threshold.
Solid (red) lines: signal shapes, dashed (green) lines: fitted background,
dotted (blue) curves: sum of both. Ranges of incident photon energies are given
at left-hand side.
}
\label{fig:eta_inv}
\end{figure}
Double $\pi^0$ production sets in with a very low cross section around 300 MeV
and loss of two of the four decay photons is unlikely. The only possible background
source is production of single $\pi^0$ off quasi-free neutrons with loss of one
decay photon and misidentification of the neutron as photon. However, the
corresponding recoil neutrons are mostly emitted to forward angles and can be
identified in TAPS with time-of-flight versus energy and pulse-shape analyses.
The important step is then the separation of the coherent reaction from breakup
reactions with emission of recoil nucleons. The suppression of such events by the
required non-detection of recoil nucleons is limited since the detection
efficiency for recoil neutrons is only on the order of 30\% (larger than 90\%
for recoil protons). We use therefore in addition the overdetermination
of the reaction kinematics of the two-body final state.
The laboratory kinetic energy of the meson $E_m^{\rm lab}$ is directly measured with
the calorimeter, and its kinetic cm-energy $E_m^{\star}$ follows from the incident
photon energy $E_{\gamma}$.
\begin{figure}[thb]
\resizebox{0.45\textwidth}{!}
\includegraphics{fig_07.eps}
}
\caption{Missing-energy spectra for events in the $\eta$ invariant mass peaks for
different ranges of incident photon energy. Notation for curves is as in
Fig.~\ref{fig:pi_inv}. Vertical dotted lines: expected positions of coherent peaks.
Left-hand side: two-photon events, right-hand side: six-photon events.
}
\label{fig:eta_misse}
\end{figure}
The mesons are boosted into the cm-system and the
difference $\Delta E_{\pi}$ of the two kinetic energies in the cm-system is
constructed as
\begin{equation}
\Delta E_m = E_m^{\star}(E_m^{\rm{lab}}) - E_m^{\star}(E_{\gamma}).
\end{equation}
\clearpage
The result of this analysis for pion production is shown in Fig.~\ref{fig:pi_misse}.
The peaks at zero missing energy correspond to coherent production and dominate the
process at low incident photon energies. At higher incident photon energies breakup
background appears at negative missing energies. The shape of the signals was
generated with a full Monte Carlo simulation of the experiment using the GEANT3
package \cite{Brun_86}. The event generator for the coherent process was based on
trivial two-body kinematics; for the breakup reaction, the momentum distribution of
the bound nucleons was taken into account. Final state interactions were not taken
into account, which explains the deviations between data and Monte Carlo in the
tails of the distributions for higher incident photon energies.
The separation of coherent and breakup
processes, which must be done in dependence on the pion angles, is straightforward
for the energy range up to $E_{\gamma}$ = 300 MeV as shown in Fig.~\ref{fig:pi_misse}.
At higher incident photon energies, the contribution from breakup reactions becomes
dominant in the angle-integrated missing-energy spectra. Across the angular
distribution missing energy spectra vary. The fraction of coherent events compared to
breakup is larger for forward angles, but due to kinematics the separation between
coherent and breakup events in missing energy is better at backward angles.
The `coherent reaction' includes incoherent excitation of the 478-keV level, which cannot
be resolved by the missing energy analysis.
The analysis for coherent $\eta$-production follows the same scheme. Invariant-mass
and missing-energy spectra are summarized in
Figs.~\ref{fig:eta_inv} and \ref{fig:eta_misse}. The main difference to $\pi^0$
production is that, near threshold, the ratio of coherent to breakup
cross sections is much less favorable. This comes from two effects discussed in
Sec.~\ref{sec:pwia}. The involved momentum transfers are much larger,
suppressing the coherent cross section via the form factor. Since furthermore
(apart from small components in the nuclear wave functions) only the $1p_{3/2}$
proton contributes, the $A^2$ factor is missing in the coherent cross section.
The invariant-mass peaks from the two-photon decays show some background
(double $\pi^0$ production with two undetected photons, single $\pi^0$ production
off quasi-free neutrons with one undetected photon and a misidentified neutron),
which must be subtracted. The invariant-mass signals of the six-photon decays are
much cleaner. In this case, the invariant masses of the three $\pi^0$-mesons
are also used to identify the reaction as discussed in \cite{Pheron_12}.
The contribution of breakup background to the missing-energy spectra is substantial.
A clean coherent signal appears only in the immediate threshold region. At higher
energies the signal can be extracted only by fitting the simulated line-shapes to
the data, which for incident photon energies above 650 MeV becomes unfeasible.
However, the simultaneous extraction of the cross section from the two different
$\eta$-decay channels gives some estimate for the typical level of uncertainty.
Absolute cross sections were extracted from the measured yields with the target
surface density, the incident photon flux, and the simulated detection efficiencies.
The latter were generated with GEANT3 \cite{Brun_86} simulations. Typical values
(depending on incident photon energy and polar angle of the meson) are 20\% - 50\%
for coherent $\pi^0$ production and 35\% - 40\% for coherent $\eta$-production
to the six-photon final state and 60\% - 70\% for the two-photon final state.
The uncertainty for the detection efficiency simulations is smaller than in
\cite{Zehr_12} for two reasons. Only photons had to be detected, for which the
response of the detector system is best understood. There is no additional uncertainty
from the properties of the event generator because in both cases only trivial two-body
kinematics is involved in the final state. We estimate the systematic uncertainty of
the detection efficiency below the 5\% level. The incident photon flux was determined
from the counting of the number of deflected electrons in the focal plane by live-time
gated scalers. The fraction of correlated photons that pass the collimator and reach
the target (tagging efficiency, $\approx$ 50\% for this experiment) was determined with
special experimental runs. A total absorbing lead-glass counter was moved into the
photon beam at reduced intensity of the primary electron beam. The intensity was reduced
at the electron source, so that no accelerator parameters differed from normal running.
In addition to these periodical absolute measurements the intensity was monitored in
arbitrary units during normal data taking with an inonization chamber at the end of
the photon-beam line. The systematic uncertainty for the flux measurement is
estimated below the 5\% level. The systematic uncertainty of the surface density of
the solid $^7$Li target is estimated as 3\% (due to a somewhat irregular shape
of the target).
The largest uncertainty is related to the separation of coherent signal and breakup
background. For coherent $\pi^0$ production we estimate a systematic uncertainty due
to this effect of 2\% - 5\% for incident photon energies from threshold to 200~MeV,
5\% - 8\% between 200~MeV and 300~MeV, and 8\% - 20\% between 300~MeV and 500~MeV.
For $\eta$-production most of this uncertainty is reflected in the statistical
uncertainties of the yields, which include the uncertainty related to the fitting
of the missing energy spectra.
\section{Results}
\label{sec:results}
The results for the two reaction channels are of different quality and intended
for different purposes. The $\pi^0$-data have excellent statistical quality.
In most figures their statistical error bars are smaller than the symbol sizes.
Although we compare them here only to PWIA approximations to discuss their most
important features, they may serve as precision tests for more advanced models,
taking into account the correct nuclear structure of $^7$Li and the nuclear effects
beyond PWIA.
The pioneering results for $\eta$-production, at a cross section level of 10 - 20 nb,
have limited statistical precision, but still allow a comparison of the threshold
behavior to the $^3$He case.
\begin{figure*}[thb]
\centerline{\resizebox{0.98\textwidth}{!}
\includegraphics{fig_08.eps}
}}
\caption{Angular distributions for coherent $\pi^0$-production for different ranges
of incident photon energy. Curves: results of PWIA model normalized in absolute scale
to experiment.
}
\label{fig:pi_ang}
\end{figure*}
\clearpage
\subsection{Coherent $\pi^0$-photoproduction}
\label{sec:pires}
Angular distributions and the total cross section for the
$\gamma+^7$Li$\rightarrow {^7}\mbox{Li}+\pi^{0}$ reaction are summarized
in Figs.~\ref{fig:pi_ang} and \ref{fig:pi_total}. We discuss first the total cross
section. The energy dependence and absolute magnitude reflect the properties of the
elementary production cross section off the nucleon, trivial factors like $A^2$ and
sin$^{2}(\Theta_{\pi}^{\star})$, the nuclear form factor, FSI effects, and possible
in-medium modifications of the involved nucleon resonances (here the $\Delta$(1232)).
We compare the data to similar results for the deuteron \cite{Krusche_99} and
$^{12}$C \cite{Krusche_02}. The systematic evolution of the $\Delta$-resonance peak
in dependence of the nuclear mass number from `almost free' production for the deuteron
to `almost nuclear density' for carbon is clearly visible.
\begin{figure}[htb]
\resizebox{0.5\textwidth}{!}
\includegraphics{fig_09.eps}
}
\caption{Total cross section for coherent $\pi^0$-production. The shaded (green) band
indicates the size of systematic uncertainty of the data.
Data for the deuteron (scaled down by factor of two) \cite{Krusche_99} and $^{12}$C
\cite{Krusche_02} for comparison.
Solid curve: PWIA results, Eq.~\ref{eq:pi_pwia}, dashed curve: predicted contribution
of spin-flip amplitude (first term of Eq.~\ref{eq:pi_sf}) scaled up by a factor of 10,
dotted curve: incoherent contribution from excitation of 478-keV level
(second term of Eq.~\ref{eq:pi_sf}) scaled up by factor of 100.
Insert: ratio of measured cross section and PWIA, dashed lines: range of systematic
uncertainty of $^7$Li data.
}
\label{fig:pi_total}
\end{figure}
One should keep in mind, as discussed in detail in \cite{Krusche_02},
that the effective position of the $\Delta$-resonance peak is determined by different
effects: the interplay between the nuclear form factor and the sin$^2(\Theta)$ term
in the PWIA approximation (see Eq.~\ref{eq:pi_no}; not valid for the $J=1$ deuteron
which is lacking the sin$^2$ term), the FSI effects in distorted-wave impulse
approximation (DWIA), and the density
dependent in-medium modification of the position and width of the resonance.
Actually, the model of Drechsel et al. \cite{Drechsel_99}, which reproduced quite
well the data for nuclei from carbon to lead \cite{Krusche_02}, predicts an
{\em upward} shift of the $\Delta$(1232) in-medium resonance position; although
the peak in the cross section appears to be {\em downward} shifted due to the
other effects. The lithium case is interesting because it is transitional
between the $\Delta$-in-vacuum and $\Delta$-in-normally-dense-matter
cases. Previous results \cite{Krusche_02} have shown that the measured cross sections
from carbon to lead can be reproduced with $\Delta$-self energies extracted from
$^{4}$He data. However, $^4$He is itself a very dense nucleus and the effective density
of $^7$Li is significantly lower than for any of the nuclei studied so far.
The extraction of $\Delta$-self energies from the lithium data will require detailed
model calculations, taking into account the FSI effects, which are not yet available
but in progress.
Here, we compare the measured cross sections to the PWIA modeling discussed in
Sec.~\ref{sec:pwia}. It is obvious from the figure that the elastic spin-flip-term
(term with $F^2_{C*}$ in Eq.~\ref{eq:pi_sf}) and the incoherent excitation of
the 478-keV state of the $^7$Li nucleus (term with $F^2_{Cx*}$ in Eq.~\ref{eq:pi_sf})
are negligible effects for the total cross section, both much smaller than
the systematic uncertainty of the data. (Note, however, the importance of the
spin-flip-term for the angular distributions discussed below.)
In the low-energy range, up to incident photon energies of $\approx$ 225 MeV, the
measured cross sections agree surprisingly well with the PWIA results (mostly within
systematic uncertainties of the data).
This demonstrates that the trivial effects of the coherent process are well understood
in PWIA and that in this regime effects from FSI and in-medium modifications of the
$\Delta$-resonance must be either both small or cancelling. In the maximum of the
$\Delta$-resonance, PWIA largely overestimates the data. This is consistent with the
expected onset of strong FSI and the in-medium damping of the $\Delta$-resonance.
At even higher incident photon energies, beyond the energy range where the elementary
cross section is dominated by the $\Delta$(1232) excitation, the model is missing
contributions from other photoproduction multipoles (e.g. from the excitation of the
P$_{11}$(1440) and D$_{13}$(1520) resonances and background terms), so that no
agreement can be expected.
The shape of the angular distributions in Fig.~\ref{fig:pi_ang} is quite well reproduced
at low incident photon energies and even reasonably well at higher energies. This is so,
because the shape is dominated by the sin$^2(\Theta)$ term and the nuclear form factor.
However, a closer inspection of the angular distributions also shows some systematic
deviations between experiment and PWIA for the energy range where FSI effects seem to
be small. For a more detailed analysis Figure~\ref{fig:ff} shows a reduced version
of the differential cross sections as a function of the squared momentum transfer
$q^2$. The cross sections have been divided by the PWIA estimate from Eq.~\ref{eq:pi_pwia},
but without the form-factor terms in Eqs.~\ref{eq:pi_no} and \ref{eq:pi_sf}. The square roots
of these ratios, shown in the figure, correspond to the nuclear mass form factor when the
PWIA is valid (and the incoherent excitation can be neglected). Shown are only the results
for pion cm-angles with $\mbox{cos}(\Theta^{\star}_{\pi})>-0.5$, where the PWIA
approach seems to be reasonable. The first important observation is that the
$q^2$-dependence of these distributions is almost independent of incident photon energy.
This is what one would expect for a $q^2$ dependence related to the nuclear form factor.
\begin{figure}[ttb]
\resizebox{0.49\textwidth}{!}
\includegraphics{fig_10.eps}
}
\caption{Form factor of $^7$Li extracted from the ratio of measured angular
distributions and PWIA results for different ranges of incident photon energy
(see text). The absolute scale corresponds to the 152 MeV data, the other data are
scaled down by successive factors of two. The solid lines correspond to fits with
Eq.~\ref{eq:owells}. The insert shows the rms mass radii (red dots) extracted from the
fits with Eq.~\ref{eq:mod_rms}. The solid line represents
the average (dotted lines statistical uncertainty), the dashed line the rms charge
radius (for point-like protons).
}
\label{fig:ff}
\end{figure}
It was then tested whether the data can be fitted with a model of the form factor.
The form corresponding to a simple harmonic oscillator shell model
\begin{equation}
F_{HO}(q^2) = d(1-cq^2)\mbox{exp}(-aq^2)
\label{eq:swell}
\end{equation}
did not give satisfying results for the whole range of momentum transfers
(see below). Much better results were obtained with the double-well form for
$s$- and $p$-orbits used in \cite{Suelzle_67}:
\begin{eqnarray}
F_{MO} & = & a_0\left[\frac{2}{3}\mbox{exp}(-q^2b_1^2/4)\right.\nonumber\\
& & \left.+ \frac{1}{3}(1-q^2a_2^2/6)\mbox{exp}(-q^2b_2^2/4)\right]\nonumber \\
b^2_i & = & a^2_i(1-1/A),\;\;\; i=1,2
\label{eq:owells}
\end{eqnarray}
where $a_0$ accounts for the overall normalization and $a_1$, $a_2$ are the
well-strength parameters of the $s$- and $p$-wells. Fits with this model form factor
are shown in Fig.~\ref{fig:ff} as solid lines. They excellently describe the data over
a large range of incident photon energies and momentum transfers with almost identical
parameters. The average values of the well-strength parameters are
\begin{eqnarray}
a_1 & = & (1.599\pm0.001)\;\; \mbox{fm}\nonumber\\
a_2 & = & (2.47\pm0.06)\;\; \mbox{fm}\;.
\end{eqnarray}
Suelzle et al. \cite{Suelzle_67} quote for the charge distribution parameters
$a_1$=(1.55$\pm$0.015)~fm and $a_2$=(2.02$\pm$0.06)~fm so that the s-well strength
is very similar for charge and mass distribution (3\% difference), while
the p-well strength is $\approx$~20\% larger for the mass distribution.
The rms radius is related to these parameters by:
\begin{equation}
r^2_{rms} = \frac{A-1}{A}\left(a_1^2+\frac{1}{2}a_2^2\right)+\frac{1}{3}a_2^2 .
\label{eq:mod_rms}
\end{equation}
The values for $r_{rms}$ obtained from the fits are shown in the insert of
Fig.~\ref{fig:ff}. They show no systematic variation with incident photon energy
and their average of $\approx$ 2.62~fm is significantly larger than the $r_{rms}$
radius of the charge distribution ($\approx$ 2.27~fm in \cite{Suelzle_67};
note that the value of 2.43~fm quoted in this reference includes the charge radius
of the proton).
\begin{figure}[thb]
\resizebox{0.49\textwidth}{!}
\includegraphics{fig_11.eps}
}
\caption{Average of the form factors from Fig.~\ref{fig:ff} compared to the charge
form factors from electron scattering (divided by proton charge form factor).
Solid (blue) line: fit to present data
($q^2<$ 3~fm$^{-2}$) with Eq.~\ref{eq:owells}. Results for fit with
Eq.~\ref{eq:series} and $N=5$ ($q^2<$ 3 fm$^{-2}$), and fit with $N=3$ for
$q^2<$0.5~fm$^{-2}$ (dashed and dotted lines) are not distinguishable from solid line;
(black) solid line: fit of electron scattering data with Eq.~\ref{eq:owells}.
}
\label{fig:radius}
\end{figure}
\begin{table*}[th]
\caption[Properties of meson]{
\label{tab_01}
Fit results for the mass rms-radius. Results are given in column (1) for fits with
the full PWIA model (Eqs.~\ref{eq:pi_no},\ref{eq:pi_sf},\ref{eq:pi_pwia}),
in column (2) for a truncated model without the spin-flip contribution
(Eq.~\ref{eq:pi_sf}), and in column (3) for a model with the spin-flip contribution
arbitrarily doubled. Column (4) shows for comparison results of fits to charge form
factor from electron scattering (divided by proton charge form factor).
First row $^{1)}$ average of the fit results from Fig.~\ref{fig:ff} with the
double-well model (Eq.~\ref{eq:owells}) over an range of incident photon energies
from 150 MeV - 360 MeV ($\chi^2$ values are averages for all fits). Second row $^{2)}$
fit with the double-well model to the
averaged form factor for $q^2<$ 3 fm$^{-2}$ (Fig.~\ref{fig:radius}). Third row $^{3)}$
fit with series (Eq.~\ref{eq:series} with $N=5$ (for the model with neglected spin-flip
term with $N=7$, since $N=5$ did not converge). Fourth row $^{4)}$ fit with series
with $N=2$ for $q^2<$ 0.5 fm$^{-2}$ (only few data for charge form factor).
}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{2}{c|}{full model}
& \multicolumn{2}{c|}{no spin-flip}
& \multicolumn{2}{c|}{spin-flip doubled}
& \multicolumn{2}{c|}{charge form factor}\\
method
& $r_{rms}^{(m)}$ [fm]
& $\chi^2$
& $r_{rms}^{(m)}$ [fm]
& $\chi^2$
& $r_{rms}^{(m)}$ [fm]
& $\chi^2$
& $r_{rms}^{(ch)}$ [fm]
& $\chi^2$\\
\hline
$^{1)}$ double well & 2.618$\pm$0.004 & 3.4 & 2.710$\pm$0.004 & 7.2 & 2.587$\pm$0.004 & 3.2 & - & -\\
$^{2)}$ double well & 2.659$\pm$0.007 & 8.6 & 2.898$\pm$ 0.003 & 24 & 2.612$\pm$0.002 & 8.4 & 2.30$\pm$0.02 & 3.7\\
$^{3)}$ series N=5 (7) & 2.635$\pm$0.002 & 8.4 & 2.981$\pm$ 0.002 & 16 & 2.575$\pm$0.002 & 8.3 & 2.17$\pm$0.04 & 1.9\\
$^{4)}$ series N=2 ($q^2<$ 0.5 fm$^{-2}$) & 2.56$\pm$0.12 & 5.9 & 3.12$\pm$0.08 & 14 & 2.398$\pm$0.15 & 7.2 & 2.2$\pm$1.2 & 1.2\\
\hline
\end{tabular}
\end{center}
\end{table*}
For a more detailed analysis the average of the distributions from Fig.~\ref{fig:ff}
for incident photon energies up to 280 MeV (after renormalization of their absolute
scales) is compared in Fig.~\ref{fig:radius} to the charge form-factor values from
\cite{Suelzle_67,Bumiller_72,Lichtenstadt_89}. It is evident that the
$q^2$-de\-pen\-dence of the electron scattering data is different from the present
results. Both data sets have been fitted for the range of $q^2< 3$~fm$^{-2}$
with Eqs.~\ref{eq:swell} and \ref{eq:owells}. The fits with the simple harmonic
oscillator model (Eq.~\ref{eq:swell}) were of much inferior quality
(reduced $\chi^2 \approx$ 880 for present data compared to $\approx$ 8 for the
double-well form Eq.~\ref{eq:owells}) and were not further considered. The fits with
the double-well form from Eq.~\ref{eq:owells} are shown in Fig.~\ref{fig:radius}
as solid blue (present data) and solid black (electron scattering data) lines.
They correspond to the following rms-radii ($r_{rms}^{(ch)}$: electron data,
$r_{rms}^{(m)}$: present data):
\begin{eqnarray}
r_{rms}^{(ch)} & = & (2.30 \pm 0.02)\;\; \mbox{fm},\\
r_{rms}^{(m)} & = & (2.66 \pm 0.01)\;\; \mbox{fm}.
\end{eqnarray}
The insert of Fig.~\ref{fig:radius} shows the ratio of the present data and this
fit (filled, red points). For $q^2$-values up to 3~fm$^{-2}$ the fit reproduces the
shape to within $\pm$0.5\%. Due to this small systematic differences between fit
curve and data, the result for the radius is almost independent on the fitted range.
If, for example, we fit only the data for incident photon energies below 225 MeV,
where agreement between data and PWIA is best, the radius changes only from 2.659~fm
to 2.653~fm.
Also shown in the insert (black, open points) is the
result from an analysis that neglected the spin-flip term (Eq.~\ref{eq:pi_sf}) in
the elementary production cross section. The influence of this term is substantial
at small $q^2$ values; the reduced $\chi^2$ of the fit rises from 8.6 to 24 if it is
omitted.
The rms radius can be also extracted from the present data without the use of a
specific model for the form factor from its slope for $q^2\rightarrow 0$, using
the expansion
\begin{equation}
F(q^2) = 1 - \frac{q^2}{6} r^2_{rms} + {\cal{O}}(q^4).
\end{equation}
The data were fitted with the ansatz
\begin{equation}
F(q^2) = \sum_{n=0}^{N}c_nq^{2n}\;,
\label{eq:series}
\end{equation}
from which the rms radius follows as
\begin{equation}
r_{rms} = \sqrt{-6c_1/c_0}\;,
\label{eq:rms}
\end{equation}
where for correctly normalized form factors $c_0$ would be unity (here it differs
by a few per cent from unity).
Different fits have been exploited. Two extreme cases are fits for the $q^2$ range
up to 3~fm$^{-2}$ with $N=5$ and with $N=2$ only for small momentum transfers
($q^2 <$ 0.5~fm$^{-2}$). The results for $r_{rms}^{(m)}$ extracted from Eq.~\ref{eq:rms}
are in agreement and close to the above value from the double-well harmonic-oscillator
model:
\begin{equation}
r_{rms}^{(m)}= (2.635\pm 0.002)\;\;\mbox{fm}
\end{equation}
for the $N=5$ fit over the full range and
\begin{equation}
r_{rms}^{(m)} = (2.56 \pm 0.12)\;\; \mbox{fm}
\end{equation}
for the slope from the low-momentum transfer $N=2$ fit (quoted uncertainties are
statistical). The fit curves are so similar to the double-well result
that they are indistinguishable from it in Fig.~\ref{fig:radius}.
The results for all model fits are summarized in Table~\ref{tab_01}. The form factors
derived in PWIA from the coherent pion data correspond to an $rms$-mass radius of
$\approx$ (2.60 - 2.65) fm$^{-2}$ (column (1) of the table), which is significantly
larger than the result for the charge radius (column (4) of the table).
The reduced $\chi^2$ of the fits is larger than unity, which is due to the
systematic structure of the form factor at the sub-percent level (see insert of
Fig.~\ref{fig:radius}), which is significant within statistical uncertainties,
but much smaller than (energy dependent) systematic uncertainties not included
in the fitting process.
One source of systematic uncertainty is the contribution of
the spin-flip-term (Eq.~\ref{eq:pi_sf}) in the PWIA approximation, which includes
only the dominant $M_{1+}$ amplitude and ignores all other production multipoles.
Its influence has been tested by model fits: excluding it completely (column (2) of
Table~\ref{tab_01}) and arbitrary doubling its strength (column (3)).
Excluding the spin-flip contribution increases significantly the $\chi^2$ of the fits
and increases the value of the radius. Enhancing the spin-flip term by a factor of
two is quite a large (probably unrealistic) variation, since in the $\Delta$-resonance
range it is strongly dominated by the well-known $M_{1+}$ multipole.
The $\chi^2$-values of these fits are similar to the standard version and the radius
becomes smaller, but is still larger than the charge radius. The comparison gives
some indication of the possible size of systematic uncertainty due to this term.
The contribution of the inelastic 478-keV excitation was ignored for the form-factor
extraction. To test its importance we subtracted the PWIA estimate for this
process from the measured angular distributions and repeated the analysis.
This removes strength at large $q^2$, which makes the form factor steeper and thus
tends to increase the radius. However, the effect is smaller than statistical
uncertainties and can be safely neglected.
\begin{figure}[thb]
\resizebox{0.49\textwidth}{!}
\includegraphics{fig_12.eps}
}
\caption{ Main picture: angular distributions for incident photon energies from
152 MeV (bottom curve) to 264 MeV (top curve, same energy bins as in
Fig.~\ref{fig:ff}) compared to PWIA results using the
fitted form factor. Absolute scales of model results normalized to data (see text).
The insert shows on a larger scale the low-energy results (144 MeV - 168 MeV).
}
\label{fig:difflin}
\end{figure}
So far no model results are available for FSI effects in $^7$Li.
Results for other light nuclei ($^4$He, $^{12}$C) \cite{Drechsel_99,Krusche_02}
have shown that they are important for the energy dependence of the
total cross section.
Nevertheless, the good agreement of the measured total cross section with the PWIA
modelling at incident photon energies below 225 MeV indicates that they must
be small for $^7$Li in this energy range. The main FSI effect depends on the pion
kinetic cm energies (and thus on the incident photon energy) but it could also modify
to some extent the shape of the angular distributions, which are the basis for the
form-factor extraction. However, the form-factor fits (see insert of Fig.~\ref{fig:ff})
give consistent results for the mass radius $r_{rms}^{(m)}$ over a wide range of incident
photon energy, over which the energy-de\-pen\-dent FSI effects change drastically,
from a few per cent between 180 MeV and 220 MeV to almost 40\% around 280 MeV
(see insert of Fig.~\ref{fig:pi_total}).
In order to explain the observed difference between the extracted form factor and the
charge form-factor data, FSI effects with a very peculiar behavior would be needed.
This is demonstrated in Fig.~\ref{fig:difflin}, where the low-energy angular
distributions are compared to a modified PWIA. The only difference to the PWIA
curves in Fig.~\ref{fig:pi_ang} is that instead of the form factor from electron
scattering the double-well parameterization of the present form factor data
from Fig.~\ref{fig:radius}, corresponding to $r_{rms}^{(m)}$=2.66~fm was used.
The absolute scales of the PWIA results were renormalized to the data in order to
remove the energy-de\-pen\-dent FSI effects. This PWIA must describe by construction
the angular distributions on average. However, it actually agrees almost
perfectly with the shapes of all individual distributions, with very different
relations between pion angles and nuclear momentum transfers. This means that an FSI
effect would be needed, which over a range of incident photon energy of more than
100 MeV has exactly the same angular and momentum-transfer dependence as a change
of the form factor from an rms radius of 2.3 fm to 2.66 fm. Although this does not
seem to be a likely scenario, reasonably sophisticated modelling of the FSI
effects is needed before a final conclusion can be drawn. However, results for
a similar analysis of coherent photoproduction off carbon, calcium, and lead nuclei
point to a small influence of FSI on the extracted radii for light nuclei.
Fully taking into account the FSI effects \cite{Krusche_05} lowered the extracted
value of the mass radius for $^{208}$Pb by 5.8\%, for $^{40}$Ca by 2.2\%,
but for the lighter $^{12}$C only by 0.9\%, while the observed difference between
charge and mass radius for $^7$Li is on the 10\% level.
\subsection{Coherent $\eta$-photoproduction}
The total cross sections extracted for the two $\eta$-decay channels, summarized
in Fig.~\ref{fig:eta_total}, are nicely consistent. They show a much smoother rise
at production threshold than the $^3$He data (cf. Fig.~\ref{fig:helium}).
For a quantitative analysis their average is compared to PWIA modelling,
based on Eq.~\ref{eq:eta} in Fig.~\ref{fig:eta_pwia}. As discussed in
Sec.~\ref{sec:pwia} the situation is much different to pion production since for
$\eta$-production the cross section is dominated by the contribution of the odd
$1p_{3/2}$ proton, which is only a small correction in the $\pi^0$ case.
\begin{figure}[thb]
\resizebox{0.49\textwidth}{!}
\includegraphics{fig_13.eps}
}
\caption{Comparison of the total cross section for coherent $\eta$-production from
the two-photon and six-photon decay of the $\eta$. The vertical dotted lines
indicate coherent and breakup thresholds.
}
\label{fig:eta_total}
\end{figure}
\begin{figure}[htb]
\resizebox{0.49\textwidth}{!}
\includegraphics{fig_14.eps}
}
\caption{Comparison of the average of the experimental two-photon and six-photon
cross sections to the PWIA results. Dashed (dotted) curves: coherent contribution
(Eq.~\ref{eq:eta} without $F^2_{Cx*}$-term) based on charge form factor
(mass form factor fitted to pion production).
Solid (dash-dotted) curves: sum of coherent and incoherent contribution (see Eq.~\ref{eq:eta})
for charge (mass form factor).
The insert shows the ratio of measured cross section and PWIA results,
open symbols only coherent part, filled symbols sum of coherent and incoherent
contributions).
}
\label{fig:eta_pwia}
\end{figure}
\begin{figure}[ttb]
\resizebox{0.49\textwidth}{!}
\includegraphics{fig_15.eps}
}
\caption{Comparison of measured angular distributions for
$\gamma$+$^7$Li$\rightarrow ^7$Li+$\eta$
{(red) dots} to PWIA results, solid lines: full PWIA, dashed lines: only coherent
part.
}
\label{fig:eta_diff}
\end{figure}
Results from PWIA, using the charge form factor or the mass form factor fitted
to the pion data (dashed, respectively dotted curves in Fig.~\ref{fig:eta_pwia}),
are similar. This is simply so because, for $\eta$ production, large momentum
transfers dominate where the two form factors agree. The relative contribution
of the incoherent excitation of the 478-keV state is significant in PWIA.
Also this had to be expected because for $\eta$-production there is no piece
with an $A^2$-term like Eq.~\ref{eq:pi_no} for coherent pion production and the
elastic and inelastic form factors are similar for large momentum transfers.
The systematic uncertainty of the PWIA results is larger than in
the pion case because the cross section is dominated by these less well-established
contributions. However, altogether the comparison of the energy dependence of the
measured total cross section and the PWIA results in Fig.~\ref{fig:eta_pwia}
shows no threshold enhancement above phase-space behavior, and thus no
indication for the formation of a quasi-bound state.
The situation is thus much different from the $^3$He case discussed in the
introduction which, apart from the incoherent excitation, has similar systematic
uncertainties in PWIA. Comparison of the two results highlights the special role
of the $\eta-^3\mbox{He}$ system.
Also the results for the angular distributions, summarized in Fig.~\ref{fig:eta_diff},
are consistent with this interpretation. They agree better with the momentum-transfer
dependence of the form factor than in the $^3$He case and show no tendency towards
isotropic behavior close to threshold.
\section{Summary and Conclusions}
Precise data have been measured for coherent photoproduction of $\pi^0$-mesons
off $^7$Li nuclei and coherent photoproduction of $\eta$-mesons off the same nucleus
has been identified for the first time. The experimental results for the pion
production are quite well reproduced at low incident photon energies by a PWIA
dominated by the spin/isospin-independent part of the elementary production
amplitude. The spin-flip amplitude from the unpaired $1p_{3/2}$ proton is considered
for the leading $M_{1+}$ multipole and the corrections applied for the incoherent
excitation of the 478-keV nuclear state in $^7$Li are insignificant. This model
reproduces quite well total cross sections and angular distributions at incident
photon energies below 225 MeV, indicating that distortion effects from final-state
interactions of the pion are small in this energy range.
After an adjustment of the nuclear form factor, which corresponds to a change
in the harmonic double-well parameterization from an rms radius of 2.3 fm
(reported for the charge form factor derived from electron scattering data) to
2.66 fm, the shape of angular distributions in this energy range is excellently
reproduced.
Exploiting the possible uncertainties due to approximations, in particular in the
spin-flip term of the PWIA, we find reasonable agreement between data and PWIA
for rms radii down to 2.5 fm, which are still larger than previously reported charge
radii and also larger than predictions for the mass radius, which are around 2.35 fm
(see e.g. \cite{Tomaselli_00,Kajino_88}). DWIA calculations with careful treatment
of possible FSI effects are needed for further analysis of this discrepancy.
Coherent photoproduction of $\eta$-mesons is quite difficult to measure and so far
only results for the deuteron \cite{Hoffmann_97,Weiss_01} and $^3$He
\cite{Pfeiffer_04,Pheron_12} had been reported. This experiment extended the mass
range to $^7$Li by measuring total cross sections on the level below 20 nb.
The results, also for the angular distributions, are in good agreement with
PWIA expectations and do not show an unexplained threshold enhancement as in the
$^3$He case, underlining the special role of $^3$He as a candidate for $\eta$-mesic
states.
\vspace*{1cm}
{\bf Acknowledgments}
We wish to acknowledge the outstanding support of the accelerator group
and operators of MAMI. We thank L. Tiator for the discussion of the plane wave approximations.
This work was supported by Schweizerischer Nationalfonds, Deutsche
Forschungsgemeinschaft (SFB 443, SFB/TR 16), DFG-RFBR (Grant No. 05-02-04014),
UK Science and Technology Facilities Council, STFC, European Community-Research
Infrastructure Activity (FP6), the US DOE, US NSF and NSERC (Canada).
|
1,116,691,498,479 | arxiv | \section*{Response to Reviewer \#1}
\vspace{-0.5em}
\noindent Thank the reviewer for appreciating our contributions to the analysis on the current problems with both model- and data-centric IQA approaches, and to the next-generation IQA dataset construction with the help of a computational framework. These two contributions are our key points in this submission. And applying the proposed framework to construct a new IQA dataset is our ongoing work.
\vspace{-0.5em}
\section*{Response to Reviewer \#2}
\vspace{-0.5em}
\noindent {\bf 1. About the conclusion for the two IQA approaches}. We respectfully disagree with the reviewer's comment, which is subject to an obvious \textbf{hindsight} bias. First of all, how to quantify the overfitting issue in deep learning is a wide open research question. We contributed a carefully designed experiment based on the gMAD competition, and for the first time, probed the overfitting issue in the field of BIQA. Second, we are the first to analyze the difficulty of current human-rate IQA datasets with a qualitative diversity analysis. We attribute this to the superficial treatment of \textbf{sample selection rather than model overfitting as commented by the Reviewer}.
Towards addressing the identified problems, we describe a simple and effective computational framework to integrate model-centric and data-centric IQA. As also pointed out by the Reviewer, we construct a specific example, where the key step is to learn a sampling-worthiness module to spot diverse failures
of ``top-performing'' IQA models. We sincerely hope the Reviewer appreciate the simplicity and effective of the proposed solution, as simplicity is the beauty of science and engineering.
As researchers in computer vision begin to aware the value of data and to make effort to learn generalizable computational models, we believe people in related fields such as object recognition and vision+language will definitely benefit from our problem formulation, our thoroughly designed experiments, and our proposed solutions.
\noindent {\bf 2. About the proposed framework and experiments on proposed module}.
We agree with the Reviewer that the proposed sampling-worthiness module is straightforward but it is effective and flexible. It is effective in that it outperforms many deep active learning methods in failure identification. It is flexible in that any current and future top-performing BIQA models can be directly leveraged to improve data value and model performance.
We also agree with the reviewer the evaluation may be simple, but this is just a proof-of-concept experiment and we kindly refer the reviewer to Response 1 for main contributions. In addition, more details of the experiments are included in the supplementary material.
\vspace{-0.5em}
\section*{Response to Reviewer \#3}
\vspace{-0.5em}
\noindent {\bf Response to Q1}. Thanks for the excellent question. The ``general'' overfitting in IQA means that the model only works for trained visual distortions (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, over exposure and motion blurring) and fails to generalize in the presence of mild distribution shift (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, out of focus blur). The ``narrow'' overfitting in IQA means that the model fails to generalize to seen visual distortions when image content varies. It is generally difficult to probe overfitting in IQA (and many fields of computer vision), and we have designed an innovative experiment based on gMAD.
Yes, we believe the overfitting issue can be solved (or at least alleviated) by collecting more data. But the key issue is what data to collect (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, sample selection), which is largely overlooked in IQA (and many fields of computer vision). Ideally, we would like to allocate expensive human labelling budget to ``worthy'' examples. A superficial treatment of sample selection (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, random selection) is exactly the cause of the easy dataset problem in IQA as identified by our carefully designed experiments, and is one of the key motivations to integrating model-centric and data-centric methods.
\noindent {\bf Response to Q2}.
In IQA, a deeper backbone doesn't necessarily resolve overfitting. As mentioned in Sec.~3.1, HyperIQA, a much simpler model, is able to outperform LinearityIQA with a more powerful backbone in our gMAD experiment (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the overfitting experiment). Moreover, integrating model-centric and data-centric IQA is addressing not only overfitting but also the easy dataset problem.
\noindent {\bf Response to Q3}.
There is little content overlapping between these datasets; they only share the same realistic camera distortions. If there were overlapping between the latter and former datasets, this only makes the latter dataset easier.
\noindent {\bf Response to Q4}.
Thanks for the excellent suggestion, for which we will definitely explore in future. The key contributions of the current paper are our problem formulations, our thoroughly designed experiments, and our proposed solutions.
\noindent {\bf Response to Q5}.
The training datasets matter less in the experiment; we can select any currently available top-performing BIQA models with arbitrary training data/strategy to facilitate the construct of a new dataset. The current experiment is a proof-of-concept for constructing a new dataset based on a specific BIQA model trained on the combination of previous datasets. We have experimented with another SOTA BIQA model, HyperIQA, and arrived at similar conclusions.
\end{document}
\section{Introduction}
\label{sec:intro}
Image quality assessment (IQA) is indispensable in a broad range of image processing and computer vision applications, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, image acquisition, compression, enhancement and display~\cite{wang2006modern}. In recent years, learning-based methods~\cite{ye2012unsupervised,tang2014blind,xu2016blind,ma2017dipiq}, especially convolutional neural networks (CNN)~\cite{kang2014convolutional,ma2017end,bosse2017deep} have significantly advanced the field of IQA. A learning-based IQA system generally has two key components: the engine ``model" and its fuel ``data." The IQA model is expected to learn to predict image quality from a large amount of human-annotated data. From this perspective, it is convenient to categorize existing IQA studies into model-centric and data-centric approaches.
\begin{figure}[t]
\centering
\addtocounter{subfigure}{-1}
\subfloat{\includegraphics[width=1\linewidth]{model/framework}}\\
\vspace{-.3cm}
\caption{Past work makes weak connections between data-centric and model-centric IQA, hindering further progress of IQA.}
\vspace{-0.3cm}
\label{fig:framework}
\end{figure}
The goal of model-centric IQA~\cite{mittal2012no,kang2014convolutional,liu2017rankiqa,talebi2018nima,su2020blindly,zhang2021uncertainty} is to build computational models (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, objective methods) that provide consistent predictions of human perception of image quality. More advanced learning-based IQA models have been developed by improving upon computational structure, objective function, and optimization technique. Specifically, the \textit{computational structures} have shifted from shallow~\cite{kang2014convolutional} to deep methods with cascaded linear and nonlinear stages~\cite{hosu2020koniq}. Effective computational modules have also been identified along the way, such as generalized divisive normalization (GDN) over half-wave rectification (ReLU)~\cite{ma2017end}, bilinear pooling over global average pooling~\cite{zhang2018blind}, and adaptive convolution over standard convolution~\cite{su2020blindly}. The \textit{objective function} mainly pertains to the formulation of IQA. It is intuitive to think of visual quality as absolute quantity, and employ the Miknowski metric to measure the prediction error. Another popular formulation is learning-to-rank~\cite{gao2015learning} by treating perceptual quality as relative quantity, which admits a family of pairwise and listwise losses~\cite{liu2011learning,ma2017dipiq}. Other loss functions, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, for accelerated convergence~\cite{li2020norm} and uncertainty quantification~\cite{zhang2021uncertainty,wang2021semi}, have also begun to emerge. The \textit{optimization techniques} in IQA benefit significantly from practical tricks to train large-scale CNNs for visual recognition~\cite{li2017scaling}. One learning strategy specific to IQA was proposed by Zhang~\emph{et al}\onedot~\cite{zhang2021uncertainty}, who enabled an IQA model to be trained on multiple IQA datasets without perceptual scale realignment.
The goal of data-centric IQA is to construct human-rated IQA datasets via psychophysical experiments for benchmarking and developing objective IQA models. A common theme in data-centric IQA~\cite{sheikh2006statistical,ciancio2010no,ghadiyaram2015massive,hosu2020koniq,fang2020perceptual} is to design efficient subjective testing methodologies to collect reliable human ratings of image quality, typically in the form of mean opinion scores (MOSs). Extensive practice~\cite{mantiuk2012comparison} seems to show that there is no free lunch in data-centric IQA: collecting more reliable MOSs generally requires more delicate and time-consuming psychophysical procedures, such as two-alternative forced choice (2AFC) in a well-controlled laboratory environment with proper instructions. Bayesian experimental design has been implemented~\cite{xu2012hodgerank,ye2014active} in an attempt to improve rating efficiency. Arguably a more crucial step in data-centric IQA is sample selection, which is, however, much under-studied. Vonikakis~\emph{et al}\onedot~\cite{vonikakis2017probabilistic} proposed a dataset shaping technique to identify image subset with uniformly distributed attributes of interest. Cao~\emph{et al}\onedot~\cite{cao2021debiased} described a sample selection method in the context of real-world image enhancement based on the principle of maximum discrepancy competition~\cite{wang2008maximum,wang2020going}.
Although the past achievements in IQA are worth celebrating, only weak connections have been made between model-centric and data-centric IQA, which we argue is the primary impediment to further progress of IQA (see Fig.~\ref{fig:framework}). From the model perspective, objective methods are optimized and evaluated on fixed (and extensively re-used) sets of data, leading to the \textit{overfitting} problem. An excellent example is in the field of full-reference IQA~\cite{wang2006modern}, where the methods are achieving higher and higher correlation numbers, but fail the na\"{i}ve test of reference image recovery~\cite{ding2021comparison}. From the data perspective, IQA datasets are generally constructed, while being blind to existing ``top-performing'' models. This may cause the \textit{easy dataset} problem~\cite{fang2020perceptual}: the newly created dataset may pose little challenge and/or expose few quality prediction failures of existing IQA models, resulting in a significant waste of expensive human labeling.
In this paper, we take one of the first steps towards integrating model-centric and data-centric IQA approaches. Our main contributions are three-folds.
\begin{itemize}
\item We design a series of experiments to probe the overfitting and easy dataset problems of blind IQA (BIQA) in real settings.
\item We describe a computational framework that integrates model-centric and data-centric IQA. The key idea is to augment the main quality predictor by an auxiliary computational module to score sampling-worthiness of candidate images for dataset construction.
\item We provide a specific example of this framework, where we start with a ``top-performing'' BIQA model - UNIQUE \cite{zhang2021uncertainty}, and jointly train a failure predictor by learning to rank the predictions of UNIQUE. Our sampling-worthiness module is then the combination of the learned failure predictor and a diversity measure computed as the semantic distance between ImageNet features~\cite{Simonyan14c}. Experiments show that the proposed sampling-worthiness module is able to spot the diverse failures of UNIQUE in comparison to state-of-the-art deep active learning methods~\cite{pop2018deep,sener2018active,burbidge2007active}. These samples are ideal to be incorporated in next-generation IQA datasets.
\end{itemize}
\section{Related Work}
\label{sec:rw}
In this section, we provide a concise overview of model-centric and data-centric IQA.
\subsection{Model-centric IQA}
We will focus on reviewing model-centric BIQA, in which we conduct experiments to demonstrate our idea.
Model-centric BIQA improves the quality prediction performance from the model perspective without reliance on original and undistorted images. Conventional BIQA models pre-defined non-learnable \textit{computational structures} to extract natural scene statistics (NSS). Learning occurred at the quality regression stage by fitting a mapping from NSS to MOSs. Commonly, NSS were extracted in transform domain, such as wavelet~\cite{moorthy2011blind} and DCT~\cite{saad2012blind}, where the statistical irregularities can be effectively characterized.
Nevertheless, transform-based methods were usually very slow, which motivated BIQA models in spatial domain, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, BRISQUE~\cite{mittal2012no} and CORNIA~\cite{ye2012unsupervised}.
With latest advances in CNNs, learnable computational structures based on stages of linear convolution, downsampling, and nonlinear activation have revolutionized the field of BIQA~\cite{liu2017rankiqa,talebi2018nima,su2020blindly,zhang2021uncertainty}. Along this line, the \textit{objective function} plays an important role in guiding the optimization of BIQA models. It is straightforward to formulate BIQA as regression, and thus the Miknowski metric is the objective function of the choice. An alternative view of BIQA is through the lens of learning-to-rank~\cite{liu2011learning}, with the goal of inferring relative quality rather than absolute quality. Pairwise learning-to-rank objectives, such as the cross entropy~\cite{ma2017dipiq} and the fidelity~\cite{tsai2007frank} losses, and listwise objectives, such as the cross entropy over permutation probabilities~\cite{ma2017dipiq} and the Pearson linear correlation~\cite{li2020norm} have been successfully adopted. One last ingredient of model-centric IQA is the selection of \textit{optimization techniques} to combat the lack of human-rated training data. Fine-tuning from pre-trained CNNs on other vision tasks~\cite{talebi2018nima}, patch-wise training~\cite{bosse2017deep}, and quality-aware pre-training \cite{su2020blindly} are practical optimization tricks in BIQA, taking inspirations from training large-scale CNNs for visual recognition~\cite{deng2009imagenet}. Of particular interest is the unified optimization strategy by Zhang~\emph{et al}\onedot~\cite{zhang2021uncertainty}, allowing a single model to learn from multiple datasets simultaneously.
The success of model-centric BIQA was established on the same IQA datasets that have been used to compare the models for quite many years. The impressive correlation numbers achieved by more sophisticated computational structures and optimization techniques may be a consequence of overfitting, and thus questionable.
\subsection{Data-centric IQA}
Data-centric IQA improves the quality prediction performance from the data perspective, and consists of two major steps: sample selection and subjective testing. The immediate outputs are a list of human-rated datasets for training and benchmarking objective IQA models. For a long time, the research focus of data-centric IQA has been designing reliable and efficient \textit{subjective testing} methodologies for MOS collection~\cite{bt500,men2021subjective}. It is generally believed a 2AFC design (also known as paired comparison) in a well-controlled laboratory environment is more reliable than single-stimulus and multiple-stimulus methods. However, the cost to exhaust all paired comparisons scales quadratically with the number of images, $M$, in the dataset, and is prohibitively expensive when $M$ is large. Several methods have been introduced to reduce the cost of the 2AFC design, including HodgeRank~\cite{xu2012hodgerank} and Bayesian experimental designs~\cite{ye2014active}. To further accelerate the process, crowdsourcing-based single-stimulus methods~\cite{chen2009crowdsourceable,ghadiyaram2015massive} have been practiced to build large-scale IQA datasets with relatively noisier MOSs.
\textit{Sample selection} is perhaps more crucial in data-centric IQA, but experiences much less success. Early datasets, such as LIVE~\cite{sheikh2006statistical}, TID2008~\cite{ponomarenko2009tid2008}, and CSIQ~\cite{larson2010most}, selected images with simulated distortions. Due to the combination of reference content, distortion type and level, the number of distinct reference images is often limited. Recent large-scale IQA datasets began to contain images with realistic camera distortions, including CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, SPAQ~\cite{fang2020perceptual}, and PaQ-2-PiQ~\cite{ying2020patches}. Such shift in sample selection provides a good test of synthetic-to-realistic generalization.
Sample diversity during dataset creation has also been taken into account. Vonikakiset~\emph{et al}\onedot~\cite{vonikakis2017probabilistic} cast sample diversity as a mixed integer linear programming, and selected a dataset with uniformly distributed image attributes, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, brightness, colorfulness, contrast, and sharpness. Euclidean distances between deep features~\cite{hosu2020koniq,cao2021debiased} are also used to measure the semantic similarity. Nevertheless, sample diversity is only one piece of the story; often the included images are too easy to challenge existing BIQA models. For example, Fang~\emph{et al}\onedot~\cite{fang2020perceptual} took nearly one year to create the SPAQ datasets, consisting of more than $11,000$ images, most of which turn out to be easy examples and pose few difficulties to current BIQA models (see Sec.~\ref{subsec:easy dataset} for details).
Active learning~\cite{settles2010active,ren2020survey} may come naturally into playing to prioritize sample selection. Wang~\emph{et al}\onedot~\cite{wang2021active} spotted the failures of a BIQA model with the help of multiple full-reference IQA methods in the setting of synthetic distortions. They~\cite{wang2021troubleshooting} later extended the idea to the authentic distortion scenario by creating a set of self-competitors via network pruning. The two methods are algorithm-specific, and may not be generalized to other existing BIQA models. Other active learning strategies such as uncertainty-based sampling~\cite{cohn1996active}, query by committee~\cite{seung1992}, expected model change maximization~\cite{cai2013maximizing}, and failure prediction~\cite{scheirer2008fusion,zhang2014predicting} may be applied, but it remains to be seen whether these are feasible and if so, how effective they are in the context of deep learning-based IQA.
\begin{table}[t]
\centering
\caption{SRCC between the performance ranking of the competing nine BIQA methods and their published time on four datasets. It is clear that the quality prediction performance ``improves'' steadily over time.}
\label{tab:srcc}
\begin{tabular}{l|cccc}
\toprule
Dataset & BID & CLIVE & KonIQ-10k & SPAQ \\
\hline
SRCC & 0.6946 & 0.7950 & 0.6695 & 0.7029 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table*}[!ht]
\caption{Ranking results of nine deep learning-based BIQA models. A smaller rank indicates better performance. ``All'' indicates that UNIQUE is trained on the combined dataset of LIVE, CSIQ, KADID-10K, BID, CLIVE, and KonIQ-10K.}
\label{tab:nr-iqa-summary}
\centering
\begin{tabular}{l|lllcccc}
\toprule
Name & Backbone & Formulation & Dataset & Time & SRCC Rank & gMAD Rank & $\Delta$ Rank \\
\hline
UNIQUE~\cite{zhang2021uncertainty} & ResNet-34 & Ranking & All & 2021.03 &1 & 5&-4 \\
KonCept512~\cite{hosu2020koniq} & InceptionResNetV2 & Regression & KonIQ-10k & 2020.01 & 2 & 1&1\\
HyperIQA~\cite{su2020blindly} & ResNet-50 & Regression & KonIQ-10k & 2020.08 &3 & 2&1 \\
LinearityIQA~\cite{li2020norm}& ResNeXt-101& Regression & KonIQ-10k & 2020.10 & 4 & 3&1\\
MetaIQA+~\cite{zhu2021generalizable} & ResNet-18 & Regression & CLIVE & 2021.04 & 5 & 8&-3\\
Fang2020~\cite{fang2020perceptual} & ResNet-50 & Regression & SPAQ &2020.08 &6 &9 &-3\\
NIMA~\cite{talebi2018nima} & VGG-16 & Distribution & AVA & 2018.04 & 7 & 4 &3\\
DeepIQA~\cite{bosse2017deep} & VGG-like CNN &Regression & LIVE &2018.01& 8 & 7 & 1\\
RankIQA~\cite{liu2017rankiqa} & VGG-16 & Ranking & LIVE & 2017.12& 9 & 6 &3\\
\bottomrule
\end{tabular}
\end{table*}
\section{Problems in the Progress of BIQA}\label{sec:problem}
In this section, we design a series of experiments to empirically prove that the \textit{overfitting} and \textit{easy dataset} problems have actually emerged in the current development of BIQA, which we attribute to the weak connections between model-centric and data-centric approaches.
\subsection{The Overfitting Problem}\label{subsec:overfitting}
As there is no standardized definition of overfitting, especially in the context of deep learning~\cite{goodfellow2016deep}, quantifying overfitting is still an open problem. Here we choose to use the group maximum differentiation (gMAD) competition~\cite{ma2018group} to probe the generalization of BIQA models to a large-scale unlabeled image set. gMAD is a discrete instantiation of MAD competition~\cite{wang2008maximum} that relies on synthesized images to optimally distinguish the models. As opposed to the \textit{average-case} performance measured on existing IQA datasets, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, by Spearman's rank correlation coefficient (SRCC), gMAD can be seen as an indication of the \textit{worst-case} performance by comparing the models using extremal image pairs that are likely to falsify them. We declare an overfitting issue of a BIQA model, if it shows stronger performance on standard IQA datasets, but weaker performance in gMAD competition.
\paragraph{Experimental Setup} We choose nine BIQA models from 2017 to 2021: RankIQA~\cite{liu2017rankiqa}, DeepIQA~\cite{bosse2017deep}, NIMA~\cite{talebi2018nima}, KonCept512~\cite{hosu2020koniq}, Fang2020~\cite{fang2020perceptual}, HyperIQA~\cite{su2020blindly}, LinearityIQA~\cite{li2020norm}, UNIQUE~\cite{zhang2021uncertainty}, and MetaIQA+~\cite{zhu2021generalizable}. Table~\ref{tab:srcc} shows the SRCC results between the published time of the algorithms and their performance ranking\footnote{As each BIQA model assumes different (and unknown) training and testing splits, for a debiased comparison, we compute the quality prediction performance on the full dataset.} on four widely used IQA datasets, BID~\cite{ciancio2010no}, CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, and SPAQ~\cite{fang2020perceptual}. We find that ``steady'' progress over the years has been made by employing more complicated computational structures and more advanced training strategies.
We now set the stage for the nine BIQA models to perform gMAD competition. Specifically, we first gather a large-scale unlabeled dataset $\mathcal{U}$, containing $100,000$ photographic images with marginal distributions nearly uniform w.r.t. several image attributes\footnote{We adopt five image attributes - JPEG compression ratio, brightness, colorfulness, contrast, and sharpness.}. Our dataset covers a wide range of realistic camera distortions, such as sensor noise contamination, motion and out-of-focus blurring, under-and over-exposure, contrast reduction, color cast, and a mixture of them. Given two BIQA models $q_i(\cdot)$ and $q_j(\cdot)$, gMAD~\cite{ma2018group} selects top-$K$ image pairs that best discriminate between them:
\begin{align} \label{eq:gmad2}
({x}^{\star}_k, y^{\star}_k) = &\mathop{\text{argmax}}_{x,y} q_i(x) - q_i(y)\nonumber\\
&\text{ s.t. } q_j(x) = q_j(y)=\alpha, \; x,y\in \mathcal{U}\setminus\mathcal{D},
\end{align}
where $\mathcal{D}=\{{x}^\star_{k'},{y}^\star_{k'}\}^{k-1}_{k'=1}$ is the current gMAD image set. The $k$-th image pair must lie on the $\alpha$-level set of $q_j$, where $\alpha$ specifies a quality level. The roles of $q_{i}$ and $q_{j}$ should be switched. $Q$ (non-overlapping) quality levels are selected to cover the full quality spectrum. By exhausting all distinct pairs of BIQA models and quality levels, we arrive at a gMAD set $\mathcal{D}$ that contains a total of $9\times8\times5\times 2 = 720$ image pairs, where we set $Q=5$ and $K=2$.
We then invite $25$ human subjects and gather perceived quality judgments of each gMAD pair using the 2AFC method. We refer the readers to the Appendix for the detailed setup of the subjective experiment. After subjective testing, we obtain the raw pairwise comparison matrix $A\in\mathbb{R}^{9\times 9}$, where $a_{ij}\in\{0,1\ldots, 250\}$ indicates the counts of $x^\star$ preferred over $y^\star$ by the $25$ subjects on the ten image pairs (by solving Problem~\eqref{eq:gmad2}). We compute, from $A$, a second matrix $B\in\mathbb{R}^{9\times 9}$, where $b_{ij} = a_{ij}/a_{ji}$ denotes the pairwise dominance of $q_i$ over $q_j$. Laplace smoothing is applied when $a_{ji}$ is close to zero. We convert the pairwise comparison
into a global ranking $r\in\mathbb{R}^9$ using Perron rank~\cite{saaty1984inconsistency}:
\begin{align}
r = \lim_{t\rightarrow \infty}\frac{1}{t}\sum_{\beta=1}^{t}\frac{B^\beta}{1^TB^\beta1},\label{eq:perron}
\end{align}
where $1$ is a $9$-dimensional vector of all ones. The solution to Eq.~(\ref{eq:perron}) is the normalized eigenvector of $B$ corresponding to the largest eigenvalue. A larger $r_i$ indicates better performance of $q_i$ in the gMAD competition.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{model-centric/hyper_linear}
\vspace{-.3cm}
\caption{Representative gMAD pairs between HyperIQA and LinearityIQA. \textbf{(a)} Fixing HyperIQA at the low quality level. \textbf{(b)} Fixing HyperIQA at the high quality level. \textbf{(c)} Fixing LinearityIQA at the low quality level. \textbf{(d)} Fixing LinearityIQA at the high quality level.}
\vspace{-0.3cm}
\label{fig:gmad2}
\end{figure*}
\paragraph{Results} Table~\ref{tab:nr-iqa-summary} compares the ranking results of the nine BIQA models in the gMAD competition and in terms of the average SRCC on the four full datasets, BID~\cite{ciancio2010no}, CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, and SPAQ~\cite{fang2020perceptual}. The primary observation is that the latest models such as UNIQUE~\cite{zhang2021uncertainty} and MetaIQA+~\cite{zhu2021generalizable} tend to overfit the peculiarities of the training sets by adopting more advanced optimization techniques.
Second, it is a little surprising that a BIQA model with a more sophisticated backbone is not likely to overfit, as evidenced by high rankings in gMAD. One exception is HyperIQA~\cite{su2020blindly}. With a modest ResNet-50 as the backbone augmented by a self-adaptive subnetwork, HyperIQA is able to outperform LinearityIQA with a more powerful ResNeXt-101 (see Fig.~\ref{fig:gmad2} for a visual comparison). Third, compared to SPAQ of a similar scale, KonIQ-10k serves as a more appropriate training set, on which more generalizable models can be learned. It is quite surprising to find that Fang2020~\cite{fang2020perceptual} underperforms NIMA and DeepIQA, which are, respectively, trained on datasets of aesthetic and synthetic image quality. In summary, the SRCC between the gMAD ranking of the algorithms and their published time is only $0.0753$, implying that the progress made by model-centric IQA might be somewhat over-estimated in terms of real-world generalization.
\begin{table}[t]
\centering
\caption{SRCC between predictions of UNIQUEs and the MOSs of different test sets. ``---'' means that the corresponding dataset is used for jointly training.}
\label{tab:srcc2}
\begin{tabular}{l|ccc}
\toprule
SRCC & UNIQUEv1 & UNIQUEv2 & UNIQUEv3 \\
\hline
CLIVE & 0.6998 & --- & --- \\
KonIQ-10k & 0.6917 & 0.7251 & --- \\
SPAQ & 0.7204 & 0.7932 & 0.8112\\
\bottomrule
\end{tabular}
\end{table}
\subsection{The Easy Dataset Problem}\label{subsec:easy dataset}
In order to reveal the easy dataset problem, it suffices to empirically prove that the newly created datasets are less effective in falsifying current BIQA algorithms.
\paragraph{Experimental Setup} We work with the same four databases - BID~\cite{ciancio2010no}, CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, and SPAQ~\cite{fang2020perceptual}. To achieve our goal, we select a state-of-the-art BIQA model - UNIQUE~\cite{zhang2021uncertainty} - that permits training on multiple datasets at the same time. We then train three UNIQUEs on the combination of available datasets in chronological order: 1) BID for training, and CLIVE, KonIQ-10k and SPAQ for testing; 2) BID and CLIVE for training, and KonIQ-10k and SPAQ for testing; 3) BID, CLIVE and KonIQ-10k for training, and SPAQ for testing. For each training setting, we randomly sample $80\%$ images from each dataset to construct the training set, leaving the remaining $10\%$ for validation and $10\%$ for testing, respectively. To reduce the bias caused by the randomness in dataset splitting, we repeat the training procedure ten times, and report the median SRCC result. More training details can be found in the Appendix.
\begin{figure*}[t]
\centering
\addtocounter{subfigure}{-1}
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_5}}\\
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_5}}\\
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_5}}\\
\vspace{-.3cm}
\caption{The top-5 difficult samples in SPAQ. \textbf{(a)}-\textbf{(e)} for UNIQUEv1. \textbf{(f)}-\textbf{(j)} for UNIQUEv2. \textbf{(k)}-\textbf{(o)} for UNIQUEv3.}
\vspace{-0.3cm}
\label{fig:diversity1}
\end{figure*}
\paragraph{Results} Table~\ref{tab:srcc2} lists the SRCC results between predictions of UNIQUEs and the MOSs of different IQA datasets as test sets. The primary observation is that as more datasets are available for training, the newly created ones are more difficult to challenge the most recent UNIQUE. For example, trained on the combination of BID, CLIVE, and KonIQ-10k, UNIQUEv3 achieves a satisfactory SRCC of $0.8112$ on SPAQ. Even for UNIQUEv1, its performance is better on SPAQ than on CLIVE, which was released much earlier. Similar conclusions can be drawn by inspecting the performance of UNIQUEv2 on KonIQ-10k and SPAQ, which are consistent with the observations in Sec.~\ref{subsec:overfitting}. What is worse, the most difficult examples for each dataset (as measured by the mean squared error (MSE) between model predictions and MOSs) often share similar visual appearances, which are simply different manifestations of the same underlying failure cause (see visual examples in Fig.~\ref{fig:diversity1}). This indicates the sample diversity may not be well imposed in existing datasets.
\section{Proposed Idea}
\label{sec:pm}
In this section, we describe a computational framework for integrating model-centric and data-centric IQA approaches, and provide a specific instance under the framework to alleviate the overfitting and easy dataset problems.
\subsection{Proposed Framework}
As shown in Fig.~\ref{fig:framework}, there is a rich body of work on how to train IQA models on available human-rated datasets, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the connection from data-centric IQA to model-centric IQA. The missing part of closing the loop is to leverage existing IQA models to guide the creation of new IQA datasets. Assuming that a subjective testing environment exists, in which reliable MOSs can be collected, the problem reduces to how to sample, from a large-scale unlabeled image set $\mathcal{U}$ with great scene complexities and visual distortions, a subset $\mathcal{D}$, whose size depends on the human labeling budget. Motivated by the experimental results in Sec.~\ref{sec:problem}, we argue that the images in $\mathcal{D}$ are sampling-worthy if they are
\begin{itemize}
\item \textit{difficult}, which best manifest themselves as dramatic failures of state-of-the-art IQA models
\item and \textit{diverse}, which test different aspects of the models, therefore exposing different erroneous behaviors.
\end{itemize}
Mathematically, sample selection corresponds to the following optimization problem:
\begin{align}\label{eq:s}
\mathcal{D} = \argmax_{\mathcal{S}\subset{\mathcal{U}}} \mathrm{Diff}(\mathcal{S};q)+\lambda\mathrm{Div}(\mathcal{S}),
\end{align}
where $\mathrm{Diff}(\cdot)$ is a difficulty measure of $\mathcal{S}$ w.r.t. the IQA model, $q(\cdot)$. We may define $\mathrm{Diff}(\cdot)$ on a set of IQA algorithms as well. $\mathrm{Div}(\cdot)$ quantifies the diversity of $\mathcal{S}$. $\lambda$ is a trade-off parameter for the two terms. As a specific case of subset selection \cite{davis1997adaptive,natarajan1995sparse}, Problem~\eqref{eq:s} is generally NP-hard unless special properties of $\mathrm{Diff}(\cdot)$ and $\mathrm{Div}(\cdot)$ can be exploited. Popular approximate solutions to subset selection include greedy algorithms and convex relaxation methods.
Once $\mathcal{D}$ is identified, we collect the MOS for each $x\in \mathcal{D}$ in the assumed subjective testing environment, which completes the connection from model-centric IQA to data-centric IQA. The newly labeled $\mathcal{D}$, by definition, exposes different aspects of failure cases of the IQA model $q(\cdot)$, which is useful for improving its generalization. We then iterate the process of model rectification, sample selection, and subjective testing, with the ultimate goal of improving learning-based IQA from both model and data perspectives. We summarize the proposed framework in Algorithm~\ref{alg:Framwork}.
\begin{algorithm}[t]
\caption{Proposed Framework to Integrate Model-centric and Data-centric IQA}
\label{alg:Framwork}
\KwIn{A training set $\mathcal{L}$, a large-scale unlabeled image set $\mathcal{U}$, a strong off-the-shelf BIQA model $q^{(0)}(\cdot)$ (with associated loss function and optimization technique), a difficulty measure $\mathrm{Diff}(\cdot)$, a diversity measure $\mathrm{Div}(\cdot)$, and the maximum iteration number $T$.}
\KwOut{$T$ new datasets $\{\mathcal{D}^{(t)}\}_{t=1}^T$, and a rectified IQA model $q^{(T)}$}
\For{$t \gets 1$ \KwTo $T$}
{
$\mathcal{U}\leftarrow\mathcal{U}\setminus\left(\bigcup_{t'=1}^{t-1}\mathcal{D}^{(t')}\right)$\\
$\mathcal{D}^{(t)}=\argmax_{\mathcal{S}\subset{\mathcal{U}}} \mathrm{Diff}(\mathcal{S};q^{(t-1)})+\lambda\mathrm{Div}(\mathcal{S})$\\
Collect the MOS for each $x\in \mathcal{D}^{(t)}$ in an assumed subjective testing environment\\
Train $q^{(t)}$ (or fine-tune $q^{(t-1)}$ to obtain $q^{(t)}$) on the combination of $\mathcal{L}$ and $\bigcup_{t'=1}^{t}\mathcal{D}^{(t')}$\\
}
\end{algorithm}
\subsection{A Specific Instance in BIQA}
In this subsection, we provide a specific instance of the proposed computational framework to demonstrate its feasibility in integrating model-centric and data-centric BIQA.
To better contrast with the results in Sec.~\ref{subsec:easy dataset} and to reduce the load of subjective testing, we use SPAQ to simulate the large-scale unlabeled image set $\mathcal{U}$. The strong off-the-shelf BIQA model, $q(\cdot)$, is again UNIQUE~\cite{zhang2021uncertainty}, which employs ResNet-34 as the backbone, and the full BID, CLIVE, and KonIQ-10k as the combined training set $\mathcal{L}$. The objective function used for optimization is the fidelity loss~\cite{tsai2007frank}. The optimization technique is a variant of stochastic gradient descent \cite{kingma2014adam}.
\begin{figure}[t]
\centering
\addtocounter{subfigure}{-1}
\subfloat[]{\includegraphics[width=1\linewidth]{model/model1}}\\
\subfloat[]{\includegraphics[width=1\linewidth]{model/failure_module}}\\
\vspace{-.3cm}
\caption{\textbf{(a)} The main quality predictor, $q(\cdot)$, and the auxiliary failure predictor, $f(\cdot)$, are jointly optimized by minimizing two fidelity losses. \textbf{(b)} The backbone of $q(x)$ is ResNet-34, composed of four residual blocks. The failure predictor $f(x)$ takes the feature maps from each block as input, and produces a scalar to indicate the difficulty of learning $x$.}
\vspace{-0.3cm}
\label{fig:model1}
\end{figure}
The core to our method is the instantiation of the sampling-worthiness module, which consists of two computational submodules to quantify the difficulty of a candidate set $\mathcal{S}$ w.r.t. to $q(\cdot)$ and the diversity of $\mathcal{S}$. Inspired by previous seminal work~\cite{welling2009herding,elhamifar2013sparse,misra2014data,yoo2019learning}, we choose to measure the difficulty through failure prediction. As shown in Fig.~\ref{fig:model1}, our failure predictor, $f(\cdot)$, has two characteristics: 1) it is an auxiliary module that incurs a small number of parameters; 2) it is jointly trained with main quality predictor. For the feature maps from each residual block, we first summarize spatial information via global average pooling and further process them with a fully connected (FC) layer, followed by ReLU nonlinearity.
After that, the four feature vectors are concatenated to pass through another FC layer to compute a scale $f(x)$ as an indication of the difficulty of learning $x$. Assuming Gaussianity of $f(x)$ with unit variance, the probability that $x$ is more difficult than $y$ can be calculated by
\begin{align}
\hat{p}(x,y)\ =\ \Phi\left(\frac{f(x)-f(y)}{\sqrt{2}}\right).
\end{align}
For the same training pair $(x,y)$, the ground-truth label can be computed by
\begin{align}
\label{eq:failure}
p(x,y) = \begin{cases}
1 \quad \mbox{if}\,\ \vert q(x)-\mu(x)\vert \geq\vert q(y)-\mu(y)\vert \\
0 \quad \mbox{otherwise},
\end{cases}
\end{align}
where $\mu(\cdot)$ represents the MOS of an image. That is, $p(x,y)=1$ indicates that $x$ is more difficult to learn than $y$, as evidenced by a high absolute error.
We learn the parameters of the failure predictor by minimizing the fidelity loss between $p(x,y)$ and $\hat{p}(x,y)$:
\begin{align} \label{eq:fidelity}
\ \ell_\mathrm{F}(x,y) = &1-\sqrt{p(x,y)\hat{p}(x,y)}\nonumber\\
&-\sqrt{(1-p(x,y))(1-\hat{p}(x,y))}.
\end{align}
A significant advantage of the learning-to-rank formulation of failure prediction is that $f(\cdot)$ is independent of the scale of $q(\cdot)$, which may oscillate over iterations~\cite{yoo2019learning}. After sufficient training, we may adopt $f(\cdot)$ to quantify the difficulty of $\mathcal{S}$:
\begin{align}\label{eq:diff}
\mathrm{Diff}(\mathcal{S}) = \frac{1}{\vert \mathcal{S}\vert} \sum_{x\in\mathcal{S}} f(x).
\end{align}
We next define the diversity of $\mathcal{S}$ as the mean pairwise distances computed from the $1,000$-dim logits of the VGGNet~\cite{Simonyan14c}:
\begin{align}\label{eq:div}
\mathrm{Div}(\mathcal{S}) = \frac{1}{\vert \mathcal{S}\vert^2}\sum_{(x,y)\in\mathcal{S}}\left\Vert\mathrm{logit}(x)- \mathrm{logit}(y)\right\Vert_2^2,
\end{align}
which provides a reasonable account for the semantic dissimilarity among natural scenes. While maximizing $\mathrm{Diff}(\mathcal{S})$ in Eq. \eqref{eq:diff} enjoys a linear complexity in the problem size, it is not the case when maximizing $\mathrm{Div}(\mathcal{S})$~\cite{kuo1993analyzing}. Thus to facilitate subset selection, we use a similar greedy method (in Eq.~\eqref{eq:gmad2}) to solve Problem~\eqref{eq:s}. Assuming $\mathcal{D} = \{x_{k'}^\star\}_{k'=1}^{k-1}$ is the (sub)-optimal subset that contains $k-1$ images, the $k$-th optimal image can be chosen by
\begin{align}
x_k^\star = \mathop{\text{argmax}}_{x\in \mathcal{U}\setminus\mathcal{D}} f(x) + \frac{\lambda}{k-1}\sum_{k'=1}^{k-1}\left\Vert\mathrm{logit}(x)- \mathrm{logit}(x_{k'}^\star)\right\Vert_2^2.
\end{align}
\begin{table}
\centering
\caption{SRCC results of the proposed sampling-worthiness module against five deep active learning methods with and without the diversity measure. The large-scale unlabeled set $\mathcal{U}$ is simulated with SPAQ~\cite{fang2020perceptual}. It is noteworthy that a lower SRCC in $\mathcal{D}$ indicate stronger capability of failure identification. RS: Random sampling. QBC: Query by committee. GS: Greedy sampling. }
\label{tab:srcc_cmp1}
\setlength{\tabcolsep}{2.2mm}{
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}[-3pt]{Method} & \multicolumn{2}{c}{Without diversity }&\multicolumn{2}{c}{With diversity}\\
\cmidrule(lr){2-3} \cmidrule(lr){4-5}
& $\mathcal{D}$ &$\mathcal{U}\setminus\mathcal{D}$ & $\mathcal{D}$ & $\mathcal{U}\setminus\mathcal{D}$\\ \hline
RS & 0.8118 & 0.8111 & 0.8091 & 0.8111 \\
QBC~\cite{burbidge2007active} & 0.5212 & 0.8120 & 0.4936 & 0.8122 \\
MC dropout~\cite{pop2018deep} & 0.6892 & 0.8096 & 0.6350 & 0.8096 \\
Core-set~\cite{sener2018active} & 0.5915 & 0.8130 & 0.6113 & 0.8128 \\
GS~\cite{bhaskara2019greedy} & 0.5707 & 0.8119 & 0.5209 & 0.8118 \\\hline
Proposed & \textbf{0.3584} &0.8121 & \textbf{0.2964} & 0.8127 \\
\bottomrule
\end{tabular}
}
\end{table}
\vspace{-.2cm}
\subsection{Experiments}\label{subsec:ER}
\paragraph{Experimental Setup}
Training is carried out by minimizing two fidelity losses, one for quality assessment and the other for failure prediction, the latter of which is downweighted by a factor of $2$ to emphasize the main task. The parameters of UNIQUE based on ResNet-34~\cite{he2016deep} are initialized with the weights pre-trained on ImageNet~\cite{deng2009imagenet}. The last FC layer of UNIQUE and the failure predictor are initialized by He's method~\cite{he2015delving}. We adopt Adam~\cite{kingma2014adam} with an initial learning rate of $10^{-4}$ and a decay factor of $10$ for every three epochs, and we train our model for twelve epochs. The first three epochs are warm-up training, where only randomly initialized layers are adjusted. We set the $\lambda$ in Eq.~\eqref{eq:s} to $10^{-5}$ in order to balance the scales between $\mathrm{Diff}(\cdot)$ and $\mathrm{Div}(\cdot)$. We use SPAQ to simulate $\mathcal{U}$, and select a subset $\mathcal{D}$ of size $100$. Similarly, we repeat the training procedure five times to reduce the influence of random initializations, and report the median results.
\paragraph{Failure Identification Results}
We compare the failure identification capability of the proposed sampling-worthiness module against several deep active learning methods, including random sampling, MC dropout~\cite{pop2018deep}, core-set selection~\cite{sener2018active}, query by committee\cite{seung1992,wang2021troubleshooting}, and greedy sampling~\cite{bhaskara2019greedy}. More details can be found in the Appendix on how these methods are implemented in conjunction with UNIQUE. Table~\ref{tab:srcc_cmp1} shows the SRCC results between the model predictions of UNIQUEs and the MOSs on the selected $\mathcal{D}$ and $\mathcal{U}\setminus\mathcal{D}$. A lower SRCC in $\mathcal{D}$ indicates better failure identification performance. We find that, for all methods except random sampling, the selected images in $\mathcal{D}$ are more difficult than the remaining ones.
The proposed sampling-worthiness module, with and without the diversity measure, delivers the best performance, identifying more difficult samples which will be useful for improve the generalizability of the quality predictor. It is interesting to note that the failure identification performance of the proposed failure predictor can be enhanced by the incorporation of the diversity measure.
\paragraph{Visual Results} Fig.~\ref{fig:diversity} shows representative top-$K$ images selected from SPAQ by the proposed sampling-worthiness module. Without the diversity constraint, our method is inclined to select difficult images of similar visual appearances, corresponding to the same underlying failure cause. When the diverse constraint is imposed, the selected images are more diverse in content and distortion. Apart from the exposure problem in the top images due to the high dynamic range of the scenes, lens blur in (e) and color cast in (f) also emerge. This indicates that the diversity measure encourages the failure predictor to expose different erroneous behaviors of UNIQUE.
\begin{figure}[t]
\centering
\addtocounter{subfigure}{-1}
\subfloat[]{\includegraphics[width=0.32\linewidth]{demo/without_diversity/without_diversity1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.32\linewidth]{demo/without_diversity/without_diversity2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.32\linewidth]{demo/without_diversity/without_diversity3}}\\
\subfloat[]{\includegraphics[width=0.32\linewidth]{demo/with_diversity/with_diversity1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.32\linewidth]{demo/with_diversity/with_diversity2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.32\linewidth]{demo/with_diversity/with_diversity3}}\\
\vspace{-.3cm}
\caption{Representative images selected from SPAQ by the proposed sampling-worthiness module \textbf{(a)}-\textbf{(c)} with and \textbf{(d)}-\textbf{(f)} without the diversity measure. Zoom in for improved distortion visibility.}
\vspace{-0.3cm}
\label{fig:diversity}
\end{figure}
\section{Conclusion and Future Work}
In this paper, we have conducted an empirical study to reveal the \textit{overfitting} and \textit{easy dataset} problems rooted in the current development of IQA. We believe these arise because of the weak connection from model to data. We have made an initial attempt to integrate model-centric and data-centric IQA, and provided a specific instance to verify its feasibility by developing a sampling-worthiness module for difficulty and diversity quantification. Our module has been proved preliminarily more effective in spotting diverse failures of the involved IQA model.
In the future, we will improve the current sampling-worthiness module by developing better difficulty and diversity measures. We may also search for more efficient discrete optimization techniques to solve the subset selection problem in our context. Moreover, we will certainly utilize the sampling-worthiness module to construct a large-scale challenging IQA dataset, with the goal of facilitating the development of more generalizable IQA models. Last, we hope the proposed framework will inspire researchers in related fields to rethink exciting future directions of IQA.
{\small
\bibliographystyle{ieee_fullname}
\section{Details of the Compared Methods}
As mentioned in the main paper, we choose nine BIQA models published from 2017 to 2021: RankIQA~\cite{liu2017rankiqa}, DeepIQA~\cite{bosse2017deep}, NIMA~\cite{talebi2018nima}, KonCept512~\cite{hosu2020koniq}, Fang2020~\cite{fang2020perceptual}, HyperIQA~\cite{su2020blindly}, LinearityIQA~\cite{li2020norm}, UNIQUE~\cite{zhang2021uncertainty}, and MetaIQA+~\cite{zhu2021generalizable}. RankIQA~\cite{liu2017rankiqa} trains a Siamese Network~\cite{chopra2005learning} which contains two identical VGG-like~\cite{Simonyan14c} network branches with a ranking loss module to rank images in terms of image quality. DeepIQA~\cite{bosse2017deep} employs a network with ten convolutional layers, five pooling layers, and two fully connected layers, which is relatively deep but not sophisticated. NIMA~\cite{talebi2018nima} predicts the distribution of human opinion scores using a convolutional neural network other than predicting the mean opinion score like other methods. The authors claim that the network can be used to score images reliably with a high correlation to human perception. KonCept512~\cite{hosu2020koniq} proposes a deep learning model based on InceptionResNet~\cite{szegedy2016rethinking} architecture, which is claimed to have an excellent generalization. For Fang2020~\cite{fang2020perceptual}, the baseline model adopts ResNet-50~\cite{he2016deep} as the backbone for the prediction. HyperIQA~\cite{su2020blindly} proposes a self-adaptive hyper network architecture to blind assess image quality in the wild, which separates the IQA procedure into three stages, including a backbone network for semantic feature extraction, a content understanding hyper network for quality perception rule learning, and a target network for quality prediction. LinearityIQA~\cite{li2020norm} explores norm in designing the loss function for faster convergence and better performance. UNIQUE~\cite{zhang2021uncertainty} develops a unified uncertainty-aware BIQA model based on ResNet-34~\cite{he2016deep} architecture for both synthetic and realistic distortions, which can be trained on multiple IQA databases simultaneously. MetaIQA+~\cite{zhu2021generalizable} proposes to learn a generalized NR-IQA model based on optimization-based deep meta-learning. More details of these methods are concluded in the Table~\ref{tab:nr-iqa-summary}.
\section{Illustrated Images of the Collected Dataset}
As mentioned in main paper, the collected images cover a wide range of realistic camera distortions, such as sensor noise contamination, motion and out-of-focus blurring, under-and over-exposure, contrast reduction, color cast, and a mixture of them. We show some visual examples in Fig.~\ref{fig:examples}, where these distortions can be observed.
\section{Detailed Setup of the Subjective Experiment}
The two-alternative forced choice(2AFC) method is adopted, allowing differentiation of subtler quality variations. Subjects are forced to choose the image with higher perceived quality with unlimited viewing time for the resulting 720 paired comparisons. We gather data from $25$ subjects, who are mostly young researchers with a computer science background but unaware of the goal of this work, including 12 male and 13 female subjects, whose ages are between 22 and 30. Participants are asked to finish the experiments in the place which has normal lighting condition without reflecting ceiling walls and floor. The LCD monitor resolution of $2560 \times 1600$ pixel is used to display all images. No time constraints were placed, and participants can take a rest at any time to avoid the influence of the fatigue effect.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{model-centric/NIMA_SPAQ}
\caption{Representative gMAD pairs between NIMA and Fang2020. (\textbf{a}) Fixing NIMA at the low quality level. (\textbf{b}) Fixing NIMA at the high quality level. (\textbf{c}) Fixing Fang2020 at the low quality level. (\textbf{d}) Fixing Fang2020 at the high quality level.}
\label{fig:gmad1}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{model-centric/hyper_metaiqa}
\caption{Representative gMAD pairs between HyperIQA and MetaIQA+. \textbf{(a)} Fixing HyperIQA at the low quality level. \textbf{(b)} Fixing HyperIQA at the high quality level. \textbf{(c)} Fixing MetaIQA+ at the low quality level. \textbf{(d)} Fixing MetaIQA+ at the high quality level.}
\label{fig:gmad2}
\end{figure*}
\section{More Training Details on Sec. 3.2}
The four real-world image databases - BID~\cite{ciancio2010no}, CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, and SPAQ~\cite{fang2020perceptual} have been adopted in our experiments. The state-of-the-art BIQA model - UNIQUE - that permits training on multiple datasets at the same time is selected for our experiments. We train three versions of the UNIQUE model on the combination of available datasets in the chronological order. For each training setting, we randomly sample $80\%$ images from each dataset to construct the training set, leaving the remaining $10\%$ for validation and $10\%$ for testing, respectively. Here are the training details for the three training setting.
\begin{itemize}
\item We first train the UNIQUE on BID and test the difficulty of the entire CLIVE, KonIQ-10k, and SPAQ, respectively. 20,000 pairs of images generated by 469 images are used for training. The trained model is named UNIQUEv1.
\item Then, we train UNIQUE on BID and CLIVE and test the difficulty of the entire KonIQ-10k and SPAQ, respectively. 60,000 pairs images, among them 20,000 are from BID, and 40,000 are generated from 930 images in CLIVE, are for training. The trained model is named UNIQUEv2.
\item At last, we train UNIQUE on BID, CLIVE, and KonIQ-10k, and test the difficulty of the entire SPAQ. 150,000 pairs of images are for training, where 20,000 are from BID, 40,000 are from CLIVE, and 90,0000 are generated from 8,058 images in KonIQ-10k. The trained model is named UNIQUEv3.
\end{itemize}
The parameters of UNIQUE based on ResNet-34~\cite{he2016deep} are initialized with the weights pre-trained on ImageNet~\cite{deng2009imagenet}. The parameters of the last fully connected layer are initialized by He’s method~\cite{he2015delving}. The initial learning rate is set to $10^{-4}$ with a decay factor of 10 for every three epochs, and UNIQUE is trained for twelve epochs in total. A warm-up training strategy is adopted: in the first three epochs, the parameters of ResNet-34 are frozen, and only the parameters of the last fully connected layer are updated. To reduce the bias caused by the randomness in the train-val-test splitting, we repeat this procedure ten times and report the median SRCC results.
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{model-centric/kon_meta}
\caption{Representative gMAD pairs between KonCept512 and MetaIQA+. (\textbf{a)}) Fixing KonCept512 at the low quality level. (\textbf{b}) Fixing KonCept512 at the high quality level. (\textbf{c}) Fixing MetaIQA+ at the low quality level. (\textbf{d}) Fixing MetaIQA+ at the high quality level.}
\label{fig:gmad3}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=1\linewidth]{model-centric/kon_SPAQ}
\caption{Representative gMAD pairs between KonCept512 and Fang2020. (\textbf{a}) Fixing KonCept512 at the low quality level. (\textbf{b}) Fixing KonCept512 at the high quality level. (\textbf{c}) Fixing Fang2020 at the low quality level. (\textbf{d}) Fixing Fang2020 at the high quality level.}
\label{fig:gmad4}
\end{figure*}
\section{Additional Experimental Results}
\paragraph{For overfitting problem}
We compare the nine IQA models using gMAD for the overfitting problem. And here we will provide more visual examples in Figs.~\ref{fig:gmad1},~\ref{fig:gmad2},~\ref{fig:gmad3} and~\ref{fig:gmad4}.
Fig.~\ref{fig:gmad1} depicts four gMAD competition results between NIMA and Fang2020. It is clear that the pairs of images in (a) and (b) exhibit similar quality, which is in disagreement with Fang2020 but consistent with NIMA. While, in (c) and (d), we observe that NIMA is able to attack Fang2020 successfully by finding its counterexamples with large perceptual gaps between the best and the worst. Fig.~\ref{fig:gmad2} shows four representative gMAD pairs between MetaIQA+ and HyperIQA. In (a) and (b), HyperIQA successfully survives the attacks from MetaIQA+, with pairs of images in (a) and (d) having similar quality according to human perception. When the roles of HyperIQA and MetaIQA+ are reversed, It is clear that the pairs of images in (c) and (d) exhibit substantially different qualities. HyperIQA correctly predicts top images to have much better quality than bottom images. Fig.~\ref{fig:gmad3} shows four representative gMAD pairs between MetaIQA+ and KonCept512. MetaIQA+ fails to attack KonCept512 according to (a) and (b), while KonCept512 favors the top images in (c) and (d), which is consistent with human judgments. Fig.~\ref{fig:gmad4} shows four representative gMAD pairs between Fang2020 and KonCept512. The pairs of images in (a) and (b) show similar quality, which means that KonCept512 survives the attacks of Fang2020. And at the same time, it can attack Fang2020 successfully by picking the failure cases of Fang2020 through (c) and (d). This further validates the overfitting of the model-centric methods.
\paragraph{For the easy dataset problem}
The most difficult examples for each dataset (as measured by the mean squared error (MSE) between model predictions and MOSs) often share similar visual appearances, which are simply different manifestations of the same underlying failure cause. Visual examples in Fig.~\ref{fig:diversityclive} and ~\ref{fig:diversitykon} are the difficult examples in CLIVE and KonIQ-10K, respectively, which indicates the sample diversity may not be well imposed in existing datasets.
\begin{figure*}[t]
\centering
\addtocounter{subfigure}{-1}
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/Clive/1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/Clive/2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/Clive/3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/Clive/4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/Clive/5}}\\
\caption{The top-5 difficult samples in CLIVE. (\textbf{a})-(\textbf{e}) for UNIQUEv1.}
\label{fig:diversityclive}
\end{figure*}
\begin{figure*}[t]
\centering
\addtocounter{subfigure}{-1}
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v1_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v1_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v1_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v1_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v1_5}}\\
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v2_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v2_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v2_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v2_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/kon/v2_5}}\\
\caption{The top-5 difficult samples in KonIQ-10k. (\textbf{a})-(\textbf{e}) for UNIQUEv1. \textbf{(f)}-\textbf{(j)} for UNIQUEv2.}
\label{fig:diversitykon}
\end{figure*}
\section{Details of Sample Selection}
We have described how to select samples with the sampling-worthiness module in the main paper. In this supplemental material, we describe how the samples are selected with other compared methods, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, random sampling, MC dropout, Core-set, query by committee, and greedy sampling.
Sample selection corresponds to the following optimization problem:
\begin{align}\label{eq:s}
\mathcal{D} = \argmax_{\mathcal{S}\subset{\mathcal{U}}} \mathrm{Diff}(\mathcal{S};q)+\lambda\mathrm{Div}(\mathcal{S}),
\end{align}
where $\mathrm{Diff}(\cdot)$ is measured by the active learning methods, and $\mathrm{Div}(\cdot)$ is measured by the $1,000$-dim logits of the VGGNet.
\paragraph{Random Sampling}
100 images are randomly sampled from the unlabeled dataset without considering any condition.
\paragraph{MC Dropout}
The BIQA model we employ is UNIQUE~\cite{zhang2021uncertainty} with the full BID, CLIVE, and KonIQ-10k as the combined training set. The total training pairs are 150,000, where 20,000 are from BID, 40,000 are from CLIVE, and 90,000 are from KonIQ-10K. The parameters of UNIQUE based on ResNet-34~\cite{he2016deep} are initialized with the weights pre-trained on ImageNet~\cite{deng2009imagenet}. The parameters of the last FC layer of UNIQUE are initialized by He's method~\cite{he2015delving}. We adopt Adam~\cite{kingma2014adam} with an initial learning rate of $10^{-4}$ and a decay factor of $10$ for every three epochs, and we train our model for twelve epochs in total. During the inference time, we randomly dropout the model for 15 times at a dropout rate $p=0.5$. After that, there will be 15 predictions for every sample, and we calculate the variance, which acts as the $\mathrm{Diff}(\cdot)$ in Eq.~\eqref{eq:s}, and can be treated as the sampling model. Specifically, the higher variance indicates the model is more uncertain of the sample. Therefore, the 100 images with the highest variance are sampled as the most difficult cases for the target model. When considering the diversity, we take the image with the highest variance as the first image. When we sample the second image, we consider both the uncertainty and the diversity. The diversity - $\mathrm{Div}(\cdot)$ in Eq.~\eqref{eq:s} can be measured by computing the distance between the $1,000$-dim logits of the VGGNet of the image $x\in \mathcal{U}\setminus\mathcal{D}$ and the mean of the $1,000$-dim logits of the VGGNet of the selected images $\mathcal{D}$. The $\lambda$ in Eq.~\eqref{eq:s} is set to $10^{-6}$ to balance the scales between $\mathrm{Diff}(\cdot)$ and $\mathrm{Div}(\cdot)$ according to the value of the variance.
\paragraph{Core-set}
We also employ UNIQUE~\cite{zhang2021uncertainty} in the core-set method. However, it is a little different from the original model. Besides the output of the original UNIQUE, the feature of the last layer will be output for further sampling. The training process is the same as the process above. When we sample the images, K-center is employed, and the features, output by the models, are employed as the input data for sampling. The newly selected image $x\in \mathcal{U}\setminus\mathcal{D}$ needs to have the maximum distance from the selected image set $ \mathcal{D}$. The distance between the image $x$ and the selected image set $ \mathcal{D}$ can be achieved by calculating the minimum distance between the image $x$ and each image of set $\mathcal{D}$. Core-set has already considered the diversity since it employed the distance between the features it learned to sample the images. As we use the $1,000$-dim logits of the VGGNet to measure the diversity in other methods, to be consistent, we replace the features learned by core-set with the $1,000$-dim logits of the VGGNet to calculate the distance for sampling.
\paragraph{Query By Committee}
The implementation of query by committee method is also based on the UNIQUE~\cite{zhang2021uncertainty}. However, different from the above cases that use UNIQUE based on ResNet-34, the UNIQUE we use here is based on ResNet-18, because the pruning process of ResNet-34 is complicated and is more time-consuming. Except for the backbone, the training processes are the same as the above cases. After training, the model $f$ will be pruned using small $l_{2}$-norms filters~\cite{Li2017PruningFF} and filter pruning via geometric median(FPGM)~\cite{he2019filter}. For both cases, the pruning process is operated for 3 times, and the pruning rates are $0.4$, $0.5$, and $0.6$. Every time after pruning, the pruned model will be trained for 12 epochs to improve its performance. There will be six predictions derived from the six pruned models for every sample, and then the variance for every sample, which acts as the $\mathrm{Diff}(\cdot)$ in Eq.~\eqref{eq:s}, will be calculated among the six predictions. The higher variance indicates the sample is with higher uncertainty. Thus, 100 images with the highest variances are sampled as the most difficult cases for the target model. When considering the diversity - $\mathrm{Diff}(\cdot)$ in Eq.~\eqref{eq:s}, the distance between the $1,000$-dim logits of the VGGNet of the image $x\in \mathcal{U}\setminus\mathcal{D}$ and the mean of the $1,000$-dim logits of the VGGNet of the selected images $\mathcal{D}$. The $\lambda$ in Eq.~\eqref{eq:s} is set to $10^{-6}$ in this case according to the value of the variance here.
\paragraph{Greedy sampling}
We employ K-center in this case, but slightly different from the original K-center method, which usually selects the furthest point from the current set of centers. In this case, we drop only one sample from the whole dataset at one time. After dropping the sample, we will do the clustering and then calculate the precision. For every sample, there will be a precision which represents the performance of the clustering method when dropping this sample. Assuming the precision is $P_{all}$, when there is no sample dropped, and the precision is $P_{i}$, when the $i$-th sample is dropped. Then, $\delta =P_{i}-P_{all}$ can represents the difficulty -$\mathrm{Diff}(\cdot)$ in Eq.~\eqref{eq:s} of the $i$-th sample. The higher $\delta$ means the sample is more difficult, and we sample the 100 images with the highest $\delta$. Similar to the diversity measure - $\mathrm{Diff}(\cdot)$ in Eq.~\eqref{eq:s} in the above methods, the distance between the VGG feature of the newly selected image and the mean of VGG feature of the selected images acts as the diversity measure here. The $\lambda$ in Eq.~\eqref{eq:s} is set to $10^{-7}$ in this case according to the value of the $\delta$.
{\small
\bibliographystyle{ieee_fullname}
\section{Introduction}\label{sec:intro}
Image quality assessment (IQA) is indispensable in a broad range of image processing and computer vision applications, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, image acquisition, compression, enhancement, and rendering~\cite{wang2006modern}. In recent years, learning-based methods~\cite{ye2012unsupervised,xu2016blind,ma2017dipiq}, especially those using convolutional neural networks (CNNs)~\cite{kang2014convolutional,ma2017end,bosse2017deep} have significantly advanced the field of IQA. A learning-based IQA system generally has two key components: the engine ``model" and its fuel ``data." The IQA model is learned to predict image quality from a large number of human-annotated data. From this perspective, it is natural to categorize IQA studies into model-centric and data-centric approaches.
\begin{figure}[!t]
\centering
\addtocounter{subfigure}{-1}
\subfloat{\includegraphics[width=\linewidth]{model/framework}}
\caption{Past work makes weak connections between data-centric and model-centric IQA, hindering the further progress of the field. The
connection from data-centric IQA to model-centric IQA embodies tremendous amount of work on how to train ``accurate'' IQA models on human-rated datasets. The missing part for closing the loop is to leverage model-centric IQA to guide the design of data-centric IQA, especially in sample selection.}
\label{fig:framework}
\end{figure}
The goal of model-centric IQA~\cite{mittal2012no,liu2017rankiqa,talebi2018nima,su2020blindly,zhang2021uncertainty} is to build computational methods (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, objective models) that provide consistent predictions on human perception of image quality. Improved learning-based IQA models have been developed from the aspects of computational structures, objective functions, and optimization techniques. Particularly, the \textit{computational structures} have shifted from shallow~\cite{kang2014convolutional} to deep methods with cascaded linear and nonlinear stages~\cite{hosu2020koniq}. Effective quality computation operators have also been identified along the way, such as generalized divisive normalization (GDN) over half-wave rectification (ReLU)~\cite{ma2017end}, bilinear pooling over global average pooling~\cite{zhang2018blind}, and adaptive convolution over standard convolution~\cite{su2020blindly}. The \textit{objective functions} mainly pertain to the formulation of IQA. It is intuitive to think of visual quality as absolute quantity, and employ the Minkowski metric to measure the prediction error. Another popular formulation is learning-to-rank~\cite{gao2015learning} by treating perceptual quality as relative quantity, which admits a family of pairwise and listwise ranking losses~\cite{liu2011learning,ma2017dipiq}. Other loss functions for accelerated convergence~\cite{li2020norm} and uncertainty quantification~\cite{zhang2021uncertainty,wang2021semi}, have also begun to emerge. The \textit{optimization techniques} in IQA benefit significantly from practical tricks to train large-scale CNNs for visual recognition~\cite{li2017scaling}.
One learning strategy specific to IQA is the ranking-based dataset combination trick~\cite{zhang2021uncertainty}, which enables an IQA model to be trained on multiple datasets without perceptual scale realignment.
The goal of data-centric IQA is to construct human-rated IQA datasets via psychophysical experiments for the purpose of benchmarking and developing objective IQA models. A common theme in data-centric IQA~\cite{sheikh2006statistical,ciancio2010no,ghadiyaram2015massive,fang2020perceptual} is to design efficient subjective testing methodologies to collect reliable human ratings of image quality, typically in the form of mean opinion scores (MOSs). Extensive practice~\cite{mantiuk2012comparison} seems to show that there is no free lunch in data-centric IQA: collecting more reliable MOSs generally requires more delicate and time-consuming psychophysical procedures, such as the two-alternative forced choice (2AFC) in a well-controlled laboratory environment with proper instructions. Bayesian experimental designs have also been implemented~\cite{xu2012hodgerank,ye2014active} in an attempt to improve rating efficiency. Arguably a more crucial step in data-centric IQA is sample selection, which is, however, much under-studied. Vonikakis~\emph{et al}\onedot~\cite{vonikakis2017probabilistic} proposed a dataset shaping technique to identify the image subset with uniformly distributed attributes of interest. Cao~\emph{et al}\onedot~\cite{cao2021debiased} described a sample selection method in the context of real-world image enhancement based on the principle of maximum discrepancy competition~\cite{wang2008maximum,wang2020going}.
Although the past achievements in IQA are worth celebrating, only weak connections have been made between model-centric and data-centric IQA, which we argue is the primary impediment to further progress of IQA (see Fig.~\ref{fig:framework}). From the model perspective, objective methods are optimized and evaluated on fixed (and extensively reused) sets of data, leading to the \textit{overfitting} problem. An excellent example occurs in the subfield of full-reference IQA~\cite{wang2006modern}, where objective methods are achieving higher and higher correlation numbers, but fail the na\"{i}ve test of reference image recovery~\cite{ding2021comparison}. From the data perspective, IQA datasets are generally constructed while being blind to existing objective models. This may cause the \textit{easy dataset} problem~\cite{fang2020perceptual}: the newly created dataset may expose few failures of existing IQA models, resulting in a significant waste of the expensive human labeling budget.
In this paper, we take initial steps towards integrating model-centric and data-centric IQA. Our main contributions are three-folds.
\begin{itemize}
\item We design a series of experiments to probe computationally the overfitting problem and the easy dataset problem of blind IQA (BIQA) in real settings.
\item We describe a computational framework that integrates model-centric and data-centric IQA. The key idea is to augment the (main) quality predictor with an (auxiliary) computational module to score the sampling-worthiness of candidate images for dataset construction.
\item We provide a specific example of this framework, where we start with a (fixed) ``top-performing'' BIQA model, and train a failure predictor by learning to rank its prediction errors. Our sampling-worthiness module is then the weighted combination of the learned failure predictor and a diversity measure computed as the semantic distance between deep content-aware features~\cite{Simonyan14c}. Experiments show that the proposed sampling-worthiness module is able to spot diverse failures of existing BIQA models in comparison to several deep active learning methods~\cite{pop2018deep,sener2018active,burbidge2007active,RDwu2019}. These samples are indeed worthy of being incorporated into next-generation IQA datasets.
\end{itemize}
\section{Related Work}\label{sec:rw}
In this section, we provide a concise overview of model-centric and data-centric IQA. We then
briefly describe the unified no-reference image quality and uncertainty evaluator (UNIQUE) \cite{zhang2021uncertainty}, as we will rely heavily on it to demonstrate the proposed framework.
\subsection{Model-Centric IQA}
We will focus on reviewing model-centric BIQA, which improves quality prediction from the model perspective without reliance on original undistorted images. Conventional BIQA models pre-defined non-learnable \textit{computational structures} to extract natural scene statistics (NSS). Learning occurred at the quality regression stage by fitting a mapping from NSS to MOSs. Commonly, NSS were extracted in the transform domain~\cite{moorthy2011blind,saad2012blind}, where the statistical irregularities can be more easily characterized. Nevertheless, transform-based methods were usually very slow, which motivated BIQA models in the spatial domain, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, BRISQUE~\cite{mittal2012no} and CORNIA~\cite{ye2012unsupervised}.
With the latest advances in CNNs, learnable computational structures based on stages of linear convolution, downsampling, and nonlinear activation have revolutionized the field of BIQA~\cite{liu2017rankiqa,talebi2018nima,su2020blindly,zhang2021uncertainty}. Along this line, the \textit{objective functions} play an important role in guiding the optimization of BIQA models. It is straightforward to formulate BIQA as regression, and the Minkowski metric is the objective function of choice. An alternative view of BIQA is through the lens of learning-to-rank~\cite{liu2011learning}, with the goal of inferring relative rather than absolute quality. Pairwise learning-to-rank objectives such as the cross entropy~\cite{ma2017dipiq} and the fidelity~\cite{tsai2007frank} losses, and listwise objectives such as the cross entropy over permutation probabilities~\cite{ma2017dipiq} and the Pearson linear correlation~\cite{li2020norm} have been successfully adopted. One last ingredient of model-centric IQA is the selection of \textit{optimization techniques} for effective model training, especially when the human-rated IQA data are scarce. Fine-tuning from pre-trained CNNs on other vision tasks~\cite{talebi2018nima}, patchwise training~\cite{bosse2017deep}, and quality-aware pre-training~\cite{su2020blindly} are practical optimization tricks in BIQA. Of particular interest is the unified optimization strategy by Zhang~\emph{et al}\onedot~\cite{zhang2021uncertainty}, allowing a single model to learn from multiple datasets simultaneously.
The success of model-centric BIQA was established on the same datasets~\cite{sheikh2006statistical,larson2010most,TID2013,ghadiyaram2015massive}, which have been used to compare BIQA models for quite many years. Thus, it is reasonable to conjecture that the impressive correlation numbers achieved by sophisticated computational structures and optimization techniques may be a consequence of overfitting, and need more cautious analysis.
\subsection{Data-Centric IQA}
Data-centric IQA improves the quality prediction performance from the data perspective, and consists of two major steps: sample selection and subjective testing. The immediate output is a human-rated dataset for training and benchmarking objective IQA models. For a long time, the research focus of data-centric IQA has been designing reliable and efficient \textit{subjective testing} methodologies for MOS collection~\cite{bt500,men2021subjective}. It is generally believed that the 2AFC design (also known as paired comparison) in a well-controlled laboratory environment is more reliable than single-stimulus and multiple-stimulus methods. However, the cost to exhaust all paired comparisons scales quadratically with the number of images in the dataset, and is prohibitively expensive when the image size is large. Several methods have been introduced to reduce the cost of the 2AFC design, including HodgeRank~\cite{xu2012hodgerank} and Bayesian experimental designs~\cite{ye2014active}. To further accelerate the process, crowdsourcing-based single-stimulus methods~\cite{chen2009crowdsourceable,ghadiyaram2015massive} have been practiced to build large-scale IQA datasets with relatively noisier MOSs.
\textit{Sample selection} is perhaps more crucial in data-centric IQA, but experiences much less success. Early datasets, such as LIVE~\cite{sheikh2006statistical}, TID2008~\cite{ponomarenko2009tid2008}, and CSIQ~\cite{larson2010most}, selected images with simulated distortions. Due to the combination of reference content, distortion type and level, the number of distinct reference images is often limited. Recent large-scale IQA datasets began to contain images with realistic camera distortions, including CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, SPAQ~\cite{fang2020perceptual}, and PaQ-2-PiQ~\cite{ying2020patches}. Such shift in sample selection provides a good test for synthetic-to-realistic generalization.
Sample diversity during dataset creation has also been taken into account. Vonikakiset~\emph{et al}\onedot~\cite{vonikakis2017probabilistic} cast sample diversity as a mixed integer linear programming, and selected a dataset with uniformly distributed image attributes (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, brightness, colorfulness, contrast, and sharpness). Euclidean distances between deep features~\cite{hosu2020koniq,cao2021debiased} are also used to measure the semantic similarity. Nevertheless, sample diversity is only one piece of the story; often, the included images are too easy to challenge existing IQA models. For example, Fang~\emph{et al}\onedot~\cite{fang2020perceptual} took nearly one year to create the SPAQ dataset consisting of more than 11,000 images, most of which turn out to be easy examples, and pose little challenge to current BIQA models (see Sec.~\ref{subsec:easy dataset}).
Active learning~\cite{settles2010active,ren2020survey} may come naturally into play to prioritize sample selection. Wang~\emph{et al}\onedot~\cite{wang2021active} spotted the failures of a BIQA model with the help of multiple full-reference IQA methods in the setting of synthetic distortions. They~\cite{wang2021troubleshooting} later extended the idea to the authentic distortion scenario by creating a set of self-competitors via network pruning. The two methods are algorithm-specific, and may not be generalized to existing BIQA models. Other active learning strategies such as uncertainty-based sampling~\cite{cohn1996active}, query by committee~\cite{seung1992}, expected model change maximization~\cite{cai2013maximizing}, and failure prediction~\cite{scheirer2008fusion,zhang2014predicting} may be applied. Nevertheless, it remains to be seen whether these strategies are feasible and, if so, how effective they are in the context of deep learning-based IQA.
\subsection{UNIQUE}
UNIQUE~\cite{zhang2021uncertainty} is a recently proposed BIQA model, which combines multiple IQA datasets as the training data. Specifically, assuming Gaussianity of the true perceptual quality $q(x)$ with mean $\mu(x)$ and variance $\sigma^2(x)$ and the independence of quality variability across images, the probability of image $x$ having higher perceptual quality than image $y$ can be calculated as
\begin{align}\label{eq:tql}
p(x,y) =\Pr(q(x) \ge q(y))= \Phi\left(\frac{\mu(x)-\mu(y)}{\sqrt{\sigma^{2}(x)+ \sigma^{2}(y)}}\right),
\end{align}
where $\Phi(\cdot)$ denotes the standard Gaussian cumulative distribution function. From the $i$-th dataset for a total of $N$ datasets, UNIQUE randomly samples $N_{i}$ pairs of images $\{(x_j^{(i)}, y_j^{(i)})\}_{j=1}^{N_i}$, and the combined training dataset is thus in the form of $\mathcal D = \{\{(x_j^{(i)}, y_j^{(i)}), p_j^{(i)}\}_{j=1}^{N_i}\}_{i=1}^{N}$.
UNIQUE aims to learn two differentiable functions $f_w(\cdot)$ and $\sigma_w(\cdot)$, parameterized by a vector $w$, for quality and uncertainty estimation. The prediction of $p(x,y)$ in Eq.~\eqref{eq:tql} can be done by replacing $\mu(\cdot)$ and $\sigma(\cdot)$ with their respective estimates:
\begin{align}\label{eq:probability}
\hat{p}(x,y) = \Phi\left(\frac{f_w(x)- f_w(y)}{\sqrt{\sigma^{2}_w(x)+ \sigma^{2}_w(y)}}\right).
\end{align}
UNIQUE minimizes the fidelity loss~\cite{tsai2007frank} between the two probability distributions $p(x, y)$ and $\hat{p}(x, y)$ for parameter optimization:
\begin{align} \label{eq:fidelity_U}
\ell(x,y,p) = &1-\sqrt{p(x,y)\hat{p}(x,y)}\nonumber\\
&-\sqrt{(1-p(x,y))(1-\hat{p}(x,y))}.
\end{align}
To resolve the scaling ambiguity in Eq.~\eqref{eq:probability} and to resemble the human uncertainty when perceiving digital images, UNIQUE adds a hinge-like regularizer during training~\cite{zhang2021uncertainty}. The original UNIQUE employs ResNet-34~\cite{he2016deep} as the backbone followed by bilinear pooling and $\ell_2$-normalization, and implements $f_w(\cdot)$ and $\sigma_w(\cdot)$ as the two outputs of a fully connected (FC) layer.
\section{Probing Problems in the Progress of BIQA}\label{sec:problem}
In this section, we design a series of experiments to probe computationally that the \textit{overfitting} problem and the \textit{easy dataset} problem have actually emerged in the current development of BIQA, which we attribute to the weak connections between model-centric and data-centric BIQA.
\begin{table}[!t]
\centering
\caption{SRCC between the performance ranking of nine BIQA methods and their publishing time on four datasets. It is clear that the quality prediction performance ``improves'' steadily over time}
\label{tab:srcc}
\begin{tabular}{lcccc}
\toprule
Dataset & BID~\cite{ciancio2010no} & CLIVE~\cite{ghadiyaram2015massive} & KonIQ-10k~\cite{ghadiyaram2015massive} & SPAQ~\cite{fang2020perceptual} \\
\midrule
SRCC & 0.6946 & 0.7950 & 0.6695 & 0.7029 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table*}[!t]
\caption{Ranking results of nine BIQA models. A smaller rank indicates better performance. ``Distribution'' in the third column means that NIMA uses the earth mover's distance to match the ground-truth and predicted 1D quality distributions. ``All'' in the fourth column indicates that UNIQUE is trained on the combined dataset of LIVE, CSIQ, KADID-10K, BID, CLIVE, and KonIQ-10K}
\label{tab:nr-iqa-summary}
\centering
\begin{tabular}{llllcccc}
\toprule
Name & Backbone & Formulation & Training Set & Publishing Time & SRCC Rank & gMAD Rank & $\Delta$ Rank \\
\midrule
UNIQUE~\cite{zhang2021uncertainty} & ResNet-34 & Ranking & All & 2021.03 &1 & 5&-4 \\
KonCept512~\cite{hosu2020koniq} & InceptionResNetV2 & Regression & KonIQ-10k & 2020.01 & 2 & 1&1\\
HyperIQA~\cite{su2020blindly} & ResNet-50 & Regression & KonIQ-10k & 2020.08 &3 & 2&1 \\
LinearityIQA~\cite{li2020norm} & ResNeXt-101& Regression & KonIQ-10k & 2020.10 & 4 & 3&1\\
MetaIQA+~\cite{zhu2021generalizable} & ResNet-18 & Regression & CLIVE & 2021.04 & 5 & 8&-3\\
Fang2020~\cite{fang2020perceptual} & ResNet-50 & Regression & SPAQ &2020.08 &6 &9 &-3\\
NIMA~\cite{talebi2018nima} & VGG-16 & Distribution & AVA & 2018.04 & 7 & 4 &3\\
DeepIQA~\cite{bosse2017deep} & VGG-like CNN &Regression & LIVE &2018.01& 8 & 7 & 1\\
RankIQA~\cite{liu2017rankiqa} & VGG-16 & Ranking & LIVE & 2017.12& 9 & 6 &3\\
\bottomrule
\end{tabular}
\end{table*}
\begin{figure*}[!t]
\centering
\includegraphics[width=\linewidth]{model-centric/hyper_metaiqa}
\caption{Representative gMAD pairs between HyperIQA and MetaIQA+. \textbf{(a)} Fixing HyperIQA at the low quality level. \textbf{(b)} Fixing HyperIQA at the high quality level. \textbf{(c)} Fixing MetaIQA+ at the low quality level. \textbf{(d)} Fixing MetaIQA+ at the high quality level.}
\label{fig:gmad1}
\end{figure*}
\subsection{Overfitting Problem}\label{subsec:overfitting}
As there is no standardized computable definition of overfitting, especially in the context of deep learning~\cite{goodfellow2016deep}, quantifying overfitting is still a wide open problem. Here we choose to use the group maximum differentiation (gMAD) competition~\cite{ma2018group} to probe the generalization of BIQA models to a large-scale unlabeled dataset. gMAD is a discrete instantiation of the MAD competition~\cite{wang2008maximum} that relies on synthesized images to optimally distinguish the models. As opposed to the \textit{average-case} performance measured on existing IQA datasets, say by Spearman's rank correlation coefficient (SRCC), gMAD can be seen as a \textit{worst-case} performance test by comparing the models using extremal image pairs that are likely to falsify them. We declare an overfitting case of a BIQA model if it shows strong average performance on standard IQA datasets but weak performance in the gMAD competition.
\paragraph{Experimental Setup} We choose nine BIQA models from 2017 to 2021: RankIQA~\cite{liu2017rankiqa}, DeepIQA~\cite{bosse2017deep}, NIMA~\cite{talebi2018nima}, KonCept512~\cite{hosu2020koniq}, Fang2020~\cite{fang2020perceptual}, HyperIQA~\cite{su2020blindly}, LinearityIQA~\cite{li2020norm}, UNIQUE~\cite{zhang2021uncertainty}, and MetaIQA+~\cite{zhu2021generalizable}. Table~\ref{tab:srcc} shows the SRCC results between the publishing time of the algorithms and their performance ranking\footnote{As each BIQA model assumes different (and unknown) training and testing splits, for a less biased comparison, we compute the quality prediction performance on the full dataset.} on four widely used IQA datasets, BID~\cite{ciancio2010no}, CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, and SPAQ~\cite{fang2020perceptual}. We find that ``steady progress'' over the years has been made by employing more complicated computational structures and more advanced optimization techniques.
We now set the stage for the nine BIQA models to perform gMAD competition. Specifically, we first gather a large-scale unlabeled dataset $\mathcal{U}$, containing 100,000 photographic images with marginal distributions nearly uniform w.r.t. five image attributes (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, JPEG compression ratio, brightness, colorfulness, contrast, and sharpness). Our dataset covers a wide range of realistic camera distortions, such as sensor noise contamination, motion and out-of-focus blur, under- and over-exposure, contrast reduction, color cast, and a mixture of them. Given two BIQA models $f_i(\cdot)$ and $f_j(\cdot)$, gMAD~\cite{ma2018group} selects top-$K$ image pairs that best discriminate between them:
\begin{align} \label{eq:gmad2}
({x}^{\star}_k, y^{\star}_k) = &\argmax_{x,y} f_i(x) - f_i(y)\nonumber\\
&\text{s.t.}\ \ f_j(x) = f_j(y)=\alpha, \; x,y\in \mathcal{U}\setminus\mathcal{D}_{k-1},
\end{align}
where $\mathcal{D}_{k-1}=\{{x}^\star_{k'},{y}^\star_{k'}\}^{k-1}_{k'=1}$ is the current gMAD image set. The $k$-th image pair must lie on the $\alpha$-level set of $f_j$, where $\alpha$ specifies a quality level. The roles of $f_{i}$ and $f_{j}$ should be switched. $Q$ (non-overlapping) quality levels are selected to cover the full quality spectrum. By exhausting all distinct pairs of BIQA models and quality levels, we arrive at a gMAD set $\mathcal{D}$ that contains a total of $9\times8\times5\times 2 = 720$ image pairs, where we set $Q=5$ and $K=2$.
We invite $25$ human subjects to gather perceived quality judgments of each gMAD pair using the 2AFC method. They are forced to choose the image with higher perceived quality for the $720$ paired comparisons. The $25$ subjects are mostly young researchers (age between $22$ and $30$) with a computer science background, but are unaware of the goal of this work, including $12$ male and $13$ female subjects. They are asked to finish the experiments in an office environment with a normal lighting condition and without reflecting ceiling walls and floors. The LCD monitor
has a resolution of $2560 \times 1600$ pixels, which is sufficient to display the image pairs simultaneously in the random spatial and temporal order. The subjects can move closer to and farther away from the display for better distortion visibility. No time constraint is enforced, and the participants can take a break at any time to minimize the influence of the fatigue effect~\cite{bt500}.
After subjective testing, we obtain the raw pairwise comparison matrix $A\in\mathbb{R}^{9\times 9}$, where $a_{ij}\in\{0,1\ldots, 250\}$ indicates the counts of $x^\star$ preferred over $y^\star$ by the $25$ subjects on the ten associated image pairs (by solving Problem~\eqref{eq:gmad2}). We compute, from $A$, a second matrix $B\in\mathbb{R}^{9\times 9}$, where $b_{ij} = a_{ij}/a_{ji}$ denotes the pairwise dominance of $f_i$ over $f_j$. Laplace smoothing~\cite{schutze2008introduction} is applied when $a_{ji}$ is close to zero. We convert the pairwise comparisons
into a global ranking $r\in\mathbb{R}^9$ using Perron rank~\cite{saaty1984inconsistency}:
\begin{align}\label{eq:perron}
r = \lim_{t\rightarrow \infty}\frac{1}{t}\sum_{\beta=1}^{t}\frac{B^\beta}{\mathbf{1}^TB^\beta\mathbf{1}},
\end{align}
where $\mathbf{1}$ is a $9$-dimensional vector of all ones. The solution to Eq.~\eqref{eq:perron} is the normalized eigenvector of $B$ corresponding to the largest eigenvalue. A larger $r_i$ indicates better performance of $f_i$ in the gMAD competition.
\begin{figure*}[!t]
\centering
\includegraphics[width=\linewidth]{model-centric/hyper_linear}
\caption{Representative gMAD pairs between HyperIQA and LinearityIQA. \textbf{(a)} Fixing HyperIQA at the low quality level. \textbf{(b)} Fixing HyperIQA at the high quality level. \textbf{(c)} Fixing LinearityIQA at the low quality level. \textbf{(d)} Fixing LinearityIQA at the high quality level.}
\label{fig:gmad2}
\end{figure*}
\begin{table*}[!t]
\centering
\caption{Detailed training specifications of UNIQUE variants for probing the easy dataset problem}
\label{tab:details}
\begin{tabular}{lll}
\toprule
Variant & [Training] / [Test] Datasets & [\#Training Pairs] / [\#Test Images] \\
\midrule
UNIQUEv1 & [BID] / [CLIVE, KonIQ-10k, SPAQ] & [20,000] / [1,162, 10,073, 11,125] \\
UNIQUEv2 & [BID, CLIVE] / [KonIQ-10k, SPAQ] & [20,000, 40,000] / [10,073, 11,125]\\
UNIQUEv3 & [BID, CLIVE, KonIQ-10k] / [SPAQ]& [20,000, 40,000, 90,000] / [11,125]\\
\bottomrule
\end{tabular}
\end{table*}
\paragraph{Results} Table~\ref{tab:nr-iqa-summary} compares the ranking results of the nine BIQA models in the gMAD competition and in terms of the average SRCC on the four full datasets, BID~\cite{ciancio2010no}, CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, and SPAQ~\cite{fang2020perceptual}. The primary observation is that the latest published models, such as UNIQUE~\cite{zhang2021uncertainty}, MetaIQA+~\cite{zhu2021generalizable}, and Fang2020~\cite{fang2020perceptual} tend to overfit the peculiarities of the training sets with advanced optimization techniques. In particular, UNIQUE learns to rank image pairs from all available datasets, MetaIQA+ adopts deep meta learning for unseen distortion generalization, while Fang2020 enables adaptive multi-task learning for incorporation of auxiliary quality-relevant information. They rank much higher in terms of average SRCC than in gMAD.
Compared to improving upon optimization techniques, selecting computational structures with more capacity\footnote{As all backbone architectures are originally proposed for ImageNet classification, we thus conveniently regard the ImageNet validation accuracy as a rough indication of model capacity.} as the backbones seems to be a wiser choice, as evidenced by KonCept512, HyperIQA, and LinearityIQA with high rankings in gMAD. Fig.~\ref{fig:gmad1} shows such a visual comparison of the representative gMAD pairs between MetaIQA+ based on ResNet-18 and HyperIQA based on ResNet-50. Pairs of images in (a) and (b) have similar quality according to human perception, which is consistent with HyperIQA. When the roles of HyperIQA and MetaIQA+ are reversed, it is clear that the pairs of images in (c) and (d) exhibit substantially different quality. HyperIQA correctly predicts top images to have much better quality than bottom images, and meanwhile, the weaknesses of MetaIQA+ in handling dark and blurry scenes have also been exposed. Nevertheless, purely increasing the capacity of the backbone does not necessarily lead to consistent improvements in quality prediction, which again may be a consequence of potential overfitting. For example, HyperIQA identifies content-adaptive convolution to be a more quality-relevant computation. With ResNet-50 as the backbone, it is able to outperform LinearityIQA with a more powerful ResNeXt-101 (see Fig.~\ref{fig:gmad2}).
In addition, compared to SPAQ, KonIQ-10k of a similar scale serves as a more suitable training set, on which more generalizable models can be learned.
More surprisingly, Fang2020~\cite{fang2020perceptual} trained on SPAQ even underperforms NIMA and DeepIQA, which are, respectively, trained on datasets of aesthetic and synthetic image quality. This provides a strong indication that how overfitting can emerge if a training set is not well prepared. This issue is also closely related to the easy dataset problem, which will be deeply investigated in the next subsection. In summary, the SRCC between the gMAD ranking of the BIQA models and their publishing time is only $0.0753$, implying that the progress made by model-centric IQA might be somewhat over-estimated in terms of real-world generalization.
\subsection{Easy Dataset Problem}\label{subsec:easy dataset}
In order to reveal the easy dataset problem, it suffices to empirically show that the newly created datasets are less effective in falsifying current BIQA models.
\paragraph{Experimental Setup} We work with the same four datasets - BID~\cite{ciancio2010no}, CLIVE~\cite{ghadiyaram2015massive}, KonIQ-10k~\cite{hosu2020koniq}, and SPAQ~\cite{fang2020perceptual}. To achieve our goal, we select a state-of-the-art BIQA model - UNIQUE~\cite{zhang2021uncertainty} - that permits training on multiple datasets. We train three UNIQUEs (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, UNIQUEv1, UNIQUEv2, and UNIQUEv3, respectively) on the combination of available datasets in chronological order. For each training setting, we randomly sample $80\%$ images from each dataset to construct the training set, leaving the remaining $10\%$ for validation and $10\%$ for testing. To reduce the bias caused by the randomness in dataset splitting, we repeat the training procedure ten times, and report the median SRCC results for UNIQUE variants. Detailed training specifications can be found in Table~\ref{tab:details}.
\begin{figure*}[!t]
\centering
\addtocounter{subfigure}{0}
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v1_5}}
\vspace{-2mm}
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v2_5}}
\vspace{-2mm}
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_4}}\hskip.3em
\subfloat[]{\includegraphics[width=0.19\linewidth]{data-centric/difficult_case/v3_5}}
\caption{The top-5 difficult samples in SPAQ, as measured by the MSEs between model predictions and MOSs: \textbf{(a)}-\textbf{(e)} for UNIQUEv1. \textbf{(f)}-\textbf{(j)} for UNIQUEv2. \textbf{(k)}-\textbf{(o)} for UNIQUEv3. We expect such samples to be visually diverse in terms of the content and distortion types in order to expose different failure scenarios of UNIQUEs, which is unfortunately not observed.}
\label{fig:diversity1}
\end{figure*}
\begin{table}[!t]
\centering
\caption{SRCC between predictions of UNIQUEs and MOSs of different test sets. ``---'' means that the corresponding dataset is used for jointly training}
\label{tab:srcc2}
\begin{tabular}{lccc}
\toprule
SRCC & UNIQUEv1 & UNIQUEv2 & UNIQUEv3 \\
\midrule
CLIVE~\cite{ghadiyaram2015massive} & 0.6998 & --- & --- \\
KonIQ-10k~\cite{hosu2020koniq} & 0.6917 & 0.7251 & --- \\
SPAQ~\cite{fang2020perceptual} & 0.7204 & 0.7932 & 0.8112\\
\bottomrule
\end{tabular}
\end{table}
\paragraph{Results} Table~\ref{tab:srcc2} lists the SRCC results between predictions of UNIQUEs and MOSs of different IQA datasets as test sets. The primary observation is that as more datasets are available for training, the newly created ones are more difficult to challenge the most recent UNIQUE. For example, trained on the combination of BID, CLIVE, and KonIQ-10k, UNIQUEv3 achieves a satisfactory SRCC of $0.8112$ on SPAQ, which is higher than $0.7932$ and $0.7204$ for UNIQUEv2 and UNIQUEv1 trained with fewer data. Although SPAQ is the latest dataset, it is easier than CLIVE and KonIQ-10k. This is supported by the highest correlations obtained by UNIQUEv1 and UNIQUEv2 on SPAQ. These results are consistent with the observations in Sec.~\ref{subsec:overfitting}, where models trained on KonIQ-10k and CLIVE rank higher than models trained on SPAQ. What is worse, the most difficult examples (as measured by the mean squared error (MSE) between model predictions and MOSs) often share similar visual appearances (see visual examples in Fig.~\ref{fig:diversity1}). This shows that the sample diversity and difficulty of existing datasets may not be well imposed in a principled way.
\section{Proposed Framework for Integrating Model-Centric and Data-Centric IQA}\label{sec:pm}
In this section, we describe a computational framework for integrating model-centric and data-centric IQA approaches, and provide a specific instance within the framework to alleviate the overfitting and easy dataset problems.
\subsection{Proposed Framework}
As shown in Fig.~\ref{fig:framework}, there is a rich body of work on how to train IQA models on available human-rated datasets, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the connection from data-centric IQA to model-centric IQA. The missing part for closing the loop is to leverage existing IQA models to guide the creation of new IQA datasets. Assuming that a subjective testing environment exists, in which reliable MOSs can be collected, the problem reduces to how to sample, from a large-scale unlabeled dataset $\mathcal{U}$ with great scene complexities and visual distortions, a subset $\mathcal{D}$, whose size is constrained by the human labeling budget. Motivated by the experimental results in Sec.~\ref{sec:problem}, we argue that the images in $\mathcal{D}$ are \textit{sampling-worthy} if they are
\begin{itemize}
\item \textit{difficult}, which best manifest themselves as dramatic failures of state-of-the-art IQA models,
\item and \textit{diverse}, which test different aspects of the models, therefore exposing different erroneous behaviors.
\end{itemize}
Mathematically, sample selection corresponds to the following optimization problem:
\begin{align}\label{eq:s}
\mathcal{D} = \argmax_{\mathcal{S}\subset{\mathcal{U}}} \mathrm{Diff}(\mathcal{S};f)+\lambda\mathrm{Div}(\mathcal{S}),
\end{align}
where $\mathrm{Diff}(\cdot)$ is a difficulty measure of $\mathcal{S}$ w.r.t. the IQA model, $f(\cdot)$. It is straightforward to define $\mathrm{Diff}(\cdot)$ on a set of IQA algorithms as well. $\mathrm{Div}(\cdot)$ quantifies the diversity of $\mathcal{S}$. $\lambda$ is a trade-off parameter for the two terms. As a specific case of subset selection~\cite{davis1997adaptive,natarajan1995sparse}, Problem~\eqref{eq:s} is generally NP-hard unless special properties of $\mathrm{Diff}(\cdot)$ and $\mathrm{Div}(\cdot)$ can be exploited. Popular approximate solutions to subset selection include greedy algorithms and convex relaxation methods.
\begin{algorithm}[!t]
\caption{Computational Framework for Integrating Model-Centric and Data-Centric IQA}
\label{alg:Framwork}
\KwIn{A training set $\mathcal{L}$, a large-scale unlabeled image set $\mathcal{U}$, an off-the-shelf BIQA model $f^{(0)}(\cdot)$ (with the associated loss function and optimization technique), a difficulty measure $\mathrm{Diff}(\cdot)$, a diversity measure $\mathrm{Div}(\cdot)$, and the maximum iteration number $T$}
\KwOut{$T$ new datasets $\{\mathcal{D}^{(t)}\}_{t=1}^T$, and a rectified IQA model $f^{(T)}$}
\For{$t \gets 1$ \KwTo $T$}
{
$\mathcal{U}\leftarrow\mathcal{U}\setminus\left(\bigcup_{t'=1}^{t-1}\mathcal{D}^{(t')}\right)$\\
$\mathcal{D}^{(t)}=\argmax_{\mathcal{S}\subset{\mathcal{U}}} \mathrm{Diff}(\mathcal{S};f^{(t-1)})+\lambda\mathrm{Div}(\mathcal{S})$\\
Collect the MOS for each $x\in \mathcal{D}^{(t)}$ in an assumed subjective testing environment\\
Train $f^{(t)}$ (or fine-tune $f^{(t-1)}$ to obtain $f^{(t)}$) on the combination of $\mathcal{L}$ and $\bigcup_{t'=1}^{t}\mathcal{D}^{(t')}$\\
}
\end{algorithm}
Once $\mathcal{D}$ is identified, we collect the MOS for each $x\in \mathcal{D}$ in the assumed subjective testing environment, which makes the connection from model-centric IQA to data-centric IQA. The newly labeled $\mathcal{D}$, by construction, exposes different failures of the IQA model $f(\cdot)$, which is in turn useful for improving its generalization. We then iterate the process of model rectification, sample selection, and subjective testing, with the ultimate goal of improving learning-based IQA from both model and data perspectives. We summarize the proposed computational framework in Algorithm~\ref{alg:Framwork}.
\subsection{A Specific Instance in BIQA}
In this subsection, we provide a specific instance of the proposed computational framework, and demonstrate its feasibility in integrating model-centric and data-centric BIQA.
To better contrast with the results in Sec.~\ref{subsec:easy dataset} and to reduce the load of subjective testing, we use SPAQ to simulate the large-scale unlabeled dataset $\mathcal{U}$. The off-the-shelf BIQA model for demonstration is again UNIQUE~\cite{zhang2021uncertainty}, which trains on the combination of the full BID, CLIVE, and KonIQ-10k as $\mathcal{L}$.
\begin{figure}[!t]
\centering
\addtocounter{subfigure}{0}
\subfloat[]{\includegraphics[width=0.95\linewidth]{model/model1}}\\
\subfloat[]{\includegraphics[width=1\linewidth]{model/model2}}\\
\caption{\textbf{(a)} The main quality predictor, $f(\cdot)$, is fixed and the auxiliary failure predictor, $g(\cdot)$, is optimized by minimizing the fidelity loss. \textbf{(b)} The backbone of $f(\cdot)$ is ResNet-34, composed of four residual blocks. The failure predictor $g(\cdot)$ accepts the pooled feature representations from the four blocks, and produces a scalar to indicate the learning difficulty of the input image.}
\label{fig:model1}
\end{figure}
\begin{table}[!t]
\centering
\caption{Convolutional architecture of ResNet-34~\cite{he2016deep} as the backbone of the quality and failure predictors. The nonlinearity and normalization are omitted for brevity}
\begin{tabular}{l |c }
\toprule
Layer Name & Network Parameter\\
\hline
Convolution & \text{7$\times$7, 64, stride 2}\\
\hline
Max Pooling & \text{3$\times$3, stride 2}\\
\hline
\multirow{3}{*}{Block 1}& \blockc{64}{1}{3}\\
& \\
&\\
\hline
\multirow{6}{*}{\shortstack{Block 2}}& \blockd{64}{128}{2}{1}{1}\\
&\\
&\\
&\blockc{128}{1}{3}\\
&\\
&\\
\hline
\multirow{6}{*}{\shortstack{Block 3}}& \blockd{128}{256}{2}{1}{1}\\
&\\
&\\
& \blockc{256}{1}{5}\\
& \\
&\\
\hline
\multirow{6}{*}{\shortstack{Block 4}}& \blockd{256}{512}{2}{1}{1}\\
& \\
&\\
& \blockc{512}{1}{2}\\
& \\
&\\
\bottomrule
\end{tabular}
\label{tab:network}
\end{table}
The core of our method is the instantiation of the sampling-worthiness module, which consists of two computational submodules to quantify the difficulty of a candidate set $\mathcal{S}$ w.r.t. to $f(\cdot)$ and the diversity of $\mathcal{S}$. Inspired by previous seminal work~\cite{welling2009herding,elhamifar2013sparse,misra2014data,yoo2019learning}, we choose to measure the difficulty through failure prediction. As shown in Fig.~\ref{fig:model1}, our failure predictor, $g(\cdot)$, has two characteristics: 1) it is an auxiliary module that incurs a small number of parameters; 2) it can either be solely trained while holding the quality predictor $f(\cdot)$ fixed or jointly trained along with $f(\cdot)$. Recall that our goal is to expose diverse failures of existing BIQA models, and thus it is preferred not to re-train or fine-tune $f(\cdot)$. For the main experiments, we choose to fix the quality predictor, and defer the case of joint training in Sec.~\ref{subsec:ER2}.
The failure predictor $g(\cdot)$ accepts the feature maps of the input image $x$ from each \textit{fixed} residual block (see details in Table~\ref{tab:network}) as inputs, and summarizes spatial information via global average pooling. Each stage of pooled features then undergo an FC layer with the same number of output channels, $C$, followed by ReLU nonlinearity. After that, the four feature vectors of the same length are concatenated to pass through another FC layer to compute a scalar $g(x)$ as the indication of the difficulty of learning $x$. Assuming Gaussianity of $g(x)$ with unit variance, the probability that $x$ is more difficult than $y$ is calculated by
\begin{align}
\hat{p}_\mathrm{F}(x,y)\ =\ \Phi\left(\frac{g(x)-g(y)}{\sqrt{2}}\right).
\end{align}
For the same training pair $(x,y)$, the ground-truth label can be computed by
\begin{align}
\label{eq:failure}
p_\mathrm{F}(x,y) = \begin{cases}
1 \quad \mbox{if}\,\ \vert f(x)-\mu(x)\vert \geq\vert f(y)-\mu(y)\vert \\
0 \quad \mbox{otherwise},
\end{cases}
\end{align}
where $\mu(\cdot)$ represents the MOS. That is, $p_\mathrm{F}(x,y)=1$ indicates that $x$ is more difficult to learn than $y$, as evidenced by a higher absolute error.
We learn the parameters of the failure predictor (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, five FC layers) by minimizing the fidelity loss between $p_\mathrm{F}(x,y)$ and $\hat{p}_\mathrm{F}(x,y)$:
\begin{align} \label{eq:fidelity}
\ \ell(x,y,p_\mathrm{F}) = &1-\sqrt{p_\mathrm{F}(x,y)\hat{p}_\mathrm{F}(x,y)}\nonumber\\
&-\sqrt{(1-p_\mathrm{F}(x,y))(1-\hat{p}_\mathrm{F}(x,y))}.
\end{align}
One significant advantage of the learning-to-rank formulation of failure prediction is that $g(\cdot)$ is independent of the scale of $f(\cdot)$, which may oscillate over iterations~\cite{yoo2019learning} if joint training is enabled. After sufficient training, we may adopt $g(\cdot)$ to quantify the difficulty of $\mathcal{S}$:
\begin{align}\label{eq:diff}
\mathrm{Diff}(\mathcal{S}) = \frac{1}{\vert \mathcal{S}\vert} \sum_{x\in\mathcal{S}} g(x),
\end{align}
where $\vert\mathcal{S}\vert$ denotes the cardinality of the set $\mathcal{S}$.
We next define the diversity of $\mathcal{S}$ as the mean pairwise distances computed from the $1,000$-dim logits of the VGGNet~\cite{Simonyan14c}:
\begin{align}\label{eq:div}
\mathrm{Div}(\mathcal{S}) = \frac{1}{\vert \mathcal{S}\vert^2}\sum_{(x,y)\in\mathcal{S}}\left\Vert\mathrm{logit}(x)- \mathrm{logit}(y)\right\Vert_2^2,
\end{align}
which provides a reasonable account for the semantic dissimilarity. While maximizing $\mathrm{Diff}(\mathcal{S})$ in Eq.~\eqref{eq:diff} enjoys a linear complexity in the problem size, it is not the case when maximizing $\mathrm{Div}(\mathcal{S})$~\cite{kuo1993analyzing}. To facilitate subset selection, we use a similar greedy method (in Eq.~\eqref{eq:gmad2}) to solve Problem~\eqref{eq:s}. Assuming $\mathcal{D} = \{x_{k'}^\star\}_{k'=1}^{k-1}$ is the (sub)-optimal subset that contains $k-1$ images, the $k$-th optimal image can be chosen by
\begin{align}
x_k^\star = \argmax_{x\in \mathcal{U}\setminus\mathcal{D}} g(x) + \frac{\lambda}{k-1}\sum_{k'=1}^{k-1}\left\Vert\mathrm{logit}(x)- \mathrm{logit}(x_{k'}^\star)\right\Vert_2^2.
\end{align}
\subsection{Experiments}\label{subsec:ER}
\paragraph{Experimental Setup}
Training is carried out by minimizing the fidelity loss in Eq.~\eqref{eq:fidelity} for failure prediction while fixing the quality predictor. The output channel of the four FC layers for feature projection is set to $C=128$. All five FC layers in $g(\cdot)$ are initialized by He's method~\cite{he2015delving}. We adopt Adam~\cite{kingma2014adam} with a mini-batch size of $32$, an initial learning rate of $10^{-4}$ and a decay factor of $10$ for every five epochs, and we train the failure predictor for fifteen epochs. During sample selection, we set the $\lambda$ in Eq.~\eqref{eq:s} to $10^{-6}$ in order to balance the scale difference between $\mathrm{Diff}(\cdot)$ and $\mathrm{Div}(\cdot)$. We use SPAQ to simulate $\mathcal{U}$, and select a subset $\mathcal{D}$ of size $100$. Similarly, we repeat the training procedure five times to reduce the influence of random initializations, and report the median results. We compare the failure identification capability of the proposed sampling-worthiness module against several deep active learning methods, including random sampling, sampling by representativeness-diversity~\cite{RDwu2019}, UNIQUE uncertainty~\cite{zhang2021uncertainty}, query by committee~\cite{seung1992}, core-set selection~\cite{sener2018active}, and MC dropout~\cite{pop2018deep}. The setups of the competing methods are summarized as follows.
\begin{itemize}
\item For random sampling as the baseline, $100$ images are uniformly sampled from the unlabeled dataset $\mathcal{U}$.
\item For sampling by representativeness-diversity, $k$-means clustering is performed in the bilinearly pooled and $\ell_2$-normalized feature space. For the $i$-th iteration where $i=\{1,\cdots,100\}$, we perform $k$-means clustering on $\mathcal{U}$, where $k=i$. We identify the largest cluster that does not contain previously selected images as the current most representative cluster\footnote{Note that such cluster always exists because the number of clusters is larger than the number of already selected images in each iteration.}, from which the image with the minimum distance to its centroid is sampled.
\item The UNIQUE model comes with a trained uncertainty estimator $\sigma_w(\cdot)$ (see Eq.~\eqref{eq:probability}). We directly maximize it for sample selection.
\item For query by committee, we obtain fifteen augmented versions of the input image by randomly flipping, resizing, and cropping. The prediction variance acts as the $\mathrm{Diff}(\cdot)$ in Eq.~\eqref{eq:s}.
\item For core-set selection, we employ the bilinearly pooled and $\ell_2$-normalized feature vector as the image representation, and iteratively select images with the maximum distances to the sampled dataset $\mathcal{D}$ in the representation space. Here, the image-to-set distance is defined as the minimum Euclidean distance between the image and all images in $\mathcal{D}$. It is important to note that the diversity measure defined in Eq.~\eqref{eq:div} differs from core-set selection in 1) feature representation (VGG-based content representation against UNIQUE-based quality representation) and 2) image-to-set distance calculation.
\item For MC dropout, we randomly dropout the bilinearly pooled and $\ell_2$-normalized feature vector for fifteen times at a dropout rate $p=0.5$. The prediction variance is used to replace $\mathrm{Diff}(\cdot)$ in Eq.~\eqref{eq:s}. Although the original UNIQUE does not include dropout during training, we empirically verify that the incorporation of dropout as post-processing does not hurt its quality prediction performance, where an SRCC of $0.8383$ between mean predictions and MOSs has been observed.
\end{itemize}
\begin{figure*}[!t]
\centering
\setcounter{subfigure}{0}
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/without_diversity/without_diversity1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/without_diversity/without_diversity2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/without_diversity/without_diversity3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/without_diversity/without_diversity4}}
\vspace{-2mm}
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/with_diversity/with_diversity1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/with_diversity/with_diversity2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/with_diversity/with_diversity3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{demo/with_diversity/with_diversity4}}
\caption{Representative images selected from SPAQ by the proposed sampling-worthiness module. \textbf{(a)}-\textbf{(d)}/\textbf{(e)}-\textbf{(h)} are selected images without/with the diversity measure. Zoom in for improved distortion visibility.}
\label{fig:diversity}
\end{figure*}
\begin{table}[!t]
\centering
\caption{SRCC results of the proposed sampling-worthiness module against six competing methods with and without the diversity measure. The large-scale unlabeled set $\mathcal{U}$ is simulated with SPAQ~\cite{fang2020perceptual}. A lower SRCC in $\mathcal{D}$ indicates a stronger capability of failure identification. RD: Representativeness-diversity}
\label{tab:srcc_cmp1}
\setlength{\tabcolsep}{2.2mm}{
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}[-3pt]{Method} & \multicolumn{2}{c}{Without diversity }&\multicolumn{2}{c}{With diversity}\\
\cmidrule(lr){2-3} \cmidrule(lr){4-5} & $\mathcal{D}$ &$\mathcal{U}\setminus\mathcal{D}$ & $\mathcal{D}$ & $\mathcal{U}\setminus\mathcal{D}$\\
\midrule
Random sampling & 0.8452 & 0.8382 & 0.8373 & 0.8383 \\
Sampling by RD~\cite{RDwu2019} & 0.5932 & 0.8383 & 0.5575 & 0.8381 \\
UNIQUE uncertainty~\cite{zhang2021uncertainty} & 0.5633 & 0.8397 & 0.5477 & 0.8400 \\
Query by committee~\cite{seung1992} & 0.5487 & 0.8395 & 0.5352 & 0.8395 \\
Core-set selection~\cite{sener2018active} & 0.4968 & 0.8396 & 0.3796 & 0.8398 \\
MC dropout~\cite{pop2018deep} & 0.4902 & 0.8376 & 0.3841 & 0.8379 \\
\midrule
Proposed & \textbf{0.1894} & 0.8362 & \textbf{0.1413} & 0.8364 \\
\bottomrule
\end{tabular}
}
\end{table}
\paragraph{Failure Identification Results}
Table~\ref{tab:srcc_cmp1} shows the SRCC results between UNIQUE predictions and MOSs on the selected $\mathcal{D}$ and the remaining $\mathcal{U}\setminus\mathcal{D}$. A lower SRCC in $\mathcal{D}$ indicates better failure identification performance. We find that, for all methods except random sampling, the selected images in $\mathcal{D}$ are more difficult than the remaining ones. The proposed sampling-worthiness module delivers the best performance, identifying significantly more difficult samples. It is interesting to note that the failure identification performance of all methods, including the proposed failure predictor, can be enhanced by the incorporation of the diversity measure\footnote{One subtlety is that each sampling strategy requires separate manual optimization of the trade-off parameter $\lambda$ due to different scales between the two terms in Eq.~\eqref{eq:s}.}.
\paragraph{Visual Results} Fig.~\ref{fig:diversity} shows representative top-$K$ images selected from SPAQ by the proposed sampling-worthiness module. Without the diversity constraint, the failure predictor alone is inclined to select difficult images of similar visual appearances, corresponding to the same underlying failure cause. When the diverse constraint is imposed, the selected images are more diverse in content and distortion.
\begin{figure*}[!t]
\centering
\setcounter{subfigure}{0}
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/nima_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/nima_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/nima_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/nima_4}}
\vspace{-2mm}
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/linearity_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/linearity_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/linearity_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/linearity_4}}
\vspace{-2mm}
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/hyper_1}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/hyper_2}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/hyper_3}}\hskip.3em
\subfloat[]{\includegraphics[width=0.24\linewidth]{failure_cases/hyper_4}}
\caption{Representative failure cases selected from SPAQ by the proposed sampling-worthiness module for \textbf{(a)}-\textbf{(d)} NIMA, \textbf{(e)}-\textbf{(h)} LinearityIQA, and \textbf{(i)}-\textbf{(l)} HyperIQA, respectively.}
\label{fig:failure cases}
\end{figure*}
\begin{table}[!t]
\centering
\caption{SRCC results of the proposed sampling-worthiness module in falsifying six BIQA models on the unlabeled dataset $\mathcal{U}$ simulated with SPAQ}
\label{tab:srcc_cmp2}
\setlength{\tabcolsep}{2.2mm}{
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}[-3pt]{Method} & \multicolumn{2}{c}{Without diversity }&\multicolumn{2}{c}{With diversity}\\
\cmidrule(lr){2-3} \cmidrule(lr){4-5} & $\mathcal{D}$ &$\mathcal{U}\setminus\mathcal{D}$ & $\mathcal{D}$ & $\mathcal{U}\setminus\mathcal{D}$\\
\midrule
NIMA~\cite{talebi2018nima} & 0.0432 & 0.4019 & 0.0236 & 0.4010 \\
KonCept512~\cite{hosu2020koniq} & 0.0358 & 0.8465 & 0.0054 & 0.8465 \\
HyperIQA~\cite{su2020blindly} & 0.2675 & 0.8431 & 0.1216 & 0.8430 \\
Fang2020~\cite{fang2020perceptual} & 0.0690 & 0.7690 & 0.0336 & 0.7683 \\
LinearityIQA~\cite{li2020norm} & 0.1242 & 0.8737 & 0.1204 & 0.8737 \\
MetaIQA+~\cite{zhu2021generalizable} & 0.1565 & 0.8470 & 0.0779 & 0.8473 \\
\bottomrule
\end{tabular}
}
\end{table}
\subsection{Ablation Studies}\label{subsec:ER2}
To verify the flexibility and effectiveness of our sampling-worthiness module, we use it to spot diverse failures of six other BIQA models, including NIMA~\cite{talebi2018nima}, KonCept512~\cite{hosu2020koniq}, Fang2020~\cite{fang2020perceptual}, HyperIQA~\cite{su2020blindly}, LinearityIQA~\cite{li2020norm}, and MetaIQA+~\cite{zhu2021generalizable}. The experimental setups and training protocols follow those described in Sec.~\ref{subsec:ER}.
Table~\ref{tab:srcc_cmp2} shows the SRCC results of the proposed sampling-worthiness module in falsifying the six BIQA models on $\mathcal{U}$ simulated with SPAQ. We find that the proposed module is able to spot difficult samples of the respective BIQA model (even for Fang2020, which is originally trained on SPAQ). Fig.~\ref{fig:failure cases} shows representative failure cases of NIMA, LinearityIQA, and HyperIQA using the learned failure predictor alone. We observe that different BIQA models have their distinctive failure modes. For example, LinearityIQA fails to handle motion blur, while NIMA is weak at penalizing motion blur by camera shake.
We last investigate one interesting question: whether the failure predictor jointly trained with UNIQUE as the quality predictor is able to falsify the original UNIQUE (with and without the diversity measure). Specifically, joint training is performed by minimizing two fidelity losses\footnote{To simplify the implementation, we do not learn the uncertainty estimator $\sigma_w(\cdot)$, and instead set it to one.}, one for quality assessment (in Eq.~\eqref{eq:fidelity_U}) and the other for failure prediction (in Eq.~\eqref{eq:fidelity}). The auxiliary task of failure prediction is downweighted by a factor of $2$ to emphasize the main task of quality assessment. The ResNet-34 backbone is initialized with the weights pre-trained on ImageNet~\cite{deng2009imagenet}. The last FC layer of UNIQUE and the failure predictor are initialized by He's method~\cite{he2015delving}. Other training protocols are the same as those described in Sec.~\ref{subsec:ER}, except that we perform a warm-up training in the first five epochs, where only randomly initialized layers are adjusted.
\begin{table}[!t]
\centering
\caption{SRCC results of the sampling-worthiness module jointly trained with UNIQUE, where $\mathcal{U}$ is simulated with SPAQ}
\label{tab:srcc_cmp_jointly}
\setlength{\tabcolsep}{2.2mm}{
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}[-3pt]{Method} & \multicolumn{2}{c}{Without diversity }&\multicolumn{2}{c}{With diversity}\\
\cmidrule(lr){2-3} \cmidrule(lr){4-5} & $\mathcal{D}$ &$\mathcal{U}\setminus\mathcal{D}$ & $\mathcal{D}$ & $\mathcal{U}\setminus\mathcal{D}$\\
\midrule
Jointly trained UNIQUE & 0.3518 & 0.8291 & 0.3046 & 0.8290 \\
Original UNIQUE & 0.3582 & 0.8386 & 0.3442 & 0.8387 \\
\bottomrule
\end{tabular}
}
\end{table}
Table~\ref{tab:srcc_cmp_jointly} shows the SRCC results of the sampling-worthiness module in falsifying the jointly trained UNIQUE and its original version. We find that the proposed module is able to expose different failures of the jointly trained UNIQUE, and such capability is transferable to falsify the original UNIQUE, although not as remarkable as the counterpart trained while fixing UNIQUE (see the last row of Table~\ref{tab:srcc_cmp1}).
\section{Conclusion and Future Work}
In this paper, we first conducted computational studies to reveal the overfitting problem and the easy dataset problem rooted in the current development of BIQA. We believe these arise because of the weak connections from model to data. Motivated by these, we have proposed a computational framework to integrate model-centric and data-centric IQA. We also provided a specific instance by developing a sampling-worthiness module for difficulty and diversity quantification. Our module has been proved flexible and effective in spotting diverse failures of BIQA models.
In the future, we will improve the current sampling-worthiness module by developing better difficulty and diversity measures. We may also search for more efficient discrete optimization techniques to solve the subset selection problem in the context of IQA. Moreover, we will certainly leverage the sampling-worthiness module to construct a large-scale challenging IQA dataset, with the goal of facilitating the development of more generalizable IQA models. Last, we hope the proposed computational framework will inspire researchers in related fields to rethink the exciting future directions of IQA.
\bibliographystyle{IEEEtran}
|
1,116,691,498,480 | arxiv |
\section{#2}\input{#1}}
\makeatother
\def\arabic{figure}{\arabic{figure}}
\def\arabic{claim}{\arabic{claim}}
\def\arabic{lemma}{\arabic{lemma}}
\newcommand{{\it\bf q}}{{\it\bf q}}
\newcommand{{\it\bf p}}{{\it\bf p}}
\newcommand{{\\it\bf M}}{{\\it\bf M}}
\newcommand{{\it\bf s}}{{\it\bf s}}
\newcommand{{\it\bf x}}{{\it\bf x}}
\newcommand{{\it\bf T}}{{\it\bf T}}
\newcommand{{\it\bf u}}{{\it\bf u}}
\newcommand{{\it\bf a}}{{\it\bf a}}
\newcommand{{\it\bf b}}{{\it\bf b}}
\newcommand{{\it\bf M}}{{\it\bf M}}
\newcommand{\ell_{ \bC}}{\ell_{ {\it\bf M}}}
\newcommand{r_{ \bC}}{r_{ {\it\bf M}}}
\newcommand{\ell_{\cap}}{\ell_{\cap}}
\newcommand{r_{\cap}}{r_{\cap}}
\newcommand{{{}_\star}}{{{}_\star}}
\title{Symbolic Dynamics of the Collinear Three-Body
Problem}
\author{Samuel R. Kaplan}
\address{Department of Mathematics, Bowdoin College\footnote{The
author is presently at the University of North Carolina at Asheville,
Asheville, NC 28801} Brunswick, ME 04011}
\begin{document}
\begin{abstract}
Solutions to the collinear three-body problem which do not end in
triple collision pass through an infinite number of binary
collisions. Given three masses, we show that four geometric quantities
generate a finite description of itineraries of binary collisions. In
the best circumstances, this description is semi-conjugate to a
Poincar\'{e} map of the flow. For other cases these quantities give
upper and lower bounds on the itineraries which can occur. In addition
to describing the dynamics of the collinear three-body problem, the
results of this paper rederives the existence of oscillatory motion in
the $N$-body problem for $N\geq 3$.
\end{abstract}
\maketitle
\secinput{ch1}{Introduction}
\secinput{ch2}{Hamiltonian Coordinates}
\secinput{ch3}{McGehee Coordinates}
\secinput{ch4}{The Stable Manifold of Triple Collision}
\secinput{ch5}{Main Results}
\secinput{ch6}{Oscillatory Motion}
\input{refs}
\end{document}
|
1,116,691,498,481 | arxiv | \section{Introduction}
Much remains to be understood concerning the "generic behavior" of solutions of the fluid equations for incompressible flows at very large Reynolds number. Kolmogorov's theory is based on the idea that every quantity scales with the power dissipated per unit time and mass. Assuming the viscosity to be negligible at the scales of observation, one needs to introduce, to explain dissipation, a transfer of energy from large to small scales where viscosity becomes significant. This approach predicts well the spectra of velocity fluctuations as a function of the wave number \cite{frisch}.
However it was quickly realized that this approach is not able to describe other observed phenomena like intermittency \cite{interm} which can be seen as the occurrence of large velocity fluctuations not describable at all by Gaussian or quasi-Gaussian statistics. Besides the property of energy conservation of the Euler fluid equations, very little of the properties of those equations is used to derive Obukhov-Kolmogorov spectra. The present work intends to explain first the idea of self-similar solution, and then to show that some of its consequences can be directly observed in the fluctuations of velocity and acceleration recorded in the highly turbulent flow of the Modane wind tunnel.
In 1934 Leray \cite{leray} published a paper on the equations for an incompressible fluid in 3D, with and without viscosity. He introduced many important ideas, among them the notion of weak solution and also the problem to be solved to show the existence (or not) of a solution becoming singular after a finite time when starting from smooth initial data.
Over the years this motivated a lot of works, mostly by mathematicians, the main effort being to try to prove or disprove the existence of such singularities assuming properties of the initial data. Other attempts have been directed toward a direct solution of the dynamical Euler equations, with the purpose of showing they have or not a finite time singularity.
It has been suggested recently \cite{YP}, \cite{modane}
to find a direct numerical solution of the Leray equations for a self-similar singularity of the Euler equations (or Euler-Leray equations).
The Euler equations read:
\begin{equation}
\frac{\partial{\bf{u}}}{\partial t} + {\bf{u}}\cdot \nabla {\bf{u}} = - \nabla p
\textrm{,}
\label{eq:Euler1}
\end{equation}
and
\begin{equation}
\nabla \cdot {\bf{u}} = 0
\textrm{,}
\label{eq:Euler2}
\end{equation}
Leray looked at the Navier-Stokes equations, which amounts to add $\nu \nabla^2 {\bf{u}}$ to the right-hand side of equation (\ref{eq:Euler1}). Specifically he looked at solutions of the self-similar type:
\begin{equation}
{\bf{u}}( {\bf{r}}, t) = (t^*- t)^{-\alpha} {\bf{U}} ( {\bf{r}}(t^*- t)^{-\beta})\textrm{,}
\label{eq:self}
\end{equation}
where $t^*$ is the time of the singularity (set to zero afterwards), where $\alpha$ and $\beta$ are positive exponents to be found and where the field with upper-case letters $ {\bf{U}}(.)$ is to be derived by solving Euler, or Navier-Stokes equations.
That such a velocity field is a solution of Euler or Navier-Stokes equations implies
\begin{equation}
\alpha + \beta =1
\textrm{,}
\label{eq:alpha1}
\end{equation}
a condition which ensures the balance between the two terms in the l.h.s. of (\ref{eq:Euler1}), which are respectively of order $t^{-(\alpha+1)}$ and $t^{-(2\alpha+\beta)}$.
Below we shall compare our predictions with experimental data taken from the Modane wind tunnel, where the velocity field is advected by a mean flow. In this case the self-similar solution (\ref{eq:self}) describes the behavior of the fluctuations of the velocity field, $ {\bf{\delta v}}= {\bf{v}}- <{\bf{v}}>.$ But the balance condition (\ref{eq:alpha1}) is still valid, because the advection term, $< {\bf{v}}>\cdot \nabla {\bf{\delta v }}$, coming from the mean flow, of order $t^{-(\alpha+\beta)}$, is smaller than the others two.
In the case of Navier-Stokes equation, the balance with the dissipative term $\nu \nabla^2 {\bf{u}}$, of order $t^{-(\alpha+2\beta)}$, imposes $\beta=1/2$, which yields the exponents found by Leray for the case of the Navier-Stokes equations
\begin{equation}
\alpha = \beta =1/2
\textrm{,}
\label{eq:alpha2}
\end{equation}
\section{ Euler-Leray's equations}
In the case of Euler equation, there are several possibilities to get a second relation between the two exponents, according to what conservation laws are considered. Let consider first the constraint of conservation of circulation. The circulation $\Gamma=\oint{u ds }$ along a closed curve carried by the flow toward the singularity, is of order $U^{2 }t^{\beta-\alpha}$. Then the condition for conservation of circulation implies $ \alpha - \beta=0$, that gives (\ref{eq:alpha2}), namely the same exponents as for Navier-Stokes case.
Moreover the velocity scales like
\begin{equation}
u(r, t) \sim (-t)^{-1/2} \Gamma^{1/2}
\textrm{,}
\label{eq:circ}
\end{equation}
near the singularity because $\Gamma \sim U^{2}$. With such a choice, the total energy of solutions of the self-similar problem is diverging, but the divergence of the energy does not imply the absence of singularity \cite{modane}.
If one imposes instead of the conservation of circulation that the energy in the collapsing domain is conserved, one must satisfy the constraint $- 2 \alpha + 3 \beta = 0$, which yields $\alpha = 3/5$ and $\beta = 2/5$, the Sedov-Taylor exponents.
No set of singularity exponents can satisfy both constraints of energy conservation and of constant circulation on carried closed curves.
We choose $\alpha = \beta = 1/2$ in the following, or
\begin{equation}
{\bf{u}}( {\bf{r}}, t) = (t^*- t)^{-\frac{1}{2}} {\bf{U}} ( {\bf{r}}(t^*- t)^{-\frac{1}{2}})
\textrm{,}
\label{eq:self2}
\end{equation}
With upper case letters for the position,
$ {\bf{R}} = {\bf{r}}(- t)^{-1/2}$, the Euler equations become the Euler-Leray equations for $ {\bf{U}}({\bf{R}})$,
\begin{equation}
\frac{1}{2}({\bf{U}} + {\bf{R}}\cdot \nabla {\bf{U}}) + {\bf{U}}\cdot \nabla {\bf{U}} = - \nabla P
\textrm{,}
\label{eq:EulerU}
\end{equation}
and
\begin{equation}
\nabla \cdot {\bf{U}} = 0
\textrm{.}
\label{eq:DivU}
\end{equation}
A singularity of the self similar type must decay at large distances in such a way that, at such large distances (in the stretched variables), it becomes independent on time. Otherwise it would depend singularly on time everywhere and so not be a point wise singularity. Moreover the solution of the Euler-Leray equations must be smooth as a function of ${\bf{R}}$. Otherwise it makes a singular solution at any time, not at a single time, like for example solutions of the type of Landau submerged jet \cite{landau jets} which are singular uniformly in time, and cannot belong to the class of solutions considered here.
The first constraint (solution independent on time at large distances) is satisfied if ${\bf{U}} \sim 1/R $ at $R$ large. Returning to the initial space-time dependence one gets the asymptotic behavior (in the stretched variable) $ u \sim (-t)^{-1/2}/ r (-t)^{-1/2} \sim 1/r$ with no time dependence. At $t =0$ (time of singularity) the velocity field of the singular solution is exactly like $1/r$ times a function of the angle to satisfy incompressibility (a property perhaps experimentally checkable by particle image velocimetry).
The $1/R$ behavior of the solution cancels the linear term in Euler-Leray (and NS-Leray as well), dominant at large distances, as it should. This leads to seek a formal Laurent expansion of this solution in inverse powers of $R$,
\begin{equation}
{\bf{U}}({\bf{R}}) = \sum_{n =1}^{\infty} \frac{1}{R^n} {{\bf{W}}_n} (\hat{\bf{R}})
\textrm{.}
\label{eq:Euler2exp}
\end{equation}
with $\hat{\bf{R}} = {\bf{R}} /R$ unit vector.
Putting this expansion of ${\bf{U}}$ into Euler-Leray, we get
\begin{equation}
\frac{1}{2} \sum_{n = 3}^{\infty}(1 - n) \frac{1}{R^n} {{\bf{W}}_n} (\hat{\bf{R}}) + {\bf{U}}\cdot \nabla {\bf{U}} = - \nabla P
\textrm{.}
\label{eq:Pom4}
\end{equation}
Knowing ${\bf{W}}_1(\hat{\bf{R}})$ (arbitrary at this step) this can be mapped in an iteration for computing ${\bf{W}}_3(\hat{\bf{R}})$, ${\bf{W}}_5(\hat{\bf{R}})$, etc. The pressure being derived from the incompressibility condition. This is practically very cumbersome. Other methods have to be found to solve Euler-Leray, as explained now: the idea is to replace the Euler-Leray equation by an iteration.
Define the $k$- Cartesian component of the nonlinear part of Euler-Leray as,
\begin{equation}
V_k({\bf{R}}) = \partial_k P + U_j \partial_j U_k
\textrm{.}
\label{eq:EulerV}
\end{equation}
If one assumes the vector ${\bf{V}}$ to be known one can solve formally Euler-Leray by integration on the modulus of ${\bf{R}}$,
\begin{equation}
U_k({\bf{R}}) = \frac{1}{R} \int_0^{R} {\mathrm{d}}R' (- 2 V_k(R', {\hat{\bf{R}}}))
\textrm{.}
\label{eq:Euler6}
\end{equation}
The basic principle for an iterative solution is to assume that the left-hand side is known, put the rest in the (non linear) right-hand side, compute this right-hand side for a given field satisfying basic constraint. This yields an estimate for the velocity field which can be put into the right-hand side and the iteration is continued in principle until it converges to the desired fixed point which is the solution of the equations one started from. However things do not work this well for a number of reasons. This runs into a number of difficulties that have yet not been got rid out. There is a fundamental point that gives some optimism for the existence of such a non trivial solution: the Euler-Leray equation has a variational formulation \cite{YP}, as the original Euler equations, and it is likely that a non trivial extremum of the corresponding functional exists.
To conclude on the Euler-Leray equation, it yields a well defined schema for the existence of solutions of the Euler equations in 3D, becoming singular in a finite time at a single point. A by-product of this analysis is the set of exponents of the singularity which may be compared with the experimental data for the large fluctuations observed in the records of time dependent velocity in a turbulent flow, as done below.
One obvious motivation for working on Euler-Leray singularities is their possible connection with the (loosely defined) phenomenon of intermittency in high Reynolds number flows. This raises several question:
1. What is the difference between Euler-Leray and Navier-Stokes-Leray singularities?
2. What is specific to Leray singularities compared to other schema for intermittency?
3. What would be specific of an Euler-Leray singularity in a time record of large Reynolds number flow?
Point 1: Difference between Euler-Leray and Navier-Stokes-Leray: little is known about it, in particular do both have nontrivial solutions, or does none has nontrivial solutions or only one has nontrivial solution?
Point 2 : If intermittency is caused by Leray-like singularities, they should show a strong positive correlation between singularities of the velocity and the acceleration (see below). Compared to predictions derived from Kolmogorov theory this (positive) correlation is a strong indication of the occurrence of Leray-like singularities near large fluctuations. It is fair to say however that Kolmogorov himself never mentions this question of finite time singularity of either Navier-Stokes or Euler equations. So it would be unfair to attribute to him any claim about those singularities.
Point 3: Both in Euler-Leray and NS-Leray cases, the velocity field at the singular time scales like $1/r$, $r$ distance to the singularity.
The scaling laws derived for the velocity-acceleration correlations in time records of velocity is fairly simple. First in the case of Euler equation, the order of magnitude of the velocity in terms of the circulation close to the singular point, equation (\ref{eq:circ}), is associated to the relation $U_{Euler} \sim \Gamma^{1/2}$, as written above.
In the case of the Navier-Stokes equation the solutions depend only on one dimensionless number, which is locally (close to the singularity) of order
\begin{equation}
Re_{s} \sim \frac{\Gamma}{\nu}
\textrm{.}
\label{eq:Re}
\end{equation}
The dissipation imposes therefore to multiply the pre-factor $\Gamma^{1/2}$ by a numerical function $f(Re_{s})$ of the Reynolds number. One
can write $U_{N-S}\sim \Gamma^{1/2} f(\Gamma/\nu) $,
that gives the relation,
\begin{equation}
u(r, t) \sim (-t)^{-1/2} \Gamma^{1/2} f(\Gamma/\nu)
\textrm{.}
\label{eq:circNS}
\end{equation}
The numerical value of $f$ depends on the precise solution we are considering, and of the value of the extension of the path $r(-t)^{(-1/2}$ defining the circulation.
As the kinematic viscosity of the fluid, $\nu$ tends to zero at fixed $\Gamma$ (or $Re_{s}$ tends to infinity), $f$ tends to the limit value $f(\infty)$ which should be a finite number. Because this limit correspond to the case of Euler equation, assuming that the solutions (for Euler and N-S equations) merge, we get $f(\infty)=1$,
which gives the order of magnitude of a self-similar solution in N-S case in this limit. But for finite values of the Reynolds number $Re_{s}$, the relation (\ref{eq:circNS}) cannot give a direct estimate of the magnitude of N-S self-similar solution because the function $f()$ is unknown at this time. In that case the local Reynolds number cannot be deduced from experimental results, see below in the next section.
From the condition of conservation of the circulation close to the singularity, and (\ref{eq:Re}), we deduce that the local Reynolds number is also constant (in order in magnitude) in the collapsing domain, because the constraint on of the circulation imposes a balance between the growth of the velocity and the stretching of the singular domain. This property contradicts the standard ideas on turbulence according to which the only relevant parameter is the power dissipated per unit mass and time.
Here we find that in such singular events, even though the length scales become very small, the Reynolds numbers typical of those small domains do not tend to zero, but instead stay constant, because the velocity grows continuously (at the same pace) as the space scale decreases.
Let us introduce the \textit{effective} circulation
\begin{equation}
\tilde{\Gamma}^{1/2} = \Gamma^{1/2} f(\Gamma/\nu)
\textrm{.}
\label{eq:effcirc}
\end{equation}
for a self-similar solution in the Euler and N-S cases, for simplicity. We can immediately derive the scaling for the acceleration $\gamma(r, t)$, that is $\gamma(r, t) \sim (-t)^{-3/2} \tilde{\Gamma}^{1/2} $. Accordingly one finds the time independent relation
\begin{equation}
u^3 \sim \gamma \tilde{\Gamma} \textrm{.}
\label{eq:cubv}
\end{equation}
Let us now turn to the Kolmogorov scaling. The starting point is the famous Kolmogorov relation $ u_r \sim (\epsilon r)^{1/3}$ where $u_r$ is the typical change of velocity over a distance $r$ and $\epsilon$ is the power dissipated per unit mass of the turbulent fluid. With those scaling and $r\sim u_{r}t$ the acceleration $\gamma\sim u_{r}/t \sim u_{r}^{2}/r$ becomes of order $\gamma \sim \epsilon^{2/3} r^{-1/3}$. Therefore the relationship between $u_r$ and $\gamma$ independent on $r$ is
\begin{equation}
\gamma u_r \sim \epsilon \textrm{.}
\label{eq:Kolm}
\end{equation}
The latter expression is in complete contradiction with (\ref{eq:cubv} ) deduced for self-similar solutions, a problem that we shall consider just below by comparing with experimental data.
\section{ Experimental results }
The two relations (\ref{eq:cubv}) and (\ref{eq:Kolm}) can be tested against the experimental results by comparing the values of the velocity fluctuations $\delta v=v-<v>$ and of the acceleration $\gamma$ recorded at the same point and the same time in the domain of large accelerations. Our aim is to use experimental data in order to look at the occurence of self-similar solutions in the turbulent flow, and more precisely to search if self-similar solutions of type (\ref{eq:self2}) exist actually in the flow. If such solutions exist, even as rare events, they should be visible at least in the large accelerations domain where one should expect to obtain a relation of type (\ref{eq:cubv}) between acceleration and velocity fluctuations. On the contrary, if Kolmogorov-scalings are the only ones which drive $\it{all}$ the dynamics, large accelerations should occur predominantly when the velocity fluctuations are close to zero.
In other words, we want to see if large accelerations occur for large or for small velocity fluctuations. We looked at the data obtained in the wind tunnel of Modane, where the turbulent Eulerian velocity was recorded by a single hot wire.
Details of the experimental set-up can be found in Refs.\cite{expmod}, \cite{exp2mod}. Let us shortly recall the conditions of this experiment : the Reynolds number is equal to $Re_{\lambda}=\sqrt{15Re}= 2500$, so that the regime is in fully developed turbulence. The measurements were made in the return vein of the wind tunnel, where turbulence is not really isotropic, but mainly resulting from the separation of an unstable boundary layer. The sampling time is $t_{s}= 1/25 ms$ ($f_{s} = 25 khz$). It is smaller than the dissipation time $ t_{\eta}=1/10.7 ms$. In the following we assume ergodicity of the velocity fluctuations, it follows that any average is calculated as a temporal average, with time running over the full data ( $13.7 \; 10^{6}$ points)
covering a total record time of about $10$ minutes.
The average velocity is $<v>= 20.545 m/s$, and the standard deviation is $\sigma_{v}= 1.69m/s$.
In figure \ref{fig:histogram} we show the 2D-histogram of velocity fluctuation and acceleration recorded at the same time, in the domain of large accelerations, $ \gamma \ge 5 \sigma_{\gamma} $. We show that there are more events for positive velocity fluctuations than for negative and null ones.
This qualitative observation is in favor of the existence of self-similar solutions, but is not sufficient to claim that they are of the form (\ref{eq:self2}). To investigate a quantitative relationship between $\gamma$ and $(\delta v)^{n}$, for various values of the integer $n$, we have calculated the conditional momentum of the velocity fluctuations $<(\delta v)^{n }>_{/\gamma}$, which are the average values of $(\delta v)^{n} $ weighted by the conditional probability of the velocity $P_{\gamma}(\delta v)$ for a given value $\gamma$ of the acceleration,
\begin{equation}
<( \delta v)^{n}>_{\gamma}= \int{ d(\delta v) (\delta v)^{n} P_{\gamma}(\delta v) }
\textrm{.}
\label{eq:moment}
\end{equation}
Setting $u=\delta v$ for ease, the probability $P(u_{i},\gamma_{j})du d\gamma$ for the set of variables $(u,\gamma)$ to be inside the domain $ [(u_{i},u_{i}+du)\cup (\gamma_{j}, \gamma_{j } + d\gamma)]$ , is given by the number of points recorded in this domain divided by the total number of recorded points, $\mathcal{N}$,
\begin{equation}
P(u_{i},\gamma_{j}) du d\gamma= N_{i,j} /\mathcal{N}
\textrm{.}
\label{eq:pij}
\end{equation}
\begin{figure}
\centerline{
\includegraphics[height=2.50in]{histog.pdf}
}
\caption{Histogram of the coupled variables $v(t_{i}) ,\gamma(t_{i})$ in the large acceleration domain, $ \gamma \ge 5\sigma_{\gamma} $. }
\label{fig:histogram}
\end{figure}
The conditional probability $ P_{\gamma_{j}}(u_{i}) du$ for the velocity to be inside the domain $[u_{i},u_{i}+du]$ if the acceleration is inside the domain $ [\gamma_{j}, \gamma_{j } + d\gamma] $, is given by $ N_{i,j} / N_{j}$, where $N_{j}=\sum_{i} N_{i,j}$. Using (\ref{eq:moment})-(\ref{eq:pij}), we get the following expression for the momenta in terms of the number of points recorded in the elementary domains,
\begin{equation}
< u^{n}>_{\gamma_{j}}= \sum_{i} u_{i}^{n} \frac {N_{i,j}} {N_{j}}
\textrm{.}
\label{eq:moment2}
\end{equation}
\begin{figure}
\centerline{
(a)\includegraphics[height=2.2in]{v3.pdf}
(b)\includegraphics[height=2.2in]{v-b.pdf}
}
\centerline{
(c)\includegraphics[height=2.0in]{v3sa-c.pdf}
(d)\includegraphics[height=2.0in]{va-d.pdf}
}
\caption{ Experimental test to show the validity of relation (\ref{eq:cubv}), see text.The
conditional averages $ <(\delta v^{3}) >_{\gamma}$ , and $<\delta v>_{\gamma}$, plotted in (a) and (b) respectively, display a monotonic growth with respect to the acceleration $\gamma$, the linear behavior of curve (a) is emphasized by the red dashed straight segments.
The conditional ratio $<(\delta v)^{3} /\gamma>_{\gamma}$ in (c) presents a quasi-constant behavior for $\gamma \gtrsim \sigma_{\gamma}$ in agreement with (\ref{eq:cubv}). In (d) the product $<(\delta v)\gamma>_{\gamma}$ shows a clear non-constant behavior, but a strong growth with respect to $\gamma$, except in a small interval close to the origin where (\ref{eq:Kolm}) is approximately valid. All curves are calculated via the probability density of the set $[\delta v, \gamma] $ taken at the same time. The velocity and acceleration are scaled to their respective standard deviation.}
\label{fig:test}
\end{figure}
To check which one of relations (\ref{eq:cubv}) or (\ref{eq:Kolm}) is compatible with experimental data of Modane, we have plotted $ <(\delta v)^{3} >/_{\gamma}$ and
$<\delta v>_{\gamma}$ in Figs.\ref{fig:test}-(a)-(b). These curves assert that Kolmogorov scalings cannot fit the experimental data in the domain of large and even moderate values of the acceleration, because the two conditional averages increases with $\gamma$ (therefore the product $ < \gamma \delta v >$ cannot stay constant). To evaluate more precisely the constant behavior of the ratio $<u^{3}/\gamma>_{\gamma}$ (Leray scalings) and the non constant behavior of the product $ < u \gamma >_{\gamma} $ (Kolmogorov scalings) versus $\gamma$, we have plotted these quantities in Figs. (c)-(d).
It appears clearly that the Kolmogorov scaling leading to relation (\ref{eq:Kolm}) is incompatible with the data of Modane, because the product $<\gamma u>_{\gamma} $ shown in Fig.(d) is definitely not constant in the large interval of acceleration we have investigated. On the opposite we find that the ratio $<u^{3}/\gamma>_{\gamma}$ , shown in Fig.(c), displays a fairly constant behavior except in the domain of small acceleration (or the order of the standard deviation $\sigma_{\gamma}$).
Now let us see if one may derive an order of magnitude of the local Reynolds number from our study.The ratio $<u^{3}>_{\gamma}/ \gamma$ is numerically equal to $\tilde{\Gamma}=1.33 10^{-3}$. From (\ref{eq:effcirc}), we have $ \tilde{\Gamma}/\nu = Re_{s} f(Re_{s})^{2}$, that gives the following relation for local Reynolds number,
\begin{equation}
Re_{s} f(Re_{s})^{2} =85.6
\textrm{.}
\label{eq:Re2}
\end{equation}
when taking the kinematic viscosity of air equal to $\nu \simeq 1.56 \;10^{-5}$, its value at room temperature. If the numerical function $f$ is of order unity, the result in (\ref{eq:Re2}) gives a local Reynolds number much smaller than the global Reynolds number which was experimentally estimated as $Re=4.2\; 10^{5}$ (see above), but much larger than unity, the value it should take if the singularity were one of the ultimate outcomes of a Kolmogorov cascade stopped by viscosity effects.
In summary, we have shown
that the linear relation (\ref{eq:cubv}) between $u^3$ and the acceleration $\gamma$ is well verified experimentally for large acceleration values, see Fig.\ref{fig:test}-(a) and (c). This result is well explained within the hypothesis of
existence of Leray-type singularities in the flow. Differently the relation (\ref{eq:Kolm}) is found to be invalid for acceleration values larger than its standard deviation, but can be approximately valid in the domain $\gamma < \sigma_{\gamma}$, namely close to small acceleration values, see Fig.\ref{fig:test}-(d). The non-validity of (\ref{eq:Kolm}) comes obviously from the fact that the original scaling by Kolmogorov, $ u_r \sim (\epsilon r)^{1/3}$, even though it describes an average property of the velocity fluctuations cannot do it for large values of $u_r$: large values of $u_r$ would have to be linked to large distances, incompatible with large accelerations which concerns short distances. Kolmogorov scaling remains compatible however with the average properties of the fluctuations, but our linear relation between $u^3$ and $\gamma$ spans a rather wide range of values of $u$ and $\gamma$ and so could be hard to reconcile with a Kolmogorov scaling even on average.
Note that such a good fit between (\ref{eq:cubv}) and experimental data is slightly unexpected because it implies that the pre-factor $\tilde{\Gamma}$ is not changing much from one large fluctuation to the other. It could indicate that this Reynolds number dependent pre-factor is such that for some unknown reason it does not change appreciably in different realization of the singularity.
Finally we may conclude from this study that besides the Kolmogorov cascade (already observed by using Modane's data \cite{modane}), it is quite probable that singularities exist in the flow, and that they could be of the Leray-form (\ref{eq:self}).
\section*{Acknowledgments}
The authors are very grateful to B\'ereng\`ere Dubrulle, Jean Ginibre, Christophe Josserand, Thierry Lehner and St\'ephane Popinet for very useful discussions.
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\bibitem{leray} J. Leray, "Essai sur le mouvement d'un fluide visqueux emplissant l'espace", Acta Math. {\bf{63}} (1934) p. 193 - 248.
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\bibitem{landau jets} L. D. Landau and E. M. Lifschitz in ''Fluid Mechanics'', Institute of Physical Problems, U.S.S.R. Academy of Sciences, Moscow. Volume 6 of Course of theoretical physics, §23. Exact solutions of the equations of motion for a viscous fluid; L. D. Landau, Dokl. Akad. Nauk. SSR, {\bf{48}} (1944) p. 289.
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\endthebibliography{}
\end{document}
|
1,116,691,498,482 | arxiv | \section{Introduction}\label{SecI}
For quantum processors built with superconducting qubits, both the control accuracy and the qubit
number have shown steady improvement over the past two decades \cite{Kjaergaard2020}.
Notably, quantum gate operations, which are generally implemented by using microwave
or baseband flux pulses \cite{Krantz2019}, with errors reaching
the fault-tolerant thresholds have been achieved in quantum
processors with several tens of qubits \cite{Arute2019,Zhu2022,Acharya2022,Kim2021}.
Nevertheless, it is known that fulfilling the full promises of quantum computing requires
the implementation of fault-tolerant schemes, which will need the high-fidelity control
of millions of qubits \cite{Fowler2012,Gidney2021}. In such large-scale superconducting
quantum processors, the needed physical source, such as control electronics and cooling power in
cryogenic systems, could be the most challenging obstacle for achieving accurate control
of qubits, let alone solving the wiring problem \cite{Frankea2019,Reilly2019,Martinis2020}
and the device yield problem \cite{Hertzberg2021,Kreikebaum2020}.
Generally, in superconducting quantum processors, the qubit parameters, such as the qubit
frequency, the qubit anharmonicity, the coupling efficiency between qubits and control
lines, and the signal attenuations in control lines, can be different from each other.
Thus, in current small-scale quantum processors, to ensure accurate qubit control, each
qubit should have its dedicated control pulse with different parameter
settings \cite{Kelly2018,Klimov2020}. This means that the microwave control pulses
could differ from each other in their amplitudes, frequencies, and phases, while for
the baseband flux pulse, their amplitudes could be different. Moreover, these control pulses
are generated at room temperature, and then delivered to qubits in the cryogenic system
through a series of attenuators and filters for suppressing harmful noises, such as
thermal noise \cite{Krantz2019,Chen2018,Krinner2019}. Considering these general arguments, the
physical sources for realizing qubit control could be highly related to the type of
employed control. To be more specific, the microwave control and its signal synthesis
are more complicated and expensive than that of the baseband flux control, for which only a
single digital-to-analog converter (DAC) per qubit is needed \cite{Krantz2019,Chen2018,Krinner2019}.
Moreover, given the limited available cooling power in cryogenic systems, the microwave
control lines generally need the attenuation of $60\,\rm dB$ (about $20\,\rm dB$ at
the mixing chamber plate (MXC), for which the available cooling power is smallest),
leading to heating loads larger than that of the baseband flux lines (about $20\,\rm dB$,
need no attenuation at the MXC stage) \cite{Arute2019,Chen2018,Krinner2019}. Additionally, in
large-scale quantum processors, microwave control requires higher-density control lines, making it
challenging to suppress the microwave crosstalk \cite{Wenner2011,Rosenberg2019,Huang2021M} and
thus to achieve high-fidelity qubit control. By contrast, with baseband flux control, there
exists only a single control parameter, potentially allowing the application of multiplexing
technologies and cross-bar technologies to address the challenges, e.g., the wiring problem, toward
large-scale quantum computing \cite{Hill2015,Vandersypen2017,Veldhorst2017,Li2018}.
Given the above discussion, when scaling up to large-scale quantum processors, implementing
baseband flux control could make requirements less stringent than that of microwave control.
However, currently, microwave control is generally the essential one for implementing qubit
addressing and single-qubit gate operations \cite{Motzoi2009,Chen2016,Krantz2019}, and even
for two-qubit gates, such as cross-resonance gates \cite{Chow2011}. In this work, we explore
theoretically the possibility of developing the baseband flux control of frequency-tunable
qubits with the help of always-on shared microwave drives. It should be noted that previous
works on studying baseband control of superconducting qubits mainly focus on low-frequency
qubits, e.g., the composite qubit \cite{Campbell2020} and heavy-fluxonium qubit \cite{Zhang2021}, here
we focus on the transmon qubits \cite{Koch2007} that have been widely used in current
superconducting quantum processors. The basic idea of our control strategy is sketched
in Fig.~\ref{fig1}(a), where two qubits are coupled via a coupler and the always-on microwave
drive (XY line) is shared by both qubits (in principle, can be extended to
multi-qubit cases), the qubit control and the single-qubit addressing can be realized only
through the flux (Z) control lines. Our work is motivated by Kane's proposal for realizing
spin-based quantum computing \cite{Kane1998}, where the spin qubit is by default
off-resonance with the global always-on microwave magnetic
field (i.e., at the idle point) and single-qubit gate operations are realized by tuning
the spin qubit on-resonance with the field (i.e., at the working point) \cite{Wolfowicz2014,Laucht2015},
as shown in Figs.~\ref{fig1}(b) and~\ref{fig1}(c). Due to the always-on
shared drive, the computational states are the basis states of the microwave-dressed
qubit \cite{Liu2006,Zhao2022,Wei2022}, and accordingly, in the present work, all the qubit
control are analyzed based on this microwave-dressed basis. As an example application
of this baseband control strategy, in a system comprising two frequency-tunable transmon
qubits coupled via a tunable coupler \cite{Yan2018}, we study the feasibility of this
strategy for achieving high-fidelity gate operations. By theoretical analysis, we will
show that:
(i) To implement single-qubit gate operations, especially, $\sqrt{X}$ gates, the
baseband Z(flux)-control provides great flexibility in the gate tune-up procedure. This
flexibility could be used to relieve stringent requirements on qubit frequency, drive strength,
and gate time for implementing single-qubit gates, and thus can even compensate for the
non-uniformity of qubit parameters, potentially allowing to perform multiplexed
control of qubits.
(ii) Since the transmon qubit has a weak anharmonicity, in the traditional microwave
control setup, leakage during gate operations can be suppressed by using the
derivative removal by adiabatic gate (DRAG) scheme \cite{Motzoi2009}. In our setup, while the DRAG scheme
can no longer be directly utilized, we show that by using a modified fast-adiabatic
scheme, the leakage can also be suppressed heavily.
(iii) While the always-on microwave drive is detuned from the qubits, it can
induce ac-Stark frequency shifts on the qubits \cite{Tuorila2010,Schneider2018,Liu2006,Zhao2022,Wei2022}.
Consequently, any fluctuations in the drive amplitude will cause qubit dephasing. By
numerical simulation, we study the effect of the amplitude-dependent noise on
the qubit and show that the fluctuation-induced dephasing can be
eliminated by tuning the qubit away from the drive. Similarly, by numerical simulation
of qubit readout dynamics, the impacts of the always-on drive on the
readout fidelity can also be neglected safely when the drive detuning is far larger
than the drive strength.
(iv) In the qubit architecture with tunable coupling, we show that with the baseband
control strategy and the modified fast-adiabatic scheme, high-fidelity single-qubit
gates are achievable. We also outline the leading error mechanisms that should be
considered carefully when applying baseband control in large-scale quantum
systems. Additionally, we further show that with the always-on microwave
drive, baseband-controlled two-qubit CZ gates can still be achieved with
high gate fidelity and short gate length.
The rest of the paper is organized as follows. In Sec.~\ref{SecII}, we provide an overview of the baseband
control scheme. In Sec.~\ref{SecIII}, we consider an example application of the
baseband control strategy for achieving high-fidelity single- and two-qubit gates in a
qubit architecture with tunable coupling. In Sec.~\ref{SecIV}, we will provide discussions of the challenges
and opportunities for realizing the baseband control strategy in superconducting quantum processors. Finally, we
provide a summary of our work in Sec.~\ref{SecV}.
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{scheme.pdf}
\end{center}
\caption{Baseband flux control of transmon qubits with a shared always-on microwave drive.
(a) Sketch of a baseband flux controlled two frequency-tunable transmon qubit system ($Q_{0}$ and $Q_{1}$) with
dedicated Z lines. The two qubits are coupled via a coupler $Q_{c}$, which could be a tunable coupler.
Through a shared XY line, the two qubits are driven simultaneously by a global and always-on microwave
drive with the frequency of $\omega_{d}$ and the constant amplitude of $\Omega_{d}$. Baseband flux
pulses are delivered to the qubits and coupler through their dedicated Z lines. (b) At the idle point, the
qubit is far detuned from the drive, i.e., the drive
detuning, $|\Delta_{d}|=|\omega_{01}-\omega_{d}|\gg\Omega_{d}$. Left: Bloch vector in the rotating frame with respect to
the drive (here, confined to qubit subspace spanned by the lowest two-energy levels of
the transmon qubit). Due to the always-on drive, the Bloch vector (dashed red arrow) at the idle point is tilted toward the
X-axis. We thus choose the logical computational states to be the dressed eigenstates defined by
the tilted Bloch vector. Right: Energy level diagram of
the qubit at the idle point. Here, $\alpha_{q}$ denotes the qubit anharmonicity. (c) When
operating the system at the working point, where the qubit is on-resonance with the drive, single-qubit
operations can be implemented. Left: Bloch vector (solid red arrow) at the working point. Since the initial Bloch vector is
slightly tilted, a small detuning $\delta_{d}$ between the qubit and the drive is needed for enabling
complete Rabi oscillations. Right: Energy level diagram of the qubit at the working point.}
\label{fig1}
\end{figure}
\section{Overview of the baseband control setup}\label{SecII}
Here, we provide an overview of the baseband control setup schematically illustrated
in Fig.~\ref{fig1}(a). In our setup, frequency-tunable transmon qubits are driven simultaneously by a
single always-on global drive with a constant amplitude and each qubit has its dedicated
flux control lines, i.e., Z lines. Same to Kane's proposal \cite{Kane1998}, single-qubit
addressing or single-qubit gate operations can be implemented by tuning the qubits
on-resonance with the global drive, as shown in Fig.~\ref{fig1}(c). By contrast, when
biasing at the idle point, as shown in Fig.~\ref{fig1}(b), the qubit is far detuned from the
drive, thus in principle, qubit readout and baseband-controlled two-qubit gates can be
realized with minimal impacts from the always-on drive. To evaluate the feasibility of the
control scheme, in the following, we first consider implementing universal control
of an ideal two-level system, which is subjected to an always-on drive, using only Z-control.
Next, we will consider a more practical case of transmon qubits, which has a weak
qubit anharmonicity, making qubits particularly susceptible to leakage during gate
operations. We will show that with a fast-adiabatic scheme \cite{Martinis2014b}, Z-controlled single-qubit gate
operations can be achieved with fast speed and low leakage. Finally, by biasing the qubit at the
idle point, we will further study the impact of the always-on drive on the qubit dephasing
and qubit readout.
\subsection{Z-control of an ideal two-level system}\label{SecIIA}
For a baseband controlled two-level system subjected to an always-on global drive, the system Hamiltonian
is (hereinafter, we set $\hbar=1$)
\begin{equation}
\begin{aligned}\label{eq1}
H_{lab}=\frac{\omega_{q}}{2}\sigma_{z}+\Omega_{d}\cos(\omega_{d}t)\sigma_{x}
\end{aligned}
\end{equation}
where $\omega_{q}$ is the bare qubit frequency and can change according to the Z
control pulse, $\omega_{d}$ and $\Omega_{d}$ are the frequency and
the amplitude of the drive, respectively. Moving into the rotating frame with respect to the
global drive and after applying the rotating wave approximation (RWA), the Hamiltonian
reads
\begin{equation}
\begin{aligned}\label{eq2}
H_{rot}=\frac{\Delta_{d}}{2}\sigma_{z}+\frac{\Omega_{d}}{2}\sigma_{x}
\end{aligned}
\end{equation}
where $\Delta_{d}=\omega_{q}-\omega_{d}$ denotes the drive detuning. Note that
unless otherwise stated, the RWA is used throughout this work.
At the idle point, the drive detuning is far large than the drive strength, thus
the Bloch vector is slightly tilted towards the X-axis, as shown in Fig.~\ref{fig1}(b).
Generally, the dressed eigenstates defined by this tilted Bloch vector are chosen to be
the logical computational states. The tilted angle and the dressed states can be quantitatively
obtained by diagonalization of the Hamiltonian in Eq.~\ref{eq2}, giving rise to
\begin{equation}
\begin{aligned}\label{eq3}
H_{diag}=\frac{\Delta}{2}Z,\,{\rm with}\,Z\equiv\cos{\theta}\sigma_{z}+\sin{\theta}\sigma_{x},
\end{aligned}
\end{equation}
where $\theta=\arctan(\Omega_{d}/\Delta_{d})$ is the tilted angle and $\Delta=\sqrt{\Delta_{d}^{2}+\Omega_{d}^{2}}$.
From the above, when $|\Delta_{d}|\gg \Omega_{d}$, one can neglect the tilt angle, as well as the
difference between the bare states and the dressed states.
As shown in Eq.~\ref{eq3}, by biasing the qubit at the idle point, the Z rotations
can be easily realized by choosing suitable delay times $\tau$ between Z pulses, i.e.,
\begin{equation}
\begin{aligned}\label{eq4}
U_{z}=e^{-i\frac{\Delta \tau}{2}Z}.
\end{aligned}
\end{equation}
Note here that compared with the traditional microwave control, Virtual-Z (VZ) gate scheme \cite{McKay2017} is not suitable for the
present baseband control. However, similar to the VZ gate, besides time delay, here, no actual control
pulses are needed for implementing Z rotations.
Generally, as shown in Fig.~\ref{fig1}(c), by tuning the qubit on-resonance with
the global drive, single-qubit X rotations can be achieved. However, we note that since
the initial Bloch vector is slightly tilted, as shown in Fig.~\ref{fig1}(b), a small drive
detuning $\delta_{d}=|\Omega_{d}^{2}/\Delta_{d}|$ is needed for enabling ideal X rotations
with respect to the initial Bloch vector defined by Eq.~\ref{eq3}. Thus, according to Z control
pulses, X rotations can be realized by tuning the qubit from the idle point to the working
point with a small overshoot \cite{Barends2019}. This fact is further illustrated by the
results shown in Fig.~\ref{fig2}. By initializing the qubit in state $|0\rangle$ and
using square pulses (results with cosine-decorated square pulses can be found in Appendix~\ref{A}),
Figure~\ref{fig2}(a) shows populations $P_{1}$, i.e., the population in
state $|1\rangle$ at the end of the applied pulse, versus the drive detuning $\delta_{d}$
and the pulse length. Here, the drive amplitude is $10\,\rm MHz$ and the detuning at the
idle point is $-100\,\rm MHz$. Similarly, given a fixed pulse length of $50\,\rm ns$, Figure~\ref{fig2}(b)
shows $P_{1}$ versus $\delta_{d}$ and $\Omega_{d}$. The optimal parameters for X rotations are
indicated by the red stars. Indeed, we find that a small frequency overshoot is needed
for X rotations.
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{uniformity_qubit.pdf}
\end{center}
\caption{Flexibility of $\sqrt{X}$ rotations. (a) Population in state $|1\rangle$ (i.e., $P_{1}$ at
the end of the pulse) versus the drive detuning and the pulse length of square pulses with the qubit prepared in
state $|0\rangle$. Here, the drive strength is $10\,\rm MHz$ and the drive detuning at the idle
point is $-100\,\rm MHz$. The red star indicates the optimal parameter set for implementing X
rotations, while the dashed and dotted lines indicate the available parameter sets for
implementing $\sqrt{X}$ rotations based on numerical simulations and analytical expression
in Eq.~\ref{eq5}, respectively. (b) same as in (a), instead showing $P_{1}$ versus the drive detuning
and drive amplitude with the fixed gate length of $50\,\rm ns$.}
\label{fig2}
\end{figure}
In the present work, note that choosing $\sqrt{X}$ gates as the native gates
could simplify the tune-up procedure of single-qubit gate operations. This is because:
(i) arbitrary single-qubit rotations can be generated by two $\sqrt{X}$ gates
and three Z gates \cite{McKay2017}, i.e., $Z_{\phi1}-\sqrt{X}-Z_{\phi2}-\sqrt{X}-Z_{\phi3}$,
with $Z_{\phi}\equiv\exp[-i\phi Z/2]$;
(ii) compared with the native X gate, the implementation
of $\sqrt{X}$ gate does not pose stringent requirements on the on-resonance condition, i.e.,
even the qubit is slightly off-resonance with the drive, $\sqrt{X}$ gate can
still be achieved. This can be captured by the analytical
expression of Rabi oscillations for qubits initialized in state $|0\rangle$, i.e., Rabi's formula
\begin{equation}
\begin{aligned}\label{eq5}
P_{1}(t)=\frac{\Omega_{d}^{2}}{\Omega_{d}^{2}+\Delta_{d}^2}\sin^2\left[\frac{t}{2}\sqrt{\Omega_{d}^{2}+\Delta_{d}^2}\right].
\end{aligned}
\end{equation}
From Eq.~\ref{eq5}, implementing $\sqrt{X}$ gates requires $P_{1}=1/2$ at the end of the applied
pulse, giving rise to the relations among the pulse length, the drive detuning $\Delta_{d}$, and the drive
amplitude $\Omega_{d}$, as illustrated by the dotted lines of Fig.~\ref{fig2}. Accordingly, the
results based on numerical simulations are also presented, as indicated by the dashed line
of Fig.~\ref{fig2}. Note here that the derivation of the analytical equation ignores
the slight tilt at the idle point, and this explains the discrepancy between the analytical
and numerical results. Both the analytical and numerical results show that compared to X rotations, the
available parameter ranges of $\sqrt{X}$ rotations can provide great flexibility in its
tune-up procedure.
Generally, due to the flexibility of $\sqrt{X}$ rotations, for tuning-up $\sqrt{X}$
gates, the above-mentioned overshoot can be ignored. In the next subsection, we will
show that following this way, given a fixed drive detuning, $\sqrt{X}$ gate can be realized
by only optimizing the ramp times of control pulses, as suggested by Fig.~\ref{fig2}(a).
Meanwhile, in large-scale quantum systems with multiplexed control, this flexibility can
be the most encouraging advantage as to mitigate single-qubit gate error due to
stray coupling between qubits and to compensate for the non-uniformity of qubit
parameters. This will be discussed in detail in Sec.~\ref{SecIV}.
\subsection{Baseband control of qubit with fast-adiabatic ramps}\label{SecIIB}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=4cm,height=4cm]{spectrum.pdf}
\includegraphics[width=4cm,height=4cm]{leakage.pdf}
\end{center}
\caption{Leakage out of the computational subspace. (a) Energy spectrum (solid lines) versus the drive detuning in the
rotating frame corresponding to the global drive. The dashed lines denote the bare energy levels (i.e., spectrum
without the drive). Here, the used system parameters are qubit
anharmonicity $\alpha_{q}/2\pi=-250\,\rm MHz$, drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$, and drive
amplitude $\Omega_{d}/2\pi=10\,\rm MHz$. (b) Bloch vector for the leakage space spanned by
states $\{|1\rangle,|2\rangle\}$. The strength of the coupling between states $|1\rangle$
and $|2\rangle$ is $\sqrt{2}\Omega_{d}$. At the idle point and in the leakage space, the
drive detuning is $\Delta_{d}+\alpha_{q}$, while at the working point, the
detuning is $\alpha_{q}$. During single-qubit X rotations, the Bloch vector in this leakage
space varies according to the drive detuning. In the present work, to avoid possible leakage
during qubit control, the qubit idle frequency is far detuned below the drive frequency.}
\label{fig3}
\end{figure}
In the above discussion, the single-qubit baseband control is discussed for an
ideal two-level system. Nervelessness, for piratical superconducting qubits, such
as the transmon qubit, the weak qubit anharmonicity makes single-qubit gate operations
particularly prone to leakage outside the qubit subspace. For one such baseband control transmon
qubit, which is driven by an always-on global drive, the system Hamiltonian is (hereafter, transmon qubits
are modeled as anharmonicity oscillators \cite{Koch2007})
\begin{equation}
\begin{aligned}\label{eq6}
H_{q}=\omega_{q}a_{q}^{\dagger}a_{q}+\frac{\alpha_{q}}{2}a_{q}^{\dagger}a_{q}^{\dagger}a_{q}a_{q}
+\frac{\Omega_{d}}{2}(a_{q}^{\dagger}e^{-i\omega_{d}t}+a_{q}e^{+i\omega_{d}t}).
\end{aligned}
\end{equation}
Here, $a_{q}\,(a_{q}^{\dagger})$ is the annihilation (creation) operator.
Figure~\ref{fig3}(a) shows the energy spectrum of the driven qubit versus the
drive detuning with qubit anharmonicity $\alpha_{q}/2\pi=-250\,\rm MHz$,
drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$, and drive
amplitude $\Omega_{d}/2\pi=10\,\rm MHz$. One can find that due to the
global drive, there exits an off-resonance coupling
between states $|2\rangle$ and $|1\rangle$, which can cause leakage to state $|2\rangle$
when performing X rotations, i.e., biasing the qubit from its idle point to the
working point. Note here that we omit the discussion of leakage from state $|0\rangle$
to state $|2\rangle$, since this channel involves second-order processes, giving negligible
leakage errors. While this leakage issue can be addressed by using the DRAG scheme
in the traditional microwave control setup, this scheme cannot be directly unutilized
for the baseband flux control setup. This is because, in the current setup, only Z control
is available.
Additionally, in principle, at the idle point, qubits could be far detuned
above or below the frequency of the always-on drive. However,
to avoid possible leakage error during qubit control, such as gate operations and
qubit initialization and readout, we prefer to bias the qubit away from the
harmful avoid crossing caused by coupling between states $|1\rangle$ and $|2\rangle$, as
shown in Fig.~\ref{fig3}(a). In this way, during the baseband-controlled gate operations,
the qubit system will not sweep through or operate nearby this harmful avoid crossing, generally
allowing the suppression of the leakage to state $|2\rangle$. Considering
this fact, hereafter, we consider biasing the qubit below the drive frequency at
the idle point. Even in the setting, during Z-controlled single-qubit gate
operations, leakage error can still occur due to the non-adiabatic error, as shown
in Fig.~\ref{fig3}(b). In the following, we will consider using a
fast-adiabatic control scheme for suppressing the leakage further.
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{leak_popu2.pdf}
\end{center}
\caption{Minimizing leakage out of the computational subspace for performing Z-controlled X rotations.
(a) Leakage as a function of the ramp times for the transmon qubit initialized in
state $|1\rangle$ with the anharmonicity $\alpha_{q}/2\pi=\{-200,\,-250,\,-300\}\rm MHz$ (
denoted by the solid line, dashed line, and dotted line, respectively). Inset shows
the typical fast-adiabatic pulse and the fast-adiabatic flat-top pulse (i.e., square pulse with
fast-adiabatic ramps) for controlling the qubit frequency from the idle point ($6.0\,\rm GHz$) to
the working point ($6.1\,\rm GHz$). The other system parameters
are: the hold time (i.e., the pulse length of the flat part) $t_{h}=20\,\rm ns$,
drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$, and drive amplitude $\Omega_{d}/2\pi=10\,\rm MHz$.
(b) Same as in (a), instead showing the population in states $|0\rangle$ and $|1\rangle$
versus the ramp times for the Z-controlled X rotations.}
\label{fig4}
\end{figure}
As shown in Fig.~\ref{fig3}(b), the leakage error occurs when one non-adiabatically varies
the driving detuning. Considering that coherence times of superconducting qubits are still limited, our target
is to find a good flux control pulse, thus the non-adiabatic error is suppressed
while maintaining a fast operation speed. Fortunately, this issue has already been addressed
successfully by using a fast-adiabatic scheme introduced in Ref.\cite{Martinis2014b}. Within
the scheme, optimal control pulses can be obtained for minimizing non-adiabatic errors for any pulse
longer than the chosen pulse length. However, we note that the original scheme
only addresses the non-adiabatic error in the pulse ramps. Thus, here, to address the leakage
issue in our setting, we consider using a square control pulse with optimal fast-adiabatic ramps, which is obtained
following the fast-adiabatic scheme \cite{Martinis2014b} (see Appendix~\ref{B} for
details). In the inset of Fig.~\ref{fig4}(a) shows the optimal ramp pulse (solid blue line), which is used for
generating our target control pulse with a flat middle part and fast-adiabatic
ramps (solid orange line). Hereafter, we refer to this pulse as the fast-adiabatic flat-top pulse.
Here, we turn to evaluate the efficiency of the proposed fast-adiabatic flat-top pulse. We consider
that the qubit idle frequency is $6.0\,\rm GHz$ and during the implementation of
single-qubit X rotations, the drive detuning $\Delta_{d}$ varies from the idle
point at $-100\,\rm MHz$ to the work point at $0\,\rm MHz$ and then coming
back, according to the fast-adiabatic flat-top pulse. By initializing the qubit in state $|1\rangle$,
Figure~\ref{fig4}(a) shows the population leakage to state $|2\rangle$ as a function of ramp times
with the hold time of 20 ns and the qubit anharmonicities $\alpha_{q}/2\pi=\{-200,\,-250,\,-300\}\rm MHz$.
For easy comparison, the results for applying only the fast-adiabatic pulse are also presented. One
can find that by using the fast-adiabatic flat-top pulse, the leakage can be suppressed
below $10^{-6}$ for ramp times longer than 10 ns, and inserting a square pulse in the
fast-adiabatic pulse does not change the efficiency of the original fast-adiabatic
scheme. In Fig.~\ref{fig4}(b), we also show the populations in $|0\rangle$
and $|1\rangle$ versus the ramp times. Additionally, Appendix~\ref{B} presents further
results for different drive strengths.
From the results shown in Fig.~\ref{fig4}, one can find that $\sqrt{X}$ rotations can be
realized with the ramp time at about 10 ns, giving rise to the total pulse length of about 30 ns.
Meanwhile, same as the case for two-level systems (in Sec.\ref{SecIIA}), here, single-qubit Z rotations
can be easily implemented by controlling the delay times between Z pulses. Therefore, we could
reasonably expect that with the help of the fast-adiabatic scheme, fast-speed single-qubit
operations could be achieved with low leakage errors (we will evaluate the single-qubit gate
performance in detail in the following section).
\subsection{Dephasing due to fluctuations in the drive amplitude}\label{SecIIC}
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{dephasing.pdf}
\end{center}
\caption{Qubit dephasing due to the fluctuations in the amplitude of the always-on
drive. (a) Time evolution of the magnitudes of the averaged off-diagonal matrix
element, i.e., $|\langle\rho_{01}(t)\rangle|$, for the qubit initialized in
state $(|0\rangle+|1\rangle)/\sqrt{2}$. Here, we assume that the drive
amplitudes ($\Omega_{d}/2\pi=10\,\rm MHz$) subject to amplitude-dependent
Gaussian noise, i.e., $\textsl{N}(0,\sigma)$, and 2000 realizations
of noise are used for obtaining $|\langle\rho_{01}(t)\rangle|$. The dashed lines are
exponential fits $[1-\exp(-t/T_{\phi})]/2$, giving rise to the dephasing
time $T_{\phi}$. (b) Dephasing time versus the noise variance. For given
noise variances, the inset shows the dephasing time versus the drive amplitude. Here, the
other parameters used are: $\Delta_{d}/2\pi=-100\,\rm MHz$
and $\alpha_{q}/2\pi=-250\,\rm MHz$.}
\label{fig5}
\end{figure}
Within the introduced baseband control setup, at the idle point, the global
always-on drive acts as an off-resonance drive and can induce ac-Stark
frequency shifts on the qubits. For the two-level system studied
in Sec.\ref{SecIIA}, the shift is given
as $\delta\omega=\Delta-\Delta_{d}\approx \Omega_{d}^{2}/(2\Delta_{d})$,
while, taking the higher energy levels of the transmon qubit into
consideration, the shift is \cite{Schneider2018}
\begin{equation}
\begin{aligned}\label{eq7}
\delta\omega\approx \frac{\alpha_{q}\Omega_{d}^{2}}{2\Delta_{d}(\Delta_{d}+\alpha_{q})}.
\end{aligned}
\end{equation}
From Eq.~\ref{eq7}, the shift has a quadratic-dependent on the drive amplitude, making
the qubit frequency more susceptible to possible amplitude noise. Therefore, fluctuations in the drive
amplitude can cause qubit dephasing, which has been recently observed in superconducting
qubits \cite{Wei2022}.
Here, to numerically study the amplitude-fluctuation-induced qubit dephasing, we generate
an amplitude-dependent noise, i.e., amplitude fluctuations are proportional to
the amplitudes. By assuming the drive subject to zero-mean Gaussian
noise, i.e., $\textsl{N}(0,\sigma)$, we numerically simulate the time evolution of the
off-diagonal matrix element $\rho_{01}(t)$ for the qubit initialized in
state $(|0\rangle+|1\rangle)/\sqrt{2}$. After averaging $\rho_{01}(t)$ over 2000
trajectories (i.e., realizations of noise), the magnitudes of the off-diagonal matrix element display a clear
exponential decay, as shown in Fig.~\ref{fig5}(a). Here, the evolution time
is $10\,\mu s$, the other used parameters are: $\Delta_{d}/2\pi=-100\,\rm MHz$,
$\Omega_{d}/2\pi=10\,\rm MHz$, and $\alpha_{q}/2\pi=-250\,\rm MHz$, giving
rise to $\delta\omega/2\pi\approx-0.36\,\rm MHz$. By fitting the decay curves
to $[1-\exp(-t/T_{\phi})]/2$, Figure~\ref{fig5}(b) shows the dephasing time $T_{\phi}$
versus the noise variance $\sigma$. Here, we also show the results for
$\Omega_{d}/2\pi=20\,\rm MHz$. Additionally, in the inset, we further
show the dephasing times versus the drive amplitudes.
From the results shown in Fig.~\ref{fig5}(b), and given the typical noise variance of $1\%$, we
can conclude that the amplitude-noise induced dephasing can be safely neglected by detuning the
qubit far from the drive frequency. Meanwhile, we note that to ensure high-fidelity gate
operations within sub-100 ns, the drive amplitude itself should be larger than $10\,\rm MHz$.
\subsection{Impact of the always-on drive on the qubit readout}\label{SecIID}
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{readout.pdf}
\end{center}
\caption{Qubit dispersive readout with the presence of the always-on drive.
(a) Histograms of the integrated readout quadrature for the qubit prepared in
states $|0\rangle$ (blue) and $|1\rangle$ (orange). The dashed blue line and
dashed orange line denote the Gaussian fits of the histograms for states $|0\rangle$
and $|1\rangle$, respectively. The intersection point of the two fitted distributions
gives the state-decision threshold. The inset shows the IQ scatter plot of the
integrated readout quadrature. (b) Same as in (a), instead showing the results
without the always-on drive. (c) Readout error $1-F$ as a function of the drive
strength.}
\label{fig6}
\end{figure}
As mentioned in Sec.~\ref{SecI} and Sec.~\ref{SecIIA}, due to the presence
of the always-on drive, in this work, the microwave dressed states are defined as the
computational states. Here, since the available control over the always-on
drive is limited, the previous method \cite{Zhao2022,Huang2021}, in which the dressed state is first mapped back to
the corresponding bare state, and then the traditional dispersive readout is employed for
inferring the qubit information \cite{Wallraff2005}, cannot be directly utilized. However, as
discussed in Sec.~\ref{SecIIA}, when the drive detuning is far-detuned from the
qubit, i.e., $|\Delta_{d}|\gg\Omega_{d}$, the difference between dressed states and
bare states can be neglected. Therefore, we expect that by keeping a large ratio of the
drive detuning to the drive amplitude, the qubit information can be directly inferred
using the traditional dispersive readout scheme.
To explore the possible impact of the always-on drive on the qubit dispersive readout, we numerically
simulate the system dynamics during the dispersive readout. By applying a 250-ns square readout pulse with
frequency $\omega$ and amplitude $\Omega$ to the readout resonator with decay rate $\kappa$, the
full system dynamics are governed by the Hamiltonian
\begin{equation}
\begin{aligned}\label{eq8}
H_{\rm read}=&H_{q}+\omega_{r}a_{r}^{\dagger}a_{r}+g(a_{q}^{\dagger}a_{r}+a_{q}a_{r}^{\dagger})
\\&+\frac{\Omega}{2}(a_{r}^{\dagger}e^{-i\omega t}+a_{r}e^{+i\omega t}),
\end{aligned}
\end{equation}
where $H_{q}$ denotes the qubit Hamiltonian given in Eq.~\ref{eq6}, $\omega_{r}$ is the frequency of
the readout resonator, $a_{r}\,(a_{r}^{\dagger})$ is the annihilation (creation) operator of
the resonator, and $g$ denotes the strength of the qubit-resonator coupling. In this following,
the qubit information is encoded into single quadrature, i.e., $I$-quadrature, by choosing the readout
frequency to be $\omega=(\omega_{r0}+\omega_{r1})/2$ \cite{Wallraff2005}. Here, $\omega_{r0}$ and $\omega_{r1}$ denote the
dressed resonator frequencies with the qubit in states $|0\rangle$ and $|1\rangle$, respectively. The other
system parameters are: $\omega_{q}/2\pi=6.0\,\rm GHz$,
$\alpha_{q}/2\pi=-250,\rm MHz$, $\omega_{d}/2\pi=6.1\,\rm GHz$, $\omega_{r}/2\pi=5.0\,\rm GHz$,
$g/2\pi=100\,\rm MHz$, $\kappa/2\pi=5\,\rm MHz$, and $\Omega/2\pi=7\,\rm MHz$.
According to Eq.~\ref{eq8}, we simulate the system dynamics based on solving the stochastic master
equation \cite{Johansson2012}. Then,
following Ref.~\cite{Walter2017}, we further caulate the integrated readout quadrature with an optimal
weight function (see Appendix~\ref{C} for details). With 5000 repetitions of the simulation for each qubit basis
state, i.e., $|0\rangle$ and $|1\rangle$, Figure~\ref{fig6}(a) shows the two histograms of the
integrated readout quadrature with the qubit prepared in states $|0\rangle$ and $|1\rangle$,
respectively. Here, the drive magnitude is $20\,\rm MHz$. For easy comparison, we also present
the result for the global drive is absent, as shown in Fig.~\ref{fig6}(b).
Fitting the histograms to Gaussian functions gives the state-decision threshold at the
intersection point of the two fitted distributions. Accordingly, the readout fidelity
can be calculated as $F=1-[P(0|1)+P(1|0)]/2$, where $P(0|1)$ ($P(1|0)$) denotes the error probability
that the qubit initialized in state $|1\rangle$ ($|0\rangle$) is identified as in
state $|0\rangle$ ($|1\rangle$). Accordingly, Figure~\ref{fig6}(c) shows the readout error $1-F$ versus the drive strength. One can
find that when increasing the drive amplitude from $0$ to $20\,\rm MHz$, while the error
shows an upward trend, the increased error is below $1\%$. Moreover, the upward trend also suggests
that by further increasing the ratio $|\Delta_{d}|/\Omega_{d}$, the increased error should be
heavily suppressed.
\section{An Application in qubit architectures with tunable coupling}\label{SecIII}
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{coupling_moving.pdf}
\end{center}
\caption{Residual coupling with varying qubit frequency. (a) Left: residual resonance XY coupling
versus the qubit frequency and the coupler frequency. Horizontal cut through (Left) denotes the
result plotted in (Right), i.e., XY coupling versus the qubit frequency with the coupler
frequency fixed at $\omega_{c}/2\pi=11.35\,\rm GHz$. (b) Left: residual ZZ coupling versus the
frequencies ($\omega_{0}$ and $\omega_{1}$) of the two qubits with the coupler frequency
fixed at $11.35\,\rm GHz$. Horizontal cut through (Left) denotes the
result plotted in (Right), i.e., ZZ coupling versus frequency of $Q_{1}$ with the $Q_{0}$'s
frequency fixed at $\omega_{0}/2\pi=6.0\,\rm GHz$. Vertical cut through (Left) denotes the result plotted
in the inset of (Right), i.e., ZZ coupling versus frequency of $Q_{0}$ with the $Q_{1}$'s
frequency fixed at $\omega_{0}/2\pi=5.9\,\rm GHz$.}
\label{fig7}
\end{figure}
Given the overview of the baseband control scheme, in this section, we will present an
example application of this scheme in a qubit architecture with tunable coupling. As depicted
in Fig.~\ref{fig1}(a), we consider that two frequency-tunable transmon qubit $Q_{0}$ and $Q_{1}$ are
coupled via a tunable coupler $Q_{c}$ (i.e., an auxiliary transmon qubit) and both
qubits are driven by an always-on global drive. After applying RWA, the system
Hamiltonian is given by
\begin{equation}
\begin{aligned}\label{eq9}
H=&\sum_{j=0,1,c}\big(\omega_{j}a_{j}^{\dagger}a_{j}+\frac{\alpha_{j}}{2}a_{j}^{\dagger}a_{j}^{\dagger}a_{j}a_{j}\big)
\\&+\sum_{\substack{k=0,1,c\\j\neq k}}g_{jk}(a_{j}a_{k}^{\dagger}+a_{j}^{\dagger}a_{k})
\\&+\sum_{i=0,1}\frac{\Omega_{d}}{2}(a_{i}^{\dagger}e^{-i\omega_{d}t}+a_{i}e^{+i\omega_{d}t}),
\end{aligned}
\end{equation}
where $\omega_{j}$ and $\alpha_{j}$ are the bare qubit frequency and the qubit anharmonicity
of $Q_{j}$, $q_{j}\,(q_{j}^{\dagger})$ is the associated annihilation (creation)
operator, and $g_{jk}$ denotes strength of the coupling between $Q_{j}$ and $Q_{k}$. Hereafter, the
system state is denoted by the notation $|Q_{0}Q_{c}Q_{1}\rangle$ and
the used system parameters are: the qubit anharmonicity $\alpha_{0}/2\pi=\alpha_{1}/2\pi=-250\,\rm MHz$,
the coupler anharmonicity $\alpha_{c}/2\pi=-200\,\rm MHz$, the direct qubit-qubit coupling
strength $g_{01}/2\pi=13\,\rm MHz$ (at $\omega_{0}/2\pi=\omega_{1}/2\pi=5.5\,\rm GHz$),
the qubit-coupler coupling strength $g_{0c}/2\pi=g_{1c}/2\pi=160\,\rm MHz$
(at $\omega_{0(1)}/2\pi=\omega_{c}/2\pi=5.5\,\rm GHz$), the drive
amplitude $\Omega_{d}/2\pi=10\,\rm MHz$, and the drive frequency $\omega_{d}/2\pi=6.1\,\rm GHz$.
Note here that the RWA is used for simplifying numerical simulation (otherwise, given the always-on
drive, Floquet methods could be employed here \cite{Huang2021,Shirley1965,Sambe1973,Petrescu2021}). However, in
the present two-qubit system with tunable coupling, the non-RWA terms in the original Hamiltonian (see Appendix~\ref{D}
for details) can significantly affect the effective coupling between qubits and can shift
the bare qubit frequency. Thus, here, considering non-RWA terms while
still working within the RWA formalism, we keep second-order corrections from the non-RWA
terms and find that with this correction (details on its derivation can be
found in Appendix~\ref{D}), the results agree well with the results without
applying the RWA. Accordingly, the corrections are taken into consideration
throughout the following discussion.
Before going into details of the baseband controlled gate operations, we give a few brief discussions of
the tunable coupling architecture. For performing gate operations in multiqubit systems, the key
benefit of the introduced tunable coupler is that the inter-qubit coupling strength can be tuned
off by biasing the coupler at a certain frequency point, i.e., zero-coupling point. However, the
zero-coupling point can change when the qubit is just biased slightly away from its idle point.
Fortunately, in the tunable coupling architecture, biasing
the qubit slightly away generally only causes a small increase in the residual inter-qubit
coupling. This can be found in Fig.~\ref{fig7}. Figure~\ref{fig7}(a) shows the strengths of the
residual resonance XY coupling versus the qubit frequency and the coupler frequency, while
Figure~\ref{fig7}(b) shows the residual ZZ coupling versus the frequencies of the two qubits with the
coupler frequency fixed at $11.35\,\rm GHz$. Here, the XY coupling and the ZZ coupling are
numerically calculated by the diagonalization of the Hamiltonian Eq.~\ref{eq9} in the rotating
frame defined by the always-on drive. To be more specific, the XY coupling is extracted as half the
energy difference between dressed eigenstates $|10\rangle$ and $|01\rangle$, while the ZZ coupling
is $\zeta_{zz}=(E_{11}-E_{10})-(E_{01}-E_{00})$. Here, $E_{ij}$ denotes the energy of
dressed eigenstate $|ij\rangle$, which is adiabatically connected to the
bare state $|i0j\rangle$ \cite{Ghosh2013}.
According to the above results, in the following discussion, we consider that at the
system idle point, the frequency of qubit $Q_{0}$ and qubit $Q_{1}$ are $\omega_{0}/2\pi=6.0\,\rm GHz$
and $\omega_{1}/2\pi=5.9\,\rm GHz$, respectively, and the frequency of coupler $Q_{c}$ is
$\omega_{c}/2\pi=11.35\,\rm GHz$. Therefore, the residual ZZ coupling is below $10\,\rm kHz$ at
the system idle point. Moreover, during the gate operations based
on slightly tuning qubit frequency, such as implementing single-qubit gates by tuning the qubit from its idle
point to the working point (e.g., at $6.1\,\rm GHz$), the residual inter-qubit ZZ coupling
can always be below $10\,\rm kHz$.
\subsection{Single-qubit gate operation}\label{SecIIIA}
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{leakage_2q.pdf}
\end{center}
\caption{Performing Z-controlled X rotations with the fast-adiabatic
flat-top pulse in the two-qubit system with tunable coupling. (a) Leakage as a function of the
times of the pulse ramp for qubit $Q_{0}$ initialized in state $|1\rangle$. Inset shows
the population in states $|0\rangle$ and $|1\rangle$ versus the ramp times for the
Z-controlled X rotations. The solid lines and dashed lines represent the results
with the coupler biased at two different idle points, i.e., $11.35\,\rm GHz$ and $11.25\,\rm GHz$, respectively.
Same as in Fig.~\ref{fig4}, here, the hold time of the utilized fast-adiabatic flat-top pulse is
fixed at $t_{h}=20\,\rm ns$. (b) Same as in (a), instead showing the results for $Q_{1}$.
Additionally, here also shows the population leakage to $Q_{0}$, as indicated by the orange lines.}
\label{fig8}
\end{figure}
Following the scheme introduced in Sec.~\ref{SecIIB}, here, Z-controlled single-qubit gates
are realized by using the fast-adiabatic flat-top pulse. Note here that in the present
two-qubit system, single-qubit gate operations for one qubit are tuned up and characterized
with the other qubit in its ground state $|0\rangle$.
Figure~\ref{fig8} shows the leakage versus the ramp times of
the pulse with the qubit initialized in its excited state $|1\rangle$. One can find that
while for $Q_{0}$, the result is in line with our theory discussed in
Sec.~\ref{SecIIB}, i.e., by increasing the ramp times, the leakage can be
further suppressed, the result of $Q_{1}$ seems unreasonable at first glance.
However, during gate operations applied to $Q_{1}$, $Q_{1}$ is
tuned from its idle point at $5.9\,\rm GHz$ to the working
point at about $6.1\,\rm GHz$, according to the fast-adiabatic pulse, while the
$Q_{0}$ is fixed at its idle point at $6.0\,\rm GHz$. Therefore, during the pulse
ramp, $Q_{1}$ will sweep through a tiny avoided crossing formed by the
residual resonance XY coupling between $Q_{0}$ and $Q_{1}$ at $6.0\,\rm GHz$. On
contrast, during single-qubit gate operations, $Q_{0}$ will not sweep through $Q_{1}$.
As shown in Fig.~\ref{fig7}(a), the strength of the residual XY coupling
is about $0.1\,\rm MHz$. Thus, sweeping through this avoided
crossing slowly will generally cause more leakage into
the nearby qubit $Q_{1}$, as shown in Fig.~\ref{fig8}(b), where the
orange solid line denotes the population of $Q_{0}$ in state $|1\rangle$. One
can find that for $t_{r}\geq 10\,\rm ns$, the leakage into $Q_{0}$ gives
the leading contributions to the total leakage error. These results
suggest that there exists a trade-off
between gate error resulting from the qubit itself and error
from spectator qubits, i.e., suppressing leakage into $|2\rangle$
favors longer gate times, while mitigating the leakage into the $Q_{0}$
favors short gate times. This observation is in agreement with
that in previous work \cite{Zhao2022b}.
To address the above issue, one can change the idle point of the coupler, thus
at the resonance point $6.0\rm GHz$, the XY coupling is further suppressed. By
biasing the coupler at $11.25\,\rm GHz$, the residual XY coupling is
suppressed below $0.01\,\rm MHz$. Accordingly, the leakage into $Q_{0}$ is indeed
suppressed heavily, as shown in Fig.~\ref{fig8}(b), where the dashed lines show
the results with the coupler biased at $11.25\,\rm GHz$.
Here, we turn to evaluate the gate performance of the baseband controlled
single-qubit gates, and use the metric of the state-average gate fidelity \cite{Pedersen2007}
in the following discussion (details on the fidelity calculation can also be found
in \cite{Zhao2022b}). As mentioned in Sec.~\ref{SecIIA}, in the present work, we focus on the
implementation of $\sqrt{X}$ gates. From the inset of Figs.~\ref{fig8}(a)
and ~\ref{fig8}(b), one can find that for both qubits, $\sqrt{X}$ gates
can be realized with a ramp time of about $10\,\rm ns$, giving rise to the total gate
time of about $30\,\rm ns$. Moreover, even by biasing the coupler
at $11.35\,\rm GHz$, Figure~\ref{fig8} shows that when the ramp time
is about $10\,\rm ns$, the leakage error can still be suppressed
below $5\times10^{-5}$ for both qubits. This is to be
expected since sweeping through the tiny avoided crossing with fast
speed could suppress leakage. By optimizing the ramp times, we find that
for both qubits, up to single-qubit Z rotations, $\sqrt{X}$ gate can be achieved
with gate fidelity exceeding $99.999\%$ (for $Q_{0}$, the gate fidelity is $99.9998\%$ and the
optimal gate time is $30.2\,\rm ns$, while for $Q_{1}$, are the $99.9996\%$
and $29.4\,\rm ns$). As mentioned before, Z gates can be easily realized by
choosing suitable time delays between flux pulses. In this way, universal
single-qubit gates can be achieved by combining Z gates and $\sqrt{X}$ gates.
\subsection{Two-qubit CZ gate}\label{SecIIIB}
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{two_qubit_gate.pdf}
\end{center}
\caption{Performing CZ gates with the fast-adiabatic flat-top pulse in the two-qubit
system with tunable coupling. During the gate operations, qubit $Q_{0}$ is fixed at its idle
point, i.e., $6.0\,\rm GHz$, qubit $Q_{1}$ and coupler $Q_{c}$ are tuned from their idle
points ($5.9\,\rm GHz$ and $11.35\,\rm GHz$) to the working points, resulting in a complete
population oscillation between states $|101\rangle$ and $|200\rangle$. (a) The cosine-decorated square
pulse with the ramp times of $10\,\rm ns$ for implementing CZ gates. Up to single-qubit phase
gates, the gate fidelity is $99.82\%$. (b) Time evolution of the qubit
state population during the gate operation with the cosine-decorated square pulse.
Here, $P_{ij}$ denotes the population in state $|i0j\rangle$ for the two-qubit system initialized
in state $i0j\rangle$. (c) The employed fast-adiabatic flat-top pulse with a hold time of $10\,\rm ns$ for
biasing coupler. while for biasing $Q_{1}$, a cosine-decorated square pulse with the ramp
time of $6\,\rm ns$ is used. Here, the total pulse length is $30.7\,\rm ns$ and up to single-qubit phase gates, the CZ gate
fidelity is $99.94\%$. (d) Time evolution of the qubit state population during the
gate operation with the fast-adiabatic pulse, showing that the population swap between
qubits is suppressed below $10^{-3}$.}
\label{fig9}
\end{figure}
Having discussed the single-qubit control, we now turn to the two-qubit case. Here, we
consider the implementation of CZ gates in the two-qubit system with an
always-on drive. During the gate operations, $Q_{0}$ is fixed at its idle
point, i.e., $6.0\,\rm GHz$, $Q_{1}$ is tuned from its idle
point ($5.9\,\rm GHz$) to the working point, where a complete oscillation between states $|101\rangle$
and $|200\rangle$ can occur. Meanwhile, the coupler is tuned from its idle
point at $11.35\,\rm GHz$ to a working point at about $7\,\rm GHz$, giving
rise to the CZ coupling strength of $20\,\rm MHz$ (see Appendix~\ref{D}).
Figure~\ref{fig9}(a) shows the typical control pulse, i.e., the pulse with a flat middle part
and cosine-shaped ramps (see Appendix~\ref{A} for details), with a pulse length of $30\rm ns$ for the CZ
implementation. By optimizing numerically the working points of $Q_{c}$
and $Q_{1}$, the gate fidelity of the implemented CZ gate (up to single-qubit Z phases) is $99.82\%$.
After inspecting the qubit dynamics during the gates, one can find that the leading error
source is the population swap between two qubits, as shown in Fig.~\ref{fig9}(b). Following
the fast-adiabatic scheme discussed in Sec.~\ref{SecIIB} and the previous work \cite{Martinis2014b,Sung2021}, here, the
fast-adiabatic flap-top pulses, as shown in Fig.~\ref{fig9}(c), with a
hold time of $10\,\rm ns$, is used to suppress the population swap. Accordingly, the
population swap is indeed largely suppressed, as shown in Fig.~\ref{fig9}(d), improving the
CZ gate fidelity to $99.94\%$ with a gate time of $30.7\,\rm ns$. Additionally, we note that
generally, by increasing the gate length, the residual gate error can be
further suppressed (see also in Appendix~\ref{E}).
The above results show that although there exists an always-on global drive in the tunable coupling
architecture, high-fidelity two-qubit gates can still be achieved in a short time. This success is
mainly based on the fact that during the gate operations, the global drive is far detuned from both
qubits and the coupler.
\section{discussion}\label{SecIV}
Given the above theoretical analysis of the implementation of the baseband flux control in
tunable coupling architecture, in the following, we will give a few discussions of the challenges and opportunities for
realizing the baseband control strategy in large-scale superconducting quantum processors.
\subsection{Practical challenges}\label{SecIVA}
While our theoretical study shows that baseband controlled gate operations
can be realized with high fidelity and fast speed, we note that besides
the qubit decoherence, there exist several practical experimental issues
that will limit the available gate performance:
(i) Flux pulse distortion. Flux pulse distortion has been demonstrated as a critical issue faced by baseband
flux-controlled gate operations \cite{Jerger2019,Rol2020,Foxen2019}. Moreover, the above-demonstrated
high-fidelity gate operations are achieved by using pulse shaping technologies, thus, the
impact of flux pulse distortion can become more prominent in our setting.
(ii) Stray coupling beyond nearest neighbors. Generally, in our setting, the always-on drive is shared
by multiple qubits. When performing single-qubit gates in parallel, multi-qubit will
be tuned on-resonance with the same drive. This means that any stray coupling between these qubits
will cause population swaps among these qubits, as discussed in Sec.~\ref{SecIIIA}, leading to additional
gate errors compared to isolated gates. While near-neighbor couplings between qubits can be controlled well
in the tunable coupling architecture, parasitic coupling beyond nearest neighbors can still exist due to, such as
stray capacitive coupling, in multi-qubit systems \cite{Barends2014,Zajac2021,Yanay2022,Zhao2022b}. This will
degrade the efficiency of the baseband control strategy in large-scale quantum processors.
(iii) Defect modes, such as TLSs \cite{Muller2019}. Same as in (ii), when performing single-qubit gates, the
working frequencies of multi-qubit are almost limited to a fixed one, i.e., the frequency of the
shared drive. This will limit the ability to mitigate the impacts from defect modes by tuning the qubit away
from the defects \cite{Klimov2018}.
(iv) Keeping track of the single-qubit phase accumulation. In our setting, the qubit
frequency at its idle point is detuned from the always-on drive. Thus, the single-qubit
phase will accumulate at the speed of the drive detuning $\Delta$ during the idle time.
While the accumulated phase can be employed to realize single-qubit Z gates,
on the other hand, when performing gate sequences or quantum circuits, the accumulated phase
should be tracked carefully over the whole time domain. Compared with the traditional microwave
control, this could complicate the implementation of quantum circuits.
In addition, we note that owing to the great flexibility of Z-controlled $\sqrt{X}$ gate, as discussed
in Sec.\ref{SecIIA}, the issues, related to (ii) and (iii), may be addressed. From the results shown in
Fig.~\ref{fig2}, we can find that given a fixed drive amplitude or a fixed pulse
length, $\sqrt{X}$ gates can be achieved with a small drive detuning, for which its magnitude can
even be compared with that of the always-on drive. Thus, when implementing isolated or paralleled
single-qubit gates, the working frequencies of qubits can be biased intentionally at
different frequency points, thus impacts of sub-MHz stay coupling can be mitigated. Similarly,
the defect's impact can be suppressed by biasing qubits away from the leading
defect modes.
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{crossbar.pdf}
\end{center}
\caption{(a) Multiplexing control of qubit lattices of frequency-tunable superconducting
qubits with share XY and Z lines and local programmable memory. During the parallel gate operations, the
local memory can be used to switch on or off the control on the individual qubit, and can provide the
static bias for compensating qubit non-uniformity. (b) Network of word lines for digital addressing the
local memory.}
\label{fig10}
\end{figure}
\subsection{Opportunities for solving challenges towards large-scale quantum processors}\label{SecIVB}
\begin{figure}[tbp]
\begin{center}
\includegraphics[keepaspectratio=true,width=\columnwidth]{uniformity_transmon.pdf}
\end{center}
\caption{Flexibility of $\sqrt{X}$ rotations on superconducting transmon qubits.
(a) Population in states $|0\rangle$ (left panel) and $|2\rangle$ (right panel) versus the
drive detuning and the drive amplitude with the qubit prepare with in state $|1\rangle$.
Here, the qubit anharmonicity is $-250\,\rm MHz$, the drive detuning at the idle point is $-100\,\rm MHz$,
and the length of the square pulse is $50\,\rm ns$. The dashed lines indicate the
available parameter sets for implementing $\sqrt{X}$ rotations. (b) same as in (a),
instead showing the case with cosine-decorated square pulses. The ramp time and the hold
time are $10\,\rm ns$ and $50\,\rm ns$, respectively.}
\label{fig11}
\end{figure}
Currently, in small-scale superconducting quantum processors, each qubit has its
dedicated control lines, such as XY lines and flux (Z) lines. Moreover, due to the
non-uniformity of qubit parameters, such as qubit frequency and anharmonicity, the
coupling efficiency between qubits and control lines, and the signal attenuations
and distortions in control lines, control pulses can differ from qubit to qubit. Thus,
generally, microwave control pulses differ with each other in their amplitudes, frequencies,
and phases, while for the baseband flux pulse, their amplitudes could be different. When
scaling up to large-scale quantum computing, such strategy is not scalable.
Given the recent progress in the pursuit of scalable spin-based quantum
computing with multiplexing technologies and crossbar
technologies \cite{Hill2015,Vandersypen2017,Veldhorst2017,Li2018}, we may also
consider how to utilize these technologies for solving the above-mentioned
challenges toward large-scale superconducting quantum processors. One possible
example is schematically illustrated in Fig.~\ref{fig10}(a), where both the XY and Z lines
are shared by multiple qubits in a square lattice of frequency-tunable qubits. Compared
with the spin qubit, it seems that the superconducting qubit can provide
more flexible control over its physical size and qubit
parameters \cite{Martinis2020,Barends2014,Zhao2020N,Mamin2022,Zhao2022c,Chow2015}, yet, it
can also show prominent non-uniformity. Unfortunately, the success of the multiplexing
technologies and crossbar technologies highly hinges on the uniformity of qubit
parameters. This can be more prominent for superconducting quantum processors based
on individual microwave control.
In the context of the implementation of multiplex control of superconducting qubits, baseband
flux control may alleviate this issue of non-uniformity. Within our baseband control
setup, and applying the multiplexing technologies shown in Fig.~\ref{fig10}(a), there are three
main leading non-uniformity issues:
(i) The non-uniform amplitude of the shared microwave
drive or (ii) flux pulse felt by qubits (caused by, such as the different coupling
efficiency between qubits and the global XY/Z line and signal attenuations in control
lines);
(iii) Independently calibrated parameters, including pulse length and pulse shape, of
flux pulse for implementing accurate control on individual qubits.
However, as discussed in Sec.\ref{SecIIA}, for two-level systems, given a fixed pulse
length and pulse shape (i.e., square shape, see Appendix~\ref{A} for results with smooth
pulses), owing to the flexibility of Z-controlled single-qubit gates (based on $\sqrt{X}$ gates),
the available parameter ranges (i.e., the drive amplitude and the drive detuning) can
be explored for compensating the non-uniform of drive amplitude. This exciting feature
is illustrated in Fig.~\ref{fig2}(b). Furthermore, similar results can also be obtained
for superconducting qubits, such as transmon qubits. Figures~\ref{fig11}(a) and~\ref{fig11}(b)
present the results for transmon qubits with $50$-$\rm ns$ square pulses and smooth pulses (i.e., cosine-decorated
square pulse with a hold time of $50\,\rm ns$ and a ramp time of $10\,\rm ns$), respectively.
Here, the qubit anharmonicity is $-250\,\rm MHz$ and the other used parameters are same as
in Fig.~\ref{fig2}. In addition, we note that to achieve uniform control pulses, the Z gate scheme based on
time-delay and the proposed fast-adiabatic scheme cannot be employed. Here, we can instead use
cosine-decorated square pulses for implementing Z gates by tuning the qubit from the idle
point, and for suppressing leakage. From Figs.~\ref{fig11}(a) and~\ref{fig11}(b), one can find that
by adding cosine-shaped ramps, the leakage error can be suppressed below $10^{-4}$, while for
the square pulse the leakage can approach $10^{-3}$. To further suppress the leakage
error, one can increase the ramp time or decrease the drive amplitude. However, this will
increase gate length and thus cause more decoherence errors.
Considering the above results, the solution to the above three issues may reduce to address the
non-uniformity issue of flux pulse amplitudes. This non-uniformity
could be removed by developing on-chip programmable memory, such as the one demonstrated by using
Single Flux Quantum (SFQ) logic \cite{Johnson2010,McDermott2010}, which could further compensate for the
remaining non-uniformity. In this way, combing with the word lines for digital
addressing, as shown in Fig.~\ref{fig10}(b), it is possible using only a few global XY and Z lines
to achieve parallel control of large numbers
of qubits \cite{Hill2015,Vandersypen2017,Veldhorst2017,Li2018}. Nevertheless,
given the practical experimental limitations, such as the limited cooling
power, the realization of the local programmable memory, which is compatible
with superconducting qubits, is still rarely explored \cite{Johnson2010}, and
undoubtedly, will be one of the most crucial challenges for implementing multiplexing
control technologies. Last, but not least, we must stress
that before solving the issue of pulse distortions, the efficiency of the above-discussed
scheme could be limited for implementing gate-based quantum computing.
\section{conclusion}\label{SecV}
In this work, we propose and theoretically study the possibility of implementing baseband
control of superconducting qubits, which are subjected to an always-on global drive. Our
results provide a general understanding and the basic principles of realizing
the baseband control scheme for superconducting qubits, such as frequency-tunable transmon
qubits. In the qubit architecture with tunable coupling, we show that high-fidelity and
fast-speed gate operations are possible by employing this baseband control scheme. Additionally,
we further describe potential challenges and opportunities for implementing such baseband
control strategy toward large-scale superconducting quantum processors.
\begin{acknowledgments}
We acknowledge the helpful discussions with Zhaohua Yang, Yanwu Gu, and Zhi-Hai Liu. This work
was supported by the National Natural Science Foundation of
China (Grants No.12204050, No.11905100, and No.11890704), the Beijing
Natural Science Foundation (Grant No.Z190012), and the
Key-Area Research and Development Program of Guang Dong Province (Grant No. 2018B030326001).
P.X. was supported by the Young Fund of Jiangsu Natural Science Foundation of
China (Grant No.BK20180750) and the National
Natural Science Foundation of China (Grant No.12105146).
\end{acknowledgments}
\emph{Note added.}-- During the preparation of this manuscript, we became aware of a recent related work \cite{Bejanin2022},
which presents the experimental demonstration of baseband-controlled single-qubit gates in superconducting transmon qubits.
|
1,116,691,498,483 | arxiv | \section{Conclusion}
We presented extensions to the tracking method Linajea~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot} to improve tracking of all cells during embryonic development.
In addition to combining deep learning to learn position and movement vectors of each cell and integer linear programming to extract tracks over time and ensure long term consistency, we integrate cell state information into the ILP, together with a method to automatically determine the weights of the ILP objective, alleviating the need for potentially suboptimal manually configured grid-search.
At the time of submission our method headed the leaderboard of the CTC for the DET and TRA scores for the \textbf{Fluo-N3DH-CE} dataset.
On two other datasets of both confocal and lightsheet recordings of \textit{C. elegans} our method outperforms the tool Starrynite, which is often used by practitioners for studies of \textit{C. elegans}, by a wide margin. Furthermore, an ablation study reveals that each of our proposed methodological advances improves upon baseline Linajea.
The low error rate achieved by our method
will
further
push down the required time for manual curation
This will facilitate studies that require a large number of samples.
More effort is still necessary in the later stages of development.
In future work we will extend the tracking all the way to the end of the embryonic development.
This poses additional challenges as the whole embryo starts to twitch, causing abrupt movements.
A second avenue of future work is to combine the two stages of the method.
Recent work~\cite{pogan2020_diffe_of_black_combi_solve} has proposed a method to incorporate black box solvers into a gradient-based end-to-end neural network learning process.
This shows great promise to increase the performance of our method even further.
\section{Results}
\label{sec:exp}
To measure the performance of our method we evaluate it on three different datasets of developing \textit{C. elegans} embryos, the \textbf{Fluo-N3DH-CE} dataset of the Cell Tracking Challenge benchmark (CTC)~\cite{ulman17_objec_compar_cell_track_algor}, three confocal recordings (\textbf{mskcc-confocal}) and three lightsheet recordings (\textbf{nih-ls}).
See Suppl.\ Table~\ref{suppl_tab:impl} for information on implementation and computational details.
\paragraph{The \textbf{Fluo-N3DH-CE} dataset}
\label{subsec:ctc_data}
~\cite{murray08_autom_analy_embry_gene_expres} consists of four 3d+time anisotropic confocal recordings until the 350 cell stage; 2 public ones for training and 2 private ones for the official evaluation.
All tracks are annotated.
The polar body filter is not used for this dataset.
Our method (named JAN-US) achieves a detection score (DET) of 0.981, and a tracking score (TRA) of 0.979, thereby outperforming the previous state of the art from Elephant~\cite{sugawara2021_track_cell_linea_in_3d_by_incre_deep_learn} (DET 0.979, TRA 0.975) at the time of submission. These results, which are listed on the challenge website, were generated using our cell state classifier (\textit{linajea+csc}).
See Suppl. Table~\ref{suppl_tab:metrics} for a short description of the metrics.
For details on the dataset, the challenge format and the metrics, please refer to~\cite{ulman17_objec_compar_cell_track_algor,matula15_cell_track_accur_measur_based}.
\textbf{Discussion. } As the labels for CTC test data are not public, a qualitative assessment of the improvement over the previous state of the art is not possible. However, the challenge scores are defined to be interpretable in terms of reduction of manual labor necessary for fixing an automated tracking solution. In this regard, our improvement in TRA over Elephant should mean that our method entails a 16\% reduction in manual curation effort as compared to Elephant (\(\frac{(1-0.975)-(1-0.979)}{(1-0.975)}=0.16\)).
Our improvement in DET and TRA scores on the challenge test data can further be put into perspective by comparison with the improvements in DET and TRA that we obtain and analyze on our other datasets (as described in the following, summarized in Table~\ref{tab:nuclei_results}).
We take this comparison as further indication that in terms of DET and TRA, an improvement in the 3rd decimal place as achieved on the CTC data can mean a considerable difference in performance.
\begin{table*}[tbp]
\begin{center}
\captionof{table}{Quantitative results on \textbf{mskcc-confocal} and \textbf{nih-ls} data. For description of error metrics see Suppl.\ Table~\ref{suppl_tab:metrics}; absolute number of errors normalized per 1000 GT edges; best value bold, value with insignificant difference to best value underlined (significance assessed with Wilcoxon's signed-rank test and \(p<0.01\)).}
\begin{tabu} to 1.0\linewidth{ X[2.8l] | X[1.0c] X[1.0c] X[1.0c] X[1.2c] X[1.2c] | X[1.2c] | X[1.0c] | X[1.6c] | X[1.6c] }
\toprule
& FP & FN & IS & FPdiv & FNdiv & div & sum & \multicolumn{1}{c|}{DET} & TRA\\
\toprule
\multicolumn{10}{c}{\textbf{mskcc-confocal} 270 frames}\\
\midrule
Starrynite
& 7.9& 13& 0.62& 0.58& 1.2& 1.8& 24 & 0.97875 & 0.97495\\
linajea
& \textbf{3.6}& \textbf{5.5}& \underline{0.062}& 0.89& \textbf{0.26}& 1.2& 10.3 & 0.99514 & 0.99418\\
lin.+csc+sSVM
& \underline{3.7}& \underline{5.6}& \textbf{0.046}& \textbf{0.053}& 0.40& \textbf{0.46}& \textbf{9.6} & \textbf{0.99570} & \textbf{0.99480}\\
\midrule
\multicolumn{10}{c}{\textbf{nih-ls} 270 frames}\\
\midrule
Starrynite
& 22 & 18 & 2.4 & 0.66 & 1.6 & 2.2 & 45 & 0.81850 & 0.81114 \\
linajea
&\textbf{12} & \underline{6.5}& \textbf{0.46}& 1.5 & \textbf{0.40}& 1.86& \underline{21} & 0.99367 & 0.99279\\
lin.+csc+sSVM
&\underline{13} & \textbf{5.3}& \underline{0.59}& \textbf{0.20}& \underline{0.49}& \textbf{0.69}& \textbf{20} & \textbf{0.99511} & \textbf{0.99433}\\
\end{tabu}
\label{tab:nuclei_results}
\end{center}
\end{table*}
\paragraph{The \textbf{mskcc-confocal} dataset}
\label{subsec:conf_data}
consists of three fully annotated 3d+time an\-iso\-tro\-pic confocal recordings (data: \url{https://doi.org/10.5281/zenodo.6460303}).
The ground truth has been created using Starrynite~\cite{bao06_autom_cell_lineag_tracin_caenor_elegan,santella14_semi_local_neigh_based_framew}, followed by manual curation (supported and verified by using the fixed \textit{C. elegans} lineage).
The annotations include the polar bodies (marked separately).
We report the uncurated Starrynite results as a baseline.
As we perform weakly-supervised training on point annotations yet Elephant requires segmentation masks we cannot compare to it on this data.
We train and evaluate all models on the first 270 frames (approximately 570 cells in the last frame and 52k in total per sample).
Per experiment we use one recording each as training, validation and test set.
We do this for all six possible combinations. For each combination, we perform three experimental runs, starting from different random weight initializations, leading to a total of 18 experimental runs (all numbers averaged).
Divisions that are off by one frame compared to the annotations are not counted as errors as the limited frame rate leads to inherent inaccuracies in the data and annotations.
Both Linajea and our extended method considerably outperform Starrynite (see Table~\ref{tab:nuclei_results}, and Suppl.\ Fig.\ \ref{fig:error_plot} for the respective box plots).
We conducted an ablation study on the \textbf{mskcc-confocal} dataset to measure the effect of the individual extensions we propose (see Table~\ref{suppl_tab:ablation}):
We report results without the ILP, and without the cell state classifier in the ILP (this matches~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}).
Both strongly suffer from false positive (FP)-type errors.
We compare results with and without sSVM for weights tuning, and find that sSVM-determined weights yield competitive results.
The sSVM finds similar weights for all experimental runs (see Suppl. Fig.\ \ref{fig:ssvm_param_dist}).
Finally we employ the polar body filter (we remove them from the ground truth, too).
This reduces FP errors.
Due to the strong tree structure of the tracks the notion of ``how many tracks are correct'' is not well defined.
To still try to quantify the intention behind it we evaluate the fraction of error-free tracklets of varying lengths~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}, see Suppl.\ Fig.\ \ref{fig:corr_segments}.
\noindent\textbf{Discussion. } We did not expect to see large differences between the results for the sSVM-de\-ter\-mined weights and for the manually configured grid search as we have gathered experience in choosing appropriate parameters for the weights grid search for this data.
Thus the explicit search is often faster as it can be parallelized trivially.
However, for other data, where this information is not at hand, the targeted sSVM is very convenient and is computationally more efficient.
Interestingly, depending on the weights, the system appears to be able to exchange FP and FN errors.
The sSVM-determined weights seem to prioritize FP errors.
By adapting the cost function \(\Delta\) one can modulate this depending on respective application-specific needs (see Suppl.\ Table~\ref{suppl_tab:delta_res}).
\begin{table*}[tbp]
\centering
\caption{Ablation study on
the \textbf{mskcc-confocal} data. We ablate solving an ILP altogether (ILP), incorporating the cell state classifier (csc), employing an sSVM for weights search (ssvm), and incorporating the polar body filter (pbf).}
\begin{tabu} to 1.0\linewidth{ X[0.5c] X[0.5c] X[0.5c] X[0.7c] | X[0.5c] X[0.5c] X[0.9c] X[1.0c] X[1.0c] | X[1.0c] | X[0.7c] | X[1.2c] | X[1.2c]}
\toprule
ILP & csc & ssvm & pbf & FP & FN & IS & FPdiv & FNdiv & div & \multicolumn{1}{c|}{sum} & DET & TRA\\
\toprule
\multicolumn{12}{c}{\textbf{mskcc-confocal} 270 frames}\\
\midrule
\xmark & \xmark & \xmark & \xmark &
5.0& \textbf{4.6}& \underline{0.048}& 1.6& \textbf{0.25}& 1.9& 11.6 & \underline{0.99567}& 0.99464 \\
\cmark & \xmark & \xmark & \xmark &
\underline{3.6}& 5.5& 0.062& 0.89& \underline{0.26}& 1.2& 10.3 & 0.99514 & 0.99418\\
\cmark & \cmark & \xmark & \xmark &
\underline{3.4}& 5.7& \textbf{0.028}& 0.11& \underline{0.27}& \textbf{0.38}& \underline{9.5} & 0.99526 & 0.99437\\
\cmark & \cmark & \cmark & \xmark &
\underline{3.7}& 5.6& \underline{0.046}& \underline{0.053}& 0.40& 0.46& \underline{9.6} & \textbf{0.99570} & \textbf{0.99480}\\
\cmark & \cmark & \cmark & \cmark &
\textbf{2.5}& 5.5& \underline{0.047}&\textbf{0.048}& 0.39& 0.44 &\textbf{8.5} & \underline{0.99533} & \underline{0.99441}
\\
\bottomrule
\end{tabu}
\label{suppl_tab:ablation}
\end{table*}
\paragraph{The \textbf{nih-ls} dataset}
\label{subsec:ls_data}
contains three fully annotated 3d+time isotropic lightsheet recordings~\cite{Moyle2021} (data: \url{https://doi.org/10.5281/zenodo.6460375}).
Our experimental setup is similar to \textbf{mskcc-confocal}.
\noindent\textbf{Discussion. }It is interesting to compare our results on \textbf{nih-ls} and \textbf{mskcc-confocal}:
Due to the isotropic resolution of \textbf{nih-ls} we expected the results to be superior, yet the error metrics we observe do not support this intuition.
A closer look at qualitative results reveals some clues that may explain part of it:
Apoptotic cells are more distinct and visible earlier in \textbf{nih-ls} (see e.g. Suppl.\ Fig.\ \ref{fig:pol_apo_bodies}) and thus have not been annotated in the ground truth.
Yet in the current state our model does not handle this transition explicitly and thus continues to track them temporarily, leading to a larger number of false positives, as indicated by the quantitative results.
As we already have a cell state classifier as part of our model, it will be straightforward to add apoptotic cells as a remedy
\section{Introduction}
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{figures/overview_figure3.pdf}
\caption{\label{fig:method} Method overview: We use a 4d U-Net to predict cell candidates and movement vectors. These are used to construct a candidate graph \(G\) with node and edge scores \(g_s\) as in~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}. We propose to integrate learnt cell state scores \(cs_s\). Graph \(G\), feature matrix \(S\), weights \(w\) and a set of feasibility constraints form an ILP that yields the cell lineage. We propose to find \(w\) via a structured SVM (sSVM); proposed changes highlighted in magenta; figure adapted from~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}.
}
\end{figure}
Advances in microscopy have made the recording of whole embryo development possible~\cite{keller2010fast,krzic2012multiview}.
However there is an inherent trade-off between frame rate, resolution and the prevention of phototoxicity~\cite{Weigert2018_CARE}.
While it is possible to capture high signal-to-noise images with high resolution, this can damage the organism, especially during early embryonic development.
Consequently, embryonic development is commonly captured at low frame rate and with low signal-to-noise ratio (SNR). Together with cell-cycle inherent signal fluctuation, low SNR renders automated cell detection challenging.
Furthermore, low frame rate renders automated tracking challenging as overlap-based tracking approaches are not applicable. Last but not least, the similar shape and appearance of distinct cell nuclei also renders similarity-based tracking approaches ineffective.
A number of automated cell tracking approaches have been developed to tackle reduced SNR as well as frame rates on the order of minutes.
Such methods have enabled a range of studies on a variety of organisms, where it wouldn't have been feasible to do tracking manually
\cite{li2019_syste_probe_and_spati_regul_of_cell_posit_varia_durin_embry,murray08_autom_analy_embry_gene_expres,cao2020establishment,medeiros2021_multisc_light_sheet_organ_imagi_frame,Wolff2018,guignard20_conta_area_depen_cell_commu_and_the_morpho_invar_of_ascid_embry}.
The \emph{Cell Tracking Challenge} (CTC)~\cite{ulman17_objec_compar_cell_track_algor}, an extensive benchmark that contains 2d+time and 3d+time datasets of different organisms recorded with a variety of microscopes, allows for a quantitative comparison of automated cell tracking methods.
Current methods for cell tracking in \textit{C.\ elegans} follow the \emph{tracking-by-detection} paradigm, which first computes (candidate) cell detections in all frames, and in a second step links matching cell detections across frames.
To this end, Starrynite~\cite{bao06_autom_cell_lineag_tracin_caenor_elegan}, which is widely used by practitioners, uses classical computer vision to detect locations of maximum signal in each frame, nearest neighbor matching for link detection, and local post-processing to resolve ambiguities that occur in case of cell divisions.
Linkage can also be achieved in a globally optimal manner by means of combinatorial optimization, as in the competitive method Baxter~\cite{magnusson15_global_linkin_cell_track_using_viter_algor} (former rank 2 on the CTC \textit{C.\ elegans} benchmark), which employs the Viterbi algorithm in the linkage step.
Related methods can also directly yield an optimal feasible cell lineage tree, in the face of over- and underdetections~\cite{schiegg2013conservation}, as well as for an overcomplete set of candidate detections~\cite{jug2014optimal,Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}.
To modernize the detection step, Cao et al.\ \cite{cao2020establishment} replaced the classical detection step of Starrynite by neural network-based cell segmentation.
Similarly, the recently proposed active learning framework Elephant~\cite{sugawara2021_track_cell_linea_in_3d_by_incre_deep_learn} (former rank 1 on the CTC \textit{C.\ elegans} benchmark) uses modern deep learning for segmentation and detection. However, both perform local linkage via nearest neighbor search (in case of Elephant optionally supported via a learned optical flow estimate).
We propose a method that unifies the individual advantages of Baxter~\cite{magnusson15_global_linkin_cell_track_using_viter_algor} and Elephant~\cite{sugawara2021_track_cell_linea_in_3d_by_incre_deep_learn}.
To this end we build upon Linajea~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}, a recent method that combines deep learning and combinatorial optimization for cell tracking.
We propose extensions of Linajea to capture properties specific to recordings of the model organism \emph{C.\ elegans}, namely relatively many cell divisions, and the presence of \emph{polar bodies}
which look similar to nuclei. Our extended method yields state of the art accuracy on benchmark \emph{C.\ elegans} data.
Besides accuracy, we also address efficiency.
We propose to use a structured SVM (sSVM) to facilitate the tuning of the weights of the ILP objective which alleviates the need for manual configuration of a grid search.
This is particularly useful in light of our extensions as they introduce additional weights and thereby would increase the dimensionality of the search.
In summary our contributions are:
\begin{itemize}
\item A learnt cell state and polar body detector, integrated into an existing approach that combines deep learning and an ILP for nuclei tracking.
\item Fully automated tuning of the weights of the ILP objective via a sSVM.
\item The new state-of-the-art for \emph{C.\ elegans} in the Cell Tracking Challenge.
\item Two new datasets made publicly available as benchmark data, namely three con\-fo\-cal and three lightsheet recordings of \textit{C.\ elegans}, all fully annotated.
\end{itemize}
\subsubsection*{Acknowledgments}
We would like to thank Anthony Santella, Ismar Kovacevic and Zhirong Bao and Ryan Christensen, Mark W. Moyle and Hari Shroff for providing us with their data and annotations, for generously allowing us to make the data public and for valuable information and feedback.
P.H. was funded by the MDC-NYU exchange program and HFSP grant RGP0021/2018-102. P.H. and D.K. were supported by the HHMI Janelia Visiting Scientist Program. A.S. was supported by grant 2019-198110 (5022) from the Chan Zuckerberg Initiative and the Silicon Valley Community Foundation.
\clearpage
{
\bibliographystyle{splncs04}
\section{Method}
Our method extends the tracking-by-detection approach Linajea~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}.
We briefly review Linajea, and then describe our extensions in detail. For more details on Linajea, please refer to~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}.
For an overview of our extended method, see Fig.~\ref{fig:method}.
\paragraph{\textbf{Linajea. }}
\label{subsec:linajea}
Linajea~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot} implements a 4d U-Net~\cite{ronneberger15_u_net,ccicek16_u_net,funke_conv4d} to predict the position and movement of each nucleus.
Position is encoded as a single-channel image of Gaussian-shaped blobs, one per nucleus, as in~\cite{hofener2018deep}.
The locations of the respective intensity maxima correspond to nuclei center points.
Movement is encoded as 3d vectors per pixel within a nucleus. Each vector points to the spatial location (center point) of the same (or parent) nucleus in the previous time frame.
Note, the backwards direction of the movement vectors simplifies tracking as cells can only divide going forward but cannot merge.
The four output channels necessary for the above encoding are trained jointly via L2 loss.
During inference, local maxima of the predicted position map serve as cell candidates.
An integer linear program (ILP) is employed to select and link a feasible subset of these cell candidates. To this end, a candidate graph is established, where nodes represent cell candidates, and edges represent cell linkage candidates. Node- and edge costs are derived from the U-Net's position- and movement prediction channels, respectively.
Solving the ILP assigns a label "selected" or "not selected" to each candidate cell (node) and each candidate link (edge) such that a cost-based objective is minimized. Linear constraints on the node- and edge labels ensure that a valid tracking solution is extracted from the graph, i.e., a binary forest where each tree only branches forward in time.
Linajea's objective is a weighted sum of the costs of the selected nodes and edges. There are four tunable weights: A constant cost \(w_{\text{node-sel}}\) for selecting a node, a factor \(w_{\text{node-cost}}\) to scale the position prediction, a factor \(w_{\text{edge-cost}}\) to scale the distance between predicted movement vector target and linked cell position, and a constant cost \(w_{\text{track-cost}}\) for each track. The track cost is incorporated via an additional binary node label "track start", with appropriate constraints to ensure consistency with the selection labels.
Linajea performs grid-search within a manually defined range to find a set of suitable weights.
We propose two extensions to Linajea: (1) An additional network to classify cell state,
and respective additional costs and feasibility constraints in the ILP, and (2) the use of a structured SVM (sSVM) to automatically find the optimal weights for the ILP objective, as described in the following.
\paragraph{\textbf{Cell State Classifier. }}
\label{subsec:cell_state}
We propose to incorporate a classifier to determine the cell state of each cell candidate similarly to~\cite{santella14_semi_local_neigh_based_framew}.
We assign to each candidate one of four classes: parent cell (i.e., a cell that is about to undergo cell division), daughter cell (cell that just divided), continuation cell (cell track that continues without division) and polar body.
We train a 3d ResNet18~\cite{he15_deep_resid_learn_image_recog} with 3d convolutions for this task.
The parent/daughter/continuation classes are incorporated into the ILP as a separate set of node labels, with their weighted prediction scores as costs.
In addition to Linajea's feasibility constraints on node and edge selection~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}, we impose novel constraints that ensure that (1) selection- and cell state labels are consistent, and (2) a parent at time \(t\) can only be linked to a daughter at time \(t\)+1 and vice versa.
Formally, for each edge \(e=(u,v)\) between nodes \(u\) and \(v\) in the graph (directed forward in time), let \(y_{\text{edge},e}, \ y_{\text{node},u} \in \{0,1\}\) denote binary variables that represent edge- and node selection as in Linajea. We introduce novel binary variables \(y_{\text{daughter},u}\), \(y_{\text{parent},u}\), \(y_{\text{continue},u}\) \(\in \{0,1\}\) that represent daughter-, parent- and continuation cell state labels per node.
To ensure that a selected node is assigned exactly one cell state, for each node, we introduce the linear equality constraint \(y_{\text{parent},u} + y_{\text{daughter},u} + y_{\text{continue},u} - y_{\text{node},u} = 0\) to be included in the ILP. To ensure that selected edges constitute feasible parent-daughter links, for each edge, we introduce novel inequality constraints \( y_{\text{parent},u} + y_{\text{edge},e} - y_{\text{daughter},v} \leq 1\) and \( y_{\text{daughter},v} + y_{\text{edge},e} - y_{\text{parent},u} \leq 1\) to be included in the ILP. Thus, e.g., if node \(u\) is labelled parent (\(y_{\text{parent},u} =1\)), and edge \(e\) is selected (\(y_{\text{edge},e} =1\)), node \(v\) has to be labelled daughter (\(y_{\text{daughter},v} =1\)).
We add the cell state predictions to Linajea's objective. This entails new weights \(w_{\text{parent}}, w_{\text{daughter}}, w_{\text{continue}}\) that serve to scale the prediction scores of respectively selected and labelled nodes. A further weight \(w_{\text{division}}\) serves as constant division cost, contributed by each selected parent.
By default, we do not perform any postprocessing on the tracks (such as removal of short tracks).
We propose one exception regarding the polar bodies:
Depending on the specific study they might not be of interest, and even if, they are often not contained in the ground truth tracks as they are not considered ``proper'' cells.
That is why we add them as an additional class to our cell state classifier.
The score for the polar body class can be used to optionally detect and remove them from the tracks. Suppl.~Fig.~\ref{fig:pol_apo_bodies} shows an exemplary polar body.
\paragraph{\textbf{Structured SVM-based weights search.}}
\label{subsec:ssvm}
Manually configured grid search for optimal weights as in~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot} can be costly and generally a dataset specific search range has to be found for each new dataset.
To alleviate the need for this manual step, we propose the use of a structured SVM (sSVM) for automatic weight selection~\cite{joachims09_predi_struc_objec_with_suppo_vecto_machin,teo10_bundl_metho_for_regul_risk_minim}.
Following~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}, our extended objective can be phrased as
\begin{equation}
\min_{\mathbf{y}} \langle S\mathbf{w},\mathbf{y} \rangle \quad s.t. \quad G(\mathbf{y}) \in \mathcal{F}_2,
\label{eq:ilp_cost}
\end{equation}
where \(\mathbf{y}\) is
a vector of all binary indicator variables, including our new ones
\[\mathbf{y} = \big[\mathbf{y}_{\text{node}}^T,\mathbf{y}_{\text{track}}^T,\mathbf{y}_{\text{parent}}^T,\mathbf{y}_{\text{daughter}}^T,\mathbf{y}_{\text{continue}}^T, \mathbf{y}_{\text{edge}}^T\big]^T \in \{0, 1\}^{5|V|+|E|}. \]
\(G(\mathbf{y})\) denotes the graph formed by the selected nodes and edges in \(\mathbf{y}\). \(\mathcal{F}_2\) denotes the set of all feasible binary forests.
\(S^{\text{dim}(y)\times \text{dim}(w)}\) is a sparse feature matrix that contains all node- and edge features. It has the following columns:
(1) 1 for \textit{node} indicators (i.e., in the first \(|V|\) rows), and 0 otherwise.
(2) Candidate cell prediction for \textit{node} indicators, and 0 otherwise.
(3) 1 for \textit{track start} indicators, 0 otherwise.
(4) 1 for \textit{parent class} indicators, 0 otherwise.
(5-7) Parent/daughter/continue class predictions for \textit{parent/daughter/continue} indicators, 0 otherwise.
(8) Edge cost
for \textit{edge} indicators, and 0 otherwise.
\(\mathbf{w}\) is the vector of weights
\[
\mathbf{w} = \big[w_{\text{node-sel}}, w_{\text{node-score}}, w_{\text{track}}, w_{\text{div}}, w_{\text{parent}}, w_{\text{daughter}}, w_{\text{continue}}, w_{\text{edge}} \big]^T.
\]
Solving the ILP~\eqref{eq:ilp_cost} yields the best feasible \(\mathbf{y}\) given some \(\mathbf{w}\).
However, appropriate values for \(\mathbf{w}\) are unknown a priori.
With the help of the ground truth annotations, what we can determine though is a ``best effort'' indicator vector \(\mathbf{y}'\).
This equates to the best possible feasible solution given the set of predicted cell candidates and movement vectors.
We thus want to find the weights \(\mathbf{w}\) such that solving \eqref{eq:ilp_cost} yields \(\mathbf{y}=\mathbf{y}'\), or as close as possible to it.
To find such weights, given \(\mathbf{y}'\), we derive a modified objective from Eq.~\ref{eq:ilp_cost} which we then minimize w.r.t.\ the weights.
We thus follow the sSVM approach with a loss-augmented objective~\cite{joachims09_predi_struc_objec_with_suppo_vecto_machin,teo10_bundl_metho_for_regul_risk_minim}. Formally, we seek a \(\mathbf{w}\) that minimizes
\begin{equation} L(\mathbf{w}) = \langle S\mathbf{w},\mathbf{y}'\rangle - \min_{\mathbf{y}:\ G(\mathbf{y}) \in \mathcal{F}_2} \big( \langle S\mathbf{w},\mathbf{y}\rangle - \Delta(\mathbf{y}',\mathbf{y}) \big) + \lambda | \mathbf{w} | ^2 \ ,
\end{equation}
with Hamming cost function \(\Delta\) to measure the deviation of the optimal \(\mathbf{y}\) for a given \(\mathbf{w}\) and the best effort \(\mathbf{y}'\), and a hyperparameter \(\lambda\geq 0\) for weighing L2 regularization on the weights (we use \(\lambda=0.001\)).
To give a brief intuition why optimizing this loss yields the desired parameters (for which we neglect the L2 regularizer for the moment): It is easy to see that \(L(\mathbf{w}) \geq 0\) because \(\Delta \geq 0\)
and
\(\min_{\mathbf{y}:\ G(\mathbf{y}) \in \mathcal{F}_2} \langle S\mathbf{w},\mathbf{y}\rangle \leq \langle S\mathbf{w},\mathbf{y}'\rangle\).
Furthermore, if \(\mathbf{w}\) yields \(\mathbf{y}'\) as minimum of the ILP, \(\arg\!\min_{\mathbf{y}:\ G(\mathbf{y}) \in \mathcal{F}_2} \langle S\mathbf{w},\mathbf{y}\rangle = \mathbf{y}'\), then \(L(\mathbf{w}) = 0\), i.e., the loss is minimized. Last but not least, if a \(\mathbf{w}\) with zero loss does not exist, the loss seeks a \(\mathbf{w}\) that yields an ILP-minimizing \(\mathbf{y}\) that is at least ``close'' to \(\mathbf{y}'\) both in terms of the Hamming loss and in terms of its objective value \(\langle S\mathbf{w},\mathbf{y}\rangle\).
For details on the sSVM optimization procedure, please refer to~\cite{joachims09_predi_struc_objec_with_suppo_vecto_machin,teo10_bundl_metho_for_regul_risk_minim}.
\section{Appendix}
\label{sec:appendix}
\FloatBarrier
\begin{center}
\captionof{table}{Quantitative results on \textbf{mskcc-confocal} data with modified \(\Delta\) cost function. The cost for each FP/FN is multiplied by 10/100 respectively.}
\begin{tabu} to 1.0\linewidth{ X[2.8l] | X[1.0c] X[1.0c] X[1.0c] X[1.2c] X[1.2c] | X[1.2c] | X[1.0c] }
\toprule
& FP & FN & IS & FPdiv & FNdiv & div & sum\\
\toprule
\multicolumn{8}{c}{\textbf{mskcc-confocal} 270 frames}\\
\midrule
lin.+csc+sSVM
& 3.7& 5.6& 0.046& 0.053& 0.40& 0.46& 9.6\\
\midrule
+FN*10
& 3.8& 5.1& 0.11& 0.33& 0.30& 0.62& 9.6\\
+FN*100
& 4.4& 4.9& 0.14& 0.95& 0.24& 1.1& 10.6\\
\midrule
+FP*10
& 3.1& 5.8& 0.03& 0.041& 0.50& 0.54& 9.5\\
+FP*100
& 2.8& 10.6& 0.011& 0.031& 0.80& 0.83& 14\\
\bottomrule
\end{tabu}
\label{suppl_tab:delta_res}
\end{center}
\begin{center}
\centering
\captionof{table}{\label{suppl_tab:metrics}Description of error metrics used for evaluation:}
\begin{tabu} to 0.9\linewidth{ X[2.0c] | X[10.0l] }
\toprule
FP/FN & false positive/negative edge\\
IS & identity switch/cross-over of tracks\\
FP/FNdiv & false positive/negative division\\
div & sum of division errors\\
sum & sum of all errors (incl. divisions)\\
DET & weighted, normalized score over how many false positive detections have to be deleted and false negatives have to be added to get from the prediction to the ground truth.\\
TRA & similar to DET but with additional terms for the addition, deletion and modification of links between objects\\
\end{tabu}
\end{center}
\begin{center}
\captionof{table}{Information about implementation and computational aspects (for more details see \url{https://github.com/funkelab/linajea}):}
\begin{longtable}{ p{.30\textwidth} p{.70\textwidth} }
\toprule
& Description\\
\midrule
Tracking Network & 4d U-Net, 3 levels; valid padding; starting with 12 feature maps; quadrupled at each level; separable transposed convolutions for upsampling; if anisotropic, don't downsample anisotropic dimension until voxel size across dimensions is roughly isotropic\\
& Input size (in voxels): [7, 40, 148, 148] (\textit{mskcc-confocal}), [7, 148, 148, 148] (\textit{nih-ls})\\
& Pixels outside of cell (determined by some roughly estimated radius) trained with loss factor \(0.01\) (\textit{mskcc-confocal})/ \(0.000001\) (\textit{nih-ls})\\
& At every iteration a random tile (of size input size) is selected: select a random cell; place input tile around it such that the cell is contained in it, at a random location within tile; depending on the location there might be many neighboring cells included, cells with more neighbors will be sampled less often, as they will also be included if their neighbors are sampled; at least 25\% of iterations have to contain a division\\
& Adam optimizer; batch size 1; learning rate 0.00005; weighted MSE loss; automatic mixed precision\\
& Trained for 400k iterations (last checkpoint used for prediction)\\
& Stochastic weight averaging every 1k after 50k\\
& Augmentations: elastic, scale, intensity, flipping, shift (of center frame)\\
& During inference maxima with a score below \(0.2\) are discarded\\
\midrule
Cell State Network & 3d ResNet18\\
& Input size (in voxels): [5, 8, 64, 64] (\textit{mskcc-confocal}), [5, 32, 32, 32] (\textit{nih-ls})\\
& Adam optimizer; batch size 64 (\textit{mskcc-confocal}), 16 (\textit{nih-ls}); learning rate 0.0005, 0.000005 after 20k iterations; cross entropy or focal loss; automatic mixed precision\\
& Dropout; no batch normalization; global average pooling\\
& Trained for 60k iterations; best checkpoint selected via validation\\
& No test time augmentation\\
& No stochastic weight averaging\\
& trained on one embryo; validated on a second embryo; inference run on cells predicted by the tracking network on third embryo (same combination for both networks)\\
& Augmentations: elastic, scale, intensity, flipping, jitter, noise (s\&p, speckle, only for \textit{mskcc-confocal})\\
\midrule
Postprocessing & Block-wise processing: tiled only temporally (multiple whole frames) \\
& After a reasonable range for values for weights had been determined validate on/search a set of 50 different combinations per experiment to find optimal ones\\
& Ground truth annotation and predicted cell have to be closer than 15 (in world units/isotropic size) to be able to count as a match\\
& Set of weights selected that results in the lowest sum of errors on the validation set\\
\midrule
Runtime & U-Net: training around 7 days, prediction a couple of hours\\
(very roughly, & ResNet: training 1 day, prediction 1 hour\\
on a single node, & Generating edges: 5min\\
steps after training & Solving with one set of ILP weights: 15min\\
can be parallelized) & Evaluation: 1-5min (depending on what scores should be computed)\\
\midrule
Hardware & Trained on one machine with one V100 (for anisotropic data a smaller GPU is sufficient, too)\\
& GPU only needed for training and prediction\\
& Weight search can be parallelized trivially and profits from many CPU cores\\
\bottomrule
\end{longtable}
\label{suppl_tab:impl}
\end{center}
\begin{minipage}{\textwidth}
\centering
\hspace*{\fill}%
\includegraphics[width=0.45\textwidth]{figures/polar_bodies}%
\hfill%
\includegraphics[width=0.45\textwidth]{figures/apoptotic_bodies}%
\hspace*{\fill}%
\captionof{figure}{\label{fig:pol_apo_bodies}Examples of polar and apoptotic bodies in \textit{C. elegans}}
\end{minipage}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/plot_errors2.png}
\captionof{figure}{\label{fig:error_plot}Box plot of the errors of the different approaches on \textbf{mskcc-confocal}.
Starrynite~\cite{bao06_autom_cell_lineag_tracin_caenor_elegan} is an often used method in the analysis of \textit{C. elegans}. \textit{greedy} refers to the method without the ILP. \textit{linajea} matches the prior work~\cite{Malin-Mayor2021_autom_recon_of_whole_embry_cell_linea_by_learn_from_spars_annot}. \textit{linajea+csc+ssvm} is our full method with automatically determined ILP weights. Each step lowers the number of errors. \textit{greedy} lowers especially the number of \textit{FP} and \textit{FN} edges, not as much the number of false divisions. The ILP on its own (\textit{linajea}) can already lower the number of false divisions a bit, but the inclusion of the classifier in \textit{linajea+csc+ssvm} lowers them drastically. For the quantitative numbers see Table~\ref{suppl_tab:ablation}.
}
\end{minipage}
\begin{minipage}{\textwidth}
\centering
\hspace*{\fill}%
\includegraphics[width=0.45\textwidth]{figures/mskcc_segments}%
\hfill%
\includegraphics[width=0.45\textwidth]{figures/nih_segments}%
\hspace*{\fill}%
\captionof{figure}{\label{fig:corr_segments}Fraction of correct tracklets of varying length for \textit{mskcc-confocal} (left, improvement of 3 percentage points at 100 frames and 6 at 200 frames of linajea+csc+ssvm over linajea) and \textit{nih-ls} (right, improvement of 5 percentage points at 100 frames and 2 at 200 frames over linajea), \(p<0.01\). Computed using a sliding window approach; for each window size in the range \([1, 200]\) we determine for every possible tracklet of that length if it is error-free or not and compute the fraction of fully correct ones.}
\end{minipage}
\begin{minipage}{\textwidth}
\centering
\includegraphics[width=\textwidth]{figures/ssvm_param_distribution.png}
\captionof{figure}{\label{fig:ssvm_param_dist}Box and whisker plot of the distribution of the automatically determined ILP weights over the 18 experimental runs of the \textbf{mskcc-confocal} dataset. The sSVM finds similar values for each respective candidate graph and with a similar ratio to each other.
\end{minipage}
|
1,116,691,498,484 | arxiv | \section{\textcolor{black}{Introduction}}
\textcolor{black}{With the development of the Fifth Generation (5G)
wireless communication systems, high-mobility scenarios such as high-speed
rail and Vehicle-To-everything (V2x) communications have gained increasingly
more interest. Due to the high-speed relative motion between the transmitter
and receiver, the transmitted signal, propagating through multiple
different paths, arrives at the receiver with different Doppler frequency
offsets (DFOs), thus resulting in a fast time-varying multipath fading
channel. In this case, the link performance such as achievable data
rate will be degraded significantly due to the Doppler-induced channel
aging effect \cite{Souden2009Robust}. To overcome this challenge,
several methods have been proposed in previous works, as elaborated
below. }
\textbf{\textcolor{black}{Direct Channel Estimation/Prediction: }}\textcolor{black}{For
line-of-sight (LOS) channels with a single LOS path, it is relatively
easy to compensate the Doppler effect and resolve the channel aging
issue by estimating the DFO parameter of the LOS path. However, when
there are multiple different paths due to rich scattering, it is challenging
to compensate the Doppler effect because different paths with different
DFOs are mixed together in the received signal. Nevertheless, some
works have proposed to directly estimate the fast time-varying channels
in the time/frequency domain \cite{Berger2010Application,Wang2018Channel}.
Existing channel estimators can be classified into two types. The
first type approximates time-varying channels using a linearly time-varying
(LTV) channel model \cite{Mostofi2005ICI,Liu2015On}. For example,
a hybrid frequency/time-domain channel estimation algorithm is proposed
in \cite{Mostofi2005ICI} based on the LTV model and two methods are
introduced to mitigate the Doppler effect. However, this algorithm
introduces a processing delay of at least one Orthogonal Frequency
Division Multiplexing (OFDM) symbol. The second type of estimators
adopt the basis expansion model (BEM) \cite{Xin2014Study} to convert
the problem of estimating the channel impulse response (CIR) to that
of estimating the basis function weights \cite{Al2010A}. For example,
in \cite{Al2010A}, the channel estimation and Doppler mitigation
are jointly considered by exploiting the correlations in time and
frequency domains, and the basis function coefficients are estimated
via the linear minimum mean squared error (LMMSE) approach. However,
accurate knowledge on the maximum DFO is required to determine the
minimum order of basis function and the computational burden is also
heavy for multi-antenna systems \cite{8558718}. Moreover, the BEM
inevitably introduces approximation error to channel estimation due
to the imperfect model assumed.}
\textbf{\textcolor{black}{Orthogonal Time Frequency Space (OTFS) Modulation:
}}\textcolor{black}{OTFS modulation \cite{Monk2016OTFS} is an emerging
technique which is able to handle the fast time-varying channels.
This method modulates transmitted symbols in the delay-Doppler domain
instead of time/frequency domain as in traditional modulation techniques
such as OFDM.. The idea is to transform the time-varying channel into
a time-invariant channel in the delay-Doppler domain. Early works
on OTFS modulation focused on the single-input single-output (SISO)
systems \cite{Monk2016OTFS}. Later, OTFS is extended to multiple-input
multiple-output (MIMO) systems by transmitting consecutive impulses
with proper guard time between two adjacent ones to distinguish different
base station (BS) antennas \cite{Ramachandran2018MIMO}. However,
the channel estimation method in \cite{Ramachandran2018MIMO} cannot
be directly applied to massive MIMO system since a large number of
antennas are required to be distinguished by transmitting such impulses,
which will lead to large pilot overhead.}
\textbf{\textcolor{black}{Angular-Domain DFO Estimation and Compensation:}}\textcolor{black}{{}
Since the different DFOs of multiple paths are resulted from their
different angles of arrival (AoAs)/angles of departure (AoDs) at the
receiver, they can be separated in the angle domain via spatial processing.
As such, angular-domain DFO estimation and compensation is another
popular approach to address the Doppler-induced channel aging issue
\cite{Souden2009Robust,Bellili2014A}. For MIMO systems, prior work
\cite{Chizhik2004Slowing} first pointed out that channel time-variation
can be slowed down through beamforming with a large number of transmit/receive
antennas. Motivated by this, a small-scale uniform circular antenna-array
(UCA) is adopted in \cite{Zhang2012Multiple,Guo2013Multiple} to separate
multiple DFOs via array beamforming. However, the DFO compensation
methods in \cite{Zhang2012Multiple,Guo2013Multiple} only apply to
scenarios with very sparse channels due to the limited spatial resolution
of small-scale MIMO. Recently, some works have exploited high-spatial
resolution provided by massive MIMO to address the high-mobility induced
challenges \cite{Liu2014On,Chen2017Directivity,Guo2017High,Guo2018High,Guo2018Angle}.
For example, the authors of \cite{8558718} propose to separate the
DFOs in angular-domain by beamforming with a large-scale uniform linear
antenna-array (ULA) at the mobile user side. After estimating and
compensating the DFO in each angle, the resultant quasi time-invariant
channel can be estimated more efficiently. However, the signaling
overhead for the maximum likelihood (ML) based joint estimation of
the massive MIMO channel matrix and DFO parameters is extremely high.
Moreover, only a single data stream is transmitted from the BS to
the mobile user via all possible channel directions, and thus it cannot
enjoy the spatial multiplexing gain as well as the huge array signal-to-noise
ratio (SNR) gain provided by massive MIMO. }
\textcolor{black}{The above works have focused on channel estimation.
It is also possible to directly search for the best beamforming vectors
without explicit channel estimation. For example, in \cite{Wang2009Beam},
the authors propose an exhaustive search (ES) scheme, which examines
all beam pairs in the codebook and determines the best pair that maximizes
a given performance metric (e.g., beamforming gain). To reduce the
training overhead of the ES scheme, the hierarchical search (HS) scheme
proposed in \cite{Alkhateeb2017Channel} utilizes hierarchical codebooks
and has a favorable performance at low SNRs. However, these codebook-based
beam search methods suffer from the quantization error caused by the
codebook and the channel aging effect. }
\textcolor{black}{In this paper, we consider high-mobility mmWave
massive MIMO systems, where both the BS and users are equipped with
massive antenna arrays to facilitate Doppler compensation, improve
the spectrum and energy efficiency, as well as overcome the large
path loss at high frequency band. As such, combining the mmWave and
massive MIMO technologies has the potential to significantly improve
the capacity and reliability of high-mobility wireless communications.
However, it is very challenging to design an efficient channel estimation
scheme. For example, since both ends have massive MIMO, the dimension
of the channel matrix is huge and conventional downlink/uplink channel
estimation or codebook-based beam search will lead to large signaling
overhead. Although various pilot overhead reduction methods such as
those based on compressive sensing (CS) have been proposed for the
estimation of slow massive MIMO fading channels \cite{Gao2016Channel}\cite{Bajwa2010Compressed},
they do not consider the Doppler effect and thus cannot be applied
to the high mobility scenario. To overcome this challenge, we propose
a novel angular-domain selective channel tracking and Doppler compensation
scheme, which exploits the dynamic sparsity of the mmWave massive
MIMO channel as well as precoded training in both downlink and uplink
to significantly reduce the signaling overhead. The main new contributions
of our paper are given as follows.}
\begin{itemize}
\item \textbf{\textcolor{black}{Angular-Domain Selective Channel Tracking:
}}\textcolor{black}{We propose a selective channel tracking scheme
to only estimate partial angular--domain channel parameters at the
user side that are sufficient for Doppler effect compensation to significantly
reduce the pilot overhead. Moreover, we propose an efficient downlink
training vector design at the BS side to strike a balance between
}\textit{\textcolor{black}{exploitation}}\textcolor{black}{{} of most
promising channel directions for array (SNR) gain and }\textit{\textcolor{black}{exploration}}\textcolor{black}{{}
of unknown channel directions. Compared to the conventional random
training vector design, the proposed design can exploit the massive
MIMO array gain to further enhance the channel tracking performance.}
\item \textbf{\textcolor{black}{Angular-Domain Selective Doppler Compensation:
}}\textcolor{black}{We propose an angular-domain selective DFO compensation
scheme at the user side which selectively converts the dominant paths
of the fast time-varying channel into a slow time-varying effective
channel. Compared to the non-selective DFO compensation scheme in
\cite{Guo2018High}, the proposed scheme can enjoy the huge array
(SNR) gain provided by the massive MIMO and also balance the tradeoff
between the CSI signaling overhead reduction and spatial multiplexing
gain maximization.}
\item \textbf{\textcolor{black}{Channel Tracking Algorithm based on Dynamic
VBI: }}\textcolor{black}{To further reduce the pilot overhead, the
proposed partial channel tracking design is formulated as a dynamic
CS problem with unknown DFO parameters in the measurement matrix.
Then, we adopt a three-layer hierarchical Markov model to capture
the dynamic sparsity of the partial angular--domain channel. The
existing methods, such as Variational Bayesian inference (VBI) \cite{Tzikas2008The}
and Sparse Bayesian Learning (SBL) \cite{Dai2018FDD}, cannot be directly
applied to this three-layer hierarchical prior. To address this challenge,
we propose a Doppler-aware-dynamic Variational Bayesian inference
(DD-VBI) algorithm, which combines the VBI and message-passing approaches
to achieve superior channel tracking performance.}
\end{itemize}
\textcolor{black}{}
\textcolor{black}{The rest of this paper is organized as follows.
In Section \ref{sec:System-Model}, we describe the system model and
frame structure. In Section \ref{sec:Angular-Domain-Selective-Channel},
we give a brief introduction of the proposed angular-domain selective
channel tracking and Doppler compensation scheme. In Sections \ref{sec:Downlink-Partial-Angular-Domain}
and \ref{sec:Doppler-Aware-Dynamic-VBI-algori}, we present the three-layer
hierarchical Markov model for partial angular channel vector and the
proposed DD-VBI algorithm. The simulation results and conclusions
are given in Sections \ref{sec:Simulation-Results} and \ref{sec:Conclusion}.}
\textit{\textcolor{black}{Notations:}}\textcolor{black}{{} For a set
of scalars $\left\{ x_{1},...,x_{N}\right\} $ and an index set $\mathcal{S}\subseteq\left\{ 1,...,N\right\} $,
we use $\left[x_{n}\right]_{n\in\mathcal{S}}$ to denote a column
vector consisting of the elements of $\left\{ x_{1},...,x_{N}\right\} $
indexed by the set $\mathcal{S}$. Similarly, for a set of column
vectors $\left\{ \mathbf{x}_{1},...,\mathbf{x}_{N}\right\} $ with
$\mathbf{x}_{n}\in\mathbb{C}^{M}$, $\left[\mathbf{x}_{n}\right]_{n\in\mathcal{S}}$
denotes a column vector consisting of the elements of $\left\{ \mathbf{x}_{1},...,\mathbf{x}_{N}\right\} $
indexed by the set $\mathcal{S}$. We use $\mathbf{X}\left(:,j\right)$
to denote the $j$-th column of a matrix $\mathbf{X}$. The key notations
are summarized in Table \ref{tab:Notations-1}.}
\textcolor{black}{}
\begin{table}
\begin{centering}
\textcolor{black}{
\begin{tabular}{|c|c|}
\hline
\textcolor{black}{Notations} & \textcolor{black}{Meaning}\tabularnewline
\hline
\hline
\textcolor{black}{$N_{p}$} & \textcolor{black}{Number of downlink training vectors}\tabularnewline
\hline
\textcolor{black}{$\mathbf{v}_{t}$} & \textcolor{black}{Downlink training vector}\tabularnewline
\hline
\textcolor{black}{$M(N)$} & \textcolor{black}{Number of antennas at the BS (user)}\tabularnewline
\hline
\textcolor{black}{{} $L_{t}$} & \textcolor{black}{Number of propagation paths}\tabularnewline
\hline
\textcolor{black}{$\alpha_{t,q}$} & \textcolor{black}{The complex path gain of the $q$-th path}\tabularnewline
\hline
\textcolor{black}{$f_{d,t}$} & \textcolor{black}{The maximum DFO}\tabularnewline
\hline
\textcolor{black}{$\xi_{t,q}(\vartheta_{t,q})$} & \textcolor{black}{The AoD (AoA) of the $q$-th path}\tabularnewline
\hline
\textcolor{black}{$\eta_{t}$} & \textcolor{black}{Rotation angle of user's antenna array}\tabularnewline
\hline
\textcolor{black}{$\theta_{T,m}(\theta_{R,m})$} & \textcolor{black}{$m$-th AoD grid (AoA grid)}\tabularnewline
\hline
\textcolor{black}{$\boldsymbol{\beta}_{T,t}(\boldsymbol{\beta}_{R,t})$} & \textcolor{black}{The AoD/AoA off-grid vector}\tabularnewline
\hline
\textcolor{black}{$N_{b}$} & \textcolor{black}{Number of RF chains at the user}\tabularnewline
\hline
\textcolor{black}{$\tilde{N}$} & \textcolor{black}{Number of AoA grid}\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\textcolor{black}{\caption{\textcolor{blue}{\label{tab:Notations-1}}\textcolor{black}{The key
notations used in the paper.}}
}
\end{table}
\section{\textcolor{black}{System Model\label{sec:System-Model}}}
\subsection{\textcolor{black}{System Architecture and Frame Structure}}
\textcolor{black}{Consider a time-division duplexing (TDD) mmWave
massive MIMO system with one static BS serving a fast-moving user}\footnote{\textcolor{black}{For clarity, we focus on a single user system. However,
the proposed selective channel tracking and Doppler compensation scheme
can be readily extended to multi-user systems.}}\textcolor{black}{. The BS is equipped with $M\gg1$ antennas. The
user is equipped with $N\gg1$ antennas. The time is divided into
frames, with each frame containing a downlink subframe and an uplink
subframe, as illustrated in Fig. 1. Each subframe contains a large
number of symbol durations.}
\textcolor{black}{}
\begin{figure}[htbp]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.8]{frame}}}
\par\end{centering}
\textcolor{black}{\caption{Illustration of frame strcture.}
}
\end{figure}
\textcolor{black}{In the $t$-th downlink subframe, there are $N_{p}$
uniformly distributed training vectors, which are set to be the same
vector denoted as $\mathbf{v}_{t}$. For convenience, we use $\mathcal{N}_{p}$
to denote the symbol index set for the $N_{p}$ training vectors.
Note that inserting $N_{p}$ identical training vectors uniformly
in the downlink subframe facilitates the estimation of AoAs and Doppler
parameters at the user, as will be explained later. Based on the estimated
AoAs and Doppler parameters, the user applies a Doppler compensation
matrix to mitigate the Doppler effect and essentially converts the
fast time-varying channel into a slow time-varying effective channel.
In the uplink subframe, there are two sets of $N_{p}^{u}$ training
vectors at the beginning and end of the uplink subframe, respectively.
The two sets of uplink training vectors are used to estimate the slow
time-varying effective channel after Doppler compensation. Specifically,
the uplink transmission (e.g., beamforming and power allocation) in
the $t$-th uplink subframe is optimized based on the slow time-varying
effective channel estimated at the $t$-th uplink subframe. On the
other hand, by making use of the channel reciprocity, the downlink
transmission in the $t$-th downlink subframe is optimized based on
the slow time-varying effective channel estimated at the end of the
$\left(t-1\right)$-th uplink subframe. Since the effective channel
after Doppler compensation changes slowly compared to the subframe
duration, such a design can effectively overcome the channel aging
issue caused by the Doppler effect.}
\subsection{\textcolor{black}{Doppler Multipath Channel Model}}
\textcolor{black}{For clarity, we focus on the case when both the
BS and mobile user are equipped with a half-wavelength space ULA and
the channel is flat fading. To incorporate the DFO with conventional
mmWave channels, the downlink channel model for the antenna pair $\left\{ n_{t},n_{r}\right\} $
is given by \cite{Bajwa2010Compressed}}
\textcolor{black}{
\begin{equation}
{\color{blue}{\color{black}h_{n_{r},n_{t}t,i}=\sum_{q=1}^{L_{t}}\alpha_{t,q}e^{j\left[2\pi f_{d,t}icos(\vartheta_{t,q}+\eta_{t})+\psi_{n_{t}}(\xi_{t,q})+\psi_{n_{r}}(\vartheta_{t,q})\right]},}}\label{eq:jake's channel model}
\end{equation}
where $t$ stands for the frame index, $i$ stands for the symbol
index, $L_{t}$ is the total number of propagation paths, $\alpha_{t,q}$
is the random complex path gain associated with the $q$-th propagation
path, $f_{d,t}$ is the maximum DFO of the $t$-th frame, $\xi_{t,q}$
and $\vartheta_{t,q}$ are the AoD and AoA of the $q$-th path, respectively,
and $\eta_{t}$ is rotation angle of the user\textquoteright s antenna
array with respect to the moving direction in the t-th frame. Here,
$\psi_{n_{t}}(\xi_{t,q})$ and $\psi_{n_{r}}(\vartheta_{t,q})$ represent
the phase shifts induced at the $n_{t}$-th transmit antenna and the
$n_{r}$-th receive antenna, respectively, which depend on the antenna
structure, position, and the direction of the path. Note that in (\ref{eq:jake's channel model}),
we have implicitly assumed that the channel parameters $L_{t},\alpha_{t,q},\xi_{t,q},\vartheta_{t,q},f_{d,t},\eta_{t}$
are fixed within each frame but may change over different frames,
which is usually true even for high-speed users \cite{Guo2018High}.
However, the channel $h_{n_{r},n_{t},t,i}$ itself may change over
different symbols at a much faster timescale due to the fast changing
phase term $2\pi f_{d,t}icos(\vartheta_{t,q}+\eta_{t})$ caused by
the Doppler effect.}
\subsection{\textcolor{black}{Angular Domain Channel Representation}}
\textcolor{black}{To obtain the angular domain channel representation,
we introduce a uniform grid of $\tilde{M}$ AoDs and $\tilde{N}$
AoAs over $\left[0,2\pi\right)$}
\textcolor{black}{
\begin{multline*}
\{\theta_{T,m}:\\
sin(\theta_{T,m})=\frac{2}{\tilde{M}}\left(m-\left\lfloor \frac{\tilde{M}-1}{2}\right\rfloor \right),m=0,\ldots,\tilde{M}-1\},
\end{multline*}
}
\textcolor{black}{
\begin{multline*}
\{\theta_{R,n}:\\
sin(\theta_{R,n})=\frac{2}{\tilde{N}}\left(n-\left\lfloor \frac{\tilde{N}-1}{2}\right\rfloor \right),n=0,\ldots,\tilde{N}-1\}.
\end{multline*}
}
\textcolor{black}{In practice, the true AoDs/AOAs usually do not lie
exactly on the grid points. In this case, there will be mismatches
between the true AoDs/AOAs and the nearest grid point. To overcome
this issue, we introduce an off-grid basis for the angular domain
channel representation, as in \cite{Dai2018FDD}. Specifically, let
$\theta_{T,m_{t,q}}$ and $\theta_{R,n_{t,q}}$ denote the nearest
grid point to $\xi_{t,q}$ and $\vartheta_{t,q}$, respectively. We
introduce the AoD off-grid vector $\boldsymbol{\beta}_{T,t}=\left[\beta_{T,t,1},\beta_{T,t,2},...,\beta_{T,t,\tilde{M}}\right]^{T}$
such that}
\textcolor{black}{
\[
\beta_{T,t,m}=\begin{cases}
\xi_{t,q}-\theta_{T,m_{t,q}}, & m=m_{t,q},q=1,2,...,L_{t}\\
0, & \text{otherwise}
\end{cases}.
\]
Similarly, let $\boldsymbol{\beta}_{R,t}=\left[\beta_{R,t,1},\beta_{R,t,2},...,\beta_{R,t,\tilde{N}}\right]^{T}$
denote the AoA off-grid vector, such that}
\textcolor{black}{
\[
\beta_{R,t,n}=\begin{cases}
\vartheta_{t,q}-\theta_{R,n_{t,q}}, & n=n_{t,q},q=1,2,...,L_{t}\\
0, & \text{otherwise}
\end{cases}.
\]
}
\textcolor{black}{For half-wavelength space ULAs, the array response
vectors at the BS and user side are given by $\boldsymbol{a}_{T}(\theta)=\frac{1}{\sqrt{M}}\left[1,e^{-j\pi sin(\theta)},e^{-j2\pi sin(\theta)},\ldots,e^{-j(M-1)\pi sin(\theta)}\right]^{T}$
and $\boldsymbol{a}_{R}(\theta)=\frac{1}{\sqrt{N}}\left[1,e^{-j\pi sin(\theta)},e^{-j2\pi sin(\theta)},\ldots,e^{-j(N-1)\pi sin(\theta)}\right]^{T}.$
For convenience, define two matrices ${\color{blue}\boldsymbol{A}_{R,i}\,(\boldsymbol{\beta}_{R,t},f_{d,t},\eta_{t})\:=\:[\tilde{\boldsymbol{a}}_{R,i}\,(\beta_{R,t,1},f_{d,t},\eta_{t}),\,\ldots\,,}$}
\noindent \textcolor{black}{${\color{blue}\tilde{\boldsymbol{a}}_{R,i}\,(\beta_{R,t,\tilde{N}},f_{d,t},\eta_{t})\,]\text{\ensuremath{\in}}\mathbb{C}^{N\text{\texttimes}\tilde{N}}}$
and $\boldsymbol{A}_{T}(\boldsymbol{\beta}_{T,t})=[\boldsymbol{a}_{T}(\theta_{T,1}+\beta_{T,t,1}),...,\boldsymbol{a}_{T}(\theta_{T,\tilde{M}}+\beta_{T,t,\tilde{M}})]\in\mathbb{C}^{M\times\tilde{M}}$,
where ${\color{blue}\tilde{\boldsymbol{a}\,}_{R,i}(\,\beta_{R,t,n},\,f_{d,t},\eta_{t})\,\,=}$}
\noindent \textcolor{black}{${\color{blue}\boldsymbol{a}_{R}\,(\theta_{R,n}+\,\beta_{R,t,n})\:\times\:e^{j2\pi f_{d,t}icos(\theta_{R,n}+\beta_{R,t,n}+\eta_{t})}}$.
Furthermore, define $\tilde{\boldsymbol{X}}_{t}\in\mathbb{C}^{\tilde{N}\times\tilde{M}}$
as the angular domain channel matrix with the $(n,m)$-th element
given by
\[
\tilde{x}_{t,n,m}=\begin{cases}
\alpha_{t,q}, & (n,m)=(n_{t,q},m_{t,q}),q=1,2,...,L_{t}\\
0, & \text{otherwise}
\end{cases}.
\]
Then, for given AoA off-grid, DFO parameter , rotation angle pair
$\boldsymbol{\varphi}_{t}=\left\{ \boldsymbol{\beta}_{R,t},f_{d,t},\eta_{t}\right\} $,
and AoD off-grid vector $\boldsymbol{\beta}_{T,t}$, $\boldsymbol{H}_{t,i}$
can be expressed in a compact form as}
\textcolor{black}{
\begin{equation}
\boldsymbol{H}_{t,i}\left(\boldsymbol{\varphi}_{t},\boldsymbol{\beta}_{T,t}\right)=\boldsymbol{A}_{R,i}(\boldsymbol{\varphi}_{t})\tilde{\boldsymbol{X}_{t}}\boldsymbol{A}_{T}^{H}(\boldsymbol{\beta}_{T,t}),\label{eq:angular domain}
\end{equation}
where t $e^{j\psi_{n_{t}}(\xi_{t,q})}$ and $e^{j\psi_{n_{r}}(\vartheta_{t,q})}$
in (\ref{eq:jake's channel model}) are implicitly contained in the
array response matrices $\boldsymbol{A}_{R,i}(\boldsymbol{\varphi}_{t})$
and $\boldsymbol{A}_{T}^{H}(\boldsymbol{\beta}_{T,t})$.}
\textcolor{black}{Note that we can also define the angular domain
representation for more general 2-dimensional (2D) antenna arrays.
In this case, the array response vector $\boldsymbol{a}_{T}(\theta,\phi)$
(or $\boldsymbol{a}_{R}(\theta,\phi)$) can be expressed as a function
of the azimuth angle $\theta$ and elevation angle $\phi$. Please
refer to \cite{Dietrich2000Adaptive} for the details.}
\section{\textcolor{black}{Angular-Domain Selective Channel Tracking and Doppler
Compensation\label{sec:Angular-Domain-Selective-Channel}}}
\textcolor{black}{In this section, we propose an efficient angular-domain
selective channel tracking and Doppler compensation scheme at the
user side. The proposed scheme can exploit both dynamic sparsity of
mmWave massive MIMO channel and high resolution of AoA at multi-antenna
mobile users to accurately estimate the downlink AoAs and maximum
DFO. Using these estimated parameters, a Doppler compensation matrix
is applied at the user to convert the fast time-varying channel into
a slow time-varying effective channel, based on which efficient downlink/uplink
transmissions can be achieved. The proposed scheme includes four key
components, namely the Angular-Domain Selective Channel Tracking,
Selective Doppler Compensation, Slow Time-Varying Effective Channel
Estimation, and Downlink Training Vector Design. Fig. 2 illustrates
a top-level diagram of the proposed scheme and the details of each
component are elaborated below. The frame index $t$ will be omitted
when there is no ambiguity.}
\textcolor{black}{}
\begin{figure}[htbp]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.77]{fivecomponentmodified}}}
\par\end{centering}
\textcolor{black}{\caption{A top-level diagram of the proposed scheme.}
}
\end{figure}
\subsection{\textcolor{black}{Outline of Angular-Domain Selective Channel Tracking
at the User}}
\textcolor{black}{This component is used to estimate the downlink
AoAs, rotation angle and maximum DFO based on the $N_{p}$ downlink
training vectors. Thanks to the high-spatial resolution provided by
the large array at the user side, the user can distinguish DFOs associated
with different AOAs from multiple active paths. However, since both
the BS and the user are equipped with the large array, the parameter
space can be very large if we attempt to estimate the full angular--domain
channel parameters (i.e., the full angular domain channel matrix $\tilde{\boldsymbol{X}}$,
rotation angle $\eta$ and maximum DFO $f_{d}$). Since the DFO only
occurs at the mobile user side, we propose to only estimate partial
angular--domain channel parameters that are just sufficient to obtain
AoAs, rotation angle and maximum DFO for Doppler compensation. }
\textcolor{black}{Specifically, the product of channel $\boldsymbol{H}_{i}$
and downlink training vector $\mathbf{v}$ can be expressed as:}
\textcolor{black}{
\begin{align}
\boldsymbol{H}_{i}\mathbf{v} & =\sum_{n=1}^{\tilde{N}}\sum_{m=1}^{\tilde{M}}\tilde{x}_{n,m}{\color{red}{\color{black}\tilde{\boldsymbol{a}}_{R,i}(\boldsymbol{\varphi})\boldsymbol{a}_{T}^{H}(\theta_{T,m}+\beta_{T,m})}}\mathbf{v},\nonumber \\
& =\sum_{n=1}^{\tilde{N}}x_{n}{\color{black}{\color{red}{\color{black}\tilde{\boldsymbol{a}}_{R,i}(\boldsymbol{\varphi})}}={\color{red}{\color{black}\boldsymbol{A}_{R,i}(\boldsymbol{\varphi})}}}\boldsymbol{x},
\end{align}
where $\boldsymbol{x}=\left[x_{1},...,x_{\tilde{N}}\right]^{T}$ with
$x_{n}=\sum_{m=1}^{\tilde{M}}\tilde{x}_{n,m}{\color{red}{\color{black}\boldsymbol{a}_{T}^{H}(\theta_{T,m}+\beta_{T,m})}}\mathbf{v}$
are called partial angular channel coefficients since $\boldsymbol{x}$
only contain partial information about the full angular channel $\tilde{\boldsymbol{X}}$.
Specifically, a non-zero $\left|x_{n}\right|^{2}$ with value larger
than the noise floor indicates that there is an active path to the
$n$-th AoA direction at the user side. Therefore, we only need to
estimate $\tilde{N}$ partial channel parameters $\boldsymbol{x}$,
the AoA off-grid vector $\boldsymbol{\beta}_{R}$, rotation angle
$\eta$ and maximum DFO $f_{d}$, which are much less than the original
$\tilde{N}\tilde{M}$ full channel parameters, the off-grid vector
, rotation angle and maximum DFO. Note that if the $N_{p}$ training
vectors are different, there will be $\tilde{N}N_{p}$ partial angular
channel coefficients, leading to a larger parameter space to be estimated.
Moreover, with uniformly distributed training vectors, the phase rotation
due to the Doppler term $e^{j2\pi f_{d}icos(\theta_{R,n}+\beta_{R,t,n}+\eta_{t})}$
is larger compared to the case when the $N_{p}$ training vectors
are squeezed in the beginning of the downlink subframe, leading to
a better estimation performance for the Doppler parameter $f_{d}$.
Therefore, such a selective channel tracking design based on $N_{p}$
uniformly distributed and identical training vectors can significantly
reduce the number of downlink training vectors $N_{p}$ required to
achieve accurate estimation of AoAs, rotation angle and Doppler parameters.}
\textcolor{black}{The received baseband pilot signal is given by}
\textcolor{black}{
\begin{equation}
\boldsymbol{y}_{i}=\boldsymbol{H}_{i}\mathbf{v}+\boldsymbol{n}_{i},\forall i\in\mathcal{N}_{p}\label{eq:rec}
\end{equation}
where $\mathbf{v}\in\mathbb{C}^{M}$ is the training vector for downlink
channel tracking, and $\mathbf{n}_{i}$ is the additive white Gaussian
noise (AWGN) with each element having zero mean and variance $\sigma^{2}$,
respectively. The exact choice of $\mathbf{v}$ is postponed to Section
\ref{subsec:Downlink-Training-Vector}.}
\textcolor{black}{The aggregate received pilot signal (channel measurements)
of all the $N_{p}$ downlink pilot symbols (training vectors) in the
$t$-th frame can be expressed in a compact form as}
\textcolor{black}{
\begin{equation}
\boldsymbol{y}=\left[\boldsymbol{H}_{i}\mathbf{v}+\boldsymbol{n}_{i}\right]_{i\in\mathcal{N}_{p}}.\label{eq:rec_vector}
\end{equation}
}
\textcolor{black}{Based on the received downlink training vectors,
the user obtains the estimated partial channel parameters $\hat{\boldsymbol{x}}$,
$\hat{\boldsymbol{\beta}}_{R}$, $\hat{\eta}$ and $\hat{f}_{d}$
using a selective channel tracking algorithm. The detailed problem
formulation and algorithm design for selective channel tracking scheme
are postponed to Section \ref{sec:Downlink-Partial-Angular-Domain}.}
\subsection{\textcolor{black}{Angular-Domain Selective Doppler Compensation at
the User}}
\textcolor{black}{This component is used to convert the fast time-varying
channel into a slow time-varying effective channel after obtaining
the estimated partial channel parameters $\hat{\boldsymbol{x}}$,
$\hat{\boldsymbol{\beta}}_{R}$, $\hat{\eta}_{t}$ and $\hat{f}_{d}$.
We first select a set of $N_{d}$ most significant AoA directions
with the largest energy, where the energy of the $n$-th AoA direction
${\color{black}\theta_{R,n}+\beta_{R,n}}$ is defined as $\left|\hat{x}_{n}\right|^{2}$.
The parameter $N_{d}\leq N$ is used to control the tradeoff between
the spatial multiplexing gain and the effective CSI signaling overhead
(i.e., the CSI signaling overhead required to obtain effective channel
$\boldsymbol{H}_{i}^{s}$ in (\ref{eq:effectH})). Let $\mathcal{N}_{d}\subseteq\left\{ 1,...,N\right\} $
denote the index set of the selected $N_{d}$ most significant AoA
directions. Then, in order to mitigate the Doppler effect and perform
per-AoA DFO compensation for each selected AoA direction, a DFO compensation
matrix $\mathbf{W}_{i}^{d}\mathbf{D}_{i}\in\mathbb{C}^{N\times N_{d}}$,
which also serves as beamforming matrix, is applied at the user side.
In this way, we can convert the fast time-varying channel $\boldsymbol{H}_{i}$
into a slow time-varying effective channel $\boldsymbol{H}_{i}^{s}\in\mathbb{C}^{N_{d}\text{\texttimes}M}$
as}
\textcolor{black}{
\begin{align}
\boldsymbol{H}_{i}^{s} & =\mathbf{D}_{i}^{H}(\mathbf{W}_{i}^{d})^{H}\boldsymbol{H}_{i},\label{eq:effectH}
\end{align}
where $\mathbf{W}_{i}^{d}=\left[\boldsymbol{a}_{R}(\theta_{R,n}+\beta_{R,n})\right]_{n\in\mathcal{N}_{d}}\in\mathbb{C}^{N\times N_{d}}$
and $\mathbf{D}_{i}=\text{Diag}\left(\left[e^{j2\pi\hat{f}_{d}icos(\theta_{R,n}+\beta_{R,n}{\color{blue}+\eta_{t}})}\right]_{n\in\mathcal{N}_{d}}\right)\in\mathbb{C}^{N_{d}\times N_{d}}$.}
\textcolor{black}{In the following, we explain why the Doppler effect
can be alleviated by applying the above DFO compensation matrix to
obtain an effective channel $\boldsymbol{H}_{i}^{s}$. For half-wavelength
space ULA, if there is no estimation error for the partial channel
parameters, we have
\begin{equation}
\boldsymbol{H}_{i}^{s}=\sum_{m=1}^{\tilde{M}}\left[\tilde{x}_{n,m}\right]_{n\in\mathcal{N}_{d}}\boldsymbol{a}_{T}^{H}(\theta_{T,m}+\beta_{T,m})+O\left(\frac{1}{\sqrt{N}}\right),\label{eq:Hsi}
\end{equation}
as $N\rightarrow\infty$ \cite{Caire_TIT13_JSDM}. From (\ref{eq:Hsi}),
$\boldsymbol{H}_{i}^{s}$ is constant within a frame if we ignore
the small order term $O\left(\frac{1}{\sqrt{N}}\right)$, i.e., the
Doppler effect can be completely eliminated for sufficiently large
$N$. This observation is also consistent with the results in \cite{Guo2018High}.}
\subsection{\textcolor{black}{Slow Time-Varying Effective Channel Estimation
at the BS}}
\textcolor{black}{The user can simply transmit $N_{d}$ orthogonal
pilots in the uplink training stage. Then the conventional Least Squares
(LS) based channel estimation method can be used at the BS to obtain
the estimated slow time-varying effective channel $\hat{\boldsymbol{H}}_{i}^{s}$.
Based on $\hat{\boldsymbol{H}}_{i}^{s}$, the BS can optimize the
precoder for both uplink and downlink transmissions. Note that the
optimization of MIMO precoder is a standard problem and there are
many existing solutions with different performance and complexity
tradeoff. Then, the optimized uplink precoder is fed back to the user
for uplink transmission. Since $N_{d}$ can be much less than $N$,
the feedback overhead for the optimized uplink precoder is acceptable
for practice.}
\subsection{\textcolor{black}{Training Vector Design at the BS\label{subsec:Downlink-Training-Vector}}}
\textcolor{black}{The training vector $\mathbf{v}_{t}$ at the BS
is designed according to the slow time-varying effective channel $\hat{\boldsymbol{H}}_{t-1}^{s}$
estimated at the end of the $(t-1)$-th uplink subframe. The basic
idea for training vector design is to strike a balance between }\textit{\textcolor{black}{exploitation}}\textcolor{black}{{}
of known channel directions (i.e., transmitting training signal over
the most promising channel directions with large effective channel
energy to achieve beamforming gain) and }\textit{\textcolor{black}{exploration}}\textcolor{black}{{}
of unknown channel directions (i.e., transmitting training signal
over other channel directions to detect unknown channel directions).
Since the effective channel $\boldsymbol{H}_{i}^{s}$ changes slowly,
the effective channel $\hat{\boldsymbol{H}}_{t-1}^{s}$ estimated
at the end of the $(t-1)$-th uplink subframe is expected to provide
valuable information for the most promising channel directions. On
the other hand, the information about the most promising channel directions
extracted from $\hat{\boldsymbol{H}}_{t-1}^{s}$ may not be perfect
due to the estimation error and CSI delay. In addition, some new direction
may arise in the next frame. Therefore, the other channel directions
should also be incorporated into the training vector to facilitate
the detection of unknown channel directions. }
\textcolor{black}{Specifically, we first project the estimated effective
channel $\hat{\boldsymbol{H}}_{t-1}^{s}$ onto an orthogonal basis
$\mathbf{B}^{s}=\left[\mathbf{b}_{1}^{s},...,\mathbf{b}_{M}^{s}\right]\in\mathbb{C^{\mathit{M\times M}}}$
to obtain the effective channel energy on each basis vector (quantized
channel direction) as $\lambda_{m}^{s}=\left\Vert \hat{\boldsymbol{H}}_{t-1}^{s}\mathbf{b}_{m}^{s}\right\Vert ^{2},\forall m$.
The basis matrix is chosen such that the projection vector $\boldsymbol{\lambda}^{s}=\left[\lambda_{1}^{s},...,\lambda_{M}^{s}\right]^{T}$
is as sparse as possible. For half-wavelength ULAs, we can simply
choose the basis matrix $\mathbf{B}^{s}$ as an $M\times M$ DFT matrix.
Then we find the index set of the most promising channel directions
as
\begin{equation}
\mathcal{M}^{*}=\text{argmin}_{\mathcal{M}}\left|\mathcal{M}\right|,\text{ s.t. }\sum_{m\in\mathcal{M}}\lambda_{m}^{s}/\sum_{m=1}^{M}\lambda_{m}^{s}\geq\mu,\label{eq:M}
\end{equation}
where $\mu$ is a threshold which is chosen to be closed to 1. In
other words, the most promising channel directions contain $\mu$
fraction of the total effective channel energy. Let $N_{s}=\left|\mathcal{M}^{*}\right|$.
Finally, the training vector is given by }
\textcolor{black}{
\begin{alignat}{1}
\mathbf{v}_{t}= & \frac{\sqrt{\rho}}{\sqrt{N_{s}}}\sum_{m\in\mathcal{M}^{*}}e^{j\theta_{m}^{s}}\mathbf{b}_{m}^{s}\nonumber \\
& +\frac{\sqrt{1-\rho}}{\sqrt{M-N_{s}}}\sum_{m\in\left\{ 1,...,M\right\} \backslash\mathcal{M}^{*}}e^{j\theta_{m}^{s}}\mathbf{b}_{m}^{s}.\label{eq:vtdesign}
\end{alignat}
where the first term in (\ref{eq:vtdesign}) exploites the information
about the most promising $N_{s}$ channel directions extracted from
$\hat{\boldsymbol{H}}_{t-1}^{s}$, $\rho$ is a system parameter which
determines the proportion of transmit power used to exploit the most
promising channel directions, the second term is used to detect the
other unknown channel directions, $\theta_{m}^{s}$ is randomly generated
from $\left[0,2\pi\right]$. }
\textcolor{black}{}
\begin{figure}[htbp]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.57]{5}}}
\par\end{centering}
\textcolor{black}{\caption{\textcolor{blue}{\label{fig:Achievable-data-rate}}Achievable data
rate versus the parameter $\rho$ with different values of the parameter
$\mu$.}
}
\end{figure}
\textcolor{black}{In Fig. \ref{fig:Achievable-data-rate}, we illustrate
how the achievable data rate is affected by changing the parameters
$\rho$ and $\mu$ to achieve different tradeoffs between the exploration
and exploitation. It can be seen that setting $\mu=0.9$ and $\rho=0.5$
can strike a good balance between }\textit{\textcolor{black}{exploitation}}\textcolor{black}{{}
and }\textit{\textcolor{black}{exploration}}\textcolor{black}{.}
\section{\textcolor{black}{Problem Formulation for Angular-Domain Selective
Channel Tracking\label{sec:Downlink-Partial-Angular-Domain}}}
\subsection{\textcolor{black}{Three-layer Hierarchical Markov Model for Partial
Angular Channel Vector\label{subsec:Three-layer-Hierarchical-Markov}}}
\textcolor{black}{The dynamic sparsity of the partial angular channel
coefficients $\boldsymbol{x}_{t}$ is captured using a three-layer
hierarchical Markov model, as illustrated in Fig. \ref{fig:Three-layer-hierarchical-Markov}.
The first layer of random variable is the channel support vector $\boldsymbol{s}_{t}\in\{0,1\}^{\tilde{N}}$,
whose $n$-th element, denoted by $s_{t,n}$, indicates whether the
channel coefficient $x_{t,n}$ is active ($s_{t,n}=1$) or not ($s_{t,n}=0$).
The second layer of random variable is the precision vector $\boldsymbol{\gamma}_{t}=[\gamma_{t,1},\cdots,\gamma_{t,\tilde{N}}]^{T}$,
where $\gamma_{t,n}$ represents the precision (inverse of the variance)
of $x_{t,n}$. The third layer of random variables are partial angular
channel coefficients $\boldsymbol{x}_{t}$. For convenience, denote
a time series of vectors $\left\{ \boldsymbol{x}_{\tau}\right\} _{\tau=1}^{t}$
as $\boldsymbol{x}_{1:t}$ (same for $\mathbf{\boldsymbol{\gamma}}_{1:t}$,
$\boldsymbol{s}_{1:t}$, $f_{d,1:t}$,${\color{black}\boldsymbol{\beta}}_{{\color{black}R,1:t}}$).
Then the three-layer hierarchical Markov prior distribution (joint
distribution of $\boldsymbol{x}_{1:t}$, $\mathbf{\boldsymbol{\gamma}}_{1:t}$
and $\boldsymbol{s}_{1:t}$ ) is given by}
\textcolor{black}{
\begin{equation}
p(\boldsymbol{x}_{1:t},\mathbf{\boldsymbol{\gamma}}_{1:t},\boldsymbol{s}_{1:t})=\prod_{\tau=1}^{t}p\left(\boldsymbol{s}_{\tau}|\boldsymbol{s}_{\tau-1}\right)p(\mathbf{\boldsymbol{\gamma}}_{\tau}|\boldsymbol{s}_{\tau})p(\boldsymbol{x}_{\tau}|\mathbf{\boldsymbol{\gamma}}_{\tau}),\label{eq:three layer}
\end{equation}
where $p\left(\boldsymbol{s}_{1}|\boldsymbol{s}_{0}\right)\triangleq p\left(\boldsymbol{s}_{1}\right)$,
the conditional probability $p\left(\boldsymbol{x}_{\tau}|\mathbf{\boldsymbol{\gamma}}_{\tau}\right)$
has a product form $p\left(\boldsymbol{x}_{\tau}|\mathbf{\boldsymbol{\gamma}}_{\tau}\right)=\prod_{n=1}^{\tilde{N}}p\left(x_{\tau,n}|\mathbf{\gamma}_{\tau,n}\right)$
and each is modeled as a Gaussian prior distribution}
\textcolor{black}{
\begin{equation}
p\left(x_{\tau,n}|\gamma_{\tau,n}\right)=CN\left(x_{\tau,n};0,\gamma_{\tau,n}^{-1}\right),\label{eq:xcondruo}
\end{equation}
}
\textcolor{black}{The conditional prior of precision vector $\boldsymbol{\gamma}_{\tau}$
is given by}
\textcolor{black}{
\begin{equation}
p\left(\mathbf{\boldsymbol{\gamma}}_{\tau}|\boldsymbol{s}_{\tau}\right)=\prod_{n=1}^{\tilde{N}}\Gamma\left(\gamma_{\tau,n};a_{\tau},b_{\tau}\right)^{s_{\tau,n}}\Gamma\left(\gamma_{\tau,n};\overline{a}_{\tau},\overline{b}_{\tau}\right)^{1-s_{\tau,n}},\label{eq:gamma}
\end{equation}
$\Gamma\left(\gamma;a_{\gamma},b_{\gamma}\right)$ is a Gamma hyperprior.
$a_{\tau},b_{\tau}$ are the shape and rate parameters of the channel
precision $\gamma_{\tau,n}$ conditioned on $s_{\tau,n}=1$ and they
should be chosen such that $\frac{a_{\tau}}{b_{\tau}}=E[\gamma_{\tau,n}]=\Theta(1)$,
since the variance $\gamma_{\tau,n}^{-1}$ of $x_{\tau,n}$ is $\Theta(1)$
when it is active ($s_{\tau,n}=1$). $\overline{a}_{\tau},\overline{b}_{\tau}$
are the shape and rate parameters , conditioned on the opposite event
(i.e., $s_{\tau,n}=0$). In this case, the shape and rate parameters
$\overline{a}_{\tau},\overline{b}_{\tau}$ of the precision $\gamma_{\tau,n}$
should be chosen such that $\frac{\overline{a}_{\tau}}{\overline{b}_{\tau}}=E[\gamma_{\tau,n}]\gg1$,
since the variance $\gamma_{\tau,n}^{-1}$ of $x_{\tau,n}$ is close
to zero when it is inactive.}
\textcolor{black}{Note that the exact channel distribution is usually
unknown in practice. In this case, it is reasonable to choose a prior
distribution such that the derived algorithm can promote sparsity
with low complexity and achieve robust performance to different channel
distributions. By controlling the parameters in the Gamma distribution
of $\gamma_{\tau,n}$, one can easily promote sparsity based on the
knowledge of channel support $\boldsymbol{s}_{\tau}$, as explained
above. Moreover, Since the Gamma distribution for $\gamma_{\tau,n}$
is the conjugate probability distribution of the Gaussian distribution
for $x_{\tau,n}$, the above hierarchical prior for $x_{\tau,n}$
and $\gamma_{\tau,n}$ facilitates low-complexity VBI algorithm design
with closed-form update equations \cite{Tzikas2008The}. Finally,
the VBI-type algorithm derived from the such a hierarchical prior
is well known to be insensitive to the true distribution of the sparse
signals \cite{Tzikas2008The,Ji2008Bayesian}. As a result, similar
hierarchical prior distribution has been widely adopted in sparse
Bayesian learning \cite{Wipf2003Bayesian}.}
\textcolor{black}{Due to the slowly changing propagation environment,
the channel supports often change slowly over time, which implies
that $s_{\tau,n}$ depends on $s_{\tau-1,n}$, e.g., if $s_{\tau-1,n}=1$,
then there is a higher probability that $s_{\tau,n}$ is also 1. Such
dynamic sparsity of support vectors can be naturally modeled as a
temporal Markov model with an initial prior distribution $p(\mathbf{s}_{1})$
and a transition probability:}
\textcolor{black}{
\begin{equation}
p(\mathbf{s}_{\tau}|\mathbf{s}_{\tau-1})=\prod_{n=1}^{\tilde{N}}p(s_{\tau,n}|s_{\tau-1,n}),\label{eq:Markov support}
\end{equation}
where the transition probability is given by $p(s_{\tau,n}=1|s_{\tau-1,n}=0)=\rho_{0,1}$,
and $p(s_{\tau,n}=0|s_{\tau-1,n}=1)=\rho_{1,0}$. The Markov parameters
$\left\{ \rho_{1,0},\rho_{0,1}\right\} $ characterize the degree
of temporal correlation of the channel support. Specifically, smaller
$\rho_{1,0}$ or $\rho_{0,1}$ lead to highly correlated supports
across time, which means the propagation environment between the user
and BS is changing slowly. Larger $\rho_{1,0}$ or $\rho_{0,1}$ can
allow support to change substantially across time, which means the
propagation environment is changing significantly. Moreover, the statistic
parameters $\left\{ \rho_{1,0},\rho_{0,1}\right\} $ could be automatically
learned based on the EM framework during the recovery process \cite{Ziniel2013Dynamic},
as detailed in Appendix \ref{subsec:Derivation-2}. The initial distribution
$p(s_{1,n}),\forall n$ is set to be the steady state distribution
of the Markov chain in (\ref{eq:Markov support}), i.e.,}
\textcolor{black}{
\[
\lambda\triangleq p(s_{1,n})=\frac{\rho_{0,1}}{\rho_{0,1}+\rho_{1,0}}.
\]
This ensures that all elements of $s_{\tau,n}$ have the same marginal
distribution $p(s_{\tau,n})=\lambda^{s_{\tau,n}}(1-\lambda)^{1-s_{\tau,n}}$.}
\textcolor{black}{In practice, the noise precision $\kappa_{\tau}=\sigma_{\tau}^{-2}$
is usually unknown and we model it as a Gamma hyperpiror $p\left(\kappa_{\tau}\right)=\Gamma\left(\kappa_{\tau};a_{\kappa,\tau},b_{\kappa,\tau}\right)$,
where we set $a_{\kappa,\tau},b_{\kappa,\tau}\rightarrow0$ as in
\cite{Dai2018FDD} so as to obtain a broad hyperprior.}
\textcolor{black}{}
\begin{figure}[htbp]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.74]{threelayer}}}
\par\end{centering}
\textcolor{black}{\caption{\label{fig:Three-layer-hierarchical-Markov}Three-layer hierarchical
Markov model for partial angular channel coefficients.}
}
\end{figure}
\subsection{\textcolor{black}{Selective Channel Tracking Formulation}}
\textcolor{black}{Using the angular domain channel representation,
the receive signal at $t$-th frame $\boldsymbol{y}_{t}\in\mathbb{C}^{NN_{p}}$
can be rewritten as a CS model with an unknown AoA off-grid and DFO
parameter pair ${\color{blue}\boldsymbol{\varphi}_{t}=\left\{ \boldsymbol{\beta}_{R,t},\eta_{t},f_{d,t}\right\} }$
in the measurement matrix as}
\textcolor{black}{
\begin{equation}
\boldsymbol{y}_{t}=\boldsymbol{F}_{t}\boldsymbol{x}_{t}+\boldsymbol{n}_{t},\label{eq:compress formulation}
\end{equation}
where the measurement matrix is given by $\boldsymbol{F}_{t}=[\boldsymbol{F}_{t,1};...;\boldsymbol{F}_{t,N_{p}}]\text{\ensuremath{\in}}C^{NN_{p}\text{\texttimes}\tilde{N}}$,
$\boldsymbol{F}_{t,i}=\boldsymbol{A}_{R,i}({\color{red}{\color{black}\boldsymbol{\varphi}_{t}}})$,
$\boldsymbol{n}_{t}=[\boldsymbol{n}_{t,i}]_{i\in\mathcal{N}_{p}}$.}
\textcolor{black}{In each frame $t$, the user needs to estimate the
partial channel parameters $\boldsymbol{x}_{t}$ , the AoA off-grid
and DFO parameter pair ${\color{blue}\boldsymbol{\varphi}_{t}=\left\{ \boldsymbol{\beta}_{R,t},\eta_{t},f_{d,t}\right\} }$,
given the observations up to $t$ frame $\boldsymbol{y}_{1:t}$ in
model (\ref{eq:compress formulation}), the estimated AoA off-grid
and DFO parameter pairs $\hat{\boldsymbol{\varphi}}_{1:t-1}=\left\{ \hat{\boldsymbol{\beta}}_{R,1:t-1},\hat{\eta}_{1:t-1},\hat{f}_{d,1:t-1}\right\} $
up to $\left(t-1\right)$ frame. In particular, for given $\boldsymbol{\varphi}_{t}$,
we are interested in computing minimum mean-squared error (MMSE) estimates
of ${x_{t,n}}$, $\hat{x}_{t,n}={\color{red}{\color{black}\mathrm{E}\left[x_{t,n}|\boldsymbol{y}_{1:t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t}\right]}}$,
where the expectation is over the marginal posterior:}
\textcolor{black}{
\begin{flalign}
{\color{red}} & {\color{black}{\color{red}{\color{black}p(x_{t,n}|\boldsymbol{y}_{1:t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t})}}}\nonumber \\
{\color{red}{\color{black}\propto}} & {\color{black}{\color{red}{\color{black}\int_{-x_{t,n}}p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t}),}}}\label{eq:posterior}
\end{flalign}
where $\boldsymbol{v}_{t}=\left\{ \boldsymbol{x}_{t},\boldsymbol{s}_{t},\boldsymbol{\gamma}_{t},\kappa_{t}\right\} $,
$-x_{t,n}$ denotes the vector collections integration over $\boldsymbol{v}_{t}$
except for the element $x_{t,n}$ and $\propto$ denotes equality
after scaling.}
\textcolor{black}{On the other hand, the optimal $\boldsymbol{\varphi}_{t}$
at the $t$-th frame is obtained by ML as follows \cite{Dai2018FDD}:}
\textcolor{black}{
\begin{alignat}{1}
\boldsymbol{\hat{\varphi}}_{t} & {\color{red}{\color{black}=\mathrm{arg}\max_{\boldsymbol{\varphi}_{t}}\ln p(\boldsymbol{y}_{1:t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t})}}\nonumber \\
{\color{red}} & {\color{red}{\color{black}\mathrm{=arg}\max_{\boldsymbol{\varphi}_{t}}\ln\int_{\boldsymbol{v}_{t}}p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t})d\boldsymbol{v}_{t}.}}\label{eq:maxlikelyhood}
\end{alignat}
}
\textcolor{black}{Once we obtain the ML estimate of $\boldsymbol{\hat{\varphi}}_{t}$,
and the associated conditional marginal posterior $p(x_{t,n}|\boldsymbol{y}_{1:t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t})$,
we can obtain the MMSE estimates of $\left\{ x_{t,n}\right\} $.}
\textcolor{black}{One challenge in computing the MMSE estimate is
the calculation of the exact posterior in (\ref{eq:posterior}) whose
factor graph has loops. In the next subsection, we propose a Doppler-aware-dynamic-VBI
(DD-VBI) algorithm to approximately calculate the marginal posteriors
${p(x_{t,n}|\boldsymbol{y}_{1:t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t})}$
by combining the message passing and VBI approaches, and use the in-exact
majorization-minimization (MM) method (which is a generalization of
the EM method) \cite{Dai2018FDD} to find an approximate solution
for (\ref{eq:maxlikelyhood}). The proposed DD-VBI algorithm is shown
in the simulations to achieve a good performance.}
\section{\textcolor{black}{Doppler-Aware-Dynamic-VBI algorithm \label{sec:Doppler-Aware-Dynamic-VBI-algori}}}
\subsection{\textcolor{black}{Decomposition and Approximation of Joint Probability
Distribution\label{subsec:Decomposition-Joint-Probability}}}
\textcolor{black}{This section is to decompose and approximate the
joint probability distribution in (\ref{eq:maxlikelyhood}) such that
the joint probability distribution at the $t$-th frame only involves
the probability density function (PDF) of the current hidden variables
$\boldsymbol{v}_{t}$, the current observation $\boldsymbol{y}_{t}$,
and the messages $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1},\hat{\boldsymbol{\varphi}}_{1:t-1})$
passed from the previous frame, based on which a more efficient algorithm
can be designed.}
\textcolor{black}{The joint probability distribution in (\ref{eq:maxlikelyhood})
and (\ref{eq:posterior}) can be written as}
\textcolor{black}{
\begin{align*}
& p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t})\\
\propto & \sum_{\boldsymbol{s}_{t-1}}p(\mathbf{s}_{t-1}|\mathbf{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})p(\boldsymbol{s}_{t}|\boldsymbol{s}_{t-1})\\
& p(\boldsymbol{y}_{t}|\boldsymbol{x}_{t},\kappa_{t};\boldsymbol{\varphi}_{t})p(\boldsymbol{x}_{t}|\mathbf{\boldsymbol{\gamma}}_{t})p(\mathbf{\boldsymbol{\gamma}}_{t}|\boldsymbol{s}_{t})p(\kappa_{t})\\
\approx & \sum_{\boldsymbol{s}_{t-1}}q(\boldsymbol{s}_{t-1}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})p(\boldsymbol{s}_{t}|\boldsymbol{s}_{t-1})\\
& p(\boldsymbol{y}_{t}|\boldsymbol{x}_{t},\kappa_{t};\boldsymbol{\varphi}_{t})p(\boldsymbol{x}_{t}|\mathbf{\boldsymbol{\gamma}}_{t})p(\mathbf{\boldsymbol{\gamma}}_{t}|\boldsymbol{s}_{t})p(\kappa_{t})\\
= & \hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})p(\boldsymbol{y}_{t}|\boldsymbol{x}_{t},\kappa_{t};\boldsymbol{\varphi}_{t})\\
& p(\boldsymbol{x}_{t}|\mathbf{\boldsymbol{\gamma}}_{t})p(\mathbf{\boldsymbol{\gamma}}_{t}|\boldsymbol{s}_{t})p(\kappa_{t}),
\end{align*}
where $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})=\sum_{\mathbf{s}_{t-1}}q(\boldsymbol{s}_{t-1}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})p(\boldsymbol{s}_{t}|\boldsymbol{s}_{t-1})$,
$q(\boldsymbol{s}_{t-1}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})$
is a tractable approximation for the posterior $p(\mathbf{s}_{t-1}|\mathbf{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})$
and $p\left(\boldsymbol{y}_{t}|\boldsymbol{x}_{t},\kappa_{t};\boldsymbol{\varphi}_{t}\right)=\mathcal{CN}\left(\boldsymbol{y}_{t};\boldsymbol{F}_{t}\boldsymbol{x}_{t},\kappa_{t}^{-1}\boldsymbol{I}\right)$.
Both $q(\boldsymbol{s}_{t-1}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})$
and $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})$
can be calculated based on the messages passed from the previous frame.
We will elaborate how to calculate $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})$
later in subsection \ref{subsec:Cauculating--passed}. When $t=1$,
$\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})$
is reduced to $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1};\hat{\boldsymbol{\varphi}}_{1})=p(\boldsymbol{s}_{1})$.}
\textcolor{black}{For simplicity, we define
\begin{alignat}{1}
& \hat{p}(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\hat{\boldsymbol{\varphi}}_{1:t-1},\boldsymbol{\varphi}_{t})\nonumber \\
= & \hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})p(\boldsymbol{y}_{t}|\boldsymbol{x}_{t},\kappa_{t};\boldsymbol{\varphi}_{t})\nonumber \\
& p(\boldsymbol{x}_{t}|\mathbf{\boldsymbol{\gamma}}_{t})p(\mathbf{\boldsymbol{\gamma}}_{t}|\boldsymbol{s}_{t})p(\kappa_{t}).\label{eq:pyvfappro}
\end{alignat}
}
\textcolor{black}{In the rest of this section, we will omit $\hat{\boldsymbol{\varphi}}_{1:t-1}$
in the PDFs when there is no ambiguity.}
\subsection{\textcolor{black}{Outline of the Doppler-Aware-Dynamic-VBI algorithm
in Frame $t$}}
\textcolor{black}{The basic idea of the DD-VBI algorithm is that,
at every frame $t$, simultaneously approximates the marginal posterior
$\left\{ p(x_{t,n}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})\right\} $
and maximizes the log-likelihood $\ln p(\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$
with respect to $\boldsymbol{\varphi}_{t}$, based on the noisy measurements
of the $t$-th frame and the messages $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1})$
passed from the previous frame. In summary, for every frame, the DD-VBI
algorithm performs iterations between the following two major steps
until convergence, as shown in Fig. \ref{fig:Interaction-between-the}.}
\begin{itemize}
\item \textcolor{black}{DD-VBI-E Step: Given $\boldsymbol{\varphi}_{t}$
at $t$-th frame and messages $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\hat{\boldsymbol{\varphi}}_{1:t-1})$
passed from the previous frame, calculate the approximate marginal
posterior of $p(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$,
denoted as $q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$,
using the sparse VBI approach, as elaborated in subsection \ref{subsec:Doppler-Turbo-OAMP-E-Step}.}
\item \textcolor{black}{DD-VBI-M Step: Given $q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})\approx p(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$,
construct a surrogate function for the objective function $\ln p(\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$,
then maximize the surrogate function with respect to $\boldsymbol{\varphi}_{t}$
as elaborated in subsection \ref{subsec:Doppler-Turbo-OAMP-M-Step}. }
\end{itemize}
\textcolor{black}{After convergence, the messages $\hat{p}(\boldsymbol{s}_{t+1}|\boldsymbol{y}_{1:t};\hat{\boldsymbol{\varphi}}_{1:t})$
are calculated based on $q(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{1:t})$
and passed to the next frame. In the following, we first elaborate
the M step, which is a variation of the in-exact MM method in \cite{Dai2018FDD}.
After that, we will elaborate how to approximately calculate the posterior
$p(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})\approx q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$
in the E step, which is required to construct the surrogate function
in the M step.}
\textcolor{black}{}
\begin{figure}[htbp]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.77]{twomodules}}}
\par\end{centering}
\textcolor{black}{\caption{I\label{fig:Interaction-between-the}interaction between the two modules
of the DD-VBI algorithm within a frame.}
}
\end{figure}
\subsection{\textcolor{black}{DD-VBI-M Step \label{subsec:Doppler-Turbo-OAMP-M-Step}}}
\textcolor{black}{It is difficult to directly maximize the log-likelihood
function $\ln p(\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$,
because there is no closed-form expression due to the multi-dimensional
integration over $\boldsymbol{v}_{t}$ as in (\ref{eq:maxlikelyhood}).
To make the problem tractable, in the DD-VBI-M Step, we adopt an in-exact
MM method in \cite{A1977Maximum,Dai2018FDD}, which maximizes a surrogate
function of $\ln p(\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$
with respect to $\boldsymbol{\varphi}_{t}$, to find an approximate
solution of (\ref{eq:maxlikelyhood}). Specifically, let $u(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t})$
be the surrogate function constructed at some fixed point $\boldsymbol{\dot{\varphi}}_{t}$,
which satisfies the following properties:}
\textcolor{black}{
\begin{gather}
u(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t})\leq\ln p(\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}),\nonumber \\
u(\boldsymbol{\dot{\varphi}}_{t};\boldsymbol{\dot{\varphi}}_{t})=\ln p(\boldsymbol{y}_{1:t};\boldsymbol{\dot{\varphi}}_{t}),\nonumber \\
\frac{\partial u(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t})}{\partial\boldsymbol{\varphi}_{t}}|_{\boldsymbol{\varphi}_{t}=\boldsymbol{\dot{\varphi}}_{t}}=\frac{\partial\ln p(\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})}{\partial\boldsymbol{\varphi}_{t}}|_{\boldsymbol{\varphi}_{t}=\boldsymbol{\dot{\varphi}}_{t}}.\label{eq:surr}
\end{gather}
Inspired by the EM algorithm \cite{A1977Maximum}, we use the following
surrogate function:}
\textcolor{black}{
\begin{equation}
u(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t})=\int q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})\ln\frac{p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})}{q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})}d\boldsymbol{v}_{t},\label{eq:surrogate function}
\end{equation}
where $q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$
is a tractable approximation of $p(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$.
In subsection \ref{subsec:Decomposition-Joint-Probability}, we have
approximated the joint probability distribution in (\ref{eq:surrogate function})
using $\hat{p}(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})$.
Therefore, the surrogate function can be approximated as}
\textcolor{black}{
\begin{equation}
\hat{u}(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t})=\int q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})\ln\frac{\hat{p}(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})}{q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})}d\boldsymbol{v}_{t},\label{eq:surr_1}
\end{equation}
When there is no approximation error for the associated PDFs, i.e.,
$q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})=p(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$
and $\hat{p}(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})=p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})$,
it can be verified that $\hat{u}(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t})$
satisfies the properties in (\ref{eq:surr}).}
\textcolor{black}{In the M step of the $j$-th iteration, we update
$\boldsymbol{\varphi}_{t}$ as
\begin{gather}
\boldsymbol{\varphi}_{t}^{j+1}=\mathrm{arg}\max_{\boldsymbol{\varphi}_{t}}\hat{u}(\boldsymbol{\varphi}_{t};\boldsymbol{\varphi}_{t}^{j}),\label{eq:max_surr}
\end{gather}
where $\left(\cdot\right)^{j}$ stands for the $j$-th iteration. }
\textcolor{black}{In our problem, $\hat{u}(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t})$
is a non-convex function and it is difficult to find its optimal solution.
Therefore, we use a simple gradient update as in \cite{Dai2018FDD},
i.e.}
\textcolor{black}{
\begin{align}
\boldsymbol{\varphi}_{t}^{j+1} & =\boldsymbol{\varphi}_{t}^{j}+\boldsymbol{\tau}^{j}\frac{\partial\hat{u}(\boldsymbol{\varphi}_{t};\boldsymbol{\dot{\varphi}}_{t}^{j})}{\partial\boldsymbol{\varphi}_{t}},\label{eq:ugrad}
\end{align}
where $\boldsymbol{\tau}^{j}$ is the step sizes determined by the
Armijo rule \cite{Dai2018FDD}.}
\textcolor{black}{The approximate posterior $q\left(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)$
has a factorized form as
\begin{multline}
q\left(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)\\
=q\left(\boldsymbol{x}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)q\left(\boldsymbol{\gamma}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)q\left(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)\text{.}\label{eq:factorized}
\end{multline}
}
\textcolor{black}{Therefore, after the convergence of the DD-VBI,
we not only obtain an approximate stationary solution $\boldsymbol{\hat{\varphi}}_{t}$
of (\ref{eq:maxlikelyhood}), but also the associated (approximate)
marginal conditional posterior $q\left(\boldsymbol{x}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{1:t}\right)\approx p(\boldsymbol{x}_{t}|\boldsymbol{y}_{1:t},\boldsymbol{\varphi}_{t})$.}
\subsection{\textcolor{black}{DD-VBI-E Step\label{subsec:Doppler-Turbo-OAMP-E-Step}}}
\textcolor{black}{DD-VBI-E Step performs the sparse VBI to approximate
the conditional marginal posteriors $p(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})$
based on the following joint prior distribution: }
\textcolor{black}{
\begin{alignat}{1}
& \hat{p}(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})\nonumber \\
= & \hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1})p(\boldsymbol{y}_{t}|\boldsymbol{x}_{t},\kappa_{t};\boldsymbol{\varphi}_{t})\nonumber \\
& p(\boldsymbol{x}_{t}|\mathbf{\boldsymbol{\gamma}}_{t})p(\mathbf{\boldsymbol{\gamma}}_{t}|\boldsymbol{s}_{t})p(\kappa_{t}).\label{eq:pyvfappro-1}
\end{alignat}
}
\textcolor{black}{The corresponding approximate posterior distributions
$q\left(\boldsymbol{v}_{t}\right)$ obtained by the sparse VBI will
be given by (\ref{eq:poster_kappa})-(\ref{eq:poster_q}).}
\subsubsection{\textcolor{black}{Outline of Sparse VBI within a Frame}}
\textcolor{black}{For convenience, we use $\boldsymbol{v}_{t,n}$
to denote an individual variable in $\boldsymbol{v}_{t}$. Let $\mathcal{H}=\left\{ n|\forall\boldsymbol{v}_{t,n}\in\boldsymbol{v}_{t}\right\} $.
Moreover, we use $q\left(\boldsymbol{v}_{t}\right)$ as a simplified
notation for $q\left(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)$
when there is no ambiguity. The approximate conditional marginal posterior
$q\left(\boldsymbol{v}_{t}\right)$ could be calculated by minimizing
the Kullback-Leibler divergence (KLD) between $p\left(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)$
and $q\left(\boldsymbol{v}_{t}\right)$ subject to a factorized form
constraint on $q\left(\boldsymbol{v}_{t}\right)$ as
\begin{align}
\mathcal{\mathscr{A}}_{\mathrm{VBI}}:\thinspace\thinspace\thinspace & q^{*}\left(\boldsymbol{v}_{t}\right)=\arg\min_{q\left(\boldsymbol{v}_{t}\right)}\int q\left(\boldsymbol{v}_{t}\right)\ln\frac{q\left(\boldsymbol{v}_{t}\right)}{p\left(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)}d\boldsymbol{v}_{t}\label{eq:KLDmin}\\
& \mathrm{s.t.}\thinspace\thinspace\thinspace\thinspace q\left(\boldsymbol{v}_{t}\right)=\prod_{n\in\mathcal{H}}q\left(\boldsymbol{v}_{t,n}\right),\label{eq:factorconstrain}\\
& \thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\thinspace\int q\left(\boldsymbol{v}_{t,n}\right)d\boldsymbol{v}_{t,n}=1,\forall n\in\mathcal{H}.
\end{align}
Problem $\mathcal{\mathscr{A}}_{\mathrm{VBI}}$ is non-convex, we
aim at finding a stationary solution (denoted by $q^{*}\left(\boldsymbol{v}_{t}\right)$)
of $\mathcal{\mathscr{A}}_{\mathrm{VBI}}$, as defined below.}
\begin{defn}
\textcolor{black}{[Stationary Solution]\label{lem:optimality-conditon-1}$q^{*}\left(\boldsymbol{v}_{t}\right)=\prod_{n\in\mathcal{H}}q^{*}\left(\boldsymbol{v}_{t,n}\right)$
is called a stationary solution of Problem $\mathcal{\mathscr{A}}_{\mathrm{VBI}}$
if it satisfies all the constraints in $\mathcal{\mathscr{A}}_{\mathrm{VBI}}$
and $\forall n\in\mathcal{H}$,
\begin{multline*}
q^{*}\left(\boldsymbol{v}_{t,n}\right)=\\
\arg\min_{q\left(\boldsymbol{v}_{t,n}\right)}\int\prod_{l\neq n}q^{*}\left(\boldsymbol{v}_{t,l}\right)q\left(\boldsymbol{v}_{t,n}\right)\ln\frac{\prod_{l\neq n}q^{*}\left(\boldsymbol{v}_{t,l}\right)q\left(\boldsymbol{v}_{t,n}\right)}{p\left(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right)}d\boldsymbol{v}_{t}.
\end{multline*}
}
\end{defn}
\textcolor{black}{By finding a stationary solution $q^{*}\left(\boldsymbol{v}_{t}\right)$
of $\mathcal{\mathscr{A}}_{\mathrm{VBI}}$, we could obtain the approximate
posterior $q^{*}\left(\boldsymbol{v}_{t,n}\right)\thickapprox p\left(\boldsymbol{v}_{t,n}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}\right),\forall n\in\mathcal{H}$. }
\textcolor{black}{A stationary solution of $\mathcal{\mathscr{A}}_{\mathrm{VBI}}$
can be obtained via alternately optimizing each individual density
$q\left(\boldsymbol{v}_{t,n}\right),n\in\mathcal{H}$. For given $q\left(\boldsymbol{v}_{t,l}\right),\forall l\neq n$,
the optimal $q\left(\boldsymbol{v}_{t,n}\right)$ that minimizes the
KLD in $\mathcal{\mathscr{A}}_{\mathrm{VBI}}$ is given by
\begin{equation}
q\left(\boldsymbol{v}_{t,n}\right)\propto\exp\left(\left\langle \ln p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})\right\rangle _{\prod_{l\neq n}q\left(\boldsymbol{v}_{t,l}\right)}\right),\label{eq:optimal_q}
\end{equation}
where $\left\langle f\left(x\right)\right\rangle _{q(x)}=\int f\left(x\right)q(x)dx$.
However, $p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})$
is intractable. Since $p(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})\approx\hat{p}(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})$
in (\ref{eq:pyvfappro}), \eqref{eq:optimal_q} can be approximated
as
\begin{equation}
q\left(\boldsymbol{v}_{t,n}\right)\propto\exp\left(\left\langle \ln\hat{p}(\boldsymbol{y}_{1:t},\boldsymbol{v}_{t};\boldsymbol{\varphi}_{t})\right\rangle _{\prod_{l\neq n}q\left(\boldsymbol{v}_{t,l}\right)}\right).\label{eq:optimal_q-1}
\end{equation}
Based on \eqref{eq:optimal_q-1}, the update equations of all variables
are given in the subsequent subsections. The detailed derivation can
be found in Appendix \ref{subsec:Derivation-1}. Note that the operator
$\left\langle \cdot\right\rangle _{\boldsymbol{v}_{t,l}}$ is equivalent
to $\left\langle \cdot\right\rangle _{q\left(\boldsymbol{v}_{t,l}\right)}$
and the expectation $\left\langle f\left(\boldsymbol{v}_{t,l}\right)\right\rangle _{q\left(\boldsymbol{v}_{t,l}\right)}$
w.r.t. its own approximate posterior is simplified as $\left\langle f\left(\boldsymbol{v}_{t,l}\right)\right\rangle $. }
\subsubsection{\textcolor{black}{Initialization of Sparse VBI\label{subsec:Initialization-of-Sparse}}}
\textcolor{black}{In order to trigger the alternating optimization
(AO) algorithm, we use the following initializations for the distribution
functions $q\left(\boldsymbol{x}_{t}\right),q(\boldsymbol{\gamma}_{t})$
in the first iteration of every frame and $q\left(\boldsymbol{s}_{1}\right)$
in the first iteration of first frame. In the rest iterations, we
initialize $q\left(\boldsymbol{x}_{t}\right)$,$q\left(\boldsymbol{s}_{t}\right)$,$q(\boldsymbol{\gamma}_{t})$
to the (approximate) posterior calculated in the previous frame.}
\begin{itemize}
\item \textcolor{black}{Initialize $q\left(\boldsymbol{s}_{1}\right)=\hat{p}(\boldsymbol{s}_{1}|\boldsymbol{y}_{1};\boldsymbol{\varphi}_{t})=\prod_{n=1}^{\tilde{N}}q\left(s_{1,n}\right)$
with $q\left(s_{1,n}\right)=\left(\tilde{\pi}_{1,n}\right)^{s_{1.n}}\left(1-\tilde{\pi}_{1,n}\right)^{1-s_{1,n}}$. }
\item \textcolor{black}{For given $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t})=\prod_{n=1}^{\tilde{N}}\left(\tilde{\pi}_{t,n}\right)^{s_{t.n}}\left(1-\tilde{\pi}_{t,n}\right)^{1-s_{t,n}}$,
initialize a gamma distribution for $\mathbf{\boldsymbol{\gamma}}_{t}$:
$q\left(\mathbf{\boldsymbol{\gamma}}_{t}\right)=\prod_{n=1}^{\tilde{N}}\Gamma\left(\gamma_{t,n};\tilde{a}_{\gamma,t,n},\tilde{b}_{\gamma,t,n}\right)$,
where $\tilde{a}_{\gamma,t,n}=\tilde{\pi}_{t,n}a_{t}+\left(1-\tilde{\pi}_{t,n}\right)\overline{a}_{t}$,
$\tilde{b}_{\gamma,t,n}=\tilde{\pi}_{t,n}b_{t}+\left(1-\tilde{\pi}_{t,n}\right)\overline{b}_{t}$.}
\item \textcolor{black}{Initialize a Gaussian distribution for $\boldsymbol{x}_{t}$:
$q\left(\boldsymbol{x}_{t}\right)=\mathcal{CN}(\boldsymbol{x}_{t};\boldsymbol{\mu}_{t},\boldsymbol{\Sigma}_{t})$
, where $\boldsymbol{\Sigma}_{t}=\left(\mathrm{diag}\left(\left\langle \mathbf{\boldsymbol{\gamma}}_{t}\right\rangle \right)+\left(\boldsymbol{F}_{t}\right)^{H}\boldsymbol{F}_{t}\right)^{-1},$
$\boldsymbol{\mu}_{t}=\boldsymbol{\Sigma}_{t}\left(\boldsymbol{F}_{t}\right)^{H}\boldsymbol{y}_{t}.$}
\end{itemize}
\subsubsection{\textcolor{black}{Update for $q\left(\kappa_{t}\right)$}}
\textcolor{black}{From (\ref{eq:optimal_q-1}), $q\left(\kappa_{t}\right)$
can be derived as
\begin{equation}
q\left(\kappa_{t}\right)=\Gamma(\kappa_{t};\tilde{a}_{\kappa,t},\tilde{b}_{\kappa,t}).\label{eq:poster_kappa}
\end{equation}
where $\tilde{a}_{\kappa,t}=a_{\kappa}+NN_{p}$, $\tilde{b}_{\kappa,t}=b_{\kappa}+\left\langle \left\Vert \boldsymbol{y}_{t}-\boldsymbol{F}_{t}\boldsymbol{x}_{t}\right\Vert ^{2}\right\rangle _{\boldsymbol{x}_{t}}=b_{\kappa}+\left\Vert \boldsymbol{y}_{t}-\boldsymbol{F}_{t}\boldsymbol{\mu}_{t}\right\Vert ^{2}+\mathrm{tr}\left(\boldsymbol{F}_{t}\boldsymbol{\Sigma}_{t}\left(\boldsymbol{F}_{t}\right)^{H}\right)$. }
\subsubsection{\textcolor{black}{Update for $q(\boldsymbol{x}_{t})$}}
\textcolor{black}{$q\left(\boldsymbol{x}_{t}\right)$ can be derived
as
\begin{equation}
q\left(\boldsymbol{x}_{t}\right)=\mathcal{CN}\left(\boldsymbol{x}_{t};\boldsymbol{\mu}_{t},\boldsymbol{\Sigma}_{t}\right).\label{eq:poster_x}
\end{equation}
$\boldsymbol{\mu}_{t}$ and $\boldsymbol{\Sigma}_{t}$ can be calculated
through
\begin{alignat}{1}
\boldsymbol{\Sigma}_{t} & =\left(\mathrm{diag}\left(\left\langle \mathbf{\boldsymbol{\gamma}}_{t}\right\rangle \right)+\left\langle \kappa_{t}\right\rangle \left(\boldsymbol{F}_{t}\right)^{H}\boldsymbol{F}_{t}\right)^{-1},\nonumber \\
{\color{black}{\color{black}}} & {\color{black}{\color{black}{\color{black}{\color{blue}=\mathbf{R}-\upsilon\mathbf{R}\left(\boldsymbol{F}_{t}\right)^{H}\left(\mathbf{I}+\upsilon\boldsymbol{F}_{t}\mathbf{R}\left(\boldsymbol{F}_{t}\right)^{H}\right)^{-1}\boldsymbol{F}_{t}\mathbf{R}.}}}}\label{eq:Sigma_x}
\end{alignat}
}
\textcolor{black}{
\begin{equation}
\boldsymbol{\mu}_{t}=\left\langle \kappa_{t}\right\rangle \boldsymbol{\Sigma}_{t}\left(\boldsymbol{F}_{t}\right)^{H}\boldsymbol{y}_{t}.\label{eq:Mu_x}
\end{equation}
}
\noindent \textcolor{black}{where $\left\langle \mathbf{\boldsymbol{\gamma}}_{t}\right\rangle =\frac{\tilde{a}_{\gamma,t,n}}{\tilde{b}_{\gamma,t,n}}$,
$\left\langle \kappa_{t}\right\rangle =\frac{\tilde{a}_{\kappa,t}}{\tilde{b}_{\kappa,t}}$,
$\mathbf{R}=\mathrm{diag}\left(\left[\frac{\tilde{b}_{\gamma,t,1}}{\tilde{a}_{\gamma,t,1}},\cdots,\frac{\tilde{b}_{\gamma,t,\tilde{N}}}{\tilde{a}_{\gamma,t,\tilde{N}}}\right]\right)$,
$\upsilon=\left\langle \kappa_{t}\right\rangle =\frac{\tilde{a}_{\kappa,t}}{\tilde{b}_{\kappa,t}}$.}
\subsubsection{\textcolor{black}{Update for $q\left(\mathbf{\boldsymbol{\gamma}}_{t}\right)$}}
\textcolor{black}{$q\left(\mathbf{\boldsymbol{\gamma}}_{t}\right)$
can be derived as}
\textcolor{black}{
\begin{equation}
q\left(\mathbf{\boldsymbol{\gamma}}_{t}\right)=\prod_{n=1}^{\tilde{N}}\Gamma\left(\gamma_{t,n};\tilde{a}_{\gamma,t,n},\tilde{b}_{\gamma,t,n}\right),\label{eq:poster_rho}
\end{equation}
where $\tilde{a}_{\gamma,t,n},\tilde{b}_{\gamma,t,n}$ are given by:
\begin{align}
\tilde{a}_{\gamma,t,n}= & \left\langle s_{t,n}\right\rangle a_{t}+\left\langle 1-s_{t,n}\right\rangle \overline{a}_{t}+1,\label{eq:a_tilde}\\
\tilde{b}_{\gamma,t,n}= & \left\langle s_{t,n}\right\rangle b_{t}+\left\langle 1-s_{t,n}\right\rangle \overline{b}_{t}+\left\langle \left|x_{t,n}\right|^{2}\right\rangle .\label{eq:b_tilde}
\end{align}
}
\noindent \textcolor{black}{where $\left\langle s_{t,n}\right\rangle =\tilde{\pi}_{t,n},\left\langle 1-s_{t,n}\right\rangle =1-\tilde{\pi}_{t,q},$
$\left\langle \left|x_{t,n}\right|^{2}\right\rangle =\left|\mu_{t,n}\right|^{2}+\Sigma_{t,n}$,
$\mu_{t,n}$ is the $n$-th element of $\boldsymbol{\mu}_{t}$, $\Sigma_{t,n}$
is the $n$-th diagonal element of $\boldsymbol{\Sigma}_{t}$.}
\subsubsection{\textcolor{black}{Update for $q\left(\boldsymbol{s}_{t}\right)$}}
\textcolor{black}{$q\left(\boldsymbol{s}_{t}\right)$ can be derived
as }
\textcolor{black}{
\begin{equation}
q\left(\boldsymbol{s}_{t}\right)=\prod_{n=1}^{\tilde{N}}\left(\pi_{t,n}\right)^{s_{t.n}}\left(1-\pi_{t,n}\right)^{1-s_{t,n}},\label{eq:poster_q}
\end{equation}
where $\pi_{t,n}$ is given by }
\textcolor{black}{
\begin{align}
\pi_{t,n}=\frac{1}{C} & \frac{\tilde{\pi}_{t,n}b_{t}^{a_{t}}}{\Gamma(a_{t})}e^{\left(a_{t}-1\right)\left\langle \ln\mathbf{\boldsymbol{\gamma}}_{t,n}\right\rangle -b_{t}\left\langle \mathbf{\boldsymbol{\gamma}}_{t,n}\right\rangle },\label{eq:pi_tilde}
\end{align}
and $C$ is the normalization constant, given by $C=\frac{\tilde{\pi}_{t,n}b_{t}^{a_{t}}}{\Gamma(a_{t})}e^{\left(a_{t}-1\right)\left\langle \ln\mathbf{\boldsymbol{\gamma}}_{t,n}\right\rangle -b_{t}\left\langle \mathbf{\boldsymbol{\gamma}}_{t,n}\right\rangle }+\frac{(1-\tilde{\pi}_{t,n})\overline{b}_{t}^{\overline{a}_{t}}}{\Gamma(\overline{a}_{t})}e^{\left(\overline{a}_{t}-1\right)\left\langle \ln\mathbf{\boldsymbol{\gamma}}_{t,n}\right\rangle -\overline{b}_{t}\left\langle \mathbf{\boldsymbol{\gamma}}_{t,n}\right\rangle }$,
$\left\langle \ln\gamma_{t,n}\right\rangle =\psi\left(\tilde{a}_{\gamma,t,n}\right)-\ln\left(\tilde{b}_{\gamma,t,n}\right)$,
$\psi\left(x\right)=\frac{d}{dx}\ln\left(\Gamma\left(x\right)\right)$
is the digamma function, defined as the logarithmic derivative of
the gamma function.}
\textcolor{black}{}
\begin{algorithm}
\textcolor{black}{\caption{\label{alg1}Doppler-Aware-Dynamic-VBI algorithm}
}
\textcolor{black}{1: }\textbf{\textcolor{black}{for}}\textcolor{black}{{}
$t=1,2,...$ }\textbf{\textcolor{black}{do}}
\textcolor{black}{2: ~~Initialize the distribution functions according
to Section }
\textcolor{black}{~~~~~~\ref{subsec:Initialization-of-Sparse}.}
\textcolor{black}{3: ~~}\textbf{\textcolor{black}{while}}\textcolor{black}{{}
not converge }\textbf{\textcolor{black}{do}}
\textcolor{black}{4: ~~}\textbf{\textcolor{black}{while}}\textcolor{black}{{}
not converge }\textbf{\textcolor{black}{do}}
\textcolor{black}{5: ~~~~}\textbf{\textcolor{black}{{} \%DD-VBI-E
Step: }}
\textcolor{black}{6: ~~~~ Update $q\left(\kappa_{t}\right)$ using
(\ref{eq:poster_kappa}).}
\textcolor{black}{7: ~~~~ Update $q(\boldsymbol{x}_{t})$ using
(\ref{eq:poster_x}).}
\textcolor{black}{8: ~~~~ Update $q\left(\mathbf{\boldsymbol{\gamma}}_{t}\right)$
using (\ref{eq:poster_rho}).}
\textcolor{black}{9: ~~~~ Update $q\left(\boldsymbol{s}_{t}\right)$
using (\ref{eq:poster_q}) .}
\textcolor{black}{10: ~~}\textbf{\textcolor{black}{end while}}
\textcolor{black}{11: ~~~~}\textbf{\textcolor{black}{{} \%DD-VBI-M
Step: }}
\textcolor{black}{12: ~~~~ Construct the surrogate function $\hat{u}$
in (\ref{eq:surr_1}) using the }
\textcolor{black}{~~~~~~~~~ output of DD-VBI-E Step $q(\boldsymbol{v}_{t}|\boldsymbol{y}_{1:t};\boldsymbol{\varphi}_{t}).$}
\textcolor{black}{13: ~~~~ Update $\boldsymbol{\varphi}_{t}$
using (\ref{eq:max_surr}).}
\textcolor{black}{14: ~~}\textbf{\textcolor{black}{end while}}
\textcolor{black}{15: ~~Let $\hat{f}_{d,t}$ denote the converged
maximum DFO for frame}
\textcolor{black}{~~~~~~~$t$. Calculate $\hat{p}(\boldsymbol{s}_{t+1}|\boldsymbol{y}_{1:t};\boldsymbol{\hat{\varphi}}_{1:t})$
using (\ref{eq:Pheads}).}
\textcolor{black}{~~~~~~~Pass messages $\hat{p}(\boldsymbol{s}_{t+1}|\boldsymbol{y}_{1:t};\boldsymbol{\hat{\varphi}}_{1:t})$
to frame $t+1$.}
\textcolor{black}{16: ~~Estimate ${x_{t,n}}$ using (\ref{eq:Mu_x}).}
\textcolor{black}{17: ~~}\textbf{\textcolor{black}{end for}}
\end{algorithm}
\subsection{\textcolor{black}{Messages $\hat{p}(\boldsymbol{s}_{t+1}|\boldsymbol{y}_{1:t};\boldsymbol{\hat{\varphi}}_{1:t})$
Passed to the Next Frame \label{subsec:Cauculating--passed}}}
\textcolor{black}{After the convergence of the DD-VBI iterations in
frame $t$, let $\hat{f}_{d,t}$ and $q(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t},;\boldsymbol{\hat{\varphi}}_{1:t})=\prod_{n=1}^{\tilde{N}}\left(\pi_{t,n}\right)^{s_{t.n}}\left(1-\pi_{t,n}\right)^{1-s_{t,n}}$
denote the converged Doppler parameter and the associated approximate
posterior for $\boldsymbol{s}_{t}$, respectively. Then, the messages
$\hat{p}(\boldsymbol{s}_{t+1}|\boldsymbol{y}_{1:t};;\boldsymbol{\hat{\varphi}}_{1:t})$
is calculated as}
\textcolor{black}{
\begin{align}
& \hat{p}(\boldsymbol{s}_{t+1}|\boldsymbol{y}_{1:t};\boldsymbol{\hat{\varphi}}_{1:t})\nonumber \\
= & \prod_{n=1}^{\tilde{N}}\sum_{\mathbf{s}_{t,n}}q\left(s_{t,n}|\boldsymbol{y}_{1:t},\boldsymbol{\hat{\varphi}}_{1:t}\right)p(s_{t+1,n}|s_{t,n})\\
= & \prod_{n=1}^{\tilde{N}}\left(\tilde{\pi}_{t+1,n}\right)^{s_{t+1,n}}\left(1-\tilde{\pi}_{t+1,n}\right)^{1-s_{t+1,n}}.\label{eq:Pheads}
\end{align}
where}
\textcolor{black}{
\[
\tilde{\pi}_{t+1,n}=\left(1-\pi_{t,n}\right)\rho_{0,1}+\pi_{t,n}(1-\rho_{1,0}).
\]
Finally, the messages $\hat{p}(\boldsymbol{s}_{t+1}|\boldsymbol{y}_{1:t};\boldsymbol{\hat{\varphi}}_{1:t})$,
as the prior to the channel support, is passed to the next frame.}
\textcolor{black}{The overall DD-VBI algorithm is summarized in Algorithm
1. Note that in the $t$-th frame of DD-VBI, the contribution of the
previous observations $\boldsymbol{y}_{1:t-1}$ on the estimation
of $\boldsymbol{v}_{t}$ and $\boldsymbol{\varphi}_{t}$ is summarized
in the messages $\hat{p}(\boldsymbol{s}_{t}|\boldsymbol{y}_{1:t-1};\boldsymbol{\hat{\varphi}}_{1:t-1})$
passed from frame $t-1$. Therefore, in the $t$-th frame, there is
no need to store all the observations $\boldsymbol{y}_{1:t-1}$ up
to frame $t-1$.}
\begin{rem}
\textcolor{black}{In practical mmWave massive MIMO systems, the number
of RF chains can be less than the number of antennas at both the BS
and user sides to reduce the hardware cost and power consumption.
The proposed scheme can be easily extended to the case with limited
RF chains. For example, suppose there are only $M_{b}<M$ RF chains
at the BS and $N_{b}<N$ RF chains at the user. In this case, when
the BS transmits the training vector $\mathbf{v}=\mathbf{g}\mathbf{F}\in\mathbb{C}^{M}$
for downlink channel tracking in the $i$-th symbol duration, the
user employs $\mathbf{U}_{i}=\mathbf{W}_{i}\mathbf{G}_{i}\in\mathbb{C}^{N\times N_{b}}$
as a combining matrix to combine the received signal into $N_{b}$
baseband channel measurements, where $\mathbf{F}\in\mathbb{C}^{M\times M_{b}}$
and $\mathbf{g}\in\mathbb{C}^{M_{b}}$ are the RF training matrix
and baseband training vector at the BS, respectively, and $\mathbf{W}_{i}\in\mathbb{C}^{N\times N_{b}}$
and $\mathbf{G}_{i}\in\mathbb{C}^{N_{b}\times N_{b}}$ are the RF
and baseband combining matrix at the mobile user in the $i$-th symbol
duration, respectively. We can still write the received pilot signal
as a CS model as in (\ref{eq:compress formulation}), but with $\boldsymbol{F}_{t}=[\boldsymbol{F}_{t,1};...;\boldsymbol{F}_{t,N_{p}}]\text{\ensuremath{\in}}C^{N_{b}N_{p}\text{\texttimes}\tilde{N}}$,
$\boldsymbol{F}_{t,i}=\boldsymbol{U}_{t,i}^{H}\boldsymbol{A}_{R,i}(\boldsymbol{\varphi}_{t})$,
$\boldsymbol{n}_{t}=[\boldsymbol{U}_{t,i}^{H}\boldsymbol{n}_{t,i}]_{i\in\mathcal{N}_{p}}$.}
\end{rem}
\subsection{\textcolor{black}{Complexity and Signaling Overhead Comparison\label{subsec:Complexity-issue-and} }}
\textcolor{black}{The computational complexity of the proposed algorithm
is dominated by the update of $q(\boldsymbol{x}_{t})$. Assuming the
arithmetic with individual elements has complexity $\mathcal{O}(1)$,
the computational complexity of matrix inversion in (\ref{eq:Sigma_x})
is $\mathcal{O}(N_{b}^{3}N_{p}^{3})$ and the total number of multiplications
to update $q(\boldsymbol{x}_{t})$ is $3\tilde{N}+2N_{b}N_{p}\tilde{N}^{2}+2(N_{b}N_{p})^{2}\tilde{N}+(N_{b}N_{p})^{2}$.
Supposing the algorithm executes $I$ iterations, the total complexity
order of the proposed method is $\mathcal{O}\left(I(N_{b}N_{p})^{2}\tilde{N}\right)$,
considering that $\tilde{N}$ is usually larger than $N_{b}N_{p}$.
In Table \ref{tab:Asymptotic-complexity-1} and \ref{tab:Asymptotic-complexity-1-2-1-2},
we assume the mmWave channel has $L_{D}$ dominant paths and compare
the complexity and signaling overhead of the proposed algorithm with
the following baseline algorithms:}
\begin{itemize}
\item \textbf{\textcolor{black}{Baseline 1}}\textcolor{black}{{} (ML) \cite{8558718}:
This is the ML based joint DFO and channel estimation algorithm proposed
in \cite{8558718}. }
\item \textbf{\textcolor{black}{Baseline 2}}\textcolor{black}{{} (ES) \cite{Wang2009Beam}:
This is the codebook based exhaustive beam search algorithm in \cite{Wang2009Beam}.
$K_{E}$ denotes the number of beamforming vectors of the codebook
in ES.}
\item \textbf{\textcolor{black}{Baseline 3}}\textcolor{black}{{} (HS) \cite{Alkhateeb2017Channel}:
This is the codebook based hierarchical beam search algorithm in \cite{Alkhateeb2017Channel}.
$S$ denotes the total level of hierarchical codebook. $K_{H}$ denotes
the number of beamforming vectors of each codebook level in HS.}
\end{itemize}
\textcolor{black}{As seen from Table \ref{tab:Asymptotic-complexity-1},
the complexity order of the proposed algorithm is similar to the baselines.
For example, $N_{b}N_{p}$ can range from $\mathcal{O}\left(L_{D}\right)$
to $\mathcal{O}\left(N\right)$ to achieve different tradeoff between
performance and complexity, and we usually have $\tilde{N}=\mathcal{O}\left(N\right)$,
$K_{E}=\mathcal{O}\left(N\right)$. For a resolution $\mathcal{O}(2\pi/N)$
of the HS scheme, we usually have $\mathcal{S=O}\left(\log_{K_{H}}(N/L_{D})\right)$
\cite{Alkhateeb2017Channel}. In this typical case, the complexity
of the proposed scheme and the baseline schemes are shown in the third
column of Table \ref{tab:Asymptotic-complexity-1}. On the other hand,
the proposed algorithm has the lowest signaling overhead for both
the general case and the typical case, as shown in Table \ref{tab:Asymptotic-complexity-1-2-1-2}.}
\textcolor{black}{}
\begin{table}
\begin{centering}
\textcolor{black}{
\begin{tabular}{|c|c|c|}
\hline
\textcolor{black}{Algorithms} & \textcolor{black}{Complexity order} & \textcolor{black}{Typical complexity order}\tabularnewline
\hline
\hline
\textcolor{black}{proposed} & \textcolor{black}{$\mathcal{O}\left(I(N_{b}N_{p})\tilde{N}^{2}\right)$} & \textcolor{black}{$\mathcal{O}\left(IL_{D}^{2}N\right)-\mathcal{O}\left(IN^{3}\right)$}\tabularnewline
\hline
\textcolor{black}{ML} & \textcolor{black}{$\mathcal{O}\left(\tilde{N}^{3}\right)$} & \textcolor{black}{$\mathcal{O}\left(N^{3}\right)$}\tabularnewline
\hline
\textcolor{black}{ES} & \textcolor{black}{$\mathcal{O}\left(L_{D}^{2}K_{E}^{4}\right)$} & \textcolor{black}{$\mathcal{O}\left(L_{D}^{2}N^{4}\right)$}\tabularnewline
\hline
\textcolor{black}{HS} & \textcolor{black}{$\mathcal{O}\left(SL_{D}^{4}K_{H}^{2}\right)$} & \textcolor{black}{$\mathcal{O}\left(\log_{K_{H}}(N/L_{D})L_{D}^{4}K_{H}^{2}\right)$}\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\textcolor{black}{\caption{\textcolor{blue}{\label{tab:Asymptotic-complexity-1}}\textcolor{black}{Complexity
orders for different schemes.}}
}
\end{table}
\textcolor{black}{}
\begin{table}
\begin{centering}
\textcolor{black}{
\begin{tabular}{|c|c|c|}
\hline
\textcolor{black}{Algorithms} & \textcolor{black}{Signaling overhead} & \textcolor{black}{Typical signaling overhead}\tabularnewline
\hline
\hline
\textcolor{black}{proposed} & \textcolor{black}{$\mathcal{O}(L_{D}/N_{b})+L_{D}$} & \textcolor{black}{$4+L_{D}$}\tabularnewline
\hline
\textcolor{black}{ML} & \textcolor{black}{$N$} & \textcolor{black}{$N$}\tabularnewline
\hline
\textcolor{black}{HS} & \textcolor{black}{$K_{H}^{2}\left(L_{D}\right)^{3}S$} & \textcolor{black}{$K_{H}^{2}\left(L_{D}\right)^{3}\log_{K_{H}}(N/L_{D})$}\tabularnewline
\hline
\textcolor{black}{ES} & \textcolor{black}{$K_{E}^{2}$} & \textcolor{black}{$N^{2}$}\tabularnewline
\hline
\end{tabular}
\par\end{centering}
\textcolor{black}{\caption{\textcolor{blue}{\label{tab:Asymptotic-complexity-1-2-1-2}}\textcolor{black}{Signaling
overhead for different schemes.}}
}
\end{table}
\section{\textcolor{black}{Simulation Results\label{sec:Simulation-Results}}}
\textcolor{black}{In this section, we compare the performance of the
proposed algorithm with the baseline algorithms described in Section
\ref{subsec:Complexity-issue-and}. For the ML baseline, we also consider
the case when only partial channel parameters are estimated as in
the proposed scheme (ML-Partial). For the proposed scheme, we also
consider the case when the training vector is generated randomly (DD-VBI-Random).
The channel parameters are based on the millimeter-wave statistical
spatial channel model (mm-SSCM) as specified in \cite{1Samimi120163},
which was developed according to the 28- and 73-GHz ultrawideband
propagation measurements in New York City. The signal bandwidth is
$50MHz$ and the frame duration is set as $T_{b}=0.5ms$. The carrier
frequency is 28GHz . The mobile user employs a ULA of $M=128$ antennas
and the inter antenna spacing is $\lambda/2$, and the BS also employs
a ULA. The user velocity is assumed to be 380km/h, which translates
to $f_{d,t}\approx10$KHz. }
\textcolor{black}{In the simulation, we will consider both cases when
the user is equipped with a full set of RF chains and limited RF chains.
For the case with limited RF chains, the number of RF chains is set
to be $N_{b}=16$. The MSE for DFO estimation and the uplink achievable
data rate are adopted as the performance metrics. The frequency MSE
and the channel MSE is defined as $\frac{\left\Vert \hat{f}_{d,t}-f_{d,t}\right\Vert ^{2}}{\left\Vert f_{d,t}\right\Vert ^{2}}$
and $\frac{\left\Vert \hat{\boldsymbol{x}}_{t}-\boldsymbol{x}_{t}\right\Vert ^{2}}{\left\Vert \boldsymbol{x}_{t}\right\Vert ^{2}}$,
respectively. The parameter $N_{d}$ is chosen to be equal to the
number of dominant AoA directions at the user side (an AoA direction
is called a dominant AoA direction if its energy is no less than 10\%
of the most significant AoA direction) and $N_{d}$ data streams are
transmitted over the $N_{d}$ dominant AoA directions in the uplink
with equal power allocation. Since the uplink transmission is only
designed based on the dominant AoA directions which are known at the
user, there is no need for the BS to feed back the slow time-varying
effective channel.}
\subsection{\textcolor{black}{Doppler Frequency and Channel MSE Performance}}
\begin{figure}[!tbph]
\begin{centering}
\textsf{\includegraphics[scale=0.5]{2}}
\par\end{centering}
\caption{\textcolor{blue}{\label{fig:Frequency-MSE-and}}Frequency MSE and
channel MSE performance versus the pilot number. Set SNR= 0 dB. (a)
full RF chains with $N=128$. (b) limited RF chains with $N=128$,
$N_{b}=16$\textcolor{black}{.}}
\end{figure}
\textcolor{black}{The Doppler frequency and channel MSE performance
of different algorithms versus the pilot number, SNR and number of
BS antennas are shown in Fig. \ref{fig:Frequency-MSE-and} and Fig.
\ref{fig:Frequency-MSE-performance}. It can be seen that the proposed
DD-VBI algorithm achieves large performance gain over all the baseline
algorithms, under both full RF chains and limited RF chains. By using
the proposed training vector design to strike a balance between }\textit{\textcolor{black}{exploitation}}\textcolor{black}{{}
of known channel directions and }\textit{\textcolor{black}{exploration}}\textcolor{black}{{}
of unknown channel directions, the DD-VBI algorithm could further
improve the MSE performance compared to the case with a random training
vector. This demonstrates that the proposed algorithm can effectively
estimate the maximum DFO by selective channel tracking and efficient
training vector design. Note that as the number of BS antennas increases,
the parameters to be estimated increase significantly and the spatial
resolution and array gain will also increase. Since the ML-Full method
does not exploit the selective channel estimation method or channel
sparsity to reduce the number of free parameters, its performance
may degrade with the number of BS antennas. However, the performance
gain of the proposed algorithm improves because in this case, the
number of parameters to be estimated does not increase with the number
of BS antennas. This demonstrates that the proposed algorithm is a
powerful method for accurate DFO estimation, even when both BS and
mobile user are equipped with massive MIMO, and the number of RF chains
at the user side is limited. }
\textcolor{black}{}
\begin{figure}[!tbph]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.5]{3}}}
\par\end{centering}
\textcolor{black}{\caption{\textcolor{black}{\label{fig:Frequency-MSE-performance}}Frequency
MSE performance versus the number of BS antennas and SNR. (a) Set
SNR= 0 dB. Full RF chains with $N=128$. (b) Set SNR= 0 dB. Limited
RF chains with $N=128$, $N_{b}=16$. (c) The number of pilots is
fixed as 5. Full RF chains with $N=128$. (d) The number of pilots
is fixed as 5. Limited RF chains with $N=128$, $N_{b}=16$.}
}
\end{figure}
\subsection{\textcolor{black}{Achievable Data Rate Performance}}
\textcolor{black}{}
\begin{figure}[tp]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.5]{1}}}
\par\end{centering}
\textcolor{black}{\caption{\textcolor{blue}{\label{fig:Achievable-data-rate-1}}Achievable data
rate versus the number of BS antennas, pilot number and SNR. (a) and
(c) Set SNR= 0 dB. Full RF chains with $N=128$. (b) and (d) Set SNR=
0 dB. Limited RF chains with $N=128$, $N_{b}=16$. (e) The number
of pilots is fixed as 5. Full RF chains with $N=128$. (f) The number
of pilots is fixed as 5. Limited RF chains with $N=128$, $N_{b}=16$.}
}
\end{figure}
\textcolor{black}{The achievable data rates of different algorithms
versus the pilot number, SNR and number of BS antennas are shown in
Fig. \ref{fig:Achievable-data-rate-1}. It can be observed that the
performance of all algorithms increases with the number of pilots,
SNR and number of BS antennas. The proposed DD-VBI algorithm can achieve
large performance gain over various baselines under both full RF chains
and limited RF chains. Moreover, by using the proposed training vector
design, the proposed DD-VBI algorithm could further improve the achievable
data rate. This verifies that the proposed selective channel tracking
and Doppler compensation scheme can also enhance the achievable data
rate with low pilot overhead, under different SNRs and numbers of
BS antennas.}
\subsection{\textcolor{black}{Complexity versus Realized Gain}}
\textcolor{black}{In practice, we can control the tradeoff between
the complexity and realized gain of the proposed algorithm by adjusting
the number of iterations. In Fig. \ref{fig:The-tradeoff-between},
we plot the achievable rate versus the CPU time. It can be seen that
the proposed algorithm can achieve a better performance than the baseline
algorithms for the same CPU time. Moreover, the performance gain increases
with the CPU time. Therefore, the proposed algorithm provides a better
and more flexible tradeoff between the performance and computational
power.}
\textcolor{black}{}
\begin{figure}[htbp]
\begin{centering}
\textsf{\textcolor{black}{\includegraphics[scale=0.5]{4}}}
\par\end{centering}
\textcolor{black}{\caption{\textcolor{blue}{\label{fig:The-tradeoff-between}}\textcolor{black}{The
tradeoff between the complexity and realized gain of the proposed
algorithm.}}
}
\end{figure}
\section{\textcolor{black}{Conclusion\label{sec:Conclusion}}}
\textcolor{black}{We propose an angular-domain selective channel tracking
and Doppler compensation scheme for high-mobility massive MIMO systems.
Firstly, we propose a selective channel tracking scheme and the associated
Doppler-aware-dynamic-VBI algorithm to accurately estimate the DFO
and partial angular--domain channel parameters with reduced pilot
overhead. Then, we propose an angular-domain selective DFO compensation
scheme to convert the dominant paths of the fast time-varying channel
into a slow time-varying effective channel, based on which efficient
uplink and downlink transmissions can be achieved. Simulations verify
that the proposed scheme not only can mitigate the Doppler and channel
aging effect with much less pilots than existing schemes, but also
can achieve a good tradeoff between the CSI signaling overhead and
spatial multiplexing/array gain. }
|
1,116,691,498,485 | arxiv |
\section{Problem Statement}
\nobi{The following starts with what Trainify does, but we should first justify why we sit on Trainify (e.g., what are the merits?). In fact, it would be better to justify why we sit on ``abstract-state-based systems''. }
\textbf{Trainify}~\cite{} aims to train a DNN-based controller to represent a deterministic policy $\pi: S_\phi \rightarrow A$ that specifies a unique action adopted in an abstract state based on $\phi$. The trained DRL system can be represented as a tuple $\mathcal{M} = (S,A,f,\pi, S^{0}, \phi)$ where $S$ is the continuous state space, $S^{0}$ is the set of initial states of the system, $f$ represents the system dynamics, $\pi$ means the policy, or it can also denotes neural network controller, $\phi$ is the abstraction function. This system is essentially an NNCS with abstract function shown in Figure \ref{fig:nncs_abs_function}. The main difference with the general NNCS is the neural network receives abstract state $\phi(s)$ instead of actual state $s$. A trajectory of the system is a sequence in which states and actions occur alternately:
\begin{equation}
s_0, a_0, s_1, a_1, s_2...
\end{equation}
where $s_0 \in S^0$ is a initial state. Each action $a_t$ is determined by state $s_t$, state abstraction function $\phi$ and policy $\pi$: $a_t = \pi(\phi(s_t))$ and $t \in \mathbb{N}$ denotes the time step. Every state $s_t\ (t>0)$ is determined by system dynamics $f$, action $a_{t-1}$ and its precursor state $s_{t-1}$: $s_t = f(s_{t-1}, a_{t-1})$.
We use $G$ and $B$ ($G \cap B = \emptyset$) to represent the set of states in target region and the set of bad states respectively. Verifying the bounded safety properties of the system is equivalent to decide whether there is a trajectory $s_0, a_0, s_1, a_1, s_2,...,s_T$ in which there exists some $0 \le t \le T$ which makes $s_t \in B$, given the time horizon $T$. For the liveness properties, we need to decide whether for every trajectory $s_0, a_0, s_1, a_1, s_2...$ with intial state $s_0 \in S^0$, there exists some $t \ge 0$ which makes $s_t \in G$. Both bounded safety properties and liveness properties can be handled by reachability analysis.
\section{Preliminaries}
\subsection{Deep Reinforcement Learning (DRL)}
\begin{wrapfigure}{r}{0.5\textwidth}
\vspace{-6mm}
\centering
\includegraphics[width=0.48\textwidth]{imgs/drl.pdf}
\caption{The DRL framework}
\vspace{-6mm}
\label{fig:drl}
\end{wrapfigure}
DRL is a deep-learning-based technique for developing intelligent agents, in which Deep Neural Networks (DNNs) are planted and trained to compute optimal actions on system states. Figure \ref{fig:drl} shows the DRL framework. An agent reacts to the environment over time, which is often discretized by a time scale $\delta$~\cite{park2021time}.
It first observes a state $s_t$ from the environment at each time step $t$ and feeds the state into the network to compute a constant action $a_t$.
Next, the agent transits to the successor state $s_{t+1}$ by performing $a_t$ on $s_t$ according to some environment dynamics and receives a reward $r_t$ from the environment. During training, the process is repeated and the parameters in the network are updated to maximize the cumulative reward. Once the training is completed, the network implements a state-action policy function $\pi$ that maps each system state to its optimal action. Note that the state space of a DRL system is usually continuous and infinite.
\begin{example}
\label{exa:DRLexample}
We consider a classic DRL task of training a two-dimensional agent to move from the initial region $x_1 \in [0.7, 0.9]$ and $x_2 \in [0.7, 0.9]$ to a goal region $x_1 \in [-0.3, 0.1]$ and $ x_2 \in [-0.35, 0.5]$.
The dynamics $f$ of the environment is defined as follows ~\cite{2019nonlinear}:
\vspace{-1mm}
\begin{align}
x'_1 = x_1 + (x_2 - x_1^{3}) \cdot \delta\quad \quad
x'_2 = x_2 + u \cdot \delta
\end{align}
\vspace{-5mm}
\noindent The sign $u$ represents the action to take on $x_1,x_2$. Starting from an initial state, the successor state $s'$ is obtained by the dynamics $f$. If $s'$ is not in the goal region, the agent receives a negative immediate reward, otherwise a positive one. A neural network is trained to guide the agent to the goal region by maximizing the cumulative reward. \qed
\end{example}
\vspace{-3mm}
\vspace{-2mm}
\subsection{Reachability Problem of DRL Systems}
\vspace{-1mm}
We formalize the reachability problem of DRL systems. A trained DRL system can be modeled as a 5-tuple ${\cal D}=\langle S,S^0,A,\pi,f\rangle$, where $S$ is the set of $n$-dimensional system states on $n$ continuous variables, $S^0\subseteq S$ is the set of initial states, $A$ is the set of system actions, $\pi:S\rightarrow A$ is a policy function implemented by a DNN, and $f:S\times A\rightarrow S$ is a non-linear continuous environment dynamics. $\cal D$ is essentially a state transition system. A trajectory of $\cal D$ is defined as a sequence
of interleaving states and actions $s_0, a_0, s_1, a_1, s_2, ...$,
where $s_0 \in S^0$ is an initial state. Each action $a_t$ is determined by current state $s_t$ and neural network $\pi$, i.e.,~$a_t = \pi(s_t)$. Every state $s_t\ (t>0)$ is determined by environment dynamics $f$, action $a_{t-1}$ and its preceding state $s_{t-1}$, i.e.,~$s_t = f(s_{t-1}, a_{t-1})$.
Given two states $s,s'\in S$, there is a one-time step transition from $s$ to $s'$, denoted by $s\overset{\pi,f}{\rightarrow} s'$, if there exists an action $a$ such that $a=\pi(s)$ and $s'=f(s,a)$.
\begin{definition}[Reachable state space]
Given a trained DRL system ${\cal D}=\langle S,S^0,A,\pi,f\rangle$, let ${\cal R}_{\cal D}$ be the least set of all the reachable states such that:
\begin{itemize}
\item $S^0\subseteq {\cal R}_{\cal D}$;
\item For all $s,s'\in S$, $s'\in {\cal R}_{\cal D}$ if $s\in {\cal R}_{\cal D}$ and $s\overset{\pi,f}{\rightarrow} s'$.
\end{itemize}
\end{definition}
The reachability problem of a DRL system $\cal D$ is to determine whether an arbitrarily given state $s$ is reachable or not from some state in $S^0$. The problem is undecidable in general because the reachability problem of most nonlinear systems are undecidable \cite{asarin2012low}.
\vspace{-1mm}
\begin{theorem}[Undecidability]
Given a trained DRL system ${\cal D}=\langle S,S^0,A,\pi,f\rangle$ and a state $s\in S$, it is undecidable whether $s\in {\cal R}_{\cal D}$ is true or not.
\end{theorem}
\vspace{-1mm}
\begin{definition}[$t$-step reachability]
Given a trained DRL system ${\cal D}=\langle S,S^0,A,\pi,f\rangle$, a state $s\in S$ and time step $t\ (t\in \mathbb{Z^+})$, $s$ is $t$-step reachable if there exists $s'\in S$ such that $s'$ is $(t-1)$-step reachable and $s=f(s',\pi(s'))$. The states in $S^0$ are $0$-step reachable.
\end{definition}
Let ${\cal R}_{\mathcal{D},t}$ be the set of all the $t$-step reachable states. The bounded reachability problem of $\cal D$ is to compute ${\cal R}_{\mathcal{D},t}$ for a given time step $t$.
When $S^0$ is finite, it is straightforward to compute ${\cal R}_{\mathcal{D},t}$ by computing a bounded list of reachable states from each initial state. However, $S^0$ in DRL is almost always infinite, represented by the intervals for the state variables. The infinity of $S^0$ makes the problem of computing ${\cal R}_{\mathcal{D},t}$ challenging.
\begin{definition}[Reach-Avoid Problem]
\label{def:reach_avoid}
For a trained DRL system $\cal D$, given a set of goal states $S_g$, a set of unsafe states $S_u$ $(S_g~\cap~S_u = \emptyset)$ and a time horizon $T$, the reach-avoid problem is to check whether there exists a trajectory $s_0, a_0, s_1, a_1, s_2, \dots, s_T$ of $\cal D$ such that $\forall \, 0 \le t \le T, s_t \notin S_g$ or $\exists \, 0 \le t \le T, s_t \in S_u$.
\end{definition}
The reach-avoid problem is to search for all trajectories and prove $\cal D$ always reaches a set of goal states and avoids unsafe states within some time horizon $T$. Through over-approximating ${\cal R}_{\mathcal{D},t}$ for $0 \le t \le T$, we can check whether the goal region is eventually reachable and the unsafe region is not reachable.
Definition \ref{def:reach_avoid} is a variant of the reach-avoid problem defined in~\cite{xue2016reach}.
\vspace{-2mm}
\section{Deep Reinforcement Learning with State Abstraction}
Over-approximating the reachable states directly on $\cal D$ is non-trivial as the dual over-approximations are inevitable. In this paper, we try to avoid the over-approximation for the neural network by introducing a specific type of abstraction functions on the basis of $\cal D$.
Given a system state space $S$, we denote $S_\phi$ as a finite set of abstract states (each abstract state represents a possibly infinite set of actual system states in $S$). Let $\phi:S\rightarrow S_\phi$ be an abstraction function that maps each actual state $s$ in $S$ to the corresponding abstract state in $S_\phi$, and $\tau:{S_\phi}\rightarrow {2^S}$ be the inverse concretization function such that $\tau({s_\phi})=\{s|s\in S,\phi(s)=s_\phi \}$.
The basic idea of DRL with state abstraction is to feed the abstract state $\phi(s)$ instead of an actual state $s$ into a neural network to calculate an optimal action. It guarantees that all the actual states in $\tau(s_\phi)$ share a unique action determined by the network~\cite{jin2022cegar}.
During the training phase, the 4-tuple $(\phi(s_t), a_t, r_t, \phi(s_{t+1}))$ is collected to update the neural network. We call a trained DRL system with state abstraction as an abstract-state based DRL (ASDRL) system .
An ASDRL system can be represented by $\mathcal{M} = (S, S^{0},A,\pi, f, \phi)$.
A trajectory of an ASDRL system is also a sequence
of interleaving states and actions $s_0, a_0, s_1, a_1, s_2, ...$. The only difference from the trajectory of $\cal D$ is that each action $a_t$ of $\mathcal{M}$ is determined by first applying $\phi$ to $s_t$ and feeding $\phi(s_t)$ into neural network $\pi$, i.e.,~$a_t = \pi(\phi(s_t))$.
\begin{wrapfigure}{r}{0.5\textwidth}
\vspace{-3ex}
\begin{center}
\includegraphics[width=0.48\textwidth]{imgs/ASDRL.pdf}
\end{center}
\vspace{-3.5ex}
\caption{The ASDRL system}
\label{fig:asdrl}
\vspace{-5.5ex}
\end{wrapfigure}
In general, the goal of state abstraction is to reduce the size or complexity of the state space by grouping together similar states~\cite{abel2019theory}. In~\cite{jin2022cegar}, another significant target of state abstraction is to facilitate the verification of reinforcement learning systems through discretizing the continuous state space $S$ into a finite set of intervals.
Specifically, the abstract state space $S_\phi$ is obtained by dividing each dimension in the original $n$-dimensional state space into a set of intervals, which means each abstract state can be represented as a $2n$-dimensional vector $(l_1,u_1,\dots,l_n,u_n)$.
We also call the $2n$-dimensional vector as an interval box.
In what follows, an interval box is used to represent a set of actual states inside its bound. That is, for a $2n$-dimensional vector $(l_1,u_1,\dots,l_n,u_n)$, we use it to represent the set of $n$-dimensional actual states $\{(x_1, \dots, x_n) \mid l_i \le x_i < u_i, \forall 1 \le i \le n\}$. In this work we divide the state space uniformly for better scalability. More specifically, let $L_i$ and $U_i$ be the lower and upper bounds for the $i$-th dimension value of $S$. We first define the abstraction granularity as an $n$-dimensional vector $\gamma=(d_1,d_2,\ldots,d_n)$. Then the $i$-th dimension will be divided evenly into $(U_i-L_i)/d_i$ intervals. With the interval-based discretization, we define the interval-based abstraction function as follows:
\begin{definition}[Interval-based Abstraction Function]
\label{def:abs_fun}
Given an $n$-dimensional continuous state space $S$ and an abstract state space $S_\phi$ obtained by discretizing $S$ based on an abstraction granularity $\gamma$, for every actual state $s=(x_1, \dots, x_n) \in S$ and abstract state $s_\phi = (l_1,u_1,\dots,l_n,u_n) \in S_\phi$, the interval-based abstraction function $\phi:S\rightarrow S_\phi$ is defined as $\phi(s) = s_\phi$ if and only if for each dimension $1 \le i \le n: l_i \le x_i < u_i$.
\end{definition}
\begin{example}
\vspace{-1ex}
Consider a simple example in Figure~\ref{fig:asdrl}, here we have two state variables $x_1$ and $x_2$ whose the lower and upper bounds are both $0$ and $0.5$, respectively. By choosing the abstraction granularity as $(0.1, 0.1)$, we can have the corresponding abstract state $(0.3, 0.4, 0.2, 0.3)$ of an actual state $(0.35, 0.25)$. With an interval-based abstraction function defined above, we can obtain an ASDRL system depicted in Figure~\ref{fig:asdrl}.
\vspace{-1ex}
\end{example}
\begin{remark}
The differences from the general DRL system lie in the interval-based abstraction function and the neural network. The interval-based abstraction function converts the $n$-dimensional actual state into an abstract state denoted by a $2n$-dimensional vector. Consequently, the number of neurons in the input layer of the neural network needs to be doubled to make $\phi(s)$ as an input. Such differences ensure that the trained network can produce the same action for all the actual states located in the same interval box.
\vspace{-2ex}
\end{remark}
\section{Appendix}
\section{Assessing Verisig 2.0 with Big Weights}
\label{subsec:big_weights}
Verisig 2.0 may fail to verify the reach-avoid problems when dealing with neural networks with big weights.
To demonstrate this,
we initialize the weights of neural network with larger values (random numbers $w_l \sim \mathbf{N}(\mu, \sigma^{2})$ with $\mu =0, \sigma = 0.1$) and show the experimental results in Figure~\ref{fig:big_weights_reachable_sets}.
We observe that the calculated reachable sets contain large over-approximation error except for B4. In Tora, Verisig 2.0 fails to calculate the complete reachable sets due to too large over-approximation error. Hence, it is fairly to say that Verisig 2.0 is sensitive to big weights.
\begin{figure}[h]
\begin{center}
\begin{tabular}{ccc}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/l_b1_Verisig_tanh_3_100.png}
\caption{B1}
\label{fig:b1}
\end{subfigure}&
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/l_b2_Verisig_tanh_3_100.png}
\caption{B2}
\label{fig:b2}
\end{subfigure}&
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/l_b3_Verisig_tanh_3_100.png}
\caption{B3}
\label{fig:b3}
\end{subfigure}
\\
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/l_b4_Verisig_tanh_3_100.png}
\caption{B4}
\label{fig:b4}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/l_b5_Verisig_tanh_4_200.png}
\caption{B5}
\label{fig:b5}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/l_tora_Verisig_tanh_4_100.png}
\caption{Tora}
\label{fig:tora}
\end{subfigure}
\end{tabular}
\end{center}
\vspace{-3ex}
\caption{Assessing Verisig 2.0 on Larger Networks with Big Weights.
Red box: reachable set; Green lines: simulation trajectories; Blue box: goal region;
Purple dashed box: unsafe region.}
\label{fig:big_weights_reachable_sets}
\end{figure}
\section{Benchmarks Setting}
\label{sec:benchmarks}
We provide the setting of seven benchmarks in Table~\ref{tab:benchmarks_setting}. The initial region and goal region are the same as the setting in~\cite{ivanov2021verisig}. The abstraction granularity is a hyper-parameter used in the abstraction-based training and the calculation of reachable sets. All the experimental results in Section~\ref{subsec:scalability} and Section~\ref{subsec:tightness} are based on the following settings.
\begin{table}[htbp]
\centering
\caption{Benchmarks Setting}
\vspace{2mm}
\label{tab:benchmarks_setting}
\renewcommand{\arraystretch}{0.9}
\setlength{\tabcolsep}{2.5pt}
\begin{tabular}{|c|c|l|l|l|}
\hline
\textbf{Task}&\centering \textbf{Abstraction Granularity}&\centering \textbf{Initial Region}&\centering \textbf{Goal Region}&\textbf{Unsafe Region} \\
\hline
B1 & [0.02, 0.02]&\makecell[l]{$x_1 \in [0.8, 0.9]$\\$ x_2 \in [0.5,0.6]$}& \makecell[l]{$x_1 \in [0, 0.2]$\\$ x_2 \in [0.05,0.3]$}&\makecell[l]{$x_1 \in [0.4, 0.7]$\\$ x_2 \in [-0.1,0.2]$}\\
\hline
B2 & [0.1, 0.1]&\makecell[l]{$x_1 \in [0.7, 0.9]$\\$x_2 \in [0.7,0.9]$}& \makecell[l]{$x_1 \in [-0.3, 0.1]$\\$ x_2 \in [-0.35,0.5]$}&\makecell[l]{$x_1 \in [0.12, 0.42]$\\$ x_2 \in [0.1,0.6]$}\\
\hline
B3 & [0.2, 0.2]&\makecell[l]{$x_1 \in [0.8, 0.9]$\\$ x_2 \in [0.4,0.5]$}& \makecell[l]{$x_1 \in [0.2, 0.3]$\\$ x_2 \in [-0.3,-0.05]$}&\makecell[l]{$x_1 \in [0.55, 0.75]$\\$x_2 \in [-0.1,0.1]$}\\
\hline
B4 & [0.2, 0.2, 0.2]&\makecell[l]{$x_1 \in [0.25, 0.27]$\\$ x_2 \in [0.08,0.1]$\\$x_3 \in [0.25, 0.27]$}& \makecell[l]{$x_1 \in [-0.05, 0.05]$\\$ x_2 \in [-0.05,0]$}&\makecell[l]{$x_1 \in [0.05, 0.1]$\\$x_2 \in [0.02,0.04]$}\\
\hline
B5 & [0.1, 0.1, 0.1]&\makecell[l]{$x_1 \in [0.38, 0.4]$\\$ x_2 \in [0.45,0.47]$\\$ x_3 \in [0.25,0.27]$}& \makecell[l]{$x_1 \in [-0.4, -0.28]$\\$ x_2 \in [0.05,0.22]$}&\makecell[l]{$x_1 \in [-0.05, 0.05]$\\$x_2 \in [0.15, 0.25]$}\\
\hline
Tora & [0.2, 0.2, 0.2, 0.2]&\makecell[l]{$x_1 \in [-0.77, -0.75]$\\$ x_2 \in [-0.45,-0.43]$\\$x_3 \in [0.51,0.54]$\\$ x_4 \in [-0.3,-0.28]$}& \makecell[l]{$x_1 \in [-0.1, 0.2]$\\$ x_2 \in [-0.9,-0.6]$}&\makecell[l]{$x_1 \in [-0.25, 0.10]$\\$x_2 \in [0.2, 0.7]$}\\
\hline
ACC & \makecell{[1, 0.1, 0.1,\\ 1, 0.1, 0.1]}&\makecell[l]{$x_1 \in [90, 91]$\\$ x_2 \in [32, 32.05]$\\ $x_4 \in [10, 11]$\\ $x_5 \in [30, 30.05]$\\$x_3, x_6 \in [0, 0]$}&\makecell{$x_2 \in [22.81, 22.87]$\\$x_5 \in [29.88, 30.02]$}&\makecell[l]{$x_2 \in [26, 29]$\\$x_5 \in [30.05, 30.15]$}\\
\hline
\end{tabular}
\end{table}
\section{Tightness Comparison}
\label{subsec:tight_com}
This section presents the tightness comparison results on B3, B4, B5 and ACC in Figure~\ref{fig:reachable_sets_b345}. For B3, B4 and B5, all methods achieve similar results. However, for ACC, a 6-dimensional environment, the sample-based approach ReachNN$^{*}$ produces huge over-approximation error.
\begin{figure}[t]
\begin{minipage}[b]{0.08\linewidth}
\begin{tikzpicture}
\tikz \node [draw] at (0.5,-1) {B3};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
{ \textbf{Our Approach}}\\
\vspace{1ex}
\includegraphics[scale=0.23]{imgs/b3_box_trace_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
{ \textbf{Verisig 2.0}}\\
\vspace{1ex}
\includegraphics[scale=0.23]{imgs/b3_Verisig_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
{ \textbf{ReachNN$^*$}}\\
\vspace{1ex}
\includegraphics[scale=0.23]{imgs/b3_reachNN_tanh_3_100.png}
\end{minipage}
\\
\vspace{1ex}
\begin{minipage}[b]{0.06\linewidth}
\begin{tikzpicture}
\tikz \node [draw] at (0.5,-1) {B4};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b4_box_trace_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b4_Verisig_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b4_reachNN_tanh_3_100.png}
\end{minipage}
\\
\vspace{1ex}
\begin{minipage}[b]{0.06\linewidth}
\begin{tikzpicture}
\tikz \node [draw] at (0.5,-1) {B5};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b5_box_trace_tanh_4_200.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b5_Verisig_tanh_4_200.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b5_reachNN_tanh_4_200.png}
\end{minipage}
\\
\vspace{1ex}
\begin{minipage}[b]{0.06\linewidth}
\begin{tikzpicture}
\tikz \node [draw] at (0.5,-1) {ACC};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/acc_box_trace_tanh_4_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/acc_Verisig_tanh_4_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/acc_reachNN_tanh_4_100.png}
\end{minipage}
\caption{Larger network with Tanh activation function. Red box: reachable set; Green lines: simulation trajectories; Blue box: goal region;
Purple dashed box: unsafe region.}
\label{fig:reachable_sets_b345}
\end{figure}
\section{Differential and Decomposing Analysis Results}
\label{subsec:diff_dec_result}
In this section, we provide the complete differential and decomposing analysis results in Figure~\ref{fig:differential_result} and Figure~\ref{fig:decompose_result}. All of these results are consistent with the conclusion in Section~\ref{subsec:discussion}.
\begin{figure}[htbp]
\begin{center}
\begin{tabular}{ccc}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{imgs/b2_agg_comparison.png}
\caption{B2}
\label{fig:b2_agg_com}
\end{subfigure}&
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{imgs/b3_agg_comparison.png}
\caption{B3}
\label{fig:b3_agg_com}
\end{subfigure}&
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{imgs/b4_agg_comparison.png}
\caption{B4}
\label{fig:b4_agg_com}
\end{subfigure}
\\
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{imgs/b5_agg_comparison.png}
\caption{B5}
\label{fig:b5_agg_com}
\end{subfigure}&
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{imgs/tora_agg_comparison.png}
\caption{Tora}
\label{fig:tora_agg_com}
\end{subfigure}&
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=\textwidth]{imgs/acc_agg_comparison.png}
\caption{ACC}
\label{fig:acc_agg_com}
\end{subfigure}
\\
\end{tabular}
\end{center}
\vspace{-3ex}
\caption{Differential Analysis Results.}
\label{fig:differential_result}
\end{figure}
\begin{figure}[t]
\begin{center}
\begin{tabular}{cccc}
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b1_tanh_abstraction_granularity_com.png}
\caption{B1 (tanh)}
\label{fig:b1_dec_tanh}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b1_relu_abstraction_granularity_com.png}
\caption{B1 (relu)}
\label{fig:b1_dec_relu}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b2_tanh_abstraction_granularity_com.png}
\caption{B2 (tanh)}
\label{fig:b2_dec_tanh}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b2_relu_abstraction_granularity_com.png}
\caption{B2 (relu)}
\label{fig:b2_dec_relu}
\end{subfigure}
\\
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b3_tanh_abstraction_granularity_com.png}
\caption{B3 (tanh)}
\label{fig:b3_dec_tanh}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b3_relu_abstraction_granularity_com.png}
\caption{B3 (relu)}
\label{fig:b3_dec_relu}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b4_tanh_abstraction_granularity_com.png}
\caption{B4 (tanh)}
\label{fig:b4_dec_tanh}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b4_relu_abstraction_granularity_com.png}
\caption{B4 (relu)}
\label{fig:b4_dec_relu}
\end{subfigure}
\\
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b5_tanh_abstraction_granularity_com.png}
\caption{B5 (tanh)}
\label{fig:b5_dec_tanh}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/b5_relu_abstraction_granularity_com.png}
\caption{B5 (relu)}
\label{fig:b5_dec_relu}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/tora_tanh_abstraction_granularity_com.png}
\caption{Tora (tanh)}
\label{fig:tora_dec_tanh}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/tora_relu_abstraction_granularity_com.png}
\caption{Tora (relu)}
\label{fig:tora_dec_relu}
\end{subfigure}
\\
\multicolumn{2}{c}{
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/acc_tanh_abstraction_granularity_com.png}
\caption{ACC (tanh)}
\label{fig:acc_dec_atnh}
\end{subfigure}}&
\multicolumn{2}{c}{
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=\textwidth]{imgs/acc_relu_abstraction_granularity_com.png}
\caption{ACC (relu)}
\label{fig:acc_dec_relu}
\end{subfigure}}
\\
\end{tabular}
\end{center}
\vspace{-3ex}
\caption{Decomposing Analysis Results. Due to the space reason, for B1-B5 and Tora, we use a scalar value $x_1$ to denote the $n$-dimensional abstraction granularity vector $\gamma = (x_1,...,x_1)$. For ACC, we use a 3-dimensional vector $(x_1, x_2, x_3)$ to denote the 6-dimensional abstraction granularity vector $\gamma = (x_1,x_2,x_3,x_1,x_2,x_3)$}
\label{fig:decompose_result}
\end{figure}
\section{Related Work and Conclusion}
\vspace{-2mm}
Many reachability analysis approaches have been developed for DRL systems. Most of them are focused on reducing the overestimation that is caused by the over-approximation of the embedded neural networks.
Tran \textit{et al.}~\cite{tran2019safety,tran2020nnv} proposed an approach called NNV by directly combining the star set analysis technique~\cite{tran2020verification} used to deal with the neural network with CORA~\cite{althoff2015introduction}, a reachability analysis tool for non-linear systems. This approach produces large over-approximation error since it omits the dependencies between the neural network inputs and outputs.
To obtain tighter reachable sets, a number of methods have been proposed to capture the input-output relation of neural networks recently. One class of these methods is based on set propagation.
Ivanov \textit{et al.}~\cite{ivanov2020verifying,ivanov2021verisig} proposed a method (i.e., Verisig and Verisig 2.0) that propagates Taylor models layer by layer through the neural network to over-approximate the input-output relation.
Schilling \textit{et al.}~\cite{schilling2022verification} utilized the zonotope propagation technique to analyze the neural network and presented a method for inter-conversion of zonotope and Taylor model, enabling the combination of the reachability tools used for neural networks and dynamical systems.
Another class of methods tries to approximate a neural network as a whole directly.
Huang \textit{et al.}~\cite{huang2019reachnn,fan2020reachnn} proposed an approach (i.e., ReachNN and ReachNN$^{*}$) to
abstract the neural networks with differentiable or non-differentiable activation functions through Bernstein polynomials. Dutta \textit{et al.} ~\cite{dutta2019reachability} proposed Sherlock based on rule generation to compute a polynomial approximation that abstracts the neural network with ReLU activation functions. Because these methods need to sample from the input space of the neural network, it becomes inefficient when handling high-dimensional environments.
We have presented \textsf{BBReach}, a tight and scalable abstraction-based reachability analysis approach for DRL systems. \textsf{BBReach}~leverages the state-of-the-art training technique for neural networks with state abstraction
and efficiently computes tight reachable sets with our proposed
adjacent interval aggregation
to avoid the over-approximation for neural networks. We have experimentally demonstrated that \textsf{BBReach}~is compatible to neural networks with arbitrary activation functions, scalable to large neural networks, and
efficient in
reducing the number of system states while
computing reachable sets.
We plan to
explore other possibilities for further reducing the over-approximation error, e.g., by using
more complex abstract domains such as zonotope.
\section{Experiments}
\vspace{-1mm}
In this section we conduct a comprehensive
assessment of \textsf{BBReach}~and the state-of-the-art tools.
Our goal is mainly threefold: to demonstrate (i) \textsf{BBReach}'s high scalability and efficiency (Section~\ref{subsec:scalability}), (ii) its tightness with respect to verification results (Section \ref{subsec:tightness}), and (iii) the effectiveness of the adjacent interval aggregation (Section~\ref{subsec:discussion}).
Additionally, we evaluate how \textsf{BBReach}~performs under
different abstraction granularity levels (Section~\ref{subsec:discussion}).
\vspace{-4mm}
\subsection{Implementation and Benchmarks}
\vspace{-1mm}
\noindent \textbf{Implementation.}
We implement our
approach in a
tool called \textsf{BBReach}~in Python. We use the
SciPy~\cite{2020SciPy-NMeth} package as an optimization solver. Additionally, we employ the parallelized computing by initial-set partition~\cite{chen2012taylor}, a standard approach used in the reachability analysis of hybrid systems to obtain tighter bounds of true reachable states.
With the initial set partitioned into $k$ subsets,
the $k$ sub-problems can be solved in parallel, which accelerates our approach
with multiple cores.
\vspace{0.5ex}
\noindent \textbf{Competitors.}
We compare our tool with the state-of-the-art tools, namely Verisig 2.0~\cite{ivanov2021verisig}, which is an approach based on Taylor model propagation, and ReachNN$^{*}$~\cite{fan2020reachnn}, which approximates neural networks through polynomial regression.
\vspace{0.5ex}
\noindent \textbf{Benchmarks and Properties.}
The selected benchmarks include 7 reinforcement learning tasks with dimensions ranging from 2 to 6 from Verisig 2.0.
For each task, we train four neural networks with different activation functions and the size of neurons and consequently obtain 28 instants totally. It is worth mentioning that for each instant, the weights in the network used for \textsf{BBReach}~ are different from those for Verisig 2.0 and ReachNN$^*$ due to the abstract training approach.
Nevertheless, we guarantee that all the trained systems can achieve the best rewards for the same task. In particular, we initialize the neural networks with smaller weights as otherwise Verisig 2.0 would introduce larger over-approximations
(see our observations in Appendix~\ref{subsec:big_weights}).
We also train the networks under different abstraction granularity levels to evaluate how the abstraction granularity affects the efficiency of \textsf{BBReach}.
We set a reach-avoid problem for each task by specifying its goal region and unsafe region. The detailed settings
are given in Table~\ref{tab:benchmarks_setting}, Appendix~\ref{sec:benchmarks}.
\vspace{0.5ex}
\noindent \textbf{Experimental Setup.}
All experiments are conducted on a workstation running Ubuntu 18.04 with 32 cores AMD Ryzen Threadripper CPU @ 3.7GHz and 128GB RAM.
\vspace{-3mm}
\subsection{Comparison on Scalability and Efficiency }
\label{subsec:scalability}
\vspace{-1mm}
\begin{table}[t]
\centering
\footnotesize
\setlength{\tabcolsep}{3pt}
\caption{Comparison on verification time (s) and result (verified or not).}
\label{tab:time_comparison}
\renewcommand{\arraystretch}{0.9}
\begin{tabular}{|c|c|r|r|r|c|r|r|c|r|c|}
\hline
\multirow{2}{*}{\textbf{Task}}& \multirow{2}{*}{\textbf{Dim}}& \multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Network}}}&\multicolumn{3}{c|}{\textbf{\textsf{BBReach}}}&\multicolumn{3}{c|}{\textbf{Verisig 2.0}}&\multicolumn{2}{c|}{\textbf{ReachNN$^{*}$}}\\
\cline{4-11}
~&~&~&\textbf{1 core}&\textbf{20 cores}&\textbf{VR}&\textbf{1 core} &\textbf{20 cores}&\textbf{VR}& \textbf{Default}&\textbf{VR}\\
\hline
\hline
\multirow{4}{*}{B1}&\multirow{4}{*}{2}&Tanh($2 \times 20$)&36.0&2.17&\checkmark&45&38&\checkmark&60 &Unk \\
~&~&Tanh($3 \times 100$)&40.7&2.34&\checkmark&413&125&\checkmark&162&\checkmark\\
\cline{3-11}
~&~&ReLU($2 \times 20$)&34.1&2.25&\checkmark&---&---&\multirow{2}{*}{N/A}&20&\checkmark\\
~&~&ReLU($3 \times 100$)&105.4&6.56&\checkmark&---&---&&330&Unk\\
\hline
\multirow{4}{*}{B2}&\multirow{4}{*}{2}&Tanh($2 \times 20$)&1.0&0.33&\checkmark&5.2&4.1&Unk&71&Unk\\
~&~&Tanh($3 \times 100$)&1.3&0.43&\checkmark&195&49&\checkmark&172&Unk\\
\cline{3-11}
~&~&ReLU($2 \times 20$)&3.5&1.04&\checkmark&---&---&\multirow{2}{*}{N/A}&4&\checkmark\\
~&~&ReLU($3 \times 100$)&5.2&1.44&\checkmark&---&---&& 6647&Unk\\
\hline
\multirow{4}{*}{B3}&\multirow{4}{*}{2}&Tanh($2 \times 20$)&10.2&0.94&\checkmark&36&28&\checkmark &115&\checkmark\\
~&~&Tanh($3 \times 100$)&10.3&0.94&\checkmark&394&102&\checkmark&93&\checkmark\\
\cline{3-11}
~&~&ReLU($2 \times 20$)&10.1&0.95&\checkmark&---&---&\multirow{2}{*}{N/A}&69&\checkmark\\
~&~&ReLU($3 \times 100$)&0.9&0.14&\checkmark&---&---&& 10321&Unk\\
\hline
\multirow{4}{*}{B4}&\multirow{4}{*}{3}&Tanh($2 \times 20$)&0.8&0.35&\checkmark&7&5.1&\checkmark&17&\checkmark \\
~&~&Tanh($3 \times 100$)&1.0&0.44&\checkmark&209&35&\checkmark&20&\checkmark\\
\cline{3-11}
~&~&ReLU($2 \times 20$)&1.0&0.44&\checkmark&---&---&\multirow{2}{*}{N/A}&7&\checkmark\\
~&~&ReLU($3 \times 100$)&0.5&0.24&\checkmark&---&---&&18&\checkmark\\
\hline
\multirow{4}{*}{B5}&\multirow{4}{*}{3}&Tanh($3 \times 100$)&14.8&0.95&\checkmark&157&44&\checkmark&27&Unk \\
~&~&Tanh($4 \times 200$)&9.6&0.67&\checkmark&2264&232&\checkmark&2344&Unk\\
\cline{3-11}
~&~&ReLU($3 \times 100$)&9.9&0.64&\checkmark&---&---&\multirow{2}{*}{N/A}&90&\checkmark\\
~&~&ReLU($4 \times 200$)&9.7&0.76&\checkmark&---&---&& 2845&Unk\\
\hline
\multirow{4}{*}{Tora}&\multirow{4}{*}{4}&Tanh($3 \times 20$)&814.3&44.25&\checkmark&69&46&\checkmark&1610&\checkmark \\
~&~&Tanh($4 \times 100$)&843.9&46.17&\checkmark&---&---&DNF&---&DNF\\
\cline{3-11}
~&~&ReLU($3 \times 20$)&230.2&14.11&\checkmark&---&---&\multirow{2}{*}{N/A}&778&\checkmark\\
~&~&ReLU($4 \times 100$)&257.1&14.72&\checkmark&---&---&&---&DNF\\
\hline
\multirow{4}{*}{ACC}&\multirow{4}{*}{6}&Tanh($3 \times 20$)&1.7&0.38&\checkmark&113&50&\checkmark&5498&Unk \\
~&~&Tanh($4 \times 100$)&1.8&0.38&\checkmark&3304&410&\checkmark&---&DNF\\
\cline{3-11}
~&~&ReLU($3 \times 20$)&2.0&0.42&\checkmark&---&---&\multirow{2}{*}{N/A}&4633&Unk\\
~&~&ReLU($4 \times 100$)&2.0&0.48&\checkmark&---&---&&---&DNF \\
\hline
\end{tabular}
\begin{tablenotes}
\small \item \textbf{Remarks.}
VR: verification result;
Tanh/ReLU$(n \times k)$: a neural network with the activation function Tanh/ReLU, $n$ hidden layers, and $k$ neurons per hidden layer; N/A: not applicable;
$\checkmark$: the reach-avoid problem is successfully verified;
Unk: the reach-avoid problem could not be verified due to large over-approximation error; DNF: the calculation did not finish; ---: no time data available due to N/A or DNF.
\end{tablenotes}
\vspace{-3ex}
\end{table}
The verification results and time costs (in seconds) are shown in Table \ref{tab:time_comparison}. It is observed that our approach succeeds in verifying all 28 instants with any neural network configuration.
Verisig 2.0 succeeds in 12 instants, and ReachNN$^{*}$ in 13 instants.
The two tools report 1 and 11 unknown cases (marked by Unk), respectively.
The unknown cases occur the tools cannot compute a set of reachable states that are completely covered by the goal region. These cases can be verified by \textsf{BBReach}, which means that our approach introduces less over-approximations than the other two tools.
Particularly in Tora case, the two tools did not finish calculating the complete reachable sets (marked by DNF) due to the huge over-approximation error.
In ACC, the process of ReachNN$^{*}$ was killed due to the memory limitation when processing larger neural networks ($4\times 100$).
Note that Verisig 2.0 is not applicable to the systems driven by ReLU neural networks, and we mark the corresponding results by N/A.
Regarding efficiency, our approach costs the least time when parallelization is enabled. Even for single core computation, it takes less time than Verisig 2.0 and ReachNN$^{*}$ in most cases. The advantage of verification time becomes more notable when dealing with larger networks. Because we treat the neural network as a black-box mapping instead of over-approximating it layer by layer, the verification time of \textsf{BBReach}~with larger neural networks is about the same as the smaller neural networks, or even less. However, the verification time of Verisig 2.0 and ReachNN$^{*}$ both increases significantly in most cases when dealing with larger neural networks.
Based on the above analysis, \textsf{BBReach}~demonstrates substantial improvement over Verisig 2.0 and ReachNN$^{*}$ with respect to computation efficiency. Furthermore, \textsf{BBReach}~scales well with the size of the neural network controller.
\vspace{-3mm}
\subsection{Comparison on the Tightness }
\label{subsec:tightness}
\vspace{-1mm}
\begin{figure}[t]
\begin{minipage}[b]{0.08\linewidth}
\begin{tikzpicture}
\tikz \node [draw] at (0.3,-1) {B1};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
{ \textbf{\textsf{BBReach}}}\\
\vspace{1ex}
\includegraphics[scale=0.23]{imgs/b1_box_trace_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
{ \textbf{Verisig 2.0}}\\
\vspace{1ex}
\includegraphics[scale=0.23]{imgs/b1_Verisig_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
{ \textbf{ReachNN$^*$}}\\
\vspace{1ex}
\includegraphics[scale=0.23]{imgs/b1_reachNN_tanh_3_100.png}
\end{minipage}
\\
\vspace{1ex}
\begin{minipage}[b]{0.06\linewidth}
\begin{tikzpicture}
\tikz \node [draw] at (0.4,-1) {B2};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b2_box_trace_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b2_Verisig_tanh_3_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/b2_reachNN_tanh_3_100.png}
\end{minipage}
\\
\vspace{1ex}
\begin{minipage}[b]{0.06\linewidth}
\begin{tikzpicture}
\tikz \node [draw] at (0.5,-1) {Tora};
\end{tikzpicture}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/tora_box_trace_tanh_4_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/tora_Verisig_tanh_4_100.png}
\end{minipage}
\begin{minipage}[b]{0.3\linewidth}
\centering
\includegraphics[scale=0.23]{imgs/tora_reachNN_tanh_4_100.png}
\end{minipage}
\vspace{-1ex}
\caption{Larger network with the Tanh activation function. Red box: reachable set; Green lines: simulation trajectories; Blue box: goal region;
Purple dashed box: unsafe region.}
\label{fig:reachable_sets}
\vspace{-4ex}
\end{figure}
We compare the tightness of the over-approximation results of true reachable states by plotting the reachable sets computed by different methods and the corresponding simulation trajectories. We present the results of B1, B2 and Tora computed by different methods when dealing with larger neural networks with the Tanh activation functions in Figure \ref{fig:reachable_sets}. The results for the other benchmarks are given in Appendix~\ref{subsec:tight_com}.
In Figure \ref{fig:reachable_sets}, the red boxes represent the reachable sets calculated by different methods at each time step. The green lines are the simulation trajectories and the blue box denotes the preset goal region. If there exists a red box completely inside the blue box and all red boxes disjoint from the purple box, the verification succeeds. The more overlap between the green trajectories and the red boxes indicates the more accurate calculation of the reachable sets. In most scenarios, our approach achieves comparable results to Verisig 2.0 and ReachNN$^{*}$, such as B1, B3, B4, and B5 (we defer the comparison results for B3, B4, and B5 to Appendix~\ref{subsec:tight_com}).
Regarding B2, our approach performs better than ReachNN$^{*}$ but worse than Verisig 2.0. This
is due to the large variation in the decisions made by the corresponding neural networks. As depicted by the simulation trajectories in Figure~\ref{fig:reachable_sets}, the agent enters the goal region from different directions, i.e., bottom for the system experimented by \textsf{BBReach}\ and right for the other one.
In this case, \textsf{BBReach}\ and Verisig 2.0 successfully verified the eventual reachability of the goal region while ReachNN$^{*}$ failed.
For Tora, our approach significantly surpasses the competitors:
both Verisig 2.0 and ReachNN$^{*}$ did not finish the calculation
due to the large over-approximation error. In particular, the resulting
bound of action, upon Verisig 2.0's termination, is (-95046891, 95048286) which is too large for Verisig 2.0 to continue the calculation.
In summary, our approach \textsf{BBReach}~is more stable with no huge over-approximation error as in Tora and sufficiently tight to verify all the properties.
\begin{comment}
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b1_box_trace_tanh_3_100.png}
\caption{B1(Our Approach)}
\label{fig:b1_1}
\end{subfigure}&
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b1_Verisig_tanh_3_100.png}
\caption{B1(Verisig 2.0)}
\label{fig:b1_2}
\end{subfigure}&
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b1_reachNN_tanh_3_100.png}
\caption{B1(ReachNN$^{*}$)}
\label{fig:b1_3}
\end{subfigure}
\\
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b2_box_trace_tanh_3_100.png}
\caption{B2(Our Approach)}
\label{fig:b2_1}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b2_Verisig_tanh_3_100.png}
\caption{B2(Verisig 2.0)}
\label{fig:b2_2}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b2_reachNN_tanh_3_100.png}
\caption{B2(ReachNN$^{*}$)}
\label{fig:b2_3}
\end{subfigure}
\\
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b3_box_trace_tanh_3_100.png}
\caption{B3(Our Approach)}
\label{fig:b3_1}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b3_Verisig_tanh_3_100.png}
\caption{B3(Verisig 2.0)}
\label{fig:b3_2}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b3_reachNN_tanh_3_100.png}
\caption{B3(ReachNN$^{*}$)}
\label{fig:b3_3}
\end{subfigure}
\\
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b4_box_trace_tanh_3_100.png}
\caption{B4(Our Approach)}
\label{fig:b4_1}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b4_Verisig_tanh_3_100.png}
\caption{B4(Verisig 2.0)}
\label{fig:b4_2}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b4_reachNN_tanh_3_100.png}
\caption{B4(ReachNN$^{*}$)}
\label{fig:b4_3}
\end{subfigure}
\\
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b5_box_trace_tanh_4_200.png}
\caption{B5(Our Approach)}
\label{fig:b5_1}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b5_Verisig_tanh_4_200.png}
\caption{B5(Verisig 2.0)}
\label{fig:b5_2}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/b5_reachNN_tanh_4_200.png}
\caption{B5(ReachNN$^{*}$)}
\label{fig:b5_3}
\end{subfigure}
\\
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/tora_box_trace_tanh_4_100.png}
\caption{Tora(Our Approach)}
\label{fig:tora_1}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/tora_Verisig_tanh_4_100.png}
\caption{Tora(Verisig 2.0)}
\label{fig:tora_2}
\end{subfigure} &
\begin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{imgs/tora_reachNN_tanh_4_100.png}
\caption{Tora(ReachNN$^{*}$)}
\label{fig:tora_3}
\end{subfigure}
\end{tabular}
\end{center}
\caption{Larger Network with Tanh Activation Function}
\label{fig:reachable_sets}
\end{figure}
\end{comment}
\begin{figure}[t]
\centering
\begin{minipage}[t]{0.49\textwidth}
\centering
\includegraphics[width=5.5cm]{imgs/b1_agg_comparison.png}
\vspace{-2mm}
\caption{Differential Analysis on B1}
\label{fig:ana_adj_agg}
\end{minipage}\hspace{0ex}
\begin{minipage}[t]{0.49\textwidth}
\centering
\includegraphics[width=5.5cm]{imgs/b1_abstraction_granularity_com.png}
\vspace{-2mm}
\caption{Decomposing analysis on B1 (Tanh)}
\label{fig:eff_ana_b1_tanh}
\end{minipage}
\vspace{-4mm}
\end{figure}
\vspace{-3mm}
\subsection{Differential and Decomposing Analysis}
\label{subsec:discussion}
\vspace{-1mm}
\noindent \textbf{Differential Analysis.}
To demonstrate the significance of the adjacent interval aggregation in Algorithm~\ref{alg:interval_agg}, we measure the growth rate of the number of interval boxes also with no aggregation.
Figure~\ref{fig:ana_adj_agg} shows the results on the B1 benchmark.
We observe that the number of interval boxes grows rapidly with no aggregation, which implies a dramatically increased verification overhead. With the adjacent interval aggregation, the number of interval boxes is extremely small and stable. Hence, it is fair to conclude that the adjacent interval aggregation is a substantial benefit to \textsf{BBReach}.
The results on the other six benchmarks are similar and provided in Appendix~\ref{subsec:diff_dec_result}.
\vspace{1ex}
\noindent \textbf{Decomposing Analysis.}
We evaluate how different abstraction granularity levels affect the performance of \textsf{BBReach}~and its components.
Abstraction granularity is a crucial hyper-parameter used in both training and calculation of reachable sets. To better understand the impact of abstraction granularity, for each benchmark, in addition to the abstraction granularity level in Table~\ref{tab:benchmarks_setting}, we choose three finer abstraction granularity levels and three coarser abstraction granularity levels, respectively, to evaluate the efficiency on both tanh and relu neural networks. We also measure the time consumed by each of the three steps, i.e., interval segmentation, interval-based over-approximation, and adjacent interval aggregation.
For illustration, we present in Figure~\ref{fig:eff_ana_b1_tanh} the results for B1 with the tanh neural network. Regarding the 1-core results, we observe that, as the abstraction granularity becomes coarse-grained, the verification time decreases and eventually stabilizes; on the other hand, a fairly fine-grained abstraction granularity, e.g., (0.002, 0.002), could result in much higher verification overhead.
We also observe that the interval-based over-approximation and the adjacent interval aggregation take most of the verification time while the
overhead of the interval segmentation is negligible. Finally, as expected, the parallelization (with 20 cores) can
significantly accelerate \textsf{BBReach}. The evaluation results are similar as in the other six cases, which we enclose in Appendix \ref{subsec:diff_dec_result}.
\section{Introduction}
Modern AI-empowered software
systems
such as autonomous driving
are typically developed by utilizing Deep Reinforcement Learning (DRL) \cite{shalev2016safe}.
In such a system,
the well-trained
Deep Neural Networks (DNNs)
determine the optimal actions during its interactions
with the surroundings.
Due to the lack of interpretability~\cite{fan2021interpretability,yampolskiy2020unexplainability} and the vulnerability to adversarial samples~\cite{szegedy2014intriguing} for
DNNs, concerns have recently been raised about the safety and reliability of DRL~\cite{gomes2016will,schmidt2021can}.
Hence, providing provable safety guarantees for DRL systems prior to their deployments is a key challenge for DRL's
application in safety-critical settings.
Reachability analysis, as one of the powerful formal methods, has been applied to the verification of continuous \cite{bertsekas1971minimax} and hybrid \cite{alur1995algorithmic} systems since the 1990s.
To cite a few, its successful applications include
invariant checking~\cite{rungger2017computing}, robust control~\cite{limon2005robust,schurmann2018reachset}, fault detection~\cite{scott2016constrained,su2017model}, and set-based predication~\cite{althoff2016set,pereira2017overapproximative}.
More recently, reachability analysis has been demonstrated to be an effective approach to the verification of DRL systems~\cite{dutta2019reachability,fan2020reachnn,ivanov2021verisig}.
The essence of reachability analysis is to compute all the reachable system states from the given initial state(s).
In particular, for a DRL system,
one must take every state $s$ from a given set
of states, feed $s$ to the planted DNN to determine the corresponding action $a$ on $s$, and compute the
subsequent state $s'$ by applying $a$ to $s$ according to usually nonlinear system dynamics.
Compared to continuous and hybrid systems, it is significantly more challenging to compute reachable states for DRL systems due to the embedded complex and inexplicable DNNs.
In addition to over-approximating the nonlinear system dynamics~\cite{chen2013flow,frehse2011spaceex,lygeros1999controllers}, one also over-approximates the embedded DNNs for computing the reachable states of DRL systems.
Specifically,
given a set $S_i$ of continuous system states at some step $i$, one (i)
overestimates a set $A_i$ of actions that are applied to
$S_i$ by over-approximating the neural network on $S_i$, and then (ii)
overestimates a set of subsequent states by applying $A_i$ to $S_i$ using the over-approximated system dynamics.
Such \emph{dual over-approximations} inevitably introduce large overestimation which accumulates step by step, resulting in a considerable number of unreachable states in the computed reachable sets.
Additionally,
the state-of-the-art (dual)
over-approximation approaches \cite{ivanov2021verisig,dutta2019reachability,fan2020reachnn} are also restricted to
certain types of neural network architectures and activation functions, as well as the network size \cite{huang2019reachnn}.
For instance,
Verisig 2.0~\cite{ivanov2021verisig} does not support the neural networks with the ReLU activation functions; Sherlock~\cite{dutta2019reachability} is applicable only to ReLU-based networks; ReachNN$^{*}$~\cite{fan2020reachnn} is not scalable with respect to the network size and more overestimation would be introduced for larger networks.
\vspace{1ex}
\noindent \textbf{Our Approach.}
In this paper we propose a novel, tight and scalable reachability analysis approach for DRL systems.
Our approach leverages the recent abstraction-based DRL training method~\cite{jin2022cegar}, by which
the DNN planted in a DRL system can admit an abstract state
and produce a unique action.
Intuitively, an abstract state represents a (probably infinite) set of actual system states. Given a system state space $S$ and a set of states represented by an interval $I\subseteq S$, our approach proceeds as follows: (i) discretize $S$ into a finite set of abstract states that are represented by interval boxes; (ii) segment $I$ into a set of sub-intervals $B_I$ and feed the abstract state enclosing each interval $\mathcal{I}$ in $B_I$ into the neural network to compute the action $a$; and (iii)
apply action $a$ to $\mathcal{I}$ to compute its successor interval $\mathcal{I'}$. The concretization of $\mathcal{I'}$ is an over-approximation set of the actual states which are successors from those inside $\mathcal{I}$.
This process allows us to treat the DNN as a black-box oracle, which consequently avoids over-approximating it to compute the action for a set of actual states.
No over-approximation to the neural network significantly reduces
the over-approximation error of computed reachable sets. Moreover, it renders our approach agnostic to the size, the architecture, and the activation function of a neural network.
However, this abstraction-based approach may still suffer from the notorious state-explosion problem with the increasing depth of system transitions. To address this challenge, we propose a novel \emph{adjacent interval aggregation} algorithm to limit the expansion of the number of intervals, thus making the calculation of reachable sets scalable over a longer time horizon. By merging the adjacent intervals, we can significantly restrain the
growth in the
number of
computed intervals, guarantee
no much overestimation introduced, and thus
reach a balance between
the number of intermediate intervals and the overestimation imposed by the aggregation.
Moreover, by partitioning the initial set, we enable a parallelizing optimization of our approach that accelerates
the problem solving.
\vspace{1ex}
\noindent \textbf{Main Contributions.}
We provide: (i) a novel abstraction-based black-box reachability analysis approach for DRL systems, together with an adjacent interval aggregation algorithm to cope with the state explosion;
(ii) an efficient prototype, called \textsf{BBReach}, with the optimization of parallelizing the computation; and (iii) an extensive assessment of \textsf{BBReach}~on a wide range of benchmarks, demonstrating
its outperformance over the state-of-the-art tools, the effectiveness of the adjacent interval aggregation algorithm, and the impact of the granularity of the abstractions.
\section{The Abstraction-based Reachability Analysis Approach}
\vspace{-1mm}
In this section we first present the overall process of over-approximating the reachable states of an ASDRL system with the abstraction function specified in Definition~\ref{def:abs_fun}. We then discuss three techniques, namely interval-based over-approximation, interval segmentation, and adjacent interval aggregation, which are used to achieve efficient calculation and to limit both the increase of computed intermediate intervals and the over-approximation error.
The three techniques are all operations defined on interval boxes, which constitute a whole abstract state space.
\vspace{-1mm}
\subsection{Overview of Our Approach}
\begin{wrapfigure}{r}{0.55\textwidth}
\vspace{-5ex}
\begin{algorithm}[H]
\caption{Reachable sets calculation}
\label{alg:reachsets_cal}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{ASDRL $\mathcal{M} = (S, S^{0},A,\pi, f, \phi)$\\ Time horizon $T$}
\Output{Reachable sets}
Compute an interval $I_0$ satisfying $S^0 \subseteq I_0$\\
$X_0 = [I_0]$\\
\For{t = 1,..., T}{
interval\_arr = []\\
\For{$I$ in $X_{t-1}$}{
segmented\_intervals = segment($I$, $\phi$)\\
\For{$\mathcal{I}$ in segmented\_intervals}{
interval\_arr.append($post(\mathcal{I})$)
}
}
$X_t$ = aggregate(interval\_arr)
}
\Return $X_0, X_1,..., X_T$
\end{algorithm}
\vspace{-5ex}
\end{wrapfigure}
Given an ASDRL system $\mathcal{M}$ and a time horizon $T$, our reachability analysis aims to compute a sequence of reachable sets $X_0, X_1, X_2,...X_T$ in which $X_{t+1} = aggregate(post(X_t))$ for $0 \le t < T$ to over-approximate the true reachable states of $\mathcal{M}$. Each reachable set $X_t$ consists of a set of interval boxes enclosing the true reachable states at time step $t$. The $post$ operator computes a set of interval boxes that encloses all reachable states after one time step from a given set of states, while it can also be used to compute the successor state of a given state.
In our approach, the $post$ operation is executed by formulating it as an optimization problem when the input is an infinite set. However, the formulated optimization problem cannot be directly solved due to the neural network component and abstraction function. To deal with this issue, we present the interval segmentation technique to transform the original optimization problem into several simplified problems. However, interval segmentation will lead to a rapid increase in the number of interval boxes, and hence, we use the $aggregate$ operation to limit the expansion of number of interval boxes to make our approach scalable over a longer time horizon.
\begin{figure}[t]
\centering
\includegraphics[width=1\textwidth]{imgs/reachability_overview.pdf}
\vspace{-5mm}
\caption{An example of over-approximating one-step reachable states}
\vspace{-4ex}
\label{fig:onestep}
\end{figure}
The overall computation process of reachable sets is presented in Algorithm~\ref{alg:reachsets_cal}. It is an iterative process where each iteration calculates the reachable set at next time step based on the current reachable set.
Figure~\ref{fig:onestep} depicts the process of one iteration. The abstraction function $\phi$ depicted by the black dotted mesh in Figure~\ref{fig:onestep} divides the state space into a set of interval boxes (i.e. the smallest black dotted boxes), each of which represents an abstract state. In Figure~\ref{fig:onestep}(a), suppose the orange box denotes the reachable set $X_{t-1} = \{I\}$. To compute $X_t$, we divide $I$ into a set of smaller interval boxes denoted by different colors according to $\phi$ illustrated in Figure~\ref{fig:onestep}(b), which is executed in line 6 of Algorithm~\ref{alg:reachsets_cal}.
Then for each interval box in Figure~\ref{fig:onestep}(b), we over-approximate the reachable states from it after one time step in line 8. For instance, through calculating the corresponding abstract state $\mathbf{s}_{\mathcal{I}^1}$ of interval box $\mathcal{I}^1$ and feed $\mathbf{s}_{\mathcal{I}^1}$ into the neural network, the action is determined as a constant without any over-approximation error for the neural network $\pi$, which renders our approach the black-box feature. After the over-approximation process for the system dynamics $f$, the over-approximation result $post(X_{t-1})$ including four interval boxes is shown in Figure~\ref{fig:onestep}(c). Finally, in line 9 we aggregate the interval boxes adjacent to each other and obtain the larger interval boxes denoted by the orange boxes in Figure~\ref{fig:onestep}(d). The aggregation result constitutes the set $X_t$. Repeating the above process can produce the sequence of reachable sets.
\vspace{-3mm}
\subsection{Over-Approximation of Environmental Dynamics}
\label{sec:overapp}
\vspace{-1mm}
Given a set of interval boxes $X_t$, we need to compute $post(X_t)$ by over-approximating the reachable states starting from all interval boxes in $X_t$ after one time step owing to the non-linearity of system dynamics $f$. By parameterizing the bound of a reachable set, one can solve an optimization problem minimizing or maximizing a specific metric under the constraint that all solutions have to be enclosed~\cite{althoff2021set}.
\vspace{-2ex}
\paragraph{\textbf{Interval-Based Over-Approximation.}} Therefore, we formulate the calculation of $post(X_t)$ as optimization problems defined on each interval box in $X_t$. Specifically, for each interval box $I_{X_t}$ in $X_t$, we can compute the upper and lower bounds of $post(I_{X_t})$ for each dimension $i$ by solving the following two optimization problems respectively:
\vspace{-1ex}
\begin{equation}
\label{eq:opequation}
\begin{aligned}
\mathop{\arg\max}\limits_{s_t\in I_{X_t}}\ \ v_i \cdot f(s_t,\pi(\phi({s_t}))) \quad \quad
\mathop{\arg\min}\limits_{s_t\in I_{X_t}}\ \ v_i \cdot f(s_t,\pi(\phi({s_t})))
\end{aligned}
\vspace{-1ex}
\end{equation}
\noindent where $v_i$ is a one-hot vector with only the $i$-th element being $1$ and the value of other elements are 0. Based on the above computation, we can obtain the reachable set after one time step as $post(X_t) = \bigcup\limits_{I_{X_t} \in X_t} post(I_{X_t})$.
However, the difficulty in these optimization problems is that the objective function $v_i \cdot f(s_t,\pi(\phi({s_t})))$ is non-differentiable and discontinuous, which implies that these optimization problems cannot be solved directly.
\vspace{-2ex}
\paragraph{\textbf{Interval Segmentation.}} As noted above, we compute the reachable sets by solving the optimization problems in expression (\ref{eq:opequation}). However, notice that in these optimization problems, $\pi(\phi(s_t))$ is related to both the neural network controller and the abstraction function which means $\pi(\phi(s_t))$ may not be a continuous and differentiable function. To tackle this problem, we propose to use interval segmentation: divide each interval box $I_{X_t}$ in $X_t$ based on abstraction function $\phi$.
\begin{wrapfigure}{r}{0.5\textwidth}
\vspace{-6ex}
\includegraphics[width=0.48\textwidth]{imgs/interval_seg.pdf}
\vspace{-1ex}
\caption{Interval segmentation.}
\label{fig:interval_segmentation}
\vspace{-6ex}
\end{wrapfigure}
Based on the abstraction function we defined, the continuous state space is divided into a set of abstract states $S_\phi$ in which each abstract state is an interval box. Each interval box $I_{X_t}$ may intersect with multiple abstract states. Considering that the neural network outputs the same action on the actual states corresponding to a same abstract state, we segment $I_{X_t}$ into a set of sub-intervals in which all the actual states within the same sub-interval correspond to a same abstract state as depicted in Figure~\ref{fig:interval_segmentation}. Specifically, we define $\mathbf{S}_{I_{X_t}} = \{s_\phi \mid s_\phi \cap I_{X_t} \neq \emptyset \}$ as the set of abstract states that intersect with the interval box $I_{X_t}$ firstly. Based on the abstraction function, we divide $I_{X_t}$ into a set of interval boxes $\mathbf{B}_{I_{X_t}} = \{\mathcal{I}_{X_t} \mid \mathcal{I}_{X_t} = s_\phi \cap I_{X_t} \wedge s_\phi \in \mathbf{S}_{I_{X_t}} \}$.
Thereafter, for each interval box $\mathcal{I}_{X_t}$ in $\mathbf{B}_{I_{X_t}}$, we can solve the following optimization problems:
\vspace{-1ex}
\begin{equation}
\label{eq:opequation2}
\begin{aligned}
\mathop{\arg\max}\limits_{s_t\in \mathcal{I}_{X_t}}\ \ v_i \cdot f(s_t,\pi(\phi({s_t})))\quad \quad
\mathop{\arg\min}\limits_{s_t\in \mathcal{I}_{X_t}}\ \ v_i \cdot f(s_t,\pi(\phi({s_t})))
\end{aligned}
\vspace{-1ex}
\end{equation}
For every interval box $\mathcal{I}_{X_t} \in \mathbf{B}_{I_{X_t}}$, we have $\forall s'_1, s'_2 \in \mathcal{I}_{X_t} : \phi(s'_1) = \phi(s'_2)$. Therefore, in these two optimization problems, $\pi(\phi({s_t}))$ can be replaced by a constant which makes these optimization problems easy to be solved. This segmentation process makes the original optimization problems defined on $I_{X_t}$ converted to $\lvert \mathbf{B}_{I_{X_t}}\rvert$ easy-to-solve optimization problems, which implies $post(I_{X_t}) = \{ post(\mathcal{I}_{X_t}) \mid \mathcal{I}_{X_t} \in \mathbf{B}_{I_{X_t}} \}$.
\vspace{-3mm}
\subsection{Adjacent Interval Aggregation}
Since the interval segmentation obtains $\lvert \mathbf{B}_{I_{X_t}}\rvert$ interval boxes for each interval box $I_{X_t}$ in $X_t$, the number of interval boxes will increase rapidly with the increase of time step $t$, which makes the computation of reachable sets extremely time-consuming. For this reason, we need to use the $aggregate$ operation which combines multiple interval boxes into their minimum bounding rectangle to reduce the number of interval boxes.
\begin{figure}[t]
\begin{center}
\begin{tabular}{cccc}
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=0.9\textwidth]{imgs/post_example.pdf}
\captionsetup{width=1.03\textwidth}
\vspace{-1ex}
\caption{$post$ Result}
\label{fig:post_example}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=0.9\textwidth]{imgs/no_agg.pdf}
\captionsetup{width=1.03\textwidth}
\vspace{-1ex}
\caption{No Aggregation}
\label{fig:no_agg}
\end{subfigure}&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=0.9\textwidth]{imgs/adj_agg.pdf}
\captionsetup{width=1.03\textwidth}
\vspace{-1ex}
\caption{Adjacent Aggregation}
\label{fig:adj_agg}
\end{subfigure}
&
\begin{subfigure}[b]{0.25\textwidth}
\includegraphics[width=0.9\textwidth]{imgs/all_agg.pdf}
\captionsetup{width=1.03\textwidth}
\vspace{-1ex}
\caption{All Aggregation}
\label{fig:all_agg}
\end{subfigure}
\end{tabular}
\end{center}
\vspace{-5ex}
\caption{The effect of adjacent interval aggregation.}
\label{fig:interval_agg}
\vspace{-3ex}
\end{figure}
Figure~\ref{fig:interval_agg} illustrates this aggregation process. Supposing $X_t$ contains only one interval box, through the interval segmentation and interval-based over-approximation, we will obtain four interval boxes in $post(X_t)$ depicted in Figure~\ref{fig:post_example}. If we skip the aggregation procedure and proceed directly to the calculation at next time step, $X_{t+1}$ will consist of four interval boxes in Figure~\ref{fig:no_agg}. This will lead to an exponential growth rate of the number of interval boxes with the increase of time step $t$. On the contrary, if all the interval boxes in $post(X_t)$ are directly aggregated into one interval box, there will produce large over-approximation error after aggregation as illustrated in Figure~\ref{fig:all_agg}. Therefore, to ensure the efficiency of the calculation without introducing large overestimation, we need to find adjacent interval boxes and aggregate them displayed in Figure~\ref{fig:adj_agg}.
\vspace{-1ex}
\begin{figure}[t]
\begin{center}
\begin{tabular}{ccc}
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=0.9\textwidth]{imgs/inclusion_rel.pdf}
\caption{Inclusion}
\label{fig:inclusion_rel}
\end{subfigure}&
\begin{subfigure}[b]{0.34\textwidth}
\includegraphics[width=0.9\textwidth]{imgs/intersection_rel.pdf}
\caption{Intersection}
\label{fig:intersection_rel}
\end{subfigure}&
\begin{subfigure}[b]{0.33\textwidth}
\includegraphics[width=0.9\textwidth]{imgs/seperation_rel.pdf}
\caption{Separation}
\label{fig:separation_rel}
\end{subfigure}
\end{tabular}
\end{center}
\vspace{-5ex}
\caption{Visualizing the adjacent relation.}
\label{fig:adjacent_relations}
\vspace{-5ex}
\end{figure}
\vspace{-1ex}
\paragraph{\textbf{Adjacent Relation.}}To perform the adjacent interval aggregation, we need to define the adjacent relation initially. In this paper, we define the adjacent relation between interval boxes $A=(l_1,u_1,\dots,l_n,u_n)$ and $B=(l'_1,u'_1,\dots,l'_n,u'_n)$ as the following three forms in which $h = (h_1,\dots,h_n)$ is the preset distance threshold. At first, for the case where one interval box is completely included in the other interval box, we need to aggregate them to avoid repetitive calculation. Then the intersection relation is defined to aggregate those interval boxes that have intersection and would not induce large over-approximation error after aggregation. The separation relation is similar to the intersection one but two interval boxes do not intersect. Figure \ref{fig:adjacent_relations} illustrates the following three forms.
\vspace{-1ex}
\begin{enumerate}
\item Inclusion Relation: \\
$\forall i \in \mathbb{N}, 1\le i \le n : (l_i \le l'_i \wedge u_i \ge u'_i) \vee
(l_i \ge l'_i \wedge u_i \le u'_i$)
\item Intersection Relation:\\
$\exists ! d \in \mathbb{N}, 1\le d \le n : l'_d \le l_d \le u'_d \le u_d \vee l_d \le l'_d \le u_d \le u'_d$;\\
$\forall i \in \mathbb{N}, i \neq d \wedge
1\le i \le n: \lvert l_i - l'_i \rvert \le h_i \wedge \lvert u_i - u'_i \rvert \le h_i$
\item Separation Relation:\\
$\exists ! d \in \mathbb{N}, 1\le d \le n : l_d - u'_d \le h_d \vee l'_d - u_d \le h_d$;\\
$\forall i \in \mathbb{N}, i \neq d \wedge 1\le i \le n : \lvert l_i - l'_i \rvert \le h_i \wedge \lvert u_i - u'_i \rvert \le h_i$
\end{enumerate}
With the definition of adjacent relation, for the adjacent interval aggregation problem,
a brute-force algorithm is to check and aggregate two adjacent interval boxes by traversing all interval boxes repeatedly until no interval boxes can be aggregated together. The time complexity of traversing $n$ interval boxes once to detect the existence of adjacent relation is $O(n^2)$. In the worst case, it requires traversing $n$ times. Hence, the time complexity of this algorithm is $O(n^3)$.
\begin{wrapfigure}{r}{0.55\textwidth}
\begin{algorithm}[H]
\footnotesize
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\Input{The interval boxes set $post(X_t)$}
\Output{The aggregation results $Arr$}
\caption{\mbox{Adjacent Interval Aggregation}}
\label{alg:interval_agg}
Initialize flag = [False, False,...], $Arr$ = [] \\
Construct the adjacency matrix $M$
\\
\For{$I_{p}$ in $post(X_t)$}{
\If{not flag[$I_p$]}{
Initialize queue = [$I_p$]\\
\While{queue is not empty}{
$I$ = queue.pop()\\
flag[$I$] = True\\
$I_{adjs}$ = getAdjacent($I$, $M$)\\
\For{item in $I_{adjs}$}{
$I_p$ = aggInterval($I_p$, item)\\
\If{not flag[item]}{
queue.put(item)
}
}
}
$Arr$.add($I_p$)\\
}
}
\Return $Arr$
\end{algorithm}
\vspace{-5ex}
\end{wrapfigure}
To achieve the adjacent interval aggregation more efficiently, we propose a novel algorithm shown in Algorithm~\ref{alg:interval_agg}. The key idea is to assume that the adjacent relation is transitive. For instance, if interval box $A$ is adjacent to interval box $B$ and $B$ is adjacent to interval box $C$, then the interval boxes $A,B,C$ can be aggregated together in our algorithm. Based on this assumption, we pre-construct an adjacency matrix to store the adjacent relations between the interval boxes in $post(X_t)$ firstly. This procedure is executed at line 2 in Algorithm~\ref{alg:interval_agg} with the time complexity $O(n^2)$.
We then implement this adjacent interval aggregation procedure using breadth-first search with the time complexity $O(n^2)$.
Hence, the holistic time complexity is optimized to $O(n^2)$.
\vspace{-1ex}
\begin{example}
Consider the two-dimensional system in Example~\ref{exa:DRLexample}. Supposing the $post(X_t)$ includes four interval boxes: $\widehat{I}_1 =(0.08, 0.16, 0.3,0.4)$, $\widehat{I}_2=(0.17,0.25, 0.32, 0.42)$, $\widehat{I}_3=(0.19,0.27,0.07,0.2)$,
$\widehat{I}_4=(0.2,0.28,0.1,0.21)$ and the distance threshold is $h = (0.02,0.02)$, according to the definition of adjacent relation, we can conclude that $\widehat{I}_1$, $\widehat{I}_2$ have separation relation and $\widehat{I}_3$, $\widehat{I}_4$ have intersection relation. Thus, after adjacent interval aggregation, we obtain a set of two interval boxes $X_{t+1} = \{I_{1,2} = (0.08,0.25,0.3,0.42), I_{3,4} = (0.19,0.28,0.07,0.21)\}$.
\end{example}
\vspace{-7mm}
\subsection{The Soundness}
\vspace{-1mm}
Our reachability analysis approach consists of the $post$ operation defined on each interval box generated from the interval segmentation, and the adjacent interval aggregation process.
Lemma~\ref{lem:seg} proves the soundness of $post$ operation defined on each interval box in $B_{I_{X_t}}$. The soundness guarantee of adjacent interval aggregation is given in lemma~\ref{lem:agg}.
\vspace{-1ex}
\begin{lemma}
\label{lem:seg}
For each interval box $\mathcal{I} \in B_{I_{X_t}}$, we have $s_{t+1} \in post(\mathcal{I})$ for $\forall s_t \in \mathcal{I}$.
\end{lemma}
\begin{proof}
\vspace{-1ex}
$post(\mathcal{I})$ is an interval box that contains a set of states and $s_{t+1} = post(s_t)$. Then we need to prove that for each dimension $i$, the $i$-dimensional value of $post(s_t)$ is included in the $i$-dimensional interval of $post(\mathcal{I})$.
According to the optimization problems in expression~\ref{eq:opequation2}. For each dimension $i$, we calculate the lower bound $l_i$ and upper bound $u_i$ of $v_i \cdot f(s,\pi(\phi(s)))$ defined on $\mathcal{I}$. Due to $s_t \in \mathcal{I}$, we have $\forall s \in \mathcal{I}, \pi(\phi(s)) = \pi(\phi(s_t)) = a$ where $a$ is a constant, and for an arbitrary dimension $i$ we obtain:
\vspace{-1ex}
\[
l_i = \min \limits_{s \in \mathcal{I}} v_i \cdot f(s, a) \le v_i \cdot post(s_t) = v_i \cdot f(s_t, a) \le \max \limits_{s \in \mathcal{I}} v_i \cdot f(s, a) = u_i.
\vspace{-2ex}
\]
\noindent It now follows that $s_{t+1} = post(s_t) \in post(\mathcal{I})$. \qed
\vspace{-1ex}
\end{proof}
\begin{lemma}
\label{lem:agg}
Given a set of interval boxes $X$, let $Arr$ denote the adjacent interval aggregation result of $X$. We have $\forall I \in X,\ \exists \widehat{I} \in Arr:\ I \subseteq \widehat{I}$.
\end{lemma}
\begin{proof}
\vspace{-1ex}
In algorithm~\ref{alg:interval_agg}, every interval box in $X$ needs to be traversed. For each interval box $I \in X$, there exist two cases: (i) $I$ is is not involved in the adjacent interval aggregation process,. In this case, $I$ will be directly added to $Arr$, thus $\exists \widehat{I} = I: I \subseteq \widehat{I}$. (ii) $I$ is aggregated into another interval box $I'$. Since the $aggregate$ operation produces the minimum bounding rectangle which encloses all interval boxes involved, we have $\exists \widehat{I} = I': I \subseteq \widehat{I}$. Consequently, we conclude that $\forall I \in X,\ \exists \widehat{I}:\ I \subseteq \widehat{I} \wedge \widehat{I} \in Arr$. \qed
\vspace{-1ex}
\end{proof}
The following theorem establishes the soundness of our reachability analysis approach, i.e., all true reachable states are contained in the reachable sets we computed.
\vspace{-1ex}
\begin{theorem}
\vspace{-2ex}
\label{correctness}
Given an ASDRL system $\mathcal{M}=(S, S^{0},A,\pi, f, \phi)$, for every trajectory $s_0, a_0, s_1, a_1, ...$ where $s_0 \in S^{0}$ and the sequence of computed reachable sets $X_0, X_1, X_2, \dots$ with $X_0 = \{I_{X_0}\} \wedge S^{0} \subseteq I_{X_0}^{1}$, we have $\forall t \in \mathbb{N}$, there exists an interval box $I, s_t \in I \wedge I \in X_t$.
\end{theorem}
\begin{proof}
\vspace{-1ex}
By induction on the
length of trajectories. Given an arbitrary trajectory $s_0, a_0, s_1,$ $a_1, ...$, we have:
\vspace{1ex}
\noindent \textbf{Base Case:} $t = 0$: Since $s_0 \in S^{0}$ and $S^{0} \subseteq I_{X_0}$, we have $s_0 \in I_{X_0} \wedge I_{X_0} \in X_0$.
\vspace{1ex}
\noindent \textbf{Induction Step:} $t \in \mathbb{Z^+}$: Assume $s_t \in I_{X_t}^{n_1} \wedge I_{X_t}^{n_1} \in X_t$ holds. Then, we just need to prove there exists an interval box $I$ satisfying $ s_{t+1} \in I \wedge I \in X_{t+1}$. Firstly, let us consider the segmentation process for $I_{X_t}^{n_1}$, we divide $I_{X_t}^{n_1}$ into a set of interval boxes $\mathbf{B}_{I_{X_t}^{n_1}} = \{ \mathcal{I}_{X_t}^{1}, \mathcal{I}_{X_t}^{2}, \dots, \mathcal{I}_{X_t}^{max}\}$ with $I_{X_t}^{n_1} = \bigcup\limits_{n=1}^{max} \mathcal{I}_{X_t}^{n}$. Thus $\exists \ n_2 \in \mathbb{Z^+}: s_t \in \mathcal{I}_{X_t}^{n_2}$. Based on Lemma~\ref{lem:seg}, we have $s_{t+1} = post(s_t) \in post(\mathcal{I}_{X_t}^{n_2})$.
After the adjacent interval aggregation process, $X_{t+1}$ consists of the aggregation result. According to Lemma~\ref{lem:agg}, we have $\exists \widehat{I}: post(\mathcal{I}_{X_t}^{n_2}) \subseteq \widehat{\mathcal{I}} \wedge \widehat{\mathcal{I}} \in X_{t+1}$.
Therefore, we have $s_{t+1} \in \widehat{\mathcal{I}} \wedge \widehat{\mathcal{I}} \in X_{t+1}$ and we can conclude that $\forall t \in \mathbb{N}$, there exists an interval box $I$ such that $ s_t \in I \wedge I \in X_t$ for every trajectory. \qed
\vspace{-1ex}
\end{proof}
\section{Related Work}
|
1,116,691,498,486 | arxiv | \section{Introduction}
Let $g$ be a Lorentzian metric with signature $(-,+,\dots,+)$ on the manifold $M$ of dimension $1+n$, $n\ge2$. Light-like geodesics $\gamma(s)$ (also called null geodesics) are the solutions of the geodesic equation $\nabla_s \dot\gamma=0$ for which $g(\dot\gamma,\dot\gamma)=0$. There is no canonical unit speed parameterization as in the Riemannian case as discussed below. For some fixed choice of it, we define the weighted light ray transform $L_\kappa f$ of a function (or a distribution) $f$ on $M$ by
\be{1.1}
L_\kappa f(\gamma) = \int \kappa(\gamma(s), \dot\gamma(s)) f(\gamma(s))\,\d s,
\ee
where $\gamma$ runs over all null geodesics. Here $\kappa$ is a weight function, positively homogeneous in its second variable of degree zero, which makes it parameterization independent.
When $\kappa=1$, we use the notation $L$.
Conditions for $\supp f$ and the interval of definition of the geodesics will be specified below but in all cases, the integration is carried over a compact interval. This transform appears in the study of hyperbolic equations when we want to recover a potential term from boundary or scattering information, see, e.g., \cite{MR1004174,Ramm-Sj, Ramm_Rakesh_91,waters2014stable, watersR_2013, Salazar_13, Aicha_15} for time dependent coefficients, and also \cite{BellassouedDSF, Carlos_12} for time-independent ones. It belongs to the class of the restricted X-ray transforms since the complex of curves is restricted to the lower dimension manifold $g(\dot\gamma, \dot\gamma)=0$.
Our goal is to study the invertibility of $L_\kappa $, including its microlocal invertibility. While the methods we develop could be used to study stable recovery of the ($C^\infty$) spacelike wave front set of $f$, we concentrate our attention here on support theorems for analytic metrics and weights.
In \cite{MR1004174}, the author showed that if $g$ is the Minkowski metric, and if $f(t,x)$ is supported in a cylinder $\R\times B(0,R)$ and has tempered growth in the time variable, then $L f$ determines $f$ uniquely, see also \cite{Ramm-Sj}. The proof was based on the fact that $L f$ recovers the Fourier transform $\hat f$ of $f$ (w.r.t.\ all variables) in the spacelike cone $|\tau|<|\xi|$ in a direct (and stable) way and since $\hat f(\tau,\xi)$ is analytic in the $\xi$ variable (with values distributions in the $\tau$ variable), then one can fill in the missing cone by analytic continuation in the $\xi$ variable. It is easy to see that there is no stable way to recover $\hat f$ in the timelike cone $|\tau|>|\xi| $ (true also in the most general Lorentzian case, see next paragraph) thus $L$ has a high degree of instability, see also \cite{Begmatov01}. From a physical point of view, this could be expected: we can recover all ``signals'' moving slower than light, and we should not expect to recover those moving faster than light; and the latter should not exist anyway expect for possible group velocity faster than light.
When the metric is not flat, it is fairly obvious that $L_\kappa f$ cannot ``see'' the wave front $\WF(f)$ in the timelike cone, this just follows from the inspection of the wave front of the Schwartz kernel of $L_\kappa $, see also Theorem~\ref{thm_M} for the Minkowski case. Recovery of $\WF(f)$ in the spacelike cone is far less obvious and certainly requires some geometric assumptions like no conjugate points or existence of a foliation of strictly convex surfaces, as we explain below. One possible approach
is to analyze the normal operator $L_\kappa 'L_\kappa $ as in \cite{Greenleaf-Uhlmann, Greenleaf_Uhlmann90, Greenleaf_UhlmannCM}. That operator is in the $I^{p,l}$ class of \PDO s with singular kernels, which are Fourier Integral Operators (FIOs), in fact, see \cite{Greenleaf-U_90} and the references there. The analysis of $L_\kappa 'L_\kappa $ in the Minkowski case for $n=2$ is presented in \cite{Greenleaf-Uhlmann, Greenleaf_Uhlmann90, Greenleaf_UhlmannCM} as an example illustrating a much more general theory. Applying the $I^{p,l}$ calculus to get more refined microlocal results however requires the cone condition which cannot be expected to hold on general Riemannian manifolds due to the lack of symmetry, as pointed out in \cite{Greenleaf_UhlmannCM}.
An alternative approach to recover the $C^\infty$ spacelike singularities can be found in \cite{LOSU-strings}.
Our main result is support theorems and injectivity of $L_\kappa $ for analytic metrics and weights (on analytic manifolds $M$). It can be viewed as an extension of the classical Helgason support theorem for Radon transforms in the Euclidean space \cite{Helgason-Radon}.
We use analytic microlocal arguments. Such techniques go back to \cite{BQ1,BQ,Boman-Helgason}. In \cite{BQ1}, the authors prove support theorems for Radon transforms (with flat geometry) and analytic weights. In \cite{BQ}, they study ``admissible line complexes'' in $R^{1+2}$ with analytic weights, and type III there includes a weighted version of $L_\kappa $ in the Minkowski case.
Their arguments however are based on the calculus of the analytic FIOs as an analytic version of the $C^\infty$ analysis in \cite{Greenleaf-Uhlmann}. Such a generalization does not exist to the best of the author's knowledge. Even the analytic \PDO\ calculus is quite delicate already, see, e.g., \cite{Treves}, and an analytic version of the FIO calculus, including the $I^{p,l}$ one, would pose even more challenges. Support theorems for the geodesic transforms on simple analytic manifolds have been proved with analytic microlocal techniques in \cite{Venky09,SV} and related results; even for tensor fields in \cite{FSU, SU-JAMS, SU-AJM}. A breakthrough was made by Uhlmann and Vasy in \cite{UV:local}; who proved a support theorem in the Riemannian case near a strictly convex point of a surface in dimensions $n\ge3$ without the analyticity condition. The X-ray transform is assumed to be zero on all geodesics close to tangent ones to the surface at that point, and $f$ is a priori supported on the concave side. Their arguments are based on application of the scattering calculus \cite{Melrose94} and the $n\ge3$ assumption is needed to guarantee ellipticity in a neighborhood of the point.
The approach we propose is simpler and avoids all the difficulties related to the singularities of the symbol of $L_\kappa 'L_\kappa $: we form smooth timelike surfaces foliated by lightlike geodesics over which one can compute a weighted Radon transform $R$ by just applying Fubini's theorem. This reduces the problem to a microlocal inversion of that (non-restricted) Radon transform known on an open set of surfaces, which in the smooth case is doable with classical microlocal techniques going back to Guillemin \cite{Guillemin85,GuilleminS}. Analytic singularities can be resolved by the local Radon transform, as well \cite{ZH_surfaces}.
On the other hand, this approach does not allow us to analyze the lightlike singularities, where some form of the $I^{p,l}$ calculus would still be needed.
In the proof of Theorem~\ref{thm_LC} those surfaces do not appear explicitly but they can be thought of as the level surfaces of the phase function $\phi$.
One would expect to be able to do the analytic microlocal inversion by treating $R'R$ as an analytic \PDO\ but it is not clear how to do that to obtain purely local results due to the delicate nature of cut-offs allowed in that calculus. Instead, we use the analytic stationary phase approach by Sj\"ostrand \cite{Sj-A} already used by the author and Uhlmann in \cite{SU-AJM}, see also \cite{FSU,Venky09,SV}.
As a simple example illustrating the reduction of the restricted ray transform $L_\kappa $ to a classical Radon transform $R$, consider the Minkowski case. Light geodesics are given then by the lines parallel to $(1,\theta)$, with $|\theta|=1$. Every timelike plane (with a normal $\nu=(\nu_t,\nu_x)$ such that $|\nu_t|<|\nu_x|$) can be represented easily as a foliation of light rays. If $L_\kappa f\in C^\infty$ (or analytic), then so is $Rf$ on the open manifold of those planes. Then we have to invert microlocally the classical Radon transform, which is well known.
This argument still works if we introduce a weight in $L_\kappa $ and/or know $L_\kappa f$ localized to an open set of light rays only, see Theorem~\ref{thm_C}.
Finally, we notice that some global conditions on the geodesic flow are clearly needed for
microlocal inversion, even in the spacelike cone. If $g = -\d t^2+ h_{ij}(x)\d x^i \d x^j$, where $h$ is a Riemannian metric on a bounded domain, then $L_\kappa $, restricted to $t$-independent function reduces to the geodesic X-ray transform $X$. It has been shown recently \cite{SU-caustics, MonardSU14} that when $n=2$, $Xf$ recovers $\WF(f)$ in a stable way if and only if there are no conjugate points. When $n\ge3$, the no conjugate points condition is sufficient \cite{SU-JAMS, SU-AJM} and there are examples of metrics of product type for which it is necessary, by the 2D results in \cite{SU-caustics, MonardSU14}. On the other hand, the support theorem in \cite{UV:local} provides global uniqueness and stability under another type of condition: existence of a foliation by strictly convex surfaces (conjugate points may exist). This implies stable invertibility when $n\ge3$ without analyticity assumptions. We assume the foliation condition but in contrast to \cite{UV:local}, here $n=2$ is allowed since for our purposes, full ellipticity (in all directions at a point) is not needed; only ellipticity at directions conormal to the foliation suffices. Also, full ellipticity does not hold in the Lorentzian case since the timelike singularities are invisible.
On the other hand, we require $g$ and $\kappa$ to be analytic. One would expect support theorems under the no-conjugate points assumption as well but that remains an open question.
\textbf{Acknowledgments.} The author thanks Manuel Guti\'errez for helpful discussions about Lorentzian geometry and to Gunther Uhlmann for numerous discussions about Integral Geometry and Inverse Problems.
\section{Main results}
\subsection{Support theorems for the Minkowski spacetime}
Let $g = -\d t^2+ (\d x^1)^2+\dots +(\d x^n)^2$ be the Minkowski metric in $\R^{1+n}$.
Future pointing lightlike geodesics are given by $s\mapsto (t+s,x+s\theta)$ with $|\theta|=1$. They can be reparameterized by shifting and rescaling $s$. Note that the notion of ``unit'' speed is not invariantly defined under Lorentzian transformations but in a fixed coordinate system, the scaling parameter $1$ (i.e., $\d t/d s=1$) is a convenient choice.
Set
\be{Ldef}
Lf(x,\theta) = \int f(s, x+s\theta)\,\d s,\quad x\in \R^{n}, \,\theta\in S^{n-1}.
\ee
This definition is based on parameterization of the lightlike geodesics (lines) by their point of intersection with $t=0$ and direction $(1,\theta)$. We will use the notation
\be{g}
\ell_{x,\theta}(s) = (s,x+s\theta).
\ee
The parameterization $(x,\theta)$ defines a natural topology and a manifold structure of the set of the future pointing lightlike geodesics.
Given a weight $\kappa\in C^\infty(\R\times\R^n\times S^{n-1})$, we can define the weighted version $L_\kappa$ of $L$ by
\[
L_\kappa f(x,\theta) = \int \kappa(s, x+s\theta,\theta) f(s, x+s\theta)\,\d s,\quad x\in \R^{n}, \,\theta\in S^{n-1}.
\]
In the terminology of relativity theory, vectors $v=(v^0,v')$ satisfying $|v_0|<|v'|$ (i.e., $g(v,v)>0$) are called \textit{spacelike}. The simplest example are vectors $(0,v')$, $v'\not=0$.
Vectors with $|v_0|>|v'|$ (i.e., $g(v,v)<0$) are \textit{timelike}; an example is $(1,0)$ which points along the time axis.
\textit{Lightlike} vectors are those for which we have equality: $g(v,v)=0$. For covectors,
the definition is the same but we replace $g$ by $g^{-1}$, which is consistent with the operation of raising and lowering the indices.
Surfaces with timelike normals (which are covectors) are spacelike, and vice versa.
\begin{definition}\label{def_slow}
Let $K$ be a subset of the Minkowski spacetime. We say that $K$ expands with speed less than one if
\be{supp}
K\subset \left\{ (t,x);\; |x|\le c|t|+R \right\}\quad \text{for some $0<c<1$, $R>0$}.
\ee
\end{definition}
Condition \r{supp} is easily seen to be invariant under Lorentz transformations. Also, it does not require $\supp f$ to be compact.
The terminology we used is a bit ambiguous. What we actually mean is that the cross-section of $K$ with any plane $t=\text{const.}$ is a bounded set contained in a ball expanding with a speed less than one. If $\partial K$ is smooth, we do not really require it to be timelike.
In the Minkowski spacetime, we have the following support theorem.
\begin{theorem}\label{thm_linesM}
Let $f\in\mathcal{D}'(\R^{1+n})$ be so that $\supp f$ expands with a speed less than one.
Let $\ell_{x_0,\theta_0}$ be a fixed lightlike line in the Minkowski spacetime
and let $U\ni(x_0,\theta_0)$ be an open and connected subset of $\R^n\times S^{n-1}$. Let $ \kappa(t,x,\theta)$ be analytic and non-vanishing for $(t,x)$ near $\supp f$ so that $(x-t\theta,\theta)\in U$.
If $L_\kappa f(x,\theta)=0$ in $U$ and if $\ell_{x_0,\theta_0}$ does not intersect $\supp f$, then none of the lines $\ell_{x,\theta}$, $(x,\theta)\in U$, does.
\end{theorem}
\subsection{Support theorems on Lorentzian manifolds}
Let $(M,g)$ be a Lorentzian manifold. Light-like (null) geodesics are defined as the geodesics $\gamma(s)$ for which $g(\dot\gamma,\dot\gamma)=0$. They exist at least locally by the ODE theory.
There is no canonical parameterization since for any linear transformation of the $s$ variable $\sigma(s) = as+b$, $a\not=0$, $\gamma\circ\sigma$ is still a null geodesic. Moreover, $a$ and $b$ may change from geodesic to geodesic. Let $S$ be a spacelike surface near a fixed lightlike geodesic $\gamma_0(s)$, intersecting $S$ for $s=0$. Then we can parameterize the lightlike geodesics in some neighborhood of $\gamma_0(0)\in S$ close to $\gamma_0$ with directions close to $\dot\gamma(0)$ with initial points $x$ on $S$ and initial lightlike directions $v$ at $x$ pointing in the direction of $\dot\gamma_0$. A choice of the scaling of the parameter $s$ along each $\gamma(s)$ can be fixed by requiring $\gamma(0)\in S$ and requiring the normal component of $\dot\gamma$ on $S$ to be a given negative function, for example $-1$. If that function is smooth/analytic when $S$ is smooth/analytic, we call the parameterization smooth/analytic. This property does not depend on the choice of $S$ and also defines a topology and a smooth/analytic structure of the lightlike geodesics defined on a fixed interval. We could use a timelike surface as initial points instead.
If $\mathcal{C}\subset M$ is closed, we call the null geodesic $\gamma(s)$ \textit{non-trapping in $\mathcal{C}$}, if $\gamma^{-1}(\mathcal{C})$ is contained in some open finite interval call it $I$ for a moment. For any local parameterization of null geodesic as above, the maximally extended null geodesic with initial points and directions close enough to $\gamma$ would leave $\mathcal{C}$ for $s$ near the ends of $I$. Some of them may return to $\mathcal{C}$ for $s\not\in I$ (even though this cannot happen to $\gamma$ but we restrict them to $I$ only. Then we consider those geodesics a neighborhood of $\gamma$, identified with the neighborhood of the initial points and directions in that parameterization. That definition of local neighborhood is independent of the chosen parameterization and defines a topology near $\gamma$ (restricted to $I$).
For any such choice of the parameterization, we then define $L_\kappa f$ locally by \r{1.1} for any $f\in C_0^\infty$, with $s$ restricted to $I$.
A different analytic parameterization would change $L_\kappa f$ (in a trivial way) but it will not change its property to be smooth or analytic, or zero.
We do not want to assume that $f$ is compactly supported but we always assume that
we integrate over a set of light geodesics non-trapping in $\supp f$. Then locally, we may cut, a smooth $f$ in a smooth way to make it compactly supported without changing $L_\kappa f$ near that geodesic. This reduces the local analysis to compactly supported functions. An example is a function supported in a cylinder $|x|\le R$ in the Minkowski case; or more general $f$ with $\supp f$ expanding with speed less than one, see \r{supp}.
This allows $L_\kappa f$ to be well defined for smooth $f$ over open sets of non-trapped light geodesics and then by duality for distributions $f$. Indeed, for every distribution $f$ in $M$ we can set locally $L_\kappa f= L_\kappa\chi f$, with a suitable $\chi\in C_0^\infty$, and the latter makes sense by duality. In other words, near a fixed null geodesic, non-trapping in $\supp f$, it is enough to study $L_\kappa$ restricted to compactly supported distributions $f$. Based on that, to simplify the formulation of the next theorem, we assume that $\supp f$ is compact.
\begin{theorem}\label{thm_L}
Let $(M,g)$ be an analytic Lorentizan manifold and let $\kappa$ be an analytic non-vanishing weight.
Let $F: M \to [0,1]$ be a smooth function. Assume that $f\in \mathcal{E}'(M)$ and
(i) $F^{-1}(0)\cap\supp f=\emptyset$,
(ii) $\d F\not=0$ on $\supp f$,
(iii) $F^{-1}(\sigma)\cap \supp f$ is strictly lightlike-convex for all $\sigma\in[0,1]$.
\noindent
Then if $L_\kappa f(\gamma)=0$ in a neighborhood of all null-geodesics with the property that each one is tangent to some of the surfaces $F^{-1}(\sigma)$, $\sigma\in [0,1]$, then $f=0$ on $F^{-1}[0,1)$.
\end{theorem}
We refer to Definition~\ref{def_convex} for the notion of lightlike convexity.
Examples of strictly lightlike-convex surfaces in the Minkowski spacetime, which cannot be spacelike at any point, include the cylinder $|x|=R$, $R>0$; more generally, the smooth part of the double cone $|x|=c|t|$, with $0<c<1$ fixed; or the hyperboloid $|x|^2=c^2t^2+C$ with $C>0$ and such a $c$. They are all timelike. We also note that we can actually require $L_\kappa f=0$ on a suitable submanifold of lightlike geodesics of dimension $1+n$ only, as it follows form the proof. Moreover, it is only enough to assume that $L_\kappa f$ is analytic there, even microlocally so, see Remark~\ref{rem_n}.
To demonstrate a typical application of Theorem~\ref{thm_L}, we will point out how one can show that if in the Minkowski space time $f$ satisfies \r{supp}, and $Lf=0$, then $f=0$, which, of course
follows from Theorem~\ref{thm_linesM} as well. We choose $F(\sigma) = \sigma+\tilde c^2(t-t_0)^2-|x-x_0|^2$ with $0<c<\tilde c<1$, $\sigma>0$. The constant $\sigma$ can be rescaled to be fit in $[0,1]$.
Choosing various $x_0$ and $t_0$ we can prove $f=0$. One can perturb the metric a little bit assuming that $\supp f$ is supported in a fixed compact set, to get examples for non-flat metrics.
\section{Analysis in the Minkowski case}
\subsection{Fourier Transform analysis} Let $M=\R^n$ and $g$ be Minkowski. By the Fourier Slice Theorem, knowing the X-ray transform for some direction $\omega$ recovers uniquely $\hat f$ on $\omega^\perp$.
More precisely, the Fourier Slice Theorem in our case can be written as
\[
\hat f|_{ \tau+\xi\cdot\theta=0 } =
\hat f(-\theta\cdot\xi,\xi) = \int_{\R^n} e^{-\i x\cdot\xi}Lf(x,\theta) \, \d x, \quad\forall\theta\in S^{n-1}.
\]
Here and below, we denote by $\zeta = (\tau,\xi)$ the dual variables to $z=(t,x)$. The proof is easy, see \r{5}. The union of all $( 1,\theta)^\perp$ for all unit $\theta$ is $\{|\tau|\le |\xi|\} = \Sigma_s \cup\Sigma_t $, that is easy to see. This correlates well with the theorems below. In particular, we see that knowing $\hat f(\zeta)$ for a distribution $f$ with a well defined Fourier transform, recovers $\hat f$ in the spacelike cone uniquely and in a stable way. Under the assumption that $\supp f$ is contained in the cylinder $|x|\le R$ for some $R$ (and temperate w.r.t. $t$), one can use the analyticity of the partial Fourier transform of $f$ w.r.t.\ $x$ to extend $\hat f$ analytically to the timelike cone, as well.
This is how it has been shown in \cite{MR1004174} that $L$ is injective on such $f$.
\subsection{The normal operator $X'X$}
We formulate here a theorem about the Schwartz kernel of the normal operator $N=L'L$. We will skip the proof because we will not use the theorem for our main results. One way to obtain it is to think of $L$ as a weighted version of the X-ray transform $X$ with a distributional weight $\delta(\tau^2-|\xi|^2)$ and use the results about the weighted X-ray transform, see, e.g., \cite{SU-Duke} and allow a singular weight there. Details will appear in \cite{SU-book}, see also \cite{LOSU-strings}.
\begin{theorem}\label{thm_M}\
(a)
\[
L'Lf = \mathcal{N}*f, \quad \mathcal{N}(t,x) = \frac{\delta(t-|x|) +\delta(t+|x|) }{|x|^{n-1}}.
\]
(b)
\[
L'Lf = C_n\mathcal{F}^{-1}\frac{(|\xi|^2-\tau^2)_+^\frac{n-3}2} {|\xi|^{n-2}} \mathcal{F}f, \quad \forall f\in \mathcal{S}(\R^{1+n}), \quad C_n:= 2\pi|S^{n-2}| .
\]
(c)
\[
h(\Box_+) f = C_n^{-1} |D_x|^{n-2}\Box_+^{\frac{3-n}{2}}X'Xf,
\]
where $h$ is the Heaviside function, and $\Box = \partial_t^2-\Delta_z$ and $\mathcal{F}$ is the Fourier transform.
\end{theorem}
Above, we used the notation $s_+=\max(s,0)$ with the convention that $s_+^0$ is the Heaviside function.
In particular, when $n=3$, we get $\sigma(X'X) = C_3|\xi|^{-1}h\left( |\xi|^2- |\xi_0|^2\right)$. Then
\[
h(\Box_+)f = C_3^{-1} |D_z|X'Xf.
\]
As we can expect, there is a conormal singularity of the symbol even away from $\xi=0$ living on the characteristic cone, and $X'X$ is elliptic outside it, and only there. The theorem shows that ``singularities traveling slower than light'' can be recovered. The ones traveling faster cannot.
\subsection{Recovery of spacelike $C^\infty$ singularities in the Minkowski case} \label{sec_2.3}
Our first theorem says that knowing $Lf$ near a lightlike geodesic $\ell_0$ allows us to recover all spacelike singularities conormal to $\ell_0$. We denote the conormal bundle of $\ell_0$ by $N^*\ell_0$. Recall that the conormal singularities to $\ell_0$ contain lightlike ones, as well. This result follows from the analysis in \cite{Greenleaf-Uhlmann, Greenleaf_Uhlmann90, Greenleaf_UhlmannCM} and the reason we present it here is to illustrate the main idea on a simpler problem where we can do explicit computations.
\begin{theorem}\label{thm_C}
Let $f$ be a distribution so that $\ell(s)\not\in\supp f$ for $|s|>1/C$ with some $C$ for all lightlike lines $\ell$ near $\ell_0$. Let $L_\kappa f(\ell)\in C^\infty$ for $\ell$ in some neighborhood $\Gamma$ of $\ell_0$. Then $\WF(f)\cap N^*\ell_0$ contains no spacelike covectors.
\end{theorem}
\begin{proof}
We construct planes foliated by lightlike geodesics with a fixed direction $(1,\theta)\in \R \times S^{n-1}$. Any such plane intersects the $t=0$ plane in a $(n-1)$-dimensional plane in the $x$ space. Let the latter be $\pi_{p,\omega}= \{x\cdot\omega=p\}$, $\omega\in S^{n-1}$, $p\in \R$. Then the plane that we denote by $ \pi_{p,\omega,\theta}$ is the flow out of the null geodesics with celestial direction $\theta$ originating from $\pi_{p,\omega}$, i.e.,
\be{pi1}
\pi_{p,\omega,\theta} = \{(t,x+t\theta);\; x\in \pi_{p,\omega}\}.
\ee
The same plane can be also described by the equation $(x-t\theta)\cdot\omega=p$, therefore,
\be{pi2}
\pi_{p,\omega,\theta} = \{(t,x);\; (t,x)\cdot (-\theta\cdot\omega,\omega)=p\}.
\ee
The dot product here is in the Euclidean sense, and can be also thought of as a pairing of a vector and a covector, in invariant terms. In particular, we see that the set of such planes coincides with the timelike ones and the lightlike ones as a borderline case.
Let $\zeta^0\not=0$ be spacelike and conormal to $\ell_0$ at a point that we can always assume that to be the origin. Applying a Lorentz transformation, we can always assume that $\zeta^0 = e^{n-1}: = (0,\dots,0,1,0)\in \R^{1+n}$ and $\ell_0= \ell_{0,e_{n}} = (s,0,\dots0,s)$. Here and below, we use the notations $e_k$ and $e^k$ to denote vectors/covectors with all entries zero instead of the $k$-th one.
Take the plane $\pi_0=\{(t,x);\; x^{n-1}=0\}$, conormal to $\zeta^0$. This is the plane constructed above with $\omega=\zeta^0= e^{n-1}$ and $\theta = e_{n}$ (we could have chosen any other $\theta\perp\omega$ but we chose this one because it is related to $\ell_0)$.
It is more convenient to extend the parameters $(p,\omega)$ by homogeneity. We allow $\omega$ to be non-unit and denote it by $\xi$. Then the planes $\pi_{p,\omega,\theta}$ are given by
\be{zeta()}
z\cdot\zeta=p, \quad \zeta= (-\theta\cdot\xi,\xi).
\ee
We will choose a suitable analytic family of $(\xi,\theta)$ near $\xi=e^{n-1}$, $\theta=e_{n}$
parameterized by an $n+1$-dimensional parameter so that the map from that parameter to the (co-)normal $\zeta$ is a local diffeomorphism.
We keep $\xi$ unrestrained and let $\theta$ depend on an 1D parameter, $q$. Then
\be{3''}
\partial \zeta/\partial \xi_k = (- \theta^k ,e^k ),\quad k=1,\dots,n, \quad \text{and}\quad \partial \zeta/\partial q|_{q=0} = (- \partial \theta/\partial q\cdot\omega|_{q=0},0).
\ee
This system of vectors is linearly independent, if and only if $\partial \theta/\partial q\cdot\omega|_{q=0} \not=0$.
Therefore, the variation $\partial \theta/\partial q$ should be chosen not parallel to $\pi_{p,\xi}$. This leaves essentially a variation in the direction of $\xi$.
Based on that, we set
\be{theta}
\theta(q) = (\cos q) e_{n}+(\sin q) e_{n-1}.
\ee
Then
\be{zeta}
\zeta(q,\xi) = (-\theta(q)\cdot\xi,\xi)
\ee
and
\be{4}
\pi_{p,\xi,\theta(q)} = \{(t,x);\; (t,x)\cdot \zeta(q,\zeta)=p\}, \quad p\in\R;\; \xi\in\R^{n}\setminus 0.
\ee
The fact that $(q,\xi)\to\zeta$ is a local diffeomorphism is also easy to verify directly. Solving \r{zeta} for $(q,\xi)$ yields $\xi_i=\zeta_i$, $i-1,\dots,n$, and
\be{4e}
\zeta_n\cos q+ \zeta_{n-1} \sin q= -\zeta_0
\ee
and the latter is uniquely solvable for $q$ near $q=0$ for $\zeta$ near $e^{n-1}$; let $q=q(\zeta)$ be the solution.
We can write the defining equation also as $(t,x)\cdot\nu=\tilde p :=p/|\zeta|$ with $\nu=\zeta(q,\xi)/|\zeta(q,\xi)|$. Then it is easy to show that with $\xi$ restricted back to unit sphere, the map $\R\times S^{n-1} \ni (q,\xi)\to \nu\in S^{n}$ is a local analytic diffeomorphism near $p=0$. In other words, $(q,\xi)$ with $\xi$ unit, parameterizes the normal $\nu$ to \r{4} in a locally diffeomorphic way.
By the support assumption of the theorem, there exists a neighborhood $U$ of $(0,e_{n})$ and $A>0$, so that all lightlike geodesic issued from $U$ leave $\supp f$ for $|s|\ge A$. Take a smooth function $\chi(x,\theta)$ supported in $U$ equal to $1$ near $(0,e_{n})$.
Since $L_\kappa f\in C^\infty$, we have $\chi L_\kappa f|_{x\in \pi_{p,\xi,\theta(q)}, \theta=\theta(q)}\in C^\infty$, as well. Integrate $[\chi L_\kappa f](x,\theta(q))$ with respect to $x$ on the plane $ \pi_{p,\xi}$ to get by Fubini's theorem:
\be{4r}
Rf(\pi_{p,\xi,\theta(q)} ) := \int_{ \pi_{p,\xi,\theta(q)}} \chi \kappa f \,\d\mu_{ {p,\xi,\theta(q)}}\in C^\infty
\ee
for some measure analytically depending on $(p,q,\xi)$, i.e. an analytic and positive multiple of the Euclidean measure on each plane. Above, the integral is taken over a compact set; moreover, we can cut $f$ to a compactly supported distribution away from where we integrate without affecting the integral. Therefore, $f$ is in the microlocal kernel of the weighted Radon transform $R$ with a weight not vanishing at $(t,x)=0$ on the plane $\pi_0$. This allows us to apply $R'$ to get an elliptic \PDO\ of order $-2$, see, e.g., \cite{BQ}. Therefore, $f$ is microlocally smooth at $(0,\zeta^0)$ as claimed.
\end{proof}
\begin{remark}\label{rem_n}
Note that we only needed to know that $L_\kappa f(x,\theta)$ vanishes (being microlocally smooth in a certain cone would suffice) for $\theta=\theta(q)$ only with $|q|\ll1$; i.e., we require knowledge of a restricted version of the already restricted $L$. This is similar to the known fact that in the Euclidean space we can invert the X-ray transform by ``slicing'' $\R^n$ into 2D planes. We could have proven our results in the Minkowski spacetime in $1+2$ dimensions only and then extended it to any dimension $1+n\ge 1+2$. The same remarks applies to the analytic case below but we need to know that $L_\kappa f$ is microlocally analytic (instead of just smooth) in some conic set. Even in the Lorentzian case, we still need to know $L_\kappa$ restricted to a certain an $(1+n)$-dimensional submanifold of geodesics.
\end{remark}
\subsection{Recovery of analytic spacelike singularities in the Minkowski case}
We show first that we can recover all spacelike analytic singularities of $f$ conormal to the lightlike lines along we integrate. For a definition of the analytic wave front set $\WFA(f)$, we refer to \cite{Sj-A} and \cite{Treves}.
\begin{lemma}\label{lemma3.1}
Let $f\in\mathcal{D}'(\R^{1+n})$ and let $\ell_{x_0,\theta_0}$ be a fixed lightlike line so that $\ell_{x,\theta}(s)\not\in\supp f$ for $|s|\ge 1/C$ with some $C$ for all $(x,\theta)$ near $(x_0,\theta_0)$. Let $ \kappa(s,x+s\theta)$ be analytic and non-vanishing for those $s,x,\theta$. If $L_\kappa f(x,\theta)=0$ near $(x_0,\theta_0)$, then $N^*\ell_{x_0,\theta_0}\cap \WFA(f)$ contains no spacelike covectors.
\end{lemma}
\begin{proof}
One would expect the proof to be a complete analog of that of Theorem~\ref{thm_C} but that proof involves smooth cutoffs along the planes we integrate over. We cannot do this in our case because that would destroy the analyticity of the weight. On the other hand, we need the localization because we know that $L_\kappa f=0$ near $\ell_{x_0,\theta_0}$ only.
We use the local coordinates in the proof of Theorem~\ref{thm_C}, where $\zeta^0=e^{n-1}$, $\theta_0 = e_n$, $z_0=0$.
Let $\chi_N\in C_0^\infty(\R^n)$ be with support in $B(0,\eps)$, $\eps>0$, with $\chi_N=1$ near $x_0=0$ so that
\be{A1}
|\partial_{x}^\alpha\chi_N|\le (CN)^{|\alpha|}, \quad \text{for $|\alpha|\le N$},
\ee
see \cite{Treves}. Then for $0<\eps\ll1$, $\lambda>0$, and $\theta$ close to $\theta_0=e_n$,
\[
0= \int e^{\i \lambda x\cdot\xi} (\chi_N L_\kappa f)(x,\theta)\,\d x=
\iint e^{\i\lambda x\cdot\xi} \chi_N (x)
\kappa(s,x+s\theta,\theta) f(s,x+s\theta)\,\d s\, \,\d x.
\]
If $(1,\theta)\cdot\zeta=0$ with $\zeta=(\tau,\xi)$, then $x\cdot\xi= (t,x+t\theta)\cdot\zeta$. Make the change of variables $x+s\theta\mapsto x$ in the integral above to get
\be{5}
\begin{split}
0 &= \int e^{\i \lambda x\cdot\xi} (\chi_N L_\kappa f)(x,\theta)\,\d x\\
& =
\iint e^{\i\lambda (t,x)\cdot\zeta} \chi_N (x-t\theta)
\kappa(t,x,\theta) f(t,x)\,\d t\, \,\d x, \quad \text{if $(1,\theta)\cdot\zeta=0$}.
\end{split}
\ee
For $\xi\in \R^n$, choose $\theta=\theta(q)$ as in \r{theta}, and set
$\zeta= (-\theta(q) \cdot\xi,\xi)$ as in \r{zeta()}. Then the orthogonality condition in \r{5} is satisfied.
To connect this with the analysis in section~\ref{sec_2.3}, notice that we can get the same result by taking the Fourier transform $\mathcal{F}_{p\to\lambda }$ in \r{4r}, with $\chi=\chi_N$. Choose now $q=q(\zeta)$ as in \r{4e}. The orthogonality condition still holds and we have
\be{5.2}
\int e^{\i\lambda z\cdot\zeta } a_N(z,\zeta) f(z)\,\d z =0 \quad \text{near $\zeta=e^{n-1}$},
\ee
where $ a_N= \chi_N(x-t\theta(q
)\kappa(t,x,\theta(q)) $, with $q=q(\zeta)$, is analytic and elliptic near $(z,\zeta) = (0,e^{n-1})$ (but not analytic away from some neighborhood of it) and satisfies pseudo-analytic estimates of the type \r{A1}.
We will apply the complex stationary phase method of Sj\"ostrand \cite{Sj-A} similarly to the way it was applied in \cite{KenigSU} to the partial data Calder\'on problem and in \cite{FSU, SU-AJM} to integral geometry ones.
Fix $0<\delta\ll\eps$, see \r{A1}. With some $w\in\R^{1+n} $, $\eta\in \mathbf{R}^{1+n}$ close to $w=0$, $\eta=e^{n-1}$, multiply the l.h.s.\ of \r{5} by
\be{chi_delta}
e^{ \i \lambda(\i ( \zeta-\eta)^2/2 - w\cdot\zeta) }
\ee
and integrate w.r.t.\ $\zeta$ in the ball $|\zeta-\eta|< \delta$ to get
\be{6}
\int_{ |\zeta-\eta|< \delta}\int e^{\i\lambda\Phi(w,z,\zeta,\eta)} a_N(z,\zeta)f(z)\,\d z\, \d \zeta=0,
\ee
where
\[
\Phi = (z-w)\cdot\zeta + \i (\zeta-\eta)^2/2.
\]
We split the $z$ integral \r{6} into two parts: over $\{z;\; |z-w|<\delta/2\}$ and then over the complement of that set. Since $|\Phi_\zeta|$ has a ($\delta$-dependent) positive lower bound for $|z-w|\ge\delta/2$, we can integrate in the outer integral in \r{6} by parts w.r.t.\ $\zeta$, see, e.g., \cite{FSU, SU-AJM} using \r{A1} and the fact that on the boundary $|\zeta-\eta|=\delta$, the factor $e^{\i\lambda \Phi}$ is exponentially small with $\lambda$. We then get
\be{7}
\Big| \iint_{|z-w|<\delta/2, \,|\zeta-\eta|\le \delta } e^{\i\lambda\Phi(w,z,\zeta,\eta)} a(z,\zeta )f(z)\,\d z\,
\d \zeta \Big| \le C(CN/\lambda)^N + CNe^{-\lambda/C}
\ee
where $ a$ equals $a$ with the $\chi_N$ factor missing, i.e., $a= \kappa(t,x,\theta(q(\zeta)) $, which is independent of $N$ because on the support of the integrand, that factor is equal to $1$ for $\delta\ll\eps$, see \r{5.2}. Choose now $N$ so that $N\le \lambda/(Ce)\le N+1$ to get an exponential error on the right.
The phase $\Phi$, as a function of $\zeta$, has a unique critical point
$
\zeta_c =\eta +\i (z-w)
$
and $|\zeta_c-\eta| \le \delta/2$ on the support of the integrand in \r{7}.
Set
\be{psi}
\psi(w,z,\eta) = \Phi|_{\zeta=\zeta_c}.
\ee
Therefore,
\[
\psi = \eta\cdot (z-w) + \i |z-w|^2 -\frac{\i}2|z-w|^2 =
\eta\cdot (z-w)+\frac{\i}2 |z-w|^2 .
\]
This is the type of phase functions that are used to test for analytic microlocal regularity.
We apply now \cite[Theorem~2.8]{Sj-A} and the remark after it to the $\zeta$-integral in \r{7}
to get
\be{8}
\Big| \int_{|z-w|<\delta/2} e^{\i\lambda\psi(w,z,\eta)} b(w,z,\eta ,\lambda)f(z)\,\d z\,
\Big| \le Ce^{-\lambda/C}.
\ee
for $(z,\eta)$ close to $(0,e^{n-1})$,
with some classical elliptic analytic symbol $b$ of order $0$ in the sense of \cite{Sj-A} near $(w,z,\eta)= (0,0,e^{n-1})$. In particular, the principal part of $b(0,0,e^n,\lambda)$ is $\beta \kappa(0,0,e_{n})$ with $\beta$ an elliptic factor depending on the phase, see \cite[Theorem~2.8]{Sj-A}.
This implies $(0,e^{n-1})\not\in \WFA(f)$, see \cite{Sj-A}.
\end{proof}
\subsection{Proofs of the support theorems in the Minkowski spacetime}
The next proposition is a unique continuation result across a timelike surface in the Minkowski case which implies Theorem~\ref{thm_linesM}.
\begin{proposition}\label{pr_M_supp}
Let $f\in\mathcal{D}'(\R^{1+n})$ and let $\ell_{x_0,\theta_0}$ be a fixed lightlike line in the Minkowski spacetime so that $\ell_{x,\theta}(s)\not\in\supp f$ for $|s|\ge 1/C$ with some $C$ for all $(x,\theta)$ near $(x_0,\theta_0)$. Let $ \kappa(s,x+s\theta)$ be analytic for those $s,x,\theta$.
If $L_\kappa f(x,\theta)=0$ near $(x_0,\theta_0)$ and if $f=0$ on one side of $S$ near $z_0$, then $f=0$ near $z_0$.
\end{proposition}
\begin{proof}
Assume that $z_0\in\supp f$. Then $(z_0,\mp \nu(z_0))\in \WFA(f)$ by the Sato-Kawai-Kashiwara Theorem, see \cite{SKK} and \cite{Sj-A}, where $\nu(z_0)$ is one of the two unit co-normals to $S$ at $z_0$. That covector is spacelike by the assumption about $S$, and is conormal to $\dot\ell_{z_0,\theta_0}(0)$. This contradicts Lemma~\ref{lemma3.1}, which completes the proof of the proposition.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm_linesM}]
Fix $(x_1,\theta_1)\in U$. Let $[0,1]\ni p\to (x_p,\theta_p)$ be a continuous family in $U$ connecting $(x_0,\theta_0)$ with $(x_1,\theta_1)$.
We can always assume that $U$ is bounded, hence $\bar U$ is compact.
Let $\mathcal{U}$ be the set of points lying on $\ell_{x,\theta}$, $(x,\theta)\in U$.
Choose $\tilde c\in(c,1)$, where $c$ is the constant in \r{supp}. Denote by $\tilde \ell_{x,\theta}(s) =(s,x+s\tilde c\theta) $, $(x,\theta)\in\R^n\times S^{n-1}$, the timelike geodesics with speed $\tilde c$. By \r{supp}, there exists $A>0$ so that $ \ell_{x,\theta}(s)\not\in\supp f$ for all $(x,\theta)\in \bar U$ and $|s|\ge A$,
and so that the same holds for $\tilde \ell_{x,\theta}(s)$ uniformly w.r.t.\ $\tilde c$ as long as $c+\mu\le \tilde c\le1$ with some fixed $\mu\in(0,1-c)$.
Let $\eps>0$ be such that the cylinder $C_{x_0,\theta_0}: = \cup_{|y-x_0|\le\eps} \ell_{y,\theta}$ is disjoint from $\supp f$ but sill lies in $\mathcal{U}$. By the arguments above, we can assume that $\supp f$ is compact since $t$ is bounded on it along the light lines under consideration.
Therefore, for the cylinder $\tilde C_{x,\theta}: = \cup_{|y-x|\le\eps/2} \tilde\ell_{y,\theta}$ we have
\be{9}
\tilde C_{x,\theta}\cap \{|t|\le A\}\subset C_{x,\theta}\cap \{|t|\le A\}
\ee
for every $(x,\theta)$ as long as $\tilde c$ is close enough to $c$ but still smaller than it. We require one more property for $\tilde c$ which actually refines \r{9}:
\be{10}
\begin{split}
&\text{$ \forall (x,\theta)\in\bar U $, $\forall z\in \tilde C_{x,\theta}\cap \{|t|\le A\}$, and every unit $\theta_1$ with $|\theta_1-\theta|\le \sqrt{1-\tilde c^2}$,}\\
&\text{the lightlike line through $z$ in the direction of $(1,\theta_1)$ stays in $C_{x,\theta}\cap \{|t|\le A\}$.}
\end{split}
\ee
This property can be guaranteed for $1-\tilde c\ll1$ by continuity and compactness.
We fix such a $\tilde c$.
Assume that the family $\{\tilde C_{x_p,\theta_p};\; p\in[0,1]\}$ has a common point with $\supp f$. Let $p_0$ be the least $p\in[0,1]$ (which exists by compactness and continuity arguments) for which $\tilde C_{x_p,\theta_p}\cap \supp f \not=\emptyset$. Then $f=0$ in the interior of $\tilde C_{x_p,\theta_p}$ and there is a point $z^\sharp$ on its boundary which is also in $\supp f$. Let $\zeta^\sharp$ be a non-vanishing conormal to that cylinder at $z^\sharp$. After normalization, we get $\zeta^\sharp = (-c\theta_p\cdot\omega,\omega)$ for some $\omega\in S^{n-1}$. Clearly, $\zeta^\sharp$ is spacelike. Let $\tilde \ell_{x^\sharp,\theta_p}$ be the line on the cylinder $\tilde C_{x_p,\theta_p}$ through $z_0$; then $\zeta^\sharp$ is conormal to it at $z^\sharp$.
To apply Proposition~\ref{pr_M_supp}, we claim that there is a lightlike line at $z^\sharp$
normal to $\zeta^\sharp$ so that this line is still in $\mathcal{U}$. Suppose for a moment that this done. Then by
by Proposition~\ref{pr_M_supp}, we would get that $f$ vanishes near that point, which would be a contradiction. Therefore, such a $p_0$ would not exist, and in particular, $f=0$ near $\ell_{x_1,\theta_1}$.
To prove the claim, we are looking for a unit $\theta^\sharp$ so that $\zeta^\sharp\cdot (1,\theta^\sharp)=0$. This is equivalent to solving $(\theta^\sharp -c\theta)\cdot\omega = 0$ for $\theta^\sharp$. It is easy to see that this is always possible to do and the solution closest to $\theta$ is at its farthest distance from $\theta$ when $\omega=\pm \theta$; then $|\theta^\sharp-c\theta|=\sqrt{1-c^2}$. This shows that $|\theta^\sharp-\theta|\le \sqrt{1-c^2}$ uniformly in $\omega$. Property \r{10} then proves the claim.
\end{proof}
\section{The Lorentzian case}
\subsection{Support theorems for analytic Lorentzian manifolds} Let $(M,g)$ be a Lorentzian manifold now.
Next theorem is an analog of Theorem~\ref{thm_C}. Since the global geometry of the null-geodesics in the Lorentzian case is non-trivial and in particular, one can have conjugate points, the assumptions are stronger.
\begin{definition}\label{def_convex}
Let $S$ be a smooth surface near a point $z\in S$ and let $F$ be a defining function so that $S=F^{-1}(0)$ near $z$, $\d F(z)\not=0$, and declare $\{F<0\}$, to be the ``interior'' of $M$ near $z$. Similarly, $\{F>0\}$ is the ``exterior'' of $M$ near $z$. We say that $S$ is \textit{strictly convex} at $z$ in the direction $v\in T_zS$, if $\nabla^2F(z)(v,v)>0$.
We call $S$ strictly lightlike-convex if it timelike, it is strictly convex at all lightlike $(z,v)\in TS$, and every maximal lightlike geodesic tangent to $S$ at some point has no other common points with $S$.
\end{definition}
Here $\nabla^2F$ is the Hessian of $F$, with $\nabla$ being the covariant derivative. This notion of convexity is equivalent to $\frac{d^2} {\d s^2}F\circ\gamma(s)<0$ for the geodesic $\gamma$ through $x$ in the direction $v$; and it is independent of the choice of $F$.
\begin{theorem}\label{thm_LC}
Let $(M,g)$ be an analytic Lorentzian manifold.
Let $S$ be a timelike surface near a fixed point $z_0\in S$. Let $\gamma_0$ be a lightlike geodesic through $z_0$ tangent to $S$ at $z_0$. Assume that $S$ is strictly convex at $z_0$ in the direction of $\dot\gamma_0$, and that $\kappa$ is analytic and non-vanishing near $(z_0,\dot\gamma_0|_{z_0})$. Let $f$ be a distribution, and let $\gamma_0$ be non-trapping in $\supp f$.
Let $L_\kappa f(\gamma)=0$ for all lightlike geodesics $\gamma$ near $\gamma_0$.
If $f=0$ in the exterior of $S$ near $z_0$, then $f=0$ near $z_0$.
\end{theorem}
\begin{proof}
By the Sato-Kawai-Kashiwara Theorem, see \cite{SKK} and \cite{Sj-A}, that we already used in the proof of Proposition~\ref{pr_M_supp}, it is enough to prove that $f$ is microlocally analytic in the direction of the conormal $\zeta_0$ to $S$ at $z_0$.
We follow the construction in Proposition~\ref{pr_M_supp}. We can consider the former proof as a linearized version of the present one, when we replace $S$ with its tangent plane at $z_0$ normal to $\zeta_0$, and the geodesics through $z_0$ by tangent lines.
By the strict convexity assumption,
for any lightlike geodesic $\gamma$ which is an $O(\eps)$, $\eps\ll1$, perturbation of $\gamma_0$ (in a fixed parameterization), the intersection of $\gamma$ with the interior of $S$, in any local chart has Euclidean length $O(\sqrt\eps)$. This allows us to work in a fixed coordinate system $(t,x)$ near $z_0$. We choose a spacelike surface $S_0$ through $z_0$. We then choose
semigeodesic coordinates $(t,x)$ near $z_0=0$ normal to $S_0$, i.e., the lines $(t,x) =(t,\text{const.})$, are future pointing (the future direction being determined by $\gamma_0$) timelike geodesics normal to $S_0$ and $g$ is given locally by $-\d t^2+ g_{\alpha\beta}\d x^\alpha \d x^\beta$, see \cite{Petrov_book}. Such coordinates are constructed by taking a normal field $v$ to $S$ normalized so that $g(v,v)=-1$ and using it as initial directions of the geodesics $x=\text{const}$. We can arrange that $z_0=0$ and $\dot\gamma_0(0)=(0,e_{n})$.
In those coordinates, future pointing geodesics near $\gamma_0$, close to $S_0$, are parameterized by their initial points $x\in S$ and the projection $\theta$ of their tangents to $TS_0$, i.e., $\gamma_{x,\theta}(s)$ is defined as the geodesic issued from $(0,x)$ with $\dot\gamma_{x,\theta}(0)=(1,\theta)$.
For $(x,\theta)$ close to $(0,e_{n})$,
we then write
\[
L_\kappa f(x,\theta) = \int \kappa(\gamma_{x,\theta}(s),\dot \gamma_{x,\theta}(s)) f(\gamma_{x,\theta}(s))\,\d s.
\]
We chose $\theta=\theta(q)$ as in \r{theta} with $|q|<\eps$, where $\eps$ is the number controlling the size of $\supp\chi_N$, see \r{A1}.
Then for $0<\eps\ll1$, $\lambda>0$,
\[
\begin{split}
0 &= \int e^{\i \lambda x\cdot\xi} (\chi_N L_\kappa f)(x,\theta(q))\,\d x\\
&=
\iint e^{\i\lambda x\cdot\xi} \chi_N (x) \kappa(\gamma_{x,\theta(q)}(s), \dot \gamma_{x,\theta(q)}(s)) f(\gamma_{x,\theta(q)}(s) )\,\d s\, \,\d x.
\end{split}
\]
For every fixed $q$ near $q=0$, which fixes $\theta=\theta(q)$, the map $(s,x)\to z =\gamma_{x,\theta}(s) $ is a local diffeomorphism near $z_0=0$ by the Implicit Function Theorem.
Let $s^\sharp(z,\theta)$, $x^\sharp(z,\theta)$ be the inverse map. Since $\gamma_{x,\theta}(s) = (s,x+s\theta)+O(s^2)$, we get the Taylor expansion
\be{10q}
s^\sharp(z,\theta(q)) = t+ O(t^2), \quad x^\sharp (z,\theta(q))= x-t \theta(q) +O(t^2),
\ee
where $z:=(t,x)$.
Those expansions can be justified by the Implicit Function Theorem.
Make the change of variables $(s,x)\to z$ above to get
\be{11}
0=
\int e^{\i\lambda\phi} \chi_N (x^\sharp (z,\theta(q))) \kappa J (q,z)f(z)\,\d z
\ee
with $\phi(z,\xi,q ) = x^\sharp (z,\theta(q))\cdot\xi$. Here, $\kappa$ is the weight in the new variables, and $J$ is the related Jacobian.
If $g$ is Minkowski, we get $\phi = (z-t\theta(q))\cdot\xi$, which is the same function as in \r{5.2}.
Set $\zeta=(q,\xi)$. Then $q=\zeta_0$, $\xi=\zeta' = (\zeta_1,\dots,\zeta_n)$ and
\be{10phi}
\phi(z,\zeta ) = x^\sharp (z,\theta(\zeta_0))\cdot\zeta'.
\ee
\begin{lemma}\label{lemma_phase}
$\det\phi_{z\zeta}(0,e^{n-1})= -1$.
\end{lemma}
\begin{proof}
To compute $\phi_{x\zeta}(0,e^{n-1})$, write first (recall that $z_0=t$)
\be{15}
\phi_{\zeta_k}|_{\zeta= e^{n-1}, z_0=0} = x^\sharp(z,e_n)|_{z_0=0} = z^k, \quad k=1,\dots,n.
\ee
Therefore,
\[
\phi_{z^i\zeta_k}(0,e^{n-1})=\delta_i^k, \quad k=1,\dots,n, \; i=0,1,\dots,n.
\]
Therefore, $\det\phi_{z\zeta}(0,e^{n-1})= \phi_{\zeta_0 z^0}(0,e^{n-1})$. One the other hand, the latter equals $-1$ as follows from \r{10q} and \r{theta}.
\end{proof}
We now get from \r{11}:
\be{12}
0=\int e^{\i\lambda \phi(z,\zeta)} a_N(z,\zeta) f(z)\, \d z=0\quad \text{near $\zeta=e^{n-1}$},
\ee
compare with \r{5.2}, with $ a_N$ elliptic and analytic near $(0,e^{n-1})$ but not for all $(z,\zeta)$. On the other hand, it satisfies a pseudo-analytic estimate of the type \r{A1}.
Similarly to the proof of Lemma~\ref{lemma3.1}, for $w$ and $\eta$ as \r{chi_delta}, in multiply \r{12} by the factor
\[
e^{\i \lambda(\i (\zeta-\eta)^2/2 - \phi(w,\zeta) ) }
\]
and integrate w.r.t.\ $\zeta$ over the ball $|\zeta-\eta|<\delta $ with $0<\delta\ll\eps$ to get \r{6} with
\[
\Phi = \phi(z,\zeta)- \phi(w,\zeta)+ \i (\zeta-\eta)^2/2 .
\]
The rest of the proof follows closely those in \cite{FSU, SU-AJM}. By Lemma~\ref{lemma_phase}, $|\Phi_\xi|$ has a lower bound outside any neighborhood of $z=w$ for $w$ localized as above and $z$ in the support of the integrand. This allows us to integrate by parts to get \r{7} in this case and choose $N\sim \lambda/(Ce)$ to make the r.h.s.\ of \r{7} exponentially small with $\lambda$.
The phase function $\Phi$ has an analytic extension for $\zeta$ in some complex neighborhood of $\zeta_0=e^{n-1}$. By Lemma~\ref{lemma_phase}, $\phi_\zeta(z,\zeta)=\phi_\zeta(w,\zeta)$ for such $\zeta$ and $z$, $w$ close to $0$ implies $z=w$. Therefore, the critical point $\zeta_c=\eta$ of $\Phi$ w.r.t.\ $\zeta$ is real only when $z=w$ and at that point, $\Im\Phi_{\zeta\zeta}>0$; and it is unique and complex otherwise, still satisfying that inequality by a perturbation argument, when $0<\delta\ll\eps$. Then we get \r{8} with $\psi$ defined as in \r{psi}. Then we conclude as in \cite{FSU, SU-AJM} that $(0,\xi_0)\not\in\WFA(f)$.
This arguments so far work if $f$ is a continuous function, for example. If $f$ is a distribution, as stated, we need to take a smooth cutoff $\chi_\delta$ and consider the $z$-integrals above in distribution sense.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm_L}]
Assume that the statement of the theorem is not true.
Let $\sigma_0\in [0,1)$ be the infimum of all $\sigma$ for which $F^{-1}(\sigma)\cap \supp f\not=\emptyset$.
Then $f=0$ in the ``exterior'' $F^{-1}(0,\sigma_0)$ of $S_0= F^{-1}(\sigma_0)$, and $\supp f$ has a common $z_0$ point with $S_0$. The latter follows from a compactness argument. In particular, $\sigma_0>0$.
Since $S_0$ is timelike by assumption, there is a lightlike geodesic $\gamma_0$ through $z_0$ tangent to $S_0$ which does not hit $S_0$ again by the strict convexity assumption. Near $z_0$, the geodesic $\gamma_0$ lies in the exterior $F^{-1}[0,\sigma_0]$ by the local part of the strict convexity assumption; and this is also true globally by the global part of that assumption.
By Theorem~\ref{thm_LC}, $f=0$ near every common point of $S_0$ and $\supp f$, which is a contradiction.
\end{proof}
|
1,116,691,498,487 | arxiv | \section{}
Venus has no known satellites \citep{2009Icar..202...12S}, but has four known co-orbitals: (322756)~2001~CK$_{32}$ \citep{2004Icar..171..102B},
2002~VE$_{68}$ \citep{2004MNRAS.351L..63M,2012MNRAS.427..728D}, 2012~XE$_{133}$ \citep{2013MNRAS.432..886D}, and 2013~ND$_{15}$
\citep{2014MNRAS.439.2970D}. These objects are temporarily trapped in a 1:1 mean motion resonance with Venus, but are not gravitationally bound
to it. It is believed that any putative primordial Venus co-orbitals (for example, Trojans) were lost early in the history of the Solar System;
present-day Venus co-orbitals are expected to be of transient nature \citep{2006Icar..185...29M}. Venus co-orbitals are very challenging targets
as they spend most of the time in the unobservable (daytime) sky, at solar elongations well below 90\degr. The minimum orbit intersection
distances (MOIDs) with the Earth of 322756, 2002 VE$_{68}$, 2012~XE$_{133}$, and 2013~ND$_{15}$ are 0.076, 0.027, 0.0019, and 0.0078 AU,
respectively. This property alone makes them objects of significant practical interest as they tend to approach the Earth from the daytime side.
Here, we present numerical evidence suggesting that 2015~WZ$_{12}$ \citep{2015MPEC....X...04C} is a possible Venus co-orbital. With these
tentative results we hope to encourage a search for precovery images of this minor body and perhaps even follow-up observations that may help in
improving its poorly determined orbit so its current dynamical nature is better understood. The orbit determination of 2015~WZ$_{12}$ currently
available (epoch JD 2458000.5) is based on 71 observations (1 Doppler) for a data-arc span of 6 d and has semi-major axis, $a$ = 0.721826$\pm$0.000012~AU,
eccentricity, $e$ = 0.41250$\pm$0.00003, inclination, $i$ = 3\fdg6261$\pm$0\fdg0005, longitude of the ascending node, $\Omega$ = 251\fdg7613$\pm$0\fdg0006,
and argument of perihelion, $\omega$ = 345\fdg7200$\pm$0\fdg0005.\footnote{\href{http://ssd.jpl.nasa.gov/sbdb.cgi}{JPL's Small-Body Database}}
Unfortunately, this Aten asteroid can experience close encounters with Mercury, Venus and the Earth--Moon system, making its orbital evolution
very chaotic; its MOID with the Earth is 0.0043 AU. With an absolute magnitude of 26.3 it may have a probable diameter of 19 m; large enough
to cause a local disturbance, not too different from that of the Chelyabinsk event, in case of impact.
We have used the heliocentric Keplerian orbital elements and 1$\sigma$ uncertainties of 2015~WZ$_{12}$ to perform a preliminary exploration of
its short-term dynamical evolution (for technical details see \citealt{2012MNRAS.427..728D,2013MNRAS.432..886D,2014MNRAS.439.2970D}). Our limited
analysis strongly suggests that its evolution becomes difficult to reconstruct or predict beyond 100 yr. Figure~\ref{fig:1} shows the evolution
backward and forward in time of several orbital elements and other relevant parameters of 2015~WZ$_{12}$ using initial conditions compatible with
the nominal orbit presented above. The top panel shows that 2015~WZ$_{12}$ experiences close encounters with the Earth at relatively short-range;
these are often low-velocity flybys as they take place at aphelion. The second to top panel shows the behavior of the so-called Kozai-Lidov
parameter that measures the evolution of the component of the orbital angular momentum perpendicular to the ecliptic; the value remains fairly
constant. The third to top panel shows the variation of its relative mean longitude; when its value oscillates, the object is engaged in a 1:1
mean motion resonance. Asteroid 2015~WZ$_{12}$ might have been until recently a transient Trojan of Venus (it lost this status after suffering a
close encounter with Mercury). Other control orbits show a somewhat different evolution, but in all cases we observe frequent switching between
the various co-orbital states and their hybrids (see \citealt{2012MNRAS.427..728D,2013MNRAS.432..886D,2014MNRAS.439.2970D}). The bottom panel
shows the evolution of the nodal distances of 2015~WZ$_{12}$; flybys with the Earth--Moon system take place at the descending node, while Mercury
is approached at the ascending node.
An object as small as 2015~WZ$_{12}$ must be a fragment of a larger body, it may have its provenance in the main asteroid belt, but it might have
been produced {\it in situ}, i.e. in the region between the orbits of Mercury and the Earth--Moon system, via super-catastrophic break-ups
\citep{2016Natur.530..303G}. Follow-up observations of this target in the near future will be difficult, though; it will reach its next favorable
perigee in 2018 November, at a solar elongation of about 85\degr with an apparent magnitude in excess of 26.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.37,angle=0]{fcon1x8_2015WZ12.eps}
\caption{Evolution of the values of the orbital elements and other relevant parameters for the nominal orbit of 2015~WZ$_{12}$.
The top panel shows the geocentric distance (Hill radius of the Earth, 0.0098~AU). The Kozai-Lidov parameter is shown
in the second to top panel. The value of the resonant angle is displayed in the third to top panel. The evolution of
the orbital elements, semi-major axis, eccentricity, inclination and argument of perihelion is shown in the fourth to
top panel and the fourth, third and second to bottom panels, respectively. The bottom panel shows the distance from
the Sun to the descending (thick line) and ascending nodes (dotted line); the aphelion and perihelion distances of
Mercury, Venus and the Earth are indicated as well.
\label{fig:1}}
\end{center}
\end{figure}
\acknowledgments
We thank S.~J. Aarseth for providing the code used in this research, A.~I. G\'omez de Castro, I. Lizasoain and L. Hern\'andez Y\'a\~nez of the
Universidad Complutense de Madrid (UCM) for providing access to computing facilities. This work was partially supported by the Spanish
`Ministerio de Econom\'{\i}a y Competitividad' (MINECO) under grant ESP2014-54243-R. Part of the calculations and the data analysis were
completed on the EOLO cluster of the UCM. EOLO, the HPC of Climate Change of the International Campus of Excellence of Moncloa, is funded by the
MECD and MICINN. This is a contribution to the CEI Moncloa. In preparation of this paper, we made use of the NASA Astrophysics Data System and
the MPC data server.
|
1,116,691,498,488 | arxiv | \section{Introduction}
\label{sec:introduction}
\setcounter{equation}{0}
Recently, our understanding of chiral symmetry on the lattice
has substantially improved.
Lattice Dirac operators
have been obtained \cite{overlap-D,fixed-point-D},
which are gauge covariant,
define local actions \cite{locality-of-overlap-D} and
satisfy the Ginsparg-Wilson relation \cite{ginsparg-wilson-rel}.
The Ginsparg-Wilson relation
\begin{equation}
\gamma_5 D + D \gamma_5 = a D R \gamma_5 D
\end{equation}
implies the exact chiral symmetry of the action
under the transformation \cite{exact-chiral-symmetry}
\begin{equation}
\label{eq:chiral-transformation-of-psi}
\delta \psi(x) = \gamma_5 \left( 1-aRD \right) \psi(x), \quad
\delta \bar \psi(x) = \bar \psi(x) \gamma_5 .
\end{equation}
The explicit gauge covariant solution of the Ginsparg-Wilson relation
has been derived by Neuberger \cite{overlap-D} from the overlap
formulation of chiral determinant \cite{overlap}.
It is defined through
the hermitian Wilson-Dirac operator $H$ with a negative mass
in a certain range
\footnote{We adopt the definition of the overlap Dirac
operator so that the normalization of the factor one half is
included. This leads to the Ginsparg-Wilson relation with $R=2$.},
\begin{equation}
\label{eq:overlap-dirac-operator}
D= \frac{1}{2a } \left( 1+\gamma_5\frac{H}{\sqrt{H^2}}\right) .
\end{equation}
The locality properties of the Dirac operator has been
examined by Hern\'andes, Jansen and L\"uscher \cite{locality-of-overlap-D}.
The issue of the practical implimentation of the Dirac operator
has been studied by Neuberger \cite{practical-D-neuberger},
Edwards, Heller and Narayanan \cite{practical-D-narayanan-etal},
Chiu \cite{practical-D-chiu}
and A.~Borici \cite{practical-D-borici}.
The domain-wall fermion \cite{domain-wall-fermion} is the basis
of the overlap Dirac operator.
In a simplified formulation
\cite{boundary-fermion,boundary-fermion-QCD,truncated-overlap},
the domain-wall fermion consists of $N$-flavor Wilson
fermions with a certain flavor-mixing mass matrix.
Due to its structure of the chiral hopping and the boundary condition
in the flavor space, a single light Dirac fermion can emerge in the
spectrum. This light fermion can be probed most suitably
by the field variables at the boundary in the flavor space,
which are referred as $q(x)$ and $\bar q(x)$ by Furman and Shamir.
It has been argued that the chiral symmetry of
the light fermion is preserved up to corrections suppressed
exponentially in the number of flavors \cite{boundary-fermion-QCD}.
This fact has been observed
numerically \cite{vranas-schwinger-model,blum-soni}
and have been found useful for the numerical simulation of
lattice QCD \cite{columbia,blum-soni-wingate,lagae-sinclair}.
The perturbative studies are found in
\cite{aoki-taniguchi,kikukawa-neuberger-yamada}.
The authors refer the reader to \cite{blum-lat98} for
recent review.
In this context, the chiral transformation of this light fermion
is defined as
\begin{equation}
\label{eq:chiral-transformation-of-q}
\delta q(x) = \gamma_5 q(x), \quad
\delta \bar q(x) = \bar q(x) \gamma_5 .
\end{equation}
The goal of this paper is to understand the chiral property
of the light fermion of the domain-wall fermion
from the point of view of the exact chiral
symmetry based on the Ginsparg-Wilson relation.
Several authors have discussed
the direct relation between the domain-wall fermion and the Dirac
fermion which is described by the overlap Dirac operator.
Vranas has shown that
in order for the subtraction of the massive modes of the domain-wall
fermion, it is suitable to introduced the $N$-flavor
Wilson-Dirac boson
with the flavor-mixing mass matrix which is anti-periodic
in the flavor space \cite{vranas-pauli-villars}.
Neuberger has shown in \cite{truncated-overlap} through the
explicit calculation of the partition function how the overlap
Dirac operator emerges in the limit of infinite number of flavors:
the subtracted partition function at a finite flavor $N$ can be written
as a single determinant of the truncated overlap Dirac operator
\begin{equation}
\label{eq:truncated-overlap-dirac-operator}
D_N=
\frac{1}{2a}\left(
1+\gamma_5 \tanh \frac{N}{2} a_5 \widetilde H \right).
\end{equation}
Note that $\widetilde H$ here is defined through the transfer matrix of the
five-dimensional Wilson fermion with a negative mass:
\begin{equation}
\widetilde H = - \frac{1}{a_5} \ln T .
\end{equation}
In the limit of infinite flavors, this reduces to the overlap
Dirac operator Eq.~(\ref{eq:overlap-dirac-operator}), in which
the hermitian Wilson-Dirac operator $H$ is replaced by $\widetilde H$.
Thus the domain-wall fermion with the subtraction
of the Pauli-Villars field is equivalent to the Dirac fermion
described by the truncated overlap Dirac operator.
The relation between the light fermion field $q(x)$ and $\bar q(x)$
of the domain-wall fermion and the fermion field
$\psi(x)$ and $\bar \psi(x)$ described by the overlap Dirac operator
has also been suggested by several authors.
Neuberger pointed out in \cite{truncated-overlap}
the correspondence between the mass term
for $q(x)$
\begin{equation}
m \, \bar q(x) q(x)
\end{equation}
and the mass term for $\psi(x)$
\begin{equation}
m \, \bar \psi(x) \left(1-\frac{a}{2}RD\right) \psi(x) .
\end{equation}
It has also been noticed by the authors of
\cite{chiu-etal,neidermayer-lat98}
that the transformation property of the operator
$\left(1-\frac{a}{2}RD\right) \psi(x)$ under the chiral transformation
Eq.~(\ref{eq:chiral-transformation-of-psi}) is same as $q(x)$:
\begin{equation}
\delta \left\{ \left(1-\frac{a}{2}RD\right) \psi(x) \right\}
= \gamma_5 \left\{ \left(1-\frac{a}{2}RD\right) \psi(x) \right\} .
\end{equation}
These facts suggest that there could be a correspondence as
\begin{eqnarray}
q(x), \quad \bar q(x)
&\Longleftrightarrow&
\left(1-\frac{a}{2}RD\right) \psi(x) , \quad \bar \psi(x)
\end{eqnarray}
at least in the limit of infinite flavors.\footnote{This
correspondence could be different if we adopt a different transformation
of the exact chiral symmetry of L\"uscher from
Eq.~(\ref{eq:chiral-transformation-of-psi}).}
In this paper, we will further examine the above correspondence
between the light fermion field of the domain-wall fermion and
the Dirac field described by the (truncated) overlap Dirac
operator.
For this purpose, we derive the low energy effective action
of the light fermion field by integrating out
$N-1$ heavy flavors of the domain-wall fermion:
\begin{equation}
S_N^{\rm eff} = a^4 \sum_x \bar q(x) \, D_N^{\rm eff} \, q(x).
\end{equation}
As easily understood, this can be achieved by calculating
the propagator of $q(x)$ and $\bar q(x)$,
because it should be given by the inverse
of the effective Dirac operator of these fields.
\footnote{ In this paper, the bra-ket symbol $\langle \cdots
\rangle$ denotes the Wick contraction of the fermion fields in it
by their propagators, not the fermionic vacuum expectation values
which must includes the weight of the fermion action.
The bra-ket symbol with the subscript $c$ denotes
the connected contraction.
}
\begin{equation}
\left\langle q(x) \, \bar q(y) \right\rangle
= \frac{1}{a^4} { D_N^{\rm eff} }^{-1}(x,y) .
\end{equation}
It turns out that
the propagator of the light fermion
is closely
related to the inverse of the truncated overlap Dirac operator as follows:
\begin{eqnarray}
\frac{a}{a_5} { D_N^{\rm eff} }^{-1}+ a \delta(x,y)
&=&
{ D_N^{\rm \phantom{f}} }^{-1} .
\end{eqnarray}
Namely,
the inverse of the effective Dirac operator
gives the inverse of the truncated overlap Dirac operator up to
a local contact term. This contact term just takes account of the
chiral symmetry breaking in the Ginsparg-Wilson relation,
which holds true for the overlap Dirac operator
in the limit of the infinite flavors.
The above relation
allows us to relate the field variables of the light fermion,
$q(x)$ and $\bar q(x)$, with the field variables
described by the truncated overlap Dirac operator,
$\psi(x)$ and $\bar \psi(x)$.
Then we can clarify the relation between the
almost preserved chiral symmetry of the domain-wall fermion under
the transformation Eq.~(\ref{eq:chiral-transformation-of-q}) and the
(would-be) exact chiral symmetry of the Dirac fermion described
by the (truncated) overlap Dirac operator under the transformation
Eq.~(\ref{eq:chiral-transformation-of-psi}).
It is also possible to
relate the low energy observables of the domain-wall
fermion which are written in terms of $q(x)$ and $\bar q(x)$
to those written in terms of $\psi(x)$ and $\bar \psi(x)$
\begin{equation}
{\cal O}_{\rm DW} [q,\bar q] = {\cal O}_N [\psi, \bar \psi; D_N ]
\end{equation}
and to examine the chiral properties of the observables
through the (would-be) exact chiral symmetry based
on the Ginsparg-Wilson relation.
This article is organized as follows.
In section~\ref{sec:effective-action-of-q},
we will derive the effective action of the light fermion field
of the domain-wall fermion by integrating out the heavy
$N-1$ flavors.
The calculation of the effective action (the propagator of the light
fermion)
is a straightforward application of that given by Neuberger
in \cite{truncated-overlap}.
In section~\ref{sec:chiral-property-q-and-psi},
we will establish the relation between the light fermion field
and the fermion field described by the truncated overlap Dirac operator.
Then we discuss the chiral properties of the light fermion
from the point of view of the exact chiral symmetry based
on the Ginsparg-Wilson relation.
In section~\ref{sec:axial-anomaly-q-and-psi},
we reexamine the axial anomaly of the domain-wall fermion
\cite{axial-anomaly-in-domain-wall}
in view of the axial anomaly associated with the exact chiral symmetry
\cite{index-theorem-at-finite-lattice,exact-chiral-symmetry,
lattice-chiral-jaccobian}.
In section~\ref{sec:Pauli-Villars-field}, the contribution of
the Pauli-Villars field to the currents and the axial anomaly
is examined.
\section{Low energy effective action of the domain-wall fermion}
\label{sec:effective-action-of-q}
\setcounter{equation}{0}
We first review briefly the domain-wall fermion and its relation
to the Dirac fermion described by the overlap Dirac operator.
Then we evaluate the low energy effective action of the light
fermion of the domain-wall fermion and discuss the result
in relation to the Ginsparg-Wilson relation.
\subsection{Light fermion of the domain-wall fermion}
The domain-wall fermion, in its simplified
formulation \cite{boundary-fermion,boundary-fermion-QCD,truncated-overlap},
consists of $N$-flavor Wilson fermions\footnote{In this paper,
we assume that the number of flavor $N$ is even.}
\begin{equation}
\label{eq:action-domain-wall-fermion}
S_{\rm DW}
=\sum_{s,t=1}^N a^4 \sum_x
\bar \psi_{s}(x)
\left\{
\gamma_\mu \frac{1}{2}\left(\nabla_\mu+\nabla_\mu^\ast\right) \delta_{st}
+ P_L M_{st} + P_R M^\dagger_{st}
\right\} \psi_{t}(x)
\end{equation}
with a certain flavor-mixing mass matrix:
in the case with $N=6$, it is given by
\begin{eqnarray}
\label{eq:flavor-mixing-mass-matrix}
M_{st}&=&
\frac{1}{a_5}
\left( \begin{array}{cccccc}
B & -1 & 0 & 0 & 0 & 0 \\
0 & B & -1 & 0 & 0 & 0 \\
0 & 0 & B & -1 & 0 & 0 \\
0 & 0 & 0 & B & -1 & 0 \\
0 & 0 & 0 & 0 & B & -1 \\
0 & 0 & 0 & 0 & 0 & B
\end{array} \right),
\end{eqnarray}
and
\begin{eqnarray}
\label{eq:operator-B}
B &=& 1 + a_5 \left(
-\frac{a}{2} \nabla_\mu\nabla_\mu^\ast - \frac{m_0}{a}
\right) .
\end{eqnarray}
Due to its structure of the chiral hopping and the boundary condition
in the flavor space, a single light Dirac fermion can emerge in the
spectrum. The exact eigenvalues and eigenvectors
of the mass matrix for the free theory at a finite flavor $N$ has
been given in \cite{truncated-overlap}.
This light fermion can be probed most suitably
by the field variables at the boundary in the flavor space
and is denoted as $q(x)$ and $\bar q(x)$ by Furman and Shamir:
\begin{equation}
q(x) = \psi_{1L}(x) + \psi_{NR}(x) , \quad
\bar q(x) = \bar \psi_{1L}(x) + \bar \psi_{NR}(x).
\end{equation}
Following Neuberger, we may change the flavor index of the left-handed
component by the chirally asymmetric parity transformation in the
flavor space:
\begin{eqnarray}
\psi^\prime_s(x) &=& \left( P_R + P_L P \right)_{st} \psi_t(x), \\
\bar \psi^\prime_s(x) &=& \bar \psi_t(x) \left( P_R P + P_L \right)_{ts} ,
\end{eqnarray}
where
\begin{equation}
P_{st} = \left( \begin{array}{cccccc}
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0
\end{array} \right) \quad (N=6). \\
\end{equation}
By this transformation, the mass matrix becomes hermitian,
\begin{eqnarray}
\label{eq:hermitian-mass-matrix}
M^{\rm H}_{st}&=& M_{st} P =P M^\dagger_{st} \nonumber\\
&=&
\frac{1}{a_5}
\left( \begin{array}{cccccc}
0 & 0 & 0 & 0 & -1& B\\
0 & 0 & 0 & -1& B & 0 \\
0 & 0 & -1& B & 0 & 0 \\
0 &-1 & B & 0 & 0 & 0 \\
-1& B & 0 & 0 & 0 & 0 \\
B & 0 & 0 & 0 & 0 & 0
\end{array} \right) , \quad (N=6). \nonumber\\
\end{eqnarray}
In this basis, it is the $N$-th flavor field that is most suitable
to probe the light fermion:
\begin{equation}
q(x) = \psi_N^\prime(x), \qquad \bar q(x) = \bar \psi_N^\prime(x) .
\end{equation}
The chiral transformation adopted by Shamir and Furman
\cite{boundary-fermion-QCD} is given in this hermitian basis as follows:
\begin{equation}
\label{eq:chiral-transformation-of-DW}
\delta \psi'_s(x)
= \left( \Gamma_5 \right)_{st} \psi'_t(x),
\end{equation}
where $\Gamma_5$ is given (for $N=6$) by
\begin{equation}
\left( \Gamma_5 \right)_{st} =
\left( \begin{array}{cccccc} -\gamma_5 & 0 & 0 & 0 & 0 & 0 \\
0 & -\gamma_5 & 0 & 0 & 0 & 0 \\
0 & 0 & -\gamma_5 & 0 & 0 & 0 \\
0 & 0 & 0 & \gamma_5 & 0 & 0 \\
0 & 0 & 0 & 0 & \gamma_5 & 0 \\
0 & 0 & 0 & 0 & 0 & \gamma_5
\end{array}
\right) \qquad (N=6).
\end{equation}
In particular, the light fermion transforms as
Eq.~(\ref{eq:chiral-transformation-of-q}).
\[
\delta q(x) = \gamma_5 q(x), \quad
\delta \bar q(x) = \bar q(x) \gamma_5 .
\]
With this definition of the chiral transformation,
the chiral symmetry is broken only by
the diagonal $\frac{N}{2}$-th element of the hermitian
mass matrix Eq.~(\ref{eq:hermitian-mass-matrix}),
i.e. the (diagonal) mass term of the $\frac{N}{2}$-th flavor:
\begin{equation}
\label{eq:chiral-property-domain-wall-D}
\left\{ \Gamma_5 D_{\rm DW}^\prime + D_{\rm DW}^\prime \Gamma_5 \right\}_{st}
=\frac{2}{a_5} \,
\gamma_5 \delta_{s \frac{N}{2}}\delta_{t \frac{N}{2}} .
\end{equation}
\subsection{Pauli-Villars field and truncated overlap Dirac operator}
In order to take the limit of infinite flavors
and to relate the domain-wall fermion to the Dirac fermion described
by the overlap Dirac operator,
it is suitable to introduce, as a Pauli-Villars field,
the $N$-flavor Wilson-Dirac bosons with the flavor-mixing
mass matrix which is anti-periodic in the flavor
space \cite{vranas-pauli-villars}:
\begin{eqnarray}
\label{eq:hermitian-mass-matrix-antiperiodic}
M^{\rm PV}_{st}&=&
\frac{1}{a_5}
\left( \begin{array}{cccccc}
0 & 0 & 0 & 0 & -1& B\\
0 & 0 & 0 & -1& B & 0 \\
0 & 0 & -1& B & 0 & 0 \\
0 &-1 & B & 0 & 0 & 0 \\
-1& B & 0 & 0 & 0 & 0 \\
B & 0 & 0 & 0 & 0 & 1
\end{array} \right) . \nonumber\\
\end{eqnarray}
Then the total action of the domain-wall fermion with the subtraction
of the Pauli-Villars field is given by
\begin{eqnarray}
\bar S_{\rm DW}
&=&\sum_{s,t=1}^N a^4 \sum_x
\bar \psi_{s}^\prime(x)
\left\{
\gamma_\mu \frac{1}{2}\left(\nabla_\mu+\nabla_\mu^\ast\right) \delta_{st}
+ M_{st}^H \right\} \psi_{t}^\prime(x)
\nonumber\\
&+&
\sum_{s,t=1}^N a^4 \sum_x
\bar \phi_{s}^\prime(x)
\left\{
\gamma_\mu \frac{1}{2}\left(\nabla_\mu+\nabla_\mu^\ast\right) \delta_{st}
+ M^{\rm PV}_{st}
\right\} \phi_{t}^\prime(x). \nonumber\\
\end{eqnarray}
As shown by Neuberger in \cite{truncated-overlap}
that the partition function of this total system at a finite
flavor $N$ can be written by a single determinant of
the truncated overlap Dirac operator:
\begin{equation}
\label{eq:subtracted-partition-function}
\bar Z_{\rm DW} = Z_{\rm DW} Z_{\rm PV}
= \det \, a D_N .
\end{equation}
The truncated overlap Dirac operator is defined by
Eq.~(\ref{eq:truncated-overlap-dirac-operator}) through
the transfer matrix, which is given explicitly
in the chiral basis \footnote{
The gamma matrices in the chiral basis are chosen as follows
in our convention,
\begin{equation}
\gamma_\mu=\left( \begin{array}{cc} 0 & \sigma_\mu \\
\sigma_\mu^\ast & 0
\end{array} \right) ,
\quad
\gamma_5=\left( \begin{array}{cc} 1 & 0 \\
0 & -1
\end{array} \right) , \quad
\sigma_\mu = \left( 1, \sigma_1, \sigma_2, \sigma_3 \right) .
\end{equation}
}
as
\begin{equation}
T \equiv e^{- a_5 \bar H} = \left(
\begin{array}{cc} \frac{1}{B} & \frac{1}{B} C \\
-C^\dagger \frac{1}{B}
& B + C^\dagger \frac{1}{B} C
\end{array} \right),
\end{equation}
where
\begin{equation}
\label{eq:operator-C}
C = a_5 \, \sigma_\mu \,
\frac{1}{2}\left(\nabla_\mu+\nabla_\mu^\ast\right)
\end{equation}
and $B$ is given by Eq.~(\ref{eq:operator-B}).
Thus the domain-wall fermion with the subtraction
of the Pauli-Villars field is equivalent to the Dirac fermion
described by the truncated overlap Dirac operator:
\begin{equation}
\label{eq:truncated-overlap-Dirac-fermion}
S_N
= a^4 \sum_x \bar \psi(x) \, D_N \, \psi(x) .
\end{equation}
The overlap Dirac operator, which is obtained from $D_N$ in the
limit of the infinite flavors (but with a finite $a_5$),
is denoted as $\widetilde D \equiv \lim_{N\rightarrow \infty} D_N$.
This reduces to $D$ of
Eq.~(\ref{eq:truncated-overlap-dirac-operator}) in the limit that
$a_5$ vanishes.
\subsection{Effective action of the light fermion of domain-wall fermion}
\label{sec:effective-action-of-q-calculation}
Now we evaluate the effective action of the light fermion
field by integrating out $N-1$ heavy flavors of the domain-wall fermion:
\begin{equation}
S_N^{\rm eff} = a^4 \sum_x \bar q(x) \, D_N^{\rm eff} \, q(x).
\end{equation}
As mentioned in the introduction, this can be achieved
by calculating the propagator of $q(x)$, because
it should be given by the inverse of the effective Dirac operator
for the field variables $q(x)$ and $\bar q(x)$.
In order to obtain the propagator of these fields, we introduce
the sources for them,
\begin{equation}
a^4 \sum_x \left\{ \bar J(x) q(x) + \bar q(x) J(x) \right\} .
\end{equation}
We first describe the case with four flavors
($N=4$) for simplicity and then generalize the result to any flavors.
In the chiral basis of the gamma matrices,
the original action
of the domain-wall fermion Eq.~(\ref{eq:action-domain-wall-fermion})
can be written in the matrix form as
\begin{equation}
S_{\rm DW} = a^4 \sum_x \bar \Psi(x) D_{\rm DW} \Psi(x), \qquad (N=4)
\end{equation}
where
\begin{eqnarray}
\bar \Psi(x)&=&
\left(\begin{array}{cccccccc}
\bar \psi_{1L} & \bar \psi_{1R} &
\bar \psi_{2L} & \bar \psi_{2R} &
\bar \psi_{3L} & \bar \psi_{3R} &
\bar \psi_{4L} & \bar \psi_{4R}
\end{array}\right), \\
\nonumber\\
D_{\rm DW}&=&
\frac{1}{a_5}
\left( \begin{array}{cccccccccc}
B & C & 0 & 0 & 0 & 0 & 0 & 0\\
-C^\dagger & B & 0 & -1 & 0 & 0 & 0 & 0\\
-1 & 0 & B & C & 0 & 0 & 0 & 0\\
0 & 0 & -C^\dagger & B & 0 & -1 & 0 & 0\\
0 & 0 & -1 & 0 & B & C & 0 & 0\\
0 & 0 & 0 & 0 & -C^\dagger & B & 0 & -1\\
0& 0 & 0 & 0 & -1 & 0 & B & C \\
0& 0 & 0 & 0 & 0 & 0 & -C^\dagger & B \\
\end{array} \right) , \\
\nonumber\\
\Psi(x) &=& \left(\begin{array}{c}
\psi_{1R} \\ \psi_{1L} \\
\psi_{2R} \\ \psi_{2L} \\
\psi_{3R} \\ \psi_{3L} \\
\psi_{4R} \\ \psi_{4L}
\end{array}\right) .
\end{eqnarray}
Following Neuberger \cite{truncated-overlap},
we then make the Dirac operator almost upper
triangle. We first exchange the right-handed component
and the left-handed component of each flavor in $\Psi(x)$:
\begin{equation}
\left( \begin{array}{c} \psi_{t R} \\
\psi_{t L} \end{array} \right)
\Longrightarrow
\left( \begin{array}{c} \psi_{t L} \\
\psi_{t R} \end{array} \right)
=
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \,
\left( \begin{array}{c} \psi_{t R} \\
\psi_{t L} \end{array} \right),
\quad ( t = 1,2,3,4).
\end{equation}
Then the Dirac operator of the domain-wall fermion becomes
\begin{eqnarray}
D_{\rm DW} &\Longrightarrow&
\frac{1}{a_5}
\left( \begin{array}{cccccccccc}
C & B & 0 & 0 & 0 & 0 & 0 & 0\\
B & -C^\dagger & -1 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & C & B & 0 & 0 & 0 & 0\\
0 & 0 & B& -C^\dagger & -1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & C & B & 0 & 0\\
0 & 0 & 0 & 0 & B & -C^\dagger & -1 & 0 \\
0& 0 & 0 & 0 & 0 & -1 & C & B \\
0& 0 & 0 & 0 & 0 & 0 & B & -C^\dagger \\
\end{array} \right) . \nonumber\\
\end{eqnarray}
We further move the first row down to the last row:
\begin{eqnarray}
&\Longrightarrow& D_{\rm DW}'' \equiv
\frac{1}{a_5}
\left( \begin{array}{cccccccccc}
B & -C^\dagger & -1 & 0 & 0 & 0 & 0 & 0\\
0 & -1 & C & B & 0 & 0 & 0 & 0\\
0 & 0 & B& -C^\dagger & -1 & 0 & 0 & 0\\
0 & 0 & 0 & -1 & C & B & 0 & 0\\
0 & 0 & 0 & 0 & B & -C^\dagger & -1 & 0 \\
0& 0 & 0 & 0 & 0 & -1 & C & B \\
0& 0 & 0 & 0 & 0 & 0 & B & -C^\dagger \\
C & B & 0 & 0 & 0 & 0 & 0 & 0\\
\end{array} \right) . \nonumber\\
\end{eqnarray}
Accordingly, the components of the domain-wall fermion fields reads
\begin{eqnarray}
\bar \Psi(x)&\Longrightarrow&
\bar \Psi'' (x)=
\left(\begin{array}{cccccccc}
\bar \psi''_{1R} & \bar \psi''_{1L}
& \bar \psi''_{2R} & \bar \psi''_{2L}
& \bar \psi''_{3R} & \bar \psi''_{3L}
& \bar \psi''_{4R} & \bar \psi''_{4L}
\end{array}\right) \nonumber\\
&& \phantom{\bar \Psi'' (x)} \equiv
\left(\begin{array}{cccccccc}
\bar \psi_{1R} &
\bar \psi_{2L} & \bar \psi_{2R} &
\bar \psi_{3L} & \bar \psi_{3R} &
\bar \psi_{4L} & \bar \psi_{4R} & \bar \psi_{1L}
\end{array}\right), \nonumber\\
\\
\Psi(x) &\Longrightarrow&
\Psi''(x) =
\left(\begin{array}{c}
\psi''_{1L} \\ \psi''_{1R} \\
\psi''_{2L} \\ \psi''_{2R} \\
\psi''_{3L} \\ \psi''_{3R} \\
\psi''_{4L} \\ \psi''_{4R}
\end{array}\right)
\equiv
\left(\begin{array}{c}
\psi_{1L} \\ \psi_{1R} \\
\psi_{2L} \\ \psi_{2R} \\
\psi_{3L} \\ \psi_{3R} \\
\psi_{4L} \\ \psi_{4R}
\end{array}\right) .
\end{eqnarray}
The sources for $q(x)$ and $\bar q(x)$ may be expressed in this
upper-triangle basis as follows:
\begin{eqnarray}
&& a^4 \sum_x \left\{ \bar J(x)
\left[
P_L \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\psi''_{1}(x)
+P_R \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\psi''_{N}(x) \right]
\right. \nonumber\\
&& \qquad\qquad \qquad\qquad \qquad\qquad \qquad\quad
\left.
+ \bar \psi''_{N} (x)
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
J(x) \right\} . \nonumber\\
\end{eqnarray}
In order to express the two by two blocks of $D''_{\rm DW}$,
we introduce the following abbreviations
\begin{eqnarray}
&&\alpha
= \frac{1}{a_5}
\left( \begin{array}{cc} B & -C^\dagger \\
0 & -1 \end{array} \right),
\quad
\beta
= \frac{1}{a_5}
\left( \begin{array}{cc}
-1 & 0 \\
C & B
\end{array} \right), \\
&&
\alpha_0
= \frac{1}{a_5}
\left( \begin{array}{cc} B & -C^\dagger \\
0 & 0 \end{array} \right),
\quad
\beta_0
= \frac{1}{a_5}
\left( \begin{array}{cc}
0 & 0 \\
C & B
\end{array} \right).
\end{eqnarray}
In this basis it is now easy to integrate fields from the first flavor down
to the last flavor.\footnote{ Note that $q_L(x)$ is
still in the first component of $\Psi''(x)$, although
$\bar q_L(x)$ is in the $N$-th component of $\bar \Psi''(x)$.
This is why we could not evaluate the effective action directly in
this basis which is convenient for the integration.}
The terms which include the field variables
of the first flavor are following
(The summation over the lattice indices $x$ with the measure factor $a^4$
is understood in the following equations.):
\begin{eqnarray}
&&
\left[
\bar J(x) P_L \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
+ \bar \psi''_N(x) \, \beta_0
\right] \, \psi''_{1}(x)
\nonumber\\
&& \qquad\qquad\quad
+ \bar \psi''_1(x) \, \alpha \, \psi''_1(x)
\nonumber\\
&& \qquad\qquad\qquad\qquad\qquad
+ \bar \psi''_1(x) \, \beta \, \psi''_2(x) .
\end{eqnarray}
After integrating the first flavor,
the terms which include the second flavor are found as follows:
\begin{eqnarray}
&&
\left[
\bar J(x) P_L \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
+ \bar \psi''_N(x) \, \beta_0
\right]
\, \left( - \alpha^{-1} \beta \right) \psi''_2(x)
\nonumber\\
&& \qquad\qquad\qquad
+ \bar \psi''_2(x) \, \alpha \, \psi''_2(x)
\nonumber\\
&& \qquad\qquad\qquad\qquad\qquad\quad
+ \bar \psi''_2(x) \, \beta \, \psi''_3(x) .
\end{eqnarray}
The integration of the second flavor leaves
the terms which include the third flavor as follows:
\begin{eqnarray}
&&
\left[
\bar J(x) P_L \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
+ \bar \psi''_N(x) \, \beta_0
\right]
\, \left( - \alpha^{-1} \beta \right)^2 \psi''_3(x)
\nonumber\\
&& \qquad\qquad\qquad
+ \bar \psi''_3(x) \, \alpha \, \psi''_3(x)
\nonumber\\
&& \qquad\qquad\qquad\qquad\qquad\quad
+ \bar \psi''_3(x) \, \beta \, \psi''_4(x) .
\end{eqnarray}
After the integration of the third flavor, only the forth flavor
remains:
\begin{eqnarray}
\label{eq:last-flavor-terms}
&&
\bar J(x) \left[
P_L \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\left( - \alpha^{-1} \beta \right)^{N-1}
+P_R \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\right] \psi''_N(x)
\nonumber\\
&& \qquad\qquad\qquad
+
\bar \psi''_N(x) \,
\left[ \alpha_0
+\beta_0 \,
\left( - \alpha^{-1} \beta \right)^{N-1} \right] \psi''_N(x)
\nonumber\\
&& \qquad\qquad\qquad\qquad\qquad\quad
+ \bar \psi''_N (x)
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
J(x), \qquad (N=4). \nonumber\\
\end{eqnarray}
As easily seen, this result holds true for any flavors $N$.
Now noting
\begin{eqnarray}
\label{eq:alpha-beta-T}
\alpha_0 \alpha^{-1} &=&
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
P_L
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \nonumber\\
\beta_0 \beta^{-1} &=&
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
P_R
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \nonumber\\
\left(-\beta \, \alpha^{-1}\right)
&=&
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\gamma_5 \, e^{a_5 \widetilde H} \, \gamma_5
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) ,
\end{eqnarray}
we can evaluate the factors of the first and second terms
of Eq.~(\ref{eq:last-flavor-terms}) as follows:
\begin{eqnarray}
\label{eq:factor-1-2}
&&
\left[
P_L \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\left( - \alpha^{-1} \beta \right)^{N-1}
+P_R \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\right] \alpha^{-1}
\nonumber\\
&& \qquad \qquad = - a_5
\left[ P_R + P_L \, e^{N a_5 \widetilde H } \right] (-\gamma_5)
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) , \\
&& \left[ \alpha_0
+\beta_0 \,
\left( - \alpha^{-1} \beta \right)^{N-1} \right] \alpha^{-1}
\nonumber\\
&& \qquad \qquad
=
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)
\left[ P_L + P_R \, e^{N a_5 \widetilde H } \right] (-\gamma_5)
\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) .
\end{eqnarray}
From these results, we obtain
\begin{eqnarray}
\left\langle q(x) \bar q(y) \right\rangle
&=& \frac{a_5}{a^4}
\left(P_R + P_L \, e^{N a_5 \widetilde H } \right)
\frac{1}{P_L + P_R \, e^{N a_5 \widetilde H}}
\nonumber\\
&=& \frac{a_5}{a^4}
\frac{1-\gamma_5 \tanh a_5 \frac{N}{2} \widetilde H }
{1+\gamma_5 \tanh a_5 \frac{N}{2} \widetilde H } .
\end{eqnarray}
Then, the effective action of the light fermion turns out to be given
by the following effective Dirac operator:
\begin{eqnarray}
D_N^{\rm eff}(x,y)
&=&
\frac{1}{a^4}\left\langle q(x) \, \bar q(y) \right\rangle ^{-1} \nonumber\\
&=& \frac{1}{a_5} \,
\frac{1+\gamma_5 \tanh \frac{N}{2} a_5 \widetilde H}
{1-\gamma_5 \tanh \frac{N}{2} a_5 \widetilde H} .
\end{eqnarray}
\subsection{The effective action of the light fermion and
the Ginsparg-Wilson relation}
We may also consider the effective action of the $N$-th flavor of the
bosonic Pauli-Villars field. As we see from
Eq.~(\ref{eq:hermitian-mass-matrix-antiperiodic}),
the difference between the mass matrices of the domain-wall fermion
and the Pauli-Villars field is only in the diagonal $N$-th
element of the latter, which takes account of the anti-periodicity
in the flavor space. Then the integration of the first $N-1$
components can be achieved just in the same way as the case of
the domain-wall fermion. The effective action of the $N$-th flavor
of the Pauli-Villars field turns out to be same as that of the
light-fermion, but in this case, with the additional mass term.
Thus the total effective action of the domain-wall fermion with the
subtraction can be written in the following form:
\begin{equation}
\label{eq:effective-action-subtracted}
\bar S_N^{\rm eff}
= a^4 \sum_x \bar q(x) \, D_N^{\rm eff} \, q(x)
+ a^4 \sum_x \bar Q(x) \left\{ D_N^{\rm eff}+\frac{1}{a_5} \right\} Q(x) ,
\end{equation}
where we have denoted the $N$-th flavor of the Pauli-Villars
field by
\begin{equation}
\label{eq:N-th-PV-field}
Q(x) = \phi_N^\prime(x), \qquad \bar Q(x) = \bar \phi_N^\prime(x) .
\end{equation}
This result is consistent with
Eq.~(\ref{eq:subtracted-partition-function}),
because the relation holds true
\begin{equation}
\label{eq:truncated-overlap-vs-effective-action-q}
\frac{ a_5 D_N^{\rm eff} }{1+ a_5 D_N^{\rm eff}}
= a D_N ,
\end{equation}
and the partition function calculated from the effective action
$\bar S_N^{\rm eff}$ ( Eq.~(\ref{eq:effective-action-subtracted}))
is identical to that from the action with
the truncated overlap Dirac operator
$S_N$ ( Eq.~(\ref{eq:truncated-overlap-Dirac-fermion})):
\begin{eqnarray}
\bar Z_{\rm DW}
&=& \int [d q d \bar q] [d Q d \bar Q]
e^{- a^4 \sum_x \bar q(x) D_N^{\rm eff} q(x)
- a^4 \sum_x \bar Q(x) \left\{ D_N^{\rm eff}
+\frac{1}{a_5} \right\} Q(x) } \nonumber\\
&=& \int [d \psi d \bar \psi]
e^{- a^4 \sum_x \bar \psi(x) D_N \psi(x) } .
\end{eqnarray}
Eq.~(\ref{eq:truncated-overlap-vs-effective-action-q}) may be written
also in the following form:
\begin{eqnarray}
\label{eq:inverse-truncated-overlap-vs-propagator-q}
\frac{1}{a_5} \, { D_N^{\rm eff} }^{-1}
+ \delta(x,y)
&=&
\frac{1}{a} \, { D_N^{\phantom{f}} }^{-1} .
\end{eqnarray}
Namely, the inverse of the effective Dirac operator
(the propagator of the light fermion)
gives the inverse of the truncated overlap Dirac operator up to
a local contact term.
This contact term just takes account of the
chiral symmetry breaking in the Ginsparg-Wilson relation,
which holds true for the overlap Dirac operator
in the limit of the infinite flavors.
This result implies that the propagator of the light fermion
reduces to
{\it the chirally symmetric part of the inverse of the
overlap Dirac operator} in the limit of the infinite flavors,
assuming that $\widetilde D$ does not have any zero mode:
\begin{equation}
\gamma_5 \, { D^{\rm eff} }^{-1}
+{ D^{\rm eff} }^{-1} \gamma_5 = 0
\end{equation}
and
\begin{equation}
\label{eq:inverse-overlap-vs-propagator-q}
{\widetilde D}^{-1}
=
\frac{a}{a_5} \, { D^{\rm eff} }^{-1}
+ a \delta(x,y) \qquad (N=\infty).
\end{equation}
It may be interesting to observe that
the infinitely many flavors of the domain-wall fermion
serve to prepare the chirally symmetric part of the inverse
of the overlap Dirac operator and that
it is the Pauli-Villars field ($N$-th flavor) that add it
the contact term of the chiral symmetry breaking
in the Ginsparg-Wilson relation.
This contrasts with the situation of the original derivation
of the Ginsparg-Wilson relation in \cite{ginsparg-wilson-rel},
where the chiral symmetry breaking is introduced
by the kernel of the block-spin transformation.
The properties of the chirally symmetric part of the inverse
of the Dirac operator which satisfies the Ginsparg-Wilson relation
have been discussed extensively
by Chiu et al. \cite{chiu-etal}.
Eq.~(\ref{eq:truncated-overlap-vs-effective-action-q})
reduces to the expression of the Dirac operator in terms
of the chirally symmetric part discussed there.
As is also emphasized by these authors,
the effective action of the light fermion, in the limit of infinite
flavors, must necessarily be nonlocal.
This can be easily checked in the free theory.
Then, the effective action of the light fermion itself
is not defined well and useful in this limit.
In fact, a subtlety appears when one attempts to calculate the axial
anomaly from the effective action, as will be discussed in the last
section.
However, as we will discuss below, the light fermion field variables
$q(x)$ and $\bar q(x)$ still remain to be a good probe
for the Dirac fermion described by the overlap Dirac operator.
They can be expressed by the Dirac fermion
field $\psi(x)$ and $\bar \psi(x)$ through a certain local
expression including the overlap Dirac operator $\widetilde D$.
In this expression, we do not encounter any singularity associated
with the nonlocal behavior of $D^{\rm eff}$.
Moreover, the low energy observables of the domain-wall fermion
which are written in terms of $q(x)$ and $\bar q(x)$
can be expressed by $\psi(x)$ and $\bar \psi(x)$.
And these observables, in fact, turn out to have
good chiral properties from the point of view of the exact chiral
symmetry based on the Ginsparg-Wilson relation.
\section{Chiral property of the light fermion}
\label{sec:chiral-property-q-and-psi}
\setcounter{equation}{0}
With the result obtained in the previous section,
we consider the relation between the light fermion field
of the domain-wall fermion
and the Dirac fermion field
described by the truncated overlap Dirac operator.
Then we discuss the chiral property of the light fermion field
and various observables written by it
in view of the exact chiral symmetry based on the Ginsparg-
Wilson relation.
\subsection{Light fermion field $q(x)$ and $\bar q(x)$}
The correspondence between the light fermion field $q(x)$ and $\bar q(x)$
and the Dirac fermion field $\psi(x)$ and $\bar \psi(x)$
which is described by the truncated overlap Dirac operator
is now easily understood. We may relate them by
\begin{eqnarray}
\label{eq:relation-q-psi}
q(x)&=& Z \, \frac{1}{1+a_5 D_N^{\rm eff}} \, \psi(x)
=Z \, \left( 1-\frac{a}{2} R D_N \right) \psi(x) , \nonumber\\
\bar q(x)&=& \bar \psi(x),
\end{eqnarray}
where
$Z=\frac{a_5}{a}$ and $R=2$.
In the functional integral of the partition function,
the Jacobian of the change of the field variable from $q(x)$
to $\psi(x)$ along this relation,
just compensates the determinant resulting from the integration
of the last flavor of the bosonic Pauli-Villars field $Q(x)$.
A few comments are in order.
As long as the equivalence of the partition functions
is concerned, there is an ambiguity
in the correspondence Eq.~(\ref{eq:relation-q-psi})
which is related to the scale transformation:
\begin{equation}
\psi(x) \longrightarrow z \, \psi(x), \qquad
\bar \psi(x) \longrightarrow z^{-1} \bar \psi(x).
\end{equation}
This scale factor may even depend on $D_N^{\rm eff}$.
We have fixed this freedom so that
the chiral transformation of $\bar q(x)$ matches with that
of $\bar \psi(x)$. As to the choice of the constant scale factor,
it is our convention for simplicity.
The correspondence Eq.~(\ref{eq:relation-q-psi})
would be different if we adopt a different transformation
of the exact chiral symmetry of L\"uscher from
Eq.~(\ref{eq:chiral-transformation-of-psi}).
\subsection{Chiral transformation of the light fermion}
The explicit relation between $q(x)$ and $\psi(x)$ helps us
to understand the chiral properties of the domain-wall
fermion in terms of the (would-be) exact chiral symmetry
of the Ginsparg-Wilson fermion described by the (truncated)
overlap Dirac operator.
First of all,
Eq.~(\ref{eq:inverse-truncated-overlap-vs-propagator-q})
relates quantitatively the chiral symmetry breaking in the domain-wall
fermion to the breaking of the Ginsparg-Wilson relation
in the truncated overlap Dirac fermion:
these breakings can be characterized by a single quantity
\begin{eqnarray}
\label{eq:breaking-delta}
\delta_N &\equiv&
Z^{-1}\left(
\gamma_5 {D_N^{\rm eff} }^{-1}
+ { D_N^{\rm eff} }^{-1} \gamma_5 \right) \\
&\equiv&
\gamma_5 { D_N }^{-1} +{ D_N }^{-1} \gamma_5 - a R \gamma_5 .
\end{eqnarray}
We may consider the chiral transformation
for the Dirac fermion described by the truncated overlap
Dirac operator a l\'a L\"uscher
\begin{equation}
\label{eq:chiral-transformation-of-psi-truncated}
\delta \psi(x) = \gamma_5 \left( 1-aRD_N \right) \psi(x), \quad
\delta \bar \psi(x) = \bar \psi(x) \gamma_5 .
\end{equation}
The chiral symmetry of the action is broken by the amount
\begin{equation}
\delta S_N = a^4 \sum_x \bar \psi(x) \Delta_N \psi(x),
\end{equation}
\begin{equation}
\label{eq:breaking-Delta}
\Delta_N \equiv
\gamma_5 \left\{ 1- \left(\tanh \frac{N}{2} a_5 \widetilde H \right)^2
\right\}
= D_N \delta_N D_N .
\end{equation}
This breaking vanishes in the limit of infinite flavors $N=\infty$,
as long as the eigenvalues of $\widetilde H$
are bounded from zero uniformly with respect to
the gauge fields in consideration.
We assume this in the following discussions.
We also assume that $\widetilde D$,
the overlap Dirac operator in the limit of the infinite flavors,
should not have any zero mode and that its inverse should exist,
in order to assure that $\delta_N$ should vanish
in this limit, too.
When we discuss the anomaly of the domain-wall fermion
in relation to the index of $\widetilde D$, we introduce
the mass term of the light fermion to assure the limit.
The chiral transformation of
Eq.~(\ref{eq:chiral-transformation-of-psi-truncated})
induces the transformation of the light fermion as follows:
\begin{eqnarray}
\delta q(x)
&=& Z \, \left( 1-\frac{a}{2} R D_N \right) \delta \psi(x)
\nonumber\\
&=& Z \, \left( 1-\frac{a}{2} R D_N \right)\,
\, \gamma_5 \left( 1-a R D_N \right)\, \psi(x)
\nonumber\\
&=& Z \,
\left\{
\gamma_5 \left( 1-a R D_N \right)
-\frac{a}{2} R D_N \, \gamma_5 \left( 1-a R D_N \right)
\right\} \, \psi(x)
\nonumber\\
&=& Z \,
\left\{
\gamma_5 \left( 1-\frac{a}{2} R D_N \right)
- \frac{a}{2} R \Delta_N
\right\} \, \psi(x)
\nonumber\\
&=&
\gamma_5 \, q(x)- a_5 \Delta_N \, \psi(x) .
\end{eqnarray}
And it reduces to the transformation of
Eq.~(\ref{eq:chiral-transformation-of-q}) in the limit
of the infinite flavors.
Thus we can see how the chiral transformation
of Eq.~(\ref{eq:chiral-transformation-of-q}),
which is adopted by Furman and Shamir
(See also Eq.~(\ref{eq:chiral-transformation-of-DW})), is related
to that of L\"uscher Eq.~(\ref{eq:chiral-transformation-of-psi})
in the limit of the infinite flavors.
\subsection{Low energy observables in terms of the light fermion}
Next we consider the low energy observables of the domain-wall fermion,
which are written in terms of $q(x)$ and $\bar q(x)$.
We discuss the relation to the observables in terms of the Dirac
fermion field which is described by the overlap Dirac operator and
whose chiral property is governed by the Ginsparg-Wilson relation.
\subsubsection{Scalar and pseudo scalar bilinear operators}
First of all, from the explicit relation
Eq.~(\ref{eq:relation-q-psi}) between the light
fermion field and the Dirac field described by the truncated
overlap Dirac operator,
we obtain the relation of the scalar and pseudo scalar bilinear operators:
\begin{eqnarray}
\bar q(x) q(x)
&=& Z \, \bar \psi(x) \left(1- \frac{a}{2}RD_N \right)\psi(x) , \\
\bar q(x) \gamma_5 q(x)
&=& Z \, \bar \psi(x) \gamma_5 \left(1- \frac{a}{2}RD_N \right)\psi(x) .
\end{eqnarray}
In this respect, it is interesting to note that the scalar and pseudo
scalar bilinear operators in the r.h.s.
consist the exact chiral multiplet of the chiral transformation
Eq.~(\ref{eq:chiral-transformation-of-psi})
in the limit of infinite flavors ,
as discussed by Niedermayer \cite{neidermayer-lat98}.
In fact, these operators can be written in the chiral components
defined by $\hat \gamma_5 = \gamma_5\left(1-a R \widetilde D\right)$ for
$\psi(x)$ and by $\gamma_5$ for $\bar \psi(x)$ as follows:
\begin{eqnarray}
\bar q(x) q(x) &\longrightarrow&
\bar \psi_L(x) \psi_R(x) + \bar \psi_R(x) \psi_L(x) , \\
\bar q(x) \gamma_5 q(x) &\longrightarrow&
\bar \psi_L(x) \psi_R(x) - \bar \psi_R(x) \psi_L(x) .
\end{eqnarray}
\subsubsection{Conserved vector currents}
The vector current of the domain-wall fermion is conserved
at a finite flavor $N$. It is expected to correspond to
the conserved vector current of the truncated overlap Dirac fermion.
We will show that the vector current of the domain-wall fermion,
if it is probed by the light fermion field $q(x)$ and $\bar q(x)$
at low energy, just corresponds to
the conserved vector current of the truncated overlap Dirac fermion.
In general, a vector current may be defined
with the kernel
\begin{equation}
V^a_\mu(x) = \sum_{y,z} \bar \psi(x) K^a_\mu(x;y,z) \psi(x) .
\end{equation}
The kernel is obtained from the Dirac operator by
introducing the auxiliary vector field
\begin{equation}
U^B_\mu(x)= \exp \left( i B^a_\mu(x) T^a \right),
\end{equation}
where $T^a$ are assumed as the generators of
the $U(N_F)$ real flavor group,
and by differentiating
the Dirac operator with respect to the vector field:
\begin{equation}
\delta D(y,z) = \sum_x B^a_\mu(x) \, K^a_\mu(x;y,z)
+ {\cal O}\left(B^2\right).
\end{equation}
By this procedure, we can obtain
the kernel for the domain-wall fermion
from $D_{\rm DW}^\prime
\equiv\gamma_\mu\frac{1}{2}\left(\nabla_\mu+\nabla_\mu^\ast\right)+
M^{\rm H} $, which we denote as $K^a_{\mu {\rm DW}}$.
The kernel for the truncated overlap Dirac fermion
is obtained from $D_N$, which we denote as $K^a_{\mu N}$.
Then, the vector current of the domain-wall fermion with
the subtraction can be written as
\begin{equation}
V^a_{\mu {\rm DW}}(x)
= \sum_{st}^N \bar \psi^\prime_s
\left\{ K^a_{\mu {\rm DW}}(x) \right\}_{st} \psi^\prime_t
.
\end{equation}
The vector current of the truncated overlap Dirac fermion can be
written as
\begin{equation}
V^a_N(x) = \bar \psi \, K^a_{\mu N}(x) \, \psi.
\end{equation}
For these two vector currents, we can infer the following identity:
\begin{equation}
\label{relation-of-vector-currents-q}
\left\langle q(y) \, V^a_{\mu {\rm DW}}(x) \,
\bar q(z) \right\rangle_c
= Z \left\langle \psi(y) \, V^a_{\mu N}(x) \, \bar \psi(z) \right\rangle_c.
\end{equation}
This identity follows from the relation between $D_{\rm DW}^{-1}$ and
$D_N^{-1}$ given by
Eq.~(\ref{eq:inverse-truncated-overlap-vs-propagator-q}):
in fact, it follows that
\begin{equation}
\label{eq:inverse-truncated-overlap-vs-propagator-q-variation}
\left\{ \delta {D_{\rm DW}'}^{ -1} \right\}_{NN}
= \delta { D_N^{\rm eff} }^{-1}
= Z \delta D_N^{-1} .
\end{equation}
Using the identity $ \delta D^{-1} = - D^{-1} \delta D D^{-1}$,
it can be expressed as Eq.~(\ref{relation-of-vector-currents-q}).
\subsubsection{Almost conserved axial vector currents}
According to Shamir and Furman \cite{boundary-fermion-QCD},
the axial vector current of the domain-wall fermion
is conserved up to the corrections suppressed exponentially
in the number of flavors. Then it is expected that
this axial vector current is related in the limit of the infinite
flavors to the conserved axial vector current of the overlap
Dirac fermion.
We will show that the axial vector current of the domain-wall fermion,
if it is probed by the light fermion field $q(x)$ and $\bar q(x)$
at low energy, reduces to
the conserved axial vector current of the overlap Dirac fermion.
The axial vector current of the domain-wall fermion
is defined with the kernel of the vector current as
\begin{equation}
A^a_{\mu {\rm DW}}(x)
=
\sum_{s,t}^N
\bar \psi_s^\prime
\left\{ K^a_{\mu {\rm DW}} (x) \, \Gamma_5 \right\}_{st} \psi_t^\prime .
\end{equation}
On the other hand, the axial vector current of the
truncated overlap Dirac fermion may be defined as
\begin{equation}
A^a_{\mu N}(x) =
\bar \psi \,
K^a_{\mu N} (x) \, \gamma_5\left(1-aRD_N\right) \, \psi ,
\end{equation}
which naturally follows from the chiral transformation
Eq.~(\ref{eq:chiral-transformation-of-psi-truncated}).
This current reduces to the Noether current associated
with the exact chiral symmetry of L\"uscher \cite{axial-current}
in the limit of the infinite flavors, which we denote as
\begin{equation}
\widetilde A^a_{\mu}(x) =
\bar \psi \,
\widetilde K^a_{\mu } (x) \,
\gamma_5\left(1-aR \widetilde D\right) \, \psi .
\end{equation}
For these two axial vector currents, we can infer the following
relation:
\begin{equation}
\label{relation-of-axial-vector-currents-q}
\lim_{N\rightarrow \infty}
\left\langle q(y) \, A^a_{\mu {\rm DW}}(x) \, \bar q(z)
\right\rangle_c
=
Z \left\langle \psi(y) \, \widetilde A^a_{\mu }(x) \,
\bar \psi(z) \right\rangle_c .
\end{equation}
This relation can be shown in the following way.
The Dirac operator of the domain-wall fermion satisfies
\begin{equation}
\left\{ \Gamma_5 D_{\rm DW}^\prime + D_{\rm DW}^\prime \Gamma_5 \right\}_{st}
=2 \gamma_5 \delta_{s \frac{N}{2}}\delta_{t \frac{N}{2}} .
\end{equation}
On the other hand, the truncated overlap Dirac operator satisfies
\begin{equation}
\label{eq:chiral-property-truncated-overlap-D}
\gamma_5 D_N + D_N \gamma_5 \left(1- a R D_N \right) = \Delta_N .
\end{equation}
Then it turns out that
the vector and axial vector currents are related each other as follows:
\begin{eqnarray}
\label{relation-of-currents-domainwall}
&&
\left\langle q(y) \, A^a_{\mu {\rm DW}}(x) \, \bar q(z) \right\rangle
+\left\langle q(y) \, V^a_{\mu {\rm DW}}(x) \, \bar q(z) \right\rangle
\gamma_5
\nonumber\\
&&=
\sum_s^N \left\{ {D_{\rm DW}}^{-1} K^a_{\mu {\rm DW}}(x) \right\}_{N,s} \cdot
\left\{ {D_{\rm DW}}^{-1} \right\}_{s,\frac{N}{2}} 2 \gamma_5
\left\{ {D_{\rm DW}}^{-1} \right\}_{\frac{N}{2},N}(y,z) , \nonumber\\
\end{eqnarray}
and
\begin{eqnarray}
\label{relation-of-currents-truncated-overlap}
&&
\left\langle \psi(y) \, A^a_{\mu N}(x) \, \bar \psi(z) \right\rangle
+\left\langle \psi(y) \, V^a_{\mu N}(x) \, \bar \psi(z)\right\rangle
\gamma_5
\nonumber\\
&& \qquad\qquad\qquad
=
\left\{ {D_N}^{-1} K_{\mu N}(x) \right\}
{D_N}^{-1} \Delta_N {D_N}^{-1}(y,z) .
\end{eqnarray}
Using Eq.~(\ref{relation-of-vector-currents-q}), we obtain
\begin{eqnarray}
\label{relation-of-axial-vector-currents-q-N}
&& \left\langle q(y) \, A^a_{\mu {\rm DW}}(x) \, \bar q(z) \right\rangle
\nonumber\\
&&\quad =
Z \left\langle \psi(y) \, A^a_{\mu N}(x) \, \bar \psi(z) \right\rangle
-Z {D_N}^{-1} K_{\mu N}(x) {D_N}^{-1} \Delta_N {D_N}^{-1}(y,z) .
\nonumber\\
&& \qquad
+
\left\{ {D_{\rm DW}}^{-1} K_{\mu {\rm DW}}(x) {D_{\rm DW}}^{-1}
\right\}_{N,\frac{N}{2}} 2 \gamma_5
\left\{ {D_{\rm DW}}^{-1} \right\}_{\frac{N}{2},N} (y,z) .
\nonumber\\
\end{eqnarray}
Now, it is possible to evaluate
the propagator between the $\frac{N}{2}$-th flavor and the $N$-th
flavor of the domain-wall fermion,
using the similar method to evaluate the propagator of the light
fermion in section~\ref{sec:effective-action-of-q}.
The calculation is described in the
appendix~\ref{sec:propagators-heavy-modes}.
The result can be expressed as
\begin{eqnarray}
\left\{ D_{\rm DW} \right\}^{-1}_{\frac{N}{2} N} (x,y)
&\equiv&
\left\langle \psi_{\frac{N}{2}}^\prime(x) \, \bar q(y) \right\rangle
\nonumber\\
&=& \frac{1}{2 \cosh \frac{N}{2} a_5 \widetilde H } \, D_N^{-1}(x,y) .
\end{eqnarray}
From this and Eq.~(\ref{eq:breaking-Delta}),
we infer that the second and third terms of the right-hand sides of
Eq.~(\ref{relation-of-axial-vector-currents-q-N})
vanish in the limit of the infinite flavors
and we finally obtain
Eq.~(\ref{relation-of-axial-vector-currents-q}).
Thus the vacuum expectation value of the axial vector current of
the domain-wall fermion with respect to the light fermion field
leads directly to that of the axial vector current associated with
the exact chiral symmetry based on the Ginsparg-Wilson relation.
Note that our results hold true with dynamical gauge fields
as long as the eigenvalues of $\widetilde H$
are bounded from zero uniformly with respect to the gauge fields.
The renormalization factor of the axial current
of the domain-wall fermion thus reduces to unity in the limit
of the infinite flavors. Our result is consistent with
the perturbative result at one-loop
obtained by Aoki and Taniguchi \cite{aoki-taniguchi}.
\subsubsection{Correspondence of various observables}
From the above results, we can obtain the
correspondence of the various observables of the light fermion
of the domain-wall fermion to the observables of
the Dirac fermion which is described by the (truncated) overlap
Dirac operator. We summarize it here.
(Note again that in this paper, the bra-ket symbol $\langle \cdots
\rangle$ denotes the Wick contraction of the fermion fields in it
by their propagators, not the fermionic vacuum expectation values
which must includes the weight of the fermion action.
The bra-ket symbol with the subscript $c$ denotes
the connected contraction. )
\begin{itemize}
\item Correlation of the vector currents:
\begin{equation}
\left\langle V^a_{\mu {\rm DW}}(x) \,\,
\bar q(y) \gamma_\nu T^b q(y) \right\rangle_c
= Z \left\langle V^a_{\mu N}(x) \,\,
\bar \psi(y) \gamma_\nu T^b \psi(y)
\right\rangle_c .
\end{equation}
\item Correlation of the pseudo scalar densities:
\begin{eqnarray}
&&\left\langle \bar q(x)\gamma_5 T^a q(x) \, \,
\bar q(y)\gamma_5 T^b q(y)
\right\rangle \nonumber\\
&& \quad =
Z^2\left\langle
\bar \psi(x)\gamma_5\left(1-\frac{a}{2}RD_N\right) T^a \psi(x)
\, \,
\bar \psi(y)\gamma_5\left(1-\frac{a}{2}RD_N\right) T^b \psi(y)
\right\rangle . \nonumber\\
\end{eqnarray}
\item Correlation of the axial vector current and pseudo scalar density:
\begin{equation}
\lim_{N\rightarrow \infty}
\left\langle A^a_{\mu {\rm DW}}(x) \,\,
\bar q(y) \gamma_5 T^b q(y) \right\rangle_c
= Z \left\langle \widetilde A^a_{\mu }(x) \,\,
\bar \psi(y) \gamma_5 T^b \psi(y)
\right\rangle_c .
\end{equation}
\item Amplitude of $K^0$-$\bar K^0$ mixing:
\begin{eqnarray}
&& \lim_{N\rightarrow \infty}
\left\langle \bar d_N s_N(y)
\left( \sum_{st}^N \, \bar s_s
\left\{ K_{\mu {\rm DW}} (x) \left(1- \Gamma_5 \right) \right\}_{st}
d_t \, \right)^2
\bar d_N s_N (z) \right\rangle
\nonumber\\
&& \qquad
= Z^2
\left\langle \bar d s(y)
\left( \, \bar s \,
\widetilde K_\mu(x) \left(1- \hat \gamma_5 \right) d \, \right)^2
\bar d s(z) \right\rangle ,
\end{eqnarray}
where $s_t(x)$ and $d_t(x)$ $(t=1,\cdots,N)$ are the
domain-wall fermion fields for s-quark and d-quark, respectively.
$s(x)$ and $d(x)$ are the Dirac fermion fields
which are described by the overlap Dirac operator $\widetilde D$
for s-quark and d-quark, respectively.
\end{itemize}
Thus we see that
the observables in the light fermion field variables
$q(x)$ and $\bar q(x)$ of the domain-wall fermion
(with the subtraction of the Pauli-Villars field)
leads directly to the observables
which have good chiral property with
respect to the exact chiral symmetry based on
the Ginsparg-Wilson relation.
\section{Axial anomaly of domain-wall fermion}
\label{sec:axial-anomaly-q-and-psi}
\setcounter{equation}{0}
The explicit chiral symmetry breaking
in the domain-wall fermion under
the chiral transformation Eq.~(\ref{eq:chiral-transformation-of-DW})
is expected to reproduce
the axial anomaly \cite{axial-anomaly-in-domain-wall}.
Here we examine this chiral symmetry breaking term
in relation to the axial anomaly
which is associated with the exact
chiral symmetry based on the Ginsparg-Wilson relation
\cite{index-theorem-at-finite-lattice,exact-chiral-symmetry},
\begin{equation}
\label{eq:anomaly-of-GW-fermion}
- a {\rm tr} \gamma_5 R \widetilde D(x,x) .
\end{equation}
\subsection{Axial anomaly}
With the definition of
the chiral transformation Eq.~(\ref{eq:chiral-transformation-of-DW}),
the chiral symmetry breaking occurs at
the diagonal
$\frac{N}{2}$-th element of the hermitian mass matrix
Eq.~(\ref{eq:hermitian-mass-matrix}), i.e.
mass term of the $\frac{N}{2}$-th flavor.
In fact,
the flavor singlet axial vector current of the domain-wall fermion
satisfies the axial Ward-Takahashi identity:
\begin{equation}
\label{eq:axial-WT-identity-DW}
\partial_\mu^\ast \left\langle A_{\mu {\rm DW}}(x) \right\rangle
=
\frac{2}{a_5}
\left\langle \bar \psi^\prime_{\frac{N}{2}}(x) \, \gamma_5 \,
\psi^\prime_{\frac{N}{2}}(x) \right\rangle .
\end{equation}
It is expected that the explicit breaking in the r.h.s.
of the above identity should reproduce the axial anomaly.
It has been shown by a perturbative calculation that
it is indeed the case \cite{axial-anomaly-in-domain-wall}.
It is actually possible to evaluate the breaking term
non-perturbatively.
Using the similar method to evaluate the propagator of the light
fermion in section~\ref{sec:effective-action-of-q}, the
propagator of the $\frac{N}{2}$-th flavor fermion
of the domain-wall fermion is evaluated explicitly as
\begin{eqnarray}
\label{eq:chiral-symmetry-breaking-domain-wall-fermion}
\left\langle \psi^\prime_{\frac{N}{2}}(x) \,
\bar \psi^\prime_{\frac{N}{2}}(y)
\right\rangle
&=& \frac{a_5}{a^4} \left(-1 + \gamma_5 \frac{1}{2} a R D_N \gamma_5
\right. \nonumber\\
&& \qquad\qquad \left.
+\frac{1}{2a} \frac{1}{\cosh \frac{N}{2} a_5 \widetilde H }
D_N^{-1} \gamma_5 \frac{1}{\cosh \frac{N}{2} a_5 \widetilde H }
\gamma_5 \right) . \nonumber\\
\end{eqnarray}
The calculation is described in the
appendix~\ref{sec:propagators-heavy-modes}.
Then it follows immediately that
the chiral symmetry breaking term can be written as
\begin{eqnarray}
\frac{2}{a_5}
\left\langle \bar \psi^\prime_{\frac{N}{2}}(x)
\gamma_5 \psi^\prime_{\frac{N}{2}}(x)
\right\rangle
&=& \frac{1}{a^4}
\left( - a {\rm tr} \gamma_5 R D_N (x,x)
\phantom{
D_N^{-1} \gamma_5 \frac{1}{\cosh^2 \frac{N}{2} a_5 \widetilde H}(x,x)
}
\right.\nonumber\\
&& \qquad \qquad \left.
- {\rm tr}
D_N^{-1} \gamma_5 \frac{1}{a}
\frac{1}{\cosh^2 \frac{N}{2} a_5 \widetilde H}(x,x)
\right)
\nonumber\\
&=& \frac{1}{a^4} \left(
- a{\rm tr} \gamma_5 R D_N (x,x) - {\rm tr} D_N^{-1} \Delta_N (x,x)
\right) .
\nonumber\\
\end{eqnarray}
We can see that it reduces to the anomaly
Eq.~(\ref{eq:anomaly-of-GW-fermion}) in the limit of the infinite flavors.
\subsection{Relation to the index of overlap Dirac operator}
We next discuss the relation of the axial anomaly
of the domain-wall fermion to
the index of the overlap Dirac operator $\widetilde D$.
For this purpose, we consider the situation
so that $\widetilde D$ has zero modes.
In order to make the limit of the infinite flavors well-defined,
we introduce the mass term of the light fermion.
\begin{equation}
m \bar q(x) q(x) .
\end{equation}
Then it follows from the axial Ward-Takahashi identity that
\begin{equation}
\label{eq:axial-WT-ideinty-DW-mqq}
a^4 \sum_x 2 \left\langle \bar \psi^\prime_{\frac{N}{2}}(x)
\gamma_5 \psi^\prime_{\frac{N}{2}}(x)
\right\rangle
+ a^4 \sum_x 2 m \left\langle \bar q (x) \gamma_5 q(x) \right\rangle = 0 .
\end{equation}
The second term of the l.h.s. is evaluated as
\begin{eqnarray}
\label{eq:anomalous-term-q}
- a^4 \sum_x 2 m \left\langle \bar q (x) \gamma_5 q(x) \right\rangle
&=& 2 m {\rm Tr} \gamma_5 \frac{1}{ D_N^{\rm eff} + m } \nonumber\\
&=& 2 m {\rm Tr} \gamma_5 \left(1-\frac{a}{2} R D_N \right)
\frac{1}{ D_N + m \left(1-\frac{a}{2} R D_N \right) } \nonumber\\
&=& - a {\rm Tr} \gamma_5 R D_N \nonumber\\
&& - (1-a m) {\rm Tr} \Delta_N
\frac{1}{ D_N + m \left(1-\frac{a}{2} R D_N \right) } . \nonumber\\
\end{eqnarray}
Therefore we obtain in the limit of the infinite flavors
\begin{equation}
\label{eq:anomalus-term-DW-and-index}
\lim_{N \rightarrow \infty}
a^4 \sum_x 2 \left\langle \bar \psi^\prime_{\frac{N}{2}}(x)
\gamma_5 \psi^\prime_{\frac{N}{2}}(x)
\right\rangle
= -a {\rm Tr} \gamma_5 R \widetilde D
= 2 \, {\rm Index} \left(\widetilde D \right).
\end{equation}
Since all the dependence on $m$ comes with $\Delta_N$,
this limit does not depend on the value of $m$.
This explicit chiral symmetry breaking term has been
evaluated numerically
by Argonne group \cite{lagae-sinclair}
as the probe of the topological charge.
We have seen how this quantity is related to
the index of the overlap Dirac operator which
satisfies the Ginsparg-Wilson relation.
\section{Contributions of the Pauli-Villars field}
\label{sec:Pauli-Villars-field}
\setcounter{equation}{0}
Finally, we will discuss the contributions of the Pauli-Villars
field to vector and axial vector currents and axial anomaly.
In the previous sections, we have discussed
the currents of the domain-wall fermion
probed by the light fermion field $q(x)$ and $\bar q(x)$.
However, if we consider the currents themselves and
their correspondence to those of the overlap Dirac fermion,
we need to include the contribution of the Pauli-Villars fields.
\subsection{Vector and axial vector currents}
The conserved vector current of the domain-wall fermion
with the subtraction of the Pauli-Villars fields is defined by
\begin{equation}
\overline{V}^a_{\mu {\rm DW}}(x)
= \sum_{st}^N \bar \psi^\prime_s
\left\{ K^a_{\mu {\rm DW}}(x) \right\}_{st} \psi^\prime_t
+ \sum_{st}^N \bar \phi^\prime_s
\left\{ K^a_{\mu {\rm DW}}(x) \right\}_{st} \phi^\prime_t .
\end{equation}
Note that we obtain the same kernel for the Pauli-Villars field
as the domain-wall fermion,
from $D_{\rm PV}^\prime
\equiv\gamma_\mu\frac{1}{2}\left(\nabla_\mu+\nabla_\mu^\ast\right)+
M^{\rm PV}$.
The almost conserved axial vector current is defined similarly by
\begin{equation}
\overline{A}^a_{\mu {\rm DW}}(x)
=
\sum_{s,t}^N
\bar \psi_s^\prime
\left\{ K^a_{\mu {\rm DW}} (x) \, \Gamma_5 \right\}_{st} \psi_t^\prime
+\sum_{s,t}^N
\bar \phi_s^\prime
\left\{ K^a_{\mu {\rm DW}} (x) \, \Gamma_5 \right\}_{st} \phi_t^\prime .
\end{equation}
For these vector and axial vector currents, we can infer the following
identities:
\begin{eqnarray}
\label{relation-of-vector-currents}
\langle \overline{V}^a_{\mu {\rm DW}}(x) \rangle
&=& \langle V^a_{\mu N}(x) \rangle , \\
\label{relation-of-axial-vector-currents}
\lim_{N\rightarrow \infty}
\left\langle \overline{A}^a_{\mu {\rm DW}}(x) \right\rangle
&=&
\left\langle \widetilde A^a_{\mu }(x) \right\rangle .
\end{eqnarray}
The identity of the vector currents follows immediately from
Eq.~(\ref{eq:subtracted-partition-function}). But we will show it
here directly for later use.
The l.h.s. of Eq.~(\ref{relation-of-vector-currents}), if
multiplied with the auxiliary vector field $B^a_\mu(x)$, can be
rewritten as follows:
\begin{eqnarray}
\label{eq:derivation-relation-of-vector-currents}
({\rm l.h.s.})&\simeq & a^4 \sum_x B^a_\mu(x)
\langle \overline{V}^a_{\mu {\rm DW}}(x) \rangle
\nonumber\\
&=& - \sum_t^N {\rm tr}
\left\{ \delta D'_{\rm DW}
\left( {D'_{\rm DW}}^{-1}- {D'_{\rm PV}}^{-1}\right) \right\}_{tt}
\nonumber\\
&=& - \sum_t^N {\rm tr}
\left\{ \delta D'_{\rm DW}
{D'_{\rm DW}}^{-1} \left(M^{\rm PV}-M^{\rm H}\right){D'_{\rm PV}}^{-1} \right\}_{tt}
\nonumber\\
&=& \sum_t^N {\rm tr}
\left\{ \left[D'_{\rm PV} -\left(M^{\rm PV}-M^{\rm H}\right)\right]
\delta {D'_{\rm DW}}^{-1} \left(M^{\rm PV}-M^{\rm H}\right){D'_{\rm PV}}^{-1} \right\}_{tt}
\nonumber\\
&=& {\rm tr}
\left\{ \delta {D'_{\rm DW}}^{-1} \right\}_{NN}
\frac{1}{a_5} \left\{1 - \frac{1}{a_5}{D'_{\rm PV}}^{-1} \right\}_{NN} .
\end{eqnarray}
In the last equality, we have noted that
\begin{equation}
\left\{M^{\rm PV}-M^{\rm H}\right\}_{st}= \frac{1}{a_5} \delta_{sN}\delta_{Nt}.
\end{equation}
The propagator of the $N$-th flavor of the Pauli-Villars field
has been evaluated in section \ref{sec:effective-action-of-q} as
\begin{equation}
\frac{1}{a_5} \left\{ {D'_{\rm PV}}^{-1} \right\}_{NN}
= \frac{a^4}{a_5} \langle Q(x) \bar Q(y) \rangle
= \frac{1}{a_5 D_N^{\rm eff}+ 1 }
= \left( 1- a D_N \right) .
\end{equation}
With this result and
Eq.~(\ref{eq:inverse-truncated-overlap-vs-propagator-q-variation}),
we obtain
\begin{eqnarray}
({\rm l.h.s.})
&=& {\rm tr} \delta {D_N}^{-1} \, D_N \nonumber\\
&=& a^4 \sum_x B^a_\mu(x) \, \langle V^a_{\mu N}(x) \rangle .
\end{eqnarray}
As to the identity of the axial vector current,
using
Eqs.~(\ref{eq:chiral-property-domain-wall-D}) and
(\ref{eq:chiral-property-truncated-overlap-D}) and
through the similar calculation as in
Eq.~(\ref{eq:derivation-relation-of-vector-currents}) ,
we obtain
\begin{eqnarray}
\label{relation-of-axial-vector-currents-N}
&& \langle \overline{A}^a_{\mu {\rm DW}}(x) \rangle \nonumber\\
&& \quad
=
\langle A^a_{\mu N}(x) \rangle
+ {\rm tr} K^a_{\mu N}(x) D_N^{-1} \Delta_N D_N^{-1}
\nonumber\\
&& \qquad
- {\rm tr}
\left\{ {D_{\rm DW}}^{-1} K^a_{\mu {\rm DW}}(x) {D_{\rm DW}}^{-1}
\right\}_{N,\frac{N}{2}}
2 \gamma_5
\left\{ {D_{\rm DW}}^{-1} \right\}_{\frac{N}{2},N} D_N . \nonumber\\
\end{eqnarray}
From this and Eq.~(\ref{eq:breaking-Delta}),
we infer that the second and third terms of the right-hand side of
Eq.~(\ref{relation-of-axial-vector-currents-N})
vanish in the limit of the infinite flavors
and we finally obtain
Eq.~(\ref{relation-of-axial-vector-currents}).
\subsection{Axial anomaly}
The flavor singlet axial vector current of the domain-wall fermion with
the subtraction of the Pauli-Villars field satisfies the
following axial Ward-Takahashi identity:
\begin{eqnarray}
\label{eq:axial-WT-identity-DW-PV}
&& \partial_\mu^\ast \left\langle \overline{A}_{\mu {\rm DW}}(x) \right\rangle
\nonumber\\
&& =
2 \left\langle \bar \psi^\prime_{\frac{N}{2}}(x) \, \gamma_5 \,
\psi^\prime_{\frac{N}{2}}(x) \right\rangle
+
2 \left\langle \bar \phi^\prime_{\frac{N}{2}}(x) \, \gamma_5 \,
\phi^\prime_{\frac{N}{2}}(x) \right\rangle
+ 2 \left\langle \bar Q(x) \, \gamma_5 \,
Q(x) \right\rangle , \nonumber\\
\end{eqnarray}
where $Q(x)$ and $\bar Q(x)$ stand for the $N$-th flavor
of the Pauli-Villars field as defined in Eq.~(\ref{eq:N-th-PV-field}).
Note that in this case three chiral symmetry breaking terms
would contribute to axial anomaly. The first term is the contribution
of the domain-wall fermion and was evaluated in
section~\ref{sec:axial-anomaly-q-and-psi}. The second and third
terms are the contributions of the Pauli-Villars field.
In order to evaluate them, we need the propagators of the
$\frac{N}{2}$-th flavor and $N$-th flavor of the Pauli-Villars field.
The propagator of $Q(x)$ and $\bar Q(x)$ was evaluated in
section~\ref{sec:effective-action-of-q}.
\begin{equation}
\langle Q(x) \bar Q(y) \rangle
= \frac{1}{a^4}\frac{1}{D_N^{\rm eff}+ \frac{1}{a_5}}
= \frac{a_5 }{a^4}\left( 1- a D_N \right) .
\end{equation}
The propagator of the
$\frac{N}{2}$-th flavor of the Pauli-Villars field
can be evaluated using the similar method to evaluate the
propagator of the
$\frac{N}{2}$-th flavor of the domain-wall fermion,
which was used in section~\ref{sec:axial-anomaly-q-and-psi}.
The calculation is described in
appendix~\ref{sec:propagators-heavy-modes}.
The result can be written as
\begin{equation}
\left\langle \phi^\prime_{\frac{N}{2}}(x)
\bar \phi^\prime_{\frac{N}{2}}(y)\right\rangle
= -\gamma_5 \frac{a_5 }{a^4} \left( 1-aD_N\right) \gamma_5 .
\end{equation}
From these results, we see that
the net contribution of the anomalous terms of
the r.h.s. of Eq.~(\ref{eq:axial-WT-identity-DW-PV}) is same
as that of the domain-wall fermion (without the subtraction of the
Pauli-Villars field) and is given by
\begin{equation}
\label{eq:axial-WT-identity-DW-PV-result}
\partial_\mu^\ast \left\langle \overline{A}_{\mu {\rm DW}}(x) \right\rangle
= \frac{1}{a^4} \left(
- a{\rm tr} \gamma_5 R D_N (x,x) - {\rm tr} D_N^{-1} \Delta_N (x,x)
\right) .
\end{equation}
This identity corresponds term by term
to the axial Ward-Takahashi identity of the truncated overlap Dirac
fermion, which is derived under the chiral transformation
Eq.~(\ref{eq:chiral-transformation-of-psi-truncated}):
\begin{equation}
\label{eq:axial-WT-identity-truncated-overlap-Dirac-fermion}
\partial_\mu^\ast \left\langle A_{\mu N}(x) \right\rangle
= \frac{1}{a^4} \left(
- a{\rm tr} \gamma_5 R D_N (x,x)
+ \left\langle \bar \psi(x) \Delta_N \psi(x) \right\rangle
\right) .
\end{equation}
And it reduces in the limit of the infinite flavors to
the anomalous axial Ward-Takahashi identity of the overlap
Dirac fermion:
\begin{equation}
\label{eq:axial-WT-identity-overlap-Dirac-fermion}
\partial_\mu^\ast \left\langle \widetilde A_{\mu }(x) \right\rangle
= \frac{1}{a^4} \left(
- a{\rm tr} \gamma_5 R \widetilde D (x,x)
\right) .
\end{equation}
\subsection{Probe for topological charge at a finite flavor}
In view of the Eqs.~(\ref{eq:axial-WT-ideinty-DW-mqq}),
(\ref{eq:anomalous-term-q}) and (\ref{eq:anomalus-term-DW-and-index}),
it seems reasonable to probe the topological charge by the anomalous term
of Eq.~(\ref{eq:axial-WT-identity-DW}), as in
the numerical calculation by Argonne group \cite{lagae-sinclair}.
In this respect,
we may probe the topological charge rather directly
using the truncated overlap Dirac operator.
\begin{equation}
Q_N= -\frac{1}{2} a {\rm Tr} \gamma_5 R D_N .
\end{equation}
The topological charge density,
$-\frac{1}{2} a {\rm tr} \gamma_5 R D_N(x,x)$ is assumed to be
a local functional of the gauge fields
in the limit of the infinite flavors
\cite{index-theorem-at-finite-lattice,
exact-chiral-symmetry,locality-of-overlap-D}.
This fact is reflected in the domain-wall fermion
in that $Q_N$ can be evaluated as
a contribution of the massive flavor to the anomalous term
of Eq.~(\ref{eq:axial-WT-identity-DW}).
We point out that $Q_N$ can also be
expressed as follows:
\begin{eqnarray}
Q_N
&=& - a^4 \sum_x m \left\langle \bar q (x) \gamma_5 q(x) \right\rangle
\Big\vert_{ma=1} \nonumber\\
&=& a^4\sum_x \left\langle \bar Q (x) \gamma_5 Q(x) \right\rangle .
\end{eqnarray}
\section{Discussion}
We have discussed the chiral property of the light fermion
of the domain-wall fermion by considering its low energy effective
action
\begin{equation}
S_N^{\rm eff} = a^4 \sum_x \bar q(x) \, D_N^{\rm eff} \, q(x),
\end{equation}
where the effective Dirac operator has a simple relation
to the truncated overlap Dirac operator as
\begin{eqnarray}
\frac{a}{a_5} { D_N^{\rm eff} }^{-1}+ a \delta(x,y)
&=&
{ D_N^{\rm \phantom{f}} }^{-1} .
\end{eqnarray}
We have argued that the chiral property of
the light fermion field is understandable also
from the point of view of the exact chiral symmetry
based on the Ginsparg-Wilson relation, which holds true
for the overlap Dirac operator.
As discussed in section \ref{sec:effective-action-of-q},
in the limit of the infinite flavors,
the effective action itself becomes chiral, but non-local.
A subtlety of the effective action
in this limit becomes clear when one attempts to calculate the axial
anomaly directly from the effective action: with such chiral and
non-local action, it is not easy to identify the source of the axial anomaly.
One possible way might be to consider the explicit breaking term
at a finite flavor $N$:
\begin{equation}
a^4 \sum_x \alpha(x) \,
\left\langle \bar q(x) \left\{ \gamma_5, D_N^{\rm eff} \right\} q (x)
\right\rangle \Big\vert_{ma} .
\end{equation}
(We have introduced a bare mass of the light fermion and
$\alpha(x)$ is an infinitesimal local parameter.)
In view of Eq.~(\ref{eq:anomalous-term-q}) and the axial
Ward-Takahashi identity,
as least for the global case
this term actually reproduces the anomaly as
the index of the overlap Dirac operator in the limit
of the infinite flavors.
We leave this question for future study.
\section*{Acknowledgments}
We would like to thank A.~Yamada, T.~Onogi,
S.~Aoki, T.~Izubuchi and Y.~Taniguchi for enlightening discussions.
Y.K. is also grateful to O.~Miyamura and A.~Nakamura
for discussions.
Y.K. is supported in part by Grant-in-Aid for Scientific Research from
Ministry of Education, Science and Culture(\#10740116,\#10140214).
|
1,116,691,498,489 | arxiv | \section{Introduction}
Double-layer two dimensional electron gas (2DEG) systems,
where electrons are confined to nearby
parallel planes, are expected to exhibit many novel phenomena
due to interlayer electron-electron
interaction. For example, in strong magnetic fields
Coulomb effects are
expected to produce new incompressible ground states
that exhibit the fractional quantum Hall effect\cite{ahm1} and
to cause the collapse of certain integer quantum Hall
effect gaps\cite{ahm2}. In zero magnetic field, it has
been suggested that Wigner crystallization in double layer
systems is favored by interlayer Coulomb interactions\cite{neilson}
and that in the case of electron-hole systems, excitonic
superfluity could result\cite{spfl} from interlayer interactions.
In recent experiments by
Gramila et al. on electron-electron double layer
systems\cite{gramila1,gramila2} and in
similar experiment by Sivan et al. on electron-hole systems\cite{e-h},
the strength of interlayer interactions
was studied directly by measuring the frictional drag of
one two-dimensional electron gas layer on another.
In these experiments a current flowing in one layer tends to
induce a current in nearby layers.
If no current is allowed to flow in the nearby layer
an electric field develops whose influence cancels the
frictional force between the layers. The transresistance, defined as
the ratio of the induced voltage in the second layer to
the applied current in the first layer, directly measures
the rate at which momentum is transferred from the current
carrying 2DEG to its neighbor.
Drag between spatially separated electron systems
due to Coulomb interactions between carriers was first considered by
Pogrebinskii\cite{sov} and Price\cite{price}.
The experiments of Gramila et al.\cite{gramila1,gramila2} and of
Sivan et al.\cite{e-h} have stimulated recent theoretical
attention, especially to the case of drag between two-dimensional
electron layers at low
temperatures\cite{gramila1,gramila2,solomon,jauho,mahan1,rojo,takis}.
The interlayer Coulomb drag is caused by fluctuations in
the density of electrons in each layer since 2D layers with
charge uniformly distributed will not exert any frictional
forces upon each other.
In this paper we examine for the
first time the possibility of enhanced frictional drag
between disordered layers due to the diffusive nature of
long-wavelength long-time electron density fluctuations.
Disorder is known\cite{aa,plee} to enhance interaction effects
and to lead to violations of Fermi liquid theory for
individual two-dimensional electron gas layers.
In the diffusive regime,
where wavelengths are longer than the mean free path of
the electrons and times are longer than the electron scattering time,
the electron density-density response function possesses a
diffusion pole\cite{forster} in momentum-frequency space.
In perturbation theory the diffusion pole arises from
dressing the electron-electron vertex\cite{plee}
with corrections arising from impurity potential scattering.
At shorter distances or shorter times the disorder vertex
corrections are not important.
We find that disorder enhances the interlayer drag
at low temperatures, changing the temperature dependence
of the drag from $T^2$ to $-T^2 \ln T$.
In very high-mobility samples (on which existing
experiments have been performed) or in samples with small
layer separations the corrections due to disorder become
important only at extremely low temperatures.
For samples with lower mobility or more widely separated
layers the influence on the interlayer scattering rate
from disorder scattering should be easily measurable.
Most\cite{gramila1,solomon,jauho} previous work on interlayer
friction has been based on Boltzmann transport theory which
cannot capture disorder enhanced interaction effects.
(An exception is the work of Vasilopoulos and co-workers\cite{takis}.)
In Section II we present a derivation of the expression
for the frictional transresistance based on the memory
function formalism\cite{forster} which is sufficiently
general to treat the case of disordered 2DEG layers.
Although phonon-mediated\cite{gramila2,takis,myphmed}
interlayer interactions are not discussed explicitly
in this paper the expression we derive in Section II
are sufficiently general that other coupling mechanisms
can be incorporated as an effective-interlayer interaction
potential. Similarly particle-particle interaction
vertex corrections, which are
probably quantitatively important especially
for electron-hole double layer systems\cite{e-h},
can also be incorporated into the results we derive in
Section II as a contribution to the effective
electron-electron interaction\cite{we}. Readers interested only
in the application to disordered double-layer
systems should proceed to Section III where we
discuss how disorder within the layers influences the interlayer
friction. The interlayer scattering rate of typical
electron-electron double layer samples is evaluated numerically
in Section IV. We show that the temperatures
below which disorder becomes important decreases very
rapidly with increasing mobility. A
brief summary of our findings concludes the paper in
Section V.
\section{Memory Function Formalism Derivation}
The memory function formalism provide a very convenient method
for deriving a flexible expression for the transresistance of double
layer 2DEG systems.
The Kubo current-current correlation function formula
for the conductivity is
first converted into a force-force
correlation function expression for the resistivity
by making use of Mori's\cite{mori} projection operator.
This force-force correlation function is then evaluated at
lowest order in the screened interlayer interaction to
obtain an approximate expression for the transresistance.
The advantage of using the force-force correlation function
rather than the current-current correlation function
is that it yields a reasonable approximation even when
evaluated at lowest order\cite{mf1,mf2}. At this level
the results are physically equivalent to
momentum-balance\cite{takis} approximations or to
relaxation-time approximations in a Boltzmann-transport
approach\cite{mahan2}. The derivation for the situation
of present interest is sketched in the following paragraphs.
For notational simplicity we restrict ourself to the case of
zero magnetic field so that the
current is in the same direction as the
applied electric field and the conductance,
therefore, forms a 2$\times$2
matrix with respect to the layer indices.
(The final expression for the transresistance
is equally valid\cite{magfield}
in the presence of a magnetic field.)
We need to consider only the long wavelength limit of the conductance.
To use the memory function formalism it is convenient to
write the Kubo formula in the form
\begin{equation}
\sigma_{ij}(\omega) = {\beta\over\nu} \int_{0}^{\infty}dt
e^{i\omega t} ( {\hat J}_i(t), {\hat J}_j )
\label{ku}
\end{equation}
where $\beta=1/k_B T$, $\nu$ is the
cross section area of the 2DEG layers, and
the indices $i$ and $j$ are layer labels.
$\sigma_{ij}(\omega)$ gives the current density induced in
layer $i$ due to an electric field in layer $j$.
$\hat J$ is the zero wavevector Fourier component of
the current density operator.
The inner product appearing in Eq.~(\ref{ku}) is defined by
\begin{eqnarray}
C_{AB}(t) &\equiv & ({\hat A(t)},{\hat B}) \nonumber \\
&\equiv & \beta^{-1}\int_{0}^{\beta}d\lambda\langle {\hat A}^\dagger(t),
{\hat B}(i\hbar\lambda)\rangle
\label{ip}
\end{eqnarray}
In Eq.~(\ref{ip}) the angle brackets denote thermal averages.
The following relationship can be used to change Eq.~(\ref{ku}) into
the more familiar form of the Kubo formula\cite{mf2}:
\begin{equation}
i\beta\partial_tC_{AB}(t) = {1\over\hbar}\langle[{\hat A}(t),{\hat
B}]\rangle.
\label{dtc}
\end{equation}
The projection operator method is now
used to obtain an expression for the matrix inverse
in layer indices of $C_{J_iJ_j}$.
We define a superoperator ${\cal P}$ which `projects' an operator
$\hat O$ onto the current density, and its complement ${\cal Q}$, by
\begin{eqnarray}
{\cal P}\hat O
&\equiv & \sum_k{ {\hat J}_k ({\hat J}_k,\hat O) \over
({\hat J}_k,{\hat J}_k ) }
\\
{\cal P} \hat O & \equiv & \hat O-{\cal Q} \hat O
\label{po}
\end{eqnarray}
It is useful to define a matrix $\chi_{ij}$:
\begin{equation}
\chi_{ij} = {\beta\over\nu} C_{J_iJ_j}(0) = ({\hat J}_i,{\hat J}_j)
= {n_ie^2\over m}\delta_{ij}.
\label{ch}
\end{equation}
Here $n_i$ is the areal density of 2D electrons in the ${i}$-th layer.
Following the usual development of the memory function
formalism\cite{forster}, we obtain an equation for the matrix
inverse of the Fourier transform of $C_{J_iJ_j}(t)$:
\begin{equation}
[C_{J_iJ_j}(z)]^{-1} = {\beta\over\nu}\chi^{-1}[-iz{\bf 1}+M(z)]
\label{em}
\end{equation}
where
\begin{equation}
M_{ij}(z) = {\beta\over\nu} (\dot{J}_i,{i\over z-{\cal Q}{\cal L}}
\dot{J}_j)\chi_{jj}^{-1}.
\label{si}
\end{equation}
A dot over an operators denotes its time derivative at $t=0$
and the Liouville superoperator is defined by ${\cal L}
{\hat O} = [\hat H,\hat O]$, where $\hat H$ is the Hamiltonian.
In obtaining Eq.~(\ref{em}),
the time reversal invariance condition ${\cal P}\dot{J}=0$
has been applied. Combining Eq.~(\ref{ku}), Eq.~(\ref{em})
and Eq.~(\ref{si}) we obtain an expression for the resistivity
matrix:
\begin{eqnarray}
\rho_{ij}(z) &=& \chi_{ii}^{-1} M _{ij}(z)\nonumber \\
&=& \chi_{ii}^{-1}\chi_{jj}^{-1}{\beta\over\nu}
\int_{0}^{\infty}dt e^{izt}(\dot{J}_i,e^{-i{\cal Q}Lt}\dot{J}_j)
\label{ro}.
\end{eqnarray}
$\rho_{ij}$ relates the electric field in layer $i$ to the
current density in layer $j$.
With the relation $\dot{J}_i = -e/m F_i$, we obtain the force-force
correlation function expression for the trans-resistance,
\begin{equation}
\rho_{LR} = {\beta\over n_Ln_Re^2\nu}\int_{0}^{\infty}dte^{izt}
(\hat F_L,e^{-i{\cal Q}{\cal L}t}\hat F_R).
\label{rolr}
\end{equation}
It is easy to demonstrate that $\rho_{LR}$ is identically zero
in the absence of interlayer coupling, since the forces in
left and right layers are uncorrelated.
The leading contribution to the force operator
from interlayer interactions
can be expressed in terms of the
interlayer interaction potential $U_e(q)$ and the electron
density $\varrho(q)$
\begin{equation}
{\bf F}_{R(L)} = \pm {i\over\nu}\sum_{\vec q}{\vec q}\varrho_L(\vec q)
\varrho_R(-\vec q) U_e(q).
\label{fe}
\end{equation}
To leading order in interlayer interactions
$ e^{-i{\cal Q}{\cal L}t}\hat F_i$ in Eq.~(\ref{rolr}) can be
replaced by $e^{-i{\cal L}t}\hat F_i$.
This replacement leads to the desired
force-force correlation function expression for the trans-resistance
\begin{equation}
\rho_{LR}(z) = {\beta\over n_Ln_Re^2\nu}
\int_{0}^{\infty}dt e^{izt} (F_L(t),F_R)_0.
\label{ffr}
\end{equation}
The subscript on the inner product in Eq.~(\ref{ffr})
indicates that it should be evaluated in the
absence of interlayer interactions. Substituting the
explicit expression for the interlayer forces gives
\begin{equation}
\rho_{LR}(z) = {\beta\over2n_Ln_Re^2\nu^3}\sum_{\vec q}q^2
\int_{0}^{\infty}e^{izt}dt|U_e(q)|^2({\hat A(t)},
{\hat A}^{\dagger}(0))_0
\label{r1}
\end{equation}
with
\begin{equation}
{\hat A(t)} = \varrho_L(-\vec q,t)\varrho_R(\vec q,t).
\label{as}
\end{equation}
The correlation function for decoupled layers appearing
in Eq.~(\ref{r1}) is related to the isolated layer density
fluctuations. Using a representation of
exact eigenstates that in the $z=0$ limit it is easy to show that
\begin{equation}
\rho_{LR} = { \pi \beta \over 2 n_L n_R e^2 }
\int {d^2 \vec q \over (2 \pi)^2 } q^2 |U_e(q)|^2
\int_{-\infty}^{\infty} d \omega
S_L(\vec q, \omega) S_R( - \vec q, - \omega)
\label{ahm1}
\end{equation}
where $S_i(\vec q,\omega) $ is the dynamic structure factor
for layer $i$:
\begin{equation}
S_i(\vec q, \omega) \equiv { 1 \over \nu }
\sum_{n,m} \exp (- \beta E_n) |\langle n | \rho_i(\vec q) | m \rangle |^2
\delta (\omega - (E_m-E_n)/\hbar )
\label{ahm2}
\end{equation}
It is usually more convenient to express the resistance in
terms of individual layer response functions rather than the dynamic
structure factor.
We relate $S_i(q,\omega)$
to the retarded density-density response function for layer
$i$, $\chi_{i}(\vec q, \omega) $ by
applying the fluctuation-dissipation theorem\cite{forster}
\begin{equation}
S_i(q,\omega) = {\hbar\over1-e^{-\hbar\omega\beta}}
{\rm Im} \chi_i(\vec q,\omega).
\label{fdearly}
\end{equation}
This gives us the final form of our expression for the drag
resistivity, which is summarized
diagrammatically in Fig.~(\ref{fa}).
\begin{equation}
\rho_{LR} = {\hbar^2\beta\over\pi n_Ln_Re^2}{1\over\nu}
\sum_{\bf q}q^2|U_e(q)|^2\int_{0}^{\infty}d\omega
{{\rm Im}\chi_{R}(q,\omega)
{\rm Im}\chi_{L}(q,\omega)\over
e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}.
\label{r3}
\end{equation}
\section{Disorder and Screening}
The electron density-density response
function $\chi(q,\omega)$ in Eq.~(\ref{r3})
can be obtained by applying many-body perturbation
theory methods to a 2DEG whose Hamiltonian
contains disorder and (or) interaction terms.
For a non-interacting disorder-free 2DEG the
response function can be evaluated analytically\cite{2degrf}.
Disorder leads to an enhancement in ${\rm Im}\chi(q,\omega)$
at low frequencies and long wavelengths and, as we discuss
below, can enhance the interlayer friction. The enhancement
reflects the increased spatial correlation of states with
nearby energies in disordered systems. The effect of
disorder on the friction can be described without
making a specific model of disorder by invoking the
Einstein relation between the conductivity and the diffusive
density-density response at long wavelengths and
low frequencies\cite{arbitrary}.
We introduce a phenomenological intralayer electron
(transport) scattering time $\tau$.
$\tau$ is related to the mobility by $\mu=e\tau/m$ and
at low temperatures is related to the mean-free-path
by $l = \tau\hbar k_f/m$.
For $q l > 1$ or $ \omega \tau > 1$ we assume that disorder is
unimportant and approximate the density-density response
function by the non-interacting electron result\cite{2degrf}.
At zero temperature and zero disorder
\begin{eqnarray}
&&\chi_i(q,\omega) \equiv
\chi_{i}^{B}(q,\omega)\nonumber \\
&=& {dn\over d\mu}{m\over q^2}\{{q^2\over m} -
C_+|(k_fq/m)^2-(\omega+\varepsilon_q)^2/\hbar^4|^{1/2}
-C_-|(k_fq/m)^2-(\omega-\varepsilon_q)^2/\hbar^4|^{1/2}\}
\label{xb}
\end{eqnarray}
\noindent where $\varepsilon_q=\hbar q^2/2m$,
$\mu$ is the chemical potential and the $C_{\pm}$ are:
\begin{eqnarray}
C_{\pm} &=& {\rm sign}(\varepsilon_q\pm\omega)\ \ \ \ \ \ {\rm if} \ \ \
(k_fq/m)^2-(\omega\pm\varepsilon)^2/\hbar^2<0,
\nonumber\\
C_{\pm} &=&\pm i\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm if}\ \ \
(k_fq/m)^2-(\omega\pm\varepsilon_q)^2/\hbar^2.
>0\nonumber
\end{eqnarray}
However for $q l < 1$ and $\omega \tau < 1$ disorder becomes
important. In this diffusive regime the electron
density-density response function is completely characterized
by the diffusion constant\cite{plee}:
\begin{eqnarray}
\chi_i(q,\omega) &\equiv &
\chi_{i}^{D}(q,\omega) \nonumber \\
&=& {dn\over d\mu}{Dq^2\over Dq^2-i\omega}
\label{xd}
\end{eqnarray}
\vskip 1ex
\centerline{ $\ \ {q<{1/l}}$\ \ \ \ \ \ \ \ \
$\omega<{1 / \tau}$}
\vskip 1ex
\noindent where $D=l^2/ 2 \tau$ is the diffusion
constant.
The derivation of the expression for the transresistivity in
the previous section is
valid up to second order in the interlayer interaction.
For the system of physical interest the interlayer interaction
is Coulombic and it is essential to include screening in
order to get qualitatively correct results. In this
paper we adopt the usual expediency of employing
the second-order expression with the interlayer interaction
replaced by a screened interlayer interaction and argue that
this includes the most important higher-order effects.
In the disorder free limit our expression for the transresistivity
then becomes identical to those derived using other approaches
in earlier work\cite{gramila1,solomon,jauho,mahan1,takis}.
The random-phase-approximation (RPA) screened interlayer
interaction is
\begin{equation}
U_e(q,\omega) = {V_e(q)\over[1+V_a(q)\chi_L
(q,\omega)][1+V_a(q)\chi_R(q,\omega)]-V_e^2(q)\chi_L(q,\omega)
\chi_R(q,\omega)}
\label{rpa}
\end{equation}
where the bare intra- and inter-layer electron-electron interaction
potentials are $V_a(q)=2\pi e^2/q$ and $V_e(q)=V_a(q)e^{-qd}$
where $d$ is the separation between the layers.
In the above expression either the ballistic or the diffusive
form for $\chi_{L(R)}$ should be used as appropriate. Note
that the interlayer interaction is cut-off by the factor
$e^{-qd}$ for $q > 1/d$. The layer separation dependence
of the friction in both diffusive and ballistic limits
results from this cutoff. Physically the cutoff reflects the
fact that charge fluctuations in one layer with a wavelength shorter
than the layer separation get averaged out when viewed from
the other layer.
The expression for the trans-resistance of Eq.~(\ref{r3})
can be split into contributions from the ballistic and diffusive
regimes. With $\rho_{LR}^{-1} \equiv n_Re^2\tau_{LR}/m$, we have
\begin{equation}
\tau_{RL}^{-1} = \tau_B^{-1} + \tau_{\Delta}^{-1}
\label{rlbd}
\end{equation}
where
\begin{equation}
\tau_B^{-1} = {\hbar^2\beta\over2\pi^2mn_L}\int_{0}^{\infty}
dqq^3\int_{0}^{\infty}d\omega |U_e|^2
{{\rm Im}[\chi_L^B(q,\omega)] {\rm Im}[\chi_R^B(q,\omega)]
\over e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}
\label{tbe}
\end{equation}
and
\begin{equation}
\tau_{\Delta}^{-1} =\tau_D^{-1} -
{\hbar^2\beta\over2\pi^2mn_L}\int_{0}^{1\over l}
dqq^3\int_{0}^{1\over\tau}d\omega |U_e|^2
{{\rm Im}[\chi_L^B(q,\omega)] {\rm Im}[\chi_R^B(q,\omega)]
\over e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}
\label{tdel}
\end{equation}
with
\begin{equation}
\tau_D^{-1} =
{\hbar^2\beta\over2\pi^2mn_L}\int_{0}^{1\over l}
dqq^3\int_{0}^{1\over\tau}d\omega |U_e|^2
{{\rm Im}[\chi_L^D(q,\omega)] {\rm Im}[\chi_R^D(q,\omega)]
\over e^{\beta\hbar\omega}+e^{-\beta\hbar\omega}-2}
\label{tde}
\end{equation}
$\tau_B^{-1}$ is the result for
a disorder-free 2DEGs and
$\tau_{\Delta}^{-1}$ is the correction due to the
the enhanced fluctuations at long wavelengths and low frequencies
in disordered systems.
In the next section we discus the evaluation of these expressions.
\section{Numerical Results and Discussion}
The dependence of the interlayer scattering rate on temperature
and on layer separation depends on whether
the interlayer scattering is dominated by $\tau_B^{-1}$ or
$\tau_D^{-1}$.
In Fig.~(\ref{fb}) and Fig.~(\ref{fc})
we show numerical results for $\tau_B$ and
$\tau_{\Delta}$ as a functions of temperature calculated
for two different values of layer separation for a high
mobility two-dimensional electron gas sample.
The data in these figures are obtained from numerical evaluation of
Eq.~(\ref{tbe}) and Eq.~(\ref{tde}) with the input
parameters taken from the experiment of ref.~\cite{gramila1}.
In Fig.~(\ref{fb}) we see that $\tau_B^{-1} \sim
T^2/d^4$ at low temperatures, as pointed out in
Ref.\cite{gramila1}. (The $d^{-4}$ dependence can be
recognized by noticing that the scattering rate decreases by
a factor of approximately four when the layer separation
increases by a factor of $\sqrt{2}$.) From Fig.~(\ref{fc}) one can see
that $\tau_{\Delta}^{-1}$ falls off more slowly
with both temperature and layer separation
as $T\rightarrow 0$. The inset to Fig.~(\ref{fc})
establishes that for $T\ll T_{\tau}$,
$\tau_D^{-1} \sim - T^2{\rm ln}T/d^2$. The dimensionless
temperature scale for Fig.~(\ref{fb}) is the Fermi
temperature ($T_F \equiv E_F / k_B$) which is about $60$K for
this sample, while
the dimensionless temperature scale for Fig.~(\ref{fc}) is
the disorder temperature
($ T_{\tau} \equiv \hbar / k_B \tau $) which is
$\sim 56$mK for this sample. Comparing Fig.~(\ref{fb})
and Fig.~(\ref{fc}) we see that for this high-mobility
sample the disorder correction is smaller than
one part in $10^5$ at temperatures above $\sim 10mK$.
The origin of the temperature and layer dependence
seen in Fig.~(\ref{fb}) and Fig.~(\ref{fc}) can be
understood by looking at the limit of large layer separations where
the interlayer scattering rates can be evaluated analytically.
The evaluation of
$\tau_{LR}^{-1}$ at large layer separations and low
temperatures for the
disorder-free limit,
where the relation $\tau_B\propto T^2/d^4$ holds,
has been carried out previously
by several authors\cite{gramila1,solomon,jauho}.
(The results quoted in Ref.~(\cite{gramila1}) and
Ref.~(\cite{solomon}) are in error by a factor of two.) We
rederive those results here to allow a comparison with the
disordered case. For $T \ll T_F$ and $ d \gg k_F^{-1}$
only the low frequency and long wavelength limit of
${\rm Im}\chi$ contributes importantly to Eq.~(\ref{r3}).
($k_F$ is the Fermi wavelength.) From Eq.~(\ref{xb}) it
follows that in this limit
$ {\rm Re} \chi (q,\omega) = dn/d\mu$, and ${\rm Im} \chi (q,\omega) =
dn/d\mu (2 \hbar \omega /E_F) (k_F/q)$. For $T \ll T_F$, we can
ignore the contribution of ${\rm Im}\chi$ to screening the
interlayer interaction and it follows from Eq.~(\ref{rpa}) that
we can replace the interlayer interaction by
\begin{equation}
U_e(q) = { \pi e^2 q \over k_{TF}^2 \sinh{qd} }
\label{uesq}
\end{equation}
where $k_{TF} \equiv 2 \pi e^2 dn/d\mu$ is the single
layer Thomas-Fermi screening wavevector. Note that for
$q \to 0$ the effective screening wavevector is $2 k_{TF}^2 d$
, which is proportional to the layer separation. (For GaAs
$k_{TF} = 0.2 {\rm nm}^{-1}$ independent of electron density.)
With these approximations the integral over frequency and
wavevector are known and we obtain
\begin{equation}
\tau_B^{-1} = {- \pi\zeta(3)(k_BT)^2\over16\hbar\varepsilon_f
(k_{TF}d)^2(k_Fd)^2}.
\label{equ:tb2}
\end{equation}
In this result two powers of $d^{-1}$ may be associated with
the enhanced screening of the interlayer interaction at large
separations and two powers of $d^{-1}$ with the combination
of phase space considerations which cause the integrand to
to vary as $q^1$ for small $q$.
For layer separations larger than the mean free path the
electron response is diffusive over the entire range of
wavevectors contributing importantly to Eq.~(\ref{r3}).
For low temperatures it follows from Eq.~(\ref{xd}) that we
may use ${\rm Re} \chi (q,\omega) = dn / d \mu$ and
${\rm Im} \chi (q,\omega) = dn / d\mu [ \omega D q^2 /
( \omega^2 + (D q^2)^2) ]$. Again we may ignore the
contribution to screening from ${\rm Im}\chi$ so that
the relevant limit of the screened interlayer interaction
is unchanged. At small $\omega$, ${\rm Im}\chi \propto q^{-2}$
compared to the $q^{-1}$ of the ballistic case. The
integrand of the wavevector integral thus goes as $q^{-1}$
at small wavevector. This logarithmically divergent wavevector
integral is cutoff at $q \sim {\omega/D}^{1/2}$.
The remaining frequency integral is elementary and we obtain
\begin{equation}
\tau_D^{-1} = {- \pi(k_BT)^2{\rm ln}(T/T_{\tau})
\over 12 \hbar\varepsilon_f(q_{TF}d)^2(k_Fl)^2}.
\label{equ:td2}
\end{equation}
The change in the layer separation from $d^{-4}$ in the
ballistic case to $d^{-2}$ in the diffusive case can be traced
directly to the change in the wavevector dependence of
${\rm Im}\chi$ from $q^{-1}$ to $q^{-2}$.
For $d \ll l$ Eq.~(\ref{equ:td2}) correctly gives the
contribution to the drag from $q \ll l^{-1}$ and
Eq.~(\ref{equ:tb2}) gives the contribution from
$l^{-1} < q < d^{-1}$. Because of the different
temperature dependence it is still true that the
contribution from the diffusive regime will dominate at
sufficiently low temperatures. Comparing Eq.~(\ref{equ:td2})
and Eq.~(\ref{equ:tb2}) we can estimate the crossover temperature:
\begin{equation}
T_c \sim T_{\tau} \exp [ - 3 (l/d)^2 / 4 \zeta (3)]
\label{tcross}
\end{equation}
For high-mobility samples the diffusive enhancement of the
drag will be observable only at extremely low temperatures.
For example, in GaAs $T_{\tau}$ is about $0.2$K and $l \sim 10 \mu$
for a sample with a mobility of $\sim 10^6cm^{2}/sV$ and
a typical density. For a layer separation of $\sim 500 A^{\circ}$
this implies that $T_c \sim 10^{-100} K$.
For the samples in the experiment of ref.~\cite{gramila1},
which are of extremely high mobility and small layer separations,
the value of $T_{\tau}$ is $56mK$ and the $d/l$ is about $10^{-3}$.
$ \tau_{\Delta}^{-1}$ is smaller than $\tau_B^{-1}$
by a factor of $10^{6\sim7}$ at $T\sim T_{\tau}$.
The correction term, $\tau_{\Delta}^{-1}$, will be
difficult to observe for accessible temperatures.
In samples with lower mobility and/or thicker
barriers between the layers, $\tau_{\Delta}^{-1}$ and $\tau_B^{-1}$
have comparable amplitudes for $T\sim T_{\tau}$ and the contribution
from the diffusive regime dominates at lower temperatures.
In Fig.~(\ref{fd}) we plot the relative contribution to
the drag from the diffusive regime vs. mobility for a layer
separation of $50 {\rm nm}$ and $n =1.5 \times 10^{11} {\rm cm}^{-2}$.
These results show that the effect of disorder
will become easily observable at typical low temperatures
for samples with mobilities below $\sim 10^{5} {\rm cm}^2 {\rm V}^{-1}
{\rm s}^{-1}$.
It is possible to fabricate double-layer systems in which
one layer is much more disordered than the other.
In particular one may have $l\gg d$ for one layer
and $l\le d$ for the other layer.
Following the same steps leading to Eq.~(\ref{equ:td2}),
it is possible to derive an expression for the low-temperature
transresistance in such a system
by using one diffusive response
function and one free electron response function:
\begin{equation}
\tau_{LR}^{-1} = {\pi^3(k_BT)^2\over72\hbar\varepsilon_f
(k_{TF}d)^2(k_Fd)(k_Fl)}.
\label{equ:tbd}
\end{equation}
The dependences on temperature and layer
separation, $\tau_{LR}^{-1} \sim
T^2/d^3$, are easily understood by comparing it to Eq.~(\ref{equ:tb2})
and Eq.~(\ref{equ:td2})
and noticing that the integrand in this case approaches a
constant at small wavevector transfers.
In Fig.~(\ref{fd}), results are shown for
the relative correction due to
disorder enhancement for the case where one layer
consists of free electrons while the other layer has
a finite mobility.
The relative correction is essentially independent of temperatures
at low temperatures and it is weaker than in
the case where both layers are disordered.
\section{summary}
Using the memory-function method,
we have derived an expression
for the transresistance of double layer systems
which is sufficiently general to treat the case of
disordered layers. The expression has been evaluated
as a function of temperature, layer separation (d)
and in-plane mobility $\mu$.
Both the case where only one layer is disordered and
the case where both layers are disordered have been considered.
We find that the drag varies as $d^{-4}$, $d^{-3}$ and $d^{-2}$
at low temperatures for clean, single-layer disorder, and
double-layer disorder cases respectively. In the case of
double-layer disorder the transresistance varies as
$ - T^2 \ln (T)$ at low temperatures, otherwise the
transresistance is proportional to $ T^2$. The low-temperature
drag is proportional to $\mu^{-2}$ and $\mu^{-1}$ for
double-layer disorder and single-layer disorder respectively.
Except for the case of extremely low temperatures the
crossover from clean to disordered regimes occurs
when the mean-free-path within a layer becomes smaller
than the layer separation.
In very high mobility two-dimensional electron gas systems,
where the transresistance has been studied
experimentally up to the present,
the effect of disorder on the drag is negligible at available
temperatures. We predict that the enhancement due to disorder
and the associated crossovers in the temperature-dependence
and layer-separation dependence will be observable at
low temperatures in moderate and low mobility samples.
\acknowledgments
The authors are grateful for stimulating
interactions with T.J. Gramila and J.P. Eisenstein and
with Jun Hu and Anthony Chan. Informative
discussions with B.I. Altshuler during the initial stages of
this work and helpful communications with A.P. Jauho,
A.G. Rojo and G.D. Mahan are also acknowledged.
This work was supported by the National Science Foundation under
Grant No. DMR-9113911.
|
1,116,691,498,490 | arxiv | \subsection{Quantum Nilpotent (Anti-)BRST Symmetries Corresponding to the Classical Gauge Symmetry Transformations}
We have listed the quantum (anti-)BRST symmetries corresponding to the {\it classical} gauge symmetry transformations (4)
in our Eqs. (7) and (8). It is elementary to check that these {\it quantum} symmetries are off-shell nilpotent
$(s_{(a)b}^2 = 0)$ of order two. The requirement of the absolute anticommutativity $(s_b\,s_{ab} + s_{ab}\,s_b = 0)$
leads to the restriction: $b + \bar b = 0 \Longrightarrow \bar b = -b$. As a consequence, we have the {\it full}
set of (anti-)BRST symmetry transformations [corresponding to the {\it classical} gauge symmetry transformations (4)] as follows:
\begin{eqnarray*}
&&s_{ab}\,x = \bar c\,p_x, \qquad s_{ab}\,p_x = 0, \qquad s_{ab}\,t = \bar c\,m, \qquad s_{ab}\,p_t = 0, \nonumber\\
&&s_{ab}\,E = \dot{\bar c}, \qquad s_{ab}\,\bar c = 0, \qquad s_{ab}\,c = - i\,b, \qquad s_{ab}\,b = 0,
\end{eqnarray*}
\begin{eqnarray}
&&s_{b}\,x = c\,p_x, \qquad s_{b}\,p_x = 0, \qquad s_{b}\,t = c\,m, \qquad s_{b}\,p_t = 0, \nonumber\\
&&s_{b}\,E = \dot{c}, \qquad s_{b}\, c = 0, \qquad s_{b}\,\bar c = i\,b, \qquad s_{b}\,b = 0.
\end{eqnarray}
It is straightforward to check that the above (anti-)BRST symmetry transformations are off-shell nilpotent $(s_{(a)b}^2 = 0)$
and absolutely anticommuting $(s_b\,s_{ab} + s_{ab}\,s_b = 0)$ in nature. The (anti-)BRST invariant Lagrangian
$L_b$ (which is the generalization of the {\it classical} $L_f$ to its {\it quantum} level) can be written as\footnote {The structure of gauge-fixing and FP-ghost
terms is {\it exactly} like the Abelian 1-form $(A^{(1)} = dx^{\mu}\,A_\mu)$ gauge theory where we have the BRST-invariant Lagrangian density:
$ {\cal L}_{b} = -\,\frac{1}{4}\, F_{\mu \nu}\,F^{\mu \nu}
+ s_b\, [-\,i\, \bar c \,(\partial_{\mu}\,A^{\mu} + \frac{b}{2})]\, \equiv \, -\,\frac{1}{4}\, F_{\mu \nu}\,F^{\mu \nu}
+ s_{ab}\, [-\,i\, c \,(\partial_{\mu}\,A^{\mu} + \frac{b}{2})] \, \equiv \, -\,\frac{1}{4}\, F_{\mu \nu}\,F^{\mu \nu}
+ s_b \, s_{ab} \, [\frac{ i}{2} \, A^\mu A_\mu - \frac{1}{2} {\bar c\, c}]$. Here $A_\mu$ is the vector potential, $F_{\mu \, \nu} = \partial_{\mu}\,A_{\nu}
- \partial_{\nu}\, A_{\mu}$ is the field strength tensor and rest of the symbols are same
as in Eqs. (10) and (11). Note that the $2$-form $F^{(2)} = d \, A^{(1)} = \frac{1}{2}\,(d\,x^\mu \wedge d\, x^\nu )\,F_{\mu \nu}$
defines the field strength tensor $F_{\mu \nu}$ (where $d = d\,x^\mu \, \partial_{\mu}$ in $ F^{(2)} = d \, A^{(1)} $ stands for the exterior derivative of the differential geometry).}:
\begin{eqnarray}
L_b &=& L_f + s_b\,\Big[-\,i\,\bar c\,\Big(\dot E + \frac{b}{2}\Big)\Big] \equiv L_f + s_{ab}\,\Big[i\,c\,\Big(\dot E + \frac{b}{2}\Big)\Big],
\nonumber\\
&\equiv& L_f + s_b\,s_{ab}\,\Big[\frac{i\,E^2}{2} - \frac{\bar c\,c}{2}\Big] \equiv L_f - s_{ab}\,s_{b}\,\Big[\frac{i\,E^2}{2}
- \frac{\bar c\,c}{2}\Big].
\end{eqnarray}
In other words, we have expressed the gauge-fixing and Faddeev-Popov (FP) ghost terms in {\it three} different ways which,
ultimately, lead to the following expression for $L_b$, namely;
\begin{eqnarray}
L_b &=& L_f + b\,\dot E + \frac{b^2}{2} - i\,\dot{\bar c}\,\dot c, \nonumber\\
&\equiv& p_x\,\dot x + p_t\,\dot t - \frac{1}{2}\,E\,(p_x^2 + 2\,m\,p_t) + b\,\dot E + \frac{b^2}{2} - i\,\dot{\bar c}\,\dot c.
\end{eqnarray}
It should be noted that we have dropped the total derivative terms in obtaining $L_b$ from (10). The above equation demonstrates that
we have obtained a {\it unique} (anti-)BRST invariant Lagrangian. This has happened because
the CF-type restriction is trivial (i.e. $b + \bar b = 0$) in our {\it simple} case of NR system. We can explicitly check that:
\begin{eqnarray}
s_b\,L_b = \frac{d}{d\,\tau}\,\Big[\frac{c}{2}\,p_x^2 + b\,\dot c\,\Big], \qquad s_{ab}\,L_b = \frac{d}{d\,\tau}\,
\Big[\frac{\bar c}{2}\,p_x^2 + b\,\dot{\bar c}\,\Big],
\end{eqnarray}
which lead to the derivation of the conserved (anti-)BRST charges $[Q_{(a)b}]$ as follows:
\begin{eqnarray}
Q_{ab} = b\,\dot{\bar c} + \frac{1}{2}\,\bar c \,(p_x^2 + 2\,m\,p_t) \equiv b\,\dot{\bar c} - \dot b\,\bar c, \nonumber \\
Q_{b} = b\,\dot{c} + \frac{1}{2}\,c \,(p_x^2 + 2\,m\,p_t) \equiv b\,\dot{c} - \dot b\,c.
\end{eqnarray}
In the last step, we have used $\dot b = -\,(1/2)\,(p_x^2 + 2\,m\,p_t)$ which emerges out as the EL-EOM from $L_b$ w.r.t. the
Lagrange multiplier variable $E(\tau)$ .
We close this sub-section with a few crucial and decisive remarks. First, we can check that the (anti-)BRST charges are
conserved $[{\dot Q}_{(a)b} = 0]$ by using the EL-EOMs. Second, the (anti-)BRST charges $[Q_{(a)b}]$ are off-shell
nilpotent $[Q_{(a)b}^2 = 0]$ of order {\it two} due to the {\it direct} observations that: $s_b\,Q_b = s_b\,[b\,\dot c - \dot b\,c] = 0$
and $s_{ab}\,Q_{ab} = s_{ab}\,[b\,\dot{\bar c} - \dot b\,\bar c] = 0$ which encode in their folds $s_b\,Q_b = -\,i\,\{Q_b, Q_b\}
= 0 \Longrightarrow Q_b^2 = 0$ {\it and} $s_{ab}\,Q_{ab} = -\,i\,\{Q_{ab}, Q_{ab}\} = 0 \Longrightarrow Q_{ab}^2 = 0$.
Third, the above nilpotency is {\it also} encoded in: $Q_b = s_b\,[b\,E + i\,\dot{\bar c}\,c]$ implying that $s_b\,Q_b = 0$
due to $s_b^2 = 0$ {\it and} we {\it also} point out that $s_{ab}\,Q_{ab} = 0$ due to the nilpotency $(s_{ab}^2 = 0)$ of $s_{ab}$ because
$Q_{ab} = s_{ab}\,[b\,E + i\,\bar c \,\dot c]$. Fourth, we observe that $s_{ab}\,Q_b = i\,\{Q_b, Q_{ab}\} \equiv -\,i\,b\,
\dot b + i\,\dot b\,b = 0$ and $s_b\,Q_{ab} = -\,i\,\{Q_{ab}, Q_b\} = i\,b\,\dot b - i\,b\,\dot b = 0$ which explicitly
lead to the conclusion that the off-shell nilpotent charges $Q_{(a)b}$ are {\it also} absolutely anticommuting $(Q_b\,Q_{ab} + Q_{ab}\,Q_b
= 0)$ in nature. Fifth, the above observation of the absolute anticommutativity can be also expressed in terms of
the nilpotency property because we observe that $Q_b = s_{ab}\,(-\,i\,\dot c\,c)$ and $Q_{ab} = s_b\,(i\,\dot{\bar c}\,\bar c)$
which imply that $s_{ab}\,Q_b = -\,i\,\{Q_b, Q_{ab}\} = 0$ and $s_{b}\,Q_{ab} = -\,i\,\{Q_{ab}, Q_{b}\} = 0$ [due to the
off-shell nilpotency $(s_{ab}^2 = 0)$ of the anti-BRST as well as the off-shell nilpotency $(s_b^2 = 0)$ of the BRST symmetry transformations].
Sixth, it can be seen that the physical space (i.e. $\mid phys>$) in the {\it total} Hilbert space of states is defined by
$Q_b \mid phys> = 0$ which implies that $b\mid phys> \equiv \Pi_{E}\, \mid phys> = 0$ and $\dot b \mid phys> \equiv (p_x^2 + 2\,m\,p_t)\,\mid phys> = 0$.
In other words, the Dirac quantization conditions (with the first-class constraints $\Pi_{E} \approx 0, \, p_x^2 + 2\,m\,p_t \approx 0$) are
beautifully satisfied. Finally, physicality criterion $Q_b \mid phys> = 0$ implies that the two physical states $\mid phys^{'}>$ and $\mid phys>$
belong to the {\it same} cohomological class
w.r.t. the nilpotent BRST charge $Q_b$ if they differ by a BRST {\it exact} state (i.e. $\mid phys^{'}> = \mid phys> + Q_b\,\mid \chi>$
for non-null $|\chi>$).
\section{Nilpotent and Anticommuting (Anti-)BRST Symmetries for the Phase Variables: MBTSA}
This section is devoted to the derivation of the transformations: $s_b\,x = C\,\dot x,\, s_b\,p_x = C\,{\dot p}_x, \, s_b\,t = C\,\dot t,
\, s_b\,p_t = C\,{\dot p}_t,\, s_{ab}\,x = \bar C\,\dot x,\, s_{ab}\,p_x = \bar C\,{\dot p}_x, \, s_{ab}\,t = \bar C\,\dot t,
\, s_{ab}\,p_t = \bar C\,{\dot p}_t$ by exploiting the theoretical tricks of MBTSA. Before we set out to perform this exercise, it is essential to pinpoint
the off-shell nilpotency and absolute anticommutativity properties of the (anti-)BRST symmetry transformations on the phase variables [cf. Eq. (6)].
It can be easily checked that the off-shell nilpotency requirement (i.e. $s_{(a)b}^{2}\,S = 0,\, S = x,\, p_x,\, t,\,p_t $) leads to the (anti-)BRST
symmetry transformations for the (anti-)ghost variables as:
\begin{eqnarray}
s_{ab}\,\bar C = \bar C \,\dot {\bar C}, \qquad\qquad s_b\,C = C\,\dot C.
\end{eqnarray}
Furthermore, the absolute anticommutativity requirement: $ \{s_b, \,s_{ab} \}S = 0$ for the generic phase variable $S = x,\, p_x,\, t,\,p_t$
leads to the following
\begin{eqnarray}
\{s_b,\, s_{ab}\}S = i\,[B + \bar B + i\,(\bar C\, \dot C - \dot {\bar C}\, C)]\,\dot S \,\quad \Longrightarrow \quad \, B +
\bar B + i\,(\bar C\, \dot C - \dot {\bar C}\, C) = 0.
\end{eqnarray}
In other words, the absolute anticommutativity property ($s_b\,s_{ab} + s_{ab}\,s_b = 0$) is satisfied if and only if we invoke the
sanctity of the CF-type restriction: $ B + \bar B + i\,(\bar C\, \dot C - \dot {\bar C}\, C) = 0$. It goes without saying that the {\it above} cited
requirements of the off-shell
nilpotency and absolute anticommutativity properties are very {\it sacrosanct} within the framework of BRST approach to gauge and/or reparameterization
invariant theories.
Against the backdrop of the above discussions, we set out to deduce the (anti-)BRST symmetry transformations: $s_{ab}\,S = \bar C\,\dot S, \, s_b\,S
= C\, \dot S$ (with $S = x,\, p_x,\, t,\,p_t$) and the CF-type restrictions: $ B + \bar B + i\,(\bar C\, \dot C - \dot {\bar C}\, C) = 0$
within the framework of MBTSA. Towards this end in our mind, first of all, we generalize the
{\it classical} function $g(\tau)$ [in $\tau \longrightarrow \tau^{'} = g(\tau)
\equiv \tau - \epsilon(\tau)$] onto a $(1, 2)$-dimensional supermanifold as
\begin{eqnarray}
g(\tau) \quad \longrightarrow \quad \tilde g (\tau, \theta, \bar\theta) = \tau - \theta\,\bar C(\tau) - \bar\theta\,C(\tau) + \theta\,\bar\theta\,k(\tau),
\end{eqnarray}
where $(\bar C)C$ variables are the (anti-)ghost variables of Eq. (6) and $k(\tau)$ is a secondary variable that has to be determined from
the consistency conditions [that include the off-shell nilpotency as well as absolute anticommutativity requirements]. It will be noted that,
due to the mappings: $s_b \leftrightarrow \partial_{\bar\theta}\mid_{\theta = 0},\, s_{ab} \leftrightarrow \partial_{\theta}\mid_{\bar\theta = 0}$ [14-16],
we have taken the coefficients of $\theta$ and $\bar\theta$ in Eq. (16) as the (anti-)ghost variables $(\bar C)C$. This has been done due to our
observation in the infinitesimal reparameterization symmetry transformation $(\delta_r)$ [where $\delta_r\,\tau = -\,\epsilon(\tau)$] at the {\it classical}
level. Following the basic tenet of BRST formalism, the infinitesimal parameter $\epsilon(\tau)$
has been replaced (in the BRST-{\it quantized} theory) by the (anti-)ghost variables thereby leading to the (anti-)BRST symmetry transformations:
$s_{ab}\,\tau = -\,\bar C,\,s_b\,\tau = -\,C$.
For our present 1D diffeomorphism (i.e. reparameterization) invariant theory, the generic variable $S(\tau)$ can be generalized to
a supervariable [$\tilde S(\tilde g(\tau, \theta, \bar\theta), \theta, \bar\theta)$] on the (1, 2)-dimensional supermanifold [22] with the following
super expansion along {\it all} the Grassmannian directions of the (1, 2)-dimensional supermanifold, namely;
\begin{eqnarray}
\tilde S \big[\tilde g(\tau, \theta, \bar\theta), \theta, \bar\theta) \big] = {\cal S} \big [\tilde g(\tau, \theta, \bar\theta) \big]
+ \theta\,\bar R \big[\tilde g(\tau, \theta, \bar\theta) \big]
+ \bar\theta\,R \big [\tilde g(\tau, \theta, \bar\theta) \big]
+ \theta\,\bar\theta\,Q \big [\tilde g(\tau, \theta, \bar\theta) \big],
\end{eqnarray}
where the expression for $\tilde g(\tau, \theta, \bar\theta)$ is given in Eq. (16). It should be noted that {\it all} the primary as well as the secondary supervariables on the
r.h.s. of (17) are function of the $(1, 2)$-dimensional super infinitesimal diffeomorphism transformation (16). At this stage, we can perform the Taylor
expansions for {\it all} the supervariables as:
\begin{eqnarray}
\theta\,\bar\theta\,Q(\tau - \theta\,\bar C - \bar\theta\,C + \theta\,\bar\theta\,k) &=& \theta\,\bar\theta\,Q(\tau), \nonumber\\
\bar\theta\,R(\tau - \theta\,\bar C - \bar\theta\,C + \theta\,\bar\theta\,k) &=& \bar\theta\,R(\tau) + \theta\,\bar\theta \, \bar C(\tau)\,\dot R(\tau),
\nonumber\\
\theta\,\bar R(\tau - \theta\,\bar C - \bar\theta\,C + \theta\,\bar\theta\,k) &=& \theta\,\bar R(\tau) - \theta\,\bar\theta \, C(\tau)\,\dot{\bar R}(\tau),
\nonumber\\
{\cal S} (\tau - \theta\,\bar C - \bar\theta\,C + \theta\,\bar\theta\,k) &=& S(\tau) - \theta\,\bar C(\tau)\,\dot S(\tau) - \bar\theta\,C(\tau)\,\dot S(\tau)
\nonumber\\
&+& \theta\,\bar\theta\, \big [k(\tau)\,\dot S (\tau) - \bar C (\tau) \, C (\tau)\,\ddot S (\tau) \big ].
\end{eqnarray}
Collecting all these terms and substituting them into (17), we obtain the following super expansion for the supervariable on the (1, 2)-dimensional supermanifold, namely;
\begin{eqnarray}
\tilde S \big [\tilde g(\tau, \theta, \bar\theta), \theta, \bar\theta \big ] &=& S(\tau) + \theta\,(\bar R - \bar C\,\dot S) + \bar\theta\,(R - C\,\dot S) \nonumber\\
&+& \theta\,\bar\theta\,\big[Q + \bar C\,\dot R - C\,\dot{\bar R} + k\,\dot S + C\,\bar C\,\ddot S\big].
\end{eqnarray}
We now exploit the horizontality condition (HC) which physically implies that {\it all} the {\it scalar} variables should {\it not} transform { \it at all} under
any kind of spacetime, internal, supersymmetric, etc., transformations. With respect to the
1D space of trajectory of the particle, all the supervariables on
the l.h.s. and r.h.s. of Eq. (18) are {\it scalars}. The HC, in our case, is:
\begin{eqnarray}
\tilde S \big [\tilde g(\tau, \theta, \bar\theta), \theta, \bar\theta) \big] = S(\tau).
\end{eqnarray}
The above equality implies that {\it all} the coefficients of $\theta$, $\bar \theta$ and $\theta\, \bar \theta$ of Eq. (19) should be set equal to zero.
In other words, we have the following:
\begin{eqnarray}
R = C \, \dot S, \qquad \bar R =\bar C \, \dot S, \qquad Q = C \, \dot {\bar R} - \bar C \, \dot R - k \, \dot S + \bar C \, C \, \ddot S.
\end{eqnarray}
Substitutions of the values of $R$ and $\bar R$ into the expression for $Q$ leads to the following:
\begin{eqnarray}
Q = -\,(\bar C \, \dot C + \dot {\bar C} \, C)\, \dot S - k \, \dot S - \bar C \, C \, \ddot S.
\end{eqnarray}
As explained before Eq. (20) (i.e. exploiting the key properties of {\it scalars}), it is evident that (17) can be {\it finally} written
(with $\tilde S \big [\tilde g(\tau, \theta, \bar\theta), \theta, \bar\theta \big ] = \tilde S (\tau, \theta, \bar \theta) $ as
\begin{eqnarray}
\tilde S\big[ \tau, \theta, \bar\theta)\big] &=& {S}(\tau) + \theta\,\bar R(\tau)
+ \bar\theta\,R(\tau) + \theta\,\bar\theta\,Q(\tau) \nonumber\\
& \equiv & {S}(\tau) + \theta \,(s_{ab}\,S) + \bar \theta \,(s_{b}\,S) + \theta \,\bar \theta \,(s_{b} \, s_{ab} \, S),
\end{eqnarray}
where, due to the well known mappings: $s_b\leftrightarrow \partial_{\bar \theta}\mid _{\theta = 0},\, s_{ab}\leftrightarrow
\partial_{\theta}\mid _{\bar \theta = 0}$ [14-16],
the coefficients of $\theta$ and $\bar \theta$ are the anti-BRST and BRST symmetry transformations [cf. Eq. (6)]. We point out that the key properties
of {\it scalars} on the r.h.s. of Eq. (17) implies that we have: ${\cal S}[\tilde g(\tau, \theta, \bar\theta)] = S(\tau), \, R[\tilde g(\tau,
\theta, \bar\theta)] = R(\tau), \, \bar R[\tilde g(\tau, \theta, \bar\theta)] = \bar R(\tau)$ and $Q[\tilde g(\tau, \theta, \bar\theta)] = Q(\tau)$.
A comparison between (21) and (23) implies that we have already derived the nilpotent (anti-)BRST symmetry transformations: $R = s_b\,S = C\,\dot S$ and
$\bar R = s_{ab}\,S = \bar C\,\dot S$. In other words, we have obtained: $s_b\,x = C\,\dot x, \, s_b\,p_x = C\,{\dot p}_x, \, s_b\,t = C\,\dot t, \,
s_b\,p_t = C\,{\dot p}_t$ and $s_{ab}\,x = \bar C\,\dot x, \, s_{ab}\,p_x = \bar C\,{\dot p}_x, \, s_{ab}\,t = \bar C\,\dot t, \,
s_{ab}\,p_t = \bar C\,{\dot p}_t$. Furthermore, it is evident that:
\begin{eqnarray}
s_b\,s_{ab}\,S = Q = -\,(\bar C\,\dot C + \dot{\bar C}\,C)\,\dot S - k\,\dot S - \bar C\,C\,\ddot S.
\end{eqnarray}
The requirement of the absolute anticommutativity: $\{s_b, s_{ab}\}\,S = 0$ implies that $s_b\,s_{ab}\,S = -\,s_{ab}\,s_b\,S$ which, in turn,
leads to the following relationships:
\begin{eqnarray}
s_b\,s_{ab}\,S &=& s_b\,\bar R = Q \equiv -\,(\bar C\,\dot C + \dot{\bar C}\,C)\,\dot S - k\,\dot S - \bar C\,C\,\ddot S, \nonumber\\
-s_{ab}\,s_b\,S &=& -\,s_{ab}\,R = Q \equiv -\,(\bar C\,\dot C + \dot{\bar C}\,C)\,\dot S - k\,\dot S - \bar C\,C\,\ddot S.
\end{eqnarray}
The explicit computation of the following, using the (anti-)BRST symmetry transformations of the phase variables in Eqs. (6) and (14), are:
\begin{eqnarray}
s_b\,\bar R &=& i\,B\,\dot S - \bar C\,\dot C\,\dot S - \bar C\,C\,\ddot S \equiv Q, \nonumber\\
-\,s_{ab}\, R &=& -\,i\,\bar B\,\dot S - \dot{\bar C}\,C\,\dot S - \bar C\,C\,\ddot S \equiv Q.
\end{eqnarray}
Equating (25) and (26), we obtain the following interesting relationship:
\begin{eqnarray}
k = -\,\dot{\bar C}\,C - i\,B \equiv i\,\bar B - \bar C\,\dot C \quad \Longrightarrow \quad B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C) = 0.
\end{eqnarray}
In other words, it is the consistency conditions of the BRST formalism that lead to the determination of $k(\tau)$ in Eq. (16) within the ambit of MBTSA. A
close look at Eqs. (25), (26) and (27) establishes that a precise determination of $Q(\tau)$ in (23) leads to (i) the validity
of the absolute anticommutativity (i.e. $\{s_b, s_{ab}\}\,S = 0$) of the off-shell nilpotent (anti-)BRST symmetries, and (ii) the deduction of the
(anti-)BRST invariant\footnote{This statement is {\it true} only when the whole theory is considered on a {\it submanifold} of the
Hilbert space of the quantum variables where the CF-type restriction: $B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C) = 0$ is satisfied. In other words,
we explicitly compute $ s_b \, [B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)] = (\frac{d}{d \tau}) \, [B + \bar B + i\,(\bar C\,\dot C -
\dot{\bar C}\,C)]\,C - [B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)]\, \dot C $ and $ s_{ab} \, [B + \bar B + i\,(\bar C\,\dot C -
\dot{\bar C}\,C)] = (\frac{d}{d \tau}) \, [B + \bar B + i\,(\bar C\,\dot C -
\dot{\bar C}\,C)]\, \bar C - [B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)]\, \dot{\bar C} $ which imply that $ s_{(a)b}\,
[B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)] = 0$ is true {\it only} on the above mentioned submanifold.} CF-type restriction
$B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C) = 0$ on our theory.
We conclude this section with the following useful and crucial remarks. First, we set out to derive the (anti-)BRST symmetry transformations
(corresponding to the classical reparameterization symmetry transformations) for the phase variables [cf. Eq. (6)]. We have accomplished
this goal in Eq. (21). Second, we have derived the CF-type restriction: $B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C) = 0$ within the
purview of MBTSA [cf. Eq. (27)] which is actually hidden in the determination of $Q(\tau)$ in Eq. (23). Third, for the application of the theoretical
potential of MBTSA, we have taken the {\it full} super expansion of the {\it generic} supervariable [cf. Eq. (17)] along {\it all} the possible
Grassmannian directions of the (1, 2)-dimensional supermanifold. Fourth, unlike the application of BT-superfield/supervariable
approach to the {\it gauge} theories [14-16] where spacetime does {\it not} change, in the case of MBTSA, the super diffeomorphism transformation
(16) has been taken into account in all the {\it basic} as well as {\it secondary} supervariables. Fifth, taking into account
the inputs from Eqs. (21) and (26), we obtain the following super expansion of the {\it generic} variable $S(\tau)$, namely;
\begin{eqnarray}
\tilde S^{(h)} (\tau, \theta, \bar\theta) &=& S(\tau) + \theta\,(\bar C\,\dot S) + \bar\theta\,(C\,\dot S) + \theta\,\bar\theta\,[i\,B\,\dot S - \bar C\,\dot C
\,\dot S - \bar C\,C\,\ddot S] \nonumber\\
&\equiv & S(\tau) + \theta\,(s_{ab}\,S) + \bar\theta\,(s_b\,S) + \theta\,\bar\theta\,(s_b\,s_{ab}\,S),
\end{eqnarray}
where $S = x, p_x, t, p_t$ and the superscript $(h)$ on the supervariable $\tilde S(\tau, \theta, \bar\theta)$ denotes that this supervariable
has been obtained after the application of HC. Finally, the standard nilpotent (anti-)BRST symmetry transformations
(8) dictate that we can have the following (anti-)chiral super expansions for the supervariables corresponding to $(\bar C)C$, namely;
\begin{eqnarray}
&&C(\tau) \quad \longrightarrow \quad {F}^{(c)}(\tau, \theta) = C (\tau) + \theta\,(i\,\bar B) \equiv C (\tau) + \theta\,(s_{ab}\,C), \nonumber\\
&&\bar C(\tau) \quad \longrightarrow \quad {{\bar F}^{(ac)}}(\tau, \bar \theta) = \bar C (\tau) + \bar\theta\,(i\,B) \equiv \bar C (\tau)
+ \bar\theta\,(s_{b}\,\bar C),
\end{eqnarray}
where the superscripts $(c)$ and $(ac)$ denote the {\it chiral} and {\it anti-chiral} supervariables. The above observation gives us a clue that we should exploit
the theoretical strength of ACSA to BRST formalism for our further discussions.\\
\section{Coupled Lagrangians and Quantum (Anti-)BRST Symmetries Corresponding to the Classical Reparameterization Symmetry Transformations}
In addition to the quantum (anti-)BRST symmetries in (6), (8) and (14), we derive {\it all} the other off-shell nilpotent and
absolutely anticommuting (anti-)BRST symmetries corresponding to the {\it classical} infinitesimal and continuous reparameterization
symmetry transformations (2). We exploit the strength of the {\it sacrosanct} requirements of off-shell nilpotency and absolute anticommutativity
properties. In this context, we point out that we have already derived $s_b\,C = C\,\dot C, \, s_{ab}\,\bar C = \bar C\,\dot{\bar C}$
by invoking the sanctity of the off-shell nilpotency $(s_{(a)b}^2 = 0)$ property for the phase variables (i.e. $s_{(a)b}^2\,S = 0,
S = x, p_x, t, p_t$). It is interesting to note the following absolute anticommutativity requirements, namely;
\begin{eqnarray}
&&\{s_b, s_{ab}\}\,C = 0 \quad \Longrightarrow \quad s_b\,\bar B = \dot{\bar B}\,C - \bar B\,\dot C, \nonumber \\
&&\{s_b, s_{ab}\}\,\bar C = 0 \quad \Longrightarrow \quad s_{ab}\,B = \dot{B}\,\bar C - B\,\dot{\bar C},
\end{eqnarray}
leads to the derivation of the $s_{b}\,{\bar B}$ and $s_{ab}\,B$. We can readily check that $s_b^2\,\bar B = 0, \, s_{ab}^2\,B = 0$ are satisfied due to our knowledge of the BRST and anti-BRST symmetry transformations:
$s_b\,C = C\,\dot C, \,
s_{ab}\,\bar C = \bar C\,\dot{\bar C}$ {\it and} the fermionic $(C^2 = {\bar C}^2 = 0, \, C\,\bar C + \bar C\,C = 0)$ nature
of the (anti-)ghost variables $(\bar C)\,C$. We further note that $\{s_b, s_{ab}\}\,B = 0$ and $\{s_b, s_{ab}\}\,\bar B = 0$.
The requirement of the absolute anticommutativity on the $E(\tau)$ variable leads to:
\begin{eqnarray}
\{s_b, s_{ab}\}\,E(\tau) = \frac{d}{d\,\tau}\,\Big[i\,\big\{B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big\}\,E(\tau)\Big].
\end{eqnarray}
Thus, we emphasize that the absolute anticommutativity property $(s_b\,s_{ab} + s_{ab}\,s_b = 0)$ on the phase variables [cf. Eq. (15)]
as well as on the Lagrange multiplier variable [cf. Eq. (31)] are satisfied if and only if the CF-type restriction is invoked. In the full blaze of glory,
the {\it quantum} (anti-)BRST symmetry transformations [corresponding to the infinitesimal reparameterization symmetry transformations (2)] are as follows:
\begin{eqnarray}
s_{ab}\,x &=& \bar C\,\dot x,\quad s_{ab}\,p_x = \bar C\,\dot{p_x},\quad s_{ab}\,t = \bar C\,\dot t,\quad s_{ab}\,p_t = \bar C\; \dot p_t,\quad
s_{ab}\,E = \frac{d}{d\,\tau}\,(\bar C\, E),\nonumber\\
s_{ab}\,C &=& i\, \bar B,\quad s_{ab}\, \bar C = \bar C\, \dot {\bar C}, \quad s_{ab}\,\bar B = 0, \quad s_{ab}\, B = \dot {B}\, \bar C - B\, \dot{\bar C},
\end{eqnarray}
\begin{eqnarray}
s_{b}\,x &=& C\,\dot x,\quad~ s_{b}\,p_x = C\,\dot{p_x},\quad~ s_{b}\,t = C\,\dot t,\quad~~ s_{b}\,p_t = C\; \dot p_t,\quad
s_{b}\,E = \frac{d}{d\,\tau}\,( C\, E),\nonumber\\
s_{b}\,\bar C &=& i\, B,\quad s_{b}\, C = C\, \dot C, \quad s_{b}\, B = 0, \quad s_{b}\, \bar B = \dot {\bar B}\, C - \bar B\, \dot C.
\end{eqnarray}
The above { \it fermionic} symmetry transformations are off-shell nilpotent and absolutely anticommuting provided the whole theory is considered on
a submanifold of the space of quantum variables where the CF-type restriction: $B + \bar B + i\, (\bar C\, \dot C - \dot {\bar C}\, C) = 0$ is satisfied.
The existence of the {\it above} CF-type restriction leads to the derivation of the coupled (but equivalent) Lagrangians (i.e. $L_B$ and $L_{\bar B}$) as follows:
\begin{eqnarray}
L_B = L_f + s_b\,s_{ab}\Big[\frac{i\,E^2}{2} - \frac{\bar C\, C}{2}\Big],\nonumber\\
L_{\bar B} = L_f - s_{ab}\,s_{b}\Big[\frac{i\,E^2}{2} - \frac{\bar C\, C}{2}\Big].
\end{eqnarray}
We point out that the terms inside the square brackets are {\it same} as in Eq. (10) for the BRST analysis of the {\it classical}
gauge symmetry transformations (4). Furthermore, in contrast to the {\it unique} (anti-)BRST invariant Lagrangian [cf. Eq. (11)] (corresponding
to the {\it classical} gauge symmetry transformations), we have obtained here a set of coupled (but equivalent) (anti-)BRST invariant
Lagrangians in Eq. (34). This has happened because of the fact that the CF-type restriction ($b + \bar b = 0 $) is {\it trivial}
in the case of the {\it former} while it is a {\it non-trivial} restriction [$B + \bar B + i\, (\bar C \, \dot C - \dot {\bar C} \, C) = 0 $] in
the context of the {\it latter}.
One can readily compute the operation of $s_{(a)b}$ on the quantities in the square brackets of Eq. (34). In the full blaze of their glory,
the coupled (but equivalent) Lagrangians $L_B$ and $L_{\bar B}$ are as follows\footnote{ It will be worthwhile to mention {\it here} that the {\it form} of the gauge-fixing and Faddeev-Popov ghost terms is {\it same} as in the cases of NSUSY (i.e. scalar) and SUSY (i.e. spinning) relativistic particles [23, 24].}
\begin{eqnarray}
L_B &=& L_f + B\,\Big[E\,\dot E -i\, (2\, \dot{\bar C}\, C + {\bar C}\,\dot C)\Big]+ \frac{B^2}{2} \nonumber\\
&&- i\,E\,\dot E\,\dot{\bar C}\,C - \,i\,E^2\,\dot{\bar C}\,\dot C - \dot{\bar C}\,{\bar C}\,\dot C\,C, \nonumber\\
L_{\bar B} &=& L_f - \bar B\,\Big[E\,\dot E - i\,(2\, {\bar C}\,\dot C + \dot{\bar C}\,C)\Big]+\frac{{\bar B}^2}{2} \nonumber\\
&&- i\,E\,\dot E\,{\bar C}\,\dot C - \,i\,E^2\,\dot{\bar C}\,\dot C - \dot{\bar C}\,{\bar C}\,\dot C\,C,
\end{eqnarray}
where the subscripts $B$ and $\bar B$ on the Lagrangians are appropriate because $L_B$ depends {\it uniquely} on the Nakanishi-Lautrup
auxiliary variable $B$ (where ${\bar B}$ is {\it not} present at all). Similarly, the Lagrangian $L_{\bar B}$ is {\it uniquely} dependent on $\bar B$.
They are coupled because the EL-EOMs with respect to $B$ and $\bar B$ from $L_B$ and $L_{\bar B}$, respectively, yield
\begin{eqnarray}
B = -\,E\,\dot E +2\,i\,\dot{\bar C}\,C + i\,\bar C\,\dot C, \qquad \bar B = E\,\dot E - 2\,i\,{\bar C}\,\dot C - i\,\dot{\bar C}\, C,
\end{eqnarray}
which lead to the deduction of the CF-type restrictions: $B + \bar B + i \,(\bar C \, \dot C - \dot {\bar C} \, C) = 0 $. Furthermore, the condition
$L_B \equiv L_{\bar B}$ also demonstrates the existence of the CF-type restriction: $B + \bar B + i \,(\bar C \, \dot C - \dot {\bar C} \, C) = 0 $
on our theory (cf. Appendix A below).
At this stage, we are in the position to study the (anti-)BRST symmetries of the Lagrangians $L_B$ and $L_{\bar B}$. It is straightforward
to note that we have the following:
\begin{eqnarray}
&&s_b\,L_B = \frac{d}{d\,\tau}\Big[C\,L_f + B^2\,C - i\,B\,\bar C\,\dot C\,C + E\,\dot E\,B\,C + E^2\,B\,\dot C \Big],
\end{eqnarray}
\begin{eqnarray}
&&s_{ab}\,L_{\bar B} = \frac{d}{d\,\tau}\Big[\bar C\,L_f + {\bar B}^2\,\bar C
- i\,\bar B\,\dot{\bar C}\,\bar C\,C - E\,\dot E\,\bar B\,\bar C - E^2\,\bar B\,\dot{\bar C} \Big].
\end{eqnarray}
The above observations demonstrate that the action integrals $S_1 = \int_{-\infty}^{\infty} d\,\tau\,L_B$ and $S_2 = \int_{-\infty}^{\infty} d\,\tau\,
L_{\bar B}$ remain invariant under the SUSY-type (i.e. fermionic) off-shell nilpotent, continuous and infinitesimal (anti-)BRST symmetry transformations for
the physical variables that vanish off at $\tau = \pm \infty$.
At this crucial juncture, we establish the {\it equivalence} of the coupled Lagrangian $L_B$ and $L_{\bar B}$ w.r.t the (anti-)BRST symmetry transformations
$[s_{(a)b}]$. In this context, we apply $s_{ab}$ on $L_B$ and $s_{b}$ on $L_{\bar B}$ to obtain the following
\begin{eqnarray}
s_{ab}\,L_{B} &=& \frac{d}{d\,\tau}\,\Big[\bar C\,L_f
+ E\,\dot E\,(i\,\dot{\bar C}\,\bar C\,C + B\,\bar C) + E^2\,(i\,\dot{\bar C}\,\bar C\,\dot C + B \,\dot{\bar C}) \nonumber\\
&+& {B}^2\,\bar C + i\,(2\,B - \bar B)\,\dot{\bar C}\,\bar C\,C \Big] \nonumber\\
&+& \big[B+ \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big]\,( 2\,i\,\dot{\bar C}\,\bar C\,\dot C
- 2\,B\,\dot{\bar C}- E\,\dot E\,\dot{\bar C} + i\,\ddot{\bar C}\,\bar C\,C ) \nonumber\\
&-& \frac{d}{d\,\tau}\big[B+ \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big]\, \big[ B\,\bar C + E^2\,\dot{\bar C} \big],
\end{eqnarray}
\begin{eqnarray}
s_b\,L_{\bar B} &=& \frac{d}{d\,\tau}\,\Big[C\,L_f + E\,\dot E\,(i\,\bar C\,C
\,\dot C - \bar B\,C) + E^2\,(i\,\dot{\bar C}\,C\,\dot C - \bar B \,\dot C) \nonumber\\
&+& {\bar B}^2\,C - i\,(2\,\bar B - B)\,\bar C\,C\,\dot C \Big] \nonumber\\
&+& \big[B+ \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big]\,( - 2\,i\,
\dot{\bar C}\,C\,\dot C - 2\,\bar B\,\dot C + E\,\dot E\,\dot C + i\,\bar C\,\ddot C\,C ) \nonumber\\
&+& \frac{d}{d\,\tau}\big[B+ \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big]\, \big[+ E^2\,\dot C - \bar B\,C \big],
\end{eqnarray}
which demonstrate that the coupled Lagrangians $L_B$ and $L_{\bar B}$ (and corresponding action integrals) respect {\it both}
(i.e. BRST and anti-BRST) symmetry transformations {\it together} provided the whole theory is considered on a supermanifold in the Hilbert space
of {\it quantum} variables where the CF-type restriction: $B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C) = 0 $ is satisfied. It should
be recalled that, under the {\it latter} restriction, we {\it also} have the absolute anticommutativity property (i.e. $\{s_b, s_{ab}\} = 0$ of the (anti-)BRST
symmetry transformations.
We end this section with the following key comments. First, the properties of the off-shell nilpotency and absolute anticommutativity are {\it sacrosanct} in
the realm of BRST approach to gauge and/or diffeomorphism invariant theories. Second, physically, the first property (i.e. off-shell nilpotency)
implies that these {\it fermionic} symmetry transformations are supersymmetric-type as they transform bosonic variables to fermionic
variables and vice-versa.
Third, the property of the absolute anticommutativity encodes the linear independence of the BRST and anti-BRST symmetry transformations.
Fourth, the absolute anticommutativity property owes its origin to the existence of the CF-type restrictions which are connected with the concepts
of gerbes [7, 8]. Fifth, as the {\it classical} gauge theory is characterized by the {\it first-class} constraints, in exactly similar fashion, the
{\it quantum} gauge and/or diffeomorphism [i.e. (anti-)BRST]
invariant theories are characterized by the existence of the CF-type restrictions {\it within} the ambit of
BRST formalism. Sixth, the coupled Lagrangians $L_B$ and $L_{\bar B}$ are {\it equivalent} because both of them respect BRST and
anti-BRST symmetry transformations as is clear from Eqs. (37)-(40) provided the {\it whole} theory is considered on the submanifold of the total Hilbert
space of the quantum variables where the CF-type restriction: $B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C) = 0 $ is satisfied.\\
\section{Quantum Off-Shell Nilpotent (Anti-)BRST Symmetries of the Other Variables: ACSA}
In this section, we derive the nilpotent (anti-)BRST symmetry transformations $[s_{(a)b}]$ for {\it all} the {\it other} variables
[cf. Eqs. (32), (33)] {\it besides} the phase space variables $(x, p_x, t, p_t)$ whose (anti-)BRST symmetries have already been derived in Sec. 3
by exploiting the theoretical potential of MBTSA. To achieve the above goal, we exploit the ideas behind ACSA to BRST
formalism [25-29]. In this context, first of all, we focus on the derivation of the BRST symmetry transformations: $s_b\, B = 0, \,
s_b\,\bar B = \dot{\bar B}\,C - \bar B \, \dot C, \, s_b\,C = C\,\dot C, \, s_b\,E = \dot E\,C + E\,\dot C$
[cf. Eq. (33)]. For this purpose, we generalize the {\it ordinary} variables $[B(\tau),\, \bar B(\tau),\, C(\tau), \,E(\tau)]$ onto
a $(1, 1)$-dimensional {\it anti-chiral} super sub-manifold as follows
\begin{eqnarray}
B(\tau) \quad &\longrightarrow& \quad {\cal B}(\tau, \bar\theta) = B(\tau) + \bar\theta\,f_1(\tau), \nonumber\\
\bar B(\tau) \quad &\longrightarrow& \quad {\cal \bar B}(\tau, \bar\theta) = \bar B(\tau) + \bar\theta\,f_2(\tau), \nonumber\\
C(\tau) \quad &\longrightarrow& \quad F(\tau, \bar\theta) = C(\tau) + \bar\theta\,b_1(\tau), \nonumber\\
E(\tau) \quad &\longrightarrow& \quad \Sigma(\tau, \bar\theta) = E(\tau) + \bar\theta\,f_3(\tau),
\end{eqnarray}
where we note that $(f_1, f_2, f_3)$ are the {\it fermionic} secondary variables and $b_1(\tau)$ is the {\it bosonic}
secondary variable because of the fermionic $({\bar\theta}^2 = 0)$ nature of the Grassmannian variable $\bar\theta$
which characterizes the {\it anti-chiral} super sub-manifold (along with the {\it bosonic} evolution parameter $\tau$).
It is elementary to note that the observation $s_b\,B = 0$ implies the following super expansion (in view of the fact that
$\partial_{\bar \theta} \,\leftrightarrow \, s_b $), namely;
\begin{eqnarray}
{\cal B}^{(b)}\,(\tau, \bar\theta) = B(\tau) + \bar\theta\,(0) \equiv B(\tau) + \bar\theta\,(s_b \, B),
\end{eqnarray}
where the superscript $(b)$ on the {\it anti-chiral} supervariable ${\cal B}\,(\tau, \bar\theta)$ denotes that the coefficient
of $\bar \theta$ yields the BRST symmetry transformation: $ s_b \, B = 0$ due to the trivial equality: ${\cal B}\,(\tau, \bar\theta) = B\,(\tau)$
which emerges from the observation that the Nakanishi-Lautrup auxiliary variable $B(\tau)$ is a BRST invariant quantity [cf. Eq. (33)]. In other
words, we have found out that the secondary variable $f_1 \, (\tau) = 0$ in the super expansions (41).
At this stage, we find {\it other} non-trivial BRST invariant quantities for the derivation of the secondary variables: $b_1, \, f_2,\,f_3$ of Eq. (41).
We observe that\footnote{We have specifically
taken here $s_b\,(C\, \dot x) = 0$ for our purpose. However, one can take the general expression: $s_b\,S = C \, \dot S \,(S = x,\,p_x,\,t
,\,p_t)$ for the derivation of $b_1 (\tau) = C \, \dot C$.}: $ s_b \, ( C \, \dot {x}) = 0, \, s_b \,[\dot{\bar B}\, C - \bar B \, \dot C] = 0, \, s_b\, [E \, \dot C + \dot E \, C] = 0$.
The basic tenets of the ACSA to BRST formalism requires that the quantities in the square brackets have to be {\it independent} of the Grassmannian variables
$\bar \theta$ when they are generalized onto a (1, 1)-dimensional super sub-manifold, namely;
\begin{eqnarray}
&F (\tau, \bar \theta) \, \dot{X}^{(h,\, ac)} \,(\tau, \bar \theta) = C \,(\tau )\, \dot x (\tau )\nonumber\\
&{\dot{\bar{\cal B}}}(\tau, \bar\theta) \,F (\tau, \bar \theta) - {\bar {\cal B}}(\tau, \bar\theta) \, \dot{F} (\tau, \bar \theta) = \dot{\bar B}
(\tau ) \, C (\tau) - {\bar B}(\tau ) \, \dot C (\tau) \nonumber\\
&{\Sigma }(\tau, \bar\theta) \,\dot F (\tau, \bar \theta) + \dot{\Sigma}(\tau, \bar\theta) \, {F} (\tau, \bar \theta) = E \,
(\tau ) \, \dot C (\tau) + \dot E(\tau) \, C (\tau),
\end{eqnarray}
where ${X}^{(h,\,ac)}$ is the {\it anti-chiral} limit of the full expansion of ${X}^{(h)} \,(\tau,
\theta, \bar \theta)$ obtained after the application of HC [cf. Eq. (28)], namely;
\begin{eqnarray}
{X}^{(h)} \,(\tau, \theta, \bar \theta) = x (\tau) + \theta \, (\bar C\, \dot x) + \bar
\theta \,(C \, \dot {x}) + \theta\, \bar \theta \,[i\,B\,\dot x - \bar C\,\dot C\,\dot x - \bar C\,C\,\ddot x],
\end{eqnarray}
which has been obtained [cf. Eq. (28)] in Sec. 3 using the theoretical strength of MBTSA. In other words, from the top entry of Eq. (43),
we have the following restrictions:
\begin{eqnarray}
&F (\tau, \bar \theta) \, \dot{X}^{(h, \, ac)} \,(\tau, \bar \theta) = C \,(\tau )\, \dot x(\tau )\nonumber\\
&\Longrightarrow \big[C\,(\tau) + \bar \theta \,b_1\,(\tau) \big]\, \big[ \dot x + \bar \theta \,(\dot C \,
\dot x + C \, \ddot x)\big] = C \,(\tau )\, \dot x(\tau ).
\end{eqnarray}
From the above relationship we obtain $b_1 (\tau) = C\,\dot C$. Thus, we have the following
\begin{eqnarray}
F^{(b)} (\tau, \bar \theta) = C\,(\tau) + \bar \theta \,(C\,\dot C) \equiv C\,(\tau) + \bar \theta \,(s_b\,C),
\end{eqnarray}
where the superscript $(b)$ on the l.h.s. of the supervariable denotes that the coefficient of $\bar\theta$ is nothing but the BRST symmetry transformation
$s_b\,C$. We have to use the above super expansion in the second entry from the top in (43) to obtain the following:
\begin{eqnarray}
{\dot{\bar {\cal B}}}(\tau, \bar\theta)\,F^{(b)}(\tau, \bar\theta) - \bar{\cal B}(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau, \bar\theta) = \dot{\bar B}(\tau)\,C(\tau)
- {\bar B}(\tau)\,{\dot C}(\tau).
\end{eqnarray}
In other words, we have the following equality
\begin{eqnarray}
[\dot{\bar B} + \bar\theta\,{\dot f}_2(\tau)]\,[C(\tau) + \bar\theta\,(C\,\dot C)] - [\bar B(\tau) + \bar\theta\,f_2(\tau)]\,[\dot C(\tau) + \bar\theta
\,(C\,\ddot C)] \nonumber\\
= \dot{\bar B}(\tau)\,C(\tau) - \bar B(\tau)\,\dot C(\tau),
\end{eqnarray}
which yields the following condition on the secondary variable $f_2$, namely;
\begin{eqnarray}
{\dot f}_2\,C - f_2\,\dot C - \dot{\bar B} \,\dot C \, C + \bar B \,\ddot C \,C = 0.
\end{eqnarray}
It is straightforward to note that $f_2 = \dot{\bar B}\,C - \bar B\,\dot C$ satisfies the above condition in a precise manner. We point
out that the last entry (from the top) of Eq. (43) can be re-written, in view of our the super expansion in Eq. (46), as follows:
\begin{eqnarray}
\Sigma(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau, \bar\theta) + \dot\Sigma(\tau, \bar\theta)\,F^{(b)}(\tau, \bar\theta) = E(\tau)\,\dot C(\tau)
+ \dot E(\tau)\,C(\tau).
\end{eqnarray}
The substitutions of expansions from (41) and (46) lead to the following condition on the secondary variable $f_3(\tau)$ [present in the
expansion of $\Sigma(\tau, \bar\theta)$], namely;
\begin{eqnarray}
f_3\,\dot C + {\dot f}_3 \,C - E\,\ddot C \, C - \dot E\,\dot C\,C = 0,
\end{eqnarray}
which is satisfied by the choice $f_3 = E\,\dot C + \dot E\,C$. Hence, we have the following super expansions [with the BRST symmetry
transformations (33) as input], namely;
\begin{eqnarray}
{\bar {\cal B}}^{(b)}(\tau, \bar\theta) &=& \bar B(\tau) + \bar\theta\,(\dot{\bar B}\,C - \bar B\,\dot C) \equiv \bar B(\tau) + \bar\theta\,(s_b\,\bar B),
\nonumber\\
\Sigma^{(b)}(\tau, \bar\theta) &=& E(\tau) + \bar\theta\,(E\,\dot C + \dot E\,C) \equiv E(\tau) + \bar\theta\,(s_b\,E),
\end{eqnarray}
where the coefficients of $\bar\theta$ (in view of $\partial_{\bar\theta} \leftrightarrow s_b$) are the BRST symmetry transformations (33).
For the convenience of the readers, we have performed the
explicit computations of $f_3 = E\,\dot C + \dot E\,C$ and $f_2 = \dot {\bar B} \, C - \bar B \, \dot C$
in our Appendix B. It is clear that we have already computed the BRST
transformations $s_b\,B = 0,\, s_b\,C = C\,\dot C, \, s_b\,\bar B = \dot{\bar B}\,C - \bar B \, \dot C, \, s_b\,E = E\,\dot C + \dot E\,C$
by exploiting the virtues of ACSA in Eqs. (42), (46) and (52).
We concentrate now on the derivation of the anti-BRST symmetry transformations (32) by exploiting the theoretical strength of ACSA to
BRST formalism. It is obvious that, in Sec. 3, we have already computed $s_{ab}\,S = \bar C\,\dot S \;(S = x, p_x, t, p_t)$ and $s_{ab}\,C = i\,\bar B$
by exploiting MBTSA to BRST formalism. Our objective in the present part of our section is to derive: $s_{ab}\,\bar B = 0, \, s_{ab}\,\bar C = \bar C\,\dot{\bar C},\,
s_{ab}\,B = \dot B\,\bar C - B\,\dot{\bar C},\, s_{ab}\,E = E\,\dot{\bar C} + \dot E\,\bar C$ by exploiting ACSA to BRST formalism. In this
context, first of all, we generalize the {\it ordinary} variables onto a $(1, 1)$-dimensional {\it chiral} super sub-manifold as
\begin{eqnarray}
\bar B(\tau) \qquad &\longrightarrow& \qquad \bar{\cal B}(\tau, \theta) = \bar B(\tau) + \theta\,{\bar f}_1(\tau), \nonumber\\
B(\tau) \qquad &\longrightarrow& \qquad {\cal B}(\tau, \theta) = B(\tau) + \theta\,{\bar f}_2(\tau), \nonumber\\
\bar C(\tau) \qquad &\longrightarrow& \qquad {\bar F}(\tau, \theta) = \bar C(\tau) + \theta\,{\bar b}_1(\tau), \nonumber\\
E(\tau) \qquad &\longrightarrow& \qquad \Sigma(\tau, \theta) = E(\tau) + \theta\,{\bar f}_3(\tau),
\end{eqnarray}
where $({\bar f}_1,\,{\bar f}_2, {\bar f}_3)$ are the {\it fermionic} secondary variables, ${\bar b}_1(\tau)$ is a
{\it bosonic} secondary variable and the above $(1, 1)$-dimensional {\it chiral} super sub-manifold is parameterized by $(\tau, \theta)$.
It is straightforward to note that $s_{ab}\,\bar B = 0$ implies that: ${\bar {\cal B}}(\tau, \theta) = B(\tau)$ and, as a consequence,
we have ${\bar f}_1(\tau) = 0$ which leads to
\begin{eqnarray}
{\bar{\cal B}}^{(ab)}(\tau, \theta) = \bar B(\tau) + \theta\,(0) \equiv \bar B(\tau) + \theta\,(s_{ab}\,\bar B),
\end{eqnarray}
where the superscript $(ab)$ on the {\it chiral} supervariable denotes that
we have obtained $s_{ab}\,\bar B = 0$ as the coefficient of $\theta$. The other useful and interesting anti-BRST invariant quantities of
our interest [cf. Eq. (32)] are:
\begin{eqnarray}
s_{ab}\,[\dot B\,\bar C - B\,\dot{\bar C}] = 0, \qquad s_{ab}\,[E\,\dot{\bar C} + \dot E \,\bar C] = 0, \qquad s_{ab}\,[\bar C\,\dot x] = 0.
\end{eqnarray}
The quantities in the square brackets can be generalized onto the $(1, 1)$-dimensional {\it chiral} super sub-manifold. Following the
fundamental requirement(s) of ACSA to BRST formalism, {\it these} quantities must be independent of the Grassmannian variable $\theta$.
In other words, we have the following restrictions on the {\it chiral} supervariables
\begin{eqnarray}
\dot{\cal B}(\tau, \theta)\,\bar F(\tau, \theta) - {\cal B}(\tau, \theta)\,\dot{\bar F}(\tau, \theta) &=& \dot B(\tau)\,\bar C(\tau) - B(\tau)\,
\dot{\bar C}(\tau), \nonumber \\
\Sigma(\tau, \theta)\,\dot{\bar F}(\tau, \theta) + \dot\Sigma(\tau, \theta)\,\bar F(\tau, \theta) &=& E(\tau)\,\dot{\bar C}(\tau) + \dot E(\tau)\,\bar C(\tau),
\nonumber\\
\bar F(\tau, \theta)\,{\dot X}^{(h, c)}(\tau, \theta) &=& \bar C(\tau)\,\dot x(\tau),
\end{eqnarray}
where $X^{(h, c)}(\tau, \theta)$ is the {\it chiral} limit of the super expansion in Eq. (44). In other words, we have the following explicit
expression for the supervariable $X^{(h, c)}(\tau, \theta)$, namely;
\begin{eqnarray}
X^{(h, c)}(\tau, \theta) = x(\tau) + \theta\,(\bar C\,\dot x).
\end{eqnarray}
Taking the expansions from (53) and (57), we find that the {\it last} entry of Eq. (56) yields: ${\bar b}_1(\tau) = \bar C\,\dot{\bar C}$.
Hence, we have obtained the following super expansion
\begin{eqnarray}
{\bar F}^{(ab)}(\tau, \theta) = \bar C(\tau) + \theta\,(\bar C\,\dot{\bar C}) \equiv \bar C(\tau) + \theta\,(s_{ab}\,\bar C),
\end{eqnarray}
where the superscript $(ab)$ on the {\it chiral} supervariable on the l.h.s. denotes that it has been derived after the application
of the anti-BRST invariant restriction in (56). The coefficient $\theta$ is nothing but the anti-BRST symmetry transformation:
$s_{ab}\,\bar C = \bar C\,\dot{\bar C}$. This equation also shows that $\partial_\theta \leftrightarrow s_{ab}$ and it leads to the
anti-BRST symmetry for $\bar C$.
We utilize now the {\it two} top entries of (56) where we use the explicit expansion for ${\bar F}^{(ab)}(\tau, \theta)$ of (58) in the following
restrictions on the supervariables, namely;
\begin{eqnarray}
\dot{\cal B}(\tau, \theta)\,{\bar F}^{(ab)}(\tau, \theta) - {\cal B}(\tau, \theta)\,\dot{\bar F}^{(ab)}(\tau, \theta) &=& {\dot B}(\tau)\,\bar C(\tau)
- B(\tau)\,\dot{\bar C}(\tau), \nonumber \\
\Sigma(\tau, \theta)\,\dot{\bar F}^{(ab)}(\tau, \theta) + \dot\Sigma(\tau, \theta)\,{\bar F}^{(ab)}(\tau, \theta) &=& E(\tau)\,\dot{\bar C}(\tau)
+ \dot E(\tau)\,\bar C(\tau).
\end{eqnarray}
The substitutions from (53) and (58) lead to:
\begin{eqnarray}
\dot{\bar f}_2\,\bar C - \bar f_2\,\dot{\bar C} - \dot B\,\dot{\bar C}\,\bar C + B\,\ddot{\bar C}\,\bar C &=& 0, \nonumber \\
{\bar f}_3\,\dot{\bar C} + \dot{\bar f}_3\,{\bar C} - E\,\ddot{\bar C}\,\bar C - E\,\dot{\bar C}\,\bar C &=& 0.
\end{eqnarray}
It is straightforward, following the theoretical tricks of Appendix B, to find out the solutions for the secondary variables ${\bar f}_2(\tau)$
and ${\bar f}_3(\tau)$ which are as follows:
\begin{eqnarray}
{\bar f}_2(\tau) = \dot B\,\bar C - B\,\dot{\bar C}, \qquad\qquad {\bar f}_3 = E\,\dot{\bar C} + \dot E\,\bar C.
\end{eqnarray}
Substitutions of these secondary variables into the super expansions (53) leads to the determination of the anti-BRST symmetry
transformations for the variables $B(\tau)$ and $E(\tau)$ as the
coefficients of $\theta $ in the following
\begin{eqnarray}
\Sigma^{(ab)} \,(\tau, \theta) = E(\tau) + \theta \, (E\, \dot{\bar C} + \dot E \, \bar C) \,\equiv \, E(\tau) + \theta \, [s_{ab}\, E (\tau)],\nonumber\\
{\cal B}^{(ab)} \, (\tau, \theta) = B (\tau) + \theta \, (\dot B \, \bar C - B \dot {\bar C}) \, \equiv \, B (\tau) + \theta \, [s_{ab}\, B (\tau)],
\end{eqnarray}
where the superscript $(ab)$ on the {\it chiral} supervariable denotes that these supervariables have been obtained after the applications of the anti-BRST
invariant restrictions (56). Moreover, the above observation establishes that: $s_{ab}\Leftrightarrow \partial_{\theta}$ which implies that the
nilpotency $(s_{ab}^2 = 0, \,\partial_{\theta}^2 = 0)$ properties of $s_{ab}$ and $\partial_\theta$ are connected with each-other. Thus, we have
obtained all the anti-BRST symmetry transformations (besides the phase variables) in our Eqs. (54), (58) and (62). This completes our discussion
on the derivation of the off-shell nilpotent and absolutely anticommuting (anti-)BRST symmetry transformations (32) and (33) within the ambit of
ACSA to BRST formalism.\\
\vskip 0.5cm
\section{Symmetry Invariance of the Lagrangians: ACSA}
In this section, we establish the {\it equivalence} of the coupled Lagrangian $L_B$ and $L_{\bar B}$ as far as the (anti-)BRST
symmetry invariance (within the purview of ACSA to BRST formalism) is concerned. We accomplish this objective by generalizing
the {\it ordinary} Lagrangians to their counterpart {\it super} Lagrangians as
\begin{eqnarray}
L_{\bar B} \rightarrow {\tilde L}_{\bar B}^{(c)}(\tau, \theta) &=& {\tilde L}_f^{(c)}(\tau, \theta) - {\bar {\cal B}}^{(ab)}(\tau, \theta)\Big[{\Sigma}^{(ab)}
(\tau, \theta)\,
{\dot {\Sigma}}^{(ab)}(\tau, \theta)
- i\,\big\{2\,{{\bar F}}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta) \nonumber\\
&+& {\dot{\bar F}}^{(ab)}(\tau, \theta)\,{F}^{(ab)}(\tau, \theta)\big \}\Big] + \frac{1}{2}\,{\bar {\cal B}}^{(ab)}(\tau, \theta)\,{\bar {\cal B}}^{(ab)}
(\tau, \theta)
\nonumber\\
&-& i\,{\Sigma}^{(ab)}(\tau, \theta)\,{\Sigma}^{(ab)}(\tau, \theta)\,{\dot{\bar F}}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta) \nonumber\\
&-& i\,{\Sigma}^{(ab)}(\tau, \theta)\,{\dot {\Sigma}}^{(ab)}(\tau, \theta)\,{{\bar F}}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta) \nonumber\\
&-& {\dot{\bar F}}^{(ab)}(\tau, \theta)\,{\bar F}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta)\,F^{(ab)}(\tau, \theta),
\end{eqnarray}
\begin{eqnarray}
L_B \rightarrow {\tilde L}_B^{(ac)}(\tau, \bar\theta) &=& {\tilde L}_f^{(ac)}(\tau, \bar\theta) + {\cal B}^{(b)}(\tau, \bar\theta)
\Big[{\Sigma}^{(b)}(\tau, \bar\theta)\,{\dot {\Sigma}}^{(b)}(\tau, \bar\theta)
- i\,\big\{2\,{\dot{\bar F}}^{(b)}(\tau, \bar\theta)\,F^{(b)}(\tau, \bar\theta) \nonumber\\
&+& {\bar F}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau, \bar\theta)\big\}\Big] + \frac{1}{2}\,{\cal B}^{(b)}(\tau, \bar\theta)\,
{\cal B}^{(b)}(\tau, \bar\theta) \nonumber\\
&-& i\,{\Sigma}^{(b)}(\tau, \bar\theta)\,{\Sigma}^{(b)}(\tau, \bar\theta)\,{\dot{\bar F}}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau, \bar\theta) \nonumber\\
&-& i\,{\Sigma}^{(b)}(\tau, \bar\theta)\,{\dot {\Sigma}}^{(b)}(\tau, \bar\theta)\,{\dot{\bar F}}^{(b)}(\tau, \bar\theta)\,{F}^{(b)}(\tau, \bar\theta) \nonumber\\
&-& {\dot{\bar F}}^{(b)}(\tau, \bar\theta)\,{\bar F}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau, \bar\theta)\,F^{(b)}(\tau, \bar\theta),
\end{eqnarray}
where ${\tilde L}_f^{(c)}$ and ${\tilde L}_f^{(ac)}$ are the generalizations of the first-order Lagrangian $(L_f)$ to its counterpart chiral
and anti-chiral {\it super} Lagrangians as
\begin{eqnarray}
{\tilde L}_f^{(c)}(\tau, \theta) &=& P_x^{(h, c)}(\tau, \theta)\,{\dot X}^{(h, c)}(\tau, \theta) + P_t^{(h, c)}(\tau, \theta)\,
{\dot T}^{(h, c)}(\tau, \theta) \nonumber\\
&-& \frac{\Sigma^{(ab)}(\tau, \theta)}{2}\,\Big[P_x^{(h, c)}(\tau, \theta)\,P_x^{(h, c)}(\tau, \theta)
+ 2\,m\,P_t^{(h, c)}(\tau, \theta)\Big], \nonumber\\
{\tilde L}_f^{(ac)}(\tau, \bar\theta) &=& P_x^{(h, ac)}(\tau, \bar\theta)\,{\dot X}^{(h, ac)}(\tau, \bar\theta) + P_t^{(h, ac)}(\tau, \bar\theta)\,
{\dot T}^{(h, ac)}(\tau, \bar\theta) \nonumber\\
&-& \frac{\Sigma^{(b)}(\tau, \bar\theta)}{2}\,\Big[P_x^{(h, ac)}(\tau, \bar\theta)\,P_x^{(h, ac)}(\tau, \bar\theta) + 2\,m\,P_t^{(h, ac)}(\tau, \bar\theta)\Big],
\end{eqnarray}
where the superscripts $(c)$ and $(ac)$ denote the {\it chiral} and {\it anti-chiral} generalizations
and the rest of the supervariables with superscripts $(b)$ and $(ab)$ have already been explained earlier in Sec. 5. The supervariables
with superscripts $(h, c)$ and $(h, ac)$ are the {\it chiral} and {\it anti-chiral} limits of the super phase
variables $(X^{(h)}, P_x^{(h)}, T^{(h)}, P_t^{(h)})$ that have been obtained after the application of HC. Thus, these are the counterparts of the ordinary
phase variables $(x, \,p_x,\,t,\,p_t)$ and they have been explained in
Sec. 3. In the above equation (65), the {\it super} phase variables with superscript $(h, c)$ and $(h, ac)$ can be expressed in terms
of the {\it generic} supervariable as follows
\begin{eqnarray}
S(\tau) \quad &\rightarrow& \quad {\cal S}^{(h, c)}(\tau, \theta) = S(\tau) + \theta\,[\bar C\,\dot S (\tau)], \nonumber\\
S(\tau) \quad &\rightarrow& \quad {\cal S}^{(h, ac)}(\tau, \bar\theta) = S(\tau) + \bar\theta\,[C\,\dot S (\tau)],
\end{eqnarray}
where the (anti-)chiral supervariables on the l.h.s. stand for the {\it super} phase variables $(X, P_x, T, P_t)$ with the
proper {\it chiral} and {\it anti-chiral} superspace coordinates $(\tau, \theta)$ and $(\tau, \bar\theta)$ as their arguments. The set of
supervariables $(X, P_x, T, P_t)$ are the generalizations of the {\it ordinary} phase variables $(x, \,p_x,\,t,\,p_t)$ to their
{\it (anti-)chiral} counterparts onto the $(1, 1)$-dimensional (anti-)chiral super submanifolds of the {\it general} $(1, 2)$-dimensional
supermanifold. It is straightforward to check that the following is true, namely;
\begin{eqnarray}
\frac{\partial}{\partial\,\theta}\,{\tilde L}_f^{(c)}(\tau, \theta) = \frac{d}{d\,\tau}[\bar C\,L_f] \quad &\Longleftrightarrow& \quad s_{ab}\,L_f =
\frac{d}{d\,\tau}\,[\bar C\,L_f], \nonumber\\
\frac{\partial}{\partial\,\bar\theta}\,{\tilde L}_f^{(ac)}(\tau, \bar\theta) = \frac{d}{d\,\tau}[C\,L_f] \quad &\Longleftrightarrow& \quad s_{b}\,L_f =
\frac{d}{d\,\tau}\,[C\,L_f].
\end{eqnarray}
In other words, we have captured the (anti-)BRST invariance of the first-order Lagrangian $(L_f)$ in view of the mappings:
$s_b \leftrightarrow \partial_{\bar\theta}, \, s_{ab} \leftrightarrow \partial_{\theta}$. Since in the {\it ordinary} space, the
(anti-) BRST symmetry transformations acting on $L_f$ produce the {\it total} derivatives [cf. Eq. (67)], the action
integral $S = \int^{+\infty}_{-\infty}\,d\,\tau\,L_f$ remains invariant under the transformations $s_{(a)b}$.
At this stage, we focus on the (anti-)BRST invariance of the coupled Lagrangians $L_B$ and $L_{\bar B}$ [cf. Eqs. (38), (37)].
We can express {\it these} invariances within the ambit of ACSA (in view of the mappings: $s_b \leftrightarrow \partial_{\bar\theta}, \,
s_{ab} \leftrightarrow \partial_\theta$), namely;
\begin{eqnarray}
\frac{\partial}{\partial\,\theta}\,{\tilde L}_{\bar B}^{(c)}(\tau, \theta) = \frac{d}{d\,\tau}\,\Big[\bar C\,L_f - e\,\dot e\,\bar B\,\bar C
- e^2\,\bar B\,\dot{\bar C} + {\bar B}^2\,\bar C - i\,\bar B\,\dot{\bar C}\,\bar C\,C \Big] = s_{ab}\,L_{\bar B},
\end{eqnarray}
\begin{eqnarray}
\frac{\partial}{\partial\,\bar\theta}\,{\tilde L}_B^{(ac)}(\tau, \bar\theta) = \frac{d}{d\,\tau}\,\Big[C\,L_f + e\,\dot e\,B\,C
+ e^2\,B\,\dot C + B^2\,C - i\,B\,\bar C\,\dot C\,C \Big] = s_b\,L_B,
\end{eqnarray}
where the {\it super} Lagrangians ${\tilde L}_{\bar B}^{(c)}(\tau, \theta)$ and ${\tilde L}_B^{(ac)}(\tau, \bar\theta)$ have been
already quoted in Eqs. (63) and (64). It is interesting to note that the r.h.s. of (68) and (69) are {\it same} as we have found
in the {\it ordinary} space [cf. Eq. (38), (37)]. To prove the {\it equivalence} of the Lagrangians $L_B$ and $L_{\bar B}$ w.r.t.
the (anti-)BRST symmetry transformations [cf. Eqs. (32), (33)] within the purview of ACSA, we generalize the {\it ordinary} Lagrangians
$L_B$ and $L_{\bar B}$ as follows
\begin{eqnarray}
L_B \rightarrow {\tilde L}_B^{(c)}(\tau, \theta) &=& {\tilde L}_f^{(c)}(\tau, \theta) + {\cal B}^{(ab)}(\tau, \theta)
\Big[\Sigma^{(ab)}(\tau, \theta)\,{\dot \Sigma}^{(ab)}(\tau, \theta)
- i\,\{2\,{\dot{\bar F}}^{(ab)}(\tau, \theta)\,F^{(ab)}(\tau, \theta) \nonumber\\
&+& {\bar F}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta)\}\Big] + \frac{1}{2}\,{\cal B}^{(ab)}(\tau, \theta)\,
{\cal B}^{(ab)}(\tau, \theta)
\nonumber \\
&-& i\,\Sigma^{(ab)}(\tau, \theta)\,\Sigma^{(ab)}(\tau, \theta)\,{\dot{\bar F}}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta) \nonumber\\
&-& i\,\Sigma^{(ab)}(\tau, \theta)\,{\dot \Sigma}^{(ab)}(\tau, \theta)\,{\dot{\bar F}}^{(ab)}(\tau, \theta)\,{F}^{(ab)}(\tau, \theta) \nonumber\\
&-& {\dot{\bar F}}^{(ab)}(\tau, \theta)\,{\bar F}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta)\,F^{(ab)}(\tau, \theta),
\end{eqnarray}
\begin{eqnarray}
L_{\bar B} \rightarrow {\tilde L}_{\bar B}^{(ac)}(\tau, \bar\theta) &=& {\tilde L}_f^{(ac)}(\tau, \bar\theta) - {\bar{\cal B}}^{(b)}(\tau,
\bar\theta)\Big[\Sigma^{(b)}(\tau, \bar\theta)\,{\dot \Sigma}^{(b)}(\tau, \bar\theta) - i\,\big\{2\,{{\bar F}}^{(b)}(\tau, \bar\theta)\,
{\dot F}^{(b)}(\tau, \bar\theta)
\nonumber\\
&+& {\dot{\bar F}}^{(b)}(\tau, \bar\theta)\,{F}^{(b)}(\tau, \bar\theta)\big \}\Big] + \frac{1}{2}\,{\bar {\cal B}}^{(b)}(\tau, \bar\theta)\,
{\bar {\cal B}}^{(b)}(\tau, \bar\theta)
\nonumber\\
&-& i\,\Sigma^{(b)}(\tau, \bar\theta)\,\Sigma^{(b)}(\tau, \bar\theta)\,{\dot{\bar F}}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau, \bar\theta)
\nonumber\\
&-& i\,\Sigma^{(b)}(\tau, \bar\theta)\,{\dot \Sigma}^{(b)}(\tau, \bar\theta)\,{{\bar F}}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau,
\bar\theta) \nonumber\\
&-& {\dot{\bar F}}^{(b)}(\tau, \bar\theta)\,{\bar F}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}(\tau, \bar\theta)\,F^{(b)}(\tau, \bar\theta),
\end{eqnarray}
where {\it all} the notations and symbols have already been explained earlier. To find out the result of the operations of $s_b$ on $L_{\bar B}$ and $s_{ab}$
on $L_B$, we observe the following (in view of the mappings: $s_b \leftrightarrow \partial_{\bar\theta},\,s_{ab} \leftrightarrow \partial_{\theta}$), namely;
\begin{eqnarray}
\frac{\partial}{\partial\,\theta}\,{\tilde L}_B^{(c)}(\tau, \theta) &=& \frac{d}{d\,\tau}\,\Big[\bar C\,L_f + E\,\dot E\,(i\,\dot{\bar C}\,
\bar C\,C + B\,\bar C) + E^2\,(i\,\dot{\bar C}\,
\bar C\,\dot C + B \,\dot{\bar C}) \nonumber\\
&+& {B}^2\,\bar C + i\,(2\,B - \bar B)\,\dot{\bar C}\,\bar C\,C\Big] \nonumber\\
&+& \big[B+ \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big]\,(2\,i\,\dot{\bar C}\,\bar C\,\dot C
- E\,\dot E\,\dot{\bar C} - 2\,B\,\dot{\bar C} + i\,\ddot{\bar C}\,\bar C\,C) \nonumber\\
&-& \frac{d}{d\,\tau}\big[B+ \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big]\,( B\,\bar C + E^2\,\dot{\bar C}) \equiv s_{ab}\,L_B,
\end{eqnarray}
\begin{eqnarray}
\frac{\partial}{\partial\,\bar\theta}\,{\tilde L}_{\bar B}^{(ac)}(\tau, \bar\theta) &=& \frac{d}{d\,\tau}\,\Big[C\,L_f
- E\,\dot E\,(i\,\bar C\,\dot C\, C + \bar B\,C)- E^2\,(i\,\dot{\bar C}\,\dot C\, C + \bar B \,\dot C) \nonumber\\
&+& {\bar B}^2\,C + i\,(2\,\bar B - B)\,\bar C\,\dot C\, C \Big] \nonumber\\
&+& \big[B+ \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C)\big]\,\big [i\,\bar C\,\ddot C\,C + 2\,i\,
\dot{\bar C}\,\dot C\, C - 2\,\bar B\,\dot C + E\,\dot E\,\dot C \big ] \nonumber\\
&+& \frac{d}{d\,\tau}\big[i\,(\bar C\,\dot C - \dot{\bar C}\,C) + B + \bar B \big]\,(E^2\,\dot C - \bar B\,C) \equiv s_b\,L_{\bar B},
\end{eqnarray}
within the framework of ACSA. It is self-evident, from the r.h.s. of (72) and (73), that we have the
BRST invariance of $L_{\bar B}$ and anti-BRST invariance of $L_B$ if and only if our whole theory is considered on
the submanifold of the Hilbert space of quantum variables where the
CF-type restriction: $B + \bar B + i\,(\bar C\,\dot C - \dot{\bar C}\,C) = 0$ is satisfied.
We end this section with the following crucial remarks. First of all, we have captured the BRST
and anti-BRST invariance of $L_B$ and $L_{\bar B}$, respectively, in the terminology of ACSA on the (anti-)chiral super submanifolds
[cf. Eqs. (68), (69)]. Second, we have {\it also} demonstrated the anti-BRST invariance of $L_B$ {\it and} BRST invariance of $L_{\bar B}$
in the superspace formalism [cf. Eqs. (72), (73)] where the theoretical techniques of ACSA have played very important roles. Third,
we have also expressed the (anti-)BRST invariance of the first-order Lagrangian $L_f$ in Eq. (67).
Finally, we have proven the {\it equivalence } of $L_B$ and $L_{\bar B}$ within the framework of ACSA in the Eqs. (68), (69), (72) and (73).\\
\section{Nilpotency and Absolute Anticommutativity Properties of the (Anti-)BRST Charges: ACSA}
Our present section is divided into two subsections. In subsection 7.1, we discuss the off-shell nilpotency
and absolute anticommutativity of the conserved (anti-)BRST charges in the {\it ordinary} space. Our subsection 7.2
deals with the {\it above} properties within the realm of ACSA to BRST formalism. In other words, we capture the off-shell
nilpotency and absolute anticommutativity of the conserved fermionic (anti-)BRST charges in the superspace by taking the theoretical inputs from ACSA.
\subsection{Nilpotency and Anticommutativity: Ordinary Space}
The perfect symmetry invariance of $L_{\bar B}$ under the anti-BRST symmetry transformations
[cf. Eq. (38)] {\it and} $L_B$ under the BRST symmetry transformations [cf. Eq. (37)] allow us to compute the Noether
conserved charges by using the standard techniques of Noether's theorem (applied to the
action integrals corresponding to the Lagrangians $L_{\bar B}$ and $L_B$) as
\begin{eqnarray}
Q_{\bar B} &=& \frac{\bar C\,E}{2}\,(p_x^2 + 2\,m\,p_t) + {\bar B}^2\,\bar C - i\,\bar B\,\dot{\bar C}\,\bar C\,C
- \bar B\,E\,\dot E\,\bar C - \bar B\,E^2\,\dot{\bar C} , \nonumber\\
Q_B &=& \frac{C\,E}{2}\,(p_x^2 + 2\,m\,p_t) + B\,E^2\,\dot C - i\,B\,\bar C\,\dot C\,C + B\,E\,\dot E\,C + B^2\,C ,
\end{eqnarray}
where conserved $({\dot Q}_{\bar B} = 0, \, {\dot Q}_B = 0)$ (anti-)BRST charges are denoted by $Q_{(\bar B)B}$.
The conservation law $({\dot Q}_{\bar B} = 0, \, {\dot Q}_B = 0)$ can be proven by using the EL-EOMs derived from the coupled Lagrangians $L_{\bar B}$
and $L_B$. For readers' convenience, we prove the conservation $({\dot Q}_B = 0)$ of the BRST charge by using the
EL-EOMs derived from $L_B$ in our Appendix C.
First of all, we concentrate on the proof of the off-shell nilpotency properties of the (anti-)BRST charges $Q_{(\bar B)B}$.
In this context, we note that the following EL-EOMs w.r.t. the variable $E$ from $L_{\bar B}$ and $L_B$, respectively, yield the following:
\begin{eqnarray}
\dot{\bar B}\,E - i\,E\,\dot{\bar C}\,\dot C + i\,E\,\bar C\,\ddot C - \frac{1}{2}\,(p_x^2 + 2\,m\,p_t) &=& 0, \nonumber\\
\dot B\,E + i\,E\,\dot{\bar C}\,\dot C - i\,E\,\ddot{\bar C}\,C + \frac{1}{2}\,(p_x^2 + 2\,m\,p_t) &=& 0.
\end{eqnarray}
The above equations can be used to recast $Q_{\bar B}$ and $Q_B$ as follows:
\begin{eqnarray}
Q_{\bar B} ^{(1)} &=& E^2\,(\dot{\bar{B}}\,{\bar{C}} - {\bar{B}}\,\dot{\bar{C}} + i\,\dot {\bar C}\,\bar{C}\,\dot C)
- i\,\bar{B}\,\dot{\bar C}\,{\bar{C}}\,C - \bar{B}\,E\,\dot E\, \bar{C} + \bar{B}^2\, \bar{C}, \nonumber\\
Q_B ^{(1)} &=& E^2\,(B\,\dot C - \dot B\,C - i\,\dot {\bar C}\,\dot C\,C) - i\,B\,\bar C\,\dot C\,C + B\,E\,\dot E\, C + B^2\, C,
\end{eqnarray}
Using the following EL-EOMs w.r.t. the variables $C$ and $\bar B$, respectively, from $L_{\bar B}$, namely;
\begin{eqnarray}
&&i\,\bar B\,\dot{\bar C} + 2\,i\,\dot{\bar B}\,\bar C - 3\,i\,E\,\dot E\,\dot{\bar C} - i\,E^2\,\ddot{\bar C} - i\,{\dot E}^2\,\bar C
- i\,E\,\ddot E\,\bar C + \ddot{\bar C}\,\bar C\,C + 2\,\dot{\bar C}\,\bar C\,\dot C = 0, \nonumber\\
&&\bar B = E\,\dot E - i\,(2\,\bar C\,\dot C + \dot{\bar C}\,C),
\end{eqnarray}
we obtain the following {\it exact} and interesting expression for the anti-BRST charge:
\begin{eqnarray}
Q_{\bar B}^{(1)} \quad \longrightarrow \quad Q_{\bar B}^{(2)} &=& E^2\,(\dot{\bar{B}}\,{\bar{C}} - {\bar{B}}\,\dot{\bar{C}}
+ i\,\dot {\bar C}\,\bar{C}\,\dot C) + i\,E^2\,\ddot{\bar C}\,\bar C\,C + 2\,i\,E\,\dot E\,\dot{\bar C}\,\bar C\,C \nonumber\\
&\equiv& s_{ab}\,[i\,E^2\,(\bar C\,\dot C - \dot{\bar C}\,C)].
\end{eqnarray}
At this juncture, we apply the basic principle behind the relationship between the continuous symmetry transformations (e.g. $s_{ab}$)
and its generator $[Q_{\bar B}^{(2)}]$ which implies that:
\begin{eqnarray}
s_{ab}\,Q_{\bar B}^{(2)} = -\,i\,\{Q_{\bar B}^{(2)}, Q_{\bar B}^{(2)}\} = 0 \quad \Rightarrow \quad [Q_{\bar B}^{(2)}]^2 = 0 \quad \Leftrightarrow
\quad s_{ab}^2 = 0.
\end{eqnarray}
Thus, we observe that the off-shell nilpotency $([Q_{\bar B}^{(2)}]^2 = 0)$ of the anti-BRST charge $Q_{\bar B}^{(2)}$ and the anti-BRST symmetry
transformations $(s_{ab})$
are {\it inter-related}. Thus, we have proven the off-shell nilpotency of the anti-BRST charge $Q_{\bar B}^{(2)}$. In exactly
similar fashion, we exploit the following EL-EOMs w.r.t. the variables $\bar C$ and $B$ from the Lagrangian $L_B$
\begin{eqnarray}
&&i\,B\,\dot C + 2\,i\,\dot B\,C + 3\,i\,E\,\dot E\,\dot C + i\,E^2\,\ddot C + i\,{\dot E}^2\,C + i\,E\,\ddot E\,C + \bar C\,\ddot C\,C
+ 2\,\dot{\bar C}\,\dot C\,C = 0, \nonumber\\
&& B = - E\,\dot E + i\,(2\,\dot{\bar C}\,C + \bar C\,\dot C),
\end{eqnarray}
to recast the BRST charge $Q_B^{(1)}$ into another interesting form (i.e. $Q_B^{(2)}$) as
\begin{eqnarray}
Q_B^{(1)} \quad \rightarrow \quad Q_B^{(2)} &=& E^2\,(B\,\dot C - i\,\dot{\bar C}\,\dot C\,C - \dot B\,C) - 2\,i\,E\,\dot E\,\bar C\,\dot C\,C
- i\,E^2\,\bar C\,\ddot C\,C
\nonumber\\
&\equiv& s_b\,[i\,E^2\,(\dot{\bar C}\,C - \bar C\,\dot C)]
\end{eqnarray}
which turns out to be an {\it exact} quantity w.r.t. $s_b$. Thus, we find that we have the following:
\begin{eqnarray}
s_{b}\,Q_{B}^{(2)} = -\,i\,\{Q_{B}^{(2)}, Q_{B}^{(2)}\} = 0 \quad \Rightarrow \quad [Q_{B}^{(2)}]^2 = 0 \quad \Leftrightarrow
\quad s_{b}^2 = 0.
\end{eqnarray}
In other words, we have proven the off-shell nilpotency $([Q_B^{(2)}]^2 = 0)$ of the BRST charge $Q_B^{(2)}$. Once again, we find that off-shell
nilpotency $(s_{b}^2 = 0)$ of the BRST symmetry transformations and the off-shell nilpotency $([Q_{B}^{(2)}]^2 = 0)$ are intertwined an intimate manner.
We now focus on the proof of the absolute anticommutativity property of the BRST charge with the anti-BRST charge and vice-versa.
First of all, let us focus on the BRST charge $Q_{B}^{(2)}$ [cf. Eq. (81)]. Using the CF-type restriction: $B + \bar B + i\,
(\bar C\,\dot C - \dot {\bar C}\, C) = 0$, we can easily check the following transformation:
\begin{eqnarray}
Q_{B}^{(2)} \quad \longrightarrow \quad Q_{B}^{(3)} &=& E^2 \,(\dot{\bar B}\,C - 2\,i\,\dot{\bar C}\,\dot C\,C - \bar B\,\dot C )
- 2\,i\,E\,\dot E\,\bar C\,\dot C\,C
\nonumber\\
&\equiv& s_{ab}\,[i\,E^2\,\dot C\,C].
\end{eqnarray}
In other words we have been able to express the above BRST charge as an {\it exact} form w.r.t. the anti-BRST symmetry transformations $(s_{ab})$.
This is an interesting observation because using the relationship between the continuous symmetry transformations and their generators, we can
obtain the following from (83), namely;
\begin{eqnarray}
s_{ab}\,Q_B^{(3)} = -i\,\{Q_B^{(3)}, Q_{\bar B}^{(3)}\} = 0 \quad \Leftrightarrow \quad s_{ab}^2 = 0.
\end{eqnarray}
Thus we have been able to demonstrate that the absolute anticommutativity of the BRST charge {\it with} the anti-BRST charge is connected with
the off-shell nilpotency $(s_{ab}^{2} = 0)$ of the anti-BRST symmetry transformation $(s_{ab})$. In exactly similar fashion, we can have a different form
of the anti-BRST charge $Q_{\bar B}^{(2)}$ [cf. Eq. (78)] by using the CF-type restriction: $B + \bar B + i\,
(\bar C\,\dot C - \dot {\bar C}\, C) = 0$. In other words, we have the following interesting transformation:
\begin{eqnarray}
Q_{\bar B}^{(3)} \quad \longrightarrow \quad Q_{\bar B}^{(3)} &=& E^2\,(B\,\dot{\bar C} + 2\,i\,\dot{\bar C}\,\bar C\,\dot C - \dot B\,\bar C ) +
2\,i\,E\,\dot E\,\dot{\bar C}\,\bar C\,C \nonumber\\
&\equiv& s_b\,[i\,E^2\, \dot{\bar C}\,\bar C].
\end{eqnarray}
It is straightforward to note that we have the following relationship:
\begin{eqnarray}
s_b\,Q_{\bar B}^{(3)} = -\,i\,\{Q_{\bar B}^{(3)}, Q_B^{(3)}\} = 0 \quad \Leftrightarrow \quad s_b^2 = 0.
\end{eqnarray}
In other words, we point out that the absolute anticommutativity of the anti-BRST charge {\it with} the BRST
charge is intimately connected with the off-shell nilpotency $(s_b^2 = 0)$ of the BRST symmetry transformations
$(s_b)$. This completes our discussions on the off-shell nilpotency and absolute anticommutativity of
the conserved (anti-)BRST charges in the {\it ordinary} space. In a subtle manner, the observations
in (83) and (85) prove the validity of the CF-type restriction: $B + \bar B + i\,
(\bar C\,\dot C - \dot {\bar C}\, C) = 0$ {\it on} our theory.
\subsection{Nilpotency and Anticommutativity: ACSA}
The key observations of subsection $7.1$ can be translated into the {\it superspace} by using the basic
terminology of ACSA. Keeping in our mind the mappings $\partial_{\bar\theta} \leftrightarrow s_b, \,
\partial_{\theta} \leftrightarrow s_{ab}$, we note that the (anti-)BRST charges $Q_{(\bar B)B}$ [cf.
Eqs. (78), (81)] can be expressed as:
\begin{eqnarray}
Q_{\bar B} = \frac{\partial}{\partial\,\theta}\,\Big[i\,E^{(ab)}(\tau, \theta)\,E^{(ab)}(\tau, \theta)\,
\big\{{\bar F}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta) - \dot{\bar F}^{(ab)}(\tau, \theta)\,F^{(ab)}
(\tau, \theta)\big\}\Big] \nonumber\\
= \int d\,\theta \,\Big[i\,E^{(ab)}(\tau, \theta)\,E^{(ab)}(\tau, \theta)\,
\big\{{\bar F}^{(ab)}(\tau, \theta)\,{\dot F}^{(ab)}(\tau, \theta) - \dot{\bar F}^{(ab)}(\tau, \theta)\,F^{(ab)}
(\tau, \theta)\big\}\Big]
\end{eqnarray}
\begin{eqnarray}
Q_{B} = \frac{\partial}{\partial\,\bar\theta}\,\Big[i\,E^{(b)}(\tau, \bar\theta)\,E^{(b)}(\tau, \bar\theta)\,
\big\{\dot{\bar F}^{(b)}(\tau, \bar\theta)\,{F}^{(b)}(\tau, \bar\theta) - {\bar F}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}
(\tau, \bar\theta)\big\}\Big] \nonumber\\
= \int d\,\bar\theta \,\Big[i\,E^{(b)}(\tau, \bar\theta)\,E^{(b)}(\tau, \bar\theta)\,
\big\{\dot{\bar F}^{(b)}(\tau, \bar\theta)\,{F}^{(b)}(\tau, \bar\theta) - {\bar F}^{(b)}(\tau, \bar\theta)\,{\dot F}^{(b)}
(\tau, \bar\theta)\big\}\Big].
\end{eqnarray}
It is straightforward to observe that we have the following:
\begin{eqnarray}
\frac{\partial}{\partial\,\theta}\,Q_{\bar B} = 0 \quad &\Leftrightarrow& \quad s_{ab}\,Q_{\bar B} = 0 \quad \Leftrightarrow
\quad Q_{\bar B}^2 = 0 \quad \Leftrightarrow \quad \partial_\theta^2 = 0, \nonumber\\
\frac{\partial}{\partial\,\bar\theta}\,Q_{B} = 0 \quad &\Leftrightarrow& \quad s_{b}\,Q_{B} = 0 \quad \Leftrightarrow
\quad Q_B^2 = 0 \quad \Leftrightarrow \quad \partial_{\bar\theta}^2 = 0.
\end{eqnarray}
Thus, the off-shell nilpotency of the (anti-)BRST charges is connected with the nilpotency
$(\partial_\theta^2 = 0,\,\partial_{\bar\theta}^2 = 0)$ of the translational generators $(\partial_\theta,
\partial_{\bar\theta})$ along the Grassmannian directions of the {\it chiral} and {\it anti-chiral} $(1, 1)$-dimensional
super sub-manifolds. This observation is consistent with our discussion of the nilpotency property
in the {\it ordinary} space if we remember the mappings: $s_b \leftrightarrow \partial_{\bar\theta},\,s_{ab}
\leftrightarrow \partial_\theta$ [14-16].
As far as the absolute anticommutativity property is concerned, we note that the expressions of the (anti-)BRST
charges in (85) and (83) can be translated into the {\it superspace} where we can exploit the theoretical tools of
ACSA. To accomplish this goal, we keep in our knowledge the mappings: $s_b \leftrightarrow \partial_{\bar\theta},\,s_{ab}
\leftrightarrow \partial_\theta$ to recast the expressions (85) and (83) as:
\begin{eqnarray}
Q_{\bar B}^{(3)} &=& \frac{\partial}{\partial\,\bar\theta}\,\Big[i\,E^{(b)}(\tau, \bar\theta)\,E^{(b)}(\tau, \bar\theta)
\,\dot{\bar F}^{(b)}(\tau, \bar\theta)\,{\bar F}^{(b)}(\tau, \bar\theta)\Big] \nonumber\\
&\equiv& \int d\,\bar\theta \,\Big[i\,E^{(b)}(\tau, \bar\theta)\,E^{(b)}(\tau, \bar\theta)
\,\dot{\bar F}^{(b)}(\tau, \bar\theta)\,{\bar F}^{(b)}(\tau, \bar\theta)\Big],\nonumber\\
Q_B^{(3)} &=& \frac{\partial}{\partial\,\theta}\,\Big[- i\,E^{(ab)}(\tau, \theta)\,E^{(ab)}(\tau, \theta)
\,{\dot F}^{(ab)}(\tau, \theta)\,{F}^{(ab)}(\tau, \theta)\Big] \nonumber\\
&\equiv& \int d\,\theta\,\Big[- i\,E^{(ab)}(\tau, \theta)\,E^{(ab)}(\tau, \theta)
\,\dot{F}^{(ab)}(\tau, \theta)\,{F}^{(ab)}(\tau, \theta)\Big].
\end{eqnarray}
It is now straightforward to check that the following are true, namely;
\begin{eqnarray}
\partial_{\bar\theta}\,Q_{\bar B}^{(3)} = 0 \quad &\Leftrightarrow& \quad s_b\,Q_{\bar B}^{(3)} = 0 \quad \Leftrightarrow \quad
\{Q_{\bar B}^{(3)}, Q_{B}^{(3)}\} = 0 \quad \Leftrightarrow \quad \partial_{\bar\theta}^2 = 0, \nonumber\\
\partial_{\theta}\,Q_{B}^{(3)} = 0 \quad &\Leftrightarrow& \quad s_{ab}\,Q_{B}^{(3)} = 0 \quad \Leftrightarrow \quad
\{Q_{B}^{(3)}, Q_{\bar B}^{(3)}\} = 0 \quad \Leftrightarrow \quad \partial_{\theta}^2 = 0,
\end{eqnarray}
which establishes the fact that the ACSA to BRST formalism distinguishes between the {\it two} types
of absolute anticommutativity properties. In other words, we observe that the absolute anticommutativity of the BRST charge {\it with}
the anti-BRST charge is connected with the nilpotency $(\partial_{\theta}^2 = 0)$ of the translational generator $(\partial_\theta)$
along the Grassmannian direction of the (1, 1)-dimensional {\it chiral} super sub-manifold. On the contrary, the absolute
anticommutativity of the anti-BRST
charge {\it with} the BRST charge is connected with the nilpotency $(\partial^2_{\bar\theta} = 0)$ of the
translational generator $(\partial_{\bar \theta})$ along
the Grassmannian direction of the (1, 1)-dimensional {\it anti-chiral} super sub-manifold.
\section{Conclusions}
In our present endeavor, we have purposely taken a reparameterization invariant NR and NSUSY system so that we
could discuss theoretical aspects that are different from our earlier works on the NSUSY relativistic
{\it scalar} and SUSY relativistic {\it spinning} particles [23, 24]. We have demonstrated, however, in our present
investigation that $(i)$ the CF-type restriction, and $(ii)$ the sum of gauge-fixing and Faddeev-Popov ghost
terms are {\it same} for our present NR and NSUSY system as have been shown by us for the relativistic particles (in
our earlier works [23, 24]). The above observations are interesting results of our present investigation which
establish the {\it universality} of the (anti-)BRST invariant CF-type restriction for the 1D diffeomorphism invariant
(i.e. reparameterization) theories.
The CF-type restriction(s) are the hallmark of a {\it quantum} theory that is BRST-{\it quantized}. In fact, for a
D-dimensional diffeomorphism invariant theory, it has been shown [22, 33] that the {\it universal} CF-type
restrictions for a BRST-{\it quantized} theory is: $B_{\mu} + {\bar B}_\mu + i\, ({\bar C}^{\rho}\,\partial_\rho\,C_\mu
+ {C}^{\rho}\,\partial_\rho\, \bar C_\mu) = 0$ where $\mu, \rho = 0, 1, 2,...D-1$, $B_\mu$ and $\bar B_\mu$ are
the Nakanishi-Lautrup auxilary fields and the (anti-)ghost fields $(\bar C_\mu)\,C_\mu$ correspond to the
D-dimensional diffeomorphism parameter $\epsilon_{\mu}(x)$ in the infinitesimal transformation: $x_\mu\rightarrow
x_\mu' = x_\mu - \epsilon_{\mu}(x)$. The {\it universality} of the above CF-type restriction implies that, for our
1D diffeomorphism (i.e. reparameterization) invariant theory, the CF-type restriction is: $B + \bar B + i\,
(\bar C\,\dot C - \dot {\bar C}\, C) = 0$. This is what we have obtained from various theoretical tricks in our
present endeavor. The existence of the CF-type restriction is very fundamental to a BRST-quantized theory as it is connected
with the geometrical objects called gerbes [7, 8]. Physically, the existence of the CF-type restriction leads to the independent
nature of the BRST and anti-BRST symmetries (and corresponding conserved charges) at the {\it quantum} level
(that are connected with a given {\it classical} local symmetry).
Our present work (and earlier works [23, 24]) can be generalized to the cases of (super)string and gravitational theories which are also
diffeomorphism invariant. In fact, in our earlier work on a bosonic string theory [34], we have shown the existence
of the CF-type restriction in the context of its BRST quantization and it has turned out to be the 2D version of the {\it universal}
CF-type restriction for the D-dimensional diffeomorphism invariant theory. It is gratifying to pinpoint the fact that
we have derived the CF-type restrictions: $B^a + \bar B^a + i \; \big (\bar C^m \; \partial_m\; C^a + C^m \; \partial_m\; \bar C^a \big ) = 0$
(with $a, m = 0, 1$) for a model of bosonic string theory [34].
This has happened because the bosonic string theory has the
2D diffeomorphism invariance on the 2D world-sheet. We have applied the beautiful blend of MBTSA and ACSA to derive {\it all}
the (anti-)BRST symmetries as well as the 2D version of the CF-type restriction in the case of a bosonic string theory of our interest [35].
In our present investigation, we have utilized only {\it two} and/or {\it one} Grassmannian variables because there are {\it only}
two nilpotent symmetries in the theory. If a theory is endowed with the nilpotent (anti-)BRST as well as (anti-)co-BRST symmetries, then, we have to
invoke {\it four} number of Grassmannian variables. We are currently exploring such kinds of possibilities.\\
\vskip 0.3cm
\begin{center}
{\bf Appendix A: The CF-Type Restriction from $L_B \equiv L_{\bar B}$ }\\
\end{center}
\vskip 0.3cm
\noindent
In this Appendix, we provide the step-by-step derivation of the CF-type restriction by requiring the equivalence of the coupled Lagrangian $L_B$ and $L_{\bar B}$
[cf. Eq. (35)]. A close look at them demonstrates that if we demand $L_B \equiv L_{\bar B}$, the terms that are common would cancel out. For instance, we have
cancellations of terms $L_f,\, -i \, E^2 \, \dot {\bar C} \, \dot C$ and $ - \, \dot {\bar C} \, \bar C \, \dot C \, C$ that are present {\it both} in $L_B$
and $L_{\bar B}$.
Thus, we are left with the following equality:
\[
\frac{B^2}{2} + B\,[E \, \dot E - i \, (2 \, \dot{\bar C} \, C + \bar C \, \dot C)] - i \, E \, \dot E \, \dot {\bar C}\, C
\]
\[
\equiv \frac{\bar B^2}{2} + \bar B\,[E \, \dot E - i \, (2 \, {\bar C} \, \dot C + \dot{\bar C} \, C)] - i \, E \, \dot E \, {\bar C}\, \dot C.
\eqno (A.1)
\]
From the above equation, it is evident that we have
\[
E \, \dot E \, [ B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C)],
\eqno (A.2)
\]
on the l.h.s. when we bring {\it all} the terms from
the r.h.s. to the l.h.s.. At this stage, excluding (A.2), the {\it left-over} terms on the l.h.s. and the r.h.s. are
\[
\frac{B^2}{2} - \frac{\bar B^2}{2} - 2 \, i \, B \, \dot {\bar C}\, C - i \, B \, \bar C \, \dot C
- 2\, i \, \bar B\, \bar C \dot C - i\, \bar B \dot {\bar C} \, C = 0.
\eqno(A.3)
\]
The above equation can be expressed as:
\[
\frac{B^2}{2} - \frac{\bar B^2}{2} - i \,(B + \bar B)\,\dot{\bar C}\, C - i\, (B + \bar B)\,\bar C \, \dot C - i\ B \, \dot {\bar C}\,C
- i\, \bar B \, \bar C \, \dot C = 0.
\eqno(A.4)
\]
The re-arrangements of the terms produce the following
\[
\frac{B^2}{2} - \frac{\bar B^2}{2} - i\,[B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C)]\,\dot {\bar C}\,C - i \,[B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C)]\, \bar C\, \dot C
\]
\[- i\, B \dot {\bar C}\, C - i\, \bar B \, \bar C\, \dot C = 0.
\eqno(A.5)
\]
Taking into account $(A.2)$, we have the following
\[
\big[E\, \dot E - i \, \dot {\bar C}\, C - i \, \bar C \, \dot C \big]\,\big[B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C)\big] + \frac{B^2}{2} - \frac{\bar B^2}{2}
\]
\[- i\ B \, \dot {\bar C}\,C - i\, \bar B \, \bar C \, \dot C = 0.
\eqno(A.6)
\]
Substituting for $ -\, i \, B\, \dot {\bar C} \,C = -\,\frac{i}{2} \, B\, \dot {\bar C} \,C - \,\frac{i}{2} \, B\, \dot {\bar C} \,C$
and $- \, i\, \bar B \bar C \dot C = -\, \frac{i}{2}\, \bar B \, \bar C \, \dot C - \frac{i}{2} \, \bar B \, \bar C \, \dot C$ and re-arranging
the terms, we end up with the following {\it final} result:
\[
\big[ B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C)\big]\,\big[E \, \dot E + \frac{1}{2}\,
\{ B - \bar B - 3\, i\,(\dot {\bar C}\,C + \bar C \, \dot C)\,\} \big] = 0.
\eqno(A.7)
\]
The above equation establishes the existence of the CF-type restriction: $B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C) = 0$
on our theory due to the equivalence of the coupled (but equivalent) Lagrangians (i.e. $L_B \equiv L_{\bar B}$).
This is due to the fact that, in {\it no} way, we can state that the other combination:
$E\,\dot E + \frac{1}{2}\,\{B - \bar B - 3\,i\,(\dot{\bar C}\,C + \bar C\,\dot C)\} = 0$. On the contrary, the CF-type
restriction: $B + \bar B + i \, (\bar C \, \dot C - \dot {\bar C} \, C) = 0$ has been proven from various angles [cf. Eq. (36)].
We end this Appendix with the concluding remark that we have derived the CF-type restriction on our theory from theoretical requirements related with the
symmetries of the coupled (but equivalent) Lagrangians and the absolute anticommutativity properties. However, our present derivation of the CF-type
restriction is more {\it direct} as well as {\it transparent}.\\
\vskip 0.5cm
\begin{center}
{\bf Appendix B: On the derivation of $f_3 = E \, \dot C + \dot E \,C$ and $f_2 = \dot{\bar B}\,C - \bar B\,\dot C$}\\
\end{center}
\vskip 0.5cm
\noindent
The theoretical content of this Appendix is, first of all, devoted to the {\it explicit} derivation of $f_3(\tau)$ in the expansion
of $\Sigma(\tau, \bar\theta)$ in the {\it anti-chiral} super expansions (41). Towards this objective in mind, we focus
on Eq. (51) where the first-order differential equation w.r.t. the evolution parameter $\tau$ for $f_3$ has been expressed. We can
re-write it as
\[
f_3\,\dot C + {\dot f}_3\,C - \dot E\,\dot C \, C - E\,\ddot C \, C = 0 \Rightarrow
\]
\[
\frac{d}{d\,\tau}\,[f_3\,C] - \frac{d}{d\,\tau}\,[(\dot E\, C + E\, \dot C)\,C] = 0, \eqno (B.1)
\]
where we have used the fermionic property $(C^2 = 0)$ of the ghost variable $(C)$. The above equation can be re-expressed in a different {\it but}
useful form as the total derivative w.r.t. $\tau$:
\[
\frac{d}{d\,\tau}\,[\{f_3 - (E\,\dot C + \dot E \, C)\} \, C] = 0. \eqno (B.2)
\]
Integrating the above equation from $\tau = - \infty$ to $\tau = + \infty$ (which are the limiting cases for $\tau$ in our theory), we
obtain the following relationship:
\[
[ f_3 - (E\,\dot C + \dot E\,C)]\, C = 0. \eqno (B.3)
\]
We would like to point out that, while deriving (B.3) from (B.2), we have assumed that {\it all} the physical variables of the Lagrangian $L_B$ and the
secondary variable $f_3 (\tau)$ vanish off at $\tau = \pm \, \infty $. For $C \ne 0$, we obtain the desired result:
$f_3 = E\,\dot C + \dot E\,C$. We have taken $C \ne 0$ because the whole set of BRST symmetry transformations
in Eq. (33) is true {\it only} when the ghost variable $C(\tau)$ has the non-trivial and non-zero value.
We now concentrate on the precise determination of $f_2 (\tau)$ of the super-expansion (41). In other words, we wish to show that
$f_2 = \dot {\bar B} \, C - \bar B \, \dot C $. For this purpose, we note that we have a first-order differential equation w.r.t. the
evolution parameter $\tau$ for the secondary variable $f_2 (\tau)$ in Eq. (49). This can be re-expressed as follows
\[
f_2\,\dot C + {\dot f}_2\,C - 2\,f_2\,\dot C - 2\, \dot {\bar B} \, \dot C \, C + \dot {\bar B} \, \dot C \, C + \bar B \, \ddot C \, C = 0
\eqno (B.4)
\]
where we have added and subtracted $f_2 \, \dot C$ and $\dot {\bar B} \, \dot C \, C$. The above equation implies that we have now
its modified form (with total derivatives) as:
\[
\frac{d}{d\,\tau}\,[f_2\,C] - 2\,(f_2\,\dot C + \dot{\bar B}\,\dot C\,C) + \frac{d}{d\,\tau}\,[\bar B\,\dot C\,C] = 0
\]
\[
\Rightarrow \quad \frac{d}{d\,\tau}\,\Big[f_2\,C + \bar B\,\dot C\,C\Big] - 2\,\Big[(f_2 - \dot{\bar B}\,C)\,\dot C\Big] = 0. \eqno (B.5)
\]
Using the fermionic $(C^2 = \dot C^2 = 0)$ property of the ghost variables $C$ and $\dot C$, we can recast the above equation in the
following interesting form where $[f_2 - (\dot {\bar B} \, C - \bar B \, \dot C)]$ appears very nicely in the individual terms of the
following difference, namely;
\[
\frac{d}{d\,\tau}\,\Big[\{f_2 - (\dot{\bar B}\,C - \bar B\,\dot C)\}\,C\Big] - 2\,\Big[\{f_2 - (\dot{\bar B}\,C - \bar B\,\dot C)\}\,\dot C\Big] = 0. \eqno(B.6)
\]
We can expand the {\it total} derivative in the {\it first} term to obtain:
\[
\frac{d}{d\,\tau}\, \Big[\{f_2 - (\dot{\bar B}\,C - \bar B\,\dot C)\} \Big]\, C - \Big[\{f_2 - (\dot{\bar B}\,C - \bar B\,\dot C)\}\,\dot C \Big] = 0. \eqno(B.7)
\]
Defining $f_2 - (\dot{\bar B}\,C - \bar B\,\dot C) = \chi$ leads us to the following
\[
\Big(\frac{d}{d\,\tau}\,\chi\Big)\,C - \chi\,\dot C = 0 \qquad \Rightarrow \qquad \dot\chi\,C = \chi\,\dot C. \eqno(B.8)
\]
Multiplying from the right by $C$ and taking into account the fermionic (i.e. $C^2 = 0$) nature of the ghost variable $C$, we obtain the following
\[
0 = \chi\,\dot C\,C \qquad \Rightarrow \qquad \chi = 0 \;\; \; \qquad \big[\text{for} \quad C\,\dot C \ne 0\big]. \eqno(B.9)
\]
It should be noted that we have the off-shell nilpotent BRST symmetry transformation: $s_b\,C = C\,\dot C$ [cf. Eq. (33)]. As a consequence,
the combination of the variables $C\,\dot C \ne 0$.
If the symmetry of a theory is the guiding principle behind {\it its beauty}, it is {\it physically} correct to assume that $s_b\,C = C\,\dot C \ne 0$.
In fact, if we take $C \dot C = 0$ the whole beauty and sacrosanct properties
(i.e. off-shell nilpotency and absolute anticommutativity) of the (anti-)BRST symmetry transformations of our present theory
will be spoiled. As a consequence, the CF-type restriction will {\it no} longer remain (anti-)BRST invariant. It should be recalled, however,
that we have invoked the CF-type restriction in proving the equivalence of the Lagrangians [cf. Eq. (35)] w.r.t. the (anti-)BRST
symmetry transformations. Hence,
our conclusion in (B.9) is {\it correct} which leads to the derivation of $f_2 = \dot{\bar B}\,C - \bar B\,\dot C$ from $\chi = 0$.\\
\vskip 0.3cm
\begin{center}
{\bf Appendix C: On the Proof of the Conservation Law}\\
\end{center}
\vskip 0.3cm
\noindent
We take up here the expression for the BRST charge $Q_B$ [cf. Eq. (74)] that has been derived using the Noether theorem [cf. Sec. 7]. We
exploit the EL-EOM derived from the Lagrangian $L_B$ to recast the expression for ${\dot Q}_B$, namely;
\[
{\dot Q}_B = \frac{\dot C\,E}{2}\,(p_x^2 + 2\,m\,p_t) + \frac{C\,\dot E}{2}\,(p_x^2 + 2\,m\,p_t) + C\,E\,(p_x\,{\dot p}_x + m\,p_t)
\]
\[
+2\,E\,\dot E\,B\,\dot C + E^2\,B\,\dot C + E^2\,B\,\ddot C + {\dot E}^2\,B\,C + E\,\ddot E\,B\,C + E\,\dot E\,\dot B\,C
\]
\[
+ E\,\dot E\,B\,\dot C + 2\,B\,\dot B\,C - i\,\dot B\,\bar C\,\dot C\,C - i\,B\,\dot{\bar C}\,\dot C\,C
- i\,B\,\bar C\,\ddot C\,C, \eqno(C.1)
\]
into the following form
\[
{\dot Q}_B = -i\,E^2\,\ddot{\bar C}\,\dot C\,C - i\,E\,\dot E\,\dot{\bar C}\,\dot C\,C + 3\,E\,\dot E\,B\,\dot C + E^2\,B\,\ddot C + {\dot E}^2\,B\,C
\]
\[
+ E\,\ddot E\,B\,C + 2\,B\,\dot B\,C + B^2\,\dot C - i\,\dot B\,\bar C\,\dot C\,C - i\,B\,\dot{\bar C}\,\dot C\,C
- i\,B\,\bar C\,\ddot C\,C, \eqno(C.2)
\]
where we have used the EL-EOM from $L_B$ as:
\[
{\dot p}_x = 0, \qquad {\dot p}_t = 0, \qquad \frac{1}{2}\,(p_x^2 + 2\,m\,p_t) = -\,\dot B\,E + i\,E\,\ddot{\bar C}\,C - i\,E\,\dot{\bar C}\,\dot C.
\eqno(C.3)
\]
The expression (C.2) can be further changed to a {\it reduced} form as follows
\[
{\dot Q}_B = i\,B\,\dot{\bar C}\,\dot C\,C - i\,\dot B\,\bar C\,\dot C\,C - i\,E\,\dot E\,\dot{\bar C}\,\dot C\,C - i\,E^2\,\ddot{\bar C}\,\dot C\,C, \eqno(C.4)
\]
if we use the EL-EOM from $L_B$ w.r.t. the variable $\bar C$ as
\[
2\,\dot B\,C + B\,\dot C + 3\,E\,\dot E\,\dot C + E^2\,\ddot C + {\dot E}^2\,C + E\,\ddot E\,C - i\,\bar C\,\ddot C\,C - 2\,i\,\dot{\bar C}\,\dot C\,C = 0.
\eqno(C.5)
\]
The expression in (C.4) can be proven to be equal to {\it zero} by using the following EL-EOM from $L_B$ w.r.t. the variable $C$, namely;
\[
i\,\dot B\,\bar C - i\,B\,\dot{\bar C} + i\,E\,\dot E\,\dot{\bar C} + i\,E^2\,\ddot{\bar C} - \ddot{\bar C}\,\bar C\,C - 2\,\dot{\bar C}\,\bar C\,\dot C
= 0. \eqno(C.6)
\]
We end this Appendix with the remark that, in exactly {\it similar} fashion, we can prove the conservation law $({\dot Q}_{\bar B} = 0)$ of the anti-BRST
charge [cf. Eq. (74)] which has been derived by exploiting the theoretical tricks of the Noether theorem.\\
\vskip 0.4cm
\noindent
{\bf Acknowledgments}\\
\noindent
Two of us (AKR and AT) gratefully acknowledge the financial support from the {\it BHU-fellowship} program of the
Banaras Hindu University (BHU), Varanasi (U. P.),
under which, the present investigation has been carried out. Fruitful and enlightening
comments by our esteemed Reviewer are thankfully acknowledged, too. \\
\vskip 0.4cm
\noindent
{\bf Data Availability}\\
\noindent
No data were used to support this study.\\
\noindent
{\bf Conflicts of Interest}\\
\noindent
The authors declare that there are no conflicts of interest
|
1,116,691,498,491 | arxiv | \section{Introduction}
Typically, more than 30\% of the fuel consumption of ocean--going ships is from making waves \cite{faltinsen05}. A resistance is felt due to the work done by the ship on the surrounding water, which propagates away in the form of wave energy. While going back over a century \cite{michell1898,havelock09,havelock14,havelock19,havelock22,wehausen73,noblesse83}, wave resistance on ships has also been the focus of recent investigations \cite{benzaquen14}.
Two of us recently showed that the wave resistance acting on a ship in steady motion can be significantly altered by the presence of a shear current beneath the water surface \cite{li16}. In conditions with no shear, wave resistance typically becomes important for Froude numbers around $0.3$ and peaks in the vicinity of $0.5$ before decreasing again as the wake becomes dominated by diverging waves. When a sub-surface shear current is present, however, both the Froude number at which wave resistance sets in, and the value at which it peaks, are in general changed, with opposite effects whether the ship travels along, against, or across the current \cite{li16}. Moreover, sub-surface shear causes the angle made by the ship waves to differ from Lord Kelvin's classic $19.47^\circ$, being smaller for shear-assisted and larger for shear-inhibited motion, and asymmetric around the line of motion when the angle with the current is oblique \cite{ellingsen14a}.
In the latter case momentum is imparted to the water at different rates to starboard and port, and the corresponding wave radiation force experienced by the ship obtains a lateral component in addition to the conventional sternward wave resistance \cite{li16}. No corresponding phenomenon exists in rectilinear motion if the current has depth-uniform velocity profile.
Our concern in this paper is to introduce realism, compared to previous studies which have considered idealised models. We study how the shear of a real, measured current may affect the wave radiation forces on actual ships. We use an example shear profile measured in the Columbia River delta. These waters are crossed by thousands of ships each year, and we study model ships with dimensions and velocities typical of different vessel types operating there.
This includes not only the forces acting during steady motion, but also transient forces from manoeuvring motions. To this end, the most general theory of linear ship waves (or waves from free--surface sources more generally) to date has been developed, and is presented here, allowing a shear current to vary arbitrarily with respect to depth both in direction and magnitude, as long as it may be considered uniform in horizontal directions.
We demonstrate in Section \ref{sec:num} how a real shear current can have a very significant effect on the wave--making forces acting on real ships. At typical Froude numbers we find for smaller boats (tugboats, fishing boats) that the wave resistance can differ by a factor $3$ or more between upstream and downstream motion at the same velocity relative to the free surface. The lateral radiation force acting when travelling across the shear is also very significant; it is typically around $20\%$ of the sternward resistance force in steady motion, but can momentarily reach more than $50\%$ of the wave resistance during maneouvring. These are by no means small effects, and will affect the seakeeping and the optimal choice of velocity and route of travel, and perhaps also cause safety issues for ships manoeuvring in proximity of each other.
This paper contains two major sections, one theoretical, one of an applied nature. The reader primarily interested in what the practical effect of shear in real--life situations might be, may wish to refer directly to the numerical results in Section \ref{sec:num} bearing in mind the system definitions in Section \ref{sec:def}. The theoretical foundations and framework is laid out in Section \ref{sec:basic}; it has been presented, as far as we have been able to, so as to be useful to readers who wish to employ the formalism for their own purposes.
Studies of transient wave resistance go back a long time. Whenever a ship undergoes changes in velocity during acceleration or manoeuvring, transient waves are emitted, and the wave radiation force correspondingly will be time dependent for the duration during which the created transient ring-wave remains in the immediate vicinity of the ship. A century ago, Havelock studied the wave resistance in 2 dimensions due to a suddenly appearing ship, modelled as a distribution of additional pressure at the water's surface, suppressing the free surface approximately as would a ship \cite{havelock17}. The resistance force was found to increase from zero to a peak value before relaxing in an oscillatory manner to its static value. The speed of relaxation was found to depend closely on the aspect ratio of the disturbance, since the bow and stern waves from a more slender ship tend to cancel, causing a quicker relaxation to steady conditions and a more stable steady wave resistance. On the other hand a circular ``ship'' with little such interference, experienced a very slow relaxation rate. A study of the resistance felt by a submerged cylinder starting suddenly from rest revealed similar results \cite{havelock49}. Studies of ships in various kinds of acceleration is a related classical problem \cite{bhattacharyya56, wehausen60}.
Approaching the problem of waves in three--dimensional systems in the presence of sheared flows, standard methods to calculate waves and motions of floating bodies must be immediately discarded, based as they are on potential theory. No satisfactory theory of creating bodies from submerged sources and sinks exist even in the simplest shear currents exists, not to mention advanced panel methods \cite{ellingsen16}. A feasible approach for our purposes is however to create a ``ship--shaped footprint'' in the free surface by introducing an external surface pressure. The approach goes back over a century \cite{havelock08} and has recently been employed in wave resistance studies \cite{benzaquen14}. Such a model, only affects the dynamic boundary condition, not the equations of motion, thus does not in principle pose any restrictions on the flow vorticity.
\subsection{Outline}
The investigated system is presented in Section \ref{sec:def} along with the basic formalism. Section \ref{sec:basic} then goes on to develop the general theory of waves from moving, time--dependent surface disturbances upon a horizontal background current which may vary arbitrarily with depth, both in direction and magnitude. In particular, a suitable formalism for working with a general (not explicitly known) dispersion relation is derived in Section \ref{sec:dispersion}, and applied to the general problem in Section \ref{sec:tr}. In Section \ref{sec:approx} practical considerations are presented concerning numerical evaluation of the dispersion relation for arbitrary velocity profiles, and the formalism for calculating wave resistance and lateral radiation force is derived and discussed in Section \ref{sec:R}.
Section \ref{sec:num} is of a more applied nature and presents numerical results for particular situations. A measured velocity profile from the Columbia River estuary is used, and pressure distributions modelling ships of realistic dimensions are employed in order to provide reasonably realistic estimates of the effect of shear in these waters while retaining some generality. For comparison, and to illustrate the effect of shear without the large number of lengthscales and parameters, corresponding results for the simple case of a linearly depth--dependent current are given in Section \ref{sec:numlin} before conclusions are drawn. Some further details on derivation and numerical procedures are found in appendices.
\subsection{System definition}\label{sec:def}
In this section the system under scrutiny is defined, along with general formalism used in the paper. The system is a generalisation of that considered in Ref.~\cite{li16b}.
\begin{figure}[tb]
\includegraphics[width=\columnwidth]{figure1.pdf}
\caption{
Schematic sketch of the system: a ship travelling with arbitrary, time--dependent velocity atop a shear current of arbitrary depth--dependence. Here a ``lab'' coordinate system is shown, fixed relative to the sea--bed.
}
\label{fig:geom}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=.9\columnwidth]{figure2.pdf}
\end{center}
\caption{(a) Illustration of shear--assisted vs shear--inhibited ship motion; shown in the ``lab'' reference frame relative to the sea bed, and (b) in the reference frame relative to the water surface. (c) Definition of angles
$\gamma$ (angle between $\mathbf{k}$ and $\mathbf{U}_0$), $\beta$ (angle between $\mathbf{U}_0$ and $x$ axis, and $\theta$ (angle between $\mathbf{k}$ and $x$-axis). The reference frame is here at rest with respect to the ship. Note: $\beta=0$ is the maximally shear assisted direction of motion, $\beta=\pi$ the maximally shear inhibited.}
\label{fig:agls}
\end{figure}
We consider infinitesimal wave amplitudes described by the surface elevation function $ \hat{\zeta}(\mathbf{r},t) $ with horizontal position $\mathbf{r}=(x,y)=r(\cos\varphi,\sin\varphi)$ and time $t$.
The waves are superimposed on a depth-varying background flow $\mathbf{U}(z)$. In our general theory in Section \ref{sec:basic}, $\mathbf{U}(z)$ may vary both in magnitude and direction, although our numerical examples in Section \ref{sec:num} will all be unidirectional. We use the shorthand $\mathbf{U}(0)=\mathbf{U}_0$. A sketch of the system is seen in Fig.\ref{fig:geom}. We assume incompressible and inviscid flow.
The three velocity components and pressure perturbation due to the waves we name $\hat{u},\hat{v},\hat{w}$, and $\hat{p}$, respectively, all functions of $\mathbf{r},z$ and $t$. Hatted quantities are considered small, and we linearise with respect to these. The flow field is thus $[\mathbf{V},P]=[\mathbf{U}(z)+\hat{u}\mathbf{e}_x+\hat{v}\mathbf{e}_y+\hat{w}\mathbf{e}_z,-\rho gz+\hat{p}]$, with $\mathbf{V}$ and $P$ the total velocity and pressure fields, respectively, $g$ the gravitational acceleration, and $\rho$ the density of the water.
The flow obeys the Euler equation
\begin{equation}\label{euler}
\partial_t\mathbf{V} + (\mathbf{V}\cdot\nabla)\mathbf{V}=-\nabla P/\rho - g\mathbf{e}_z.
\end{equation}
We neglect surface tension.
The physical quantities are defined in Fourier space of the horizontal plane as $[\hat{\zeta},\hat{u},\hat{v},\hat{w},\hat{p}](\mathbf{r},z,t)\leftrightarrow [\zeta,u,v,w,p](\mathbf{k},z,t)$ as
\begin{equation}\label{fourier}
[\hat{\zeta},\hat{u},\hat{v},\hat{w},\hat{p}](\mathbf{r},z,t)=\int\frac{\mathrm{d}^2 k}{(2\pi)^2} [\zeta,u,v,w,p](\mathbf{k},z,t)\mathrm{e}^{\rmi \mathbf{k}\cdot\mathbf{r}}
\end{equation}
so that $ \mathbf{k}=(k_x,k_y)=(k\cos\theta,k\sin\theta) $ is the wave vector (It is understood that $\hat{\zeta}$ and $\zeta$ do not depend on $z$).
The water depth $h$ is constant, and may be allowed to tend to $\infty$.
In the system sketched in Fig.~\ref{fig:geom} no less than three different reference frames are natural, depending on the question under consideration. Fig.~\ref{fig:geom} shows the ``lab'' reference frame, i.e., as seen by an observer on shore. A second frame of reference which we use in Section \ref{sec:tr} is that which is fixed on the moving model ship. Finally, in section \ref{sec:num} we will sometimes work in the frame of reference in which the water surface is at rest.
For this reason the oft used terms `upstream' and `downstream' are ambiguous as denotations of directions of motion.
We will instead use the terms `shear--assisted' and `shear--inhibited' to describe directions of ship motion or wave motion relative to the sub--surface current. The motion is assisted by the current if, in a reference system where the water surface is at rest, the sub-surface current has a component along the direction of motion (this corresponds to the ship travelling upstream in the case of e.g.\ a river). Correspondingly, for shear--inhibited motion the sub--surface current has positive component against the ship's motion, in a system where the free surface is at rest (corresponds to downstream motion on a river). These concepts are visualised in Fig.~\ref{fig:agls}a and b. They are only strictly well defined only for velocity profiles that do not change direction or sign relative to the free surface, yet this is sufficient for our present purposes.
In later sections we shall make use of polar coordinates in the horizontal plane, which we define in figure \ref{fig:agls}c, for a system in which the ship is at rest. Note that the angle $\beta$ differs by $\pi$ from that used in \cite{ellingsen14a,li16}, where a reference system relative to the water surface was used.
The angle between $\mathbf{k}$ and $\mathbf{U}_0$ is $\gamma$.
\section{Theory: linear surface waves from an arbitrary time--varying wave source, propagating on an arbitrary shear current}
\label{sec:basic}
In this section we present a theoretical framework for calculating waves from arbitrary wave sources on the free surface, in flows with arbitrary dispersion relation $\omega(\mathbf{k})$, affected by sub-surface currents that may vary both with depth and direction. To our knowledge no theory this general has ever been presented. As a special case the theory provides a procedure for calculation and analysis of ship waves on arbitrary horizontal shear currents.
From the linearised Euler equations and continuity equation in $\mathbf{k}$-space we have the relations (cf.\ e.g.\ the procedure of \cite{shrira93})
\begin{subequations}
\begin{align}
(\partial_t+\mathrm{i} \bk\cdot\bU)w'(z,t)-\mathrm{i} \bk\cdot\bU'w(z,t) =& -k^2 p(z,t)/\rho, \label{eq:p}\\
(\partial_t+\mathrm{i} \bk\cdot\bU)w(z,t) =& - p'(z,t)/\rho, \label{eq:dp}
\end{align}
\end{subequations}
where a prime denotes differentiation with respect to $z$, and the dependence on $\mathbf{k}$ of $p$ and $w$ is suppressed here and henceforth.
\subsection{General form of surface wave dispersion relation}
\label{sec:dispersion}
We will now present a general, implicit form of the dispersion relation for waves atop a general depth--dependent shear flow $\mathbf{U}(z)$. The relation allows us to derive general expressions for surface waves from an arbitrary free--surface source in Section \ref{sec:tr}. Determining $\omega(\mathbf{k})$ for a specific situation is the topic of Section \ref{sec:approx}.
We use the physical values $\omega_\pm(\mathbf{k})$ to express the free--surface elevation for a given $\mathbf{k}$-component as:
\begin{equation}\label{zeta}
\zeta(\mathbf{k},t) = Z_+(\mathbf{k})\mathrm{e}^{-\mathrm{i} \omega_+ t} + Z_-(\mathbf{k})\mathrm{e}^{-\mathrm{i} \omega_- t}
\end{equation}
where $Z_\pm$ are unknown coefficients to be determined.
Also the other perturbed quantities $u,v,w$ and $p$ will have time dependence $\propto \exp(-\mathrm{i} \omega_\pm t)$.
If the values of $Z_\pm$ are known from initial conditions, the full time dependent solution to the free--surface elevation can be found from \eqref{zeta}.
The phase velocities $\omega_+(\mathbf{k})/k$ and $\omega_-(\mathbf{k})/k$ correspond to partial waves propagating in directions $\mathbf{k}$ and $-\mathbf{k}$, respectively. They satisfy the relation
\begin{equation}\label{omrel}
-\omega_-(\mathbf{k})=\omega_+(-\mathbf{k}).
\end{equation}
Hence there is a unique, positive phase velocity $\omega_+(\mathbf{k})$ in propagation direction $\mathbf{k}$, and the integral over all $\mathbf{k}$ effectively accounts for each mode twice. The relation \eqref{omrel} is general and holds for any shear current.
We show in \ref{appx-1} that the dispersion relation for a plane wave of small amplitude on a depth--dependent flow may be written
\begin{align} \label{dispR}
\Delta_R&(\mathbf{k}, \omega) \equiv (1+I_g) (\omega - \bk\cdot\bU_0)^2 + \notag\\
& (\omega - \bk\cdot\bU_0)\bk\cdot\bU'_0{\tanh kh}/{k}-gk \tanh kh = 0,
\end{align}
where $\Delta_R$ is defined for later reference, and
\begin{equation}
I_g(\mathbf{k}) =\int\limits_{-h}^{0}\rmd z \dfrac{\bk\cdot\bU''(z) w(z,0) \sinh k(z+h)}{k[\bk\cdot\bU(z)-\omega] w(0,0)\cosh kh}.
\end{equation}
The implicit dispersion relation \eqref{dispR} is extremely useful for analytical purposes. It is not itself closed, since both $\omega(\mathbf{k})$ and $w(z,t)$ are unknowns.
The two roots of the equation $\Delta_R=0$ are $\omega=\omega_\pm(\mathbf{k})$. It is found e.g.\ in \cite{smeltzer17} that the zeros of $\Delta_R$ are simple, hence Eq.~\eqref{dispR} may be written on the form
\begin{equation} \label{Domg}
\Delta_R(\mathbf{k}, \omega) = (1+I_g) (\omega-\omega_+)(\omega-\omega_-)=0.
\end{equation}
\subsection{Waves from an arbitrary, time-dependent pressure distribution} \label{sec:tr}
We wish to find a solution to the surface pattern
resulting from a time-dependent externally applied pressure distribution $\hat{p}_\mathrm{ext}(\mathbf{r},t)\leftrightarrowp_\mathrm{ext}(\mathbf{k},t)$ at the free surface.
The pressure, when positive, depresses the water surface thus modelling a moving wave source such as a ship. Using an applied surface pressure as wave source rather than e.g.\ potential theory with submerged sources such as are often used in the theory of ship motions \cite{faltinsen90}, is advantageous since only the boundary conditions are directly affected. This is necessary in our system, since the flow we consider is inescapably rotational and potential theory is inapplicable.
It should be noted that the relation between the shape of the applied pressure and the resulting surface depression is not altogether trivial for a moving source, and has some Froude number dependence. This introduces a certain quantitative uncertainty in the results presented in section \ref{sec:num}; this is a question we intend to address in the near future.
By superposition, the response $G(\mathbf{k},t)$ of the system to an arbitrary time-dependent pressure distribution can be expressed as a time-integral of pressure pulses emitted at all previous times,
\begin{equation}\label{evolution}
G(\mathbf{k},t) = \int_{-\infty}^t \mathrm{d} \tau p_\mathrm{ext}(\mathbf{k},\tau)H(\mathbf{k},t-\tau).
\end{equation}
$H(\mathbf{k},t-t_0)$ is the system's response to an impulsive pressure rate
$p_I(t) = I\delta(t)$
which imparts a finite impulse to the free surface during an infinitesimally short time.
$I$ equals unity in units of pressure. $G$ and $H$ physically may represent any of the perturbation quantities $u,v,w,p$ or $\zeta$.
Mathematically $H$ plays the role of a Green's function.
We now proceed to finding the response of the free surface to a pressure impulse. In Eq.~\eqref{evolution} we let $G\to\zeta$, and the correspondng response function we call $H_\zeta(\mathbf{k},t)$.
The full time evolution $\zeta(\mathbf{k},t)$ for $t>0$ is then calculated from \eqref{evolution} as
\begin{equation}\label{zetat}
\zeta(\mathbf{k},t)=\int_{-\infty}^t \mathrm{d} \tau p_\mathrm{ext}(\mathbf{k},\tau)H_\zeta(\mathbf{k},t-\tau)
\end{equation}
with $H_\zeta$ derived in the following, given in \eqref{Hz}.
The prescribed impulsive pressure enters the equation system via the dynamic free surface boundary condition, which can be written
\begin{align}
\mathrm{i} \bk\cdot\bU^\prime_0w - \left(\partial_t + \mathrm{i} \bk\cdot\bU_0 \right) w^\prime - k^2g\zeta & = k^2I\delta(t)/\rho.\\
\left(\partial_t + \mathrm{i} \bk\cdot\bU_0 \right) \zeta & = w, \label{kbc}
\end{align}
with $w,w'$ evaluated at $z=0$.
Here $\mathbf{U}_0$ is surface velocity, and a prime denotes differentiation with respect to $z$.
Integration over an infinitesimal time interval
$t = 0_-$ to $0_+$ yields the following relations for $w(z,t)$ and $\zeta(t)$,
\begin{subequations} \label{eq:t=0}
\begin{align}
w^\prime(0,0_+) & = -k^2I/\rho, \label{wI}\\
\zeta(0_+) & = 0, \\
\dot{\zeta}(0_+) &= w(0,0_+) ,
\end{align}
\end{subequations}
using the assumptions that the system is completely at rest for
$t<0$ and that all physical quantities have finite values at $t>0$, at $ t=0_+ $ in particular.
We suppress the dependence of $w$ and $\omega$ on $\mathbf{k}$ in this subsection.
When a current of arbitrary depth--variation is present, the primary challenge is that analytical expressions for $\omega_\pm(\mathbf{k})$ and $ w(z,t) $ cannot be found. We show in \ref{appx-1} the relations
\begin{subequations}\label{eq:dwF}
\begin{align}
w^\prime(0,0_+) & = k(1+I_g )w(0,0_+)\coth kh, \\
& = -\dfrac{\bk\cdot\bU^\prime_0\tilde{\omega}+gk^2}{\tilde{\omega}^2}w(0,0_+),\\
&= \frac{k}{F(\mathbf{k})}w(0,0_+), \label{Fdef}
\end{align}
\end{subequations}
where $ \omega $ can be either of the roots of $\Delta_R=0$, i.e.\ $ \omega_+ $ or $ \omega_- $, and the intrinsic frequency is $\tilde{\omega}=\omega - \bk\cdot\bU_0$.
Eq.~\eqref{Fdef} defines the quantity $F(\mathbf{k})$ for later reference.
We note that $F(\mathbf{k})$ can be written in several different forms,
\begin{subequations}
\begin{align}\label{Falt}
F(\mathbf{k}) &= \frac{k w(0,0_+)}{w'(0,0_+)}\\
&=\frac{\tanh kh}{1+I_g}\label{Fth}\\
&=\frac{(\omega-\omega_-)(\omega-\omega_+)}{\Delta_R}\tanh kh\\
&=\frac{k\tilde{\omega}(\mathbf{k})^2}{gk^2-\bk\cdot\bU_0'\tilde{\omega}(\mathbf{k})}.\label{Fom}
\end{align}
\end{subequations}
Which form of $F(\mathbf{k})$ is most convenient is different in different cases. The final form \eqref{Fom} has the advantage that only the value of $\omega(\mathbf{k})$ is required when $\mathbf{U}(z)$ is known.
From \eqref{zeta}, \eqref{eq:t=0} and \eqref{eq:dwF} then follows
\begin{subequations}
\begin{align}
Z_+ + Z_- =&0;\\
\omega_+Z_+ + \omega_-Z_- =& -\mathrm{i} Ik F(\mathbf{k})/\rho
\end{align}
\end{subequations}
Solving for $Z_\pm$ and inserting into
\eqref{zeta} yields the surface elevation $H_\zeta$ from an impulsive pressure pulse as
\begin{equation} \label{Hz}
H_\zeta(\mathbf{k},t) = \frac{\mathrm{i} k F(\mathbf{k})
}{2\rho\omega_\text{div}(\mathbf{k})}(\mathrm{e}^{-\mathrm{i}\omega_-t}-\mathrm{e}^{-\mathrm{i}\omega_+t}),
\end{equation}
where the ``divergence frequency'' is, using \eqref{omrel},
\begin{equation}
\omega_\text{div}(\mathbf{k}) = \frac12[\omega_+(\mathbf{k}) - \omega_-(\mathbf{k})]=\frac12[\omega_+(\mathbf{k}) + \omega_+(-\mathbf{k})],
\end{equation}
so that $\omega_\text{div}/k$ is the phase speed with which oppositely propagating waves move apart.
\subsubsection{Suddenly appearing ship}\label{sec:sudden}
As a step towards modelling a ship during manoeuvring or acceleration in a simple manner, we
consider the special case where $p_\mathrm{ext}$ is constant for $t>0$ and zero at $t<0$, i.e., a ``ship'' that is launched at $t=0$ already having its final velocity and continuing in steady motion thereafter. This is the system considered long ago by Havelock \cite{havelock17}. It is an artificial situation, but one which can be used as a building block to model more realistic situations. Turning the arrow of time yields instead a suddenly disappearing ship, and adding at the same instance the appearence of the same ship but with a slightly different velocity, say, is a simple model of a rapidly turning and/or accelerating ship. In numerical examples we will consider the more realistic case of a suddenly starting ship.
We use a reference frame following the ship, so that the motion of the ship relative to the water surface is contained in the surface current velocity $\mathbf{U}_0$ as measured in this system.
The time integral in \eqref{zetat} can be solved explicitly, and $\zeta$ splits naturally into a steady and a transient contribution
\begin{subequations}\label{generalz}
\begin{align}
\hat{\zeta}(\mathbf{r},t) =& \lim_{\epsilon\to 0}[\zeta_s(\mathbf{r}) + \zeta_t(\mathbf{r},t)],\\
\hat{\zeta}_s(\mathbf{r}) =&\frac1\rho\int\frac{\mathrm{d}^2 k}{(2\pi)^2} \frac{kp_\mathrm{ext}(\mathbf{k}) F(\mathbf{k}) }{(\omega_+-\mathrm{i}\epsilon)(\omega_--\mathrm{i}\epsilon)}\mathrm{e}^{\rmi \mathbf{k}\cdot\mathbf{r}},\label{staticGen}\\
\hat{\zeta}_t(\mathbf{r},t)=&\frac1\rho\int\frac{\mathrm{d}^2 k}{(2\pi)^2} \frac{kp_\mathrm{ext}(\mathbf{k}) F(\mathbf{k}) \mathrm{e}^{\rmi \mathbf{k}\cdot\mathbf{r}}}{2\omega_\text{div}(\mathbf{k})}\notag \\
&\times\left(\frac{\mathrm{e}^{-\mathrm{i} \omega_+ t}}{\omega_+-\mathrm{i}\epsilon}\right.\left.-\frac{\mathrm{e}^{-\mathrm{i} \omega_- t}}{\omega_--\mathrm{i}\epsilon}\right). \label{eq:tranW
\end{align}
\end{subequations}
Subscripts $s$ and $t$ denote stationary and transient, respectively. Upon splitting into $\zeta_s$ and $\zeta_t$ it was necessary to employ a radiation condition by adding a small imaginary part $-\mathrm{i}\epsilon$ to wave frequencies, whereby $\omega_\pm\to\omega_\pm-\mathrm{i} \epsilon$ (see, e.g., \cite{li16})
assuring that waves can only be radiated away from the source. Mathematically this moves the poles to complex values of $\mathbf{k}$, rendering the integrals definite. Physically, it introduces an arrow of time by implying the time--independent $\hat{\zeta}_s$ was ``switched on'' some time in the far past, and consequently likewise the transient contribution which exactly cancels the steady one for $t<0$.
Given a value for $\omega_\pm(\mathbf{k})$ (using any of various approximation schemes described below), equation \eqref{zetat} now produces $\hat{\zeta}(\mathbf{r},t)$ at all times; the Fourier transform is taken as in equation \eqref{fourier}, for example using a fast Fourier transform (FFT) algorithm.
\subsubsection{Stationary ship waves}
The simplest case is the classical situation of a ship which has been travelling at constant velocity for a long time.
The wave pattern in this case is readily obtained from \eqref{zetat} when taking the limit $ t \to \infty $, which yields
\begin{align}
\hat{\zeta}(\mathbf{r})&=\hat{\zeta}_s(\mathbf{r})\notag \\
&= \lim_{\epsilon\to0}\int\frac{\mathrm{d}^2 k}{(2\pi)^2}\frac{kp_\mathrm{ext}(\mathbf{k}) \tanh kh }{\rho\Delta_R(\mathbf{k},\omega+\mathrm{i} \epsilon)}\mathrm{e}^{\rmi \mathbf{k}\cdot\mathbf{r}},\label{shipW}
\end{align}
Using $\Delta_R$ on the form \eqref{Domg} is instructive.
Transient waves described by \eqref{eq:tranW} vanish at large times $ t\to\infty $, as will be further discussed in \S\ref{sec:asymp}.
Eq.~\eqref{shipW} is exactly the expression for ship waves from a ship moving with velocity $-\mathbf{U}_0$ relative to the water surface, as derived in \cite{li16} (note that angle $\beta$ differs by $\pi$ from that of \cite{li16,ellingsen14a}), generalised to the case of general dispersion.
\subsubsection{Suddenly starting ship: wave patterns and asymptotics}
\label{sec:asymp}
\begin{figure}[thb]
\includegraphics[width=\columnwidth]{figure3.pdf}
\caption{
Super-Gaussian model ship pressure distributions from Eq.~(\ref{p}). Aspect ratios (left to right) $W=3,5,8$.
}
\label{fig:hulls}
\end{figure}
\begin{figure*}[htb]
\begin{center}
\includegraphics[width=.95\textwidth]{figure4.pdf}
\end{center}
\caption{
Wave patterns of model ship suddenly set in motion from rest at $T=0$, at increasing nondimensional time $T=t\sqrt{g/L}$ where $L$ is the ship length. The ship is modelled as a super--Gaussian of aspect ratio $L/b=6$; see Eq.~\eqref{p}. First row: no shear; Second row: shear--assisted ($\beta = 0$); Third row: side--on shear ($\beta=\pi/2$); Fourth row: shear--inhibited ($\beta=\pi$). The shear Froude number is $\mathrm{Fr} _s=S|\mathbf{U}_0|/g=0.8$ with $\mathbf{U}_0$ the ship velocity relative to the water surface. The reference system is relative to the ship, rotated so that ship motion is the same in all cases.
}
\label{fig:patterns}
\end{figure*}
We consider now the model of a ship which starts suddenly from rest. Formally this situation is created from the ``suddenly appearing ship" model in Section \ref{sec:sudden} by superposing the ring wave from a ship at rest suddenly disappearing at $t=0$, and reappearing in the same instance with velocity $\mathbf{U}_0$ relative to the water surface.
As a simple model ``ship'' we use an elliptical super-Gaussian pressure distribution with length $L$ and beam (width) $b$ of the form
\begin{equation}\label{p}
p_\mathrm{ext}(\mathbf{r},t) = p_0\exp\left\{ -\pi^2\left[\left(2x_\beta/L\right)^2 + \left(2y_\beta/b\right)^2\right]^3 \right\},
\end{equation}
where
\begin{subequations}
\begin{align}
x_\beta(t) &= [x-x_0(t)]\cos\beta(t) + [y-y_0(t)]\sin\beta(t), \\
y_\beta(t) &= -[x-x_0(t)]\sin\beta(t) + [y-y_0(t)]\cos\beta(t)
\end{align}
\end{subequations}
expressed along the major and minor axes of the ellipse in a reference system (e.g.\ relative to the water surface) where the ship's position may be time--dependent.
In a reference system fixed on the ship, $x_0=y_0=0$ and $[x_\beta,y_\beta] = r[\cos(\varphi-\beta),\sin(\varphi-\beta)]$. The Froude number is $\mathrm{Fr} = |\mathbf{U}_0|/\sqrt{gL}$.
The super-Gaussian is a fairly realistic model of the submerged part of a hull shape, while avoiding having to specialise to a particular type of hull. Model ``ship'' pressure distributions for some aspect ratios are shown in Fig.~\ref{fig:hulls}.
When first set in motion, the ship creates an initial ring wave which propagates away. After some time the transient ring wave, $\zeta_t$, has disappeared from sight and only a stationary ship wave pattern behind the travelling ship, $\zeta_s$, remains. This is clear from Fig.~\ref{fig:patterns}, where the wave patterns are shown for increasing times after appearence, for different directions of motion atop a linear shear profile
in deep water.
For large times the transient surface wave $\hat{\zeta}_t$ at some point far from the origin will vanish as $t^{-1/2}$. This can be shown rigorously with path integral methods and the stationary phase approximation, but is also physically clear from noting that the full transient wave energy will eventually radiate through any vertical, circular control surface of radius $R$, and wave energy must thus fall off as $R_r(r)^{-1}$ for a ring wave of radius $\sim R_r$. Since wave energy of each Fourier mode moves outward in the far-field at a constant, $k$-dependent group velocity, $R_r\sim c_g t$, and since wave energy is $\propto \hat{\zeta}^2$, the time dependence $\hat{\zeta}_t\sim t^{-1/2}$ follows for large $t$.
\subsection{Practical calculation techniques for arbitrary velocity profiles}
\label{sec:approx}
To calculate the free--surface elevation
\eqref{zetat}
one needs to find the roots $\omega_\pm(\mathbf{k})$ of \eqref{dispR}, which is itself not closed since both $ \omega $ and $ w(z) $ are unknowns. Analytical results are in general not available, except for the simplest current varying linearly with depth.
There are several numerical or semi--analytical techniques that allow calculation of $\omega_\pm(\mathbf{k})$ for an arbitrary $\mathbf{U}(z)$ which we briefly review in this section.
\subsubsection{Simplest case: linear profile}\label{sec:linear}
Consider first the simplest case of a linearly depth--dependent current. This is the only known case where an explicit, analytical dispersion relation is available for all $\mathbf{k}$. This idealised case is therefore instructive for analysis since analytical results can be derived.
To calculate ship waves during steady motion, say, one might work in a frame of reference where the model ship is at rest, and the ship's velocity relative to the water surface is $-\mathbf{U}_0$ where $\mathbf{U}_0=[U_0,V_0]=|\mathbf{U}_0|[\cos\beta,\sin\beta]$ (see also Fig.~\ref{fig:agls}c). The current is unidirectional, i.e.,
$\mathbf{U}(z)=\mathbf{U}_0+S z\mathbf{e}_x$.
(This corresponds a ship moving in direction $\beta+\pi$ relative to the water surface. )
We define \cite{ellingsen14a}
\begin{equation}
\mathrm{Fr} _s=\frac{|\mathbf{U}_0|S}{g}.
\end{equation}
For the linear shear profile one obtains \cite{li15a,ellingsen14b}
\begin{subequations}\label{linrel}
\begin{align}
\omega_\pm =& \omega_1 \pm \sqrt{\omega_1^2+\omega_2^2};\\
\omega_1 =& \bk\cdot\bU_0 -{\textstyle\frac1{2}} S\tanh kh\cos\theta;\\
\omega_2^2 =& (S\cos\theta\bk\cdot\bU_0 +gk)\tanh kh -(\bk\cdot\bU_0)^2,
\end{align}
\end{subequations}
hence $\omega_+\omega_- = -\omega_2^2$, and
\begin{align}\label{linrel2}
\omega_\text{div} &=
\sqrt{gk\tanh kh+(S \tanh kh\cos\theta/2)^2}
\end{align}
Since $I_g=0$ when $\mathbf{U}''(z)=0$, \eqref{Fth} simply gives $F(\mathbf{k})=\tanh kh$. Determining $F$ and $\omega_\pm$ is sufficient for calculating all cases considered above, the most general case being \eqref{zetat} with \eqref{Hz}.
\subsubsection{The piecewise--linear approximation} \label{sec:pla}
A useful numerical scheme to this end is the piecewise--linear approximation (PLA), which was analysed in Refs.~\cite{smeltzer17,zhang05}, and which we will use herein to obtain numerical results. As described herein the PLA is restricted to unidirectional $\mathbf{U}(z)$; extension to shear currents changing direction is relatively straightforward. Alternative approximations to the dispersion relation are thereafter briefly discussed in section \ref{sec:altapprox}.
The piecewise--linear approximation (PLA), sometimes called the $ N-$layer model, utilises the fact that explicit solutions are available when the velocity profile is linear as discussed above. A smooth velocity profile $u(z)$ is approximated by a series of linear segments inside $N$ artificial layers, allowing the solution to the vertical velocity to be expressed explicitly within each layer and matched at the artificial layer boundaries. We provide further details in \ref{appx-2}. Following the derivation process in \cite{smeltzer17}, within the top layer the vertical velocity satisfies
\begin{align} \label{eq:w1}
w (\mathbf{k},t)
= & A_1 \sinh k(z+h_1) + B_1 \cosh k(z+h_1), \notag \\
& \text{for} -h_1<z<0,
\end{align}
in which $ h_1 $ is the thickness of the top layer and $ A_1$ and $ B_1 $ are coefficients depending on $\mathbf{k}$ and $t$, which are determined by the matching conditions at the $N-1$ layer interfaces and from free--surface and bottom boundary conditions. Inserting \eqref{eq:w1} into the first form of $ F(\mathbf{k}) $ in \eqref{Falt} yields
\begin{equation}\label{FPLA}
F(\mathbf{k}) = \dfrac{A_1\sinh k h_1 + B_1\cosh k h_1}{A_1\cosh k h_1 + B_1\sinh k h_1}
\end{equation}
evaluated at $t=0$.
The next essential step is to obtain solutions for $ \omega_\pm $, exact or approximate, and to determine $ A_1 $ and $ B_1 $ via the PLA procedure \cite{smeltzer17}. The PLA is particularly suitable for problems which are solved in the Fourier plane since it provides a rapid and accurate solution to the dispersion relation $\omega(\mathbf{k})$ equally well for all wavelengths, converging to the exact value as $N$ increases \cite{smeltzer17,zhang05}. For our numerical demonstrations we find that $4$-$5$ layers are typically enough at the $~1\%$ accuracy level.
\subsubsection{Alternative approximations to the dispersion relation}\label{sec:altapprox}
A simpler approach than the PLA can be obtained by evaluating $\omega_+(\mathbf{k})$ using an explicit, approximate dispersion relation. The accuracy of such approximations is not so easily predicted, however, and is different in different areas of the $\mathbf{k}$ plane.
A much used approximation which is accurate to within a few percent for all $\mathbf{k}$ in many cases, is the relation by Kirby \& Chen \cite{kirby89}
\begin{equation}\label{kirby}
\omega_+(\mathbf{k}) \approx \omega_0(k) + \int_{-h}^0\mathrm{d} z\frac{2\mathbf{k}\cdot\mathbf{U}(z)\cosh 2k(z+h)}{\sinh 2kh}
\end{equation}
where $h$ is the total depth of the flow and $\omega_0=\sqrt{gk\tanh kh}$ (note that this 3D generalization of the Kirby \& Chen expression also allows the direction of $\mathbf{U}$ to vary with $z$).
The approximate value for $\omega_+(\mathbf{k})$ is inserted into equations \eqref{zetat} via \eqref{Fom}.
We recently made progress on the question of analytical approximations to dispersion relations, deriving error estimates for \eqref{kirby} and also presenting a more robust alternative to \eqref{kirby} in Ref.~\cite{ellingsen17}. Two of us (YL \& S\AA E) have also developed and implemented another numerical method, a simple and promising alternative to the PLA based on direct integration of \eqref{eq:p} and \eqref{dispR} (manuscript in preparation).
\subsection{Transient wave resistance and radiation force}\label{sec:R}
A travelling ship imparts momentum to the water around it to create waves, giving rise to a wave radiation force acting on the ship in the opposite direction. In the absence of shear the wave radiation force always points sternwards for ships in rectilinear motion, and is called wave resistance, or wave--making resistance. Wave resistance typically accounts for more than 30\% of the energy consumption of ocean going vessels \cite{faltinsen05}.
We work in a reference frame where the ship is at rest, and the water surface moves at velocity $\mathbf{U}_0$ as shown in Fig.~\ref{fig:agls}c.
Following Havelock \cite{havelock17} the wave radiation force created by a travelling pressure distribution is the force exerted by the external pressure $\hat{p}_\text{ext}(\mathbf{r},t)$ acting on vertical projections of the moving surface $\hat{\zeta}(\mathbf{r},t)$. The force along unit vector $\mathbf{e}_f$ acting on horizontal area $\mathrm{d}^2 r$ at $\mathbf{r}$ is thus
\begin{equation}
\mathrm{d} f(\mathbf{r},t) = \hat{p}_\text{ext}(\mathbf{r},t)(\mathbf{e}_f\cdot\nabla)\hat{\zeta}(\mathbf{r},t)\mathrm{d}^2 r.
\end{equation}
A ship travelling at an oblique angle with a sub--surface shear current will in general radiate waves asymmetrically around its line of motion, and the radiation force will consequently have both a sternward and a lateral component. The two components are derived with the methods laid out in \cite{li16}, to yield
\begin{align}\label{R}
\begin{array}{c}R_{\parallel}(t)\\R_{\perp}(t)\end{array}=&-\frac 1{U_0} \int \mathrm{d}^2r \hat{p}_\text{ext}(\mathbf{r},t)
\left(\begin{array}{c}\mathbf{U}_0\\\mathbf{e}_z\times\mathbf{U}_0\end{array}\right)\cdot\nabla \hat{\zeta}(\mathbf{r},t)\notag\\
=&-\mathrm{i}\int\frac{\mathrm{d}^2 k}{(2\pi)^2}
\left( \begin{array}{c}k\cos\gamma\\k\sin\gamma\end{array}\right)
p_\mathrm{ext}^*(\mathbf{k},t)\zeta(\mathbf{k}
\end{align}
where an asterisk denotes the complex conjugate and $\parallel$ and $\perp$ denote sternward resistance and lateral radiation force towards starboard (towards the right), respectively.
The transient radiation forces may thus be evaluated by inserting $\zeta_t(\mathbf{r},t)$ from \eqref{eq:tranW} into \eqref{R}, giving
\begin{subequations}
\begin{align}\label{Rt}
\begin{array}{c}R_{\parallel,t}(t)\\R_{\perp,t}(t)\end{array}&=-\frac \mathrm{i}{8\pi^2\rho}\lim_{\epsilon\to 0}\int_{-\pi}^\pi \mathrm{d} \gamma I(\gamma,t); \\
I(\gamma,t)&=\int_0^\infty \mathrm{d} k \frac{k^3
|p_\text{ext}(\mathbf{k},t)|^2
F(\mathbf{k}) }{\omega_\text{div}(\mathbf{k})} \left( \begin{array}{c} \cos\gamma\\\sin\gamma\end{array}\right)
\notag \\
&\times \left(\frac{\mathrm{e}^{-\mathrm{i}\omega_+t}}{\omega_+-\mathrm{i} \epsilon}-\frac{\mathrm{e}^{-\mathrm{i}\omega_-t}}{\omega_--\mathrm{i} \epsilon}\right) .\label{I}
\end{align}
\end{subequations}
Expressing radiation forces on the form \eqref{Rt} is useful for analytical purposes. For numerical purposes we use \eqref{R} more directly using a fast Fourier transform (FFT) method.
The static part of the wave resistance is obtained by inserting $\zeta_s$ into \eqref{R}. We refer to \cite{li16} for further details on the evaluation of the static part of the wave resistance.
\begin{figure}[htb]
\includegraphics[width=.9\columnwidth]{figure5.pdf}
\caption{The intrinsic wave frequency $\tilde{\omega}_+(k_0(\gamma),\gamma)$, in units of $\sqrt{L/g}$, which solves the dispersion relation in direction $\gamma$ under different conditions of a linear shear profile. }
\label{fig:omega}
\end{figure}
\subsubsection{Wave resistance oscillations}\label{sec:Rosc}
The transient behaviour of the wave resistance after the ship is set in motion, is to oscillate around its ultimate static value, at a frequency which varies greatly with direction or motion as well as shear strength. We will now explain what decides the oscillation frequency.
The integral \eqref{I} is given solely by the contribution from the poles (infinitesimally close to) where $\omega_\pm(\mathbf{k})$ are zero. Since $\omega_+(\mathbf{k})$ and $\omega_-(\mathbf{k})$ are related through relation \eqref{omrel}, and we are free to replace $\mathbf{k}\leftrightarrow-\mathbf{k}$ under the integral sign, considering the zeros of the positive frequency $\omega_+(\mathbf{k})$ is sufficient. Taking the $k$ integral first as written out in \eqref{Rt}, the pole picks out a value $k_0(\gamma)$ so that
\begin{equation}
\omega_+(k_0(\gamma),\gamma) = 0.
\end{equation}
Thus the particular frequency is picked out which satisfies the dispersion relation, which is to say that only waves which are able to propagate towards infinity along direction $\gamma$ may contribute to the wave resistance.
When $t$ grows large (while keeping $r$ constant), the exponential factor $\exp [-\mathrm{i}\omega_+(k_0(\gamma),\gamma)t]$ in the integrand of \eqref{I}, and is therefore dominated by the contribution from the value of $\gamma$ where the phase is stationary, that is, the value of $\gamma$ where
\begin{equation}
\partial_\gamma\omega_+(k_0(\gamma),\gamma) = 0.
\end{equation}
Some time after $t=0$, the transient contribution to the wave resistance will therefore oscillate in time with the frequency of a stationary point, a maximum or minimum of $\omega_+(k_0(\gamma),\gamma)$ with respect to $\gamma$.
Let intrinsic frequencies be denoted with a tilde,
\begin{equation}
\tilde\omega = \omega-\bk\cdot\bU_0 .
\end{equation}
For the case of a linear shear current, we plot
$\tilde\omega_+(k_0(\gamma),\gamma)$ in units of $\sqrt{g/L}$
as a function of $\gamma$ in Fig.~\ref{fig:omega};
$L$ is a characteristic length of the wave disturbance to be specified in particular examples below.
We see that in all cases there is a stationary point at $\gamma=-\pi$. In the most shear--assisted direction ($\beta=0$), this frequency is enhanced compared to no shear, giving a faster oscillation of the wave resistance, whereas the opposite is the case in the maximally shear inhibited direction ($\beta=\pi$), where the oscillation can become very slow. For shear--assisted motion there are also two other stationary phase points at angles either side of $\gamma=\pi$, as is evident in Fig.~\ref{fig:omega}. Notably, the presence of shear which inhibits motion can dramatically decrease the oscillation frequency compared to still water, even at moderate shear.
\section{Numerical results}
\label{sec:num}
In this section we present numerical calculations of transient wave resistance on different model ships. While retaining generality by not specialising to particular real hull shapes, we have emphasised realism: a reasonably realistic model is used for the shape of the ship hull, and calculations are performed for a real velocity profile measured in the Columbia River estuary, where there is high traffic of vessels of many types. Parameters for vessel length and beam are taken from real ships known to travel in these waters.
The choice of the Columbia River delta for our data is primarily due to the excellent shear profile data available \cite{kilcher10}, although the location is also particularly apt for studies of ship wave effects. Thousands of ships ranging from carrier ships of more than $1000$ ft to small boats, are piloted up and down the Columbia river each year, in waters which are considered particularly trecherous, sometimes referred to as the Graveyard of the Pacific.
Following \cite{li16,benzaquen14} we plot wave resistance relative to the constant
\[
R_0 = \frac{p_0^2}{2\pi^3 \rho g}.
\]
\subsection{Linear velocity profile}\label{sec:numlin}
In order to better highlight the underlying physics of the effect of shear on wave resistance, we begin by considering the simplest shear flow, which varies linearly as a function of depth, $U(z)=U_0+Sz$. Realistic shear profiles are considered in Section \ref{sec:genprof}.
\begin{figure*}[tb]
\begin{center}
\includegraphics[width=.8\textwidth]{figure6.pdf}
\end{center}
\caption{Transient wave resistance on a ship set suddenly in motion as a function of nondimensional time $T=t\sqrt{L/g}$, for different cases where a linearly depth-dependent shear current is present in deep water.
The ``ship'' is modelled as an ellipsoidal, super-Gaussian surface pressure of aspect ratio $6$ and $L=1$ (arbitrary units), moving with $\mathrm{Fr} =0.3$. Note that the abcissa is scaled differently for $T>20$. Note furthermore that $R_\parallel$ scales linearly with $L$, which is arbitrary in this scale--free system, hence so is the scaling of the ordinate axis.}
\label{fig:resistance}
\end{figure*}
\begin{figure}[tb]
\includegraphics[width=\columnwidth]{figure7.pdf}
\caption{Same as Fig.~\ref{fig:resistance}, but for the transient lateral wave radiation force $R_\perp$, for motion normal to a the shear current in a reference frame following the water surface; $\beta=\pi/2$ (see Fig.~\ref{fig:geom}). The scaling of the ordinate is arbitrary (see Fig.~\ref{fig:resistance}).}
\label{fig:lateral}
\end{figure}
\subsubsection{Suddenly starting ship}
For the simplest, linearly varying velocity profile considered in section \ref{sec:linear} we calculate the transient wave resistance for a ship modelled as in equation \eqref{p},
whose velocity goes suddenly from zero to a constant value $V$. While idealised, this models a starting ship without the need for further parameterisation of the acceleration phase. An example of what the transient wave resistance looks like is shown in figure \ref{fig:resistance}. The model ship is elliptical with aspect ratio $6$ and length $L=1$ (arbitrary units since the problem is intrinsically scale--free), and calculation is performed for $\mathrm{Fr} =0.3$ and shear strengths varying from $\mathrm{Fr} _s = 0$ to $0.8$.
The oscillation frequencies of the transient wave resistance are found to agree well with the stationary phase values of $\omega_+(k_+(\gamma),\gamma)$ in figure \ref{fig:omega} as expected.
The transient wave resistance is seen to go through a sharp peak shortly after the ship is set in motion, and then relax in an underdamped manner towards its steady--motion value. For shear--assisted motion ($\beta=0$), the initial peak can be much higher than its static value, whereas this effect is weaker in the case without shear ($\mathrm{Fr} _s=0$) and for shear--inhibited ship motion. Letting the shear vary from strongly inhibiting (high $\mathrm{Fr} _s$, $\beta=\pi$) via no shear to fairly strongly assisting, we see that the transient oscillations increase both in amplitude and frequency, whereas the static wave resistance decreases. An interesting observation is that for very strongly motion-assisting current ($\mathrm{Fr} _s=0.8$ in this case), the total wave resistance can actually be negative during some time intervals, since oscillation amplitudes are large and the static wave resistance correspondingly small.
Both the difference in oscillation frequency and the magnitude of the steady motion wave resistance can be understood by considering the relative values of phase velocity and group velocity in different directions of wave propagation. A detailed discussion of this may be found in Ref.~\cite{ellingsen14b}. For a linear shear current, where the dispersion relation \eqref{linrel} is known analytically, one finds that in a reference frame following the free surface, the group velocity is quite similar in all directions of motion, whereas phase velocity can differ greatly. In shear--inhibited directions dispersion is weakened and an emitted wave group will retain its initial shape and width to a greater extent than in quiescent water. The opposite is the case for shear--assisted wave propagation; here the phase velocity can far exceed the group velocity, so wave groups quickly spread and have a rapidly changing, volatile appearence.
When the ship suddenly starts, an initial ring wave is emitted, as seen in Fig.~\ref{fig:patterns}. Wave resistance will continue to oscillate for as long as this ring wave remains in the ship's near--zone. The fact that group velocity is fairly isotropic means that it takes approximately the same time for the ring wave to disappear from sight, matching the observation that the oscillations in Fig.~\ref{fig:resistance} die off at a similar rate in all cases. The frequency of oscillation, however, depends on the phase speed of the transient waves \emph{within} the ring wave group, and the higher phase velocity for shear--assisted propagation means faster oscillations, as also observed in Fig.~\ref{fig:resistance}, and explained in connection with Fig.~\ref{fig:omega}.
Finally, we found in Ref.~\cite{li16} that the effect on shear on wave resistance is, in a rough sense, to effectively change the Froude number to a value based on the ship velocity relative to some depth--average current speed rather than its surface value.
The Froude number is effectively lowered in shear--assisted motion, and increased in shear--inhibited motion.
A detailed discussion is found in section \ref{sec:realsudd} where we compare a real velocity profile to a linear approximation in this respect. Since the general trend is that wave resistance increases with increasing $\mathrm{Fr} $ for $\mathrm{Fr} \sim0.3$, this explains why the resistance in steady motion is typically decreased for shear--assisted motion and increased for shear--inhibited motion. However, this does not always hold true, due to interference effects between waves from bow and stern.
We go on to calculate the transient lateral radiation force for the same ship, shown in Fig.~\ref{fig:lateral}.
The Froude numbers $0.25$ and $0.3$ are chosen as realistic examples. The ship motion is now across the shear current, $\beta=\pi/2$ as defined in figure \ref{fig:geom}. For an aspect ratio of $6$ the lateral force is roughly half the magnitude of the sternward force. We find the relative magnitude of lateral to sternward force to vary strongly with Froude number and aspect ratio, as indicated for the former case by the large effect of lowering $\mathrm{Fr} $ from $0.30$ to $0.25$.
\subsection{General, realistic velocity profiles}\label{sec:genprof}
\begin{figure}[tb]
\includegraphics[width= .9\columnwidth]{figure8.pdf}
\caption{Transient wave resistance for motion in the shear assisted (a) and inhibited (b) directions atop the measured current in the Columbia River delta, as a function of nondimensional time $T=t\sqrt{g/L}$. Three ships are modelled with equation \eqref{p} with dimensions as given in Table \ref{tbl:ships}.
The wave resistance in quiescent waters is shown for comparison. Inset to (a): measured Columbia River velocity profile $U_{\mathrm{RISE}}(z)$ \cite{kilcher10} approximated with a $6$th order polynomial, in a reference frame moving with the surface current. The legend applies to both a) and b).}
\label{fig:R0}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=.9\columnwidth]{figure9.pdf}
\caption{Wave resistance force in steady motion for Ship 2 (tugboat) as a function of Froude number for the maximally shear assisted ($\beta=0$) and inhibited ($\beta=\pi$) directions of motion. a) The Columbia River velocity profile, b) Linearly varying profile $U(z)=U_0+Sz$ with $\mathrm{Fr}_s\equiv U_0S/g=0.4$.}
\label{fig:RofF}
\end{figure}
We now compute the transient wave resistance using a real, measured velocity profile. The shear current is that measured by the RISE project, a tidal current in the mouth of the Columbia River \cite{kilcher10}\footnote{Since measurements begin at $2$m depth, we presume this point to be at the surface, thus offsetting all data by $2$m. This should be a conservative procedure since shear strength increases closer to the surface.}. Buoyant fresh water from the river creates a strong surface jet as it enters the salt water of the Pacific Ocean. We approximate the measured data with a 6th order polynomial which is then subjected to the piecewise-linear procedure to calculate the dispersion relation numerically, as described in section \ref{sec:basic} and detailed in \ref{appx-2}.
The current profile $U_{\mathrm{RISE}}(z)$ in a reference frame where the surface current is zero is shown in the inset of figure \ref{fig:R0}a. We model various ships using Eq.~\eqref{p} with dimensions $L$ (length) and $b$ (beam) representative of typical vessels traveling at the Columbia River mouth, tabulated in Table \ref{tbl:ships}.
\begin{table}[tb]
\begin{center}
\begin{tabular}{cllllll}
\hline
ID&Ship Type&Length & Beam & Speed & Aspect \\
&& $L$ [m] & $b$ [m] & [Knots] & ratio\\ \hline
1& Bulk carrier & 170 & 28 & 11.9 & 6.07 \\
2& Tugboat & 32 & 10.4 & 10.3 & 3.08 \\
3& Fishing boat & 19 & 6 & 8.0 & 3.17 \\ \hline
\end{tabular}
\caption{Parameters of the modeled ships, chosen as representative dimensions from boat traffic on the Columbia River. Real-time data on vessels in these waters is available at http://www.columbiariverbarpilots.com. Froude numbers for ships $1,2,3$ are $0.15,0.3$ and $0.3$, respectively.}
\label{tbl:ships}
\end{center}
\end{table}
\subsubsection{Suddenly starting ship}\label{sec:realsudd}
Results for transient sternward wave resistance for a ship starting suddenly in maximally shear-assisted and shear-inhibited directions of motion (corresponding to upstream and downstream motion in the Columbia delta, respectively) are shown in figure \ref{fig:R0}.
Two ships are modelled,
a bulk carrier ship, and a smaller vessel typical of a tugboat; Ships 1 and 2 in Table \ref{tbl:ships}, respectively.
The wave resistance in quiescent waters is shown for comparison.
The behaviour
of the smaller ship ('Ship 2')
is similar to that observed for the simple linear shear current, with wave resistance exhibiting a sharp peak shortly after the ship is set in motion, whereupon it relaxes in an underdamped way to the steady motion value with a frequency which is higher for shear assisted than for shear inhibited motion. Fluctuations are stronger for shear assisted (upstream) motion as was also noted in Fig.~\ref{fig:resistance}, and amount to transient variations in the order of $10\%$ of the static value in this case. The wave resistance of the larger vessel (`Ship 1') approaches an insignificantly small value at large times, attributed to the lower Froude number ($0.15$) for this modelled vessel.
The most interesting observation made in Fig.~\ref{fig:R0} might concern the steady motion value of wave resistance. Untypically, wave resistance is increased compared to quiescent waters both for shear--assisted and shear--inhibited ship motion. This appears to run counter to lessons learned from a previous, much simpler and less realistic model study \cite{li16}, where shear--assisted motion was always found to decrease wave resistance in this Froude number range. The reason is that our present, more realistic ship model \eqref{p} has a sharper bow and stern than the circular ``ship'' considered in \cite{li16}, leading to interference effects between bow and stern waves such as are found for real ships. Indeed these interferences must be taken into account when choosing optimal operational speed in ship design \cite{schneekluth87}. We plot the Froude number dependence of the steady motion wave resistance for different $\mathrm{Fr}$ for the Colubia current profile in Fig.~\ref{fig:RofF}a for the tugboat (Ship 2). The plot clearly demonstrates that wave resistance in steady motion depends very strongly on direction and Froude number. $\mathrm{Fr} =0.3$, the speed of Ship 2 in Fig.~\ref{fig:R0}, is a special case where shear increases wave resistance in both directions. Increasing the velocity a little to $\mathrm{Fr} =0.33$, a very different conclusion is reached: here, shear-inhibited wave resistance (ship travelling downstream) is more than a factor $3$ greater than in the opposite direction.
\begin{figure}[htb]
\includegraphics[width= .9\columnwidth]{figure10.pdf}
\caption{
a) Transient lateral radiation force per unit ship length $R_\perp/(R_0L)$ for motion normal to the measured current in the Columbia River delta, in a reference system where the free surface is at rest ($\beta=\pi/2$), as a function of nondimensional time $T=t\sqrt{g/L}$. Three ships are modeled using \eqref{p} with dimensions $L$ and $b$ and Froude number $Fr$ as indicated. b) Transient lateral radiation force relative to transient wave resistance for the two smaller modeled ships. The legend applies to both a) and b).
}
\label{fig:R90}
\end{figure}
\begin{figure*}[htb]
\includegraphics[width= 2.0\columnwidth]{figure11.pdf}
\caption{
Transient wave resistance as a function of nondimensional time $T=t\sqrt{g/L}$ for a ship beginning a circular manoeuvring motion atop the measured shear current in the Columbia River delta. The ship dimensions are typical of a tugboat operating in these waters, `Ship 2' in Table \ref{tbl:ships}, initiating a turn of radius $4L$ at $T=0$, from having traveled in a straight path in the shear--assisted direction (upstream). The path is circular thereafter, as seen in a reference system where the water surface is at rest. The angle the ship has turned is shown above the figure. The situation at $90^\circ$ is shown in the inset for illustration. Also shown is the same manoeuvre in quiescent waters ($U(z) = 0$).
}
\label{fig:turn90}
\end{figure*}
Corresponding results for the lateral radiation force for motion across the shear current ($\beta=\pi/2$, measured in a reference frame in which the water surface is at rest) is shown in figure \ref{fig:R90}
for the three different ships in Table \ref{tbl:ships}. In order to make the values comparable, we divide the force by the length of the ship. The lateral radiation force shows similar oscillations for short times as the sternward resistance in Fig.~\ref{fig:R0}a, with the exception of the large carrier ship (Ship 1) which displays far stronger transient oscillations initially. Indeed, while the sternward resistance force is likely to be negligible for Ship 1, this needs not be the case for the early transient shortly after start.
In Fig.~\ref{fig:R90}b we show the lateral radiation force relative to the sternward resistance force for cross--current motion, for Ships 2 and 3. The relative strength has very weak oscillations, but a highly conspicuous trait is how the relative strength of the transient force is more than twice as strong just after appearance of the ``ship'' compared to its asymptotic value, about $50-60\%$ percent of the transient sternward force at the time of the initial peak that is present in both force components. Again this indicates that the transient behavior of the lateral force could well have a bearing on seakeeping performance during manoeuvering, when transient waves will be emitted by the ship.
We note furthermore that when stationary conditions have been reached, the radiation force is approximately $20\%$ of the sternward component. This is a significant laterally directed force which must be compensated by steering (it is not to be confused, of course, with the lateral drag force which will also be present due to the shear flow between surface level and the ship's draught, a separate question not studied here. With no shear there is neither a net lateral drag nor radiation force when $\beta=\pi/2$.)
The simplicity of working with the linearly dependent velocity profile as a model for a real current makes it tempting in practice to eschew the need to calculate $\omega(\mathbf{k})$ for a general shear flow, and instead approximate the real profile by a linear one with a representative constant shear. However, if we were to approximate the Columbia profile by a linear profile with a shear approximately that at the water surface --- giving $\mathrm{Fr} _s\sim 0.4$ for our parameters --- one could make a very great error in calculating the steady-motion wave resistance. In Fig.~\ref{fig:RofF}b we plot the steady--motion wave resistance as a function of $\mathrm{Fr}$ using this model. It is clear that while the trend and general behaviour is similar, the rapid variation of $R_\parallel$ with $\mathrm{Fr}$ for $0.2\lesssim\mathrm{Fr}\lesssim 0.4$ means the error can be several hundred percent. Clearly a better job can be made with a better choice of $\mathrm{Fr}_s$, yet choosing a sufficiently good value in practice (if such exists) will require the use of knowledge of the full velocity profile and moreover be specific to each vessel. In our opinion this may not be any simpler nor numerically cheaper than a full calculation such as we have performed, and for which an effective calculation tool is already now developed.
We note, however, the possibility that a two--layer model might be a compromise which is the best of both worlds. In such a model a surface layer is given one constant shear value, and deeper waters another. It is well suited for modelling a surface shear layer due to wind or tides for many practical purposes. Such a model is analytically tractable while containing the key parameter of the vertical extent of the surface shear layer, whose relation to the ship length is a determining parameter. Analysis of such a model in the context studied here is beyond our present scope; the dispersion relation that can be used directly in the formalism of Section \ref{sec:basic} may however be found in Ref.~\cite{smeltzer17b}.
\subsubsection{Turning ship}
Analysis of a suddenly moving ship yields insight into transient wave resistance forces
due to sudden changes in velocity along a straight course.
It is of interest to consider another
example of a ship manoeuvre; a turning motion. Figure \ref{fig:turn90} shows the wave resistance for a ship initially traveling along a straight path upstream in the Columbia River delta (shear--assisted direction), which begins a circular turning manoeuvre of radius $4L$ at $T = 0$. The forward velocity $\mathrm{Fr}=0.3$ remains unchanged through the manoeuvre.
We consider as example a typical tugboat operating in these waters, Ship 2 in Table \ref{tbl:ships}.
In order to given an impresson of all different directions of motion, we let the ship do a full $360^\circ$ turn; a snapshot at $90^\circ$ is shown in the inset. The same ship manoeuvre in quiescent waters is shown for comparison.
All graphs display certain oscillations at different times during the manoeuvre, due to the sudden change in lateral acceleration after $T=0$, and later because the ship encounters its own previously emitted waves.
In quiescent water the lateral radiation force fluctuates around a constant value of, in this case, approximately $40R_0$ due to the now asymmetric wave field; another way of seeing it is that the turning ship must accelerate water towards the centre of the arc, resulting in an outwardly directed lateral added mass force. The sternward force without shear also fluctuates around a constant as it should.
A different behaviour is observed for both force components, however, when the measured Columbia River shear current is present. Both resistance and lateral force vary greatly throughout, both peaking at around twice their quiescent value, and the lateral force at times dropping to zero and even small negative values. For a ship to follow such a path with precision will thus require considerably greater skill than in quiescent water, having to account for the changing lateral and sternward forces. The lateral force can also reach more than $50\%$ of the resistance force for a part of the circle with our parameters, typical of boat traffic in the area, by no means a small force in a manoeuvring context.
\section{Conclusions}
We have studied the wave radiation forces, including wave--making resistance, for different model ships in a real, measured current in the Columbia River delta. We calculate transient wave resistance on a ``ship'' modeled as a traveling pressure distribution in the form of an elliptic super--Gaussian.
Choosing values of length/beam typical of smaller vessels (tugboats, fishing boats) we find that wave resistance can vary drastically depending on direction of motion, upstream or downstream, showing a strong dependence on Froude number. For typical Froude numbers --- $\mathrm{Fr}\sim 0.2$ to $0.4$ --- we find that wave resistance can differ by more than a factor $3$ between upstream and downstream motion. Appropriate choice of vessel velocity can thus make a large difference to resistance in strongly sheared waters.
When there is an oblique angle between the ship's line of motion and the shear current, the emitted ship wave pattern will be asymmetric, with more waves propagating to one side than the other. The total wave radiation (or wave--making) force then also has a lateral component. For our example model ships representative of tugboats or fishing boats, the lateral force was found to be approximately $20\%$ of the sternward resistance force for a ship in steady motion.
We also study the transient behaviour of wave radiation forces acting on ships which change their velocity. As a simple example we consider ships that are set suddenly in motion. Both components of the wave radiation force undergo an initial peak as an initial ring wave is created, whereupon they oscillate in an underdamped manner towards their steady--motion values. For motion across the shear current the lateral force is found to have a stronger initial peak, and the lateral force momentarily reaches more than 50\% of the value of the sternward force just after motion commences.
The general trend for typical small--ship operational Froude numbers is that compared to quiescent water, wave resistance decreases for upstream (shear--assisted) ship motion, and increases for downstream (shear--inhibited) motion, although interference effects between bow waves and stern waves can alter this for certain Froude numbers.
We also considered a circular manoeuvring motion atop the Columbia River current seen from a reference system following the water surface, for a small ship (tugboat). Unlike on quiescent water were both resistance and lateral force are constant through the motion (modulo small oscillations due to encountering the ship's own waves), these vary greatly through the circular path on the Columbia River mouth. Variations of amplitude of approximately $100\%$ of the quiescent values of the forces are found. For a ship to follow such a path with precision will thus require considerably greater skill. The lateral force can also reach more than $50\%$ of the resistance force for a part of the circle with our parameters, typical of boat traffic in the area.
The second main achievement reported in this manuscript is the development of a theory that allows calculation of waves from a general, time-dependent applied surface pressure acting on the free surface of a horizontally directed shear current which may vary arbitrarily with depth in both direction and magnitude. We present a framework which provides the means to effectively calculate ship waves and wave resistance without undue difficulty. The theory is based on deriving the response of a water surface satisfying an arbitrary dispersion relation, to an impulsive applied pressure. The wave pattern is then calculated as the integral of emitted waves at all previous times. It is necessary to devise a scheme to obtain the dispersion relation numerically; in this paper we used the piecewise--linear approximation \cite{smeltzer17}, but several other options are available.
\subsection*{Acknowledgements}
S{\AA}E is funded by the Norwegian Research Council (FRINATEK), project number 249740. We are grateful to Peter Maxwell for improvements to the PLA numerical code.
|
1,116,691,498,492 | arxiv | \section{Introduction}
Phase resolved energy distributions of pulsars' signals
are important to obtain information on the
radiation processes and geometry of the emission regions in the
magnetosphere.
At $\gamma$-ray energies, at variance with other energy bands, the
three brightest sources (Vela, Crab and Geminga) show remarkably
similar pulse structures with two main peaks at a large phase
separation ranging from 0.4 to 0.5.
This pattern is confirmed by the very recent discovery of pulsed
emission from PSR~J2021+3651 (Halpern et~al. 2008), that has a peak
separation of 0.47.
The double peak structure of the Crab Pulsar (PSR B0531+21) is well
observed across the entire electromagnetic spectrum, but the intensity
ratio between the two peaks changes with energy.
In particular, in the X and soft $\gamma$-ray ranges the emission of the
second peak (P2) becomes higher than the first one (P1), and a relevant
emission from the region between the two peaks (interpeak or bridge, Ip)
increases like P2.
This behaviour is observed up to a few MeV, where the pulse shape turns
almost sharply to be similar, although not equal, to the optical light
curve.
On the basis of a large collection of data, covering
the frequency interval from the optical to the GeV band,
we proposed a model (Massaro et al. 2006a, hereafter MCCM) able to
describe the spectral and phase distributions by means of a double
two-component model.
The energy spectra of these components are not described by a simple
power law, because of the continuous spectral steepening towards high
energies.
We found that a very satisfactory model is a parabolic law in a
double-logarithmic plot, corresponding to a log-normal spectral
distribution.
A useful test to verify the goodness of this multicomponent model, and
in particular the existence of the two high energy components, is the
study of the pulse shape at energies higher than a few GeV, where P2
is expected to be again the dominant feature as in hard X/soft $\gamma$
rays.
Some hints in this direction are given by the EGRET pulse profile
(Thompson 2004), but statistics above 5 GeV are so poor that no firm
conclusion can be obtained.
Very recently, the MAGIC telescope (e.g. Lorenz 2004) has detected
pulsed emission from the Crab (Aliu et al. 2008),
at energies above 25 GeV and with 6.4 standard deviations significance,
showing the very well established two peak profile, with similar
amplitudes of the two peaks.
In this Note we compare the MAGIC data to the predictions of the MCCM
model, and use them to constrain the high energy components.
The model is synthetically described in Sect. 2, the extension to MAGIC
results is presented in Sect. 3 and some hypotheses about the origin of
these components on the basis of some recent works on the high energy
emission from young pulsars are discussed in Sect. 4.
\section{The multicomponent model of Crab pulsar }
\begin{figure*}
\includegraphics[width=0.5\textwidth]{fig1.eps}
\includegraphics[width=0.5\textwidth]{fig2.eps}
\caption{
\emph{Left panel:} The ratio between the fluxes of P2 and P1 phase regions (P1:
-0.06--0.04; P2: 0.32--0.43), compared to the predictions of the model.
The black data points come from various experiments (Kuiper et al. 2001),
while the red point at GeV energies has been obtained from MAGIC data.
The two extrapolations above 1 GeV correspond to different values
of the cut-off energy of the $C_{O\gamma}$ spectrum: 15 GeV (solid red
line) and 11 GeV (dashed blue line). The $C_{X\gamma}$ cut-off energy
is fixed at 15 GeV.
\emph{Right panel:} Broadband spectral energy distribution of the total averaged pulse
and of the four components:
dashed line: $C_{O}$; dash-dotted line: $C_{X}$; dot-dot-dashed line:
$C_{O\gamma}$; dash-dash-dotted line: $C_{X\gamma}$.
Data points and SED of individual components are the same of the
MCCM model with the addition of the new MAGIC data
(black square, where the shaded area represents the systematic error on the spectral fits performed by Aliu et al., 2008).}
\label{figure1}
\end{figure*}
\subsection{The two-component model: optical to hard X-rays}
As early presented in Massaro et~al. (2000), the spectral and phase
changes of Crab X-ray pulse shape are well reproduced by two components.
The first component, called $C_O$, is assumed to have the same pulsed
profile observed at optical frequencies, while the second component,
$C_X$, is described by an analytical model to reproduce the observed
pulse profiles.
The latter component dominates at the interpeak (Ip) and second peak
(P2) phase regions.
This choice can be justified also from theorical models like that proposed
by Eastlund et~al. (1997), who considered the synchrotron emission from
marginally clamped electrons in a shell at the boundary of the closed
magnetosphere and calculated pulse profile very similar to $C_O$.
By means of spectral fits of BeppoSAX data, we obtained that the $C_O$ and $C_X$ components
were described by a log-parabolic spectral law,
\begin{equation}
F(E) = KE^{-(a+b~\mathrm{Log}E)}
\end{equation}
where $E$ is the energy in keV, $K$ is the photon flux at 1~keV and
the parameter $b$ describes the ``curvature'' of the log-parabola.
Best fit estimate of the parameters allowed to derive the peak energies
of the spectral energy distributions (SED), $E_p = 10^{(2-a)/2b}$,
that were found at 12 keV and 178 keV for $C_O$ and $C_X$,
respectively, and $b$ was found the same for both components and equal
to 0.16.
\subsection{The model at $\gamma$-ray energies}
Observations performed by the COMPTEL and EGRET experiments onboard the
\emph{Compton Gamma Ray Observatory} (CGRO; Kuiper et~al. 2001; Thompson
2004) provided light curves above 10 MeV of a good statistical quality,
which show that the pulse shape returns to be similar to that of $C_O$,
i.e. the optical/soft X-ray one, although some minor differences are
present, for instance in the shape of P2.
At energies higher than $\sim$500 MeV there is some indication that the
emission from Ip and P2 appears to increase, in analogy with the X-ray
band.
Left panel of Figure \ref{figure1} shows the energy evolution of the P2/P1 flux ratio,
computed in the phase intervals defined by Kuiper et~al. (2001) that are
reported in the caption.
To model the $\gamma$-ray emission and the change of the pulse shape,
MCCM assumed that there are two more, high-energy spectral components,
$C_{O\gamma}$ and $C_{X\gamma}$, with the same pulse shape of the
corresponding lower-energy components and with spectral distributions
also given by Eq. (1).
This extended model has six new parameters, i.e. the peak energies,
curvatures and normalizations of the $C_{O\gamma}$ and $C_{X\gamma}$
components, that should be determined by data fitting.
Given the statistical quality of CGRO data, the resulting estimates
are much more uncertain than in the X-rays.
Therefore, we assumed that the curvature parameters were equal
to the $C_O$ and $C_X$ ones ($b = 0.16$), and adjusted the normalizations
and peak energies to reproduce the observed total (phase-averaged)
spectrum.
Peak energies of $C_{O\gamma}$ and $C_{X\gamma}$ were found to lie
around 300 MeV and 2 GeV, respectively.
To be consistent with the upper limits to the TeV pulsed emission
(e.g. Lessard et~al. 2000) we added also an exponential cutoff to both
$C_{O\gamma}$ and $C_{X\gamma}$, at the energy $E_c=15$ GeV.
This value was not compelled by observational or physical reasons,
but was simply a guess to take into account the upper limits.
The MCCM model was then able to reproduce both the broadband energy
spectrum of the total pulse (Figure~\ref{figure1}, right panel), and the spectra of
the P1, Ip and P2 phase intervals.
In the left panel of Figure~\ref{figure1} we plotted also the fitted and extrapolated
P2/P1 ratio for two values of the $C_{O\gamma}$ cut-off energy, i.e.
11 and 15 GeV.
This ratio depends on the normalizations and shapes of the $C_O$ and
$C_X$ components in the proper phase ranges, and this is not immediately
apparent in the phase averaged spectrum of Figure \ref{figure1} (see MCCM
for details).
\section{The pulse shape and spectrum in the GeV band}
The pulse shape observed above 25 GeV by the MAGIC telescope
(Aliu et al. 2008) is shown in Figure~\ref{magic},
where it is normalized to the counts of the first peak.
Despite the statistical quality of data, is well apparent that
P2 has an amplitude comparable and possibly higher than P1,
in agreement with the trend barely apparent from the EGRET data
(Thompson 2004).
In Figure~\ref{magic} it is also plotted the pulse shape obtained
from our model, with the same parameters' values used in MCCM, for
an energy of 25 GeV.
The cut-off energy (15 GeV) is the same for both the $C_{O\gamma}$
and $C_{X\gamma}$ components.
The amplitude of the peaks is very well reconstructed; the pulse
shape in the MAGIC band predicted by our model has small variations
with energy, with the height of P2 that varies by about 15\% between
25 and 60 GeV.
The spectral distribution in the MAGIC range is still poorly determined,
mainly because of the uncertainties in its energy scale.
Aliu et al. (2008) combined the very high energy MAGIC data with the
COMPTEL and EGRET points and evaluated that a possibile exponential cut-off
can be at $E_{c} = 17.7 \pm 2.8 \pm 5.0$ GeV, thus compatible to the value assumed
in the MCCM model.
However, the CGRO-MAGIC data are compatible with both an exponential or a super-exponential shape,
with a preference for the former.
We reported the flux estimate in our SED model (Figure~\ref{figure1}, right panel)
and found that it is in a substantial agreement with the extrapolation which was
introduced to take properly into account the EGRET data.
Finally, note that in the MCCM model the flux above $\sim$10 GeV
is mainly due to the $C_{X\gamma}$ component (Figure~2).
This is not in contrast with the evidence of P1 in the pulse profile,
because its flux is concentrated in a quite narrow phase interval,
and $C_{X\gamma}$ has however a wide pedestal extending below P1
(Figure~\ref{magic}).
The good matching of the predicted pulse shape with the MAGIC one
implies that the cut-off energy should be similar for $C_{O\gamma}$
and $C_{X\gamma}$, because a lower cut-off for $C_{O\gamma}$ would
imply an higher P2/P1 ratio (as shown in Fig.~\ref{figure1}) and
therefore a higher amplitude for the second peak.
One of the assumptions of the MCCM model was that the phase
distributions of the various components do not change with the energy.
It was essentially motivated by the need to have a low number of
parameters and supported by the satisfactory agreement of the
computed pulse profiles with data.
However, in Massaro et al. (2000) it was already shown that the
introduction of a mild energy dependence of two parameters of the analytical description of $C_X$
would improve the pulse profile modelling, particularly in the Ip region.
Our extrapolation at 25 GeV predicts a higher Ip than that given by the MAGIC data, especially
in the leading wing of P2.
We expect a number of detected events in the Ip phase interval,
in excess to the off-pulse level, around 2600,
whereas the measured excess is $\sim$900 counts, but still compatible with the off-pulse emission, due to the large uncertainty.
Note that the S/N ratio does not allow to reach any firm conclusion: there is, for example, a dip at phase $0.3$ (another one is at phase $-0.15$), well
below the mean off-pulse level, that reduces the Ip content.
The assumption that the shape of $C_{X\gamma}$ is equal to that of
$C_{X}$ could not be exactly verified at such different energies.
They could be different, with a shallower profile for the high-energy
counterpart, as already suggested in MCCM upon considering the Ip spectrum.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig3.eps}}
\caption{MAGIC observed pulse shape above 25 GeV, normalized at the count
rate of the first peak, compared with the prediction of MCCM (solid line).}
\label{magic}
\end{figure}
\section{Discussion}
On the basis of a large collection of data, from the optical band
to $\gamma$-ray energies, we developed a model (MCCM) that describes
with a good accuracy the spectral and phase distributions.
This model has essentially an heuristic value, being aimed to
establish a consistent scenario for the development of a realistic
physical model.
The recent detection of the pulsed emission from Crab at energies
above 25 GeV by the MAGIC collaboration is a relevant result that
completes our knowledge on the spectral properties of this important
source.
We have shown here that these results are in a good agreement with
the expectations from the MCCM model and confirm its validity.
It would be interesting to investigate theoretically the physical
plausibility for the presence of two couples of emission components
producing the observed spectra and phase distributions.
A possibility to be further developed arise from the similar curvature
parameter for the X-ray and $\gamma$-ray components, that suggests a
synchrotron-self-Compton (SSC) mechanism for the emission (Morini 1983,
Cheng \& Wei 1995).
A stream of relativistic electrons moving along the magnetic field
lines would undergo a Compton upscattering on the synchrotron photon
field.
A different location and azimuthal distribution in the magnetosphere
would explain the different pulse shape for the $C_{O}$ and $C_{X}$
components.
An alternative hypothesis is that the $\gamma$-ray components are
photons emitted by primary electrons via the curvature radiation
mechanism, and the softer ones are the synchrotron emission from
secondary pairs.
However, in this case one would expect a more pronounced curvature
of the log-parabolic spectra for the latter components.
In fact, if the electron spectrum is a log-parabola with a curvature
parameter $r$, in the $\delta$-approximation the synchrotron curvature
$b$ would be equal to $r/4$ (Massaro et~al., 2006b), while for
curvature radiation this parameter would be $r/9$, because of the
dependence on the electrons' Lorentz factor as $\gamma^{3}$ instead
of $\gamma^{2}$ for the synchrotron process.
About the log-parabolic (or log-normal) energy distribution of electrons,
we recall that it can originate by stochastic acceleration processes or
when the acceleration probability is energy-dependent.
Several models have been proposed to explain the high-energy emission
from pulsars, both in the outer gap (Cheng et al., 1986, 2000)
and in the polar/slot gap (Muslimov \& Harding 2003, 2004) frameworks.
A common feature of these models is that the broadband emission
comes from components originating from different physical processes.
Takata \& Chang (2007) developed a 3D outer gap model based on the 2D
analytical solution of the accelerating field and particle motion
by Takata et al. (2004, 2006) and Hirotani (2006).
In this model the X-ray emission of P1 is due to two separate
components, with curved, roughly log-parabolic, spectra and originating by synchrotron emission
of secondary pairs in different regions of the outer gap, i.e. below
and beyond the null-charge surface.
Their explanation of the $\gamma$-ray spectrum invokes other
two components due to Inverse Compton scattering of secondary pairs and
curvature radiation from primary electrons.
Their modelling of P2 and Ip, unlike the MCCM model, has different weights
for these two components.
The increase of P2 and Ip with respect to P1 in this model is due
to the fact that the emission comes from regions in the lower
magnetosphere, where the high magnetic field produces harder spectra.
More recently, Harding et al. (2008) performed a 3D simulation of the
emission from the Crab pulsar, assuming a Slot Gap accelerator,
in which the emission is due to two distribution of particles,
primary electrons and electron-positron pairs.
The X-ray emission is due to the synchrotron radiation
from the pairs, while the $\gamma$-rays are curvature
radiation and the synchrotron radiation resulting from the resonant
cyclotron absorption of radio beam photons by the primary particles.
Future observations, in particular with the Fermi Gamma-ray Space Telescope,
will be very useful to confirm the spectral cut-off observed
by MAGIC, to improve the estimate of $E_c$, and to bridge the gap between the 100--500
MeV EGRET lightcurves and the $\ge$25 GeV MAGIC one.
It will be possible to follow in detail the evolution of the P2/P1 flux ratio and to
verify with a much higher statistics whether the Ip emission is actually related to that of P2.
Another future interesting test will come from phase-resolved X-ray
polarimetry, where MCCM expect that the hard X-rays polarization properties
of P2 should become increasingly similar to those of Ip,
because of the higher contribution of the $C_{X}$ component.
|
1,116,691,498,493 | arxiv | \section{Introduction}
For any missing notation or reference let us refer to \cite{BraLeSpi1999}.
For a graph $G$, let $V(G)$ ($E(G)$, respectively) denote its vertex set (edge set, respectively).
For a subset $U \subseteq V(G)$, let $N_G(U) = \{v \in V(G) \setminus U:$ $v$ is adjacent to some
$u \in U\}$ be the {\em neighborhood of U in G}, and $A_G(U) = V \setminus (U \cup N(U))$ be the {\em anti-neighborhood of U in G}.
If $U = \{u_1,\ldots,u_k\}$, then let us simply write $N_G(u_1,\ldots,u_k)$ instead of $N_G(U)$, and $A_G(u_1,\ldots,u_k)$ instead of $A_G(U)$.
For $U \subseteq V(G)$ let $G[U]$ denote the subgraph of $G$ induced by $U$. For a vertex $v \in V(G)$ and for a subset $U \subset V(G)$
(with $v \not \in U$), let us say that {\em $v$ contacts $U$} if $v$ is adjacent to some vertex of $U$, and {\em $v$ dominates $U$} if $v$ is adjacent to each
vertex of $U$. A {\em component of $G$} is the vertex set of a maximal connected subgraph of $G$.
An {\em independent set} (or a {\em stable set})
of a graph $G$ is a subset of pairwise nonadjacent vertices of $G$. An independent set of $G$ is {\em maximal} if it is not properly contained
in any other independent set of $G$.
For a given graph $H$, a graph $G$ is {\em $H$-free} if none of its induced subgraphs is isomorphic to $H$; in particular, $H$ is called a {\em forbidden induced subgraph of $G$}. Given two graphs $G$ and $F$, $G + F$ denotes the disjoint union of $G$ and $F$; in particular, $2G=G+G$ and in general, for $l \ge 2$, $lG$ denotes the disjoint union of $l$ copies of $G$.
The following specific graphs are mentioned later. A {\em chordless path} $P_k$ has vertices $v_1,v_2,\ldots,v_k$ and
edges $v_jv_{j+1}$ for $1 \le j < k$. A {\em chordless cycle} $C_k$, $k \ge 4$, has vertices $v_1,v_2,\ldots,v_k$ and edges $v_jv_{j+1}$
for $1 \le j < k$ and $v_kv_1$. A $K_n$ is a complete graph of $n$ vertices. A $K_{1,n}$ is a complete
bipartite graph whose sides respectively have one vertex, called the {\em center} of $K_{1,n}$, and $n$ vertices,
called the {\em leaves} of $K_{1,n}$ (if $n = 1$ then there are two trivial centers). $K_{1,3}$ is also called {\em claw}.
A {\em fork} (sometimes called {\em chair}) has vertices $a,b,c,d,e$, and edges $ab,ac,ad,de$ (thus, a fork contains a claw as an induced subgraph). An {\em apple} is formed by a $C_k$, $k \geq 4$, plus one vertex adjacent to exactly one vertex of the $C_k$.
For indices $i,j,k \ge 0$, let $S_{i,j,k}$ denote the graph with vertices $u,x_1,\ldots,x_i$, $y_1,\ldots,y_j$, $z_1,\ldots,z_k$ such that the subgraph induced by $u,x_1,\ldots,x_i$ forms a $P_{i+1}$ $(u,x_1,\ldots,x_i)$, the subgraph induced by $u,y_1,\ldots,y_j$ forms a $P_{j+1}$ $(u,y_1,\ldots,y_j)$, and the subgraph induced by $u,z_1,\ldots,z_k$ forms a $P_{k+1}$ $(u,z_1,\ldots,z_k)$, and there are no other edges in $S_{i,j,k}$. Thus, {\em claw} is $S_{1,1,1}$, and $P_k$ is isomorphic to e.g. $S_{0,0,k-1}$.
Let $G$ be a given graph and let $w$ be a weight function on $V(G)$. For an independent set $I$, its weight is $w(I):= \Sigma_{v \in I} w(v)$.
Let $\alpha_w(G):= \max \{w(I): I$ independent in $G\}$ denote the maximum weight of any independent set of $G$.
The {\em Maximum Weight Independent Set} ({\em MWIS}) problem asks for an independent set of $G$ of maximum weight.
If all vertices $v$ have the same weight $w(v) = 1$, $\alpha_w(G)=\alpha(G)$ and MWIS is called the {\em MIS} problem.
MWIS is NP-hard \cite{GarJoh1979} and remains NP-hard under various restrictions,
such as for triangle-free graphs \cite{Polja1974} and more generally for graphs without chordless cycle
of given length \cite{Murph1992}, for cubic graphs \cite{GarJoh1977} and more generally for $k$-regular
graphs \cite{FriHedJac1998}, and for planar graphs \cite{GarJohSto1976}.
It can be solved in polynomial time for various graph classes, such as for $P_4$-free graphs \cite{CorLerSte2004} and more generally
perfect graphs \cite{GroLovSch1984}, for claw-free graphs \cite{FaeOriSta2011,Minty1980,NakTam2001,NobSas2011,Sbihi1980} and
more generally fork-free graphs \cite{Aleks2004/1,LozMil2008} and apple-free graphs \cite{BraLozMos2010,BraKleLozMos2008},
for $2K_2$-free graphs \cite{Farbe1989} and more generally $lK_2$-free graphs for any constant $l$ (by combining an algorithm
generating all maximal independent sets of a graph \cite{TsuIdeAriShi1977} and a polynomial upper bound on the number of
maximal independent sets in $lK_2$-free graphs \cite{Aleks1991,FarHujTuz1993,Prisn1995}), $K_2+$claw-free graphs \cite{LozMos2005},
and $2P_3$-free graphs \cite{LozMos2012}. Furthermore MWIS can be solved in polynomial time for $P_5$-free graphs as recently proved in \cite{LokVatVil2014}.
The first two polynomial time algorithms for MWIS on claw-free graphs were introduced in 1980 by Minty \cite{Minty1980} and independently by Sbihi \cite{Sbihi1980}, then revisited by Nakamura and Tamura \cite{NakTam2001}, and recently improved by Faenza, Oriolo, and Stauffer \cite{FaeOriSta2011,FaeOriSta2014}, and by Nobili and Sassano \cite{NobSas2011,NobSas2015} with the best known time bound in \cite{NobSas2015}.
\begin{theo}\label{theo:claw}{\bf \cite{NobSas2015}}
For claw-free graphs, the MWIS problem can be solved in time $O(n^2 \log n)$. \qed
\end{theo}
Obviously, for every graph $G$ the following holds:
$$\alpha_w(G) = max \{w(v) + \alpha_w(G[A(v)]): v \in V\}$$
Thus, for any graph $G$, MWIS can be reduced to the same problem for the anti-neighborhoods of all vertices of $G$. Then we have:
\begin{prop}\label{K1}
For any graph $F$, if M(W)IS can be solved for $F$-free graphs in polynomial time then M(W)IS can be solved for $K_1 + F$-free graphs in polynomial time. \qed
\end{prop}
Let us report the following result due to Alekseev \cite{Aleks1983,Aleks2004/2}. Let us say that a graph is of {\em type $T$}
if it is graph $S_{i,j,k}$ for some indices $i,j,k$.
\begin{theo}\label{theo:Alekseev}{\bf \cite{Aleks1983}}
Let ${\cal X}$ be a class of graphs defined by a finite set $M$ of forbidden induced subgraphs.
If $M$ does not contain any graph every connected component of which is of type $T$, then the M(W)IS problem is {\em NP}-hard for the class ${\cal X}$.
\end{theo}
Alekseev's result implies that M(W)IS is NP-hard for $K_{1,4}$-free graphs $-$ the fact
that M(W)IS is NP-hard for $K_{1,4}$-free graphs is already mentioned in \cite{Minty1980}.
Unless P = NP, Alekseev's result implies that for any graph $F$, if M(W)IS is polynomial time solvable for $F$-free graphs,
then each connected component of $F$ is of type $T$. By Proposition \ref{K1}, for any graph $F$,
if M(W)IS can be solved in polynomial time for $F$-free graphs then for any constant $l$, M(W)IS can be solved in polynomial time for $lK_1 + F$-free graphs.
It follows that, since for any constant $l$, M(W)IS can be solved in polynomial time for $lK_2$-free graphs \cite{Aleks1991,FarHujTuz1993,Prisn1995,TsuIdeAriShi1977}, for fork-free graphs \cite{Aleks2004/1,LozMil2008}, for $K_2+$claw-free graphs \cite{LozMos2005}, for $2P_3$-free graphs \cite{LozMos2012}, and for $P_5$-free graphs \cite{LokVatVil2014}, the minimal graphs $F$ of type $T$ for which the complexity of M(W)IS for $F$-free graphs was open are: $P_6$, $S_{1,1,3}$, $S_{1,2,2}$, $K_2+P_4$, $2K_2+P_3$, $P_3$+claw, and thus, the minimal graph classes, defined by forbidding one induced subgraph, for which the complexity of M(W)IS was open are:
\begin{itemize}
\item[ ] $P_6$-free graphs, $S_{1,1,3}$-free graphs, $S_{1,2,2}$-free graphs, $K_2+P_4$-free graphs, $2K_2+P_3$-free graphs, $P_3$+claw-free graphs.
\end{itemize}
In this manuscript, we show that for any constant $l$, MWIS can be solved for $l$claw-free graphs in polynomial time. This extends the known results for MWIS on claw-free graphs, $lK_2$-free graphs for any constant $l$, $K_2$+claw-free graphs, $2P_3$-free graphs, and solves the open question for MWIS on $2K_2+P_3$-free graphs and on $P_3$+claw-free graphs.
Our approach is based on Farber's approach showing that every $2K_2$-free graph has ${\cal O}(n^2)$ maximal independent sets \cite{Farbe1989} (reported in Section \ref{2K2fr}), which directly leads to a polynomial time algorithm to solve MWIS for $2K_2$-free graphs by dynamic programming.
\section{Maximal Independent Sets in $2K_2$-Free Graphs}\label{2K2fr}
In this section let us refer to Algorithm ${\cal A}$ (subsequently called Algorithm Alpha) from \cite{LozMos2005} which formalizes the aforementioned approach by Farber \cite{Farbe1989}; our subsequent approach for MWIS on $l$claw-free graphs is based on this algorithm.
For a $2K_2$-free input graph $G$, Algorithm Alpha produces a family ${\cal S}$ of independent sets of $G$, which can be computed in time
${\cal O}(n^3)$ and which contains ${\cal O}(n^2)$ members such that each maximal independent set of $G$ is contained in some member of ${\cal S}$.
For a graph $G=(V,E)$ with $|V|=n$, a {\em vertex ordering} $(v_1,v_2,\ldots,v_n)$ of $G$ is a total ordering of the vertex set $V$ of $G$.
For such a vertex ordering $(v_1,v_2,\ldots,v_n)$ of $G$, let $G_i:=G[\{v_1,v_2,\ldots,v_i\}]$ denote the subgraph of $G$ induced by the first $i$ vertices, $i \ge 1$.
Given a vertex ordering $(v_1,v_2,\ldots,v_n)$, at each loop $i$, $1 \le i \le n$, Algorithm Alpha provides a family ${\cal S}_i$ of subsets of $\{v_1,v_2,\ldots,v_i\}$ (by modifying ${\cal S}$ at loop $i$ by extending some of its members or by adding new members) such that each maximal independent set of $G_i$ is contained in some member of ${\cal S}_i$, and finally returns the family ${\cal S}_n = {\cal S}$.
\noindent {\bf Algorithm Alpha}\\
{\bf Input:} A $2K_2$-free graph $G$ and a vertex ordering $(v_1,v_2,\ldots,v_n)$ of $G$.\\
{\bf Output:} A family ${\cal S}$ of subsets of $V(G)$.\\
\\
${\cal S}:=\{\emptyset\}$;\\
{\bf For} $i=:1$ {\bf to} $n$ {\bf do}\\
{\bf begin}\\
1. [Extension of some members of ${\cal S}$]\\
\hspace*{0.5cm} {\bf For each} $H \in {\cal S}$ {\bf do}\\
\hspace*{1cm} {\bf If} $H \cup \{v_i\}$ is an independent set {\bf then} $H:=H\cup \{v_i\}$.\\
2. [Addition of new members to ${\cal S}$]\\
\hspace*{0.5cm} {\bf For each} $K_2$ of $G_i$ containing $v_i$ (i.e., for each edge $uv_i$ of $G_i$) {\bf do}\\
\hspace*{1cm} $H:=\{v_i\}\cup A_{G_i}(u,v_i)$;\\
\hspace*{1cm} ${\cal S} := {\cal S} \cup \{H\}$.\\
{\bf end.}
Then the MWIS problem can be solved for $2K_2$-free graphs by the following algorithm.
\noindent {\bf Algorithm $2K_2$-Free-MWIS}\\
{\bf Input:} A $2K_2$-free graph $G$.\\
{\bf Output:} A maximum weight independent set of $G$.
\begin{itemize}
\item[(1)] Execute Algorithm Alpha for $G$. Let ${\cal S}$ be the resulting family of subsets of $V(G)$.
\item[(2)] For each $H \in {\cal S}$, compute a maximum weight independent set of $G[H]$ (note that each $H \in {\cal S}$ is an independent set since $G$ is $2K_2$-free). Then choose a best solution, i.e., one of maximum weight.
\end{itemize}
Then one obtains the following result.
\begin{theo}{\bf \cite{Farbe1989}}
For $2K_2$-free graphs, the MWIS problem can be solved in time ${\cal O}(n^{4})$ by Algorithm $2K_2$-Free-MWIS. \qed
\end{theo}
\section{Maximal Independent Sets in Claw+Claw-Free Graphs}\label{2Clawfr}
\subsection{A Basic Lemma}\label{BasicLemma}
First let us introduce a preparatory result.
For each $k \in \{1,\ldots,14\}$, let $L_k$ be the graph drawn in the subsequent figure.
Note that each $L_k$ contains an induced claw. For each $k \in \{1,\ldots,14\}$, let $W(L_k)$ denote the set of white vertices of $L_k$, let $B(L_k)$ denote
the set of black vertices of $L_k$, and let $top(L_k)$ denote the (white) vertex at the top of $L_k$.
\begin{figure}\label{FigLk}
\centering
\includegraphics[width=\textwidth]{FigureClaw.eps}
\caption{Graphs $L_k$ for $k = 1,\ldots,14$}
\end{figure}
\begin{lemm}\label{lemm:claw}
For a graph $G$, assume that $v \in V(G)$ is a vertex such that $v$ is contained in an induced claw of $G$ and $G[V(G) \setminus \{v\}]$ is claw-free.
Then for each maximal independent set $I$ of $G$ with $v \in I$, there is a $k \in \{1,\ldots,14\}$ such that $I \subseteq W(L_k) \cup A_{G}(L_k)$
for an induced subgraph $L_k$ of $G$ with $v$ = top$(L_k)$.
\end{lemm}
{\bf Proof.} Let $K$ be a claw in $G$ with, say, $V(K)=\{v,a,b,c\}$. Let $I$ be a maximal independent set of $G$ containing $v$, and let $I' := I \setminus \{v\}$. Then for $H:=V(G) \setminus \{v\}$, $I'$ is a maximal independent set of $G[H \setminus N(v)]$. Let us distinguish between the following cases.
{\bf Case 1} $G[H]$ is connected.
By assumption, $v$ is contained in an induced claw of $G$. Let us distinguish between two subcases.
{\bf Case 1.1 $v$ is the center of $K$}.
Since $G[H]$ is claw-free, each of $a,b,c$ has at most two neighbors in $I'$.
{\bf Case 1.1.1} If a vertex of $a,b,c$, say $a$, has two neighbors in $I'$, say $s_1,s_2$ then $I \subseteq W(L_1) \cup A_{G}(L_1)$ with $V(L_1)=\{v,a,s_1,s_2\}$, $W(L_1)=\{v,s_1,s_2\}$, and $v$ = top$(L_1)$.
{\bf Case 1.1.2} If none of $a,b,c$ has a neighbor in $I'$ then $I \subseteq W(L_2) \cup A_{G}(L_2)$ with $V(L_2)=\{v,a,b,c\}$ and $v$ = top$(L_2)$.
{\bf Case 1.1.3} Now assume that Cases 1.1.1 and 1.1.2 are excluded. This means that one of $a,b,c$, say without loss of generality $a$, has exactly one neighbor in $I'$ and $b$ and $c$ have at most one neighbor in $I'$. Let $as_1 \in E$ for $s_1 \in I'$. Note that not both of $b$ and $c$ are adjacent to $s_1$ since $H$ is claw-free, and in general, $a,b$ and $c$ do not have any common neighbor in $I'$.
If $N(b) \cap I'=N(c) \cap I'=\emptyset$ then we have $I \subseteq W(L_3) \cup A_{G}(L_3)$ with $V(L_3)=\{v,a,b,c,s_1\}$ and $v$ = top$(L_3)$.
If $b$ has exactly one neighbor in $I'$, say $s_2$, and $N(c) \cap I' = \emptyset$ then
if $s_1 \neq s_2$, we have $I \subseteq W(L_4) \cup A_{G}(L_4)$ with $V(L_4)=\{v,a,b,c,s_1,s_2\}$ and $v$ = top$(L_4)$, and
if $s_1 = s_2$, we have $I \subseteq W(L_6) \cup A_{G}(L_6)$ with $V(L_6)=\{v,a,b,c,s_1\}$ and $v$ = top$(L_6)$, and similarly for the case when $c$ has exactly one neighbor in $I'$, and $N(b) \cap I' = \emptyset$.
Finally, assume that both $b$ and $c$ have a neighbor in $I'$, i.e., there are $s_2,s_3 \in I'$ with $bs_2 \in E$ and $cs_3 \in E$.
If $s_1,s_2,s_3$ are pairwise distinct then we have $I \subseteq W(L_5) \cup A_{G}(L_5)$ with $V(L_5)=\{v,a,b,c,s_1,s_2,s_3\}$ and $v$ = top$(L_5)$.
Now assume that $|\{s_1,s_2,s_3\}|=2$ (recall that $|\{s_1,s_2,s_3\}|=1$ is impossible). Without loss of generality, let $s_1=s_2$. Then we have
$I \subseteq W(L_7) \cup A_{G}(L_7)$ with $V(L_7)=\{v,a,b,c,s_1,s_3\}$ and $v$ = top$(L_7)$.
{\bf Case 1.2 $v$ is a leaf of $K$}.
Without loss of generality, let $b$ be the center of $K$.
Since $G[H]$ is claw-free, $b$ has at most two neighbors in $I'$, and if $a \notin I'$ ($c \notin I'$, respectively), the same holds for $a$ ($c$, respectively).
The following subcases are exhaustive by symmetry.
{\bf Case 1.2.1} If $a,c \in I'$ then $I \subseteq W(L_1) \cup A_{G}(L_1)$ with $V(L_1)=\{v,a,b,c\}$ and $v$ = top$(L_1)$.
{\bf Case 1.2.2} If exactly one of $a,c$ is in $I'$, say without loss of generality, $a \in I'$ and $c \not \in I'$ (and more generally, only one of the neighbors of $b$ is in $I'$ - otherwise we have Case 1.2.1) then $c$ has a neighbor in $I'$, say $s$, since $I'$ is a maximal independent set of $G[H \setminus N(v)]$. Then clearly, $s$ is nonadjacent to $a$ and $v$ and is nonadjacent to $b$ (otherwise $b$ would have two neighbors in $I'$). Then $I \subseteq W(L_8) \cup A_{G}(L_8)$
with $V(L_8)=\{v,a,b,c,s\}$ and $v$ = top$(L_8)$.
{\bf Case 1.2.3} Now assume that Cases 1.2.1 and 1.2.2 are excluded. Thus, $a,c \not \in I'$. Then both $a$ and $c$ must have a neighbor in $I'$ since $I'$ is a maximal independent set of $G[H \setminus N(v)]$.
If no neighbor of $a$ or $c$ in $I'$ is adjacent to $b$ then both $a$ and $c$ have exactly one neighbor in $I'$, else a claw in $G[H]$ would arise involving $b$.
Let $s_1,s_2 \in I'$ with $as_1 \in E$, $cs_2 \in E$.
If $s_1 \neq s_2$ then $I \subseteq W(L_9) \cup A_{G}(L_9)$ with $V(L_9)=\{v,a,b,c,s_1,s_2\}$ and $v$ = top$(L_9)$.
If $s_1=s_2$ then $I \subseteq W(L_{10}) \cup A_{G}(L_{10})$ with $V(L_{10})=\{v,a,b,c,s_1\}$ and $v$ = top$(L_{10})$.
Now assume that, without loss of generality, a neighbor $s \in I'$ of $a$ is adjacent to $b$. We claim:
(i) $s$ is adjacent to $c$, since otherwise Case 1.2.2 holds with $s$ instead of $a$;
(ii) $a$ and $c$ have at most one more neighbor in $I'$, and such a neighbor is non-adjacent to $b$, since otherwise Case 1.2.1 holds (i.e., $b$ has two neighbors in $I'$).
If neither $a$ nor $c$ have another neighbor in $I'$ then $I \subseteq W(L_{11}) \cup A_{G}(L_{11})$ with $V(L_{11})=\{v,a,b,c,s\}$ and $v$ = top$(L_{11})$.
If there is $s_1 \in I'$ with $s_1 \neq s$, $as_1 \in E$ and the only neighbor of $c$ in $I'$ is $s$ then $I \subseteq W(L_{12}) \cup A_{G}(L_{12})$ with $V(L_{12})=\{v,a,b,c,s,s_1\}$ and $v$ = top$(L_{12})$, and similarly if $c$ has two neighbors $s,s_1 \in I'$ and $a$ has only neighbor $s \in I'$.
Finally, if $a$ and $c$ have another neighbor in $I'$, say $s_1,s_2 \in I'$, $s \neq s_1, s \neq s_2$ with $as_1 \in E$ and $cs_2 \in E$ then we have:
If $s_1 \neq s_2$ then $I \subseteq W(L_{13}) \cup A_{G}(L_{13})$ with $V(L_{13})=\{v,a,b,c,s,s_1,s_2\}$ and $v$ = top$(L_{13})$, and
if $s_1 = s_2$ then $I \subseteq W(L_{14}) \cup A_{G}(L_{14})$ with $V(L_{14})=\{v,a,b,c,s,s_1\}$ and $v$ = top$(L_{14})$.
{\bf Case 2}: $G[H]$ is not connected.
This case can be treated similarly as Case 1 in which $G[H]$ is connected. If $G$ is not connected then we can solve MWIS separately for each component of $G$. If $G$ is connected and $v$ is the leaf of a claw then obviously, $G[H]$ is connected. Thus, we can assume that $v$ is the center of a claw $K$, and we can follow the arguments of Case 1. For brevity let us omit the proof, which can be split into the subcases in which $G[H]$ has two or three components.
Finally we have $I \subseteq W(L_k) \cup A_{G}(L_k)$, for some $k \in \{1,\ldots,7\}$, with $v$ = top$(L_k)$.
\qed
\subsection{MWIS for Claw+Claw-Free Graphs}\label{claw+clawfr}
Now we show that for claw+claw-free graphs, MWIS can be solved in time ${\cal O}(n^{10})$.
For this, we need the following notion:
\begin{defi}\label{defi: good family}
Let $G$ be a graph and let ${\cal S}$ be a family of subsets of $V(G)$. Then ${\cal S}$ is a {\em good claw-free family of} $G$ if the following holds:
\begin{enumerate}
\item[$(i)$] Each member of ${\cal S}$ induces a claw-free subgraph in $G$.
\item[$(ii)$] Each maximal independent set of $G$ is contained in some member of ${\cal S}$.
\item[$(iii)$] ${\cal S}$ contains polynomially many members and can be computed in polynomial time.
\end{enumerate}
\end{defi}
The basic step is the subsequent Algorithm Gamma(2) (based on the corresponding Algorithm Alpha of Section \ref{2K2fr}) which, for any claw+claw-free (i.e., 2claw-free) input graph $G$, computes a good claw-free family ${\cal S}$ of $G$. The approach is based on Farber's idea for MWIS on $2K_2$-free graphs described in Algorithm Alpha of Section \ref{2K2fr}.
{\bf Algorithm Gamma(2)}\\
{\bf Input:} A claw+claw-free graph $G$ and a vertex-ordering $(v_1,v_2,\ldots,v_n)$ of $G$.\\
{\bf Output:} A good claw-free family ${\cal S}$ of $G$.\\
${\cal S}:=\{\emptyset\}$;\\
{\bf For} $i=:1$ {\bf to} $n$ {\bf do}\\
{\bf begin}\\
1. [Extension of some members of ${\cal S}$]\\
\hspace*{0.5cm} {\bf For each} $H \in {\cal S}$ {\bf do}\\
\hspace*{0.5cm} {\bf If} $G[H \cup \{v_i\}]$ is claw-free {\bf then} $H:=H\cup \{v_i\}$.\\
2. [Addition of new members to ${\cal S}$]\\
\hspace*{0.5cm} {\bf For each} induced $L_k$ of $G_i$, $k \in \{1,\ldots,14\}$, with $v_i$ = top$(L_k)$ {\bf do}\\
\hspace*{1cm} Compute a good claw-free family, say ${\cal F}$, of $G[A_{G_i}(L_k)]$. \\
\hspace*{1cm} {\bf For each} $F \in {\cal F}$, set ${\cal S}:= {\cal S} \cup \{W(L_k) \cup F\}$. \\
{\bf end.}
\begin{prop}\label{prop:claw+claw}
Step $2$ of Algorithm Gamma$(2)$ is well defined, i.e., $G[A_{G_i}(L_k)]$ is claw-free and has a good claw-free family $($formed by one member, namely,
$A_{G_i}(L_k))$ which can be computed in constant time.
\end{prop}
{\bf Proof.}
Subgraph $G[A_{G_i}(L_k)]$ is claw-free since $G$ is assumed to be claw+claw-free, each $L_k$ contains an induced claw and $A_{G_i}(L_k)$ is defined as the anti-neighborhood of $V(L_k)$. Then the subgraph $G[A_{G_i}(L_k)]$ has a good claw-free family (formed by one member, namely, $A_{G_i}(L_k)$) which can be computed in constant time.
\qed
For proving the correctness and the time bound of Algorithm Gamma(2), we need the following lemmas.
\begin{lemm}\label{lemmC:1}
Let $G$ be a claw+claw-free graph and let ${\cal S}$ be the result of Algorithm Gamma$(2)$. Then we have:
\begin{itemize}
\item[$(i)$] Each member of ${\cal S}$ induces a claw-free subgraph of $G$.
\item[$(ii)$] Each maximal independent set of $G$ is contained in some member of ${\cal S}$.
\end{itemize}
\end{lemm}
{\bf Proof.} $(i)$: Each member of ${\cal S}$ is created either in the initialization step as the empty set or in Step 1 or Step 2 of some loop. Clearly, each member $H\cup \{v_i\}$ created in Step~1 induces a claw-free subgraph in $G$ since each member of ${\cal S}$ is extended in Step 1 only if the extension preserves its claw-freeness.
According to Step 2 and to Proposition~\ref{prop:claw+claw}, each member of ${\cal S}$ created in Step 2 is the disjoint union of a vertex subset of a claw-free subgraph, namely $W(L_k)$, and of a claw-free subgraph representing its anti-neighborhood $A_{G_i}(L_k)$, namely a member of a good claw-free family. Therefore, each member of ${\cal S}$ created in Step 2 induces a claw-free graph. This completes the proof of statement $(i)$.
$(ii)$: By ${\cal S}_i$, let us denote the family ${\cal S}$ resulting by the $i$-th loop of Algorithm Gamma(2).
Let us show that for all $i \in \{1,\ldots,n\}$, each maximal independent set of $G_i$ is contained in a member $H$ of ${\cal S}_i$.
The proof is done by induction. For $i=1$, the statement is trivial. Then let us assume that the statement holds for $i-1$ and prove that it holds for $i$.
Let $I$ be a maximal independent set of $G_i$.
If $v_i \not\in I$, then by the induction assumption, $I$ is contained in some member of ${\cal S}_{i-1}$, and thus of ${\cal S}_i$, since each member of ${\cal S}_{i-1}$ is a (not necessarily proper) subset of a member of ${\cal S}_i$.
If $v_i \in I$, then let us consider the following argument. By the induction assumption, let $H \in {\cal S}_{i-1}$ with $I \setminus \{v_i\} \subseteq H$. Note that for all $j$, $1 \le j \le n$, each member of ${\cal S}_j$ induces a claw-free graph, as one can easily verify by an argument similar to the proof of statement $(i)$. Thus, $G[H]$ is claw-free.
Then let us consider the following two cases which are exhaustive by definition of Algorithm Gamma(2).
{\bf Case 1}: $G[H \cup \{v_i\}]$ is claw-free.
Then $I$ is contained in the set $H \cup \{v_i\}$, which is a member of ${\cal S}_i$ since it is generated by Step 1 of the algorithm at loop $i$.
{\bf Case 2}: $G[H \cup \{v_i\}]$ is not claw-free.
Then by Lemma \ref{lemm:claw}, since $G[H]$ is claw-free, there is a $k \in \{1,\ldots,14\}$ such that $I \subseteq W(L_k) \cup A_{G_i}(L_k)$ for an induced subgraph $L_k$ of $G_i$ with $v_i$ = top$(L_k)$, and $W(L_k) \cup A_{G_i}(L_k)$ is contained in ${\cal S}_i$ since it is generated by Step 2 of Algorithm Gamma(2) at loop $i$.
\qed
\begin{lemm}\label{lemmC:3}
The family ${\cal S}$ produced by Algorithm Gamma$(2)$ contains ${\cal O}(n^{7})$ members and can be computed in ${\cal O}(n^{9})$ time, which is also the time bound of Algorithm Gamma$(2)$.
\end{lemm}
{\bf Proof.}
The members of ${\cal S}$ are created either in the initialization step or in Step 2 of all the loops of Algorithm Gamma(2) and then are possibly (iteratively) extended in Step 1 of Algorithm Gamma(2).
Concerning the member created in the initialization step, i.e., the empty set: This member is created in constant time and is possibly (iteratively) extended by Step 1 of each loop in ${\cal O}(n)$ time (and the number of loops is $n$). Then this member can be computed in ${\cal O}(n^{2})$ time.
Concerning the members created in Step 2 of all the loops: Such members are created with respect to all induced $L_k$, $1 \le k \le 14$ (the maximum number of vertices in any $L_k$ is 7), of $G_i$, i.e., with respect to a family of ${\cal O}(n^7)$ subsets of $G_i$ (in fact the algorithm produces the anti-neighborhoods of all $L_k$ for $k \in \{1,\ldots,14\}$ of $G_i$ just once since at loop $i$ all such $L_k$ contain $v_i$). Then for the respective anti-neighborhood, namely $A_{G_i}(L_k)$, of each such subset the algorithm computes a good claw-free family. By Proposition \ref{prop:claw+claw}, $G[A_{G_i}(L_k)]$ is claw-free and has a good claw-free family (which contains one member and can be computed in constant time). Therefore the cardinality of the family of such members is ${\cal O}(n^7)$ and all such members can be created in ${\cal O}(n^7)$ time (since each such member can be created in Step 2 in constant time). Then such members are possibly (iteratively) extended in Step 1 in ${\cal O}(n)$ time (and the number of loops is $n$). Then such members can be computed in ${\cal O}(n^9)$ time.
Therefore, $S$ contains ${\cal O}(n^{7})$ members and can be computed in ${\cal O}(n^{9})$ time, which is also the time bound of Algorithm Gamma(2).
\qed
Note that Lemmas \ref{lemmC:1} and \ref{lemmC:3} directly imply the following.
\begin{coro}\label{coro:claw+claw}
Every claw+claw-free graph has a good claw-free family which can be computed by Algorithm Gamma$(2)$. \qed
\end{coro}
Then the MWIS problem can be solved for claw+claw-free graphs by the following algorithm.
{\bf Algorithm MWIS(2)}\\
{\bf Input:} A claw+claw-free graph $G$.\\
{\bf Output:} A maximum weight independent set of $G$.
\begin{itemize}
\item[(1)] Execute Algorithm Gamma(2) for $G$. Let ${\cal S}$ be the resulting family of subsets of $V(G)$.
\item[(2)] For each $H \in {\cal S}$, compute a maximum weight independent set of $G[H]$. Then choose a best solution, i.e., one of maximum weight.
\end{itemize}
\begin{theo}
Algorithm MWIS$(2)$ is correct and can be done in ${\cal O}(n^{10})$ time.
\end{theo}
{\bf Proof.} {\em Correctness}: By Lemma \ref{lemmC:1} $(ii)$, Algorithm MWIS(2) is correct.
{\em Time bound}:
By Lemma \ref{lemmC:3}, step (1) can be executed in ${\cal O}(n^{9})$ time.
By Lemma \ref{lemmC:3}, the family ${\cal S}$ contains ${\cal O}(n^{7})$ members.
Then, by Lemma \ref{lemmC:1} $(i)$ and Theorem \ref{theo:claw}, step (2) can be executed in ${\cal O}(n^{10})$ time.
Thus, Algorithm MWIS(2) can be executed in time ${\cal O}(n^{10})$.
\qed
Then one obtains the following result.
\begin{coro}\label{theo:WIS-claw+claw}
For claw+claw-free graphs, the MWIS problem can be solved in time ${\cal O}(n^{10})$ by Algorithm MWIS$(2)$. \qed
\end{coro}
\section{MWIS for $l$Claw-Free Graphs}\label{MWISlclaw-fr}
In this section we show that for any fixed $l \ge 2$, MWIS for $l$claw-free graphs can be solved in polynomial time.
For this, we first describe the subsequent Algorithm Gamma($l$), which for any $l$claw-free input graph $G$ computes a good claw-free family ${\cal S}$ of $G$. The approach recursively uses Algorithm Gamma$(l-1)$ for Algorithm Gamma($l$), starting with Algorithm Gamma(2) of subsection \ref{claw+clawfr}.
{\bf Algorithm Gamma$(l)$}\\
{\bf Input:} An $l$claw-free graph $G$ and a vertex-ordering $(v_1,v_2,\ldots,v_n)$ of $G$.\\
{\bf Output:} A good claw-free family $S$ of $G$.\\
${\cal S}:=\{\emptyset\}$;\\
{\bf For} $i=:1$ {\bf to} $n$ {\bf do}\\
{\bf begin}\\
1. [Extension of some members of ${\cal S}$]\\
\hspace*{0.5cm} {\bf For each} $H \in {\cal S}$ {\bf do}\\
\hspace*{0.5cm} {\bf If} $G[H \cup \{v_i\}]$ is claw-free {\bf then} $H:=H\cup \{v_i\}$.\\
2. [Addition of new members to ${\cal S}$]\\
\hspace*{0.5cm} {\bf For each} induced $L_k$ of $G_i$, $k \in \{1,\ldots,14\}$, with $v_i$ = top$(L_k)$ {\bf do}\\
\hspace*{1cm} Compute a good claw-free family, say ${\cal F}$, of $G[A_{G_i}(L_k)]$ by\\
\hspace*{1cm} Algorithm Gamma$(l-1)$. \\
\hspace*{1cm} {\bf For each} $F \in {\cal F}$, set ${\cal S}:= {\cal S} \cup \{W(L_k) \cup F\}$. \\
{\bf end.}
{\bf Assumption 1.} To prove the subsequent Proposition \ref{prop:lclaw}, Lemmas \ref{lemmC:1l} and \ref{lemmC:3l}, and Corollary \ref{coro:lclaw}, we need to consider them as a {\em unique} result, in order to give a proof by induction on $l$. For $l$ = 2, the proof of Proposition \ref{prop:lclaw}, of Lemmas \ref{lemmC:1l} and \ref{lemmC:3l}, and of Corollary \ref{coro:lclaw} is respectively that of Proposition \ref{prop:claw+claw}, of Lemmas \ref{lemmC:1} and \ref{lemmC:3}, and of Corollary \ref{coro:claw+claw}.
Then let us assume that the subsequent Proposition \ref{prop:lclaw}, Lemmas \ref{lemmC:1l} and \ref{lemmC:3l}, and Corollary \ref{coro:lclaw} hold for $l-1$ and let us show that they hold for $l$.
\begin{prop}\label{prop:lclaw}
Step $2$ of Algorithm Gamma$(l)$ is well defined, i.e., $G[A_{G_i}(L_k)]$ is $(l-1)$claw-free and has a good claw-free family which can be computed by Algorithm
Gamma$(l-1)$.
\end{prop}
{\bf Proof.}
Subgraph $G[A_{G_i}(L_k)]$ is $(l-1)$claw-free since $G$ is $l$claw-free and since $A_{G_i}(L_k)$ is defined as the anti-neighborhood of $L_k$ containing an induced claw. Then by Assumption 1, i.e., by Corollary \ref{coro:lclaw} with respect to $l-1$, subgraph $G[A_{G_i}(L_k)]$ has a good claw-free family which can be computed by Algorithm Gamma$(l-1)$.
\qed
For proving the correctness and the time bound of Algorithm Gamma$(l)$, we need the following lemmas.
\begin{lemm}\label{lemmC:1l}
Let $G$ be an $l$claw-free graph and let ${\cal S}$ be the result of Algorithm Gamma$(l)$. Then we have:
\begin{itemize}
\item[$(i)$] Each member of ${\cal S}$ induces a claw-free subgraph of $G$.
\item[$(ii)$] Each maximal independent set of $G$ is contained in some member of ${\cal S}$.
\end{itemize}
\end{lemm}
{\bf Proof.}
According to Assumption~1, the proof is similar to that of Lemma~\ref{lemmC:1}, with Proposition~\ref{prop:lclaw} instead of Proposition~\ref{prop:claw+claw} and with Algorithm Gamma($l$) instead of Algorithm Gamma(2).
\qed
\begin{lemm}\label{lemmC:3l}
The family ${\cal S}$ produced by Algorithm Gamma$(l)$ contains polynomially many members and can be computed in polynomial time, which is also the time bound of Algorithm Gamma$(l)$.
\end{lemm}
{\bf Proof.}
The members of ${\cal S}$ are created either in the initialization step or in Step 2 of all the loops of Algorithm Gamma($l$) and then are possibly (iteratively) extended in Step 1 of Algorithm Gamma($l$).
Concerning the member created in the initialization step, i.e., the empty set, this member is created in constant time and is possibly (iteratively) extended by Step 1 of each loop in ${\cal O}(n)$ time (the number of loops is $n$). Then this member can be computed in ${\cal O}(n^{2})$ time.
Concerning the members created in Steps 2 of all the loops, such members are created with respect to all induced $L_k$ of $G$, i.e., with respect to a family of ${\cal O}(n^7)$ subsets of $G$ (in fact, the algorithm produces the anti-neighborhoods of all $L_k$ for $k \in \{1,\ldots,14\}$ of $G_i$ just once since at loop $i$ all such $L_k$ contain $v_i$ as their top vertex). Then for the respective anti-neighborhood, namely $A_{G_i}(L_k)$, of each such subset the algorithm computes a good claw-free family. By Proposition \ref{prop:lclaw}, $G[A_{G_i}(L_k)]$ is $(l-1)$claw-free and has a good claw-free family. Therefore the cardinality of the family of such members is bounded by a polynomial and all such members can be created in polynomial time (since each such member can be created in Step~2 in polynomial time). Then such members are possibly (iteratively) extended in Step~1 in ${\cal O}(n)$ time (the number of loops is $n$). Thus, such members can be computed in polynomial time.
Therefore, ${\cal S}$ can be computed in polynomial time, which is also the time bound of Algorithm Gamma($l$). \qed
Note that Lemmas \ref{lemmC:1l} and \ref{lemmC:3l} directly imply the following.
\begin{coro}\label{coro:lclaw}
For any fixed $l$, each $l$claw-free graph has a good claw-free family which can be computed via Algorithm Gamma$(l)$. \qed
\end{coro}
Then for $l$claw-free graphs, the MWIS problem can be solved by the following algorithm. \\
{\bf Algorithm MWIS($l$)}\\
{\bf Input:} An $l$claw-free graph $G$.\\
{\bf Output:} A maximum weight independent set of $G$.
\begin{itemize}
\item[(1)] Execute Algorithm Gamma$(l)$ for $G$. Let ${\cal S}$ be the resulting family of subsets of $V(G)$.
\item[(2)] For each $H \in {\cal S}$, compute a maximum weight independent set of $G[H]$. Then choose a best solution, i.e., one of maximum weight.
\end{itemize}
\begin{theo}
Algorithm MWIS$(l)$ is correct and can be executed in polynomial time.
\end{theo}
{\bf Proof.} {\em Correctness}: By Lemma \ref{lemmC:1l} $(ii)$, Algorithm MWIS$(l)$ is correct.
{\em Time bound}: By Lemma \ref{lemmC:3l}, step (1) can be executed in polynomial time.
By Lemma~\ref{lemmC:3l}, the family ${\cal S}$ contains polynomially many members.
Then by Lemma \ref{lemmC:1l} $(i)$ and by Theorem \ref{theo:claw}, step (2) can be executed in polynomial time.
Thus, Algorithm MWIS$(l)$ can be executed in polynomial time.
\qed
Then one obtains the following result.
\begin{coro}\label{theo: WIS-lclaw}
For any fixed $l$, the MWIS problem can be solved in polynomial time for $l$claw-free graphs by Algorithm MWIS$(l)$. \qed
\end{coro}
\begin{footnotesize}
\renewcommand{\baselinestretch}{0.4}
|
1,116,691,498,494 | arxiv | \section{Astronomy in our societies}
We, astronomers and people around us now, tend to see in astronomy an activity that is mainly, if not solely devoted to the understanding of the Universe and the objects to be found within. This is an intellectual quest that is particularly fruitful since the beginning of the space age and the opening of the electro-magnetic spectrum from radio waves to gamma rays to the observations of the sky. This is certainly a valid point of view now. The quest for understanding as an activity for itself has, however, certainly not been the main activity of astronomers over history.
Most of the astronomical workforce has been invested not so much in a cultural endeavour, but rather in a tedious quest for the measurement and keeping of time. Agriculture requires that seeds are planted well in advance of the season of growth. It is, therefore, necessary to know when to plant. This time cannot be estimated on the current weather, but requires advance planning for which the only useful information is based on the positions of Sun, Moon and stars. Since the Moon month and the year do not have a simple relation, the quest for timekeeping during the year is a complex one. Astronomers have, therefore, spent large efforts world wide to solve this problem.
The same is true for other needs of society, navigation certainly requires seaworthy ships, it also requires the capacity to locate oneself on the surface of the Earth in unknown territories and on sea. Again, this is provided by a knowledge of the respective positions of Sun, Moon, planets and stars, together with a good mastering of time keeping. All of this knowledge is based on astronomy.
The daily needs of human society also require some level of time keeping, be it only to be able to meet at given time and place. Here again the local timekeeping in many organised societies has relied on astronomical observations. The results being then relayed to the population by bells, canons and other signals.
Astronomy has thus been a most practical endeavour for most of the human history. The present conference amply demonstrates this not only in the European culture, but also in other ones. The benefits of astronomy for society cannot be overestimated. Agriculture, navigation and the organisation of societies have depended crucially on astronomical observations for almost the whole of the development of human civilisations. The development of a "Weltbild" should be seen in this context as an, important, side benefit.
The practical importance of astronomy has only rather recently ceased to dominate our activities. The observatory of Geneva has, as one example, been in charge of certifying chronometers for the local manufactures until the late 1960's. It is also interesting to learn that the observatory of Besan\c{c}on, in France, North of the Jura mountains, was founded in 1878 not so much to contribute to the great human quest for knowledge about the world than to help the local watch manufacturers who considered themselves at a disadvantage compared to their competitors of the South of the Jura mountains, in Neuch\^atel and Geneva, who had access to astronomical observatories.
It is, therefore, only very recently that Astronomy as become a mainly cultural quest, the practical benefits of which are only minor and indirect, to be found in the knowledge gained in making complex instruments capable of observing in remote and hostile environments.
\section{Space Sciences and The Development of Modern Societies}
\subsection{The Past}
Space sciences have been a prime mover, eventhough not the only one, in the development of space activities. Astronomy, as a major part of space sciences, has thus had a leading role in the build up of space capacities in the different parts of the world that developed them in the last decades. This role, as well as the more nationalistic connotations of the development, is well illustrated by the following section of the Moon speech given by president Kennedy in Houston in 1962:
{\it
"Those who came before us made certain that this country rode the first waves of the industrial
revolutions, the first waves of modern invention, and the first wave of nuclear power, and this
generation does not intend to founder in the backwash of the coming age of space. We mean to
be a part of it--we mean to lead it. For the eyes of the world now look into space, to the moon and
to the planets beyond, and we have vowed that we shall not see it governed by a hostile flag of
conquest, but by a banner of freedom and peace. We have vowed that we shall not see space
filled with weapons of mass destruction, but with instruments of knowledge and understanding.
Yet the vows of this Nation can only be fulfilled if we in this Nation are first, and, therefore, we
intend to be first. In short, our leadership in science and in industry, our hopes for peace and
security, our obligations to ourselves as well as others, all require us to make this effort, to solve
these mysteries, to solve them for the good of all men, and to become the world's leading spacefaring
nation."
}
In short, let space be devoted to peaceful scientific activities, but let this development be a US development.
In Europe the role of science in the development of space activities has even been more prominent. This is certainly due to the fact that Europe is not, yet?, identified as such in a nationalistic sense. National pride or interests have therefore had less influence than elsewhere around the Earth.
Since the early developments of the 1960's, space activities have become a very important element of the infrastructures of our societies. Their importance is often not quite valued at the right level. Suffice it to say as illustration here that Earth observations, be they for the management of world natural or agricultural ressources, or for meteorological observations on one side and navigation on another side both play an increasing role in our daily lives. Modern transportation on land, at sea or in the air all depend on space capacities, so does our ability to predict and cope with extreme natural events like tropical storms or Earthquakes.
Clearly not all space activities are concerned with peaceful activities meant to increase the well being of the world population. A lot of them are used to increase the dominance of some societies, be it on an economical, political, cultural or military levels. "Societies" as the word is used here means an ensemble that is large enough to develop space activities. Typically such societies are at the continental level, rather than at the national level, at least in the present Europe.
\subsection{The Present}
The dependence of our societies on space based tools means that each independent society must master these tools in order to remain independent and to foster the well being of its citizens. Failure to do so in a society implies an important dependence on the tools that other societies are willing to share. Such willingness can only be granted as long as the interests of the granting society are preserved or re-inforced.
At present the US dominate the world space activities, their space budget being about 8 times that of Europe. The military fraction of the US space budget is probably slightly over 1/2 of the total US space expenditures. It is beyond the scope of this paper to present and discuss world space budgets. The discrepancy between the European and US space budgets is, however, sufficiently large that the argument does not need further details. The large difference in expenditures means that Europe is certainly dependent on US space products and technologies to a very high degree. This is painfully illustrated by the lack of a European navigation system. Several other "societies", like India or China, have taken steps to increase their space budgets in a massive way and will therefore gain independence on this aspect of their development, possibly before Europe.
Whereas space sciences have played a major role in the development of space based technologies that now form a vital element of the infrastructures of our societies, their present role is much less evident. the council of the European Space Agency (ESA) has organised a review of ESA's science programme. This lead to the "SPRT" report on the Science programme of 2007 in which one finds:
{\it
"Nowadays space science helps us to understand the evolution of the Universe and the
solar system including Earth. Space science in Europe has initially been the main driver for
the development of space technologies, which were later the basis for many applications serving
a wide range of societal needs. It provides tools and insights, which are of direct interest to mankind."
}
This means in short that whereas space science was an important element of the development of space technologie, it is not considered important anymore. Indeed policy makers, like industrial companies, require now immediate benefit from their investments in space, the days of the pioneers are long gone. Science does not offer immediate return on investment, knowledge is difficult to "count" and consequently, the space science budgets decline.
\subsection{The Future}
There is still an enormous volume of the parameter space in physics, astrophysics, cosmology and probably biology and elsewhere to be studied from space. To cite only one example, we are now in the very early steps in the understanding of the links between accretion onto supermassive black holes in the center of galaxies and the properties of the stellar populations that form the bulk of these galaxies. This study will require deep observations of the accretion itself, the X-ray background that represents probably the integrated accretion activity and the galaxies themselves. A study that can only be done using space instruments.
The space science community in many parts of the world has the intellectual power to propose and to implement many original ideas leading to new missions and instruments. This is illustrated by the cosmic vision process in Europe and the decadal surveys in the US. The pool of knowledge and expertise available in the space science communities of the world is enormously valuable, it is, however, fragile and vulnerable and will not survive a longer decline in the space science budgets.
Space science developments are in general very demanding. Observations or measurements must be optimised to the very end of the technological capacities to respond to the progressing quest for knowledge. Space sciences are therefore a unique area in which innovation is continuously pushed.
Space sciences are for all these reasons a prime application to develop, train and master space tools. It should become the engine with which "societies" can gain their independence in the space sector. Space science investments are, therefore, of prime importance to shape the relationships between "societies" in the coming decades.
Space science investments offer not only a central opportunity to gain independence in all sectors of modern life, they also offer a number of side benefits. While training and mastering space application, one also learns about space and the world, one thus gains on two sides. The knowledge acquired is relatively easy to share with the public, the endeavour is therefore not confined to engineers and scientists, but is relevant for society as a whole. This knowledge also contributes to a rational approach to the world, something that is often very necessary.
Space science developments are appropriate for cooperative endeavours. This means that seeking independence in space matters is not an aggressive process. This requires, however, that cooperation be made in a balanced way. Cooperation in this light is no "Ersatz" for ones own development and cannot replace large investments. Cooperation must be seen as a welcome added bonus, but may not become an essential feature of a project.
\section{Conclusion}
Astronomy has had an immense practical impact on the development of our civilisations for 1000's years and space sciences, that naturally include astronomy, have played a major part in the shaping of the modern world. Space science is, however, now not perceived as important for our development anymore. This should be massively changed and space science should become, again, a prime mover in space developments. This would allow societies to gain independence on all the areas in which space plays a role. The effort to be done in this domain will not only be important for space applications, it will also be fruitful and and contribute to the peaceful development of the world.
\end{document}
|
1,116,691,498,495 | arxiv | \section{Introduction}
Multiwavelenght follow-up studies of GRBs in the last decade have
established that a significant fraction of long GRBs arise from the
simultaneous collapse of a massive star (e.g. Woosley \& Bloom 2006,
Della Valle 2006). Best examples of this are the SN/GRB associations
so far discovered in the local universe, 2006aj with GRB 060218 (Pian
{\it et al.\/}~2006, Campana {\it et al.\/}~2006, Modiaz et al. 2006) and SN 1998bw with
980425 (Galama {\it et al.\/}~1998). Further evidence ($0.1<z<0.2$) comes from
SN 2003dh with GRB 030329 (Stanek {\it et al.\/}~2003; Hjorth {\it et al.\/}~2003) and
GRB 031203, which have been found to be associated with bright
broad-lined type Ic SNe (Malesani {\it et al.\/}~2004). This type of GRB/SN
association applies to a significant fraction of long GRBs, but not
all of them (Della Valle et al. 2006a, Fynbo et al. 2006, Gal-Yam et
al. 2006, Gehrels et al. 2006). At larger redshifts the association
GRB-SNe relies on about a dozen of rebrightenings (``bumps'') observed
during the late decay stages of the GRB afterglow light curve (see Zeh
{\it et al.\/}~2004 and references therein). They have been interpreted as due
to the emergence of the optical contribution of an underlying SN
(Bloom {\it et al.\/}~1999). In two cases, GRB 021211/SN 2002lt (Della Valle
{\it et al.\/}~2003) and GRB 050525A/SN 2005nc (Della Valle {\it et al.\/}~2006b),
spectroscopic observations obtained during these ``bumps'' are
suggestive of the presence of SN components. In spite of these
remarkable achievements, there are still many uncertainties hampering
our knowledge the rate of these events. Particularly, the recent
discovery of GRB 060218 at z=0.03 has raised the question whether or
not a population of ``local'' and ``Low-Luminosity'' GRBs (LL-GRBs,
i.e. $L<10^{48\div 49}$ erg/sec) with different properties from the
energetically ``High-Luminosity'' GRBs (HL-GRBs) does exist (e.g. Cobb
{\it et al.\/}~2006, Soderberg et al. 2006a, Amati et al. 2006).
The aim of this Letter is twofold: first we derive the
rate for the local ($z\lsim 0.1$) and LL-GRBs,
with two independent approaches. Then, we investigate the existence
of two classes of GRBs by measuring the ratios of the rates of LL-GRBs
and HL-GRBs to SNe-Ibc progenitors.
\section{Estimate of the Rate of sub-luminous GRBs}
We consider three LL-GRBs, GRB 980425 (z=0.008), GRB
060218 (z=0.03) and GRB 031203 (z=0.105) detected by {\it BeppoSax}
Wide Field Camera (WFC), {\it Swift} Burst Alert Telescope (BAT) and
INTEGRAL respectively and infer from them an empirical rate for
subluminous GRBs. Following Soderberg et al. 2006a we
estimate the peak flux in the range 1-1000 keV and compare it with
the threshold peak flux calculated by Band (2003, 2006). In this way
we derive the maximum distance ($D_{\rm max}$) the event could be
detected and therefore the maximum volume ($V_{\rm max}$). Considering
the sky coverage ($S_{\rm cov}$) and the number of years of operation
($T$) of each individual the detector the rate of LL-GRBs
similar to the GRBs given above is:
\begin{equation}
R_{GRB}=\frac{1}{V_{\rm max}}\frac{1}{S_{\rm cov}}\frac{1}{T}
\end{equation}
{\bf GRB 980425} This burst was detected by {\it BeppoSAX} WFC (Pian
{\it et al.\/}~1999) and by BATSE (Kippen {\it et al.\/} 1998). The peak flux in the
50-300 keV band was $F_{50-300}=4.48$ ph cm$^{-2}$ s$^{-1}$.
The redshift of this burst is z=0.0085, implying an isotropic peak luminosity
$L_{980425}\sim 5\times
10^{46}$ erg s$^{-1}$ several orders of magnitude smaller than typical
long HL-GRBs that have $L\sim 10^{51}$ erg$s^{-1}$ on average. By using the
spectrum given by Jimenez, Band \& Piran (2001) we extrapolate the peak
flux $F_{1-1000}=7.6$ ph cm$^{-2}$ s$^{-1}$. Given the threshold
$F_T=0.8$ ph cm$^{-2}$ s$^{-1}$ we find $D_{\rm max}=120$ Mpc. The BeppoSax
Sky coverage was about 0.08 and the operation time $\sim$ 4 yrs. Therefore
the rate of GRB 980425-like events is $R\sim 430$ Gpc$^{-3}$
yr$^{-1}$.
{\bf GRB 060218}. This burst was detected by {\it Swift}/BAT. The peak
flux was $F_{15-150}=2\times 10^{-8}$ erg cm$^{-2}$ s$^{-1}$ and the
spectral index 1.5 (Campana {\it et al.\/}~2006).
The redshift is, z=0.03, implying an isotropic peak luminosity
of the order of $L_{980425}$.
The extrapolated peak flux is
$F_{1-1000}=1.37$ ph cm$^{-2}$ s$^{-1}$. Given the threshold $F_T=1$
ph cm$^{-2}$ s$^{-1}$ we find $D_{max}=160$ Mpc. The {\it Swift}
Sky coverage was 0.17 and the operation time is one year.
Therefore we estimate that the rate of GRB 060218-like events
is $R\sim 350$ Gpc$^{-3}$ yr$^{-1}$.
{\bf GRB 031203} This burst was detected by INTEGRAL (Malesani
{\it et al.\/}~2004). The peak flux in the 20-200 keV band was
$F_{20-200}=1.2$ ph cm$^{-2}$ s$^{-1}$ and the spectral index is 0.8
($F_{\nu}=\nu^{-0.8}$).
The redshift of this burst is, z=0.105, implying an
isotropic peak luminosity $L_{031203}\sim 3\times
10^{48}$ erg s$^{-1}$ that is three order of magnitude smaller than
canonical long bursts but two order higher than the previous ones.
The extrapolated flux is $F_{1-1000}=3.35$ ph
cm$^{-2}$ s$^{-1}$. Given the threshold $F_T=0.7$ ph cm$^{-2}$ s$^{-1}$
(Mereghetti \& G\"otz 2005) we find $D_{max}=950$ Mpc. The INTEGRAL
Sky coverage is 0.5\% and the operation time was 3 yrs. Therefore the rate of
GRB 031203-like events is $R\sim 25$ Gpc$^{-3}$ yr$^{-1}$.
The redshift and the luminosity of this burst are
intermediate between LL-GRBs and HL-GRBs and this suggest a
continuity between the two classes.
The occurrence of two events such as GRB 980425 and GRB 060218, within
$\sim 150$ Mpc distance implies a rate of
$380^{+620}_{-225}$ LL-GRB Gpc$^{-3}$ yr$^{-1}$ in agreement with
Soderberg et al. 2006a result. The attached errors represent $1\sigma$
Poissonian standard deviation (Gherels 1986). Another potential source
of uncertainty (which is not included) is the correction for
beaming. However, we point out that LL-GRBs may be much less
collimated than typical HL-GRBs (see next section).
One way to estimate the jet opening angle, $\theta$, is to look
for ``achromatic-breaks'' in the afterglow light curve.
The steepening of the afterglow light curve, break, occurs
when the bulk Lorentz factor of the relativistic outflow becomes lower
than $1/\theta$.
From the data on the afterglow of GRB 980425 and GRB 060218 it seems
that the presence of breaks in the light curves is very late,
if existent at all, implying very large $\theta$.
For example radio observations of GRB 060218
(Soderberg {\it et al.\/}~2006a) have set $\theta \gsim 70^{\circ}$
which corresponds to a beaming factor ($f_b=1/(1-cos\theta)$),
$f_b^{-1}\lsim1.5$.
By interpreting the break in
the X-ray light curve of GRB 031203 as due to a jet, we obtain $\theta
\sim 30^{\circ}$, which corresponds to $f_b^{-1}\sim 8$.
The beaming corrected rate would thus be $R\sim 25\times 8 \lsim 200$ event
Gpc$^{-3}$ yr$^{-1}$, consistent with the value reported above.
\section{Comparison with the cosmological high luminous
and low-z low-luminous GRB rates}
The above estimate of the LL-GRBs rate may be affected
by the small number of available events, therefore an independent
check is in order. To do this we compare the empirical rate with
the rate inferred from the large GRB samples obtained by
both BATSE and Swift. As suggested by Schmidt (2001), the GRB local
rate can be estimated by using the BATSE peak flux distribution.
This is given by the convolution of two unknown quantities, the
luminosity function (LF) and the GRB formation rate. Following
Guetta, Piran \& Waxman (2005) and Guetta et al. (2004), we assume
that GRBs trace the star formation history and adopt the
Rowan-Robinson star formation rate (SFR), $ R_{GRB}(z) = \dot{n}_\circ \, {\rm
min}(10^{0.75},10^{0.75\,z})$, (Rowan-Robinson 1999).
Other rate evolutions have been analized in detail in Firmiani et al.
2004. Then we find
the best fit to the LF by comparing the BATSE observed peak flux
distribution with the predicted one, assuming a form for the LF
(i.e. a power law with a minimum and maximum luminosities). We show
in Fig. 1 that a single power-law LF, $\Phi(L)=L^{-1.6}$, with a
minimum ($L_{\rm min}=5\times 10^{49}$ erg s$^{-1}$) and maximum
luminosity (L$_{\rm max}=5\times 10^{52}$ erg s$^{-1}$) fits the BATSE peak
flux distribution very well. To obtain the observed local
rate of GRBs per unit volume, $\dot{n}_\circ$, we need to estimate the effective
full-sky coverage of our GRB sample. The BATSE catalog represents
3.185 years of BATSE full sky coverage implying a rate of 692 GRB per
year. Using our LF we find $\dot{n}_\circ\sim 1.1
\,$Gpc$^{-3}$yr$^{-1}$, that is much
smaller than what would be required in order to explain the local rate of
LL-GRBs\footnote{Note that for a different SFR like
for example the SF2-SFR of Porciani and Madau (2001) we still find a
good fit and a local rate smaller by a factor $\sim 2$.}.
However the local rate discussed in the previous paragraph was derived
under the hypothesis that classical bursts exceed by far the
luminosity of GRB 980425, of GRB 060218 and of GRB 031203 (i.e. in the
estimate of the rate Guetta, Piran \& Waxman (2005)
considered a minimum luminosity for the
luminosity function (LF) $L_{\rm min}\gtrsim 5\times 10^{49}$ erg/sec):
as such, this rate cannot be compared with the
empirical one derived for LL-GRBs. A self consistent estimate can
only be obtained with a LF that extends down to luminosities as low as that
of GRB 980425 and GRB 060218.
It is important to realize that most of the bursts below $L\sim
10^{48\div 49}$ erg s$^{-1}$ (LL-GRBs) are undetectable by current detectors,
unless they are extremely close (z$<0.1$).
On the basis of the observables that we have (i.e. the peak flux
distribution) we cannot constrain the minimum value of the LF,
therefore we find reasonable to take the minimum luminosity
observed in GRBs ($L_{980425}$) as our minimum value for the LF.
If we repeat the same procedure given above using the same
LF and L$_{\rm max}$ but a different $L_{\rm min}=L_{980425}$,
we find that the BATSE peak flux distribution is fitted
very well and the rate increases as $\dot{n}_\circ\sim 200\,$Gpc$^{-3}$yr$^{-1}$.
The increase in the rate of LL-GRB is due to the fact that
the luminosity function increases rapidly with decreasing luminosity
from $5\times 10^{49}$ ergs/sec down to $L_{\rm min}=L_{980425}$
(Guetta \& Piran 2006).
If we further extend the LF considering $L_{\rm min}$ in the range between
$0.1\,L_{980425}-L_{980425}$ we obtain $\dot{n}_\circ=200-1800$
Gpc$^{-3}$ yr$^{-1}$ consistent with the empirical rate determined in
section 2.
This luminosity function fits well also the Swift peak flux
distribution. Taking into account that Swift has detected 122 long
GRBs in 1.6 yr and covers 1/6 of the sky, applying the same method
described above, we find a rate of $\dot{n}_\circ=110-1200$ for $L_{\rm min}$ in
the range between $0.1\,L_{980425}-L_{980425}$ consistent with the
rate derived from the BATSE data and with the empirical rate.
\section{Rate of SNIbc, Hypernovae and GRBs}
The fraction of SNe-Ibc that produces GRBs can be measured as follows.
A rate of $\sim 2 \times 10^4$ SNe-Ibc Gpc$^{-3}$ yr$^{-1}$ is
derived by combining the local density of B luminosity of $\sim
1.2\times 10^8 L_{B,\odot}$ Mpc$^{-3}$ (e.g. Madau, Della Valle \&
Panagia 1998) with the rate of SNe-Ibc observed in Sbc--Irr
Hubble types (these morphological types are appropriate to represent
the GRB hosts) of 0.16 SNe per century and per 10$^{10}$ $L_{B,\odot}$
(SNu units, Cappellaro, Evans \& Turatto 1999). This SN rate has to be
compared with the rate of HL-GRBs of $\sim 1.1$ GRB Gpc$^{-3}$
yr$^{-1}$ (Guetta, Piran \& Waxman 2005) after rescaling for the jet
beaming factor, $f_b$. There exist different estimates for this
parameter: from $\sim 75$ (Guetta, Piran \& Waxman 2005) to $\sim 500$
(Frail {\it et al.\/}~2001) corresponding to beaming angles $\sim
10^\circ$--$4^\circ$, respectively. Taking these figures at their face
value, we find the ratio HL-GRB/SNe-Ibc to be in the range: $\sim
0.4\%-3\%$ and the ratio LL-GRBs/SNe-Ibc: $\sim 1\%-9\%$ (for
$f_b^{-1}=1$). Radio surveys give independent and consistent
constraints: Berger {\it et al.\/}~(2003) find that the incidence of SN
1998bw-like events, in the nearby universe, is $\lsim 3\%$,
Soderberg et al. (2006c) find HL-GRB/SNe-Ibc $<10\%$.
The computation of the GRB/HNe ratio requires a further step. The
measurement of the SN rate is based on the control-time methodology
(Zwicky 1938) that implies the systematic monitoring of galaxies of
known distances and the use of appropriate templates for the light
curves of each SN type (see Cappellaro {\it et al.\/}~1993 for bias and
uncertainties connected with this procedure). Unfortunately all
Hypernovae reported in Tab. I have not been discovered during time
`controlled' surveys, and therefore any attempt to derive an absolute
value of the rate of Hypernovae should be taken with caution. One
possibility is to compute the frequency of occurrence of all SNe-Ib/c
and Hypernovae in a limited distance sample of objects. We extracted
193 SNe-Ib/c from an upgraded version of the Asiago catalog
(\texttt{http://web.pd.astro.it/supern}), 19 of these have been
spectroscopically classified as Hypernovae.
In Fig. 2 we show that the cumulative distributions of the recession
velocities of the hosts of HNe (i.e. broad-lined SNe-Ib/c) and
``standard'' SNe-Ibc (dotted line) are statistically indistinguishable
(KS probability=0.73) for $cz< 10,000$ km/s. Within this volume we
find 158 objects, 12 of which are HNe. After excluding SN 1998bw
and 2006aj, because they were not serendipitously discovered and
assuming that the host galaxies of both `normal' SNe Ib/c and
Hypernovae were monitored with a comparable level of efficiency, we
infer that the fraction of Hypernovae is about $10/156 \simeq 7\%$ of
the total number of SNe Ib/c. Therefore the HN rate turns out to be
$\sim$ 0.015 SNe per century and per 10$^{10}$ $L_{B,\odot}$.
Note that this value is about an order of magnitude larger than
reported by Podsiadlowski et al. (2004). These authors find a similar
ratio HNe/SNe-Ibc (about 5\%), however they assume that HNe are on
average 4 times brighter than ``standard'' SNe-Ibc, and therefore they
can be detected in a volume about 10 times larger, therefore they
adopt an ``actual'' ratio HNe/SNe-Ibc $\sim 0.5\%$. However their
assumption on the luminosity at maximum of HNe does not seem supported
by either our Fig. 2 and the results of Soderberg et al. (2006b) who
find the magnitude distribution of GRB/XRF-SNe and local SNe-Ib/c (see
Richardson et al. 2006) to be statistically indistinguishable.
Finally we check whether our result is consistent with the lack of a
single HN detection in the Cappellaro {\it et al.\/}'s~(1999) sample. In
particular, from Poisson statistics we find that the
probabilities of obtaining a null result are $\sim$50\% and $\sim$5\% when the
expected HN numbers are 0.7 and 3.0. After using the control time of
Cappellaro {\it et al.\/}~(1999) we derive $\sim 0.012$ and $0.05$ HNe per
century per 10$^{10}$ $L_{B,\odot}$. Then, one should expect that the
HN rate is likely similar to the 50\% probability value (0.012 SNu)
and hardly higher than the 5\% probability value (0.05 SNu). The rate
of 0.015 SNu derived in this paper is fully consistent with the limits
set by the poissonian statistics.
Together these data imply a ratio LL-GRBs/HNe in the range
$\sim 1\%-10\%$ and beaming factors $f_b^{-1}\lsim 10$ (or $\theta \gsim
25^\circ$). In view of these findings, we shall assume, in the following,
that LL-GRBs emit (almost) isotropically.
It is interesting to compare the beaming
corrected rate of HL-GRBs with the rate of LL-GRBs. For every
observed HL-GRB there are $f_b^{-1}$ bursts that are not observed,
thus the true rate is $R\sim 100-550$ Gpc$^{-3}$
yr$^{-1}$. For isotropic LL-GRBs our analysis yields comparable rates:
$R\lsim 150-600$ Gpc$^{-3}$ yr$^{-1}$ (see section 2), $R\lsim
100-1200$ Gpc$^{-3}$ yr$^{-1}$ (from Swift data, section 3) and
$R\lsim 200-1800$ Gpc$^{-3}$ yr$^{-1}$ (from BATSE data, section
3). These results indicate that if there are two populations of GRBs
(HL-GRBs and LL-GRBs)
the respective frequency of occurrence are
comparable within a factor of $\sim 3$. However the data do not exclude the
case that only one population of GRBs is responsible for
giving rise to both the highly collimated component (preferentially
observable at high z) and to the almost isotropic components (detectable
only in nearby GRBs) (e.g. Woosley \& Heger 2006).
\begin{table}
\begin{tabular}{crr}
\hline
\tablehead{1}{r}{b}{SN}
& \tablehead{1}{r}{b}{cz~km/s}
& \tablehead{1}{r}{b}{References}\\
\hline
1997dq & 958 & Mazzali {\it et al.\/}~2004\\
1997ef & 3539& Filippenko 1997\\
1998bw & 2550& Galama {\it et al.\/}~1998 \\
1999as & 36000& Hatano {\it et al.\/}~2001\\
2002ap & 632 & Mazzali {\it et al.\/}~2002, Foley {\it et al.\/}~2003\\
2002bl & 4757& Filippenko {\it et al.\/}~2002\\
2003bg & 1320& Filippenko \& Chornack 2003a\\
2003dh & 46000& Stanek {\it et al.\/}~2003, Hjorth {\it et al.\/}~2003\\
2003jd & 5635& Filippenko {\it et al.\/}~2003b; Matheson {\it et al.\/}~2003\\
2003lw & 30000& Malesani {\it et al.\/}~2004\\
2004af & 16800& Riello {\it et al.\/}~2004\\
2004bu & 5549& Foley {\it et al.\/}~2004 \\
2005da & 4495& Modjaz {\it et al.\/}~2005\\
2005fk & 63500& Frieman {\it et al.\/}~2005a\\
2005kr & 36484& Frieman {\it et al.\/}~2005b\\
2005ks & 28500& Frieman {\it et al.\/}~2005b\\
2005kz & 8117& Filippenko, Foley \& Matheson 2005\\
2005nb & 7127& Roman \& Rostopchin 2006\\
2006aj & 9000& Masetti {\it et al.\/}~2006\\
\hline
\end{tabular}
\caption{Hypernovae}
\end{table}
\begin{figure*}[h]
\centering
\includegraphics{f1ref.eps}
\caption{\label{all}
Predicted logN-logP distribution (dashed-line) for the best fit LF with a
Rowan-Robinson SFR (1999) vs. the observed logN-logP taken from the BATSE
catalog. }
\end{figure*}
\begin{figure*}[h]
\centering
\includegraphics[width=13cm,angle=-90]{f2ref.ps}
\caption{\label{all} Cumulative distribution of the radial
velocities of the hosts of ``standard'' SN-Ibc (solid-line) and HNe
(dashed-line), within $cz < 10000$ km/s,
extracted from an upgraded version of the Asiago SN
Catalog (\texttt{http://web.pd.astro.it/supern}). A K-S test shows
that the probability that the two SN samples originate form the same
population is $\sim 73\%$.}
\end{figure*}
\section{Conclusions}
From the available information on GRBs and SNe-Ibc rates a
number of interesting results emerge:
i) we have computed the HN rate, on robust empirical grounds, and
found that this sub-class of SNe-Ibc, characterized by broad-lined
spectra, includes about 7\% of SNe-Ibc. An analysis of the
cumulative distribution of the recession velocities of the respective
hosts, does not suggest that HNe are intrinsically more luminous (on
average) than standard SNe-Ibc. All together these facts imply a
HN rate of $1.5\times 10^{-4}$ HNe $yr^{-1} 10^{10}L_{B\odot}$.
ii) the ratio HL-GRBs/HNe is smaller than 1, possibly in the range
0.04--0.3. The ratio LL-GRB/HNe is in the range 0.1--1.
iii) if we assume that all HNe are able to produce LL-GRBs, then the
LL-GRB/HNe ratio allows us to constrain the beaming factor to be
smaller than $f_b^{-1}\lsim 10$.
iv) from the analysis of two nearest GRBs we have derived a
LL-GRB rate of $380^{+620}_{-225}$ GRBs Gpc$^{-3}$ yr$^{-1}$. The
attached errors represent $1\sigma$ Poissonian standard
deviation. This result is consistent with the rates of 200--1800 and
100--1200 LL-GRB Gpc$^{-3}$ yr$^{-1}$, derived from the BATSE and
Swift GRBs.
v) the frequencies of occurrence of HL-GRBs (HL-GRB/SNe-Ibc $\lsim
3\%$) and LL-GRBs (LL-GRB/SNe-Ibc $\lsim 9\%$) may suggest the
existence of two physically distinct classes of GRBs (e.g. Cobb et
al. 2006, Soderberg et al. 2006a, Pian et al. 2006, Amati et al. 2006)
in which LL-GRBs are (intrinsically) more frequent events than
HL-GRBs. However, due to the uncertainties, we cannot exclude that a
single population of GRBs originates both the isotropic and
sub-energetic component, detectable only in nearby GRBs, and the
highly collimated component, observable by sampling huge volumes of
space, and thus mainly detectable in high-z GRBs.
\section{Acknowledgements}
We thank the anonymous referee for useful comments which have improved
the presentation of this paper and Sandro Mereghetti, Daniele Malesani
and Lorenzo Amati for useful discussions. The authors are also
indebted with Luigi Stella and Mario Vietri for their critical reading
of the manuscript. M.D.V. is grateful to the University of Tokyo for
the friendly hospitality and creative atmosphere. MDV acknowledges
the PRIN-INAF 2005 grant for financial support.
\newpage
|
1,116,691,498,496 | arxiv |
\subsection{Video QA task}
Video question and answering (QA) task is a challenging computer vision task
where a computer needs to answer questions given with input videos.
Though challenging, the task is worth studying since
it is an effective way to evaluate
how well a model understands the content of videos;
we can form any kinds of questions to test our QA models,
from naive ones (e.g., what, where, etc.)
to more profound ones (e.g., how, why, etc.).
Most video QA methods have included subtitles or scripts
,as well as the visual cue
in that actors' lines in the text, are crucial to
grasp essential information on videos.
Therefore, to solve the video QA task,
the system needs to extract proper features
from both visual inputs (i.e., RGB frames and optical flow) and
textual inputs (i.e., subtitle, query, and answer candidates),
and adequately correlate those features to infer correct answers.
When compared to image QA task,
video QA task has more challenges
in that it needs to additionally deal with
the temporal domain of visual information
and connect each feature from different modalities temporally.
Even setting aside the multi-modality,
extracting good visual features rich in temporal information
itself is difficult and has been actively studied in the field of video recognition.
Researchers have been approached video QA task from various perspectives.
Na et al.~\cite{na2017read} and Kim
et al.~\cite{Kim2017DeepStoryVS} propose
a deep model based on memory network architectures
for embedding the story of videos and reasoning the correct answer.
Zhu et al.~\cite{zhu2017uncovering} adopt a GRU
encoder-decoder
for modeling the temporal structure of a video and apply
a scoring mechanism for choosing the correct answer.
Various techniques~\cite{jang2017tgif, xu2017video, mun2017marioQA, gao2018motion}
adopt a spatiotemporal attention mechanism to select important features from the appearance
and motion information to solve the questions.
Also, 3D ConvNet~\cite{jang2017tgif,xu2017video,mun2017marioQA} is
commonly used for extracting temporal features from RGB video frames.
Many previous approaches use the ImageNet~\cite{imagenet_cvpr09} pre-trained
network for extracting spatial features and use
LSTM~\cite{hochreiter1997long}, GRU~\cite{chung2014empirical}, or
C3D~\cite{tran2015learning} for extracting temporal features from the sequence
of videos.
However, it has been shown for action recognition tasks that two-stream
method~\cite{simonyan2014two,wang2016temporal,fan2017identifying} that
utilizes optical flows for temporal cues has been more successful in terms of
video understanding than other methods. Therefore, departing from previous
video QA work, we adopt a two-stream network structure for extracting useful
spatiotemporal features from the sequence of video frames.
\Skip{
The reason why we adopted a two-stream network structure instead of other previous methods
is \YOON{give a short reason here first, instead of linking}.
This will be elaborated in the subsequent section.
}
\subsection{Two-stream network structure}
Thanks to its strength in processing the spatiotemporal domain, the two-stream
network structure has been widely used in the action classification field.
Simonyan et al.~\cite{simonyan2014two
, Wang et al.~\cite{wang2016temporal}, and Fan et al.~\cite{fan2017identifying}
use a two-stream ConvNet that gets two kinds of inputs:
one is a single frame of a video for the spatial stream ConvNet,
and the other is a multi-frame optical flow of the
video for the temporal stream ConvNet.
The two-stream network structure suggested by Carreira et al.~\cite{carreira2017quo}
also gets two kinds of inputs
but takes both the sequence of RGB and optical flow, respectively.
Simonyan et al.~\cite{simonyan2014two}, Wang et al.~\cite{wang2016temporal}, and Carreira et al.~\cite{carreira2017quo}
show higher accuracy on action recognition tasks
over single-stream architectures or
recurrent neural networks in dealing with the temporal domain of videos.
Fan et al.~\cite{fan2017identifying} show its capability of
processing spatiotemporal domain features with
identifying a camera wearer from a third-person view camera scene.
In this work, we propose to use the two-stream ConvNet
for video QA task,
focusing on its ability
to process spatiotemporal domain features.
\subsection{Attention mechanism}
Attention mechanism has been widely used for various applications including
image search~\cite{Kim18, kalantidis2016cross, noh2017large}.
Since the queries of the video QA task generally ask about a specific object
or event at a specific timing in a story, solving the video QA task needs to
focus on the important information
that is closely related to the queries from the story.
Seo et al.~\cite{seo2016bidirectional} present an attention flow layer that makes both context-aware query and query-aware context vectors by computing a similarity matrix and using it as an attention mask.
Lei et al.~\cite{lei2018tvqa} adopt the attention flow layer
as a context matching module and feed the context-aware vectors into bidirectional LSTM for jointly modeling the context and query.
Departing from the vector-wise attention methods, Hu et al.~\cite{hu2018squeeze} present a channel-wise attention method,
Squeeze-and-Excitation (SE) structure, that can be applied to any given transformation
$F_{tr} : X \mapsto U$, where $X \in \mathbb{R}^{H^{'} \times W^{'} \times C^{'}},
U \in \mathbb{R}^{H \times W \times C}$. SE block can be easily attached to the existing ConvNet models such as ResNet~\cite{he2016deep}
or GoogLeNet~\cite{szegedy2015going
, and the ConvNet with SE block shows a better result than ConvNet without SE block at the image classification task.
In this work, we apply the SE structure
to our two-stream network
for extracting features with
the channel-wise attention of the spatiotemporal domain.
We also utilize the context matching module
for matching the spatiotemporal features from the video frames and the queries.
\subsection{Two-stream I3D with SE structure}
The visual stream of our method
starts with the feature extraction stage.
In this work, we adopt two-stream I3D~\cite{carreira2017quo}, which shows its capacity for processing
video frames in action classification task.
However, unlike simple classification task,
visual features for QA task are highly required to focus on
salient objects and disregard others
since it has to be correlated (i.e. context matching) to
textual features which are
relatively more focused on necessary context by its nature.
Therefore, we adapt Squeeze-and-Excitation (SE)~\cite{hu2018squeeze} structure for temporal inputs
and add it
to several layers of I3D extractor
to generate more refined and attended visual features.
Our visual feature extractor (see Fig.~\ref{fig:theme_ex})
is based on
I3D pretrained on ImageNet~\cite{imagenet_cvpr09}
and Kinetics~\cite{kay2017kinetics} dataset.
It produces
a tuple of visual features
$V^S = \{V_{spt}^{S},V_{tpr}^{S}\} $
which includes spatial feature $V_{spt}^{S}$ from
RGB frames $F_{RGB} = \{a_0, a_1, ..., a_n\}$ and
temporal feature $V_{tpr}^{S}$ from
flow frames $F_{flow} = \{b_0, b_1, ..., b_{n-1}\}$,
where $n$ is the number of frames in a sequence,
$a_i \in \mathbb{R}^{224\times224\times3}$
and $b_i \in \mathbb{R}^{224\times224\times2}$.
Here, we need to have temporal sequence preserved
in extracted features
since in context matching stage, the features are
temporally matched and attended with query features.
Therefore, different from I3D~\cite{carreira2017quo}
producing two 400-dimensional vectors
from RGB and flow
(i.e.
$\{V_{spt}, V_{tpr} \}$, where
$V_{spt}, V_{tpr}\in\mathbb{R}^{400}$
),
we remove temporal pooling layers
to preserve the temporal sequence
so that we get
$V_{spt}^{S} \in \mathbb{R}^{n \times 400}$
and
$V_{tpr}^{S}\in\mathbb{R}^{(n-1) \times 400}$;
the preserved temporal sequence
is utilized in the context matching phase.
\Skip{
The two-stream I3D architecture gets a sequence of RGB frames, $F_{RGB} =
\{a_0, a_1, ..., a_n\} \in \mathbb{R}^{224\times224\times3}$, and optical flow
frames, $F_{flow} = \{b_0, b_1, ..., b_{n-1}\} \in
\mathbb{R}^{224\times224\times2}$, as inputs, where $n$ is the number of
video frames. The two-stream I3D in our method is pre-trained on
ImageNet~\cite{imagenet_cvpr09} and Kinetics~\cite{kay2017kinetics} dataset,
and is fine-tuned with the sequence of video frames, which are included in
the target video QA task.
The original two-stream I3D produces a set of spatiotemporal feature vectors
$V = \{V_{spt}, V_{tpr}|V_{spt}\in\mathbb{R}^{400},
V_{tpr}\in\mathbb{R}^{400}\}$, where $spt$ and $tpr$ indicate spatial and
temporal parts\YOON{Am I right?}, respectively.
Since the sequence length of spatiotemporal feature vectors from the original two-stream I3D \YOON{unclear..} is reduced to one by global average pooling,
we modified the original network to produce the spatiotemporal feature vectors $V^{S}$ which keeps their sequence length, XXX\YOON{use the notation here (define earlier if necessary); unclear to know which sequence that
you are talking here..} \YOON{What is the definition of the sequence?}, so the spatiotemporal feature vectors with the
sequence, $V^{S} = \{V_{spt}^{S},V_{tpr}^{S}|V_{spt}^{S}\in\mathbb{R}^{n_{RGB} \times 400},
V_{tpr}^{S}\in\mathbb{R}^{n_{flow} \times 400}\}$, where $n_{RGB}$ is the
number of RGB frame sequence and $n_{flow}$ is the number of optical flow
frame sequence, \YOON{odd English structure. break the long sentence into two sentences.} can be matched and attended with the query features in the
context matching layer.
}
The video frames included in the video QA task
commonly have unnecessary information
(e.g. background clutters or
unrelated objects)
which degrades the performance of a video QA model.
The textual queries and subtitles, however,
have less clutters
because textual information is usually
well focused on related objects or actions.
This kind of gap between two modalities
raise a difficulty in context matching stage
where visual and textual information is merged.
To make our feature extractor
concentrate more on the crucial objects
in the video frames,
we utilize the Squeeze-and-Excitation (SE)
structure~\cite{hu2018squeeze}
and integrate it within the two-stream I3D,
where a
feature vector $X$ is transformed into
another feature vector $U$ by the
inception
module~\cite{szegedy2015going, carreira2017quo}.
Fig.~\ref{fig:theme_ex} shows
how Sqeeuze-and-Excitation structure
is merged into the visual feature extractor.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{figure/theme_example.pdf}
\end{center}
\caption{
Our two-stream I3D with the Squeeze-and-Excitation structure. Both
RGB frames and optical-flow frames are processed by video feature
extractor. $N$ X I\&S means $N$ different Inception \& SE
block
modules.
}
\label{fig:theme_ex}
\end{figure}
In the SE block structure of two-stream I3D,
the \textit{Squeeze} operation
embeds global spatiotemporal information of output feature vectors of
the Inception module into a channel descriptor,
$z \in \mathbb{R}^{1\times 1\times 1\times C}$,
with global average
pooling (Fig.~\ref{fig:se_block}).
The
\textit{Excitation} operation generates the attended channel descriptor $s$ by two
fully-connected layers with activation functions:
\begin{gather}
s=F_{ex}(z,W)=\sigma(g(z,W))=\sigma(W_2\delta(W_{1}z)),
\end{gather}
where $z$ is the squeezed feature vectors, $\sigma$ is the sigmoid
function, $\delta$ is the ReLU~\cite{nair2010rectified} function, and
$W_1\in\mathbb{R}^{{C\over r}\times C}$
and $W_2\in\mathbb{R}^{C \times{C\over r}}$
are
the weight parameters of the fully-connected
dimensionality-reduction layer
and dimensionality-increasing layer, respectively.
The FC layers include a parameter $r$ for the ratio of dimensionality reduction.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1.0\linewidth]{figure/se_block.pdf}
\end{center}
\caption{
Our Squeeze-and-Excitation structure
}
\label{fig:se_block}
\end{figure}
By introducing the scale function:
\begin{equation}
F_{scale}:
\mathbb{R}^{H \times W \times L \times C}\times
\mathbb{R}^{1 \times 1 \times 1 \times C}
\rightarrow
\mathbb{R}^{H \times W \times L \times C},
\end{equation}
which operates on the channel descriptor $s$ and
the output feature vector of the
Inception block $U$,
we get the channel-wise attended feature vector
$\tilde{X}$:
\begin{equation}
\tilde{X} = F_{scale}(U, s) = U \odot s,
\end{equation}
where $\odot$ is the element-wise product
with shape broadcasting.
The SE block structure can be added to each inception module
without changing the core structure of I3D
to produce better features.
\subsection{Multimodal joint embedding and context matching}
To better express the needed visual context
conditioned by queries,
extracted spatiotemporal features
should be correlated
and jointly embedded with them.
For the multimodal joint embedding,
we adopt an attention flow layer proposed by Seo et
al.~\cite{seo2016bidirectional} and Lei et al.~\cite{lei2018tvqa}.
By the method, the spatiotemporal feature vectors and the textual feature vectors of the query are
jointly embedded to form a set of context-aware feature vectors.
We use GLoVe~\cite{pennington2014glove}
word embedding to vectorize
each word in the query.
Query sentences composed of multiple GLoVe vectors
are processed by bidirectional
LSTM~\cite{schuster1997bidirectional,hochreiter1997long}
which constructs our
textual features
$H_{query}\in\mathbb{R}^{n_{query} \times 2d}$,
from the sequence of words in the query, where $n_{query}$
is the number of words in query consisting of the question and the answer
candidates.
Although visual features attended by SE blocks,
visual I3D feature vectors $V^{S}$ and
textual GloVe vectors $H_{query}$
have a different level of information,
and there is also innate domain gap between them.
This gap of information level can disturb the joint embedding of
multimodal features and deliver the obscure information to the
prediction layer.
For these reasons,
we put an information level adjusting layer to remove
the gap of information level
between the two types of feature vectors.
The adjusting layer consists of
a learnable fully-connected
layer to each feature vectors that have
$W\in\mathbb{R}^{400\times 400}$ as weight and Leaky ReLU~\cite{xu2015empirical} as an
activation function.
\Skip{For
adjusting the level of information between the two types of feature vectors, we
design a calibration layer with FC layer that have $W\in\mathbb{R}^{400}$
as a weight and Leaky ReLU~\cite{xu2015empirical} as an activation function.}
Our information-level adjusting layer produces
a set of level-adjusted
spatiotemporal feature
vectors
$V_{spt}^{'S}\in\mathbb{R}^{n_{RGB} \times 400},
V_{tpr}^{'S}\in\mathbb{R}^{n_{flow} \times 400}$ and the calibrated textual
query feature vectors $H_{query}^{'}\in\mathbb{R}^{n_{query} \times 400}$.
The calibrated spatiotemporal feature vectors of video frames and the textual
query feature vectors are jointly modeled to produce a context-aware query
feature $G \in\mathbb{R}^{n_{video} \times 400}$ in a context matching layer
\cite{seo2016bidirectional, lei2018tvqa}:
\begin{align}
G &= S {H'}_{query}
\\
S &= softmax(V^{'S}_{\cdots} {H'}_{query}^{T})
\end{align}
where $S \in\mathbb{R}^{n_{video} \times {n_{query}}}$ is a similarity matrix.
We calculate the similarity matrix $S$ with the matrix multiplication operation
to link each sequence of video feature vectors and query feature vectors, and
the softmax function to emphasize the important information in the similarity
matrix.
The calibrated spatiotemporal feature vectors $V^{'S}$ and the context-aware
query feature vector $G$ are fused together to form a multimodal joint embedded
feature vector $M_{video, i}=\{M_{spt}, M_{tpr}\}$:
\begin{gather}
M_{video, i} = [V^{'S};G_{q};G_{a_{i}};V^{'S} \odot G_{q};V^{'S} \odot
G_{a_{i}}]\\M_{video, i}\in\mathbb{R}^{n_{video} \times 2000},
\end{gather}
where $n_{video}$ is the number of RGB or optical flow frames, $q$ is a
question and $a_{i}$ is the $i$-th answer candidate.
The dimension of $M_{video, i}$ is $n_{video}$ times 2,000, since we
concatenate five vectors, each of which is in 400 dimensions.
To predict the correct answer from $M_{video, i}$, we convert the fused
feature vectors $M_{video, i}$ to scalars of probability score
${p_{video,i}}=\{p_{spt}, ~p_{tpr}\}$ with an FC layer and a temporal
max-pooling layer, which can choose the most important information from the
spatiotemporal-fused feature.
\Skip{
\YOON{there have been many details here and there, but not much on the
importance or intellectual contributions. Please point them out in proper
places. Also, talk about differentiation over prior methods.}
}
\subsection{Processing of textual context and answer prediction}
We also perform the joint modeling of the query and the context encoded in another
textual context (e.g. subtitles),
which is already included in the video QA task.
The textual context is processed by bidirectional
LSTM~\cite{schuster1997bidirectional, hochreiter1997long} and fused with the
query feature vectors to form the joint embedded feature $M_{text, i}$.
The fused feature $M_{text, i}$ is encoded with bidirectional LSTM again to
extract the temporal information and max-pooled in the temporal domain to get
the answer probability score ${p_{text,i}}$.
\Skip{Similar to the method for processing the textual query, the textual context in video QA task is embedded with GLoVe~\cite{pennington2014glove} word embedding, and encoded with the bidirectional LSTM~\cite{schuster1997bidirectional, hochreiter1997long} with $2d$ hidden states.
The attention flow layer~\cite{seo2016bidirectional, lei2018tvqa} in Sec3.2 is also adopted for jointly embedding the features of textual context and the features of textual query, and produces the jointly embedded feature $M_{text, i}\in\mathbb{R}^{n_{text} \times 10d}$,
where $n_{text}$ is the number of words in the corresponding textual context of QA task
and $d$ is the number of dimension in word embedding.
The fused vector $M_{text, i}$ is converted to the}
Finally, we normalize the answer probability score $p_{video, i}$ and
$p_{text, i}$ with the softmax function and sum up to get the final answer
probability score.
To make the correct answer candidate's score is higher than any other wrong
answer candidates,
we adopt the log-sum-exp pairwise (LSEP) function~\cite{li2017improving}, which
is a smooth approximation of the marginal hinge ranking loss, as an object
function:
\begin{equation}
l_{LSEP} = \log\left(1 + \sum_{v\notin Y_{i}}\sum_{u\in Y_{i}}\exp(p_{x,v}-p_{x,u}) \right),
\end{equation}
where $Y_{i}$ is the correct answer.
By this smoothed ranking loss function, we can pose a margin between the
wrong answer candidates and the correct answer candidates in the feature space.
\subsection{Dataset}
The TVQA dataset~\cite{lei2018tvqa} includes 152,545 QA pairs from 21,793
TV show clips. The QA pairs are split into the ratio of 8:1:1
for training, validation, and test sets.
The TVQA dataset provides the sequence of video frames extracted at 3 FPS, the
corresponding subtitles with the video clips, and the query consisting of a
question and four answer candidates. Among the four answer candidates, there
is only one correct answer.
Since the TVQA dataset provides only the sequence of RGB video frames for
the visual context, we computed the optical flow frames with TV-L1
algorithm~\cite{wedel2009improved} for our two-stream spatiotemporal feature
extractor.
The dataset also provides the timestamps for each query, so we trained and
tested our model with the timestamps to localize the video and subtitle data.
\subsection{Implementation details}
We train the textual context processing channel and the visual context
processing channel separately for reducing the variances of neural networks and
achieving the effectiveness of the model
ensemble~\cite{dietterich2000ensemble}.
\Skip{By doing this, we can validate how
prediction result from the visual context processing channel affect the final
result,
which is a summation of the $p_{text, i}$ and $p_{video, i}$.}
When training the textual channel, we use the Adam optimizer~\cite{kingma2014adam}, where an initial learning rate is 0.0003, a momentum parameter $\beta_1$ is 0.9, a momentum parameter $\beta_1$ is 0.999, numerical stability parameter $\epsilon$ is $1e^{-8}$, and an exponential decay rate for 0.9 at every five epochs.
The model is trained for 100 epoch with the early stopping
method~\cite{caruana2001overfitting}, where the patience value is three, for
preventing the overfitting problem. We train our textual channel on a
machine, which has Intel Xeon CPU E5-2650 v4 @ 2.20GHz, 64GB of RAM, and four
Nvidia GTX 1080Ti GPU. The mini-batch sizes of each GPU are set to 32, and we take five days for training.
When training the visual channel, we import the pre-trained two-stream
I3D~\cite{carreira2017quo}, which was trained with the Kinetics
dataset~\cite{kay2017kinetics} that includes 25fps videos, and then fine-tune it
with the TVQA dataset. We use two kinds of optimizers, which are Adam~\cite{kingma2014adam} and SGD~\cite{bottou2010large},
because the two-stream I3D feature extractor
empirically requires a higher learning rate~\cite{carreira2017quo}
than other layers (e.g. context matching layer or calibration layer) in the
visual channel. We set the maximum number of frames to 69 per question due to
the limitation of VRAM in our GPU. We use SGD
optimizer~\cite{bottou2010large} for
training the two-stream I3D spatiotemporal feature extractor, where an initial
learning rate is 0.02, a momentum parameter for 0.9, and an exponential decay
rate for 0.9 at every five epochs. Each stream of two-stream I3D is trained
separately, and their predictions are combined
at the inference time.
For training the calibration layer and scoring layer in the visual channel, we
use the Adam optimizer, where an initial learning rate is 0.0003, a momentum
parameter $\beta_1$ is 0.9, a momentum parameter $\beta_1$ is 0.999, numerical
stability parameter $\epsilon$ is 0.1, and an exponential decay rate for 0.9 at
every five epochs. The model is trained for 40 epochs with the early stopping
method. A machine which we train our visual channel with has Intel Xeon CPU
E5-2630 v4 @ 2.20GHz, 1TB of RAM, and eight Nvidia RTX 2080Ti GPU. The
mini-bach sizes of each GPU are set to 4 and it takes three weeks for training.
Our model is implemented with Tensorflow~\cite{abadi2016tensorflow}.
\subsection{Experiment results on TVQA dataset}
Table~\ref{tab:overall_result} shows the results from baseline methods to our
model on the experiment of TVQA dataset.
All experiments are tested with the timestamp option in the dataset,
which is used for localizing the video and subtitle related to the query.
We test three baseline results adopted from the work of Lei et
al.~\cite{lei2018tvqa}, which are tested with three kinds of video features:
image indicates the ImageNet~\cite{imagenet_cvpr09} pre-trained
ResNet101~\cite{he2016deep} features, which are extracted from convolutional
layer5 after pooling and has 2048 dimension, region uses the pre-trained
Faster-RCNN~\cite{anderson2018bottom, krishna2017visual} features, and concept
uses the detected object labels of the pre-trained Faster-RCNN.
Our Image, RGB-I3D, Flow-I3D, and Two-stream I3D shown in Table~\ref{tab:overall_result} are variations of our model:
Image uses the pre-trained ResNet101 features, whose dimension is reduced to
400 for fitting to our model,
RGB-I3D uses the spatial video features from the sequence of video RGB frames,
Flow-I3D uses the temporal video features from the sequence of optical-flow
frames, and Two-stream I3D uses both spatiotemporal video feature. The random
selection model gets 20\% accuracy because the TVQA dataset has one correct
answer among five answer candidates.
\Skip{V+Q in the second column of table means that the model predict the answer with the video and query as the source of information,
and S+V+Q in the third column of table means that the model predict the answer with the video feature, subtitle, and query as the source of information.}
When we evaluate the models with subtitle and query information, our model gets
better results than the tested baseline methods. We assume that the LSEP loss
function, which we adopt instead of cross-entropy loss, helps our model to
learn better features and solve more difficult questions than the baseline
model.
We expected that the features from the two-stream network show higher accuracy
than using the ImageNet feature, but the ImageNet feature from ResNet101, Our Image,
gets the highest accuracy when considering both
video and query among the tested four different types of video features
evaluated with our model.
Furthermore, the ImageNet feature in our model shows a lower result than the
baseline methods with the ImageNet feature. This is caused by the reduction of
dimension on the ImageNet feature to fit our model to the context matching,
degrading the amount of information in the feature.
Under the test setting of S+V+Q, we find that all kinds of video features show
lower accuracy than the baseline, and they even degrades the performance of S+Q
setting. Especially, the video features from I3D much degrades the performance
of S+Q setting than the ImageNet feature (Our Image). To find out why our
video features get an inferior result over the baseline and degrade the
accuracy of the text-only setting, we analyze the
baseline and our approaches' accuracy of each question type.
\Skip{Among the three of our video features, the spatiotemporal feature from
two-stream I3D gets the highest accuracy when considering both video and query
information.
This is mainly because the spatial features from the RGB-I3D stream and the
temporal features from the Flow-I3D stream complement each other and
provide relatively
rich information than each single stream feature.
However, our video features do not work well compared to the baseline and get a lower accuracy under the
test setting of S+Q. To find out why our video features get an inferior result
over the baseline and degrade the accuracy of the text-only setting, we analyze the
baseline and our approaches' accuracy of each question type.}
Table~\ref{tab:6w-result} shows the accuracy of different methods under each
question type. We split the questions into six types of what, who, where, how,
why, and others.
As shown in S+Q column, our text only model improves accuracy in every question
types except for 'why' and 'others' which account for 10\% of the total
questions. Specifically, our model improves the most in 'where' questions,
where the original model shows the lowest accuracy.
On the contrary to the result of S+Q, our model shows lower accuracy than
baseline methods in every question types except for 'others' in V+Q. Our
hypothesis of this result is that our spatiotemporal feature extractor with the
two-stream I3D has difficulty in extracting the feature from the provided video
data, because 3fps, which is a frame rate of provided video data, is too low
for extracting sufficiently dense optical-flow for our two-stream I3D, which is
originally trained on 25fps videos. Nevertheless, our V+Q model shows the
highest accuracy on 'where' question type, which our S+Q model shows the lowest
accuracy. This result
shows the possibility that our visual model can complement the textual model.
\subsection{The effectiveness of the SE structure}
To see the effectiveness of our SE structure, we perform ablation study w/ and
w/o the SE block structure in Table~\ref{tab:se_result}.
We find that the models with the SE structure show higher accuracy than the
models without the SE structure in S+V+Q. Based on this result, we can see that
the SE structure in our method helps to
extract the complementing spatiotemporal features to the textual feature.
\subsection{Limitations and future work}
Since the visual channel in our model spent too much time for training, we
needed more than a week to check the experiment result. This heavy
requirement of computational power led to the insufficient amount of attempts
for finding a proper architecture design and hyper-parameters.
In future work, we will train and evaluate our model in a more efficient way
and search for the cause of the malfunction in the visual
channel, such as modifying the structure of I3D to work with low frame videos or
checking the context module whether it works properly to reduce the gap of
information level.
\section{Introduction}
\input{content/1_introduction.tex}
\section{Related Work}
\input{content/2_related_work.tex}
\section{Our Approach}
\input{content/3_methodology.tex}
\section{Experiment}
\input{content/4_experiment_and_result.tex}
\section{Conclusion}
\input{content/5_conclusion.tex}
\bibliographystyle{ieeetr}
|
1,116,691,498,497 | arxiv |
\section{Proofs of Corollaries}\label{sec:corpf}
\subsection{Proof of Corollary \ref{cor:shortestpath}}\label{apdx:shortestpath}
Since LP \eqref{lp:shortestpath} always has an integral solution,
it suffices to show that the max-product BP on GM \eqref{gm:shortestpath} converges to the solution of LP.
The proof of Corollary \ref{cor:shortestpath} can be done by using Theorem \ref{thm:main}.
From GM \eqref{gm:shortestpath}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}.
Suppose there are $v$ and $x_{\delta(v)}$ with $\psi_v(x_{\delta(v)})=1$.
Consider the case when there is $e\in\delta^i(v)$ with $x_{e}=1\ne x^*_{e}$.
If ${e^\prime}\in\delta^i(v)$ with $x_{e^\prime}=0\ne x^*_{e^\prime}$ exists, choose such ${e^\prime}$.
If not, choose ${e^\prime}\in\delta^o(v)$ with $x_{e^\prime}=1\ne x^*_{e^\prime}$.
On the other hand, consider when there is $e\in\delta^i(v)$ with $x_{e}=0\ne x^*_e$. If ${e^\prime}\in\delta^o(v)$ with $x_{e^\prime}=1\ne x^*_{e^\prime}$ exists, choose such ${e^\prime}$.
If not, choose ${e^\prime}\in\delta^i(v)$ with $x_{e^\prime}=0\ne x^*_{e^\prime}$. Then,
$$
\psi_v(x^\prime_{\delta(v)})=1,\qquad
\mbox{
where
$x^\prime_{e^{\prime\prime}} = \begin{cases}
x_{e^{\prime\prime}}~&\mbox{if}~e^{\prime\prime}\ne e,e^\prime\\
x^*_{e^{\prime\prime}}~&\mbox{otherwise}
\end{cases}$.}$$
$$
\psi_v(x^{\prime\prime}_{\delta(v)})=1,\qquad
\mbox{
where
$x^{\prime\prime}_{e^{\prime\prime}} = \begin{cases}
x_{e^{\prime\prime}}~&\mbox{if}~e^{\prime\prime}= e,e^\prime\\
x^*_{e^{\prime\prime}}~&\mbox{otherwise}
\end{cases}$.}$$
We can apply similar argument for the case when $e\in\delta^o(v)$, $v=s~\mbox{or}~t$.
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:shortestpath} is unique, the max-product BP on GM \eqref{gm:shortestpath} converges to the solution of LP \eqref{lp:shortestpath}.
\subsection{Proof of Corollary \ref{cor:matching}}\label{apdx:matching}
The proof of Corollary \ref{cor:matching} can be done by using Theorem \ref{thm:main}.
From GM \eqref{gm:modmatching}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}. Suppose there are $v$ and $x_{\delta(v)}$ with $\psi_v(x_{\delta(v)})=1$.
Consider the case when there is $e_i\in\delta(v)$ with $x_{e_i}=1\ne x^*_{e_i}$.
Then, there is $e^\prime_j\in\delta(v)$ with $x_{e^\prime_j}=0\ne x^*_{e^\prime_j}$.
Choose such $e^\prime_j$.
On the other hand, consider when there is $e_i\in\delta(v)$ with $x_{e_i}=0\ne x^*_{e_i}$.
Then, there is $e^\prime_j\in\delta(v)$ with $x_{e^\prime_j}=1\ne x^*_{e^\prime_j}$.
Choose such $e^\prime_j$.
Then,
$$
\psi_v(x^\prime_{\delta(v)})=1,\qquad
\mbox{
where
$x^\prime_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k\ne e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
$$
\psi_v(x^{\prime\prime}_{\delta(v)})=1,\qquad
\mbox{
where
$x^{\prime\prime}_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k= e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:modmatching} is unique, the max-product BP on GM \eqref{gm:modmatching} converges to the solution of LP \eqref{lp:modmatching}.
\subsection{Proof of Corollary \ref{cor:oddmatching}}\label{apdx:oddmatching}
From GM \eqref{gm:oddmatching}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}.
For $v\in V$, we can apply same argument as the maximum weight matching case.
Suppose there are $v_C$ and $y_{\delta(v_C)}$ with $\psi_C(y_{\delta(v_C)})=1$.
Consider the case when there is ${(u_1,v_C)}\in\delta(v_C)$ with $y_{(u_1,v_C)}=1\ne y^*_{(u_1,v_C)}$.
As a feasible solution $y_{\delta(v_C)}$ forms a disjoint even paths \citep{shin2013graphical},
check edges along the path contains $u_1$. If there is $u_2\in V(C)$ in the path with $y_{(u_2,v_C)}=1\ne y^*_{(u_2,v_C)}$ exists, choose such ${(u_1,v_C)}$.
If not, choose $(u_2,v_C)\in V(C)$ with $y_{(u_2,v_C)}=0\ne y^*_{(u_2,v_C)}$ at the end of the path.
On the other hand, consider the case when there is ${(u_1,v_C)}\in\delta(v_C)$ with $y_{(u_1,v_C)}=0\ne y^*_{(u_1,v_C)}$.
As a feasible solution $y_{\delta(v_C)}$ form a disjoint even paths,
check edges along the path contains $u_1$. If there is $u_2\in V(C)$ in the path with $y_{(u_2,v_C)}=0\ne y^*_{(u_2,v_C)}$ exists, choose such ${(u_1,v_C)}$.
If not, choose $(u_2,v_C)\in V(C)$ with $y_{(u_2,v_C)}=1\ne y^*_{(u_2,v_C)}$ at the end of the path.
Then, from disjoint even paths point of view, we can check that
\begin{align*}
&\psi_C(y^\prime_{\delta(v_C)})=1,\\
&\qquad\mbox{
where
$y^\prime_{(u,v_C)} = \begin{cases}
y_{(u,v_C)}~&\mbox{if}~u\ne u_1,u_2\\
y^*_{(u,v_C)}~&\mbox{otherwise}
\end{cases}$.}\\
&\psi_C(y^{\prime\prime}_{\delta(v_C)})=1,\\
&\qquad\mbox{
where
$y^{\prime\prime}_{(u,v_C)} = \begin{cases}
y_{(u,v_C)}~&\mbox{if}~u=u_1,u_2\\
y^*_{(u,v_C)}~&\mbox{otherwise}
\end{cases}$.}
\end{align*}
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:oddmatching} is unique and integral, the max-product BP on GM \eqref{gm:oddmatching} converges to the solution of LP \eqref{lp:oddmatching}.
\subsection{Proof of Corollary \ref{cor:vertexcover}}\label{apdx:vertexcover}
The proof of Corollary can be done by using Theorem \ref{thm:main}.
From GM \eqref{gm:vertexcover}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}.
Suppose there are $v$ and $x_{\delta(v)}$ with $\psi_v(x_{\delta(v)})=1$.
Consider the case when there is $e_i\in\delta(v)$ with $x_{e_i}=1\ne x^*_{e_i}$.
If there is $e^\prime_j\in\delta(v)$ with $x_{e^\prime_j}=0\ne x^*_{e^\prime_j}$, choose such $e^\prime_j$.
If not, choose $e^\prime_j=e_i$
On the other hand, consider when there is $e_i\in\delta(v)$ with $x_{e_i}=0\ne x^*_{e_i}$.
If there is $e^\prime_j\in\delta(v)$ with $x_{e^\prime_j}=1\ne x^*_{e^\prime_j}$, choose such $e^\prime_j$.
If not, choose $e^\prime_j=e_i$
Then,
$$
\psi_v(x^\prime_{\delta(v)})=1,\qquad
\mbox{
where
$x^\prime_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k\ne e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
$$
\psi_v(x^{\prime\prime}_{\delta(v)})=1,\qquad
\mbox{
where
$x^{\prime\prime}_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k= e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:vertexcover2} is unique, the max-product BP on GM \eqref{gm:vertexcover} converges to the solution of LP \eqref{lp:vertexcover2}.
\subsection{Proof of Corollary \ref{cor:ec}}\label{apdx:modec}
The proof of Corollary can be done by using Theorem \ref{thm:main}.
From GM \eqref{gm:modec}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}.
Suppose there are $v$ and $x_{\delta(v)}$ with $\psi_v(x_{\delta(v)})=1$.
Consider the case when there is $e_i\in\delta(v)$ with $x_{e_i}=1\ne x^*_{e_i}$.
If there is $e^\prime_j\in\delta(v)$ with $x_{e^\prime_j}=0\ne x^*_{e^\prime_j}$, choose such $e^\prime_j$.
If not, choose $e^\prime_j=e_i$
On the other hand, consider when there is $e_i\in\delta(v)$ with $x_{e_i}=0\ne x^*_{e_i}$.
If there is $e^\prime_j\in\delta(v)$ with $x_{e^\prime_j}=1\ne x^*_{e^\prime_j}$, choose such $e^\prime_j$.
If not, choose $e^\prime_j=e_i$
Then,
$$
\psi_v(x^\prime_{\delta(v)})=1,\qquad
\mbox{
where
$x^\prime_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k\ne e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
$$
\psi_v(x^{\prime\prime}_{\delta(v)})=1,\qquad
\mbox{
where
$x^{\prime\prime}_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k= e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:modec} is unique, the max-product BP on GM \eqref{gm:modec} converges to the solution of LP \eqref{lp:modec}.
\subsection{Proof of Corollary \ref{cor:tsp}}\label{apdx:tsp}
The proof of Corollary \ref{cor:tsp} can be done by using Theorem \ref{thm:main}.
From GM \eqref{gm:tsp}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}.
Suppose there are $v$ and $x_{\delta(v)}$ with $\psi_v(x_{\delta(v)})=1$.
Consider the case when there is $e\in\delta(v)$ with $x_{e}=1\ne x^*_{e}$.
By formulation of GM, there exists ${e^\prime}\in\delta(v)$ with $x_{e^\prime}=0\ne x^*_{e^\prime}$. Choose such ${e^\prime}$.
On the other hand, consider when there is $e\in\delta(v)$ with $x_{e}=0\ne x^*_e$.
There exists ${e^\prime}\in\delta(v)$ with $x_{e^\prime}=1\ne x^*_{e^\prime}$. Choose such ${e^\prime}$.
Then,
$$
\psi_v(x^\prime_{\delta(v)})=1,\qquad
\mbox{
where
$x^\prime_{e^{\prime\prime}} = \begin{cases}
x_{e^{\prime\prime}}~&\mbox{if}~e^{\prime\prime}\ne e,e^\prime\\
x^*_{e^{\prime\prime}}~&\mbox{otherwise}
\end{cases}$.}$$
$$
\psi_v(x^{\prime\prime}_{\delta(v)})=1,\qquad
\mbox{
where
$x^{\prime\prime}_{e^{\prime\prime}} = \begin{cases}
x_{e^{\prime\prime}}~&\mbox{if}~e^{\prime\prime}= e,e^{\prime\prime}\\
x^*_{e^{\prime\prime}}~&\mbox{otherwise}
\end{cases}$.}$$
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:tsp} is unique and integral, the max-product BP on GM \eqref{gm:tsp} converges to the solution of LP \eqref{lp:tsp}.
\subsection{Proof of Corollary \ref{cor:maxvertexpacking}}\label{apdx:maxvertexpacking}
The proof of Corollary \ref{cor:maxvertexpacking} can be done by using Theorem \ref{thm:main}.
From GM \eqref{gm:maxvertexpacking}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}.
Suppose there are $v$ and $x_{\delta(v)}$ with $\psi_v(x_{\delta(v)})=1$.
Consider the case when there is $e\in\delta(v)$ with $x_{e}=1\ne x^*_{e}$.
If ${e^\prime}\in\delta(v)$ with $x_{e^\prime}=0\ne x^*_{e^\prime}$ exists. Choose such ${e^\prime}$.
If not, there exists ${e^\prime}\in\delta(v)$ with $x_{e^\prime}=1\ne x^*_{e^\prime}$. Choose such ${e^\prime}$.
On the other hand, consider when there is $e\in\delta(v)$ with $x_{e}=0\ne x^*_e$.
If ${e^\prime}\in\delta(v)$ with $x_{e^\prime}=1\ne x^*_{e^\prime}$ exists. Choose such ${e^\prime}$.
If not, there exists ${e^\prime}\in\delta(v)$ with $x_{e^\prime}=0\ne x^*_{e^\prime}$. Choose such ${e^\prime}$.
Then,
$$
\psi_v(x^\prime_{\delta(v)})=1,\quad
\mbox{
where $
\begin{cases}
&x^\prime_{e^{\prime\prime}} = \begin{cases}
x_{e^{\prime\prime}}~&\mbox{if}~e^{\prime\prime}\ne e,e^\prime\\
x^*_{e^{\prime\prime}}~&\mbox{otherwise}
\end{cases}\\
&y^\prime_{v} = y^*_v
\end{cases}$
.}$$
$$
\psi_v(x^{\prime\prime}_{\delta(v)})=1,\quad
\mbox{
where $
\begin{cases}
&x^{\prime\prime}_{e^{\prime\prime}} = \begin{cases}
x_{e^{\prime\prime}}~&\mbox{if}~e^{\prime\prime}= e,e^{\prime\prime}\\
x^*_{e^{\prime\prime}}~&\mbox{otherwise}
\end{cases}\\
&y^\prime_{v} = y_v
\end{cases}$
.}$$
Case of $y$ variable can be done in similar manner.
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:maxvertexpacking} is unique and integral, the max-product BP on GM \eqref{gm:maxvertexpacking} converges to the solution of LP \eqref{lp:maxvertexpacking}.
\subsection{Proof of Corollary \ref{cor:networkflow}}\label{apdx:networkflow}
The proof of Corollary can be done by using Theorem \ref{thm:main}.
From GM \eqref{gm:networkflow}, each variable is connected to two factors ({\em C2} of Theorem \ref{thm:main}). Now, lets check {\em C3} of Theorem \ref{thm:main}.
Suppose there are $v$ and $x_{\delta(v)}$ with $\psi_v(x_{\delta(v)})=1$.
Consider the case when there is $e_i\in\delta^o(v)$ with $x_{e_i}=1\ne x^*_{e_i}$.
If there is $e_j\in\delta(v)^i$ with $x_{e_j}=1\ne x^*_{e_j}$, choose such $e_j$.
Otherwise, there is $e_j\in\delta(v)^o$ with $x_{e_j}=0\ne x^*_{e_j}$.
Choose such $e_j$.
On the other hand, consider when there is $e_i\in\delta^o(v)$ with $x_{e_i}=0\ne x^*_{e_i}$.
If there is $e_j\in\delta(v)^i$ with $x_{e_j}=0\ne x^*_{e_j}$, choose such $e_j$.
Otherwise, there is $e_j\in\delta(v)^o$ with $x_{e_j}=1\ne x^*_{e_j}$.
Choose such $e_j$.
One can apply similar argument for $x_i\in\delta(v)^i$.
Then,
$$
\psi_v(x^\prime_{\delta(v)})=1,\qquad
\mbox{
where
$x^\prime_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k\ne e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
$$
\psi_v(x^{\prime\prime}_{\delta(v)})=1,\qquad
\mbox{
where
$x^{\prime\prime}_{e^{\prime\prime}_k} = \begin{cases}
x_{e^{\prime\prime}_k}~&\mbox{if}~e^{\prime\prime}_k= e_i,e^\prime_j\\
x^*_{e^{\prime\prime}_k}~&\mbox{otherwise}
\end{cases}$.}$$
From Theorem \ref{thm:main}, we can conclude that
if the solution of LP \eqref{lp:networkflow2} is unique, the max-product BP on GM \eqref{gm:networkflow} converges to the solution of LP \eqref{lp:networkflow2}.
\section{Applications of Theorem \ref{thm:main}}\label{sec:applications}
In the following sections, we introduce concrete instances of LPs satisfying the conditions of Theorem \ref{thm:main}
so that BP correctly converges to its optimal solution.
Specifically, we consider LP formulations associated to several
combinatorial optimization problems including
shortest path, maximum weight perfect matching,
traveling salesman, maximum weight disjoint vertex cycle packing, vertex/edge cover and network flow.
We note that the shortest path result, i.e., Corollary \ref{cor:shortestpath},
is known \citep{ruozzi2008st}, where we rediscover it as a corollary of Theorem \ref{thm:main}.
Our other results, i.e., Corollaries \ref{cor:matching}-\ref{cor:networkflow},
are new and what we first establish in this paper.
\subsection{Example I: Shortest Path}\label{sec:shortest}
Given a directed graph $G=(V,E)$ and non-negative edge weights $w=[w_e:e\in E]\in \mathbb R_+^{|E|}$,
the shortest path problem
is to find the shortest path from the source $s$ to the destination $t$:
it minimizes the sum of edge weights along the path. One can naturally design the following LP for this problem:
\begin{equation}\label{lp:shortestpath}
\begin{split}
&\mbox{minimize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad \sum_{e\in \delta^o(v)} x_{e}-\sum_{e\in\delta^i(v)} x_{e}
=
\begin{cases}
&1~~~\mbox{if}~v=s\\
&-1~\mbox{if}~v=t\\
&0~~~\mbox{otherwise}
\end{cases}\\
&\qquad\qquad\qquad~ x=[x_{e}]\in [0,1]^{|E|}.
\end{split}
\end{equation}
where $\delta^i(v),\delta^o(v)$ are the set of incoming, outgoing edges of $v$.
It is known that the above LP always has an integral solution, i.e., the shortest path from $s$ to $t$.
We consider the following GM for LP \eqref{lp:shortestpath}:
\begin{equation}\label{gm:shortestpath}
\Pr[X=x]~\propto~\prod_{e\in E} e^{-w_{e} x_{e}}\prod_{v\in V} \psi_{v} (x_{\delta(v)}),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e\in\delta^o(v)} x_{e}-\sum_{e\in\delta^i(v)} x_{e}\\
&\qquad=
\begin{cases}
&1~~~\mbox{if}~v=s\\
&-1~\mbox{if}~v=t\\
&0~~~\mbox{otherwise}
\end{cases}\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:shortestpath}, one can easily check Conditions {\em C2, C3}
of Theorem \ref{thm:main} hold and
derive the following corollary whose
formal proof is presented in Section \ref{apdx:shortestpath}.
\begin{corollary}\label{cor:shortestpath}
If the shortest path from $s$ to $t$, i.e., the solution of the shortest path LP \eqref{lp:shortestpath}, is unique, then the max-product BP on GM \eqref{gm:shortestpath} converges to it.
\end{corollary}
The uniqueness condition in the above corollary is easy to guarantee by adding small random noises to edge weights.
\subsection{Example II: Maximum Weight Perfect Matching}\label{subsec:matching}
Given an undirected graph $G=(V,E)$ and non-negative edge
weights $w=[w_e:e\in E]\in \mathbb{R}_+^{|E|}$ on edges, the maximum weight perfect matching problem is to find a set of edges such that each vertex is connected to exactly one edge in the set and the sum of edge weights in the set is maximized. One can naturally design the following LP for this problem:
\begin{equation}\label{lp:matching}
\begin{split}
&\mbox{maximize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad \sum_{e\in\delta(v)} x_e= 1\\%,\quad\forall\, v\in V\\
&\qquad\qquad\qquad~ x=[x_e]\in [0,1]^{|E|}.
\end{split}
\end{equation}
where $\delta(v)$ is the set of edges connected to a vertex $v$.
If the above LP has an integral solution, it corresponds to the solution of the maximum weight perfect matching problem.\\\\
It is known that
the maximum weight matching LP \eqref{lp:matching}
always has a half-integral solution $x^*\in \{0,\frac12,1\}^{|E|}$. We will design BP for obtaining the half-integral solution.
To this end, duplicate each edge $e$ to $e_1,e_2$ and define a new graph $G^\prime=(V,E^\prime)$ where $E^\prime=\{e_1,e_2:e\in E\}$.
Then, we suggest the following equivalent LP that always have an integral solution:
\begin{equation}\label{lp:modmatching}
\begin{split}
&\mbox{maximize}\qquad~ w^\prime\cdot x\\
&\mbox{subject to}\qquad \sum_{e_i\in\delta(v)} x_{e_i}= 2\\%\qquad\forall\,v\in V\\
&\qquad\qquad\qquad~ x=[x_{e_i}]\in [0,1]^{|E^\prime|}.
\end{split}
\end{equation}
where $w^\prime_{e_1}=w^\prime_{e_2}=w_e$.
One can easily observe that solving LP \eqref{lp:modmatching} is equivalent to solving LP \eqref{lp:matching}
due to our construction of $G^\prime$ and $w^\prime$.
Now, construct the following GM for LP \eqref{lp:modmatching}:
\begin{equation}\label{gm:modmatching}
\Pr[X=x]~\propto~\prod_{e_i\in E^\prime} e^{w^\prime_{e_i} x_{e_i}}\prod_{v\in V} \psi_{v} (x_{\delta(v)}),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e_i\in\delta(v)} x_{e_i}= 2\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:modmatching},
we derive the following corollary of Theorem \ref{thm:main}
whose
formal proof is presented in Section \ref{apdx:matching}.
\begin{corollary}\label{cor:matching}
If the solution of the maximum weight perfect matching LP \eqref{lp:modmatching} is unique, then
the max-product BP on GM \eqref{gm:modmatching} converges it.
\end{corollary}
Again, the uniqueness condition in the above corollary is easy to guarantee by adding small random noises to edge weights $[w^\prime_{e_i}]$.
We note that it is known \citep{bayati2011belief} that BP converges to
the unique and integral solution of LP \eqref{lp:matching}, while Corollary \ref{cor:matching}
implies that BP can solve it without the integrality condition.
We note that one can easily obtain a similar result for the maximum weight (non-perfect) matching problem, where
we omit the details in this paper.
\subsection{Example III: Maximum Weight Perfect Matching with Odd Cycles}
In previous section we prove that BP converges to the optimal (possibly, fractional)
solution of LP \eqref{lp:modmatching}, equivalently LP \eqref{lp:matching}.
One can add odd cycle (also called Blossom) constraints and make those LPs tight i.e. solves the maximum weight perfect matching problem:
\begin{equation}\label{lp:matchingcycle}
\begin{split}
&\mbox{maximize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad \sum_{e\in\delta(v)} x_e= 1,\quad\forall\, v\in V\\
&\qquad\qquad\qquad~ \sum_{e\in C} x_e\le \frac{|C|-1}{2}, \quad\forall C\in\mathcal C,\\
&\qquad\qquad\qquad~ x=[x_e]\in [0,1]^{|E|}.
\end{split}
\end{equation}
where $\mathcal C$ is a set of odd cycles in $G$.
The authors \citep{shin2013graphical} study BP for solving
LP \eqref{lp:matchingcycle} by replacing $\sum_{e\in\delta(v)} x_e= 1$ by $\sum_{e\in\delta(v)} x_e\leq 1$, i.e.,
for the maximum weight (non-perfect) matching problem.
Using Theorem \ref{thm:main}, one can extend the result to
the maximum weight perfect matching problem, i.e., solving LP \eqref{lp:matchingcycle}.
To this end, we follow the approach
\citep{shin2013graphical} and
construct the following graph $G^\prime=(V^\prime,E^\prime)$ and weight $w^\prime=[w^\prime_e:
e\in E^{\prime}]\in\mathbb R^{|E^\prime|}$ given set $\mathcal C$ of disjoint odd cycles:
\begin{align*}
&V^\prime=V\cup\{v_C:C\in\mathcal{C}\}\\
&E^\prime=\{(u,v_C):u\in C,C\in\mathcal{C}\}\cup E\setminus\{e\in C:C\in\mathcal{C}\}
\end{align*}
\begin{align*}
&w^\prime_e=
\begin{cases}
\frac{1}{2}\sum_{e^\prime\in E(C)}(-1)^{d_C(u,e^\prime)}w_{e^\prime}&\mbox{if}~e={(u,v_C)}\\
&\mbox{for some}~C\in\mathcal{C}\\
w_e&\mbox{otherwise}
\end{cases},
\end{align*}
where
$d_C(u,e^\prime)$ is the graph distance between $u,e^\prime$ in cycle $C$.
Then, LP \eqref{lp:matchingcycle} is equivalent to the following LP:
\begin{equation}\label{lp:oddmatching}
\begin{split}
&\mbox{maximize}\qquad~ w^\prime\cdot y\\
&\mbox{subject to}\quad~ \sum_{e\in\delta(v)} y_e= 1,\qquad\qquad\qquad\qquad\forall\, v\in V\\
&\qquad\qquad\quad \sum_{u\in V(C)}(-1)^{d_C(u,e)}y_{(v_C,u)}\in [0,2],~\forall e\in E(C)\\
&\qquad\qquad\quad \sum_{e\in\delta(v_C)}y_e\le |C|-1, \qquad\qquad\quad~~\forall C\in\mathcal{C}\\
&\qquad\qquad\qquad y=[y_e]\in [0,1]^{|E^\prime|}.
\end{split}
\end{equation}
Now, we construct the following GM from the above LP:
\begin{equation}\label{gm:oddmatching}
\Pr[Y=y]~\propto~\prod_{e\in E} e^{w_e y_e}\prod_{v\in V} \psi_{v} (y_{\delta(v)})\prod_{C\in\mathcal{C}} \psi_{C} (y_{\delta(v_C)}),
\end{equation}
where the factor function $\psi_v$, $\psi_C$ is defined as
\begin{align*}
&\psi_{v}(y_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e\in\delta(v)} y_e= 1\\
0&\mbox{otherwise}
\end{cases},\\
&\psi_{C}(y_{\delta(v_C)})=
\begin{cases}
1&\mbox{if}~ \sum_{u\in V(C)}(-1)^{d_C(u,e)}y_{(v_C,u)}\in\{0,2\}\\
&\quad\sum_{e\in\delta(v_C)}y_e\le |C|-1\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:oddmatching}, we derive the following corollary of Theorem \ref{thm:main}
whose formal proof is presented in Section \ref{apdx:oddmatching}.
\begin{corollary}\label{cor:oddmatching}
If the solution of the maximum weight perfect matching with odd cycles LP \eqref{lp:oddmatching} is unique and integral, then
the max-product BP on GM \eqref{gm:oddmatching} converges to it.
\end{corollary}
We again emphasize that
a similar result for the maximum weight (non-perfect) matching problem was established in \citep{shin2013graphical}.
However, the proof technique in the paper does not extend to the perfect matching problem. This is in essence because
presumably the perfect matching problem is harder than the non-perfect matching one.
Under the proposed generic criteria of Theorem \ref{thm:main}, we overcome the technical difficulty.
\subsection{Example IV: Vertex Cover}\label{subsec:vc}
Given an undirected graph $G=(V,E)$ and non-negative integer vertex weights $b=[b_v:v\in V]\in \mathbb{Z}_+^{|V|}$, the vertex cover problem is to find a set of vertices
minimizes the sum of vertex weights in the set such that each edge is connected to at least one vertex in it.
This problem is one of {Karp's 21 NP-complete problems} \citep{karp1972reducibility}.
The associated LP formulation to the vertex cover problem is as follows:
\begin{equation}\label{lp:vertexcover0}
\begin{split}
&\mbox{minimize}\qquad~ b\cdot y\\
&\mbox{subject to}\qquad y_u+y_v\ge 1\\%, (u,v)\in E\\
&\qquad\qquad\qquad~ y=[y_v]\in [0,1]^{|V|}.
\end{split}
\end{equation}
However, if we design a GM from the above LP,
it does not satisfy conditions in Theorem \ref{thm:main}.
Instead, we will show that BP can solve the following dual LP:
\begin{equation}\label{lp:vertexcover}
\begin{split}
&\mbox{maximize}\qquad~ \sum_{e\in E}x_e\\
&\mbox{subject to}\qquad \sum_{e\in\delta(v)} x_e\le b_v\\%,\quad\forall\, v\in V\\
&\qquad\qquad\qquad~ x=[x_e]\in \mathbb R_+^{|E|}.
\end{split}
\end{equation}
Note that the above LP always has a half-integral solution.
As we did in Section \ref{subsec:matching},
one can duplicate edges, i.e., $E^\prime=\{e_1,\dots,e_{2b_{\max}}:e\in E\}$ with
$b_{\max}=\max_v b_v$, and design the following equivalent LP having an integral solution:
\begin{equation}\label{lp:vertexcover2}
\begin{split}
&\mbox{maximize}\qquad~ w^\prime\cdot x\\
&\mbox{subject to}\qquad \sum_{e_i\in\delta(v)} x_{e_i}\le 2b_v,\quad\forall\, v\in V\\
&\qquad\qquad\qquad~ x=[x_{e_i}]\in [0,1]^{|E^\prime|}
\end{split},
\end{equation}
where $w^\prime_{e_i}=w_e$ for $e\in E$ and its copy $e_i\in E^{\prime}$.
From the above LP, we can construct the following GM:
\begin{equation}\label{gm:vertexcover}
\Pr[X=x]~\propto~\prod_{e_i\in E^\prime} e^{w_{e_i}^\prime x_{e_i}}\prod_{v\in V} \psi_{v} (x_{\delta(v)}),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e_i\in\delta(v)} x_{e_i}\le2 b_v\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:vertexcover}, we derive the following corollary of Theorem \ref{thm:main}
whose formal proof is presented in Section \ref{apdx:vertexcover}.
\begin{corollary}\label{cor:vertexcover}
If the solution of the vertex cover dual LP \eqref{lp:vertexcover2} is unique, then the
max-product BP on GM \eqref{gm:vertexcover} converges it.
\end{corollary}
Again, the uniqueness condition in the above corollary is easy to guarantee by adding small random noises to edge weights $[w^\prime_{e_i}]$.
We further remark that if
the solution of the primal LP \eqref{lp:vertexcover0} is integral, then
it can be easily found from the solution of the dual LP \eqref{lp:vertexcover2}
using the strictly complementary slackness condition \citep{bertsimas1997introduction} .
\iffalse
The proof of Corollary \ref{cor:vertexcover} is presented in the appendix.
From the solution of dual LP \eqref{lp:vertexcover} and strictly complementary slackness \citep{bertsimas1997introduction} 1,
we can directly find the solution of LP \eqref{lp:vertexcover0} if its solution is unique and integral.
\begin{corollary}\label{cor:vertexcover0}
If the solution of LP \eqref{lp:vertexcover0} is unique and integral, then the solution of LP \eqref{lp:vertexcover0}
can be found from the solution of its dual LP \eqref{lp:vertexcover}.
\end{corollary}
\begin{proof}
Let $[y^*_v]$ be a solution of vertex cover LP \eqref{lp:vertexcover0} and $[x^*_e]$ be a solution of dual LP \eqref{lp:vertexcover}.
By strictly complementary slackness condition and a uniqueness and intragrality of $[y^*_v]$,
if $\sum_{e\in\delta(v)}x_e=b_v$, then $y^*_v=1$ and
if $\sum_{e\in\delta(v)}x_e<b_v$, then $y^*_v=0$.
\end{proof}
\fi
\iffalse
\subsection{Minimum Vertex Disjoint Cycle Cover}
Given a directed graph $G=(V,E)$, the minimum vertex disjoint cycle cover problem is to find a set of minimum number of cycles
such that each vertex is contained in exactly one cycle. This problem is known to be NP-Hard problem.
Let define $E^\prime=\{e_1,\dots,e_n|e\in E\},~n=|V|$ be a union of $n$ duplications of $E$ and define $G^\prime=(V,E^\prime)$.
Then, solving the minimum vertex disjoint cycle cover problem on $G$ is equivalent to solving the following IP in $G^\prime$.
\begin{equation}\label{ip:mincyclecover}
\begin{split}
&\mbox{minimize}\quad~~~ \sum_{e_1\in E^\prime}x_{e_1}\\
&\mbox{subject to}\quad~ \sum_{e_{1}\in\delta^o(v)} x_{e_{1}}+\sum_{e_{i+1}\in\delta^o(v)} x_{e_{i+1}}\\
&\qquad\qquad\quad-\sum_{e_{i}\in\delta^i(v)} x_{e_{i}}=0,~\forall i\ne 1\\
&\qquad\qquad\qquad \sum_i\sum_{e_i\in\delta^i(v)} x_{e_i}=1\\
&\qquad\qquad\qquad \sum_i\sum_{e_i\in\delta^o(v)} x_{e_i}=1,~\forall v\in V\\
&\qquad\qquad\quad x=[x_{e_i}]\in \{0,1\}^{|E^\prime|}.
\end{split}
\end{equation}
\subsection{Example V: Maximum Edge Disjoint Cycle Packing}
Given a directed graph $G=(V,E)$, the maximum edge disjoint cycle packing problem is to find a maximum number of edge disjoint cycles
in $G$. This problem is known to be NP-Hard problem.
Let define $E^\prime=\{e_1,\dots,e_n|e\in E\},~n=|V|$ be a union of $n$ duplications of $E$ and define $G^\prime=(V,E^\prime)$.
Then, solving the maximum edge disjoint cycle packing problem on $G$ is equivalent to solving the following IP in $G^\prime$.
\begin{equation}\label{ip:maxcyclepacking}
\begin{split}
&\mbox{maximize}\quad~~~ \sum_{e_1\in E^\prime}x_{e_1}\\
&\mbox{subject to}\quad~ \sum_{e_{1}\in\delta^o(v)} x_{e_{1}}+\sum_{e_{i+1}\in\delta^o(v)} x_{e_{i+1}}\\
&\qquad\qquad\quad-\sum_{e_{i}\in\delta^i(v)} x_{e_{i}}=0,~\forall i\ne 1\\
&\qquad\qquad\qquad \sum_i\sum_{e_i\in\delta^i(v)} x_{e_i}=1\\
&\qquad\qquad\quad~ \sum_{e_1\in\delta^i(v)} x_{e_1}=0\\
&\qquad\qquad\qquad \sum_i\sum_{e_i\in\delta^o(v)} x_{e_i}=1,~\forall v\in V\\
&\qquad\qquad\quad x=[x_{e_i}]\in \{0,1\}^{|E^\prime|}.
\end{split}
\end{equation}
\fi
\subsection{Example V: Edge Cover}\label{subsec:ec}
Given an undirected graph $G=(V,E)$ and non-negative edge
weights $w=[w_e:e\in E]\in \mathbb{R}_+^{|E|}$ on edges, the minimum weight edge cover problem is to find a set of edges such that
each vertex is connected to at least one edge in the set and the sum of edge weights in the set is minimized. One can naturally design
the following LP for this problem:
\begin{equation}\label{lp:ec}
\begin{split}
&\mbox{minimize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad \sum_{e\in\delta(v)} x_e\ge 1\\%,\quad\forall\, v\in V\\
&\qquad\qquad\qquad~ x=[x_e]\in [0,1]^{|E|}.
\end{split}
\end{equation}
where $\delta(v)$ is the set of edges connected to a vertex $v$.
If the above LP has an integral solution, it corresponds to the solution of the minimum weight edge cover problem.
Similarly as the case of matching,
it is known that
the minimum weight edge cover LP \eqref{lp:ec}
always has a half-integral solution $x^*\in \{0,\frac12,1\}^{|E|}$. We will design BP for obtaining the half-integral solution.
To this end, duplicate each edge $e$ to $e_1,e_2$ and define a new graph $G^\prime=(V,E^\prime)$ where $E^\prime=\{e_1,e_2:e\in E\}$.
Then, we suggest the following equivalent LP that always have an integral solution:
\begin{equation}\label{lp:modec}
\begin{split}
&\mbox{minimize}\qquad~ w^\prime\cdot x\\
&\mbox{subject to}\qquad \sum_{e_i\in\delta(v)} x_{e_i}\ge 2\\%\qquad\forall\,v\in V\\
&\qquad\qquad\qquad~ x=[x_{e_i}]\in [0,1]^{|E^\prime|}.
\end{split}
\end{equation}
where $w^\prime_{e_1}=w^\prime_{e_2}=w_e$.
One can easily observe that solving LP \eqref{lp:modec} is equivalent to solving LP \eqref{lp:ec}
due to our construction of $G^\prime$ and $w^\prime$.
Now, construct the following GM for LP \eqref{lp:modec}:
\begin{equation}\label{gm:modec}
\Pr[X=x]~\propto~\prod_{e_i\in E^\prime} e^{-w^\prime_{e_i} x_{e_i}}\prod_{v\in V} \psi_{v} (x_{\delta(v)}),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e_i\in\delta(v)} x_{e_i}\ge 2\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:modec},
we derive the following corollary of Theorem \ref{thm:main}
whose
formal proof is presented in Section \ref{apdx:modec}.
\begin{corollary}\label{cor:ec}
If the solution of the minimum weight edge cover LP \eqref{lp:modmatching} is unique, then
the max-product BP on GM \eqref{gm:modec} converges it.
\end{corollary}
Again, the uniqueness condition in the above corollary is easy to guarantee by adding small random noises to edge weights $[w^\prime_{e_i}]$.
\subsection{Example VI: Traveling Salesman}\label{sec:tsp}
Given a directed graph $G=(V,E)$ and non-negative edge weights $w=[w_e:e\in E]\in \mathbb{R}_+^{|E|}$,
the traveling salesman problem (TSP) is to find the minimum weight Hamiltonian cycle in $G$.
The natural LP formulation to TSP is following:
\begin{equation}\label{lp:tsp}
\begin{split}
&\mbox{minimize}\qquad w\cdot x\\
&\mbox{subject to}\qquad \sum_{e\in\delta(v)} x_{e}=2\\%~\forall v\in V\\
&\qquad\qquad\quad x=[x_{e}]\in [0,1]^{|E|}.
\end{split}
\end{equation}
From the above LP, one can construct the following GM:
\begin{equation}\label{gm:tsp}
\Pr[X=x]~\propto~\prod_{e\in E} e^{-w_e x_e}\prod_{v\in V} \psi_{v} (x_{\delta(v)}),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e\in\delta(v)} x_{e}=2\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:tsp}, we derive the following corollary of Theorem \ref{thm:main}
whose formal proof is presented in Section \ref{apdx:tsp}.
\begin{corollary}\label{cor:tsp}
If the solution of the traveling salesman LP \eqref{lp:tsp} is unique and integral,
then the max-product BP on GM \eqref{gm:tsp} converges it.
\end{corollary}
Again, the uniqueness condition in the above corollary is easy to guarantee by adding small random noises to edge weights.
\iffalse
\begin{table*}[t]
\begin{center}{\footnotesize
\caption{Experimental results for small size complete graph and
each number is the average among 100 samples.
For example, Greedy+BP means that the Greedy algorithm using edge weights as BP beliefs
as we describe in Section \ref{sec:exp}.
The left value is the approximation ratio, i.e.,
the average weight ratio between the heuristic solution and the exact solution.
The right value is the average weight of the heuristic solutions.
The last row is a ratio of tight TSP LP \eqref{lp:tsp}.}
\vspace{0.1in}
\begin{tabular} {|l|r|r|r|r|r|}
\hline
Size &5 &10 &15 &20 &25\\[0.01cm] \hline
Greedy &1.07~/~1.84&1.20~/~2.25&1.33~/~2.58&1.51~/~2.85&1.51~/~3.04 \\[0.01cm] \hline
Greedy+BP &1.00~/~1.75&1.06~/~2.01&1.14~/~2.25&1.18~/~2.26&1.19~/~2.39 \\[0.01cm] \hline
Christofides &1.06~/~1.85&1.38~/~2.56&1.67~/~3.20&1.99~/~3.75&2.16~/~4.32 \\[0.01cm] \hline
Christofides+BP &1.00~/~1.75&1.07~/~2.04&1.17~/~2.30&1.22~/~2.34&1.26~/~2.53 \\[0.01cm] \hline
Insertion &1.03~/~1.79&1.29~/~2.38&1.53~/~2.95&1.72~/~3.26&1.89~/~3.77 \\[0.01cm] \hline
Insertion+BP &1.00~/~1.75&1.34~/~2.47&1.54~/~2.98&1.85~/~3.48&2.00~/~4.01 \\[0.01cm] \hline
N-Neighbor &1.07~/~1.84&1.27~/~2.39&1.42~/~2.74&1.55~/~2.96&1.64~/~3.30 \\[0.01cm] \hline
N-Neighbor+BP &1.00~/~1.75&1.06~/~2.01&1.11~/~2.19&1.15~/~2.20&1.13~/~2.27 \\[0.01cm] \hline
2-Opt &1.00~/~1.75&1.08~/~2.04&1.12~/~2.21&1.24~/~2.36&1.28~/~2.57 \\[0.01cm] \hline
2-Opt+BP &1.00~/~1.75&1.04~/~1.96&1.07~/~2.11&1.09~/~2.10&1.12~/~2.27 \\[0.01cm] \hline
Tight LPs &100\%&93\%&88\%&87\%&84\% \\[0.01cm] \hline
\end{tabular}
\label{table:tspsmallval}
}
\end{center}
\begin{center}{\footnotesize
\caption{Experimental results for sparse Erdos-Renyi graph with fixed average vertex degrees
and
each number is the average among 1000 samples.
The left value is the ratio that a heuristic finds the Hamiltonian cycle without penalty edges.
The right value is the average weight of the heuristic solutions.}
\begin{tabular} {|l|r|r|r|r|r|r|}
\hline
Size&\multicolumn{3}{|c|}{100}&\multicolumn{3}{|c|}{200}\\[0.01cm] \hline
Degree&10&25&50&10&25&50\\[0.01cm] \hline
Greedy &0\%~/~7729.43 &0.3\%~/~2841.98 &13\%~/~1259.08 &0\%~/~15619.9 &0\%~/~5828.88 &0.3\%~/~2766.07 \\[0.01cm] \hline
Greedy+BP &14.4\%~/~1612.82 &21.7\%~/~1110.27 &44.1\%~/~622.488 &6.4\%~/~2314.95 &10.4\%~/~1687.29 &16.4\%~/~1198.48 \\[0.01cm] \hline
Christoifeds &0\%~/~19527.3 &0\%~/~16114.3 &0\%~/~10763.7 &0\%~/~41382.5 &0\%~/~37297.0 &0\%~/~32023.1 \\[0.01cm] \hline
Christofides+BP &14.2\%~/~2415.73 &20\%~/~1663.47 &34.9\%~/~965.775 &6.1\%~/~3586.77 &9.2\%~/~2876.35 &12.9\%~/~2183.80 \\[0.01cm] \hline
Insertion &0\%~/~12739.2 &84.5\%~/~198.099 &100\%~/~14.2655 &0\%~/~34801.6 &0.9\%~/~3780.71 &99.6\%~/~44.1293 \\[0.01cm] \hline
Insertion+BP &0\%~/~13029.0 &76.2\%~/~283.766 &100\%~/~14.6964 &0\%~/~34146.7 &0.3\%~/~4349.11 &99.9\%~/~41.2176 \\[0.01cm] \hline
N-Neighbor &0\%~/~9312.77 &0\%~/~3385.14 &7.6\%~/~1531.83 &0\%~/~19090.7 &0\%~/~7383.23 &0.3\%~/~3484.82 \\[0.01cm] \hline
N-Neighbor+BP &16\%~/~1206.95 &26.2\%~/~824.232 &50.06\%~/~509.349 &6.9\%~/~1782.17 &12.4\%~/~1170.38 &24.3\%~/~888.421 \\[0.01cm] \hline
2-Opt &34.5\%~/~1078.03 &100\%~/~14.6873 &100\%~/~7.36289 &2\%~/~3522.78 &100\%~/~35.8421 &100\%~/~18.6147 \\[0.01cm] \hline
2-Opt+BP &76.7\%~/~293.45 &100\%~/~13.5773 &100\%~/~6.53995 &33.7\%~/~1088.79 &100\%~/~34.7768 &100\%~/~17.4883 \\[0.01cm] \hline
Tight LPs &62.7\% &62.3\% &63.0\% &52.2\% &55.1\% &52.2\% \\[0.01cm] \hline
\end{tabular}
\label{table:tspsparseratio}
}
\end{center}
\end{table*}
\fi
\subsection{Example VII: Maximum Weight Cycle Packing}
Given an undirected graph $G=(V,E)$ and non-negative edge weights $w=[w_e:e\in E]\in \mathbb{R}_+^{|E|}$,
the maximum weight vertex disjoint cycle packing problem is to find the maximum weight set of cycles with no common vertex.
It is easy to observe that it is equivalent to find
a subgraph maximizing the sum of edge weights on it such that each vertex of the subgraph has degree 2 or 0.
The natural LP formulation to this problem is following:
\begin{equation}\label{lp:maxvertexpacking}
\begin{split}
&\mbox{maximize}\qquad w\cdot x\\
&\mbox{subject to}\qquad \sum_{e\in\delta(v)} x_{e}=2y_v\\
&\qquad\qquad x=[x_{e}]\in [0,1]^{|E|},y=[y_v]\in[0,1]^{|V|}.
\end{split}
\end{equation}
From the above LP, one can construct the following GM:
\begin{equation}\label{gm:maxvertexpacking}
\Pr[X=x,Y=y]~\propto~\prod_{e\in E} e^{w_e x_e}\prod_{v\in V} \psi_{v} (x_{\delta(v)},y_v),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)},y_v) =
\begin{cases}
1&\mbox{if}~ \sum_{e\in\delta(v)} x_{e}=2y_v\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:maxvertexpacking}, we derive the following corollary of Theorem \ref{thm:main}
whose formal proof is presented in Section \ref{apdx:maxvertexpacking}.
\begin{corollary}\label{cor:maxvertexpacking}
If the solution of maximum weight vertex disjoint cycle packing LP \eqref{lp:maxvertexpacking} is unique and integral,
then the max-product BP on GM \eqref{gm:maxvertexpacking} converges it.
\end{corollary}
Again, the uniqueness condition in the above corollary is easy to guarantee by adding small random noises to edge weights.
\iffalse
\subsection{Maximum Weight Edge Disjoint Cycle Packing}
Given a graph $G=(V,E)$ and weight $w=[w_e:e\in E]\in \mathbb{R}^{|E|}$,
the maximum weight edge disjoint cycle packing problem is to find a maximum weight set of cycles with no common edge.
Then, the maximum weight edge disjoint cycle packing problem on $G$ is to find
a subgraph maximizes the sum of edge weights such that each vertex has even degree.
\begin{equation}\label{lp:maxedgepacking}
\begin{split}
&\mbox{maximize}\qquad \sum_{e\in E}w_e x_{e}\\
&\mbox{subject to}\qquad \sum_{e\in\delta(v)} x_{e}=\sum_{i=1}^{deg(v)}2y_{v_i}\\
&\qquad\qquad~~ x=[x_{e}]\in \{0,1\}^{|E|},y=[y_{v_i}]\in[0,1]^{{2|V||E|}}.
\end{split}
\end{equation}
Where, $\delta(v)$ is a set of edges connected to a vertex $v$.
From above LP, we can construct GM
\begin{equation}\label{gm:maxedgepacking}
\Pr[X=x,Y=y]~\propto~\prod_{e\in E} e^{w_e x_e}\prod_{v\in V} \psi_{v} (x_{\delta(v)}),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e\in\delta(v)} x_{e}=\mbox{even}\\
0&\mbox{otherwise}
\end{cases}.\\
\end{align*}
Let us introduce Corollary from above GM.
\begin{corollary}\label{cor:maxedgepacking}
If the maximum weight edge disjoint cycle packing LP \eqref{lp:maxedgepacking} has a unique and integral solution,
then max-product BP on GM \eqref{gm:maxedgepacking} converges to the solution of LP \eqref{lp:maxedgepacking}
\end{corollary}
The proof of Corollary \ref{cor:maxedgepacking} is presented in the appendix.
\fi
\subsection{Example VIII: Minimum Cost Network Flow}
Given a directed graph $G=(V,E)$, supply/demand $d=[d_v]\in\mathbb{Z}_+^{|E|}$ and
capacity $c=[c_e:e\in E]\in\mathbb{Z}_+^{|E|}$,
the minimum cost network flow problem can be forumlated by the following LP.
\begin{equation}\label{lp:networkflow}
\begin{split}
&\mbox{minimize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad \sum_{e\in \delta^o(v)} x_{e}-\sum_{e\in\delta^i(v)} x_{e}=d_v\\
&\qquad\qquad\qquad~ x_e\le c_e\\
&\qquad\qquad\qquad~ x=[x_{e}]\in \mathbb{R}_+^{|E|},
\end{split}
\end{equation}
where $\delta^i(v),\delta^o(v)$ are the set of incoming, outgoing edges of $v$.
It is known that the above LP always has an integral solution.
We will design BP for obtaining the solution of LP \eqref{lp:networkflow}.
To this end, duplicate each edge $e$ to $e_1,\dots,e_{c_e}$ and define a new graph $G^\prime=(V,E^\prime)$ where $E^\prime=\{e_1,\dots,e_{c_e}:e\in E\}$.
Then, we suggest the following equivalent LP that always have an integral solution:
\begin{equation}\label{lp:networkflow2}
\begin{split}
&\mbox{minimize}\qquad~ w^\prime\cdot x\\
&\mbox{subject to}\qquad \sum_{e_i\in \delta^o(v)}-\sum_{e_i\in\delta^i(v)} x_{e_i}=d_v\\
&\qquad\qquad\qquad~ x=[x_{e_i}]\in[0,1]^{|E^\prime|}.
\end{split}
\end{equation}
where $w^\prime_{e_1}=\dots=w^\prime_{e_{c_e}}=w_e$. One can easily observe that solvin
LP \eqref{lp:networkflow} is equivalent to solving LP \eqref{lp:networkflow2} due to our construction
of $G^\prime$ and $w^\prime$.
Now, construct the following GM for LP \eqref{lp:networkflow2}:
\begin{equation}\label{gm:networkflow}
\Pr[X=x]~\propto~\prod_{e_i\in E^\prime}e^{-w^\prime_{e_i} x_{e_i}}\prod_{v\in V}
\psi_{v} (x_{\delta(v)}),
\end{equation}
where the factor function $\psi_v$ is defined as
\begin{align*}
&\psi_{v}(x_{\delta(v)}) =
\begin{cases}
1&\mbox{if}~ \sum_{e_i\in\delta^o(v)} x_{e_i}
-\sum_{e_i\in\delta^i(v)} x_{e_i}=d_v\\
0&\mbox{otherwise}
\end{cases}.
\end{align*}
For the above GM \eqref{gm:networkflow}, one can easily check Conditions {\em C2, C3}
of Theorem \ref{thm:main} hold and
derive the following corollary whose
formal proof is presented in Section \ref{apdx:networkflow}.
\begin{corollary}\label{cor:networkflow}
If the shortest path from $s$ to $t$, i.e., the solution of the network flow LP \eqref{lp:networkflow}, is unique, then the max-product BP on GM \eqref{gm:networkflow} converges to it.
\end{corollary}
\section{Conclusion}
The BP algorithm has been the most popular algorithm for solving inference problems arising graphical models,
where its distributed implementation, associated ease of programming
and strong parallelization potential are the main reasons for its growing popularity.
In this paper, we aim for designing BP algorithms solving LPs,
and provide sufficient conditions for its correctness and convergence.
We believe that our results provide
new interesting directions on designing efficient distributed (and parallel) solvers for large-scale LPs.
\iffalse
where is fast and can be easily parallelized.
Solving LP by BP may reduce the computational cost in many practical applications.
Previously, BP analysis were very sensitive to the problem setup and
there was no general conditions on BP correctness and convergence.
Our theoretical result contributes to unify the general framework of BP analysis covers prior results.
In section \ref{sec:applications}, we show various applications of our result.
We show that our theoretical result covers previous works \cite{sanghavi2011belief,NIPS2013_4949,bayati2007belief,ruozzi2008st,huang2007loopy}
and new problems such as traveling salesman, longest path.
Furthermore, in \ref{subsec:matching}, we provide examples that BP can solve LP even if LP is not tight
while prior works on BP correctness and convergence require the LP tightness.
Our result broaden the class of LPs that solution can be found by BP.
We provide the new direction that BP can solve exact LP by using graphical transformation.
\fi
\section{Introduction}
\iffalse
Graphical model (GM) has been one of powerful paradigms for succinct representations of joint probability distributions in variety of scientific fields \citep{yedidia2005constructing,richardson2008modern,mezard2009information,wainwright2008graphical}.
GM represents a joint distribution of some random vector to a graph structured model
where each vertex corresponds to a random variable and each edge captures to a conditional independence between random variables.
In many applications involving GMs, finding maximum-a-posteriori (MAP) assignment in GM is an important inference task, which
is known to be computationally intractable (i.e., NP-hard) in general \citep{chandrasekaran08com}.
\fi
The max-product belief propagation (BP) is the most popular heuristic for approximating a maximum-a-posteriori (MAP) assignment\footnote{In general,
MAP is NP-hard to compute exactly \citep{chandrasekaran08com}.} of given
Graphical model (GM) \citep{yedidia2005constructing,richardson2008modern,mezard2009information,wainwright2008graphical}, where
its performance has been not well understood in loopy GMs, i.e., GM with cycles.
Nevertheless, BP often shows remarkable performances even on loopy GM.
Distributed implementation, associated ease of programming
and strong parallelization potential are the main reasons for the growing popularity of the BP
algorithm. For example, several software architectures for implementing parallel BPs were recently proposed \citep{LowGKBGH10graph, gonzalez2010parallel, ma2012task}.
In the past years, there have been made extensive research efforts to understand BP performances on loopy GMs
under connections to combinatorial optimization \cite{bayati2005maximum, sanghavi2011belief, huang2007loopy, salez2009belief, bayati2011belief, shin2013graphical, ruozzi2008st, gamarnik2012belief, chandrasekaran2011counting, bandyopadhyay2006counting, sanghavi2009message}.
In particular, it has been studied about the BP convergence to the correct answer
under a few classes of loopy GM formulations of combinatorial optimization
problems: matching \citep{bayati2005maximum, sanghavi2011belief, huang2007loopy, salez2009belief}, perfect matching \citep{bayati2011belief},
matching with odd cycles \citep{shin2013graphical}, shortest path \citep{ruozzi2008st} and network flow \cite{gamarnik2012belief}.
The important common feature of these instances is that BP converges to a correct MAP assignment if
the Linear Programming (LP) relaxation of the MAP inference problem is tight, i.e., it has
no integrality gap. In other words, BP can be used an efficient distributed solver for those LPs, and is presumably of better choice than
classical centralized LP solvers such as simplex methods \citep{dantzig1998linear}, interior point methods \citep{thapa2003linear} and ellipsoid methods \citep{khachiyan1980polynomial}
for large-scale inputs.
However, these theoretical results on BP are very sensitive to underlying structural properties depending on specific problems
and it is not clear what extent they can be generalized to, e.g.,
the BP analysis for matching problems \citep{bayati2005maximum, sanghavi2011belief, huang2007loopy, salez2009belief}
do not extend to even for perfect matching ones \citep{bayati2011belief}.
In this paper, we overcome such technical difficulties for
enhancing the power of BP as a LP solver.
\subsection{Contribution}
We establish a generic criteria for GM formulations of given LP so that
BP converges to the optimal LP solution.
By product, it also provides a sufficient condition for unique BP fixed point.
As one can naturally expect given prior results,
one of our conditions requires the LP tightness. Our main contribution is finding
other sufficient generic conditions so that BP converges to the correct MAP assignment of GM.
First of all, our generic criteria can rediscover all prior BP results on this line,
including matching \citep{bayati2005maximum, sanghavi2011belief,huang2007loopy}, perfect matching \citep{bayati2011belief},
matching with odd cycles \citep{shin2013graphical} and shortest path \citep{ruozzi2008st}, i.e.,
we provide a unified framework on establishing the convergence and correctness of BPs in relation to associated LPs.
Furthermore, we provide new instances under our framework: we show that BP can solve LP formulations associated to other popular
combinatorial optimizations including perfect matching with odd cycles,
traveling salesman, cycle packing, network flow and vertex/edge cover, which are not known in the literature.
Here, we remark that the same network flow problem was already studied using BP by Gamarnik et al. \cite{gamarnik2012belief}.
However,
our BP is different from theirs and much simpler
to implement/analyze: the authors study BP on continuous GMs,
and we do BP on discrete GMs.
While most prior known BP results on this line focused on the case when the associated LP has an integral solution,
the proposed criteria naturally guides the BP design to compute fractional LP solutions as well (see Section \ref{subsec:matching}
and Section \ref{subsec:vc} for details).
Our proof technique is build upon on that of \cite{sanghavi2011belief} where
the authors construct
an alternating path in the computational tree induced by BP to analyze its performance
for the maximum weight matching problem. Such a trick needs specialized case studies depending on the associated LP
when
the path reaches a leaf of the tree, and this is one of main reasons why it is not easy to generalize
to other problems beyond matching.
The main technical contribution of this paper is providing a way to avoid the issue in the BP analysis
via carefully analyzing associated LP polytopes.
The main appeals of our results are providing not only tools on BP analysis, but also
guidelines on BP design for its high performance, i.e.,
one can carefully design a BP given LP so that it satisfies the proposed criteria.
Our results provide not only new tools on BP analysis and design,
but also new directions on efficient distributed (and parallel) solvers for large-scale LPs
and combinatorial optimization problems.
\subsection{Organization}
In Section \ref{sec:pre}, we introduce necessary backgrounds for the BP algorithm.
In Section \ref{sec:main}, we provide the main result of the paper as well as its several concrete applications to popular combinatorial optimizations.
Proofs are presented in Section \ref{sec:mainpf} and Section \ref{sec:corpf}.
\section{Convergence and Correctness of Belief Propagation}
\label{sec:main}
\subsection{Convergence and Correctness Criteria of BP}
In this section, we provide the main result of this paper: a convergence and correctness criteria of BP.
Consider the following GM: for $x=[x_i]\in \{0,1\}^n$ and $w=[w_i]\in \mathbb R^n$,
\begin{equation}
\Pr[X=x]~\propto~\prod_{i} e^{-w_i x_i}\prod_{\alpha\in F} \psi_{\alpha} (x_\alpha),\label{eq:gm1}
\end{equation}
where $F$ is the set of non-variable factors and
the factor function $\psi_\alpha$ for $\alpha\in F$ is defined as
\begin{align*}
&\psi_{\alpha}(x_{\alpha}) =
\begin{cases}
1&\mbox{if}~ A_{\alpha} x_{\alpha}\ge b_{\alpha},~ C_{\alpha}x_{\alpha}=d_{\alpha}\\
0&\mbox{otherwise}
\end{cases},
\end{align*}
for some
matrices $A_{\alpha}, C_{\alpha}$ and vectors $b_{\alpha}, d_{\alpha}$.
Now we consider the Linear Programming (LP) corresponding the above GM:
\begin{equation}\label{eq:lp1}
\begin{split}
&\mbox{minimize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad
\psi_\alpha(x_\alpha)=1,\quad \forall \alpha\in F\\
&\qquad\qquad\qquad~ x=[x_i]\in [0,1]^n.
\end{split}
\end{equation}
One can easily observe that the MAP assignments for GM \eqref{eq:gm1} corresponds to the (optimal) solution of LP \eqref{eq:lp1}
if the LP has an integral solution $x^*\in \{0,1\}^n$. As stated in the following theorem, we establish other sufficient conditions
so that the max-product BP can indeed find the LP solution.
\iffalse
To establish the performance of BP on GM \eqref{eq:gm1} for solving IP \eqref{eq:ip1}, we also consider the following
the LP (Linear Programming) relation to IP \eqref{eq:ip1}:
\begin{equation}\label{eq:lp1}
\begin{split}
&\mbox{minimize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad
\psi_\alpha(x_\alpha)=1,\quad \forall \alpha\in F\\
&\qquad\qquad\qquad~ x=[x_i]\in [0,1]^n.
\end{split}
\end{equation}
\fi
\begin{theorem}\label{thm:main}
The max-product BP on GM \eqref{eq:gm1} with arbitrary initial message converges to the solution of LP \eqref{eq:lp1} if the following conditions hold:
\begin{itemize}
\item[C1.] LP \eqref{eq:lp1} has a unique integral solution $x^*\in\{0,1\}^n$, i.e., it is tight.
\item[C2.] For every $i\in \{1,2,\dots, n\}$, the number of factors associated with $x_i$ is at most two, i.e., $|F_i|\leq 2.$
\item[C3.] For every factor $\psi_\alpha$, every $x_\alpha\in\{0,1\}^{|\alpha|}$ with $\psi_\alpha(x_\alpha)=1$, and
every $i\in\alpha$ with $x_i\neq x^*_i$,
there exists
$\gamma\subset \alpha$
such that
$$|\{j \in\{i\}\union \gamma:|F_j|=2\}|\le 2$$
$$
\psi_\alpha(x^\prime_\alpha)=1,\qquad
\mbox{
where
$x^\prime_k = \begin{cases}
x_k~&\mbox{if}~k\notin \{i\}\union \gamma\\
x^*_k~&\mbox{otherwise}
\end{cases}$.}$$
$$
\psi_\alpha(x^{\prime\prime}_\alpha)=1,\qquad
\mbox{
where
$x^{\prime\prime}_k = \begin{cases}
x_k~&\mbox{if}~k\in\{i\}\union \gamma\\
x^*_k~&\mbox{otherwise}
\end{cases}$.}$$
\end{itemize}
\end{theorem}
Since the above theorem holds for arbitrary initial messages,
the conditions {\em C1, C2, C3} also provides the uniqueness of BP fixed points, as stated in what follows.
\begin{corollary}\label{cor:fixedpoint}
The max-product BP on GM \eqref{eq:gm1} has a unique fixed point if conditions C1, C2, C3 hold.
\end{corollary}
The conditions {\em C2, C3} are typically easy to check given GM \eqref{eq:gm1} and
the uniqueness in {\em C1} can be easily guaranteed via adding random noises,
On the other hand, the integral property in {\em C1} requires to analyze LP \eqref{eq:lp1}, where it has been extensively studied
in the field of combinatorial optimization \citep{schrijver2003combinatorial}. Nevertheless, Theorem \ref{thm:main}
provides important guidelines to design BP algorithms, irrespectively of the LP analysis.
\section{Preliminaries}\label{sec:pre}
\subsection{Graphical Model}
A joint distribution of $n$ (binary) random variables $Z=[Z_i]\in \{0,1\}^n$ is called a Graphical Model (GM) if it factorizes as follows: for $z=[z_i]\in \Omega^n$,
\begin{equation*}
\Pr[Z=z]~\propto~\prod_{i\in\{1,\dots,n\}}\psi_i(z_i)\prod_{\alpha\in F} \psi_{\alpha} (z_\alpha),\label{eq:generic_gm}
\end{equation*}
where $\{\psi_i,\psi_{\alpha}\}$ are (given) non-negative functions, the so-called factors;
$F$ is a collection of subsets
$$F=\{\alpha_1,\alpha_2,...,\alpha_k\}\subset 2^{\{1,2,\dots, n\}}$$
(each $\alpha_j$ is a subset of $\{1,2,\dots, n\}$ with $|\alpha_j|\ge 2$); $z_\alpha$ is the projection of $z$ onto
dimensions included in $\alpha$.\footnote{For example, if $z=[0,1,0]$ and $\alpha=\{1,3\}$, then $z_\alpha=[0,0]$.}
In particular, $\psi_i$ is called a variable factor.
Figure~\ref{fig:startup} depicts the
the graphical relation between factors $F$ and variables $z$.
\begin{figure}[h]
\centering
\footnotesize
\begin{tikzpicture}[node distance = 2cm, auto]
\node [block] (a-one) {$\alpha_1$};
\node [block, right of=a-one] (a-two) {$\alpha_2$};
\node [block, right of=a-two] (a-three) {$\alpha_3$};
\node [cloud, below left of=a-one] (z-one) {$z_1$};
\node [cloud, right of=z-one] (z-two) {$z_2$};
\node [cloud, right of=z-two] (z-three) {$z_3$};
\node [cloud, right of=z-three] (z-four) {$z_4$};
\path [line] (a-one) -- (z-one);
\path [line] (z-one) -- (a-one);
\path [line] (a-one) -- (z-three);
\path [line] (z-three) -- (a-one);
\path [line] (a-two) -- (z-one);
\path [line] (z-one) -- (a-two);
\path [line] (a-two) -- (z-two);
\path [line] (z-two) -- (a-two);
\path [line] (a-two) -- (z-four);
\path [line] (z-four) -- (a-two);
\path [line] (a-three) -- (z-two);
\path [line] (z-two) -- (a-three);
\path [line] (a-three) -- (z-three);
\path [line] (z-three) -- (a-three);
\path [line] (a-three) -- (z-four);
\path [line] (z-four) -- (a-three);
\end{tikzpicture}
\caption{
Factor graph for the graphical model
$\Pr[z] \propto \psi_{\alpha_1}(z_1,z_3)\psi_{\alpha_2}(z_1,z_2,z_4)\psi_{\alpha_3}(z_2,z_3,z_4)$,
i.e., $F=\{\alpha_1,\alpha_2,\alpha_3\}$ and $n=4$.
Each $\alpha_j$ selects a subset of $z$. For example, $\alpha_1$ selects
$\{z_1,z_3\}$.
}
\label{fig:startup}
\normalsize
\end{figure}
Assignment ${z}^*$ is called a maximum-a-posteriori (MAP) assignment if ${z}^*$ satisfies
${z}^*=\arg\max_{{z}\in\{0,1\}^n} \Pr[{z}].$
This means that computing a MAP assignment requires us to compare
$\Pr[{z}]$ for all possible $z$, which is typically
computationally intractable (i.e., NP-hard) unless the induced bipartite graph
of factors $F$ and variables $z$, so-called factor graph, has a bounded treewidth \citep{chandrasekaran08com}.
\subsection{Max-Product Belief Propagation}
The (max-product) BP algorithms are popular heuristics for approximating the MAP assignment in a graphical model.
BP is an iterative procedure; at each iteration $t$, there are four
messages
$$\{m^{t}_{\alpha\rightarrow i}(c),
m^{t}_{i\rightarrow\alpha}(c):
c\in\{0,1\} \}$$
between
every variable $z_i$ and every associated $\alpha\in F_i$, where
$F_i:= \{\alpha\in F: i \in \alpha\}$; that is, $F_i$ is
a subset of $F$ such that all $\alpha$ in $F_i$ include the $i^{th}$
position of $z$ for any given $z$.
Then, messages are updated as follows:
\iffalse
\begin{figure}[ht]
\centering
\subfigure[$m^{t}_{i\rightarrow\alpha}(0)$]{
\begin{tikzpicture}[node distance = 2cm, auto]
\node [block] (a-one) {$\alpha_1$};
\node [cloud, below left of=a-one] (z-one) {$z_1$};
\node [cloud, right of=z-one] (z-three) {$z_3$};
\path [line] (z-one) --
node[auto,left] {$m^{t}_{1\rightarrow\alpha_1}(0,1)$}
(a-one);
\path [line] (z-three) --
node[auto,right] {$m^{t}_{3\rightarrow\alpha_1}(0,1)$}
(a-one);
\end{tikzpicture}
\label{fig:i_to_alpha}
}
\subfigure[$m^{t}_{\alpha\rightarrow{}i}(0)$]{
\begin{tikzpicture}[node distance = 2cm, auto]
\node [block] (a-one) {$\alpha_1$};
\node [cloud, below left of=a-one] (z-one) {$z_1$};
\node [cloud, right of=z-one] (z-three) {$z_3$};
\path [line] (a-one) --
node[auto,left] {$m^{t}_{\alpha_1\rightarrow1}(0,1)$}
(z-one);
\path [line] (a-one) --
node[auto,right] {$m^{t}_{\alpha_1\rightarrow3}(0,1)$}
(z-three);
\end{tikzpicture}
\label{fig:alpha_to_i}
}
\caption{
A graph depicting the messages that are passed between $\alpha_1$ and it's
connected variables $z_1$ and $z_3$ at time $t$.
The entire graph is depicted in Figure~\ref{fig:startup}.
}
\label{fig:messages}
\end{figure}
\fi
\begin{align}
&\quad m^{t+1}_{\alpha\rightarrow i}(c) ~=~ \max_{z_{\alpha}:z_i=c}
\psi_\alpha (z_{\alpha})
\prod_{j\in \alpha\setminus i} m_{j\rightarrow \alpha}^t (z_j)\label{eq:msg:alpha_to_i}\\
&\quad m^{t+1}_{i\rightarrow\alpha}(c) ~=~
\psi_i(c)\prod_{\alpha^{\prime}\in F_i\setminus \alpha} m_{\alpha^{\prime}
\rightarrow i}^t (c)\label{eq:msg:i_to_alpha}.
\end{align}
First, we note that each $z_i$ only sends messages to $F_i$; that is,
$z_i$ sends messages to $\alpha_j$ only if $\alpha_j$ selects/includes $i$.
The outer-term in the message computation \eqref{eq:msg:alpha_to_i}
is maximized over all possible $z_\alpha\in\{0,1\}^{|\alpha|}$ with $z_i=c$.
The inner-term is a product that only depends on the variables $z_j$ (excluding $z_i$) that
are connected to $\alpha$.
The message-update \eqref{eq:msg:i_to_alpha} from variable $z_i$ to factor $\psi_\alpha$ is a product
which considers all messages received by $\psi_\alpha$ in the previous iteration,
except for the message sent by $z_i$ itself.
One can reduce the complexity of messages by combining
\eqref{eq:msg:alpha_to_i} and \eqref{eq:msg:i_to_alpha} as:
\begin{align*}
m^{t+1}_{i\rightarrow\alpha}(c) =
\psi_i(c)
\prod_{\alpha^{\prime}\in F_i\setminus \alpha} \max_{z_{\alpha^\prime}:z_i=c}
\psi_{\alpha^\prime} (z_{\alpha^\prime})
\prod_{j\in \alpha^\prime\setminus i} m_{j\rightarrow \alpha^\prime}^t (z_j),
\end{align*}
which we analyze in this paper.
Finally, given a set of messages $\{m_{i\to\alpha}(c)$, $m_{\alpha\to
i}(c):c\in\{0,1\}\}$, the so-called BP marginal beliefs are computed as follows:
\begin{eqnarray}\label{eq:bpdecision}
b_i[z_i]&=
\prod_{\alpha\in F_i} m_{\alpha\to i}(z_i).\label{eq:marginalbelief}
\end{eqnarray}
Then, the BP algorithm outputs $z^{BP}=[z_i^{BP}]$ as
$$
z_i^{BP}=\begin{cases}
1&\mbox{if}~ b_i[1]>b_i[0]\\
?&\mbox{if}~b_i[1]=b_i[0]\\
0&\mbox{if}~ b_i[1]<b_i[0]
\end{cases}.
$$
It is known that $z^{BP}$ converges to a MAP assignment after a large enough number of iterations,
if the factor graph
is a tree and the MAP assignment is unique. However, if the graph has loops in it, the BP
algorithm has no guarantee to find a MAP assignment in general.
\iffalse
\subsection{Integer Programming and Linear Programming Relaxation}
Integer programming (IP) is an optimization problem in which all variables are integer and its objective function and constraints are linear.
We can formulate general IP as below:
\begin{equation}\label{eq:ip0}
\begin{split}
&\mbox{minimize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad Ax\le b\\
&\qquad\qquad\qquad~ x=[x_i]\in \mathbb{Z}^n\\
&\qquad\qquad\qquad~ A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m.
\end{split}
\end{equation}
However, solving IP is NP-Hard. Hence, we usually solve relaxed problem called linear programming (LP) rather than IP.
In LP, we relax the integer constraint of variables as below.
\begin{equation}\label{eq:lp0}
\begin{split}
&\mbox{minimize}\qquad~ w\cdot x\\
&\mbox{subject to}\qquad Ax\le b\\
&\qquad\qquad\qquad~ x=[x_i]\in \mathbb{R}^n\\
&\qquad\qquad\qquad~ A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m.
\end{split}
\end{equation}
\fi
\section{Proof of Theorem \ref{thm:main}}\label{sec:mainpf}
To begin with, we define some necessary notation.
We let $\mathcal P$ denote the polytope of feasible solutions of LP \eqref{eq:lp1}:
$$\mathcal P :=\left\{x\in [0,1]^n\,:\, \psi_\alpha(x_\alpha)=1,~\forall\,\alpha\in F \right\}.$$
Similarly, $\mathcal P_\alpha$ is defined as
$$\mathcal P_\alpha :=\left\{x\in [0,1]^{|\alpha|}\,:\, \psi_\alpha(x_\alpha)=1 \right\}.$$
We first state the following key technical lemma.
\begin{lemma}\label{lemma:c4}
There exist universal constants $K,\eta>0$ for LP \eqref{eq:lp1} such that
if $z\in[0,1]^n$ and $0<\varepsilon<\eta$ satisfy the followings:
\begin{itemize}
\item[1.]
There exist at most two violated factors for $z$, i.e.,
$\left|\{\alpha\in F\,:\, z_\alpha\notin \mathcal{P}_\alpha\}\right|\leq 2.$
\item[2.] For each violated factor $\alpha$,
there exist $i\in \alpha$ such that
$z^\dagger_\alpha\in \mathcal P_\alpha,$
where $z^\dagger= z + \varepsilon e_i$
or $z^\dagger = z - \varepsilon e_i$ and
$e_i\in\{0,1\}^n$ is the unit vector whose $i$-th coordinate is $1$,
\end{itemize}
then there exists $z^\ddagger\in \mathcal P$ such that $\|z-z^\ddagger\|_1 \leq \varepsilon K$.
\end{lemma}
The proof of Lemma \ref{lemma:c4} is presented in Section \ref{sec:pflemma:c4}.
Now, from Condition {\em C1}, it follows that
there exists $\rho>0$ such that
\begin{equation}
\rho:=\inf_{x\in\mathcal{P}\setminus x^*} \frac{w\cdot x- w\cdot x^*}{\|x-x^*\|_1}>0.
\end{equation}
We let $\hat{x}^t\in\{0,1,?\}^n$ denote the BP estimate at the $t$-th iteration for the MAP computation.
We will show that under Conditions {\em C1-C3},
$$\hat{x}^t = x^*,\qquad\mbox{for}~t > \left(\frac{w_{\max}}\rho+1\right)K,$$
where $w_{\max}=\max_j |w_j|$ and $K$ is the universal constant in Lemma \ref{lemma:c4}.
Suppose the above statement is false, i.e., there exists $i\in\{1,2,\dots, n\}$ such that
$\hat{x}^t_i \neq x^*_i$ for $t > \left(\frac{w_{\max}}\rho+1\right)K$.
Under the assumption, we will reach a contradiction.
\iffalse
We will prove that given tightness and uniqueness of LP, there is a belief propagation algorithm which correctly converges to the solution.
To prove the correct convergence, we need to show that for large enough iteration $t$, the estimate $\hat{x}^t_e$ satisfies
\begin{equation}
\begin{aligned}
&\hat{x}_i=1~\mbox{if}~x^*_i=1\\
&\hat{x}_i=0~\mbox{if}~x^*_i=0\\
\end{aligned}
\end{equation}
\fi
Now we construct a tree-structured GM $T_i(t)$, popularly known as the computational tree \citep{weiss2001optimality}, as follows:
\begin{itemize}
\item[1.] Add $y_i\in \{0,1\}$ as the root variable with variable factor function $e^{-w_i y_i}$.
\item[2.] For each leaf variable $y_j$ and for each $\alpha\in F_j$ and $\psi_\alpha$ is not associated with $y_j$ in the current tree-structured GM, add
a factor function $\psi_\alpha$ as a child of $y_j$.
\item[3.] For each leaf factor $\psi_\alpha$ and for each variable $y_k$ such that $k\in\alpha$ and $y_k$ is not associated with $\psi_\alpha$ in
the current tree-structured GM, add a variable $y_k$ as a child of $\psi_\alpha$ with variable factor function $e^{-w_k y_k}$.
\item[4.] Repeat Step 2, 3 $t$ times
\end{itemize}
\iffalse
\textcolor{red}{
In general, the initial message of BP is set as 1.
The change of the initial message of BP on GM \eqref{eq:gm1} affects to the weight of the leaf variables.
However, for arbitrary initial message,
the parent factor of the leaf variable decides the weight of the leaf variable
i.e. the weight of the leaf variable might be asymmetric.
By condition {\em C2}, the leaf variable can have at most two different weights.
Let define $w^\prime_{i1},w^\prime_{i2}$ be weights of the leaf variable $i$ for some initial message.
Let define $w^\prime_{\max}=\max_i (|w_{i1}|,|w_{i2}|)$.
}
\fi
Suppose the initial messages of BP are set by 1, i.e., $m_{j\to\alpha}(\cdot)^0 =1$.
Then, if $\hat{x}^t_i\in\{0,?\}$, it is known \citep{weiss1997belief}
that there exists a MAP configuration
$y^{MAP}$ on $T_i(t)$ with $y^{MAP}_i=0$ at
the root variable.
For other initial messages, one can guarantee the same property under
changing weights of leaf variables of the tree-structured GM.
Specifically,
for a leaf variable $k$ with $|F_k=\{\alpha_1,\alpha_2\}|=2$ and $\alpha_1$
being its parent factor in $T_i(t)$,
we reset its variable factor by $e^{-w_k^\prime y_k}$, where
\begin{equation}
w^\prime_{k}=w_k-\log\frac{\max_{z_{\alpha_2}:z_k=1}\psi_{\alpha_2}(z_{\alpha_2})\Pi_{j\in\alpha_2\setminus k}m_{j\rightarrow\alpha_2}^0(z_j)}
{\max_{z_{\alpha_2}:z_k=0}\psi_{\alpha_2}(z_{\alpha_2})\Pi_{j\in\alpha_2\setminus k}m_{j\rightarrow\alpha_2}^0(z_j)}.\label{eq:otherinitial}
\end{equation}
This is the reason why our proof of Theorem \ref{thm:main} goes through for arbitrary initial messages.
For notational convenience, we present the proof for the standard initial message of $m_{j\to\alpha}^0(\cdot) =1$,
where it can be naturally generalized to other initial messages using \eqref{eq:otherinitial}.
Now we construct a new valid assignment $y^{NEW}$ on the computational tree $T_i(t)$
as follows:
\begin{itemize}
\item[1.] Initially, set $y^{NEW} \leftarrow y^{MAP}$.
\item[2.] Update the value of the root variable of $T_i(t)$ by $y^{NEW}_i\leftarrow x^*_i$.
\item[3.] For each child factor $\psi_\alpha$ of root $i\in\alpha$, choose $\gamma\subset\alpha$ according to Condition {\em C3}
and update the associated variable by $y^{NEW}_{j}\leftarrow x^*_{j}~~\forall j\in\gamma$.
\item[4.] Repeat Step 2,3 recursively by substituting $T_i(t)$ by the subtree of $T_i(t)$ of root $j\in\gamma$ until the process stops (i.e., $i=j$) or the leaf of $T_i(t)$ is reached
(i.e., $i$ does not have a child).
\end{itemize}
One can notice that the set of revised variables in Step 2 of the above procedure forms a path structure $Q$ in the tree-structured GM.
We first, consider the case that both ends of the path $Q$ touch leaves of $T_i(t)$, where other cases can be argued in a similar manner.
Define
$\zeta_j$ and $\kappa_j$ be the number of copies of $x_j$ in path $Q$ with
$x^*_{j}=1$ and $x^*_{j}=0$, respectively, where
$\zeta=[\zeta_j] ,\kappa=[\kappa_j]\in \mathbb Z_+^{n}$ .
Then, from our construction of $y^{NEW}$, one can observe that
\begin{align*}
&y^{NEW}=y^{MAP}+\zeta-\kappa\\
&w \cdot
y^{MAP}-w\cdot
y^{NEW}= w\cdot (\kappa-\zeta).
\end{align*}
If we set $z=x^*+\varepsilon(\kappa-\zeta)$ where $0<\varepsilon<\min\{1/2t,\eta\}$,
then
one can check that $z$ satisfies the conditions of Lemma \ref{lemma:c4} using
Conditions {\em C2, C3}.
Hence, from Lemma \ref{lemma:c4},
there exists $z^\ddagger \in \mathcal P$ such that
\begin{align*}
&\|z^\ddagger - z\|_1 \leq \varepsilon K\\
&\|z^\ddagger - x^*\|_1 \geq \varepsilon (\|\zeta\|_1+\|\kappa\|_1 - K) \geq \varepsilon (t-K).
\end{align*}
where $z = x^* + \varepsilon (\kappa-\zeta)$.
\iffalse
Choose $0<\varepsilon <\min_{i\in \{1,2,..n\}}(\frac{1}{\zeta_i},\frac{1}{\kappa_i})$. Then,
\begin{itemize}
\item $z=x^*+\varepsilon (\kappa -\zeta)\in[0,1]^n$
\item There exists at most two $\zeta_1,\zeta_2\in F$ does not satisfy
$$A_{\zeta_l}z_{\zeta_l}\ge b_{\zeta_l},~ C_{\zeta_l}z_{\zeta_l}=d_{\zeta_l},~l\in\{1,2\}$$
as $A(\kappa -\zeta)\ge 0$ except for leaves by condition 3. i.e. factor condition \eqref{eq1} is violated only for leaf factors
$\psi_{\zeta_1},\psi_{\zeta_2}\in P$.
\item We can choose $i^\prime\in \zeta_l,~l\in\{1,2\}$ touches the leaf of $P$. Then
$x = z - \varepsilon e_{i^\prime} \in [0,1]^n$ satisfies
$$A_{\zeta_l}z_{\zeta_l}\ge b_{\zeta_l},~ C_{\zeta_l}z_{\zeta_l}=d_{\zeta_l}$$
\end{itemize}
Now, we can apply condition 4. From condition 4, there exist $z^\prime\in\mathcal{P}$ and universal constant $K$ such that
$$A_{\zeta}z^\prime_{\zeta}\ge b_{\zeta},~ C_{\zeta}z^\prime_{\zeta}=d_{\zeta},\quad \forall \zeta\in F.$$
From lemma\eqref{lemma:1},
\fi
Hence, it follows that
\iffalse
\begin{equation*}
\begin{split}
0<\delta&\le\frac{w(z^\ddagger)-w(x^*)}{\|z^\ddagger-x^*\|_1}\\
&\le\frac{w(z)+\varepsilon w_{\max} K-w(x^*)}{\varepsilon(2t-K)}\\
&=\frac{\varepsilon w(\kappa-\zeta)+\varepsilon w_{max}K}{
\varepsilon(2t-K)}\\
&=\frac{w(\kappa-\zeta)+w_{max}K}{2t-K}
\end{split}
\end{equation*}
Therefore, we can bound $w(\kappa-\zeta)$ as
\begin{equation*}
w(\kappa-\zeta)\ge 2\delta t -(w_{\max} +\delta) K>0~
\end{equation*}
for $t > K (w_{\max}/\delta+1)$.
The above inequality leads to the contradiction to the fact that $y^{MAP}$ is a MAP configuration in $T_i(t)$ since
$$ w\cdot y^{NEW} = w (\zeta-\kappa) + w\cdot y^{MAP} < w\cdot y^{MAP}.$$
This completes the proof of Theorem \ref{thm:main}.
\fi
\begin{equation*}
\begin{split}
0<\rho&\le\frac{w\cdot z^\ddagger-w\cdot x^*}{\|z^\ddagger-x^*\|_1}\\
&\le\frac{w\cdot z+\varepsilon w_{\max} K-w\cdot x^*}{\varepsilon(t-K)}\\
&=\frac{\varepsilon w\cdot (\kappa-\zeta)+\varepsilon w_{\max}K}{
\varepsilon(t-K)}\\
&=\frac{w\cdot(\kappa-\zeta)+w_{\max}K}{t-K}
\end{split}
\end{equation*}
Furthermore, if $t > \left(\frac{w_{\max}}{\rho}+1\right)K$, the above inequality implies that
\begin{align*}
w\cdot y^{MAP}-w\cdot y^{NEW}
&=w\cdot (\kappa-\zeta)\\
&\ge \rho t -(w_{\max} +\rho) K~>~0.
\end{align*}
This is the contradiction to the fact that
$y^{MAP}$ is a MAP configuration.
\iffalse
and threrfore $w\cdot y^{MAP}-w\cdot y^{NEW}>0$.
From the choice of $y^{MAP}$, it must be $\Pr[Y=y^{MAP}]\ge\Pr[Y=y^{NEW}]$.
However, as $w\cdot (\kappa-\zeta)>0$
\begin{align*}
\Pr[Y=y^{NEW}]&=e^{-w\cdot y^{MAP}+w\cdot (\kappa-\zeta)}\\
&>e^{-w\cdot y^{MAP}}\\
&=\Pr[Y=y^{MAP}]
\end{align*}
and it leads to contradiction.
Therefore, $\hat{x}^t=1$ and
\fi
This completes the proof of Theorem \ref{thm:main}.
\subsection{Proof of Lemma \ref{lemma:c4}}\label{sec:pflemma:c4}
One can write
$\mathcal{P}=\{x\,:\,Ax\ge b\}\subset [0,1]^n$ for some matrix $A\in \mathbb R^{m\times n}$ and vector $b\in \mathbb R^m$,
where without loss of generality, we can assume that $\|A_i\|_2=1$ where $\{A_i\}$ is the set of row vectors of $A$.
We define
\begin{align*}
&\mathcal{P}_{\varepsilon}=\{x\,:\,Ax\ge b-\varepsilon\mathbf{1}\},
\end{align*}
where $\mathbf{1}$ is the vector of ones.
Then, one can check that $z\in\mathcal{P}_\varepsilon$ for $z,\varepsilon$ satisfying conditions of Lemma \ref{lemma:c4}.
Now we aim for finding a universal constant $K$ satisfying
$${{\tt dist}}(\mathcal{P},\mathcal{P}_\varepsilon):=\max_{x\in\mathcal{P}_\varepsilon}(\min_{y\in\mathcal{P}} \|x-y\|_1) \le \varepsilon K,$$
which leads to the conclusion of Lemma \ref{lemma:c4}.
To this end, for
$\xi\subset[1,2,\dots, m]$ with $|\xi|=n$,
we let $A_\xi$ be
the square sub-matrix of $A$ by choosing
$\xi$-th rows of $A$ and $b_\xi$ is the $n$-dimensional subvector of $b$ corresponding $\xi$.
Throughout the proof, we only consider $\xi$ such that $A_\xi$ is invertible.
Using this notation, we first claim the following.
\begin{claim}\label{clm1}
If $A_{\xi}$ is invertible and $v_\xi:=A_{\xi}^{-1} b_{\xi}\in \mathcal P$, then
$v_\xi$ is a vertex of
polytope $\mathcal{P}$.
\end{claim}
\begin{proof}
Suppose $v_\xi$ is not a vertex of $\mathcal P$,
i.e. there exist $x,y\in\mathcal{P}$ such that $x\neq y$ and
$v_\xi=\lambda x+(1-\lambda)y$ for some $\lambda\in(0,1/2]$.
Under the assumption, we will reach a contradiction.
Since $\mathcal{P}$ is a convex set,
\begin{equation}
\frac{3\lambda}2x+\left(1-\frac{3\lambda}2\right)y\in\mathcal{P}.\label{eq1:pfclm1}
\end{equation}
However, as $A_\xi$ is invertible,
\begin{equation}
A_\xi\left(\frac{3\lambda}2x+\left(1-\frac{3\lambda}2\right)y\right)\ne b_\xi.\label{eq2:pfclm2}
\end{equation}
From \eqref{eq1:pfclm1} and \eqref{eq2:pfclm2}, there exists a row vector $A_i$ of $A_\xi$
and the corresponding element $b_i$ of $b_\xi$ such that
$$A_i\cdot \left(\frac{3\lambda}2x+\left(1-\frac{3\lambda}2\right)y\right)>b_i.$$
Using the above inequality and $A_i\cdot (\lambda x+(1-\lambda)y)=b_i,$ one can conclude that
$$A_i\cdot \left(\frac{\lambda}2 x+\left(1-\frac{\lambda}2\right)y\right)<b_i,$$
which contradict to $\frac{\lambda}2x+\left(1-\frac{\lambda}2\right)y\in\mathcal{P}$.
This completes the proof of Claim \ref{clm1}.
\end{proof}
We also note that if $v$ is a vertex of polytope $\mathcal P$, there exists $\xi$ such that
$A_{\xi}$ is invertible and $v=A_{\xi}^{-1} b_{\xi}$.
We define the following notation:
\begin{align*}
&
\mathcal I=\{\xi\,:\, A_\xi^{-1}b_\xi\in\mathcal{P}\}
\quad
\mathcal I_{\varepsilon}=\{\xi\,:\,A_\xi^{-1}(b_\xi-\varepsilon\mathbf{1})\in\mathcal{P}_\varepsilon\},
\end{align*}
where Claim \ref{clm1} implies that
$\{v_\xi:=A_{\xi}^{-1} b_{\xi}\,:\,\xi\in \mathcal I\}$ and $\{u_{\xi,\varepsilon} :=A_\xi^{-1}(b_\xi-\varepsilon\mathbf{1})\,:\,\xi\in \mathcal I_{\varepsilon}\}$
are sets of vertices of $\mathcal{P}$ and $\mathcal{P}_\varepsilon$, respectively.
Using the notation, we show the following claim.
\begin{claim}\label{claim:c4}
There exists $\eta>0$ such that $\mathcal I_{\varepsilon}\subset \mathcal I$
for all $\varepsilon\in (0,\eta)$.
\end{claim}
\begin{proof}
\iffalse
To begin with, define
\begin{align*}
&c=\min_{x\in\{v_\xi\,:\,v_\xi\notin\mathcal{P}\},y\in\mathcal{P}}\|x-y\|_1
\end{align*}
Set $\eta=c/2K$.
Let define $\mathcal{C}_\varepsilon$ be a convex hull of $\{A_i^{-1}(b_i-\varepsilon\mathbf{1})|i\in I_\mathcal{P}\}$.
\fi
Suppose $\eta>0$ satisfying the conclusion of Claim \ref{claim:c4} does not exist.
Then, there exists a strictly decreasing sequence $\{\varepsilon_k>0:k=1,2,\dots\}$ converges to 0 such that
$\mathcal{I}_{\varepsilon_k}\cap\{\xi\,:\,\xi\notin\mathcal{I}\}\neq\emptyset .$
Since $|\{\xi:\xi\subset [1,2,\dots, m]\}|<\infty$, there exists $\xi^\prime$ such that
\begin{equation}
|\mathcal K:=\{k\,:\,\xi^\prime\in\mathcal{I}_{\varepsilon_k}\cap\{\xi\,:\,\xi\notin\mathcal{I}\}\}|=\infty .\label{eq1:pfclaimc4}
\end{equation}
For any $k\in \mathcal K$,
observe that
the sequence $\{u_{\xi^\prime,\varepsilon_\ell}:\ell\geq k,\ell\in \mathcal K\}$ converges to $v_{\xi^\prime}$.
Furthermore, all points in the sequence are in $\mathcal{P}_{\varepsilon_k}$
since $\mathcal{P}_{\varepsilon_\ell}\subset\mathcal{P}_{\varepsilon_k}$ for any $\ell\geq k$.
Therefore,
one can conclude that $v_{\xi^\prime}\in\mathcal{P}_{\varepsilon_k}~\mbox{for all}~k\in \mathcal K,$
where we additionally use the fact that $\mathcal{P}_{\varepsilon_k}$ is a closed set.
Because $\mathcal{P}=\bigcap_{k\in \mathcal K}\mathcal{P}_{\varepsilon_k}$, it must be that $v_{\xi^\prime}\in\mathcal{P}$, i.e.,
$v_{\xi^\prime}$ must be a vertex of $\mathcal{P}$ from Claim \ref{clm1}.
This contradicts to the fact
$\xi^\prime\in\{\xi\,:\,\xi\notin\mathcal{I}\}$.
This completes the proof of Claim \ref{claim:c4}.
\end{proof}
From the above claim, we observe that
any $x\in\mathcal{P}_\varepsilon$ can be expressed as a convex combination of $\{u_{\xi,\varepsilon}\,:\,\xi\in \mathcal{I}\}$,
i.e., $x=\sum_{\xi\in I}\lambda_\xi u_{\xi,\varepsilon}$ with $\sum_{\xi\in I}\lambda_\xi =1$ and $\lambda_\xi\geq 0$.
For all $\varepsilon\in(0,\eta)$ for $\eta>0$ in Claim \ref{claim:c4}, one can conclude that
\begin{align*}
{\tt dist}(\mathcal{P},\mathcal{P}_\varepsilon)
&\le\max_{x\in\mathcal{P}_\varepsilon}\|\sum_{\xi\in\mathcal{I}}\lambda_\xi u_{\xi,\varepsilon}-\sum_{\xi\in\mathcal{I}}\lambda_\xi v_\xi\|_1\\
&=\max_{x\in\mathcal{P}_\varepsilon}\varepsilon\|\sum_{\xi\in\mathcal{I}}\lambda_\xi A_\xi^{-1}\mathbf{1}\|_1\\
&\le\varepsilon \max_\xi \|A_\xi^{-1}\mathbf{1}\|_1,
\end{align*}
where we choose $K=\max_\xi \|A_\xi^{-1}\mathbf{1}\|_1$.
This completes the proof of Lemma \ref{lemma:c4}.
\iffals
First one can check that it suffices to show
$\{u_\xi\,:\, v_\xi\notin\mathcal{P}\}\cap\mathcal{P}_\varepsilon=\emptyset$.
To this end, we observe that
any $x\in\mathcal{C}_\varepsilon$ can be expressed as a convex combination of $\{u_\xi\,:\,\xi\in \mathcal{I}_\mathcal{P}\}$,
i.e.
$$x=\sum_{\xi\in I_\mathcal{P}}\lambda_\xi u_\xi,$$
where $\sum_{\xi\in\mathcal{I}_\mathcal{P}}\lambda_\xi=1$ for some non-negative values $\{\lambda_\xi\}$.
Hence, we have
\begin{align*}
{\tt dist}(\mathcal{P},\mathcal{C}_\varepsilon)
&\le\max_{x\in\mathcal{C}_\varepsilon}\|\sum_{\xi\in\mathcal{I}_\mathcal{P}}\lambda_\xi u_\xi-\sum_{\xi\in\mathcal{I}_\mathcal{P}}\lambda_\xi v_\xi\|_1\\
&=\max_{x\in\mathcal{C}_\varepsilon}\varepsilon\|\sum_{\xi\in\mathcal{I}_\mathcal{P}}\lambda_\xi A_\xi^{-1}\mathbf{1}\|_1\\
&\le\varepsilon K,
\end{align*}
where we use definitions of ${\tt dist}(\cdot,\cdot)$, $u_\xi, v_\xi$ and $K$.
Now suppose there exists $x\in\{u_\xi\,:\, v_\xi\notin\mathcal{P}\}\cap\mathcal{P}_\varepsilon$,
and we will reach a contradiction.
By the definition of $K$
$$\max_{\xi\,:\,v_\xi\notin\mathcal{P}}\|u_\xi-v_\xi\|_1\le\varepsilon K.$$
Since,
\begin{align*}
{\tt dist}(\mathcal{P},\mathcal{C}_\varepsilon)+\max_{\xi\,:\, v_\xi\notin\mathcal{P}}\varepsilon\|A_\xi^{-1}\mathbf{1}\|_1
&< c
\end{align*}
from the definition of $c$, if $x\in\{u_\xi\,:\,v_\xi\notin\mathcal{P}\}$, then $x\notin\mathcal{C}_\varepsilon$ and vice versa.
$x\in\{u_\xi\,:\, v_\xi\notin\mathcal{P}\}\cap\mathcal{C}_\varepsilon=\emptyset$ and therefore
$\mathcal{P}_\varepsilon\subset\mathcal{C}_\varepsilon$. We can conclude that
$\{u_\xi\,:\, v_\xi\notin\mathcal{P}\}\cap\mathcal{P}_\varepsilon=\emptyset$.
\end{proof}
In the proof of Claim \ref{claim:c4}, we showed that $\mathcal{P}_\varepsilon\subset\mathcal{C}_\varepsilon$
for $\varepsilon\in(0,\eta)$. Therefore,
$${\tt dist}(\mathcal{P},\mathcal{P}_\varepsilon)\le {\tt dist}(\mathcal{P},\mathcal{C}_\varepsilon)\le\varepsilon K$$
for $\varepsilon\in(0,\eta)$ and this completes the proof of Lemma \ref{lemma:c4}.
\f
\iffalse
To help the proof, let us introduce a Claim.
\begin{claim}\label{claim:c4}
For any polytope
$$\{x\in [0,1]^n|Ax\ge b,c\cdot x\ge d\}$$
with $A\in\mathbb{R}^{m\times n},b\in\mathbb{R}^m,c\in\mathbb{R}^n,d\in\mathbb{R}$,
there exists $K$ such that for all $\varepsilon >0$,
$$dist(\mathcal{P},\mathcal{P}_\epsilon):=\max_{x\in\mathcal{P}_\epsilon}(\min_{y\in\mathcal{P}} \|x-y\|_2) \le \varepsilon K$$
where
$\mathcal{P} =\{x\in [0,1]^n|Ax\ge b,c\cdot x=d\},~\mathcal{P}_\epsilon=\{x\in [0,1]^n|Ax\ge b,d\ge c\cdot x\ge d -\varepsilon\}$\\
\end{claim}
\begin{proof}
The proof of above Claim is following. For a polytope $\{x\in [0,1]^n|Ax\ge b,c\cdot x\le d\}$, there are finite number of vertices.
Choose $\delta$ such that there is no vertex $v$ of $\{x\in [0,1]^n|Ax\ge b,c\cdot x\le d\}$ satisfying $d-\delta\le c\cdot v<d$.
As $\mathcal{P}_\delta$ is bounded, we can choose $K$ such that
$$\max_{z\in\mathcal{P}_\delta,y\in\mathcal{P}}\|z-y\|_2\le\delta K$$
Consider the case when $\varepsilon>\delta$. For any $x\in\{x\in [0,1]^n|Ax\ge b,c\cdot x=d-\varepsilon\},y\in\mathcal{P}$,
set $z=\frac{\delta}{\varepsilon}x+(1-\frac{\delta}{\varepsilon})y$ Then, $z\in\{x\in [0,1]^n|Ax\ge b,c\cdot x=d-\delta\}$,
$$\|x-y\|_2=\frac{\varepsilon}{\delta}\|z-y\|_2\le\varepsilon K$$
and it implies $\mbox{dist}(\mathcal{P},\mathcal{P}_\epsilon)\le\varepsilon K$ for $\varepsilon>\delta$\\
Now, consider the case when $\varepsilon<\delta$. From the choice of $\delta$, there is no vertex $v$ of $\mathcal{P}_\delta$ satisfying $d-\delta<c\cdot v<d$.
Let $v_1,\dots,v_m,u_1,\dots,u_k$ be vertices of $\mathcal{P}_\delta$ with $c\cdot v_i=d-\delta$ and $c\cdot u_j=d$.
From the property of the polytope, for any $x\in\{x\in [0,1]^n|Ax\ge b,c\cdot x=d-\varepsilon\}\subset\mathcal{P}_\delta$ can be expressed as a convex combination of
$v_1,\dots,v_m,u_1,\dots,u_k$. It implies that there are $y\in\mathcal{P},z\in\{x\in [0,1]^n|Ax\ge b,c\cdot x=d-\delta\}$
such that $x$ is a convex combination of $y$ and $z$. Therefore,
$$\|x-y\|_2=\frac{\varepsilon}{\delta}\|z-y\|_2\le\varepsilon K$$
and it implies $\mbox{dist}(\mathcal{P},\mathcal{P}_\epsilon)\le\varepsilon K$ for $\varepsilon<\delta$.
\end{proof}
There are finite number of vertex candidates
By Claim \ref{claim:c4} and the mathematical induction,
we can find $K$ such that there exists $z^\ddagger\in \mathcal P$ with $\|z-z^\ddagger\|_1 \leq \varepsilon K$.
\fi
|
1,116,691,498,498 | arxiv | \section{A New GHRS Study of Ly$\alpha$ Clouds -- Motivation}
Quasar absorption lines provide a powerful tool for probing the evolution
of the universe from $z$ = 0 to 5, but to correctly interpret this information,
one must understand the nature of the absorbers.
Studies with the GHRS have yielded important results
on the nature of low redshift ``Ly$\alpha$ clouds,'' low column density
gas clouds which produce {\sc H i} Ly$\alpha$ absorption lines
in the spectra of background QSOs (see Morris 1996 for a
brief review). GHRS observations of 3C273 shortly after the deployment of
HST revealed that there are considerably more Ly$\alpha$ clouds at low redshift
than expected based on the observed evolution of the number of clouds per
unit redshift ($dN/dz$) at high $z$ (Morris et al. 1991; see also the
FOS study of Bahcall et al. 1991). This abundance of low $z$ clouds provides an opportunity to learn
about the nature of the absorbers by {\it directly} studying the environment
(i.e., galaxies, galaxy clusters, voids, etc.) where the Ly$\alpha$ clouds
are found. Studies of the relationship between low $z$ Ly$\alpha$ clouds and
galaxies
using the GHRS have been carried out by Morris et al. (1993), Stocke et al.
(1995), and Shull, Stocke, \& Penton (1996). These programs find that (1)
Ly$\alpha$ clouds are not randomly distributed with respect to galaxies, but
the absorber-galaxy correlation is not as strong as the galaxy-galaxy
correlation, and (2) some Ly$\alpha$ clouds are found in galaxy voids, although
overall the clouds tend to ``avoid the voids.'' In
general, the Ly$\alpha$ clouds studied with GHRS do not have nearby
associated galaxies; Morris et al. (1993) report that there are no galaxies
observed within 230 kpc of any of the 3C273 Ly$\alpha$ clouds, and Stocke et al. find
that there are no galaxies within 450 kpc of their Ly$\alpha$ absorbers. These
GHRS results seem to be in conflict with HST FOS studies
of Ly$\alpha$ clouds. For example, based on a redshift survey of galaxies
near QSOs observed with the FOS, Lanzetta et al. (1995) find that 32-60\% of
the Ly$\alpha$ clouds in their sample are associated with luminous galaxies
within $\sim$ 160 kpc of the QSO sight lines. To reconcile these discordant
results, it has been
suggested that there are two populations of Ly$\alpha$ clouds at low
redshift: (1) strong Ly$\alpha$ clouds with $N$({\sc H i}) $\geq \ 10^{14}$
cm$^{-2}$ which dominate the Ly$\alpha$ cloud sample of Lanzetta et al. and
mostly occur in large halos of luminous galaxies, and (2) lower column density
absorbers which are less closely tied to galaxies and are, in some cases,
truly intergalactic gas clouds
(the Morris et al. and Stocke et al. Ly$\alpha$ clouds have
$N$({\sc H i}) $\leq \ 5\times 10^{13}$ cm$^{-2}$.) Currently this suggestion
cannot be rigorously tested, however, because the sample of weaker Ly$\alpha$
clouds is small.
\begin{figure}
\plotfiddle{trippghrs_f1.eps}{3.0in}{0}{60}{60}{-180}{-135}
\caption{A portion of the high S/N spectrum of the QSO H1821+643 obtained
with the GHRS using the G140L grating.}
\end{figure}
To significantly improve the statistics of low redshift {\it weak} Ly$\alpha$
clouds, we are conducting a program to study the relationship between low $z$
Ly$\alpha$ clouds and galaxies in the direction of three QSOs using the GHRS
and the
WIYN multiobject spectrograph (HYDRA). Using the G140L grating, we will obtain
GHRS spectra with S/N $\approx$ 100:1, adequate for detection of Ly$\alpha$
clouds with $W_{\lambda}$ = 50 m\AA\ (4$\sigma$), and this will increase the
{\it weak} Ly$\alpha$ cloud sample size by a factor of 4-5. The WIYN HYDRA
will be used to measure the redshifts of galaxies in the $\sim 1^{\circ}$ fields
centered on the QSOs. Figure 1 shows a portion of the GHRS spectrum of
H 1821+643 ($z_{\rm em}$ = 0.297) obtained for this program. The WIYN galaxy
redshift survey for this field has been completed, and the full analysis of
the GHRS and WIYN data for this sight line will be presented in a subsequent
paper. Some preliminary results are summarized below.
\section{Highly Ionized Oxygen at {\it z} = 0.225}
The {\sc O vi} 1031.9, 1037.6 \AA\ doublet is well-detected at $z_{\rm abs}$
= 0.2250 in the H 1821+643 spectrum (see Figure 1) along with {\sc H i}
Ly$\alpha$ and Ly$\beta$, and we
have G160M (FWHM = 15 km s$^{-1}$) GHRS observations of the
{\sc O vi} 1031.9 \AA\ and {\sc H i} Ly$\beta$ absorption profiles
obtained in the Galactic ISM program of Savage et al. (1995).
These G160M profiles are well-described by Voigt profiles with the parameters
listed in Table 1. The width of the {\sc O vi} profile indicates that
$T \ \leq \ 1.8\times 10^{6}$K, and the redshift difference between the
{\sc H i} and {\sc O vi} profiles implies a 40 km s$^{-1}$ centroid shift
between the neutral and highly ionized gas. The {\sc H i} Ly$\alpha$ profile
shows complex component structure (see Figure 2), and some of this structure
could be due to a broad hot {\sc H i} component associated with the {\sc O vi}.
\begin{table}
\caption{O VI Absorber -- Profile Fitting Results}
\begin{tabular}{lccc}
\tableline
Species & Redshift & log $N$ (cm$^{-2}$) & $b$ (km s$^{-1}$) \\ \tableline
H I Ly$\beta$ & 0.224892$\pm$0.000008 & 15.32$\pm$0.07 & 50.7$\pm$3.2 \\
O VI & 0.225026$\pm$0.000010 & 14.29$\pm$0.03 & 42.8$\pm$3.5 \\
\tableline \tableline
\end{tabular}
\end{table}
Schneider et al. (1992) have detected an emission line galaxy at $z$ = 0.2256
within 90 kpc of the sight line. Our WIYN redshift survey has discovered
another emission line galaxy close to the sight line at $z$ = 0.2263.
Therefore it is possible that the {\sc O vi} absorption at $z$ = 0.2250
originates in the intracluster medium of a group of galaxies.
\section{Clustered Ly$\alpha$ Clouds?}
The absorption profiles of three out of the four strongest {\sc H i} Ly$\alpha$
lines contain complex component structure with a main strong component and
several weaker outlying components spanning $\sim$1000-1500 km s$^{-1}$ (see
Figure 2). Even the Ly$\alpha$ line at $z_{\rm abs}$ = 0.1699 which does not
show resolved weak components has an asymmetric profile which is evidence
of unresolved profile components. This seems to suggest that there is some
clustering of weak Ly$\alpha$ clouds at low redshift. However,
we have calculated the two-point
correlation function using all of the Ly$\alpha$ clouds detected in the
H 1821+643 GHRS spectrum, and this does not show evidence of clustering on
any scale, but the sample of Ly$\alpha$ lines may be too small for adequate
statistics.
\begin{figure}
\plotone{trippt2.eps}
\caption{Absorption profiles of the four strongest {\sc H i} Ly$\alpha$ lines
detected in the GHRS spectrum of H 1821+643.}
\end{figure}
\acknowledgements Our WIYN galaxy redshift program is a collaboration with
Buell Jannuzi. This work is supported by NASA through grant NAG5-1852.
|
1,116,691,498,499 | arxiv | \section{Introduction}
The quest to unravel the nature of space and time is one of the oldest and most challenging ones in the history of intellectual endeavors. Quite intriguingly, combining fundamental principles of quantum mechanics with an information theoretic approach to spacetime allows making statements about the structure of space and time way beyond experimentally accessible scales \cite{spin,string,string2}. In this work we use arguments based on entanglement entropy and unitarity to conclude that spatial slices of spacetime must be path-connected topological spaces in order to avoid running into conflicts with these concepts.
In Sect.\ \ref{II.1} we review the notion of entanglement entropy and quantum mutual information. In Sect.\ \ref{II.2} we derive a constraint that is set by the unitarity principle of quantum mechanics on the mutual information shared by two regions of space. In Sect.\ \ref{II.3}, we argue that space cannot consist of (microscopic) disconnected structures without violating this principle. Finally, a summary is provided.
\section{Quantum Information, Entanglement and Spacetime} \label{II}
\subsection{A Brief Introduction} \label{II.1}
Consider a 3-dimensional spatial region $C$ together with its complement $D$ in a 4-dimensional spacetime. Thus $C$ and $D$ are disjoint except for the 2-dimensional boundary shared by them. Suppose the total information about the geometry of region $E=C\cup D$ is stored in a density matrix $\rho_{\rm CD}$. The entangelement entropy between the two regions can then be defined via the von Neumann formula as
\begin{equation}\label{ent}
S_{C} =- \Tr [\rho_{C} \log{\rho_{C}}] \, \, , \qquad S_{D} =- \Tr [\rho_{D} \log{\rho_{D}}] \, .
\end{equation}
Here $\rho_{C}$ and $\rho_{D}$ are the reduced density matrices constructed out of $\rho_{CD}$ by tracing out $D$ and $C$:
\begin{equation}\label{rho}
\rho_{C} = \Tr_{D} [\rho_{CD}] \, \, , \qquad \rho_{D} = \Tr_{C} [\rho_{CD}] \, .
\end{equation}
The mutual information $I_{\rm CD}$ is given by \cite{qminfo}
\begin{equation}\label{main}
I_{\rm CD}\equiv S_{C}+S_{D}-S_{\rm CD} \geq 0
\end{equation}
where $S_{\rm CD} \equiv - \Tr [\rho_{CD} \log{\rho_{CD}}]$ is the total entanglement of the $(C ,\, D)$-system. In a general state, $I_{CD}$ is a measure of correlation between $C$ and $D$. It essentially indicates how much region $C$ knows about $D$ and vice versa. If all the correlation is quantum, i.e. $S_{\rm CD}=0$, then we have a pure state $S_{C}=S_{D}$.
\subsection{Unitarity and Mutual Information} \label{II.2}
The question we seek to answer is: "How does unitarity constrain the mutual information shared by two regions of space?". To address this question we start from the Callan-Symanzik equation \footnote{For example one could consider a massive scalar field living on a predescribed background spacetime.}. This equation implies the following differential equation for $n$-point correlation functions $\Gamma^{(n)}(p_j, g(\mu), \mu)$ \cite{mag}
\begin{equation} \label{cz}
\left[\mu\,\partial_{\mu}+\beta(g(\mu)) \frac{\partial}{\partial {g(\mu)}}-\frac{n}{2}\eta\left(g(\mu)\right) \right]\Gamma^{(n)}(p_j, g(\mu), \mu)=0 .
\end{equation}
Here $\mu$ is the renormalization scale, $\eta\left(g(\mu)\right)$ is the anomalous dimension of the fundamental fields appearing in the action, $g(\mu)$ is a set of coupling constants and $g(\mu)$ satisfies the renormalization group equation $\beta(g(\mu))=\mu \frac{d}{d \mu} g(\mu)$. The background field formalism allows to extend these renormalization group techniques also to the deep quantum gravity regime \cite{Reuter:1996cp} by treating quantum fluctuations of the spacetime metric akin to a quantum field theory in curved spacetime \cite{Niedermaier:2006wt,robertobook}. The general solution to ``\eqref{cz}'' can be written as
\begin{equation} \label{solcz}
\Gamma^{(n)}(p_j, g(\mu), \mu)=exp\left[\frac{n}{2}\int_{\mu_0}^{\mu}\frac{d\mu^{'}}{\mu^{'}}\eta\left(g(\mu^{'})\right)\right]\, \Gamma^{(n)}(p_j, g(\mu_0), \mu_0) \, .
\end{equation}
The crucial observation is that for a generic \textbf{unitary} quantum field theory (QFT) it is safe to assume $\eta(g(\mu))\geq 0$ \cite{unit,unit2}. If we then define operators $O_C$ and $O_D$ with support on $C$ and $D$ respectively, the connected two-point function $\langle O_C O_D \rangle_c$ is positive semi-definite $\braket{O_{C}\,O_{D}}_c \geq 0$. Consequently, using ``\eqref{solcz}'', we get $\partial_{\mu}\braket{O_{C}\,O_{D}}_c \geq 0$. A rigorously proven example of this feature is the celebrated c-theorem in conformal field theory where unitarity implies the positivity of the central charge and consequently $\partial_{\mu}\braket{O_{C}\,O_{D}}_c \geq 0$ \cite{cft,cft2}.
On the other hand, for any two operators $O_{C}$ and $O_{D}$, one can use the well-established properties of the conditional probabilities and certain results of quantum information science to show that the mutual information between the two regions $C$ and $D$ obeys \cite{qminfo1}:
\begin{equation}\label{infobound}
I_{\rm CD} \geqslant \frac{\braket{O_{C}\,O_{D}}_c^2}{2|O_C|^2\,|O_{D}|^2}.
\end{equation}
In more physical terms, ``\eqref{infobound}'' indicates that the quantum mutual information shared by two regions of space is the maximum amount of correlation they can have with each other.
Regarding ``\eqref{infobound}'' it is crucial to distinguish two cases: firstly, ``\eqref{infobound}'' can be applied to "discrete" spaces coming with a hard UV-cutoff. Such spaces would be path-disconnected by definition and the operator norms $|O|$ are finite by construction. This is the case considered in van Raamsdonk's work \cite{van}, where it was concluded that if the mutual information between $C$ and $D$ goes to zero, then all correlations must decrease to zero also. Secondly, one can consider continuum space which is path connected and leads to infinite values of $|O|$. As a consequence vanishing mutual information $I_{CD}$ does not automatically imply the vanishing of the correlation function. In the sequel, we will work in the first setting and construct a contradiction between the unitarity principle and a discrete space.
Any physically interesting theory should have at least one correlation function which has support on both $C$ and $D$ and is non-zero at some (finite) scale. Focusing on such a correlation function, we can make the following argument: assume the value of shared information between regions $C$ and $D$ is zero in the ultraviolet (UV) limit, i.e. we have $I_{\rm CD}^{UV} = 0$. From unitarity we have $\braket{O_{C}\,O_{D}}_c \geq 0$ and thus eq.\ ``\eqref{infobound}'' entails that $\langle O_C O_D \rangle_c^{UV} = 0$. Furthermore, if the unitarity principle is satisfied, we should have $\partial_{\mu}\braket{O_{C}\,O_{D}}_c \geq 0$. If we take $\partial_{\mu}\braket{O_{C}\,O_{D}}_c = 0$ then the correlation function vanishes on all scales contradicting our premise above. So unitarity and the existence of non-vanishing correlations imply $\partial_{\mu}\braket{O_{C}\,O_{D}}_c > 0$. This results in $\langle O_C O_D \rangle_c^{\rm IR} < 0$ which conflicts with unitarity. The bottom line of this argument is that the requirement of unitarity and eq.\ ``\eqref{infobound}'' implies $I_{\rm CD}^{UV} > 0$. In Sect.\ \ref{II.3} we argue that $I_{\rm CD}^{UV} > 0$ is in conflict with a disconnected space.\footnote{ Our statement is actually compatible with the situation where both regions $C$ and $D$ support non-trivial, unitary QFTs which do not interact with each other. In this case the two-point correlator $\langle \cO_C \, \cO_D \rangle = \langle \cO_C \rangle \langle \cO_D \rangle$ factorizes, implying the vanishing of the connected correlation function $\langle \cO_C \cO_D \rangle_c = \langle \cO_C \, \cO_D \rangle - \langle \cO_C \rangle \langle \cO_D \rangle \stackrel{!}{=} 0$.}
An alternative derivation of this conclusion builds on the $I_{theorem}$ \cite{I-theorem}. In this case one can use $\partial_{\mu} I_{CD} > 0$ to argue that $I_{CD}^{UV} = 0$ implies $I_{CD}^{IR} < 0$ and hence leads to the violation of entanglement subadditivity \footnote{Subadditivity of the entanglement of two regions $C$ and $D$ means $\rm{Entanglement}_{C \cup D} \leq \rm{Entanglement}_{C}+\rm{Entanglement}_{D}$. The entanglement entropy always satisfies this property \cite{qminfo}.} in the IR. At this stage, there is no consensus on the existence of the $I_{theorem}$ though.
\subsection{Space Must be Path-Connected} \label{II.3}
According to Van Raamsdonk \cite{van} the relation between the entanglement entropy of two regions of space, their shared area and the geodesic distance between them has a very remarkable interpretation. The original argument is presented in an AdS/CFT \cite{adscft,van} setup; here we present a simpler version using properties of the entanglement entropy in QFT. References \cite{sred,sred2,bia} suggest that to leading order in the UV cutoff the entanglement entropy is proportional to
the shared area (area of the entangling surface) between a region and its complement in space. Thus, reducing the entanglement entropy and mutual information is equivalent to reducing the shared area.
Furthermore, for a typical massive field theory with mass $m$ we have $\braket{O_{C}(\vec{x}) \,O_{D}(\vec{y})}_c \propto e^{-m\, L}$ where $\vec{x}$ and $\vec{y}$ are spatial positions in $C$ and $D$ and $L$ indicates the geodesic distance between these points. In the light of ``\eqref{infobound}'', this means $I \ll 1$ if $L \gg \frac{1}{m}$.
Combining last two paragraphs, one can summarize Van Raamsdonk's result as \cite{van}\footnote{Notably, the holographic entanglement entropy upon which the original Van Raamsdonk argument is based produces the ordinary QFT area law to leading order in the UV cutoff \cite{holo}.}:
\begin{equation}\tag{S1}\label{S1}
\parbox[c]{0.8\linewidth}{\it Zero entanglement entropy is equivalent to $C$ and $D$ being disconnected regions of space. Hence, the connectivity of the classical geometry has its roots in the properties of quantum entanglement entropy.}
\end{equation}
The core of our argument, essentially Sect.\ \ref{II.2}, survives the transition to a fundamental theory of nature as long as this theory admits an effective QFT description that is unitary and respects the basic rules of quantum information physics. If QFT itself survives the generalization, we have a scenario like asymptotic safety \cite{asym,robertobook,martin2}. If QFT is an effective description, we have something like a string theory effective description of the bulk \cite{adscft,string}. Regarding a disconnected space, according to ``\eqref{S1}'' we should have $I_{CD}^{UV}=0$ if regions $C$ and $D$ are disconnected atoms of space sharing no boundary. On the other hand, we know from the previous section that according to ``\eqref{infobound}'' the only way to reconcile $I_{CD}^{UV}=0$ with unitarity is by all correlation functions involving products of operators with support on $C$ and $D$ vanishing at all scales. If the disconnected regions are microscopic, this is of no interest in any physics scenario. So if one insists on a disconnected space in the UV and a QFT describing the universe, one has only two options: 1) Accept unitarity and consequently violate ``\eqref{infobound}'' which means certain basic rules of quantum information physics are violated. 2) Drop the unitarity principle and accept the possibility of information loss. Both options are in tension with principles of quantum mechanics. This suggests that space cannot be disconnected. A prototypical example for a physical process ruled out by these considerations is the creation and pinching-off of baby universes from our spacetime: if a quantum field has support on the region of spacetime that is pinched-off the information is lost entailing a violation of unitarity.
\section{Summary} \label{III}
In this manuscript we assumed that the universe admits a unitary QFT description, at least at the effective level. On this basis we argued that such a description together with rules of conditional probability theory and quantum information science implies that space cannot emerge from the coarse graining of a patchwork of disconnected structures. This is a consequence of ``\eqref{S1}'', the unitarity principle, ``\eqref{infobound}'' and the requirement of a non-trivial correlation function. To put it in other words, in the light of ``\eqref{S1}'' it is obvious that $I_{CD}^{UV} = 0$ is a necessary condition for having a disconnected space. Thus unitarity and ``\eqref{infobound}'' put conditions on the topological properties of space, demanding $I_{CD}^{UV} > 0$. Notably, our analysis does not imply that on a space made of path-disconnected regions, any unitary QFT is necessary trivial: every path-connected region may support its own unitary QFT coming with non-trivial correlation functions. If the regions are Planck-sized this is not a physically interesting scenario though.
An interesting situation motivating a non-trivial generalization of the ideas advocated in this work is the BKL scenario \cite{BKL}. In this model spatial points decouple close to the cosmic singularity. This should lead to vanishing correlation functions in the UV. At the same time, spacetime remains connected, thereby violating the assumption of fundamental discreteness made in our derivation. It would be interesting to understand the interplay between the information theoretic concepts underlying our work and unitarity in such a situation as well. We also note that our argument may not be applicable to (apparently discrete) models of quantum gravity as, e.g., causal dynamical triangulations\cite{Ambjorn:2012jv,Loll:2019rdj}
or group field theories \cite{Freidel:2005qe,Carrozza:2016vsq},
since in this case the ``discreteness scale'' plays the role of a UV regulator which is to be removed by taking a suitable continuum limit.
As a final remark, we note that the incompatibility between the unitarity of QFT and ``\eqref{infobound}'' with a disconnected space only depends on a set of basic properties of conditional probabilities and the unitarity of QFT. This result may restrict the class of (quantum) theories of gravity that are compatible with unitarity and basic rules of quantum information theory.
\section*{Acknowledgement}
We would like to thank Carlo Rovelli, Slava Rychkov, Tatsuma Nishioka and Horacio Casini for fruitful discussions and Francesco Caravelli, Andrea Stergiou and Andrea Trombettoni for critical comments. Special thanks to Alfio Bonanno for providing insightful comments on the draft. A.K.\ profited from illuminating discussion during the {\it Functional and Renormalization-Group methods} \text{(FRGIM 2019)} conference in Trento, Italy where the work was presented. Attending the FRGIM 2019 was made possible by the generous financial support from the “ACRI-Young Investigators Training Program”.
|
1,116,691,498,500 | arxiv | \section{Introduction}
\label{intro}
Recent advance in experiments using ultracold atoms have led to
great progress in understanding non-equilibrium dynamics of closed
quantum systems \cite{bloch1,blochrev,dipoleexp1}. These experiments
have shed considerable light on the many-body dynamics of strongly
interacting bosons in the presence of an experimentally applied tilt
or equivalently a synthetic electric field \cite{dipoleexp1}. More
recently, similar set of experiments have been carried out on a
chain of Rydberg atoms \cite{scarref1}. Such a chain consists of an
one-dimensional (1D) array of ultracold $^{87}{\rm Rb}$ atoms which
can be excited to a metastable excited Rydberg state by application
of a suitably designed laser. Two such atoms in their excited state
experience repulsive dipolar interaction between them. The strength
of this interaction can be tuned in these experiments; in
particular, it is possible to reach a regime where the existence of
two Rydberg atoms on neighboring sites is practically forbidden. In
addition, it is also possible to tune the on-site energy for
creating a Rydberg excitation. The low energy properties of the
Rydberg chain can therefore be described by the Hamiltonian
\begin{eqnarray} H_0 &=& \sum_j [(-\Omega |g_j\rangle\langle e_j| + {\rm H.c.})
+ \Delta \hat n_j] +\sum_{ij} V_{i-j} \hat n_i \hat n_j, \nonumber\\
\label{ryd1} \end{eqnarray} where $j$ denotes the site index,
$|g\rangle = \prod_j |g_j\rangle$ is the ground state, $|e_j\rangle$
denotes the state at site $j$ with a Rydberg excitation, ${\hat
n}_j$ denotes the number operator for the Rydberg excitations, and
$V_{i-j}= V^0/|i-j|^3$ is the interaction potential between the
excited atoms. In the regime of interest, $V^0$ is chosen such that
$V_1 \gg \Omega, \Delta \gg V_{n>1}$. In this regime the interaction
term in $H_0$ can be replaced by the constraint $\hat n_j \hat
n_{j+1}=0$ for all sites. For large negative $\Delta$, the ground
state of the system corresponds to Rydberg excitations on all
alternate sites; this state is dubbed as $|\mathbb{Z}_2\rangle$
since it breaks $\mathbb{Z}_2$ symmetry (there are two such states,
and they are related to each other by a translation by one site). In
contrast, for large positive $\Delta$, the ground state is the
vacuum of Rydberg excitation and is termed as $|0\rangle$. These two
ground states are separated by an Ising quantum phase transition at
$ \Delta=-1.31 \Omega$ \cite{subir1,subir2}. Both the $|0\rangle$
and $|\mathbb{Z}_2\rangle$ states have been observed
experimentally~\cite{scarref1}. Similar observations have also been
carried out using systems of 1D ultracold bosons in synthetic
electric fields~\cite{dipoleexp1}.
The experiments in Ref.\ \onlinecite{scarref1} also studied quench
dynamics of the Rydberg atoms starting from the $|\mathbb{Z}_2\rangle$ state
and found persistent coherent oscillatory dynamics of the Rydberg
excitations when the system is allowed to evolve after a sudden
quench of $\Delta \to 0$. This behavior constitutes a violation of the
eigenstate thermalization hypothesis (ETH) which is one of the
central paradigms for understanding out-of-equilibrium dynamics of
closed non-integrable quantum systems \cite{rev1a,rev1b,rev1c,rev1d,
rev2,deutsch1,srednicki1,rigol1}. It predicts eventual
thermalization for non-equilibrium dynamics of a generic many-body
state \cite{rev2}. This hypothesis is strongly violated in certain
cases such as 1D disordered electrons in their many-body localized
phase~\cite{mblref1,mblref2}, but was expected to hold in disorder-free
systems. The observed weaker failure of ETH was later understood as being
due to the presence of quantum scars in the eigenstates of $H_0$ (with
$\Delta=0$)~\cite{scarrefqm1,scarref1,scarref2a,scarref2b,scarref2c,
scarref2d,scarref2e,scarref3a,scarref3b,scarref3c,scarref3d,scarref3e}.
These quantum scars, which have large overlap with the initial
$|\mathbb{Z}_2\rangle$ state, are eigenstates with finite energy
density but anomalously low entanglement
entropy~\cite{scarref1,scarref2b,scarref2c,scarref2d,scarref3b}.
They form an almost closed subspace in the system's Hilbert space
under the action of its Hamiltonian and lead to persistent coherent
oscillatory dynamics of correlation functions starting from initial
states that have a high overlap with scars. This provides an
observable consequence of their presence as verified in recent
experiments on quench dynamics of a chain of ultracold Rydberg atoms
\cite{scarref1}. Such scar states, having high overlap with the
$|\mathbb{Z}_2\rangle$ state (hence the name $\mathbb{Z}_2$-scar)
have been theoretically studied using a forward-scattering
approximation (FSA) which reproduces the scar-manifold via a Lanczos
iteration starting from a $|\mathbb{Z}_2 \rangle$
state\cite{scarref2a,scarref2c,scarref2d}. The effect of $\mathbb
{Z}_2$ scars on the dynamics of a periodically driven Rydberg chain has
also been studied; it was found that the drive frequency can be used
as a tuning parameter to induce transitions between ETH violating
oscillatory and ETH obeying thermal regimes \cite{scarfl1}. Such
transitions were also shown to occur for a class of noisy and
quasiperiodic drives \cite{scarfl2}.
Analogous studies on quench dynamics starting from the $|0\rangle$
state find an expected thermalization which is consistent with ETH. However,
for periodically driven chains, where the drive frequency can act as a tuning
parameter, such dynamics has not been studied so far. In this work, we carry
out such a study by using a square pulse protocol for $\Delta$
\begin{eqnarray} \Delta &=& -\Delta_0 \quad {\rm for} \quad 0\le t\le T/2,
\nonumber\\
&=& \Delta_0 ~~\quad {\rm for} \quad T/2 < t \le T, \label{protocol1}
\end{eqnarray}
where $T=2\pi/\omega_D$ is the time period of the drive and $\omega_D$ is the
drive frequency. In what follows, we shall compute the correlation function
\begin{eqnarray} C_{j \ell} &=& \langle \psi(nT)|\hat n_j \hat n_{j+\ell}|
\psi(nT) \rangle, \label{corrfn1} \end{eqnarray}
where $|\psi(nT)\rangle$ is the state of the system after $n$ drive
cycles starting from the $|0\rangle$ state. To this end, we use
exact diagonalization for finite-sized chains of length $L\le 26$.
Our numerical results will be supplemented by an analytical, albeit
qualitative, explanation of the main features of the dynamics of the
system using a perturbative Floquet Hamiltonian. We derive this
Hamiltonian using Floquet perturbation theory with $\Omega/\Delta_0$
as the perturbation parameter \cite{dsen1,thomas1}. We note that
such a derivation is distinct from the standard Magnus or $1/\omega_D$
expansion; the Floquet Hamiltonian we obtain explains the qualitative
behavior of the system at both high and low frequency limits.
The central results that we obtain from such a study are as follows.
First, we show that for a range of drive frequencies, $C_{22}$
displays coherent oscillatory dynamics and does not thermalize. Such
a behavior constitutes a weak violation of ETH and has been reported
earlier for dynamics starting from the $|\mathbb{Z}_2\rangle$ state for
both quench and periodic driving~\cite{scarref1,scarfl1}. Its origin
in the earlier known cases has been shown to be due to the presence
of quantum scars which have high overlaps with the initial
$|\mathbb{Z}_2\rangle$ state. In our work, we show that an analogous
behavior for dynamics starting from the $|0\rangle$ state originates
from the existence of a different set of scar states in the Floquet
eigenstates of the system. These scars have high overlaps with the
$|0\rangle$ state (and hence are termed as $|0\rangle$ scars) and
coexists with the $|\mathbb{Z}_2\rangle$ scars for a range of
frequencies which we chart out. We study the properties of these scars
using the FSA reformulated using a different (compared to the
$\mathbb{Z}_2$ case) decomposition of an effective Hamiltonian which
qualitatively resembles the Floquet Hamiltonian of the driven chain.
Our analysis brings out the importance of higher spin terms for the
stability of the scar-induced oscillations. Second, we identify
specific drive frequencies at which the $|0\rangle$ state barely
evolves. This constitutes an example of dynamical
freezing~\cite{adas1,pekker1} in an experimentally realizable
non-integrable many-body system. We provide an analytic
understanding of this phenomenon by using the perturbative Floquet
Hamiltonian and by performing an exact analytical calculation for
small system sizes which predicts the freezing frequency almost
exactly. Finally, we show that lowering the drive frequency from the
dynamical freezing point with the highest frequency, we find a regime
where the system reaches a steady state with sub-thermal values of
$C_{22}$. We note that such steady states provide a new route to
weak ETH violation for finite-sized chains; it does not feature
coherent persistent oscillations and has no analogs for quench or
periodic dynamics starting from the $|\mathbb{Z}_2\rangle$ initial
state. Our numerics indicates that such behavior may persist as
a prethermal phase of thermodynamically large Rydberg chains up to a
large but finite number of drive cycles.
The rest of the paper is organized as follows. In Sec.\
\ref{floquet}, we introduce the basic model which we use for our
computations and derive the Floquet Hamiltonian for the system. This
is followed by Sec.\ \ref{numerics}, where we present our numerical
results and interpret them using the Floquet Hamiltonian. Finally,
in Sec.\ \ref{diss}, we summarize our results, discuss experiments
which can test them, and conclude. Further details of our
calculations on the derivation of the analytical form of the
perturbative Floquet Hamiltonian and analytic results regarding
dynamic freezing are presented in Appendices \ref{appA} and
\ref{appB} respectively. The details of the FSA calculation is
presented in App.\ \ref{appC}.
\section{Floquet perturbation theory}
\label{floquet}
The Hamiltonian describing the properties of an ultracold Rydberg atomic chain,
given by Eq.\ \eqref{ryd1}, can be directly mapped to a simple spin
model in the regime where $V_1 \gg \Delta,\Omega \gg V_{n>1}$. In
this regime the interaction term can be replaced by a hard
constraint on Rydberg excitations on neighboring sites. Such a
mapping is achieved by writing $\hat n_j = (\sigma_j^z + 1)/2$ and
$|e_j\rangle \langle g_j| = \sigma^+_j$, where $\sigma_j^{\alpha}$
denotes spin-1/2 Pauli matrices at site $j$ for $\alpha=x,y,z$, and
$\sigma^{\pm}_j = (\sigma_j^x \pm i \sigma_y)/2$. The constraint is
implemented by a local projection operator $P_j = (1-\sigma_j^z)/2$
\cite{scarref2c,scarref2d}. The resulting spin Hamiltonian can be
written, ignoring an unimportant constant, as~\cite{scarfl1}
\begin{eqnarray} H_{\rm spin} &=& \sum_j \left(-w \tilde \sigma_j^x
+ \frac{\lambda}{2}\sigma_j^z \right), \label{hamspin1}
\end{eqnarray}
where $\tilde \sigma^{\alpha}_j = P_{j-1} \sigma^{\alpha}_j P_{j+1}$
for $\alpha=x,y,z$. It can be easily seen that $H_{\rm spin}$ may be
identified with $H$ in Eq.\ \eqref{ryd1}, with $\Omega=w$ and
$\lambda=\Delta$. We note that $H_{\rm spin}$ also constitutes a
spin representation of the dipole model introduced in Ref.\
\onlinecite{subir1} and can thus be realized in experiments involving the
tilted Bose-Hubbard model~\cite{dipoleexp1}. For $\lambda=0$, $H_{\rm spin}$
yields the PXP model which is known to host $|\mathbb{Z}_2\rangle$
scars~\cite{scarref2a,scarref2b,scarref2c,scarref2d,scarref2e}.
In what follows, we will study the periodic dynamics of this
model using a square pulse protocol
\begin{eqnarray}
\lambda (t) &=& -\lambda \quad {\rm for} \quad 0\le t\le T/2, \nonumber\\
&=& \lambda ~~\quad {\rm for} \quad T/2 < t \le T, \label{protocol2}
\end{eqnarray}
which is identical to the protocol mentioned in Eq.\
\eqref{protocol1}. We shall be interested in the correlation function
\begin{eqnarray} O_{j \ell} &=& \frac{1}{4} \langle \psi(nT)|
(1+\sigma^z_j)(1+\sigma^z_{j+\ell})|\psi(nT) \rangle \label{corrfn2}
\end{eqnarray}
with $|\psi(0)\rangle= |0\rangle$. We note that $O_{j \ell}$ is
identical to $C_{j \ell}$ (Eq.\ \eqref{corrfn1}) for Rydberg atoms.
In the rest of this section, we shall derive a perturbative Floquet
Hamiltonian for $H_{\rm spin}$ (Eq.\ \eqref{hamspin1}) driven by the
protocol given in Eq.\ \eqref{protocol2} in the high drive amplitude
limit $\lambda/w \gg 1$ but without any approximation about the
drive frequency. In doing so, we shall use the formalism developed
in Refs.\ \onlinecite{dsen1} and \onlinecite{thomas1} and will closely follow
the approach of Ref.\ \onlinecite{thomas1}.
We treat the term $H_1=-w \sum_j \tilde \sigma_j^x$ in the
Hamiltonian as a perturbation and note that for $w=0$, the exact
evolution operator for the system can be written as (here and in the
rest of this work we set $\hbar=1$)
\begin{eqnarray} U_0 (t,0) &=& e^{i \lambda t \sum_j \sigma_j^z /2} \quad
{\rm for} \quad t\le T/2, \nonumber\\
&=& e^{i \lambda (T-t) \sum_j \sigma_z^j/2} \quad {\rm for} \quad
T/2 \le t \le T. \label{u0eq} \end{eqnarray}
$U_0$ is diagonal in the eigenbasis of $\sigma_j^z$. For simplicity
of calculation, we denote $|m\rangle$ to be set of states for which
$n_{\uparrow}-n_{\downarrow} = m$, where $n_{\uparrow(\downarrow)}$
is the number of spins with spin $\uparrow(\downarrow)$. For such
states, which form a complete basis, we find that
\begin{eqnarray} \langle m|U_0(t,0)|n\rangle &=& \delta_{mn} e^{i m \lambda
t/2} \quad {\rm for} \quad t\le T/2, \label{u0exp} \\
&=& \delta_{mn} e^{i \lambda (T-t) m/2} \quad {\rm for} \quad T/2
\le t \le T, \nonumber \end{eqnarray}
with $-L \le m\le L$ for a chain of size $L$. Note that the set
$|m\rangle$ has, in general, a large degeneracy for $w=0$ since a
particular $m$ may originate from many arrangements of spins on the
sites of the lattice. However, $m=-L$ corresponds to a
non-degenerate all down-spin state. This state is denoted as
$|0\rangle$ and shall be the initial state for this study. In this
language, the $|\mathbb{Z}_2\rangle$ state, which is doubly degenerate within
the constrained Hilbert space, corresponds to $m=0$.
Next, we compute the ${\rm O} (w)$ contribution to the evolution operator
for one time period, $U(T,0)$. To this end we compute the matrix
element of
\begin{eqnarray} U_1(T,0) &=& -i \int_0^T dt H_I(t) \label{firstorderu}
\end{eqnarray}
between states $|m\rangle$ and $|n\rangle$, where $H_I(t) =
U_0^{\dagger} (t,0) H_1 U_0(t,0)$ is the perturbation Hamiltonian in
the interaction picture. A straightforward calculation leads to
\begin{eqnarray}
\langle m|U_1(T,0)|n\rangle &=& \delta_{m,n+s} \frac{2w}{\lambda s}
\left( e^{i \lambda s T/2}-1\right), \label{matel1} \end{eqnarray}
where $s=\pm 1$. Thus in the $|m\rangle$ basis, we can write
\begin{eqnarray} U_1(T,0) &=& \sum_{m} \sum_j \sum_{s_j=\pm 1} c^{(1)}_{s_j}
|m\rangle\langle m+s_j|, \nonumber\\
c_s^{(1)} &=& \frac{4iw}{\lambda} \sin\left(\lambda T/4\right) e^{i
\lambda T s/4}, \label{u1exp} \end{eqnarray}
where the additional up or down spin in $|m+s_j\rangle$ resides on
the $j^{\rm th}$ site. Next, we note that the states $|m\rangle$ and
$|m\pm 1_j\rangle$ are connected by the projected ladder operators
$\tilde \sigma_j^{\pm} = (\tilde \sigma_j^x \pm i \tilde
\sigma_j^y)/2$ as $\tilde \sigma_j^{\pm}|m\rangle=|m\pm 1_j \rangle$.
This allows us to write the first-order Floquet Hamiltonian
$H_F^{(1)} = (i/T) U_1(T,0)$ (since $U_0(T,0)=1$ here) as~\cite{scarfl1}
\begin{eqnarray}
H_F^{(1)} &=& -w \frac{\sin(\gamma)}{\gamma} \sum_j ~[ \cos(\gamma)
\tilde \sigma_j^x + \sin(\gamma) \tilde \sigma_j^y], \label{fl1} \end{eqnarray}
where $\gamma= \lambda T/4$. We find that $H_F^{(1)}$ is identical
to the PXP model up to a global rotation and a overall
renormalization coefficient $\sin (\gamma)/\gamma$. We note that Eq.\
\eqref{fl1} was derived in Ref.\ \onlinecite{scarfl1} following a
slightly different approach~\cite{dsen1} and was used to explain the
ergodic-non-ergodic transitions for dynamics starting from the
$|\mathbb{Z}_2\rangle$ state. However, the present method allows us to derive
higher order terms in $H_F$, which, as we shall see, are crucial for
explaining the dynamics starting from the $|0\rangle$ state.
The second term in $U(T,0)$ can be obtained in a similar manner by
evaluating the matrix elements of
\begin{eqnarray} U_2(T,0) &=& (-i)^2 \int_0^T dt_1 H_I(t_1) \int_0 ^{t_1} dt_2
H_I(t_2). \label{secondorderu} \end{eqnarray}
A calculation, similar to the one carried out before and detailed in
the App.\ \ref{appA} yields
\begin{eqnarray} && U_2(T,0) = \sum_{j,j'} \sum_m \sum_{s_1,s_2=\pm}
c^{(2)}_{s_1 s_2} \tilde \sigma_j^{s_1} \tilde \sigma_{j'}^{s_2},
\label{u2exp} \\
&& c^{(2)}_{\pm \pm} = (c_{\pm}^{(1)})^2/2, ~~~{\rm and}~~~ c^{(2)}_{+-} =
c^{(2)}_{-+}= c^{(1)}_{+} c^{(1)}_-/2. \nonumber \end{eqnarray} Eq.\
\eqref{u2exp} implies that $U_2(T,0)= [U_1(T,0)]^2/2$. This ensures
that the second-order contribution to the Floquet Hamiltonian,
$H_F^{(2)}=0$. In fact, as pointed out in Ref.\
\onlinecite{scarfl1}, it can be shown that this is a consequence of
the fact that $H_F$ must satisfy the anticommutation relation $\{
\prod_{j=1, \cdots, L} \sigma_j^z, H_F \} =0$; this implies that
$H_f^{(2n)}=0$ for all integer $n$ since terms with an even number
of $\tilde{\sigma}^{+/-}$ cannot appear in $H_F$.
Finally we proceed to obtain the third order term in $H_F$. The
corresponding evolution operator is given by
\begin{eqnarray} U_3(T,0) &=& (-i)^3 \int_0^T dt_1 H_I(t_1) \int_0 ^{t_1} dt_2
H_I(t_2) \nonumber\\
&& \times \int_0^{t_2} dt_3 H_I(t_3). \label{thirdorderu} \end{eqnarray}
As shown in App.\ \ref{appA}, the matrix elements of $U_3(T,0)$ between
any two arbitrary states $|m\rangle$ and $|n\rangle$ can be obtained
after a somewhat detailed calculation. This yields
\begin{eqnarray} U_3(T,0) &=& \sum_{j,j',j"} \sum_m \sum_{s_1,s_2,s_3=\pm}
c^{(3)}_{s_1 s_2 s_3} \tilde \sigma_{j}^{s_1} \tilde \sigma_{j'}^{s_2}
\tilde \sigma_{j"}^{s_3}, \nonumber\\
c^{(3)}_{+++} &=& (c_+^{(1)})^3/6, \quad c^{(3)}_{---} = (c_-^{(1)})^3/6,
\nonumber\\
c^{(3)}_{+--} &=& \left[ e^{3i\lambda T/2} + e^{i\lambda T/2}
(3-i\lambda T) -2 (1+ e^{i\lambda T}) \right] \nonumber\\
&& \times \frac{w^3 e^{-i\lambda T}}{\lambda^3} = c^{(3)}_{--+}, \nonumber\\
c_{-+-}^{(3)} &=& (c_{-}^{(1)})^2 c_{+}^{(1)}/2 - 2c^{(3)}_{+--},
\quad c^{(3)}_{+ - +} = c^{(3) \ast}_{- + -}, \nonumber\\
c^{(3)}_{+ + -} &=& c^{(3) \ast}_{--+},\quad c^{(3)}_{-++} = c^{(3)
\ast}_{+--}. \label{u3exp} \end{eqnarray}
Next, we compute the contribution to the Floquet Hamiltonian from
Eq. \eqref{u3exp} which comes from non-zero terms in $U_3(T,0)
-[U_1(T,0)]^3/6$. First we note, from the expressions for
the $c^{(3)}_{+++}$ and $c^{(3)}_{---}$ terms in Eq.\ \eqref{u3exp},
that all non-zero contribution to $H_F^{(3)}$ must come from terms
which have at most two $\tilde \sigma^+$ or $\tilde \sigma^-$
operators acting on different sites. All terms in $U_3(T,0)$ having
three $\tilde \sigma^+$ or $\tilde \sigma^-$ operators cancel with
similar terms from $[U_1(T,0)]^3/6$. Furthermore, the class of terms
for which the sites where the spins reside are not nearest
neighboring or same sites (so that the $\tilde \sigma^{\pm}$ on
these sites commute) do not lead to non-zero terms in $H_F^{(3)}$.
The coefficients of all such terms can be rearranged so that they
exactly cancel with similar terms from $[U_1(T,0)]^3/6$. The terms
which provide non-zero coefficient to $H_F^{(3)}$ are found to be of
three types. The first involves three spin operators on neighboring
sites such that the constraint is respected, while the second
consists of three spin operators out of which two act on the same
site. The third involves three spin operators which act on the same
site. A careful analysis of these terms leads to the third order
Floquet Hamiltonian
\begin{eqnarray} H_F^{(3)} &=& \sum_{j} ~( A_0 ~[(\tilde \sigma_{j-1}^+
\tilde \sigma_{j+1}^+ + \tilde \sigma_{j+1}^+ \tilde \sigma_{j-1}^+) \tilde
\sigma_j^- ~-~ 6 \tilde \sigma_j^+] \nonumber \\
&& ~~~~~~~+ ~{\rm H.c.} ), \nonumber\\
A_0 &=&\left[ e^{3i\lambda T/2} +3 e^{i\lambda T/2} (1+ i \lambda T)
+2 (1-3 e^{i\lambda T}) \right] \nonumber\\
&& \times \frac{w^3 e^{-i\lambda T}}{3 i \lambda^3 T}.
\label{fl3} \end{eqnarray}
We note that the first term in $H_F^{(3)}$ involves multiple spin
operators and generates the lowest order non-PXP terms in $H_F$. The
second term of $H_F^{(3)}$ is of the same form as in $H_F^{(1)}$ and
simply leads to a ${\rm O} (w^3)$ renormalization of its coefficients.
The former set of terms will be shown to be crucial for explaining
several properties of dynamics starting from the $|0\rangle$ state
which cannot be explained by a PXP-like Floquet Hamiltonian. The
latter class of terms will be useful for an accurate determination of
the freezing frequencies which we shall discuss in the next section.
\section{Results}
\label{numerics}
In this section, we present our numerical results on the dynamics of $O_{22}$
using exact diagonalization. To this end, we first note that for the chosen
protocol (Eq.\ \eqref{protocol2}), the evolution operator is given by
\begin{eqnarray} U(T,0) = e^{-i H_{\rm spin} [\lambda]T/2} e^{-i H_{\rm
spin}[-\lambda] T/2}, \label{uevola} \end{eqnarray}
and can thus be written as
\begin{eqnarray} U(T,0) &=& \sum_{\alpha \beta} e^{-i(\epsilon_{\beta}^+ +
\epsilon_{\alpha}^-)T/2} c_{\alpha \beta}^{-+} |\alpha^-\rangle
\langle \beta^+| \label{uevol1}, \end{eqnarray}
where $\epsilon_{\alpha}^{+(-)}$ and $|\alpha^{+(-)}\rangle$ are
eigenstates and eigenfunctions of $H_{\rm spin} [+(-) \lambda]$, and
$c_{\alpha \beta}^{-+} = \langle \alpha^-|\beta^+\rangle$ denotes
eigenstate overlaps between eigenstates of $H[\lambda]$ and
$H[-\lambda]$. These eigenvalues, eigenfunctions, and the overlaps
are obtained via exact diagonalization (ED) of $H_{\rm spin}[\pm
\lambda]$ for finite system sizes $L \le 26$. This also allows us to
obtain the Floquet spectrum via diagonalization of $U(T,0)$ for
$L\le 26$. Using these, we compute the spin correlation function
$O_{22} = \langle 0|(U^{\dagger} (T,0))^n
(1+\sigma_2^z)(1+\sigma_{4}^z) U^n(T,0)|0\rangle/4$ after $n$ drive
cycles. In the limit of $n\to \infty$, the system approaches its
steady state; the value of $O_{22}$ in the steady state can be
computed using a diagonal ensemble (DE)~\cite{reimann1}. Denoting
the eigenstates of $U(T,0)$ by $|\chi_n \rangle$, it is easy to see that the
DE value of the correlator is given by
\begin{equation} O_{22}^{\rm DE} = \frac{1}{4} \sum_n |\langle 0|\chi_n\rangle
|^2 \langle \chi_n|(1+\sigma_2^z)(1+\sigma_{4}^z)|\chi_n\rangle.
\label{decorr1} \end{equation} We note that ETH predicts a steady
state value $O_{22}^{\rm DE}= 1/(\varphi^2+\varphi^4) \simeq 0.106$,
where $\varphi=(\sqrt{5}+1)/2$ is the golden ratio, which equals the
infinite temperature ensemble (ITE) value of $O_{22}$ in the
constrained Hilbert space\cite{scarfl1}.
\begin{figure*}[!]
{\includegraphics*[width=\columnwidth]{figmain1a.pdf}}%
{\includegraphics*[width=\columnwidth]{figmain1b.pdf}}
\caption{(a) Plots of $O_{22}^{\rm DE}$ as a
function of the drive frequency starting from an initial state $|0\rangle$ for
$L=14, 18, 24$. The blue dashed line indicates the infinite temperature
thermal value of $O_{22}^{\rm DE}$ as predicted by ETH. The plots clearly
indicate super-thermal and sub-thermal values of $O_{22}^{\rm DE}$
over a range of frequencies and dynamical freezing at specific
frequencies where $O_{22}^{\rm DE} \simeq 0$. (b) Similar plot for
$O_{22}^{DE}$ as a function of $\omega_D$ for $L=14$ as obtained
using the analytic perturbative Floquet Hamiltonian given by Eqs.\
\eqref{fl1} and \eqref{fl3}. All energies and frequencies are scaled in
units of $w/\sqrt{2}$, $\hbar=1$, and $\lambda=15$ in rescaled units
for all plots.} \label{fig1} \end{figure*}
A plot of $O_{22}^{\rm DE}$, computed from the exact evolution
operator, is shown in Fig.\ \ref{fig1} (a) as a function of the
drive frequency $\omega_D$. The corresponding plot, obtained
starting from the analytical Floquet Hamiltonian at
$\mathcal{O} (w^3)$ (Eqs.\ \eqref{fl1} and \eqref{fl3}), is shown in
Fig.\ \ref{fig1} (b). From these plots, we note the following
features. First, we find that $O_{22}^{\rm DE}$ obtained using the
analytic Floquet Hamiltonian provides a qualitative match with that
obtained from exact numerics. This brings out the importance of the
multiple-spin term in Eq.\ \eqref{fl3}; the PXP Floquet Hamiltonian
(Eq.\ \eqref{fl1} and the single spin term in Eq.\ \eqref{fl3}), for
dynamics starting from the $|0\rangle$ state, predict a featureless
thermal value of $O_{22}^{\rm DE}$ as a function of $\omega_D$.
Second, we note that $O_{22}^{\rm DE}$ reaches the expected infinite
temperature thermal steady state value predicted by ETH (blue dashed
line in Fig.\ \ref{fig1}) for high frequencies. This clearly
indicates that $|\mathbb{Z}_2\rangle$ scars do not play a role in
the dynamics. In contrast, at finite $\omega_D$, there are several
non-ETH like features present as a function of the drive frequency
at least up to $L=24$ (Fig.\ \ref{fig1} (a)). Third, for $8 \le
\omega_D \le 12$, Fig.\ \ref{fig1} (a) shows that $O_{22}^{DE}$
reaches super-thermal values; this phenomenon constitutes a violation
of ETH for finite-sized chains $L \le 24$. We shall discuss this
feature in detail in Sec.\ \ref{sup1}. Fourth, for $\omega_D \simeq
7.88, 3.94 ..$, $O^{\rm DE}_{22}$ remains pinned to its initial
value ($=0$); this constitutes an example of dynamical freezing at
specific drive frequencies which we discuss in Sec.\ \ref{freez1}.
Finally, for $5 \le \omega_D \le 7.5$, we find that $O_{22}^{\rm
DE}$ exhibits sub-thermal steady-state values. This constitutes
another class of violation of ETH for finite chains which we discuss
in Sec.\ \ref{sub1}. We note that the time evolution of $O_{22}$ as
a function of the number of drive cycles, shown in Fig.\ \ref{fig2a}, in
these three regimes shows qualitatively distinct behaviors which can
be discerned in realistic experiments involving Rydberg atom chains.
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain2.pdf}
\caption{Plots of $O_{22}$ as a function of the number of drive cycles $n$
(stroboscopic time) starting from an initial state
$|0\rangle$ for (a) $\omega_D=100$, (b) $\omega_D=8.5$, (c)
$\omega_D=7.88$, and (d) $\omega_D=7.26$. The blue dashed line indicates the
ETH predicted thermal value of $O_{22}$. Here $L=26$, $\lambda=15$, and all
units are the same as in Fig.\ \ref{fig1}.} \label{fig2a} \end{figure}
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain3.pdf}
\caption{Plots of $O_{22}$ as a function of the number of drive cycles $n$
(stroboscopic time) starting from an initial state $|Z_2\rangle$ for
(a) $\omega_D=100$, (b) $\omega_D=8.5$, (c) $\omega_D=7.88$, and
(d) $\omega_D=7.26$. The blue dashed line indicates the ETH predicted
thermal value of $O_{22}$. Here $L=26$, $\lambda=15$, and all
units are the same as in Fig.\ \ref{fig1}} \label{fig2b} \end{figure}
\subsection{Super-thermal steady state value}
\label{sup1}
The stroboscopic time evolution of $O_{22}$ starting from the
$|0\rangle$ state is shown in Fig.\ \ref{fig2a} for $L=26$. The
corresponding behavior of the same correlator starting from the
$|\mathbb{Z}_2\rangle$ state is shown in Fig.\ \ref{fig2b}. First,
we note that for high frequencies such as $\omega_D=100$, Fig.\
\ref{fig2a} (a) shows expected thermalization while Fig.\
\ref{fig2b} (a) shows scar-induced oscillations. This behavior is
consistent with earlier studies of the PXP
model~\cite{scarref2a,scarref2b,scarref2c,scarref2d,scarfl1} which
reported thermalization for dynamics starting from the $|0\rangle$
state in cases of both quench and periodic protocol at high drive
frequency. Panel (b) for Figs.\ \ref{fig2a} and \ref{fig2b}, in
contrast, indicate the presence of persistent oscillations for
$\omega_D=8.5$ for dynamics starting from {\it both} $|0\rangle$ and
$|\mathbb{Z}_2\rangle$ states. This leads to weak violation of ETH
and super-thermal value of $O_{22}^{\rm DE}$ for dynamics starting
from the $|0\rangle$ state.
To understand the origin of these oscillations, we show the
half-chain entanglement entropy $S_{L/2}$ of the eigenstates
$|\chi_m\rangle$ of the Floquet Hamiltonian for
$\omega_D=8.5$ in Fig.\ \ref{fig3} (a).
The details of this computation have been charted out in Ref.\
\onlinecite{scarfl1}. Fig.\ \ref{fig3} (b) shows the value of
\begin{eqnarray} O_{22}^m = \frac{1}{4} \langle \chi_m|(1+\sigma_2^z)(1+
\sigma^z_4)|\chi_m\rangle \label{eigenexp} \end{eqnarray}
for all Floquet eigenstates $|\chi_m\rangle$ as a function of the
Floquet quasienergies $E_F$. The dotted line in this plot
indicates the ETH value of $O_{22}$ at a temperature
$T_0(E_F)$ as a function of these quasienergies.
Here $T_0(E_F)$ is defined such that the average
quasienergy equals $E_F$ for a canonical ensemble
with temperature $T_0(E_F)$.
Fig.\ \ref{fig3} (a) shows the usual thermal ETH band with large
$S_{L/2}$ along with sub-thermal states with lower values of
$S_{L/2}$. The states $|\chi_m\rangle$ with $|\langle
0|\chi_m\rangle|^2 (|\langle Z_2 |\chi_m\rangle|^2)>0.01$ which
control the dynamics starting from the $|0\rangle (|Z_2\rangle)$
state is shown in red (green) circles in both panels. From Fig.\
\ref{fig3} (a), we find that the low-entropy eigenstates of $H_F$
which control the dynamics are distinct for $|0\rangle$ and
$|\mathbb{Z}_2\rangle$ initial states; at $\omega_D=8.5$, these
states coexist with each other. Furthermore, the low-entropy
eigenstates with large overlap with the $|0\rangle$ show values of
$O_{22}^m$ closer to the ETH line compared to their counterpart for
the $|\mathbb{Z}_2\rangle$ state as can be clearly seen from Fig.\
\ref{fig3} (b); thus we expect $O_{22}^{\rm DE}$ starting from the
$|0\rangle$ state to be closer to the ETH value compared to its
$|\mathbb{Z}_2\rangle$ counterpart. Nevertheless, a finite number of
these eigenstates contributing to the $|0\rangle$ dynamics are not
thermal as can be seen from Fig.\ \ref{fig3} (a). They have
significantly lower values of $S_{L/2}$ compared to the eigenstates
in the thermal band, and lead to persistent coherent oscillatory
dynamics of $O_{22}$ starting from the $|0\rangle$ state. We
therefore dub these states as $|0\rangle$ scars. Our findings
indicate that there are at least two distinct types of scars in the
Floquet spectrum of $H_{\rm spin}$ driven by the square pulse
protocol given in Eq.\ \eqref{protocol2}; this phenomenon has no
analog in the PXP model studied earlier where only
$|\mathbb{Z}_2\rangle$ scars exist. We further note that the energy
spacings between these $|0\rangle$ scar states are non-uniform
unlike their $|\mathbb{Z}_2\rangle$ counterparts; this causes a
strong beating phenomenon in the oscillation of $O_{22}$ (Fig.\
\ref{fig2a} (b)) which is much weaker for the corresponding
$|\mathbb{Z}_2\rangle$ dynamics (Fig.\ \ref{fig2b} (b)). This can be
more clearly seen in Fig.\ \ref{FT} where the Fourier transform of
$O_{22}$ starting from $|0\rangle$ (Fig.\ \ref{fig2a} (b)) and from
$|\mathbb{Z}_2\rangle$ (Fig.\ \ref{fig2b} (b)) are shown in Fig.\
\ref{FT} (a) and Fig.\ \ref{FT} (b) respectively.
As $\omega_D$ is increased, we find that the $|0\rangle$ scars merge
with the thermal band and cannot be distinguished from them for
$\omega_D>12$ where $O_{22}$ starts displaying thermal behavior
consistent with ETH (Fig.\ \ref{fig2a} (a)); in contrast, the
$|\mathbb{Z}_2\rangle$ scars persist at arbitrary high frequency
(Fig.\ \ref{fig2b} (a)). This clearly demonstrates that the
$|0\rangle$ scars require higher spin terms in $H_F$ such as the
first term of Eq.\ \eqref{fl3}; they are not eigenstates of the high
frequency Floquet Hamiltonian which constitutes a renormalized PXP
model. Finally, it is important to note here that the $|0\rangle$
scars have a higher entanglement entropy compared to their
$|\mathbb{Z}_2 \rangle$ counterparts (Fig.\ \ref{fig3} (a)) and thus
they may be more fragile to increasing system sizes.
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain4a.pdf}}\\
{\includegraphics*[width=\linewidth]{figmain4b.pdf}}
\caption{(a) Plot of $S_{L/2}$ for the eigenstates of $H_F$ for $L=26$ and
$\omega_D =8.5$ at $\lambda=15$. The eigenstates with overlap $>0.01$
with $|0\rangle ~(|\mathbb{Z}_2\rangle)$ are shown using red (green) circles.
These states are distinct and coexist at this drive frequency.
(b) Plot of $O_{22}^m$ as a function of Floquet eigenstate
quasienergies $E_F$. The violet dashed line indicates the ETH
predicted value of $O_{22}$ at a temperature $T_0 (E_F)$. All
units are the same as in Fig.\ \ref{fig1}.} \label{fig3} \end{figure}
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain5.pdf}} \caption{Plots of
$|O_{22} (\omega)|^2$ obtained from the Fourier transforms of $O_{22} (n)$
for $L=26$, $\omega_D=8.5$ and $\lambda=15$ for the initial state being
(a) $|0\rangle$ and (b) $|\mathbb{Z}_2 \rangle$ respectively. All units are
the same as in Fig.\ \ref{fig1}.} \label{FT} \end{figure}
The role of the three or higher-spin terms in $H_F$ for the
stability of $|0\rangle$ scars and the consequent coherent
oscillations can be qualitatively understood using the FSA. To this end,
we consider an effective Hamiltonian
\begin{eqnarray} H_1= -\sum_{j} \tilde \sigma^x_j + h \sum_{j} ( \tilde
\sigma^+_{j} \tilde \sigma^-_{j-1} \tilde \sigma^-_{j+1} + {\rm H.c.}),
\label{fsaham1} \end{eqnarray}
which qualitatively mimics $H_F$ found in Sec.\ \ref{floquet},
albeit with real valued coefficient $h$. Here we use $h$ as a tuning
parameter and study the properties of the scar-induced oscillations
within the FSA starting from $|0\rangle$. For this, we write $H_1= H^+ + H^-$
(with $H^-= (H^+)^\dagger$) and choose $H^- = -w \sum_j \tilde \sigma^{-}_j
+ h \sum_j \tilde \sigma^+_{j} \tilde \sigma^-_{j-1} \tilde \sigma^-_{j+1}$
so that $H^-|0\rangle=0$. The repeated application of $H^+$ on
$|0\rangle$ (forward scattering) then generates a closed Krylov
subspace. Following standard procedure, we designate a particular
forward scattering step to be exact when the action of $H^+$ on the
Krylov vector in that step can be totally reversed by the action of
$H^-$. The inexact FSA steps generate errors which we aim to
minimize. The details of this analysis is charted out in App.\
\ref{appC}. The main results that come out of this analysis are as
follows. First, we find that in the bare PXP model ($H_1(h=0)$) all
forward scattering action are inexact after the first two FSA steps;
these errors cannot be minimized for $h=0$. This shows that the FSA
predicts instability of the $|0\rangle$ scars within the
PXP model. Second, we find that $h$ provides a control knob which
can minimize the FSA errors at different steps, although there is no
common value of $h$ for which errors in all the FSA steps are
simultaneously minimized. Our analysis finds that the errors for the
third FSA step (which is also the first error generating FSA step)
is minimized for $h^{\rm min} \simeq 0.3$; furthermore, errors in
other FSA steps are minimum close to (but not exactly at) $h=h^{\rm
min}$. The details of this procedure and the $L$ dependence of this
result is detailed out in App.\ \ref{appC}. Our analysis thus brings out
the importance of higher-spin terms in $H_1$ (and $H_F$) for
the stability of $|0\rangle$ scars. Finally, we find that the addition
of further terms such as a five-spin term to $H_1$ (see App.\
\ref{appC}) can lead to further amplification of scar-induced
oscillations and chart out the values of coefficients which achieves
such amplification. The Floquet Hamiltonian $H_F$ provides a natural
setting for generating such longer-ranged terms as the drive
frequency $\omega_D$ is lowered.
\subsection{Dynamical freezing}
\label{freez1}
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain6.pdf}} \caption{Plot of
$S_{L/2}$ for the eigenstates of $H_F$ for $L=26$ and
$\omega_D=7.8835$ at $\lambda=15$. The state encircled in red shows
the presence of a quantum scar with a very high overlap ($>0.9999$)
with the $|0\rangle$ state. All units are the same as in Fig.\
\ref{fig1}.} \label{figsinglescar} \end{figure}
For $\omega_D=7.88$, we find from Fig.\ \ref{fig2a} (c), that the
$|0\rangle$ state, in spite of not being an eigenstate of $H_F$,
does not exhibit almost any time evolution. This phenomenon is also
found for other lower, subharmonic, drive frequencies as seen in
Fig.\ \ref{fig1} (a) where $O_{22}^{\rm DE}$ exhibits a sharp dip to
$0$. This is in sharp contrast to the evolution of the
$|\mathbb{Z}_2\rangle$ state which shows thermalization at this
frequency (Fig.\ \ref{fig2b} (c))~\cite{scarfl1}. This behavior
constitutes an example of dynamical freezing which we now discuss.
At these dynamic freezing frequencies, a quantum scar state with
vanishingly small entanglement has an almost perfect overlap ($>
0.9999$ till $L=26$) with the $|0\rangle$ state as shown by the
behavior of $S_{L/2}$ in Fig.\ \ref{figsinglescar}.
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain7.pdf}}
\caption{Plot of the norm ${\mathcal N}$ as a function of $\omega_D$. Here
$\lambda=15$ and all units are the same as in Fig.\ \ref{fig1}. See text for
details.} \label{fig4} \end{figure}
We first obtain a qualitative understanding of this phenomenon using
the Floquet Hamiltonian given by Eqs.\ \eqref{fl1} and \eqref{fl3}. To
this end we note that $\tilde \sigma_j^- |0\rangle = 0$ for any $j$.
Thus the first term in Eq.\ \eqref{fl3} (and any higher order terms in
$H_F$ which have $\tilde \sigma_j^-$) annihilates the state
$|0\rangle$. Consequently, the only non-trivial terms in $H_F$
contributing to the evolution of the $|0\rangle$ state are the single
spin terms charted out in Eqs.\ \eqref{fl1} and \eqref{fl3}. These single
spin terms can be written as
\begin{eqnarray} H_F^{\rm single} &=& -\sum_j \Big[\left(w
\frac{\sin(2\gamma)}{2\gamma} +
6 {\rm Re}[A_0] \right) \tilde \sigma_j^x \nonumber\\
&& \quad \quad \quad + \left(w \frac{\sin^2 \gamma}{\gamma} - 6 {\rm Im}[A_0]
\right) \tilde \sigma^y_j\Big] \nonumber\\
&=& \sum_j ~(C_1 \tilde \sigma_j^x + C_2 \tilde \sigma_j^y). \end{eqnarray}
For drive frequencies where the norm ${\mathcal N}=
\sqrt{C_1^2+C_2^2}$ of these terms is close to zero, we expect
$|\psi(T)\rangle =U(T,0)|0\rangle \simeq |0\rangle$. We find, as
shown in Fig.\ \ref{fig4}, that ${\mathcal N}$ comes very close to zero
(although it does not vanish, in contrast to exact numerics) around
$\omega_D \simeq 7.9$ which is remarkably close to the
freezing frequency observed in exact numerics.
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain8.pdf}}
\caption{Plot of $\Delta_0/2$ (black solid line) and the highest freezing
frequency $\omega_D^{\rm freeze}$ (red solid line) as a function of $L$. Here
$\lambda=15$ and all units are the same as in Fig.\ \ref{fig1}.} \label{fig5}
\end{figure}
The above perturbative analysis indicates that multiple spin terms
do not play an essential role in the freezing phenomenon since, at
least to $O(w^3)$, all of them annihilate the $|0\rangle$ state.
Thus it is natural to expect that this phenomenon can also be
qualitatively understood by focussing on small system sizes where
the small size of the Hilbert space allows for an exact analytical
calculation. To this end, we consider a $L=3$ system. In the $k=0$
sector, there are two states in its constrained Hilbert space. These
are $|0\rangle=|\downarrow \downarrow \downarrow \rangle$, and
$|1\rangle = (|\uparrow \downarrow \downarrow \rangle + |\downarrow
\uparrow \downarrow \rangle + |\downarrow
\downarrow \uparrow \rangle)/\sqrt{3}$. In the space of these states,
the Hamiltonian can be written, up to an irrelevant constant term, as
\begin{equation} H[\pm\lambda] =\left(\begin{array}{cc}
0 & -\sqrt{3}w\\
-\sqrt{3}w & \pm \lambda\\
\end{array}\right). \end{equation}
The Floquet Hamiltonian for this system can be computed exactly and is given by
\begin{eqnarray} H_{F}^{\rm exact}&=& \frac{ic}{T\sin (c)}\Big[ \frac{2\sqrt{3}
w \sin(\Delta_0 T/2)}{\Delta_0} \tau_x \nonumber\\
&& \quad \quad \quad \quad -\frac{4\sqrt{3}w\lambda\sin^2(\Delta_0 T/4)}{
\Delta_0^2} \tau_y \Big], \end{eqnarray}
where $\tau^{x,y}$ denotes Pauli matrices in the space of states
$|0\rangle$ and $|1\rangle$, $\Delta_0= \sqrt{12 w^2+\lambda^2}$ is
the static energy gap between the states $|0\rangle$ and $|1\rangle$
states, and $c=\cos^{-1} (1-24 w^2\sin^2(\Delta_0 T/4)/\Delta_0^2)$.
This leads to the expression for the matrix element between the
states $|0\rangle$ and $|1\rangle$ as
\begin{eqnarray} |\bra{0}H_F^{\rm exact} \ket{1}|= \frac{\omega_D}{2\pi}
\cos^{-1}\left[\frac{\lambda ^2+12 w^2 \cos \left(\Delta_0 T/2\right)}{
\Delta_0^2} \right]. \label{2state1} \end{eqnarray}
This shows that for $\omega_D= \omega_D^{\rm freeze}=\Delta_0/(2m)$,
where $m$ is an integer, the matrix element between $|0\rangle$ and
$|1\rangle$ exactly vanishes. Consequently, $|0\rangle$ does not
evolve at these frequencies. Thus the freezing frequencies are directly
related to the static gap $\Delta_0$ between the $|0\rangle$ and the
single up-spin ($|1\rangle$) states. We note that our analytic
expression for $\omega_D^{\rm freeze}$ provides a natural explanation
for the subharmonic structure of the lower freezing
frequencies. These frequencies turn out to match almost exactly with
ED based numerical computation for finite $L \le 26$. This is shown
in Fig.\ \ref{fig5} where $\Delta_0/2$ and the highest
$\omega_D^{\rm freeze}$ ($m=1$) is plotted as a function of $L$. The
reason for this near-perfect match is that multiple spin terms in
$H_F$ do not contribute to this phenomenon as explained earlier. We
also supplement this numerical check by an explicit analytic
calculation for $L=4$ in App.\ \ref{appB}.
\subsection{Sub-thermal steady state value}
\label{sub1}
In this section, we address the sub-thermal behavior of the system
as seen, for example, in the frequency range $ 5 \le
\omega_D \le 7.5$. Throughout this range, the dynamics
of the system, starting from the $|0\rangle$ state, is qualitatively
identical to that shown in Fig.\ \ref{fig2a} (d). It shows a rapid
approach to a steady state where $O_{22}$ assumes a sub-thermal
value; in addition, there are no persistent coherent oscillations, in
contrast to the $|\mathbb{Z}_2\rangle$ dynamics shown in Fig.\ \ref{fig2b} (d).
This behavior constitutes a novel route to a violation of ETH in
finite-sized systems for two reasons. First, we do not see here the
persistent oscillations usually seen in dynamics controlled by quantum scars,
and second, $O_{22}$ assumes sub-thermal, in contrast to
super-thermal, values in the steady state.
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain9a.pdf}}\\
{\includegraphics*[width=\linewidth]{figmain9b.pdf}}
\caption{(a) Plot of $S_{L/2}$ for the eigenstates of $H_F$
for $L=26$ and $\omega_D =7.26$. The eigenstates with overlap
$>0.01$ with $|0\rangle ~(|Z_2\rangle)$ are shown using red (green) circles.
(b) Plot for $O_{22}^m$ as a function of Floquet eigenstate quasienergies
$\epsilon_F$. The violet dashed line indicates the ETH predicted value of
$O_{22}$ at a temperature $T_0 (E_F)$. Here $L=26$, $\lambda=15$ and
all units are the same as in Fig.\ \ref{fig1}.} \label{fig6} \end{figure}
To understand this phenomenon, we plot $S_{L/2}$ for the eigenstates
of $H_F$ for $\omega_D = 7.26$ in Fig.\ \ref{fig6} (a) and
$O_{22}^m$ as a function of Floquet quasienergies in Fig.\
\ref{fig6} (b). From both these panels, we note that there are
relatively few sub-thermal states with high overlaps with
the $|0\rangle$ state; in particular, there are no thermal
states with overlap $|\langle \chi_m|0\rangle|^2
>0.01$ at this frequency. Thus the weight of the
$|0\rangle$ state is distributed among a few sub-thermal and a
relatively large set of thermal states. The dynamics starting from
the $|0\rangle$ state within this frequency range is analogous to a
quantum system coupled to a bath which features a large range of
incommensurate natural frequencies; the presence of a large number
of thermal states which have small but finite overlaps with
$|0\rangle$ mimics the effect of a bath in the present context. The
presence of such a bath leads to fast decoherence of the oscillations
and leads to a steady state. This behavior is in stark contrast to
the dynamics at $\omega_D=8.5$ where a few sub-thermal states with
large overlaps control the dynamics.
The sub-thermal values of $O_{22}$ in this steady state are more
difficult to explain. Numerically, from Fig.\ \ref{fig6} (b), we
find that the Floquet eigenstates with relatively large overlaps with
the $|0\rangle$ state have sub-thermal values of $O_{22}^m$, and this
feature is opposite to that for states with large overlaps with
$|\mathbb{Z}_2\rangle$. From Eq.\ \eqref{decorr1}, we expect that
this feature will lead to sub-thermal values of $O_{22}$ in the
steady state. However, beyond this observation, we do not have a
more analytical explanation for this phenomenon. We also note that
such sub-thermal values of $O_{22}$ in the steady state are clearly
a finite-size effect; for $L \to \infty$, the number of thermal
Floquet eigenstates with finite overlap with $|0\rangle$ will be
exponentially larger than the sub-thermal states and their
contribution is expected to lead to the ETH predicted thermal value
of $O_{22}$ in the steady state. However, for all $L \le 26$ we do
not find thermal behavior; moreover for this range of system sizes,
the steady state value of $O_{22}$ remains almost constant as can be
seen from the values of $O_{22}^{DE}$ for $L=14, ~20,\,{\rm and}\,
24$ in Fig.\ \ref{fig1}. This indicates that a restoration of ETH is
expected only for $L \gg 26$; for $L\le 26$, we find a qualitatively
distinct and experimentally discernible characteristic of $O_{22}$
which is different from both scar-induced persistent oscillations
and ETH predicted thermalization. Furthermore, the behavior of
$O_{22}$ as a function of $n$ upto $1000$ drive cycles for different
system sizes (Fig.\ \ref{subthermal}) suggests that this non-ETH
sub-thermal behavior can persist as a prethermal regime for
reasonably large $n$ before the system eventually flows to an ITE
even for much larger system sizes. We therefore believe that this
phenomenon provides a novel route to ETH violation in a finite-sized
Rydberg chain.
\begin{figure}
{\includegraphics*[width=\linewidth]{figmain10.pdf}
\caption{Plots of $O_{22}$ as a function of the number of drive cycles $n$
(stroboscopic time) starting from an initial state $|0\rangle$ for $L=20$
(green), $L=24$ (red), and $L=26$ (black), for $\omega_D=7.26$ and $\lambda
=15$. The blue dashed line indicates the ETH predicted thermal value of
$O_{22}$. All units are the same as in Fig.\ \ref{fig1}.} \label{subthermal}
\end{figure}
\section{Discussion}
\label{diss}
In this work, we have studied the dynamics of a periodically driven
Rydberg chain starting from the $|0\rangle$ state using a square
pulse protocol. Our study involves exact numerics on finite-sized
chains and a perturbative Floquet Hamiltonian based analysis whose
analytic expression is derived using Floquet perturbation theory in
the high drive amplitude limit.
Our study indicate three distinct behaviors of such dynamics. First,
we show that dynamics starting from the $|0\rangle$ state can
exhibit scar-induced persistent oscillations over a range of drive
frequencies $8 \le \omega_D \le 12$. These scars are distinct from
their $\mathbb{Z}_2$ counterparts; they are absent in the high drive
frequency limit and are not eigenstates of the (renormalized) PXP
Hamiltonian studied earlier in the literature~\cite{scarfl1,scarref1,
scarref2a,scarref2b,scarref2c,scarref2d,scarref2e}. They coexist
with the $|\mathbb{Z}_2\rangle$ scars in the above-mentioned drive
frequency range. These $|0\rangle$ scars have higher entanglement
than their $|\mathbb{Z}_2\rangle$ counterparts and seem more fragile
to increasing system sizes. It will be worthwhile to understand
perturbations that may further reduce the entanglement of these
$|0\rangle$ scars.
Second, for specific drive frequencies $\omega_D^{\rm freeze} \simeq
7.88, 3.94, \cdots$, we find that the system exhibits dynamic freezing.
We provide an analytic, albeit qualitative, explanation of this
phenomenon using the perturbative Floquet Hamiltonian and supplement
it with exact analytic calculation at small system sizes $L=3$ and 4.
Our analysis relates the freezing frequencies with the energy gap
$\Delta_0$ between the $|0\rangle$ and $|1\rangle$ states:
$\omega_D^{\rm freeze}=\Delta_0/2m$ for $m=1,2,3,\cdots$. This
provides a natural explanation for the relation between several
freezing frequencies and also provides a reasonably accurate
estimate of these frequencies as can be seen by comparing the
analytic result with exact numerics for all $L \le 26$. We note that
such a behavior has no analogue for dynamics starting from the
$|\mathbb{Z}_2\rangle$ state. Such dynamical freezing should also be
visible in the thermodynamic limit as prethermal freezing which only
flows to an ITE after an exceptionally long time scale.
Third, we show that for $5 \le \omega_D \le 7.5$, the system reaches
a steady state with sub-thermal values of $O_{22}$ for all $L \le
26$ which constitutes a violation of ETH. In contrast to the
scar-induced weak violation of ETH, in this regime, the system does
not exhibit persistent oscillations for $O_{22}$. We relate this
behavior to the presence of small overlaps of a large number of
Floquet eigenstates with the $|0\rangle$ state; this leads to the
fast decay of coherent oscillations in $O_{22}$. Moreover,
numerically we find that $O_{22}^m$ for many of the Floquet
eigenstates which have high overlap with $|0\rangle$ assumes
sub-thermal values; this leads to sub-thermal values of $O_{22}$ in
the steady state. We note that such a violation of ETH is distinct
from its counterpart in the dynamics starting from the $|\mathbb{Z}_2\rangle$
state; it does not feature persistent oscillations
and leads to steady states with sub-thermal, rather than
super-thermal, values of $O_{22}$. Our numerical results for
finite-sized chains also suggest that this sub-thermal behavior can
survive as a prethermal phase for finite but large number of drive
cycles before the system eventually flows to an ITE in the
thermodynamic limit.
All the above three features should be observable in realistic
experiments with a Rydberg chain. The differences of our proposal with
experiments already carried out in Ref.\ \onlinecite{scarref1} are
two-fold. First, for our proposal, we need to start from the
$|0\rangle$ state. This is not difficult to implement since this
state turns out to be the ground state of $H$ in Eq.\ \eqref{ryd1} for
large positive $\Delta$. Second, we need to implement a periodic
variation of $\Delta$ according to the protocol given in Eq.\ \eqref{protocol1}
instead of a quench. Our prediction is that the dynamics starting from the
$|0\rangle$ state will show persistent scar-induced oscillations, dynamic
freezing and novel steady states featuring sub-thermal values of $C_{22} \equiv
O_{22}$ in such experiments.
In conclusion, we have shown that dynamics starting from the
$|0\rangle$ state in a finite-sized periodically driven Rydberg
chain shows scar-induced oscillations, dynamic freezing, and steady
states with sub-thermal value of correlators for various ranges of
drive frequencies which we have charted out. The first feature shows
that the Floquet Hamiltonian hosts two sets of coexisting scars; the
last two phenomena have no analogs in periodic or quench dynamics
involving the $|\mathbb{Z}_2\rangle$ initial state studied earlier. We have
provided an analytic, albeit perturbative, Floquet Hamiltonian which
explains these features qualitatively and have suggested experiments
which can test our theory.
On a broader level, our work suggests the possibility of interesting
prethermal Floquet phases at moderate and low drive frequencies in
the high drive amplitude limit~\cite{Vajna2018}. Such a
prethermalization mechanism is quite distinct from the well-known
long preheating times generated in Floquet systems when the driving
frequencies are much bigger than the local energy
scales~\cite{Saito_etal} and should lead to richer possibilities.
\begin{acknowledgments}
The work of A.S. is partly supported through
the Max Planck Partner Group program between the Indian Association
for the Cultivation of Science (Kolkata) and the Max Planck
Institute for the Physics of Complex Systems (Dresden). D.S. thanks
DST, India for Project No. SR/S2/JCB-44/2010 for financial support.
\end{acknowledgments}
|
1,116,691,498,501 | arxiv | \section*{}
\section{Introduction}
The one-time pad, also known as the Vernam Cipher, invented and
patented\cite{verpat,perfcpr} by Gilbert Vernam in 1917, is a
perfect cipher.\cite{verpat,perfcpr,otp} The core idea is that any message
can be transformed into any cipher (of the same length) by a pad, such that all
transformations are equally likely. The one-time pad encryption scheme is
provably unbreakable if (i) the key is at least the size of the plaintext,
(ii) the key is truly random, and(iii) the key is used only once. The requirement
of only one time key usage makes the one-time pad impractical when the amount of
date to be securely communicated is huge.
The more practical ciphers are the block ciphers, like DES, 3DES, AES\cite{aes},
which operate on a block of message depending on the key size.
A given key is reused for mutiple message blocks. This makes the cipher prone to various kinds
of attacks, like the cloning attack, cryptanalysis attack, chosen ciphertext/plaintext attack etc.
The security of these ciphers depend mainly upon the strength of the key
(randomness) and the algorithm used for encryption and decryption in terms of
{\em confusion} and {\em diffusion} created.
In 1984, Charles H. Bennett and Gilles Brassard described the first completely
secure quantum key distribution algorithm often known as the BB84 algorithm.\cite{qkd}
And in 1997, Peter Shor\cite{qfac} provided a quantum algorithm that can break
the widely used RSA system, using quantum computers, with remarkable ease.
Together the developments showed that a completely secure, efficient, and fast
means of delivering confidential information is achievable using the laws of
quantum mechanics and quantum computers.
Ciphers based on quantum states, therefore, would be more appropriate because
of the property that an unknown state cannot be copied. Hence, none of the
attacks in classical cryptography would be applicable here. In recent times
there are many attempts and constructions for quantum ciphers. One such
attempt was made by Abdullha, A. A. et. al.\cite{qrng}, where they used quantum
random number generator and half adder for encryption and transmit with the
help of BB84 protocol.
In this paper, a quantum version of the cipher is proposed, which
utilizes the benefit of superior delivery efficiency provided by modern
telecommunication, and snoop-detection capability of the BB84 algorithm.
The quantum version exploits the fact that (1) an unknown quantum state
cannot be cloned and (2) its relative phase cannot be measured. The message
to be sent is encoded into a quantum state by altering the relative phase
using a pre-established shared key, via BB84 or any other quantum key
distribution protocol. The information transmitted is a quantum
superposition state with uniform probability and relative phase distribution.
\section{The algorithm}
The proposed algorithm requires a pre-established shared key between the
communicating parties, which can be achieved by the BB84 or similar QKD
(Quantum Key Distribution)
protocol. It is assumed that QKD is secured enough and it would be hard
to know any information about key.
The components of the proposed cryptosystem (encoder and decoder)
\begin{enumerate}
\item Hadamard transformation.
\item Oracle, $\Lambda_{k}$ defined in section 1.2, responsible for key state phase inversion.
\item Another oracle, $\Upsilon_{k,d}$ defined in section 1.3, for multiple phase inversion.
\end{enumerate}
are captured in the following Figure 1.\\
\begin{tikzpicture}
\draw (0,1.1) node {message};
\draw (0,.8) node {$|m\rangle$};
\draw (1,2.2) -- (1.5,2.2);
\draw (1,2.0) -- (1.5,2.0);
\draw (1,1.8) -- (1.5,1.8);
\draw (1,1.6) -- (1.5,1.6);
\draw (1,1.4) -- (1.5,1.4);
\draw (1,1.2) -- (1.5,1.2);
\draw (1,1.0) -- (1.5,1.0);
\draw (1,.8) -- (1.5,.8);
\draw (1,.6) -- (1.5,.6);
\draw (1,.4) -- (1.5,.4);
\draw (1,.2) -- (1.5,.2);
\draw (1.5,0) rectangle (2.5,2.5) node [midway=center] {$H^{\otimes n}$};
\draw (2.5,2.2) -- (3,2.2);
\draw (2.5,2.0) -- (3,2.0);
\draw (2.5,1.8) -- (3,1.8);
\draw (2.5,1.6) -- (3,1.6);
\draw (2.5,1.4) -- (3,1.4);
\draw (2.5,1.2) -- (3,1.2);
\draw (2.5,1.0) -- (3,1.0);
\draw (2.5,.8) -- (3,.8);
\draw (2.5,.6) -- (3,.6);
\draw (2.5,.4) -- (3,.4);
\draw (2.5,.2) -- (3,.2);
\draw (3,0) rectangle (6,2.5) node [midway=center] {\begin{tabular}{c} Key Phase \\ Inversion \\ $\Lambda_{k}$\end{tabular}};
\draw (6,2.2) -- (6.5,2.2);
\draw (6,2.0) -- (6.5,2.0);
\draw (6,1.8) -- (6.5,1.8);
\draw (6,1.6) -- (6.5,1.6);
\draw (6,1.4) -- (6.5,1.4);
\draw (6,1.2) -- (6.5,1.2);
\draw (6,1.0) -- (6.5,1.0);
\draw (6,.8) -- (6.5,.8);
\draw (6,.6) -- (6.5,.6);
\draw (6,.4) -- (6.5,.4);
\draw (6,.2) -- (6.5,.2);
\draw (6.5,0) rectangle (9.5,2.5) node [midway=center] {\begin{tabular}{c} Multiple Phase \\ Inversion \\ $\Upsilon_{k.d}$\end{tabular}};
\draw (9.5,2.2) -- (10,2.2);
\draw (9.5,2.0) -- (10,2.0);
\draw (9.5,1.8) -- (10,1.8);
\draw (9.5,1.6) -- (10,1.6);
\draw (9.5,1.4) -- (10,1.4);
\draw (9.5,1.2) -- (10,1.2);
\draw (9.5,1.0) -- (10,1.0);
\draw (9.5,.8) -- (10,.8);
\draw (9.5,.6) -- (10,.6);
\draw (9.5,.4) -- (10,.4);
\draw (9.5,.2) -- (10,.2);
\draw (10.5,1.0) node {$|\psi_{c}\rangle$};
\end{tikzpicture}
\begin{tikzpicture}
\draw (0,1.2) node {$|\psi_{c}\rangle$};
\draw (.5,2.2) -- (1,2.2);
\draw (.5,2.0) -- (1,2.0);
\draw (.5,1.8) -- (1,1.8);
\draw (.5,1.6) -- (1,1.6);
\draw (.5,1.4) -- (1,1.4);
\draw (.5,1.2) -- (1,1.2);
\draw (.5,1.0) -- (1,1.0);
\draw (.5,.8) -- (1,.8);
\draw (.5,.6) -- (1,.6);
\draw (.5,.4) -- (1,.4);
\draw (.5,.2) -- (1,.2);
\draw (1,0) rectangle (4,2.5) node [midway=center] {\begin{tabular}{c} Multiple Phase \\ Inversion \\ $\Upsilon_{k,d}$\end{tabular}};
\draw (4,2.2) -- (4.5,2.2);
\draw (4,2.0) -- (4.5,2.0);
\draw (4,1.8) -- (4.5,1.8);
\draw (4,1.6) -- (4.5,1.6);
\draw (4,1.4) -- (4.5,1.4);
\draw (4,1.2) -- (4.5,1.2);
\draw (4,1.0) -- (4.5,1.0);
\draw (4,.8) -- (4.5,.8);
\draw (4,.6) -- (4.5,.6);
\draw (4,.4) -- (4.5,.4);
\draw (4,.2) -- (4.5,.2);
\draw (4.5,0) rectangle (7.5,2.5) node [midway=center] {\begin{tabular}{c} Key Phase \\ Inversion \\ $\Lambda_{k}$\end{tabular}};
\draw (7.5,2.2) -- (8,2.2);
\draw (7.5,2.0) -- (8,2.0);
\draw (7.5,1.8) -- (8,1.8);
\draw (7.5,1.6) -- (8,1.6);
\draw (7.5,1.4) -- (8,1.4);
\draw (7.5,1.2) -- (8,1.2);
\draw (7.5,1.0) -- (8,1.0);
\draw (7.5,.8) -- (8,.8);
\draw (7.5,.6) -- (8,.6);
\draw (7.5,.4) -- (8,.4);
\draw (7.5,.2) -- (8,.2);
\draw (8,0) rectangle (9,2.5) node [midway=center] {$H^{\otimes n}$};
\draw (9,2.2) -- (9.5,2.2);
\draw (9,2.0) -- (9.5,2.0);
\draw (9,1.8) -- (9.5,1.8);
\draw (9,1.6) -- (9.5,1.6);
\draw (9,1.4) -- (9.5,1.4);
\draw (9,1.2) -- (9.5,1.2);
\draw (9,1.0) -- (9.5,1.0);
\draw (9,.8) -- (9.5,.8);
\draw (9,.6) -- (9.5,.6);
\draw (9,.4) -- (9.5,.4);
\draw (9,.2) -- (9.5,.2);
\draw (9.5,0) rectangle (10.6,2.5);
\draw (10.6,1) arc (30:150:.6cm);
\draw [->,>=stealth] (10.1,.6) -- (10.3,1.5);
\draw (10.6,2.2) -- (10.8,2.2);
\draw (10.6,2.0) -- (10.8,2.0);
\draw (10.6,1.8) -- (10.8,1.8);
\draw (10.6,1.6) -- (10.8,1.6);
\draw (10.6,1.4) -- (10.8,1.4);
\draw (10.6,1.2) -- (10.8,1.2);
\draw (10.6,1.0) -- (10.8,1.0);
\draw (10.6,.8) -- (10.8,.8);
\draw (10.6,.6) -- (10.8,.6);
\draw (10.6,.4) -- (10.8,.4);
\draw (10.6,.2) -- (10.8,.2);
\draw (11.2,1.2) node {$|m\rangle$};
\end{tikzpicture}
The message $|m\rangle$ to be sent is first passed through a Hadamard transformation
to create an equal superposition state. Next, the phase inversion is applied to
invert the key state phase. Finally the multiple phase inversion transformation
(another Oracle) is applied to invert the phases of exactly half of the basis states.
The outcome of this is an encrypted quantum state $|\psi_{c}\rangle$ with a uniform
probability and relative phase distributions. The transmitted quantum state act as a
carrier for the message, analogous to the FM transmission, where the audio wave to be
transmitted is encoded in the frequency of a high frequency carrier wave.
The recipient can apply the transformation to $|\psi_{c}\rangle$ in reverse, the multiple
phase inversion followed by the key phase inversion and finally the Hadamard
transformation, to retrieve the original message $|m\rangle$.
\subsection{Hadamard transformation}
The Hadamard transformation, irrespective of the input, creates an equal
superposition state, i.e. a uniform distribution of all possible n-qubit states
of the message space, say {\em M}, of size $N = 2^{n}$.
\begin{equation}
|\psi_{m}\rangle = H^{\otimes n} (|m\rangle) = \frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} (-1)^{m.x}|x\rangle
\end{equation}
\subsection{Key phase inversion}
The phase inversion operator, ($\Lambda_{k}$), acts as an oracle and is defined as
\begin{equation}
\Lambda_{k} = I - 2| k\rangle\langle k|
\end{equation}
where $|k\rangle$ is the quantum key state derived from the shared key $k$ and
$I$ the identity operator.
The application of this operator on the input state $|\psi_{m}\rangle$, equation (1),
creates a coupling between the input state and the key state.
\begin{align}
|\psi_{c}^{'}\rangle = \Lambda_{k} |\psi_{m}\rangle & = I|\psi_{m}\rangle - 2\frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} (-1)^{m.x} |k\rangle\langle k|x\rangle \nonumber \\
& = |\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k}|k\rangle
\end{align}
This marking of the key state accomplishes the encoding.
The application of the same operator retrieves the input state $|\psi_{m}\rangle$,
\begin{align}
\Lambda_k |\psi_{c}^{'}\rangle &= (I - 2 |k\rangle\langle k|)(|\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k}|k\rangle)\nonumber \\
&= |\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k}|k\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k} |k\rangle + \frac{4}{\sqrt{N}} (-1)^{m.k} |k\rangle \nonumber \\
&= |\psi_{m}\rangle
\end{align}
\subsection{Multiple phase inversion}
The multiple phase inversion transformation, say $\Upsilon_{k,d}$, performs phase
inversion of multiple basis states as follows.
\begin{enumerate}
\item Select one $r$, where $r|N$ ($r$ divides $N$) and $r<\frac{N}{2}$. Now say,
$d = \frac{N}{r}$, where $N = 2^{n}$ for $n$ qubits.
\item Start with the key $k$ position of the state $|\psi_{c}^{'}\rangle$ and invert the
phases of next $d$ consecutive states, i.e. states $|(k + pd) \mod N\rangle$ to
$|(k + (p+1)d - 1) \mod N\rangle$. Skip the next $d$ states from
$|(k + (p+1)d) \mod N\rangle$ to $|(k + (p+2)d - 1) \mod N\rangle$. Here
$0 \le p \le \frac{N}{2d}$ and $p \in {N \cup {0}}$, here ${N}$ is set of natural
numbers.
\end{enumerate}
The transformation can be defined as,
\begin{equation}
|\psi_{c}\rangle = \Upsilon_{k,d} |\psi_{c}^{'}\rangle
\end{equation}
The distribution of inverted vs. non-inverted phase states ($|\psi_{c}\rangle$, being the complete superposed states) will vary
on each unique choice of key state and $d$. Hence, the guessing of the state distribution is not
possible in this construction. It can be easily visualized that with this algorithm,
phases of half of the basis states will be inverted. $'d'$ can be uniquely defined
for a given key and can be arrived at as part of the key exchange process.
\subsection{Security}
The key state phase inversion operator, $\Lambda_k$ defined by equation (2), when applied to the state
$|\psi_{m}\rangle$, resulted in the inversion of the key state $|k\rangle$ as given by equation (3).
It can be rewritten as follows,
\begin{align}
|\psi_{c}^{'}\rangle &= \Lambda_{k} |\psi_{m}\rangle \nonumber \\
& = |\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k}|k\rangle \nonumber \\
& = |\psi_{m-k}\rangle - \frac{1}{\sqrt{N}} (-1)^{m.k}|k\rangle
\end{align}
where $|\psi_{m-k}\rangle = |\psi_{m}\rangle - \frac{1}{\sqrt{N}}|k\rangle$ is the superposition
of all the basis states, except the key state.
The probability of the phase inverted key state be $P_{k} = \frac{1}{N}$ and each of
the remaining $(N-1)$ non-inverted states $(\forall x\in |x\rangle, k\notin x)$ be $P_{x} = \frac{1}{N}$.
The ratio of the probability of inverted and non-inverted states
\begin{equation}
\frac{P_{k}}{P_{x}} = \frac{N}{N} = 1
\end{equation}
is probabilistically indistinguishable in this case.
The adversary can only see the transmitted state as given equation {3}, he/she has the power to
apply Hadamard transform on the transmitted state. However, since adversary does not know the
shared key state $|k\rangle$, $\Lambda_k$ remains private, applying Hadamard transform cannot
return back the original message state $|\psi_{m}\rangle$. The adversary can only guess the
construction of $\Lambda_k$ in $O(\sqrt{N})$ running time.
Alternatively, adversary can set up chosen plaintext attack by apply inversion against mean
operator (Ref: Grover's inversion against mean) to check if it could leak some information,
or can guess any inherent biasness.
Let us introduce the inversion against mean operator defind as
$\mu_{m} = (2|\psi_{m}\rangle\langle \psi_{m}| - I)$, where $|\psi_{m}\rangle$ is total state.
The inner product of key state and total state is given by $\psi_{m}|k\rangle = \langle k|\psi_{m}\rangle = \frac{1}{\sqrt{N}}$.
The application of $\mu_{m}$ to $|\psi_{c}^{'}\rangle$ the will result in
\begin{align}
|\phi_{m}\rangle = \mu_{m} |\psi_{c}^{'}\rangle &= (2|\psi_{m}\rangle\langle \psi_{m}| - I)(|\psi_{m}\rangle - \frac{2}{\sqrt{N}}|k\rangle) \nonumber \\
&= 2|\psi_{m}\rangle\langle \psi_{m}|\psi_{m}\rangle - 2\frac{2}{\sqrt{N}}|\psi_{m}\rangle\langle \psi_{m}|k\rangle - (|\psi_{m}\rangle + \frac{2}{\sqrt{N}}|k\rangle) \nonumber \\
&= (1 - \frac{4}{N}) |\psi_{m}\rangle + \frac{2}{\sqrt{N}} |k\rangle
\end{align}
Let us choose one use case having message having all 0 i.e. $|\psi_{0}\rangle = H^{\otimes n} (|00...0\rangle)$.
Equation 8 can then be rewritten as:
\begin{align}
|\phi_{0}\rangle = (1 - \frac{4}{N}) |\psi_{0}\rangle + \frac{2}{\sqrt{N}} |k\rangle
\end{align}
To extract out non-inverted phase states, equation 6 is used in equation 9 and we get
\begin{align}
|\phi\rangle &= (1 - \frac{4}{N}) |\psi_{0}\rangle + \frac{2}{\sqrt{N}} |k\rangle \nonumber \\
&= (1 - \frac{4}{N}) |\psi_{0-k}\rangle + \frac{(1 - \frac{4}{N})}{\sqrt{N}}|k\rangle + \frac{2}{\sqrt{N}} |k\rangle \nonumber \\
&= (1 - \frac{4}{N}) |\psi_{0-k}\rangle + \frac{3N - 4}{N\sqrt{N}}|k\rangle
\end{align}
The probability of the phase inverted state is therefore given by $P_{k} = (\frac{3N - 4}{N\sqrt{N}})^{2}$
and each of the remaining $(N-1)$ non-inverted states ($\forall x\in |x\rangle , k\notin x$) by
$P_{x} = \frac{(1 - \frac{4}{N})^{2}}{N}$. Looking for the same probability ratio yields
\begin{align}
\frac{P_{k}}{P_{x}} &= \frac{(\frac{3N - 4}{N\sqrt{N}})^{2}}{\frac{(1 - \frac{4}{N})^{2}}{N}} \nonumber \\
&= (\frac{3N - 1}{N - 4})^{2} \nonumber \\
&= (\frac{3 - \frac{4}{N}}{1 - \frac{4}{N}})^{2}
\end{align}
which, for large $N$, will reduce to $\lim \frac{P_{k}}{P_{x}} \rightarrow 9$.
It is noticed that there is a biasness of probability distribution between phase inverted
state and non-inverted state. The adversary has the freedom to apply $\mu_{0}$ again on the
output of first $\mu_{0}$ operation, if doing so, it will be observed from the below result
that the encrypted transmitted state would be emerged.
\begin{align}
|\phi\rangle &= (2|\psi_{0}\rangle\langle \psi_{0}| - I)(1 - \frac{4}{N}) |\psi_{0}\rangle + \frac{2}{\sqrt{N}} |k\rangle \nonumber \\
&= (1 - \frac{4}{N}) |\psi_{0}\rangle + \frac{4}{N}|\psi_{0}\rangle - \frac{2}{\sqrt{N}} |k\rangle \nonumber \\
&= |\psi_{0}\rangle - \frac{2}{\sqrt{N}} |k\rangle
\end{align}
So, with the repeated use of $\mu_{0}$, would emerge the above alternative pattern (alternative
repeation of eqution 8 equation 12). Though, the adversary cannot be able to make out any
useful infomarion but it shows a little biasness in probability distribution of the inverted
key states over rest of the individual (non-inverted) message states. This violates the
Shannon's secrecry clause for encryption\cite{shan}.
In order to solve the biasness problem, we could phase invert $M$ states and $M > 1$. The
objective is to show that if $M = \frac{N}{2}$ the biasness can be eliminated and we could
show the inverted phase states and the rest of the states will be indistinguishable.
Multiple phase inversion is actually a chain of single phase inversion of $M$ times with
the corresponding phase inversion states are $|k_{1}\rangle, |k_{2}\rangle, ... |k_{m}\rangle$,
these states remain private between the communicating parties.
\begin{align}
&(I - 2| k_{1}\rangle\langle k_{1}|) |\psi_{m}\rangle = |\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k_{1}}|k_{1}\rangle \nonumber \\
&(I - 2| k_{2}\rangle\langle k_{2}|) (|\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k_{1}}|k_{1}\rangle)
= |\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k_{1}}|k_{1}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k_{2}}|k_{2}\rangle \nonumber \\
& ... \nonumber
\end{align}
similarly goes on up to $M$ states and the final would look like
\begin{align}
(I - 2| k_{m}\rangle\langle k_{m}|) (|\psi_{m}\rangle &- \frac{2}{\sqrt{N}} (-1)^{m.k_{1}}|k_{1}\rangle \nonumber\\
&- \frac{2}{\sqrt{N}} (-1)^{m.k_{2}}|k_{2}\rangle \nonumber\\
&... \nonumber \\
&- \frac{2}{\sqrt{N}} (-1)^{m.k_{m-1}}|k_{m-1}\rangle) \nonumber \\
&= |\psi_{m}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k_{1}}|k_{1}\rangle - \frac{2}{\sqrt{N}} (-1)^{m.k_{2}}|k_{2}\rangle \nonumber\\
&... - \frac{2}{\sqrt{N}} (-1)^{m.k_{m}}|k_{m}\rangle
\end{align}
Without any loss of generality, this can be expressed as
\begin{align}
|\phi\rangle = \Lambda_k|\psi_{m}\rangle &= |\psi_{m}\rangle - \frac{2}{\sqrt{N}} \sum_{i=0}^{M-1} (-1)^{m.k_{i}}|k_{i}\rangle \nonumber\\
&= |\psi_{m-k}\rangle - \frac{1}{\sqrt{N}} \sum_{i=0}^{M-1} (-1)^{m.k_{i}}|k_{i}\rangle
\end{align}
Now, the probability of the $M$ consolidated phase inverted key states is $P_{k} = M.(\frac{1}{\sqrt{N}})^{2}$ and
$(N-M)$ non-inverted states $P_{x} = (N - M).(\frac{1}{\sqrt{N}})^{2}$ and the ratio
\begin{equation}
\frac{P_{k}}{P_{x}} = \frac{M}{(N - M)}
\end{equation}
which reduces to $\frac{P_{k}}{P_{x}} = 1$ when $M = \frac{N}{2}$.
Thus proved that there is no biasness in probability distribution between $M$ phase inverted
states with the rest of the non-inverted states. Since adversary does not know the shared key
state $|k\rangle$, $\Lambda_k$ remains private, applying Hadamard transform on the transmitted
message will not reveal any information. The adversary, however can chose to apply inversion
against mean operator (Ref: Grover's inversion against mean) to launch 'chosen plaintext attack',
taking the similar argumental approach as used during single phase inversion analysis (changing
notation of $\psi_{m}$ to $\psi_{0}$),
\begin{align}
|\phi\rangle = \mu_{0} |\psi_{0}\rangle &= (2|\psi_{0}\rangle\langle \psi_{0}| - I)(|\psi_{0}\rangle - \frac{2}{\sqrt{N}} \sum_{i=0}^{M-1} (-1)^{m.k_{i}}|k_{i}\rangle \nonumber \\
&= 2|\psi_{0}\rangle\langle \psi_{0}|\psi_{0}\rangle - 2\frac{2}{\sqrt{N}}\sum_{i=0}^{M-1} (-1)^{m.k_{i}}|\psi_{0}\rangle\langle \psi_{0}|k_{i}\rangle - (|\psi_{0}\rangle + \frac{2M}{\sqrt{N}}|k\rangle) \nonumber \\
&= (1 - \frac{4M}{N}) |\psi_{0}\rangle + \frac{2}{\sqrt{N}} \sum_{i=0}^{M-1} (-1)^{m.k_{i}}|k_{i}\rangle
\end{align}
Since $|\psi_{0}\rangle = \frac{N - M}{\sqrt{N}}|\psi_{0-k}\rangle + \frac{M}{\sqrt{N}}|k\rangle$,
to extract inverted and non-inverted phase states, equation 16 can be re-written as:
\begin{align}
|\phi\rangle &= (1 - \frac{4M}{N}) |\psi_{0}\rangle + \frac{2}{\sqrt{N}} \sum_{i=0}^{M-1} (-1)^{m.k_{i}}|k_{i}\rangle \nonumber\\
&= (1 - \frac{4M}{N})|\psi_{m-k}\rangle + ((1 - \frac{4M}{N})\frac{1}{\sqrt{N}} + \frac{2}{\sqrt{N}})\sum_{i=0}^{M-1} (-1)^{m.k_{i}}|k_{i}\rangle
\end{align}
The probability of consolidated $M$ phase inverted states is then
$P_{k} = M((1 - \frac{4M}{N}).\frac{1}{\sqrt{N}} + \frac{2}{\sqrt{N}})^{2}$ and for the
$N-M$ non-inverted states $P_{x} = (N - M).((1 - \frac{4M}{N}).\frac{1}{\sqrt{N}})^{2}$.
The ratio of probabilities is thus
\begin{equation}
\frac{P_{k}}{P_{x}} = \frac{M((1 - \frac{4M}{N}).\frac{1}{\sqrt{N}} + \frac{2}{\sqrt{N}})^{2}}{ (N - M).((1 - \frac{4M}{N}).\frac{1}{\sqrt{N}})^{2}}
\end{equation}
and when $M = \frac{N}{2}$, it reduces to
\begin{equation}
\frac{P_{k}}{P_{x}} = |\frac{\frac{N}{2}}{\frac{N}{2}}| = 1
\end {equation}
With this construction, we can show that again, the relative probability distribuion is not altered by no mens and hence no biasness.
The above algorithm is secure against 'chosen plaintext' attack. Even if there will
be single phase inversion, being phase inversion operator as private, key is safe to use
$O(\sqrt{N})$ times in a session. The transmitted message is an equal superposition
state with some co-relation to the key $k$. Without the knowledge of the key
nothing can be inferred about the message ${\it M}$.
To make it completely hardened, approximately half of the total phases of the
Hadamard transformed message state should be inverted.
\subsection{Steps of the algorithm}
The algorithm has the following steps:
\begin{enumerate}
\item Begin with the BB84 (Bennett and Brassard) quantum key distribution (QKD)
method to establish a shared key $k$ and $d$ between two communicating parties
(say A and B). Let the key state be $|k\rangle$.
\item To each message, $m \in M$, 'A' will apply the Hadamard transform to create
an equal superposition state $|\psi_{m}\rangle$.
\[|\psi_{m}\rangle = H^{\otimes n} (|m\rangle) = \frac{1}{\sqrt{N}} \sum_{x=0}^{N-1} (-1)^{m.x}|x\rangle\]
\item Apply the operator $\Lambda_{k}$ to $|\psi_{m}\rangle$ to mark the key state.
\[ \psi_{c}^{'} = \Lambda_{k} |\psi_{m}\rangle = |\psi_{m}\rangle - 2|k\rangle \]
\item Apply the multiple phase inversion operator $\Upsilon_{k,d}$ to invert
phases of multiple basis states.
\[ |\psi_{c}\rangle = \Upsilon_{k,d} |\psi_{c}^{'}\rangle \]
\item Send the resulting encrypted quantum state $|\psi_{c}\rangle$ to 'B' using a quantum channel.
\item 'B' will perform the reverse operation to retrieve the message $m$ from $|\psi_{c}\rangle$.
\end{enumerate}
\subsection{Application of proposed cipher}
Some applications of the proposed cipher construction are as follow.
\subsubsection{Authentication}
Anyone can utilize this algorithm to encode and send his/her signature
(public identity) as the message. The intended peer can decode the signature
and verify against the known one. Any tampering of the message would result
in a different signature, i.e. only a entity with the share key in
possession can generate the encoded signature. For authenticated encrypted
message however, total message length will be $2n$ with one part of $n$
bearing the identity for signature verification.
\subsubsection{Quantum Teleportation}
This can be used during quantum teleportation. 'Alice' no longer needs to use
phone or email to communicate 'Bob' her state of operation. Instead, she can
send our cipher to Bob and Bob can 'decrypt' to get the message what Alice had
performed and act accordingly to get the teleported message. Thus, we can
eradicate all classical entities involved in quantum teleportation.
\subsubsection{Rekey}
This can be used to refresh the shared key established by the BB84 QKD. The
newly generated random key $|k_{1}\rangle$ can be communicated to the peer as a
message in our cipher construction using the existing key $|k\rangle$. Aferwards,
$|k_{1}\rangle$ will be the new key and will be used for next set of message
encryption and decryption. In the whole process, BB84 QKD protocol is used only
once.
\section*{Conclusion}
The proposed quantum cipher is proved mathematically secured against known
attacks (more relevant in the current context 'chosen plaintext attack')
and can be versatile in application. The requirement of a well secure cipher,
namely diffusion and confusion, is satisfied by the Hadamard and multiple phase
inversion transformations respectively. The same idea can be extended and
similar approach can be used for multi party (multi peers) commuication securely.
\section*{References}
|
1,116,691,498,502 | arxiv | \section*{Executive Summary}
\input{sections/executive_summary}
\section{Motivation}
\input{sections/motivation}
\section{Goal}
\label{sec:goal}
\input{sections/goal}
\section{Status of Current Initiatives}
\input{sections/current_initiative}
\section{Deliverable}
\label{sec:deliv}
\input{sections/deliverable}
\section{Business Model Requirements}
\label{sec:bm}
\input{sections/business_model}
\section{Mutual Impacts between HEP and Microelectronics Industry}
\label{sec:mutual benefits}
\input{sections/mutual_benefits}
\bibliographystyle{JHEP}
\subsection{Meetings with CAD Companies}
DOE Office of High Energy Physics hosted initial meetings with major CAD and EDA tool companies including Ansys, Cadence, Google, Keysight, Siemens, and Synopsys in 2021. Business collaboration models and recommendations are presented and discussed. Here's a summary of major recommendations:
\begin{itemize}
\item Consider the concept of a DOE Collaborative Innovation Hub scoped for cooperative across the team shared access to CAD/EDA tools, training, and support.
\item Establish a dedicated cloud-based communal participation between academia, DOE national labs, and CAD/EDA companies.
\item Leverage successful solution frameworks (e.g. DARPA Innovation Package, Europractice IC Service, DOD Cloud Access Rights) through the efficiencies of shared access.
\item Incorporate some aspects of CAD/EDA companies' academia policies for research projects at national labs.
\item Leverage the academic network and cultivate talents to advance and promote innovations in semiconductor technologies.
\item The solutions need to keep intact the premise of CAD/EDA companies' contributions, with special arrangements for commercializing research results.
\item Build an Ecosystem including the CAD/EDA tools, available technologies, vendor support, and business models.
\end{itemize}
\subsection{DARPA Conversations}
DARPA Toolbox is an Agency-wide effort to provide open licensing opportunities with commercial technology vendors to the researchers behind DARPA programs. DARPA Toolbox provides easy, low-cost, scalable access to state-of-the-art tools and intellectual property (IP) under predictable legal terms and streamlined acquisition procedures. The goal is to reduce performer reliance on low-quality, low-cost tools and IP that increase execution risks and complicate post-DARPA transitions \cite{DarpaToolbox}.
The DARPA team has created a mutually beneficial value proposition with the industry, exploiting the DARPA brand association by allowing highly visible public announcements. The vendors also benefit from adapting tools for cutting edge research programs which have high potential of becoming mainstream solutions in the long term.
They have created a two-tier system where DARPA negotiates low-cost uniform pricing with vendors by utilizing a light weight contract. Then DARPA performers can choose any vendors from the toolbox to buy a subset of licenses to create their own package required for their program. The Terms and Conditions are individually negotiated by the participants with guaranteed pre-determined low-cost prices.
DARPA is currently also looking at deploying an ``all of federal government" approach to broaden the impact and serve national interest.
DOE ICPT team is engaged with DARPA to learn from their experience while tailoring the concept to DOE needs and programs.
\subsection{ICPT Engagement}
The Integrated Contractor Purchasing Team's (ICPT) objective is to provide ``Guiding Principles" that incorporate best practices in the selection of both commodities/services and suppliers. The strategic selection of commodities/services and suppliers will enhance long-term relationships with suppliers and maximize process and cost savings within the DOE Complex. The ICPT philosophy is to 1) leverage the buying power of the DOE Complex to achieve the most favorable purchasing arrangements and pricing; 2) avoid unnecessary duplication in the acquisition process; 3) establish long-term relationships with quality suppliers, to optimize the number of suppliers per commodity as much as practicable;, and 4) primarily focus on the acquisition of commercial off-the-shelf commodities and commercial services through small, small disadvantaged, woman owned, Hubzone, Veteran-Owned, Service Disabled Veteran-Owned or other minority business enterprises.
To facilitate microelectronics support for DOE Complex Wide Site and Facility Contractors, ICPT aims to start the process with three strategic agreements to standardize the CAD tool procurement and contracts across DOE. ICPT hopes to negotiate general terms and conditions and fixed pricing discounts with the CAD vendors. Individual DOE M\&O Contractors will be able to select specific titles based upon their individual site needs from the portfolio offered by the CAD companies under their ICPT Agreement. To date, ICPT has initiated conversations with three major CAD companies. In addition, ICPT has been working actively with procurement professionals across DOE complex. Mutual benefits to DOE complex and to the CAD companies to support microelectronics are expected, through streamlining the cumbersome negotiation and procurement processes in the past.
In the long term, a dynamic process of adding new CAD tool vendors, IP vendors and Foundries will also be established.
|
1,116,691,498,503 | arxiv | \section{Arabic Subjectivity/Sentiment Analysis}
In this section, almost all the work done on Arabic is covered. Here, a synopsis about Arabic (e.g., the countries where it is spoken, the number of Arabic speakers) is provided. Following that, the available resources on Arabic sentiment analysis are introduced. Finally, the Arabic subjectivity and sentiment analysis methods are reviewed.
\subsection{Arabic Language}
Arabic is the official language of 22 Arab countries. There are more than 300 million native speakers of Arabic. The growth rate (i.e., 2,501.2\%) of Arabic Internet users was ranked the fastest in 2010 by Internetworldstats (http://www.internetworldstats.com/stats7.htm) compared to 1,825.8 \% growth rate for Russian, 1,478.7 \% for Chinese and 301.4 \% for English. Arabic users represent 18.8\% (more than 65 million users) of Interent users.
The Arabic language is a collection of different variants where there is only one formal written standard variety in the media and education through the Arab world~\cite{habash2010introduction}. This variant is called Modern Standard Arabic (MSA), while others are called Arabic dialects. There is a high degree of difference between MSA and Arabic dialects. One interesting fact is that the MSA is none of any Arab's native languages.
MSA is the official language of the Arab world and it is syntactically, morphologically, and phonologically based on Classical Arabic (CA)~\cite{habash2010introduction}.
Classical Arab is the language of the Qur'an (Islam's Holy Book). While Arabic dialects are true native language forms, they are used in informal daily communication and they are not taught in schools or standardized~\cite{habash2010introduction}. In contrast to Dialects, MSA is usually written not spoken language. Arabic dialects are poorly related to Classical Arabic. There are many Arabic dialects and they are different in many aspects, mainly geography and social classes. One way for dividing Arab dialects is based on the geographic aspect~\cite{habash2010introduction} as follow:
\begin{itemize}
\item The most common dialect is Egyptian Arabic, which covers the Nile valley (Egypt and Sudan)
\item Levantine Arabic covers the dialects of Syria, Lebanon, Jordan, Palestine and Israel.
\item Gulf Arabic includes the dialects of Gulf countries (United Arab Emirates, Saudi Arabia, etc.).
\item Maghrebi (North African) Arabic which cover dialects of Algeria, Tunisia, and Morocco.
\item Iraqi Arabic covers Iraq and combines elements of Levantine and Gulf dialects.
\item Yemenite Arabic.
\end{itemize}
Each dialects group are completely homogeneous linguistically.
Arabic is a semitic language~\cite{versteegh1997arabic} which has a very rich inflectional system and is considered one of the richest languages in terms of morphology~\cite{HabashOwenRoth09}. Arabic sentential forms is divided into two types, nominal and verbal constructions~\cite{farra2010sentence}. In the verbal domain, Arabic has two word order patterns (i.e., Subject-Verb-Object and Verb-Subject-Object). In the nominal domain, a normal pattern would consist of two consecutive words, a noun (i.e., subject) then an adjective (subject descriptor).
\subsection{Resources: Corpora and lexicons}
Here, most of the available corpora and lexicons created for Arabic language are revised.
\textbf{Opinion corpus for Arabic (OCA):} OCA is an opinion corpus for Arabic with a parallel English version (EVOCA)~\cite{rushdibilingual2011, rushdi2011oca}. Rushdi-Saleh et al. extracted the OCA corpus from different movie-review web sites. It consists of 500 reviews, which are divided equally into two parts: 1) positive reviews, and 2) negative reviews. There are some issues related to the design and application of the corpus:
\begin{itemize}
\item Non-related comments (i.e, People might be giving comments on things not related to the movie or they may be commenting on previous threads).
\item Romanization of Arabic is another problem. English characters are commonly used to write Arabic words. Such practice results in the presence of multiple versions for every word.
\item The web sites used to create the corpus contains comments in many different languages.
\item Each web site has its own rating system. Some reviews are rated in a range between 1 and 10, others have a rating range from 1 to 5, and still others have a binary rating of bad or good.
\item Culture and political emotions play an important role in ratings. For instance, the ``Antichrist" movie has a rating of 6.7 in IMDB, but has a rating of 1 in reviews of the Arabic blog
\item Arabic speaking participants use different ways to report the name of movies and actors in reviews. While they sometimes keep the English version, they use the Arabic version of the names at other times.
\end{itemize}
Generating the OCA corpus is a three-step process: 1) Preprocessing, 2) Reviewing, and 3) Generating \textit{N}-grams. To illustrate, in the Preprocessing stage, the HTML page is prepared by removing HTML tags, correcting spelling mistakes, and deleting special characters. The Review process consists of tokenizing and stemming words, filtering stop words and tokens of length $< 3$. Finally, generate unigrams, bigrams, and trigrams are generated. The same process is adopted to generate EVOCA.
\textbf{MPQA subjective lexicon \& Arabic opinion holder corpus:}
Another corpus for Arabic opinion holder and subjectivity lexicon is proposed by Elaranoty et al.~\cite{elarnaoty3machine}. The authors crawled 150 MB of Arabic news and manually annotated 1 MB (available at - http://altec-center.org/) of the corpus for opinion holder. The opinion holder corpus was annotated by three different persons. Any conflict emerging because of different annotations was solved using majority voting. For prepossessing the corpus Research and Development International (RDI) tool (http://www.rdi-eg.com) was used to handle the morphological analysis of Arabic sentences and assign parts of speech (POS) tags. Finally, semantic analysis of the words were done. Arabic Named Entity Recognition (ANER)~\cite{abdelrahman2010integrated} was used for extracting names from documents. The proposed Arabic subjectivity lexicon contains strong as well as weak subjective clues by manually translating the MPQA lexicon~\cite{wilson2005recognizing}.
\textbf{Arabic Lexicon for Business Reviews:}
A sentiment lexicon for Arabic business review was proposed by Elhawary and Elfeky~\cite{elhawary2010mining}. The authors used the similarity graph to build an Arabic lexicon. The similarity graph is a type of graph where the two words or phrases would have an edge if they are similar on polarity or meaning. The weight of the edge represents the degree of similarity between two nodes. Usually, this graph is built in an unsupervised manner
based on lexical co-occurrence from large Web corpora. Here, the researchers initially used a small set of seeds then performed label propagation on an Arabic similarity graph. For building the Arabic similarity graph, a list of seeds (600,900,100) for (positive, negative and neutral) are used. The Arabic lexicon created from the similarity graph consists of two columns where the first column is the word or phrase and the second column represent the score of the word which is the sum of the scores of all edges connected to this node (word/phrase). They applied filtering rules to avoid both the sparseness of the data and garbage nodes. Garbage nodes caused the top 200 positive words to be non-positive. They removed nodes with a high number of weighted edges and kept the 25 top ranked synonyms of the word. The top 25 synonyms of positive words are 90\% positive. This ratio became 50-60\% when considering all synonyms. The sentiment of the review is computed based on the sentiment of the sentences. That is, the sentence boundary detection is used, and negation is also used, to flip the sentiment score from positive to negative and vice versa. There are around 20 Arabic words for negation. Sentences greater than 120 character (i.e., long distance) are neglected. The results show that the created Arabic lexicon has high precision but has low recall.
Another subjectivity lexicon is proposed by El-Halees~\cite{elarabic2011Halees}. This lexicon is built manually based on two resources, the SentiStrength project and an online dictionary. They translated the English list from SentiStrenght project and then manually filtered it. Common Arabic words were added to the lexicon.
AWATIF is another Arabic corpus proposed by Abdul-Mageed and Diab~\cite{abdul2012AWATIF, abdul2011subjectivitylex}. AWATIF is a multiple-genre corpus for MSA subjectivity and sentiment analysis. AWATIF is extracted from three different resources: The first resource is Penn Arabic Treebank (PATB) part 1 version 3. They used around 54.5\% from (PATB1 V3) which represents 400 documents. These documents are a collection of news wire stories from different domains (e.g., economic, sports, politics). The second resource used is Wikipedia Talk Pages (WTP). They collected around 5,342 sentences from 30 talk page covering topics from politically and social domains. The 30 pages were selected from a larger pool of 3,000 talk pages. The third resource is from the Web Forum (WF) genre and comprises 2,532 conversation threads from seven different web forums. They also used different conditions to annotate the corpus using two types of annotation, simple (SIMP) and linguistically-motivated and genre-nuanced(LG). In SIMP, they introduced simple information to annotators such as examples of positive, negative, and neutral sentence. The required task was to label each sentence with one of the tags from the set \textit{\{POSITIVE, NEGATIVE, NEUTRAL\}}. In the LG type, they introduced a linguistic background for annotators and explained the nuances of the genre for each data. Also, Abdul-Mageed and Diab manually created an adjective polarity lexicon of 3,982 adjectives where each adjective has a tag from the set \textit{\{POSITIVE, NEGATIVE, NEUTRAL\}}.
\subsection{Subjectivity and Sentiment Systems and Methods for Arabic}
Here,the different methods applied to Arabic are discussed.~\cite{rushdibilingual2011, rushdi2011oca} build machine learning classifiers exploiting both the OCA and EVOCA corpora. They use both SVMs and an NB classifier and report 90\% \textit{F}-measure on OCA and 86.9\% on EVOCA using SVMs. The point out that SVMs outperform the NB classifier, which is common in text classification tasks.
~\cite{rushdi2011oca}'s results show that there is no difference between using term frequency (tf) and term frequency-inverse document frequency (tf-idf) as weighting schemes.
Different approaches for extracting the opinion holder in Arabic are proposed in~\cite{elarnaoty3machine}. Their approach is based on both pattern matching and machine learning. They extract three different types of opinion holders. The First type of opinion holder is opinion holder for speech events, which is defined as a subjective statement said directly by someone or claimed to be said by someone. In this way, they combine the direct speech event and indirect speech event in this type. The second type of opinion holder is defined as related to an opinion holder that expresses sentiment towards certain opinion subject. The third type is defined as related to expressive subjective elements (e.g., emotions, sarcasm) expressed implicitly. Definitely the third type is the hardest type to extract because it depends on the meaning of the words rather than the structures.
The first approach~\cite{elarnaoty3machine} use to extract opinion holders is based on pattern matching. They manually extract 43 patterns where the morphological inflections of the words are neglected. Examples of these patterns are `` And $<$holder$>$ expressed his objection about ...." Another example is `` And adds $<$holder$>$...." A pattern-based opinion holder classifier is built using the extracted patterns. The following rule as to extracting an opinion holder are followed: The opinion holder is retrieved if it contains a subjective statement or a named entity and its containing statement is classified as objective or subjective using a high-precision classifier.
While the first approach is based on pattern matching, the second and third approaches are based on machine learning. Authors formulate the opinion holder problem as a classification problem where each word in the corpus is classified as ``Begining of a holder (B-holder)'', ``Inside a holder (I-holder)'' or ``Non holder''. A conditional random field (CRF) probabilistic discriminative model is used for classification.
Authors build the CRF classifier based on a set of lexical, morphological, and semantic features. Pattern matching is used as a source for additional features for training the classifier in the third approach. Syntactic features are not used because of a lack of a robust general Arabic parser. The lexical features used are the focus word itself and window of size 3 around it (i.e., previous and next three words). The second type of features, i.e., semantic field features, are generated by grouping the semantically related words and giving them the same probability. In that way the handling of a missing word of the group in training data will not affect the performance if any word of the group appeared in the test data. The third feature type used is POS Tags generated by the RDI morphological analyzer. The set of tags generated by the RDI analyzer is reduced to a small set of tags and this small set are used as features. In addition, base phrase chunk and named entity recognition features are used. Finally, a feature based on pattern matching is used such that it is detected whether any word is part of the patterns extracted manually in the first approach or not.
Experimental results on the Arabic Opining Holder corpus show that machine learning approaches based on CRF achieve better results than the pattern matching approach. The authors report 85.52\% precision, 39.49\% recall, and 54.03\% \textit{F}-measure. Authors justify the performance degradation of the system by stating that it is due to the lower performance of Arabic NLP tools compared to those of English as well as the absence of a lexical parser.
\begin{comment}
Arabic news corpus of 150 MB is crawled and manually annotated 1 MB (available at - http://altec-center.org/) of the corpus as opinion holder corpus by three different persons where conflicting on the annotation solved using majority voting. For prepossessing the corpus Research and Development International(RDI) tool (http://www.rdi-eg.com) is used to handle the morphological analysis of Arabic sentences and extract part of speech (POS) tagging and finally semantic analysis of the words. ANER Arabic Named Entity Recognition~\cite{abdelrahman2010integrated}. Authors also created subjectivity lexicon for Arabic contains strong subjective clues and weakly subjective clues by manually transiting MPQA lexicon (http://www.cs.pitt.edu/mpqa/subj\_lexicon.html)~\cite{wilson2005recognizing}.
\end{comment}
Another system for Arabic sentiment analysis is proposed bu Elhawary and Elfeky~\cite{elhawary2010mining}. Their system is designed to mine Arabic business reviews. They tried to provide the Google search engine with annotated documents containing the sentiment score. The system has several components. The first component classifies whether an Internet page is a review or not. For this component, they extend an in-house multi-label classifier to work for Arabic such that its task is to assign a tag from the set \textit{\{REVIEW, FORUM, BLOG, NEWS and SHOPPING STORE\}} to a document. To build an Arabic review classifier data set, 2000 URLs are collected and more than 40\% of them are found to be reviews. This data set is collected by searching the web using keywords that usually exist in reviews e.g " the camera is very bad". Authors translate the lists of keywords collected and add to them a list of Arabic keywords that usually appear in the opinionated Arabic text. The final list contained 1500 features and was used to build an AdaBoost classifier. The data is broken down into 80\% training and 20\% testing. After a document is classified for belonging to the Arabic review class or lack thereof, a second component of the system is used. The second component analyzes the document for sentiment. They build an Arabic lexicon based on a similarity graph for use with the sentiment component. The final component of the system is designed to provide the search engine with the snapshot of the sentiment score assigned to a document during the search.
A combined classification approach is proposed by El-Halees~\cite{elarabic2011Halees} for document level sentiment classification. He applied different classifiers in a sequence manner. A lexicon based classifier is used during a first stage. This Lexicon based classifier identifies the sentiment of a document based on an aggregation of all the opinion words and phrases in the document.
Due to the lack of enough opinion words in some documents, it is not possible to classify all documents using the lexicon based classifier. All classified documents from first classifier are used as the training set for the next classifier that is based on Maximum Entropy. The Maximum Entropy classifier is used to compute the probability that the document belongs to a certain sentiment class: If the probability is greater than a threshold of 0.75, then the document is assigned a class, otherwise the document is passed to the next stage. The next classifier for the final stage is a \textit{k-}nearest neighbors (KNN) classifier is used to find k nearest neighbors for the unannotated document using the training set coming from the previous two classifiers.
The corpus they use for evaluation consist of 1134 collected from different domains (e.g., education, politics, and sports) and has 635 positive documents (with 4375 positive sentences) and 508 negative documents (with 4118 negative sentences). Preprocessing is applied to document HTML tags and non-textual contents are removed. Alphabets are normalized and some misspelled words are corrected. Sentences are tokenized, stop words are removed, and an Arabic light stemmer is used for stemming the words, and \textit{TF-IDF} is used for term weighting.
~\cite{elarabic2011Halees} report 81.70\% \textit{F}-measure averaged over all domains for positive documents and 78.09\% \textit{F}-measure for negative documents. The best \textit{F}-measure is obtained in the education domain (85.57\% for the positive class and 82.86\% for the negative class).
Another system for Arabic sentence level classification is proposed by Farra et al.~\cite{farra2010sentence}, where two different approaches (a syntactic and a semantic approach) for sentence classification are adopted. The grammatical approach proposed by Farra et al.~\cite{farra2010sentence}, is based on Arabic grammatical structure and combines the verbal and nominal sentence structures in one general form based on the idea of actor/action. In this approach, the subjects in verbal and nominal sentences are actors and verbs are actions. They manually label Action/Actors tags to sentence structures and used such tags as features. Their feature vector constitutes the following: Sentence Type, Actor, Action, Object, Type of Noun, Adjective, type of pronoun and noun, Transition , word polarity and Sentence class.
\textit{sentence type} features determine the type of the sentence (i.e., Verbal or Nominal), the \textit{transition} feature determines the type of the word which link the current sentence with the previous sentence, the \textit{word} polarity feature determines the polarity of the word (i.e., positive, negative or neutral).
The second approach proposed by Farra et al.~\cite{farra2010sentence} combines syntactic and semantic features by extracting some features like the frequency of positive, negative, and neutral words; the frequency of special characters (e.g., ``!"); the frequency of emphasis words (e.g., ``really " and ``especially"); the frequency of conclusive and contradiction words; etc. For extracting the semantics of the words,~\cite{farra2010sentence} build a semantic interactive learning dictionary which stores the semantic polarity of word roots extracted by stemmer.
For evaluation of the grammatical approach, only 29 sentences are annotated manually for POS tags.~\cite{farra2010sentence} report 89.3\% accuracy using an SVM classifier with 10 fold-cross validation. Sentences from 44 random documents are used for evaluating the semantic and syntactic approach using a J48 decision tree classifier.~\cite{farra2010sentence} report 80\% accuracy when the semantic orientation of the words extracted and assigned manually is used, and 62\% when the dictionary is used. Farra et al.~\cite{farra2010sentence}, also classified the documents by using all sentence features and chunking the document into different parts. They report 87\% accuracy rate. with an SVM classifier.
Abdul-Mageed et al. in~\cite{abdul2011subjectivity, abdul2010automatic, abdul2012AWATIF, abdul2011subjectivitylex} created sentence level annotated Arabic corpora and built subjectivity and sentiment analysis systems exploiting them. In their systems these authors exploit various types of features, including language independent features, Arabic-specific morphological features, and genre-specific features.
Abdul-Mageed et al.~\cite{abdul2011subjectivity, abdul2010automatic} report efforts for classifying MSA news data at the sentence level for both subjectivity and sentiment. They use a two-stage SVM classifier, where a subjectivity classifier is first used to tease apart subjective from objective sentence. In a second stage, subjective sentences are classified into positive and negative cases, with an assumption that neural sentences will be treated in a future system. These authors make use of two main types of features: (1) \textit{language independent} features and (2) \textit{Arabic-specific} features. The language independent features include a \textit{domain} feature indicating the domain (e.g., politics, sports) of the document from which a sentence is derived, a \textit{unique} feature where all words with a frequency threshold of $< 4$ is replaced by the token ``UNIQUE'', \textit{N-gram} features where all \textit{N}-grams of frequency threshold of $< 4$, and an \textit{adjective} feature where adjectives indicating the occurrence of a polarized adjective based on a pre-developed polarity lexicon of 3982 entries.
95.52\% results are reported using stemming, morphological feature and adjective Results showed that adjective feature is very important it improved the accuracy by more than 20\% and unique and domain features are helpful.
~\cite{abdulmageed-kuebler-diab:2012:wassa} present SAMAR, an SVM-based system for Subjectivity and Sentiment Analysis (SSA) for Arabic social media genres. They tackle a number of research questions, including how to best represent lexical information, whether standard features are useful, how to treat Arabic dialects, and, whether genre specific features have a measurable impact on performance. The authors exploit data from four social media genres: Wikipedia Talk Pages, Web forums, chat, and Twitter tweets. The data is in both MSA and dialectal Arabic. These authors break down their data into 80\% training, 10\% development, and 10\% testing and exploit standard SSA features (e.g., the ``unique'' feature, a wide coverage polarity lexicon), social and genre features (e.g., the gender of a user), and a binary feature indicating whether a sentence is in MSA or dialectal Arabic. They are able to significantly beat their majority class baselines with most data sets and results suggest that they need individualized solutions for each domain and task, but that lemmatization is a feature in all the best approaches.
Table~\ref{tab:arab_sys} summarizes the SSA systems which are described
above.
\begin{table}
\caption{Summary of different Arabic SSA systems.}
\label{tab:arab_sys}
{
\fontsize{10pt}{10pt}
\selectfont
\begin{adjustwidth}{-1in}{-1in}
\begin{tabular}{|c|c|p{1.4cm}|c|p{2cm}|c|p{4cm}|}
\hline
& & & SSA & & & \tabularnewline
System & Type & Features & level & Corpus & Advantages & Disadvantages\tabularnewline
\hline
\hline
~\cite{Abbasi:2008} & ML &
\begin{minipage}[t]{1.5cm}
Stylistic +\\
LF
\end{minipage}
& Doc &
\begin{minipage}[t]{1.5cm}
Movie \\
reviews, \\
web \\
forums \\
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Language independence \\
-- Effective feature selection
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- High computational cost
\end{minipage}
\\
\hline
~\cite{ahmad2006multi,almas2007note} & NC &
\begin{minipage}[t]{1.5cm}
domain-specific\\
lexical \\
features \\
\end{minipage}
& Phr &
\begin{minipage}[t]{1.5cm}
Financial news \\
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Simple method \\
-- Language independence
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- No sentiment classification \\
(only phrase extraction)
\end{minipage}
\tabularnewline
\hline
~\cite{rushdibilingual2011,rushdi2011oca} & ML &
\begin{minipage}[t]{1.5cm}
LF
\end{minipage}
& Doc &
\begin{minipage}[t]{1.5cm}
Web\\
reviews \\
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Simple features \\
-- Introduces OCA corpus
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- No Arabic-specific features
\end{minipage}
\tabularnewline
\hline
~\cite{elhawary2010mining} & ML &
\begin{minipage}[t]{1.5cm}
LF
\end{minipage}
& Doc &
\begin{minipage}[t]{1.5cm}
Business\\
reviews\\
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Builds large-scale lexicon \\
-- Computes soft sentiment score \\
(in addition to hard classification) \\
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- No Arabic-specific features
\end{minipage}
\tabularnewline
\hline
~\cite{elarabic2011Halees} & \begin{minipage}[t]{0.5cm}LC+
\\ML\end{minipage} &
\begin{minipage}[t]{1.5cm}
LF
\end{minipage}
& Doc&
\begin{minipage}[t]{1.5cm}
Multi-\\
domain
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Combines lexical and ML \\
-- Multi-domain \\
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- No Arabic-specific features
\end{minipage}
\tabularnewline
\hline
~\cite{farra2010sentence} & ML &
\begin{minipage}[t]{1.5cm}
Syntactic \& \\ LF \\
\end{minipage}
&
\begin{minipage}[t]{0.5cm}
Sen+\\
Doc
\end{minipage}
&
\begin{minipage}[t]{1.5cm}
News
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Combines LF \& syntactic
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Evaluated on small dataset
\end{minipage}
\tabularnewline
\hline
~\cite{abdul2011subjectivity, abdul2010automatic, abdul2012AWATIF, abdul2011subjectivitylex} & ML &
\begin{minipage}[t]{1.5cm}
LF, \\
syntactic \& \\
genre-specific, \\
social media features \\
\end{minipage}
& Sen &
\begin{minipage}[t]{1.5cm}
News,\\
social \\
media
\end{minipage}
&
\begin{minipage}[t]{4cm}
-- Combines language-independent and Arabic-specific features \\
-- Incorporates dialectal Arabic \\
-- Employs a wide-coverage \\
polarity lexicon \\
\end{minipage}
&
\begin{minipage}[t]{4cm}
--Some genre and social media \\
features are costly to acquire
\end{minipage}
\tabularnewline
\hline
\end{tabular}
\end{adjustwidth}
\textbf{Legend} \\
Classification types: ML=Machine Learning, CL=Rule or lexicon-based classifiers, NC=No classification. \\
Features: LF=Lexical features. \\
SSA level: Doc=Document-level, Phr=Phrase-level, Sen=Sentence-level classification. \\
}
\end{table}
\begin{comment}
\begin{minipage}[t]{1.5cm}
\end{minipage}
\end{comment}
\section{Language-Independent Feature Selection/Extraction Methods}
One way of performing sentiment analyses for languages other than English or building systems workable for multiple languages is to extract and select features that do not depend on these languages. Different approaches have been followed to select and extract these features: (1) Weighted Entropy Genetic algorithms, (2) Feature Subsumption, (3) Local Grammar based methods, (4) Positional Features and (5) Common seeds word methods. Here, each feature selection/ extraction approach is described separately.
\subsection{ Entropy Weighted Genetic Weighted}
Genetic Algorithm is an optimization technique that can be used for feature selection. Entropy Weighted Genetic Weighted (EWGA) combines Information Gain (IG) and genetic algorithms (GA) to select the features. EWGA proposed in~\cite{Abbasi:2008} was used to select features of Arabic and English. IG is combined with each step in the genetic algorithms process. It is used to select the initial set of features for the initial stage. Also, it is applied during the cross-over and mutation stages. Abbasi et al. ~\cite{Abbasi:2008} presented sentiment analysis system for Web forums in multiple language based on EWGA. They used two types of features, stylistics features and lexical features. Semantic features were avoided because they are language deepened and need lexicon resources while the limitation of their data prevents the use of linking features. They evaluate their system on a benchmark testbed of movie reviews consisting of 1000 positive and 1000 negative movie reviews
~\cite{Whitelaw:2005, Pang+Lee:04a, Mullen04sentimentanalysis,Pang+Lee+Vaithyanathan:02a}.
Importantly, their system which is based on feature selection method outperforms systems in~\cite{Whitelaw:2005,Pang+Lee:04a,Mullen04sentimentanalysis,Pang+Lee+Vaithyanathan:02a}. Using this system, they achieved an accuracy rate of 91\% while other systems achieved accuracy rates between 87-90\% on the movie reviews data set. They were also able to achieve 92\% accuracy rate on Middle Eastern forums and 90\% on US forums using EWGA feature selection method.
\subsection{Feature Subsumption for Sentiment Classification in Multiple Languages}
Another method for extracting and selecting the features is proposed by Zhai et al. ~\cite{zhai:2010}. The authors proposed the feature ``subsumption'' method to extract and select substring-group features. This method was applied to Chinese, English and Spanish. The system designed by Zhai et al. consists of four processes: (1) Substring feature extraction, (2) term weighting, (3) feature selection, and (4) classification. For extracting substring-group features, they built a suffix tree with incorporating transductive learning through considering unlabeled test documents for building the suffix tree. They applied four different weighting schemes (binary, three, tf and tfidf-c) and The "tfidf-c" outperforms all other approaches. The "tfidf-c" is extended form the standrad "tfidf" and is defined as follows ~\ref{tfidf-c-eq}
\begin{equation}
tfidf-c=\frac{tf(t_{k},d_{j})\times log\left(N/df(t_{k})\right)}{\sqrt{{\displaystyle \sum_{t\in d_{j}}(tf(t_{k},d_{j})\times log\left(N/df(t_{k})\right))^{2}}}}
\label{tfidf-c-eq}
\end{equation}
where $t_{k}$ represents the term corresponding to the single feature and $tf(t_{k},d_{j})$ is the term frequency for the term k in document d, $ df(t_{k})$ is the number of documents containing the term and $N$ is the total number of documents. Term presence usually outperforms term frequency~\cite{Pang:2002,zhai:2010}.
Zhai et al. ~\cite{zhai:2010} applied document frequency method as a feature selection technique by keeping the top $N$ features with highest document frequency scores. They tested the proposed system on three data sets: 1) an English data set of movie reviews, 2) a Chinese data set of hotel reviews, and 3) a Spanish data set of reviews on cars, hotels, and other products. The accuracy rates achieved were 94.0\%, 84.3\% and 78.7\% for Chinese, English, and Spanish respectively. This system is a success if compared to systems in~\cite{Pang:2002,Jun:2007} which are used for the English and Chinese data sets. However, it was outperformed by Abbasi and et al. ~\cite{Abbasi:2008} on the English data set described in the previous section.
\subsection{Local Grammar Methods}
Local Grammar is another method that can be used to extract sentiment features. It is used to extract sentiment phrases in the financial domain ~\cite{ahmad2006multi , agic2010towards}.
Ahmed et al.~\cite{ahmad2006multi} proposed this approach for financial news domain. They identified the interesting key words by comparing the distribution of words in financial news corpus with the distribution of the same words in general language corpus. Using the context around these words they built a local grammar to extract sentiment bearing phrases. They applied their approach to Arabic, English, and Chinese. They evaluated the system manually and achieved accuracy rates between 60-75\% for extracting the sentiment bearing phrases. Importantly, the proposed system could be used to extract the sentiment phrases in the financial domain for any language.
Agi{\'c} et al.~\cite{agic2010towards} used local grammar to extract sentiment phrases of financial articles. They demonstrated that there is a relation between the number of polarized phrases and the overall sentiment of the article. They built a ``Golden sentiment analysis data set" of financial domain for Croatian. They manually annotated the articles with positive/negative annotation. Some of the articles were annotated at the phrase level.
Importantly, while Bollen et al.~\cite{bollen2011twitter} showed that there is a correlation between collective mood states extracted from large-scale Twitter feeds on one hand and the value of the Dow Jones Industrial Average (DJIA) over time on the other, Agi{\'c} et al. demonstrate that there is a statistically significant correlation between the total market trend on the Zagreb Stock Exchange and the number of positively and negatively annotated articles within the same periods. The corpus used for this analysis is collected from two different resources: online newspapers specialized on finance and a large forum discussing the Zagreb Stock Exchange (CROBEX). For CROBEX two long periods of time are chosen, one for positive articles between 2007-01-02 and 2007-05-21 and the other for negative ones published between 2008-01-07 and 2008-04-16. Of course, the financial news documents are selected randomly from the corpus for the same two periods and annotated manually.
\subsection{Positional Feature Methods}
Positional information of the words and sentences has been used to build sentiment systems. Raychev and Nakov~\cite{raychev2009language} proposed the language independent method which is based on subjectivity and positional information.
Specifically, they weighted unigram and bigram terms based on their position in documents. They incorporate the subjectivity information by removing non-subjective sentences and then they moved the subjective sentences to the end of the documents by computing the likelihood of sentence subjectivity. This was done by training a Naive Bayes classifier on subjective data set and sorting the sentences based on their Likelihood subjectivity score. They evaluate their method on the standard movie reviews data set used in~\cite{Whitelaw:2005}~\cite{Pang+Lee:04a}~\cite{Mullen04sentimentanalysis}~\cite{Pang+Lee+Vaithyanathan:02a}.
They achieved 89.85\% accuracy rate using unigrams, bigrams, subjectivity filter, and subjectivity sorting.
\subsection{Common seed words methods}
Using very few common words like ``very, '' ``bad, '' and `` good'' in English, a sentiment analysis system is built by Lin et al.~\cite{lin2011language}. The authors proposed a multilingual sentiment system using few seed words which could be applied to any language because it is language independent and does not depend on features of any language. First, they extracted opinion words based on the assumption that there is an adverb of degree on each language( e.g ``very" in English). They extracted words by heuristic information based on patterns like ``word behind very" and removing stop words based on frequency. The next step after extracting opinion words is to cluster opinion words into positive and negative clusters.
To cluster the words, they proposed a simple and effective method consisting of three steps: (1) Labeling all samples and words based on two seed words ``good and bad", (2) Computing exclusive polarity for each opinion word using KL-divergent to solve disambiguation for words appearing in positive and negative examples,
and (3) Computing the new labels for samples based on the computed polarity of words.
After creating lexicons of positive and negative words, they introduced Semi-supervised learning to build sentiment classifier. They evaluated the system using hotel reviews data sets for many languages (French, German, Spanish, and Dutch). Their system achieved accuracy rates (80.37\%, 79.13\%, 80.05\%, and 81.33\%) corresponding to (French, German, Spanish, and Dutch).
They compared their system to two baseline systems ``Sentiment lexicon based methods'' and ``Machine translation based methods''. While the translation based system outperforms the lexicon based system, the proposed system outperforms the two baselines.
\section{ Monolingual Subjectivity and Sentiment Methods}
Here, the sentiment/subjectivity analysis systems designed specifically for single languages other than English are reviewed. The systems reviewed here are done for Chinese, Urdu, Spanish, German, and French.
\subsection{Chinese}
Zhang et al.~\cite{zhang2009sentiment} proposed a sentiment analysis system for Chinese depending on rule-based system with no-annotation cost for Chinese articles in multiple domains. Their approach is based on using the sentiment lexicon and the syntactic structure of each sentence. Their method consists of two main steps: The first step is computing the sentiment of the sentences. The second step is aggregating the sentiment of the sentences to get the score of the sentiment of the document sentiment.
The sentiment of the document has been defined using equation\ref{ch-eq1}:
\begin{equation}
S_{D}=\sum_{i}^{n}p(S_{i})*W_{i}
\label{ch-eq1}
\end{equation}
where Document $D=\{S_{1}S_{2}...S_{n}\}$ and $W_{i}$ represent the importance of the sentence in the document and $p(S_{i})$ is the polarity of the sentence and $S_{\}D}$ is the sentiment of the document.
Here, the objective sentence are excluded by scanning the document for subjectivity sentences only using the occurrence of the subjective words. HowNet, a bilingual English-Chinese Lexicon, provides a dictionary of Chinese subjective words contains 3,730 positive words, 3.116 negative words, 836 positive affective words (e.g., love), 1.254 negative affective words (e.g., sad) and 219 degree adverb (e.g., very). HowNet also provides the quantify of the degree adverb.
To compute the polarity of each sentence, the researchers depend on computing the modified polarity of the words.
Generally, polarity of words could be divided into three types. The first type is prior polarity which represents the general polarity of the word. The second type is the modified polarity which represents the polarity of the words based on the modifiers surrounding the word such as negations and degree adverbs. The third type is the dynamic polarity which represents the context polarity(e, g., Unpredictable camera Vs. Unpredictable movie). Dynamic polarity is topic and domain dependent.
Zhang et al. also, proposed a heuristic based on some linguistic rules considering two factors:The relation between the word and its children in the dependency tree and the type of children negation or modifiers used in order to compute the modified polarity of each word. Polarity of a sentence is determined by calculating the polarity of the root in the dependency tree of the sentence in a recursive manner.
The second step to be performed in the system is to aggregate the sentiment of the sentences to compute the sentiment of the document as a whole.
The five independent domain features used for measuring the importance of the sentence are:\begin{itemize}
\item Position of the sentence $i$ which is computed using \begin{equation}\frac{1}{min(i,N-i+1)}\end{equation} where $N$ is the number of sentences in the document. This gives the initial sentences and the last sentences in the document higher weight. This is mainly because these sentences are thematic sentences and thus are regraded as the most important sentences in the document.
\item The Term-Weight which enables the determination of the importance of sentences containing important terms. The tf-isf~\ref{tf-isf:eq} `` term frequency - Inverse sentence frequency" which is the same as the tf-idf but working on the sentence level.
\begin{equation}
tf-isf=\sum_{t\in s}tf(t,s)*log\frac{N}{sf(t)}
\label{tf-isf:eq}
\end{equation}
\item The similarity between the sentence and the headline using cosine similarity.
\item The occurrence of keywords in the sentence.
\item The first-person mode which is a binary feature indicating if the sentence contains the first person pronoun or not.
\end{itemize}
The importance of the sentence is computed using equation~\ref{sent-imp}
\begin{equation}
\sum_{i=1}^{5}\lambda_{i}F_{i}
\label{sent-imp}
\end{equation}
where $\lambda_{i}$ is the weight of the feature and the $F_{i}$ is the score of the feature.
In~\cite{zhang2009sentiment}, the system is evaluated using two different data sets. The first data set consists of 851 articles about euthanasia related discussions collected from various web sites. This data set is manually reviewed and annotated under positive and negative labels. It contains 502 positive articles and 349 negative articles. The second data set is AmazonCN. It contains 458.522 reviews for six different products(books, music, movies, electrical appliances, digital products and camera). The data set contains 310.390 negative articles vs. 29.540 negative articles.
They reported an average accuracy rate for all data sets of 76.33\%.The proposed rule-based method has been compared to three standard machine learning methods (SVM, NB and Decision trees) whose accuracy rates were 75.31\%, 68.1\% and 65.87\%, respectively. This means that the rule based method significantly outperforms NB and Decision trees(P $<$ 0.001) This also means that there was not significant difference between the rule based method and the SVM (P=0.582).
Here, ML methods are trained using different feature sets (e.g., bag of words, words/POS and appraisal features). Appraisal features consist of a triplet of subjective words, modifiers and negated forms.
Another method for Chinese sentiment analysis is proposed in~\cite{zhang2008sentiment} where Zhang et al. use SVM with kernel methods to classify Chinese reviews. They used ``bag of words" and ``appraisal phrase" as training features. Appraisal phrase indicates the feeling towards objects. Appraisal phrases are extracted using HowNet lexicon. They evaluated the method using AmazonCN review data set. In addition to using SVM with string kernels, they used Naive Bayes Multinomial and Decision Tree. They found out that the best accuracy rate is obtained when using the bag of words and the appraisal phrase features using Information Gain as a feature selection method and SVM (with string kernels) as a classifier.
\subsection{Urdu}
A Sentiment Analysis system is proposed for Urdu in~\cite{syed2010lexicon}. This system has very special characteristics related to the language itself as Urdu is written from right to left. Also, Urdu Orthography is context sensitive and word boundaries are not determined by space. One word may contain space and two different words might be written without space. Udru has a complex morphology as it contains inflections, derivations, compounding and duplications. For example, plural formation is determined by many different ways.
In~\cite{syed2010lexicon}, Sayed et. al propose a system based on SentiUnits lexicon which is generated specifically for Urdu. SentiUnits has two types of adjectives, a single adjective phrase as well as multiple adjective phrase. Each unit in SentiUnits can be described by five attributes (Adjective, modifier, Orientation, Intensity, Polarity and Negation). Urdu adjectives can be divided into two types, one for describing the quality and quantity while the other is for describing people and could be divided into marked adjectives. These adjectives can either be inflected or unmarked as they are originally Persian loan words. Modifiers are divided into absolute, comparative and superlative.
Here, Sayed et al. used SentiUnits to build the classifier for Urdu text. The system consists of three main steps (Preprocessing, Shallow Parsing and Classification). Preprocessing is used to prepare the text by processing HTML and applying word segmentation techniques. Shallow parsing
is used to extract entities (senti-units) as well as negation. Classification is done by computing the sentiment of the sentence by comparing the extracted senti-unit obtained in the Shallow Parsing step and the lexicon. Sayed et al. evaluated the system using two different domains (Movies and Products corpus). This corpus consists of 435 movie reviews and 318 product reviews. They reported and accuracy rate of 72 \% on Movie Reviews and 78\% on Products reviews.
\subsection{Spanish}
A semantic orientation calculator (SO-CAL) designed for analyzing Spanish sentiment analysis is used in~\cite{brooke2009cross}. Brooke et al.~\cite{brooke2009cross} used lexical dictionaries where each word has a score in a range of -5 and 5. They use shifting to handle negation. They shifted the value of the score of negated word by 4 (added toward the origin). For intensifiers, each intensifier was assigned a value. The score of the accompanying words to the intensifier is multiplied by the intensifier's value to get their sentiment score. It was observed that there is a bias towards the negative in lexical based sentiment classifier. In order to avoid this bias, the authors added a fixed value to the final score of each negative expression.
Three different ways are used for building the Spanish dictionary: (1) using automated translation for English dictionary using bi-lingual dictionary (www.spanishdict.com) and Google Translate, (2) modifying the translated lists from bi-lingual dictionaries manually, and (3) building dictionaries from scratch manually.
The manually created dictionary includes a vast amount of informal and slang words if compared to the automated ones while the automated ones contain more formal words. That is why manually built dictionaries was considered advantageous if compared to outperform the automated ones.
In evaluation, the SO-CAL method outperforms the SVM classifier trained on uni-gram. The authors show that in spite of the fact that translation of corpus and resources causes a loss of some information, it is a good baseline. They also noted that the best way for long term sentiment analysis is the incorporation of Language-specific knowledge and resources. Vilares et al. ~\cite{vilares2015megaphone} apply lexical based approach on social media to analysis Spanish political tweets. They enrich SentiStrength Spanish dictionary which contained 1,409 subjective terms mainly obtained from ~\cite{pennebaker2001linguistic}. Then, the improved dictionary is used to analyze tweets about the main political parties of Spain.
\subsection{German}
In~\cite{remus2010sentiws}, Remus et al. built a SentimentWortschatz(SentiWS) which is an important resource for German. SentiWS is a publicly available German resource for sentiment analysis . It contains 16,406 positive and 16,328 negative word forms coming from 650 negative and 1,818 positive words. Each word has a POS and a weighted score between -1 and 1. The authors build this lexicon using three different resources:(1) General Inquirer~\cite{stone1966general}, (2) Co-occurrence analysis and (3) German Collocation Dictionary. To calculate polarity weighting, point-wise mutual information is used~\ref{point-wiseMI}.
\begin{equation}
PW(w)=\sum_{i\in P}log_{2}(\frac{p(w,i)}{p(w).p(i)})-\sum_{i\in N}log_{2}(\frac{p(w,i)}{p(w).p(i)})
\label{point-wiseMI}
\end{equation}
where P is a seed of positive words and N is seed of negative words.
During the evaluation of SentiWs against a data set consisting of 480 sentences annotated by two humans for each adj, adv, noun and verb in the sentence, it achieved 96\% precision, 74\% recall and 84\% F-measure.
\subsection{French}
A sentiment supervised classification system is proposed for French movie reviews in~\cite{ghorbel2011sentiment}. Ghorbel et al., use SVM classifier. They used three types of features (lexical, morpho-syntactic and semantic features). For lexical features, Unigrams are used. A stop word list is used to improve the unigram performance as French contains a lot of stop words(e.g., je, la, me, de, aux). Grouping all inflected forms of words (i.e., Lemmatization) is used to reduce the number of unigrams features. While unigrams are used as lexical features, the POS tags (a morpho-syntactic feature) are used to enrich unigrams with morpho-syntactic information to solve disambiguation and to enable handling negation. SentiWordNet is used here as an external resource for the semantic feature. Specifically, SentiWordNet has been used to translate French words to English words in order to compute the polarity of words. When evaluated, the system achieved around 93.25\% accuracy rate using a combination of the three types of features mentioned above. The common type of errors of classification were caused by misspelling, neutral, mixed reviews, Ironic expressions and translation errors.
\section*{Abstract}
Subjective and sentiment analysis have gained considerable attention recently. Most of the resources and systems built so far are done for English. The need for designing systems for other languages is increasing. This paper surveys different ways used for building systems for subjective and sentiment analysis for languages other than English. There are three different types of systems used for building these systems. The first (and the best) one is the language specific systems. The second type of systems involves reusing or transferring sentiment resources from English to the target language. The third type of methods is based on using language independent methods. The paper presents a separate section devoted to Arabic sentiment analysis.
\section{Introduction}
Nowadays, the Web has become a read and write platform where users are no longer consumers of information but producers of it as well. User-generated content written in natural language with unstructured free text is becoming an integral part of the web mainly because of the dramatic increase of social network Web sites, video sharing Web sites, news portals, online reviews sites, and online forums and blogs. Because of this proliferation of user-generated content, Web Content Mining is gaining considerable attention due to its importance for many businesses, governmental agencies, and institutions.
Sentiment analysis (also referred to as opinion mining) is a computational study of attitudes, views, and emotions found in texts. The texts could be any document (e.g., comments, feedback, reviews or blogs). Sentiment analysis can be viewed as a classification process that aims at determining whether a certain document/text was written to pass a positive or a negative opinion about a certain topic, product, or person. This process regards each document as a basic information unit. The process has been referred to as ``the document level sentiment classification" where the document is seen as an opinionated product. The analysis or classification of sentiment on the sentential level is referred to as ``sentence-level sentiment classification"~\cite{pang2008opinion}.
Sentiment analysis is gaining vast attention because of the potentiality of using opinion summary of a large number of population in industry as well as in other fields. For instance, having this opinion summary available can enhance businesses as business owners would have access to consumer opinions. Individuals can benefit from this information as they would be able to compare products. Thus, sentiment analysis makes it possible to summarize the opinion of people towards products as well as politicians.
Performing this type of analysis (either on the sentential level or the document level) has been done using two types of classifiers, Rule-based classifier~\cite{denecke2008using,kim2009conveying,zhang2009sentiment,brooke2009cross,elarnaoty3machine}, and Machine learning classifiers~\cite{Abbasi:2008,pang2008opinion,liu2010sentiment,rushdibilingual2011, rushdi2011oca,mihalcea2007learning,wan2009co}. Currently, most of these systems are built for English~\cite{pang2008opinion,liu2010sentiment}.
The current paper attempts to explore sentiment/subjective analysis systems created generally for languages other than English. A special attention is given to Arabic. The paper aims at providing the reader with information about the methods used for building sentiment analysis systems.
After surveying the different ways used for building sentiment analysis systems for languages other than English, the paper concludes with a suggestion about the optimum method(s) to be followed. The best method is the employment of tools that have to do with language-specific features. The main problem with this method is that it costs a lot to build resources for each language. The second method is transferring the sentiment knowledge from English into the target language. The final way is to use language independent methods.
The paper is divided into four parts. The first part covers the language independent methods. The second section surveys sentiment transfer methods created to transfer the sentiment from English to other languages. The third section explores systems done specifically for languages other than English. The last part focuses on the methods used for Arabic.
\input{lang_ind}
\input{translate}
\input{mono}
\input{arabic}
\section{Conclusion}
This paper surveyed different methods for building sentiment analysis systems for languages other than English. Here, it is suggested that the optimum method to be followed in building a sentiment analysis system should include the employment of tools with language-specific features. While this suggestion might be seen as problematic as it costs a lot to build resources for each language, it is the most accurate route to be followed. Alternatives to this method as previously explained would be transferring the sentiment knowledge from English into the target language or to use language independent methods.
\bibliographystyle{abbrv}
{\small
\section{Sentiment Translation Methods}
Transferring Sentiment Translation techniques of well-studied languages to new ones is another
way for building sentiment/subjectivity systems. Simply, these methods are based on using machine translation techniques to translate the resources (corpora) to the new languages.
Here, various sentiment/subjectivity methods based on machine translation will be surveyed. The techniques used to solve the problems resulting from non-accurate machine translation processes will be tackled. Other methods based on graph methods and used to translate sentiment will also be presented.
\subsection{Machine Translation}
Machine translation (henceforth MT) has been used as a simple tool to create sentiment systems
for multiple languages. In these systems, MT has been used~\cite{mihalcea2007learning, wan2008using, denecke2008using} to translate corpora of different languages into English. Following the translation, subjectivity/sentiment classifiers are built in English. The simplicity of using MT stems from the availability of its techniques and the availability of English Resources. Also, MT is used to generate resources and corpora for language other than English. Using it, sentiment lexicons and corpora have been generated for Bengali, Czech, French, German, Hungarian, Italian, Romanian, and Spanish~\cite{steinbergermultilingual,steinberger2011creating,banea2008multilingual,das2009subjectivity,brooke2009cross}.
\subsubsection{Machine Translation SSA systems:}
Due to the simplicity and availability of MT, Kerstin~\cite{denecke2008using} proposed a pipeline system based on SenitwordNet~\cite{esuli2006sentiwordnet} and MT techniques for multi-languages. The proposed system is a pipeline system consisting of the following steps:
\begin{itemize}
\item Language classification where the LingPipe language identifier is used for language
Classification;
\item Translation of the document from the identified language to English;
\item text preparation by stemming; and
\item Classification of the sentiment.
\end{itemize}
Simplicity and variability are attributes of the different ways used in building the classifiers. For instance, three different ways were used in building the classifiers in~\cite{denecke2008using}. These ways are machine learning classifiers, lingpipe and rule based classifiers.
Comparison of the three methods of classifier building shows that, the classifier based on machine learning provides the most accurate rates (the scores of SentiwordNet were 62\% on MPQA corpus~\cite{wiebe2005creating} and 66\% for German movie reviews). In~\cite{denecke2008using}, the proposed system is simple and could be applied to any language.
Similarly, MT techniques are also used to build sentiment systems for Chinese~\cite{wan2008using}. Wan et al.~\cite{wan2008using} used the automatic machine translation technique to translate Chinese documents into English. The English lexicon is used afterwards. Many methods to assemble the results from both languages were suggested. These methods include average, weighted average, min, max and majority voting. The semantic orientation method has also been used to compute the score of the documents as well as the window size in order to enable handling negation.
The obtained English results showed that using the translated reviews give better results if compared to the original Chinese ones. This situation stands in contrast to what might have been expected: The original language using original lexicon should have given better results if compared to the translated one. Also, the ensemble methods improve the obtained results.
Another usage of MT is incorporating features from many languages to improve the classifiers’ accuracy. Banea et al.~\cite{banea2010multilingual} integrated features from multiple languages when building a high precision classifier using majority vote. A basic single language trained classifier was used as a basis for this high precision classifier. The system was evaluated on MPQA corpus. The integrated feature technique was used for six different languages (Arabic, French, English, German, Romanian, and Spanish). Two types of feature-sets (monolingual and multilingual) were used. The feature vector for each sentence of the monolingual feature set consists of unigrams for this language while the feature vector of the multilingual feature set consists of combinations of monolingual unigrams.
Importantly, results show that using English annotated data sets can build successful classifiers for other languages by leveraging the annotated data set. The created classifiers have macro-accuracy between 71.30\% to 73.89\% for Arabic and English. Here, the English classifier outperformed those for other languages. Non-English based classifier results show that using the multilingual data set can improve the accuracy of the classifier for the source language as well as classifiers for the target languages. Specifically, the best results are obtained when a classifier trained over the combination of all six languages was used~\cite{banea2010multilingual}.
This suggests that using multi language data sets can enrich the features and reduce ambiguity. In addition, the English classifier achieved the best accuracy rate among all monolingual classifiers. Also, when investigating the combination of any two-language from the six languages, the German and Spanish classifier achieved the best results. Performance increased when Romanian and Arabic were added. Adding English and French did not improve the results. Indeed, these results suggest that Spanish and German expanded the dimensionality covered in English, Arabic and Romanian by adding high quality features for the classification task. They also showed that the majority voting classifier could be used as a high precision classifier with acceptable recall level by combining all monolingual classifiers.
\subsubsection{Machine Translation as a Resource Generator}
In addition to using MT as a technique in building sentiment/subjectivity systems as previously explained, it was used to create resources and dictionaries for the analyses of sentiment in multiple languages. Mihalcea et al. employ two different ways to generate resources for subjectivity in languages by leveraging tools and resources of English. The first method is translating an existing English lexicon to the target language using bi-lingual dictionary. The second method is a corpus based approach where the annotated corpus in the target language is built using a projection from the source language~\cite{mihalcea2007learning}.
In the first method, authors translate the target language lexicon using two bi-lingual dictionaries~\cite{wiebe2005creating}. Some problems emerged with this approach. First, some words lost their subjectivity in this process. For example, when translating into Romanian, the word memories lost its subjectivity as it was translated into the power of retaining the information. Second, there were cases of lack on the sense of the individual entries in the lexicon and the bilingual dictionary. Third, some multi-word expressions were not translated accurately. Consequently, this led to losing the subjectivity of some of these multi word expressions after translation.
Trials to solve the first problem have been introduced. In~\cite{das2009subjectivity}, researchers overcame this obstacle by clustering the words that have the same Root. Then, the root itself is checked against the English lexicon. If the root exists then the word is kept in the list which will be translated. To overcome the second problem, heuristic approaches are used. Examples of these heuristic approaches are using the ”“most frequent” technique in~\cite{mihalcea2007learning} and First type is First Word (FW)~\cite{kim2009conveying}. In the third problem, a simple way for solving the multi-word expression issue is using word-by-word approach~\cite{mihalcea2007learning} and using the Web to validate the translation by checking its occurrence in the Web.
Evaluation of the method of translating the lexicon using bilingual dictionaries reflects that, the translated lexicon is less reliable than the English one. The rule based classifier is used to evaluate the lexicon. This classifier uses a simple heuristic. It labels the sentence as subjective if it contains two or more strong subjective expressions and as objective if it contains at most two weak subjective expressions (no strong subjective expressions at all). Other than that, the sentence is labeled as unknown. This type of classifiers generally has high precision and low recall so it could be used to collect sentences from unlabeled corpus.
Importantly, the rule-based classifier performs badly in the objective task. One reason is that, weak subjectivity clues lose its subjectivity during the translation process. In~\cite{mihalcea2007learning}, researchers worked on a manual annotation study which showed that a small fraction of the translated words keep its subjectivity after translation.
The second method is the corpus-based approach where the annotated corpus in the target language is built using projection from the source language. Then Machine learning classifiers are trained on the labeled data. The experimental results obtained in applying this method show that generally machine learning classifiers outperform the rule-based classifier.
To overcome challenges met in cases where no bilingual dictionary or parallel corpora are available, Banea et al.\cite{banea2008multilingual} extend the work in~\cite{mihalcea2007learning} by employing multiple ways to perform automatic translation from English. This is basically done to generate resources in the new language using English resources. They designed three experiments to evaluate whether automatic translation is a good tool for generating new resources. The first and second types of experiments are done by translating the training source into the target language. In the first experiment, the training data is manually annotated. In the second one, opinion finder classifiers are used to annotate the corpus when the annotation done is in the sentence level. The obtained results show that the automatic annotated corpus is working better than the manually annotated corpus. This suggests that the clues used by researchers to annotate the data might be lost during the translation process while the clues used by classifiers are kept during this process. In the third experiment, the target language is translated into the source language then the opinion finder tool is used to label the sentences. Following that, the sentences are projected back to the target language. Finally, the classifier is trained. The authors evaluate the MT methods used for Romanian and Spanish. Results show that manually or automated labeled data are sufficient to build tools for subjectivity analysis in the new language. Furthermore, the results show comparable results to manually translated corpora.
MT has also been used to generate resources, in ~\cite{steinbergermultilingual,steinberger2011creating} parallel corpora for seven languages of “sentiment towards entities” are built. Specifically, the Gold standard sentiment data is built in English then projected into other languages (Czech, French, German, Hungarian, Italian, and Spanish). Here, a general and simple sentiment computing method has been used by counting the number of subjectivity words within the window for a given entity~\cite{steinbergermultilingual}. The resources used were sentiment dictionary available into 15 languages~\cite{steinberger2011creating}. Negation is handled by adding 4 to each sentiment score of negated words (the sentiment score of each word is between -5 and 5).
Importantly, this system is language-independent because it depends only on the lexicons. The
system employing the golden standard data achieved accuracy rates from 66\% (Italian) to
74\% for (English and Czech).
As in \cite{banea2008multilingual}, MT is used to translate English lexicon (a product of merging SentiWord English lexicon~\cite{esuli2006sentiwordnet} and Subjectivity Word List~\cite{wiebe2005creating}) into Bengali~\cite{das2009subjectivity}. Das and Bandyopadhyay~\cite{das2009subjectivity} used machine learning classifiers with many features such as part of speech tagging and chunking to divide each document into beginning, intermediate, and end. Each sentence is then given a feature indicating whether it belongs to the beginning, intermediate or end. They also used lexicon scores as features to give subjectivity scores of the word, stemming, frequency, position of subjectivity clue in the document, the title of the document, the first paragraph and the last two sentences. The overall accuracy rate of the system is found to be 76.08\% precision rate (PR) and 83.33\% recall rate (RR) for MPQA data set~\cite{wiebe2005creating}, and 79,90 (PR) and 86,55 (RR) for IMDP corpus, and 72.16\% (PR) and 76.00\% (RR) for Bengali News corpus and 74.6\% (PR) and 80.4\% (RR) for Blog corpus.
To recap, in this section different methods for using MT to build subjectivity and sentiment systems were reviewed. The main issues that emerged in the experimentation of MT have also been highlighted. In the next section, the methods done to improve machine translation SSA systems will be reviewed.
\subsection{Improving Machine Translation-based Systems}
Two methods have been built to improve machine translation SSA systems.
Mainly this section describes the co-training~\cite{wan2009co} and the structural corresponding learning~\cite{wei2010cross} methods.
\subsubsection{Co-training Method}
In~\cite{wan2009co}, Wan introduces the co-training algorithm to overcome the problem of low performance of MT methods used in [25]. The proposed co-training framework uses unlabeled data in the training process. Specifically, the co-training method manipulates two views for learning, the source language view as well as the target language view. Here, two separate classifiers are trained on labeled data, one for the source language and the other for the target one. Using of unlabeled data comes after having the two classifiers. This is done by adding the most confident samples to the labeled set on each view if the two classifiers agree. Then, the classifiers are retrained. The outcome would be two different classifiers. The prediction of the sentiment will be based upon the score of the two classifiers (e.g average of the scores of the two classifiers).
The obtained experimental results show that the co-training algorithm outperforms the inductive and transductive classifiers~\cite{wan2009co}. This framework was tested on sentiment classification for Chinese reviews. The features used are unigrams and bigrams. The term-frequency is used to weight the features which works better than tf-idf in their empirical experiments.
\subsubsection{The Structural Corresponding Learning Method}
In~\cite{wei2010cross}, researchers try to overcome the noise coming from MT methods
used in~\cite{mihalcea2007learning} by using structural corresponding learning (SCL) to find shared important features in the two languages. SCL is used for domain adaptations. Here, the authors suggest that the sentiment classification of the cross-lingual could be considered as a domain adaption problem. To use SCL, the first step is to find the set of pivot features. These features/words have the same manner on the source and target language (e.g ``very good" and ``perfect").
SCL works as follows: First, it starts by generating the weighted matrix based on the co-occurrence between pivot features and ordinary features. Second, singular vector decompression is used to select the top eigenvector features to create the mapping matrix from original domain to lower dimension domain. Third, the mapping matrix will be used with the new features in the new language/domain to train the classifier. The authors kept only the pivot features on the translation process and then used weighted matrix from source language in addition to using the new translated pivot features to train the classifier. For the selection of the pivot, some words are selected according to their occurrence. Following that these words/features are ranked according to their conditional probabilities that are computed on the labeled data.
Importantly, an evaluation of the SCL is done for the same data set used on~\cite{wan2009co} The results show that SCL outperforms the co-training~\cite{wan2009co} in terms of F-measure (reported to be 85\% in this case).
\subsection{Graph Methods for Translating Sentiment}
In addition to using MT for translating and transferring sentiment from one language to another, Graph methods have been used. Scheible et al.~\cite{scheible2010sentiment, scheible2010sentiment2} uses the graph-based approach to transfer sentiment from English to German. They built graphs containing two types of relations (coordinations and adjective-noun modifications). They specifically chose these types of relations as they contain clues for sentiment. The graph contains adjectives and nouns as nodes and relations is represented by edges. They built two graphs one for English and the other for German. To compute sentiment of the target language, SimRank algorithm is used. SimRank computes the similarity between nodes in the two graphs. SimRank algorithm is an iterative process that measures the similarity between all nodes in the graph. SimRank assumes that the two nodes are similar if their neighbors are similar. Similarity between two nodes a and b are described by equation ~\ref{sim-eq}
\begin{equation}
sim(a,b)=\frac{v}{|N(a)||N(b)|}\sum_{i\in N(a),,j\in N(b)}sim(i,j)
\label{sim-eq}
\end{equation}
Where $N(a)$ is the neighborhood group of $a$ and $v$ is a weighted vector to determine the effect of distance of neighbors of $a$ and initially the $sim(a,a)=1$.
In this method, the bi-lingual lexicon is used to get the seeds between the two graphs. The experiments are done on English and German versions of Wikipedia. The results show that this method works better than the Semantic Orientation with Point-Wise Mutual Information (SO-PMI). One problem of SimRank is that the words with high sentiment score are not the exact translation but they are semantically related.
Another graph method based on link analysis and bilingual dictionary is proposed in~\cite{kim2009conveying} to create a sentiment lexicon in the target language. The English sentiment lexicon and link analysis algorithm are used to refine the ranking scores of lexicons in both the source and target languages. In order to create a sentiment lexicon in Korean, kim et al. proposed a three-step method: (1) translating the English lexicon into Korean (or any target language) using bi-lingual dictionary, (2) refining the translated lexicon using link analysis algorithm, and (3) normalizing the sentiment scores for the lexicon items.
Here, as with any translated lexicon, the main difficulty is that many words would lose their subjectivity meaning in translation~\cite{mihalcea2007learning}. In~\cite{mihalcea2007learning} as previously explained, Mihalcea et al. used a simple heuristic based on the frequency of the word by using the first sense to overcome the challenge of translation. In this way they make use of an attribute of the bilingual dictionary in which word translations are ordered by the most frequently used then the less frequently used.
Kim et al. employ four types of heuristics to overcome this limitation. The first heuristic is using the First Type is the First Word(FW) which assign the sentiment score for the English word to only the first word of the first sense. While this type of heuristic filter uncertain words, it makes the translated lexicon smaller. The second type is reemployment of a technique used in~\cite{mihalcea2007learning} which assign the sentiment of English word to all words of the first sense. The third type (All Sense (AS)) is to assign the sentiment score of the English words to all the translated words which generate the maximum number of the words in the target language but with less reliability. The last type of heuristic is Sense Rank in which the sentiment score for the translated words is assigned according to their rank. Here, the words with higher sense rank will have higher score. The link analysis algorithm is used to refine the rank of the entities in the two lexicons ( English and the target language's).
In~\cite{kim2009conveying}, Kim et al., created a bipartite graph where there are two sets of vertices, one set is for the source language (English) and the other set is for the target language words (Korean). In this graph the edges go in either one of two directions (Korean words and their English counterparts or English words and their Korean counterparts). HITS algorithm is used to rank the vertices on the graph. To explain further, HITS has two types of nodes Hubs and Authorities. Hubs are the nodes connected to many Authority nodes while the Authority is a node connected to many Hubs. In order to refine the score of the Korean lexicon, the sentiment score of each English node is considered as equivalent to its hubness and the authority of the Korean node is considered as equivalent to as its connectness to the nodes with high hubness. Equations~\ref{auth-eq} and~\ref{hub-eq} describe how the authority of the Korean words and hubs for the English words are computed~\cite{kim2009conveying}.
\begin{equation}
Auth(w_{t})=(1-\alpha)*e_{t}+\alpha\sum_{s\in T(w_{t})}Hub(s)
\label{auth-eq}
\end{equation}
\begin{equation}
Hub(w_{s})=(1-\gamma)*e_{s}+\alpha\sum_{t\in T(w_{s})}Auth(t)
\label{hub-eq}
\end{equation}
when $\alpha$ is damping factor for the Korean and $\gamma$ is damping factor for English and
$e_{t} , e_{s}$ are the initial scores for Korean and English nodes and $T(w_{i})$ is the set of the translated words for $i$.
After refining the sentiment score for the Korean lexicon, equation~\ref{auth-eq} and ~\ref{hub-eq} could be used to refine the source lexicon (the English lexicon) by considering English words as authorities and Koreans words as hubs. After refining the ranking of the words of the lexicons, the next step would be normalizing the sentiment score to get 1 as a product of the summation of the negative, positive and neutral score of each word.
To evaluate the translated lexicon, the $p-normalized Kendall \tau distance$ equation~\ref{p-dis-eg} is used. This distance measure computes the distance between the two ordered lists:
\begin{equation}
\tau=\frac{N_{i}+1/2*N_{j}}{N}
\label{p-dis-eg}
\end{equation}
where $N_{i}$ is the number of discordant pairs and $N_{j}$ is the number of the ordered pair in the first list (source lexicon - original list) and tied in the predicted list while ${N}$ is the total number of ordered pairs in the original list.
The results show that the heuristic of translating reliable words has low $\tau$ distance while the heuristic of translating many words (less reliable words) had large$\tau$ distance.
To summarize, in this section we looked at different methods that generate lexicons and resources into different languages(English not included) by using machine translation techniques. Also, methods to improve the output of machine translation techniques have been represented.The next section will explore some of the sentiment/subjectivity analysis systems built specifically to analyze single languages other than English.
\begin{table}[]
\centering
\caption{Summary of different Sentiment Systems.}
\label{my-label}
\fontsize{10pt}{10pt}
\selectfont
\begin{adjustwidth}{-1.0in}{-0.1in}
\begin{tabular}{|p{2.5cm}|p{4.9cm}|p{3cm}|p{2cm}|p{2cm}|p{2.3cm}|}
\hline
Methodology& Pros & Cons & Sub/Sent & SSA Level & Examples \\
\hline
\hline
Weighted Entropy Genetic Algorithms (EWGA) &
\begin{tabular}[c]{@{}l@{}}
\\
- Optimize feature selection \\
- Achieved high accuracy \\ with multi languages \\
- Language independent
\end{tabular}
&
- High Computational Cost & Sentiment & Document & ~\cite{Abbasi:2008} \tabularnewline\hline
Feature Subsumption &
\begin{tabular}[c]{@{}l@{}}
\\
- Language independent\\
- Can leverage different\\
weighting techniques
\\
\end{tabular}
& - EWGA outperform Feature Subsumption & Sentiment & Document & ~\cite{zhai:2010} \tabularnewline\hline
Local Grammer Methods &
\begin{tabular}[c]{@{}l@{}}
\\
- Can extract sentiment phrases in \\
any language for financial domain\\
- Obtain high accuracy in \\ financial domain \\
\end{tabular} &
- No sentiment classification, only extract sentiment phrases & Sentiment & Phrase & ~\cite{ahmad2006multi , agic2010towards} \tabularnewline\hline
Positional Features & \begin{tabular}[c]{l} \\- Language independent \end{tabular}
& - High Computational Cost & Subjectivity & Sentence & ~\cite{raychev2009language} \tabularnewline\hline
Machine Translation & \
\begin{tabular}[c]{@{}l@{}} - Simple\\ - Flexible
\end{tabular} & - Translation may affect the sentiment meaning of a word or phrase. &
\begin{tabular}[c]{l} Sentiment\\
Subjectivity
\end{tabular} &
\begin{tabular}[c]{l} word\\phrase\\document \end{tabular} & ~\cite{denecke2008using,wan2008using,banea2010multilingual} \tabularnewline\hline
Machine Translation as a Resource Generator & - Automate the generation of lexicons and dictionaries in multi languages & - Not accurate as manual labeling & \begin{tabular}[c]{l}Sentiment \\ Subjectivity \end{tabular} & word/phrase & ~\cite{mihalcea2007learning,wiebe2005creating,das2009subjectivity,banea2008multilingual} \tabularnewline \hline
Co-Training & \begin{tabular}[c]{l} - Use unlabeled data \\ - Use two view learning
\end{tabular} & \begin{tabular}[c]{l} \\- Need labeled \\
data to train \\
initial classifiers \end{tabular} &Sentiment & Document & ~\cite{wan2009co}
\tabularnewline \hline
SCL & \begin{tabular}[c]{l} \\- Outperform co-training \\ - Formulate cros-langual \\
as cross-domain problem \\ - Reduce noisy coming from MT \\ \\
\end{tabular}
&
\begin{tabular}[c]{l} - Need to decide \\
about pivot features
\end{tabular}
& Sentiment & Document & ~\cite{wei2010cross} \tabularnewline\hline
Graph Methods & \begin{tabular}[c]{l} \\ - Used to transfer sentiment
\\from lang. to another \end{tabular} & - Need bi lingual lexicon for seed nodes & Sentiment & Document & ~\cite{scheible2010sentiment, scheible2010sentiment2}
\tabularnewline \hline
\end{tabular}
\end{adjustwidth}
\end{table}
|
1,116,691,498,504 | arxiv | \section{\bf Introduction}
Let $X$ be a compact complex manifold equipped with an area form $\omega$. Let $f:\mathbb{C}\longrightarrow X$ be a
nonconstant entire holomorphic curve. An associated {\sl Ahlfors current} of $f$ is a positive closed current of bidimension $(1,1)$ obtained as the weak limit of a certain sequence of positive currents of bounded masses
\begin{equation*}
\label{radius sequence}
\bigg\{
\dfrac{[f(\mathbb{D}_{r_n})]}{\area_{\omega}f(\mathbb{D}_{r_n})}
\bigg\}_{n\geqslant 1},
\end{equation*}
where $\mathbb{D}_{r_n}$ are discs of increasing radii $r_n\nearrow \infty$ centered at the origin. Here, to ensure that such a limit current is closed, the sequence $\{r_n\}$ is chosen in such a way that the lengths of boundaries of the discs are asymptotically negligible compared with their areas, namely
\[
\lim_{n\rightarrow\infty}
\dfrac{\leng_{\omega}(f(\partial\mathbb{D}_{r_n}))}{\area_{\omega}(f(\mathbb{D}_{r_n}))}
=
0.
\]
By Ahlfors' lemma (c.f. \cite{brunella1999, Nevanlinna1970}),
for each positive number $\epsilon>0$, the set
\[
\bigg\{
r> 0:\,\dfrac{\leng_{\omega}(f(\partial\mathbb{D}_{r}))}{\area_{\omega}(f(\mathbb{D}_{r}))}
\geqslant \epsilon
\bigg\}
\]
is of finite measure with respect to $\frac{\dif r}{r}$. Hence the above {\it ``length-area'' condition} is satisfied for most choices of increasing radii. Moreover, given a sequence of radii $\{r_n\}_{n\geqslant 1}$ with $r_n\nearrow\infty$,
after some small perturbation by scaling and extracting a subsequence,
one can always obtain an Ahlfors current for $f$.
Ahlfors currents and their analogs obtained by taking the logarithmic average $\int\frac{
\dif t}{t}(\cdot)$, called {\sl Nevanlinna currents}, are fundamental tools in studying complex hyperbolicity, value distribution theory and complex dynamical systems. Notably, they played a crucial role in the work McQuillan \cite{Mcquillan1998} on Green-Griffiths'
conjecture for algebraic surfaces of general type having positive Segre class (see also~\cite{brunella1999} for a simplified proof by Brunella). By employing Ahlfors currents,
Duval \cite{Duval2008} gave a quantitative version of the classical Brody's Lemma and obtained a characterization of complex hyperbolicity in terms of linear isoperimetric inequality for holomorphic discs. Using such currents, some geometric refinement of the classical Cartan's Second Main Theorem \cite{Duval-Huynh2018}, as well as the high dimensional Weierstrass-Casorati Theorem \cite{Huynh-Vu2020} were obtained. The reader is also referred to \cite{Dinh-Sibony2018} for recent key applications in complex dynamical systems.
Since Ahlfors currents and Nevanlinna currents encode geometric information of their original entire curves, several results in value distribution theory can be presented in terms of intersections of corresponding cohomology classes. For example, the First Main Theorem of Nevanlinna theory can be expressed as an inequality between the algebraic intersection and the geometric intersection (c.f.~\cite{Duval-Huynh2018}).
Note that in certain specific situations, Ahlfors currents (or Nevanlinna currents) from some holomorphic curve are unique \cite{Dinh-Sibony2014, Duval-Huynh2018, Dinh-Sibony2018, Dinh-Vu2020}, which subsequently leads to several interesting results.
Therefore, it is natural and fundamental to ask generally
\begin{ques}
\label{question 1}
Are all Ahlfors currents associated to the same entire curve cohomologically equivalent?
\end{ques}
The study of such currents is itself of independent interest. By Siu's decomposition Theorem~\cite{Siu1974}, an Ahlfors current $T$ can be written as the sum $T=T_{\sing}+T_{\diff}$, where the singular part $T_{\sing}=\sum_{\ell\in I}c_{\ell}\cdot[C_\ell]$ is some positive linear combination ($c_\ell> 0$; $I\subset \mathbb{Z}_+$, could be $\varnothing$) of
currents of integration on irreducible algebraic curves $C_\ell$, and where the diffuse part $T_{\diff}$ is a positive closed $(1,1)$--current having zero Lelong number along any algebraic curve. If the singular part $T_{\sing}$ is nontrivial, Duval~\cite{Duval2006} showed that any irreducible curve $C_\ell$ above must be rational or elliptic (see also~\cite{Duval2017} for a local version). In \cite{dacosta2013}, da Costa gave an example of entire curve in the projective plane whose associated Ahlfors current is supported in some line. This construction can be modified to produce Ahlfors currents supported on a rational or an elliptic curve \cite[Theorem 2.6.1]{Huynh2016}. On the other hand, we would like to mention the following unsolved question, which had been considered by
Brunella~\cite[page~200]{brunella1999}.
\begin{ques}
Is there any Ahlfors current from an entire curve such that both of its singular part and diffuse part are nontrivial?
\end{ques}
In this paper, we answer the above two questions by constructing explicit examples.
\begin{thm}
\label{thm 1}
There exists an entire curve producing
cohomologically different
Ahlfors currents.
\end{thm}
By Siu's decomposition, Ahlfors currents with nontrivial singular parts can be distinguished as different types by the data
$(|I|\in \mathbb{Z}_+\cup \{\infty\}, T_{\diff} \text{ is trivial / nontrivial})$.
\begin{thm}
\label{thm 2}
There exists an entire curve producing all types of Ahlfors currents with nontrivial singular parts.
\end{thm}
\begin{rmk}
The above two results also hold true for Nevanlinna currents, see Subsection~\ref{Singular Nevanlinna currents on X}.
\end{rmk}
Lastly,
it is natural to seek Ahlfors (Nevanlinna) currents with trivial singular part. Examples of such currents
are known to exist on $\mathbb{P}^2(\mathbb{C})$,
by looking at the Levi-flat real hypersurface
in $(\mathbb{C}^*)^2$ defined by the equation $|x|=|y|^{\alpha}$, where $\alpha$ is an irrational real number (c.f.~\cite[page~262]{Duval-Berteloot2001}). Indeed, this real hypersurface
is foliated by entire curves, while its closure in $\mathbb{P}^2(\mathbb{C})$ contains no algebraic curve.
In \cite{Dinh-Sibony2014}
there are more examples of holomorphic curves
whose associated Ahlfors (Nevanlinna) currents are diffuse and unique. In Section~\ref{section: examples}, we show new examples of diffuse Ahlfors currents on the product of two elliptic curves and on $\mathbb{P}^2(\mathbb{C})$, see Propositions~\ref{diffuse in Abelian surface}, \ref{diffuse Ahlfors currents in Cp2}.
\bigskip
We now outline the ideas and the structure of this paper. As a matter of fact, our source of inspiration is an example of da Costa \cite{dacosta2013} about a nondegenerate entire curve clustering to a line in $\mathbb{P}^2(\mathbb{C})$ (see also~\cite{Duval-Huynh2018} for more discussions). In Section \ref{construction}, we start with an elliptic curve $\mathcal{C}=\mathbb{C}/\Gamma$ equipped with a negative line bundle $\mathcal{L}$. For some large integer $m\gg 1$, we construct a section $s_m$ of $\pi_0^*\mathcal{L}^m$ having large exponential growth of order $2$, where $\pi_0:\mathbb{C}\rightarrow\mathcal{C}$ is the canonical projection. The surface $X$ is obtained by taking the geometric projectivization $\mathbb{P}(\mathcal{L}^m\oplus\mathbb{C})=:X$ of the vector bundle $\mathcal{L}^m\oplus\mathbb{C}$ on $\mathcal{C}$. Thus the section $s_m$ induces a holomorphic map $f_0: \mathbb{C}\rightarrow X$ clustering to the curve $\mathcal{C}_{\infty}$ corresponding to the ``infinity section'' of $\pi_0^*\mathcal{L}^m$. To generate Ahlfors currents with larger singular supports, we hence modify the original section $s_m$ by multiplying it with a Weierstrass canonical product
$
\psi(z)=\prod_{\lambda\in\Lambda}\Big(1-\frac{z}{\lambda}\Big)e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
$, whose zero locus $\Lambda$ is distributed in a delicate pattern,
to make sure that the new section $\psi\cdot s_m$ induces an entire curve $f:\mathbb{C}\longrightarrow X$ producing Ahlfors currents with more singularities. Indeed, for every $\lambda\in\Lambda$, since $\psi\cdot s_m(\lambda)=0$,
$f(\lambda)$ touches the curve $\mathcal{C}_0:=\mathcal{C}\times [0 \oplus 1]$, defined by the zero section of $\pi_0^*\mathcal{L}^m$, at $([\lambda], [0 \oplus 1])$.
The idea is that, the image of a small neighborhood of $\lambda\in \Lambda\subset \mathbb{C}$ by $f$ shall contribute moderate area $O(1)$ near the fiber $\mathbb{P}^1_{[\lambda]}\subset X$ over $[\lambda]\in \mathcal{C}$, and once there are sufficiently many $\lambda'\in \Lambda$ mapping to the same class $[\lambda]$ by $\pi_0$,
the area of the image of $f$ should spend a positive portion about $\mathbb{P}^1_{[\lambda]}$, hence the Ahlfors currents should charge positive mass there. See the picture below for illustration.
\begin{center}
\scalebox{.85}{\input{picture1.pdf_t}}
\end{center}
Nevertheless, to make sense of this idea, we need to show, first of all, that the growth of $\psi$ is neither too rapid nor too slow, which will be accomplished in Section \ref{section prep},
by means of the Stirling formula as well as the symmetry of the lattice $\Gamma$. Consequently, in Section~\ref{section: estimate area of discs}, we can manipulate Jensen's formula
to evaluate various areas, which distinguish the singularities of the Ahlfors currents.
In Section~\ref{section: algorithm}, we present an algorithm for constructing the zero locus $\Lambda$, which is designed for the proofs of the main theorems in Section~\ref{section: proof}.
In Section~\ref{section: examples}, we provide new examples of diffuse Ahlfors currents. Moreover, we show
cohomologically elaborate Ahlfors currents
on surfaces obtained by blowing-up $X$.
\bigskip\noindent
{\sl Convention:} Throughout this paper, $\mathsf{K}$ denotes positive numbers which are uniformly bounded from both sides
$0<K_1<\mathsf{K}<K_2<\infty$.
Further, notation $\mathsf{K}_{\star_1, \star_2, \star_3}$
indicates dependence on parameters $\star_1, \star_2, \star_3$. The notation
$\mathbb{D}(a, r)$ means the disc centered at $a$, in $ \mathbb{C}$ or in the elliptic curve $ \mathbb{C}/\Gamma$,
with the radius $r$. When $a=0\in \mathbb{C}$, we write
$\mathbb{D}_{r}$ instead of $\mathbb{D}(0, r)$. The differential operator $\dif^c$ is short for $\frac{\sqrt{-1}}{4\pi}(\overline{\partial}-\partial)$.
\vspace*{0.5cm}
\noindent{\bf Acknowledgements.} We would like to address our profound gratitude to Professor Julien Duval for introducing us to the subject and for many fruitful discussions from which we learned rich ideas.
We are grateful to Professor Sibony for sharing the reference~\cite{Dinh-Sibony2014} and for suggesting related problems. We thank Professor Yusaku Tiba for his question about cohomology classes of Ahlfors currents. We
thank Professor Jo\"el Merker, Dr. Duc-Viet Vu and Dr. Ruiran Sun for valuable comments and suggestions.
S.-Y. Xie is partially supported by the NSFC Grant No.~11688101.
Part of this paper was written during his visit to Tianyuan Mathematical Center in Southeast China / Xiamen University invited by Professor Chunhui Qiu. He is grateful to the colleagues there for hospitality and excellent working conditions. D. T. Huynh is
grateful to Academy of Mathematics and Systems Science in Beijing for enhanced scientific ambience. He also wants to acknowledge partial support from the Core Research Program of Hue University, Grant No. NCM.DHH.2020.15.
\section{\bf Construction}
\label{construction}
Fix
a smooth elliptic curve $\mathcal{C}=\mathbb{C}/\Gamma$, where the lattice
$\Gamma:=\mathbb{Z}
\oplus \mathbb{Z}\sqrt{-1}$
is chosen to affiliate the arguments later.
We can find a negative line bundle $\mathcal{L}$ on $\mathcal{C}$ equipped with some hermitian metric $h'$ having strictly negative curvature. Now comparing with the K\"ahler form $\dif\dif^c\big(|z|^2\big)$ on $\mathcal{C}$ descending from the canonical projection $\pi_0:\mathbb{C}\longrightarrow \mathbb{C}/\Gamma$,
the curvature of $h'$ is cohomologous to $-2\alpha \dif\dif^c\big(|z|^2\big)$
for some positive constant $\alpha$, namely their difference is of the form $\dif\dif^c\varphi$ for some smooth real function $\varphi$ on $\mathbb{C}/\Gamma$. Therefore, replacing the initial metric $h'$ by $h'e^{\varphi}=:h$, the curvature becomes
$
\Theta_h
=
-2\alpha\, \dif\dif^c\big(|z|^2\big)$.
Noting that the line bundle
$\pi_0^*\mathcal{L}$ on $\mathbb{C} $ is holomorphically trivial,
it has a nowhere vanishing holomorphic section $k$, which by Lelong-Poincar\'e equation satisfies that
$
\dif\dif^c\big(\mathrm{log}\,\|k\|^2_{\pi_0^*h}\big)
=
2\alpha\,\dif\dif^c\big(|z|^2\big)
$.
Hence $\mathrm{log}\,\|k\|^2_{\pi_0^*h}-2\alpha\,|z|^2$ is a harmonic function on $\mathbb{C}$, hence can be written as the real part of some holomorphic function $g$. Therefore, the modified section
$s:=e^{-g/2}\, k$
of $\pi_0^*\mathcal{L}$ has exponential growth of order two
$\|s\|_{\pi_0^*h}
\,
=\,
\|e^{-g/2}\, k\|_{\pi_0^*h}
\,
=
\,
e^{\alpha |z|^2}$.
The above construction is based on an idea of~\cite{dacosta2013}.
We now amplify the negativity of $\mathcal{L}$, by
introducing
$\mathcal{L}_m:=\mathcal{L}^{\otimes m}$ for some big multiplicity $m\geqslant 1$ to be determined. For the metric $h_m:=h^{\otimes m}$ of $\mathcal{L}_m$, the section $s_m:=s^{\otimes m}$ has large exponential growth
\begin{equation}
\label{exponential growth of s}
\|s_m\|_{h_m}
\,
=
\,
e^{m\alpha |z|^2}.
\end{equation}
Now we introduce the complex surface $X:=\mathbb{P}(\mathcal{L}_m\oplus \mathbb{C})$ obtained by the geometric projectivization of the rank $2$ vector bundle $\mathcal{L}_m\oplus \mathbb{C}$
over $\mathcal{C}$. Denote by $\pi_1: X\longrightarrow \mathcal{C}$ the canonical projection.
By
the fiberwised identification $\mathcal{L}_m\cong \mathcal{L}_m\oplus 1\subset \mathbb{P}(\mathcal{L}_m\oplus \mathbb{C})$,
the tautological space of $\mathcal{L}_m$ can be embedded into $X
$ as an open subset, whose complement is the elliptic curve $\mathcal{C}_{\infty}:=\mathcal{C}\times [1 \oplus 0]\subset \mathbb{P}(\mathcal{L}_m\oplus \mathbb{C})$ at ``infinity''.
Next, we introduce an auxiliary holomorphic function
\[
\psi(z):=\prod_{\lambda\in\Lambda}\Big(1-\frac{z}{\lambda}\Big)e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
\]
obtained by Weierstrass canonical product,
where the zero locus $\Lambda$ will be chosen carefully by the following sophisticated reasoning, to make sure that the global section $\psi\cdot s_m$
of $\pi_0^*\mathcal{L}_m$ together with the inclusion $\iota: \mathcal{L}_m\hookrightarrow X$
induce an entire curve $f: \mathbb{C}\longrightarrow X$ producing complicated Ahlfors currents.
First of all, we would like to have the estimate $\mathrm{log}\, |\psi(z)|\leqslant O(|z|^2)$,
at least for $|z|$ around $r_i$ for some specific radii $r_i\nearrow \infty$, in order to bound the area of ${f}(\mathbb{D}_{r_i})$
by $O(r_i^2)$.
Secondly, we require that the cardinality $ |\Lambda\cap \mathbb{D}_{r_i}|=O(r_i^2)$, so that the image ${f}(\mathbb{D}_{r_i})$ intersects the curve $\mathcal{C}_0:=\mathcal{C}\times [0 \oplus 1]\subset X$ defined by the zero section of $\mathcal{L}_m$ frequently enough.
Lastly, we require that each time when the image of the entire curve $f$ intersects $\mathcal{C}_0$ for $\lambda\in \Lambda$ with $|\lambda|\gg1$, it contributes $O(1)$ area near the fiber $\mathbb{P}^1_{[\lambda]}:=\pi_1^{-1}([\lambda])$.
\smallskip
Thus we declare that
\begin{enumerate}
\item[(i)]
near each annulus $\mathbf{A}_{r_i}:=\{z\in\mathbb{C}: \frac{r_2}{2}\leqslant |z|\leqslant r_i\}$, the zero locus $\Lambda$ is a mild perturbation of $ \mathbf{A}_{r_i}\cap c\,\Gamma$, where $c\geqslant 5$ is some positive integer to be determined. More precisely
\[
\Lambda=\cup_{i
\geqslant 1}
\,
B_{r_i}
\qquad
\text{where}\qquad
{B}_{r_i}
:=
\cup_{\mu\in \mathbf{A}_{r_i}\cap c\Gamma}\,
\{\mu+x_{\mu}\}.
\]
Here at the moment we only tell that all $x_{\mu}$'s take values in the fundamental domain
\[
\mathcal{D}:=\{x+y\sqrt{-1}\,:\,
0\leqslant x, y<1
\},
\]
and later in Section~\ref{section: algorithm}, we will elaborate on the choices of $x_{\mu}$'s for delicate reasons.
\smallskip
\item[(ii)]
$\{r_i\}_{i\geqslant 1}$ grow very rapidly, say
\begin{equation}
\label{r_i grow rapid}
r_1\geqslant 2020\cdot c,
\qquad
{r_{i+1}}\geqslant {r_i}^4
\quad
{\scriptstyle
(\forall\, i\,\geqslant\,1)
}.
\end{equation}
\end{enumerate}
\section{\bf Preparations}
\label{section prep}
\begin{lem}
\label{lemma 1}
One has a uniform estimate
$
\vert
\sum_{\lambda\in B_{r_i}}\,
\frac{1}{\lambda}
\vert
\leqslant
\mathsf{K}/{c^2}
$
for all $i=1, 2, \dots$.
\end{lem}
\begin{proof}
In the special case that all $x_{\mu}=0$, by the
symmetry of $\Gamma$ that $(-1)\cdot \Gamma=\Gamma$
and that of $\mathbf{A}_r$,
the sum
$\sum_{\mu\in \mathbf{A}_{r_i}\cap c\Gamma}\,\frac{1}{\mu+0}$
is always $0$.
In general,
for every $\mu\in \mathbf{A}_{r_i}
\cap c\Gamma$, one has the estimate $|\frac{1}{\mu+x_{\mu}}-\frac{1}{\mu+0}|\leqslant \mathsf{K}/r_i^2$.
Noting that the cardinality $ |B_{r_i}|\leqslant
\mathsf{K}\cdot (r_i/c)^2$, we conclude that
$\vert
\sum_{\lambda\in B_{r_i}}\,
\frac{1}{\lambda}
\vert
\leqslant
\mathsf{K}\cdot (r_i/c)^2
\cdot
\mathsf{K}/r_i^2
=
\mathsf{K}
/
c^2
$.
\end{proof}
\begin{lem}
\label{lemma 2}
One has a uniform estimate
$
\vert
\sum_{\lambda\in B_{r_i}}\,
\frac{1}{\lambda^2}
\vert
\leqslant
\mathsf{K}/{(c^2\,r_i)}
$
for all $i=1, 2, \dots$.
\end{lem}
\begin{proof}
The argument goes much the same way as the preceding one, by using the rotational symmetry of $\Gamma$ that
$\sqrt{-1}\cdot\Gamma=\Gamma$.
Indeed, we have the identity
$\sum_{\mu\in \mathbf{A}_{r_i}\cap c\Gamma}\,\frac{1}{
(\mu+0)^2}=0$. Moreover, for every $\mu\in \mathbf{A}_{r_i}
\cap c\Gamma$,
we have
$|\frac{1}{
(\mu+x_{\mu})^2}-\frac{1}{
(\mu+0)^2}|
\leqslant
\mathsf{K}/r_i^3
$.
The remaining argument is clear.
\end{proof}
We make a convention that $\mathrm{log}\, 0=-\infty$.
\begin{pro}
For every $i\geqslant 2$
and for $r_i/3\leqslant |z|\leqslant 3r_i$, one has
\begin{equation}
\label{psi<}
\mathrm{log}\, |\psi(z)|\leqslant \mathsf{K}
\cdot r_i^2/c^2.
\end{equation}
\end{pro}
To bound the area of $f(\mathbb{D}_{r_i})$ by $\mathsf{K}\cdot r_i^2$, it is crucial to have the above estimate. In fact, by classical complex analysis (c.f.~\cite[Chapter 4]{Levin1996}), we can check that $\psi$
is well-defined and that the infinite product is uniformly convergent in bounded domains, and that the exponential growth order of $\psi$ is, by applying Borel's formula~\cite[page~30, Theorem 3]{Levin1996}, exactly $2$, {\em i.e.}
$\mathrm{log}\, |\psi(z)|\leqslant \mathsf{K}_{\epsilon}\cdot |z|^{2+\epsilon}$ for any $\epsilon>0$ and for large $|z|$.
Nevertheless, for the critical case that $\epsilon=0$ we need more effort.
\begin{proof}
We first study $I:=\prod_{\ell=1}^{i-1}\prod_{\lambda\in B_{r_\ell}}(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}$ concerning smaller annuli compared with $\mathbf{A}_{r_i}$. Note that
$\prod_{\ell=1}^{i-1}\prod_{\lambda\in B_{r_\ell}}
|1-\frac{z}{\lambda}|\leqslant (1+3r_i)^{ |B_{r_1}|+\cdots +| B_{r_{i-1}}|}\leqslant
(1+3r_i)^{\mathsf{K}\cdot r_i/c^2}
$. Hence by Lemmas~\ref{lemma 1},~\ref
{lemma 2}, we receive that
$\mathrm{log}\, |I|\leqslant \mathrm{log}\, (1+3r_i)^{\mathsf{K}\cdot r_i/c^2}
+ \sum_{\ell=1}^{i-1}
(|\sum_{\lambda\in B_{r_\ell}} \frac{1}{\lambda}||z|
+
|\sum_{\lambda\in B_{r_\ell}} \frac{1}{\lambda^2}||\frac{z^2}{2}|)
\leqslant
\mathsf{K}\cdot r_i^2/c^2$.
Secondly, we observe $II:=\prod_{\lambda\in B_{r_i}}(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{3\lambda^2}}$ concerning the annulus $\mathbf{A}_{r_i}$. Note that each term $|1-\frac{z}{\lambda}|\leqslant \mathsf{K}$ by our construction, and that $| B_{r_i}|\leqslant \mathsf{K}\cdot (r_i/c)^2$. Now using Lemmas~\ref{lemma 1},~\ref
{lemma 2}, we receive that
$\mathrm{log}\, |II|\leqslant \mathsf{K}\cdot r_i^2/c^2$.
Lastly, we analyze $III:=\prod_{\ell\geqslant i+1}\prod_{\lambda\in B_{r_\ell}}(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}$ concerning larger annuli compared with $\mathbf{A}_{r_i}$.
Now the key point is that, for each
$\lambda\in B_{r_\ell}$, one has $|\frac{z}{\lambda}|\leqslant \frac{3r_i}{r_{\ell}/3}\ll 1$. Therefore we can apply the Taylor expansion of
$\mathrm{log}\, (1-\frac{z}{\lambda})$ to
achieve desired estimates. Indeed, noting that
$
\mathrm{log}\,\big( (1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}\big)
=
-\sum_{n\geqslant 3}
\frac{1}{n}
(\frac{z}{\lambda})^n$,
hence
\[
\Big|
\mathrm{log}\,
\prod_{\lambda\in B_{r_\ell}}(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
\Big|
\leqslant
\sum_{\lambda\in B_{r_\ell}}
\sum_{n\geqslant 3}
\frac{1}{n}
\Big|
\frac{z}{\lambda}
\Big|^n
\leqslant
\mathsf{K}\cdot r_\ell^2/c^2
\cdot
\sum_{n\geqslant 3}
\frac{1}{n}
\Big|
\frac{3r_i}{r_{\ell}/3}
\Big|^n
\leqslant
\mathsf{K}/c^2
\cdot
\frac{r_i^3}{r_\ell}
\cdot \mathsf{K}.
\]
Since the sequence $\{r_{\ell}\}_{\ell>i}$ grows very rapid by our construction~\eqref{r_i grow rapid}, there holds
$\sum_{\ell>i} \frac{r_i^3}{r_\ell}
<\mathsf{K}
$. Thus the above estimate yields $|\mathrm{log}\, III|\leqslant \mathsf{K}/c^2$.
Combining all the above estimates about $I, II, III$, we receive the desired inequality~\eqref{psi<}.
\end{proof}
By our construction of $\Lambda$, it intersects each disc $\mathbb{D}(z, 1)$ at most once.
Therefore we introduce
\begin{equation}
\label{psi_1}
\psi_1(z):=\prod_{\lambda\in\Lambda\setminus \mathbb{D}(z, 1)}\Big(1-\frac{z}{\lambda}\Big)e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
\end{equation}
to capture the asymptotic behavior of $\psi$ away from its zero locus $\Lambda$.
\begin{pro}
\label{the most difficult estimate of psi}
For every $z\in \mathbb{C}$ with large $|z|$,
one has
\begin{equation}
\label{psi>}
\mathrm{log}\, |\psi_1(z)
|
\geqslant -\mathsf{K}\cdot |z|^2/c^2.
\end{equation}
\end{pro}
\begin{proof}
Fix a positive small number $\eta=\frac{1}{100}$. For every $i\geqslant 1$,
we introduce the slightly larger annulus
$\widetilde{\mathbf{A}}_{r_i}:=\{x\in \mathbb{C} : (1-\eta )r_i/2\leqslant|x|
\leqslant (1+\eta)r_i\}\supseteq \mathbf{A}_{r_i}$, to make sure that $\widetilde{\mathbf{A}}_{r_i}\supseteq B_i$.
Case $(i)$: $|z|$ large with $z\notin \cup_{i
\geqslant} \widetilde{\mathbf{A}}_{r_i}$. Then $z$ lies between some two consequent annuli $\widetilde{\mathbf{A}}_{r_j}$
and $\widetilde{\mathbf{A}}_{r_{j+1}}$, i.e.,
$(1+\eta)r_j< |z|< (1-\eta)r_{j+1}/2$, and it is clear that $\psi(z)=\psi_1(z)$. Firstly,
for each $\lambda\in \cup_{\ell=1}^j B_{r_\ell}$,
we have
$|1-\frac{z}{\lambda}|\geqslant |\frac{z}{\lambda}|-1\geqslant \eta':=\eta/2$.
Thus
$\prod_{\ell=1}^{j}\prod_{\lambda\in B_{r_\ell}}
|1-\frac{z}{\lambda}|\geqslant \eta'^{\sum_{\ell=1}^j |B_{r_{\ell}}|}\geqslant
\eta'^{\mathsf{K}\cdot r_j^2/c^2}
\geqslant \eta'^{\mathsf{K}\cdot |z|^2/c^2}$. Next, thanks to
Lemmas~\ref{lemma 1},~\ref
{lemma 2}, we have
$\prod_{\ell=1}^{j}\prod_{\lambda\in B_{r_\ell}}
|e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}|
\geqslant
e^{-|z|\sum_{\ell=1}^j\mathsf{K}/c^2-|z|^2\sum_{\ell=1}^j\mathsf{K}\cdot /(c^2 r_{\ell})}\geqslant
e^{-\mathsf{K}\cdot |z|^2/c^2}$.
Lastly, by mimicking the estimate of $III$ in the preceding proof, we receive that
\[
\mathrm{log}\,
|
\prod_{\ell\geqslant j+1}\prod_{\lambda\in B_{r_\ell}}(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
|
\geqslant
-\sum_{\ell\geqslant j+1}\sum_{\lambda\in B_{r_\ell}}
\sum_{n\geqslant 3}
\frac{1}{n}
|\frac{z}{\lambda}|^n
\geqslant
-\mathsf{K}\cdot
|z|^2/c^2.
\]
Summarizing the above estimates,
we conclude that
$\mathrm{log}\, |\psi_1(z)
|
=
\mathrm{log}\, |\psi(z)|
\geqslant -\mathsf{K}\cdot |z|^2/c^2$.
Case $(ii)$: $|z|$ large with $ z\in\widetilde{\mathbf{A}}_{r_j}$ for some $j$.
By repeating the same arguments as above, we can show that, first of all,
$
\mathrm{log}\,
|
\prod_{\ell=1}^{j-1}\prod_{\lambda\in B_{r_\ell}}(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
|
\geqslant
-\mathsf{K}\cdot |z|^2/c^2$,
and secondly,
$
\mathrm{log}\,
|
\prod_{\ell\geqslant j+1}\prod_{\lambda\in B_{r_\ell}}(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
|
\geqslant
-\mathsf{K}\cdot |z|^2/c^2$.
By
Lemmas~\ref{lemma 1},~\ref
{lemma 2}, we receive
$
\mathrm{log}\,
|
\prod_{\lambda\in B_{r_j}\setminus \mathbb{D}(z, 1)}e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}
|
\geqslant
-\mathsf{K}\cdot |z|^2/c^2$.
Thus the remaining problem is to show that
$
\mathrm{log}\,
|
\prod_{\lambda\in B_{r_j}\setminus \mathbb{D}(z, 1)}(1-\frac{z}{\lambda})
|
\geqslant
-\mathsf{K}\cdot |z|^2/c^2$.
To start with, we find a point $\mu_0$ in $c\Gamma$ having the least Euclidean distance to $z$.
Then for every $\lambda=\mu+x_{\mu}\in B_{r_j}$, we have
$
|\lambda-z|
\geqslant
|\mu-z|
-|x_{\mu}|
\geqslant
\frac{1}{2}
(|\mu-z|+|\mu_0-z|)
-\sqrt{2}
\geqslant
\frac{1}{2}
|\mu-\mu_0|-\sqrt{2}
$. If moreover assume that $\mu\neq \mu_0$,
then we can continue to estimate
$|\lambda-z|
\geqslant
\frac{1}{2}|\mu-\mu_0|-\sqrt{2}
\geqslant
\frac{1}{4}
|\mu-\mu_0|$, whence
$|1-\frac{z}{\lambda}|=\frac{|\lambda-z|}{|\lambda|}\geqslant \frac{1}{8}\frac{|\mu-\mu_0|}{|\mu|}$. Since $(\frac{1}{8})^{|B_{r_j}|}\geqslant \exp(-\mathsf{K}\cdot |r_j|^2/c^2)$,
we only need to show that
\begin{equation}
\label{core estimate}
\mathrm{log}\,
\prod_{\mu_0\neq\mu\in \mathbf{A}_{r_j}\cap c\Gamma}\frac{|\mu-\mu_0|}{|\mu|}\geqslant
-\mathsf{K} \cdot |r_j|^2/c^2
\geqslant
-\mathsf{K} \cdot |z|^2/c^2.
\end{equation}
For any positive number $r'$, denote by $\Gamma_{\leqslant r'}\subset \Gamma$ the subset of points whose real and imaginary parts have absolute value $\leqslant r'$. Note that, for every $\mu\in \mathbf{A}_{r_j}\cap c\Gamma\setminus(
\mu_0+\Gamma_{\leqslant \eta r_j})$ far away from $\mu_0$, we have
\begin{equation}
\label{easy estimate}
\frac{|\mu-\mu_0|}{|\mu|}
>
\frac{\eta r_j}{r_j}
= \eta.
\end{equation}
Thus these $\mu$'s, having cardinality
$|\mathbf{A}_{r_j}\cap c\Gamma\setminus(
\mu_0+\Gamma_{\leqslant \eta r_j})|\leqslant \mathsf{K}\cdot |r_j|^2/c^2$,
cause no trouble for~\eqref{core estimate}.
Lastly, we handle $\mu\in \mathbf{A}_{r_j}\cap c\Gamma\cap (
\mu_0+\Gamma_{\leqslant \eta r_j})$ simultaneously.
Note that
$c\Gamma\cap
\Gamma_{\leqslant \eta r_j}\setminus \{0\}$ can be decomposed
into $2$ horizontal parts consisting of $\pm\{\ell\cdot c
+0\cdot \sqrt{-1}\}_{\ell=1}^{[\frac{\eta}{c}r_j]}$, plus the remaining $4[\frac{\eta}{c} r_j]+2$ vertical parts consisting of
$\pm\{i\cdot c
+\ell\cdot c\sqrt{-1}\}_{\ell=1}^{[\frac{\eta}{c}r_j]}$
for $i=0, \pm 1, \pm 2, \dots, \pm [\frac{\eta}{c} r_j]$.
Each part
contains
consequential $[\frac{\eta}{c} r_j]$ points having absolute values $\geqslant 1\cdot c, 2\cdot c, \dots, [\frac{\eta}{c} r_j]\cdot c$ respectively.
Hence
\begin{equation}
\label{square matrix}
\prod_{0\neq \mu'\in c\Gamma\cap
\Gamma_{\leqslant \eta r_j}}\,|\mu'|\geqslant
([\frac{\eta}{c} r_j]!\cdot c^{[\frac{\eta}{c} r_j]})^{4[\frac{\eta}{c} r_j]+2+2}.
\end{equation}
Now it is time to apply the Stirling formula that
for every positive integer $n$, one has
\[
n!=n^n e^{-n} \sqrt{2\pi n}\,e^{\rho_n/12n}
\]
for some $|\rho_n|\leqslant 1$.
A straightforward computation yields
{\footnotesize
\begin{align}
\label{Stirling}
\mathrm{log}\,
\prod_{\mu_0\neq\mu
\in c\Gamma\cap (
\mu_0+\Gamma_{\leqslant \eta r_j})}
\frac{|\mu-\mu_0|}{|\mu|}
&
\geqslant
\mathrm{log}\,
\prod_{0\neq \mu'\in c\Gamma\cap
\Gamma_{\leqslant \eta r_j}}\,
\frac{|\mu'|}{2r_j}
\notag
\\
\text{[by~\eqref{square matrix}]}\qquad
&
\geqslant
\mathrm{log}\,
\Big(
\big(
[\frac{\eta}{c} r_j]!
\cdot c^{[\frac{\eta}{c} r_j]}
\big)^{4[\frac{\eta}{c} r_j]+4}
\Big)
-
\mathrm{log}\,
\Big(
(2r_j)^{4[\frac{\eta}{c} r_j]^2+4[\frac{\eta}{c} r_j]}
\Big)
\notag
\\
&
=
\bigg[
\mathrm{log}\,
\Big(
\big(
[\frac{\eta}{c} r_j]!
\big)^{4[\frac{\eta}{c} r_j]+4}
\Big)
-
\mathrm{log}\,
\Big(
\big(\frac{\eta}{c} r_j
\big)^{4[\frac{\eta}{c} r_j]^2+4[\frac{\eta}{c} r_j]}
\Big)
\bigg]
+
\mathrm{log}\,
\Big(
\big(\frac{\eta}{2}
\big)^{4[\frac{\eta}{c} r_j]^2+4[\frac{\eta}{c} r_j]}
\Big)
\
\notag
\\
\text{[by the Stirling formula]}\qquad
&
\geqslant
-\mathsf{K} \cdot |r_j|^2/c^2.
\end{align}
}
Now the remaining problem is that $c\Gamma\cap (
\mu_0+\Gamma_{\leqslant \eta r_j})$ might exceed $\mathbf{A}_{r_j}$.
Let us decompose $c\Gamma\cap (
\mu_0+\Gamma_{\leqslant \eta r_j})$ with respect to $\mathbf{A}_{r_j}$ into two parts $(\mu_0+ P_{in})\cup (\mu_0+P_{out})$,
where the first (resp. second) part lies entirely in (resp. outside) $\mathbf{A}_{r_j}$.
Since $\eta$ is small,
we can find some point $y\in \mathbf{A}_{r_j}\cap c\Gamma$ such that $y+P_{out}\subset \mathbf{A}_{r_j}$
stays away from $\mu_0+\Gamma_{\leqslant 8\eta r_j}$. See the picture below for illustration.
\begin{center}
\scalebox{.70}{\input{picture2.pdf_t}}
\end{center}
Lastly, we decompose
$\mathbf{A}_{r_j}\cap c\Gamma$ into $3$ disjoint parts,
$\mu_0+ P_{in}$,
$y+ P_{out}$ and $R$ the remaining. Note that $\prod_{\mu\in y+P_{out}}\frac{|\mu-\mu_0|}{|\mu|}
\geqslant
\prod_{\mu\in \mu_0+P_{out}}\frac{|\mu-\mu_0|}{|\mu|}
$, because each factor on the left-hand-side
$\geqslant \frac{8\eta r_j}{r_j}=8\eta$, while each factor on the right-hand-side
$\leqslant \frac{\sqrt{2}\eta r_j}{r_i/4}=4\sqrt{2}\eta$. Thus
$
\prod_{\mu_0\neq\mu\in \mathbf{A}_{r_j}\cap c\Gamma}\frac{|\mu-\mu_0|}{|\mu|}
\geqslant
\prod_{\mu_0\neq\mu
\in c\Gamma\cap (
\mu_0+\Gamma_{\leqslant \eta r_j})}
\frac{|\mu-\mu_0|}{|\mu|}
\cdot
\prod_{\mu\in R}
\frac{|\mu-\mu_0|}{|\mu|}
$.
By the estimates~\eqref{easy estimate},~\eqref{Stirling} and that the cardinality $| R|\leqslant \mathsf{K}\cdot r_j^2/c^2$,
we conclude the proof.
\end{proof}
\section{\bf Estimates}
\label{section: estimate area of discs}
\subsection{$f(z)$ is close to $\mathcal{C}_{\infty}$ unless $z$ is near $\Lambda$}
\label{f(z) near infinity curve}
Recalling~\eqref{psi_1},
we first rewrite
\[
||\psi\cdot s_m(z)||_{h_m}
=
||\psi_1\cdot s_m(z)||_{h_m}
\cdot
|\diamondsuit|
\]
to concentrate positivity to the first factor,
where
if $z\in \mathbb{D}(\lambda, 1)$ for certain $\lambda\in \Lambda$ then $\diamondsuit=(1-\frac{z}{\lambda})e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda^2}}$, otherwise $\diamondsuit=1$.
Now thanks to~\eqref{exponential growth of s},~\eqref{psi>},
the left part satisfies that
\[
||\psi_1\cdot s_m(z)||_{h_m}
\geqslant
\exp
\big(
(m\cdot\alpha-\mathsf{K}/c^2)
\cdot
|z|^2
\big).
\]
A key trick in this paper is that we choose $m$ and $c$ such that
\begin{equation}
\label{key choices of m, k}
m\cdot\alpha-\mathsf{K}/c^2
>0.
\end{equation}
Thus
for any small $\epsilon>0$,
for all sufficiently large $|z|\gg 1$ with $\text{dist}(z, \Lambda)\geqslant \epsilon$,
there holds
\begin{equation}
\label{psi.s large}
||\psi\cdot s_m(z)||_{h_m}
\gg
1,
\end{equation}
i.e., $f(z)$ is very close to $\mathcal{C}_{\infty}$.
Indeed, if $\text{dist}(d, \Lambda)\geqslant 1$, then $\diamondsuit=1$, and there is nothing to proof; otherwise $\epsilon\leqslant |z-\lambda|< 1$ for some $\lambda\in \Lambda$,
hence
$
|\diamondsuit|
\geqslant
\frac{|z-\lambda|}{|\lambda|}
\cdot
\exp(-\frac{|z|}{|\lambda|}-\frac{|z|^2}{2|\lambda|^2})
\geqslant
\frac{\epsilon}{|z|+1}
\cdot
\exp(-\mathsf{K})
$, thus
$||\psi\cdot s(z)||_{h_m}=||\psi_1\cdot s(z)||_{h_m}
\cdot
|\diamondsuit|$ is very large when $|z|\gg 1$.
\subsection{Bound the area $\int_{\mathbb{D}_{2r_i}}{f}^*\omega_{X}$ from above}
Fix a K\"ahler form $\omega_{\mathcal{C}}=\dif\dif^c\big(|z|^2\big)$ on $\mathcal{C}$.
The metric $h_m=h^{\otimes m}$ of $\mathcal{L}_m=\mathcal{L}^{\otimes m}$ together with the standard Euclidean metric
$|dz|$ on $\mathbb{C}$ provide a metric for the vector bundle $\mathcal{E}:=\mathcal{L}_m\oplus \mathbb{C}$, and therefore it induces a metric on the tautological line bundle $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)$ on $\mathbb{P}(\mathcal{E})=X$.
Restricting to any fiber of
$\pi_1: X\longrightarrow \mathcal{C}$,
the curvature form $\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}$ is strictly negative due to the property of the Fubini-Study metric for the tautological line bundle $\mathcal{O}_{\mathbb{P}^1}(-1)$. Therefore, by standard compactness argument, for sufficiently small $\epsilon_1>0$, we receive a
K\"ahler form on $X$ of the shape
\begin{equation}
\label{kahler form on X}
\omega_X
:=
\pi_1^*\omega_{\mathcal{C}}-\epsilon_1\, \Theta_{{O}_{\mathbb{P}(\mathcal{E})}(-1)}.
\end{equation}
We can identify the tautological space $\mathcal{L}_m=\{(z, \xi): \xi\in \mathcal{L}_m|_{z}\}$ with an open set of $ \mathbb{P}(\mathcal{L}_m\oplus \mathbb{C})$, by
mapping $(z, \xi)\mapsto (z, [\xi\oplus 1])$. Thus in the local coordinates $(z, \xi)$,
the curvature
\[
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}=
-\dif\dif^c\big(
\mathrm{log}\,(\|\xi\|_{h_m}^2+1)
\big)
\]
is of Fubini-Study shape.
For $r$ lies in $[\frac{1}{3}r_i, 3r_i]$ for some $i\geqslant 1$, by Jensen's formula
we receive
{\footnotesize
\begin{align}
\label{Jenson formula for Funiby-Study}
\int_{1}^r\frac{\dif t}{t}\int_{\mathbb{D}_t}
f^*
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
&
=
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m}+1)(re^{i\theta})\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m}+1)(e^{i\theta})\dif \theta
\notag
\\
\text{[use~\eqref{exponential growth of s},~\eqref{psi<}]}\qquad
&
\geqslant
-\mathsf{K}\cdot r_i^2
\notag,
\end{align}
}
whence the Nevanlinna order function
satisfies the estimate
\begin{equation}
\label{order function estimate <=}
T_{f, {r}}(\omega_X)
=\int_{1}^{r}\frac{\dif t}{t}\int_{\mathbb{D}_t}f^*\omega_X
=
\int_{1}^{r}\frac{\dif t}{t}\int_{\mathbb{D}_t}f^*\pi_1^*\,\omega_{\mathcal{C}}
-
\epsilon_1\,
\int_{1}^{r}\frac{\dif t}{t}\int_{\mathbb{D}_t}f^*\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
\leqslant
\mathsf{K}\cdot r_i^2.
\end{equation}
Here is a useful observation
\begin{equation}
\label{easy trick}
T_{{f},{3r_i}}(\omega_{X})
=\int_{1}^{{3r_i}}\frac{\dif t}{t}\int_{\mathbb{D}_t}{f}^*\omega_{X}
\geqslant \int_{{2r_i}}^{{3r_i}}\frac{\dif t}{t}\int_{\mathbb{D}_{2r_i}}{f}^*\omega_{X}
=
\ln (3/2) \cdot
\int_{\mathbb{D}_{2r_i}}
{f}^*\omega_{X}.
\end{equation}
Combing the two estimates above, we conclude that
\begin{equation}
\label{disc image area is bounded}
\int_{\mathbb{D}_{2r_i}}{f}^*\omega_{X}\leqslant\mathsf{K}\cdot r_i^2.
\end{equation}
\subsection{Bound the area $\int_{\mathbb{D}_{r_i/3}}{f}^*\omega_{X}$ from below}
By Jensen's formula,
we have
{\footnotesize
\begin{align}
\label{tricky Jensen}
\int_{r_i/4}^{r_i/3}\frac{\dif t}{t}\int_{\mathbb{D}_t}
f^*
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
&
=
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m}+1)(\frac{r_i}{3}e^{i\theta})\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m}+1)(\frac{r_i}{4}e^{i\theta})\dif \theta
\notag
\\
\text{[for $r_i\gg 1$]}
\qquad
&
=
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m})(\frac{r_i}{3}e^{i\theta})\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m})(\frac{r_i}{4}e^{i\theta})\dif \theta
+
o(1).
\end{align}
}
Since the holomorphic function $\psi$ is nowhere vanishing on $\overline{\mathbb{D}}_{r_i/3}\setminus \mathbb{D}_{r_i/4}$ by our construction,
$\mathrm{log}\, |\psi|^2$
is harmonic on $\overline{\mathbb{D}}_{r_i/3}\setminus \mathbb{D}_{r_i/4}$. Thus
\[
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\, |\psi|^2(\frac{r_i}{3}e^{i\theta})\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\, |\psi|^2(\frac{r_i}{4}e^{i\theta})\dif \theta
=
0.
\]
Hence we can simplify~\eqref{tricky Jensen} as
{\footnotesize
\begin{align*}
\label{tricky Jensen 2}
\int_{r_i/4}^{r_i/3}\frac{\dif t}{t}\int_{\mathbb{D}_t}
f^*
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
&
=
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\| s_m\|^2_{h_m})(\frac{r_i}{3}e^{i\theta})\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\| s_m\|^2_{h_m})(\frac{r_i}{4}e^{i\theta})\dif \theta
+
o(1)
\notag
\\
\text{[by~\eqref{exponential growth of s}]}\qquad
&
=
-\frac{1}{4\pi}\,
\int_{0}^{2\pi}
2m\alpha\bigg|\frac{r_i}{3}\bigg|^2
+\frac{1}{4\pi}\,
\int_{0}^{2\pi}
2m\alpha\bigg|\frac{r_i}{4}\bigg|^2
+o(1)
\notag\\
&
=
-
\frac{7m\alpha}{144}r_i^2
+
o(1).
\end{align*}
}
Hence
{\footnotesize
\begin{equation}
\label{key estimates about big dics}
\int_{r_i/4}^{r_i/3}\frac{\dif t}{t}\int_{\mathbb{D}_t}f^*\omega_X
=
\int_{r_i/4}^{r_i/3}\frac{\dif t}{t}\int_{\mathbb{D}_t}f^*\pi_1^*\omega_{\mathcal{C}}
-
\epsilon_1
\int_{r_i/4}^{r_i/3}\frac{\dif t}{t}\int_{\mathbb{D}_t}
f^*
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
=
\frac{7}{144}
(\frac{\pi}{2}+
\epsilon_1 m\alpha)r_i^2
+
o(1).
\end{equation}
}
Noting that
\[
\int_{r_i/4}^{r_i/3}\frac{\dif t}{t}\int_{\mathbb{D}_t}f^*\omega_X
\leqslant
\int_{r_i/4}^{r_i/3}\frac{\dif t}{t}\int_{\mathbb{D}_{r_i/3}}f^*\omega_X
=
\mathrm{log}\,(4/3)\cdot
\int_{\mathbb{D}_{r_i/3}}f^*\omega_X,
\]
we conclude that for $i\gg 1$ there holds
\begin{equation}
\label{disc area is large enough}
\int_{\mathbb{D}_{r_i/3}}f^*\omega_X
\geqslant
\mathsf{K}\cdot r_i^2.
\end{equation}
\subsection{Estimates of $
\int_{\mathbb{D}(\lambda,\epsilon)}f^*\omega_X
$}
Mark the curve $\mathcal{C}_0\subset X$ induced by the zero section of $\mathcal{L}_m$. Note that $\mathcal{C}_0$ and $\mathcal{C}_{\infty}$ are disjoint. Contrasting to the observation in Subsection~\ref{f(z) near infinity curve},
for every $\lambda\in \Lambda$, since $\psi(\lambda)=0$, $f(\lambda)$ must lies in $\mathcal{C}_0$, which keeps certain positive distance to $\mathcal{C}_{\infty}$.
Let $\epsilon$
be a small positive radius.
Denote by
$\mathbb{D}([\lambda], \epsilon)\subset \mathcal{C}$ the disc centered at $[\lambda]$ with the radius $\epsilon$. Then the image of ${f}\big(\mathbb{D}({\lambda, \epsilon})\big)$ is contained in the neighborhood
$U_{[\lambda], \epsilon}:=
\pi_1^{-1}
(\mathbb{D}([\lambda], \epsilon))
$
of $\mathbb{P}^1_{[\lambda]}:=\pi_1^{-1}([\lambda])$.
Recall that the metric $\omega_X$ on $X$ is given in~\eqref{kahler form on X}. Firstly, since $\pi_1 \circ f=\pi_0$, where $\pi_0:\mathbb{C}\rightarrow\mathbb{C}/\Gamma$ is the canonical projection, we have
$
\int_{\mathbb{D}(\lambda,\epsilon)}f^*\pi_1^*\omega_{\mathcal{C}}
=
\pi\epsilon^2$.
Next, we estimate $
\int_{\mathbb{D}(\lambda,\epsilon)}f^*\Theta_{{O}_{\mathbb{P}(\mathcal{E})}(-1)}
$.
Recalling~\eqref{psi_1},
we can rewrite
$
\psi(z)=(
\psi_1\cdot e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda}}
)
\cdot
(
1
-
\frac{z}{\lambda}
)
$,
where the factor $\psi_2:= \psi_1\cdot e^{\frac{z}{\lambda}+\frac{z^2}{2\lambda}}$
is nonvanishing for $z\in \mathbb{D}(\lambda, \epsilon)$. Thus $\mathrm{log}\,|\psi_2|$ is a harmonic function on $\mathbb{D}(\lambda, \epsilon)$. Now, using Jensen's formula, for $|\lambda|\gg 1$ large, we can estimate
{\footnotesize
\begin{align}
\label{area of disc D(lambda,t), O P()}
\int_{\epsilon}^{2\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,t)}
f^*
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
&=
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m}+1)(\lambda+2\epsilon e^{i\theta})\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi^2\cdot s_m\|^2_{h_m}+1)(\lambda+\epsilon e^{i\theta})\dif \theta\notag\\
\text{[by~\eqref{psi.s large}, as $|\lambda|\gg 1$]}
\qquad
&
=
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi\cdot s_m\|^2_{h_m})(\lambda+2\epsilon e^{i\theta})\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\|\psi^2\cdot s_m\|^2_{h_m})(\lambda+\epsilon e^{i\theta})\dif \theta +o(1)
\notag\\
&=
\bigg[-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\| s_m\|^2_{h_m})(\lambda+2\epsilon e^{i\theta})\dif \theta+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(\| s_m\|^2_{h_m})(\lambda+\epsilon e^{i\theta})\dif \theta\bigg]
\\
&
\qquad
+
\bigg[-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,| \psi|^2(\lambda+2\epsilon e^{i\theta})\dif \theta+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,| \psi|^2(\lambda+\epsilon e^{i\theta})\dif \theta\bigg]
+o(1).
\notag
\end{align}
}
By \eqref{exponential growth of s}, the first bracket $[\cdots]$ in \eqref{area of disc D(lambda,t), O P()} can be computed as
\begin{align}
\label{first term, norm s}
-\frac{1}{4\pi}\,
\int_{0}^{2\pi}
2m
\alpha|\lambda+2\epsilon e^{i\theta}|^2\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
2m
\alpha|\lambda+\epsilon e^{i\theta}|^2\dif \theta
=
\dfrac{-3 m
\alpha\epsilon^2}{2\pi}.
\end{align}
Now we separate $\mathrm{log}\,|\psi|(z)=\mathrm{log}\,|\psi_2|\cdot \mathrm{log}\,|1-\frac{z}{\lambda}|$. Thanks to the harmonicity of
$\mathrm{log}\,|\psi_2|$, we have
\[
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(| \psi_2|^2)(\lambda+2\epsilon e^{i\theta})\dif \theta+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,(| \psi_2|^2)(\lambda+\epsilon e^{i\theta})\dif \theta
=
0.
\]
Thus we can calculate the second bracket $[\cdots]$ of \eqref{area of disc D(lambda,t), O P()} as
\begin{align}
\label{second term, norm psi}
-
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,
\bigg|
1
-
\frac{\lambda+2\epsilon e^{i\theta}}{\lambda}
\bigg|^2
\dif \theta
+
\frac{1}{4\pi}\,
\int_{0}^{2\pi}
\mathrm{log}\,
\bigg|
1
-
\frac{\lambda+\epsilon e^{i\theta}}{\lambda}
\bigg|^2
\dif \theta
=
-\mathrm{log}\, 2.
\end{align}
Hence it follows from \eqref{area of disc D(lambda,t), O P()}, \eqref{first term, norm s}, \eqref{second term, norm psi} that
\[
\int_{\epsilon}^{2\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,t)}
f^*
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
=
-\dfrac{3m\alpha\epsilon^2}{2\pi}
-\mathrm{log}\, 2
+
o(1)
\qquad
{\scriptstyle(\text{for }|\lambda|\gg 1)}.
\]
Therefore
\begin{equation}
\label{estimate are of disc lambda epsilon, epsilon--2epsilon}
\int_{\epsilon}^{2\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,t)}
f^*
\omega_X
=
\pi\epsilon^2
+\epsilon_1
\bigg(
\dfrac{3m\alpha\epsilon^2}{2\pi}
+
\mathrm{log}\, 2
+
o(1)
\bigg).
\end{equation}
By the same trick as~\eqref{easy trick}, we have
\[
\int_{\epsilon}^{2\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,t)}
f^*
\omega_X
\geqslant
\int_{\epsilon}^{2\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X
\geqslant
\mathrm{log}\, 2\cdot \int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X.
\]
Combining the above two estimates, we conclude
\begin{equation}
\label{small disc high bound}
\int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X
\leqslant
\dfrac{1}{\mathrm{log}\, 2}
\bigg(
\pi\epsilon^2
+\epsilon_1
\Big(
\dfrac{3m\alpha\epsilon^2}{2\pi}
+
\mathrm{log}\, 2
+
o(1)
\Big)
\bigg)
\qquad
{\scriptstyle(\text{for }|\lambda|\gg 1)}.
\end{equation}
Next, we provide an lower bound for $\int_{\mathbb{D}(\lambda,\epsilon)}
f^*\omega_X$. Substituting $\epsilon$ by $\frac{\epsilon}{2}$ in \eqref{estimate are of disc lambda epsilon, epsilon--2epsilon}, we receive that
\begin{equation}
\label{estimate are of disc lambda epsilon, 1/2epsilon--epsilon}
\int_{\frac{\epsilon}{2}}^{\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,t)}
f^*
\Theta_{\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)}
=
\frac{\pi\epsilon^2}{4}
+\epsilon_1
\bigg(
\dfrac{3m\alpha\epsilon^2}{8\pi}
+
\mathrm{log}\, 2
+
o(1)
\bigg).
\end{equation}
Note that
\[
\int_{\frac{\epsilon}{2}}^{\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,t)}
f^*
\omega_X
\leqslant
\int_{\frac{\epsilon}{2}}^{\epsilon}
\frac{\dif t}{t}\int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X
=
\mathrm{log}\, 2\cdot \int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X.
\]
Hence it follows from the above two estimates that
\begin{equation}
\label{small disc low bound}
\int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X
\geqslant
\dfrac{1}{\mathrm{log}\, 2}
\bigg(
\frac{\pi\epsilon^2}{4}
+\epsilon_1
\Big(
\dfrac{3m\alpha\epsilon^2}{8\pi}
+
\mathrm{log}\, 2
+
o(1)
\Big)
\bigg)
\qquad
{\scriptstyle(\text{for }|\lambda|\gg 1)}.
\end{equation}
\subsection{Area estimates of $f(\mathbb{C})$ near horizontal curves}
\label{estimate area near horizontal curves}
An irreducible algebraic curve $D\subset X$
is said to be vertical if
$\pi_1(D)$ is a point; otherwise it is called horizontal, in the sense that $\pi_1(D)=\mathcal{C}$.
Firstly, for a vertical curve $\mathbb{P}^1_{[y]}=\pi_1^{-1}([y])$, by the estimates~\eqref{small disc high bound} and~\eqref{small disc low bound}, the area of $f(\mathbb{D}_{r})$ near $\mathbb{P}^1_{[y]}$,
as $r\rightarrow \infty$,
is mostly
determined by
asymptotic
growth of $|\mathbb{D}_r\cap \Lambda|$.
Next, for the horizontal curve $D=\mathcal{C}_{\infty}$,
by Subsection~\ref{f(z) near infinity curve},
$f(\mathbb{D}_{r})$ shall concentrate a large portion of area near $\mathcal{C}_{\infty}$
as $r\rightarrow \infty$.
Lastly, for any other irreducible horizontal curve $D\not= \mathcal{C}_{\infty}$,
we devote this subsection
to prove that,
roughly speaking,
every time when
$f(\mathbb{D}_r)$ intersects with $D$, it contributes negligible area about there.
\smallskip
To start with, we take a general point $d_0\in D\setminus \mathcal{C}_{\infty}$
such that $\pi_1|_{D}$
is regular at $d_0$, i.e., some small open neighborhood $U\subset D $ of $d_0$ is a graph over $\pi_1(U)$ containing $ \pi_1(d_0)=:c_0$.
By shrinking $U$ we may assume that $U$ stays away from $\mathcal{C}_{\infty}$, that $\pi_1(U)$ is a small disc $\mathbb{D}(c_0, 3\delta)$ for some $\delta>0$, and that
the line bundle $\mathcal{L}_m$ locally has a trivialization
$\mathcal{L}_m|_{\mathbb{D}(c_0, 3\delta)}
\cong
\mathbb{D}(c_0, 3\delta)\times \mathbb{C}$, which extends to an identification
\begin{equation}
\label{local chart of X}
\vartheta:
\pi_1^{-1}(\mathbb{D}(c_0, 3\delta))
\xrightarrow[]{\cong}
\mathbb{D}(c_0, 3\delta)\times \mathbb{P}^1(\mathbb{C})
\end{equation}
by fiberwised
compactification
$\mathbb{C}\hookrightarrow \mathbb{P}^1(\mathbb{C})$ sending $z \mapsto [z: 1]$.
Hence we can read the coordinates of $U$ in the chart $ \mathbb{D}(c_0, 3\delta)\times \mathbb{C}$
as the graph of
some holomorphic map
$u: \mathbb{D}(c_0, 3\delta)\rightarrow \mathbb{C}$.
Let $p_1, p_2$ be the projections of
$\mathbb{D}(c_0, 3\delta)\times \mathbb{P}^1(\mathbb{C})$ to the two factors.
Let $\omega_{\FS}$ be the Fubini-Study form on $\mathbb{P}^1(\mathbb{C})$.
By compactness argument,
the metric $p_1^*\omega_{\mathcal{C}}+p_2^*\omega_{\FS}$ is comparable with $(\vartheta^{-1})^*\omega_X$ on $\mathbb{D}(c_0, \frac{5\delta}{2})\times \mathbb{P}^1(\mathbb{C})$, namely
\begin{equation}
\label{comparable metrics}
\mathsf{K}_{c_0, \delta, \vartheta}^{-1}\cdot(p_1^*\omega_{\mathcal{C}}+p_2^*\omega_{\FS})\leqslant
(\vartheta^{-1})^*\omega_X
\leqslant \mathsf{K}_{c_0, \delta, \vartheta}\cdot(p_1^*\omega_{\mathcal{C}}+p_2^*\omega_{\FS}).
\end{equation}
Fix a positive number
$\epsilon\ll \delta$.
Then the neighborhood $\pi_1^{-1}(\mathbb{D}(c_0, \delta))\cap D$ of $d_0$ in the coordinates reads as
\[
U_1
:=
\{
(z, w) :
z\in \mathbb{D}(c_0, \delta),
w=u(z)
\},
\]
whose $\epsilon$--open neighborhood is
\[
U_{1}^{\epsilon}
:=
\{(z, w): z\in\mathbb{D}(c_0, \delta+\epsilon), |w-u(z)|<\epsilon\}.
\]
Fix a positive small number
$\delta'<\delta/2$.
By Subsection~\ref{f(z) near infinity curve},
for $|z|\gg 1$ large with
dist$(z, \Lambda)>\delta'$, one sees that
$f(z)$ is very close to $\mathcal{C}_{\infty}$,
hence it is outside $U_1^{\epsilon}$.
Thus for bounding the area of
$f(\mathbb{D}_r)\cap U_1^{\epsilon}$ from above by $o(1)\cdot r^2$,
we only need to show that,
as $\lambda\in \Lambda$ with $|\lambda|\gg 1$, the area
$f(\mathbb{D}(\lambda, \delta'))\cap U_1^{\epsilon}$ is negligible $o(1)$.
\begin{obs}
\label{shrink radius}
Put $f_2:=p_2\circ \vartheta\circ f$.
For $\lambda\in\Lambda$ with $|\lambda|\gg 1$ and
$[\lambda]\in \mathbb{D}(c_0, \delta+\delta')$,
one has
\[
(\vartheta\circ f)^{-1}(U_1^{\epsilon})
\cap
\mathbb{D}(\lambda, \delta')
\subset
{f_2}^{-1}
\Big(\mathbb{D}
\big(
u([\lambda]), 2\epsilon\big)
\Big)
\cap
\mathbb{D}(\lambda, \delta').
\]
\end{obs}
\begin{proof}
By continuity of $u$ and by compactness of $\overline\mathbb{D}(c_0, \frac{5}{2}\delta)$, there exists
some positive number $\delta_{\epsilon}<\delta'$
such that, for any two points $x_1, x_2\in
\overline\mathbb{D}(c_0, \frac{5}{2}\delta)$ with $|x_1-x_2|< \delta_{\epsilon}$,
there holds
$|u(x_1)-u(x_2)|<\epsilon$. By Subsection~\ref{f(z) near infinity curve},
for all $\lambda
\in \Lambda$ with $|\lambda|\gg1$, the image of
$\mathbb{D}(\lambda, \delta')\setminus
{\mathbb{D}}(\lambda, \delta_{\epsilon})$
under $\vartheta\circ f$ is outside $U_1^{\epsilon}$, therefore
\[
(\vartheta\circ f)^{-1}(U_1^{\epsilon})
\cap
\mathbb{D}(\lambda, \delta')
\subset
(\vartheta\circ f)^{-1}(U_1^{\epsilon})
\cap
\mathbb{D}(\lambda, \delta_{\epsilon}).
\]
By definition, every element
$z$ in the right-hand-side satisfies that
$|f_2(z)-u([z])|<\epsilon$ and
$|z-\lambda|< \delta_{\epsilon}$.
Thus $|f_2(z)-u([\lambda])|
\leqslant
|f_2(z)-u([z])|+ |u([z])-u([\lambda])|
< \epsilon+\epsilon=2\epsilon$,
which concludes the proof.
\end{proof}
Now for every $v\in \mathbb{C}$ having absolute value $|c|\leqslant R:=\max\{|u(z)|+\epsilon:
z\in\overline\mathbb{D}(c_0, \frac{5}{2}\delta)\}<\infty$, for $\lambda\in \Lambda$ with $|\lambda|\gg1$ and
$[\lambda]\in \mathbb{D}(c_0, \delta+\delta')$, consider the restricted holomorphic function
\[
f_2: \mathbb{D}(\lambda, \delta')\longrightarrow \mathbb{C}.
\]
Since $|f_2|\gg 1$ (in particular $|f_2|>R$) on $\partial \mathbb{D}(\lambda, \delta')$ by Subsection~\ref{f(z) near infinity curve}, by the Argument Principle, the number of solutions of the equation
$f_2(y)=v$ on the disc $\mathbb{D}(\lambda, \delta')$, counting multiplicities, equals to
\[
\frac{1}{2\pi \sqrt{-1}}
\int_{z\in \partial \mathbb{D}(\lambda, \delta')}
\frac{(f_2-v)'}{f_2-v}(z)
\dif z.
\]
Noting that the above quantity takes integer value,
and that it varies continuously with respect to $v\in \mathbb{D}_R$,
it must be a constant for every $v\in \mathbb{D}_R$. Now checking the special value $v=0$, we see that
the number of solution is just $1$. Thus Observation~\ref{shrink radius} implies that, for
$\lambda\in\Lambda$
with $|\lambda|$ sufficiently large and $[\lambda]\in \mathbb{D}(c_0, \delta+\delta')$,
we have
\begin{equation}
\label{inequality 1}
\area\Big(
(\vartheta\circ f)^{-1}(U_1^{\epsilon})
\cap
\mathbb{D}(\lambda, \delta')
\Big)_{f_2^*\omega_{\FS}}
\leqslant
\area
\big(
u([\lambda]), 2\epsilon\big)_{\omega_{\FS}}
\leqslant
\mathsf{K}\cdot \epsilon^2.
\end{equation}
Also, by Subsection~\ref{f(z) near infinity curve},
for every positive number $\epsilon'>0$,
for $\lambda\in \Lambda$ with sufficiently large $|\lambda|$,
we have
\[
(\vartheta\circ f)^{-1}(U_1^{\epsilon})
\cap
\mathbb{D}(\lambda, \delta')
\subset
\mathbb{D}(\lambda, \epsilon').
\]
Therefore
\begin{equation}
\label{inequality 2}
\area\Big(
(\vartheta\circ f)^{-1}(U_1^{\epsilon})
\cap
\mathbb{D}(\lambda, \delta')
\Big)_{\pi_1^*\omega_{\mathcal{C}}}
\leqslant
\area
\Big(
\mathbb{D}(\lambda, \epsilon')
\Big)_{\pi_1^*\omega_{\mathcal{C}}}
=
\pi\cdot\epsilon'^2.
\end{equation}
Summarizing~\eqref{comparable metrics},~\eqref{inequality 1},~\eqref{inequality 2}, for
$\lambda \in \Lambda$ with $|\lambda|\gg1$ and $[\lambda]\in \mathbb{D}(c_0, \delta+\delta')$, we have
\begin{equation}
\label{very subtle estimate}
\area\Big(
(\vartheta\circ f)^{-1}(U_1^{\epsilon})
\cap
\mathbb{D}(\lambda, \delta')
\Big)_{f^*\omega_{X}}
\leqslant
\mathsf{K}_{c_0, \delta, \vartheta}\cdot
(\mathsf{K}\,
\epsilon^2
+
\pi\,\epsilon'^2
).
\end{equation}
\subsection{Area of $f$ near $\lambda\in \Lambda$ revisit}
An alternative way to interpret~\eqref{small disc low bound} is the following
\begin{obs}
\label{obs 4.2}
Let $\delta'>0$ be a small positive number. Let
$U$ be an open neighborhood of $\mathcal{C}_{\infty}$ such that its closure $\overline{U}$ stays away from $\mathcal{C}_0$.
Then one has the estimate
\begin{equation}
\label{very subtle estimate again}
\area\Big(
(X\setminus U)
\cap
f\big(
\mathbb{D}(\lambda, \delta')
\big)
\Big)_{\omega_{X}}
\geqslant
\mathsf{K}_U
\qquad
{
(\forall \lambda\in \Lambda\text{ with } |\lambda|
\gg 1)}.
\end{equation}
\end{obs}
This strengthens~\eqref{small disc low bound}
in a qualitative sense,
and will be helpful for discussing diffuse parts later. Before going to the proof of the above result, recall the following special case of Wirtinger's inequality.
\begin{pro}
\label{curve in ball area}
\text{(c.f.~ \cite[page~7]{Duval2017-2})}
Let $C$ be a proper holomorphic curve in the ball
$B(0, \epsilon)\subset \mathbb{C}^n$
passing through $0$.
Then with the standard Euclidean metric, one has
$
\area(C)\geqslant \pi \epsilon^2.$
\qed
\end{pro}
\begin{proof}[Proof of Observation~\ref{obs 4.2}]
By compactness, $\mathcal{C}_0$ can be covered by finitely many open neighborhoods $U_i$, being disjoint with $U$, with charts
$\vartheta_i: U_i\rightarrow V_i\subset \mathbb{C}^2$.
By shrinking $U_i$'s if necessary, we can assume that every pull-back by $\vartheta_i$ of the standard Euclidean metric on $\mathbb{C}^2$ is comparable with $\omega_X$. Again by the compactness of $\mathcal{C}_0$,
for every point $c\in \mathcal{C}_0$, certain chart $V_i$ of
$U_i\owns c$ contains a sufficiently large ball centered at
$\vartheta_i(c)$
with a uniform radius $r>0$.
Now, by Subsection~\ref{f(z) near infinity curve}, for $\lambda\in \Lambda$ with $|\lambda|\gg1$,
for $c=f(\lambda)\in \mathcal{C}_0$, in the chart $V_i$ we see that
$\vartheta_i\big( f(
\mathbb{D}(\lambda, \delta')
)\cap U_i\big)$
contains a proper holomorphic curve in the ball
$B(\vartheta_i(c), r)$,
having positive area $\geqslant \pi r^2$
by Proposition~\ref{curve in ball area}. The desired conclusion then follows from the comparability of metrics.
\end{proof}
\section{\bf Algorithm}
\label{section: algorithm}
First of all, we require that $m, c$ satisfy the condition~\eqref{key choices of m, k}.
Next, we choose distinct points in
a strip of $\mathcal{D}$
\begin{equation}
\label{choose y_i}
\{y_i\}_{i\in \mathbb{Z}_+}
\subset
\{x+y\sqrt{-1}:
1/6\leqslant x<1/3, 0\leqslant y<1\}.
\end{equation}
A collection of $N\geqslant 1$ points
\[
b_1,\dots, b_N\in\mathcal{D}_{\mathsf{R}}
:=
\{x+y\sqrt{-1}:
1/2\leqslant x<1, 0\leqslant y<1\},
\]
located in the right-half of $\mathcal{D}$,
is said to be {\sl distributed sparsely}, if for any disc $\mathbb{D}(a,r)$, the following cardinality estimate holds
\begin{equation}
\label{distribute sparsely}
\big|\mathbb{D}(a,r)
\cap \{b_1,\dots,b_N\}\big|
\leqslant\max\{1, \mathsf{K}\cdot r^2 N\}.
\end{equation}
For instance, this can be reached by choosing distinct points
\[
b_1, \dots, b_N
\in
\Big\{
\frac{[\sqrt{N}]+1+\ell_1}{2[\sqrt{N}]+2}
+
\frac{\ell_2}{[\sqrt{N}]+1}\sqrt{-1}
\Big\}_{0\leqslant\ell_1,\ell_2\leqslant[\sqrt{N}]
}.
\]
Let $\mathcal{S}=\{I\subset \mathbb{Z}_{+}: I\text{ is a finite nonempty set, or } I=\varnothing, \text{ or } I=\mathbb{Z}_{+}\}$. Then $\mathcal{S}$ is countable, i.e., there exists some bijection $\sigma:\mathcal{S}\rightarrow\mathbb{Z}_{+}$. On the other hand, we can decompose $\mathbb{Z}_{+}$ into some infinite disjoint union $\cup_{i\in\mathbb{Z}_{+}}\mathcal{Z}_i$, where each component $\mathcal{Z}_i$ contains infinitely many integers. For every $I\in\mathcal{S}$, write all the elements of $\mathcal{Z}_{\sigma(I)}$ in the increasing order as
$
Z_1^{I}<{Z}_2^{I}<{Z}_3^{I}<\cdots
$.
Thus we can rearrange
\begin{equation*}
\label{decompostion of Z+}
\mathbb{Z}_{+}
=
\cup_{I\in \mathcal{S}}\mathcal{Z}_{\sigma(I)}
=\cup_{I\in\mathcal{S}}
\cup_{j\geqslant 1}
\{{Z}_j^{I}\}.
\end{equation*}
For every positive integer $i=Z^I_j$,
we now choose all the $x_{\mu}\in \mathcal{D}$ for $\mu\in
\mathbf{A}_{r_i}\cap c\Gamma$ as follows.
\begin{itemize}
\item{Case I}: $I=\varnothing$.
We require that all the $x_{\mu}$'s
are distributed sparsely in $\mathcal{D}_{\mathsf{R}}$.
\smallskip
\item{Case II}: $I=\{i_1,\dots,i_k\}$ is some finite set of $k\geqslant 1$ elements, and $j\geqslant 1$ is an odd integer.
We choose all
$x_{\mu}$ from $\{y_{i_1}, \dots, y_{i_k}\}$, such that, for every
$\ell=1,\dots, k$, $x_{\mu}=y_{i_\ell}$ for at least
$[\frac{|\mathbf{A}_{r_i}\cap c\Gamma|}{k}]$ times.
\smallskip
\item{Case II'}: $I=\{i_1,\dots,i_k\}$ is some finite set of $k\geqslant 1$ elements, and $j\geqslant 1$ is an even integer.
We choose some $x_{\mu}=y_{i_\ell}$ for
$[\frac{|\mathbf{A}_{r_i}\cap c\Gamma|}{2k}]$ times, where $\ell=1,\dots, k$,
and we require the remaining $x_{\mu}$'s to be distributed sparsely in $\mathcal{D}_{\mathsf{R}}$.
\smallskip
\item{Case III}: $I=\mathbb{Z}_{+}$, and $j\geqslant 1$ is odd.
Fix a sequence of positive numbers $\{\alpha_j\}_{j=1}^{\infty}$ with $\sum_{j=1}^{\infty}\alpha_j=1$.
We choose all
$x_{\mu}$ from $\{y_{\ell}\}_{\ell=1}^{\infty}$, such that, for every
$\ell\geqslant 1$,
$x_{\mu}=y_{\ell}$ for at least
$[\alpha_{\ell}\cdot |\mathbf{A}_{r_i}\cap c\Gamma|]$ times.
\smallskip
\item{Case III'} $I=\mathbb{Z}_{+}$, and $j\geqslant 1$ is even.
For every $\ell\geqslant 1$,
we choose some $x_{\mu}=y_{\ell}$ for
$[\frac{\alpha_{\ell}}{2}\cdot|\mathbf{A}_{r_i}\cap c\Gamma|]$ times;
and we choose the remaining $x_{\mu}$'s
to be distributed sparsely in $\mathcal{D}_{\mathsf{R}}$.
\end{itemize}
\section{\bf Proofs}\label{section: proof}
We are now in position to prove the main results. Recall that from a given sequence of discs of increasing radii $r_i\nearrow \infty$, after a perturbation and passing to some subsequence, we can always receive an Ahlfors current for $f$.
\begin{obs}
\label{obs 6.1}
From the sequence of radii $\{\frac{r_i}{3}\}_{i\geqslant 1}$, one receives a singular Ahlfors current $T$ of the shape
\[
T=c_{\infty}\cdot[\mathcal{C}_{\infty}].
\]
\end{obs}
\begin{proof}
Note that
all points in
$\mathbb{D}_{r_i/3}\setminus \mathbb{D}_{r_{i-1}+2}$
keep positive distance
$\geqslant 2-\sqrt{2}$ to $\Lambda$. Thus for any small open neighborhood $U$ of $\mathcal{C}_{\infty}$,
by Subsection~\ref{f(z) near infinity curve}, for
$i\gg 1$, for every
$z\in\mathbb{D}_{r_i/3}\setminus \mathbb{D}_{r_{i-1}+2}$, we have $f(z)\in U$.
Note that for $i\gg 1$, by~\eqref{disc image area is bounded} and~\eqref{disc area is large enough}, one see that the area of $f(\mathbb{D}_{r_{i-1}+2})$ is
negligible comparing with that of $f(\mathbb{D}_{r_{i}/3})$, namely
\[
\int_{\mathbb{D}_{r_{i-1}+2}}f^*\omega_X
\leqslant
\mathsf{K}\cdot r_{i-1}^2
=
o(1)
\cdot
r_i^2
\leqslant
o(1)
\cdot
\int_{\mathbb{D}_{r_{i}/3}}f^*\omega_X.
\]
Thus $T$ charges zero mass outside $U$. Since this holds true for any open neighborhood $U\supset \mathcal{C}_{\infty}$, we conclude that $T$ must be supported on $\mathcal{C}_{\infty}$.
\end{proof}
\begin{obs}
\label{obs 6.2}
From the sequence of radii $\{r_{
Z^{\varnothing}_{j}}\}_{j\geqslant 1}$, one gets an Ahlfors current $T$ having the shape
\[
T=a_{\infty}\cdot[\mathcal{C}_{\infty}]+ T_{\diff},
\]
where $a_{\infty}$ is some positive number and $T_{\diff}$ is a nontrivial diffuse part.
\end{obs}
\begin{proof}
{\sl Step 1}: $T$ charges positive mass along $\mathcal{C}_{\infty}$.
Indeed,
for any open neighborhood $U$ of $\mathcal{C}_{\infty}$,
it follows from the preceding proof and the estimate~\eqref{disc area is large enough} that, for $j\gg 1$ and for $i=Z^{\varnothing}_{j}$, we have
$f(\mathbb{D}_{r_i/3}\setminus \mathbb{D}_{r_{i-1}+2})\subset U$
and
$
\int_{
\mathbb{D}_{r_i/3}\setminus \mathbb{D}_{r_{i-1}+2}
}f^*\omega_X
\geqslant
\mathsf{K}
\cdot
r_i^2$.
On the other hand, by~\eqref{disc image area is bounded} we know that
$\int_{
\mathbb{D}_{r_i}
}f^*\omega_X
\leqslant
\mathsf{K}\cdot r_i^2$.
Thus $T$ charges $U$ by positive mass $\geqslant \mathsf{K}>0$. Since this holds true for any $U$, we conclude that
$T$ charges positive mass along $\mathcal{C}_{\infty}$.
\smallskip
{\sl Step 2}: $T$ does not charge any other algebraic curve.
If $D\neq \mathcal{C}_{\infty}$ is an irreducible horizontal curve, using the same notations as that of Subsection~\ref{estimate area near horizontal curves},
by the
estimate~\eqref{very subtle estimate},
and by choosing $\epsilon'\leqslant \epsilon$, we know that
$T$ charges the neighborhood
$\vartheta^{-1}(U_1^{\epsilon})$ by a small mass $\leqslant \mathsf{K}_{c_0, \delta, \vartheta}\cdot {\epsilon}^2$.
Letting $\epsilon\rightarrow 0$, we receive that $T$ charges no mass on $U_1\subset D$. Thus $T$ cannot charge positive mass on $D$.
If $D=\mathbb{P}^1_{a}$ is an irreducible vertical curve, for an open neighborhood $U$ of $\mathcal{C}_{\infty}$,
we claim that
$T$ charges no mass on $D\setminus U$. Indeed,
for any small $\epsilon>0$,
by Subsection~\ref{f(z) near infinity curve},
for $|z|\gg 1$ with
$f(z)\in \pi_{1}^{-1}(\mathbb{D}(a, \epsilon))\setminus U$, there must be some $\lambda\in\Lambda$ such that
$z\in \mathbb{D}(\lambda, \epsilon)$. Note also that $\pi_1(z)=[z]$, we get
$[\lambda]\in \mathbb{D}(a, 2\epsilon)$.
By~\eqref{distribute sparsely}, we have
\[
|
\pi_0^{-1}(\mathbb{D}(a, 2\epsilon))\cap \Lambda\cap \mathbb{D}_{r_{Z^{\varnothing}_j}}
|
\leqslant
\mathsf{K}
\epsilon^2
\cdot
r_{Z^{\varnothing}_j}^2.
\]
Hence by the estimates~\eqref{small disc high bound} and~\eqref{disc area is large enough},
such points $z\in \mathbb{D}_{r_{Z^{\varnothing}_j}}$ with
$f(z)\in \pi_{1}^{-1}(\mathbb{D}(a, \epsilon))\setminus U$ constitute
only small portion of area measured by $f^*\omega_X$, thus
$T$ charges mass $\leqslant \mathsf{K}\cdot \epsilon^2$
over $\pi_{1}^{-1}(\mathbb{D}(a, \epsilon))\setminus U$. Letting $\epsilon\rightarrow 0$, we conclude the claim. Since this holds for any open neighborhood $U$, we receive that $T$ does not charge $\mathbb{P}_a^1$, which finishes this step.
\smallskip
{\sl Step 3}: $T$ has positive mass outside $\mathcal{C}_{\infty}$.
Take any small open neighborhood $U$ of
$\mathcal{C}_\infty$ such that $\overline{U}\cap \mathcal{C}_0$ is empty.
Note that for every $\lambda\in \Lambda\cap \mathbf{A}_{r_{Z^{\varnothing}_j}}$ where $j\gg 1$, by~\eqref{very subtle estimate again}, the image of $f$ about $\lambda$
contributes $\geqslant \mathsf{K}$ area outside $U$.
Moreover, by our construction
$|\Lambda\cap \mathbf{A}_{r_{Z^{\varnothing}_j}}|\geqslant\mathsf{K}\cdot r_{Z^{\varnothing}_j}^2$,
thus the total area of
$f(\mathbb{D}_{r_{Z^{\varnothing}_j}})\setminus U$ is
$\geqslant \mathsf{K}\cdot r_{Z^{\varnothing}_j}^2$. Lastly, by~\eqref{disc image area is bounded}, we conclude that
$T$ charges positive mass outside $U$.
\end{proof}
\begin{obs}\label{obs 6.3}
Fix any finite subset $I=\{i_1,\dots,i_k\}\subset \mathbb{Z}_+$ having cardinality $k\geqslant 1$.
Then from the sequence of radii $\{r_{Z^{I}_{2j-1}}\}_{
j\geqslant 1}$, one receives an Ahlfors current $T$
having the shape
\[
T=a_{\infty}\cdot[\mathcal{C}_{\infty}]+
\sum_{\ell=1}^k
a_{i_{\ell}}\cdot
[\mathbb{P}^1_{[y_{i_{\ell}}]}],
\]
where $a_{\infty}, a_{i_1}, \dots, a_{i_k}$ are some positive numbers.
\end{obs}
\begin{proof}
For any small open neighborhood $U$ of $\mathcal{C}_{\infty}\cup \mathbb{P}^1_{[y_{i_{1}}]}
\cup
\cdots
\cup \mathbb{P}^1_{[y_{i_{k}}]}$,
by Subsection~\ref{f(z) near infinity curve}, for
$j\gg 1$,
for all points
$z\in\mathbb{D}_{r_{Z^{I}_{2j-1}}}\setminus \mathbb{D}_{r_{Z^{I}_{2j-1}-1}+2}$
we have $f(z)\in U$.
Indeed, choose a very small $\epsilon>0$ such that
$U$ contains $\pi_1^{-1}(\mathbb{D}([y_{i_\ell}], \epsilon))$ for every $\ell=1, \dots, k$. Then if $\text{dist}(z, \Lambda)\geqslant \epsilon$, we know that $f(z)$ is very close to $\mathcal{C}_{\infty}$, whence $f(z)\in U$;
otherwise $\text{dist}(z, \Lambda)< \epsilon$,
that is $z\in \mathbb{D}(\lambda, \epsilon)$
for some $\lambda\in \pi_0^{-1}([y_{i_\ell}])$ ($\ell=1, \dots, k$) by our construction of $\Lambda$, hence $f(z)\in \pi_1^{-1}(\mathbb{D}([y_{i_\ell}], \epsilon))\subset U$.
Therefore, by the same argument as that of Observation~\ref{obs 6.1}, one sees that $T$ is supported in
$\mathcal{C}_{\infty}\cup \mathbb{P}^1_{[y_{i_{1}}]}
\cup
\cdots
\cup \mathbb{P}^1_{[y_{i_{k}}]}$. It remains to check that $T$ charges positive mass in each of these components.
Indeed, first of all, our algorithm guarantees that
\[
|\Lambda\cap \mathbf{A}_{r_{Z^{I}_{2j-1}}}
\cap
\pi_0^{-1}([y_{i_\ell}])
|\geqslant\mathsf{K}\cdot r_{Z^{I}_{2j-1}}^2
\qquad(\ell=1, \dots, k).
\]
By~\eqref{small disc low bound}, for any fixed small $\epsilon>0$, for large $j\gg 1$ and $i=r_{Z^{I}_{2j-1}}$,
for any $\lambda\in \Lambda\cap \mathbf{A}_{r_{i}}
\cap
\pi_0^{-1}([y_{i_\ell}])$, the holomorphic disc $f(\mathbb{D}(\lambda,\epsilon))$
is contained in $\pi_1^{-1}(\mathbb{D}([y_{i_\ell}], \epsilon))$ with area
$\int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X\geqslant \mathsf{K}$ bounded from below by some uniformly positively constant independent of $\epsilon$.
Thus the total area of such discs is $\geqslant \mathsf{K}\cdot r_i^2$. Noting that ~\eqref{disc image area is bounded} implies $\int_{\mathbb{D}(r_i)}\leqslant \mathsf{K} \cdot r_i^2$,
thus $T$ charges mass $\geqslant \mathsf{K}$ on $\pi_1^{-1}(\mathbb{D}([y_{i_\ell}], \epsilon))$.
Letting $\epsilon\rightarrow 0$, we conclude that $T$ charges positive mass on $\mathbb{P}_{[y_{i_\ell}]}^1$. Lastly, by the same argument as the Step 1 of Observation~\ref{obs 6.2}, we see that
$T$ charges positive mass on $\mathcal{C}_{\infty}$.
Thus we conclude the proof.
\end{proof}
\begin{obs}
Fix any finite subset $I=\{i_1,\dots,i_k\}\subset \mathbb{Z}_+$ having cardinality $k\geqslant 1$.
Then from the sequence of radii $\{r_{Z^{I}_{2j}}\}_{
j\geqslant 1}$, one receives an Ahlfors current $T$
having the shape
\[
T=a_{\infty}\cdot[\mathcal{C}_{\infty}]+
\sum_{\ell=1}^k
a_{i_{\ell}}\cdot
[\mathbb{P}^1_{[y_{i_{\ell}}]}]
+
T_{\diff},
\]
where $a_{\infty}$, $a_{i_{\ell}}$ ($1\leqslant \ell\leqslant k$) are some positive constants and where $T_{\diff}$ is a nontrivial diffuse part.
\end{obs}
\begin{proof}
{\sl Step 1}: $T$ charges positive mass along $\mathcal{C}_{\infty}, \mathbb{P}^1_{[y_{i_{1}}]}, \cdots, \mathbb{P}^1_{[y_{i_{k}}]}$.
This follows from the same arguments as in the preceding proof.
\smallskip
{\sl Step 2}: $T$ does not charge any other algebraic curve.
We can check it by using the same arguments as in the Step 2 of Observation~\ref{obs 6.2}.
\smallskip
{\sl Step 3}: $T$ has positive mass outside $\mathcal{C}_{\infty}\cup \mathbb{P}^1_{[y_{i_{1}}]}
\cup
\cdots
\cup \mathbb{P}^1_{[y_{i_{k}}]}$.
The argument is similar to the Step 3 of Observation~\ref{obs 6.2}. The key point is that, by our algorithm,
\[
|
(\mathcal{D}_{\mathsf{R}}+\Gamma)\cap \Lambda\cap \mathbb{D}_{r_{Z^{I}_{2j}}}
|
\geqslant
\mathsf{K}
\cdot
r_{Z^{I}_{2j}}^2,
\]
and for every $\lambda\in (\mathcal{D}_{\mathsf{R}}+\Gamma)\cap \Lambda\cap \mathbb{D}_{r_{Z^{I}_{2j}}}$,
$[\lambda]$ keeps uniform distances $\geqslant \epsilon>0$ to $[y_{i_1}], \dots, [y_{i_k}]$.
Assuming moreover that $j\gg 1$, in the same notation as Observation~\ref{obs 4.2} we receive
\[
\area\Big(
(X\setminus U)
\cap
f\big(
\mathbb{D}(\lambda, \epsilon/2)
\big)
\Big)_{\omega_{X}}
\geqslant
\mathsf{K}_U.
\]
Note that
$
f\big(
\mathbb{D}(\lambda, \epsilon/2)
\big)$
stays away from
$\pi_1^{-1}(\mathbb{D}([y_{i_\ell}], \epsilon/2))$ for all $\ell=1, \dots, k$. Thus by the same argument as the preceding proof,
we see that $T$ charges positive mass outside $U\cup \big(\cup_{\ell=1}^k \pi_1^{-1}(\mathbb{D}([y_{i_\ell}], \epsilon/2))\big)$.
\end{proof}
By much the same arguments, we have the following two results.
\begin{obs}
From the sequence of radii $\{r_{Z^{\mathbb{Z}_{+}}_{2j-1}}\}_{
j\geqslant 1}$, one receives an Ahlfors current $T$
having the shape
\[
T=a_{\infty}\cdot[\mathcal{C}_{\infty}]+
\sum_{\ell= 1}^{\infty}
a_{{\ell}}\cdot
[\mathbb{P}^1_{[y_{{\ell}}]}],
\]
where $a_{\infty}$, $a_{{\ell}}$ ($\ell\geqslant 1$) are positive numbers.
\qed
\end{obs}
\begin{obs}
\label{obs 6.6}
From the sequence of radii $\{r_{Z^{\mathbb{Z}_{+}}_{2j}}\}_{
j\geqslant 1}$, one receives an Ahlfors current $T$
having the shape
\[
T=a_{\infty}\cdot[\mathcal{C}_{\infty}]+
\sum_{\ell= 1}^{\infty}
a_{{\ell}}\cdot
[\mathbb{P}^1_{[y_{{\ell}}]}]
+
T_{\diff},
\]
where $a_{\infty}$, $a_{{\ell}}$ ($\ell\geqslant 1$) are positive numbers,
and where $T_{\diff}$ is a nontrivial diffuse part.
\qed
\end{obs}
Thus we prove Theorems~\ref{thm 1},~\ref{thm 2}.
\section{\bf Examples}
\label{section: examples}
\subsection{Diffuse Ahlfors currents}
Let $\mathcal{A}=\mathbb{C}/\Lambda\times\mathbb{C}/\Lambda$ be the surface obtained as the product of two elliptic curves where $\Lambda$ is a lattice.
Fix a reference metric $\omega_{\mathcal{A}}:=
\dif \dif^c |z_1|^2
+
\dif \dif^c |z_2|^2$ on ${\mathcal{A}}$.
Choose an irrational number $\lambda\in\mathbb{R}\setminus\mathbb{Q}$. Consider the holomorphic curve $f:\mathbb{C}\longrightarrow\mathcal{A}$ given by $f(z)=([z],[\lambda z])$.
\begin{pro}
\label{diffuse in Abelian surface}
Any Ahlfors current $T$ of $f$ is diffuse.
\end{pro}
\begin{proof}
Since there is no nonconstant holomorphic map from $\mathbb{P}^1(\mathbb{C})$ to an elliptic curve, $\mathcal{A}$ contains no rational curve.
Hence
by a theorem of Duval \cite{Duval2006}, it suffices to check that $T$ charges zero mass along any elliptic curve in $\mathcal{A}$.
\smallskip
\noindent
{\bf Fact}
(c.f. \cite[Prop.~1.3.2]{Diamond-Shurman2005}).
{\it
Let $\Phi
:
\mathbb{C}/\Gamma_1
\longrightarrow
\mathbb{C}/\Gamma_2
$
be a holomorphic map between
complex tori. Then there exist complex numbers $m, b$ with
$m\Gamma_1\subset \Gamma_2$,
such that
$\Phi([z])
= [mz+b]$.
}
\smallskip
Therefore, any nonconstant holomorphic map $\iota: \mathbb{C}/\Gamma_3\rightarrow \mathbb{C}/\Lambda\times\mathbb{C}/\Lambda$ from an elliptic curve
to $\mathcal{A}$ can be written explicitly as
$\iota([z])=([m_1 z+b_1], [m_2 z+ b_2])$
for some complex numbers $m_1, m_2, b_1, b_2$, such that $(m_1, m_2)\neq (0, 0)$ and $m_1\Gamma_3, m_2\Gamma_3\subset \Gamma$. Hence $m_2-\lambda m_1\neq 0$.
We claim that the intersection numbers
\begin{equation}
\label{counting intersection nubmers on tori}
|\iota(\mathbb{C}/\Gamma_3)\cap f(\mathbb{D}_{r})|\leqslant \mathsf{K}\cdot r^2
\end{equation}
for $r\gg 1$.
Indeed, we can find
a large disc $\mathbb{D}_R$ containing a fundamental domain of $\Gamma_3$.
Then for $z\in \mathbb{D}_r, y\in \mathbb{D}_R$ with $([z], [\lambda z])=([m_1 y+b_1], [m_2 y+ b_2])$,
we receive that
\begin{equation}
\label{solve the linear equation}
z-(m_1 y+b_1)=\lambda_1, \qquad
\lambda z-(m_2 y+ b_2)=\lambda_2
\end{equation}
for some $\lambda_1, \lambda_2\in\Lambda$ having absolute values less than $ r+\mathsf{K}, |\lambda|\cdot r+\mathsf{K}$ respectively.
Since $m_2-\lambda m_1\neq 0$, we can solve the linear equation~\eqref{solve the linear equation} as
\[
z=\frac{m_2(\lambda_1+b_1)-m_1(\lambda_2+b_2)}{m_2-\lambda m_1},
\qquad
y=
\frac{\lambda(\lambda_1+b_1)-(\lambda_2+b_2)}{m_2-\lambda m_1}.
\]
Noting that $y\in \mathbb{D}_R$,
for any fixed $\lambda_1$, the cardinality of possible choices
of
\[
\lambda_2
\in
\big(
(-m_2+\lambda m_1)\cdot
\mathbb{D}_R
+
\lambda(\lambda_1+b_1)-b_2
\big)
\cap \Lambda
\] is
$\leqslant \mathsf{K}$.
Thus the cardinality of possible choices of such $(\lambda_1, \lambda_2)\in \Lambda\times \Lambda$
is $\leqslant \mathsf{K}\cdot (r+\mathsf{K})^2\cdot \mathsf{K}\leqslant \mathsf{K} \cdot r^2$. Hence the estimate~\eqref{counting intersection nubmers on tori} is proved.
By the compactness of $\iota(\mathbb{C}/\Gamma_3)$, and by shrinking neighborhood $U$ of $\iota(\mathbb{C}/\Gamma_3)$ if necessary, each intersection point corresponds to a small area $o(1)$ component of $f(\mathbb{D}_r)\cap U$, thus
the total area of $f(\mathbb{D}_r)\cap U$
is $\leqslant o(1)\mathsf{K}\cdot r^2$.
However, the area growth of $f(\mathbb{D}_r)$ is $\mathsf{K}_{\lambda}\cdot r^2$. Thus any obtained Ahlfors current of $f$ charges mass
$\leqslant o(1)\mathsf{K}$ on $U$. By shrinking $U$, we know that $T$ charges zero mass
along $\iota(\mathbb{C}/\Gamma_3)$.
Hence
we conclude the proof.
\end{proof}
Take a holomorphic surjective map $\pi_2: \mathcal{A}\longrightarrow \mathbb{P}^2(\mathbb{C})$, which induces an entire curve
\[
f_2:=\pi_2\circ f:\,\, \mathbb{C}\,\,\longrightarrow\,\, \mathbb{P}^2(\mathbb{C}).
\]
Since $\pi_2^*\omega_{\FS}\geqslant 0$ is closed,
by the geometry of $\mathcal{A}$, in the cohomology class $[\pi_2^*\omega_{\FS}]$
we can find a harmonic representative
\[
\omega=a_1 \sqrt{-1}\dif z_1\wedge \dif \overline{z_1}
+
a_2 \sqrt{-1}
\dif z_2\wedge \dif \overline{z_2}
+
a_3 \sqrt{-1}\dif z_1\wedge \dif \overline{z_2}
+
a_4 \sqrt{-1}\dif z_2\wedge \dif \overline{z_1}
\geqslant 0
\]
for some constants $a_1, a_2, a_3, a_4$. Thus
$f^*\omega=K \sqrt{-1}\dif z\wedge\dif \overline{ z}
\geqslant 0$ where
\[
K=a_1+a_2 \lambda^2+\lambda(a_3+a_4)\geqslant 0.
\]
Since
$a_1, a_2, a_3+a_4$ cannot vanish simultaneously,
at most one $\lambda$ in
$ \mathbb{R}\setminus \mathbb{Q}$ can make $K=0$. Now we only choose $\lambda\in \mathbb{R}\setminus \mathbb{Q}$
such that $K>0$.
\begin{pro}
\label{diffuse Ahlfors currents in Cp2}
Any Ahlfors current $T_2$ of $f_2$ is diffuse.
\end{pro}
It is interesting to see that $f_2$ is tangent to a multi-valued vector field
induced by the push-forward of the constant vector field $(1, \lambda)$ on $\mathcal{A}$.
\begin{proof}
Assume that $T_2$ is obtained by an increasing radii $\{r_i\}_{i\geqslant 1} \nearrow \infty$.
By our chosen metric $\omega_{\mathcal{A}}$, the ``Length-Area'' condition of Ahlfors' lemma is automatically satisfied, thus by passing to some subsequence
$\{r_{i_k}\}_{k\geqslant 1}$
we can receive an Ahlfors current $T$ of $f$.
Noting that $f^*\omega=K \sqrt{-1}\dif z\wedge\dif \overline{ z}
> 0$,
by closeness of $T$
and by $\area(f(\mathbb{D}_{r}))_{\omega_{\mathcal{A}}}=\mathsf{K}\cdot r^2$, we receive
\[
T(\pi_2^*\omega_{\FS})=
T(\omega)\geqslant \mathsf{K}
>0.
\]
By the construction of $T$,
we receive that
\[
\area\big(f_2(\mathbb{D}_{r_{i_k}})\big)_{\omega_{\FS}}\geqslant \mathsf{K}\cdot r_{i_k}^2 \qquad
{\scriptstyle(k\,\gg\, 1)}.
\]
Fix some
$\mathsf{K}$ such that $\pi_2^*\omega_{\FS}\leqslant \mathsf{K}\cdot \omega_{\mathcal{A}}$. For any irreducible curve $C\subset \mathbb{P}^2(\mathbb{C})$, for any open neighborhood $U$ of $C$,
we have
\[
\area\big(f_2(\mathbb{D}_{{r_{i_k}}})\cap U\big)_{\omega_{\FS}}
=
\area\big(f(\mathbb{D}_{r_{i_k}})\cap \pi_2^{-1}(U)\big)_{\pi_2^*\omega_{\FS}}
\leqslant
\mathsf{K}\cdot
\area\big(f(\mathbb{D}_{r_{r_{i_k}}})\cap \pi_2^{-1}(U)\big)_{\omega_{\mathcal{A}}}.
\]
Since $T$ charges no mass along $\pi_2^{-1}(C)$ by Proposition~\ref{diffuse in Abelian surface}, by shrinking $U$,
the right-hand-side above is
$\leqslant
o(1)\cdot r_{i_k}^2$.
Thus $T_2$ charges no mass along $C$. Since $C$ is arbitrary,
we conclude the proof.
\end{proof}
\subsection{Singular Nevanlinna currents on $X$}
\label{Singular Nevanlinna currents on X}
Replacing ``Ahlfors currents'' by ``Nevanlinna currents'' in Observations~\ref{obs 6.1} -- \ref{obs 6.6},
the same statements still hold true by much the same arguments. Indeed, every upper or lower bound about $\int_{\mathbb{D}(\lambda,\epsilon)}
f^*
\omega_X$ or $\int_{\mathbb{D}_{r_i}}
f^*
\omega_X$ has a corresponding one
about order function.
A remaining technical detail we would like to mention is the following
\begin{obs}
For every $\ell\geqslant 1$,
there exists some positive $\beta_\ell<1$
such that, for $j\gg 1$ and $i=Z^{\mathbb{Z}_+}_j$,
\[
|\Lambda\cap \mathbf{A}_{r_i}
\cap \mathbb{D}_{\beta_\ell r_i}\cap \pi_0^{-1}([y_{\ell}])
|
\geqslant
\mathsf{K}_{\ell}
\cdot
r_i^2.
\]
\end{obs}
It will be helpful to show that certain Nevanlinna currents of $f$ charge positive mass along $\mathbb{P}^1_{[y_\ell]}$.
\begin{proof}
Note that for $j\gg 1$ we have
$
|\Lambda\cap \mathbf{A}_{r_i}
\cap \pi_0^{-1}([y_{\ell}])
|
>
\frac{\alpha_{\ell}}{3}
\cdot
\mathsf{K}
\cdot
r_i^2
$. Moreover, for any fixed $\beta<1$,
for $j\gg 1$, we have
$
|\Lambda\cap \mathbf{A}_{r_i}
\setminus
\mathbb{D}_{\beta r_i})|
\leqslant \mathsf{K}\cdot (1-\beta) r_i^2
$. By these two estimates, we can conclude the proof.
\end{proof}
Therefore we can replace ``Ahlfors currents'' by ``Nevanlinna currents'' in the statements of Theorems~\ref{thm 1},~\ref{thm 2}. Also, by much the same proofs,
Propositions~\ref{diffuse in Abelian surface}, \ref{diffuse Ahlfors currents in Cp2} also hold true for Nevanlinna currents.
\subsection{Singular Ahlfors currents on blow-ups of $X$}
We sketch a construction of elaborate (in the sense of cohomology classes) singular Ahlfors currents,
on the blow-ups of $X$ having Picard numbers $\geqslant 3$.
For any given positive integer $n\geqslant 1$, recall the collection of points $y_1, \dots, y_n$ given in~\eqref{choose y_i}, let
$\mathcal{X}$ be
the blow-up of $X$ at these points with the corresponding exceptional divisors $E_1, \dots, E_n$.
Let $\mathsf{p}: \mathcal{X}\rightarrow X$ be the projection.
We now use the section $\psi^2\cdot s_m$ instead of $\psi\cdot s_m$
to induce an entire curve $\mathsf{f}:\mathbb{C}\longrightarrow X$. By lifting we thus receive an entire curve $\zeta: \mathbb{C}\longrightarrow \mathcal{X}$. We strengthen our choices of $m, c$ in~\eqref{key choices of m, k}
by the condition $m\cdot\alpha-2\mathsf{K}/c^2
>0$, to make sure that the same clustering phenomenon as in Subsection~\ref{f(z) near infinity curve}
holds true
for $\mathsf{f}$
and
$\mathcal{C}_{\infty}$.
Let $e_i$ be the intersection point of the strict transformation $\widetilde{\mathcal{C}_0}$ of $\mathcal{C}_0$ with $E_i$ ($i=1,\dots, n$).
The purpose of using $\psi^2$ instead of $\psi$ is to make sure that, for $\lambda\in \Lambda$ with $[\lambda]=[y_i]$,
we have the certain value $\zeta(\lambda)=e_i$.
It is well-known that,
there exist some hermitian metrics $h_{i}$ of the line bundles $\mathcal{O}(-E_i)$ and some small positive constant $\epsilon_2\ll 1$
such that
$\omega_{\mathcal{X}}:=\mathsf{p}^*\omega_X+\epsilon_2\sum_{\ell=1}^{n}\Theta_{h_{\ell}
}
$
is a K\"ahler form on $\mathcal{X}$
(c.f. \cite[Proposition 3.24]{Voisin2007-I}).
Moreover,
comparing the lifting
$\zeta: \mathbb{C}\longrightarrow \mathcal{X}$ with $\mathsf{f}: \mathbb{C}\longrightarrow X$,
we have
\[
T_{\zeta,r}(\omega_{\mathcal{X}})
:=
\int_{1}^r\frac{\dif t}{t}\int_{\mathbb{D}_t}
\zeta^*\omega_{\mathcal{X}}
\leqslant
T_{\mathsf{f},r}(\omega_X)+O(1),
\]
(c.f. \cite[page~64, Observation 2.5.1]{Huynh2016}).
Thus we can use the same arguments for~\eqref{disc image area is bounded} to conclude that
\[
\int_{\mathbb{D}_{2r_i}}{\zeta}^*\omega_{\mathcal{X}}\leqslant\mathsf{K}\cdot r_i^2.
\]
For $\lambda\in \Lambda$ with $[\lambda]=[y_i]$,
computing in local coordinates around $e_i$,
for any small $\epsilon>0$,
for any open neighborhood $U$ of $E_i$,
assuming further that $|\lambda|\gg 1$, then the area of $\zeta(\mathbb{D}(\lambda, \epsilon))\cap U$
is uniformly positively bounded (independent of $U$ and $\epsilon$) from below by using Propositions ~\ref{the most difficult estimate of psi} and~\ref{curve in ball area}.
Therefore, as an analogue of Observation~\ref{obs 6.3},
for the finite subset $I=\{1, \dots, n\}$,
from the sequence of radii $\{r_{Z^{I}_{2j-1}}\}_{
j\geqslant 1}$, after a perturbation and passing to a subsequence, we can receive an Ahlfors current
\[
T=a_{\infty}\cdot[
\widetilde{\mathcal{C}_{\infty}}]+
\sum_{\ell=1}^n
a_{{\ell}}\cdot
[\widetilde{\mathbb{P}^1_{[y_{\ell}]}}]
+
\sum_{\ell=1}^n
b_{{\ell}}\cdot
[E_i],
\]
where $\widetilde{\mathcal{C}_{\infty}}$, $\widetilde{\mathbb{P}^1_{[y_{\ell}]}}$ stand for the strict transformations of $\mathcal{C}_{\infty}$, $\mathbb{P}^1_{[y_{\ell}]}$, and where $a_{\infty}, a_{\ell}, b_{\ell}>0$ ($\ell=1, \dots, n$) are some positive numbers.
Similarly, we have the counterparts of other Observations~\ref{obs 6.1} -- \ref{obs 6.6}.
\begin{center}
\bibliographystyle{alpha}
|
1,116,691,498,505 | arxiv | \section{Introduction}\label{sec_introduction}
The analysis of roll call data of legislative bodies has
attracted a lot of attention both in the political science and statistical
literature. For political scientists, such data allow to study broad
issues such as party cohesion as well as more specific ones such as
coalition formation; see, for example, the books by \citet
{EnelowHinich84,MatthewsStimson75,Morton99,PooleRosenthal97}.
A popular tool in political science is the ideal point model [\citet
{ClintonJackmanRivers04}] that posits a one-dimensional latent political
space along which legislators and bills they vote for are aligned.
A~legislator's position corresponds to an ideal point, where bills
coinciding with that position maximize his/her utility. These ideal
points reveal legislators' preferences and it is of interest to
infer them from roll call data. An extension of this model that
incorporates information about the text of the bills being voted upon
is discussed in \citet{Gerrish2011}, while the impact of absenteeism is
examined in \citet{Han2007}.
\begin{figure}
\includegraphics{700f01.eps}
\caption{Multidimensional scaling projection of
roll call data of the U.S. Senate for the period 2005--2006 (Republicans
shown in red and Democrats in blue).} \label{mds-results}
\end{figure}
A statistical challenge is how to best model and present the roll call
data in a way that makes interesting patterns apparent and facilitates
subsequent analyses. A number of techniques have been employed
including principal components analysis (PCA) [\citet{deLeeuw06}],
multidimensional scaling (MDS) [\citet{DiaconisGoelHolmes08}], Bayesian
spatial voting models [\citet{ClintonJackmanRivers04}], and
graphical models for binary data [\citet{BanerjeeGhaouidAspremont08}].
Dimension reduction techniques such as PCA and MDS aim at constructing
a ``map,'' with the
members of the legislative body positioned relative to their peers
according to their voting pattern.
A typical example of such a map of the
U.S. Senate members in the 109th Congress (2005--2006)
using multidimensional scaling for selected votes is shown in
Figure~\ref{mds-results}; for a detailed
description of the data see Section~\ref{sec_real_example}. A clear
separation between
members of the two parties is seen (Republicans to the left of the map
and Democrats to the right), together with some members exhibiting a
voting pattern deviating from their party, for example,
Nelson (Democrat of Nebraska), and Collins and Snow (Republicans of
Maine), while the independent Jeffords (shown in purple) votes like a
Democrat. More interestingly, the voting patterns within both parties
form distinct subclusters. While the nature of this division is
impossible to infer from an MDS or a PCA representation such as the one
shown in Figure~\ref{mds-results}, our subsequent analysis will show
that this difference is driven
by votes on defense/security and healthcare issues.
This finding suggests that treating all votes as homogeneous, that is,
assuming that they represent the same underlying relationship
between senators, may mask more subtle patterns which depend on the
issues being voted upon. Therefore, treating votes as heterogeneous is
more accurate and can provide further insight into the voting behavior
of different groups of senators on different issues. In this paper, we
focus on voting records on three types of bills: defense and national
security, environment and energy, and healthcare issues. Voting on the
latter category is typically more partisan than voting on defense and
national security and, thus, we expect to see different connections in
different categories.
The voting records of the U.S. Senate from the 109th Congress covering
the period 2005--2006 were obtained directly from the Senate's website
(\surl{www.senate.gov}). We chose the 109th Congress because its voting
patterns have been previously analyzed in the literature [see, e.g.,
\citet{BanerjeeGhaouidAspremont08}], but as we have discovered, the
version of the data previously analyzed was contaminated with voting
records from the 1990s (when the set of senators would have been
different). Thus, we collected the data ourselves, on all the 645 votes
that the Senate deliberated and voted on during that period, which
include bills, resolutions, motions, debates and roll call votes. To
study the potential heterogeneity in the voting patterns, we focused on
the three largest meaningful (i.e., excluding purely procedural votes)
categories of votes extracted from bills, resolutions and motions:
(1) defense and security issues; (2) environment and energy issues; (3)
health and medical care issues. The categories were extracted by a
combination of text analysis of bill names and manual labeling.
A~complete analysis of this data set will be presented in Section~\ref
{sec_real_example}.
Our goal in this paper is to develop a statistical model for studying
dependence patterns in such situations: there is some overall structure
present (party affiliation, which affects everything) and there are
also distinct categories with their own individual structures. Since we
are dealing with voting data, we use Markov network models to capture
the dependence structure of binary or categorical random variables.
Similar to Gaussian graphical models,
nodes in a Markov network correspond to (categorical) variables, while
edges represent dependence between nodes conditional on all other variables.
Graphical models are an exploratory data analysis tool used in a number
of application areas to explore the dependence structure between
variables, including bioinformatics [\citet{Airoldi07}], natural language
processing [\citet{JungParkChoiKim96}] and image analysis [\citet{Li01}].
In the case of Gaussian graphical models, which assumes the variables
are jointly normally distributed, the structure of the underlying graph
can be fully determined from the corresponding inverse covariance
(precision) matrix, the off-diagonal elements of which are proportional
to partial correlations between the variables. A number of methods have
been recently proposed in the literature to fit \emph{sparse} Gaussian
graphical models [see, e.g.,
\citet
{MeinshausenBuhlmann06,YuanLin07,BanerjeeGhaouidAspremont08,RothmanBickelLevinaZhu08,
RavikumarWainwrightRaskuttiYu08,PengWangZhouZhu09} and references
therein]. Sparse Markov networks for binary data (Ising models) have
been studied by \citet
{HoeflingTibshirani09,GuoLevinaMichailidisZhu09jointIsingtechreport,
RavikumarWainwrightLafferty10, Anandkumar12, Xue12}. These methods do
not allow for different categories within the data.
To allow for heterogeneity, we develop a framework for fitting
different Markov models for each category that are nevertheless \emph
{linked}, sharing nodes and some common edges across all categories,
while other edges are uniquely associated with a particular category.
This will allow us to borrow strength across categories instead of
fitting them completely separately. For the Gaussian case, this type of
joint graphical model was first studied by \citet
{GuoLevinaMichailidisZhu09CGM}, who proposed a joint likelihood based
estimation method that borrowed strength across categories. Several
other papers have proposed alternative algorithms for the Gaussian case
[\citet{Danaher,Yang12,Hara201323}]. We note that a context-specific
graphical model was proposed for count data in the form of contingency
tables by \citet{Hojsgaard04}, but contingency tables are not suitable
for high-dimensional data and the context-specific model is not sparse.
The advantage of using a Markov graphical model in this context is that
it quantifies the degree of conditional dependence between the senators based
on their voting record, and hence the obtained network, and is directly
interpretable. Techniques like multidimensional
scaling and principal components analysis represent relative
similarities between senators' voting records on the map and, hence,
the distance between any two senators can be interpreted as a
quantitative measure of similarity between their
voting records. However, unlike in a Markov network, these distances
are not interpretable in the context of a generative probability model.
The remainder of the paper is organized as follows. Section~\ref
{sec_methodology} introduces the Markov network and addresses
algorithmic issues, and Section~\ref{sec_simulation} briefly
illustrates the performance of the joint estimation method on simulated
data. A detailed analysis of the U.S. Senate's
voting record from the 109th Congress is presented in Section~\ref
{sec_real_example}. Some concluding remarks are drawn in Section~\ref
{sec_conclusion}, and the \hyperref[app]{Appendix} presents results on the asymptotic
properties of the method. The electronic supplementary material
contains a detailed investigation of missing data imputation methods
for the Senate vote data.
\section{Model and estimation algorithm}\label{sec_methodology}
In this section we present the Markov model for heterogeneous data,
focusing on the special case of binary variables (also known as the
Ising model).
The extension to general categorical variables is briefly discussed in
Section~\ref{sec_conclusion}. We start by discussing estimation of
\emph{separate} models for
each category and then develop a method for joint estimation.
The main technical challenge when estimating the likelihood of Markov
graphical models is its computational intractability
due to the normalizing constant. To overcome this difficulty, different
methods employing computationally tractable approximations to the
likelihood have been proposed in the literature;
these include methods based on surrogate likelihood [\citet
{BanerjeeGhaouidAspremont08,KolarXing08}] and pseudo-likelihood [\citet
{HoeflingTibshirani09,RavikumarWainwrightLafferty10,GuoLevinaMichailidisZhu10JOSE}].
\citet{HoeflingTibshirani09} also proposed an iterative algorithm that
successively approximates the original likelihood through a series of
pseudo-likelihoods, while \citet{RavikumarWainwrightLafferty10} and
\citet{GuoLevinaMichailidisZhu10JOSE} established asymptotic
consistency of their respective methods.
\subsection{Problem setup and separate estimation}\label{sec_review_SE}
We start from setting up notation and reviewing previous work on
estimating a single Ising model, which can be used to estimate the
graph for each category separately. Suppose that data have been
collected on $p$ \emph{binary} variables in $K$ categories, with $n_k$
observations in the $k$th category, $k = 1, \ldots, K$.
Let $\V{x}_{i}^{(k)} = (x_{i,1}^{(k)},\ldots, x_{i,p}^{(k)})$ denote
a $p$-dimensional row vector containing the data for the $i$th
observation in the $k$th category and
assume that it is drawn independently from an exponential family with
the probability mass function
\begin{equation}\qquad
\label{joint_density_Ising} \F{f}_k(X_1,\ldots, X_p) =
\frac{1}{\F{Z}(\MM{\Theta}^{(k)})} \exp \Biggl(\sum_{j=1}^p
\theta_{j,j}^{(k)} X_j + \sum
_{1 \le j < j'
\le p} \theta_{j,j'}^{(k)} X_j
X_{j'} \Biggr).
\end{equation}
The partition function $\F{Z}(\MM{\Theta}^{(k)})= \sum_{X_j \in\{
0,1\}, j} \exp(\theta_{j,j}^{(k)} X_j + \sum_{j < j'} \theta
_{j,j'}^{(k)} X_j X_{j'})$ ensures that the probabilities
in \eqref{joint_density_Ising} add up to one.
The parameters $\theta_{j,j}^{(k)}$, $1 \le j \le p$ correspond to the
main effect for variable $X_j$ in the $k$th category, and
$\theta_{j,j'}^{(k)}$ is the interaction effect between variables
$X_j$ and $X_{j'}$, $1 \le j < j' \le p$. The underlying network
associated with the $k$th category is determined by the symmetric
matrix $\MM{\Theta}^{(k)} = (\theta_{j,j'}^{(k)})_{p \times p}$.
Specifically, if $\theta_{j,j'}^{(k)}=0$, then $X_j$ and $X_{j'}$ are
conditionally independent in the $k$th category given all the remaining
variables
and, hence, their corresponding nodes are \emph{not} connected. For
each category, \eqref{joint_density_Ising} is referred to as the
Markov network in the machine learning literature and as the log-linear
model in the statistics literature, where $\theta_{j,j'}^{(k)}$ is
also interpreted as the conditional log odds ratio between $X_j$ and
$X_{j'}$ given the other variables. Although general Markov networks
allow higher order interactions (3-way, 4-way, etc.), \citet
{RavikumarWainwrightLafferty10} pointed out that in principle one can
consider only the pairwise
interaction effects without loss of generality, since higher order
interactions can be converted to pairwise ones by introducing
additional variables
[\citet{WainwrightJordan08}]. For the rest of this paper, we only
consider models with pairwise interactions of the original binary variables.
The simplest way to deal with heterogenous data is to estimate $K$
separate Markov models, one for each category. If one further assumes
sparsity for the $k$th category, the structure of the underlying graph
can be estimated by regularizing the log-likelihood using an $\ell_1$ penalty:
\begin{eqnarray}
\label{separate_method} &&\max_{\MM{\Theta}^{(k)}} \frac{1}{n_k} \sum
_{i=1}^{n_k} \Biggl\{\sum_{j=1}^p
\theta_{j,j}^{(k)} x_{i,j}^{(k)} + \sum
_{j < j'} \theta _{j,j'}^{(k)}
x_{i,j}^{(k)} x_{i,j'}^{(k)} \Biggr\}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&\qquad{}- \log
\F{Z}\bigl(\M {\Theta}^{(k)}\bigr) - \lambda\sum
_{j < j'} \bigl|\theta_{j, j'}^{(k)}\bigr|.
\end{eqnarray}
The $\ell_1$ penalty shrinks some of the interaction effects $\theta
_{j,j'}^{(k)}$ to zero and $\lambda$ controls the degree of sparsity.
However, estimating
\eqref{separate_method} directly is computationally infeasible due to
the nature of the partition function. A standard approach in such a
situation is to replace the likelihood with a pseudo-likelihood [\citet
{besag86}], which has been shown to work well in a range of situations. Here,
we use a pseudo-likelihood estimation method for Ising models [\citet
{HoeflingTibshirani09,GuoLevinaMichailidisZhu10JOSE}], based on
\begin{eqnarray}
\label{pseudo_separate_method}&& \max_{\MM{\Theta}^{(k)}} \frac{1}{n_k} \sum
_{i=1}^{n_k} \sum_{j=1}^p
\biggl[x_{i,j}^{(k)} \biggl(\theta_{j,j}^{(k)}
+ \sum_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr)
\nonumber
\\
&&\hspace*{46pt}\qquad{} - \log \biggl\{1 + \exp \biggl(\theta_{j,j}^{(k)} + \sum
_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr) \biggr\} \biggr]
\\
&& \qquad{}- \lambda\sum_{j < j'} \bigl|\theta_{j, j'}^{(k)}\bigr|,\nonumber
\end{eqnarray}
where $\MM{\Theta}^{(k)}$ is restricted to be symmetric. Criterion
\eqref{pseudo_separate_method} can be efficiently maximized using the
modified coordinate descent algorithm of \citet{HoeflingTibshirani09}.
\subsection{Joint estimation of heterogeneous networks}
The separate estimation methods reviewed in the previous section do not
take advantage of the shared nodes among the categories and potential
common structure. Our goal here is to explicitly include this into the
estimation procedure. We start by reparameterizing each~$\theta
_{j,j'}^{(k)}$ as
\begin{equation}
\label{reparameterize_omega} \theta_{j,j'}^{(k)} = \phi_{j,j'}
\gamma_{j,j'}^{(k)}, \qquad 1 \le j \neq j' \le p; 1 \le k
\le K.
\end{equation}
To avoid sign ambiguities between $\phi_{j,j'}$ and $\gamma
_{j,j'}^{(k)}$, we restrict $\phi_{j,j'} \ge0$, $1 \le j < j' \le p$.
To preserve the symmetry of $\MM{\Theta}^{(k)}$, we also require $\phi
_{j,j'} = \phi_{j',j}$ and $\gamma_{j,j'}^{(k)}=\gamma
_{j',j}^{(k)}$, for all $1 \le j < j' \le p$ and $1 \le k \le K$.
Moreover, for identifiability reasons, we restrict the diagonal
elements $\phi_{j,j}=1$ and $\gamma_{j,j}^{(k)}=\theta_{j,j}^{(k)}$.\vspace*{1pt}
Note that $\phi_{j,j'}$ is a common factor across all $K$ categories
that\vspace*{-1pt} controls the occurrence of common links shared across categories,
while $\gamma_{j,j'}^{(k)}$ is an individual factor specific to the
$k$th category.
The proposed joint estimation method maximizes the following penalized
criterion:
\begin{eqnarray}
\label{model_cgm_eta12} \max_{\{\MM{\Phi}^{(k)}, \MM{\Gamma}^{(k)}\}_{k=1}^K} && \sum_{k=1}^K
\frac{1}{n_k} \sum_{i=1}^{n_k} \sum
_{j=1}^p \biggl[x_{i,j}^{(k)}
\biggl(\theta_{j,j}^{(k)} + \sum_{j' \neq j}
\theta_{j,j'}^{(k)} x_{i,j'}^{(k)} \biggr)
\nonumber
\\
&&\hspace*{43pt} \qquad{}- \log \biggl\{1 + \exp \biggl(\theta_{j,j}^{(k)} + \sum
_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr) \biggr\} \biggr]
\\
&&\qquad{} - \eta_1 \sum_{j < j'}
\phi_{j,j'} - \eta_2 \sum_{j < j'}
\sum_{k=1}^K \bigl|\gamma_{j,j'}^{(k)}\bigr|,\nonumber
\end{eqnarray}
where $\MM{\Phi}^{(k)}=(\phi_{j,j'})_{p \times p}$ and $\MM{\Gamma
}^{(k)} = (\gamma_{j,j'}^{(k)})_{p \times p}$. The tuning parameter
$\eta_1$ controls sparsity of the common structure across the $K$
networks. Specifically, if $\phi_{j,j'}$ is shrunk to zero, all
$\theta_{j,j'}^{(1)},\ldots, \theta_{j,j'}^{(K)}$ are also zero
and, hence, there is no link between nodes $j$ and $j'$ in any of the
$K$ graphs. Similarly,
$\eta_2$ is a tuning parameter controlling sparsity of links in
individual categories. Due to the nature of the $\ell_1$ penalty, some
of $\gamma_{j,j'}^{(k)}$'s will be shrunk to zero,\vspace*{1pt} resulting in a
collection of graphs with individual differences. Note that this
two-level penalty was originally proposed by \citet{ZhouZhu07} for
group variable selection in linear regression.
The criterion \eqref{model_cgm_eta12} achieves the stated goal of
estimating common structure and hence borrows strength across the $K$
data categories,
but requires the selection of two tuning parameters. However, there is
an equivalent criterion presented next that only involves a single
tuning parameter,
thus simplifying the estimation task
\begin{eqnarray}
\label{model_cgm_lambda} &&\max_{\{\MM{\Theta}^{(k)}\}_{k=1}^K} \sum_{k=1}^K
\frac{1}{n_k} \sum_{i=1}^{n_k} \sum
_{j=1}^p \biggl[x_{i,j}^{(k)}
\biggl(\theta_{j,j}^{(k)} + \sum_{j' \neq j}
\theta_{j,j'}^{(k)} x_{i,j'}^{(k)} \biggr)
\nonumber
\\
&&\hspace*{79pt}\qquad{} - \log \biggl\{1 + \exp \biggl(\theta_{j,j}^{(k)} + \sum
_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr) \biggr\} \biggr]
\\
&&\qquad{} - \lambda\sum_{1 \le j < j' \le p} \sqrt{\sum
_{k=1}^K \bigl| \theta _{j,j'}^{(k)}\bigr|},\nonumber
\end{eqnarray}
where $\lambda=2\sqrt{\eta_1 \eta_2}$.
The optimization problems given by \eqref{model_cgm_eta12} and \eqref
{model_cgm_lambda} are equivalent in the sense that for each pair of
$(\eta_1,\eta_2)$ there is a $\lambda$ that gives the same solution
and vice versa. Their equivalence
can be formalized as follows (here $\M{A} \cdot\M{B}$ denotes the
Schur--Hadamard element-wise product of two matrices):
\begin{proposition}\label{lemma_etatolambda}
Let $\{\widehat{\MM{\Theta}}^{(k)}\}_{k=1}^K$ be a local maximizer of
\eqref{model_cgm_lambda}. Then there exists a local maximizer of
\eqref{model_cgm_eta12}, $(\widehat{\MM{\Phi}}, \{\widehat{\M
{\Gamma}}^{(k)}\}_{k=1}^K)$, such that $\widehat{\MM{\Theta}}^{(k)}
= \widehat{\MM{\Phi}} \cdot\widehat{\MM{\Gamma}}^{(k)}$, for all
$1 \le k \le K$. On the other hand, if $(\widehat{\MM{\Phi}}, \{
\widehat{\MM{\Gamma}}^{(k)}\}_{k=1}^K)$ is a local maximizer of
\eqref{model_cgm_eta12}, then there also exists a local maximizer of
\eqref{model_cgm_lambda}, $\{\widehat{\MM{\Theta}}^{(k)}\}_{k=1}^K$,
such that $\widehat{\MM{\Theta}}^{(k)} = \widehat{\MM{\Phi}} \cdot
\widehat{\MM{\Gamma}}^{(k)}$, for all $1 \le k \le K$.
\end{proposition}
The proof of this proposition is similar to the proofs of Lemma~1 and
Theorem~1 in \citet{ZhouZhu07} and is omitted here. Note that even
though choosing a single tuning parameter $\lambda$ corresponds to a
particular path in the $(\eta_1, \eta_2)$ space, this restriction
affects only the individual estimates $\phi_{j,j'}$ and $\gamma
_{j,j'}$, but not their product $\theta_{j,j'}$.
\subsection{Algorithm and model selection}\label{sec_algorithm_models}
Criterion \eqref{model_cgm_lambda} leads to an efficient estimation
algorithm based on the local linear approximation. Specifically,
letting $(\theta_{j,j'}^{(k)})^{[t]}$ denote the estimates from the
$t$th iteration, we approximate $\sqrt{\sum_{k=1}^K | \theta
_{j,j'}^{(k)}|} \approx\sum_{k=1}^K | \theta_{j,j'}^{(k)}| / \sqrt
{\sum_{k=1}^K |(\theta_{j,j'}^{(k)})^{[t]}|}$, when $\theta
_{j,j'}^{(k)} \approx(\theta_{j,j'}^{(k)})^{[t]}$.
Thus, at the $(t+1)$th iteration, problem \eqref{model_cgm_lambda} is
decomposed into $K$ individual
optimization problems:
\begin{eqnarray}
\label{model_LLA}&& \max_{\MM{\Theta}^{(k)}} \frac{1}{n_k} \sum
_{i=1}^{n_k} \sum_{j=1}^p
\biggl[x_{i,j}^{(k)} \biggl(\theta_{j,j}^{(k)}
+ \sum_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr)
\nonumber
\\
&&\hspace*{48pt}\qquad{} - \log \biggl\{1 + \exp \biggl(\theta_{j,j}^{(k)} + \sum
_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr) \biggr\} \biggr]
\\
&& \qquad{}- \lambda\sum_{1 \le j < j' \le p} \Biggl(\sum
_{k=1}^K \bigl| \bigl(\theta _{j,j'}^{(k)}
\bigr)^{[t]}\bigr| \Biggr)^{-1/2} \bigl|\theta_{j,j'}^{(k)}\bigr|\nonumber.
\end{eqnarray}
Note that criterion
\eqref{model_LLA} is a variant of criterion \eqref
{pseudo_separate_method} with a weighted $\ell_1$ penalty and hence
can be solved by the algorithm of \citet{HoeflingTibshirani09}. For
numerical stability, we
threshold $\sqrt{\sum_{k=1}^K |(\theta_{j,j'}^{(k)})^{[t]}|}$ at
$10^{-10}$. The algorithm is summarized as follows:
\begin{longlist}[\textit{Step} 1.]
\item[\textit{Step} 1.] Initialize $\widehat{\theta}_{j,j'}^{(k)}$'s ($1 \le
j,j' \le p;1 \le k \le K$) using the estimates from the separate
estimation method;
\item[\textit{Step} 2.] For each $1 \le k \le K$, update $\widehat{\theta
}_{j,j'}^{(k)}$'s by solving \eqref{model_LLA} using the
pseudo-likelihood algorithm \citet
{HoeflingTibshirani09,GuoLevinaMichailidisZhu10JOSE}.
\item[\textit{Step} 3.] Repeat step 2 until convergence.
\end{longlist}
The tuning parameter $\lambda$ in \eqref{model_cgm_lambda} controls
the sparsity of the resulting estimator and can be selected using
cross-validation. Specifically, for each $1 \le k \le K$, we randomly
split the data in the $k$th category into $D$ subsets of similar sizes
and denote the index set of the observations in the $d$th subset as
$\mathcal{T}_{d}^{(k)}$, $1 \le d \le D$. Then $\lambda$ is selected
by maximizing
\begin{eqnarray}
\label{cv_cgm} &&\frac{1}{D} \sum_{d=1}^D
\sum_{k=1}^K \frac{1}{|\mathcal
{T}_{d}^{(k)}|} \sum
_{i \in\mathcal{T}_{d}^{(k)}} \sum_{j=1}^p
x_{i,j}^{(k)} \biggl\{\bigl(\widehat{\theta}_{j,j}^{(k)}
\bigr)^{[-d]}(\lambda) + \sum_{j' \neq j} \bigl(
\widehat{\theta}_{j,j'}^{(k)}\bigr)^{[-d]}(\lambda)
x_{i,j'}^{(k)} \biggr\}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&\qquad{}- \log \biggl[1 + \exp \biggl\{\bigl(\widehat{\theta }_{j,j}^{(k)}
\bigr)^{[-d]}(\lambda) + \sum_{j' \neq j} \bigl(
\widehat{\theta }_{j,j'}^{(k)}\bigr)^{[-d]}(\lambda)
x_{i,j'}^{(k)} \biggr\} \biggr],
\end{eqnarray}
where $|\mathcal{T}_{d}^{(k)}|$ is the cardinality of $\mathcal
{T}_{d}^{(k)}$ and $(\widehat{\theta}_{j,j'}^{(k)})^{[-d]}(\lambda)$
is the joint estimate of $\theta_{j,j'}^{(k)}$ based on all
observations except those in $\mathcal{T}_{d}^{(1)}\cup\cdots
\cup\mathcal{T}_{d}^{(K)}$, as well as the tuning parameter
$\lambda$.
\section{Simulation study}\label{sec_simulation}
Before turning our attention to examining the U.S. Senate voting patterns,
we evaluate the performance of the joint estimation method on three
synthetic examples, each with $p=100$ variables and $K=3$ categories.
The network structure in each example is composed of two parts:\vadjust{\goodbreak} the
common structure across all categories and the individual structure
specific to a category. The common structures in these examples are a
chain graph, a nearest neighbor graph and a scale-free graph. These
graphs are generated as follows:
\begin{longlist}[Example~1:]
\item[Example~1:] \textit{Chain graph}.
A chain graph is generated by connecting nodes 1 to~$p$ in increasing
order, as shown in Figure~\ref{fig_illustrate}(A1).
\item[Example~2:] \textit{Nearest neighbor graph.}
The data generating mechanism of the nearest neighbor graph is adapted
from \citet{LiGui06}. Specifically, we generate $p$ points randomly on
a unit square, calculate all $p(p-1)/2$ pairwise distances, and find
three nearest neighbors of each point in terms of these distances. The
nearest neighbor network is obtained by linking any two points that are
nearest neighbors of each other. Figure~\ref{fig_illustrate}(B1)
illustrates a nearest-neighbor graph.
\item[Example~3:] \textit{Scale-free graph.}
A scale-free graph has a power-law degree distribution and can be
simulated by the Barabasi--Albert algorithm\break [\citet{BarabasiAlbert99}].
A realization of a scale-free network is depicted in Figure~\ref
{fig_illustrate}(C1).
\end{longlist}
\begin{figure}
\includegraphics{700f02.eps}
\caption{The networks used in three simulated examples. The black
lines represent the common structure, whereas the red, blue and green
lines represent the individual links in the three categories. $\rho$
is the ratio of the number of
individual links to the number of common links.}\label{fig_illustrate}\vspace*{-5pt}
\end{figure}
In each example, the network for the $k$th category ($k=1,\ldots,K$)
is created by randomly adding links to the common structure. The
individual links in different categories are disjoint and have the same
degree of sparsity, measured by $\rho$, the ratio of the number of
individual links to the number of common links. In particular, $\rho
=0$ corresponds to identical networks for all three categories. In the
simulation study, we consider $\rho=0$, $1/4$ and 1, gradually
increasing the proportion of individual links (Figure~\ref
{fig_illustrate}). Given the graphs, the symmetric parameter matrix $\M
{\Theta}^{(k)}$ is generated as follows. Each $\theta
_{j,j'}^{(k)}=\theta_{j',j}^{(k)}$ corresponding to an edge between
nodes $j$ and $j'$ is uniformly drawn from $[-1, -0.5]\cup[0.5,1]$,
whereas all other elements are set to zero. Then we generate the data
using Gibbs sampling. Specifically, suppose the $i$th iteration sample
has been drawn and is denoted as $(x_1^{(k)})^{[t]},\ldots,
(x_p^{(k)})^{[t]}$; then, in the $(t+1)$th iteration, we draw
$(x_j^{(k)})^{[t+1]}$, $1 \le j \le p$, from the Bernoulli distribution:
\begin{equation}
\bigl(x_j^{(k)}\bigr)^{[t+1]} \sim\operatorname{Bernoulli}
\biggl(\frac{\exp(\theta
_{j,j}^{(k)} + \sum_{j' \neq j} \theta_{j,j'}^{(k)}
(x_{j'}^{(k)})^{[t]})}{1 + \exp(\theta_{j,j}^{(k)} + \sum_{j' \neq
j} \theta_{j,j'}^{(k)} (x_{j'}^{(k)})^{[t]})} \biggr).
\end{equation}
To ensure that the simulated observations are close to i.i.d. samples
from the target distribution, the first 1,000,000 rounds are discarded
(burn-in) and the data are collected every 100 iterations from the
sampler. In the simulation study, we consider a balanced scenario and
an unbalanced scenario. The former consists of $n_k=300$ observations
in each category, whereas the latter has three unbalanced categories
with sample sizes $n_1=200$, $n_2=300$ and $n_3=400$.
\begin{figure}[t]
\includegraphics{700f03.eps}
\caption{Results for the balanced scenario ($n_1=n_2=n_3=300$) and
dimension $p = 100$. Black solid curve: joint estimation; red dashed
curve: separate estimation. The ROC curves are averaged over 10
replications. $\rho$ is the ratio between
the number of individual links and the number of common links.}\label
{fig_sim_subplot_K3}
\end{figure}
\begin{figure}[b]
\includegraphics{700f04.eps}
\caption{Results for the unbalanced scenario ($n_1= 200$, $n_2=300$,
$n_3=400$) and dimension
$p = 100$. Black solid curve: joint estimation; red dashed curve:
separate estimation.The ROC curves are averaged over 10
replications. $\rho$ is the ratio between the number of individual
links and the number of common links.}\vspace*{-3pt}\label{fig_sim_subplot_unbalance}
\end{figure}
We compared the structure estimation results of the joint estimation
method and the separate estimation method using ROC curves, which
dynamically characterize the sensitivity (proportion of correctly
identified links) and the specificity (proportion of correctly excluded
links) by varying the tuning parameter $\lambda$. Figure~\ref
{fig_sim_subplot_K3} shows the ROC curves averaged over 10 replications
from the three examples in the balanced scenario, where the joint
estimation method
dominates separate estimation when the proportion of individual links
is low. As $\rho$ increases, the structures become more different, and
the joint and separate methods move closer together. This is expected,
since the joint estimation method is designed to take advantage of
common structure. The results in the unbalanced scenario exhibit a
similar pattern (Figure~\ref{fig_sim_subplot_unbalance}).\looseness=-1
\section{Analysis of the U.S. Senate voting records}
\label{sec_real_example}
We applied the proposed joint estimation method to the voting records
of the U.S. Senate from the 109th Congress covering the period
2005--2006. The $p=100$ variables correspond to the senators.
The Senate held 645 votes in that period, from which we extracted
$n=222$ votes in the three largest categories, namely, defense and
security (141), environment and energy (34), and healthcare (47).
The votes are recorded as ``yes'' (encoded as ``1'') and ``no''
(encoded as ``0''). The assumption of our model is that bills within a
category are an i.i.d. sample from the same underlying Ising model.
In reality, the voting process may be more complex, with possible
temporal factors and further dependencies among bills, possibly
reflecting backroom deals. Neverthless, this is an improvement on
previous analyses of such data, which treated all bills in all
categories as i.i.d. [\citet{BanerjeeGhaouidAspremont08}], and is a
reasonable trade-off for an exploratory data analysis tool.
There were missing observations, as not all senators vote on all bills.
The number of bills containing at least one missing vote was
98 out of 141 for defense and security, missing a total of 2.26\% of
all votes; 24 out of 34 for environment and energy, missing a total of
3.23\% of votes; and 20 out of 47 for healthcare, missing 2.38\% of all
votes. While the number of bills that are missing at least one
Senator's vote is relatively high, the overall proportion of missing
observations is quite low and, thus, we do not expect it to create a
major problem in the analysis. Nevertheless, we have investigated
multiple strategies for imputing the missing data in the electronic
supplement; specifically, we considered replacing the missing vote by
the party's majority, by the majority vote of the five most similar
Senators and, to test robustness to the imputation method, also by the
opposite party's majority and at random. We found that the main
conclusions of the analysis are not very sensitive to missing data
imputation methods.
In the subsequent analysis, we replace a missing vote for a Senator by
his/her party's majority vote on the bill; for the Independent Senator
Jeffords, we take the Democratic majority vote.
After the imputation, the bills with a ``yes/no'' proportion greater
than 90\% or less than 10\% were excluded from the analysis, as these
typically correspond to procedural votes. This left 97, 29 and 40 bills
in the three categories, respectively. Given that two of the
sample sizes are fairly small (29 and 40), we added an $\ell_2$
penalty with a small tuning parameter $\lambda_2=0.01$.
This approach, known as the elastic net, has been shown to help avoid
extremely sparse networks in such situations [\citet{ZouHastie05}].
The main tuning parameter for our method was selected through
cross-validation. Following \citet{LiGui06}, we used a bootstrap
procedure for
final edge selection, estimating the network for 100 bootstrap samples
of the same size, and only retained
edges that appeared more that $\alpha$ percent of the time. This
procedure is similar to stability selection [\citet{MeinshausenBuhlmann10}].
\begin{figure}[b]
\includegraphics{700f05.eps}
\caption{The estimated graphical models for the three categories in
the Senate voting data with an inclusion cutoff value of
0.4 and tuning parameter value of 0.5. Edges common to all three
categories are shown under the heading
``common structure''; all other edges are shown on category-specific
graphs. The nodes represent the 100 senators,
with red, blue and purple node colors corresponding to Republican,
Democrat or Independent (Senator Jeffords),
respectively. A solid line corresponds to a positive interaction effect
and a dashed line to a negative interaction effect.
The width of a link is proportional to the magnitude of the
corresponding overall interaction effect.}
\label{fig_lambda05_cutoff04}
\end{figure}
The network representation, depicting both the common and the
individual structures with a cutoff value for inclusion
$\alpha=0.4$ and a value of $\lambda=0.05$, is depicted in
Figure~\ref{fig_lambda05_cutoff04}.
Note that unlike techniques such as principal components analysis and
multidimensional scaling that directly embed the senators in a
two-dimensional map, the proposed method estimates the edges and
constructs the adjacency matrix of the graph of Senators; subsequently,
we employed a graph drawing program to visualize this graph.
The common network structure estimated by the joint estimation method
is shown in the top left panel of Figure~\ref{fig_lambda05_cutoff04}.
For the individual
categories, we only plot the edges associated with the category that is
not part of the common network, to enhance the readability of the
graphs. As expected, members of the two political parties are clearly
separated. For both tuning parameter values, there are strong positive
associations between senators of the same party and selected strong
negative associations between senators of opposite parties. Obviously,
at the higher tuning parameter value the common dependence structure
becomes sparser. Of particular interest is the finding that at both
tuning values there are many more associations between Democratic
senators than Republican ones and
this pattern holds for both the common and individual structures. One
possible explanation may be that during that period the Democrats were
in the minority and thus voting more frequently as a block. Further,
the Independent
Senator Jeffords is associated with the Democrats, while the moderate
Republicans Collins, Snowe, Chafee and Specter (who switched to the
Democratic party in early 2009) are not strongly associated with their
Republican colleagues,
thus confirming results of previous analyses by
\citet{ClintonJackmanRivers04} and \citet{deLeeuw06} (albeit based on
data from the 105th Congress). The conservative Democrat Nelson
(Nebraska) is also not closely associated with his party, as well as
the very conservative Republican
de Mint (South Carolina). Also, the analysis suggests that Senator
Lieberman had a solid Democratic voting record before becoming an
Independent in 2008.
Other interesting patterns emerging from
the analysis are that the more moderate members of the two parties are
located closer to the center of their respective ``clouds'' (e.g.,
Warner, Frist,
Voinovich and Smith on the Republican side, and Levin, Reid, Mikulski
and Rockefeller on the Democratic side), the cluster of
economic conservatives on the Republican side (McConnell, Domenici,
Crapo, Inhofe),
the close ties of the liberal Democrats Kennedy, Boxer and Nelson
(Florida), the
close voting records of senators from the same state (Schumer and
Clinton from New York,
Murkowski and Stevens from Alaska, Snowe and Collins from Maine,
Cantwell and Murray from Washington). There is also a strong dependence
between Durbin, Corzine, Lincoln, Harkin and
Dodd on the Democratic side.
Examining the individual networks for the three categories shown in
Figure~\ref{fig_lambda05_cutoff04},
we note that additional positive associations among Democrats emerge,
primarily for defense and healthcare categories, thus
indicating a stronger ideological cohesion on these issues. Further, a
number of stable negative associations emerge in the environment and
healthcare categories, indicating a stronger ideological divide between
senators.
On defense, some additional strong ties emerge between more liberal
leaning Democrats (Stabenow, Biden, Leahy, Kerry, Boxer),
while a strong cluster on environmental issues arises between
Republican senators from energy producing states
(Murkowski and Stevens
from Alaska, Thune from South Dakota, Hutchison from Texas, but also
Bond from Missouri, Chambliss from Georgia, Craig from
Idaho and Roberts from Kansas with their unwavering support for
offshore drilling).
On health and medical issues, a number of additional strong positive
associations emerge among Democratic senators, possibly reflecting the
fact that the 109th Congress dealt with issues ranging
from veterans affairs, to medical malpractice to food safety and
especially on health savings accounts legislation to reduce
medical insurance costs.
Different imputation strategies for missing data were also examined and
the analysis results are given in Figures 1--3 in the Supplement
for the same values of the cutoff $\alpha$ and tuning parameter
$\lambda$. It can be
seen that similar patterns emerge, although alternative methods of
imputation may lead to the emergence of a few more
associations. Nevertheless, the main findings seem to be robust to the
examined choices of the imputation mechanism, although at very high
levels of absenteeism this may not hold [\citet{Han2007}].
For comparison purposes, separate multidimensional scaling analyses are
shown in Figure~\ref{mds-analyses} for all the votes together and for
the three categories separately. MDS (or PCA or factor analysis) is one
of the commonly taken approaches in social sciences when graphical
modeling is not considered. Figure~\ref{mds-analyses} suggests that
the overall vote clustering in the two parties is driven to a large
extent by the corresponding clustering in the defense and health categories.
On the other hand, voting on environmental issues creates a clear
separation between the two parties, although the moderate Republicans
Chafee, Collins and Snowe are shown to have a voting record similar to
the Democrats, while the Democrats
Nelson (Nebraska) and Landrieu are closer to the Republicans. At a high
level, MDS-based findings are similar to ours, which is a satisfactory
result, but they do not provide explicit clusters or edges, nor do they
provide a way to quantify the amount of dependence between individual
pairs (visualized via edge thickness in Figure~\ref{fig_lambda05_cutoff04}).
\begin{figure}
\includegraphics{700f06.eps}
\caption{Multidimensional scaling analysis for all the votes together,
and the three individual
categories. The nodes represent the 100 senators, with red, blue and
purple node colors corresponding to Republican,
Democrat or Independent (Senator Jeffords), respectively.}\label{mds-analyses}
\end{figure}
Another relevant comparison is to fitting a separate graphical model to
each of the three categories, as could have been done with any of the
previously developed methods for fitting the Ising model. The results
are shown in Figure~\ref{separate-data-analysis}, in the same format
as in Figure~\ref{fig_lambda05_cutoff04}, with edges common to all
three categories shown under ``common structure,'' and all other edges
under their own category. We followed the same tuning procedure as we
did for joint estimation, bootstrapping the data 100 times for
stability selection and selecting the value of the tuning parameter on
a validation data set. Even with the cutoff set at 1 (we included only
the edges appearing in all the bootstrap replications), the graphs are
dense and difficult to interpret. Similar to MDS, they capture party
cohesion through strong
positive associations between members of the same party for all three
categories and some negative associations between members of opposite
parties. However, different voting patterns between categories are not
clear, although the results suggest a more cohesive voting record for
both parties for the defense category. Note that since this is
exploratory data analysis, it is hard to verify which set of results is
``better.'' Nevertheless, those obtained from the joint estimation
method are more nuanced and interpretable and therefore provide
better insights into voting strategies of members of Congress.
\begin{figure}
\includegraphics{700f07.eps}
\caption{The estimated graphical models for the three categories in
the Senate voting data fitted via separate estimation.
Edges common to all three categories are shown under the heading
``common structure''; all other edges
are shown on category-specific graphs. The cutoff value is 1 (only
edges appearing in all bootstrap
replications are included). The nodes represent the 100 senators, with
red, blue and purple node colors
corresponding to Republican, Democrat or Independent (Senator Jeffords),
respectively. A solid line corresponds to a positive interaction effect
and a dashed line to a negative
interaction effect. The width of a link is proportional to the
magnitude of the corresponding overall interaction effect.}\label
{separate-data-analysis}
\end{figure}
\section{Concluding remarks}\label{sec_conclusion}
We have proposed a joint estimation method for the analysis of
heterogenous Markov networks motivated by the need to jointly estimate
heterogeneous networks, such as those of the Senate voting patterns.
The method improves estimation of the networks' common structure by borrowing
strength across categories, and allows for individual differences.
Asymptotic properties of the method have been established. In
particular, we show that the convergence rate is similar to the rate
for Gaussian graphical models in a similar context [\citet
{GuoLevinaMichailidisZhu10JOSE}]. The proposed method can be extended
to deal with general categorical data with more than two levels using
the strategy described in \citet{RavikumarWainwrightLafferty10} and
\citet{GuoLevinaMichailidisZhu10JOSE}.
The most interesting feature emerging from the analysis of the Senate
voting records is the existence of more stable associations for the
Democrats, both in terms of the common structure
and in the healthcare and defense categories.
There are other techniques suitable for analyzing roll call data.
Dimension reduction techniques
create maps, where the relative positioning of the senators allows one
to infer similarity in their voting patterns. They provide
a useful visual tool to capture broad patterns and relationships. On
the other hand, a Markov network model aims directly at estimating
the associations between the senators and thus provides an alternative
view of the voting patterns, which together with the
thresholding technique employed gives a measure of the stability of
such associations. Further, the joint estimation method
allows one to separately study the overall voting patterns and those
driven by specific issues. In our view, both sets of techniques
are useful, with dimension reduction providing a global perspective and
the Markov model revealing more nuanced patterns.
\begin{appendix}
\section*{Appendix: Asymptotic properties}\label{app}
In this section we study the asymptotic properties of the proposed
joint estimation method. Since the structure of the underlying network
only depends on the interaction effects, we focus on a variant of the
model without main effects. Specifically, we solve
\begin{eqnarray}
\label{objfun_joint_nomaineffect}\quad&& \max_{\{\MM{\Theta}^{(k)}\}_{k=1}^K} \sum_{k=1}^K
\frac{1}{n_k} \sum_{i=1}^{n_k} \sum
_{j=1}^p \biggl[x_{i,j}^{(k)}
\biggl(\sum_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr) \nonumber\\
&&\hspace*{103pt}{}- \log \biggl\{1 + \exp \biggl(\sum
_{j' \neq j} \theta_{j,j'}^{(k)}
x_{i,j'}^{(k)} \biggr) \biggr\} \biggr]
\\
&&\qquad{} - \lambda\sum_{j < j'} \sqrt{\sum
_{k=1}^K\bigl | \theta _{j,j'}^{(k)}\bigr|}.\nonumber
\end{eqnarray}
We will show that the estimator in criterion \eqref
{objfun_joint_nomaineffect} is consistent in terms of both parameter
estimation and model selection, when $p$ and $n$ go to infinity and the
tuning parameter $\lambda$ goes to zero at some appropriate rate. We
note that our results are pointwise rather than uniform in $\Theta$,
as is standard in the literature. Some interesting implications of
nonuniform bounds for sparse estimators in linear regression have
recently been discussed by \citet{LeebPoetscher2008,PoetscherLeeb2009},
although their conclusions do not apply to graphical models.
Before stating the main results, we introduce necessary notation and
regularity conditions. For each $k=1,\ldots, K$, denote $\VV{\theta
}^{(k)} = (\theta_{1,2}^{(k)},\ldots, \theta_{j,j'}^{(k)},\ldots,\break
\theta_{p-1,p}^{(k)})$ as a $p(p-1)/2$-dimensional vector, recording
all upper triangular elements in $\MM{\Theta}^{(k)}$. Let $\overline
{\VV{\theta}}^{(k)}$ be the true value of $\VV{\theta}^{(k)}$. Let
$\overline{\M{Q}}^{(k)}$ be the population Fisher information matrix
of the model in criterion \eqref{objfun_joint_nomaineffect} (see the
\hyperref[app]{Appendix} for a precise definition) and let $\mathcal{X}_{(i)}^{(k)}$
be a matrix with $p$ rows and $p(p-1)/2$ columns,
whose $(j,j')$th column is composed of zeros except for the $j$th
($j'$th) component being $x_{i,j'}$ ($x_{i,j}$). In addition, we define
$\overline{\M{U}}^{(k)} = E[\T{\MM{\mathcal{X}}_{(i)}^{(k)}} \M
{\mathcal{X}}_{(i)}^{(k)}]$. To index the zero and nonzero elements,
let $S_k = \{(j,j')\dvtx \theta_{j,j'}^{(k)} \neq0, 1 \le j < j' \le p\}$
and $S_k^c = \{(j,j')\dvtx \theta_{j,j'}^{(k)} = 0, 1 \le j < j' \le p\}$,
and let $S_{\cap} = \bigcap_{k=1}^K S_k$, $S_{\cup} = \bigcup_{k=1}^K S_k$. The cardinalities of $S_k$ and $S_{\cup}$ are denoted
by $q_k$ and $q$, respectively. For any matrix $\M{W}$ and subsets of
row and
column indices $\mathcal{U}$ and $\mathcal{V}$, let
$\M{W}_{\mathcal{U}, \mathcal{V}}$ be the matrix consisting of
rows $\mathcal{U}$ and columns $\mathcal{V}$ in $\M{W}$. Finally, let
$\Lambda_{\min}(\cdot)$ and $\Lambda_{\max}(\cdot)$ denote the
smallest and largest eigenvalue of a matrix, respectively.
The asymptotic properties of the joint estimation method rely on the
following regularity conditions:
\begin{enumerate}[(A)]
\item[(A)] Nonzero elements bounds: There exist positive constants
$\gamma_{\min}$ and $\gamma_{\max}$ such that:
\begin{enumerate}[(ii)]
\item[(i)] $\min_{1 \le k \le K} \min_{(j,j') \in S_k} |\overline
{\theta}_{j,j'}^{(k)}| \ge\gamma_{\min}$;
\item[(ii)] $\max_{1 \le k \le K} \max_{(j,j') \in S_k \setminus
S_{\cap}} |\overline{\theta}_{j,j'}^{(k)}| \le\gamma_{\max}$.
\end{enumerate}
\item[(B)] Dependency: There exist positive constants $\tau_{\min}$
and $\tau_{\max}$ such that for any $k=1,\ldots, K$,
\begin{equation}
\Lambda_{\min} \bigl(\overline{\M{Q}}_{S_k, S_k}^{(k)}
\bigr) \ge \tau_{\min
} \quad\mbox{and} \quad\Lambda_{\max} \bigl(\overline{
\M{U}}_{S_k,S_k}^{(k)}\bigr) \le\tau_{\max}.
\end{equation}
\item[(C)] Incoherence: There exists a constant $\tau\in(1-\sqrt
{\gamma_{\min}/4\gamma_{\max}},1)$ such that for any $k=1,\ldots, K$,
\begin{equation}
\bigl\| \overline{\M{Q}}_{S_k^c, S_k}^{(k)} \bigl(\overline{
\M{Q}}_{S_k,
S_k}^{(k)}\bigr)^{-1}\bigr\|_{\infty} \le1 -
\tau.
\end{equation}
\end{enumerate}
Condition (A) enforces a lower bound on the magnitudes of all nonzero
elements, as well as an upper bound on the magnitudes of those nonzero
elements associated with individual links. Conditions (B) and (C) bound
the amount of
dependence and the influence that the nonneighbors can have
on a given node, respectively. Conditions similar to (B) and (C) were also
assumed by \citet{MeinshausenBuhlmann06},
\citet{RavikumarWainwrightLafferty10},
\citet{PengWangZhouZhu09} and \citet{GuoLevinaMichailidisZhu10JOSE}.
Our conditions are most closely related to those of \citet
{GuoLevinaMichailidisZhu10JOSE}, but here they are extended to the
heterogenous data setting.
\begin{theorem}[(Parameter estimation)]\label{thm_estimation_consistency}
Suppose all regularity conditions hold. If the tuning
parameter $\lambda= C_{\lambda} \sqrt{(\log p)/ n}$ for some
constant $C_{\lambda} > (8-4\tau)\sqrt{\gamma_{\min}}/(1-\tau)$
and if $\min\{n/q^3, n_1/q_1^3,\ldots, n_K/q_K^3\} > (4/C) \log p$
for some constant $C=\min\{\tau_{\min}^2 \tau^2 / 288 (1-\tau)^2,
\tau_{\min}^2 \tau^2/72, \tau_{\min}\tau/48\}$, then there exists
a local maximizer of the criterion \eqref{objfun_joint_nomaineffect},
$\{\widehat{\MM{\theta}}^{(k)}\}_{k=1}^K$, such that, with
probability tending to 1,
\begin{equation}
\sum_{k=1}^K \bigl\| \widehat{\VV{
\theta}}^{(k)} - \overline{\VV{\theta }}^{(k)} \bigr\|_2
\le M \sqrt{\frac{q \log p}{n}},
\end{equation}
for some constant $M > (2K C_{\lambda} / \tau_{\min} \sqrt{\gamma
_{\min}}) (3-2\tau)/(2-\tau)$.
\end{theorem}
\begin{theorem}[(Structure selection)]\label{thm_modelselection_consistency}
Under conditions of Theorem~\ref{thm_estimation_consistency}, with probability tending to 1, the
maximizer $\{\widehat{\MM{\theta}}^{(k)}\}_{k=1}^K$ from Theorem~\ref{thm_estimation_consistency} satisfies
\begin{eqnarray*}
\widehat{\theta}_{j,j'}^{(k)} &\neq& 0\qquad\mbox{for all }
\bigl(j,j'\bigr) \in S_k, k=1,\ldots, K;
\\
\widehat{\theta}_{j,j'}^{(k)} &=& 0\qquad \mbox{for all }
\bigl(j,j'\bigr) \in S_k^c, k=1,\ldots, K.
\end{eqnarray*}
\end{theorem}
Theorems \ref{thm_estimation_consistency} and \ref
{thm_modelselection_consistency} establish the consistency in terms of
parameter estimation and
structure selection, respectively.
The main idea of the proofs is closely related to \citet
{GuoLevinaMichailidisZhu10JOSE}, and some strategies for dealing with
the joint estimation are borrowed from \citet{GuoLevinaMichailidisZhu09CGM}.
We introduce notation first. For the $k$th category, we define the
log-likelihood as
\[
l\bigl(\VV{\theta}^{(k)}\bigr)=\frac{1}{n_k} \sum
_{i=1}^{n_k} \sum_{j=1}^p
\biggl[x_{i,j}^{(k)} \biggl(\sum_{j' \neq j}
\theta_{j,j'}^{(k)} x_{i,j'}^{(k)}\biggr) -
\log\biggl\{1 + \exp\biggl(\sum_{j' \neq j}
\theta_{j,j'}^{(k)} x_{i,j'}^{(k)} \biggr)\biggr
\}\biggr],
\]
whose first derivative and second derivative are denoted by $\nabla
l(\VV{\theta}^{(k)})$ and $\nabla^2 l(\VV{\theta}^{(k)})$,
respectively. Note that $\nabla l(\VV{\theta}^{(k)})$ is a
$p(p-1)/2$-dimensional vector and $\nabla^2 l(\VV{\theta}^{(k)})$ is
a $p(p-1)/2 \times p(p-1)/2$ matrix.
Then, the population Fisher information matrix of
the model in \eqref{objfun_joint_nomaineffect} at
$\overline{\VV{\theta}}$ can be defined as
$\overline{\M{Q}}^{(k)} = -\F{E}[\nabla^2 l(\overline{\VV{\theta
}}^{(k)})]$, and its sample
counterpart is
$\widehat{\M{Q}}^{(k)} = -\nabla^2 l(\overline{\VV{\theta
}}^{(k)})$. We also write
$\widehat{\M{U}}^{(k)} = 1/n \sum_{i=1}^n \T{\MM{\mathcal{X}}_{(i)}^{(k)}}
\MM{\mathcal{X}}_{(i)}^{(k)}$ for the sample counterpart of $\overline
{\M{U}}^{(k)}$. Let $\underline{\VV{\theta}}^{(k)} = (\underline
{\theta}_{1,2}^{(k)},\ldots, \underline{\theta}_{j,j'}^{(k)},\ldots, \underline{\theta}_{p-1,p}^{(k)})$ be the same as $\VV
{\theta}^{(k)}$ except that all elements in $S_k^c$ are set to zero
and write $\VV{\delta}^{(k)} = \VV{\theta}^{(k)} - \overline{\VV
{\theta}}^{(k)}$ and $\underline{\VV{\delta}}^{(k)} = \underline{\VV
{\theta}}^{(k)} - \overline{\VV{\theta}}^{(k)}$. Finally, let
$\mathcal{W}$ be a subset of the index set $\{1, 2,\ldots,
p(p-1)/2\}$. For a $p(p-1)/2$-dimensional vector $\VV{\beta}$, we define
$\VV{\beta}_{\mathcal{W}}$ as the vector consisting of the
elements of $\VV{\beta}$ associated with
$\mathcal{W}$.
Next, we introduce a variant of criterion \eqref
{objfun_joint_nomaineffect} by restricting all true zeros in $\{\VV
{\theta}^{(k)}\}_{k=1}^{K}$ to be estimated as zero. Specifically, the
restricted criterion is formulated as follows:
\begin{equation}
\label{obj_restricted_Ising} \max_{\{\underline{\VV{\theta}}^{(k)}\}_{k=1}^K} \sum_{k=1}^K
l\bigl(\underline{\VV{\theta}}^{(k)}\bigr) - \lambda\sum
_{1 \le
j < j' \le p} \sqrt{\sum_{k=1}^K
\bigl|\underline{\theta}_{j,j'}^{(k)}\bigr|},
\end{equation}
and its maximizer is denoted by $\{\widehat{\underline{\VV{\theta
}}}^{(k)}\}_{k=1}^{K}$.
In addition, we consider the sample versions of regularity conditions
(B) and (C):
\begin{longlist}[(B$^{\prime}$)]
\item[(B$^{\prime}$)] \textit{Sample dependency}: There exist positive
constants $\tau_{\min}$ and $\tau_{\max}$ such that for any
$k=1,\ldots, K$,
\begin{equation}
\Lambda_{\min} \bigl(\widehat{\M{Q}}_{S_k, S_k}^{(k)}
\bigr) \ge \tau_{\min
} \quad\mbox{and} \quad\Lambda_{\max} \bigl(\widehat{
\M{U}}_{S_k,S_k}^{(k)}\bigr) \le\tau_{\max}.
\end{equation}
\item[(C$^{\prime}$)] \textit{Sample incoherence}: There exists a constant
$\tau\in(1-\sqrt{\gamma_{\min}/4\gamma_{\max}},1)$ such that for
any $k=1,\ldots, K$,
\begin{equation}
\bigl\|\widehat{\M{Q}}_{S_k^c, S_k}^{(k)} \bigl(\widehat{
\M{Q}}_{S_k,
S_k}^{(k)}\bigr)^{-1}\bigr\|_{\infty} \le1 -
\tau.
\end{equation}
\end{longlist}
For convenience of the readers, the proof of our main result is divided
into two parts: Part I presents the main idea of the proof by listing the
important propositions and the proofs of Theorems
\ref{thm_estimation_consistency} and
\ref{thm_modelselection_consistency}, whereas part II
contains additional technical details and proofs of propositions in
part I.
\subsection*{Part I: Propositions and proof of Theorems \protect\ref
{thm_estimation_consistency} and \protect\ref{thm_modelselection_consistency}}
The proof consists of the following
steps. Proposition~\ref{thm_restricted_consistency} shows that, under
sample regularity conditions (B$^{\prime}$) and
(C$^{\prime}$), the conclusions of Theorems \ref
{thm_estimation_consistency} and \ref{thm_modelselection_consistency}
hold for the
local maximizer of the restricted problem \eqref
{obj_restricted_Ising}. Next, Proposition~\ref{prop_equival_sample_population}
proves that the population regularity conditions (B) and (C) give rise
to their sample counterparts (B$^{\prime}$) and (C$^{\prime}$)
with probability tending to one, hence, the conclusions of
Proposition~\ref{thm_restricted_consistency} also hold with the
population regularity conditions. Last, we show that the local
maximizer of \eqref{obj_restricted_Ising} is also a local maximizer of
the original model \eqref{objfun_joint_nomaineffect}.
This is established via Proposition~\ref{KKT_conditions}, which sets
out the Karush--Kuhn--Tucker (KKT) conditions for the local maximizer of
criterion \eqref{objfun_joint_nomaineffect}, and Proposition~\ref
{prop_restricted_KKT}, which
shows that, with probability tending to one, the local maximizer of
\eqref{obj_restricted_Ising} satisfies these KKT conditions.
\begin{proposition}\label{thm_restricted_consistency}
Suppose condition \textup{(A)} and the sample conditions \textup{(B$^{\prime}$)} and
\textup{(C$^{\prime}$)} hold.
If the tuning parameter $\lambda= C_{\lambda} \sqrt{(\log p)/ n}$
for some constant $C_{\lambda} > (8-4\tau)\sqrt{\gamma_{\min
}}/(1-\tau)$ and
$q\sqrt{(\log p)/n} = o(1)$, then with probability tending to one,
there exists a local maximizer of the restricted criterion, $\{\widehat
{\underline{\VV{\theta}}}^{(k)}\}_{k=1}^K$, satisfying:
\begin{longlist}[(ii)]
\item[(i)] $\sum_{k=1}^K \| \widehat{\underline{\VV{\theta
}}}^{(k)} - \overline{\VV{\theta}}^{(k)} \|_2 \leq M \sqrt{q (\log p)/n}$
for some constant $M > (2K C_{\lambda} /\break \tau_{\min} \sqrt
{\gamma_{\min}}) [(3-2\tau)/(2-\tau)]$;
\item[(ii)] For each $k=1,\ldots, K$, $\widehat{\underline{\theta
}}_{j,j'}^{(k)}
\neq0$ for all $(j,j') \in S_k$ and $\widehat{\underline{\theta
}}_{j,j'}^{(k)} =0$
for all $(j,j') \in S_k^c$.
\end{longlist}
\end{proposition}
\begin{proposition}\label{prop_equival_sample_population}
Suppose the regularity conditions \textup{(B)} and \textup{(C)} hold, then for any
$\varepsilon> 0$, the following inequalities hold with probability
tending to one for all $k=1,\ldots, K$:
\begin{longlist}[(iii)]
\item[(i)] $\F{P} \{\Lambda_{\mathrm{min}} (\widehat{\M{Q}}_{S_k,S_k}^{(k)}) \le
\tau_{\mathrm{min}} - \varepsilon\} \le2 \exp\{-(\varepsilon^2/2) (n_k/q_k^2) +
2 \log q_k\}$;
\item[(ii)] $\F{P}\{\Lambda_{\max} (\widehat{\M{U}}_{S_k,S_k}^{(k)}) \ge
\tau_{\max} + \varepsilon\} \le2 \exp\{-(\varepsilon^2/2) (n_k/q_k^2) +
2 \log q_k\}$;
\item[(iii)] $\F{P}[\|\widehat{\M{Q}}_{S_k^c,S_k}^{(k)} (\widehat
{\M{Q}}_{S_k, S_k}^{(k)})^{-1}\|_{\infty} \ge1
- \tau/2] \le12 \exp(-C n_k/q_k^3 + 4\log p)$, for some constant $C
= \min\{\tau_{\mathrm{min}}^2
\tau^2 / 288 (1-\tau)^2, \tau_{\mathrm{min}}^2 \tau^2 / 72, \tau_{\mathrm{min}}
\tau/ 48\}$.
\end{longlist}
\end{proposition}
\begin{proposition}\label{KKT_conditions}
$\{\widehat{\VV{\theta}}\}_{k=1}^K$ is a local maximizer of problem
\eqref{objfun_joint_nomaineffect} if and only if the following
conditions hold for all $k=1,\ldots, K$:
\begin{eqnarray}
\nabla_{j,j'} l \bigl(\widehat{\VV{
\theta}}^{(k)}\bigr) &=& \lambda \sgn\bigl(\widehat{\theta}_{j,j'}^{(k)}
\bigr) \Big/ \Biggl(\sum_{k=1}^K \bigl|\widehat {
\theta}_{j,j'}^{(k)}\bigr|\Biggr)^{1/2}\qquad \mbox{if }
\widehat{\theta}_{j,j'}^{(k)} \neq0;
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
\bigl|
\nabla_{j,j'} l \bigl(\widehat{\VV{\theta}}^{(k)}\bigr)\bigr| &<&
\lambda\Big/ \Biggl(\sum_{k=1}^K \bigl|\widehat{
\theta}_{j,j'}^{(k)}\bigr|\Biggr)^{1/2} \qquad \mbox{if }
\widehat{\theta}_{j,j'}^{(k)}=0.
\end{eqnarray}
\end{proposition}
\begin{proposition}\label{prop_restricted_KKT} Under all conditions of
Proposition~\ref{thm_restricted_consistency},
with probability tending to one, we have, for each $k=1,\ldots,K$,
\begin{eqnarray}
\nabla_{j,j'} l \bigl(\widehat{\underline{\VV{
\theta}}}^{(k)}\bigr) &=& \lambda \sgn\bigl(\widehat{\underline{
\theta}}_{j,j'}^{(k)}\bigr) \Big/ \Biggl(\sum
_{k=1}^K \bigl| \widehat{\underline{\theta}}_{j,j'}^{(k)}\bigr|
\Biggr)^{1/2} \qquad \mbox{for all }\bigl(j,j'\bigr) \in
S_k;
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
\bigl|\nabla_{j,j'} l \bigl(\widehat{
\underline{\VV{\theta}}}^{(k)}\bigr)\bigr| &<& \lambda\Big/ \Biggl(\sum
_{k=1}^K \bigl| \widehat{\underline{\theta
}}_{j,j'}^{(k)}\bigr|\Biggr)^{1/2} \qquad \mbox{for all }
\bigl(j,j'\bigr) \in S_k^c.
\end{eqnarray}
\end{proposition}
\begin{pf*}{Proof of Theorems \protect\ref
{thm_estimation_consistency} and \protect\ref{thm_modelselection_consistency}}
The condition $\min\{n/q^3, n_1/q_1^3,\ldots, n_K/\break q_K^3\} > (4/C)
\log p$ implies that, for each $k=1,\ldots, K$, we have $-C n_k /
q_k^3 + 4 \log p < 0$ and $-(\varepsilon^2/2) (n_k / q_k^2) + 2 \log q_k
< 0$ when $q_k$ is large enough. This condition also implies $q \sqrt
{(\log p)/n}=o(1)$. In addition, by Proposition~\ref
{prop_equival_sample_population}, the sample conditions (B$^{\prime}$)
and (C$^{\prime}$) hold with probability tending to one when
regularity conditions (B) and (C) hold. Therefore, by Proposition~\ref
{thm_restricted_consistency}, with probability tending to one, the
solution of the restricted problem $\{\widehat{\underline{\VV{\theta
}}}^{(k)}\}_{k=1}^K$ satisfies both parameter estimation consistency
and structure selection consistency. On the other hand, by Proposition~\ref{prop_restricted_KKT}, with probability tending to one, $\{
\widehat{\underline{\VV{\theta}}}^{(k)}\}_{k=1}^K$ also satisfies
the KKT conditions in Proposition~\ref{KKT_conditions}, thus, it is a
local maximizer of criterion \eqref{objfun_joint_nomaineffect}. This
proves Theorems \ref{thm_estimation_consistency} and \ref
{thm_modelselection_consistency}.
\end{pf*}
\subsection*{Part II: Proofs of propositions}
Before proving the propositions, we state a few lemmas which will be
used in the proofs. These lemmas are variants of Lemmas 1, 2 and 5 in
\citet{GuoLevinaMichailidisZhu10JOSE}, adapted to the settings of the
heterogenous model and, thus, the proofs are omitted here. Likewise,
the proof of Proposition~\ref{prop_equival_sample_population} is very
similar to the proof of Propositions 3 and 4 in \citet
{GuoLevinaMichailidisZhu10JOSE} and is omitted.
\begin{lemma}\label{lemma_Bound_first_derivative}
For each $k=1,\ldots, K$, with probability tending to 1, we have
$\|\nabla l
(\overline{\VV{\theta}}^{(k)})\|_{\infty} \le C_{\nabla} \sqrt
{(\log p)/ n}$ for
some constant $C_{\nabla} > 4$.
\end{lemma}
\begin{lemma}\label{lemma_Bound_residual}
If the sample dependency condition \textup{(B$^{\prime}$)} holds and\break
$q\sqrt{(\log p)/n} = o(1)$, then for any $\alpha_k \in[0, 1]$,
$k=1,\ldots, K$, the following inequality holds with probability
tending to 1:
\begin{equation}
-\sum_{k=1}^K \T{\VV{
\delta}_{S_k}^{(k)}} \bigl[\nabla^2 l \bigl(
\overline {\VV{\theta}}^{(k)} + \alpha_k \underline{\VV{
\delta}}^{(k)}\bigr)\bigr]_{S_k,S_k} \VV{\delta}_{S_k}^{(k)}
\ge \frac{1}{2} \tau_{\min} \sum_{k=1}^K
\bigl\|\underline{\VV{\delta }}^{(k)}\bigr\|_2^2.
\end{equation}
\end{lemma}
\begin{lemma}\label{lemma_bound_r}
Suppose the sample dependency condition \textup{(B)} holds. For any $\alpha_k
\in[0,1]$, $k=1,\ldots, K$, the following inequality holds with
probability tending to one:
\begin{equation}
\bigl\|\bigl[\nabla^2 l \bigl(\overline{\VV{\theta}}^{(k)} +
\alpha_k \underline {\VV{\delta}}^{(k)}\bigr) -
\nabla^2 l \bigl(\overline{\VV{\theta}}^{(k)}\bigr)\bigr]
\underline{\VV{\delta}}^{(k)}\bigr\| _{\infty} \le\tau_{\max} \bigl\|
\underline{\VV{\delta}}^{(k)}\bigr\|_2^2.
\end{equation}
\end{lemma}
\begin{pf*}{Proof of Proposition \protect\ref{thm_restricted_consistency}}
The main idea of the proof was first introduced in this context in
\citet{RothmanBickelLevinaZhu08} and has since been used by many
authors. Define
\begin{eqnarray}
\label{fun_G} &&\F{G}\bigl(\bigl\{\underline{\VV{\delta}}^{(k)}\bigr
\}_{k=1}^K\bigr)\nonumber\\
&&\qquad= -\sum_{k=1}^K
\bigl[l\bigl(\overline{\VV{\theta}}^{(k)} + \underline{\VV{
\delta}}^{(k)}\bigr) - l\bigl(\overline{\VV{\theta}}^{(k)}\bigr)
\bigr]
\\
&&\qquad\quad{}+ \lambda\sum_{1 \le j<j' \le p} \Biggl\{ \Biggl(\sum
_{k=1}^K \bigl|\overline{\theta}_{j,j'}^{(k)}
+ \underline{\delta}_{j,j'}^{(k)}\bigr|\Biggr)^{1/2} -
\Biggl(\sum_{k=1}^K \bigl|\overline {
\theta}_{j,j'}^{(k)}\bigr|\Biggr)^{1/2}\Biggr\}.\nonumber
\end{eqnarray}
It can be seen from \eqref{obj_restricted_Ising} that $\{\underline
{\widehat{\VV{\delta}}}^{(k)}\}_{k=1}^{K}$ minimizes $\F{G}(\{
\underline{\VV{\delta}}^{(k)}\}_{k=1}^K)$ and\break $\F{G}(\{\V{0}\}
_{k=1}^K) = 0$. Thus, we must have $\F{G}
(\{\widehat{\underline{\VV{\delta}}}^{(k)}\}_{k=1}^K) \le0$. If we
take a closed set
$\mathcal{A}$ which contains $\{\V{0}\}_{k=1}^K$ and show that $\F
{G}$ is
strictly positive everywhere on the boundary $\partial
\mathcal{A}$, then it implies that $\F{G}$ has a local minimum
inside $\mathcal{A}$, since $\F{G}$ is continuous and
$\F{G}(\{\V{0}\}_{k=1}^K)=0$. Specifically, we define\break
$\mathcal{A}=\{\{\underline{\VV{\delta}}^{(k)}\}_{k=1}^K\dvtx \sum_{k=1}^K \| \underline{\VV{\delta}}^{(k)} \|_2 \le Ma_n \}$,
with boundary
$\partial\mathcal{A}= \{\{\underline{\VV{\delta}}^{(k)}\}_{k=1}^K\dvtx\break \sum_{k=1}^K \| \underline{\VV{\delta}}^{(k)} \|_2 = Ma_n \}$, for
some constant $M > (2K C_{\lambda} / \tau_{\min} \sqrt{\gamma
_{\min}}) [(3-2\tau)/\break(2-\tau)]$ and $a_n = \sqrt{q (\log p) / n}$.
For any $\{\underline{\VV{\delta}}^{(k)}\}_{k=1}^K \in\partial
\mathcal{A}$, the Taylor
series expansion gives $\F{G}(\{\underline{\VV{\delta}}^{(k)}\}
_{k=1}^K) = I_1 + I_2 + I_3$,
where
\begin{eqnarray}
I_1 &=& -\sum_{k=1}^K
\bigl[\nabla l\bigl(\overline{\VV{\theta }}^{(k)}\bigr)
\bigr]_{S_k}^{\mathsf{T}} \VV{\delta}_{S_k}^{(k)},
\nonumber
\\
\qquad I_2 &=& -\sum_{k=1}^K \T{\VV{
\delta}_{S_k}^{(k)}} \bigl[\nabla^2 l\bigl(
\overline{\VV{\theta}}^{(k)} + \alpha_k \underline{\VV{
\delta}}^{(k)}\bigr)\bigr]_{S_k,S_k} \VV{\delta }_{S_k}^{(k)}\qquad
\mbox{for some } \alpha_k \in[0,1],\hspace*{-35pt}
\\
I_3 &=& \lambda\sum_{(j,j') \in S_{\cup}} \Biggl\{
\Biggl(\sum_{k=1}^K \bigl|\overline{
\theta}_{j,j'}^{(k)} + \underline{\delta}_{j,j'}^{(k)}\bigr|
\Biggr)^{1/2} - \Biggl(\sum_{k=1}^K
\bigl|\overline {\theta}_{j,j'}^{(k)}\bigr|\Biggr)^{1/2}\Biggr\}.\nonumber
\end{eqnarray}
Since $C_{\lambda} > (8-4\tau)\sqrt{\gamma_{\min}}/(1-\tau)$, we
have $[(1-\tau)/(2-\tau)]
C_{\lambda}/\sqrt{\gamma_{\min}} > 4$. By Lemma~\ref
{lemma_Bound_first_derivative},
\begin{eqnarray}
|I_1| &\le&\sum_{k=1}^K \bigl\|
\bigl[\nabla l\bigl(\overline{\VV{\theta }}^{(k)}\bigr)
\bigr]_{S_k}\bigr\|_{\infty} \bigl\|\VV{\delta}_{S_k}^{(k)}
\bigr\|_1
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&\le& \bigl[(1-\tau)C_{\lambda}M\gamma_{\min}^{-1/2}
/ (2-\tau)\bigr] (q \log p)/n.
\end{eqnarray}
In addition, by condition $q \sqrt{(\log p) / n}=o(1)$, Lemma~\ref
{lemma_Bound_residual} holds and, thus,
\begin{equation}
I_2 \ge(\tau_{\min}/2) \sum_{k=1}^K
\bigl\|\underline{\VV{\delta }}^{(k)}\bigr\|_2^2 \ge\bigl[
\tau_{\min}/(2K)\bigr] M^2 q (\log p) / n.
\end{equation}
Finally, by the triangular inequality and regularity condition (A),
\begin{eqnarray}
|I_3| &\le& \lambda\sum_{(j,j') \in S_{\cup}}\sum
_{k=1}^K \frac
{||\overline{\theta}_{j,j'}^{(k)} +
\underline{\delta}_{j,j'}^{(k)}| - |\overline{\theta
}_{j,j'}^{(k)}||}{(\sum_{k=1}^K |\overline{\theta}_{j,j'}^{(k)} +
\underline{\delta}_{j,j'}^{(k)}|)^{1/2} + (\sum_{k=1}^K |\overline
{\theta}_{j,j'}^{(k)}|)^{1/2}}
\nonumber
\\
&\le& \bigl(\lambda\gamma_{\min}^{-1/2}\bigr) \sum
_{k=1}^K \sum_{(j,j') \in
S_{\cup}} \bigl|
\underline{\delta}_{j,j'}^{(k)}\bigr| \le\bigl(\lambda
q^{1/2} \gamma_{\min}^{-1/2}\bigr) \sum
_{k=1}^K\bigl \|\underline{\VV{\delta }}^{(k)}
\bigr\|_2
\\
&\le& \bigl(M C_{\lambda} \gamma_{\min}^{-1/2}\bigr) \bigl
\{q (\log p)/n\bigr\}.\nonumber
\end{eqnarray}
Then we have
\begin{equation}\qquad
\F{G}\bigl(\bigl\{\underline{\VV{\delta}}^{(k)}\bigr
\}_{k=1}^K\bigr) \ge M^2 \frac{q
\log p}{n}
\biggl(\frac{\tau_{\min}}{2K} - \frac{(1-\tau) C_{\lambda
}}{(2-\tau)M\gamma_{\min}^{1/2}} - \frac{C_{\lambda}}{M\gamma_{\min}^{1/2}} \biggr) > 0.
\end{equation}
The last inequality uses the condition $M > (2K C_{\lambda} / \tau
_{\min} \sqrt{\gamma_{\min}}) [(3-2\tau)/(2-\tau)]$. Therefore,
with probability
tending to 1, we have $\sum_{k=1}^K \| \widehat{\underline{\VV
{\theta}}}^{(k)} - \overline{\VV{\theta}}^{(k)} \|_2 \leq M \sqrt{q
(\log p) /n}$, and consequently claim (i) in Proposition~\ref
{thm_restricted_consistency} holds.
On the other hand, by the definition of $\widehat{\underline{\VV
{\theta}}}^{(k)}$, we have $\widehat{\underline{\theta}}_{j,j'}^{(k)}=0$
for all $(j,j') \in S_k^c$. By regularity condition (A) and
Proposition~\ref{thm_restricted_consistency}(i), for any $(j,j') \in
S_k$, $k=1,\ldots,K$, we have $|\widehat{\underline{\theta
}}_{j,j'}^{(k)}| \ge|\overline{\theta}_{j,j'}^{(k)}| - |\widehat
{\underline{\theta}}_{j,j'}^{(k)} - \overline{\theta}_{j,j'}^{(k)}|
\ge\gamma_{\min}/2 > 0$, when $n$ is large enough.
\end{pf*}
\begin{pf*}{Proof of Proposition \protect\ref{prop_restricted_KKT}}
By Proposition~\ref{thm_restricted_consistency}, with
probability tending to one, we have $\widehat{\underline{\theta
}}_{j,j'} \neq0$ for all $(j,j') \in S_k$.
Since $\{\widehat{\underline{\VV{\theta}}}^{(k)}\}_{k=1}^K$ is a
local maximizer of the restricted problem \eqref
{obj_restricted_Ising}, with
probability tending to one, $\nabla_{j,j'} l (\widehat{\underline{\VV
{\theta}}}^{(k)}) = \lambda\sgn(\widehat{\underline{\theta
}}_{j,j'}^{(k)}) / (\sum_{k=1}^K | \widehat{\underline{\theta
}}_{j,j'}^{(k)}|)^{1/2}$, for all $(j,j') \in S_k$.
To show the second claim, we apply the mean value theorem and write
$\nabla
l(\widehat{\underline{\VV{\theta}}}^{(k)}) = \nabla
l(\overline{\VV{\theta}}^{(k)}) + \V{r}^{(k)} -
\widehat{\M{Q}}^{(k)} \underline{\widehat{\VV{\delta}}}^{(k)}$,
where $\V{r}^{(k)} =
\{\nabla^2 l(\overline{\VV{\theta}}^{(k)}+\alpha_k
\widehat{\underline{\VV{\delta}}}^{(k)}) - \nabla^2
l(\overline{\VV{\theta}}^{(k)})\}
\widehat{\underline{\VV{\delta}}}^{(k)}$. After some
simplifications, we have
\begin{eqnarray}\qquad
\bigl[\nabla l\bigl(\widehat{\underline{\VV{\theta}}}^{(k)}\bigr)
\bigr]_{S_k^c} &=& \bigl[\nabla l\bigl(\overline{\VV{\theta}}^{(k)}
\bigr)\bigr]_{S_k^c} + \V{r}_{S_k^c}^{(k)}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&{}- \bigl[
\widehat{\M{Q}}_{S_k^c,S_k}^{(k)} \bigl(\widehat{\M{Q}}_{S_k,S_k}^{(k)}
\bigr)^{-1}\bigr] \bigl\{\bigl[\nabla l\bigl(\overline{\VV{\theta
}}^{(k)}\bigr)\bigr]_{S_k} + \V{r}_{S_k}^{(k)}
- \bigl[\nabla l\bigl(\widehat{\underline {\VV{\theta}}}^{(k)}\bigr)
\bigr]_{S_k}\bigr\}
\end{eqnarray}
and, thus,
\begin{eqnarray}
&&\bigl\|\bigl[\nabla l\bigl(\widehat{\underline{\VV{\theta}}}^{(k)}\bigr)
\bigr]_{S_k^c}\bigr\|_{\infty}\nonumber \\
&&\qquad\le \bigl\|\bigl[\nabla l\bigl(\overline{\VV{
\theta}}^{(k)}\bigr)\bigr]_{S_k^c}\bigr\|_{\infty} + \bigl\|
\V{r}_{S_k^c}^{(k)}\bigr\|_{\infty}
\nonumber
\\
&&\qquad\quad{}+ \bigl\|\widehat{\M{Q}}_{S_k^c, S_k}^{(k)} \bigl(\widehat{
\M{Q}}_{S_k,
S_k}^{(k)}\bigr)^{-1}\bigr\|_{\infty}\nonumber\\
&&\qquad\quad{}\times \bigl\{
\bigl\|\bigl[\nabla l\bigl(\overline{\VV{\theta}}^{(k)}\bigr)
\bigr]_{S_k}\bigr\|_{\infty} + \bigl\|\V{r}_{S_k}^{(k)}
\bigr\|_{\infty} + \bigl\|\bigl[\nabla l\bigl(\widehat{\underline{\VV{
\theta}}}^{(k)}\bigr)\bigr]_{S_k}\bigr\|_{\infty}\bigr\}
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&&\qquad\le (2-\tau)\bigl \|\nabla l\bigl(\overline{\VV{\theta}}^{(k)}\bigr)
\bigr\|_{\infty} + (2- \tau)\bigl\|\V{r}^{(k)}\bigr\|_{\infty} + (1-\tau) \bigl\|
\bigl[\nabla l\bigl(\widehat{\underline{\VV{\theta}}}^{(k)}\bigr)
\bigr]_{S_k}\bigr\|_{\infty}
\\
&&\qquad\le \bigl[(1-\tau)C_{\lambda}/\sqrt{\gamma_{\min}}\bigr] \sqrt{(
\log p)/n} + (2-\tau) \tau_{\max} M^2 q (\log p) / n
\nonumber
\\
&&\qquad\quad{}+ (1-\tau) \lambda\Big/ \min_{(j,j') \in S_k} \Biggl[\sum
_{k=1}^K \bigl|\widehat{\underline{\theta}}_{j,j'}\bigr|
\Biggr]^{1/2}
\nonumber
\\
&&\qquad\le \bigl[2(1-\tau)/\sqrt{\gamma_{\min}}\bigr] \lambda+
o_p(\lambda).\nonumber
\end{eqnarray}
On the other hand, $\lambda/ [\sum_{k=1}^K
|\widehat{\underline{\theta}}_{j,j'}^{(k)}|]^{1/2} = +\infty$ when
$(j,j') \in S_{\cup}^c$. Otherwise, if
$(j,j') \in S_{\cup} \setminus S_k$, then
\begin{eqnarray*}
\lambda\Big/ \Biggl(\sum_{k=1}^K \bigl|\widehat{
\underline{\theta}}_{j,j'}\bigr|\Biggr)^{1/2} &\ge&\lambda\Big/ \Biggl\{\sum
_{k=1}^K \bigl|\widehat{\underline{
\theta}}_{j,j'} - \overline{\theta}_{j,j'}\bigr| + \bigl|\overline{
\theta}_{j,j'}\bigr|\Biggr\}^{1/2}\\
& \ge&\lambda/ \sqrt{\gamma
_{\max}} \ge(2-2\tau) \lambda/ \sqrt{\gamma_{\min}}.
\end{eqnarray*}
Thus, for any $(j,j') \in S_k^c$ ($k=1,\ldots,K$), we have
\begin{eqnarray}
\bigl|\nabla_{j,j'} l\bigl(\widehat{\underline{\VV{\theta}}}^{(k)}
\bigr)\bigr| &\le& \max_{1 \le k \le K}\max_{(j,j') \in S_k^c}\bigl |
\nabla_{j,j'} l\bigl(\widehat{\underline{\VV{\theta}}}^{(k)}
\bigr)\bigr|
\nonumber
\\[-8pt]
\\[-8pt]
\nonumber
&< &\min_{1 \le k \le K} \min_{(j,j') \in S_k^c} \lambda\Big/ \sqrt
{\sum_{k=1}^K \bigl|\widehat{
\underline{\theta}}_{j,j'}^{(k)}\bigr|} \le \lambda\Big/ \sqrt{
\sum_{k=1}^K \bigl|\widehat{\underline{\theta
}}_{j,j'}^{(k)}\bigr|}.
\end{eqnarray}
\upqed\end{pf*}
\end{appendix}
|
1,116,691,498,506 | arxiv | \section{Introduction}
If planets and their central star form from the same nebular reservoir, their chemical compositions must correlate in some fashion.
However, observed planet/planetesimal compositions do not completely match those of their central stars. For example, in our solar system,
all rocky
planets and planetesimals contain near-solar proportions of
refractory elements but are depleted in volatile elements
\citep{Asplund:2005,Asplund:2009}.
Such volatile depletion is thought
to result from planetesimal formation in the solar nebula
\citep[][and see \citet{Li:2020} for a summary]{Cassen:1994,Cassen:1996,Cassen:2001,Ciesla:2008,Bond:2010a,Bond:2010b,Elser:2012,Moriarty:2014,Pignatale:2016}
where they form by coalescence of condensed mineral
dust
(See \citet{Johansen:2014} for a review). The planetesimal composition at a given location depends upon the initial chemical composition and the evolution of the disc \citep{Cassen:1996,Ciesla:2008,Li:2020}.
The same is likely true for
exoplanetary systems.
As the initial, cold molecular cloud cores collapse, discs form due to the existing angular momentum of the system \citep[e.g.][]{Nakamoto:1994,Sui:2019}. Subsequent heating from the forming central star and from the viscosity of the disc can vaporise primordial condensed material -- forcing the condensation sequence to begin anew, starting under high temperature and pressure conditions that gradually lessen as the disc evolves and dissipates. However, some elements condense at relatively high temperatures \citep[e.g.][]{Li:2020}. Depending upon the properties of the initial molecular cloud core (MCC) and the disc that forms therefrom, the temperatures that arise in the disc may not be sufficient to vaporise the most refractory of the pre-existing material. This situation presents a problem when trying to understand the observed chemistry of planet-forming materials since certain minerals contained in planetesimals may form in different conditions, times, and places.
Using an analytical approach, \citet{Cassen:1996} showed that the depletion of moderate volatile elements observed in CM, CO and CV carbonaceous chondrites can be reproduced using a simple disc model.
Alternatively, \citet{Yin:2005} suggested that the observed depletion pattern was inherited from the molecular cloud in which the gas is depleted in refractory elements that have already condensed into grains. Modeling the migration of solids in the viscously evolving discs, \citet{Ciesla:2008} found that the temperature should be higher than $\sim$ 1350 K at 2 to 4 AU to explain the
volatile depletion patterns in CM, CO and CV chondrites from the asteroid belt, which is difficult to achieve in a traditional alpha protoplanetary disc model.
In a subsequent paper, \citet{Yang:2012} investigated the mixing process of the grains in the solar nebula. They found that, during the evolution of solar nebula, the fraction of refractory elements is the largest when the infall from the molecular cloud core stops (at age of several $10^5$ years).
More recently, \citet{Li:2020} re-visited the \citet{Cassen:1996} results by combining the original disc P-T evolution with a chemical equilibrium model \citep[GRAINS,][]{Petaev:2009}. They reproduced the enrichment in refractory and depletion in volatile elements observed in CM, CO and CV chondrites. However,
the temperature of the disc model used in their work and by \citet{Cassen:1996} may be unrealistically high (see also \citet{Ciesla:2008}). Here we examine the temperature and pressure histories of the midplanes of protoplanetary discs that form from the collapse of molecular cloud cores as a function of the initial properties of cores to gain a better understanding of its potential effects on the dust condensation process. We select our initial conditions from the observed MCC properties to provide a realistic estimate for the properties of the resulting discs. Specifically, we consider the distribution of the initial
temperature and angular velocity of the cores, and viscosity in the disc throughout their evolution.
Our initial cores self-consistently collapse to form stars with surrounding protoplanetary discs that evolve under the influence of stellar irradiation and
viscosity -- which may arise from gravitation instability (GI) at early stage of disc evolution \citep{Lin:1987,Lin:1990,Gammie:2001,Clarke:2009,Rice:2009,Rice:2010}, magnetorotational instability \citep[MRI;][]{Balbus:1991} and hydrodynamic processes \citep{Dubrulle:1993,Dubrulle:2005}. We use a constant $\alpha$ for the viscosities in this paper.
We calculate the temperature and pressure conditions in these discs and compare them to the condensation temperatures of a variety of elements calculated by \citet{Li:2020}. From this information we make statements regarding the origins of some compounds that are or could be observed in planetary systems, including the solar system.
Our constant alpha disc model provides a benchmark for comparing with previous studies \citep[e.g.][]{Cassen:1996,Ciesla:2008} and future studies.
A more realistic, layered accretion model is left for elsewhere.
This paper is organized as follows. We begin by outlining our parameters for the initial molecular cloud cores and the evolution model of the protoplanetary disc in Section \ref{sec:clouddisc}. We show the evolution of the disc in Subsection \ref{sec:discevo}. The dependence of the maximum temperature in discs on the properties of MCC and the statistical properties of the maximum temperatures are shown in \ref{sec:maxtem}. We discuss the implications of our results on the elemental composition of chondrites and the formation of species in Section \ref{sec:discussion} and give our concluding remarks in Section \ref{sec:conclusion}.
\section{Molecular Cloud Core and Disc Evolution} \label{sec:clouddisc}
Our simulations begin with the collapse of a molecular cloud core, which self-consistently evolves into a protoplanetary disc. For all of these simulations we consider cores of one solar mass. The initial temperature ($T_{\rm C}$) and angular velocity ($\omega_{\rm C}$) of each core is chosen from the observed distributions of these properties shown in Figure \ref{fig:tem} \citep[see also Figures 1 and 2 of][]{Li:2015}. The temperatures of the cores range from roughly 7 to 40 Kelvins with the median value of 15 K \citep{Jijina:1999}. The angular velocities of the cores range from 0.3 to 13 $\times \ 10^{-14}\rm\ s^{-1}$ \citep{Goodman:1993}. Since very little mass either escapes the system or is incorporated into planets, we assume that the final mass of the central star is equal the mass of the MCC.
\begin{figure}
\includegraphics[width=\columnwidth]{T-distribution-eps-converted-to.pdf}
\includegraphics[width=\columnwidth]{w-distribution-eps-converted-to.pdf}
\caption{Distribution of $T_{\rm C}$ (upper panel) and $\omega_{\rm C}$ (lower panel) of observed molecular cloud cores. This results are from 211 temperature values \citep{Jijina:1999} and 43 angular velocities \citep{Goodman:1993}.}
\label{fig:tem}
\end{figure}
The evolution of the protoplanetary disc that forms from the collapse of an MCC is governed by \citep{Li:2015}
\begin{equation}\label{equ.diff}
\begin{aligned}
\frac{\partial \Sigma(R,t)}{\partial t}
=&\frac{3}{R} \frac{\partial}{\partial R} \left[ R^{1/2} \frac{\partial}{\partial R} (\Sigma \nu R^{1/2}) \right] +S(R,t)\\
&+S(R,t)\left\{2-3\left[\frac{R}{R_{d}(t)}\right]^{1/2}+\frac{R/R_{d}(t)}{1+[R/R_{d}(t)]^{1/2}}\right\}.
\end{aligned}
\end{equation}
Here $\Sigma(R,t)$ is the gas surface density of the disc at radius $R$ and time $t$ while $\nu$ is the kinematic viscosity.
The third term on the right hand side of Equation (\ref{equ.diff}) is due to the difference of the specific angular momentum between the infalling material and the material in the disc.
The mass influx onto the disc and protostellar system, $S(R,t)$, is \citep{Nakamoto:1994}:
\begin{equation}\label{equ.inf}
S\left(R,t\right)=
\begin{cases}
{\displaystyle \frac{\dot{M}}{4\pi R R_{\mathrm{d}}\left(t\right)}
\left[1-\frac{R}{R_{\mathrm{d}}\left(t\right)}\right]^{-1/2} } &{\rm if\ } {\displaystyle \frac{R}{R_{\mathrm{d}}\left(t\right)}<1; } \\
{}\\
0 &{\rm otherwise},
\end{cases}
\end{equation}
where $\dot{M}$ is the mass infall rate \citep{Shu:1977}
\begin{equation}\label{equ.mdot}
\dot{M} = \frac{0.975}{G}\left(\frac{\mathcal{R}}{\mu}\right)^{3/2}T_{\rm C}^{3/2},
\end{equation}
where $G$ is the gravitational constant, $\mathcal{R}$ is the gas constant, $\mu = 2.33$ is the mean molecular mass. Time $t=0$ is set to be the time when MCC starts to collapse.
The centrifugal radius is
\begin{equation}\label{equ.rd}
R_{\mathrm{d}}(t) = 31
\left(\frac{\omega_{\rm C}}{10^{-14} {\rm\ s} ^{-1} } \right)^{2}
\left(\frac{T_{\rm C}}{10 {\rm\ K}} \right)^{1/2}
\left(\frac{t}{5\times10^{5} {\rm\ yr}} \right)^{3} {\rm AU},
\end{equation}
where $\omega_{\rm C}$ is the angular momentum of the MCC.
We use the $\alpha$-prescription \citep{Shakura:1973} to calculate the viscosity, $\nu=\alpha c_s H$, where $\alpha$ is a dimensionless parameter less than 1, $H$ is the half thickness of the gas disc, $c_s=\sqrt{\mathcal{R} T/\mu}$ is the sound speed, and $T$ is the temperature of the mid-plane of the disc.
Observations indicate that the $\alpha$ value has a large range and there is no preferred value of it \citep{Rafikov:2017}.
For our purposes, each simulation uses a constant $\alpha$, and the suite of simulations examines values equal to $10^{-1}$, $10^{-2}$, $10^{-3}$, $10^{-4}$, and $10^{-5}$.
The disc surface temperature is determined by the thermal equilibrium between the cooling and heating fluxes of the disc, which is \citep{Hueso:2005}
\begin{equation}\label{equ.ts}
\sigma T_{s}^{4}=\frac{1}{2}
\left(1+\frac{1}{2\tau_{P}}\right)(\dot{E}_{\nu}+\dot{E}_{s})+\sigma
T_{\rm ir}^{4} +\sigma T_{\rm C}^{4},
\end{equation}
where $\sigma$ is the Stefan–Boltzmann constant, $\tau_{P}=\kappa_{P}\Sigma$ is the Planck mean optical depth, where $\kappa_{P}$ is the Planck mean opacity. $\dot{E}_{\nu}$ is the viscous dissipation rate and $\dot{E}_{s}$ is the energy generation rate by shock heating
-- the energy difference between the cloud core and the disc when the infalling material merges with the disc.
\citep{Nakamoto:1994}. $T_{\rm ir}$ is the effective temperature due to the irradiation from the protostar.
The midplane temperature in the disc is
\begin{equation}\label{equ.tm}
\sigma T^{4}=\frac{1}{2}
\left[\left(\frac{3}{8}\tau_{R}+\frac{1}{2\tau_{P}}\right)\dot{E}_{\nu}
+\left(1+\frac{1}{2\tau_{P}}\right)\dot{E}_{s}\right]
+\sigma T_{\rm ir}^{4} +\sigma T_{\rm C}^{4},
\end{equation}
where $\tau_{R}=\kappa_{R}\Sigma$ is the Rosseland mean optical depth, and $\kappa_{R}$ is the Rosseland mean opacity. Here $\kappa_{P}=2.39\kappa_{R}$ \citep{Nakamoto:1994}.
We use the same method as in \citet{Armitage:2001} to calculate the opacity, which comes from \citet{Bell:1994} for high temperature and \citet{Bell:1997} for low temperature.
The inner radius of the subsequent disc is 0.3 AU. At the inner radius, we set $\Sigma$ to be 0, and the material is accreted to the central star \citep{Bath:1981,Lin:1987}. The outer radius is 1.25e5 AU -- which allows the disc to expand freely.
\section{Results}
\label{sec:results}
\subsection{Disc evolution}
\label{sec:discevo}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{sur-eps-converted-to.pdf}
\includegraphics[width=0.8\columnwidth]{tem-eps-converted-to.pdf}
\includegraphics[width=1.15\columnwidth]{temrt-eps-converted-to.pdf}
\caption{Evolution of (a) surface density and (b) midplane temperature as a function of radius and time. (c) Contour plot of temperature as a function of radius and time. Here, $M_{\rm C}=1\ M_\odot$, $T_{\rm C}=15$ K, $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-3}$. Note that the temperature gets its maximum value, 1350 K, at $t=$ 324870 yr at 0.43 AU.}
\label{fig:discevo}
\end{figure}
The initial conditions of the MCC and the chosen $\alpha$ value determine the evolution of the disc surface density, midplane temperature, and the disc evolution timescale. Much of the discussion that follows uses a fiducial model for the properties of the MCC with a mass $M_\text{C} = 1\ M_\odot$, initial temperature of $T_\text{C} = 15$ K, an initial angular velocity of $\omega_\text{C} = 1 \times 10^{-14} \ \text{s}^{-1}$, and $\alpha=1\times 10^{-3}$.
Figure \ref{fig:discevo} shows how the disc surface density and midplane temperature evolve with time. At very early times, the surface density (Figure \ref{fig:discevo}a) increases at all radii as material is supplied by the collapsing MCC. It reaches the maximum value in the inner region (within 10 AU) at around the time when the collapse ceases (3.47$\times 10^5$ years). After that, it decreases quickly in the inner region and increases in the outer region as the disc expands from the effects of its viscosity.
The general trends of the temperature evolution (Figure \ref{fig:discevo}b) are similar to that of the surface density except that the temperatures at radii larger than 20 AU do not change much with time. Temperatures at larger radii peak at a later time than those at smaller radii (see the direction of contour in Figure \ref{fig:discevo}c). For the fiducial case, the temperature in the disc reaches its maximum value of 1350 K at 0.43 AU and $t=$ 324870 yr (which is $0.937 \simeq 1$ times the collapse timescale of the MCC).
\subsection{Maximum disc temperature as a function of molecular cloud core properties}
\label{sec:maxtem}
We first consider the maximum temperatures in the disc as a function of the initial MCC properties -- specifically temperature and angular velocity for a solar mass core. As the properties of the MCC change, the mass infall rate and the centrifugal radius change (See Equations \ref{equ.inf} and \ref{equ.rd}). Therefore, the evolution of the disc and the maximum temperature in the disc also change. For our study, we examine initial temperatures ranging from 7 to 39 K, and initial angular velocities ranging from 1 to 13 $\times 10^{-14}$ s$^{-1}$, and $\alpha$ values ranging from $10^{-5}$ to $10^{-1}$
\citep{Rafikov:2017}.
Figure \ref{fig:T-t} shows the maximum midplane temperature reached by a disc as a function of $T_{\rm C}$ for different $\alpha$ where the initial MCC angular velocity is fixed at $\omega_{\rm C}=3\times 10^{-14} \rm s^{-1}$. The maximum temperature increases with $T_{\rm C}$, but decreases with the viscosity. For each $\alpha$, as $T_{\rm C}$ increases, the mass infall rate increases (Equation \ref{equ.inf}), and there will be more material at the inner region of the disc. Thus, the surface density in the inner region and the maximum temperature increase. As $\alpha$ decreases, the efficiency of the expansion of the disc decreases, leaving more materials in the inner region of the disc and also a higher maximum temperature. However, the dependence of the maximum temperature on viscosity is relatively weak -- changing by only $\sim$ 50\% over four orders of magnitude in $\alpha$. (Though the resulting differences happen to span an important range near the condensation temperatures of many elements.) For the highest temperature MCCs, we see a significant increase in the highest disc temperatures
as there is more material in the inner regions of the disc -- raising the opacity and keeping the heat within the disc.
For all the cases, the maximum temperatures are between $\sim$750 and 2200 K.
Figure \ref{fig:T-w} shows the maximum disc temperature for an initial MCC temperature of 15 K, as a function of the initial MCC angular velocity -- again for a variety of $\alpha$. For each $\alpha$, as $\omega_{\rm C}$ increases, the centrifugal radius increases (Equation \ref{equ.rd}), there will be less material in the inner regions relative to the outer regions, and the maximum midplane temperature decreases. As with Figure \ref{fig:T-t}, the disc with lower $\alpha$ has higher maximum temperature, and the max temperatures range from 900 to 1600 K.
\begin{figure}
\includegraphics[width=\columnwidth]{T-Tc-eps-converted-to.pdf}
\caption{Maximum temperature as a function of $T_{\rm C}$ for different $\alpha$. Here $M_{\rm C}=1\ M_\odot$ and $\omega_{\rm C}=3\times 10^{-14} \rm s^{-1}$.}
\label{fig:T-t}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{T-w-eps-converted-to.pdf}
\caption{Maximum temperature as a function of $\omega_{\rm C}$ for different $\alpha$. Here, $M_{\rm C}=1\ M_\odot$, $T_{\rm C}=15$ K.}
\label{fig:T-w}
\end{figure}
\subsection{Statistical properties of maximum disc temperatures}
We now examine the maximum midplane temperatures that would arise in planet-forming discs, given initial conditions that match the observed properties of MCCs. For temperature $T_{\rm C}$, we choose values from 7 to 39 K with intervals of 2 K. For $\omega_{\rm C}$, the values are from 1 to 13 $\times\ 10^{-14} \rm\ s^{-1}$ with an interval of 2 $\times\ 10^{-14} \rm\ s^{-1}$ \citep{Li:2015}.
For $\alpha$, we adopt $10^{-1}$, $10^{-2}$, $10^{-3}$, $10^{-4}$, or $10^{-5}$, distributed equally among the 595 simulations \citep{Rafikov:2017}. In all of our simulations, the maximum midplane temperatures are reached at radii less than 1 AU and at early times following the collapse of the MCC. Given the distributions of $T_{\rm C}$, $\omega_{\rm C}$, and $\alpha$, the distribution of maximum temperatures in the disc, weighted by these initial conditions (Figure \ref{fig:tem}), is shown in Figure \ref{fig:N_tmax}.
For the majority of discs in the regime of these parameters, the maximum temperature is between 1000 and 1500K, with the median value being around 1250 K. This value of the maximum midplane temperature is lower than the 50\% condensation temperature of silicon at a total pressure of $10^{-4}$ bars
\citep[$\sim$ 1300 K,][]{Lodders:2003,Li:2020}.
This result has important implications for the evolution
of primordial MCC dust because all refractory elements with higher condensation temperatures will not evaporate and fractionate from each other. Meanwhile, the moderately and highly volatile elements would be evaporated and fractionated from the condensed elements if gaseous and condensed phases are physically separated.
The resulting heterotemporal condensation sequence, barring an additional heating mechanism, means that all refractory elements and most moderately volatile elements would be locked into dust from the beginning.
Only highly volatile elements would be affected at later times, during the evolution of the disc. In addition, the low maximum midplane temperatures inferred from our model may not even allow refractory inclusions, including CAIs, to form inside the evolving protoplanetary disc, since many refractory inclusion require much higher temperatures (>2000 K) to form fractionated rare-earth-element patterns \citep[e.g.,][]{MacPherson:2003,Petaev:2009b}.
Another interesting quantity to consider is the fraction of systems that would achieve a maximum temperature above a given temperature, beginning with the observed MCC distributions. In Figure \ref{fig:p_tmax}, we plot the accumulated probability of the maximum temperatures in our discs
as a function of radius. Figure \ref{fig:p_tmax}a shows that 5\% of the discs reach maximum temperatures lower than 935 K and another 5\% reach maximum temperatures higher than 1635 K (near the 50\% condensation temperature of Al at $10^{-4}$ bar).
Note that different $\alpha$ results in different maximum temperatures,
and therefore the distribution of $\alpha$ will affect the distribution of maximum temperatures.
Few discs in our sample reach maximum temperatures higher than 2000 K. These temperatures only occur with the highest temperature and lowest viscosity MCCs. We also plot the 50\% condensation temperature at $10^{-4}$ bar for several elements for comparison purposes. From these results, we see that the maximum temperatures for most discs are less than the 50\% condensation temperature of the most refractory elements, such as Ca, Al, and Os. Moreover, these high temperatures occur at distances less than one AU -- often much less.
Figure \ref{fig:p_tmax}b shows that the maximum temperatures, and the proportions of the temperatures higher than specific values generally decrease with radius.
\begin{figure}
\includegraphics[width=\columnwidth]{N_tmax_all_alpha.pdf}
\caption{Stacked histogram of maximum temperatures achieved in our ensemble of discs.
Different color means contribution from different $\alpha$.}
\label{fig:N_tmax}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{p_tmax-eps-converted-to.pdf}
\includegraphics[width=1.0\columnwidth]{prt-eps-converted-to.pdf}
\caption{(a) Fraction of discs that reach a maximum temperature higher than a given value. The vertical lines are the 50\% condensation temperatures at $10^{-4}$ bar for different elements \citep{Li:2020}.
(b) Fraction of discs that reach specific peak temperatures as a function of radius.}
\label{fig:p_tmax}
\end{figure}
There is a peak in Figure \ref{fig:p_tmax}b for small radii ($\textless 0.5$ AU) which means there is a small subset of systems with high temperatures there. The main reason for this effect is that with high $T_{\rm C}$ and low $\omega_{\rm C}$, the infall rate from the MCC to the disc is high. With low viscosity in the disc, a lot of material accumulates in the inner region of the disc. The resulting high surface density raises the opacity and absorption of energy from the central star, resulting in high temperatures (See also Figure \ref{fig:T-t} and \ref{fig:tmax_diff}).
\section{Discussion}
\label{sec:discussion}
\subsection{Influence on composition of chondrites and terrestrial planets}
Observations show that the compositions of CM, CO, and CV chondrites and terrestrial planets are substantially depleted in volatile and somewhat enriched in refractory elements relative to the composition of the Sun \citep[][see also Figure 1 in \citet{Li:2020}]{Asplund:2005,Asplund:2009}. Partial condensation of the elements in a cooling disc from very high temperature are thought to be responsible for this depletion pattern \citep{Grossman:1972,Cassen:1996,Li:2020}.
These studies assume that the initial temperature in the disc is higher than the condensation temperature of most refractory elements -- resulting in nearly complete vaporization of presolar MCC dust. The chemistry then evolves through a single condensation sequence from initially gaseous material. However, our calculations indicate that the maximum temperatures in most discs are lower than the condensation temperature of most refractory elements. This means that, the refractory and some moderately volatile elements in most discs will remain in the primordial MCC dust.
These elements would not fractionate, and the relative elemental abundances of the dust would match the stellar composition.
However, the prediction that moderately volatile elements are not fractionated from refractory elements in the condensed phases (planetesimals) is not consistent with what we observe in the solar system. In order to reproduce the observations of solar system objects, there must be some process or event that alters the condensation sequence. We, therefore, examine these discs in more detail to see what observations can be explained with our models, and what observations require some new process.
In our calculations, we treat the fact that the dust may not condense at the same time as follows. The chemical evolution of the disc uses the method described in \citet{Li:2020}.
In all our models, the mid-plane temperatures are not hot enough to completely evaporate all of the dust, so that both the dust and gas exist at $t=0$. We assume that the combination of the condensed phases and the gas immediately reaches chemical equilibrium, but the dust isolation (decoupling) from the gas phase takes time. This decoupling of the dust is modelled with a decoupling timescale (see \citet{Li:2020}). With this approach, the assumption is that the infall dust grains from the MCC are small enough to quickly reach chemical equilibrium under the $P-T$ conditions of the disc.
With the approach outlined above, only a small subset of our initial MCC conditions are capable of reproducing the depletion pattern of the chondrites and terrestrial planets (and then, the results are still a poor fit to the observed element patterns of CM, CV, and CO chondrites). These results imply that the initial conditions of the solar system are either rare, or we need other energy sources to heat the disc to a very high temperature to reset the condensation sequence.
Moreover, the maximum temperatures in our simulations occur at radii less than 1 AU. Not only is it rare to reach high enough temperatures to reset the condensation sequence, but the composition of planets beyond 1 AU can only be affected by that condensation if sufficient materials are transported outwards from the inner $\sim 0.5$AU. For many discs (including our fiducial model) some material is indeed transported outward from the interior to 1 AU (see Figure \ref{fig:discevo}). But, that material would mix with previously condensed and decoupled materials in the outer regions -- somewhat diluting the signatures of the chemical evolution of the inner region.
\subsection{Influence on the formation of compounds}
\begin{figure}
\includegraphics[width=0.5\columnwidth]{t31w1m1al5a.pdf}
\includegraphics[width=0.5\columnwidth]{t15w1m1al5a.pdf}
\includegraphics[width=0.5\columnwidth]{t15w1m1al3a.pdf}
\includegraphics[width=0.5\columnwidth]{t15w3m1al3a.pdf}
\caption{Temperature evolution for different initial conditions. (a) $T_{\rm C}=31$ K, $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-5}$,
(b) $T_{\rm C}=15$ K, $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-5}$,
(c) $T_{\rm C}=15$ K, $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-3}$, and
(d) $T_{\rm C}=15$ K, $\omega_{\rm C}=3\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-3}$.
For all the cases,
$M_{\rm C}=1\ M_\odot$.
The time for the first line here are set to be the time when the temperature gets its maximum value. The time interval is 20\% of the collapse time of MCC for $M_{\rm C}=1\ M_\odot$, $T_{\rm C}=10$ K, and $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$.
}
\label{fig:tmax_diff}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{t31w1m1a1e5ri-eps-converted-to.pdf}
\includegraphics[width=0.8\columnwidth]{t15w1m1a1e5ri-eps-converted-to.pdf}
\includegraphics[width=0.8\columnwidth]{t15w1m1a1e3ri-eps-converted-to.pdf}
\includegraphics[width=0.8\columnwidth]{t15w3m1a1e3ri-eps-converted-to.pdf}
\caption{Elemental abundances at different radii for three moments in time, from the end of the collapse to 5$t_{\rm dec}$ from then. Initial conditions are the same as in Figure \ref{fig:tmax_diff}.
}
\label{fig:Riele}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.85\columnwidth]{t31w1m1a1e5t-ratio-eps-converted-to.pdf}
\includegraphics[width=0.85\columnwidth]{t15w1m1a1e5t-ratio-eps-converted-to.pdf}
\includegraphics[width=0.85\columnwidth]{t15w1m1a1e3t-ratio-eps-converted-to.pdf}
\includegraphics[width=0.85\columnwidth]{t15w3m1a1e3t-ratio-eps-converted-to.pdf}
\caption{Evolution of the 10 most abundant condensed species, normalized to Mg$_2$SiO$_4$. The initial conditions are the same as in Figure \ref{fig:tmax_diff}. Solid lines represent species that are common to all panels while broken lines are unique to a specific panel.
}
\label{fig:species_diff_ratio}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.85\columnwidth]{mt-eps-converted-to.pdf}
\caption{Mass evolution of Mg$_2$SiO$_4$ for the four initial conditions used in the other figures.
}
\label{fig:mt}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{1abunt-eps-converted-to.pdf}
\includegraphics[width=1.0\columnwidth]{2abunt-eps-converted-to.pdf}
\includegraphics[width=1.0\columnwidth]{3abunt-eps-converted-to.pdf}
\includegraphics[width=1.0\columnwidth]{4abunt-eps-converted-to.pdf}
\caption{Relative element abundances (RA, using solar abundance as the reference) in the central star as a function of time.
The initial conditions are the same as in Figure \ref{fig:tmax_diff}.
(a) $T_{\rm C}=31$ K, $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-5}$,
(b) $T_{\rm C}=15$ K, $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-5}$,
(c) $T_{\rm C}=15$ K, $\omega_{\rm C}=1\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-3}$, and
(d) $T_{\rm C}=15$ K, $\omega_{\rm C}=3\times 10^{-14} \rm\ s^{-1}$, and $\alpha=10^{-3}$.
For all the cases,
$M_{\rm C}=1\ M_\odot$. The initial time is the beginning of the evolution of the discs.
}
\label{fig:elet}
\end{figure}
To see the effects that these maximum temperatures have on the composition of the condensed material (i.e., planetesimals) we calculate their composition for discs that form from our MCCs assuming initial solar abundances. We calculate the chemical equilibrium of the 33 elements \citep{Petaev:2009,Li:2020} at each radius
and each time in the evolving disc. We use a decoupling time-scale of 1.5$\times 10^{4}$ yr, which is consistent with the standard value used by \citet{Li:2020}. While the decoupling timescale is two orders-of-magnitude shorter than the disc evolution timescale, the compositions of the decoupled materials are significantly affected by the radial transport \citep{Li:2020}.
The temperature evolution of four discs with different initial conditions are shown in Figure \ref{fig:tmax_diff}. The first lines are set to the times when the temperatures reach their maximum values. The time intervals for different lines are 20\% of the typical collapse timescale. The temperatures in Figure \ref{fig:tmax_diff} reach their maximum values at distances between 0.38 and 0.43 AU.
Figure \ref{fig:Riele} shows the relative elemental abundances of decoupled elements normalized to the Solar abundance and Si as a function of 50\% condensation temperature at different radii. Generally, the relative abundances of refractory elements tend to be high in discs with a high maximum temperature (Figure \ref{fig:Riele}) and their abundances decrease with time as more volatile elements condense and decouple as the temperature decreases \citep{Li:2020}. The relative abundance of elements with 50\% condensation temperatures higher than the maximum temperature at this radius are equal because they have never been vaporized (they are not fractionated from each other). There are some non-monotonic changes for temperatures between 1200 to 1400 K in panels (a) to (c) (around the element Mg). These arise because the elements with lower 50\% condensation temperatures may have higher relative abundances at certain disc midplane temperatures (See Figure 4 in \citet{Li:2020}). The abundance beyond 1 AU rarely strays from unity, even in the hottest disc models (except panel a). This occurs because, beyond the 1 AU, the disc temperature never reaches 1500 K.
Compared with previous results that use simple, analytic disc models, these more realistic disc models cannot reproduce the observed volatile element depletion patterns in CM, CO, and CV (Figure \ref{fig:Riele}). To reproduce the observations, either additional heating events are needed or the volatile element depletion patterns in these chondrites were produced outside the protoplanetary disc, and they are simply a record of whatever falls into the protoplanetary disc.
Figure \ref{fig:species_diff_ratio} shows how the amounts of the 10 most abundant (by mass) condensed species change with time relative to Mg$_2$SiO$_4$ for discs shown in Figure \ref{fig:tmax_diff}
while the mass evolution of Mg$_2$SiO$_4$ is shown in Figure \ref{fig:mt}. Most of these species are the same across all panels, but there are some differences. For example, panel (a) includes Ca$_2$Al$_2$SiO$_7$, which is stable at the 1363 - 1505 K temperature range, while the other panels do not. The relative amounts of the species also differ from disc to disc, especially in (a) and (b).
We also see that the relative amount of each species tends to be stable for each disc -- which may help constrain the thermal history of the discs.
As the amount of condensed materials change for different MCC properties, the relative abundance of the final central star should also be different. Figure \ref{fig:elet} shows the relative chemical abundance of the central star as a function of element and time if we assume that all decoupled materials stay in the region where they condense. Initially, all of them have solar abundance. Over time, more elements condense and the relative abundances of the decoupled material decreases. The abundance for the refractory elements are lower than the volatile elements, which is consistent with the fact that more refractory elements decouple from the disc. This indicates that the chemical abundances in stars will differ somewhat from the abundances of the MCC from which they form as material becomes trapped in the planets.
A consequence of this result is that, when we use the stellar composition as a proxy for the composition of planets, we may need to use a composition somewhat enriched in refractory elements.
\section{Conclusions}\label{sec:conclusion}
Our results show that the properties of the initial MCCs have a significant effect on the maximum temperatures that are reached within the resulting discs. The disc temperatures, in turn, affect the composition of chondrites and planets that form. These simulations lead to the following conclusions:
1. The temperature in the inner region ($\textless10$ AU) of the disc increases first and then decreases with time. It reaches its maximum value around the end of the MCC collapse (several times $10^5$ years).
2. The maximum temperature of the disc increases with the initial temperature of the MCC and decreases with its angular velocity. The maximum temperature in the disc also increases if the viscosity in the disc decreases since the additional material in the inner region traps more heat in the disc.
3. The resulting composition of the central stars are similar to the initial composition of the MCC. However, the central stars might be slightly depleted in some of the most refractory elements -- by up to 10\% compared to the initial composition. Consequently, stellar composition may or may not be a good approximation to the initial composition of the MCC from which it formed, depending upon the situation.
4. 90\% of the simulations that use the observed properties of MCCs predict peak temperatures between 935 and 1635 K, with a median value of 1250 K. Less than 1\% of the discs from our simulations reach temperatures higher than 2000 K and are capable of vaporizing all elements. Even in these simulations, those temperatures only arise in the innermost portion of the disc.
Note that these peak temperatures depend upon our chosen distribution of $\alpha$ values.
5. Most discs reach peak temperatures that are lower than the 50\% condensation temperature of Mg. To match the depletion patterns of CM, CO, and CV chondrites and terrestrial planets, one needs either rare initial conditions of the proto-solar MCC, or other energy sources to heat the disc to a very high temperature in order to
reprocess the moderately volatile to refractory elements.
\section*{Acknowledgements}
JHS, ML, and SH are supported by the NASA grant NNX16AK08G and NSF grant AST-1910955. ZZ acknowledges support from the National Science Foundation under CAREER Grant Number AST-1753168 and Sloan Research Fellowship.
\section*{DATA AVAILABILITY}
The data underlying this article are available in the article.
\bibliographystyle{mnras}
|
1,116,691,498,507 | arxiv | \section{Introduction and Notations}
\subsection{Introduction}\lb{intr}
Weyl points are singular points on the 3-dimensional spectral bands of an operator with periodic coefficients, at which two distinct bands intersect conically. Much attention has been paid to looking for such fundamental singularities in various physical systems in the past few decades \cite{Ablowitz09,Allaire08,Castro07,Novoselov11}. They are the hallmark of many novel phenomena. Many materials such as graphene exhibit such unusual singular points on their energy bands \cite{Castro07,Novoselov11}. These singular points carry topological charges and play essential roles on the formation of topological states, for instance chiral edge states or surface states \cite{Avila2012,Goldman2012,Graf2012,Perez2014}. In the past decade, constructing and engineering the conically degenerate spectral points become one of the major research subjects in many fields. Accordingly, understanding the existence of these points on the energy bands and their connections to interesting physical phenomena are extremely important in both theoretical and applied fields.
How to obtain and justify the existence of such degenerate points become urgent in various physical systems. For instance, it is well known that honeycomb structures give rise to the existence of Dirac points in 2-D systems. The existence of Dirac points in the periodic system was first reported by Wallace in the tight-binding model and demonstrated in the continuous systems by numerical and asymptotic approaches \cite{Ablowitz2013,Ablowitz2012,Avila2012,Wallace1947}. However, the rigorous justification on the existence of Dirac points for 2-D Schr\"odinger equation with a generic honeycomb potential was recently given by Fefferman and Weinstein \cite{fefferman2012honeycomb}. They used very simple conditions to characterize honeycomb potentials and developed a framework to rigorously justify the existence of Dirac points. Their framework paved the way for the mathematical analysis on such degenerate points, and their method has been successfully extended to other 2-D wave systems \cite{Drouot2020ubiquity,Drouot2020,Keller2018,Lee-Thorp2017}. There are also other rigorous approaches to demonstrate the existence of Dirac points. Lee treated the case where the potential is a superposition of delta functions centered on sites of the honeycomb structure \cite{Lee2016}. Berkolaiko and Comech used the group representation theory to justify the existence and persistence of Dirac points \cite{Berkolaiko14}. The low-lying dispersion surfaces of honeycomb Schr\"odinger operators in the strong binding regime, and its relation to the tight-binding limit, was studied in \cite{Weinstein2016}. Ammari et al. applied the layer potential theory to honeycomb-structured Minnaert bubbles \cite{Ammari2018honeycomblattice}. Based on the rigorous justification of the existence of Dirac points, a lot of rigorous explanations on the related physical phenomena have been extensively investigated. For example, the effective dynamics of wave packets associated with Dirac points were studied in \cite{Ammari2020,Fefferman2014,hu2020linear,Watson18,Xie2019,xie2020wave}. The existence of edge states and associated dynamics are studied in \cite{Bal2019,Drouot2020,Fefferman2015}.
Despite successful applications on the aforementioned analysis of the Dirac points in 2-D systems, the advances in applications such as materials sciences, condensed matter physics, placed new theoretical demands that are not entirely met. Just as Kuchment pointed out in a recent overview article on periodic elliptic operators \cite{Kuchment2015}, "the story does not end here". One important missing piece is the analysis of 3-dimensional degenerate points which are referred to as Weyl points. Another piece is the conical points with higher order multiplicities. In the literature, some special structures are proposed to admit Weyl points \cite{leng2020multiband,Tang17,Yang2017,Lu2020Double}. However, most constructions and demonstrations are based on either tight-binding models, numerical computations or formal asymptotic expansions. To the best of our knowledge, no similar construction and rigorous analysis as aforementioned literature have been given for Weyl points with higher-order multiplicities. Due to the importance and potential applications of Weyl points in quantum mechanics, photonics and mechanics, such generic analysis is highly desired. This is the goal of our current work.
This work is concerned with the $L^{2}(\R^3)$-spectrum of the following 3-dimensional Schr\"odinger equation
\begin{equation} \lb{HV}
H=-\Delta + V(\bx), \q \bx\in \R^3,
\end{equation}
where the potential $V(\bx)$ is real-valued and periodic. By Floquet-Bloch theory \cite{Kuchment1993,Kuchment2015,Kuchment2000}, the spectrum of $H$ in $L^{2}(\R^3)$ is the union of all energy bands $E_b(\bk), ~b\ge 1$ for all $\bk$ in the Brillouin zone. For some specific $V(\bx)$, two energy bands may intersect with each other conically at some $\bk_*$. This degenerate point $\bk_*$ in the three dimensional energy bands is called a Weyl point. There are different types of Weyl points depending on the multiplicity of degeneracy.
In this work, we shall give a simple construction of three-fold Weyl points, i.e., two energy bands intersect conically with an extra band between them. We shall also rigorously justify the existence of such degenerate points by using the strategies developed in \cite{fefferman2012honeycomb}. More specifically, we first propose a very general class of admissible potentials which are characterized by several symmetries. Different from honeycomb potentials in which the inversion and $2\pi/3$-rotational symmetries are the indispensable ingredients, the potentials proposed in this paper have two rotational symmetries in addition to the inversion symmetry. These three symmetries together guarantee the three-fold degeneracy at certain high symmetry points and ensure conical structures in their vicinities. Our analysis in this work involves many novel arguments on the eigenstructures at the high symmetry points in order to explain how the 3-fold degeneracy is protected by the underline symmetries and why the Fermi velocities corresponding to different branches are the same. These important arguments are relatively trivial in the honeycomb case \cite{fefferman2012honeycomb}. Our current work not only extends the theory developed in \cite{fefferman2012honeycomb} to 3-dimensional systems but also shines some light on the analysis of singular points with higher multiplicities. Our analysis also provides the starting point of future theoretical analysis on these higher order Weyl points, such as the existence of chiral surface states, Fermi arcs and so on\cite{Yang2017,Lu2020Double}.
This work is organized as follows. In Section 2, we first introduce the lattice $\Lambda$ and its dual lattice $\Lambda^{*}$ together with their
fundamental cells $\Omega$ and $\Omega^{*}$, and then we precisely discuss the existence of high symmetry points $\bk_h$ in $\Omega^{*}$. Section 2 concludes with the Fourier analysis of $\Lambda$-periodic functions. Section 3 contains the definition of the admissible potentials characterized by several symmetries. We also review the relevant Floquet-Bloch theory for Schr\"odinger operators $H=-\Delta+V(\bx)$. In Section 4, we first propose required conditions of eigenstructures at high symmetry point $\bK$ for some eigenvalue $\mu_{*}$, i.e., $\textbf{H1-H2}$ and their consequences. We then prove the energy bands in the vicinity form a conical structure with an extra band in the middle. In Section 5, we justify that the required conditions $\textbf{H1-H2}$ do hold for nontrivial shallow admissible potentials. Specifically, we clearly show the significance of the $\mathcal{R}$ and $\mathcal{T}$ symmetries to preserve the multiplicity of eigenvalues of $H^{\e}=-\Delta+\e V(\bx)$ at $\bK$ while $\e$ is sufficiently small. Moreover, the justification is extended to generic admissible potentials. Section 6 discusses the instability of the Weyl points and perturbations of dispersion bands of when $V(\bx)$ is violated by an odd potential $W(\bx)$. Section 7 provides detailed numerical simulations of the energy bands and Weyl points in different cases for a special choice of admissible potential. In Appendix A, we present the proofs of certain Propositions and Lemmas in Section 4 and Section 5.
\subsection{Notations and conventions} \lb{nota}
Without specifications, we use the following notations and definitions.
\bu For $z\in \C$, $\ol{z}$ denotes the complex conjugate of $z$.
\bu For $\bx,\ \by\in\C^{n}$, $\langle \bx,\by \rangle:=\ol{\bx}\cdot\by=\ol{x}_{1} y_{1}+...+\ol{x}_{n}y_{n}$, and $|\bx|:= \sqrt{\langle \bx,\bx\rangle}$.
\bu For a matrix or a vector $A$, $A^{t}$ is its transpose and $A^{*}$ is its conjugate-transpose.
\bu $\Lambda\in\mathbb{R}^3$ denotes the lattice, and $\Lambda^{*}\subset(\mathbb{R}^3)^{*}=\mathbb{R}^{3}$ denotes the dual lattice of $\Lambda$. Moreover, $\bv_j,\ j=1,2,3$ are the basis vectors of $\Lambda^{*}$, while $\bq_{\ell},\ \ell=1,2,3$ are the dual basis vectors of $\Lambda^{*}$, which are chosen to satisfy $\bv_{j}\cdot \bq_{\ell}=2\pi \delta_{\ell j}$.
\bu $\langle f,g \rangle_{D}=\int_{D} \overline{f}g$ is the $L^{2}(D)$ inner product. In this work, the region $D$ of integration is assumed to be the fundamental cell $\Omega$ if it is not specified.
\bu $\nabla=(\partial_{x_{1}},\partial_{x_2},\partial_{x_3})^{T}$.
\bu $I$ denotes the $3\times3$ identity matrix.
\bu For $\ka=(\ka_x ,\ka_y ,\ka_z)\in \mathbb{R}^{3}$, $\ka^{\arg}$ represents $\frac{\ka_x \ka_y \ka_z}{|\ka|^{3}}$.
\section{Preliminaries} \lb{prel}
\subsection{The lattice $\Lambda$ and the rotation $R$} \lb{La-R}
Consider the following linearly independent vectors in $\R^3$
\[
\bv_1=\frac{a}{\sqrt{3}}\begin{pmatrix}1\\-1\\-1\end{pmatrix}, \q \bv_2=\frac{a}{\sqrt{3}}\begin{pmatrix}1\\1\\1\end{pmatrix},\q \bv_3=\frac{a}{\sqrt{3}}\begin{pmatrix}-1\\1\\-1\end{pmatrix}.
\]
Here $a>0$ is the lattice constant. Define the lattice as follows
$$
\La=\mathbb{Z}\bv_{1} \oplus \mathbb{Z}\bv_{2} \oplus \mathbb{Z}\bv_{3}:=\{n_{1}\bv_{1} +n_{2}\bv_{2} +n_{3}\bv_{3}: n_{1}, n_{2}, n_{3}\in \mathbb{Z}\}.
$$
The parameter $a$ then gives the distance between nearest neighboring sites. The fundamental period cell of $\La$ is
\be\lb{la
\Om:= \{x_{1}\bv_{1}+ x_{2}\bv_{2}+ x_{3}\bv_{3}: 0\leq x_{i}\leq1,\ i=1,2,3\}.
\ee
Let $\bq_{1}$, $\bq_{2}$, $\bq_{3}\in \R^3$ be the dual vectors of $\bv_{1}$, $\bv_{2}$, $\bv_{3}$, in the sense that
\
\bq_{\ell}\cdot\bv_{j}=2\pi \delta_{\ell j}, \q \ell,j=1,2,3.
\]
Explicitly,
\
\bq_{1}=q\begin{pmatrix}1\\0\\-1\end{pmatrix},\q
\bq_{2}=q\begin{pmatrix}1\\1\\0\end{pmatrix},\q
\bq_{3}=q\begin{pmatrix}0\\1\\-1\end{pmatrix},
\]
where $q=\frac{\sqrt{3}\pi}{a}$. Then the dual lattice of $\La$ is defined as
\[
\Las=\mathbb{Z}\bq_{1}\oplus\mathbb{Z}\bq_{2}\oplus\mathbb{Z}\bq_{3}:=\{m_{1}\bq_{1}+m_{2}\bq_{2}+m_{3}\bq_{3}:m_{1},m_{2},m_{3}\in \mathbb{Z}\}.
\]
The fundamental period cell of $\Las$ is chosen to be
\begin{equation*}
\Om^{*}:=\{c_1 \bq_1 + c_2 \bq_2 + c_3 \bq_3 :c_i \in \z(-1/2,1/2\y],i=1,2,3\}.
\end{equation*}
In this work, we are interested in the following rotation transformation $R$ in $\mathbb{R}^{3}$
\be\label{matrix R}
R= \begin{pmatrix} 0&-1&0\\1&0&0\\0&0&-1\\ \end{pmatrix}.
\ee
Obviously, $R^t R = R R^t = I$. Moreover,
\be\label{matrix}
R^{*}= R^t=R^{-1}=\begin{pmatrix} 0&1&0\\-1&0&0\\0&0&-1\\ \end{pmatrix},
\andq R^4=I.
\ee
By direct calculations, we can conclude the following proposition.
\begin{prop}\lb{rem21}
\noindent$(1)$ The eigenvalues of $R$ is $i^{\ell},\ell=1,2,3$, with the corresponding eigenvectors
\be\lb{R-ev}
\om_1=\f{1}{\sqrt{2}}(1,-i,0)^t, \qq \om_2=(0,0,1)^t,\qq \om_3=\f{1}{\sqrt{2}}(1,+i,0)^t=\ol{\om_1}.
\ee
$(2)$ $R^{*}$ and $R$ satisfy
\be\label{Rq}
\bb{split}
& R^{*}\bv_{1}=\bv_{4},\q R^{*}\bv_{2}=\bv_{1},\q R^{*}\bv_{3}=\bv_{2},\q R^{*}\bv_{4}=\bv_{3}, \\
& R\bq_{1}=\bq_{2}-\bq_{1},\q R\bq_{2}=\bq_{3}-\bq_{1},\q R\bq_{3}=-\bq_{1},\\
& R^* \bq_{1}=-\bq_{3},\q R^{*}\bq_{2}=\bq_{1}-\bq_{3},\q R^{*}\bq_{3}=\bq_{2}-\bq_{3}.
\end{split}
\ee
Thus both $R$ and $R^*$ leave $\La$ and $\Las$ invariant.
\end{prop}
\begin{defn}\lb{hsp}
A point $\bk_{h}\in\R^3$ is defined to be a high symmetry point with respect to $R$ if
\[
R\bk_{h}-\bk_{h}\in\Las.
\]
\end{defn}
\bb{rem}\lb{k+q}
By understanding $\Las_{\bk_h}:=\bk_h+\Las$ as shifted lattices, we know that $\bk_h$ is a high symmetry point if and only if $R$ leaves $\Las_{\bk_{h}}$ invariant, i.e.,
\[
R(\Las_{\bk_h}) = \Las_{\bk_{h}}.
\]
\ifl{\bf
Define, for $\bk\in \R^3$, the shifted lattice
\[
\Las_\bk:= \bk+\Las=\z\{ \bk+\bq: \bq \in \Las\y\}.
\]
Then $\bK$ is a high symmetry point if and only if
\be \lb{hsp11}
R(\Las_\bK) = \Las_\bK,
\ee
i.e. $R$ leaves $\Las_\bK$ invariant.}
\fi
\end{rem}
The following lemma indicates that inside the fundamental period cell $\Om^*$, there exist precisely four high symmetry points.
\bb{lem} \lb{fps}
A point $\bk_h=c_1\bq_{1}+c_2\bq_{2}+c_3\bq_{3}\in \Omega^{*} $
is a high symmetry point with respect to $R$ if and only if the coefficients $(c_1,c_2,c_3)$ take the following $4$ cases
\be\lb{4k}
(c_1,c_2,c_3)=(0,0,0), \
\z(1/2,1/2,1/2\y), \
\z(1/4,1/4,1/4\y), \
\z(-1/4,-1/4,-1/4\y).
\ee
\end{lem}
\Proof By (\ref{Rq}), we have
\[
R\bk_h=(-c_1-c_2-c_3)\bq_{1}+c_1\bq_{2}+c_2\bq_{3}.
\]
Then
\be \lb{K1}
R\bk_h-\bk_h=(-2c_1-c_2-c_3)\bq_{1}+(c_1-c_2)\bq_{2}+(c_2-c_3)\bq_{3}\in\Las
\ee
is the same as
\
\z(-2c_1-c_2-c_3,\, c_1-c_2,\, c_2-c_3\y)\in\mathbb{Z}^3.
\
Due to the restrictions $c_i\in (-1/2,1/2]$, \x{K1} has four solutions listed as in \x{4k}. \qed
In this work, we only focus on the following specific high symmetry point
\
\bK:= -\frac{1}{4}(\bq_{1}+\bq_{2}+\bq_{3})=q\z(-\frac12,-\frac12,\frac12\y).
\]
It follows from \x{Rq} and \x{K1} that
\begin{equation}\lb{RK}
R^4\bK=\bK, \andq R^\ell\bK=\bK+\bq_{\ell}\mbox{ for } \ell=0,1,2,3.
\end{equation}
Here $\bq_0:=\bz$.
\ifl
The study for the following high symmetry point is analogous
\[
\bK':=\frac{1}{4}(\bq_{1}+\bq_{2}+\bq_{3})=q(\frac12,\frac12,-\frac12)^t.
\]
\begin{equation}\lb{RK}
R\bK=q(1,-1,-1)^{t}=\bK+\bq_{1},\q
R^{2}\bK=q(1,1,1)^{t}=\bK+\bq_{2},\q
R^{3}\bK=q(-1,1,-1)^{t}=\bK+\bq_{3}.
\end{equation}
Introduce the zone $\Om$ by denoting its vertices as:
\begin{equation}
A\equiv \pi(1,1,-1)^t,\q B\equiv\pi(1,-1,-1)^{t},\q C\equiv\pi(1,1,1)^{t},\q D\equiv\pi(-1,1,-1)^{t}.
\end{equation}
Then $R^{*}$ maps $\Om$ to itself.
\fi
\subsection{$\La$-periodic, $\La$-pseudo-periodic functions and Fourier expansions} \lb{Per-QPer}
We say that a function $f(\bx): \R^3\to \C$ is $\La$-periodic if
\be \lb{per}
f(\bx +\bv) = f(\bx)\q \forall \bx\in \R^3, \ \bv\in \La.
\ee
More generally, given a quasi-momentum $\bk\in \R^3$, we say that a function $F(\bx): \R^3\to \C$ is $\La$-pseudo-periodic with respect to $\bk$ if
\be \lb{kper}
F(\bx +\bv) =e^{i \bk \d \bv} F(\bx)\q \forall \bx\in \R^3, \ \bv\in \La.
\ee
Let us introduce the Hilbert space
\[
L^2_\kla:= \z\{F(\bx)\in L^2_{\rm loc}(\R^3,\C): F(\bx)\mbox{ satisfies \x{kper}} \y\},
\]
where the inner product is
\[
\inn{F}{G}:= \int_\Om \ol{F(\bx)} G(\bx) d\bx \mbox{ for } F, \ G \in L^2_\kla.
\]
Similarly, we define
\[
H^{s}_{\bk,\Lambda}=\{F(\bx)\in H^{s}(\mathbb{R}^3,\mathbb{C}):\ F(\bx)\ \mbox{satisfies \x{kper}}. \}
\]
In particular, for $\bk={\bf 0}$,
\[
L^2_\per= L^2(\R^3/\La) := \z\{f(\bx)\in L^2_{\rm loc}(\R^3,\C): f(\bx)\mbox{ staisfies \x{per}} \y\}
\]
is the space of square-integrable $\La$-periodic functions. Obviously, $F(\bx)\in L^2_\kla$ if and only if
\[
f(\bx):= e^{-i \bk \d \bx} F(\bx) \in L^2_\per.
\]
That is, the mapping
\be \lb{ff}
f(\bx) \longmapsto F(\bx) :=e^{i \bk \d \bx} f(\bx)
\ee
gives a one-to-one correspondence between $L^2_\per$ and $L^2_\kla$. Moreover, it is easy to see that
\[
\inn{F}{G} =\inn{f}{g} \qqf f, \ g\in L^2_\per.
\]
That is, the mapping \x{ff} is an isometry from $L^2_\per$ to $L^2_\kla$.
Due to the $\La$-periodicity of functions $f(\bx)\in L^{2}_\per$, they can be expanded as Fourier series of the form
\be\lb{fep}
f(\bx) = \sum_{\bq\in \Las} \hat f_\bq e^{i \bq\d \bx},
\ee
where $\z\{\hat f_\bq\y\}_{\bq \in \Las}\subset l^{2}(\Lambda)$ is the sequence of Fourier coefficients, indexed using the discrete indexes $\bq$ from $\Las$. Explicitly,
\be \lb{fc0}
\hat f_\bq := \f{1}{|\Om|} \int_\Om f(\bx) e^{-i \bq \d \bx} d\bx,
\ee
where $|\Om|$ denotes the volume of the cell $\Om$. Such a form \x{fep} of Fourier expansions is consistent with
Example \ref{example-potential} and is more convenient for later uses. Note that
\[
\z\{\hat f_\bq\y\}_{\bq \in \Las}\in l^2_{\Las},
\]
the Hilbert space of square-summable complex sequences over the dual lattice $\Las$.
\bb{rem}\lb{k-exp}
Given $\bk\in \R^3$, pseudo-periodic functions $F(\bx)=e^{i \bk \d \bx} f(\bx) \in L^2_\kla$ can be expanded as
\be \lb{F-e}
F(\bx) = e^{i\bk\cdot\bx}\sum_{\bq\in \Las} \hat f_\bq e^{i \bq \d \bx}=\sum_{\bq\in \Las} \hat f_\bq e^{i (\bk+\bq) \d \bx},
\ee
where $\{\hat f_\bq\}$ is as in \x{fc0}.
\end{rem}
Rotations $R$ and $R^{*}$ in \x{matrix R} and (\ref{matrix}) can yield a transformation $\mcr$ for functions $F(\bx) \in L^2_\kla$ by
\[
\mcr[F](\bx):=F(R^{*}\bx)\q\mbox{for } \bx\in \R^3.
\]
\bb{lem}\lb{iso}
Let $\bk_h$ be a high symmetry point w.r.t. $R$. Then
\bu $\mcr$ maps $L^2_{\bk_h,\Lambda}$ to itself as a unitary operator.
\bu Define an affine transformation $\Mfr: \Las\to \Las$ by
\begin{equation}\label{eq:mfr}
R_{\bk_h} (\bq) := R\bq+R\bk_h-\bk_h\q\mbox{for } \bq \in \Las.
\end{equation}
Then, for any $\ell\in\Z$, one has
\be \lb{mfr-l}
R_{\bk_h}^\ell (\bq) =R^\ell \bq + R^\ell \bk_h-\bk_h = R^\ell(\bq+\bk_h)- \bk_h \q\mbox{for} \bq \in \Las.
\ee
In particular,
\be \lb{R4}
R_{\bk_h}^4 (\bq) = \bq \q\mbox{for}\ \bq \in\Las.
\ee
\bu The action $\mcr$ on $L^2_{\bk_h,\Lambda}$ is given by
\be \lb{RR}
\mcr\z[\sum_{\bq\in \Las} \hat f_\bq e^{i (\bk_h+\bq) \d \bx}\y] =\sum_{\bq\in \Las} \hat f_\bq e^{i R(\bk_h+\bq) \d \bx}
=\sum_{\bq\in \Las} \hat f_\bq e^{i (\bk_h+R_{\bk_h}(\bq)) \d \bx}\ .
\ee
\end{lem}
\Proof \bu For $F(\bx)=e^{i\bK \d \bx} f(\bx)\in L^2_{\bk_h,\Lambda}$, we can use expansion \x{F-e} to obtain
\be \lb{RF1}
\begin{split}
\mcr&\z[e^{i\bk_h \d \bx} f(\bx)\y] = \sum_{\bq\in \Las} \hat f_\bq e^{i (\bk_h+\bq) \d R^*\bx}= \sum_{\bq\in \Las} \hat f_\bq e^{i R(\bk_h+\bq) \d \bx}
\\&=e^{i\bk_h\cdot\bx} \sum_{\bq\in \Las} \hat f_\bq e^{i (R\bq +R\bk_h-\bk_h) \d \bx}\ .
\end{split}
\ee
As $R$ leaves $\Las$ invariant and $R\bk_h-\bk_h\in\Las$, we have $R_{\bk_h} (\bq)=R\bq +R\bk_h-\bk_h\in \Las$ for all $\bq \in \Las$. Thus
\[
\sum_{\bq\in \Las} \hat f_\bq e^{i (R\bq +R\bk_h-\bk_h) \d \bx}\in L^2_\per \andq \mcr[F]\in L^2_{\bk_h,\Lambda}.
\]
Moreover, for $F(\bx)=e^{i\bk_h \d \bx} f(\bx),\ G(\bx)=e^{i\bk_h \d \bx} g(\bx)\in L^2_{\bk_h,\Lambda}$, one has
\[
\bb{split}
& \inn{\mcr[F]}{\mcr[G]}=\int_\Om \ol{F(R^* \bx)} G(R^* \bx)d\bx \\
& = \int_\Om \ol{f(R^*\bx)} g(R^*\bx)d\bx= \int_{R^*(\Om)} \ol{f(\by)} g(\by)d\by =\inn{f}{g}\\
& =\inn{F}{G},
\end{split}
\]
because $R^*$ is an orthogonal transformation and both $f(\by)$ and $g(\by)$ are $\La$-periodic in $\by\in \R^3$. This shows that $\mcr$ is unitary.
\bu Let us check \x{mfr-l} only for $\ell\in \N$. By \x{eq:mfr}, we have for $\bq\in \Las$
\[
R_{\bk_h}^\ell(\bq)=R^\ell\bq + \sum_{j=0}^{\ell-1} R^j(R\bk_h - \bk_h)= R^\ell\bq + R^\ell \bk_h-\bk_h = R^\ell(\bq+\bk_h)-\bk_h,
\]
the desired equalities in \x{mfr-l}.
By letting $\ell=4$ in \x{mfr-l}, we obtain \x{R4} because $R^4=I$.
\bu Using \x{F-e} and \x{eq:mfr}, equality \x{RF1} can be written as \x{RR}.\qed
\bb{rem}\lb{k-K}
For $\bk_{h}=\bz$, one has $R_\bz=R$. For $\bk_h=\bK$, one has from \x{RK} that
\[
R_{\bK}(\bq)\equiv R \bq + \bq_1\q\mbox{for } \bq \in \Las.
\]
\end{rem}
\subsection{Decompositions of periodic and pseudo-periodic functions} \lb{Four}
In the following discussions we only consider the special high symmetry point $\bK$.
Notice from \x{RK} that $R_\bK^4= I$ on $\Las$, and
\[
R_\bK^\ell \ne I \q \mbox{on $\Las$\q for $\ell=1,2,3.$}
\]
Each orbit of the action $R_\bK$ on $\Las$ consists of precisely four points. Let us introduce
\[
\ssk:= \Las/R_\bK = \Las/\z\{\bq \sim \bq': \bq, \ \bq'\in \Las, \ \bq'= R_\bK^{\ell} (\bq) \mbox{ for some } \ell\in \Z \y\}\hookrightarrow \Las.
\]
Then functions $F(\bx)=e^{i\bK \d \bx} f(\bx)\in L^2_\Kla$ can be decomposed into
\be\label{S-l}
\begin{split}
F(&\bx) =\sum_{\bq\in\Las}\hat f_{\bq}e^{i(\bK+\bq)\cdot\bx}= \sum_{\bq\in\ssk} \sum^{3}_{\ell=0} \hat f_{R_\bK^\ell (\bq)} e^{i (\bK+R_\bK^\ell (\bq)) \cdot\bx}\\&= \sum_{\bq\in\ssk} \sum^{3}_{\ell=0} \hat f_{R_\bK^\ell (\bq)} e^{i R^\ell(\bK+\bq) \cdot\bx}
= \sum_{\bq\in\ssk} (\hat f_{\bq}e^{i(\bK+\bq)\cdot\bx}+\hat f_{R_\bK(\bq)}e^{iR(\bK+\bq)\cdot\bx}
\\&+ \hat f_{R_\bK^{2}(\bq)} e^{iR^{2}(\bK+\bq)\cdot\bx} +\hat f_{R_\bK^{3}(\bq)}e^{iR^{3}(\bK+\bq)\cdot\bx}).
\end{split}
\ee
Since $R^4=I$ and $R^{*4}=I$, one has $\mcr^{4}=I$ on $L^2_\per$. Hence eigenvalues $\si$ of the unitary operator $\mcr$ must satisfy $\si^4=1$. In fact, one has
\begin{equation}\lb{si}
\si= i^\ell, \qq \ell=0,1,2,3.
\end{equation}
Then we have an orthogonal decomposition for $L^{2}_\per$
\begin{equation}\lb{dec0}
L^{2}_\per=L^{2}_{\bz,1}\oplus L^{2}_{\bz,i}\oplus L^{2}_{\bz,-1}\oplus L^{2}_{\bz,-i},
\end{equation}
where the eigenspaces are
\
L^{2}_{\bz,i^\ell}:=\z\{f\in L^{2}_\per:\mcr[f]=i^\ell f \y\}, \qq \ell=0,1,2,3.
\
Note that \x{dec0} also yields an orthogonal decomposition for the space $L^2_\Kla$
\
L^{2}_\Kla=L^{2}_{\bK,1}\oplus L^{2}_{\bK,i}\oplus L^{2}_{\bK,-1}\oplus L^{2}_{\bK,-i},
\
where
\[
L^2_{\bK,i^\ell} :=
\left\{ e^{i \bK \d \bx} f(\bx): f(\bx ) \in L^2_{\bz,i^\ell}\right\} \equiv \z\{F\in L^2_\Kla: \mcr[F] = i^\ell F \y\}, \ \ell=0,1,2,3.
\]
Let $\si$ be as in \x{si} and
\[
F(\bx) = \sum_{\bq\in\Las}\hat f_{\bq}e^{i(\bK+\bq)\cdot\bx}\in L^2_{\bK, \si}\ .
\]
Then
\be \lb{Fl}
\mcr^\ell[F]= \si^\ell F\qqf \ell\in \Z.
\ee
By \x{RR}, we have
\[
\mcr^\ell[F](\bx) =\sum_{\bq\in \Las} \hat f_\bq e^{i R^\ell(\bK+\bq) \d \bx}
=\sum_{\bq\in \Las} \hat f_\bq e^{i (\bK+R_\bK^\ell\bq) \d \bx}\equiv \sum_{\bq\in \Las} \hat f_{R_\bK^{-\ell}(\bq)} e^{i (\bK+\bq) \d \bx} .
\]
Since
\[
\si^\ell F(x) = \sum_{\bq\in \Las} \si^\ell \hat f_\bq e^{i (\bK+\bq) \d \bx},
\]
we deduce from \x{Fl} that the Fourier coefficients $\hat f_\bq$ satisfy
\[
\hat f_{R_\bK^{-\ell}(\bq)}=\si^\ell \hat f_\bq \qqf \bq\in \Las,
\]
i.e.,
\be \lb{FCr}
\hat f_{R_\bK^{\ell}(\bq)}=\si^{-\ell} \hat f_\bq \qqf \bq\in \Las,\ \ell\in \Z.
\ee
Combining with general decomposition \x{S-l}, we have the following results.
\begin{lem}\label{prop:expressions-of-4Phi}
Let $\si$ be as in \x{si}.
\bu $F(\bx)\in L^{2}_{\bK,\si}$ if and only if there exists $\{ \hat f _{\bq} \} _{\bq\in \ssk} \in l^{2}_{\ssk}$ such that
\begin{align}
F(\bx)&= \sum_{\bq\in\ssk} \z(\hat f_{\bq}\sum^{3}_{\ell=0}\si^{-\ell}e^{iR^{\ell}(\bK+\bq)\cdot\bx}\y) \lb{decomp1}\\
&=\sum_{\bq\in \ssk}\hat f _{\bq}\z(e^{i(\bK+\bq)\cdot\bx}+\ol{\si}e^{iR(\bK+\bq)\cdot\bx}
+ \si^{2} e^{iR^{2}(\bK+\bq)\cdot\bx} +\si e^{iR^{3}(\bK+\bq)\cdot\bx}\y) \lb{decomp2}.
\end{align}
\bu If $F(\bx)\in L^{2}_{\bK,\si}$, then $\overline{F(-\bx)}\in L^{2}_{\bK,\bar \si}$\ .
\end{lem}
\Proof
\bu Note that $R_\bK^4=I$ and $\si$ satisfies $\si^4=1$. Substituting relations \x{FCr}, $\ell=0,1,2,3$, into \x{S-l}, we obtain equality \x{decomp1}.
As for equality \x{decomp2}, we need only to notice in \x{decomp1} that
\[
\si^0=1, \qq \si^{-1}= \bar \si, \qq \si^{-2} = \si^2, \qq \si^{-3}= \si\ .
\]
\bu We use expansion \x{decomp1} for $F(\bx)$ to obtain
\[
\begin{split}
\overline{F(-\bx)}&= \ol{\sum_{\bq\in\ssk} \z(\hat f_{\bq}\sum^{3}_{\ell=0}\si^{-\ell}e^{iR^{\ell}(\bK+\bq)\cdot(-\bx)}\y)}\\
&= \sum_{\bq\in\ssk} \z(\ol{\hat f_{\bq}}\sum^{3}_{\ell=0}\bar\si^{-\ell}e^{-iR^{\ell}(\bK+\bq)\cdot(-\bx)}\y)\\
&= \sum_{\bq\in\ssk} \z(\ol{\hat f_{\bq}}\sum^{3}_{\ell=0}\bar\si^{-\ell}e^{iR^{\ell}(\bK+\bq)\cdot \bx}\y),
\end{split}
\]
which is in $L^{2}_{\bK,\bar \si}$, following from the characterization \x{decomp1} for the eigenvalue $\bar \si$. \qed
\section{Eigenvalues of periodic Schr\"odinger operators} \label{FBT}
\subsection{Admissible potentials} \lb{Adm}
In this work, we introduce the following admissible potentials.
\begin{defn}\label{def:definition-of-lattice}
{\rm (Admissible Potentials)} Let $V(\bx)\in C^{\infty}(\R^{3})$ be real-valued. We say that $V(\bx)$ is an admissible {\rm potential} with respect to $\La$ if $V(\bx)$ satisfies
(1) $V(\bx)$ is $\La$-periodic, $V(\bx+\bv)=V(\bx)$ for all $\bx\in \R^{3}$ and $\bv\in\La$.
(2) $V(\bx)$ is real-valued and even, i.e., $\ol{V(\bx)}=V(\bx)$, $V(-\bx)\equiv V(\bx)$ for $\bx \in \R^3$.
(3) $V(\bx)$ is $\mcr$-invariant, i.e.,
\[
\mcr[V](\bx)= V(R^{*}\bx)\equiv V(\bx)\mbox{ for } \bx\in \R^3.
\]
(4) $V(\bx)$ is $\mathcal{T}$-invariant, i.e.,
\[
\mathcal{T}[V](\bx)\equiv V(T^{*}\bx)=V(\bx),
\]
where $T$ is the following matrix
\be\lb{mat}
T=
\begin{pmatrix}
-1&0&0\\
0&0&-1\\
0&1&0\\
\end{pmatrix}.
\ee
\end{defn}
We remark that the requirements (2) in Definition \ref{def:definition-of-lattice} are the so-called $\mathfrak{PT}$-symmetry. Moreover, requirement (4) is a novel symmetry for $3$-dimensional potentials which will play an important role in the later analysis for Weyl points. Admissible potentials have the following properties.
\begin{cor}
Let $V(\bx)$ be an admissible potential. Then its Fourier coefficients $\hat V_\bq$ satisfy
\[
\hat V_{-\bq}=\hat V_\bq\in \R\q \forall \bq\in \Las,
\]
and
\[
\hat V_{R^\ell \bq} = \hat V_\bq, \q \hat V_{T^\ell \bq} = \hat V_\bq \q \forall \bq\in \Las,\ \ell\in \Z.
\]
\end{cor}
\begin{rem}
Let us consider the orthogonal matrix $T$ in $\x{mat}$. It is easy to see that $T$ maps the lattice $\Lambda^{*}$ to itself and $T^{*}=T^{-1}$. Moreover, $T$ acts on $\Lambda^{*}$ as follows
\
\begin{split}
&T\bq_1=\bq_3-\bq_1, \qq T\bq_2=-\bq_1, \qq T\bq_3=\bq_2-\bq_1,\\
&T\bK=\bK+\bq_1, \qq T^2 \bK=\bK+\bq_3, \qq T^3 \bK=\bK+\bq_2.
\end{split}
\]
\end{rem}
Typical admissible potentials can be constructed using Fourier expansions.
\begin{exa}\label{example-potential}
{\rm
Let us define real, even potentials
\begin{equation*}
\begin{split}
V_1(\bx) &:=\cos(\bq_{1}\cdot\bx)+\cos((\bq_{2}-\bq_{1})\cdot\bx) +\cos((\bq_{3}-\bq_{2})\cdot\bx)+\cos(\bq_{3}\cdot\bx), \\
V_2(\bx) &:= \cos(\bq_{2}\cdot\bx)+\cos((\bq_{3}-\bq_{1})\cdot\bx).
\end{split}
\end{equation*}
It is easy to see that these $V_i(\bx)$ are $\mathcal{R}$-invariant potentials. Thus, for any real coefficients $c_i$, the potential
\begin{equation*
V(\bx)= \sum_{i=1}^2 c_i V_i(\bx)
\end{equation*}
is also $\mathcal{R}$-invariant. However, $V(\bx)$ is, in general, not $\mathcal{T}$-invariant. In fact, by noting that $T\bq_1 =\bq_1 -\bq_3$, we know that $V(\bx)$ is $\mathcal{T}$-invariant if and only if $c_1 =c_2$. Therefore
\[
V(\bx):=c(V_1(\bx)+V_2(\bx))
\]
is an admissible potential as in Definition \ref{def:definition-of-lattice} for any nonzero real number $c$.
\qed }
\end{exa}
The role of the $\mathcal{R}$- and $\mathcal{T}$-invariance of admissible potentials $V(\bx)$ can be stated as the following commutativity with the Schr\"odinger operator $H$ of \x{HV} we are going to study.
\begin{lem}\lb{prot}
$(1)$ Transformations $\mathcal{R}$ and $\mathcal{T}$ are isometric, i.e.,
\begin{equation*}
\langle\mathcal{R}f(\bx),\mathcal{R}g(\bx)\rangle=\langle f(\bx), g(\bx)\rangle,\andq
\langle\mathcal{T}f(\bx),\mathcal{T}g(\bx)\rangle=\langle f(\bx), g(\bx)\rangle
\end{equation*}
for all $f(\bx),\ g(\bx)\in L^{2}_{\bK,\Lambda}$.
$(2)$ The commutators $[H,\mathcal{R}]:=H \mathcal{R}-\mathcal{R}H$ and $[H,\mathcal{T}]:=H \mathcal{T}-\mathcal{T}H$ vanish on $H^{2}_{\bK,\Lambda}$.
\end{lem}
The proofs are direct.
\subsection{Periodic Schr\"odinger operators and Floquet-Bloch theory} \lb{FBT1}
Let $\La$ be the lattice defined in \x{la} and $V: \R^3\to \R$ be an admissible potential in the sense of Definition $\ref{def:definition-of-lattice}$.
For each quasi-momentum $\bk\in\R^{3}$, we consider the Floquet-Bloch eigenvalue problem
\begin{equation}\lb{HV-k}
\bb{split}
H\Phi(\bx,\bk) &=\mu(\bk)\Phi({\bx,\bk}),\q \bx\in\R^{3},\\
\Phi(\bx+\bv,\bk)& =e^{i\bk \cdot\bv}\Phi(\bx,\bk),\q \bx\in\R^{3},\ \bv\in\La,
\end{split}
\end{equation}
where $\mu(\bk)$ is the eigenvalue and the second condition is the pseudo-periodic condition for $\Phi(\bx,\bk)$. By setting
\[
\Phi(\bx,\bk)=e^{i\bk\cdot \bx}\phi(\bx,\bk), \q\mbox{or} \q \phi(\bx,\bk)=e^{-i\bk\cdot \bx}\Phi(\bx,\bk),
\]
we know that problem \x{HV-k} is converted into the following periodic eigenvalue problem
\begin{equation}\lb{HV-k2}
\begin{split}
H(\bk)\phi(\bx,\bk) &=\mu(\bk)\phi(\bx,\bk),\q \bx\in\R^{3},\\
\phi(\bx+\bv,\bk) & =\phi(\bx,\bk),\q \bx\in\R^{3},\ \bv\in\La.
\end{split}
\end{equation}
Here the shifted Schr\"odinger operator $H(\bk)$ is defined via
\begin{equation*}\lb{HVk}
\begin{split}
\nabla_\bk \phi(\bx) := & e^{-i \bk\d \bx} \nabla\z(e^{i \bk\d \bx} \phi(\bx)\y)=\nabla \phi(\bx) +i \bk \phi(\bx) =\z(\nabla +i \bk \y) \phi(\bx),\\
H(\bk)\phi(\bx) := & e^{-i \bk\d \bx} \Delta\z(e^{i \bk\d \bx} \phi(\bx)\y) +V(\bx) \phi(\bx)\\
= & -(\nabla+i\bk)\cdot(\nabla+i\bk)\phi(\bx)+V(\bx)\phi(\bx) \\
\equiv & -\nabla_\bk\d \nabla_\bk \phi(\bx) +V(\bx)\phi(\bx)\ .
\end{split}
\end{equation*}
The general properties of the Schr\"odinger operator with a periodic potential is given by the Floquet-Bloch theory. We end this section by listing some most important conclusions of this theory without including their proofs. We refer readers to \cite{Eastham1973,fefferman2012honeycomb,Kuchment2015,Kuchment2000,method1972Reed} for details.
\begin{prop} \rm{(Floquet-Block theory)} \lb{FT}
$(1)$ For any $\bk\in \Omega^{*}$, the Floquet-Bloch eigenvalue problem \x{HV-k2} has an ordered discrete spectrum
\begin{equation*}
\mu_{1}(\bk)\leq\mu_{2}(\bk)\leq\mu_{3}(\bk)\leq\ldots
\end{equation*}
such that $\mu_{b}(\bk)\rightarrow+\infty$ as $b\rightarrow+\infty$. Furthermore, there exist eigenpairs $\{\phi_{b}(\bx,\bk),$ $\mu_{b}(\bk)\}_{b\in \N}$ for each $\bk\in\Omega^{*}$ such that $\z\{ \phi_{b}(\bx,\bk)\y\}_{b\geq1}$ can be taken to be a complete orthonormal basis of $L^{2}_\per.$
Accordingly, problem \x{HV-k} has eigenpairs $\z\{\Phi_{b}(\bx,\bk),\ \mu_{b}(\bk)\y\}_{b\in \N}$, where
\[
\z\{ \Phi_{b}(\bx,\bk):= e^{i \bk \d \bx}\phi_{b}(\bx,\bk)\y\}_{b\in \N}
\]
is a complete orthonormal basis of $L^{2}_\Kla.$
$(2)$ The eigenvalues $\mu_{b}(\bk)$, referred as dispersion bands, are Lipschitz continuous functions of $\bk\in \Omega^{*}$.
$(3)$ For each $b\geq1$, $\mu_b(\bk)$ sweeps out a closed real interval $I_{b}$ over $\bk\in\Omega^{*}$, and the union of $I_b$ composes of the spectrum of $H$ in
$L^{2}_\per$:
\[
{\rm spec}(H)=\bigcup_{b\geq1,\bk\in\Omega^{*}}I_{b}, \qq \mbox{where } I_b=\bigl[\min_{\bk\in\Omega^{*}}\mu_{b}(\bk), \max_{\bk\in\Omega^{*}}\mu_{b}(\bk)\bigr]\ .
\]
$(4)$ Given $ \bk\in\Omega^{*}$, $\Phi_{b}(\bx,\bk)$ is smooth in $\bx\in\Omega$. Moreover, the set of eigenfunctions $\bigcup_{b\geq1, \bk\in\Omega^{*}} {\Phi_{b}(\bx,\bk)}$ is a complete orthonormal set of $L^{2}(\mathbb{R}^3)$. Consequently, any $f(\bx)\in L^{2}(\mathbb{R}^3)$ can be written in the summation form
\be\lb{fx1}
f(\bx)=\frac{1}{|\Omega^{*}|}\sum_{b\geq1}\int_{\Omega^{*}}\wi{f_b}(\bk)\Phi(\bx,\bk)\mathrm{d}\bk,
\ee
where
\[
\wi{f_b}(\bk)=\inn{\Phi_b(\bx,\bk)}{f(\bx)}=\int_{R^{3}}\ol{\Phi_{b}(\bx,\bk)}f(\bx)\mathrm{d}\bx.
\]
Here the summation \x{fx1} is convergent in the $L^2$-norm.
\end{prop}
\section{Weyl points and conical intersections}
\label{construction}
In this section, we are going to prove the existence of Weyl points on the energy bands of Schr\"odinger operators with admissible potentials that we propose in Definition \ref{def:definition-of-lattice}. The strategy used in this work is inspired by the framework that Fefferman and Weinstein developed for Dirac points in 2-D honeycomb structures \cite{fefferman2012honeycomb}. More specifically, (1) we first propose required conditions of eigen structure at $\bK$ for some eigenvalue $\mu_{*}$, i.e., the conditions \textbf{H1-H2} below; (2) we then prove the energy bands in the vicinity form a conical structure with an extra band in the middle under these conditions; (3) we justify that the required conditions \textbf{H1-H2} do hold for nontrivial shallow admissible potentials; (4) we extend the justification of required conditions to generic admissible potentials.
Compared to the study on Dirac points for the 2-D honeycomb case, the main difficulties of our current work arise from two perspectives: higher dimension and higher multiplicity. To the best of our knowledge, we have not found rigorous analysis on such degenerate points in the literature. Higher dimension makes the calculations more cumbersome. On the other hand, the higher multiplicity forces us to deal with a larger bifurcation matrix which has more freedoms which we need to reduce, for instance, the relations among the entries of the matrix. Some new symmetry arguments are introduced to conquer these difficulties.
\subsection{Spectrum structure at the high symmetry point $\bK$} \label{s41}
In this section, we are interested in the three-fold degeneracy of the high symmetry point $\bK$. So let us consider the $\bK$-quasi periodic eigenvalue problem
\begin{equation}\lb{mapro}
\begin{split}
H\Phi(\bx,\bK)&\equiv [-\Delta+V(\bx)]\Phi(\bx,\bK)=\mu_{*}\Phi(\bx,\bK),\q \bx\in\R^{3},\\
\Phi({\bf x+v,\bK})&=e^{i\bK\cdot \bv}\Phi({\bf x,\bK}),\q \bx\in\R^{3}, \bv\in \La.\\
\end{split}
\end{equation}
We first assume that there exists an eigenvalue $\mu_*$ such that the following assumption is fulfilled.
\begin{asm1}\lb{h1}
$\mu_{*}$ is a three-fold eigenvalue of $H$ in problem \x{mapro} with the corresponding eigenspace $\mathcal{E}_{\mu_*}$ such that
\[
\mathcal{E}_{\mu_*} \perp L^{2}_{\bK,1},\q \mbox{and} \q \dim\{\mathcal{E}_{\mu_*}\cap L^{2}_{\bK,i}\}=1.
\]
\end{asm1}
Then the following proposition characterizes the fine structure of the eigenspace $\mathcal{E}_{\mu_*}$.
\begin{prop}\lb{asphi}
Assume that $\mbox{\rm{\textbf{H1}}}$ holds. Then there exist functions $\Phi_{\ell}(\bx) \in L^{2}_{\bK,i^{\ell}},$ $ ~j=1,2,3$ such that $\{\Phi_1(\bx),\ \Phi_2(\bx),\ \Phi_3(\bx)=\overline{\Phi_1(-\bx)}\}$ form an orthonormal basis of $\mathcal{E}_{\mu_*}$.
\end{prop}
A direct consequence of above proposition is that $\mu_{*}$ is an $L^{2}_{\bK,i^{\ell}}$-eigenvalue of multiplicity $1$ for each $\ell=1,2,3$.
In order to keep the structure of the paper, the detailed proof of Proposition \ref{asphi} is placed in Appendix A.
\subsection{Bifurcation matrices} \lb{bif}
Under the assumption $\textbf{H1}$, we always can find an orthonormal basis $\{\Phi_1(\bx),\Phi_2(\bx),$ $\Phi_3(\bx)\}$ for $\mathcal{E}_{\mu_*}$ as in Proposition \ref{asphi}. However, the choice is not unique and a gauge freedom for each eigenfunction $\Phi_{\ell}(\bx)$ is allowed.
Giving such a basis, let us define a complex-valued matrix $M(\ka)$ for $\ka \in \mathbb{R}^3/\{0\}$ by
\[
M(\ka):=
\begin{pmatrix}
\langle\Phi_1,2i\ka\cdot\nabla\Phi_1 \rangle&\langle\Phi_1,2i\ka\cdot\nabla\Phi_2 \rangle&\langle\Phi_1,2i\ka\cdot\nabla\Phi_3 \rangle\\
\langle\Phi_2,2i\ka\cdot\nabla\Phi_1 \rangle&\langle\Phi_2,2i\ka\cdot\nabla\Phi_2 \rangle&\langle\Phi_2,2i\ka\cdot\nabla\Phi_3 \rangle\\
\langle\Phi_3,2i\ka\cdot\nabla\Phi_1 \rangle&\langle\Phi_3,2i\ka\cdot\nabla\Phi_2 \rangle&\langle\Phi_3,2i\ka\cdot\nabla\Phi_3 \rangle\\
\end{pmatrix}.
\]
It is called the bifurcation matrix which appears naturally in the eigenvalue problem. We shall see in the later section that the leading order structure of the eigenvalues of $H(\bk)$ for $\bk$ in the vicinity of $\bK$ is closely related to $M(\ka)$. In this subsection, the main properties of $M(\ka)$ and their justifications are provided.
We want to remark that $M(\ka)$ depends on the choice of the basis set $\{\Phi_1(\bx),\Phi_2(\bx),\Phi_3(\bx)\}$ due to the gauge freedom. It is evident that $M(\ka)$ is Hermitian since $2i\ka\cdot \nabla$ is self-adjoint.
We consider the admissible potential $V(\bx)$ in the sense of Definition \ref{def:definition-of-lattice}. Recall that $[H, \mathcal{T}]=0$ can imply $\mathcal{T}\mathcal{E}_{\mu_*}=\mathcal{E}_{\mu_*}$. In other words, there exists a $3\times 3$ matrix $Q_\mathcal{T}$ such that
\
\begin{pmatrix}
\mathcal{T}\Phi_1\\
\mathcal{T}\Phi_2\\
\mathcal{T}\Phi_3\\
\end{pmatrix}
=Q_{\mathcal{T}}\begin{pmatrix}
\Phi_1\\
\Phi_2\\
\Phi_3\\
\end{pmatrix}
=
\begin{pmatrix}
c_{11}&c_{12}&c_{13}\\
c_{21}&c_{22}&c_{23}\\
c_{31}&c_{32}&c_{33}\\
\end{pmatrix}
\begin{pmatrix}
\Phi_1\\
\Phi_2\\
\Phi_3\\
\end{pmatrix}.
\]
Recall from Lemma \ref{prot} that $\mathcal{T}: L^2_{\bK,\Lambda}\to L^2_{\bK,\Lambda}$ preserves the inner product, i.e., $\langle\mathcal{T}F,\mathcal{T}G \rangle=\langle F,G \rangle$ for all $f,g\in L^2_{\bK,\Lambda}$. It immediately follows that $Q_{\mathcal{T}}$ is unitary, i.e., $Q_{\mathcal{T}}^{*}Q_{\mathcal{T}}=I$. In other words, $\{\mathcal{T}\Phi_1, \mathcal{T}\Phi_2, \mathcal{T}\Phi_3\}$ is also an orthonormal basis of $\mathcal{E}_{\mu_*}$ which defines a new bifurcation matrix $M^{\mathcal{T}}(\ka)$. Namely,
\[
M^{\mathcal{T}}(\ka)\equiv \begin{pmatrix}
\langle\mathcal{T}\Phi_1,2i\ka\cdot\nabla\mathcal{T}\Phi_1 \rangle&\langle\mathcal{T}\Phi_1,2i\ka\cdot\nabla\mathcal{T}\Phi_2 \rangle&\langle\mathcal{T}\Phi_1,2i\ka\cdot\nabla\mathcal{T}\Phi_3 \rangle\\
\langle\mathcal{T}\Phi_2,2i\ka\cdot\nabla\mathcal{T}\Phi_1 \rangle&\langle\mathcal{T}\Phi_2,2i\ka\cdot\nabla\mathcal{T}\Phi_2 \rangle&\langle\mathcal{T}\Phi_2,2i\ka\cdot\nabla\mathcal{T}\Phi_3 \rangle\\
\langle\mathcal{T}\Phi_3,2i\ka\cdot\nabla\mathcal{T}\Phi_1 \rangle&\langle\mathcal{T}\Phi_3,2i\ka\cdot\nabla\mathcal{T}\Phi_2 \rangle&\langle\mathcal{T}\Phi_3,2i\ka\cdot\nabla\mathcal{T}\Phi_3 \rangle\\
\end{pmatrix}.
\]
Similarly, by using the symmetry $\mathcal{R}$, we can define another bifurcation matrix $M^{\mathcal{R}}(\ka)$ and the corresponding unitary transformation $Q_{\mathcal{T}}$. In fact, it is easy to obtain
\be \lb{QR}
Q_{\mathcal{R}}=\begin{pmatrix}
i&0&0\\
0&-1&0\\
0&0&-i\\
\end{pmatrix}.
\ee
However, the explicit form for $Q_{\mathcal{T}}$ is unknown to us.
One has the following relations for these bifurcation matrices.
\begin{prop}\lb{indu}
For any $\ka\in \mathbb{R}^3/\{0\}$, there hold
\begin{align}
M(\ka)=M^{\mathcal{R}}(R\ka)=Q_{\mathcal{R}}^{*}M(R\ka)Q_{\mathcal{R}}, \lb{tmar}\\
M(\ka)=M^{\mathcal{T}}(T\ka)=Q_{\mathcal{T}}^{*}M(T\ka)Q_{\mathcal{T}}, \lb{tmat}
\end{align}
where $R$ and $T$ are the orthogonal matrices in \x{matrix R} and \x{mat}.
\end{prop}
\Proof
We only give the proof to \x{tmat}, while the proof of \x{tmar} is similar.
By Lemma 2 in $\cite{Lee-Thorp2017}$, one has for $\ell,m=1,2,3$,
\[
\langle\Phi_{\ell},\nabla\Phi_{m}\rangle=\langle\mathcal{T}\Phi_\ell,\mathcal{T}\nabla\Phi_m \rangle=\langle \mathcal{T}\Phi_\ell,T^{*}\nabla\mathcal{T}\Phi_m\rangle.
\]
Therefore
\begin{equation}\lb{mi}
\begin{split}
(M(\ka))_{\ell m}&=\langle\Phi_{\ell},2i\ka\cdot \nabla\Phi_m\rangle= \langle\mathcal{T}\Phi_\ell,2i\ka\cdot T^{*}\nabla \mathcal{T}\Phi_m \rangle
=\langle\mathcal{T}\Phi_\ell,2iT\ka\cdot\nabla\mathcal{T}\Phi_m\rangle\\&= \sum\limits^{3}_{\ell=1}\sum\limits^{3}_{m=1}\ol{c_{i\ell}}c_{jm}\langle\Phi_{\ell},2i(T\ka)\cdot\nabla\Phi_{m}\rangle=(M^{\mathcal{T}}(T\ka))_{\ell m}.
\end{split}
\end{equation}
By recalling that $Q_{\mathcal{T}}=(c_{ij})$, we know that \x{mi} is equality \x{tmat}.
\qed
By substituting \x{QR} into \x{tmar}, we obtain
\be\lb{eqi}
\begin{split}
M^{\mathcal{R}}&(R\ka)=Q^{*}_{\mathcal{R}}
\begin{pmatrix}
\langle\Phi_1,2iR\ka\cdot\nabla\Phi_1\rangle&\langle\Phi_1,2iR\ka\cdot\nabla\Phi_2\rangle&\langle\Phi_1,2iR\ka\cdot\nabla\Phi_3\rangle\\
\langle\Phi_2,2iR\ka\cdot\nabla\Phi_1\rangle&\langle\Phi_2,2iR\ka\cdot\nabla\Phi_2\rangle&\langle\Phi_2,2iR\ka\cdot\nabla\Phi_3\rangle\\
\langle\Phi_3,2iR\ka\cdot\nabla\Phi_1\rangle&\langle\Phi_3,2iR\ka\cdot\nabla\Phi_2\rangle&\langle\Phi_3,2iR\ka\cdot\nabla\Phi_3\rangle\\
\end{pmatrix}
Q_{\mathcal{R}}\\
&=\begin{pmatrix}
\langle\Phi_1,2i\ka\cdot R^{*}\nabla\Phi_1\rangle&i\langle\Phi_1,2i\ka\cdot R^{*}\nabla\Phi_2\rangle&-\langle\Phi_1,2i\ka\cdot R^{*}\nabla\Phi_3\rangle\\
-i\langle\Phi_2,2i\ka\cdot R^{*}\nabla\Phi_1\rangle&\langle\Phi_2,2i\ka\cdot R^{*}\nabla\Phi_2\rangle&i\langle\Phi_2,2i\ka\cdot R^{*}\nabla\Phi_3\rangle\\
-\langle\Phi_3,2i\ka\cdot R^{*}\nabla\Phi_1\rangle&-i\langle\Phi_3,2i\ka\cdot R^{*}\nabla\Phi_2\rangle&\langle\Phi_3,2i\ka\cdot R^{*}\nabla\Phi_3\rangle\\
\end{pmatrix}.\\
\end{split}
\ee
Recall the transformation $R:\C^3 \to \C^3$ has eigenpairs listed in \x{R-ev}. We can then obtain the following structural result for the bifurcation matrix $M(\ka)$.
\bb{thm} \lb{phaseK}
There exist $\ups_1, \ups_2, \ups_3 \in \mathbb{C}$ such that
\be \lb{Mka}
M(\ka)=\begin{pmatrix}
0&\ka\cdot\upsilon_1\om_1&\ka\cdot\ol{\ups_{3}}\om_2\\
\ka\cdot\ups_{2}\om_1&0&\ka\cdot\ups_{2}\om_1\\
\ka\cdot\ups_{3}\om_2&\ka\cdot\ol{\ups_{2}}\om_3&0
\end{pmatrix},
\ee
where $\om_{j},\ j=1,2,3$ are eigenvectors of $R$ listed in \x{R-ev}. Moreover, there have
\be \lb{ups1}
|\upsilon_1|=|\upsilon_2|=|\upsilon_3|,
\ee
\be \lb{ups2}
\upsilon_1 \upsilon_2 \upsilon_3+\ol{\upsilon_1 \upsilon_2 \upsilon_3}=0.
\ee
\end{thm}
\Proof The proof is split into several steps.
1. Entries of $M(\ka)$. Note $(M(\ka))_{\ell j}=(M^{R}(R\ka))_{\ell j}$ $=(Q^{*}_{\mathcal{R}}M(R\ka)Q_{\mathcal{R}})_{\ell j}$ holds for $\ell,j=1,2,3$. By comparing the elements in $M^{\mathcal{R}}(\ka)$ displayed in \x{eqi} with $M(\ka)$, it is easily seen that for $\ka\in\mathbb{R}^3$, one has
\[
i^{j-\ell}\langle\Phi_{\ell},2i\ka\cdot R^{*} \nabla\Phi_{j}\rangle= \langle\Phi_{\ell},2i\ka\cdot\nabla\Phi_{j}\rangle \Longrightarrow
\ka\cdot i^{j-\ell}\langle\Phi_{\ell},2i R^{*}\nabla\Phi_{j}\rangle=\ka\cdot \langle\Phi_{\ell},2i\ka\cdot\nabla\Phi_{j}\rangle.
\]
Since $\ka\in \R^3$ is arbitrary, we claim that
\be\lb{Phi14}
R\langle\Phi_{\ell},2i \nabla\Phi_{j}\rangle=i^{j-\ell} \langle\Phi_{\ell},2i\nabla\Phi_{j}\rangle.
\ee
Equalities in \x{Phi14} have shown that, for each pair $(\ell,j)$, $\langle\Phi_{\ell},2i \nabla\Phi_{j}\rangle$ is either the zero vector or an eigenvector of $R$ associated with the eigenvalue $i^{j-\ell}$.
If $\ell=j\in \{1,2,3\}$, we know that $i^{j-\ell}=1$ is not an eigenvalue of $R$ and therefore
\[
\langle\Phi_{\ell},2i \nabla\Phi_{\ell}\rangle=0 \q \mbox{for } \ell=1,2,3.
\]
On the other hand, the other six equalities of \x{Phi14} imply that there exist constants $\ups_\ell,\ \tl \ups_\ell\in \C$ such that
\be \lb{ups4}
\z\{ \ba{l}
\langle\Phi_{1}, 2i\nabla\Phi_{2}\rangle= \ups_1 \om_1, \qq \langle\Phi_{2}, 2i\nabla\Phi_{1}\rangle= \tl\ups_1 \om_3, \\
\langle\Phi_{2}, 2i\nabla\Phi_{3}\rangle= \ups_2 \om_1, \qq \langle\Phi_{3}, 2i\nabla\Phi_{2}\rangle= \tl\ups_2 \om_3, \\
\langle\Phi_{3}, 2i\nabla\Phi_{1}\rangle= \ups_3 \om_2, \qq \langle\Phi_{1}, 2i\nabla\Phi_{3}\rangle= \tl\ups_3 \om_2.
\ea \y.
\ee
Since $M(\ka)=(M(\ka))^{*}$ and $\ol{\om_3}=\om_1$, we have necessarily
\(
\tl \ups_\ell=\ol{\ups_\ell}
\)
for $\ell=1,2,3.$
2. Proof of $|\upsilon_1|=|\upsilon_2|$.
According to the definition of $\Phi_2(\bx) \in L^{2}_{\bK,-1}$, we have
\[
\mathcal{R}[\Phi_2](\bx)=\Phi_2(R^{*}\bx)=-\Phi_2(\bx).
\]
Thus
\[
\bb{split}
&\Phi_2(R^{*}(-\bx))=-\Phi_2(-\bx),\\
&\ol{\Phi_2(R^{*}(-\bx))}=-\ol{\Phi_2(-\bx)},\\
& \mathcal{R}[\ol{\Phi_2(-\bx)}]=-\ol{\Phi_2(-\bx)}.
\end{split}
\]
The last equality means that $\ol{\Phi_2(-\bx)}\in L^{2}_{\bK,-1}$. Since $\dim(\mathcal{E}_{\mu_{*}}\bigcap L^{2}_{\bK,-1})=1$ by $\textbf{H1}$ and $\ol{\Phi_2(-\bx)}$ is also $L^{2}$-normalized, therefore
\[
\Phi_2(\bx)\equiv e^{i\theta}\ol{\Phi_2(-\bx)}\q\mbox{for some }\th\in \R.
\]
From this, we deduce that $\nabla\Phi_{2}(\bx) \equiv -e^{i\theta}\ol{\nabla \Phi_2(-\bx)}$ and
\begin{align*}
\langle\Phi_{1}, 2i\nabla\Phi_{2}\rangle&=\ol{\int {\Phi_1(\bx)} \d 2i e^{-i\theta}\nabla\Phi_2(-\bx) d\bx}\\
&= \ol{\int {\Phi_1(-\bx)} \d 2i e^{-i\theta}\nabla\Phi_2(\bx) d\bx}\\
&= e^{i\theta}\ol{\inn{\Phi_{3}}{2i\nabla\Phi_{2}}} \qq \mbox{(by changing $\bx$ to $-\bx$)}\\
&= e^{i\theta}\inn{\Phi_{2}}{2i\nabla\Phi_{3}}.
\end{align*}
From the definition of $\ups_\ell$ in \x{ups4}, we obtain $\upsilon_1 \om_1=e^{i\theta} \upsilon_2 \om_2$ and $|\upsilon_1|=|\upsilon_2|$.
3. Proof of \x{ups1} and \x{ups2}.
The proof of $|\upsilon_2|=|\upsilon_3|$ is different. For any $\ka\in \R^3$, we consider the characteristic polynomial of the bifurcation matrix $M(\ka)$
\[
p(a,\ka):=\det \z(aI+M(\ka)\y)=
\det \begin{pmatrix}
a&\langle\Phi_1,2i\ka\cdot\nabla\Phi_2 \rangle&\langle\Phi_1,2i\ka\cdot\nabla\Phi_3 \rangle\\
\langle\Phi_2,2i\ka\cdot\nabla\Phi_1 \rangle&a&\langle\Phi_2,2i\ka\cdot\nabla\Phi_3 \rangle\\
\langle\Phi_3,2i\ka\cdot\nabla\Phi_1 \rangle&\langle\Phi_3,2i\ka\cdot\nabla\Phi_2 \rangle&a\\
\end{pmatrix}.
\]
It is a cubic polynomial of $a$ with coefficients depending on $\ka$. Since $Q_{\mathcal{T}}$ is unitary, it follows from \x{tmat} that
\[
\begin{split}
p(a,\ka)&=\det \z(a I+ Q_{\mathcal{T}}^{*}M(T\ka)Q_{\mathcal{T}}\y) =\det\z( Q_{\mathcal{T}}^{*}\z(a I+M(T\ka)\y)Q_{\mathcal{T}}\y) \\ &=\det \z( a I+M(T\ka)\y).
\end{split}
\]
Thus $p(a,\ka)$ satisfies the following invariance
\be\lb{con1}
p(a,\ka)\equiv p(a,T\ka).
\ee
\ifl
By Lemma $\ref{indu}$, $\det M(a,\ka)=0$ imply $\det C^{*}(aI+M(T\ka))C=0$, since $C$ is invertible, we claim
\[
\det M(a,\ka)=0 \Longleftrightarrow \det (aI+M(T\ka))=0.
\]
Therefore, for fixed $\ka\in\mathbb{R}^3$, the three roots of $p(a,\ka)=\det M(a,\ka)$ solve $p(a,\ka)=\det (aI+M(T\ka))=0$, since $p(a,\ka)$ is also a cubic equation with highest order term $a^3$, we have
\fi
In particular, by taking $\ka=e_2:=(0,1,0)$ in \x{con1}, one has from \x{Mka} that
\be \lb{p1}
p(a,e_2)\equiv a^3-|\upsilon_2|^{2} a.
\ee
Similarly, one has $T e_2= (0,0,1)=e_3$ and by using \x{Mka} again, we have
\be \lb{p2}
p(a,T e_2)\equiv a^3-|\upsilon_3|^{2} a+\frac{1}{2}(\upsilon_{1} \upsilon_{2} \upsilon_{3}+\ol{\upsilon_{1} \upsilon_{2} \upsilon_{3}}).
\ee
By comparing the coefficients of \x{p1} and \x{p2}, we deduce from the invariance \x{con1} that there hold $|\upsilon_2|=|\upsilon_3|$ and equality \x{ups2}. Together with equality $|\upsilon_1|=|\upsilon_2|$ in the above step, we have obtained all equalities in \x{ups1} and \x{ups2}. \qed
\ifl
Therefore, the coefficients of each term of polynomial $p(a,\ka)$ are identical with $p(a,T\ka)$. Set $\ka_1=(0,1,0)$, then
$T\ka_1=(0,0,1)$, by substituting $\ka_1$ and $T\ka_1$ into \x{eq:equality_threelambda}, we obtain
\begin{align}
&p(a,\ka_1)=a^3-|\upsilon_2|^{2} a=0,\\
&p(a,T\ka_1)=a^3-|\upsilon_3|^{2} a=\frac{1}{2}(\upsilon_{1} \upsilon_{2} \upsilon_{3}+\ol{\upsilon_{1} \upsilon_{2} \upsilon_{3}}).
\end{align}
By comparing the coefficients between $p(a,\ka)$ and $p(a,T\ka)$, we proceed to derive
\begin{align}
&|\upsilon_1|=|\upsilon_2|=|\upsilon_3|=a,\\
&\upsilon_{1} \upsilon_{2} \upsilon_{3}+\ol{\upsilon_{1} \upsilon_{2} \upsilon_{3}}=0. \lb{argc}
\end{align}
\x{argc} indicates $\arg(\upsilon_{1} \upsilon_{2} \upsilon_{3})=\frac{\pi}{2}+k\pi\ k=0,1$.
\fi
We have also the following gauge invariance for $\upsilon_1 \upsilon_2 \upsilon_3$ and $|\upsilon_\ell|$.
\begin{cor}\lb{inv-g}
$(1)$ The quantity $\upsilon_1 \upsilon_2 \upsilon_3$ is gauge invariant in the sense that it does not depend on the choice of the orthonormal basis of $\mathcal{E}_{\mu_{*}}$.
$(2)$ The quantity $|\upsilon_1|=|\upsilon_2|=|\upsilon_3|$ is also gauge invariant.
\end{cor}
\Proof
Let $\z\{\Phi_\ell(\bx): \ell=1,2,3\y\}$ and $\z\{ \hat \Phi_\ell(\bx): \ell=1,2,3\y\}$ be two sets of orthonormal eigenfunctions as in Proposition \ref{asphi}. Then there exist $\tau_\ell\in \R$ such that \(
\tau_3 = -\tau_1,
\)
and
\[
\hat \Phi_\ell(\bx)= e^{i\tau_{\ell}}\Phi_{\ell}(\bx), \q \ell=1,2,3.
\]
By direct calculations, one has
\[
\begin{split}
&\hat\ups_1\om_1 =\langle\hat \Phi_1(\bx),2i\nabla\hat \Phi_2(\bx)\rangle =e^{-i\tau_1+i\tau_2}\langle\Phi_{1}(\bx),2i\nabla\Phi_2(\bx)\rangle,\\
&\hat\ups_2\om_1=\langle\hat \Phi_2(\bx),2i\nabla\hat \Phi_3(\bx)\rangle =e^{-i\tau_2-i\tau_1}\langle\Phi_{2}(\bx),2i\nabla\Phi_3(\bx)\rangle,\\
&\hat\ups_3\om_2=\langle\hat \Phi_3(\bx),2i\nabla\hat \Phi_1(\bx)\rangle =e^{i\tau_1+i\tau_1}\langle\Phi_{3}(\bx),2i\nabla\Phi_1(\bx)\rangle.
\end{split}
\]
Therefore
$$\hat\ups_1=e^{-i\tau_1+i\tau_2}\upsilon_1, \q
\hat\ups_2=e^{-i\tau_2-i\tau_1}\upsilon_2, \q
\hat\ups_3=e^{2i\tau_1}\upsilon_3.
$$
These yield the invariance
\be \lb{vvs}
\hat \ups_1 \hat \ups_2 \hat \ups_3 = \ups_1 \ups_2 \ups_3.
\ee
For (2), by taking the norms in \x{vvs} and using equalities \x{ups1}, we obtain
\[
|\hat \upsilon_\ell|^3=|\upsilon_\ell|^3.
\]
This leads to the desired invariance of $|\upsilon_\ell|$.
\qed
Due to the equalities in Theorem \ref{phaseK} and the invariance in Corollary \ref{inv-g}, let us define
\be \lb{upsf}
\upsilon_{_{\mathcal{F}}} :=|\upsilon_\ell|\in[0,+\oo),\ \ell=1,2,3.
\ee
The quantity $\upsilon_{_{\mathcal{F}}}$ of \x{upsf} is referred to as {\it the Fermi velocity} in quantum mechanics.
Now we introduce another standing assumption in this paper, which can be simply stated as
\begin{asm2}\lb{h2}
$\upsilon_{_{\mathcal{F}}}\neq 0$.
\end{asm2}
\subsection{Conical structure of the spectrum near $\bK$}\lb{cs1}
With the eigenstructure at $\bK$, we are able to obtain the corresponding eigenstructure when quasi-momentum $\bk$ is near $\bK$. The results are stated as follows.
\begin{thm}\label{thm:3+1}
Suppose that $V(\bx)$ is an admissible potential in the sense of Definition $\ref{def:definition-of-lattice}$ and consider the Schr\"odinger operator $H= -\Delta+V(\bx)$. Assume that there exists $b>1$ such that $\mu_{b-1}=\mu_{b}=\mu_{b+1}=\mu_*$ is an $L^2_{\bK,\Lambda}$-eigenvalue of $H$ and the assumptions \rm{\textbf{H1-H2}} are fulfilled.
Then, for sufficiently small but nonzero $(\ka_x,\ka_y,\ka_z) \in \R^3$, eigenvalues of $H$ satisfy
\be\lb{mu1}
\begin{split}
\mu_{b+1}(\bK+\ka)&=\mu_{*}+\xi_{+}\upsilon_{_{\mathcal{F}}}|\ka|+ o(|\ka|),\\
\mu_{b}(\bK+\ka)&=\mu_{*}+\xi_{0}\upsilon_{_{\mathcal{F}}}|\ka|+ o(|\ka|),\\
\mu_{b-1}(\bK+\ka)&=\mu_{*}+\xi_{-}\upsilon_{_{\mathcal{F}}}|\ka|+ o(|\ka|),
\end{split}
\ee
where $\upsilon_{_{\mathcal{F}}}$ is the Fermi velocity defined before, and $\xi_{+}\geq\xi_0\geq\xi_{-}$ are the three (real) roots of the following cubic equation
\be \lb{cubic}
\xi^3-\xi+2\ka^{\arg}=0,\qq \ka^{\arg}:=\frac{\ka_x \ka_y \ka_z}{|\ka|^{3}}.
\ee
\end{thm}
\Proof The proof is based on the Lyapunov-Schmidt reduction. Thanks to the eigenstructure at $\bK$ and the explicit form of the bifurcation matrix which we established in last section, we now only need to do a perturbation expansion and a rigorous justification. Compared to the 2-D honeycomb case \cite{fefferman2012honeycomb}, we encounter more complicated computations on the bifurcation. We complete it in several steps.
1. Decomposition of spaces. For $\bk=\bK$, we have
\[
\phi_{\ell}(\bx)=e^{-i\bK\cdot\bx}\Phi_{\ell}(\bx,\bK) \in L^2_{\bz, i^\ell}\subset L^2_\per,\qq \ell=1,2,3,
\]
such that
\[
H(\bK) \phi_\ell =\mu^{(0)} \phi_\ell,\qq \ell=1,2,3,
\]
where $\mu^{(0)}:= \mu_{*}$. These define a space
\[
\mcx=\mcx_\bK := {\rm span} \{\phi_1,\phi_2,\phi_3\}
\]
Consider perturbation $\bk=\bK+\ka$, where $\ka\in \R^3$ is small enough. From the defining equalities in \x{HVk}, one has
\[
H(\bK+\ka)
= H(\bK) -2i\ka\cdot(\nabla+i\bK)+\ka\cdot\ka
=H(\bK) -2i\ka\cdot \nabla_\bK+\ka\cdot\ka.
\]
To study eigenvalue problem \x{HV-k2}, let us decompose
\[
\psi(\bx,\bK+\ka)= \psi^{(0)}(\bx) + \psi^{(1)}(\bx), \qq \psi^{(0)} \in \mcx, \ \psi^{(1)}\in \mcx^\perp,
\]
and write
\[
\mu(\bK+\ka) = \mu^{(0)}+\mu^{(1)}, \qq \mu^{(1)}\in \R.
\]
Here the orthogonal complement $\mcx^\perp$ is taken from $L^2_\per$. Then
\[
H(\bK+\ka) \psi(\bx,\bK+\ka)=\mu(\bK+\ka)\psi(\bx,\bK+\ka)
\]
can be expanded as
\begin{equation}\label{eq:3+1problem-system}
\begin{split}
&\z(H(\bK)-\mu^{(0)}I\y) \psi^{(1)}= F^{(1)}= F^{(1)}(\ka,\mu^{(1)}, \psi^{(0)}, \psi^{(1)}) \\
:=& \z(2i\ka\cdot\nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y) \psi^{(1)}
+\z(2i\ka\cdot\nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y) \psi^{(0)}.
\end{split}
\end{equation}
2. Splitting of the equation using the Lyapunov-Schmidt strategy. To solve Eq. \x{eq:3+1problem-system} using such a strategy, let us introduce the orthogonal projections
\[
\mathscr{Q}_{\parallel}: H^2(\R^3/\La) \to \mcx= {\rm span} \{\phi_1,\phi_2,\phi_3\}\ \mbox{and}\
\mathscr{Q}_{\perp}:= I-\mathscr{Q}_{\parallel}: H^2(\R^3/\La) \to \mcx^\perp.
\]
Applying $\mathscr{Q}_{\parallel}$ and $\mathscr{Q}_{\perp}$ to Eq. \x{eq:3+1problem-system}, we obtain an equivalent system
\begin{align}
(H(\bK)-\mu^{(0)}I)\psi^{(1)}&= Q_{\perp} F^{(1)}(\ka,\mu^{(1)}, \psi^{(0)}, \psi^{(1)}),\lb{S1}\\
0 &= Q_{\parallel}F^{(1)} (\ka,\mu^{(1)}, \psi^{(0)}, \psi^{(1)}), \lb{S2}
\end{align}
because
\begin{equation}\label{relation:Q(perp-para)}
\mathscr{Q}_{\parallel}\psi^{(0)}=\psi^{(0)},\qq \mathscr{Q}_{\perp}\psi^{(1)}=\psi^{(1)}\andq \mathscr{Q}_{\parallel}\psi^{(1)}=\mathscr{Q}_{\perp}\psi^{(0)}=0.
\end{equation}
By using \x{eq:3+1problem-system} for $F^{(1)}$, we have
\begin{align}
\mathscr{Q}_{\perp}F^{(1)}
&=\mathscr{Q}_\perp\z(2i\ka \cdot \nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y)\psi^{(1)}+ \mathscr{Q}_\perp\z(2i\ka\d \nabla_\bK\y)\psi^{(0)},\lb{F1}\\
\mathscr{Q}_{\parallel} F^{(1)} &= \mathscr{Q}_{\parallel}\z(2i\ka\d\nabla_\bK\y)\psi^{(1)} + \mathscr{Q}_{\parallel}\z(2i\ka\d\nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y)\psi^{(0)}.\lb{F2}
\end{align}
\ifl
A solution to (\ref{eq:3+1problem-system}) could be obtained by solving the following system for $\psi^{(1)}$ and $\mu^{(1)}$
\begin{equation}
\begin{split}
(H(\bK)-\mu^{(0)}I)\psi^{(1)}&=\mathscr{Q}_{\perp}F^{(1)}(\alpha,\beta,\gamma,\ka,\mu^{(1)},\psi^{(1)}),\\
0&=\mathscr{Q}_{\parallel}F^{(1)}(\alpha,\beta,\gamma,\ka,\mu^{(1)},\psi^{(1)}).
\end{split}
\end{equation}
Specifically, that is
\begin{equation}\label{eq:(H(K)-mu0)I-psi1}
\begin{split}
(H(\bK)-\mu^{(0)}I)\psi^{(1)}=&\mathscr{Q}_\perp(2i\ka \cdot \nabla_\bK-\ka\cdot\ka+\mu^{(1)})\psi^{(1)}+\\&\mathscr{Q}_\perp(2i\ka\nabla_\bK)\psi^{(0)},
\end{split}
\end{equation}
\begin{equation}
\mathscr{Q}_{\parallel}(2i\ka\nabla_\bK-\ka\cdot\ka+\mu^{(1)})\psi^{(0)}+\mathscr{Q}_{\parallel}(2i\ka\nabla_\bK)\psi^{(1)}=0.
\end{equation}
\fi
By the assumptions of the theorem on eigenfunctions of $H(\bK)$, one knows that, when restricted to $\mcx^\perp$, $H(\bK)-\mu^{(0)}I$ has a bounded inverse
\begin{equation*}
\mathscr{E}=\mathscr{E}(\bK,\mu^{(0)}) =(H(\bK)-\mu^{(0)}I)^{-1}: \mcx^\perp \to \mathscr{Q}_\perp H^{2}(\R^{2}/\La).
\end{equation*}
By (\ref{relation:Q(perp-para)}) and \x{F1}-\x{F2}, equation (\ref{S1}) is equivalent to
\[
\psi^{(1)} =\mathscr{E} \mathscr{Q}_\perp\z(2i\ka \cdot \nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y)\psi^{(1)}= \mathscr{E} \mathscr{Q}_\perp\z(2i\ka\d \nabla_\bK\y)\psi^{(0)},
\]
i.e.
\be \lb{Eq11}
\z(I - \mathscr{E} \mathscr{Q}_\perp\z(2i\ka \cdot \nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y)\y)\psi^{(1)}= \mathscr{E} \mathscr{Q}_\perp\z(2i\ka\d \nabla_\bK\y)\psi^{(0)}.
\ee
Due to the regularity, the mapping
\[
f\mapsto \mathscr{T} \mathscr{Q}_\perp\z(2i\ka \cdot \nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y) f
\]
is a bounded operator defined on $H^{s}(\R^{2}/\La)$ for any $s$.
In the following we assume that $|\ka|+|\mu^{(1)}|$ is sufficiently small. Then the left-hand side of \x{Eq11} is invertible. Given any $\psi^{(0)}\in \mcx$, Eq. \x{Eq11} has then the unique solution in $\mathscr{Q}_{\perp}H^{2}(\R^{2}/\La):$
\begin{equation}\lb{psi11}
\psi^{(1)}=\mcp_0 \psi^{(0)}:=\z(I - \mathscr{T} \mathscr{Q}_\perp\z(2i\ka \cdot \nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y)\y)^{-1}\mathscr{T} \mathscr{Q}_\perp\z(2i\ka\d \nabla_\bK\y)\psi^{(0)}.
\end{equation}
Here $\mcp_0=\mcp_0(\mu^{(1)},\ka): \mcx \to \mathscr{Q}_{\perp}H^{2}(\R^{2}/\La)$ is a bounded linear operator. Substituting \x{psi11} into equation (\ref{S2}) and making use of \x{F2}, we obtain an equation for the unknowns $(\mu^{(1)},\psi^{(0)})$
\be\lb{A3}
\mcm(\mu^{(1)},\ka)\psi^{(0)}+\wi{\mcm}(\mu^{(1)},\ka)\psi^{(0)}=0,
\ee
where $\mcm(\mu^{(1)},\ka),\ \wi{\mcm}(\mu^{(1)},\ka): \mcx \to \mathscr{Q}_{\perp}H^{2}(\R^{2}/\La)$ are
\begin{align*}
\mcm(\mu^{(1)},\ka)&:= \mathscr{Q}_{\parallel}\z(2i\ka\d\nabla_\bK\y)\mcp_0(\mu^{(1)},\ka), \\
\wi{\mcm}(\mu^{(1)},\ka)&:= \mathscr{Q}_{\parallel}\z(2i\ka\d\nabla_\bK-\ka\cdot\ka+\mu^{(1)}\y).
\end{align*}
Note that \x{A3} is a linear system mapping from $\psi^{(0)}\in \mcx$ to $\mcx$, with an unknown parameter $\mu^{(1)}\in \R$.
Since $\mcx$ is 3-dimensional, we can write $\psi^{(0)}$ in
\be \lb{psi0}
\psi^{(0)}= \sum_{\ell=1}^3 \al_\ell \phi_\ell,\qq \al_\ell\in \C.
\ee
In order that \x{A3} has a nonzero solution $\psi^{(0)}\in \mcx$, it is necessary and sufficient that the corrections $\mu^{(1)}=\mu(\bK+\ka) -\mu(\bK)$ for eigenvalues are determined by
\begin{equation}\lb{Mmk}
\det E(\mu^{(1)},\ka)=0,
\end{equation}
where $E(\mu^{(1)},\ka)$ is the $3\tm 3$ representation of left-hand side of \x{A3}, using the coordinates for $\psi^{(0)}$ as in \x{psi0}. Precisely,
\be\lb{M01}
E(\mu^{(1)},\ka) \equiv M(\mu^{(1)},\ka)+\wi{M}(\mu^{(1)},\ka),
\ee
where
\[
\begin{split}
&M(\mu^{(1)},\ka)=\z(M_{m,\ell j}(\mu^{(1)},\ka)\y)_{3\tm 3} := \z(\inn{\phi_\ell} {\mcm(\mu^{(1)},\ka)\phi_j}\y)_{3\tm 3},\\
&\wi{M}(\mu^{(1)},\ka)=\z(M_{m,\ell j}(\mu^{(1)},\ka)\y)_{3\tm 3} := \z(\inn{\phi_\ell} {\wi{\mcm}(\mu^{(1)},\ka)\phi_j}\y)_{3\tm 3}.
\end{split}
\]
3. Explicit computation for nondegeneracy condition. We need to give a more explicit computation for equation \x{Mmk}.
To this end, by using \x{psi0} for $\psi^{(0)}$, we have from \x{psi11}
\begin{equation}\label{eq:expression-psi}
\psi^{(1)}(\bx)=\psi^{(1)}(\bx,\ka,\mu^{(1)}) \equiv \sum_{\ell=1}^3\alpha_\ell c^{(\ell)}(\bx,\ka,\mu^{(1)}),
\end{equation}
where $c^{(\ell)}(\bx,\ka,\mu^{(1)})$, $\ell=1,2,3$, are bounded by
\begin{equation}\lb{cl}
\z\|c^{(\ell)}(\d,\ka,\mu^{(1)})\y\|_{H^{2}}\leq C(|\ka|+|\mu^{(1)}|)\q \mbox{for } |\ka|+|\mu^{(1)}|\ll 1.
\end{equation}
Let us define
\[
C^{(\ell)}(\bx)=C^{(\ell)}(\bx,\ka,\mu^{(1)}):= e^{i\bK\cdot\bx}c^{(\ell)}(\bx,\ka,\mu^{(1)}).
\]
Recall that
\[
\Phi_\ell(\bx)=e^{i \bK \d \bx} \phi_\ell(\bx),\q \nabla_{\bK}\phi_{j}(\bx)
=e^{-i\bK\cdot\bx}\nabla\Phi_{j}(\bx),\q \inn{\Phi_{\ell}}{\Phi_{j}}= \inn{\phi_{\ell}}{\phi_{j}}= \delta_{\ell j}.
\]
Moreover, as $\mathscr{Q}_{\parallel}\psi^{(1)}=0$, we have from \x{eq:expression-psi} that $\mathscr{Q}_{\parallel}c^{(\ell)}=0$, i.e., $c^{(\ell)}\in \mcx^\perp$. Thus
\[
\inn{\Phi_{\ell}}{C^{(j)}}=\inn{\phi_\ell}{c^{(j)}}=0.
\]
The matrix-valued functions $M(\mu^{(1)},\ka)$ and $\wi{M}(\mu^{(1)},\ka)$ in \x{M01} are
\
\begin{split}
&M(\mu^{(1)},\ka)=\\
&\begin{pmatrix}
\mu^{(1)}+\langle \Phi_{1},2i\ka\d\nabla\Phi_{1}\rangle
&\langle \Phi_{1},2i\ka\d\nabla\Phi_{2}\rangle
&\langle \Phi_{1},2i\ka\d\nabla\Phi_{3}\rangle\\
\langle \Phi_{2},2i\ka\d\nabla\Phi_{1}\rangle
&\mu^{(1)}+\langle\Phi_{2},2i\ka\d\nabla\Phi_{2}\rangle
&\langle \Phi_{2},2i\ka\d\nabla\Phi_{3}\rangle\\
\langle \Phi_{3},2i\ka\d\nabla\Phi_{1}\rangle
&\langle \Phi_{3},2i\ka\d\nabla\Phi_{2}\rangle
& \mu^{(1)}+\langle\Phi_{3},2i\ka\d\nabla\Phi_{3}\rangle\\
\end{pmatrix}\\
&=\mu^{(1)}I+M(\ka),
\end{split}
\]
\[
\begin{split}
&\wi{M}(\mu^{(1)},\ka)=\\
&\begin{pmatrix}
\ka\cdot\ka+\langle \Phi_{1},2i\ka\cdot\nabla C^{(1)}\rangle
&\langle \Phi_{1},2i\ka\cdot\nabla C^{(2)}\rangle
&\langle \Phi_{1},2i\ka\cdot\nabla C^{(3)}\rangle\\
\langle \Phi_{2},2i\ka\cdot\nabla C^{(1)}\rangle
&\ka\cdot\ka+\langle \Phi_{2},2i\ka\cdot\nabla C^{(2)}\rangle
&\langle \Phi_{2},2i\ka\cdot\nabla C^{(3)}\rangle\\
\langle \Phi_{3},2i\ka\cdot\nabla C^{(1)}\rangle
&\langle \Phi_{3},2i\ka\cdot\nabla C^{(2)}\rangle
&\ka\cdot\ka+\langle \Phi_{3},2i\ka\cdot\nabla C^{(3)}\rangle
\end{pmatrix}.
\end{split}
\]
By noticing \x{cl}, we know that
\[
\wi{M}(\mu^{(1)},\ka)_{_{\ell j}}=\mathcal{O}(|\ka|\cdot|\mu^{(1)}|+|\ka|^{2}).
\]
4. Bifurcation of eigenvalues. By the results in Theorem \ref{phaseK}, $M(\mu^{(1)},\ka)$ simplifies to
\begin{align*}
\mu^{(1)}I+M(\ka)&= \begin{pmatrix}
\mu^{(1)}&\langle \Phi_{1},2i\ka\d\nabla\Phi_{2}\rangle&\langle \Phi_{1},2i\ka\d\nabla\Phi_{3}\rangle\\
\langle \Phi_{2},2i\ka\d\nabla\Phi_{1}\rangle&\mu^{(1)}&\langle \Phi_{2},2i\ka\d\nabla\Phi_{3}\rangle\\
\langle \Phi_{3},2i\ka\d\nabla\Phi_{1}\rangle&\langle \Phi_{3},2i\ka\d\nabla\Phi_{2}\rangle&\mu^{(1)}\\
\end{pmatrix}\\
&=
\begin{pmatrix}
\mu^{(1)}&\upsilon_{1}(\ka_{x}-i\ka_{y})&\ol{\upsilon_{3}}(\ka_{z})\\
\ol{\upsilon_{1}}(\ka_{x}+i\ka_{y})&\mu^{(1)}&\upsilon_{2}(\ka_{x}-i\ka_{y})\\
\upsilon_{3}(\ka_{z})&\ol{\upsilon_{2}}(\ka_{x}+i\ka_{y})&\mu^{(1)}\\
\end{pmatrix}.
\end{align*}
Thus the bifurcation equation \x{Mmk} is
\begin{equation}\label{eq:equality_threelambda}
\z(\mu^{(1)}\y)^{3}-\mu^{(1)}\z[|\upsilon_{3}|^{2}\ka_{z}^{2}+\frac{|\upsilon_{1}|^{2}+ |\upsilon_{2}|^{2}}{2} (\ka_{x}^{2}+\ka_{y}^{2})\y] -h(\ka)-g(\mu^{(1)},\ka)=0,
\end{equation}
where
\[
\begin{split}
&h(\ka)= \frac{1}{2}\upsilon_{1} \upsilon_{2} \upsilon_{3}(\ka^{2}_{x}-2i\ka_{x} \ka_{y}-\ka^{2}_{y}) \ka_{z}+\ol{\upsilon_{1}\upsilon_{2} \upsilon_{3}}(\ka^{2}_{x}+2i\ka_{x}\ka_{y}-\ka^{2}_{y})\ka_{z},\\
&g(\mu^{(1)},\ka)=\mathcal{O}(|\ka|^{\alpha}|\mu^{(1)}|^{\beta}),\q \alpha+\beta\geq4.
\end{split}
\]
The definitions and properties of $\upsilon_i$ are displayed in Theorem $\ref{phaseK}$. Note
that $\upsilon_1 \upsilon_2 \upsilon_3$ is purely imaginary, thus we may set $\arg(\upsilon_1 \upsilon_2 \upsilon_3)=\frac{3\pi}{2}$, because the case $\arg(\upsilon_1 \upsilon_2 \upsilon_3)=\frac{\pi}{2}$ is similar. Hence (\ref{eq:equality_threelambda}) simplifies to
\begin{equation}\lb{pka}
(\mu^{(1)})^3-\mu^{(1)}\upsilon^{2}_{_{\mathcal{F}}}|\ka|^2=-2\upsilon^{3}_{_{\mathcal{F}}}\ka_x \ka_y \ka_z+g(\mu^{(1)},\ka).
\end{equation}
We then follow the arguments as done for Proposition 4.2 in \cite{fefferman2012honeycomb}. Setting $\mu^{(1)}=\xi \ups_{\mathcal{F}}|\ka|+o(|\ka|)$ and substituting into \x{pka}, we observe that $\xi$ solves the cubic equation \x{cubic}.
\ifl \begin{equation}\label{eq:eq-of-eig}
\xi^{3}+2\ka^{\arg}-\xi=0,
\end{equation}
where $\ka^{\arg}=\frac{\ka_x \ka_y \ka_z}{|\ka|^3}$.
\fi
By the Cauchy-Schwarz inequality, one has $|\ka^{\arg}|\leq\frac{\sqrt{3}}{9}$. We therefore conclude that equation \x{cubic} has precisely three real solutions $\xi_{+}\geq\xi_{0}\geq\xi_{-}$ by using the discriminant of cubic equations. Moreover, $\xi_{+}+ \xi_{0}+ \xi_{-}=0$. Actually, the Floquet-Bloch eigenvalue problem has three dispersion hypersurfaces
\[
\begin{split}
\mu_{b+1}&=\mu_{*}+\xi_{+}\upsilon_{_{\mathcal{F}}}|\ka|+o(|\ka|),\q\\
\mu_{b}&=\mu_{*}+\xi_{0}\upsilon_{_{\mathcal{F}}}|\ka|+o(|\ka|), \q\\
\mu_{b-1}&=\mu_{*}+\xi_{-}\upsilon_{_{\mathcal{F}}}|\ka|+o(|\ka|).\\
\end{split}
\]
Consequently, we have the desired results \x{mu1} and the proof of the theorem is complete.
\qed
From the Theorem \ref{thm:3+1}, we see that the three bands intersect at the degenerate point $(\bK, \mu_*)$.
Note that the roots $\xi_*=\xi_*(\ka/|\ka|)$ of equation \x{cubic} depend only on the directions of $\ka$, not on the sizes $|\ka|$ of the quasi-momenta $\ka$.
We want to point out that there is a special direction along which two energy bands adhere to each other to leading order. Specifically, if $\ka \in \mathbf{n_*}\mathbb{R}_+$ with $\mathbf{n_*}=\z(\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}\y)$, the solutions of (\ref{cubic}) take the form
\[
\xi_{+}=\xi_{0}=\frac{\sqrt{3}}{3},\qq \xi_{-}=-\frac{2\sqrt{3}}{3}.
\]
The result indicates that the three-fold degeneracy splits into a two-fold eigenvalue and a simple eigenvalue in the vicinity of the Weyl point $\bK$. We remark here that it is not clear whether the double degeneracy persists by including higher order terms of $|\ka|$. This is an interesting problem but is beyond the scope of the current work.
At the end of this section, we characterize the lower dimensional structure of the three energy bands near the Weyl point $\bK$. According to the expressions of dispersion bands $\mu(\ka)$ in \x{mu1}, we study a special case of dispersion equation \x{cubic} as follows. If $\ka^{\arg}=0$, or equivalently, either of $\ka_x,\ \ka_y,\ \ka_z$ vanishes, the bifurcation equation (\ref{cubic}) has solutions
\[
\xi_{+}=1,\qq\xi_{0}=0,\qq \xi_{-}=-1.\qq
\]
In the transverse plane which is perpendicular to one axis direction, the three dispersion surfaces form a standard cone with a flat band in the middle, see Figure 2 in Section 7. This is exactly the band structure of the Lieb lattice in the tight binding limit \cite{Keller2018,Mukherjee14}. To the best of our knowledge, this structure has not been rigorously proved. We demonstrate its existence for our potentials in lower reduced planes.
Generally speaking, in the reduced plane, the three dispersion bands do not behave the same as the above case. Note that $(-\ka)^{\arg} =-\ka^{\arg}$. Let us fix a direction $\textbf{n}$. Then
\[
\xi^{\textbf{n}}_{+}=-\xi^{-\textbf{n}}_{-},\qq \xi^{\textbf{n}}_{0}=-\xi^{-\textbf{n}}_{0},\qq
\xi^{\textbf{n}}_{-}=-\xi^{-\textbf{n}}_{+},
\]
where the superscripts indicate the different choices of bifurcation equations depending on the directions $\textbf{n}$ or $-\textbf{n}$. We can actually construct three analytical branches of dispersion curves and each branch is a straight line to leading order. In fact, let us define
\
\begin{split}
E_{1}(\lambda)&=\mu_{b+1}(\bK+\lambda\textbf{n})=\mu_{*}+\xi^{\textbf{n}}_{+}\lambda\upsilon_{_{\mathcal{F}}}+o(|\lambda|), \\
E_{2}(\lambda)&=\mu_{b}(\bK+\lambda\textbf{n})=\mu_{*}+\xi^{\textbf{n}}_{0}\lambda\upsilon_{_{\mathcal{F}}}+o(|\lambda|),\\
E_{3}(\lambda)&=\mu_{b-1}(\bK+\lambda\textbf{n})=\mu_{*}+\xi^{\textbf{n}}_{-}\lambda\upsilon_{_{\mathcal{F}}}+o(|\lambda|).\\
\end{split}
\]
Then for a fixed direction $\textbf{n}$, the three branches $E_j(\lambda),~ j=1,2,3$ are analytical in $\lambda$.
Next we allow $\textbf{n}$ to vary in a transverse plane. Namely, let $\textbf{n}_1$ and $\textbf{n}_2$ be two orthonormal vectors and consider the dispersion surfaces in the plane spanned by $\textbf{n}_1$ and $\textbf{n}_2$. Then
\[
\mu(\bK+\lambda_1 \textbf{n}_1 +\lambda_2\textbf{n}_2)=\mu_{*}+\lambda\ups_{_\mathcal{F}}\xi^{\hat{\lambda}}_{i}+o(|\lambda|),
\]
where $|\lambda|$ denotes the length of $(\lambda_1,\lambda_2)$. Note, while $\lambda$ is fixed, $\ka^{\arg}$ is a continuous variable with respect to $\frac{\lambda_1}{\lambda}$, thus $\xi^{\hat{\lambda}}_{i}$ depends on $\frac{\lambda_1}{\lambda}$ continuously. Consequently, \x{mu1} exactly admits a cone (may not be standard and isotropic) adhered by an extra surface in the middle (see Section 7 for related figures).
\section{Justification of Assumptions $\textbf{H1}$ and $\textbf{H2}$}
\label{justif}
Theorem \ref{thm:3+1} states that as long as H1-H2 hold, the Schr\"{o}dinger operator with an admissible potential always admits a 3-fold Weyl point at the high symmetry point $\bK$. In this section, we shall justify the two assumptions \textbf{H1-H2} can actually hold generally. We first examine shallow potentials in which case we can treat the small potential as a perturbation to the Laplacian operator. Then we can conduct the perturbation theory. The main difficulty is to prove the 3-fold degeneracy persists at any order of the asymptotic expansion. We remark that in the 2-D honeycomb case \cite{fefferman2012honeycomb}, the 2-fold degeneracy is naturally protected by the inversion symmetry. But that is not enough for higher multiplicity. What are the required arguments on the 3-fold degeneracy? We will answer this question in our analysis by imposing novel symmetry arguments.
\subsection{Weyl points in shallow potential case}
We first consider the Floquet-Bloch eigenvalue problem for the operator $H^{\e}=-\Delta+\e V(\bx)$, where $\e$ is possibly small and $V(\bx)$ is a nonzero admissible potential. Without loss of generality, we consider the case that $\e$ is positive. Then the $\bK$-pseudo-periodic eigenvalue problem on the four eigenspaces of $L^{2}_{\bK,i^{\ell}},\ \ell\in\{1,2,3,4\}$ takes the form
\begin{equation}\label{eq:potential-system}
\begin{split}
H^{\e}\Phi_{\ell}(\bx,\bK)&\equiv [-\Delta+\e V(\bx)]\Phi^{\e}_{\ell}(\bx,\bK)=\mu^{\e}(\bK)\Phi^{\e}_{\ell}(\bx,\bK),\q\bx\in\R^{3} ,\\
\Phi^{\e}_{\ell}({\bf x+v,K})&=e^{i\bK\cdot \bv}\Phi^{\e}_{\ell}({\bf x,K}),\q\bx\in\R^{3},\ \bv \in \La,\\
\mcr[\Phi^{\e}_{\ell}(\bx,\bK)]&=i^{\ell}\Phi^{\e}_{\ell}(\bx,\bK), \q \ell\in\{1,2,3,4\}.\\
\end{split}
\end{equation}
We first study the special case that $\e=0$.
Note that $R$ is orthogonal and
\[
|\bK|=|R\bK|=|R^{2}\bK|=|R^{3}\bK|=\frac{3}{4} q^2.
\]
By letting $\mu^{(0)}=|\bK|^2=\frac{3}{4} q^2$, we know that $e^{iR^\ell \bK\cdot\bx}$ are eigenfunctions associated with $\mu^{(0)}$. Thus $\mu^{(0)}$ is an eigenvalue of $H^{0}$ of multiplicity at least 4. To show that the multiplicity of $\mu^{(0)}$ is exactly 4, for $\bm=(m_{1},m_{2},m_{3})\in \Z^3$ and $\bq=m_{1}\bq_{1} +m_{2}\bq_{2}+ m_{3}\bq_{3}\in \Las$, the equation
\[
|\bK +\bq|^2 = |\bK|^2
\]
will lead to
\[
[(2m_{1}+2m_{2}-1)^{2}+(2m_{1}+2m_{3}-1)^{2}+(2m_{2}+2m_{3}-1)^{2}]q^{2}=3q^{2}.
\]
Since $m_{1},\ m_{2},\ m_{3}$ are integers, it is
\[
(2m_{1}+2m_{2}-1)^{2}=(2m_{1}+2m_{3}-1)^{2}=(2m_{2}+2m_{3}-1)^{2}=1,
\]
with the precisely 4 solutions
\[
\bm=(0,0,0),\q (1,0,0),\q(0,1,0),\q(0,0,1).
\]
For these $\bm$, $\bK+\bq$ correspond to $R^\ell\bK=\bK+\bq_\ell$, $\ell=1,2,3,4$, cf. \x{RK}.
Summarizing the above calculations, we have
\bb{prop}\lb{multi}
The Laplacian $H^0\equiv-\Delta$ admits a real four-fold eigenvalue $\mu^{(0)}=|\bK|^2=\frac{3}{4} q^2$ at $\bK$, with the eigenspace spanned by $\z\{e^{iR^{\ell}\bK\cdot\bx}:\ell=1,2,3,4\y\}$.
\end{prop}
Notice from \x{RK} that $R^\nu \bK = \bK+ \bq_\nu$. Let us take the following eigenfunctions associated with $\mu^{(0)}$
\be \lb{Phi-l0}
\begin{split}
\Phi_{\ell}^{0}(\bx)&= \Phi_\ell^{0}(\bx,\bK):=\frac{1}{\sqrt{4|\Omega|}}\z(e^{i\bK\cdot\bx}+\ol{i^\ell}e^{iR\bK\cdot\bx}+ i^{2\ell} e^{iR^{2}\bK\cdot\bx} +i^\ell e^{iR^{3}\bK\cdot\bx}\y)\nn\\
&= \frac{1}{\sqrt{4|\Omega|}}\sum_{\nu=0}^3 i^{-\ell \nu} e^{i R^\nu \bK\cdot\bx} \in L^{2}_{\bK,i^\ell}=
\frac{1}{\sqrt{4|\Omega|}}\sum_{\nu=0}^3 i^{-\ell \nu} e^{i(\bK+ \bq_\nu)\cdot\bx},
\end{split}
\ee
where $\ell=1,2,3,4$, cf. \x{decomp2}. It is easily seen that
\[
\inn{\Phi_\ell^{0}}{\Phi_j^{0}} = \da_{\ell j}, \qq \ell,\, j=1,2,3,4.
\]
Based on the results in Proposition \ref{multi}, we can justify Assumptions $\textbf{H1}$ and $\textbf{H2}$ when $\e>0$ is sufficiently small.
\begin{thm}\label{thm:dispersion-3+1}
Let $V(\bx)$ be an admissible potential. Suppose that the Fourier coefficient $V_{1,0,0}>0$. Then there exists a constat $\varepsilon_0>0$ such that for any $\e\in (0,\e_0)$, $H^\e=-\Delta+\e V(\bx)$ fulfills the assumptions {\rm\textbf{H1}} and {\rm\textbf{H2}}. Moreover, one has
\begin{align}
\mu_{*}=\mu^{\e}_{\ell}&= |\bK|^{2}+\e(V_{0,0,0}-V_{1,0,0})+\mathcal{O}(\e^{2}),\q \ell=1,2,3, \lb{muss}\\
\z|\ups^{\e}_{_\mathcal{F}}\y|&= q+\mathcal{O}(|\varepsilon|)>0.\lb{la-i1}
\end{align}
Hence the lowest three energy bands intersect at the three-fold Weyl point $(\bK,\mu_{*})$.
\end{thm}
\begin{rem}
The requirement $V_{1,0,0}>0$ in Theorem $\ref{thm:dispersion-3+1}$ can be replaced by $V_{1,0,0}<0$. In the latter case, one has the second, third and fourth bands intersect at the Weyl point $(\bK,\mu_{*})$.
\end{rem}
The proof of Theorem \ref{thm:dispersion-3+1} is inspired by the methods in \cite{Lee-Thorp2017}, where the 2-fold Dirac points in the $2$-D honeycomb structure is studied. The main difficulty in the present case is the justification of the three-fold degeneracy of the perturbed eigenvalue $\mu_{*}$ at $\bK$. Recall that the two-fold degeneracy is protected by the $\mathfrak{PT}$-symmetry of $V(\bx)$ in the $2$-D honeycomb case. The potential in our work also possesses the $\mathfrak{PT}$-symmetry so that a two-fold eigenvalue $\mu_{*}$ at $\bK$ is guaranteed. However, this is not adequate to admit the three-fold degeneracy of $\mu_{*}$. In fact, we need to combine $\mathcal{T}$-symmetry to ensure that another eigenvalue is the same as $\mu_{*}$ at $\bK$. This is the main difference compared to the analysis of the previous work. In the following proof, we only list the key calculations and point out the new ingredients.
\bigskip
We begin to prove Theorem \ref{thm:dispersion-3+1}.
1. Recall that $\mu^{(0)}=|\bK|^{2}$ is the eigenvalue of the Laplacian $-\Delta$ of multiplicity $4$. Moreover, $\mu^{(0)}$ is also a simple $L^{2}_{\bK,i^{\ell}}$-eigenvalue for $\ell\in\{ 1,2,3,4\}$, with the corresponding eigenstates $\Phi^{0}_{\ell}$. Let us decompose $\Phi^{\e}_{\ell}(\bx,\bK)\in L^{2}_{\bK,i^{\ell}}$ as
\[
\Phi^{\e}_{\ell}(\bx,\bK) =\Phi^{(0)}_{\ell}(\bx,\bK) +\e\Phi^{(1)}_{\ell}(\bx,\bK).
\]
Similar to \cite{xie2020wave}, by applying Lyapunov-Schmidt reduction to (\ref{eq:potential-system}), we obtain the expression for $\mu^{\e}$ for sufficiently small $\e$
\begin{equation}\label{eq:expansion-eigenvalue}
\mu^{\e}=\mu_\ell^\e \equiv \mu^{(0)}+\e\langle\Phi^{(0)}_{\ell},V(\bx)\Phi^{(0)}_{\ell}\rangle+\mathcal{O}(\e^{2}), \quad \ell\in\{1,2,3,4\} .
\end{equation}
We now turn to the calculation of $\langle\Phi^{(0)}_{\ell},V(\bx)\Phi^{(0)}_{\ell}\rangle$. By using the $\mathcal{R}$-invariance of $V(\bx)$, it follows that
\begin{equation}\label{cv}
\begin{split}
V_{0,0,0}&=\langle e^{i\bK\cdot\by},V(\by)e^{i\bK\cdot\by}\rangle=\langle e^{iR\bK\cdot\by},V(\by)e^{iR\bK\cdot\by}\rangle\\
&=\langle e^{iR^{2}\bK\cdot\by},V(\by)e^{iR^{2}\bK\cdot\by}\rangle
=\langle e^{iR^{3}\bK\cdot\by},V(\by)e^{iR^{3}\bK\cdot\by}\rangle,\\
V_{1,0,0}&=\langle e^{i\bK\cdot\by},V(\by)e^{iR\bK\cdot\by}\rangle=\langle e^{iR\bK\cdot\by}, V(\by)e^{iR^{2}\bK\cdot\by}\rangle\\
& =\langle e^{iR^{2}\bK\cdot\by},V(\by)e^{iR^{3}\bK\cdot\by}\rangle=\langle e^{iR^{3}\bK\cdot\by},V(\by)e^{i\bK\cdot\by}\rangle,\\
V_{0,1,0}& =\langle e^{i\bK\cdot\by},V(\by)e^{iR^{2}\bK\cdot\by}\rangle=\langle e^{iR\bK\cdot\by},V(\by)e^{iR^{3}\bK\cdot\by}\rangle\\
& =\langle e^{iR^{2}\bK\cdot\by},V(\by)e^{i\bK\cdot\by}\rangle=\langle e^{iR^{3}\bK\cdot\by},V(\by)e^{iR\bK\cdot\by}\rangle,\\
V_{0,0,1} &=\langle e^{i\bK\cdot\by},V(\by)e^{iR^{3}\bK\cdot\by}\rangle=\langle e^{iR\bK\cdot\by},V(\by)e^{i\bK\cdot\by}\rangle\\
& =\langle e^{iR^{2}\bK\cdot\by},V(\by)e^{iR\bK\cdot\by}\rangle=\langle e^{iR^{3}\bK\cdot\by},V(\by)e^{iR^{2}\bK\cdot\by}\rangle,
\end{split}
\end{equation}
where
$$
V_{\alpha, \beta, \gamma}=\int_{\Om}e^{-i(\alpha\bq_{1}+\beta\bq_{2}+\gamma\bq_{3})}V(\by)d\by.
$$
By inserting the expansion $(\ref{Phi-l0})$ of $\Phi^{(0)}_{\ell}(\bx,\bK)$ and the coefficients (\ref{cv}) into (\ref{eq:expansion-eigenvalue}), and noticing that $V(\bx)$ is even, it follows that
\begin{equation}\lb{mue}
\mu^{\e}_\ell=
\begin{cases}
&\mu^{(0)}+\e(V_{0,0,0}-V_{1,0,-1})+\mathcal{O}(\e^{2}), \q \ell=1,3,\\
&\mu^{(0)}+\e(V_{0,0,0}+V_{1,0,-1}-2V_{1,0,0})+\mathcal{O}(\e^{2}), \q \ell=2,\\
&\mu^{(0)}+\e(V_{0,0,0}+V_{1,0,-1}+2V_{1,0,0})+\mathcal{O}(\e^{2}), \q \ell=4.
\end{cases}
\end{equation}
2.
Since $V(\bx)$ is $\mathcal{T}$-invariant, we have $V_{1,0,-1}=V_{1,0,0}>0$. In particular, \x{mue} is simplified to
\begin{equation}\lb{mue1}
\begin{split}
\mu^{\e}_1 =&\mu^{\e}_3=\mu^{(0)}+\e(V_{0,0,0}-V_{1,0,0})+\mathcal{O}(\e^{2}), \q \ell=1,3,\\
\mu^{\e}_2=&\mu^{(0)}+\e(V_{0,0,0}-V_{1,0,0})+\mathcal{O}(\e^{2}), \q \ell=2,\\
\mu^{\e}_4=&\mu^{(0)}+\e(V_{0,0,0}+3V_{1,0,0})+\mathcal{O}(\e^{2}), \q \ell=4.
\end{split}
\end{equation}
Here one shall notice that the $\mathcal{O}(\e^2)$ terms in $\mu^{\e}_{1,2}$ and $\mu^{\e}_{3}$ are undetermined. This means that we could not assert that $\mu^{\e}_{1,3}=\mu^{\e}_{2}$. However, it follows from \x{mue1} that these eigenvalues are ordered so that
\[
\mu^{\e}_{1}=\mu^{\e}_{3}\approx \mu^{\e}_{2}<\mu^{\e}_{4}\ .
\]
Let $\mathcal{E}_{\mu^{\e}_1}$ denote the eigenspace of $H^{\e}\Phi^{\e}=\mu^{\e}_{1}\Phi^{\e}$. Then the above analysis shows that
\be\lb{prope}
\mathcal{E}_{\mu^{\e}_{1}}\subset L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}\oplus L^{2}_{\bK,-1},\andq
2\leq\dim \mathcal{E}_{\mu^{\e}_1}\leq3\ .
\ee
The next step is to verify that $\mu^{\e}_{1}$ is really a three-fold eigenvalue, i.e., $\dim \mathcal{E}_{\mu^{\e}_{1}}=3$, with the help of the following lemma.
\begin{lem}\lb{strut}
We assert that
$\mathcal{T}\Phi^{\e}_{1}\notin L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$ for $\e$ is sufficiently small.
\end{lem}
The detailed proof of Lemma \ref{strut} is displayed in Appendix B.
We continue the proof for Theorem \ref{thm:dispersion-3+1}. Recall that $[H,\mathcal{T}]=0$. Thus
\[
\mathcal{T}(-\Delta+\e V(\bx))\Phi^{\e}_{1}=(-\Delta+\e V(\bx))\mathcal{T}\Phi^{\e}_{1}=\mu^{\e}_{1}\mathcal{T}\Phi^{\e}_{1}\ .
\]
Therefore
$\mathcal{T}\Phi^{\e}_{1}\in \mathcal{E}_{\mu^{\e}_{1}}$. By Lemma $\ref{strut}$, we deduce that $\mathcal{T}\Phi^{\e}_{1}\notin L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$.
Hence
$$
\{\Phi^{\e}_{1}(\bx),\ \ol{\Phi^{\e}_{1}(-\bx)},\ \mathcal{T}\Phi^{\e}_{1}(\bx)\}
$$
are linearly independent eigenfunctions in $\mathcal{E}_{\mu^{\e}_{1}}$. Thus $\dim \mathcal{E}_{\mu^{\e}_{1}}\geq3$. By \x{prope}, we conclude that $\dim \mathcal{E} _{\mu^{\e}_{1}}=3$ and $\mu^{\e}_{1}$ is a three-fold eigenvalue. Moreover, result \x{muss} follows from \x{mue1}.
3. We then embark on the proof of \x{la-i1}. In analogy with the construction of $\ups_{\ell}$ in Theorem \ref{phaseK}, we introduce $\ups^{(0)}_{\ell}$ by
\[
\ups^{(0)}_{1}\om_1=\langle\Phi^{(0)}_{1},2i\nabla\Phi^{(0)}_{2} \rangle \q \ups^{(0)}_{2}\om_1=\langle\Phi^{(0)}_{2},2i\nabla\Phi^{(0)}_{3}\rangle \q \ups^{(0)}_{3}\om_3=\langle\Phi^{(0)}_{3},2i\nabla\Phi^{(0)}_{1}\rangle\ .
\]
Actually in the following we will present the full calculations for each $\ups^{(0)}_{\ell}$ under the above choice of $\Phi^{(0)}_{\ell}$. By discussions given in \cite{fefferman2012honeycomb}, it is standard to apply the Lyapunov-Schmidt reduction to approximate $\upsilon_{\ell}$ while $0<\e\ll 1$. The result is
$$
\langle \Phi_{\ell},2i\nabla\Phi_j\rangle= \langle \Phi_{\ell}^{(0)},2i\nabla\Phi_j^{(0)} \rangle +\mathcal{O}(|\e|)\ .
$$
Here
\[
\Phi^{(0)}_{\ell}(\bx)=\frac{1}{\sqrt{4|\Omega|}}\sum_{\nu=0}^3 i^{-\ell \nu} e^{i(\bK+ \bq_\nu)\cdot\bx}, \qq \ell=1,2,3,4.
\]
Thus we can directly deduce that
\begin{equation*}
\begin{split}
\ol{\Phi_{\ell}^{(0)}(\bx)} &=\frac{1}{\sqrt{4|\Omega|}} \sum_{\nu=0}^3 (-i)^{-\ell \nu} e^{-i(\bK+ \bq_\nu)\cdot\bx},\\
\nabla \Phi_{j}^{(0)}(\bx) & = \frac{-2}{\sqrt{4|\Omega|}} \sum_{\mu=0}^3 i^{-j \mu} e^{i(\bK+ \bq_\mu)\cdot\bx}(\bK+ \bq_\mu),\\
\ol{\Phi_{\ell}^{(0)}(\bx)} 2i \nabla \Phi_{j}^{(0)}(\bx) & =\frac{-2}{4|\Omega|}\sum_{\nu=0}^3 \sum_{\mu=0}^3(-i)^{-\ell \nu} i^{-j \mu} e^{i(\bq_\mu-\bq_\nu)\cdot\bx}(\bK+ \bq_\mu)\ .
\end{split}
\end{equation*}
Therefore, by setting $d_{\ell j} :=(-i)^{-\ell} i^{-j}\equiv i^{\ell-j}$, one has
\begin{align*}
\inn{\Phi_{\ell}^{(0)}}{2i \nabla \Phi_{j}^{(0)}}&= -\f{1}{2} \sum_{\nu=0}^3 (d_{\ell j})^\nu(\bK+ \bq_\nu)\\
&= -\f{1}{2} \z(\sum_{\nu=0}^3 (d_{\ell j})^\nu\y)\bK -\f{1}{2} \sum_{\nu=1}^3 (d_{\ell j})^\nu \bq_\nu\\
&\equiv -\f{1}{2} \sum_{\nu=1}^3 (d_{\ell j})^\nu \bq_\nu,
\end{align*}
where $(\ell,j)=(1,2), \ (2,3), \ (3,1)$. Since
\(
d_{12}=d_{23}=-i
\)
and
\(
d_{31}=-1,
\)
one has
\be\lb{vis}
\begin{split}
&\inn{\Phi_{1}^{(0)}}{2i \nabla \Phi_{2}^{(0)}}= \inn{\Phi_{2}^{(0)}}{2i \nabla \Phi_{3}^{(0)}}= -\f{1}{2}\sum_{\nu=1}^3 (-i)^\nu \bq_\nu\\ &=\f{1}{2} \z(i\bq_1+\bq_2-i \bq_3 \y)=\frac{\sqrt{2}}{2}(1+i)q\om_1,\\
&\inn{\Phi_{3}^{(0)}}{2i \nabla \Phi_{1}^{(0)}}= -\f{1}{2} \sum_{\nu=1}^3 (-1)^\nu \bq_\nu=\f{1}{2} \z(\bq_1-\bq_2+\bq_3 \y)=-q\om_3\ .
\end{split}
\ee
From these we directly obtain $|\ups^{(0)}_{1}|=|\ups^{(0)}_{2}|=|\ups^{(0)}_{3}|=q$. This completes the proof of \x{la-i1}.
\qed
\begin{rem}
Note from \x{vis} that $\ups^{(0)}_{1}=\ups^{(0)}_2 =e^{i\frac{\pi}{4}}q$ and $\ups^{(0)}_{3}=-q$. Thus
\begin{equation}\lb{argup}
\frac{\ups_1 \ups_2 \ups_3}{|\ups_1 \ups_2 \ups_3|}=\frac{\ups^{(0)}_{1} \ups^{(0)}_2 \ups^{(0)}_{3}+\mathcal{O}(|\e|)}{|\ups_1 \ups_2 \ups_3|}=
\frac{-q^3+\mathcal{O}(|\e|)}{q^3+\mathcal{O}(|\e|)}=-1+\mathcal{O}(|\e|)\ .
\end{equation}
Recall that $\ups_1 \ups_2 \ups_3$ is gauge invariant and $\arg(\ups_1 \ups_2 \ups_3)$ is either $\frac{\pi}{2}$ or $\frac{3\pi}{2}$ by Theorem $\ref{phaseK}$. Thus we assert from \x{argup} that
\[
\arg(\ups_1 \ups_2 \ups_3)=\frac{3\pi}{2}
\]
when $\e$ is sufficiently small.
\end{rem}
\subsection{Remark on Weyl points in generic admissible potentials}\lb{generic}
Theorem \ref{thm:dispersion-3+1} studies the 3-fold Weyl points for the Schr\"odinger operator with shallow admissible potentials: $H^\e=-\Delta +\e V(\bx)$ for $\e\neq 0$ and small. In this subsection we make some remarks on the extension of these results to generic potentials, i.e., $\e=\mathcal{O}(1)$. Following the arguments established by Fefferman and Weinstein for the existence of Dirac points in 2-D honeycomb potentials, see \cite{fefferman2012honeycomb,Fefferman2014}, we claim that the assumptions {\bf H1} and {\bf H2} hold for some $(\bK, \mu_*)$ except for $\e$ in a discrete set $\mathcal{C}$ of $\R$.
Consequently, the conclusions of Theorem \ref{thm:dispersion-3+1} also hold, i.e., there always exists a 3-fold Weyl point, for the Schr\"odinger operator $H^\e=-\Delta +\e V(\bx)$ if $\e$ is not in the discrete set $\mathcal{C}$.
The main idea is based on an analytical characterization of $L^2_{\bK, \lambda}$-eigenvalue of $H^\e$. By a similar argument on the analytic operator theory and complex function theory strategy \cite{fefferman2012honeycomb,Fefferman2014}, it is possible to establish the analogous result.
Due to the length of this work, we omit the details and refer interesting readers to \cite{Fefferman2014,fefferman2012honeycomb}.
\section{Instability of the Weyl point under symmetry-breaking perturbations }\lb{sec6}
In the preceding sections, we have demonstrated that the admissible potentials generically admit a 3-fold Weyl point at $\bK$. The admissible potentials are characterized by the inversion symmetry, the $\mathcal{R}$-symmetry and the $\mathcal{T}$-symmetry. Actually we have seen the 3-fold degeneracy at $\bK$ and conical structure in its vicinity are consequence of combined actions of these symmetries. In this section, we shall discuss the instability of the 3-fold Weyl point $(\bK,\mu_*)$ if some symmetry is broken. More specifically, we only show the case where the inversion symmetry is broken which can be compared with the results to the 2-fold Dirac points in 2-D honeycomb case. The calculation of the case where $\mathcal{T}$-symmetry is broken is very cumbersome and we shall not give detailed discussion and only give numerical examples in Section 7.
Consider the perturbed eigenvalue problem
\begin{equation}\label{eq:stable}
(H+\delta V_{p}(\bx))\Psi^{\delta}(\bx,\bK)=\mu^{\delta}\Psi^{\delta}(\bx,\bK),
\end{equation}
where $V_{p}(\bx)$ is real and odd,
and $\delta$ is the perturbation parameter which is assumed to be small.
We expand $\mu^{\delta}$ and $\Psi^{\delta}(\bx)$ near the 3-fold Weyl point $(\bK, \mu_*)$ as
\[
\Psi^{\delta}(\bx)=\Psi^{(0)}(\bx)+\Psi^{(1)}(\bx), \andq \mu^{\delta}=\mu_{*}+\mu^{(1)},
\]
where $\Psi^{(0)}$ is the unperturbed eigenfunction corresponding to the the unperturbed eigenvalue $\mu_*$. We have stated in Theorem $\ref{thm:3+1}$ that
\[
\Psi^{(0)}(\bx)=\sum\limits^{3}_{i=1}\alpha_{\ell} \Phi_{\ell}(\bx)\ .
\]
Calculations analogous to those in the proof of Theorem \ref{thm:3+1} can lead to a system of homogeneous linear equations for $\alpha_1, \alpha_2,\alpha_3$
\[
(\mu^{(1)}I-\mathcal{M}_{1}-\mathcal{M}_{2})
\begin{pmatrix}
\alpha_1\\ \alpha_2\\ \alpha_3
\end{pmatrix}
=0,
\]
where
\[
\mathcal{M}_{1}=
\begin{pmatrix}
\langle \Phi_{1}(\bx) ,\delta V_{p}(\bx)\Phi_{1}(\bx) \rangle&\langle \Phi_{1}(\bx) ,\delta V_{p}(\bx)\Phi_{2}(\bx) \rangle&\langle \Phi_{1}(\bx) , \delta V_{p}(\bx)\Phi_{3}(\bx) \rangle\\
\langle \Phi_{2}(\bx) ,\delta V_{p}(\bx)\Phi_{1}(\bx) \rangle&\langle \Phi_{2}(\bx) ,\delta V_{p}(\bx)\Phi_{2}(\bx) \rangle&\langle \Phi_{2}(\bx) ,\delta V_{p}(\bx)\Phi_{3}(\bx) \rangle\\
\langle \Phi_{3}(\bx) ,\delta V_{p}(\bx)\Phi_{1}(\bx) \rangle&\langle \Phi_{3}(\bx) ,\delta V_{p}(\bx)\Phi_{2}(\bx) \rangle&\langle \Phi_{3}(\bx) ,\delta V_{p}(\bx)\Phi_{3}(\bx) \rangle\\
\end{pmatrix},
\]
and $\mathcal{M}_2$ includes higher order terms.
Therefore $\mu^{\delta}$ is the solution for the perturbed eigenvalue problem \x{eq:stable} if and only if $\mu^{(1)}$ solves
\be\lb{truc}
\det(\mu^{(1)}I-\mathcal{M}_{1}-\mathcal{M}_{2})=0\ .
\ee
Following a standard perturbation theory for Floquet-Bloch eigenvalue problems, we obtain that the solutions of \x{truc} satisfy
\[
\mu^{(1)}=\wi{\mu}+o(\delta),
\]
where $\wi{\mu}$ is the leading order effect of the perturbation which solves the equation
\begin{equation}\label{eq:stable-equation}
\det(\wi{\mu}I-\mathcal{M}_{1})
=0\ .
\end{equation}
To understand the problem, it is key to compute the explicit form of $\mathcal{M}_1$. Note that
\begin{equation}\lb{pvp1}
\begin{split}
&\langle \Phi_{1}(\bx) ,V_{p}(\bx)\Phi_{3}(\bx) \rangle=-\langle \Phi_{1}(-\by) ,V_{p}(\by)\Phi_{3}(-\by) \rangle\\
=& -\langle \ol{\Phi_{3}(\by)} ,V_{p}(\by)\ol{\Phi_{1}(\by)} \rangle=-\langle \Phi_{1}(\by)V_{p}(\by),\Phi_{3}(\by)\rangle,\\
&\langle \Phi_{2}(\bx) ,V_{p}(\bx)\Phi_{2}(\bx) \rangle=-\langle \Phi_{2}(-\by) ,V_{p}(\by)\Phi_{2}(-\by) \rangle\\
= &-\langle \ol{\Phi_{2}(\by)} ,V_{p}(\by)\ol{\Phi_{2}(\by)} \rangle=-\langle \Phi_{2}(\by)V_{p}(\by),\Phi_{2}(\by)\rangle\ .
\end{split}
\end{equation}
Therefore
$$
\langle \Phi_{1}(\bx) ,V_{p}(\bx)\Phi_{3}(\bx) \rangle=\langle \Phi_{3}(\bx) ,V_{p}(\bx)\Phi_{1}(\bx) \rangle=\langle \Phi_{2}(\bx) ,V_{p}(\bx)\Phi_{2}(\bx) \rangle=0\ .
$$
Similarly,
\begin{equation}\lb{pvp2}
\begin{split}
&\langle \Phi_{1}(\bx) ,V_{p}(\bx)\Phi_{1}(\bx) \rangle=-\langle \Phi_{1}(-\by) ,V_{p}(\by)\Phi_{1}(-\by)\rangle\\
=& -\langle \ol{\Phi_{3}(\by)} ,V_{p}(\by)\ol{\Phi_{3}(\by)}\rangle=-\langle \Phi_{3}(\by)V_{p}(\by),\Phi_{3}(\by)\rangle,\\
&\langle \Phi_{1}(\bx) ,V_{p}(\bx)\Phi_{2}(\bx) \rangle=-\langle \Phi_{1}(-\by) ,V_{p}(\by)\Phi_{2}(-\by) \rangle\\
=&-\langle \ol{\Phi_{3}(\by)} ,V_{p}(\by)\ol{\Phi_{2}(\by)}\rangle=-\langle \Phi_{2}(\by),V_{p}(\by)\Phi_{3}(\by)\rangle\ .\\
\end{split}
\end{equation}
Combining \x{pvp1} and \x{pvp2}, we obtain
\be\label{M1per}
\mathcal{M}_1=
\delta\begin{pmatrix}
-\ups^{\sharp}_1&\ups^{\sharp}_2&0\\
\ol{\ups^{\sharp}_2}&-\ups^{\sharp}_2\\
0&-\ol{\ups^{\sharp}_2}&+\ups^{\sharp}_1
\end{pmatrix}
,
\ee
where $\ups^{\sharp}_1$ and $\ups^{\sharp}_2$ represent $\langle \Phi_{1}(\bx) ,V_{p}(\bx)\Phi_{1}(\bx) \rangle$ and $\langle \Phi_{1}(\bx) ,V_{p}(\bx)\Phi_{2}(\bx) \rangle$ respectively. Obversely, $\ups^{\sharp}_1$ is real.
Let us assume that both $\ups^{\sharp}_1$ and $\ups^{\sharp}_2$ are nonzero. Substituting \eqref{M1per} into \eqref{eq:stable-equation}, we obtain
\begin{equation}\lb{eqss}
\wi{\mu}(\wi{\mu}^{2}-\delta^2(\ups^{\sharp}_1)^2)=2\delta^2 |\ups^{\sharp}_2|^2\wi{\mu}\ .
\end{equation}
Then we can conclude from \x{eqss} that the 3-fold degenerate point $(\bK, \mu_*)$ splits into 3 simple eigenvalues under an inversion-symmetry-broken perturbation. More precisely,
\[
\begin{split}
&\mu^{\delta}_1=\mu_{*}+\delta\sqrt{(\ups^{\sharp}_1)^2+2|\ups^{\sharp}_2|^2}+o(\delta), \\
&\mu^{\delta}_2=\mu_{*}+o(\delta), \\
&\mu^{\delta}_3=\mu_{*}-\delta\sqrt{(\ups^{\sharp}_1)^2+2|\ups^{\sharp}_2|^2}+o(\delta)\ .
\end{split}
\]
The above analysis implies that the 3-fold Weyl point does not persist if the inversion symmetry of the system is broken. We also include the numerical simulations for a typical admissible potential with an inversion-symmetry-broken perturbation in Section 7, see Figure 2. It is seen that the 3 bands do not intersect at $\bK$ and there exist two local gaps.
We remark that if $\mathcal{T}$-symmetry is broken and inversion-symmetry persists, the 3-fold degenerate point split into a 2-fold eigenvalue and a simple eigenvalue, see Figure 3 in Section 7. The reason is that the inversion symmetry naturally protects the 2-fold degeneracy which is similar to the 2-D honeycomb case. Due to the length of this work, we shall not include the detailed calculations while some of main ingredients can be found in our analysis to the bifurcation matrix $M(\ka)$ in Section 4.
\section{Numerical results}\label{subsec:3+1}
\lb{sec7}
In this section, we use numerical simulations to demonstrate our analysis. The numerical method that we use is the Fourier Collocation Method \cite{Yang2010}. The potential that we choose is
\begin{equation}\lb{cVx}
\begin{split}
V(\bx)= 5(&\cos(\bq_{1}\cdot\bx)+\cos((\bq_{2}-\bq_{1})\cdot\bx)
+\cos((\bq_{3}-\bq_{2})\cdot\bx)+\cos(\bq_{3}\cdot\bx)\\
&+\cos(\bq_{2}\cdot\bx)+\cos((\bq_{3}-\bq_{1})\cdot\bx)).
\end{split}
\end{equation}
It is evident that $V(\bx)$ is an admissible potential in the sense of Definition $\ref{def:definition-of-lattice}$.
According to our analysis--Theorem \ref{thm:3+1} and Theorem \ref{thm:dispersion-3+1}, the first three energy bands intersect conically at $\bK$. In the following illustrations, we plot the figures of first three energy bands in vicinity of $\bK$. Since the full energy bands are defined in $\mathbb{R}^3$, it is not easy to visualize such high dimensional structure. We just show the figures in the reduced parameter space, i.e., energy curves with the quasi-momentum being along certain specific directions and energy surfaces with the quasi-momentum being in a plane.
We plot dispersion bands $\mu(\bk)$ near $\bK$ along a certain direction $\textbf{n}$, i.e.,
\be\lb{bandc}
\mu(\lambda)=\mu(\bK+\lambda\textbf{n}).
\ee
The dispersion curves $\mu(\lambda)$ along three different directions are displayed on the top panel of Figure \ref{fig:3+1any direction} where we choose three different directions
$$
\textbf{n}=(1,0,0),\q \z(\frac{2}{3},\frac23,\frac13\y),\q \z(\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3}\y).
$$
In the first two cases, we see that the three straight lines intersect at $\lambda=0$, i.e., at the Weyl point. In the last example, we only see two straight lines intersect since one straight line is two-fold degenerate to leading order, see discussions in Section 5. The numerical simulations agree with our analysis given in Theorem \ref{thm:3+1}.
We also plot the energy surfaces with the quasi-momentum varying in along two directions, i.e.,
\[
\mu(\lambda_1,\lambda_2)=\mu(\bK+\lambda_1\textbf{n}_1+\lambda_2\textbf{n}_2).
\]
The dispersion surfaces $\mu(\lambda_1,\lambda_2)$ are displayed on the bottom panel of Figure \ref{fig:3+1any direction} where in all cases $\textbf{n}_1=(1,0,0)$ and
$$
\textbf{n}_2=(0,0,1),\q \z(0,\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\y),\q\z(0,\frac35,\frac45\y)
$$
respectively. From the figure, we see that the three dispersion surfaces intersect at the Weyl point. The first and third bands conically intersect each other with the second band in the middle. This result also agrees well with our analysis.
\begin{figure}[htbp]
\centering
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{k1_direction.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{k1_k1_ko5_direction.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{k1_k1_k1_direction.eps}
\end{minipage}
}
\quad
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{k1_k2picture.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{1-1-o5-surface.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{1-1-1-surface.eps}
\end{minipage}
}
\caption{The first three energy bands of $H=-\Delta +V(\bx)$ with $V(\bx)$ in \eqref{cVx} . {\bf Top Panel:} Energy curves $\mu(\bK+\lambda \textbf{n})$ in \x{bandc} with the quasi-momentum along a fixed direction $\textbf{n}$ being (a)$(1,0,0)$;(b)$\z(\frac{2}{3},\frac23,\frac13\y)$;(c)$ \z(\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3},\frac{\sqrt{3}}{3}\y)$ respectively.
{\bf Bottom Panel:} Energy surfaces $\mu(\lambda_1,\lambda_2)$ with the quasi-momentum along two directions $\textbf{n}_1,\ \textbf{n}_2$, where $\textbf{n}_1$ is chosen to be $(1,0,0)$ and $\textbf{n}_2$ equals (d)$(0,0,1)$; (e)$\z(0,\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\y)$; (f)$\z(0,\frac35,\frac45\y)$. The three energy bands intersect conically at the origin, i.e., at the Weyl point.}\label{fig:3+1any direction}
\end{figure}
We next verify the instability of conical singularity under certain symmetry-breaking perturbations. A perturbation is added to the above admissible potential \eqref{cVx}. In other words, we consider the Schr\"{o}dinger
operator $H^{\delta}_{i}=-\Delta+V(\bx)+\delta V_{p_{i}}(\bx)$, where $V_{p_i}(\bx),\ i=1,2$ denote the perturbation potential and $\delta$ a small parameter. In our simulations, we choose $\delta=0.01$.
\bu We first examine the role of $\mathfrak{PT}$-symmetry. The perturbation that we choose is
\begin{equation}\lb{Vpx}
\begin{split}
V_{p_1}(\bx)= &\sin(\bq_{1}\cdot\bx)+\sin((\bq_{2}-\bq_{1})\cdot\bx)
+\sin((\bq_{3}-\bq_{2})\cdot\bx)+\sin(\bq_{3}\cdot\bx)\\
&+\sin(\bq_{2}\cdot\bx)+\sin((\bq_{3}-\bq_{1})\cdot\bx).
\end{split}
\end{equation}
Obviously, $V_{p_1}$ is odd and thus breaks $\mathfrak{PT}$-invariance of $V(\bx)$. We plot the same energy band functions of $H^{\delta}_{1}$ as shown in Figure \ref{fig:instability}. We see that the three energy band functions separate with each other and two gaps open.
\begin{figure}[htbp]
\centering
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{ins-1-0-0.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{ins-1-1-o8.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{ins-1-1-1.eps}
\end{minipage}
}
\quad
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{instability1.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{instability2.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{instability3.eps}
\end{minipage}
}
\caption{The first three energy bands of $H^\delta=-\Delta +V(\bx)+\delta V_p(\bx)$ with the inversion-symmetry-breaking potential $V_p(\bx)$ in \eqref{Vpx}. The setup is the same as that in Fig. \ref{fig:3+1any direction}. The 3-fold degenerate point disappears and two local gaps open. }\label{fig:instability}
\end{figure}
\bu To see the significance of $\mathcal{T}$-invariance in the formation of three-fold conical structures, we consider the perturbation $V_{p}(\bx)$ which breaks $\mathcal{T}$-invariance. In our simulation, we choose
\begin{equation}\label{Vp2}
\begin{split}
V_{p_2}(\bx)= &\cos(\bq_{1}\cdot\bx)+\cos((\bq_{2}-\bq_{1})\cdot\bx)
+\cos((\bq_{3}-\bq_{2})\cdot\bx)+\cos(\bq_{3}\cdot\bx).
\end{split}
\end{equation}
Obviously, the perturbed potential \eqref{Vp2} possess $\mathcal{R}$-invariance and $\mathfrak{PT}$-invariance, but does not have the $\mathcal{T}$-symmetry since $T\bq_1=\bq_3-\bq_1$.
As before, we display the energy curves and surfaces near $\bK$ in Figure $\ref{tbrk}$. It is shown that the original three-fold degenerate cone structure disappears and breaks into one simple and one double eigenvalue. The nearby structure near the double eigenvalue is not naturally conical. It may correspond to other interesting phenomena but is beyond the scope of this paper.
\begin{figure}[htbp]
\centering
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{xdir.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{zdir.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3cm]{111dir.eps}
\end{minipage}
}
\quad
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{xysur.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{xzsur.eps}
\end{minipage}
}
\subfigure[]{
\begin{minipage}{0.3\textwidth}
\includegraphics[width=0.95\textwidth,height=3.5cm]{111.eps}
\end{minipage}
}
\caption{The first three energy bands of $H^\delta=-\Delta +V(\bx)+\delta V_p(\bx)$ with the $\mathcal{T}$-symmetry-breaking potential $V_p(\bx)$ in \eqref{Vp2}. The setup is the same as that in Fig. \ref{fig:3+1any direction}. The 3-fold degenerate point splits into a two-fold and a simple eigenvalues. The two-fold degeneracy comes from the inversion-symmetry of the system which we keep. There is no general conical structure near the perturbed two-fold degenerate point. }\label{tbrk}
\end{figure}
\begin{appendix}
\renewcommand{\thesection}{\Alph{section}}
\section{Proof of Proposition \ref{asphi}}
\lb{app-a}
The purpose of this appendix is to give a detailed proof to Proposition \ref{asphi}. We first prove the following lemma.
\begin{lem}\lb{even}
Let $\mu_{*}$ be an eigenvalue of $H(\bK)$ of eigenvalue problem $\x{mapro}$ with the corresponding eigenspace $\mathcal{E}_{\mu_{*}}$. If $\mathcal{E}_{\mu_{*}}\subset L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$, then $\dim\mathcal{E}_{\mu_{*}}$ is even.
\end{lem}
\Proof Let $\Phi\in\mathcal{E}_{\mu}\subset L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$. Then $\Phi(\bx)=c_1\Phi_1(\bx)+c_3\Phi_3(\bx)$ for some constants $c_1, \ c_3$, where $\Phi_1 \in L^{2}_{\bK,i}$ and $\Phi_3 \in L^{2}_{\bK,-i}$. We distinguish the following two cases.
\bu $c_1\cdot c_3=0$, say $c_3=0$ for instance. Then $\Phi(\bx)=c_1\Phi_1(\bx)$. Note that $\ol{\Phi(-\bx)}=c_1 \ol{\Phi_1(-\bx)}\in L^{2}_{\bK,-i}$ is linearly independent of $\Phi(\bx)$. Recall that $\{\ol{\Phi(-\bx)},\mu(\bK)\}$ is also an eigenpair of eigenvalue problem \x{HV-k}. We directly obtain $\ol{\Phi_1(-\bx)}\in \mathcal{E}_{\mu_{*}}$.
\bu $c_1\cdot c_3\neq0$. Applying $\mathcal{R}$ to $\Phi(\bx)$, one has $\mathcal{R}[\Phi](\bx)=ic_1\Phi_1(\bx)-ic_3\Phi_3(\bx)\in \mathcal{E}_{\mu_{*}}$. In the present case, it is easy to see that $\mathcal{R}[\Phi](\bx)$ is linearly dependent of $\Phi(\bx)$.
By the above analysis, we conclude that $\dim\mathcal{E}_{\mu}$ is even.\qed
Now we are ready to prove Proposition \ref{asphi}. Since $\Phi_{1}(\bx)\in L^{2}_{\bK,i}$ solves the Floquet-Bloch problem \x{mapro}, then $\Phi_{3}(\bx):=\ol{\Phi_{1}(-\bx)}\in L^{2}_{\bK,-i}$ is also an eigenfunction.
As $\dim\mathcal{E}_{\mu_{*}}=3$, there exists $\Phi'(\bx)\in\mathcal{E}_{\mu_{*}}$ such that $\Phi'(\bx)\notin L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$. Hence
\be\lb{phi} \Phi'(\bx)=c_1 \Phi''_1(\bx)+c_2 \Phi''_2(\bx)+c_3 \Phi''_3(\bx)\in \mathcal{E}_\mu,
\ee
where $c_2\neq 0$ and $\Phi''_{\ell}\in L^{2}_{\bK,i^{\ell}}$ satisfy $\mathcal{R}[\Phi''_{\ell}]=
i^{\ell}\Phi''_{\ell}$ for $\ell=1,2,3$.
We aim at proving the assertion that
$\Phi''_2(\bx)\in \mathcal{E}_{\mu_{*}}$.
\bu Case 1: $c_1=c_3=0$. In this case the assertion is trivial from \x{phi}.
\bu Case 2: One of $c_1$ and $c_3$ is nonzero and another is zero, say $c_1\ne 0$ and $c_3=0$. Then we have equalities
\[
\begin{split}
& \Phi'(\bx)=c_1\Phi''_1(\bx)+c_2\Phi''_2(\bx)\in \mathcal{E}_{\mu}, \\
& \mathcal{R}[\Phi'](\bx)=ic_1\Phi''_1(\bx)-c_2\Phi''_2(\bx)\in \mathcal{E}_{\mu},\\
& \mathcal{R}[\Phi'](\bx)+\Phi'(\bx)=(i+1)c_1\Phi''_1(\bx)\in L^{2}_{\bK,i}\cap \mathcal{E}_{\mu}\ .
\end{split}
\] Since the multiplicity in $L^{2}_{\bK,i}$ is one, we conclude from the last equality that
$\Phi''_1(\bx)=\al \Phi_1(\bx)$ for some $\al$. Consequently, we conclude from the first equality that $\Phi''_2(\bx)\in\mathcal{E}_{\mu}$.
\bu Case 3: Both $c_1$ and $c_3$ are nonzero. Basing on the decomposition \x{phi}, one has
\[
\mathcal{R}[\Phi'](\bx)=ic_1\Phi''_1(\bx)-c_2\Phi''_2(\bx)-ic_3\Phi''_3(\bx).
\]
Then
\be\lb{Phi''}
\Phi''(\bx):=\mathcal{R}[\Phi'](\bx)+\Phi'(\bx)=k_1\Phi''_1(\bx)+k_2\Phi''_3(\bx)\in L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}
\ee
where $k_1=c_1(i+1)$ and $k_2=c_3(1-i)$.
Assume that $\Phi''\not\in {\rm span}\{\Phi_1,\Phi_3\}$. Without loss of generality, we assume that $\Phi''_1(\bx)$ is linearly independent of $\Phi_1(\bx)$. Then $\mathcal{R}[\Phi''](\bx)+i\Phi''(\bx)=2ik_1\Phi''(\bx)\in L^{2}_{\bK,i}$, which would imply that $\mu(\bK)$ is not a three-fold eigenvalue. Thus we have shown that $\Phi''\in {\rm span}\{\Phi_1,\Phi_3\}$ is a linear combination of $\Phi_1$ and $\Phi_3$. It then follows from \x{Phi''} that $\Phi''_1=\al \Phi_1$ and $\Phi''_3=\bt \Phi_3$ for some constants $\al, \ \bt$. Going back to \x{phi}, we obtain
$\Phi'(\bx)=c'_1\Phi_1(\bx)+c_2\Phi''_2(\bx)+c'_3\Phi_3(\bx)$. This leads to the assertion $\Phi''_2(\bx)\in\mathcal{E}_{\mu_{*}}$.
The proof is complete. \qed
\section{Proof of Lemma $\ref{strut}$ }
In this appendix, we actually give a proof of a stronger conclusion. Assume that $\Phi^{c}(\bx)\in L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$ has the form
\[\Phi^{c}(\bx)=\Phi_1(\bx)+\Phi_3(\bx),\]
where $\Phi_1(\bx)\in L^{2}_{\bK,i}$ and $\Phi_3(\bx)\in L^{2}_{\bK,-i}$ are of the form \eqref{decomp2}. That is,
\[
\begin{split}
\Phi_1(\bx)&=c_1 \Phi^{(0)}_1 (\bx)+\Phi^{h}_1(\bx)= c_1 \Phi^{(0)}_1 (\bx)+\sum_{\bq\in\ssk\setminus \{(0,0,0)\}}\Phi_{\bq}(e^{i(\bK+\bq)\cdot\bx}-ie^{iR(\bK+\bq)\cdot\bx}
\\ &- e^{iR^{2}(\bK+\bq)\cdot\bx} +ie^{iR^{3}(\bK+\bq)\cdot\bx}),\\
\Phi_3(\bx)&=c_3 \Phi^{(0)}_1 (\bx)+\Phi^{h}_3(\bx)=c_3 \Phi^{(0)}_3 (\bx)+\sum_{\bq\in\ssk\setminus \{(0,0,0)\}}\Phi_{\bq}(e^{i(\bK+\bq)\cdot\bx}+ie^{iR(\bK+\bq)\cdot\bx}\\
&-e^{iR^{2}(\bK+\bq)\cdot\bx} +i e^{iR^{3}(\bK+\bq)\cdot\bx})\ .
\end{split}
\]
By the symmetry, we have the following conclusion.
\begin{lem}\lb{tsrt}
If $|c_1|+|c_3|>0$, then $\mathcal{T}\Phi^{c}(\bx)\notin L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$.
\end{lem}
Note that Lemma $\ref{strut}$ is just a direct consequence of above conclusion. Indeed, let us recall that $\Phi^{\e}_1(\bx)=(1+\mathcal{O}(\e))\Phi^{(0)}_1(\bx)+\Phi^{h}_{1}(\bx)$. Thus for sufficiently small $\e$, $\Phi^{\e}_1(\bx)$ satisfies the conditions of Lemma $\ref{tsrt}$, i.e., $c_1=1+O(\e)\neq 0$. So $\mathcal{T}\Phi^{\e}_1(\bx)\notin L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$.
\qed
It remains to prove Lemma $\ref{tsrt}$. We begin the proof by considering the action
$\mathcal{T}$ on $\Phi_\ell^{(0)}(\bx)$. By employing $\mathcal{T}$ on $e^{iR^{\ell}\bK\cdot\bx}$ accordingly, we obtain
\[
\begin{split}
&\mathcal{T}\Phi^{(0)}_1=\frac{1}{\sqrt{4|\Omega|}}(-e^{i\bK\cdot\bx}+e^{iR\bK\cdot\bx}+ie^{iR^2\bK\cdot\bx}-ie^{iR^3\bK\cdot\bx}),\\
&\mathcal{T}\Phi^{(0)}_2=\frac{1}{\sqrt{4|\Omega|}}(e^{i\bK\cdot\bx}+e^{iR\bK\cdot\bx}-e^{iR^2\bK\cdot\bx}-e^{iR^3\bK\cdot\bx}),\\
&\mathcal{T}\Phi^{(0)}_3=\frac{1}{\sqrt{4|\Omega|}}(-e^{i\bK\cdot\bx}+e^{iR\bK\cdot\bx}-ie^{iR^2\bK\cdot\bx}+ie^{iR^3\bK\cdot\bx}),\\
&\mathcal{T}\Phi^{(0)}_4=\frac{1}{\sqrt{4|\Omega|}}(e^{i\bK\cdot\bx}+e^{iR\bK\cdot\bx}+e^{iR^2\bK\cdot\bx}+e^{iR^3\bK\cdot\bx}).
\end{split}
\]
Obviously $\mathcal{T}\Phi^{(0)}_4(\bx)=\Phi^{(0)}_4(\bx)$. By direct calculations, we have the following relations between $\{\mathcal{T}\Phi^{(0)}_{\ell}(\bx)\}^{3}_{\ell=1}$ and $\{\Phi^{(0)}_{\ell}(\bx)\}^{3}_{\ell=1}$
\be\lb{Tact}
\begin{pmatrix}
\mathcal{T}\Phi^{(0)}_{1}\\
\mathcal{T}\Phi^{(0)}_2\\
\mathcal{T}\Phi^{(0)}_3
\end{pmatrix}
=
Q^{0}_{\mathcal{T}}
\begin{pmatrix}
\Phi^{(0)}_{1}\\
\Phi^{(0)}_2\\
\Phi^{(0)}_3
\end{pmatrix},
\ee
where
\be\lb{Cm}
Q^{0}_{\mathcal{T}}=
\begin{pmatrix}
-\frac{1}{2}&-\frac{1}{2}+\frac{i}{2}&-\frac{i}{2}\\
\frac{1}{2}+\frac{i}{2}&0&\frac{1}{2}-\frac{i}{2}\\
\frac{i}{2}&-\frac{1}{2}-\frac{i}{2}&-\frac{1}{2}
\end{pmatrix}.
\ee
Assume that $\mathcal{T}\Phi^{c}\in L^{2}_{\bK,i}\oplus L^2_{\bK,-i}$. Then
\[
\mathcal{T}\Phi^{c}=\mathcal{T}\Phi_1+\mathcal{T}\Phi_3 =c_1\mathcal{T}\Phi^{(0)}_{1}+c_3\mathcal{T}\Phi^{(0)}_3+\mathcal{T}\Phi^{h}_1+\mathcal{T}\Phi^{h}_3\ .
\]
By the relations in \x{Tact} and \x{Cm}, we have
\[
c_1(-\frac12 +\frac{i}{2})+c_3(-\frac{1}{2}-\frac{i}{2})=-(c_1+c_3)\frac{1}{2}+\frac{i}{2}(c_1-c_3)=0\ .
\]
This implies $c_1 =c_3 =0$, which contradicts with $|c_1|+|c_3|>0$. Therefore, $\mathcal{T}\Phi^{c}(\bx)\notin L^{2}_{\bK,i}\oplus L^{2}_{\bK,-i}$.
\qed
\end{appendix}
|
1,116,691,498,508 | arxiv | \section{Introduction}
Modern Cosmology has found itself at an impasse. The latest and most precise estimates of the Hubble constant ($H_0$) obtained from {\em Planck} (`Standard Ruler' measurements) \citep{Planck2020} disagree at the 4.2\,$\sigma$ level with those obtained from the cosmic distance ladder using `Standard Candles' \citep[e.g.][]{Freedman2012, Riess2022}. This disagreement, dubbed the `Hubble Tension', is one of the greatest challenges facing modern cosmology. A true divergence of these measurements would require us to reevaluate key models underpinning our understanding of the Universe, such as the $\Lambda$CDM model and the number of neutrinos in the standard model \citep{Bernal2016, DiValentino2017}. Given these implications, it is essential to rule out systematic uncertainties in either experiment as the source of the Hubble tension.
The majority of distance ladder studies use Cepheids in the Milky Way and/or Magellanic Clouds as their `bottom rung'. In recent years, the shift to near- and mid-infrared wavelengths, where the intrinsic dispersion in the Leavitt law (LL) is smaller, has improved the error budget for Cepheid distances \citep[e.g.][]{Freedman2012, Riess2022}, though the impact of metallicity \citep[][and references therein]{Ripepi2020, Romaniello2022} and the potential non-linearity of the LL remain open questions \citep[e.g.][]{Kodric2015,Chown2021}. These must be answered before the question of the Hubble tension can be addressed, as systematics at the base of the distance ladder will be compounded with each further step towards $H_0$ \citep{Freedman2021}.
In a \textit{Spitzer}-IRAC study of Galactic and Magellanic Cloud Cepheids, \citet[][hereafter S16]{Scowcroft2016_CO} found that Cepheids with $0.8 \lesssim \log P \lesssim 1.8$~d exhibit a clear period-colour (PC) relation in the mid-infrared (mid--IR), with longer period objects having bluer colours. \citetalias{Scowcroft2016_CO} hypothesised that the cyclical variation in mid-IR colour (their figure 2) and the PC relation are due to the significant CO vibration-rotation band-head at $4.6~\mu$m, first identified in stars by \citet{Ayres1994} and \citet{Wiedemann1994}. At the typical temperature range of Cepheids ($\sim$ 4000-6000\,K) CO is readily destroyed by chemical processes such as H + CO $\rightarrow$ C + OH. The rate of these reactions varies strongly within the temperature range of Cepheids, from $10^{-17}$ to $10^{-15}~\text{cm}^{3}~\text{s}^{-1}$ at their coolest and hottest temperatures, respectively, while the formation rate of CO through radiative association remains relatively constant at 10$^{-16}$\,cm$^{3}$\,s$^{-1}$.
The effect of metallicity on Cepheid luminosities is expected to be reduced at longer wavelengths. However, \citetalias{Scowcroft2016_CO} identified a significant metallicity dependence in their mid-IR colours.
Combining their \textit{Spitzer} photometry with spectroscopic metallicities from \citep{Genovali2014}, \citetalias{Scowcroft2016_CO} showed that the deviation from the LMC PC can be used as a photometric metallicity indicator, producing [Fe/H] estimates with accuracies competitive with spectroscopic measurements \citepalias[their figure 7]{Scowcroft2016_CO}. As one of the challenges faced by previous metallicity studies has been the limited number of Cepheids for which accurate stellar metallicity measurements are available, a robust photometric metallicity indicator for Cepheids would be a significant step forward in elimitating the metallicity systematic from the distance ladder.
Here we present the first results of an IRAM-30m pilot study to confirm the validity of the CO mechanism that \citetalias{Scowcroft2016_CO} propose as the source of the mid-IR colour-metallicity relation. We show for the first time that CO is present in Cepheid atmospheres, providing observational evidence for the cyclical CO destruction and formation mechanism discussed in \citetalias{Scowcroft2016_CO}, further strengthening the case for the use of mid-IR colour as a metallicity indicator for Cepheids.
\section{Sample}
Prior to the observing run, a subset of 52 Galactic Cepheids with \textit{Gaia} time-series photometry \citep{Gaia_2016, Gaia_2022_DR3} was identified, with the condition that each target would be observable at least 50\% of the time during the IRAM run, and that mid-IR photometry from \citet{Monson2012} and/or spectroscopic metallicities from \citet{Genovali2014} were available.
On each day of observations, targets were selected and prioritised based on the predicted phase from the \textit{Gaia} light curve at the time of observation. Targets approaching minimum light were selected; according to \citetalias{Scowcroft2016_CO}'s proposed mechanism the highest CO production is expected in this region, thus these phases represented the highest potential for CO detection. A secondary cut, removing sources close to the galactic plane was implemented to avoid contamination of the beam by intervening gas clouds. Finally, each candidate source was required to have a bright pointing source within 20 arcsec at the time of the observations to ensure accurate pointing of the telescope and maximise the sensitivity to emission from the source.
\section{Observations}
The observations were carried out with the IRAM 30m telescope at Pico Veleta, Spain, in one run during 8$^{\text{th}}$ -- 12$^{\text{th}}$ February 2023. Observations of the CO(1-0) and CO(2-1) lines were conducted simultaneously using the E090 and E230 receivers respectively in parallel. The broadband EMIR receivers were used to observe in dual polarisation mode using a single side band and a total bandwidth of 4 GHz. Wobbler switching mode was used for background subtraction with a phase of 2 seconds.
Due to technical issues with the Wobbler the throw had to be reduced from 60 to 30 arcsecs for the final two days of observations.
The receivers were tuned to the rest frequency of CO(1-0) \& CO(2-1) (115.27 and 230.54 GHz respectively) and the tuning was tested each day by observing the line calibration source W3(OH). The detected fluxes and peak antenna temperature measured for this source are consistent with the values presented in the line calibration catalogue \citep{mbg98} and were consistent between observing days to within the accepted tolerance of the telescope ($<$10\% variation, Table \ref{tab:line_cal}).
\begin{table}
\caption{Peak and total line flux measured for the line calibrator (W3HO) on each day of observations. Reference values peak at $\sim$23\,K (181.7\,Jy) and $\sim$18\,K (180.0\,Jy) respectively \citep{mbg98}. Our measured peaks are consistent with these values on each day and the total flux in the lines varies by $\sigma$=4.5\% (max=10.3\%) and $\sigma$=2.9\% (max=6.2\%) of the mean.}
\label{tab:line_cal}
\centering
\begin{tabular}{c|c|c|c|c}
\hline
{\bf Date} & {\bf CO(1-0)} & {\bf CO(1-0)} & {\bf CO(2-1)} & {\bf CO(2-1)} \\
& {\bf Peak} & {\bf Flux} & {\bf Peak} & {\bf Flux} \\
\hline
08/02 & 26.8\,K & 258\,K\,km\,s$^{-1}$ & 21.5\,K & 297\,K\,km\,s$^{-1}$ \\
& 161.3\,Jy & 1553\,Jy\,km\,s$^{-1}$ & 129.4\,Jy & 1788\,Jy\,km\,s$^{-1}$ \\
09/02 & 27.6\,K & 263\,K\,km\,s$^{-1}$ & 23.9\,K & 306\,K\,km\,s$^{-1}$ \\
& 166.2\,Jy & 1583\,Jy\,km\,s$^{-1}$ & 143.9\,Jy & 1842\,Jy\,km\,s$^{-1}$ \\
11/02 & 26.2\,K & 251\,K\,km\,s$^{-1}$ & 22.6\,K & 316\,K\,km\,s$^{-1}$ \\
& 157.7\,Jy & 1511\,Jy\,km\,s$^{-1}$ & 136.0\,Jy & 1902\,Jy\,km\,s$^{-1}$ \\
12/02 & 25.3\,K & 237\,K\,km\,s$^{-1}$ & 21.3\,K & 298\,K\,km\,s$^{-1}$ \\
& 152.3\,Jy & 1426\,Jy\,km\,s$^{-1}$ & 128.2\,Jy & 1794\,Jy\,km\,s$^{-1}$ \\
\hline
\end{tabular}
\end{table}
The FTS200 and WILMA backends were used in parallel to sample the data at 0.2 and 2\,MHz respectively. Due to technical issues, poor weather, and the availability of pointing sources total integration times per source differed significantly. As we expect significant variability of each source over its pulsation period, stacking observations from different days was not considered a viable solution to increase observation depth. We present the total integration time per source, the RMS noise reached, and the average atmospheric opacity (Tau) during each source's observations in Table. \ref{tab:sou}. Pointing \& Focus measurements were repeated at least every 1.5 hours.
The line emission is measured in antenna temperature T$^*_A$ and converted to fluxes (Jy) using the point source sensitivity conversion S/T$_A^*$ = 6.02\,Jy/K (half power beam width of 21.4 arcsec at 115\,GHz) and S/T$_A^*$ = 7.8\,Jy/K (half power beam width of 10.7 arcsec at 230\,GHz) for the E090 and E230 bands respectively. The data were reduced with the \textsc{class/gildas} software, and the spectra were smoothed to a maximum of $\sim$40km\,s$^{-1}$ consistent with the maximum linewidth expected to result from the movement of the stellar surface as it pulsates. We expect our sources to have small line of sight velocities so set our velocity zero-point at the rest frame frequency of the respective lines.
\begin{table*}
\caption{Details of the observations. T$_{\text{int}}$ is the total integration time spent observing the source. $\Delta$V is the velocity resolution used for smoothing, with a maximum of $\sim$40km\,s$^{-1}$ used to be consistent with expected linewidths induced by Cepheid pulsations.
RMS is the noise level obtained at the given smoothing. Opacity is the mean Tau value measured during the observations. All observation dates are 2023.}
\label{tab:sou}
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
\hline
{\bf Source} & {\bf T$_{\text{int}}$} & {\bf Backend} & $\Delta$V & \multicolumn{2}{c}{\bf RMS \@ 115\,GHz} & \multicolumn{2}{c}{\bf RMS \@ 230\,GHz} & \multicolumn{2}{c}{\bf Opacity (Tau)} & {\bf Date} & {\bf Period}\\
& {\bf (min)} & & {\bf (km\,s$^{-1}$}) & {\bf (mK)} & {\bf (mJy)} & {\bf (mK)} & {\bf (Jy)} & {\bf 115\,GHz} & {\bf 230\,GHz} & & {\bf Days} \\
\hline
GY~Sge & 40.2 & FTS & 32.5 & 2.93 & 17.6 & 3.04 & 23.9 & 0.373 & 0.232 & 08/02 & 51.05 \\
& & WILMA & 41.6 & 2.28 & 13.8 & 3.07 & 23.9 & & & \\
BZ~Cyg & 78.5 & FTS & 32.5 & 1.58 & 9.5 & 1.16 & 9.0 & 0.378 & 0.242 & 08/02 & 10.14 \\
& & WILMA & 41.6 & 1.13 & 6.8 & 1.07 & 8.3 & & & \\
CH~Cas (1) & 40.3 & FTS & 32.4 & 2.12 & 12.8 & 1.95 & 15.2 & 0.380 & 0.252 & 08/02 & 15.09 \\
& & WILMA & 41.6 & 1.69 & 10.2 & 1.52 & 11.9 & & & \\
CP~Cep & 91.4 & FTS & 32.5 & 1.67 & 10.1 & 0.93 & 7.3 & 0.345 & 0.072 & 11/02 & 17.86 \\
& & WILMA & 41.6 & 1.33 & 8.0 & 0.67 & 5.3 & & \\
RY~Cas & 81.1 & FTS & 32.5 & 1.64 & 9.9 & 0.83 & 6.5 & 0.344 & 0.079 & 11/02 & 12.14 \\
& & WILMA & 41.6 & 1.29 & 7.8 & 0.89 & 6.9 & & \\
KX~Cyg & 50.7 & FTS & 32.5 & 1.79 & 10.8 & 1.01 & 7.9 & 0.343 & 0.075 & 11/02 & 20.04 \\
& & WILMA & 41.6 & 1.75 & 10.5 & 1.53 & 11.9 & & \\
CH~Cas (2) & 61.0 & FTS & 32.5 & 2.04 & 12.3 & 2.30 & 18.5 & 0.420 & 0.363 & 12/02 & 15.09 \\
& & WILMA & 41.6 & 1.98 & 11.9 & 2.00 & 15.6 & & \\
CY~Cas & 61.0 & FTS & 16.3 & 3.21 & 19.4 & 3.52 & 27.5 & 0.403 & 0.327 & 12/02 & 14.38 \\
& & WILMA & 20.8 & 3.04 & 18.3 & 2.71 & 21.1 & & \\
V396~Cyg & 91.3 & FTS & 32.5 & 1.07 & 6.4 & 1.48 & 11.5 & 0.398 & 0.325 & 12/02 & 33.25 \\
& & WILMA & 41.6 & 1.07 & 6.4 & 1.43 & 11.2 & & \\
\hline
\end{tabular}
\end{table*}
\section{Results}
\subsection{CO detections}
Of the eight Cepheid Variables observed, four were detected in emission at $>$3$\sigma$ significance in at least one of the backends (see Table \ref{tab:detec}). A further two showed unexpected strong absorption features, the nature of which is discussed in detail in \S \ref{sec:abs}. Just one source (CH~Cas) had repeat observations on multiple days to check for variability, this is discussed in \S \ref{sec:var}. Spectra for each source detected are presented in Figure \ref{fig:detections} showing the CO(1-0) emission as seen in the backend that provided the highest signal-to-noise detection.
Considering the four sources detected in emission, they show an average flux of 2.9\,Jy\,km\,s$^{-1}$ with a range of 1.78 -- 5.37\,Jy\,km\,s$^{-1}$. Of these GY~Sge has by far the highest flux at more than twice that measured for the other three sources. In line with the model presented in \citetalias{Scowcroft2016_CO} CO production is expected to start at a phase of $\sim$0.1 and continue until $\sim$0.65 after which it is quickly destroyed as the light curve rises again. Because of this cyclical formation and destruction of CO Cepheids with longer periods will have substantially more time to build up CO in their atmospheres than Cepheids with shorter periods. As such GY~Sge having significantly more flux than the other detected sources is consistent with the model presented by \citetalias{Scowcroft2016_CO} as its period is significantly longer (51.05 days) than any of the other detected sources (CP~Cep is the next longest with 17.86 days).
GY~Sge is also observed at a phase of 0.25 putting it on the falling part of its light curve where CO production is expected to be at a maximum in line with the model of \citetalias{Scowcroft2016_CO}. By contrast, BZ~Cyg, CP~Cep, and RY~Cas are all observed close to the bottom of their optical light curves where CO production is expected to transition to CO destruction. As there remains some ambiguity in the exact point at which this transition occurs (it is related to the star's temperature variations rather than its flux variations so it is not possible to make direct comparisons between stars with different periods) it is possible that some of these sources where caught during the rapid CO destruction phase resulting in lower measured fluxes. In order to fully characterise this cycle of CO creation and destruction it will be necessary to regularly monitor the CO flux from Cepheid Variables with a range of periods throughout the entirety of their period, and ideally over multiple pulsations.
All detections have velocity measurements that are close to rest (0\,km\,s$^{-1}$) and are well within the typical range found for Cepheids within the {\em Gaia DR3} catalogue (100--300\,km\,s$^{-1}$). The measured FWHM for BZ~Cyg, CP~Cep, and RY~Cas are consistent with the maximum line with expected to be induced by stellar pulsations (FWHM$\sim$40\,km\,s$^{-1}$). GY~Sge has a FWHM of 93\,km\,s$^{-1}$ which is considerably higher than this. It is also substantially higher than the majority of measured line broadening parameters for Cepheids in the {\em Gaia DR3} catalogue (typically $<$50\,km\,s$^{-1}$) though it remains well within the range measured ($<$200\,km\,s$^{-1}$). The subtracted baseline was checked to ensure that an undersubtraction in the region of the line was not responsible for the detected feature, but the baseline was flat and constant over the range (200--100\,km\,s$^{-1}$) indicating that the feature must have been present prior to the baseline subtraction.
As such the four sources GY~Sge, BZ~Cyg, CP~Cep, and RY~Cas all show significant detections of CO(1-0) making them the first detections of CO emission from Cepheid Variable Stars. RY~Cas also shows CO(2-1) emission detected at 3.8$\sigma$ with a velocity and FWHM consistent with the CO(1-0) line. The none detections of KX~Cyg and V396~Cyg can potentially be attributed to the short integration time of KX~Cyg and the extremely poor conditions in which V396~Cyg was observed. Regardless, these detections represent an important result in the field indicating that CO is indeed present in the atmospheres of Cepheid Variables and can thus explain the colour variations seen by \citetalias{Scowcroft2016_CO}.
\begin{table*}
\caption{Details of the detections made for each source. Details of both backends are listed for comparison when a $>$3$\sigma$ detection was made with either. The parameters of Gaussian fits to the line features are listed. Some lines where unresolved at the binning used, as indicated by an error on the FWMH which is significantly greater than the measured FWHM. Phase is the point in the light curve at which the observation was taken, measured from the peak.}
\label{tab:detec}
\centering
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}
\hline
{\bf Source} & {\bf Line} & {\bf Backend} & \multicolumn{2}{c}{\bf Flux} & {\bf Velocity} & {\bf FWHM} & \multicolumn{2}{c}{\bf Peak} & {\bf Phase} & {\bf S/N} \\
& & & {\bf (K\,km\,s$^{-1}$)} & {\bf (Jy\,km\,s$^{-1}$)} & {\bf (km\,s$^{-1}$)} & {\bf (km\,s$^{-1}$)} & {\bf (mK)} & {\bf (mJy)} & {\bf $\phi$} & \\
\hline
GY~Sge & CO(1-0) & FTS & 0.89$\pm$0.23 & 5.37$\pm$1.39 & -38$\pm$13 & 93$\pm$25 & 9.0$\pm$2.9 & 54.1$\pm$17.6 & 0.25 & 3.87 \\
& & WILMA & 0.81$\pm$0.21 & 4.89$\pm$1.26 & -55$\pm$12 & 93$\pm$25 & 8.1$\pm$2.3 & 49.0$\pm$13.8 & & 3.86 \\
BZ~Cyg & CO(1-0) & FTS & 0.35$\pm$0.13 & 2.11$\pm$0.78 & 190$\pm$15 & 80$\pm$35 & 4.1$\pm$1.58 & 24.7$\pm$9.5 & 0.48 & 2.69 \\
& & WILMA & 0.28$\pm$0.08 & 1.69$\pm$0.48 & 188$\pm$12 & 52$\pm$30 & 5.0$\pm$1.13 & 30.1$\pm$6.8 & & 3.50 \\
CH~Cas (1) & CO(1-0) & FTS & -1.60$\pm$0.13 & -9.63$\pm$0.78 & -8$\pm$2 & 51$\pm$4 & -29.2$\pm$2.12 & -175.8$\pm$12.8 & 0.43 & 12.31 \\
& & WILMA & -1.58$\pm$0.14 & -9.51$\pm$0.84 & -8$\pm$2 & 62$\pm$10 & -23.7$\pm$1.69 & -142.7$\pm$10.2 & & 11.29 \\
& CO(2-1) & FTS & -0.49$\pm$0.11 & -3.82$\pm$0.86 & -10$\pm$6 & 40$\pm$9 & -11.5$\pm$1.95 & -89.7$\pm$15.2 & & 4.45 \\
& & WILMA & -0.38$\pm$0.09 & -2.96$\pm$0.70 & -10$\pm$4 & 41$\pm$111 & -8.5$\pm$1.52 & -66.3$\pm$11.9 & & 4.22 \\
CP~Cep & CO(1-0) & FTS & 0.39$\pm$0.103 & 2.35$\pm$0.62 & 59$\pm$7 & 46$\pm$12 & 8.00$\pm$1.67 & 48.2$\pm$10.1 & 0.70 & 3.79 \\
& & WILMA & 0.225$\pm$0.102 & 1.35$\pm$0.61 & 53$\pm$16 & 56$\pm$23 & 3.80$\pm$1.33 & 22.9$\pm$8.0 & & 2.21\\
RY~Cas & CO(1-0) & FTS & 0.295$\pm$0.098 & 1.78$\pm$0.59 & -136$\pm$10 & 52$\pm$16 & 5.38$\pm$1.64 & 32.4$\pm$9.9 & 0.56 & 3.01\\
& & WILMA & 0.300$\pm$0.082 & 1.81$\pm$0.49 & -133$\pm$5 & 42$\pm$272 & 6.77$\pm$1.29 & 40.8$\pm$7.8 & & 3.66 \\
& CO(2-1) & FTS & 0.208$\pm$0.055 & 1.62$\pm$0.43 & -143$\pm$8 & 58$\pm$16 & 3.38$\pm$0.83 & 26.4$\pm$6.5 & & 3.78 \\
& & WILMA & 0.147$\pm$0.057 & 1.45$\pm$0.44 & -138$\pm$6 & 42$\pm$239 & 3.32$\pm$0.89 & 25.9$\pm$6.9 & & 2.94 \\
KX~Cyg & \multicolumn{8}{c}{No Detection} & 0.61 \\
CH~Cas (2) & CO(1-0) & FTS & -0.43$\pm$0.15 & -2.59$\pm$0.90 & 2$\pm$8 & 55$\pm$27 & -7.28$\pm$2.04 & -43.8$\pm$12.3 & 0.75 & 2.87\\
& & WILMA & -0.44$\pm$0.17 & -3.43$\pm$1.32 & 19$\pm$17 & 75$\pm$31 & -4.85$\pm$1.98 & -37.8$\pm$18.5 & & 2.59 \\
& CO(2-1) & & \multicolumn{4}{c}{\hspace{0.65cm} No Detection} \\
CY~Cas & CO(1-0) & FTS & -1.09$\pm$0.08 & -6.56$\pm$0.48 & 2$\pm$1 & 19$\pm$9 & -53.88$\pm$3.21 & -324.4$\pm$12.3 & 0.29 & 13.63 \\
& & WILMA & -0.976$\pm$0.10 & -5.88$\pm$0.60 & 2$\pm$2 & 26 $\pm$10 & -35.68$\pm$3.04 & -214.8$\pm$11.9 & & 9.60 \\
& CO(2-1) & FTS & -0.37$\pm$0.07 & -2.89$\pm$0.55 & 1$\pm$1 & 18$\pm$72 & -21.54$\pm$3.52 & -168.0$\pm$27.5 & & 5.29 \\
& & WILMA & -0.26$\pm$0.08 & -2.03$\pm$0.62 & 2$\pm$5 & 21$\pm$10 & -12.75$\pm$2.71 & -99.5$\pm$21.1 & & 3.25 \\
V396~Cyg & \multicolumn{8}{c}{No Detection} & 0.74 \\
\hline
\end{tabular}
\end{table*}
\subsection{Absorption features}
\label{sec:abs}
Of the eight sources observed two (CH~Cas and CY~Cas) showed exceptionally strong absorption features in both CO(1-0) and CO(2-1).
Stars are not expected to have significant continuum emission at 115\,GHz and 230\,GHz \citep{pal96p} making it difficult to explain these features in the context of CO in the stellar atmospheres. Likewise, an intervening gas cloud along the line of sight would still require a 'backlight' from the star to produce the absorption feature so suffers the same difficulty. While the spectra do show baselines that are sufficiently high to accommodate the absorption features the observations were not set up to detect continuum emission and there are many factors which can contribute to the baseline level so it is not possible to equate the baseline directly to continuum emission from the source.
While the detections in both CO(1-0) and CO(2-1) make it unlikely, the individual scans were inspected one by one to rule out the possibility of a strong spike in one scan producing this feature, however evidence of the absorption feature was seen in each scan ruling out this possibility. An oversubtraction of the baseline was also ruled out as these strong features are clearly visible even prior to the baseline subtraction. As such while this does not explain the origin of these absorption features, it confirms that they are real and not an artefact of the observations.
Considering the location of CH~Cas and CY~Cas, they are very close to each other on the sky and are both within the galactic plane of the galaxy. This raises the possibility that the 'off' position used by the wobbler to remove the sky emission contained a Galactic molecular cloud with emission in CO(1-0) and CO(2-1) which was then subtracted from the emission from the source. The features in both CH~Cas and CY~Cas have a velocity consistent with zero and show very narrow lines which is consistent with this hypothesis. Additionally, the ratio of the CO(1-0) and CO(2-1) flux measurements is consistent with a higher population at J=1 than J=2, suggesting that the gas producing this feature is cold, and certainly colder than the 4000--6000\,K expected in the atmospheres of Cepheid Variables which rules out the absorption features coming from the stellar atmospheres. As such a cold molecular cloud present at the wobblers 'off' position is the most likely explanation for these absorption features, though it is not possible currently to rule out an absorbing cloud along the line of sight. \citet{jos98} encountered a similar issue and devised a method using position-switching mode to remove the absorption features which can be applied to future observations of Cepheids which show significant absorption.
\subsection{Variability}
\label{sec:var}
While ideally each Cepheid would have been observed at multiple epochs to test for variability during its pulsation cycle, due to poor weather and technical issues this was only possible for one source (CH~Cas). Unfortunately, CH~Cas is one of the two sources which show CO in absorption which is difficult to attribute to CO in its stellar atmosphere. Despite this the second observation of CH~Cas, taken four days after the first when the CO is expected to be in its destruction phase, shows significantly weaker features than on the first observation when it was well within the CO creation phase. The CO(1-0) shows a day to day flux ratio of S$_{\phi = 0.43}$/S$_{\phi = 0.75}$ = 3.7 and the CO(2-1) is not detected at $\phi$ = 0.75. While this is consistent with the expected behaviour of CO based on the model proposed by \citetalias{Scowcroft2016_CO} care must be taken when considering this variation as evidence that the CO absorption originates form the stellar atmosphere. The two observations were taken at different times of day meaning the offset direction of the wobbler will have been different. Due to technical issues with the wobbler, the throw had to be reduced from 60 to 30 arcsec between the two observations. As a result of these two points in the scenario where the absorption features results from sky subtraction (see \S \ref{sec:abs}) the location of the offset position will be different for the two observations, and the sky position may have caught a different region of the molecular gas cloud, resulting in different levels of 'absorption' being seen.
\begin{figure*}
\centering
\includegraphics[width=0.995\textwidth]{figure1.png}
\vspace{-4.0cm}
\begin{minipage}[b]{0.55\textwidth}
\end{minipage}
\hfill
\begin{minipage}[b]{0.45\textwidth}
\caption{Plots showing all of our CO(1-0) detections at a signal-to-noise of greater than 3 produced using \textsc{class/gildas} for the backend which provided the highest signal-to-noise. The individual plots are labelled to indicate which observation/object each refers to. The yellow bars show she observed spectra in antenna Temperature (K), smoothed to the velocity resolution indicated in Table \ref{tab:sou}. Fits to detected features are shown as a red line, the parameters of these fits can be found in Table \ref{tab:detec}. The repeat observations of CH~Cas is shown next to the first observation despite not reaching a signal-to-noise of $>$3 for comparison. }
\label{fig:detections}
\end{minipage}
\vspace{0.4cm}
\end{figure*}
\section{Conclusions}
CO(1-0) emission is detected by IRAM 30m in four of the eight Cepheid variable stars targeted by this study representing the first direct detections of CO gas in Cepheid Variables. This provides strong evidence for the mechanism proposed by \citetalias{Scowcroft2016_CO} to explain the unexpected mid-IR colour curves, and further supports the use of mid-IR colours for direct metallicity estimates of Cepheids. Given the temperature of our sources, we expect the $4.6~\mu$m band-head absorption to be substantially stronger than our CO(1-0) detections and should be easily detected with {\em JWST} NIRSpec.
Two sources are detected in CO absorption, most likely caused by a molecular gas cloud being present at the offset location used for the sky subtraction. To avoid this issue in future attempts to observe CO in Cepheids in the galactic plane we will use IRAM 30m's position switching mode, directed to a region of sky free from molecular gas clouds. The detections of CO presented in this paper have important consequences for future projects in this field: The measured fluxes will allow future studies at 1~mm \& 3~mm to more accurately estimate the observing time needed to detect emission from these stars. Finally, the confirmation of CO in Cepheid atmospheres highlights the need to study the variability of the absorption feature at 4.6$~\mu$m proposed by \citetalias{Scowcroft2016_CO} which can only be achieved with dedicated observations by {\em JWST}.
\section*{Data Availability}
The data underlying this article were obtained through observations at the IRAM 30m telescope (project 154-22). Raw data are available from the IRAM Archive on direct request to IRAM, and metadata can be viewed through the TAPAS archive at \url{https://tapas.iram.es/tapas/}. Reduced data and spectra will be shared on reasonable request to the corresponding author.
\section*{Acknowledgements}
The team would like to thank the IRAM 30m Astronomer on Duty, Pablo Torne, for the extensive support and assistance they provided during the difficult conditions encountered during the observing run. This publication has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 101004719 (ORP). This work was supported by the Science and Technology Facilities Council [ST/S000526/1]. Ardern is supported by an STFC studentship.
\bibliographystyle{mnras}
|
1,116,691,498,509 | arxiv | \section{Introduction}\label{sec:introduction}}
\IEEEPARstart{T}{he} success of deep neural networks (DNNs) generally depends on the elaborate design of DNN architectures. In large-scale machine learning, especially for tasks such as image and speech recognition, most DNN-based models are over-parameterized to extract the most salient features and to ensure generalization. Such cumbersome models are usually very deep and wide, which require a considerable amount of computation for training and are difficult to be operated in real-time. Thus, to achieve faster speeds, many researchers have been trying to utilize the cumbersome models that are trained to obtain lightweight DNN models, which can be deployed in edge devices. That is, when the cumbersome model has been trained, it can be used to learn a small model that is more suitable for real-time applications or deployment \cite{hinton2015distilling} as depicted in Fig.~\ref{fig:overview_fig}(a).
On the other hand, the performance of DNNs is also heavily dependent on very large and high-quality labels to training datasets. For such a reason, many endeavours have been taken to reduce the amount of labeled training data without hurting too much the performance of DNNs. A popular approach for handling such a lack of data is to \textit{transfer knowledge} from one source task to facilitate the learning on the target task. One typical example is semi-supervised learning in which a model is trained with only a small set of labeled data and a large set of unlabeled data.
Since the supervised cost is undefined for the unlabeled examples, it is crucial to apply consistency costs or regularization methods to match the predictions from both labeled and unlabeled data. In this case, knowledge is transferred within the model that assumes a dual role as \textit{teacher} and \textit{student} \cite{tarvainen2017mean}. For the unlabeled data, the student learns as before; however, the teacher generates targets, which are then used by the student for learning. The common goal of such a learning metric is to form a better teacher model from the student without additional training, as shown in Fig.~\ref{fig:overview_fig}(b). Another typical example is self-supervised learning, where the model is trained with artificial labels constructed by the input transformations (\textit{e}.\textit{g}., rotation, flipping, color change, cropping). In such a situation, the knowledge from the input transformations is transferred to supervise the model itself to improve its performance as illustrated in Fig.~\ref{fig:overview_fig}(c).
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{figures/kt_illustration.pdf}
\caption{Illustrations of KD methods with S-T frameworks. (a) for model compression and for knowledge transfer, \textit{e}.\textit{g}., (b) semi-supervised learning and (c) self-supervised learning.
}
\label{fig:overview_fig}
\end{figure*}
This paper is about \textit{knowledge distillation (KD) and student-teacher (S-T) learning}, a topic that has been actively studied in recent years. Generally speaking, KD is widely regarded as a primary mechanism that enables humans to quickly learn new complex concepts when given only small training sets with the same or different categories \cite{gutstein2008knowledge}. In deep learning, KD is an effective technique that has been widely used to transfer information from one network to another network whilst training constructively. KD was first defined by \cite{bucilua2006model} and generalized by Hinton \textit{et al}. ~\cite{hinton2015distilling}. KD has been broadly applied to two distinct fields: model compression (refer to Fig.~\ref{fig:overview_fig}(a)) and knowledge transfer (refer to Fig.~\ref{fig:overview_fig} (b) and (c)). For model compression, a smaller student model is trained to mimic a pretrained larger model or an ensemble of models. Although various forms of knowledge are defined based on the purpose, one common characteristic of KD is symbolized by its\textit{ S-T framework}, where the model providing knowledge is called the teacher and the model learning the knowledge is called the student.
In this work, we focus on analyzing and categorizing existing KD methods accompanied by various types of S-T structures for model compression and knowledge transfer. We review and survey this rapidly developing area with particular emphasis on the recent progress.
Although KD has been applied to various fields, such as visual intelligence, speech recognition, natural language processing (NLP), etc.,
this paper mainly focuses on the KD methods in the vision field, as most demonstrations have been done on computer vision tasks. KD methods used in NLP and speech recognition can be conveniently explained using the KD prototypes in vision.
As the most studied KD methods are for model compression, we systematically discuss the technical details, challenges, and potentials. Meanwhile, we also concentrate on the KD methods for knowledge transfer in semi-supervised learning, self-supervised learning, etc., and we highlight the techniques that take S-T learning as a way of learning metric.
We explore some fundamental questions that have been driving this research area. Specifically, what is the theoretical principle for KD and S-T learning? What makes one distillation method better than others? Is using multiple teachers better than using one teacher? Do larger models always make better teachers and teach more robust students? Can a student learn knowledge only if a teacher model exists? Is the student able to learn by itself? Is off-line KD always better than online learning?
With these questions being discussed, we incorporate the potentials of existing KD methods and prospect the future directions of the KD methods together with S-T frameworks. We especially stress the importance of recently developed technologies, such as neural architecture search (NAS), graph neural networks (GNNs), and gating mechanisms for empowering KD. Furthermore, we also emphasize the potential of KD methods for tackling challenging problems in particular vision fields such as
360$^\circ$ vision and event-based vision.
The main contributions of this paper are three-fold:
\begin{itemize}
\item We give a comprehensive overview of KD and S-T learning methods, including problem definition, theoretical analysis, a family of KD methods with deep learning, and vision applications.
\item We provide a systematic overview and analysis of recent advances of KD methods and S-T frameworks hierarchically and structurally and offer insights and summaries for the potentials and challenges of each category.
\item We discuss the problems and open issues and identify new trends and future direction to provide insightful guidance in this research area.
\end{itemize}
The organization of this paper is as follows. First, we explain why we need to care about KD and S-T learning in Sec.2. Then, we provide a theoretical analysis of KD in Sec.3. Section 3 is followed by Sec.4 to Sec.8, where we categorize the existing methods and analyze their challenges and potential. Fig.~\ref{fig:taxonomy_KD} shows the taxonomy of KD with S-T learning to be covered in this survey in a hierarchically-structured way. In Sec.9, based on the taxonomy, we will discuss the answers to the questions raised in Sec.1. Section 10 will present the future potentials of KD and S-T learning, followed by a conclusion in Sec.11.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.98\textwidth]{figures/taxonomy_revised.pdf}
\caption{Hierarchically-structured taxonomy of knowledge distillation with S-T learning.}
\label{fig:taxonomy_KD}
\end{figure*}
\section{What is KD and Why Concern it? }
\noindent \textbf{What's KD?} Knowledge Distillation (KD) was first proposed by \cite{bucilua2006model} and expanded by \cite{hinton2015distilling}. KD refers to the method that helps the training process of a smaller student network under the supervision of a larger teacher network. Unlike other compression methods, KD can downsize a network regardless of the structural difference between the teacher and the student network.
In \cite{hinton2015distilling}, the knowledge is transferred from the teacher model to the student by minimizing the difference between the logits (the inputs to the final softmax) produced by the teacher model and those produced by the student model.
However, in many situations, the output of softmax function on the teacher's logits has the correct class at a very high probability, with all other class probabilities very close to zero. In such a circumstance, it does not provide much information beyond the ground truth labels already provided in the dataset. To tackle such a problem, \cite{hinton2015distilling,intellabs_github} introduced the concept of 'softmax temperature', which can make the target to be 'soft.' Given the logits $z$ from a network, the class probability $p_i$ of an image is calculated as \cite{intellabs_github}:
\begin{equation}
p_i = \frac{\exp (\frac{z_i}{\rho})}{\sum_j\exp(\frac{z_i}{\rho})}
\label{soft_label}
\end{equation}
where $\rho$ is the temperature parameter. When $\rho=1$, we get the standard softmax function. As $\rho$ increases, the probability distribution produced by the softmax function becomes softer, providing more information as to which classes the teacher found more similar to the predicted class. The information provided in the teacher model is called \textit{dark knowledge} \cite{hinton2015distilling}. It is this dark knowledge that
affects the overall flow of information to be distilled. When computing the distillation loss, the same $\rho$ used in the teacher is used to compute the logits of the student.
For the images with ground truth, \cite{hinton2015distilling} stated that it is beneficial to train the student model together with the ground truth labels in addition to the teacher's soft labels. Therefore, we also calculate the 'student loss' between the student's predicted class probabilities and the ground truth labels. The overall loss function, composed of the student loss and the distillation loss, is calculated as:
\begin{equation}
\begin{split}
\mathcal{L}_{KD} = \alpha * H(y, \sigma(z_s)) +
\beta * H(\sigma(z_t;\rho), \sigma(z_s; \rho)) \\
= \alpha * H(y, \sigma(z_s) +
\beta * [KL(\sigma(z_t; \rho),
\sigma (z_s,\rho))
+ H(\sigma(z_t))]
\end{split}
\label{loss_student}
\end{equation}
where
$H$ is the loss function, $y$ is the ground truth label, $\sigma$ is the softmax function parameterized by the temperature $\rho$ ($\rho \ne 1$ for distillation loss), and $\alpha$ and $\beta$ are coefficients. $z_s$ and $z_t$ are the logits of the student and teacher respectively.
\noindent \textbf{Why concern KD? } KD has become a field in itself in the machine learning community, with broad applications to computer vision, speech recognition, NLP, etc. From 2014 to now, many research papers \cite{frunk_github,dkozlov_github} have been presented in the major conferences, such as CVPR, ICCV, ECCV, NIPS, ICML, ICLR, etc., and the power of KD has been extended to many learning processes (\textit{e}.\textit{g}., few-shot learning) except to model compression. The trend in recent years is that KD with S-T frameworks has become a crucial tool for knowledge transfer, along with model compression. The rapid increase in scientific activity on KD has been accompanied and nourished by a remarkable string of empirical successes both in academia and industry. The particular highlights on some representative applications are given in Sec.9, and
in the following Sec.\ref{theoretical_analysis}, we provide a systematic theoretical analysis.
\section{A theoretical analysis of KD}
\label{theoretical_analysis}
Many KD methods have been proposed with various intuitions. However, there is no commonly agreed theory as to how knowledge is transferred, thus making it difficult to effectively evaluate the empirical results and less actionable to design new methods in a more disciplined way.
Recently, Ahn \textit{et al}.~\cite{Ahn_2019_CVPR}, Hegde \textit{et al}. ~\cite{hegde2019variational} and Tian \textit{et al}. ~\cite{tian2019contrastive} formulate KD as a maximization of mutual information between the representations of the teacher and the student networks. Note that the representations here can refer to either the logits information or the intermediate features.
From the perspective of representation learning and information theory, the mutual information reflects the joint distribution or mutual dependence between the teacher and the student and quantifies how much information is transferred. We do agree that maximizing the mutual information between the teacher and the student is crucial for learning constructive knowledge from the teacher. We now give a more detailed explanation regarding this.
Based on Bayes's rule, the mutual information between two paired representations can be defined as:
\begin{equation}
\centering
\begin{split}
I(T;S) = H(R(T)) - H(R(T)|R(S)) \\
= -\mathbb{E}_T[\log p(R(T))] + \mathbb{E}_{T,S}[\log p((R(T)|R(S))]
\end{split}
\label{mutual_info}
\end{equation}
where $R(T)$ and $R(S)$ are the representations from both the teacher and the student, and $H(\cdot)$ is the entropy function. Intuitively, the mutual information is to increase degree of certainty in the information provided in $R(T)$ when $R(S)$ is known. Therefore, maximizing $\mathbb{E}_{T,S}[\log p((R(T)|R(S))]$ w.r.t. the parameters of the student network $S$ increases the lower bound on mutual information. However, the true distribution of $p((R(T)|R(S))$ is unknown, instead it is desirable to estimate $p((R(T)|R(S))$ by fitting a variations distribution $q((R(T)|R(S))$ to approximate the true distribution $p((R(T)|R(S))$. Then Eqn.~\ref{mutual_info} can be rewritten as:
\begin{equation}
\centering
\begin{split}
I(T;S) = H(R(T)) + \mathbb{E}_{T,S}[\log p(R(T)|R(S))] \\
= H(R(T)) + \mathbb{E}_{T,S}[\log q((R(T)|R(S))] + \\ \mathbb{E}_S[KL(p(R(T)|R(S))||q(R(T)|R(S))] \\
\end{split}
\label{vid_mutual}
\end{equation}
Assuming there is sufficiently expressive way of modeling $q$, Eqn.~\ref{vid_mutual} can be updated as:
\textbf{\begin{equation}
\centering
\begin{split}
I(T;S) \ge H(R(T)) + \mathbb{E}_{T,S}[\log q((R(T)|R(S))]
\end{split}
\label{lower_bound}
\end{equation}}
Note that the last term in Eqn.~\ref{vid_mutual} is non-negative since $KL(\cdot)$ function is non-negative and $H(R(T))$ is constant w.r.t the parameters to be optimized.
By modeling $q$, it is easy to quantify the amount of knowledge being learned by the student. In general, $q$ can be modeled by a Gaussian distribution, Monte Carlo approximation, or noise contrastive estimation (NCE). We do believe that theoretically explaining how KD works is connected to representation learning, where the correlations and higher-order output dependencies between the teacher and the student are captured. The critical challenge is increasing the lower bounds of information, which is also pointed out in \cite{tian2019contrastive}.
In summary, we have theoretically analyzed how KD works and mentioned that the \textit{representation} of knowledge is crucial for the transfer of knowledge and learning of the student network. Explicitly dealing with the representation of knowledge from the teacher is significant and challenging, because the knowledge from the teacher expresses a more general learned information (\textit{e}.\textit{g}. feature information, logits, data usage, etc.) that is helpful for building up a well-performing student. In the following sections, we will provide a hierarchically-structured taxonomy for the KD methods regarding how the information is transferred for both teacher and student, how knowledge is measured, and how the teacher is defined.
\section{KD based on the number of teachers}
\subsection{Distillation from one teacher}
\label{one_teacher}
\noindent \textbf{Overall insight:} \textit{Transferring knowledge from a large teacher network to a smaller student network can be achieved using either the logits or feature information from the teacher.}
\subsubsection{Knowledge from logits}
\noindent\textbf{Softened labels and regularization.}
Hinton \textit{et al}. ~\cite{hinton2015distilling} and Ba and Caruana ~\cite{ba2014deep} propose to shift the knowledge from teacher network to student network by learning the class distribution via softened softmax (also called 'soft labels') given in Eqn. (\ref{soft_label}). The softened labels are in fact achieved by introducing temperature scaling to increase of small probabilities. These KD methods achieved some surprising results on vision and speech recognition tasks. Recently, Mangalam \textit{et al}. ~\cite{mangalam2018compressing} introduce a special method based on class re-weighting to compress U-net into a smaller version. Re-weighting of the classes, in fact, softens the label distribution by obstructing inherent class imbalance. Compared to \cite{hinton2015distilling}, some works such as Ding \textit{et al}. ~\cite{ding2019adaptive}, Hegde \textit{et al}.~\cite{hegde2019variational}, Tian \textit{et al}.~\cite{tian2019contrastive}, Cho \textit{et al}. ~\cite{cho2019building} and Wen \textit{et al}. ~\cite{wen2019preparing}, point out that the trade-off (see Eqn.~\ref{loss_student}) between the soft label and the hard label is rarely optimal, and since $\alpha$, $\beta$ and $T$ are fixed during training time, it lacks enough flexibility to cope with the situation without the given softened labels. Ding \textit{et al}. ~\cite{ding2019adaptive} instead propose \textit{residual label} and \textit{residual loss} to enable the student to use the erroneous experience during the training phase, preventing over-fitting and improving the performance. Similarly, Tian \textit{et al}. ~\cite{tian2019contrastive} formulate the teacher's knowledge as structured knowledge and train a student to capture significantly more \textit{mutual information} during contrastive learning. Hegde \textit{et al}. ~\cite{hegde2019variational} propose to train a \textit{variational} student by adding sparsity regularizer based on variational inference, similar to the method in \cite{Ahn_2019_CVPR}. The sparsification of the student training reduces over-fitting and improves the accuracy of classification. Wen \textit{et al}. ~\cite{wen2019preparing} notice that the knowledge from the teacher is useful, but uncertain supervision also influences the result. Therefore, they propose to fix the incorrect predictions (knowledge) of the teacher via smooth regularization and avoid overly uncertain supervision using dynamic temperature.
On the other hand, Cho \textit{et al}. ~\cite{cho2019efficacy}, Yang \textit{et al}. ~\cite{yang2019snapshot} and Liu \textit{et al}. ~\cite{liu2019knowledge} focus on different perspectives of regularization to avoid under-/over-fitting. Cho \textit{et al}. ~\cite{cho2019efficacy} discover that early-stopped teacher makes a better student especially when the capacity of the teacher is larger than the student's. Stopping the training of the teacher early is akin to regularizing the teacher, and stopping knowledge distillation close to convergence allows the student to fit the training better. Liu \textit{et al}. ~\cite{liu2019knowledge} focus on modeling the distribution of the parameters as prior knowledge, which is modeled by aggregating the distribution (logits) space from the teacher network. Then the prior knowledge is penalized by a sparse recording penalty for constraining the student to avoid over-regularization. Mishra \textit{et al}. ~\cite{mishra2017apprentice} combine network quantization with model compression by training an apprentice using KD techniques and showed that the performance of low-precision networks could be significantly improved by distilling the logits of the teacher network. Yang \textit{et al}. ~\cite{yang2019snapshot} propose a snapshot distillation method to perform S-T (similar network architecture) optimization in \textit{one generation}. Their method is based on a cycle learning rate policy (refer to Eqn.~\ref{loss_student} and Eqn.~\ref{loss_student_onegenration}) in which the last snapshot of each cycle (\textit{e}.\textit{g}.,$W_{T}^{l-1}$ in iteration $l-1$) serves as a teacher in the next cycle (\textit{e}.\textit{g}., $W_{T}^{l}$ in iteration $l$). Thus, the idea of snapshot distillation is to extract supervision signals in earlier epochs in the same generation to make sure the difference between teacher and student is sufficiently large to avoid under-fitting. The snapshot distillation loss can be described as:
\begin{equation}
\begin{split}
\mathcal{L}(x, W_{l-1}) = \alpha * H(y, \sigma(z_s^{l-1} ; \rho=1) + \\
\beta * H(\sigma(z_t^{l}; \rho=\tau), \sigma (z_t^{l-1}, \rho=\tau))
\end{split}
\label{loss_student_onegenration}
\end{equation}
where the $W_{l-1}$ is the weights of student at iteration $l-1$. $z_s^{l-1}$ and $z_t^{l-1}$ represent the logits of student and teacher at iteration $l-1$. \textit{More detailed analysis for the methods with mutual information and one generation will be discussed in Sec.~\ref{online_kd}}.
\begin{table*}[t!]
\caption{A taxonomy of KD methods using logits. The given equations here are the generalized objective functions, and they may vary in individual work. }
\small
\begin{center}
\begin{tabular}{c|c|c|c|c}
\hline
Method & Sub-category & Description & \thead{KD objective \\function}&References \\
\hline \hline
\multirow{8}{*}{\thead{KD from\\ logits}}& \thead{Softened labels and\\ regularization} & \thead{Distillation using soft labels \\ and add regularizatio to \\avoid under-/over-fitting} & Eqn.~\ref{loss_student_onegenration}& \thead{\cite{hinton2015distilling, ba2014deep,mangalam2018compressing,hegde2019variational, tian2019contrastive, cho2019building, wen2019preparing} \\ \cite{ hegde2019variational,Ahn_2019_CVPR,wen2019preparing,cho2019efficacy,yang2019snapshot,liu2019knowledge,mishra2017apprentice,mishra2017apprentice}} \\ \cline{2-5}
& \thead{Learning from \\noisy labels} & \thead{Adding noise \\or using noisy data } & \thead{Eqn.~\ref{loss_student_onegenration} \\or Eqn.~\ref{loss_student_noise}} & \cite{li2017learning, xie2019self, xu2019positive,sarfraz2019noisy,srivastava2014dropout}\\ \cline{2-5}
& Imposing strictness & \thead{Adding optimization methods \\to teacher or student} & \thead{ Eqn.~\ref{loss_tsd} or \\ Eqn.~\ref{loss_student_onegenration}} & \cite{yang2019training,yu2019learning, arora2019knowledge, park2019relational, peng2019correlation,furlanello2018born, wang2016relational}\\ \cline{2-5}
& Ensemble of distribution & \thead{Estimating model or\\ data uncertainty} & Eqn.~\ref{loss_student_ensemble} & \cite{mirzadeh2019improved, cho2019efficacy, malinin2019ensemble, zhang2018deep,phuong2019distillation}\\ \hline
\end{tabular}
\end{center}
\label{table:logit_comp}
\end{table*}
\noindent\textbf{Learning from noisy labels} \cite{li2017learning,xie2019self,xu2019positive,sarfraz2019noisy} propose methods that utilize the similar knowledge (softened labels) as in \cite{hinton2015distilling} but focus on data issue. Specifically, \cite{li2017learning} assume that there is a small clean dataset ${D}_c$ and a large noisy dataset ${D}_n$, while \cite{xie2019self} and \cite{xu2019positive} use both labeled and unlabeled data to improve the performance of student. In \cite{li2017learning}, the aim of distillation is to use the large amount of noisy data ${D}_n$ to augment the small clean dataset ${D}_c$ to learn a better visual representation and classifier. That is, the knowledge is distilled from the small clean dataset ${D}_c$ to facilitate a better model from the entire noisy dataset ${D}_n$. The method is essentially different from \cite{hinton2015distilling} focusing on inferior model instead of inferior dataset. The same loss function in Eqn. ~\ref{loss_student} is used, except $z_t= \sigma[f_{{D}_c}(x)]$, where $f_{{D}_c}$ is an auxiliary model trained from the clean dataset ${D}_c$. Furthermore, a risk function on the unreliable label $\bar{y}$ is defined as $R_{ \bar{y}} = \E_{{D}_t}[||\bar{y}- y^*||]^2$, where $y^*$ is the unknown ground truth label and ${D}_t$ is the unseen test dataset. $R_{ \bar{y}}$ is an indicator that measures the level of noise in the distillation process.
Xu \textit{et al}.~\cite{xu2019positive} probes a positive-unlabeled classifier for addressing the problem of requesting the entire original training data, which can not be easily uploaded to the cloud. \cite{xie2019self} trains a \textit{noisy} student by next three steps: 1) train a teacher model on labeled data, 2) use the teacher to generate pseudo labels on unlabeled images, and 3) train a student model on the combination of labeled images and pseudo labeled images while injecting noise (adversarial perturbation) to the student for better generalization and robustness. This way, the student generalizes better than the teacher.
Similarly, \cite{sarfraz2019noisy} study adversarial perturbation and consider it as a crucial element in improving both the generalization and the robustness of the student. Based on how humans learn, two learning theories for the S-T model are proposed: fickle teacher and soft randomization. The fickle teacher model is to transfer the teacher's uncertainty to the student using Dropout \cite{srivastava2014dropout} in the teacher model. The soft randomization method is to improve the adversarial robustness of student model by adding Gaussian noise in the knowledge distillation. In this setting, the original distillation objective for the student in Eqn.~\ref{loss_student} can be updated as:
\begin{equation}
\begin{split}
\mathcal{L}(x+\delta, W) = \alpha * {H}(y, \sigma(z_s ; \rho=1) + \\
\beta * {H}(\sigma(z_t; \rho=\tau), \sigma (z_s, \rho=\tau))
\end{split}
\label{loss_student_noise}
\end{equation}
where $\delta$ is the variation of adversarial perturbation. It is shown that using the teacher model trained on clean images to train the student model with adversarial perturbation can retain the adversarial robustness and mitigate the loss in generalization.
\noindent\textbf{Imposing strictness in distillation.} In contrast, Yang \textit{et al}. ~\cite{yang2019training}, Yu \textit{et al}. ~\cite{yu2019learning}, Arora \textit{et al}. ~\cite{arora2019knowledge}, RKD \cite{park2019relational} and Peng \textit{et al}. ~\cite{peng2019correlation} shift to a new perspective focusing more on putting \textit{strictness} to the distillation process via optimization (\textit{e}.\textit{g}., distribution embedding, mutual relations, etc). In particular, \cite{yang2019training} initiates to put strictness on the teacher while \cite{yu2019learning} proposes two teaching metrics to impose strictness on the student. Yang \textit{et al}.~ observe that, except learning \textit{primary class} (namely, the ground truth), learning \textit{secondary class} ( high confidence scores in the \textit{dark knowledge} in \cite{hinton2015distilling}) may help to alleviate the risk of the student over-fitting. They thus introduce a framework of optimizing neural networks in \textit{generations} (namely, iterations), which requires training a patriarchal model ${M}^0$ only supervised by the dataset. After $m$ generations, the student ${M}^m$ is trained by $m$-th generation with the supervision of a teacher ${M}^{m-1}$. Since the secondary information is crucial for training a robust teacher, a fixed integer $K$ standing for the semantically similar class is chosen for each image, and the gap between the confidence scores of the primary class and other $K-1$ classes with highest scores is computed, This can be described as:
\begin{equation}
\begin{split}
\mathcal{L}(x, W^T) = \alpha * {H}(y, \sigma(z_t ; \rho=1) + \\
\beta * [f_{a_1}^T - \frac{1}{K-1} \sum_{k=2}^{K}f_{a_k}^T]
\end{split}
\label{loss_tsd}
\end{equation}
where $f_{a_{k}}$ indicates the $k$-th largest elements of the output (logits) $z_t$.
Note that this S-T optimization is similar to BAN \cite{furlanello2018born}; however, the goal here is to help the student learn inter-class similarity and prevent over-fitting. Different from the teacher in \cite{furlanello2018born}, the teacher here is deeper and larger than the student.
\cite{yu2019learning} extends \cite{hinton2015distilling} for metric learning by using embedding networks to project the information (logits) learned from images to the embedding space. The embeddings are typically used to perform distance computation between the data pairs of a teacher and a student. From this point of view, the knowledge computed based on the embedding network is the actual knowledge as it represents the data distribution. They design two different teachers: absolute teacher and relative teacher. For the absolute teacher, the aim is to minimize the distance between the teacher and student embeddings while the aim for the relative teacher is to enforce the student to learn any embedding as long as it results in a similar distance between the data points. They also explore hints \cite{hinton2015distilling} and attention \cite{zagoruyko2016paying} to strengthen the distillation of embedding networks. \textit{We will give more explicit explanations of these two techniques in Sec.~\ref{sec_feamap_dis}.}
\cite{arora2019knowledge} proposes an embedding module that captures interactions between query and document information for question answering. The embedding of the output representation (logits) includes a simple attention model with a query encoder, a prober history encoder, a responder history encoder, and a document encoder. The attention model minimizes the summation of cross-entropy loss and KL-divergence loss, inspired by \cite{hinton2015distilling}.
On the other hand, \cite{wang2016relational} and RKD \cite{park2019relational} consider another type of strictness, namely the \textit{mutual relation or relation knowledge} of the two examples in the learned representations for both the teacher and the student. This approach is very similar to the relative teacher in ~\cite{yu2019learning} since both aim to measure the distance between the teacher's and the student's embeddings. However, RKD \cite{park2019relational} also considers the angle-wise relational measure, similar to persevering secondary information in ~\cite{yang2019training}.
\noindent\textbf{Ensemble of distribution.}
Although various methods have been proposed to extract knowledge from logits, some works \cite{mirzadeh2019improved, cho2019efficacy, malinin2019ensemble, zhang2018deep} show that KD is not always practical due to knowledge uncertainty. The performance of the student degrades when the gap between the student and the teacher is large. \cite{malinin2019ensemble} points out that estimating the model's uncertainty is crucial since it ensures a more reliable knowledge to be transferred. They stress on the ensemble approaches to estimate the data uncertainty and the distributional uncertainty. To estimate the distributional uncertainty, an ensemble distribution distillation approach anneals the temperature of the softmax to not only capture the mean of ensemble soft labels but also the diversity of the distribution. Meanwhile,~\cite{phuong2019distillation} proposes a similar approach of matching the distribution of distillation-based multi-exit architectures, in which a sequence of feature layers is augmented with early exits at different depths. By doing so, the loss defined in Eqn.~\ref{loss_student} becomes:
\begin{equation}
\begin{split}
\mathcal{L}(x, W) = \frac{1}{K}\sum_{k=1}^K[ \alpha * {H}(y, \sigma(p_s^k ; \rho=1) + \\
\beta *{H}(\sigma(p_t^k; \rho=\tau), \sigma (p_s^k, \rho=\tau))]
\end{split}
\label{loss_student_ensemble}
\end{equation}
where $K$ indicates the total number of exits, and $p_s^k$ and $p_t^k$ represent the $ k$-th probabilistic output at exit $k$.
Conversely, \cite{you2017learning, zhang2018deep, papernot2016semi,sau2016deep, arora2019knowledge,tarvainen2017mean, furlanello2018born, lan2018knowledge, song2018collaborative, radosavovic2018data, tan2019multilingual, Vongkulbhisal_2019_CVPR,wu2019distilled, dvornik2019diversity, yang2019model, park2019feed, lee2019stochasticity, chen2019online, mirzadeh2019improved} propose to add more teachers or other auxiliaries, such as teaching assistant and small students, to improve the robustness of ensemble distribution. We will explicitly analyze these approaches in the following Sec.~\ref{multi_teach}.
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{figures/feature_distillation_fig.pdf}
\caption{An illustration of general feature-based distillation.
}
\label{fig:fea_dis_fig}
\end{figure}
\noindent\textbf{Summary.} Table.~\ref{table:logit_comp} summarizes the KD methods that use logits or `soft labels'. We divide these methods into four categories. In overall, distillation using logits needs to transfer the dark knowledge to avoid over-/under-fitting. Meanwhile, the gap of model capacity between the teacher and the student is also very crucial for effective distillation. Moreover, the drawbacks of learning from logits are obvious. First, the effectiveness of distillation is limited to softmax loss and relies on the number of classes. Second, it is impossible to apply these methods to the KD problems in which there are no labels (\textit{e}.\textit{g}., low-level vision).
\noindent \textbf{Open challenges:} The original idea in \cite{hinton2015distilling} is in its apparent generality: any student can learn from any teacher; however, it is shown that this promise of generality is hard to be achieved on some datasets \cite{zagoruyko2016paying, cho2019efficacy} (\textit{e}.\textit{g}., ImageNet \cite{deng2009imagenet}) even when regularization or strictness techniques are applied. When the capacity of the student is too low, it is hard for the student to incorporate the logits information of the teacher successfully. Therefore, it is expected to improve the generality and provide a better representation of logits information, which can be easily absorbed by the student.
\subsubsection{Knowledge from the intermediate layers}
\textbf{Overall insight:} \textit{Feature-based distillation enables learning richer information from the teacher and provides more flexibility for performance improvement. }
\label{sec_feamap_dis}
Apart from distilling knowledge from the softened labels, Romero \textit{et al}. ~\cite{romero2014fitnets} initially introduce \textit{hint} learning rooted from \cite{hinton2015distilling}. A hint is defined as the outputs of a teacher's hidden layer, which helps guide the student's learning process. The goal of student learning is to learn a feature representation that is the optimal prediction of the teacher's intermediate representations. Essentially, the function of hints is a form of regularization; therefore, a pair of hint and guided (a hidden layer of the student) layer has to be carefully chosen such that the student is not over-regularized.
Inspired by \cite{romero2014fitnets}, many endeavours have been taken to study the methods to choose, transport and match the hint layer (or layers) and the guided layer (or layers) via various layer transform (\textit{e}.\textit{g}., transformer \cite{heo2019comprehensive, kim2018paraphrasing}) and distance (\textit{e}.\textit{g}., MMD \cite{huang2017like}) metrics. Generally, the hint learning objective can be written as:
\begin{equation}
\mathcal{L}(F_T, F_S) = D({TF}_t(F_T), {TF}_s(F_S))
\label{fea_dis_loss}
\end{equation}
Where $F_T$ and $F_S$ are the selected hint and guided layers of teacher and student. ${TF}_t$ and ${TF}_s$ are the transformer and regressor functions for the hint layer of teacher and guided layer of student. $D(\cdot)$ is the distance function(\textit{e}.\textit{g}., $l_2$) measuring the similarity of the hint and the guided layers.
Fig.~\ref{fig:fea_dis_fig} depicts the general paradigm of feature-based distillation. It is shown that various intermediate feature representations can be extracted from different positions and are transformed with a certain type of regressor or transformer. The similarity of the transformed representations is finally optimized via an arbitrary distance metrics $D$ (\textit{e}.\textit{g}., $L_1$ or $L_2$ distance). In this paper, we carefully scrutinize various design considerations of feature-based KD methods and summarize four key factors that are often considered: \textit{transformation of the hint, transform of the guided layer, position of the selected distillation feature, and distance metric} \cite{heo2019comprehensive}. In the following parts, we will analyze and categorize all existing feature-based KD methods concerning these four aspects.
\begin{table*}[t!]
\caption{A taxonomy of knowledge distillation from the intermediate layers (feature maps). KP incidates knowledge projection.
}
\small
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
\hline
Method & Teacher's ${TF}_t$ & Student's ${TF}_s$& Distillation position & Distance metric & Lost knowledge \\
\hline\hline
FitNet \cite{romero2014fitnets} &None& $1 \times 1$ Conv & Middle layer & $L_1$ & None \\ \hline
AT \cite{zagoruyko2016paying} &Attention map& Attention map & End of layer group& $L_2$ & Channel dims \\ \hline
KP \cite{Zhang2017KnowledgePF} & Projection matrix & Projection matrix &Middle layers & $L_1$ + KP loss & Spatial dims\\ \hline
FSP \cite{yim2017gift} &FSP matrix & FSP matrix &End of layer group & $L_2$ & Spatial dims \\ \hline
FT \cite{kim2018paraphrasing} &Encoder-decoder& Encoder-decoder & End of layer group & $L_1$ & Channel + Spatial dims \\ \hline
AT \cite{zagoruyko2016paying} &Attention map& Attention map & End of layer group& $L_2$ & Channel dimensions \\ \hline
MINILM \cite{wang2020minilm} &Self-ttention & Self-attention & End of layer group& KL & Channel dimensions \\ \hline
Jacobian \cite{srinivas2018knowledge} & Gradient penalty & Gradient penalty & End of layer group & $L_2$ & Channel dims \\ \hline
SVD \cite{yim2017gift} & Truncated SVD & Truncated SVD & End of layer group & $L_2$ & Spatial dims \\ \hline
VID \cite{Ahn_2019_CVPR} &None& $1 \times 1$ Conv & Middle layers & $KL$ & None \\ \hline
IRG \cite{liu2019knowledge} &Instance graph & Instance graph & Middle layers & $L_2$ & Spatial dims \\ \hline
RCO \cite{jin2019knowledge} &None & None & Teacher's train route & $L_2$ & None\\ \hline
SP \cite{tung2019similarity} &Similarity matrix & Similarity matrix & Middle layer & Frobenius norm & None\\ \hline
MEAL \cite{shen2019meal} &Adaptive pooling & Adaptive pooling &End of layer group & $L_{1/2}$/KL/$L_{GAN}$ & None\\ \hline
Heo \cite{shen2019meal} &Margin ReLU & $1 \times 1$ Conv & Pre-ReLU & Partial $L_2$ & Negative features \\ \hline
AB \cite{heo2018knowledge} &Binarization & $1 \times 1$ Conv & Pre-ReLU & Margin $L_2$ & feature values \\ \hline
Chung \cite{chung2020featuremaplevel} &None & None & End of layer & $L_{GAN}$ & None \\ \hline
Wang \cite{wang2019distilling} & None & Adaptation layer & Middle layer & Margin $L_1$ & Channel + Spatial dims \\ \hline
KSANC \cite{changyong2019knowledge} & Average pooling & Average pooling & Middle layers & $L_2$ + $L_{GAN}$ & Spatial dims \\ \hline
Kulkarni \cite{kulkarni2019stagewise} & None & None &End of layer group & $L_2$ & None \\ \hline
IR \cite{aguilar2019knowledge} & Attention matrix & Attention matrix &Middle layers & KL+ Cosine & None \\ \hline
Liu \cite{liu2019knowledge} & Transform matrix & Transform matrix &Middle layers & KL & Spatial dims \\ \hline
NST \cite{huang2017like} & None & None & Intermediate layers & MMD & None \\ \hline
Gao \cite{gao2020residual} & None & None & Intermediate layers & $L_2$ & None \\
\hline
\end{tabular}
\end{center}
\label{table:fea_comp}
\end{table*}
\noindent\textbf{Transformation of hints}
As pointed in \cite{Ahn_2019_CVPR}, the knowledge of teacher should be easy to learn as the student. To do this, teacher's hidden features are usually converted by a transformation function $T_t$. Note that the transformation of teacher's knowledge is a very crucial step for feature-based KD since there is a risk of losing information in the process of transformation. The transformation methods of teacher's knowledge in AT \cite{zagoruyko2016paying}, MINILM \cite{wang2020minilm}, FSP \cite{yim2017gift}, ASL\cite{li2019layer}, Jacobian \cite{srinivas2018knowledge}, KP \cite{Zhang2017KnowledgePF}, SVD \cite{lee2018self}, SP \cite{tung2019similarity}, MEAL \cite{shen2019meal}, KSANC \cite{changyong2019knowledge}, and NST \cite{huang2017like} cause the knowledge to be missing due to the reduction of feature dimension. Specifically, AT \cite{kim2018paraphrasing} and MINILM \cite{wang2020minilm} focus on attention mechanisms (\textit{e}.\textit{g}., self-attention \cite{vaswani2017attention}) via an attention transformer $T_t$ to transform the activation tensor $F \in \mathbb{R}^{C \times H \times W}$ to $C$ feature maps $F \in \mathbb{R}^{H \times W}$. FSP \cite{yim2017gift} and ASL \cite{li2019layer} calculate the information flow of the distillation based on the Gramian matrices, through which the tensor $F \in \mathbb{R}^{C \times H \times W}$ is transformed to $G \in \mathbb{R}^{C \times N}$, where $N$ represents the number of matrices. Jacobian \cite{srinivas2018knowledge} and SVD \cite{lee2018self} map the tensor $F \in \mathbb{R}^{C \times H \times W}$ to $G \in \mathbb{R}^{C \times N}$ based on Jacobians using first-order Taylor series and truncated SVD, respectively, inducing information loss. KP \cite{Zhang2017KnowledgePF} projects $F \in \mathbb{R}^{C \times H \times W}$ to $M$ feature maps $F \in \mathbb{R}^{M \times H \times W}$, causing loss of knowledge. Similarly, SP \cite{tung2019similarity} proposes a similarity-preserving knowledge distillation method based on the observation that semantically similar inputs tend to elicit similar activation patterns. To achieve this goal, the teacher's feature $F \in \mathbb{R}^{B \times C \times H \times W}$ is transformed to $G \in \mathbb{R}^{B \times B}$, where $B$ is the batch size. The $G$ encodes the similarity of the activations at the teacher layer, but leads to an information loss during the transformation. MEAL \cite{shen2019meal} and KSANC \cite{changyong2019knowledge} both use \textit{pooling} to align the intermediate map of the teacher and student, leading to an information loss when transforming the teacher's knowledge. NST \cite{huang2017like} and PKT \cite{passalis2018learning} match the distributions of neuron selectivity patterns and the affinity of data samples between the teacher and the student networks. The loss functions are based on minimizing the maximum mean discrepancy (MMD) and Kullback-Leibler (KL) divergence between these distributions respectively, thus causing information loss when selecting neurons.
On the other hand, FT \cite{kim2018paraphrasing} proposes to extract good \textit{factors} through which transportable features are made. The transformer ${TF}_t$ is called the \textit{paraphraser} and the transformer ${TF}_s$ is called the \textit{translator}. To extract the teacher factors, an adequately trained paraphraser is needed. Meanwhile, to enable the student to assimilate and digest the knowledge according to its own capacity, a user-defined paraphrase ratio is used in the paraphraser to control the factor of the transfer. Heo \textit{et al}. ~\cite{heo2018knowledge} use the original teacher's feature in the form of binarized values, namely via a separating hyperplane (activation boundary (AB)) that determines whether neurons are activated or deactivated. Since AB only considers the activation of neurons and not the magnitude of neuron response, there is information loss in the feature binarization process. Similar information loss happens in IRG \cite{liu2019knowledge}, where the teacher's feature space is transformed to a graph representation with vertices and edges where the relationship matrices are calculated. IR \cite{aguilar2019knowledge} distills the internal representations of the teacher model to the student model. However, since multiple layers in the teacher model are compressed into one layer of the student model, there is information loss when matching the features. Heo \textit{et al}. ~\cite{heo2019comprehensive} design ${TF}_t$ with a margin ReLU function to exclude the negative (adverse) information and to allow positive (beneficial) information. The margin $m$ is determined based on batch normalization \cite{ioffe2015batch} after $1 \times 1$ convolution in the student's transformer ${TF}_s$.
Conversely, FitNet \cite{romero2014fitnets}, RCO \cite{jin2019knowledge}, Chung \textit{et al}. ~\cite{chung2020featuremaplevel}, Wang \textit{et al}. ~\cite{wang2019distilling}, Gao \textit{et al}. ~\cite{gao2020residual} and Kulkarni \textit{et al}. ~\cite{kulkarni2019stagewise} do not add additional transformation to the teacher's knowledge; this leads to no information loss from teacher's side. However, not all knowledge from the teacher is beneficial for the student. As pointed by \cite{heo2019comprehensive}, features include both adverse and beneficial information. For effective distillation, it is important to impede the use of adverse information and to avoid missing the beneficial information.
\noindent\textbf{Transformation of the guided features}
The transformation ${TF}_s$ of the guided features (namely, student transform) of the student is also an important step for effective KD. Interestingly, the SOTA works such as AT \cite{zagoruyko2016paying}, MINILM \cite{wang2020minilm}, FSP \cite{yim2017gift}, Jacobian \cite{srinivas2018knowledge}, FT \cite{kim2018paraphrasing}, SVD \cite{lee2018self}, SP \cite{tung2019similarity}, KP \cite{ Zhang2017KnowledgePF}, IRG \cite{liu2019knowledge}, RCO \cite{jin2019knowledge},MEAL \cite{shen2019meal}, KSANC \cite{changyong2019knowledge}, NST \cite{huang2017like}, Kulkarni \textit{et al}. ~\cite{kulkarni2019stagewise}, Gao \textit{et al}. ~\cite{gao2020residual} and Aguilar \textit{et al}. ~\cite{aguilar2019knowledge} use the same ${TF}_s$ as the ${TF}_t$, which means the same amount of information might be lost in both transformations of the teacher and the student.
Different from the transformation of teacher, FitNet \cite{romero2014fitnets}, AB \cite{heo2018knowledge}, Heo \textit{et al}. ~\cite{heo2019comprehensive}, and VID \cite{Ahn_2019_CVPR} change the dimension of teacher's feature representations and design ${TF}_s$ with a `bottleneck' layer ($1\times1$ convolution) to make the student's features match the dimension of the teacher's features. Note that Heo \textit{et al}. ~\cite{heo2019comprehensive} add a batch normalization layer after a $1\times1$ convolution to calculate the margin of the proposed margin ReLU transformer of the teacher. There are some advantages of using $1 \times 1$ convolution in KD. First, it offers a channel-wise pooling without a reduction of the spatial dimensionality. Second, it can be used to create a one-to-one linear projection of the stack of feature maps. Lastly, the projection created by $1 \times 1$ convolution can also be used to directly increase the number of feature maps in the distillation model. In such a case, the feature representation of student does not decrease but rather increase to match the teacher's representation; this does not cause information loss in the transformation of the student.
Exceptionally, some works focus on a different aspect of the transformation of student's feature representations. Wang \textit{et al}. ~\cite{wang2019distilling} make the student imitate the fine-grained local feature regions close to object instances of the teacher's representations. This is achieved by designing a particular adaptation function ${TF}_s$ to fulfill the imitation task. IR \cite{aguilar2019knowledge} aims to let the student acquire the abstraction in a hidden layer of the teacher by matching the internal representations. That is, the student is taught to know how to compress the knowledge from multiple layers of the teacher into a single layer. In such a setting, the transformation of the student's guided layer is done by a self-attention transformer. Chung \textit{et al}. ~\cite{chung2020featuremaplevel}, on the other hand, propose to impose no transformation to both student and teacher, but rather add a discriminator to distinguish the feature map distributions of different networks (teacher or student).
\noindent\textbf{Distillation positions of features}
In addition to the transformation of teacher's and student's features, distillation position of the selected features is also very crucial in many cases. Earlier, FitNet \cite{romero2014fitnets}, AB \cite{heo2018knowledge}, and Wang \textit{et al}. ~\cite{wang2019distilling} use the end of an arbitrary middle layer as the distillation point. However, this method is shown to have poor distillation performance. Based on the definition of layer group \cite{zagoruyko2016wide}, in which a group of layers have same spatial size, AT \cite{zagoruyko2016paying}, FSP \cite{yim2017gift}, Jacobian \cite{srinivas2018knowledge}, MEAL \cite{shen2019meal}, KSANC \cite{changyong2019knowledge}, Gao \textit{et al}.~\cite{gao2020residual} and Kulkarni \textit{et al}.~\cite{kulkarni2019stagewise} define the distillation position at the end of each layer group, in contrast to FT \cite{kim2018paraphrasing} and NST \cite{huang2017like} where the position lies only at the end of last layer group. Compared to FitNet, FT achieves better results since it focuses more on informational knowledge. IRG \cite{liu2019knowledge} considers all the above-mentioned critical positions; namely, the distillation position lies not only in the end of earlier layer group but also in the end of the last layer group. Interestingly, VID \cite{Ahn_2019_CVPR}, RCO \cite{jin2019knowledge}, Chung \textit{et al}. ~\cite{chung2020featuremaplevel}, SP \cite{tung2019similarity}, IR \cite{aguilar2019knowledge}, and Liu \textit{et al}. ~\cite{liu2019knowledge} generalize the selection of distillation positions by employing variational information maximization \cite{barber2003algorithm}, curriculum learning \cite{bengio2009curriculum}, adversarial learning \cite{goodfellow2014generative}, similarity-presentation in representation learning \cite{horn2016learning}, muti-task learning \cite{clark2019bam}, and reinforcement learning \cite{mnih2016asynchronous}. We will discuss more for these methods in later sections.
\noindent\textbf{Distance metric for measuring distillation}
The quality of KD from teacher to student is usually measured by various distance metrics. The most commonly used distance function is based on $L_1$ or $L_2$ distance. FitNet \cite{romero2014fitnets}, AT \cite{zagoruyko2016paying}, NST \cite{huang2017like}, FSP \cite{yim2017gift}, SVD \cite{lee2018self}, RCO \cite{jin2019knowledge}, FT \cite{kim2018paraphrasing}, KSANC \cite{changyong2019knowledge}, Gao \textit{et al}.~\cite{gao2020residual} and Kulkarni \textit{et al}.~\cite{kulkarni2019stagewise} are mainly based on $L_2$ distance, whereas MEAL \cite{shen2019meal} and Wang \textit{et al}. ~\cite{wang2019distilling} mainly use $L_1$ distance.
On the other hand, Liu \textit{et al}.~\cite{liu2019knowledge} and IR~\cite{aguilar2019knowledge} utilize KL-divergence loss to measure feature similarities. Furthermore, a cosine-similarity loss is adopted by IR \cite{aguilar2019knowledge} and RKD \cite{park2019relational} to regularize the context representation on the feature distributions of teacher and student.
Some works also resort to the adversarial loss for measuring the quality of KD. MEAL \cite{shen2019meal} shows that the student learning the distilled knowledge with discriminators is better optimized than the original model, and the student can learn distilled knowledge from a teacher model with arbitrary structures. Among the works focusing on feature-based distillation,
KSANC \cite{jin2019knowledge} adds an adversarial loss at the last layer of both teacher and student networks, while MEAL adds multi-stage discriminators in the position of every extracted feature representation. It is worth mentioning that using adversarial loss has shown considerable potential in improving the performance of KD. We will explicitly discuss the existing KD techniques based on adversarial learning in the following Sec.~\ref{kd_gan}.
\noindent\textbf{Potentials and open challenges}
Table.~\ref{table:fea_comp} summarizes the existing feature-based KD methods. It is shown that most works employ feature transformations for both the teacher and the student. $L1$ or $L2$ loss is the most commonly used loss for measuring KD quality.
A natural question one may ask is what's wrong with directly matching the features of teacher and student?
If we consider the activation of each spatial position as a feature, the flattened activation map of each filter is a sample of the space of selected neurons with dimension $HW$, which reflects how DNN learns an image \cite{huang2017like}. Thus, when matching distribution, it is less desirable to directly match the samples since the sample density can be lost in the space, as pointed in \cite{romero2014fitnets}. Although \cite{gao2020residual} proposes to distill knowledge by directly matching feature maps, a teaching assistant is introduced to learn the residual errors of between the feature maps of the student and teacher. This approach better mitigates the performance gap between the teacher and student, thus improving generalization capability.
\noindent \textbf{Potentials:} Feature-based methods show more generalization capabilities and quite promising results. In the next research, more flexible ways of determining the representative knowledge of features are expected. The approaches used in representation learning (\textit{e}.\textit{g}., parameter estimation, graph models) might be reasonable solutions for these problems. Additionally, neural architecture search (NAS) techniques may better handle the selection of features. Furthermore, feature-based KD methods are possible for use in cross-domain transfer and low-level vision problems.
\noindent \textbf{Open challenges:}
Although we have discussed most existing feature-based methods, it is still hard to say which one is best. First, it is difficult to measure the different aspects in which information is lost. Additionally, most works randomly choose intermediate features as knowledge, and yet do not provide a reason as to why they can be the representative knowledge among all layers. Third, the distillation position of features is manually selected based on the network or the task. Lastly, multiple features may not represent better knowledge than the feature of a single layer. Therefore, better ways to choose knowledge from layers and to represent knowledge could be explored.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{figures/multiple_teachers.pdf}
\vspace{-15pt}
\caption{Graphical illustration for KD with multiple teachers. The KD methods can be categorized into six types: (a) KD from the ensemble of logits, (b) KD from the ensemble of feature representations via some similarity matrices, (c) unifying various data sources from the same network (teacher) model A to generate various teacher models, (d) obtaining hierarchical or stochastic sub-teacher networks given one teacher network; (e) training a versatile student network from multiple heterogeneous teachers, (f) online KD from diverse peers via ensemble of logits.}
\label{fig:multiple_teachers}
\vspace{-5pt}
\end{figure*}
\subsection{Distillation from multiple teachers}
\label{multi_teach}
\textbf{Overall insight:} \textit{The student can learn better knowledge from multiple teachers, which are more informative and instructive than a single teacher.}
While impressive progress has been achieved under
the common S-T KD paradigm, where knowledge is transferred from one high-capacity teacher network to a student network. The knowledge capacity in this setting is quite limited \cite{park2019feed}, and knowledge diversity is scarce for some special cases, such as cross-model KD \cite{zhang2018better}. To this end, some works probe learning a portable student from \textit{multiple} teachers or an ensemble of teachers. The intuition behind this can be explained in analogy with the cognitive process of human learning. In practice, a student does not solely learn from a single teacher but learn a concept of knowledge better provided with instructive guidance from multiple teachers on the same task or heterogeneous teachers on different tasks. In such a way, the student can amalgamate and assimilate various illustrations of knowledge representations from multiple teacher networks, and build a comprehensive knowledge system \cite{you2017learning, shen2019customizing, sau2016deep}. As a result, many new KD methods \cite{you2017learning, papernot2016semi, ruder2017knowledge, sau2016deep, tarvainen2017mean, furlanello2018born, zhang2018deep, zhu2018knowledge, song2018collaborative, radosavovic2018data,tan2019multilingual, vongkulbhisal2019unifying, wu2019distilled, dvornik2019diversity, yang2020model, park2019feed, lee2019stochasticity, tran2020hydra, ruiz2020distilled, wu2020multi, zhang2018better, fukuda2017efficient, mehak2018knowledge, jung2019distilling, sun2019patient, lan2018knowledge, liu2019attentive, liu2019improving, mirzadeh2019improved, he2018multi, liu2019knowledge,shen2019amalgamating, ye2019student, luo2019knowledge, shen2019customizing, anil2018large, zhou2019m2kd,xiang2020learning, gao2020residual} have been proposed.
Although these works vary in various distillation scenarios and assumptions, they share some standard characteristics that can be categorized into five types: ensemble of logits, ensemble of feature-level information, unifying data sources, and obtaining sub-teacher networks from a single teacher network, customizing student network from heterogeneous teachers and learning a student network with diverse peers, via the ensemble of logits. We now explicitly analyze each category and provide insights on how and why they are valuable for the problems.
\subsubsection{Distillation from the ensemble of logits} Model ensemble of logits is one of the popular methods in KD from multiple teachers as shown in Fig.~\ref{fig:multiple_teachers}(a).
In such a setting,
the student is encouraged to learn
the softened output of the assembled teachers' logits (dark knowledge)
via the cross-entropy loss as done in \cite{you2017learning, wu2020multi, dvornik2019diversity, anil2018large, tran2020hydra, furlanello2018born, jung2019distilling, lan2018knowledge, liu2019attentive, liu2019improving, mehak2018knowledge, zhou2019m2kd, mirzadeh2019improved, papernot2016semi, tan2019multilingual, tarvainen2017mean, yang2020model}, which can be generalized into:
\begin{equation}
\label{ensemble_logits}
\mathcal{L}_{Ens}^{logits}= {H}(\frac{1}{m}\sum_{i}^{m}N_{T_i}^{\tau}(x), N_S^{\tau}(x))
\end{equation}
where $m$ is the total number of teachers, ${H}$ is the cross-entropy loss, $N_{T_i}^{\tau}$ and $N_{S}^{\tau}$ are the $i$-th teacher's and $i$-th student's logits (or softmax ouputs), and $\tau$ is the temperature. The averaged softened output serves as the incorporation of multiple teacher networks in the output layer. Minimizing Eqn.~\ref{ensemble_logits} achieves the goal of KD at this layer. Note that the averaged softened output is more objective than that of any of the individuals, because it can mitigate the unexpected bias of the softened output existing in some of the input data.
Unlike the methods as mentioned above, \cite{ruder2017knowledge,lan2018knowledge,xiang2020learning, zhang2018better,fukuda2017efficient} argue that taking the average of individual prediction may ignore the diversity and importance variety of the member teachers of an ensemble. Thus, they propose to learn the student model by imitating the summation of the teacher's predictions with a gating component. Then, Eqn.~\ref{ensemble_logits} becomes:
\begin{equation}
\label{ensemble_logits_gating}
\mathcal{L}_{Ens}^{logits}= {H}(\sum_{i}^{m}g_iN_{T_i}^{\tau}(x), N_S^{\tau}(x))
\end{equation}
where $g_i$ is the gating parameter. In \cite{ruder2017knowledge}, the $g_i$ is the normalized similarity $sim(D_{S_i}, D_T)$ of the source domain $D_{S}$ and target domain $D_T$.
\textbf{Summary:} Distilling knowledge from the ensemble of logits mainly depends on taking the average or the summation of individual teacher’s logits. Taking the average alleviates the unexpected bias, but it may ignore the diversity of individual teachers of an ensemble. The summation of the logits of each teacher can be balanced by the gating parameter $g_i$, but ways to determine better values of $g_i$ is an issue worth studying in further works.
\subsubsection{Distillation from the ensemble of features}
Distillation from the ensemble of feature representations is more flexible and advantageous than from the ensemble of logits, since it can provide more affluent and diverse cross-information to the student.
However,
distillation from the ensemble of features \cite{park2019feed, wu2019distilled, liu2019knowledge, mehak2018knowledge, zhou2019m2kd, sun2019patient, zhang2018better} is more challenging, since each teacher's feature representation at specific layers is different from the others. Hence, transforming the features, and forming an ensemble of the teachers' feature-map-level representations becomes the key problem, as illustrated in Fig.~\ref{fig:multiple_teachers}(b).
To address this issue, Park \textit{et al}. ~\cite{park2019feed} proposed feeding the student’s feature map into some nonlinear layers (called transformers). The output is then trained to mimic the final feature map of the teacher network. In this way, the advantages of the general model ensemble and feature-based KD methods, as mentioned in Sec.~\ref{sec_feamap_dis}, can both be incorporated. The loss function is given by:
\begin{equation}
\label{ensemble_feature_feed}
\mathcal{L}_{Ens}^{fea}= \sum_{i}^{m}||\frac{x_{T_i}}{||x_{T_i}||_2} - \frac{{TF}_{i}(x_S)}{||{TF}_{i}(x_S)||_2}||_1
\end{equation}
where $x_{T_i}$ and $x_S$ are $i$-th teacher's and $i$-th student's feature maps respectively, and $TF$ is the transformer (\textit{e}.\textit{g}., $3\times3$ convolution layer) used for adapting the student’s feature with that of the teacher.
In contrast, Wu \textit{et al}. ~\cite{wu2019distilled} and Liu \textit{et al}. ~\cite{liu2019knowledge} proposed letting the student model to imitate the learnable transformation matrices of the teacher models. This approach is an updated version of a single teacher model \cite{tung2019similarity}. For the $i$-th teacher and student network in \cite{wu2019distilled}, the similarity between the feature maps is computed based on the Euclidean metric as:
\begin{equation}
\mathcal{L}_{Ens}^{fea}= \sum_{i}^{m}\alpha_i|| \log(A_S) - log (A_{T_i})||_F^2
\end{equation}
where $\alpha_i$ is the teacher weight for controlling the contribution of the $i$-th teacher, and $\alpha_i$ should satisfy $\sum_i^{m}\alpha_i=1$. $A_S$ and $A_{T_i}$ are the similarity matrices of the student and the $i$-th teacher, respectively. These can be computed by $A_S=x_S^{\intercal} x_S$ and $A_T = x_{T_i}^{\intercal} x_{T_i}$, respectively.
\noindent\textbf{Open challenges:} Based on our review, it is evident that only a few studies propose distilling knowledge from the ensemble of feature representations. Although \cite{park2019feed, wu2020multi} proposed to let the student directly mimic the ensemble of feature maps of the teachers via either non-linear transformation or similarity matrices with weighting mechanisms, there still exist some challenges. First, how can we know which teacher’s feature representation is more reliable or more influential among the ensemble? Second, how can we determine the weighting parameter $\alpha_i$ for each student in an adaptive way? Third, instead of summing all feature information together, is there any mechanism of selecting the best feature map of one teacher from the ensemble as the representative knowledge?
\subsubsection{Distillation by unifying data sources }
Although the above mentioned KD methods using multiple teachers are good in some aspects, they assume that the target classes of all teacher and student models are the same. In addition, the dataset used for training is often scarce, and teacher models with high capacity are limited. To tackle these problems, some recent works \cite{vongkulbhisal2019unifying,wu2019distilled,gong2018teaching,radosavovic2018data, sau2016deep,xiang2020learning} propose data distillation by unifying data sources from multiple teachers, as illustrated in Fig.~\ref{fig:multiple_teachers}(c). The goal of these methods is to generate labels for the unlabeled data via various data processing approaches (\textit{e}.\textit{g}., data augmentation) to train a student model.
Vongkulbhisal \textit{et al}. ~\cite{vongkulbhisal2019unifying} proposed to unify an ensemble of\textit{ heterogeneous classifiers} (teachers) which may be trained to classify different sets of target classes and can share the same network architecture. To generalize distillation, a probabilistic relationship connecting the outputs of the heterogeneous classifiers with that of the unified (ensemble) classifier is proposed. Similarly,
Wu \textit{et al}. ~\cite{wu2019distilled} and Gong \textit{et al}. ~\cite{gong2018teaching} also explored transferring knowledge from teacher models trained in existing data to a student model by using unlabeled data to form a decision function.
Besides, some works utilize the potential of data augmentation approaches to build multiple teacher models from a trained teacher model.
Radosavovic \textit{et al}. ~\cite{radosavovic2018data} proposed a distillation method via \textit{multiple transformations} on the unlabeled data to build diverse teacher models sharing the same network structure. The technique consists of four steps. First, a single teacher model is trained on manually labeled data. Second, the trained teacher model is applied to multiple transformations of the unlabeled data. Third, the predictions on the unlabeled data are converted into an ensemble of numerous predictions. Fourth, the student model is trained on the union of the manually labeled data and the automatically labeled data. Sau \textit{et al}. ~\cite{sau2016deep} proposed an approach to simulate the effect of multiple teachers by injecting noise to the training data, and perturbing the logit outputs of a teacher. In such a way, the perturbed outputs not only simulate the setting of multiple teachers, but also generate noise in the softmax layer, thus regularizing the distillation loss.
\textbf{Summary:} Unifying data sources using data augmentation techniques and unlabeled data from a single teacher model to build up multiple sub-teacher models is also valid for training a student model. However, it requires a high-capacity teacher with more generalized target classes, which could confine the application of these techniques. In addition, the effectiveness of these techniques for some low-level vision problems should be studied further based on feature representations.
\subsubsection{From a single teacher to multiple sub-teachers}
It has been shown that students could be further improved with multiple teachers used as ensembles or used separately. However, using multiple teacher networks is resource heavy, and delays the training process. Following this, some methods \cite{he2018multi, you2017learning, wu2020multi, ruder2017knowledge, lee2019stochasticity, song2018collaborative, tran2020hydra} have been proposed to generate multiple sub-teachers from a single teacher network, as shown in Fig.~\ref{fig:multiple_teachers}(d). Lee \textit{et al}. ~\cite{lee2019stochasticity} proposed stochastic blocks and skip connections to teacher networks, so that the effect of multiple teachers can be obtained in the same resource from a single teacher network. The sub-teacher networks have reliable performance, because there exists a valid path for each batch. By doing so, the student can be trained with multiple teachers throughout the training phase. Similarly, Ruiz \textit{et al}. ~\cite{ruiz2020distilled} introduced hierarchical neural ensemble by employing a \textit{binary-tree} structure to share a subset of intermediate layers between different models. This scheme allows controlling the inference cost on the fly, and deciding how many branches need to be evaluated. Tran \textit{et al}. ~\cite{tran2020hydra}, Song \textit{et al}. ~\cite{song2018collaborative} and He \textit{et al}. ~\cite{ he2018multi} introduced \textit{multi-headed architectures} to build multiple teacher networks, while amortizing the computation through a shared heavy-body network. Each head is assigned to an ensemble member, and tries to mimic the \textit{individual predictions} of the ensemble member.
\noindent\textbf{Open challenges:} Although network ensembles using stochastic or deterministic methods can achieve the effect of multiple teachers and online KD, many uncertainties remain. Firstly, it is unclear how many teachers are sufficient for online distillation. Secondly, which structure is optimal among the ensemble of sub-teachers is unclear? Thirdly, balancing the training efficiency and accuracy of the student network is an open issue. These challenges are worth exploring in further studies.
\subsubsection{Customizing student form heterogeneous teachers}
\label{customize_student}
In many cases, well-trained deep networks (teachers) are focused on different tasks, and are optimized for different datasets. However, most studies focus on training a student by distilling knowledge from teacher networks on the same task or on the same dataset. To tackle these problems, \textit{knowledge amalgamation} has been initialized by recent works \cite{ye2019student, shen2019customizing, shen2019amalgamating, ye2019amalgamating, gao2017knowledge, rusu2015policy, dvornik2019diversity, liu2019knowledge, luo2019knowledge, zhou2019m2kd} to learn a versatile student model by distilling knowledge from the expertise of all teachers, as illustrated in Fig.~\ref{fig:multiple_teachers}(e). Shen \textit{et al}. ~\cite{shen2019customizing}, Ye \textit{et al}. ~\cite{ye2019student}, Luo \textit{et al}. ~\cite{luo2019knowledge} and Ye \textit{et al}. ~\cite{ye2019amalgamating} proposed training a student network by customizing the tasks without accessing human-labeled annotations. These methods rely on schemes such as branch-out \cite{ahmed2016network} or selective learning \cite{galvan2001selective}. The merits of these methods lie in their ability to reuse deep networks pre-trained on various datasets of diverse tasks to build a tailored student model based on the user demand. The student inherits most of the capabilities of heterogeneous teachers, and thus can perform multiple tasks simultaneously. Shen \textit{et al}. ~\cite{shen2019amalgamating} and Gao \textit{et al}. ~\cite{gao2017knowledge} utilized a similar methodology, but focused on same task classification, with two teachers specialized in different classification problems. In this method, the student is capable of handling comprehensive or fine-grained classification. Dvornik \textit{et al}. ~\cite{dvornik2019diversity} attempted to learn a student that can predict unseen classes by distilling knowledge from teachers via few-shot learning. Rusu \textit{et al}. ~\cite{rusu2015policy} proposed a multi-teacher single-student policy distillation method that can distill multiple policies of reinforcement learning agents to a single student network for sequential prediction tasks.
\noindent\textbf{Open challenges:} Studies such as the ones mentioned above have shown considerable potential in customizing versatile student networks for various tasks. However, there are some limitations in such methods. Firstly, the student may not be compact due to the presence of branch-out structures. Secondly, current techniques mostly require teachers to share similar network structures (\textit{e}.\textit{g}., encoder–decoder), which confines the generalization of such methods. Thirdly, training might be complicated because some works adopt a dual-stage strategy, followed by multiple steps with fine-tuning. These challenges open scopes for future investigation on knowledge amalgamation.
\begin{table*}[t!]
\caption{A taxonomy of KD with multiple teachers.
$L_{CE}$ is for cross-entropy loss, $L_{Ens}$ is for the KD loss between the ensemble teacher and the student, $L_{KD}$ indicates the KD loss between individual teacher and the student, $L_{KD_{fea +logits}}$ means KD loss using feature and logits, KL indicates the KL divergence loss for mutual learning, $L_{GAN}$ is for adversarial loss, MMD means mean maximum discrepancy loss, $L_{reg}$ is the regression loss, N/A means not available. \textit{Note that the losses summarized are generalized terms, which may vary in individual works}. }
\vspace{-15pt}
\small
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c|c}
\hline
Method & \thead{ Ensemble \\Logits} & \thead{Ensemble of \\Features} & \thead{Unifying \\data sources} & \thead{Customize \\student} & \thead{Extending\\ teacher} & \thead{Online \\KD} & \thead{Mutual \\learning} & \thead{Major Loss \\ functions} \\
\hline\hline
Anil \cite{anil2018large} & \checkmark & \xmark & \xmark & \xmark & \xmark & \checkmark & \checkmark& $L_{CE}$+$L_{Ens}$ \\ \hline
Chen \cite{chen2019online} &\checkmark& \checkmark& \xmark & \xmark & \xmark & \checkmark & \checkmark& $L_{CE}$+$L_{Ens}$+$L_{KD}$ \\ \hline
Dvornik \cite{dvornik2019diversity} &\checkmark& \xmark& \xmark & \checkmark & \xmark & \checkmark & \checkmark& $L_{CE}$+$L_{Ens}$+$L_{KD}$\\ \hline
Fukuda \cite{fukuda2017efficient} & \checkmark& \xmark & \xmark &\xmark & \xmark & \xmark & \xmark& $L_{CE}$+$L_{Ens}$ \\ \hline
Furlanello \cite{furlanello2018born} &\checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{Ens}$ \\ \hline
He \cite{he2018multi} &\xmark &\checkmark & \xmark & \xmark & \checkmark & \xmark &\xmark & $L_1$+$L_{KD}$ \\ \hline
Jung \cite{jung2019distilling} &\checkmark & \xmark & \xmark& \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD}$ \\ \hline
Lan \cite{lan2018knowledge} & \checkmark & \xmark & \xmark& \xmark & \xmark & \checkmark & \xmark & $L_{CE}$+$L_{Ens}$ \\ \hline
Lee \cite{lee2019stochasticity} & \checkmark & \checkmark & \xmark & \xmark & \checkmark & \checkmark & \checkmark & N/A \\ \hline
Liu \cite{liu2019knowledge} &\xmark& \checkmark & \xmark & \checkmark & \xmark & \checkmark & \xmark & $L_{CE}$+$L_{KD}$ \\ \hline
Luo \cite{luo2019knowledge} &\checkmark & \checkmark & \xmark &\checkmark& \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD_{fea +logits}}$\\ \hline
Zhou \cite{zhou2019m2kd} & \checkmark & \checkmark &\xmark & \checkmark & \xmark & \xmark & \xmark & MMD+$L_{KD}$ \\ \hline
Mirzadeh \cite{mirzadeh2019improved} & \checkmark & \xmark &\xmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD}$\\ \hline
Papernot \cite{papernot2016semi} &\checkmark & \xmark & \checkmark & \xmark & \xmark & \checkmark & \xmark & $L_{KD}$ \\ \hline
Park \cite{park2019feed} &\xmark & \checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD}$ \\ \hline
Radosavovic \cite{radosavovic2018data} &\checkmark & \xmark & \checkmark & \xmark & \xmark & \xmark & \xmark & N/A \\ \hline
Ruder \cite{ruder2017knowledge} &\checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD}$ \\ \hline
Ruiz \cite{ruiz2020distilled} &\checkmark & \xmark & \xmark & \xmark & \checkmark & \xmark & \xmark & $L_{CE}$+$L_{Ens}$ \\ \hline
Sau \cite{sau2016deep} &\checkmark & \xmark & \checkmark & \xmark & \xmark &\xmark & \xmark & $L_2$ (KD) \\ \hline
Shen \cite{shen2019amalgamating} &\checkmark & \checkmark & \xmark & \checkmark & \xmark & \xmark & \xmark & $L_{KD}$+$L_{PL}$ \\ \hline
Shen \cite{shen2019customizing} &\checkmark & \checkmark& \xmark & \checkmark & \xmark & \xmark & \xmark & $L_{KD_{fea +logits}}$+$L_{reg}$\\ \hline
Song \cite{song2018collaborative} & \checkmark & \xmark & \xmark & \xmark & \checkmark & \checkmark & \xmark & $L_{CE}$+$L_{KD}$\\\hline
Tarvaninen \cite{tarvainen2017mean} &\checkmark & \xmark & \xmark & \xmark & \xmark & \checkmark & \xmark & $L_{KD}$ \\ \hline
Tran \cite{tran2020hydra} &\checkmark & \xmark & \xmark & \xmark & \checkmark & \xmark & \checkmark & $L_{CE}$+$L_{Ens}$+KL \\ \hline
Vongkulbhisal \cite{vongkulbhisal2019unifying} &\checkmark & \xmark & \checkmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{Ens}$ \\ \hline
Wu \cite{wu2019distilled} &\xmark& \checkmark & \checkmark & \xmark & \xmark & \xmark & \xmark & $L_{KD}$ \\ \hline
Wu \cite{wu2020multi} &\checkmark & \xmark & \xmark & \xmark & \xmark & \xmark& \xmark & $L_{CE}$+$L_{Ens}$ \\ \hline
Yang \cite{yang2020model} &\checkmark & \xmark & \xmark& \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD}$ \\ \hline
Ye \cite{ye2019student} &\checkmark & \checkmark & \xmark & \checkmark & \xmark & \xmark & \xmark & $L_{KD}$+$L_{KD}$ \\ \hline
You \cite{you2017learning} &\checkmark & \checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD_{fea +logits}}$\\ \hline
Zhang \cite{zhang2018better} &\checkmark & \checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD_{fea +logits}}$\\ \hline
Zhang \cite{zhang2018deep} &\checkmark & \xmark & \xmark & \xmark & \xmark & \checkmark & \checkmark & $L_{CE}$+KL \\ \hline
Zhu \cite{zhu2018knowledge} &\checkmark & \xmark & \xmark & \xmark & \xmark & \checkmark & \checkmark & $L_{CE}$+KL+$L_{Ens}$ \\ \hline
Chung \cite{chung2020featuremaplevel} & \checkmark & \checkmark & \xmark &\xmark & \xmark & \checkmark & \checkmark & $L_{CE}$+$L_{GAN}$+KL \\ \hline
Kim \cite{kim2019feature} & \checkmark & \checkmark & \xmark & \xmark & \xmark & \checkmark & \checkmark &$L_{CE}$+$L_{Ens}$+KL \\ \hline
Hou \cite{hou2017dualnet} & \xmark & \checkmark & \xmark & \xmark & \xmark & \checkmark & \xmark & $L_{CE}$+$L_{Ens}$ \\ \hline
Xiang \cite{xiang2020learning} & \checkmark &\xmark & \checkmark & \xmark & \xmark & \xmark & \xmark & $L_{CE}$+$L_{KD}$ \\
\hline
\end{tabular}
\end{center}
\vspace{-15pt}
\label{table:kd_multipleTeachs}
\end{table*}
\subsubsection{Mutual learning with ensemble of peers}
One problem with conventional KD methods using multiple teachers is their computation cost and complexity, because they require pre-trained high-capacity teachers with two-stage (also called offline) learning. To simplify the distillation process, one-stage (online) KD methods \cite{anil2018large, zhang2018deep, lan2018knowledge, chen2019online, chung2020featuremaplevel, kim2019feature, hou2017dualnet, zhu2018knowledge, zhang2018better} have been developed,
as shown in Fig.~\ref{fig:multiple_teachers}(f). Instead of pre-training a static teacher model, these methods train a set of student models simultaneously by making them learn from each other in a peer-teaching manner. There are some benefits of such methods. First, these approaches merge the training processes of teachers and student models, and use peer networks to provide teaching knowledge. Second, these online distilling strategies can improve the performance of models of any capacity, leading to generic applications. Third, such a peer-distillation method can sometimes outperform teacher-based two-stage KD methods. For KD with mutual learning, the distillation loss of \textit{two peers} is based on the KL divergence, which can be formulated as:
\begin{equation}
\mathcal{L}_{Peer}^{KD}= KL(z_1, z_2) + KL(z_2, z_1)
\end{equation}
where $KL$ is the KL divergence function, and $z_1$ and $z_2$ are the predictions of peer one and peer two, respectively.
In addition, Lan \textit{et al}. ~\cite{lan2018knowledge} and Chen \textit{et al}. ~\cite{chen2019online} also constructed a multi-branch variant of a given target (student) network by adding auxiliary branches to create a local ensemble teacher (also called a group leader) model from all branches. Each branch was trained with a distillation loss that aligns the prediction of that branch with the teacher’s prediction. Mathematically, the distillation loss can be formulated by minimizing the KL divergence of $z_e$ (prediction of the ensemble teacher) and prediction $z_i$ of the $i$-th branch peer:
\begin{equation}
\mathcal{L}_{Ens}^{KD}= \sum_{i=1}^mKL(z_e, z_i)
\end{equation}
where the prediction $z_e= \sum_{i=1}^m g_iz_i$. $g_i$ is the weighting score or attention-based weights \cite{chen2019online} of the $i$-th branch peer $z_i$.
Although most of these methods only consider using logit information, some works also exploit feature information. Chung \textit{et al}. ~\cite{chung2020featuremaplevel} proposed a feature-map-level distillation by employing adversarial learning (discriminators). Kim \textit{et al}. ~\cite{kim2019feature} introduce a feature fusion module to form an ensemble teacher. However, the fusion is based on the concatenation of the features (output channels) from the branch peers. Moreover, Liu \textit{et al}. ~\cite{liu2019knowledge} presented a knowledge flow framework which moves the knowledge from the features of multiple teacher networks to a student.
\textbf{Summary:} Compared to two-stage KD methods using pre-trained teachers, distillation from student peers has many merits. The methods are based on mutual learning of peers, and sometimes on ensembles of peers. Most studies rely on logit information; however, some works also exploit feature information via adversarial learning or feature fusion. There is room for improvement in this direction. For instance, the number of peers most optimal for KD processing is worth investigating. In addition, the possibility of using both the online and offline methods simultaneously when the teacher is available is intriguing. Reducing the computation cost without sacrificing accuracy and generalization is also an open issue. We will discuss the advantages and disadvantages of online and offline KD in Sec.~\ref{online_kd}.
\noindent\textbf{Potentials}
Table.~\ref{table:kd_multipleTeachs} summarizes the KD methods with multiple instructors. Overall, most methods rely on the ensemble of logits. However, the knowledge of feature representations has not been taken into account much. Therefore, it is possible to exploit the knowledge of the ensemble of feature representations by designing better gating mechanisms. Unifying data sources and extending teacher models are two effective methods for reducing individual teacher models; however, their performances are degraded. Thus, overcoming this issue needs more research. Customizing a versatile student is a valuable idea, but existing methods are limited by network structures, diversity, and computation costs, which must be improved in future works.
\begin{table*}[t!]
\caption{A taxonomy of data-free knowledge distillation.
}
\vspace{-15pt}
\small
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c}
\hline
Method & \thead{Original \\ data needed} & \thead{ Metadata or \\prior info.} & \thead{Number of\\ generators} & Inputs & Discriminator & \thead{Multi-task \\ distillation} \\
\hline\hline
Lopes \cite{lopes2017data} & \checkmark & \thead{Activations \\of all layers} & \xmark &Image shape &\xmark &\xmark\\ \hline
Bhardwaj \cite{bhardwaj2019dream} & \checkmark & \thead{Activations of \\ pooling layer}& \xmark & Image shape & \xmark & \xmark \\ \hline
Haroush \cite{haroush2019knowledge} & \checkmark & \thead{Batch \\normalization layer} & \xmark & Image shape & \xmark & \xmark\\ \hline
Nayak \cite{nayak2019zero} & \xmark & Class similarities & \xmark & \thead{Class label+ Number of DIs} & \xmark & \xmark \\ \hline
Chen \cite{chen2019data} & \xmark &\xmark& One & Noise & Teacher & \xmark \\ \hline
Fang \cite{fang2019data} &\xmark& \xmark& One& Noise/images & Teacher + student &\xmark \\ \hline
Ye \cite{ye2020datafree} & \xmark &\xmark & Three & Noise & Teachers & \checkmark \\ \hline
Yoo \cite{yoo2019knowledge} &\xmark &\xmark & One & Noise + class labels & Teacher & \xmark \\ \hline
Yin \cite{yin2019dreaming} &\xmark& \xmark& One & Noise & Teacher & \xmark \\ \hline
Micaelli \cite{micaelli2019zero} & \xmark & \xmark & One & Noise & Teacher & \xmark \\ \hline
\end{tabular}
\end{center}
\vspace{-10pt}
\label{table:datafree_comp}
\end{table*}
\section{Distillation based on data format}
\subsection{Data-free distillation}
\label{data_free}
\noindent\textbf{Overall insight:} \textit{ Can we achieve KD when the original data for the teacher or (un)labeled data for training student are not available?}
One major limitation of most KD methods such as \cite{hinton2015distilling, park2019relational, park2019feed, romero2014fitnets} is that they assume the training samples of the original networks (teachers) or of target networks (students) to be available. However, the training dataset is sometimes unknown in real-world applications owing to privacy and transmission concerns \cite{lopes2017data}. To handle this problem, some representative data-free KD paradigms \cite{lopes2017data, nayak2019zero, ye2020datafree, chen2019data, fang2019data, bhardwaj2019dream, yin2019dreaming, haroush2019knowledge, yoo2019knowledge,micaelli2019zero, kulkarni2017knowledge} are newly developed. A taxonomy of these methods are summarized in Table.~\ref{table:datafree_comp}, and detailed technical analysis is provided as follows.
\subsubsection{Distillation based on metadata}
To the best of our knowledge,
Lopes \textit{et al}. ~\cite{lopes2017data} initially proposed to reconstruct the original training dataset using only teacher model and it is \textit{metadata} recorded in the form of \textit{precomputed activation statistics}. Thus, the objective is to find the set of images whose representation best matches the one given by the teacher network. Gaussian noise is randomly passed as input to the teacher, and the gradient descent (GD) is made to minimize the difference between the metadata and the representations of noise input. To better constrain the reconstruction, the metadata of \textit{all} layers of the teacher model are used and recorded to train the student model with high accuracy. Bhardwaj \textit{et al}. ~\cite{bhardwaj2019dream} demonstrated that metadata from a \textit{single} layer (average-pooling layer) using $k$-means clustering is sufficient to achieve high student accuracy. In contrast to \cite{lopes2017data, bhardwaj2019dream} requiring sampling the activations generated by real data, Haroush \textit{et al}. ~\cite{haroush2019knowledge} proposed using metadata (\textit{e}.\textit{g}., channel-wise mean and standard deviation) from Batch Normalization (BN) \cite{ioffe2015batch} layer with synthetic samples.
The objective of metadata-based distillation can be formulated as:
\begin{equation}
X^* = \arg\min_{X\sim R ^{H\times W}} L(\Phi(X), \Phi_0)
\end{equation}
where $X^*$ is the image (with width $W$ and height $H$) to be found, $\Phi$ is the representation of $X$, $\Phi_0$ is the representation of the metadata, and $L$ is the loss function (\textit{e}.\textit{g}., $l_2$).
\subsubsection{Distillation based on class-similarities}
Nayak \textit{et al}. ~\cite{nayak2019zero} argued that the metadata used in \cite{lopes2017data, bhardwaj2019dream} are actually not complete data-free approaches, since the metadata is formed using the training data itself.
They instead proposed a zero-shot KD approach, in which no data samples and no metadata information are used. In particular, the approach obtains useful prior information about the underlying data distribution in the form of \textit{class similarities} from the model parameters of the teacher. This prior information can be further utilized for crafting data samples (also called data impressions (DIs)) by modeling the output space of the teacher model as a Dirichlet distribution. The class similarity matrix, similar to \cite{tung2019similarity}, is calculated based on the softmax layer of the teacher model. The objective of the data impression $X_i^k$ can be formulated based on the cross-entropy loss:
\begin{equation}
X_i^k =\arg\min_X L_{CE}(y_i^{k}, T(X, \theta_T, \tau))
\end{equation}
where $y_i^{k}$ is the sampled $i$-th softmax vector, and $k$ is the certain class.
\subsubsection{Distillation using generator}
Considering the limitation of metadata and similarity-based distillation methods, some works \cite{ye2020datafree, fang2019data, chen2019data, yin2019dreaming, yoo2019knowledge, micaelli2019zero} propose novel data-free KD methods via adversarial learning \cite{goodfellow2014generative, wang2020deceiving, wang2019event}. Although the tasks and network structures vary in these methods, most are built on a common framework. That is, the pretrained teacher network is fixed as a discriminator, while a generator is designed to synthesize training samples, given various input source (\textit{e}.\textit{g}., noise \cite{chen2019data,yin2019dreaming, yoo2019knowledge, ye2020datafree}). However, slight differences exist in some studies. Fang \textit{et al}. ~\cite{fang2019data} point out the problem of taking the teacher as the discriminator since the information of the student is ignored, and generated samples cannot be customized without the student. Thus, they take both the teacher and the student as the discriminator to reduce the discrepancy between them, while a generator is trained to generate some samples to adversarially enlarge the discrepancy. In contrast, Ye \textit{et al}. ~\cite{ye2020datafree} focus more on strengthening the generator structure, and three generators are designed and subtly used. Specifically, a group-stack generator is trained to generate the images originally used for pre-training the teachers, and the intermediate activations. Then, a dual generator takes the generated image as the input, the dual part is taken as the target network (student), and regrouped for multi-label classifications. To compute the adversarial loss for both the generated image and the intermediate activations, multiple group-stack discriminators (multiple teachers) are also designed to amalgamate multi-knowledge into the generator. Yoo \textit{et al}. ~\cite{yoo2019knowledge} make the generator take two inputs: a sampled class label $y$, and noise $z$. Meanwhile, a decoder is also applied to reconstruct the noise input $z'$ and class label $y'$ from the fake data $x'$ generated by the generator from the noise input $z$, and class label $y$. Thus, by minimizing the errors between $y$ and $y'$ and between $z$ and $z'$, the generator generates more reliable data. Although adversarial loss is not used in \cite{yin2019dreaming}, the generator (called DeepInversion) taking an image prior regularization term to synthesize images is modified from DeepDream \cite{mordvintsev2015inceptionism}.
\subsubsection{Open challenges for data-free distillation}
Although data-free KD methods have shown considerable potential and new directions for KD, there still exist many challenges. First, the recovered images are unrealistic and low-resolution, which may not be utilized in some data-captious tasks (\textit{e}.\textit{g}., semantic segmentation). Second, training and computation of the existing methods might be complicated due to the utilization of many modules. Third, diversity and generalization of the recovered data are still limited, compared with the methods of data-driven distillation. Fourth, the effectiveness of such methods for low-level tasks (\textit{e}.\textit{g}., image super-resolution) needs to be studied further.
\subsection{Distillation with a few data samples}
\label{distill_fewdata}
\textbf{Overall insight:} \textit{How to perform efficient knowledge distillation with only a small amount of training data?}
Most KD methods with S-T structures, such as \cite{hinton2015distilling, kim2018paraphrasing, park2019feed, chung2020featuremaplevel}, are based on matching information (\textit{e}.\textit{g}., logits, hints) and optimizing the KD loss with the fully annotated large-scale training dataset. As a result, the training is still data-heavy and processing-inefficient. To enable efficient learning of the student while using small amount of training data, some works
\cite{liu2019semantic, li2018few, kimura2018few, bai2019few, kulkarni2017knowledge} propose few-sample KD strategies. The technical highlight of these methods is based on generating pseudo training examples, or aligning the teacher and the student with layer-wise estimation metrics.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{figures/cross_modal_distillation.pdf}
\caption{Graphical illustration of cross-modal KD methods. (a) supervised cross-modal KD from the teacher with one modality to the student with another modality. (b) unsupervised cross-modal KD with one teacher. (c) unsupervised cross-modal KD with multiple teachers, each of which is transferring the discriminative knowledge to the student.}
\label{fig:cross_modal_dis}
\end{figure*}
\subsubsection{Distillation via pseudo examples}
\textbf{Insight:} \textit{If training data is insufficient, try to create pseudo examples for training the student.}
\cite{kimura2018few, liu2019semantic, kulkarni2017knowledge} focus on creating pseudo training examples when training data is scarce and leading to overfitting of the student network. Specifically, Kimura \textit{et al}. ~\cite{kimura2018few} adopt the idea of inducing points \cite{snelson2006sparse} to generate pseudo training examples, which are then updated by applying adversarial examples \cite{szegedy2015going,goodfellow2014explaining}, and further optimized by an imitation loss. Liu \textit{et al}. ~\cite{liu2019semantic} generate pseudo ImageNet \cite{deng2009imagenet} labels from a teacher model (trained with ImageNet), and also utilize the semantic information (\textit{e}.\textit{g}., words) to add a supervision signal for the student. Interestingly, Kulkarni \textit{et al}. ~\cite{kulkarni2017knowledge} create a `mismatched' unlabeled stimulus (\textit{e}.\textit{g}., soft labels of MNIST dataset \cite{lecun1998gradient} provided by the teacher trained on CIFAR dataset \cite{krizhevsky2009learning}), which are used for augmenting a small amount of training data to train the student.
\subsubsection{Distillation via layer-wise estimation}
\textbf{Insight:} \textit{Layer-wise distillation from the teacher network via estimating the accumulated errors on the student network can also achieve the purpose of few-example KD.}
In Bai \textit{et al}. ~\cite{bai2019few} and Li \textit{et al}. ~\cite{li2018few}, the teacher network is first compressed to create a student via network pruning \cite{zhu2017prune}, and layer-wise distillation losses are then applied to reduce the estimation error on given limited samples. To conduct layer-wise distillation, Li \textit{et al}. ~\cite{li2018few} add a $1\times1$ layer after each pruned layer block in the student, and estimate the least-squared error to align the parameters with the student. Bai \textit{et al}. ~\cite{bai2019few} employ cross distillation losses to mimic the behavior of the teacher network, given its current estimations.
\subsubsection{Challenges and potentials}
Although KD methods with a small number of examples inspired by the techniques of data augmentation and layer-wise learning are convincing, these techniques are still confined by the structures of teacher networks. This is because most methods rely on network pruning from teacher networks to create student networks. Besides, the performance of the student is heavily dependent on the amount of the crafted pseudo labels, which may impede the effectiveness of these methods. Lastly, most works focus on generic classification tasks, and it is unclear whether these methods are effective for tasks without class labels (\textit{e}.\textit{g}., low-level vision tasks).
\begin{table*}[t!]
\caption{A taxonomy of cross-modal knowledge distillation methods.
}
\vspace{-15pt}
\small
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Method & Use GT & \thead{Source \\modality} & \thead{Target \\modality} & \thead{Number of \\teachers} & Online KD & Knowledge & \thead{Model \\compression} \\
\hline\hline
Ayter \cite{aytar2016soundnet} & \xmark & RGB frames & Sound & Two &\xmark &Logits & \xmark \\ \hline
Su \cite{su2016adapting} & \checkmark & HR image map& LR image & One & \xmark & Soft labels & \checkmark \\ \hline
Nagrani \cite{nagrani2018learnable} & \checkmark & RGB frames & Voice & One & \checkmark & Soft labels& \xmark \\ \hline
Nagrani \cite{nagrani2018seeing} & \checkmark & Voice/face & Face/voice & Multiple & \checkmark & Features &\xmark \\ \hline
Hoffman \cite{hoffman2016cross} & \checkmark &RDG images& Depth images & One & \xmark & Features & \xmark \\ \hline
Afouras \cite{afouras2019asr} & \xmark &Audio& Video & One & \xmark & Soft labels & \xmark \\ \hline
Albanie \cite{albanie2018emotion} &\xmark& Video frames& Sound & One & \xmark & Logits & \xmark\\ \hline
Gupta \cite{gupta2016cross} &\xmark & RGN images & Depth images & One & \xmark & Soft labels & \xmark \\ \hline
Salem \cite{salem2019learning} &\xmark& \thead{Scene \\ classification, \\object detection}& Localization & Three & \xmark & Soft labels & \xmark \\ \hline
Thoker \cite{thoker2019cross} & \xmark & RGB video & Skeleton data & One & \xmark & Logits & \xmark\\ \hline
Zhao \cite{zhao2018through} & \xmark & RGB frames & Heatmaps & One & \xmark &\thead{Confidence\\ maps} & \xmark\\ \hline
Owens~ \cite{owens2016ambient} & \xmark & Sound & Video frames & One & \xmark & Soft labels & \xmark\\ \hline
Arandjelovic \cite{arandjelovic2017look} & \xmark & Video frames & Audio & One & \xmark & Features & \xmark\\ \hline
Do \cite{do2019compact} & \checkmark & \thead{Image,\\Questions,\\Answer info.} & Image questions & Three & \xmark & Logits & \checkmark\\ \hline
Aytar \cite{aytar2017see} & \checkmark & Image & \thead{Sound, \\Image, Text} & Three & \xmark & Features & \xmark\\ \hline
Kim \cite{kim2018learning} & \xmark & Sound/images & Images/sound & One & \xmark & Features & \xmark\\ \hline
Dou \cite{dou2020unpaired} & \xmark & CT images & MRI images & One & \checkmark & Logits & \checkmark\\ \hline
Hafner \cite{hafner2018cross}& \xmark & Depth images & RGB images & One & \xmark & Embeddings & \xmark\\ \hline
Gan \cite{gan2019self} & \checkmark & Video frame & Sound & One & \xmark & \thead{Feature \\ soft labels} & \xmark \\ \hline
Perez \cite{perez2020audio} & \checkmark & \thead{RGB video\\Acoustic images} & Audio & One & \xmark & \thead{Soft labels} & \xmark \\
\hline
\end{tabular}
\end{center}
\vspace{-10pt}
\label{table:cross_modal_KD}
\end{table*}
\subsection{Cross-modal distillation}
\label{cross_modal_sec}
\textbf{Overall insight:} \textit{KD for cross-modal learning is typically performed with network architectures containing modal-specific representations or shared layers, utilizing the training images in correspondence of different domains.}
One natural question we ask is if it is possible to transfer knowledge from a pre-trained teacher network for one task to a student learning another task, while the training examples are in correspondence across domains. \textit{Note that KD for cross-modal learning is essentially different from that for domain adaptation, in which data are drawn independently from different domains, but the tasks are the same.}
Compared to previously mentioned KD methods focused on transferring supervision within the same modality between the teacher and the student, cross-modal KD uses the teacher's representation as a supervision signal to train the student learning another task. In this problem setting, the student needs to rely on the visual input of the teacher to accomplish its task. Following this, many novel cross-modal KD methods \cite{aytar2016soundnet,albanie2018emotion, gupta2016cross, zhao2018through, do2019compact, thoker2019cross,salem2019learning, afouras2019asr, dou2020unpaired, hafner2018cross,su2016adapting, hoffman2016cross, owens2016ambient,arandjelovic2017look,aytar2017see,nagrani2018learnable,nagrani2018seeing, zhao2020knowledge} have been proposed. We now provide a systematic analysis of the technical details, and point the challenges and potential of cross-domain distillation.
\subsubsection{Supervised cross-modal distillation}
Using the ground truth labels for the data used in the student network is the common way of cross-modal KD, as shown in Fig.~\ref{fig:cross_modal_dis}(a).
Do \textit{et al}. ~\cite{do2019compact}, Su \textit{et al}. ~\cite{su2016adapting}, Nagrani \textit{et al}. ~\cite{nagrani2018learnable}, Nagrani \textit{et al}. ~\cite{nagrani2018seeing} and Hoffman \textit{et al}. ~\cite{hoffman2016cross} rely on supervised learning for cross-modal transfer. Several works \cite{nagrani2018seeing,nagrani2018learnable, afouras2019asr, perez2020audio} leverage the synchronization of visual and audio information in the video data, and learn a joint embedding between the two modalities. Afouras \textit{et al}. ~\cite{afouras2019asr} and Nagrani \textit{et al}. ~\cite{nagrani2018seeing} transfer the voice knowledge to learn a visual detector, while Nagrani \textit{et al}. ~\cite{nagrani2018learnable} utilize visual knowledge to learn a voice detector (student). In contrast, Hoffman \textit{et al}. ~\cite{hoffman2016cross}, Do \textit{et al}. ~\cite{do2019compact} and Su \textit{et al}. ~\cite{su2016adapting} focus on different modalities in the visual domain only. In particular, Hoffman \textit{et al}. ~\cite{hoffman2016cross} learn a depth network by transferring the knowledge from an RGB network, and fuse the information across modalities. This improves the object recognition performance during the test time. Su \textit{et al}. ~\cite{su2016adapting} utilize the knowledge from high-quality images to learn a classifier with better generalization on low-quality image (paired).
\subsubsection{Unsupervised cross-modal distillation} Most cross-modal KD methods exploit unsupervised learning, since the labels in target domains are hard to get. Thus, these methods are also called distillation `\textit{in the wild}'. In this setting, the knowledge from the teacher's modality provides \textit{supervision} for the student network. To this end, some works \cite{afouras2019asr,albanie2018emotion,aytar2016soundnet,gupta2016cross,hafner2018cross,salem2019learning,thoker2019cross,zhao2018through,owens2016ambient,aytar2017see,arandjelovic2017look,dou2020unpaired,kim2018learning, zhao2020knowledge} aimed for cross-modal distillation in an unsupervised manner.
\subsubsection{Learning from one teacher} Afouras \textit{et al}. ~\cite{afouras2019asr}, Albanie \textit{et al}. ~\cite{albanie2018emotion}, Gupta \textit{et al}. ~\cite{gupta2016cross}, Thoker \textit{et al}. ~\cite{thoker2019cross}, Zhao \textit{et al}. ~\cite{zhao2018through}, Owens \textit{et al}. ~\cite{owens2016ambient}, Kim \textit{et al}. ~\cite{kim2018learning}, Arandjelovic \textit{et al}. ~\cite{arandjelovic2017look}, Gan \textit{et al}. ~\cite{gan2019self}, and Hafner \textit{et al}. ~\cite{hafner2018cross} focus on distilling knowledge from one teacher (see Fig.~\ref{fig:cross_modal_dis}(b)), and mostly learn a single student network. Thoker \textit{et al}. ~\cite{thoker2019cross} and Zhao \textit{et al}. ~\cite{zhao2018through} learn two students. Especially, Thoker \textit{et al}. ~ refer to mutual learning \cite{zhang2018deep}, where two students learn from each other based on two KL divergence losses. In addition, Zhao \textit{et al}. ~\cite{zhao2018through} exploit the feature fusion strategy, similar to \cite{kim2019feature, ke2019dual} to learn a more robust decoder. Do \textit{et al}. ~\cite{do2019compact} focuses on unpaired images of two modalities, and learns a semantic segmentation network (student) using the knowledge from the other modality (teacher).
\subsubsection{Learning from multiple teachers} Aytar \textit{et al}. ~\cite{aytar2016soundnet}, Salem \textit{et al}. ~\cite{salem2019learning}, Aytar \textit{et al}. ~\cite{aytar2017see} and Do \textit{et al}. ~\cite{do2019compact} exploit the potential of distilling from multiple teachers as mentioned in Sec.~\ref{multi_teach}. Most methods rely on concurrent knowledge among visual, audio, and textual information, as shown in Fig.~\ref{fig:cross_modal_dis}(c). However, Salem \textit{et al}. ~\cite{salem2019learning} focus on the visual modality only, where teachers learn the information of object detection, image classification, and scene categorization via a multi-task approach, and distill the knowledge to a single student.
\subsubsection{Potentials and open challenges}
\noindent \textbf{Potentials:} Based on the analysis of the existing cross-modal KD techniques in Table.~\ref{table:cross_modal_KD}, we can see that cross-modal KD expands the generalization capability of the knowledge learned from the teacher models. Cross-domain KD has considerable potential in \textit{relieving the dependence} for a large amount of labeled data in one modality or both. In addition, cross-domain KD is more \textit{scalable}, and can be \textit{easily} applied to \textit{new} distillation tasks. Moreover, it is advantageous for learning multiple modalities of data `in the wild', since it is relatively easy to get data with one modality based on other data. In visual applications, cross-modal KD has the potential to distill knowledge among images taken from different types of cameras. For instance, one can distill knowledge from an RGB image to event streams (stacked event images from event cameras) \cite{su2016adapting, wang2020eventsr}.
\noindent \textbf{Open challenges:} Since the knowledge is the transferred representations (\textit{e}.\textit{g}., logits, features) of teacher models, ensuring the robustness of the transferred knowledge is crucial. We hope to transfer the good representations, but negative representations do exist. Thus, it is imperative that the supervision provided by the teachers is complementary to the target modality. Moreover, existing cross-modal KD methods are highly dependent on data sources (\textit{e}.\textit{g}., video, images), but finding data with paired (\textit{e}.\textit{g}., RGB image with depth pair) or multiple modalities (class labels, bounding boxes and segmentation labels) is not always an easy task. We are compelled to ask if it is possible to come up with a way for data-free distillation or distillation with a few examples? In other words, is it possible to just learn a student model with the data from the target modality based on the knowledge of the teacher, without referencing the source modality?
Moreover, existing cross-modal KD methods are mostly offline methods, which are computation-heavy and memory-intensive. Thus, it would be better if an online KD strategy is considered. Lastly, some works (\textit{e}.\textit{g}., \cite{salem2019learning,aytar2017see}) learn a student model using the knowledge from multiple teachers. However, the student is less versatile or modality-dependent. Inspired by the analysis of Sec.~\ref{customize_student}, we open a research question: Is it possible to learn a versatile student that can perform tasks from multiple modalities?
\begin{figure*}[t!]
\centering
\includegraphics[width=0.9\textwidth]{figures/onlline_disllation.pdf}
\caption{An illustration of online KD methods. (a) online KD with individual student peer learning from each other, (b) online KD with student peers sharing trunk (head) structure, (c) online KD by assembling the weights of each student to form a teacher or group leader.}
\label{fig:online_distillation}
\end{figure*}
\section{Online and Teacher-free distillation}
\subsection{Online distillation}
\label{online_kd}
\textbf{Overall insight:} \textit{With the absence of a pre-trained powerful teacher, simultaneously training a group of student models by learning from peers’ predictions is an effective substitute for two-stage (offline) KD}
In this section, we provide a deeper analysis of online (one-stage) KD methods in contrast to the previously discussed offline (two-stage) KD methods. Offline KD methods often require pre-trained high-capacity teacher models to perform one-way transfer \cite{hinton2015distilling, Ahn_2019_CVPR, kim2018paraphrasing, you2017learning,fukuda2017efficient, hou2017dualnet, mullapudi2019online, gao2019multistructure, lin2019mod}. However, it is sometimes difficult to get such ‘good’ teachers, and the performance of the student gets degraded when the gap of network capacity between the teacher and the student is significant. In addition, two-stage KD requires many parameters, resulting in higher computation costs. To overcome these difficulties, some studies focus on online KD that simultaneously trains a group of student peers by learning from the peers’ predictions.
\subsubsection{Individual student peers}
Zhang \textit{et al}. ~\cite{zhang2018deep}, Gao \textit{et al}. ~\cite{gao2019multistructure} and Anil \textit{et al}. ~\cite{anil2018large} focus on online \textit{mutual learning} \cite{zhang2018deep} (also called codistilation) in which a pool of untrained student networks with the same network structure simultaneously learns the target task. In such a peer-teaching environment, each student learns the average class probabilities from the other (see Fig.~\ref{fig:online_distillation}(a)). However, Chung \textit{et al}. ~\cite{chung2020featuremaplevel} also employ individual students, and additionally design a feature map-based KD loss via adversarial learning. Hou \textit{et al}. ~\cite{hou2017dualnet} proposed DualNet, where two individual student classifiers were fused into one fused classifier. During training, the two student classifiers are locally optimized, while the fused classifier is globally optimized as a mutual learning method. Other methods such as \cite{mullapudi2019online, cioppa2019arthus}, focus on online video distillation by periodically updating the weights of the student, based on the output of the teacher. Although codistillation achieves parallel learning of students, \cite{zhang2018deep, anil2018large, chung2020featuremaplevel, hou2017dualnet} do not consider the ensemble of peers' information as done in other works such as \cite{chen2019online,gao2019multistructure}.
\subsubsection{Sharing blocks among student peers} Considering the training cost of employing individual students, some works propose sharing network structures (\textit{e}.\textit{g}., head sharing) of the students with branches as shown in Fig.~\ref{fig:online_distillation}(b). Song \textit{et al}. ~\cite{song2018collaborative} and Lan \textit{et al}. ~\cite{lan2018knowledge} build the student peers on multi-branch architectures \cite{szegedy2015going}. In such a way, all structures together with the shared trunk layers (often use head layers) can construct individual student peers, and any target student peer network in the whole multi-branch can be optimized.
\subsubsection{Ensemble of student peers}
While using codistillation and multi-architectures can facilitate online distillation, knowledge from all student peers is not accessible. To this end, some studies \cite{lan2018knowledge, kim2019feature, chen2019online,lin2019mod, gao2019multistructure} proposed using the assembly of knowledge (logits information) of all student peers to build an on the fly teacher or group leader, which is in turn distilled back to all student peers to enhance student learning in a closed-loop form, as shown in Fig.~\ref{fig:online_distillation}(c). Note that in ensemble distillation, the student peers can either be independent, or share the same head structure (trunk). The ensemble distillation loss is given by Eqn.~\ref{ensemble_logits_gating} of Sec.~\ref{multi_teach}, where a gating component $g_i$ is added to balance the contribution of each student. Chen \textit{et al}.~\cite{chen2019online} obtain the gating component $g_i$ based on the self-attention mechanism \cite{vaswani2017attention}.
\subsubsection{Summary and open challenges}
\textbf{Summary:} Based on the above analysis, we have determined that codistillation, multi-architectures, and ensemble learning are three main techniques for online distillation. There are some advantages of online KD compared with offline KD. Firstly, online KD does not require pre-training teachers. Secondly, online learning provides a simple but effective way to improve the learning efficiency and generalizability of the network, by training together with other student peers. Thirdly, online learning with student peers often results in better performance than offline learning.
\noindent \textbf{Open challenges:} There are some challenges in online KD. Firstly, there is a lack of theoretical analysis for why online learning is sometimes better than offline learning. Secondly, in online ensemble KD, simply aggregating students’ logits to form an ensemble teacher restrains the diversity of student peers, thus limiting the effectiveness of online learning. Thirdly, existing methods are confined problems in which ground truth (GT) labels exist (\textit{e}.\textit{g}., classification). However, for some problems (\textit{e}.\textit{g}., low-level vision problems), ways for the student peers to form effective ensemble teachers need to be exploited.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{figures/self_kd_category.pdf}
\caption{An illustration of self-distillation methods. (a) born-again distillation. Note that $T$ and $S_1, \cdots, S_n$ can be multi-tasks. (b) distillation via `deep' supervision where the deepest branch (B$_n$) is used to distill knowledge to shallower branches. (c) distillation via data augmentation (\textit{e}.\textit{g}., rotation, cropping). (d) distillation with network architecture transformation (\textit{e}.\textit{g}., changing convolution filters). }
\label{fig:self_kd}
\end{figure*}
\subsection{Teacher-free distillation}
\label{self_distillation}
\textbf{Overal insight:} \textit{Is it possible to enable the student to distill knowledge by itself to achieve plausible performance?}
The conventional KD approaches \cite{hinton2015distilling, romero2014fitnets, tung2019similarity, kim2019feature, yim2017gift} still have many setbacks to be tackled, although significant performance boost has been achieved.
First of all, these approaches have low efficiencies,
since student models scarcely exploit all knowledge from the teacher models. Secondly, designing and training high-capacity teacher models still face many obstacles. Thirdly, two-stage distillation requires high computation and storage costs. To tackle these challenges, several novel self-distillation frameworks \cite{mobahi2020self,xu2019data,hahn2019self,lee2019rethinking, furlanello2018born, crowley2018moonshine, zhang2019your, hou2019learning, clark2019bam, luan2019msd, yang2019training, lan2018self} have been proposed recently. The goal of self-distillation is to learn a student model by distilling knowledge in itself without referring to other models. We now provide a detailed analysis of the technical details for self-distillation.
\begin{table*}[t!]
\caption{A taxonomy of self-distillation methods. Logits and hints indicate the knowledge to be distilled. `Deep' supervision is for self-distillation from deepest branch (or layer) of the student network. One-stage KD is checking whether self-distillation is achieved in one step. \checkmark/ \xmark is for yes/no.}
\vspace{-15pt}
\small
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
Method & Logits & Hints & \thead{Data \\ augmentation} & \thead{'Deep' \\supervision} & One-stage KD & \thead{Multi-task KD} & \thead{Architecture \\transformation} \\
\hline\hline
Clarm \cite{clark2019bam} & \checkmark & \xmark & \xmark &\xmark &\xmark &\checkmark & \xmark \\ \hline
Chowley \cite{crowley2018moonshine} & \xmark & Attention map& \xmark & \xmark & \xmark & \xmark & \checkmark \\ \hline
Furlanello \cite{furlanello2018born} & \checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & \xmark \\ \hline
Hahn \cite{hahn2019self} & \checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & \xmark \\ \hline
Hou \cite{hou2019learning} & \xmark &Attention maps& \xmark & \checkmark & \checkmark & \xmark & \xmark \\ \hline
Luan \cite{luan2019msd} & \checkmark &Feature maps& \xmark & \checkmark& \checkmark & \checkmark &\xmark \\ \hline
Lee \cite{lee2019rethinking} &\checkmark& \xmark& \checkmark& \xmark & \xmark & \xmark & \xmark\\ \hline
Xu \cite{xu2019data} &\checkmark & Feature maps & \checkmark & \xmark & \checkmark & \xmark &\xmark \\ \hline
Zhang \cite{zhang2019your} &\checkmark& Feature maps& \xmark & \checkmark & \checkmark & \xmark &\xmark \\ \hline
Yang \cite{yang2019training} & \checkmark & \xmark & \xmark & \xmark & \xmark & \xmark & \xmark\\
\hline
\end{tabular}
\end{center}
\vspace{-10pt}
\label{table:selfKD_comp}
\end{table*}
\subsubsection{Born-again distillation}
\textbf{Insight}: \textit{Sequential self-teaching of students enables them to become masters, and outperform their teachers significantly.}
Furlanello \textit{et al}. ~\cite{furlanello2018born} initializde the concept of self-distillation, in which the students are parameterized identically to their teachers, as shown in Fig.~\ref{fig:self_kd}(a). Through sequential teaching, the student is continuously updated, and at the end of the procedure, additional performance gains are achieved by an ensemble of multiple student generations. Hahn \textit{et al}. ~\cite{hahn2019self} apply born-again distillation \cite{furlanello2018born} to natural language processing. Yang \textit{et al}. ~\cite{yang2019training} observe that it remains unclear how S-T optimization works, and they then focus on putting strictness (adding an extra term to the standard cross-entropy loss) to the teacher model, such that the student can better learn inter-class similarity, and potentially prevent over-fitting. Instead of learning a single task, Clark \textit{et al}. ~\cite{clark2019bam} extend \cite{furlanello2018born} to the multi-task setting, where single-task models are distilled sequentially to teach a multi-task model. Since the born-again distillation approach is based on the multi-stage training, it is less efficient and computation-heavy compared to the following methods.
\subsubsection{Distillation via `deep' supervision}
\textbf{Insight:} \textit{The deeper layers (or branches) in the student model contains more useful information than those of shallower layers.}
Among the methods, Hou \textit{et al}. ~\cite{hou2019learning}, Luan \textit{et al}. ~\cite{luan2019msd} and Zhang \textit{et al}. ~\cite{zhang2019your} propose similar approaches where the target network (student) is divided into several shallow sections (branches) according to their depths and original structures (see Fig.~\ref{fig:self_kd}(b)). As the deepest section may contain more useful and discriminative feature information than shallower sections, the deeper branches can be used to distill knowledge to the shallower branches. In contrast, in \cite{hou2019learning}, instead of directly distilling features, attention-based methods used in \cite{zagoruyko2016paying} are adopted to force shallower layers to mimic the attention maps of deeper layers. Luan \textit{et al}. ~\cite{luan2019msd} make each layer branch (ResNet block) a classifier. Thus, the deepest classifier is used to distill earlier the classifiers' feature maps and logits.
\subsubsection{Distillation based on data augmentation}
\label{dis_dataaug}
\textbf{Insight:} \textit{Data augmentation (\textit{e}.\textit{g}., rotation, flipping, cropping, etc) during training forces the student network to be invariant to augmentation transformations via self-distillation}.
Although most methods focus on how to better supervise student in self-distillation,
data representations for training the student are not fully excavated and utilized. To this end, Xu \textit{et al}. ~\cite{xu2019data} and Lee \textit{et al}. ~\cite{lee2019rethinking} focus on self-distillation via data augmentation of the training samples, as shown in Fig.~\ref{fig:self_kd}(c). There are some advantages to such a framework. First, it is efficient and effective to optimize a single student network without branching or the assistance of other models. Second, with data-to-data self-distillation, the student learns more inherent representations for generalization. Third, the performance of the student model is significantly enhanced with relatively low computation cost and memory load.
Xu \textit{et al}. ~\cite{xu2019data} apply random mirror and cropping to the batch images from the training data. Besides, inspired by mutual learning \cite{zhang2018deep}, the last feature layers and softmax outputs of the original batch image and distorted batch images are mutually distilled via MMD loss \cite{huang2017like} and KL divergence loss, respectively. In contrast, Lee \textit{et al}. ~\cite{lee2019rethinking} consider two types of data augmentation (rotation and color permutation to the same image), and the ensemble method used in \cite{lan2018knowledge, chen2019online, zhu2018knowledge} is employed to aggregate all logits of the student model to one, which is in turn is used by the student to transfer the knowledge to itself.
\subsubsection{Distillation with architecture transformation}
\textbf{Insight:} \textit{A student model can be derived by changing the convolution operators in the teacher model with any architecture change.}
In contrast with all the above-mentioned self-distillation methods, Crowley \textit{et al}. ~\cite{crowley2018moonshine} proposes structure model distillation for memory reduction by replacing standard convolution blocks with cheaper convolutions, as shown in Fig.~\ref{fig:self_kd}(d). In such a way, a student model that is a simple transformation of the teacher's architecture is produced. Then, attention transfer (AT) \cite{huang2017like} is applied to align the teacher's attention map with that of the student's.
\subsubsection{Summary and open challenges}
\textbf{Summary:} In Table.~\ref{table:selfKD_comp}, we summarize and compare different self-distillation approaches. Overall, using logits/feature information and two-stage training for self-distillation with `deep' supervision from the deepest branch are main stream. Besides, data augmentation and attention-based self-distillation approaches are promising. Lastly, it is shown that multi-task learning with self-distillation is also a valuable direction, deserving more research.
\noindent\textbf{Open challenges:}
There still exist many challenges to tackle. First, theoretical support laks in explaining why self-distillation works better. Mobahi \textit{et al}. ~\cite{mobahi2020self} provide theoretical analysis for born-again distillation \cite{furlanello2018born} and find that self-distillation may reduce over-fitting by loop-over training, thus leading to good performance. However, it is still unclear why other self-distillation methods (\textit{e}.\textit{g}., online `deep' supervision \cite{luan2019msd, zhang2019your, hou2019learning}) work better.
In addition, existing methods focus on self-distillation with certain types of group-based network structures (\textit{e}.\textit{g}., ResNet group). Thus, the generalization and flexibility of self-distillation methods need to be probed further. Lastly, all existing methods focus on classification-based tasks, and it is not clear whether self-distillation is effective for other tasks (\textit{e}.\textit{g}., low-level vision tasks).
\section{Label-required/-free distillation}
\textbf{Overall Insight:} \textit{It is possible to learn a student without referring to the labels of training data?}
\vspace{-5pt}
\subsection{Label-required distillation} The success of KD relies on the assumption that labels provide the required level of semantic description for the task at hand \cite{hinton2015distilling, bucilua2006model}. For instance, in most existing KD methods for classification-related tasks \cite{hinton2015distilling, park2019relational, szegedy2015going, park2019feed, hegde2019variational, yuan2019revisit, Ahn_2019_CVPR}, image-level labels are required for learning student network. Meanwhile, some works exploit the pseudo labels when training data are scarce. We now provide a systematic analysis for these two types of methods.
\subsubsection{KD with original labels.} Using the ground truth labels for the data used in the student network is the common way for KD. As depicted in Eq.\ref{loss_student}, the overall loss function is composed of the student loss and the distillation loss. The student loss is heavily dependent on the ground truth label $y$. Following this fashion, main-stream methods mostly utilize the original labels and design better distillation loss terms to achieve better performances \cite{chen2019online, wang2019heterogeneous, zhou2018graph, chen2017learning, guo2018learning, huang2017like}. This convention has also been continually adopted in recent KD methods, such as online distillation \cite{chen2019online, mullapudi2019online}, teacher-free distillation \cite{gan2019self,zhang2018better, zhang2019your}, and even cross-modal learning \cite{gupta2016cross, su2016adapting, nagrani2018learnable, nagrani2018seeing, do2019compact, aytar2017see}. While using labels expand the generalization capability of knowledge for learning student network, such approaches fail when labels are scarce or unavailable.
\subsubsection{KD with pseudo labels.} Some works also exploit the pseudo labels. The most common methods can be discomposed into two groups. The first one aims to create noisy labels. \cite{li2017learning, xie2019self, xu2019positive, sarfraz2019noisy} propose to leverage large number of noisy labels to augment small amount of clean labels, which turns to improve the generalization and robustness of student network. The second group of methods focus on creating pseudo labels via metadata \cite{lopes2017data}, class similarities \cite{nayak2019zero} or generating labels \cite{ye2020datafree,fang2019data}, etc.
\vspace{-5pt}
\subsection{Label-free distillation} However, in real-world applications, labels for the data used in the student network is not always easy to obtain. Hence, some attempts have been taken for label-free distillation. We now provide more detailed analysis for these methods.
\subsubsection{KD with dark knowledge.} This has inspired some works to exploit KD without the requirement of labels. Based on our review, label-free distillation is mostly achieved in cross-modal learning, as discussed in Sec.~\ref{cross_modal_sec}. With paired modality data (\textit{e}.\textit{g}., video and audio), where the label of modality (\textit{e}.\textit{g}., video) is available, the student learns the end tasks only based on the distillation loss in Eq.\ref{loss_student} \cite{aytar2016soundnet, afouras2019asr, salem2019learning, owens2016ambient, arandjelovic2017look, hafner2018cross}. That is, in this situation, the dark knowledge of teacher provides `supervision' for the student network.
\subsubsection{Creating meta knowledge.} Recently, a few methods \cite{ye2020datafree, fang2019data,yin2019dreaming} propose data-/label-free frameworks for KD. The core technique is to craft samples with labels by using either feature or logits information, which are also called meta knowledge. Although these methods point out an interesting direction for KD, there still exist many challenges to achieve reasonable performance.
\subsection{Potential and challenges.} Label-free distillation is a promising since it relieves the need for data annotation. However, the current status of research shows that there still exist many uncertainties and challenges in this direction. The major concern is how to ensure that the `supervision' provided by teacher is reliable enough. As some works interpret the knowledge as a way a label regularization \cite{yuan2019revisit} or class similarities \cite{cheng2020explaining}, it is crucial to guarantee that the knowledge can be captured by the student.
Another critical challenge of label-required distillation is that, in Eq.~\ref{loss_student}, the KD loss term never involves any label information although the student loss (\textit{e}.\textit{g}., cross-entropy loss) uses labels. As labels provide informative knowledge for the student learning, it is worthwhile to find a way to use labels for the KD loss to further improve the performance. While it is generally acknowledged that a pretrained teacher has already mastered sufficient knowledge about the label information, its predictions still have a considerable gap with the ground truth labels. Based on our literature review, there exists some difficulties to bring label information to the distillation loss. That is, to bring the label information, the teacher might need to be updated or fine-tuned, which may cause additional computation cost. However, with some recent attempts based on meta learning or continual learning, it is possible to learn the label information with only a few examples. Besides, it might be possible to learn a bootstrap representation based on the labels, as done in \cite{grill2020bootstrap}, and further incorporate the information to the KD loss. We do believe this direction is promising in real-world applications and thus expect future research might move towards this direction.
\section{KD with novel learning metrics}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.85\textwidth]{figures/kd_based_on_GAN.pdf}
\caption{ An illustration of GAN-based KD methods.
(a) KD based on GAN \cite{goodfellow2014generative} where discriminator $D$ discerns the feature/logits of $T$ and $S$; (b) KD based on conditional GAN (CGAN) \cite{mirza2014conditional} where the input also functions as a condition to $D$; (c) KD based on TripleGAN \cite{chongxuan2017triple} where the classifier $C$, teacher $T$ and $D$ play a minmax game. }
\label{fig:kd_based_on_GAN}
\end{figure*}
\subsection{Distillation via adversarial learning}
\label{kd_gan}
\textbf{Overall Insight:} \textit{GAN can help learn the correlation between classes and preserve the multi-modality of S-T framework, especially when student has relatively small capacity.}
In Sec.~\ref{one_teacher}, we have discussed the two most popular approaches for KD. However, the key problem is that it is difficult for the student to learn the true data distribution from the teacher, since the teacher can not perfectly model the real data distribution. Generative adversarial networks (GANs) \cite{goodfellow2014generative, chongxuan2017triple,wang2019event, wang2020deceiving, wang2020eventsr} have been proven to have potential in learning the true data distribution in image translation. To this end, recent works \cite{wang2018kdgan, xu2017training, belagiannis2018adversarial, liu2018teacher, liu2018ktan, wang2018adversarial, roheda2018cross, xu2018training, chen2019data, shen2019meal, heo2019knowledge, liu2019exploiting, goldblum2019adversarially, hong2019gan, zhai2019lifelong, wang2019minegan,gao2019adversarial,shen2019adversarial,liu2019structured, chung2020featuremaplevel,aguinaldo2019compressing, li2020gan} have tried to explore adversarial learning to improve the performance of KD.
These works are, in fact, built on three fundamental prototypes of GANs \cite{goodfellow2014generative, li2017learning, mirza2014conditional}. Therefore, we formulate the principle of these three types of GANs, as illustrated in Fig.~\ref{fig:kd_based_on_GAN}, and analyze the existing GAN-based KD methods.
\subsubsection{A basic formulation of GANs in KD}
The first type of GAN, as shown in Fig.~\ref{fig:kd_based_on_GAN}(a), is proposed to generate continuous data by training a generator $G$ and discriminator $D$, which penalizes the generator $G$ for producing implausible results. The generator $G$ produces synthetic examples $G(z)$ (\textit{e}.\textit{g}., images) from the random noise $z$ sampled using a specific distribution (\textit{e}.\textit{g}., normal) \cite{goodfellow2014generative}. These synthetic examples are fed to the discriminator $D$ along with the real examples sampled from the real data distribution $p(x)$. The discriminator $D$ attempts to distinguish the two inputs, and both the generator $G$ and discriminator $D$ improve their respective abilities in a minmax game until the discriminator $D$ is unable to distinguish the fake from the real. The objective function can be written as follows:
\begin{equation}
\begin{split}
\min_G\max_D J(G,D) = \E_{x \sim p(x)}[log(D(x))] + \\
\E_{z \sim p(z)}[log(1-D(G(z)))]
\end{split}
\label{adv_loss}
\end{equation}
where $p_z(z)$ is the distribution of noise (\textit{e}.\textit{g}., uniform or normal).
The second type of GAN for KD is built on conditional GAN (CGAN) \cite{mirza2014conditional,isola2017image, wang2019event, wang2020deceiving}, as shown in Fig.~\ref{fig:kd_based_on_GAN}(b). CGAN is trained to generate samples from a class conditional distribution $c$. The generator is replaced by useful information rather than random noise. Hence, the objective of the generator is to generate realistic data, given the conditional information. Mathematically, the objective function can be written as:
\begin{equation}
\begin{split}
\min_G\max_D J(G,D) = \E_{x \sim p(x)}[log(D(x|c))] + \\
\E_{z \sim p(z)}[log(1-D(G(z|c)))]
\end{split}
\label{cgan_loss}
\end{equation}
Unlike the above-mentioned GANs, triple-GAN \cite{li2017learning} (the third type) introduces a three-player game where there is a classifier $C$, generator $G$, and discriminator $D$, as shown in Fig.~\ref{fig:kd_based_on_GAN}(c). Adversarial learning of generators and discriminators overcomes some difficulties \cite{goodfellow2014generative}, such as not having a good optimization and the failure of the generator to control the semantics of generated samples. We assume that there is a pair of data $(x,y)$ from the true distribution $p(x,y)$. After a sample $x$ is sampled from $p(x)$, $C$ assigns a pseudo label $y$ following the conditional distribution $p_c(y|x)$, that is, $C$ characterizes the conditional distribution $p_c(y|x) \approx p(y|x)$.The aim of the generator is to model the conditional distribution in the other direction $p_g(x|y) \approx p(x|y)$, while the discriminator determines whether a pair of data (x,y) is from the true distribution $p(x,y)$. Thus, the minmax game can be formulated as:
\begin{equation}
\begin{split}
\min_{C,G}\max_D J(C,G,D) = \E_{(x,y) \sim p(x,y)}[log(D(x,y))] + \\
\alpha \E_{(x,y) \sim p_c(x,y)}[log(1-D(x,y))] +\\
(1-\alpha) \E_{(x,y)\sim p_g(x,y)}[log(1-D(G(y,z),y))]
\end{split}
\label{tripe_gan_loss}
\end{equation}
where $\alpha$ is a hyper-parameter that controls the relative importance of $C$ and $G$.
\begin{table*}[t!]
\caption{Taxonomy of KD based on adversarial learning.
}
\small
\begin{center}
\begin{tabular}{c|c|c|c|c|c}
\hline
Method & GAN type & Purpose & Inputs of $D$ & Number of $D$ & Online KD \\
\hline\hline
Chen \cite{chen2019data} & First type &Classification& Logits & One & No \\ \hline
Belagiannis \cite{belagiannis2018adversarial} &First type& Classification& Features & One & No \\ \hline
Liu \cite{liu2018ktan} & First type &\thead{Classification \\ Object detection} & Features &One & No \\ \hline
Hong \cite{hong2019gan} &First type &Object detection &Features & Six & No \\ \hline
Wang \cite{wang2018adversarial} &First type& Classification& Features & One & No \\ \hline
Aguinaldo \cite{aguinaldo2019compressing} & First type & Classification & Features& One & No \\ \hline
Chung \cite{chung2020featuremaplevel} & First type (LSGAN \cite{mao2017least}) & Classification & Features & Two/Three & Yes \\ \hline
Wang \cite{wang2019minegan} & First type(WGAN-GP \cite{gulrajani2017improved}) & Image generation & Features & One/Multiple & Yes \\ \hline
Chen \cite{chen2020distilling} &First/Second type& Image translation & Features & Two & No \\ \hline
Liu \cite{liu2019structured} &Second type (WGAN-GP \cite{gulrajani2017improved})& Semantic segmentation& Features & One & No \\ \hline
Xu \cite{xu2017training} &Second type& Classification & Logits & One & No \\ \hline
Roheda \cite{roheda2018cross} &Second type & \thead{Cross-domain \\ surveillance} & Features & One & Yes\\ \hline
Zhai \cite{zhai2019lifelong} &Second type (BicyleGAN \cite{zhu2017toward}) & Image translation & Features & One & Yes\\ \hline
Liu \cite{liu2018teacher} &Second (AC-GAN \cite{odena2017conditional}) & Image translation &Features & One & No\\ \hline
Wang \cite{wang2018kdgan} &Third type & Image translation & Features & One & No \\ \hline
Li \cite{li2020gan} & First/Second type & Image translation & Features & One & No \\ \hline
Fang \cite{fang2019data} & First type & \thead{Classification\\ Semantic segmentation} & Logits & One & No \\ \hline
Yoo \cite{yoo2019knowledge} & Second type & Classification & Logits & One & No \\
\hline
\end{tabular}
\end{center}
\label{table:gan_kd_comp}
\end{table*}
\subsubsection{How does GAN boost KD?}
Based on the aforementioned formulation of GANs, we analyze how they are applied to boost the performance of KD with S-T learning.
\noindent\textbf{KD based on the conventional GAN (first type)} Chen \textit{et al}. \cite{chen2019data} and Fang \textit{et al}. \cite{fang2019data} focused on distilling the knowledge of \textit{logits} from teacher to student via the first type of GAN, as depicted in Fig.~\ref{fig:kd_based_on_GAN}(a).
\footnote{\cite{chen2019data,fang2019data} are data-free KD methods, which will be explicitly discussed in Sec. \ref{data_free}.} There are several benefits of predicting logits based on the discriminator. First, the learned loss, as described using Eqn~\ref{adv_loss}, can be effective in image translation tasks \cite{isola2017image, wang2019event, wang2020deceiving}. The second benefit is closely related to the multi-modality of the network output; therefore, it is not necessary to exactly mimic the output of one teacher network to achieve good student performance as it is usually done \cite{hinton2015distilling, romero2014fitnets}.
However, the low-level feature alignment is missing because the discriminator only captures the high-level statistics of the teacher and student outputs (logits).
In contrast, Belagiannis \textit{et al}. ~\cite{belagiannis2018adversarial}, Liu \textit{et al}. ~\cite{liu2018ktan}, Hong \textit{et al}. ~\cite{hong2019gan}, Aguinaldo \textit{et al}. ~\cite{aguinaldo2019compressing}, Chung \textit{et al}. ~\cite{chung2020featuremaplevel}, Wang \textit{et al}. ~\cite{wang2019minegan}, Wang \textit{et al}. ~\cite{wang2018adversarial}, Chen \textit{et al}. ~\cite{chen2020distilling}, and Li \textit{et al}. ~\cite{li2020gan} aimed to distinguish whether the \textit{features} come from the teacher or student via adversarial learning, which effectively pushes the two distributions close to each other. \footnote{Note that in \cite{chung2020featuremaplevel}, least squares GAN (LSGAN) \cite{mao2017least} loss was used and in \cite{wang2019minegan}, Wasserstein GAN-gradient penalty (WGAN-GP) loss \cite{gulrajani2017improved} was used to stabilize training.} The features of the teacher and student are used as inputs to the discriminator because of their \textit{dimensionality}. The feature representations extracted from the teacher are high-level abstract information and easy for classification, which lowers the probability for the discriminator to make a mistake \cite{liu2018ktan}.
However, the GAN training in this setting is sometimes unstable and even difficult to converge, particularly when the model capacity between the student and teacher is large. To address this problem, some regularization techniques such as dropout \cite{srivastava2014dropout} or $l_2$ or $l_1$ regularization \cite{belagiannis2018adversarial} are added to Eqn.~\ref{adv_loss} to confine the weights.
\noindent\textbf{KD based on CGAN (second type)}
Xu \textit{et al}. ~\cite{xu2017learning} and Yoo \textit{et al}. ~\cite{ yoo2019knowledge} employed CGAN \cite{mirza2014conditional} for KD, where the discriminator was trained to distinguish whether the \textit{label distribution} (logits) was from the teacher or the student. The student, which was regarded as the generator, was adversarially trained to deceive the discriminator. Liu \textit{et al}. ~\cite{liu2018teacher} also exploited CGAN for compressing image generation networks. However, the discriminator predicted the class label of the teacher and student, together with an auxiliary classifier GAN \cite{odena2017conditional}.
In contrast, Roheda \textit{et al}. ~\cite{roheda2018cross}, Zhai \textit{et al}. ~\cite{zhai2019lifelong}, Li \textit{et al}. ~\cite{li2020gan}, Chen \textit{et al}. ~\cite{chen2020distilling}, and Liu \textit{et al}. ~\cite{liu2019structured} focused on discriminating the \textit{feature} space of the teacher and student in the CGAN framework. Interestingly, Chen \textit{et al}. ~\cite{chen2020distilling} deployed two discriminators, namely, the teacher and student discriminators,for compressing image translation networks. To avoid model collapse, Liu \textit{et al}. ~\cite{liu2019structured} used Wasserstein loss \cite{gulrajani2017improved} to stabilize training.
\noindent\textbf{KD based on TripleGAN (third type)} In contrast to the distillation methods based on conventional GAN and CGAN , Wang \textit{et al}. ~\cite{wang2018kdgan} proposed a three-player game named KDGAN, consisting of a classifier (tent), a teacher, and a discriminator (similar to the prototype in TripleGAN \cite{li2017learning}), as shown in Fig.~\ref{fig:kd_based_on_GAN}(c). The classifier and the teacher learn from each other via distillation losses, and are adversarially trained against the discriminator via the adversarial loss defined in Eqn.~\ref{tripe_gan_loss}. By simultaneously optimizing the distillation and the adversarial loss, the classifier (student) learns the true data distribution at equilibrium.
\subsubsection{Summary and open challenges}
In Table~\ref{table:gan_kd_comp}, we summarize existing GAN-based knowledge distillation methods regarding the practical applications, input features of the discriminator $D$, the number of discriminators used, and whether it is one-stage (without the need for the teacher to be trained first). In general, most methods focus on classification tasks based on the first type of GAN (conventional GAN) \cite{goodfellow2014generative} and use the features as the inputs to the discriminator $D$. Besides, it is worth noting that most methods use only one discriminator for discerning the student from the teacher. However, some works such as \cite{chung2020featuremaplevel}, \cite{wang2019minegan} and \cite{chen2020distilling} employ multiple discriminators in their KD frameworks. One can see that most methods follow a two-stage KD paradigm where the teacher is trained first, and then knowledge is transferred to the student via KD loss. In contrast, studies such as \cite{chung2020featuremaplevel, wang2019minegan, roheda2018cross, zhai2019lifelong} also exploit online (one-stage) KD, without the necessity of pre-trained teacher networks. \textit{More detailed analyses of KD methods with respect to online/two-stage distillation and image translation are described in Sec.~\ref{online_kd} and Sec.~\ref{img_trans}, respectively.}
\noindent \textbf{Open challenges:} The first challenge for GAN-based KD is the stability of training, especially when the capacity between the teachers and the students is large. Secondly, it is less intuitive whether using only logits or only features or both as inputs to the discriminator is good because there lacks theoretical support. Thirdly, the advantages of using multiple discriminators are less clear and what features in which position are suitable for training GAN also needs to be further studied.
\subsection{Distillation with graph representations}
\label{kd_graph}
\noindent \textbf{Overall insight:} \textit{Graphs are the most typical locally connected structures that capture the features and hierarchical patterns for KD.}
Up to now, we have categorized and analyzed the most common KD methods using either logits or feature information. However, one critical issue regarding KD is data. In general, training a DNN requires embedding a high-dimensional dataset to facilitate data analysis. Thus, the optimal goal of training a teacher model is not only to transform the training dataset into a low-dimensional space, but also to analyze the intra-data relations \cite{lee2019graph,liu2019knowledgegraph}. However, most KD methods do not consider such relations. Here, we introduce the definitions of the basic concepts of graph embedding and knowledge graph based on \cite{cai2018comprehensive,hamilton2017representation}. We provide an analysis of existing graph-based KD methods, and discuss new perspectives about KD.
\noindent \subsubsection{Notation and definition}
\begin{defn}
A \textbf{graph} can be depicted as $\mathcal{G} = (V, E)$, where $v \in V$ is a node and $e \in E$ is an edge. A graph $\mathcal{G}$ is associated with a node type mapping function $F_v$: $V \to \mathcal{T}^v$, and an edge type mapping function $F_e$: $E \to \mathcal{T}^e $.
\end{defn}
Here, $\mathcal{T}^v$ and $\mathcal{T}^e$ denote the node types and edge types, respectively. For any $v_i \in V$, there exists a particular mapping type: $F_v(v_i) \in \mathcal{T}^v$. Similar mapping comes to any $e_{ij} \in E$, which is mapped as $F_e(e_{ij}) \in \mathcal{T}^e$, where $i$ and $j$ indicate the $i$-th and $j$-th nodes.
\begin{defn}
A \textbf{homogeneous graph (directed graph)}, depicted as $\mathcal{G}_{hom} = (V, E)$, is a type of graph in which $|\mathcal{T}^v| = |\mathcal{T}^e|=1$. All nodes and edges in this graph embedding are of one type.
\end{defn}
\begin{defn}
A \textbf{knowledge graph}, defined as $\mathcal{G}_{kn} = (V, E)$, is an instance of a directed heterogeneous graph whose nodes are \textit{entities}, and edges are subject-property-object triplets. Each edge has the form: head entity, relation, tail entity, denoted as $<h,r,t>$, indicating a relationship from a head $h$ to a tail $t$.
\end{defn}
$h, t \in V$ are entities and $r \in E$ is the relation. Hereby, we note $<h, r, t>$ as a triplet for knowledge graph. An example is shown in Fig.~\ref{fig:knoweldge_graph}. The knowledge graph includes two triplets $<h, r, t>$: $< Los Angeles, IsCityOf, California>$ and $<California, isStateOf, US>$.
\begin{table}[t!]
\centering
\caption{A summary of notations used in Sec.~\ref{kd_graph}.}
\begin{tabular}[width=\columnwidth]{l|c}
\hline
Notations & Descriptions\\
\hline\hline
$|\cdot|$ & The cardinally of a set \\ \hline
$\mathcal{G}= (V,E)$ & Graph $\mathcal{G}$ with a set of node $V$ and set of edge $E$ \\ \hline
$v_i$, $e_{ij}$ & A node $v_i \in V$ and an edge $e_{ij}$ linking $v_i$ and $v_j$ \\ \hline
$\textbf{x}_{v_i}$, $\textbf{x}_{e[v_i]}$ & Features of $v_i$ and features of edges of $v_i$ \\ \hline
$\textbf{h}_{ne[v_i]}$, $\textbf{x}_{ne[v_i]}$ & Features of states and of neighboring nodes of $v_i$ \\ \hline
$F_v(v_i)$, $F_e(e_{ij})$ & Mapping of node type $v_i$ and edge type $e_{ij}$ \\ \hline
$\mathcal{T}^v$, $\mathcal{T}^e$ & The set of node types and set of edge types \\ \hline
$<h, r, t>$ & Head, relation and tail in knowledge graph \\ \hline
$N$ & Number of nodes in the graph \\ \hline
$\textbf{h}_{v_i}$ & Hidden state of $i$-th node $v$ \\ \hline
$f_t$, $f_o$ & local transition and output functions \\ \hline
$F_t$, $F_o$ & global transition and output functions \\ \hline
$\textbf{H}$, $\textbf{O}$, $\textbf{X}$ & Stack of all hidden states, outputs, features \\ \hline
$\textbf{H}^{t}$ & Hidden state of $t$-th iteration of $\textbf{H}$ \\ \hline
\hline
\end{tabular}
\label{tab:comp_table1}
\end{table}
\begin{figure}[t!]
\centering
\includegraphics[width=\columnwidth]{figures/graph.pdf}
\caption{An example of knowledge graph.}
\label{fig:knoweldge_graph}
\end{figure}
\noindent \textbf{Graph neural networks.} A graph neural network (GNN) is a type of DNN that operates directly on the graph structure. A typical application is about node classification \cite{scarselli2008graph}. In the node classification problem, the $i$-th node $v_i$ is characterized by its feature $x_{v_i}$, and ground truth $t_{v_i}$. Thus, given a labeled graph $\mathcal{G}$, the goal is to leverage the labeled nodes to predict the unlabeled ones. It learns to represent each node with a $d$ dimensional vector state $h_{v_i}$ containing the information of its neighborhood. Specifically speaking, $h_{v_i}$ can be mathematically described as \cite{zhou2018graph}:
\begin{equation}
\textbf{h}_{v_i} = f_t(\textbf{x}_{v_i}, \textbf{x}_{co[v_i]}, \textbf{h}_{ne[v_i]}, \textbf{x}_{ne[v_i]})
\end{equation}
\begin{equation}
\textbf{o}_{v_i} = f_o(\textbf{h}_{v_i}, \textbf{x}_{v_i})
\end{equation}
where $\textbf{x}_{co[v_i]}$ denotes the feature of the edges connected with $v_i$, $\textbf{h}_{ne[v_i]}$ denotes the embedding of the neighboring nodes of $\textbf{v}_i$, and $\textbf{x}_{ne[v_i]}$ denotes the features of the neighboring nodes of $v_i$. The function $f_t$ is a transition function that projects these inputs onto a $d$-dimensional space, and $f_o$ is the local output function that produces the output. Note that $f_t$ and $f_o$ can be interpreted as the feedforward neural networks. If we denote $\textbf{H}$, $\textbf{O}$, $\textbf{X}$ and $\textbf{X}_N$ as the concatenation of the outputs of stacking all the states, all the outputs, all the features, and all the node features, respectively, then $\textbf{H}$ and $\textbf{O}$ can be formulated as:
\begin{equation}
\textbf{H}= F_t(\textbf{H}, \textbf{X})
\label{global_general}
\end{equation}
\begin{equation}
\textbf{O}= F_o(\textbf{H}, \textbf{X})
\end{equation}
where $F_t$ is the global transition function and $F_o$ is the global output function. Note that $F_t$ and $F_o$ are the stacked functions of $f_t$ and $f_o$, respectively, in all nodes $V$ in the graph.
Since we are aiming to get a unique solution for $\textbf{h}_{v_i}$, in \cite{scarselli2008graph, zhou2018graph}, a neighborhood aggregation algorithm is applied, such that:
\begin{equation}
\textbf{H}^{t+1}= F_t(\textbf{H}^t, \textbf{X})
\label{globale_state}
\end{equation}
where $\textbf{H}^t$ denotes $t$-th iteration of $\textbf{H}$.
Given any initial state $\textbf{H}(0)$, $\textbf{H}^{t+1}$ in Eqn.~\ref{globale_state} convergences exponentially to the solution in Eqn.~\ref{global_general}. Based on the framework, $f_t$ and $f_o$ can be optimized via supervised loss when the target information $t_v^i$ is known:
\begin{equation}
\mathcal{L} = \sum_{i=1}^N(t_v^i - o_v^i)
\end{equation}
where $N$ is the total number of supervised nodes in the graph.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{figures/graph_comp.pdf}
\vspace{-15pt}
\caption{A graphical illustration of graph-based KD methods. GKD \cite{lassance2019deep}, IRG \cite{liu2019knowledgegraph}, KTG \cite{minami2019knowledge}, MHKD \cite{lee2019graph} all focus on graph-based knowledge distillation for model compression. GFL \cite{yao2019graph} and HGKD \cite{wang2019heterogeneous} aim to improve semi-supervised node classification via graph-based knowledge transfer, whereas GRL \cite{ma2019graph} exploits graph-based knowledge for multi-task learning.}
\label{fig:gkd}
\end{figure*}
\subsubsection{Graph-based distillation}
Based on the above explanation regarding the fundamentals of graph representations and GNN, we now delve into the existing graph-based distillation techniques. To our knowledge, Liu \textit{et al}. ~\cite{liu2011cross} first introduced a graph-modeling approach for visual recognition task in videos. Videos are action models modeled initially as bag of visual words (BoVW), which is sensitive to visual changes. However, some higher-level features are shared across views, and enable connecting the action models of different views. To better capture the relationship between two vocabularies, they construct a bipartite graph $\mathcal{G} = (V, E)$ to partition them into visual-word clusters. Note that $V$ is the union of vocabularies $V_1$ and $V_2$, and $E$ are the weights attached to nodes. In this way, knowledge from BoVW can be transferred to visual-word clusters, which are more discriminative in the presence of view changes. Luo \textit{et al}. ~\cite{luo2018graph} consider incorporating rich, privileged information from a large-scale multimodal dataset in the source domain, and improve the learning in the target domain where training data and modalities are scarce. Regarding using S-T structures for KD, to date, there are several works such as \cite{liu2019knowledgegraph,lee2019graph,ma2019graph, lassance2019deep, wang2019heterogeneous,minami2019knowledge, yao2019graph, yang2020distillating}.
GKD \cite{lassance2019deep} and IRG \cite{liu2019knowledgegraph} consider the geometry of the perspective feature spaces by reducing intra-class variations, which allow for dimension-agnostic transfer of knowledge. This perspective is the opposite of Liu \textit{et al}. ~\cite{liu2019knowledgegraph} and RKD \cite{park2019relational}. Specifically, instead of directly exploring the mutual relation between data points in students and teachers, GKD \cite{lassance2019deep} regards this relation as a geometry of data space (see Fig.~\ref{fig:gkd}(a)). Given a batch of inputs $\textbf{X}$, we can compute the inner representation $\textbf{X}_l^S=[\textbf{x}_l^S, \textbf{x} \in \textbf{X}$ and $\textbf{X}_l^T = [\textbf{x}_l^T, \textbf{x} \in \textbf{X}$ at layer $l$ ($l \in \Lambda$) of the teacher and student networks. Using cosine similarity metric, these representations can be used to build a $k$-nearest neighbor similarity graph for the teacher $\mathcal{G}_l^T(\textbf{X})=<\textbf{X}_l^T, \textbf{W}_l^T>$, and for the student $\mathcal{G}_l^S(\textbf{X})=<\textbf{X}_l^S, \textbf{W}_l^S>$. Note that $\textbf{W}_l^T$ and $\textbf{W}_l^S$ represent the edge weights, which represent the similarity between the $i$-th and $j$-th elements of $\textbf{X}_l^T$ and $\textbf{X}_l^S$. Based on graph representation for both the teacher and the student, the KD loss in Eqn.~\ref{fea_dis_loss} can be updated as follows:
\begin{equation}
\mathcal{L}= \sum_{l \in \Lambda} D\left(\mathcal{G}_l^S(\textbf{X}), \mathcal{G}_l^T(\textbf{X})\right)
\label{GKD_loss}
\end{equation}
where the distance metric $D$ is based on $L_2$ distance.
\begin{table*}[t!]
\caption{A summary of KD methods via graph representations.
}
\vspace{-17pt}
\small
\begin{center}
\begin{tabular}[width=\textwidth]{c|c|c|c|c|c}
\hline
Method & Purpose & Graph type & Knowledge type & Distance metric & Graph embedding \\
\hline\hline
GKD \cite{lassance2019deep} &Model compression& Heterogeneous graph & Layer-wise feature & $L_2$ & GSP \cite{ortega2018graph} \\ \hline
IRG \cite{liu2019knowledgegraph} &Model compression& Knowledge graph & Middle layers& $L_2$ & Instance relations\\ \hline
MHKD \cite{lee2019graph} &Model compression & Knowledge graph & Middle layers & KL & SVD \cite{lee2018self} + Attention \\ \hline
KTG \cite{minami2019knowledge} & Model compression & Directed graph & Network model & $L_1$ + KP loss & -- \\ \hline
GFL \cite{yao2019graph} &Few-shot learning& GNN & Class of nodes & Frobenius norm & HGR \cite{ying2018hierarchical} \\ \hline
HGKD \cite{wang2019heterogeneous} &Few-shot learning&GNN & Class of nodes& Wasserstein& GraphSAGE \cite{hamilton2017inductive} \\ \hline
GRL \cite{ma2019graph} & Multi-task leaning & GNN & Class of nodes & Cross-entropy & HKS \cite{li2016deepgraph} \\ \hline
Yang \cite{yang2020distillating} & Model compression & GNN & Topological info. & KL & Attention \\ \hline
\hline
\end{tabular}
\end{center}
\label{table:fea_comp}
\end{table*}
IRG \cite{liu2019knowledgegraph} essentially is similar to GKD \cite{lassance2019deep} in the construction of the graph, however, IRG also takes into account the instance of graph transformations. The aim of introducing feature space transformation across layers is because there may be too tight or dense constraints and descriptions fitting on the teacher’s instance features at intermediate layers. The transformation of the instance relation graph is composed of vertex transformation and edge transformation
from the $l_1l$-th layer to $l_2$-th layer, as shown in Fig.~\ref{fig:gkd} (b). Thus the loss in Eqn.~\ref{GKD_loss} can be extended to:
\begin{equation}
\begin{split}
\mathcal{L}= \sum_{l \in \Lambda} D_1(\mathcal{G}_l^S(\textbf{X}), \mathcal{G}_l^T(\textbf{X})) + \\
D_2 \left((\Theta_T(\mathcal{G}_l^S(\textbf{X})), \Theta_S(\mathcal{G}_l^T(\textbf{X}))\right)
\end{split}
\end{equation}
where $\Theta_T$ and $\Theta_S$ are the transformation functions for the teacher and the student, respectively, and $D_1$ and $D_2$ are the distance metrics for instance relation and instance translation.
MHKD \cite{lee2019graph} is a method that enables distilling data based knowledge from a teacher network to a graph using an attention network (see Fig.~\ref{fig:gkd}(d)). Like IRG \cite{liu2019knowledge}, feature transformation is also considered to capture the intra-data relations. The KD loss is based on the KL-divergence loss using the embedded graphs from the teacher and the student. KTG \cite{minami2019knowledge} also exploits graph representation; however, it focuses on a different perspective of KD. The knowledge transfer graph provides a unified view of KD, and has the potential to represent diverse knowledge patterns. Interestingly, each node in the graph represents the direction of knowledge transfer. On each edge, a loss function is defined for transferring knowledge between two nodes linked by each edge. Thus, combining different loss functions can represent collaborative knowledge learning with pair-wise knowledge transfer. Fig.~\ref{fig:gkd}(c) shows the knowledge graph of diverse collaborative distillation with three nodes, where $L_{s,t}$ represents the loss function used for the training node.
In addition, GFL \cite{yao2019graph}, HGKT \cite{wang2019heterogeneous}, GRL \cite{ma2019graph} and MHGD \cite{lee2019graph} all resort to GNN for the purpose of KD. HGKT and GFL focus on transferring knowledge from seen classes to unseen classes in few-shot learning \cite{kipf2016semi,sung2018learning}. GFL \cite{yao2019graph} leverages the knowledge learned by the auxiliary graphs to improve semi-supervised node classification in the target graph. As shown in Fig.~\ref{fig:gkd}(e), GFL learns the representation of a whole graph, and ensures the transfer of a similarly structured knowledge. Auxiliary graph reconstruction is achieved by using a graph autoencoder. HGTK aims to build a heterogeneous graph focusing on transferring intra-class and inter-class knowledge simultaneously. Inspired by modeling class distribution in adversarial learning \cite{goodfellow2014generative, wang2020deceiving, wang2018kdgan, heo2019knowledge, xu2017training, haidar2019textkd}, in which instances with the same class are expected to have the same distribution, the knowledge is transferred from seen classes to new unseen classes based on learned aggregation and embedding functions, as shown in Fig.~\ref{fig:gkd} (f). GRL \cite{ma2019graph} builds a multi-task KD method for representation learning based on DeepGraph \cite{li2016deepgraph}.This knowledge is based on GNN, and maps raw graphs to metric values. The learned graph metrics are then used as auxiliary tasks, and the knowledge of the network is distilled into graph representations (see Fig.~\ref{fig:gkd}(g)). The graph representation structure is learned via a CNN by feeding the graph descriptor to it. We denote pairs of graph and graph-level labels as $\{(G_i, y _i)\}_{i=1}^N$, where $G_i \in \mathcal{G}$, $y_i \in \mathcal{Y}$, and $\mathcal{G}$, $\mathcal{Y}$ are the cardinally of all possible graphs and labels respectively. Then, the loss for learning the model parameters are described as:
\begin{equation}
\mathcal{L} = \E [D(y_i, f(G_i;\theta))]
\end{equation}
where $\theta$ are the model parameters.
\noindent \textbf{Open challenges}
Graph representations are of significant importance for tackling KD problems because they better capture the hierarchical patterns in locally connected structures. However, there are some challenges. Firstly, graph representations are difficult to generalize because they are limited to structured data or specific types of data. Secondly, it is challenging to measure graph distances appropriately, since existing distance measure (\textit{e}.\textit{g}., $l_2$) may not fit well. Thirdly, layer-wise distillation is difficult to achieve in graph KD, because graph representation models and network structures in such cases are limited.
\subsection {KD for semi-/self-supervised learning}
\textbf{Overall insight:} \textit{KD with S-T learning aims to learn a rich representation by training a model with a large number of unlabeled datasets, and limited amount of labeled data.}
Semi-supervised learning usually handles the problem of over-fitting due to the lack of high-quality labels of training data. To this end, most methods apply S-T learning that assumes a dual role as a teacher and a student. The student model aims to learn the given data as before, and the teacher learns from the noisy data and generates predicted targets. These are then transferred to the student model via consistency cost. In self-supervised learning, the student itself generates knowledge to be learned via various approaches, and the knowledge is then transferred by the student to itself via distillation losses. We now provide a detailed analysis of the technical details of the existing methods.
\subsubsection{Semi-supervised learning} The baseline S-T frameworks for semi-supervised learning was initialized by Laine \textit{et al}. ~\cite{laine2016temporal} and Tarvainen \textit{et al}. ~\cite{tarvainen2017mean}, as illustrated in Fig.~\ref{fig:overview_fig}(b).
The student and the teacher models have the same structures, and the teacher learns from noise and transfers knowledge to the student via consistency cost. Interestingly, in \cite{tarvainen2017mean}, the teacher's weights are updated using the earth moving average (EMA) of the student's weights. Inspired by \cite{tarvainen2017mean}, Luo \textit{et al}. ~\cite{luo2018smooth}, Zhang \textit{et al}. ~\cite{zhang2019pairwise}, French \textit{et al}. ~\cite{french2017self}, Choi \textit{et al}. ~\cite{choi2019self}, Cai \textit{et al}. ~\cite{cai2019exploring} and Xu \textit{et al}. ~\cite{xu2019self} all employ similar frameworks where the teacher's weights are updated using exponential moving average (EMA) of the student. However, Ke \textit{et al}. \cite{ke2019dual} mention that using a coupled EMA teacher is not sufficient for the student, since the degree of coupling increases as the training goes on. To tackle this problem, the teacher is replaced with another student, and two students are optimized individually during training while a stabilization constraint is provided for knowledge exchange (similar to mutual learning \cite{zhang2018deep}).
Instead of taking independent weights between the teacher and the student, Hailat \textit{et al}. ~\cite{hailat2018teacher} employ weight-sharing, in which the last two fully connected layers of the teacher and the student are kept independent. The teacher model plays the role of teaching the student, stabilizing the overall model, and attempting to clean the noisy labels in the training dataset. In contrast, Gong \textit{et al}. ~\cite{gong2018teaching} and Xie \textit{et al}. ~\cite{xie2019self} follow the conventional distillation strategy proposed by \cite{hinton2015distilling}, where a pretrained teacher is introduced to generate learnable knowledge using unlabeled data, and utilizes it as privileged knowledge to teach the student on labeled data. However, during learning of the student, Xie \textit{et al}. \ inject noise (\textit{e}.\textit{g}., dropout) to the student such that it learns better than the teacher.
Papernot \textit{et al}. ~\cite{papernot2016semi} propose to distill from multiple teachers (an ensemble of teachers) on a disjoint subset of sensitive data (augmented with noise) and to aggregate the knowledge of teachers to guide the student on query data.
\subsubsection{Self-supervised learning} Distilling knowledge for self-supervised learning aims to preserve the learned representation for the student itself, as depicted in Fig.~\ref{fig:overview_fig}(c). Using pseudo labels is the most common approach, as done in \cite{lee2018self,noroozi2018boosting}. Specifically, Lee \textit{et al}. ~\cite{lee2018self} adopt self-supervised learning for KD, which not only ensures the transferred knowledge does not vanish, but also provides an additional performance improvement. In contrast, Noroozi \textit{et al}. ~\cite{noroozi2018boosting} propose to transfer knowledge by reducing the learned representation (from a pretrained teacher model) to \textit{pseudo-labels} (via clustering) on the unlabeled dataset,
which are then utilized to learn a smaller student network. Another approach is based on data augmentation (\textit{e}.\textit{g}., rotation, cropping, color permutation) \cite{lee2019rethinking, xu2019data,xu2020knowledge}, which has been mentioned in Sec.~\ref{dis_dataaug}. In contrast to making the `positive' and `negative' (augmented) examples, BYOL \cite{grill2020bootstrap} directly bootstraps the representations with two neural networks, referred to as online and target networks, that interact and learn from each other. This spirit is somehow similar to mutual learning \cite{zhang2018deep}; however, BYOL trains its online network to predict the target network’s representation of another augmented view of the same image. The promising performance of BYOL might point out a new direction of KD with self-supervised learning via representation boostrap rather than using negative examples.
\subsubsection{Potentials and open challenges}
Based on the technical analysis for the KD methods in semi-/self-supervised learning, it is noticeable that online distillation is the mainstream. However, there are several challenges. First, as pointed by \cite{ke2019dual}, using EMA for updating teacher's weights might lead to less optimal learning of knowledge. Second, no methods attempt to exploit the rich feature knowledge from teacher models. Third, data augmentation methods in these distillation methods are less effective compared to those proposed in Sec.~\ref{self_distillation}, in which the advantages of adversarial learning are distinctive. Fourth, the representations of knowledge in these methods are limited and less effective. BYOL \cite{grill2020bootstrap} opens a door for representation boost, and there exists a potential to further bind this idea with KD in the further research. Moreover, it has the potential to exploit a better-structured data representation approach, such as in GNNs.
With these challenges, the future directions of KD for semi-/self-supervised learning could gain inspirations from exploiting feature knowledge and more sophisticated data augmentation methods together with more robust representation approaches.
\subsection{Few-shot learning}
\textbf{Insight:}\textit{Is it possible possible to learn an effective student model to classify unseen classes (query sets) by distilling knowledge from a teacher model with the support set?}
In contrast to the methods discussed in Sec.~\ref{distill_fewdata} focusing on distillation with a few samples for training a student network (without learning to generalize to new classes), this section stresses on analyzing the technical details of few-shot learning with KD. Few-shot learning is to classify new data, having seen from only a few training examples. Few-shot learning itself is a meta-learning problem in which the DNN learns how to learn to classify, given a set of training tasks, and evaluate using a set of test tasks. Here, the goal is to discriminate between $N$ classes with $K$ examples of each (so-called $N$-way-$K$-shot classification). In this setting, these training examples are known as the \textit{support set}. In addition, there are further examples of the same classes, known as a \textit{query set}. The approaches for learning prior knowledge of a few-shot are usually based on \textit{three} types: prior knowledge about similarity, prior knowledge about learning procedure, and prior knowledge about data.
We now analyze the KD methods for few-shot learning \cite{dvornik2019diversity,flennerhag2018transferring,park2019relational,liu2019semantic,jin2019learning} that have been recently proposed.
\noindent \textbf{Prior knowledge about similarity:} Park \textit{et al}. ~\cite{park2019relational} propose distance-wise and angle-wise distillation losses. The aim is to penalize the structural differences in relation to the learned representations between the teacher and the student for few-shot learning.
\noindent \textbf{Prior knowledge about learning procedure:} \cite{flennerhag2018transferring,jin2019learning} tackle the second type of prior knowledge, namely learning procedure. To be specific, Flennerhag \textit{et al}. ~\cite{flennerhag2018transferring} focuses on transferring knowledge across the learning process, in which the information from previous tasks is distilled to facilitate the learning on new tasks. However, Jin \textit{et al}. ~\cite{jin2019learning} address the problem of learning a meta-learner that can automatically learn what knowledge to transfer from the source network to where in the target network.
\noindent \textbf{Prior knowledge about data:} Dvornik \textit{et al}. ~\cite{dvornik2019diversity} and Liu \textit{et al}. ~\cite{liu2019semantic} address the third type of prior knowledge, namely data variance. To be specific, in \cite{dvornik2019diversity}, en ensemble of several teacher networks is elaborated to leverage the variance of the classifiers and encouraged to cooperate while encouraging the diversity of prediction. However, in \cite{liu2019semantic}, the goal is to preserve the knowledge of the teacher (\textit{e}.\textit{g}., intra-class relationship) learned at the pretraining stage by generating pseudo labels for training samples in the fine-tuning set.
\subsubsection{What's challenging?}
Based on our analysis, the existing techniques actually expose crucial challenges. First, the overall performance of KD-based few-shot learning is convincing, but the power of meta-learning is somehow degraded or exempted. Second, transferring knowledge from multi-source networks is a potential, but identifying what to learn and where to transfer is heavily based on the meta-learner, and selecting which teacher to learn is computation-complex. Third, all approaches focus on a task-specific distillation, but the performance drops as the domain shifts. Thus, future works may focus more on handling these problems.
\subsection{Incremental Learning}
\textbf{Overall insight:} \textit{KD for incremental learning mainly deals with two challenges: maintaining the performance on old classes, and balancing between old and new classes.}
Incremental learning investigates learning the new knowledge continuously to update the model's knowledge, while maintaining the existing knowledge \cite{wu2019large}. Many attempts \cite{hou2019learning, michieli2019knowledge,castro2018end,wu2019large, zhou2019m2kd,shmelkov2017incremental, zhai2019lifelong} have been made to utilize KD in addressing the challenge of maintaining the old knowledge. Based on the number of teacher networks used for distillation, these methods can be categorized into two types: distillation from a single teacher and distillation from multiple teachers.
\subsubsection{Distillation from a single teacher}
Shmelkov \textit{et al}. ~\cite{shmelkov2017incremental}, Wu \textit{et al}. ~\cite{wu2019large}, Michieli \textit{et al}. ~\cite{michieli2019knowledge} and Hou \textit{et al}. ~\cite{hou2019learning} focus on learning student networks for new classes, by distilling knowledge (logits information) from pretrained teachers on old-class data.
Although these methods vary in tasks and distillation process, they follow similar S-T structures. Usually, the pretrained model is taken as the teacher, and the same network or a different network is employed to adapt for new classes. Michieli \textit{et al}. \ exploit the intermediate feature representations and transfer them to the student.
\subsubsection{Distillation from multiple teachers}
Castro \textit{et al}. ~\cite{castro2018end}, Zhou \textit{et al}. ~\cite{zhou2019m2kd} and Ammar \textit{et al}. ~\cite{ammar2015autonomous} concentrate on learning an incremental model with multiple teachers. Specifically, Castro \textit{et al}. \ share the same feature extractor between teachers and the student. The teachers contain old classes, and their logits are used for distillation and classification. Interestingly, Zhou \textit{et al}. \ propose a multi-model and multi-level KD strategy in which all previous model snapshots are leveraged to learn the last model (student). This approach is similar to born-again KD methods, as mentioned in Sec. \ref{self_distillation}, where the student model at the last step, and is updated using the assembled knowledge from all previous steps. However, the assembled knowledge also depends on the intermediate feature representations. Ammar \textit{et al}. \ develop a cross-domain incremental RL framework, in which the transferable knowledge is shared and projected to different task domains of the task-specific student peers.
\subsubsection{Open challenges}
The existing methods rely on multi-step training (offline). However, it will be more significant if the online (one-step) distillation approaches can be utilized to improve the learning efficiency and performance. Moreover, existing methods require accessing the previous data to avoid ambiguities between the update steps. However, the possibility of data-free distillation methods remains open. Furthermore, existing methods only tackle the incremental learning of new classes in the same data domain, but it will be fruitful if cross-domain distillation methods can be applied in this direction.
\subsection{Reinforcement learning}
\textbf{Overall insight:} \textit{KD in reinforcement learning is to encourage polices (such as students) in the ensemble to learn from the best policies (such as teachers), thus enabling rapid improvement and continuous optimization.}
Reinforcement learning (RL) is a learning problem that trains a policy to interact with the environment in way that yields maximal reward. To use the best policy to guide other policies, KD has been employed in \cite{ashok2017n2n,liu2019knowledge,flennerhag2018transferring,burda2018exploration,hong2020periodic,xue2020transfer,lin2017collaborative, rusu2015policy}. Based on the specialties of these methods, we divide them into three categories, and provide an explicit analysis. \textit{We assume familiarity with basics of RL, and skip the definitions of Deep Q-network and A3C.}
\subsubsection{Collaborative distillation}
Xue \textit{et al}. ~\cite{xue2020transfer}, Hong \textit{et al}. ~\cite{hong2020periodic}, and Lin \textit{et al}. ~\cite{lin2017collaborative} focus on collaborative distillation, which is similar to mutual learning \cite{zhang2018deep}. In Xue \textit{et al}., the agents teach each other based on the reinforcement rule, and teaching occurs between the value function of the agents (students and teachers). Note that the knowledge is provided by a group of student peers periodically, and assembled to enhance the learning speed and stability as in \cite{hong2020periodic}. However, Hong \textit{et al}. \cite{hong2020periodic} periodically distill the best-performing policy to the rest of the ensemble. Lin \textit{et al}. \ stress on collaborative learning among heterogeneous learning agents, and incorporate the knowledge into online training.
\subsubsection{Model compression with RL-based distillation}
Ashok \textit{et al}. ~\cite{ashok2017n2n} tackle the problem of model compression via RL. The method takes a larger teacher network, and outputs a compressed student network derived from it. In particular, two recurrent policy networks are employed to aggressively remove layers from the teacher network, and to carefully reduce the size of each remaining layer. The learned student network is evaluated by a reward, which is a score based on the accuracy and compression of the teacher.
\subsubsection{Random network distillation}
Burda \textit{et al}. ~\cite{burda2018exploration} focus on a different perspective where the prediction problem is randomly generated. The approach involves two networks: the target (student) network, which is fixed and randomly initialized, and a predictor (teacher) network trained on the data collected by the agent. With the knowledge distilled from the predictor, the target network tends to have lower prediction errors. Rusu \textit{et al}. ~\cite{rusu2015policy} also apply random initialization for the target network. However, they focus more on online learning action policies, which can be either single-task or muti-task.
\subsubsection{Potentials of RL-based KD}
We have analyzed existing RL-based KD methods in detail. Especially, we notice that model compression via RL-based KD is promising due to its extraordinary merits. First, RL-based KD better addresses the problem of \textit{scalability} of network models. This is similar to neural architecture search (NAS). Moreover, the reward functions in RL-based KD \textit{better balance} the accuracy-size trade-off. It is also possible to transfer knowledge from a \textit{smaller model to a larger model}, which is a distinctive advantage over other KD methods.
\section{Applications for visual intelligence}
\subsection{Semantic and motion segmentation}
\textbf{Insight:} \textit{Semantic segmentation is a structured problem, and structure information (\textit{e}.\textit{g}., spatial context structures) needs to be taken into account when distilling knowledge for semantic segmentation networks.}
Semantic segmentation is a special classification problem that predicts the category label in a pixel-wise manner. As existing the state-of-the-art (SOTA) methods such as fully convolutional
networks \cite{long2015fully} have large model sizes and high computation costs, some methods
\cite{dou2020unpaired, liu2019structured, michieli2019knowledge, shan2019distilling, he2019knowledge,chen2018road,xie2018improving, fang2019data, mullapudi2019online} have been proposed to train lightweight networks via KD. Although these methods vary in their learning methods, most of them share the same distillation frameworks. Particularly, Xie \textit{et al}. ~\cite{xie2018improving}, Shan \textit{et al}. ~\cite{shan2019distilling}, and Michieli \textit{et al}. ~\cite{michieli2019knowledge} focused on pixel-wise, feature-based distillation methods. Moreover, Liu ~\textit{et al}. \cite{liu2019structured} and He \textit{et al}. ~\cite{he2019knowledge} both exploited affinity-based distillation strategy using intermediate features. Liu \textit{et al}.~ also employed pixel-wise and holistic KD losses via adversarial learning. In contrast, Dou \textit{et al}. ~\cite{dou2020unpaired} focused on unpaired multi-modal segmentation, and proposed an online KD method via mutual learning \cite{zhang2018deep}. Chen \textit{et al}. ~\cite{chen2018road} proposed a target-guided KD approach to learn the real image style by training the student to imitate a teacher trained with real images. Mullapudi \textit{et al}. ~\cite{mullapudi2019online} trained a compact video segmentation model via online distillation, in which a teacher's output was used as a learning target to adapt the student and select the next frame for supervision.
\subsection{KD for visual detection and tracking}
\textbf{Insight:} \textit{Challenges such as regression, region proposals, and less voluminous labels must be considered when distilling visual detectors.}
Visual detection is a crucial high-level task in computer vision. Speed and accuracy are two key factors for visual detectors. KD is a potential choice to achieve sped-up and lightweight network models. However, applying distillation methods to detection is more challenging than applying classification methods. First, detection performance degrades seriously after compression. Second, detection classes are not equally important, and special considerations for distillation have to be taken into account. Third, domain and data generalization has to be considered for a distilled detector. To overcome these challenges, several impressive KD methods \cite{chen2017learning,wang2019distilling,hong2019gan,hao2019end,chen2019new,chen2019learning,tang2019learning,jin2019learning,saputra2019distilling,jin2020uncertainty,lee2019teaching,xu2019training,kruthiventi2017low,ge2018low,luo2016face,feng2019triplet} have been proposed for compressing visual detection networks. We categorize these methods according to their specialties (\textit{e}.\textit{g}., pedestrian detection).
\subsubsection{Generic object detection} \cite{chen2017learning,hong2019gan,chen2019new,hao2019end,jin2020uncertainty,tang2019learning,wang2019distilling, felix2020squeezed, liu2019teacher} aimed to learn lightweight object detectors with KD. Among these works, Chen \textit{et al}. ~\cite{chen2019new} and Hao \textit{et al}. ~\cite{hao2019end} highlighted learning a class-incremental student detector by following the generic KD framework (from a pretrained teacher). However, novel object detection losses were adopted as strong impetus for learning new classes. These losses handled classification results, location results, the detected region of interest, and all intermediate region proposals. Moreover, Chen \textit{et al}. ~\cite{chen2017learning} learned a student detector by distilling knowledge from the intermediate layer, logits, and regressor of the teacher, in contrast to \cite{wang2019distilling}, in which only the intermediate layer of the teacher was utilized based on fine-grained imitation masks to identify informative locations. Jin \textit{et al}. ~\cite{jin2020uncertainty}, Tang \textit{et al}. ~\cite{tang2019learning}, and Hong \textit{et al}. \cite{hong2019gan} exploited multiple intermediate layers as useful knowledge. Jin \textit{et al}.~ designed an uncertainty-aware distillation loss to learn the multiple-shot features from the teacher network. However, Hong \textit{et al}. and Tang \textit{et al}. \ were based on one-stage KD (online) via adversarial learning and semi-supervised learning, respectively. In contrast, Liu \textit{et al}. ~\cite{liu2019teacher} combined single S-T learning and mutual learning of students for learning lightweight tracking networks.
\subsubsection{Pedestrian detection} While pedestrian detection is based on generic object detection, various sizes and aspect ratios of pedestrians under extreme illumination conditions are challenges. To learn an effective lightweight detector, Chen \textit{et al}. ~\cite{chen2019learning} suggested using the unified hierarchical knowledge via multiple intermediate supervisions, in which not only the feature pyramid (from low-level to high-level features) and region features, but also the logits information were distilled. Kruthiventi \textit{et al}. ~\cite{kruthiventi2017low} learned an effective student detector in challenging illumination conditions by extracting dark knowledge (both RGB and thermal-like hint features) from a multi-modal teacher network.
\subsubsection{Face detection} Ge \textit{et al}. ~\cite{ge2018low} and Karlekar \textit{et al}. ~\cite{karlekar2019deep} compressed face detectors to recognize low-resolution faces via selective KD (last hidden layer) from teachers which were initialized to recognize high-resolution faces. In contrast, Jin \textit{et al}. ~\cite{jin2019learning}, Luo \textit{et al}. ~\cite{luo2016face}, and Feng \textit{et al}. ~\cite{feng2019triplet} used single type of image. Jin \textit{et al}. focused on compressing face detectors by using the supervisory signal from the classification maps of teacher models and regression maps of the ground truth. They identify that it is better to learn a classification map of a larger model than that of smaller models. Feng \textit{et al}. presented a triplet KD method to transfer knowledge from a teacher model to a student model, in which a triplet of samples, the anchor image, positive image, and negative image, was used. The purpose of the triplet loss was to minimize the feature similarity between the anchor and positive images, while maximizing that between the anchor and negative images. Luo \textit{et al}. addressed the importance of neurons at the higher hidden layer of the teacher, and a neuron selection method was applied to select neurons that were crucial for teaching the student. Dong \textit{et al}. ~\cite{dong2019teacher} concentrated on the interaction between the teacher and the students. Two students learned to generate pseudo facial landmark labels, which were filtered and selected as the qualified knowledge by the teacher.
\subsubsection{Vehicle detection and driving learning} Lee \textit{et al}. ~\cite{lee2019teaching}, Saputra \textit{et al}. ~\cite{saputra2019distilling}, and Xu \textit{et al}. ~\cite{xu2017training} focused more on detection tasks for autonomous driving. In particular, Lee \textit{et al}. focused on compressing a vehicle maker classification system based on cascaded CNNs (teacher) into a single CNN structure (student). The proposed distillation method used the feature map as the transfer medium, and the teacher and student were trained in parallel (online distillation). Although the detection task was different, Xu \textit{et al}. ~ build a binary weight Yolo vehicle detector by mincing the feature maps of the teacher network from easy tasks to difficult ones progressively.
Zhao \textit{et al}. ~\cite{zhao2019lates} exploited an S-T framework to encourage the student to learn the teacher's sufficient and invariant representation knowledge (based on semantic segmentation) for driving.
\subsubsection{Pose detection}
Distilling human pose detectors has several challenges. First, lightweight detectors have to deal with arbitrary person images/videos to determine joint locations with unconstrained human appearances. Second, the detectors must be robust in viewing conditions and background noises. Third, the detectors should have fast inference speeds, and be memory-efficient. To this end, \cite{zhang2019fast,martinez2019efficient,thoker2019cross,hwang2020lightweight,xu2020integral,wang2019distill, nie2019dynamic} formulated various distillation methods. Zhang \textit{et al}. ~\cite{zhang2019fast} achieved effective knowledge transfer by distilling the joint confidence maps from a pre-trained teacher model, whereas Huang \textit{et al}. ~\cite{hwang2020lightweight} exploited the heat map and location map of a pretrained teacher as the knowledge to be distilled. Furthermore, Xu \textit{et al}. ~\cite{xu2020integral}, Thoker \textit{et al}. ~\cite{thoker2019cross}, and Martinez \textit{et al}. ~\cite{martinez2019efficient} focused on multi-person pose estimation. Thoker \textit{et al}.~ addressed cross-modality distillation problems, in which a novel framework based on mutual learning \cite{zhang2018deep} of two students supervised by one teacher was initialized. Xu \textit{et al}. ~\cite{xu2020integral} learned the integral knowledge, namely, the feature, logits, and structured information via a discriminator under the standard S-T framework. Martinez \textit{et al}. ~\cite{martinez2019efficient} trained the student to mimic the confidence maps, feature maps, and inner-stage predictions of a pre-trained teacher with depth images. Wang \textit{et al}. ~\cite{wang2019distill} trained a 3D pose estimation network by distilling knowledge from non-rigid structure from motion using only 2D landmark annotations. In contrast, Nie \textit{et al}. ~\cite{nie2019dynamic} introduced an online KD in which the pose kernels in videos were distilled by leveraging the temporal cues from the previous frame in a one-shot learning manner.
\subsection{Domain adaptation}
\textbf{Insight:} \textit{Is it possible to distill knowledge of a teacher in one domain to a student in another domain?}
Domain adaptation (DA) addresses the problem of learning a target domain with the help of a different but related source domain \cite{ao2017fast}. Since Lopez \textit{et al}. ~\cite{lopez2015unifying} and Gupta \textit{et al}. ~\cite{gupta2016cross} initially proposed the technique of transferring knowledge between images from different modalities (called generalized distillation), it is natural to ask if this novel technique be used to address the problem of DA. The challenge of DA usually comes with transferring knowledge from the source model (usually with labels) to the target domain with unlabeled data. To address the problem, several KD methods based on S-T frameworks \cite{ao2017fast,hoffman2017cycada,meng2019domain,xu2019self,choi2019self,chen2019crdoco,tsai2018learning,cai2019exploring,deng2019cluster} have been proposed recently. Although these methods are focused on diverse tasks, technically, they can be categorized into two types: unsupervised and semi-supervised DA via KD.
\subsubsection{Semi-supervised DA}
French \textit{et al}. ~\cite{french2017self}, Choi \textit{et al}. ~\cite{choi2019self}, Cai \textit{et al}. ~\cite{cai2019exploring}, Xu \textit{et al}. ~\cite{xu2019self}, and Cho \textit{et al}. ~\cite{cho2019large} proposed similar S-T frameworks for semantic segmentation and object detection. These frameworks were the updated methods of Mean-Teacher \cite{tarvainen2017mean}, which is based on self-ensemble of the student networks (teacher and student models have the same structure). Note that the weights of the teacher models in these methods are the EMAs of the weights of the student models. In contrast, Choi \textit{et al}. added a target-guided generator to produce augmented images instead of stochastic augmentation, as in \cite{xu2019self, french2017self,cai2019exploring}. Cai \textit{et al}. \ also exploited the feature knowledge from the teacher model, and applied region-level and intra-graph consistency losses instead of the mean square error loss.
In contrast, Ao \textit{et al}. ~\cite{ao2017fast} proposed a generalized distillation DA method by applying the generalized distillation information \cite{lopez2015unifying} to multiple teachers to generate soft labels, which were then used to supervise the student model (this framework is similar to online KD from multiple teachers as mentioned in Sec.~\ref{multi_teach}). Cho \textit{et al}. ~\cite{cho2019large} proposed an S-T learning framework, in which a smaller depth prediction network was trained based on the supervision of the auxiliary information (ensemble of multiple depth predictions) obtained from a larger stereo matching network (teacher).
\subsubsection{Unsupervised DA} Some methods such as \cite{chen2019crdoco, hoffman2017cycada} distill the knowledge from the source domain to the target domain based on adversarial learning \cite{goodfellow2014generative} and image translation \cite{isola2017image, wang2020deceiving, wang2019event}. Technically, images in the source domain are translated to images in the target domain as data augmentation, and cross-domain consistency losses are adopted to force the teacher and student models to produce consistent predictions. Tsai \textit{et al}. ~\cite{tsai2018learning} and Deng \textit{et al}. ~\cite{deng2019cluster} focused on aligning the feature similarities between teacher and student models, compared with Meng \textit{et al}. ~\cite{meng2019domain}, who focused on aligning softmax outputs.
\subsection{Depth and scene flow estimation}
\textbf{Insight:} \textit{The challenges for distilling depth and flow estimation tasks come with transferring the knowledge of data and labels.}
Depth and optical flow estimations are low-level vision tasks aiming to estimate the 3D structure and motion of the scene. There are several challenges. First, in contrast to other tasks (\textit{e}.\textit{g}., semantic segmentation), depth and flow estimations do not have class labels. Thus, applying existing KD techniques directly may not work well.
Moreover, learning a lightweight student model usually requires a large amount of labeled data to achieve robust generalization capability. However, acquiring these data is very costly.
To address these challenges, Guo \textit{et al}.
~\cite{guo2018learning}, Pilzer \textit{et al}. ~\cite{pilzer2019refine}, and Tosi \textit{et al}. ~\cite{tosi2019learning} proposed distillation-based approaches to learn monocular depth estimation. These methods were focused on handling the second challenge, namely, data distillation. Specifically, Pilzer \textit{et al}. ~\cite{pilzer2019refine} proposed an unsupervised distillation approach, where the left image was translated to the right via the image translation framework \cite{isola2017image, wang2020deceiving}. The inconsistencies between the left and right images were used to improve depth estimation, which was finally used to improve the student network via KD. In contrast, Guo \textit{et al}.~ and Tosi \textit{et al}.~ focused on cross-domain KD, which aimed to distill the \textit{proxy} labels obtained from the stereo network (teacher) to learn a student depth estimation network. Choi \textit{et al}. ~\cite{cho2019large} learned a student network for monocular depth inference by distilling the knowledge of depth predictions from a \textit{stereo} teacher network via the data ensemble strategy.
Liu \textit{et al}. ~\cite{liu2019ddflow} and Aleotti \textit{et al}. ~\cite{aleotti2019learning} proposed data-distillation methods for scene flow estimation. Liu \textit{et al}. \ distilled reliable predictions from a teacher network with unlabeled data, and used these predictions (for non-occluded pixels) as annotations to guide a student network to learn the optical flow. They proposed to leverage on the knowledge learned by the teacher networks specialized in stereo to distill proxy annotations, which is similar to the KD method for depth estimation in \cite{guo2018learning, tosi2019learning}. Tosi \textit{et al}. ~\cite{tosi2020distilled} learned a compact network for predicting holistic scene understanding tasks including depth, optical flow, and motion segmentation, based on distillation of proxy semantic labels and semantic-aware self-distillation of optical information.
\subsection{Image translation}
\label{img_trans}
\textbf{Insight}:\textit{ Distilling GAN frameworks for image translation has to consider three factors: large number of parameters of the generators, no ground truth labels for training data, and complex framework (both generator and discriminator)}.
Attempts were made in several works to compress GANs for image translation with KD. Aguinaldo \textit{et al}. ~\cite{aguinaldo2019compressing} focused on unconditional GANs, and proposed to learn a smaller student generator by distilling knowledge from the generated images of a larger teacher generator using mean squared error (MSE). However, the knowledge incorporated in the teacher discriminator was not investigated. In contrast, Chen \textit{et al}. ~\cite{chen2020distilling} and Li \textit{et al}. ~\cite{li2020gan} focused on conditional GANs, and exploited the knowledge from the teacher discriminator. Specifically, Chen \textit{et al}. included a student discriminator to measure the distances between real images and images generated by the student and teacher generators. The student GAN was then trained under the supervision of the teacher GAN. In particular, Li \textit{et al}. ~\cite{li2020gan} adopted the discriminator of the teacher as the student discriminator, and fine-tuned the discriminator together with the compressed generator, which was automatically found with significantly lower computation cost and fewer parameters, by using NAS. In contrast, Wang \textit{et al}. ~\cite{wang2020collaborative} focused on compressing encoder-decoder based neural style transfer network via collaborative distillation (between the encoder and its decoder), where the student was restricted to learn the linear embedding of the teacher's output.
\subsection{KD for Video understanding}
\subsubsection{Video classification and recognition}
Bhardwaj \textit{et al}. ~\cite{bhardwaj2019efficient} and Wang \textit{et al}. ~\cite{wang2019progressive} employed the general S-T learning framework for video classification. The student was trained with processing only a few frames of the video, and produced a representation similar to that of the teacher. Gan \textit{et al}. ~\cite{gan2016you} focused on video concept learning for action recognition and event detection by using web videos and images. The learned knowledge from teacher network (Lead network) was used to filter out the noisy images. These were then used to fine-tune the teacher network to obtain a student network (Exceeding network). Gan \textit{et al}. ~\cite{gan2018geometry} explored geometry as a new type of practical auxiliary knowledge for self-supervised learning of video representations. Fu \textit{et al}. ~\cite{fu2019ultrafast} propose focused on video attention prediction by leveraging both spatial and temporal knowledge. Farhadi \textit{et al}. ~\cite{farhadi2019tkd} distill the temporal knowledge from a teacher model over the selected video frames to a student model.
\subsubsection{Video captioning}
\cite{zhang2020object, pan2020spatio} exploited the potential of graph-based S-T learning for image captioning. Specifically, Zhang \textit{et al}. ~\cite{zhang2020object} leveraged the object-level information (teacher) to learn the scene feature representation (student) via a spatio-temporal graph. Pan \textit{et al}. ~\cite{pan2020spatio} highlighted the importance of the relational graph connecting for all the objects in the video, and forced the caption model to learn the abundant linguistic language via teacher-recommended learning.
\section{Discussions}
In this section, we discuss some fundamental questions and challenges that are crucial for better understanding and improving KD.
\subsection{Are bigger models better teachers?}
The early assumption and idea behind KD are that soft labels (probabilities) from a trained teacher reflect more about the distribution of data than the ground truth labels \cite{hinton2015distilling}. If this is true, then it is expected that as the teacher becomes more robust, the knowledge (soft labels) provided by the teacher would be more reliable and better capture the distribution of classes. That is, a more robust teacher provides constructive knowledge and supervision to the student. Thus, the intuitive approach for learning a more accurate student is to employ a bigger and more robust teacher. However, based on the experimental results in \cite{cho2019efficacy}, it is found out that a bigger and more robust model does not always make a better teacher. As the teacher's capacity grows, the student's accuracy rises to some extent, and then begins to drop. We summarize two crucial reasons behind the lack of theoretical support for KD, based on \cite{cho2019efficacy, phuong2019distillation}.
\begin{itemize}
\item The student is able to follow the teacher, but it cannot absorb useful knowledge from the teacher. This indicates that there is a mismatch between the KD losses and accuracy evaluation methods. As pointed in \cite{phuong2019distillation}, the optimization method used could have a large impact on the distillation risk. Thus, optimization methods might be crucial for significant KD to the student.
\item Another reason comes from when the student is unable to follow the teacher due to the large model capacity between the teacher and the student. It is stated in \cite{heo2019comprehensive, hinton2015distilling} that the S-T similarity is highly related to how well the student can mimic the teacher. If the student is similar to the teacher, it will produce outputs similar to the teacher.
\end{itemize}
Intermediate feature representations are also effective knowledge that can be used to learn the student \cite{romero2014fitnets, kim2018paraphrasing}. The common approach for feature-based distillation is to transfer the features into a type of representation that the student can easily learn. In such a case, are bigger models are better teachers? As pointed in \cite{romero2014fitnets}, feature-based distillation is better than the distillation of soft labels, and deeper students perform better than shallower ones. In addition, the performance of the student increases upon increasing the number of layers (feature representations) \cite{kim2018paraphrasing}. However, when the student is fixed, a bigger teacher does not always teach a better student. When the similarity between the teacher and student is relatively high, the student tends to achieve plausible results.
\subsection{Is a pretrained teacher important?}
While most works focus on learning a smaller student based on the pretrained teacher, the distillation is not always efficient and effective. When the model capacity between the teacher and the student is large, it is hard for the student to follow the teacher, thus inducing the difficulty of optimization. Is a pretrained teacher important for learning a compact student with plausible performance? \cite{zhang2018deep, lan2018knowledge} propose learning from student peers, each of which has the same model complicity. The greatest advantage of this distillation approach is efficiency, since the pretraining of a high capacity teacher is exempted. Instead of teaching, the student peers learn to cooperate with each other to obtain an optimal learning solution. Surprisingly, learning without the teacher even enables improving the performance. The question of why learning without the teacher is better has been studied in \cite{tarvainen2017mean}. Their results indicate that the compact student may have a less chance of overfitting. Moreover, \cite{cho2019efficacy} suggests that early stopping of training on ImageNet \cite{deng2009imagenet} achieves better performance. The ensemble of students pool their collective predictions, thus helping to converge at a more robust minima as pointed in \cite{zhang2018deep}.
\subsection{Is born-again self-distillation better?}
Born-again network \cite{furlanello2018born} is the initial self-distillation method in which the student is trained sequentially, and the later step is supervised by the earlier generation. At the end of the procedure, all the student generations are assembled together to get an additional gain. So is self-distillation in the \textit{generations} better? \cite{cho2019efficacy} finds that network architecture heavily determines the success of KD in generations. Although the ensemble of the student models from all the generations outperforms a single model trained from scratch, the ensemble does not outperform an ensemble of an equal number of models trained from scratch.
Instead, recent works \cite{zhang2019your, xu2019data, mobahi2020self} shift the focus from sequential self-distillation (multiple stages) to the one-stage (online) manner. The student distills knowledge to itself without resorting to the teacher and heavy computation. These methods show more efficiency, less computation costs, and higher accuracy. The reason for such better performance has been pointed out in \cite{zhang2019your, mobahi2020self}. They have figured out that that online self-distillation can help student models converge to flat minima. Moreover, self-distillation prevents student models from the `vanishing gradient’ problem. Lastly, self-distillation helps to extract more discriminative features. In summary, online self-distillation shows significant advantages than sequential distillation methods and is more generalizable.
\subsection{Single teacher vs multiple teachers}
It is noticeable that recent distillation methods turn to exploit the potential of learning from multiple teachers. Is learning from multiple teachers really better than learning from a single teacher? To answer this question, \cite{you2017learning} intuitively identified that the student can fuse different predictions from multiple teachers to establish its own comprehensive understanding of the knowledge. The intuition behind this is that by unifying the knowledge from the ensemble of teachers, the relative similarity relationship among teachers is maintained. This provides a more integrated dark knowledge for the student. Similar to mutual learning \cite{zhang2018deep, lan2018knowledge}, the ensemble of teachers collects the individual predictions (knowledge) together, thus converging at minima that are more robust. Lastly, learning from multiple teachers relieves training difficulties such as vanishing gradient problems.
\subsection{Is data-free distillation effective enough?}
In the absence of training data, some novel methods \cite{chen2019data, ye2020datafree, lopes2017data, yoo2019knowledge} have been proposed to achieve plausible results. A theoretical explanation for why such methods are robust enough for learning a portable student has yet to be proposed. These methods are focused only on classification, and the generalization capability of such methods is still low. Most works employ generators to generate the `latent' images from noise via adversarial learning \cite{goodfellow2014explaining, wang2020deceiving}, but such methods are relatively hard to train and computationally expensive.
\subsection{Logits vs features}
The knowledge defined in existing KD methods is from three aspects: logits, feature maps (intermediate layers), and both. However, it is still unclear which one of these represents better knowledge. While works such as \cite{romero2014fitnets, kim2018paraphrasing, huang2017like, heo2019comprehensive, tung2019similarity} focus on better interpretation of feature representations and claim that features might contain richer information; some other works \cite{hinton2015distilling, zhang2018deep, nayak2019zero, wen2019preparing} mention that softened labels (logits) could represent each sample by a class distribution, and a student can easily learn the intra-class variations. However, it is noticeable that KD via logits has obvious drawbacks. First, its effectiveness is limited to the softmax loss function, and it relies on the number of classes (cannot be applied to low-level vision tasks). Secondly, when the capacity between the teacher and the student is big, it is hard for the student to follow the teacher's class probabilities \cite{cho2019efficacy}. Moreover, as studied in \cite{tung2019similarity}, semantically similar inputs tend to elicit similar activation patterns in teacher networks, indicating that the similarity-preserving knowledge from intermediate features express not only the representation space, but also the activations of object category (similar to class distributions). Thus, we can clearly see that features provide more affluent knowledge than logits, and generalize better to the problems without class labels.
\subsection{Interpretability of KD} In Sec.~\ref{theoretical_analysis}, we provided a theoretical analysis of KD based on the information maximization theory. It is commonly acknowledged that the teacher model's dark knowledge provides the privileged information on similarity of class categories to improve the learning of students \cite{hinton2015distilling, bucilua2006model}. However, why KD works is also an important question. There are some methods that explore the principles of KD from the view of label smoothing \cite{yuan2019revisit}, visual concepts \cite{cheng2020explaining}, category similarity \cite{hinton2015distilling}, etc. Specifically, \cite{yuan2019revisit} found that KD is a learned label smoothing regularization (LSR), and LSR is an ad-hoc KD. Even a poorly-trained teacher can improve the student's performance, and the weak student could improve the teacher. However, the findings in \cite{yuan2019revisit} only focus on classification-related tasks, and these intriguing results do not apply to the tasks without labels \cite{li2020gan, chen2020distilling}.
In contrast, \cite{cheng2020explaining} claims that KD makes DNN learn more task-related visual concepts and discard task-irreverent concepts to learn discriminative features. From a general perspective, the quantification of visual concepts in \cite{cheng2020explaining} provides a more intuitive interpretation for the success of KD. However, there exists a strong need that more intensive research needs to be done in this direction.
\subsection{Network architecture vs effectiveness of KD.} It has been demonstrated that distillation position has a significant impact on the effectiveness of KD \cite{cho2019efficacy, heo2019comprehensive}. Most methods demonstrate this by deploying the same network for both teacher and student. However, many fail to transfer across very different teacher and student architectures. Recently, \cite{tian2019contrastive} found that \cite{yim2017gift, ba2014deep, romero2014fitnets} perform poorly even on very similar student and teacher architectures. \cite{yuan2019revisit} also reported an intriguing finding that a poorly trained teacher also can improve the student's performance. It is thus quite imperative to excavate how network architecture affects the effectiveness of KD and why KD fails to work when the network architectures of student and teacher are different.
\section{New outlooks and perspectives}
In this section, we provide some ideas and discuss future directions of knowledge distillation. We take the latest deep learning methods (\textit{e}.\textit{g}., neural architecture search (NAS), graph neural network (GNN)), novel non-Euclidean distances (\textit{e}.\textit{g}., hypersphere), better feature representation approaches, and potential vision applications, such as 360$^\circ$ vision \cite{lee2019spherephd} and event-based vision \cite{wang2020eventsr} into account.
\subsection{Potential of NAS}
In recent years, NAS has become a popular topic in deep learning.
NAS has the potential of automating the design of neural networks. Therefore, it can be efficient for searching more compact student models. In such a way,
NAS can be incorporated with KD for model compression.
This has been recently demonstrated for GAN compression \cite{li2020gan,bashivan2019teacher}. It is shown to be effective for finding efficient student model from the teacher with lower computation costs and fewer parameters. It turns out that NAS improves the compression ratio and accelerates the KD process. A similar approach is taken by \cite{ashok2017n2n} learning to remove layers of teacher network based on reinforcement learning (RL). Thus, we propose that NAS with RL can be a good direction of KD for model compression. This might significantly relieve the complexity and enhance the learning efficiency of existing methods, in which the student is manually designed based on the teacher.
\subsection{Potential of GNN}
Although GNN has brought progress in KD learning under the S-T framework, some challenges remain. This is because most methods rely on finding structured data on which graph-based algorithms can be applied. \cite{liu2019knowledge} considers the instance features and instance relationships as instance graphs, and \cite{ma2019graph} builds an input graph representation for multi-task knowledge distillation. However, in knowledge distillation, there exists \textit{non-structural} knowledge in addition to the structural knowledge (\textit{e}.\textit{g}., training data, logits, intermediate features, and outputs of teacher), and it is necessary to construct a flexible knowledge graph to tackle the \textit{non-structural} distillation process.
\subsection{Non-Euclidean distillation measure}
Existing KD losses are mostly dependent on Euclidean loss (\textit{e}.\textit{g}., $l_1$), and have their own limitations. \cite{derezinski2014limits} has shown that algorithms that regularize with Euclidean distance, (\textit{e}.\textit{g}. MSE loss) are easily confused by random features. The difficulty arises when the model capacity between the teacher and the student is large. Besides, $l_2$ regularization does not penalize small weights enough. Inspired by a recent work \cite{park2019sphere} on GAN training, we propose that it is useful to exploit the information of higher-order statistics of data in non-Euclidean spaces (\textit{e}.\textit{g}., hypersphere). This is because geometric constraints induced by the non-Euclidean distance might make training more stable, thus improving the efficiency of KD.
\subsection{Better feature representations}
Existing methods that focus on KD with multiple teachers show potential for handling cross-domain problems or other problems where the ground truth is not available. However, the ensemble of feature representations \cite{park2019feed, chung2020featuremaplevel, hou2017dualnet} is still challenging in some aspects. One critical challenge is fusing the feature representations and balancing each of them with robust gating mechanisms. Manually assigning weights to each component may hurt the diversity and flexibility of individual feature representations, thus impairing the effectiveness of ensemble knowledge. One possible solution is attention gates, as demonstrated in some detection tasks \cite{li2019attention, schlemper2019attention}. The aim of this approach is to highlight the important feature dimensions and prune feature responses to preserve only the activations relevant to the specific task. Another approach is inspired by the gating mechanism used in long short-term memory(LSTM) \cite{hochreiter1997long,zhao2016regional}. That is, this gate unit in KD is elaborately designed to remember features across different image regions and to control the pass of each region feature as a whole by their contribution to the task (\textit{e}.\textit{g}., classification) with the weight of importance.
\subsection{A more constructive theoretical analysis}
While KD shows impressive performance improvements in many tasks, the intuition behind it is still unclear. Recently, \cite{cho2019efficacy} explained conventional KD \cite{hinton2015distilling} using linear models, and \cite{hegde2019variational, Ahn_2019_CVPR, tian2019contrastive} focus on explaining feature-based KD. Mobahi \textit{et al}.
\cite{mobahi2020self} provides theoretical analysis for self-distillation. However, the mechanism behind data-free KD and KD from multiple teachers is still unknown. Therefore, further theoretical studies on explaining the principles of these methods should be undertaken.
\subsection{Potentials for special vision problems}
While existing KD techniques are mostly developed based on vision problems (\textit{e}.\textit{g}., classification), they rarely exploit some special vision fields such as 360$^\circ$ vision \cite{lee2019spherephd} and event-based vision \cite{wang2020eventsr, wang2019event, mostafavilearning}. The biggest challenge for both these vision fields is the lack of labeled data, and learning in these needs a special change of inputs for neural networks. Thus, the potential of KD, particularly cross-modal KD, for these two fields is promising. By distilling knowledge from the teacher trained with RGB images or frames to the student network specialized in learning to predict 360$^\circ$ images or stacked event images, it not only handles the problem of lack of data, but also achieves desirable results in the prediction tasks.
\subsection{Integration of vision, speech and NLP.}
As a potential, it is promising to apply KD to the integrated learning problems of vision, speech, and NLP. Although recent attempts of cross-modal KD \cite{albanie2018emotion, zhao2018through, thoker2019cross, owens2016ambient} focus on transferring the knowledge from one modality (\textit{e}.\textit{g}., video) to the other (\textit{e}.\textit{g}., sound) on the end tasks, it is still challenging to learn the end tasks for the integration of the three modalities. The major challenge may come from collecting paired data of three modalities; however, it is possible to apply GAN or representation learning methods to unsupervised cross-modal KD for learning effective end tasks.
\section{Conclusion}
This review of KD and S-T learning has covered major technical details and applications for visual intelligence. We provide a formal definition of the problem, and introduce the taxonomy methods for existing KD approaches. Drawing connections among these approaches, we identify a new active area of research that is likely to create new methods that take advantage of the strengths of each paradigm. Each taxonomy of the KD methods shows the current technical status regarding its advantages and disadvantages. Based on explicit analyses, we then discuss methods to overcome the challenges, and break the bottlenecks by exploiting new deep learning methods, new KD losses, and new vision application fields.
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\section*{Acknowledgments}
\else
\section*{Acknowledgment}
\fi
This work was supported by the National Research
Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (NRF-2018R1A2B3008640) and Institute of Information \& Communications Technology Planning \& Evaluation(IITP) grant funded by Korea government(MSIT) (No.2020-0-00440, Development of Artificial Intelligence Technology that Continuously Improves Itself as the Situation Changes in the Real World).
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1,116,691,498,510 | arxiv | \subsubsection{Proof of Lemma~\ref{lemma_riggedgamescharacterizeresilience}}
\riggedgamescharacterizeresilience*
\begin{proof}
We begin by introducing translations between plays that are useful in all three cases.
First, we translate a play prefix~$w$ in $\mathcal{A}$ into a play prefix~$t'(w)$ in $\mathcal{A}_\mathrm{rig}$ satisfying the following invariant: $t'( (v_0, b_0) \cdots (v_j, b_j) )$ starts in $v_0$ and ends in $v_j$.
We proceed by induction starting with $t'(v_0, b_0) = (v_0, 0)$.
For the induction step, consider a play prefix~$(v_0, b_0) \cdots (v_j, b_j) (v_{j+1}, b_{j+1})$ such that $t'((v_0, b_0) \cdots (v_j, b_j))$ is already defined, which ends in $v_j$ due to our invariant.
We consider several cases:
\begin{itemize}
\item If $b_{j+1} = 1$, then $(v_j, v_{j+1})$ is a disturbance edge, which is simulated in $\mathcal{A}_\mathrm{rig}$ by Player~$1$ taking control at $v_j$, moving to $(v_j, v_{j+1})$ and then to $v_{j+1}$.
Hence, we define
\[
t'((v_0, b_0) \cdots (v_j, b_j) (v_{j+1}, b_{j+1})) = t'((v_0, b_0) \cdots (v_j, b_j) ) \cdot ((v_j,v_{j+1}),0) (v_{j+1},0).
\]
\item If $b_{j+1} = 0$ and $v_j \in V_0$, then $(v_j, v_{j+1})$ is a non-disturbance edge picked by Player~$0$, which is simulated in $\mathcal{A}_\mathrm{rig}$ by Player~$1$ ceding control at $v_j$ to Player~$0$ by moving $\overline{v_j}$, from where Player~$0$ can then move to $v_{j+1}$.
Hence, we define
\[
t'((v_0, b_0) \cdots (v_j, b_j) (v_{j+1}, b_{j+1})) = t'((v_0, b_0) \cdots (v_j, b_j) ) \cdot (\overline{v_j},0) (v_{j+1},0).
\]
\item If $b_{j+1} = 0$ and $v_j \in V_1$, then $(v_j, v_{j+1})$ is a non-disturbance edge picked by Player~$1$, which is simulated in $\mathcal{A}_\mathrm{rig}$ by Player~$1$ directly moving to $v_{j+1}$.
Hence, we define
\[
t'((v_0, b_0) \cdots (v_j, b_j) (v_{j+1}, b_{j+1})) = t'((v_0, b_0) \cdots (v_j, b_j) ) \cdot ((v_j, v_{j+1}),0) (v_{j+1},0).
\]
\end{itemize}
In each case, the invariant is satisfied and $t'((v_0, b_0) \cdots (v_j, b_j) (v_{j+1}, b_{j+1}))$ is indeed a play prefix due to $t'((v_0, b_0) \cdots (v_j, b_j))$ ending in $v_j$.
Furthermore, we extend $t'$ to infinite plays by defining $t'((v_0,b_0)(v_1,b_1)(v_2,b_2) \cdots)$ to be the unique play~$\rho'$ in $\mathcal{A}_\mathrm{rig}$ such that $t'((v_0,b_0)\cdots(v_j,b_j))$ is a prefix of $\rho'$ for every $j \in \omega$.
Let $\rho = (v_0,0) (v_1,0) (v_2,0) \cdots$ be a play in $\mathcal{A}$.
Then we have $t'(\rho) = (v_0, 0) (a_0,0) (v_1, 0) (a_1,0) (v_2, 0) (a_2,0)\cdots$ for auxiliary vertices~$a_0a_1a_2\cdots$ and $\disturbances(\rho) = \size{\set{ j \mid a_j \in D }}$, i.e., the number of disturbances during a play~$\rho$ in $\mathcal{A}$ is equal to the number of vertices from $D \subseteq A$ occurring in $t'(\rho)$.
Finally, we can use the translation~$t'$ to transform a strategy~$\sigma'$ for Player~$0$ in $\mathcal{A}_\mathrm{rig}$ to a strategy~$\sigma$ for her in $\mathcal{A}$.
To this end, let $b^-$ denote the homomorphism from $(V' \times \set{0,1})^*$ to $(V')^*$ that removes the second component.
Then, we define
\[
\sigma(v_0 \cdots v_j) = \sigma'( b^-( t'( (v_0, b_0) \cdots (v_j, b_j) ) ) \cdot \overline{v_j})
\]
where $b_0 = 0$ and for every $0< j' \le j$, $b_{j'} = 1 $ if and only if $v_{j'-1} \in V_0$ and $v_{j'} \neq \sigma(v_0 \cdots v_{j'-1}) $, i.e., we reconstruct the consequential disturbances with respect to $\sigma$ as defined thus far.
A simple induction shows that a play~$\rho$ in $\mathcal{A}$ being consistent with $\sigma$ implies that $t'(\rho)$ in $\mathcal{A}_\mathrm{rig}$ is consistent with $\sigma'$.
Now, we consider the other direction and translate a play prefix~$w$ in $\mathcal{A}_\mathrm{rig}$ into a play prefix~$t(w)$ in $\mathcal{A}$.
Here, we only consider play prefixes~$w$ starting and ending in a vertex from $V' \setminus A$, i.e., only play prefixes
that do not start or end in one of the auxiliary vertices.
This satisfies the following invariant: $t( (v_0, 0) \cdots (v_j, 0) )$ starts in $v_0$ and ends in $v_j$ (recall that $\mathcal{A}_\mathrm{rig}$ has no disturbance edges, which implies that all bits~$b_j$ in $w$ are equal to zero).
Again, we proceed by induction and start with $t(v_0,0) = (v_0,0)$.
For the induction step, consider a play prefix~$(v_0, 0) \cdots (v_j, 0)(a_j,0) (v_{j+1}, 0)$ such that $t((v_0, b_0) \cdots (v_j, b_j))$ is already defined, which ends in $v_j$ due to our invariant.
\begin{itemize}
\item If the prefix is of the form~$(v_0,0) \cdots (v_j,0)((v_j,v_{j+1}),0)(v_{j+1},0) $ with $v_j \in V_0$, then the last move simulated during the play prefix is the disturbance edge~$(v_j,v_{j+1}) \in D$.
Hence, we define
\[
t((v_0,0) \cdots (v_j,0)((v_j,v_{j+1}),0)(v_{j+1},0) ) = t((v_0,0) \cdots (v_j,0) ) \cdot (v_{j+1},1).
\]
\item If the prefix is of the form~$(v_0,0) \cdots (v_j,0)(\overline{v_j},0)(v_{j+1},0) $, then the last move simulated during the play prefix is the non-disturbance edge~$(v_j,v_{j+1}) \in E$ with $v_j \in V_0$.
Hence, we define
\[
t((v_0,0) \cdots (v_j,0)(\overline{v_j},0)(v_{j+1},0) ) = t((v_0,0) \cdots (v_j,0) ) \cdot (v_{j+1},0).
\]
\item If the prefix is of the form~$(v_0,0) \cdots (v_j,0)((v_j,v_{j+1}),0)(v_{j+1},0) $ with $v_j \in V_1$, then the last move simulated during the play prefix is the non-disturbance edge~$(v_j,v_{j+1}) \in E$.
Hence, we define
\[
t((v_0,0) \cdots (v_j,0)(v_j,v_{j+1})(v_{j+1},0) ) = t((v_0,0) \cdots (v_j,0) ) \cdot (v_{j+1},0).
\]
\end{itemize}
In each case, the invariant is satisfied and the extension is indeed a play prefix due to $t((v_0,0) \cdots (v_j,0) ) $ ending in $v_j$.
Again, we extend $t$ to infinite plays by defining $t((v_0,0)(v_1,0)(v_2,0) \cdots)$ to be the unique play~$\rho$ in $\mathcal{A}$ such that $t((v_0,0)\cdots(v_j,0))$ is a prefix of $\rho$ for every $j \in \omega$.
Let $\rho' = (v_0, 0) (a_0, 0) (v_1, 0) (a_1,0) (v_2, 0) (a_2,0) \cdots$ be a play in $\mathcal{A}_\mathrm{rig}$ starting in $V$.
This provides that $t(\rho') = (v_0,b_0) (v_1, b_1) (v_2, b_2) \cdots$ for some bits~$b_j$, and $\size{\set{ j \mid a_j \in D }} = \disturbances(t(\rho')) $, i.e., the number of vertices from $D \subseteq A$ occurring in $\rho'$ is equal to the number of disturbances during the play~$t(\rho')$ in $\mathcal{A}$.
To conclude, we again show that we can use the translation~$t$ to transform a strategy~$\sigma$ for Player~$0$ in $\mathcal{A}$ to a strategy~$\sigma'$ for her in $\mathcal{A}$.
Here, let $b^-$ denote the homomorphism from $(V \times \set{0,1})^*$ to $V^*$ that removes the second component in each letter.
Now, we define
\[
\sigma'(v_0 \cdots v_j\overline{v_j}) = \sigma( b^-( t( (v_0, 0) \cdots (v_j, 0) ) )).
\]
Finally, a simple induction shows that a play~$\rho'$ in $\mathcal{A}_\mathrm{rig}$ being consistent with $\sigma'$ implies that $t(\rho')$ in $\mathcal{A}$ is consistent with $\sigma$.
After these preparations, the proof of the three characterizations is straightforward employing the transformation of strategies described above.
\ref{lemma_riggedgamescharacterizeresilience_omegaplusone}.)
Let $v \in \mathcal{W}_0(\mathcal{A}_\mathrm{rig},\mathrm{Win}_\mathrm{rig})$, i.e., Player~$0$ has a winning strategy~$\sigma'$ from $v$.
Let the strategy~$\sigma$ for Player~$0$ in $\mathcal{A}$ be obtained from $\sigma'$ as described above.
We claim that $\sigma$ is $(\omega+1)$-resilient from $v$.
To this end, let $\rho = (v_0, b_0) (v_1, b_1) (v_2, b_2)\cdots$ be a play in $\mathcal{G}$ that starts in $v$, is consistent with $\sigma$, and has an arbitrary number of disturbances.
We need to show that $\rho$ is winning for Player~$0$, i.e., $v_0 v_1 v_2 \cdots \in \mathrm{Win}$.
As argued above, the play~$t'(\rho)$ in $\mathcal{A}_\mathrm{rig}$ is of the form~$(v_0, 0) (a_0, 0)(v_1, 0) (a_1, 0)(v_2, 0) (a_2, 0) \cdots$, starts in $v$, and is consistent with $\sigma'$.
This implies $t'(\rho) \in \mathrm{Win}_\mathrm{rig}$.
Thus, by definition of $\mathrm{Win}_\mathrm{rig}$, we have indeed $v_0v_1v_2\cdots \in \mathrm{Win}$.
Now, assume Player~$0$ has an $(\omega+1)$-resilient strategy~$\sigma$ for $\mathcal{G}$ from $v$.
Let the strategy~$\sigma'$ for Player~$0$ in $\mathcal{A}_\mathrm{rig}$ be obtained from $\sigma$ as described above.
We claim that $\sigma'$ is a winning strategy from $v$ in the game~$(\mathcal{A}_\mathrm{rig},\mathrm{Win}_\mathrm{rig})$.
To this end, let $\rho' = (v_0,0) (a_0,0) (v_1,0) (a_1,0) (v_2,0) (a_2,0)\cdots$ be a play in $\mathcal{A}_\mathrm{rig}$ starting in $v$ and consistent with $\sigma'$.
We need to show that $\rho'$ is winning for Player~$0$.
As argued above, the play~$t(\rho') = (v_0,b_0) (v_1,b_1) (v_2,b_2)\cdots $ in $\mathcal{A}$ starts in $v$ and is consistent with $\sigma$.
Since $\sigma$ is $(\omega+1)$-resilient from $v$, $t(\rho')$ is winning for Player~$0$, as it has at most $\omega$ disturbances.
Thus, $v_0 v_1 v_2 \cdots \in \mathrm{Win}$.
Hence, $\rho' \in \mathrm{Win}_\mathrm{rig}$ by definition of $\mathrm{Win}_\mathrm{rig}$, i.e., $\rho'$ is indeed winning for Player~$0$.
\ref{lemma_riggedgamescharacterizeresilience_omega}.)
As this proof is a refinement of the previous one, we only sketch the differences.
First, let $v \in \mathcal{W}_0(\mathcal{A}_\mathrm{rig},\mathrm{Win}_\mathrm{rig} \cup \mathrm{B\ddot{u}chi}(D))$, i.e., Player~$0$ has a winning strategy~$\sigma'$ from $v$ which induces a strategy~$\sigma$ for her in $\mathcal{A}$.
We show that $\sigma$ is $\omega$-resilient from $v$.
To this end, let $\rho = (v_0, b_0) (v_1, b_1) (v_2, b_2)\cdots$ be a play in $\mathcal{G}$ that starts in $v$, is consistent with $\sigma$, and has a finite number of disturbances.
We need to show that $\rho$ is winning for Player~$0$.
Again, the play~$t'(\rho)$ in $\mathcal{A}_\mathrm{rig}$ starts in $v$ and is consistent with $\sigma'$.
Now, we additionally have that $t'(\rho)$ visits vertices in $D$ only finitely often, as the number of these visits is equal to the number of disturbances in $\rho$, as argued above.
Hence, $t'(\rho)$ is not in $\mathrm{B\ddot{u}chi}(D)$, which implies $t'(\rho) \in \mathrm{Win}_\mathrm{rig}$, as $t'(\rho)$ is consistent with the winning strategy~$\sigma$.
This allows us, as before, to conclude that $\rho$ is indeed winning for Player~$0$.
Now, assume Player~$0$ has an $\omega$-resilient strategy~$\sigma$ for $\mathcal{G}$ from $v$ and let $\sigma'$ be the induced strategy for her in $\mathcal{A}_\mathrm{rig}$.
We show that $\sigma'$ is winning from $v$ in the game~$(\mathcal{A}_\mathrm{rig},\mathrm{Win}_\mathrm{rig})$, i.e., every play $\rho' = (v_0,0)(a_0,0) (v_1,0)(a_1,0) (v_2,0)(a_2,0)\cdots$ in $\mathcal{A}_\mathrm{rig}$ starting in $v$ and consistent with $\sigma'$ is winning for Player~$0$.
If $v_0 a_0 v_1 a_1 v_2 a_2 \cdots$ is in $\mathrm{B\ddot{u}chi}(D)$, then $\rho'$ is winning for Player~$0$.
Thus, assume it is not.
Then, consider the play~$t(\rho') = (v_0,b_0) (v_1,b_1) (v_2,b_2)\cdots $ in $\mathcal{A}$.
It starts in $v$, is consistent with $\sigma$, and has the same finite number of disturbances as $\rho'$ has visits to vertices in $D$.
Hence, as $\sigma$ is $\omega$-resilient from $v$, $t(\rho')$ is winning for Player~$0$.
From this we can conclude, as before, that $\rho'$ is indeed winning for Player~$0$.
\ref{lemma_riggedgamescharacterizeresilience_k}.)
Analogously to the previous one arguing about \myquot{less than $k$ disturbances} instead of \myquot{finitely many disturbances}.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_resiliencevaluessafety}}
\resiliencevaluessafety*
\begin{proof}
\ref{lemma_resiliencevaluessafety_infbranch}.)
Let $\mathcal{G} = (\mathcal{A}, \mathrm{Safety}(F))$.
Towards a contradiction assume that there is a vertex~$v \in V$ with $r_\mathcal{G}(v) = \omega$ and that there is a strategy~$\sigma$ that is $\omega$-resilient from $v$.
Due to $r_\mathcal{G}(v) < \omega+1$, $\sigma$ is not $(\omega+1)$-resilient from $v$.
Thus, there is a play~$\rho = (v_0, b_0)(v_1, b_1)(v_2, b_2) \cdots$ that starts in $v$, is consistent with $\sigma$, satisfies $\disturbances(\rho) < \omega+1$ (which is a tautology), and such that $v_0 v_1 v_2 \cdots \notin \mathrm{Safety}(F)$, i.e., there is a $j$ such that $v_j \in F$.
Consider a play of the form $\rho' = (v_0, b_0) \cdots (v_j, b_j) \rho''$ that is consistent with $\sigma$ and such that $(v_j, b_j)\rho''$ is disturbance-free.
Such a play exists, as each vertex in $V_0$ has a non-disturbance successor.
The play~$\rho'$ starts in $v$, is consistent with $\sigma$, satisfies $\disturbances(\rho') \le j$, as disturbances can only occur in the prefix~$(v_0, b_0) \cdots (v_j, b_j)$, but violates the safety condition, as $v_j \in F$ is visited by $\rho'$.
Therefore, $\sigma$ is not $(j+1)$-resilient from $v$, and in particular not $\omega$-resilient from $v$, which contradicts our assumption.
\ref{lemma_resiliencevaluessafety_finbranch}.)
Let $\mathcal{G} = (\mathcal{A}, \mathrm{Safety}(F))$ with finitely branching $\mathcal{A} = ( V, V_0, V_1, E, D )$ and let the corresponding values~$r(v)$ and the set~$S$ be defined as on Page~\pageref{page_r}.
We claim $r_\mathcal{G}(v) \le r(v)$ for every $v \in S$ and $r_\mathcal{G}(v) = \omega +1 $ for every $v \notin S$, which proves our claim.
Fix a vertex~$v \in S$.
To show $r_\mathcal{G}(v) \le r(v)$, we need to show for every strategy $\sigma$ for Player~$0$ that there is a play that starts in $v$, is consistent with $\sigma$, has at most $r(v)$ disturbances, and is losing for Player~$0$, i.e., it visits $F$ at least once.
We fix any strategy $\sigma$ and construct such a play inductively starting with the play prefix~$(v_0, b_0) = (v,0)$.
During the construction, we ensure that the prefix constructed thus far is consistent with $\sigma$ and that it ends in $S$.
Thus, assume we have constructed a play prefix~$w = (v_0, b_0) \cdots (v_j,b_j)$ satisfying the invariant.
To extend it, we distinguish two cases:
\begin{enumerate}
\item Assume $r(v_j) = 0$, i.e., $v_j \in S_0 = \att(F)$.
Then, consider the unique disturbance-free play~$(v_j,0) \rho$ consistent with $\sigma$ and the attractor strategy for Player~$1$ associated with $\att(F)$.
We extend $w$ by $\rho$ to complete the construction of the desired play.
The resulting play~$w\rho$ is consistent with $\sigma$ due to our invariant and the choice of $\rho$, and contains a vertex from $F$.
\item Assume $r(v_j) > 0$, i.e., $v_j \in S_{r(v_j)} = \att( S_{{r(v_j)}-1}\cup \bndr(S_{{r(v_j)}-1}))$.
Consider the unique disturbance-free play~$(v_j,0) \rho$ consistent with $\sigma$ and the attractor strategy for Player~$1$ associated with $\att( S_{{r(v_j)}-1}\cup \bndr(S_{{r(v_j)}-1}))$.
Let $(v_j,0) (v_{j+1},0) \cdots (v_{j+j'},0) $ be the minimal prefix of $(v_j,0) \rho$ such that $v_{j+j'} \in S_{{r(v_j)}-1}\cup \bndr(S_{{r(v_j)}-1})$.
If $v_{j+j'} \in S_{{r(v_j)}-1}$ (which implies $j' >0$ due to $v_j \notin S_{r(v_j)-1}$) then we extend $w$ to $w(v_{j+1},0) \cdots (v_{j+j'},0)$ to obtain the next prefix in our inductive construction.
If $v_{j+j'} \in \bndr(S_{{r(v_j)}-1})$, then there is a vertex~$v_{j+j'+1} \in S_{{r(v_j)}-1}$ and $(v_{j+j'},v_{j+j'+1}) \in D$ due to the definition of the $D$-boundary.
Thus, we extend $w$ to $w(v_{j+1},0) \cdots (v_{j+j'},0)(v_{j+j'+1},1)$ to obtain the next prefix in our inductive construction.
The resulting prefix is consistent with $\sigma$ and its last vertex is in $S_{{r(v_j)}-1} \subseteq S$, i.e., our invariant is satisfied.
\end{enumerate}
Now, let $v_{j_0}, v_{j_1},v_{j_2},\ldots$ be the sequence of last vertices of the prefixes obtained during the construction.
In particular, $v_{j_0} = v$.
By construction, we have $r(v_{j_0}) > r(v_{j_1}) > r(v_{j_2}) \cdots$.
Hence, we apply the second case at most $r(v_{j_0})$ many times and then have to apply the first case.
Hence, we indeed obtain an infinite play~$\rho$ starting in $v$, which is consistent with $\sigma$ due to our invariant, and which visits $F$, as the first case is eventually applied.
Finally, $\rho$ has at most $r(v_{j_0}) = r(v)$ many disturbances, as each application of the second case adds at most one disturbance edge and the first case adds none.
Thus, $\rho$ witnesses that $\sigma$ is not $(r(v)+1)$-resilient from $v$.
As we have picked $\sigma$ arbitrarily, we conclude $r_\mathcal{G}(v) \le r(v)$ as desired.
It remains to show $r_\mathcal{G}(v) = \omega +1 $ for every $v \notin S$.
We start by listing some properties of such vertices:
\begin{enumerate}
\item\label{propF} $v \notin F$, as $F \subseteq \att(F) = S_0 \subseteq S$.
\item\label{prop0} If $v \in V_0$, then there is a $v'$ with $(v,v') \in E$ and $v' \notin S$.
Towards a contradiction, assume there is no such $v'$.
Then, all successors of $v$ are in $S$.
As $v$ has only finitely many successors by assumption on $\mathcal{A}$, there is a $j$ such that all these successors are in $S_j$.
Hence, $v \in \att(S_j) \subseteq S_{j+1}\subseteq S$, which contradicts $v \notin S$.
\item\label{prop1} If $v \in V_1$, then all $v'$ with $(v,v') \in E$ satisfy $v' \notin S$.
Towards a contradiction, assume there is a successor of $v'$ in $S$.
Then, $v'$ is in some $S_j$ and $v \in \att(S_j) \subseteq S_{j+1}\subseteq S$, which contradicts $v \notin S$.
\item\label{propD} If $v \in V_0$ and $(v,v') \in D$, then $v' \notin S$.
Again, towards a contradiction assume there is a disturbance edge leading from $v$ to $v'$ in $S$.
Then, $v' $ is in some $S_j$ and $v \in \bndr(S_j) \subseteq S_{j+1}\subseteq S$, which contradicts $v \notin S$.
\end{enumerate}
Thus, due to Property~\ref{prop0}, Player~$0$ must have a positional strategy~$\sigma$ that moves from any vertex~$v \notin S$ to some successor~$v' \notin S$.
Now, consider a play~$\rho$ that starts in a vertex~$v \notin S$, is consistent with $\sigma$, and has an arbitrary number of disturbances.
It starts outside of $S$, Player~$0$ does not move into $S$ by definition of $\sigma$, Player~$1$ cannot due to Property~\ref{prop1}, and disturbances do not lead into $S$ due to Property~\ref{propD}. Hence, $\rho$ never visits $S$ and thus also avoids $F$, due to Property~\ref{propF}.
Hence, $\rho$ is winning for Player~$0$.
As $v$ and $\rho$ are arbitrary, we have shown $r_\mathcal{G}(v) = \omega+1$ for every $v \notin S$.
\end{proof}
\subsubsection{Proof of Theorem~\ref{theorem_optimalstrategiesinfinitelybranchinggames}}
\optimalstrategiesinfinitelybranchinggames*
\begin{proof}
Let $\mathcal{G} = (\mathcal{A}, \mathrm{Safety}(F))$ with finitely branching $\mathcal{A} = ( V, V_0, V_1, E, D )$, and let the values~$r(v)$ and the set~$S$ be defined as on Page~\pageref{page_r}.
We have shown $r_\mathcal{G}(v) \le r(v)$ for every $v \in S$ and $r_\mathcal{G}(v) = \omega +1 $ for every $v \notin S$ in the proof of Lemma~\ref{lemma_resiliencevaluessafety}.\ref{lemma_resiliencevaluessafety_finbranch}.
We now show $r_\mathcal{G}(v) \ge r(v)$ for every $v \in S$.
To simplify our notation, let $X_0 = F$ and $X_{j+1} = S_j \cup \bndr(S_j)$, i.e., $S_j = \att(X_j)$ for every $j$.
Now, for every $j \in \omega$, let $\sigma_j$ be the trap strategy for Player~$0$ associated with $S_j = \att(X_j)$, i.e., every disturbance-free play that starts in $V \setminus S_j$ and is consistent with $\sigma_j$ never visits $X_j$.
Recall that we defined~$r(v) = \min\set{j \mid v \in S_j}$ for all $v \in S$.
Thus, if $r(v) > 0$, then $v \notin S_{j-1}$.
We define a positional strategy~$\sigma$ for Player~$0$ as follows:
\begin{itemize}
\item If $v \in V_0 \cap S$ with $r(v) > 0$ then $\sigma(v) = \sigma_{r(v) - 1}(v)$.
\item If $v \in V_0 \cap S$ with $r(v) = 0$ then $\sigma(v) = v'$ for some arbitrary successor~$v'$ of $v$.
\item If $v \in V_0 \setminus S$ then $\sigma(v) = v'$ for some successor~$v'$ of $v$ with $v' \notin S$. We have argued
in the proof of Lemma~\ref{lemma_resiliencevaluessafety}.\ref{lemma_resiliencevaluessafety_finbranch}, that such a successor always exists if $v \notin S$.
\end{itemize}
Fix some $v \in S$ and consider a play~$\rho = (\rho_0,b_0) (\rho_1,b_1) (\rho_2,b_2) \cdots $ starting in $v \in S$, consistent with $\sigma$, and with $k < r(v)$ disturbances.
A straightforward induction on $j\ge 0$ shows that $r(\rho_j) \ge r(v)- \disturbances((\rho_0,b_0) \cdots (\rho_j,b_j) )$ for every $j$.
Thus, $r(\rho_j) \ge r(v) - k > 0 $, which implies $\rho_j \notin F \subseteq S_0$, i.e., $\rho$ is winning for Player~$0$.
Therefore, $\sigma$ is $r(v)$-resilient from every $v \in S$.
Conversely, in the proof of Lemma~\ref{lemma_resiliencevaluessafety}.\ref{lemma_resiliencevaluessafety_finbranch}, we have shown $r_\mathcal{G}(v) \le r(v)$.
Hence, $r(v) = r_\mathcal{G}(v)$, i.e., $\sigma$ is $r_\mathcal{G}(v)$-resilient from every $v \in S$.
Furthermore, the arguments presented in the proof of Lemma~\ref{lemma_resiliencevaluessafety}.\ref{lemma_resiliencevaluessafety_finbranch} for vertices~$v \notin S$ show that $\sigma$ is $(\omega+1)$-resilient from every $v \notin S$.
Altogether, $\sigma$ is optimally resilient.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_allresiliencevaluesinreachability}}
\allresiliencevaluesinreachability*
\begin{proof}
Consider the one-counter reachability game~$\mathcal{G}$ presented in Figure~\ref{fig_reachresilvalues} where the reachability condition is induced by the doubly-lined vertices, i.e., Player~$0$ wins if and only if a doubly-lined vertex is visited.
For every $\alpha \in \omega +2$, there is a vertex~$v$ with $r_\mathcal{G}(v) = \alpha$, where the lower vertex of resilience~$\omega$ has a uniform witness while the upper row of vertices does not, for reasons that are analogous to the ones presented in Example~\ref{example_safetyallpossibleresiliencevalues}:
Essentially, the upper row of vertices implements the fresh vertex~$v$ to obtain $\mathcal{G}'$.
\begin{figure}
\centering
\begin{tikzpicture}[ultra thick,label distance=-1.2mm]
\foreach \i in {0,1,...,7}{
\node[p0s,label=below right:$\i$] (l\i) at (\i*1.5,-1.5) {};
\node[p0s,double,label =below right:$\omega+1$] (a\i) at (\i*1.5,-2.6) {};
\node[p0s,label=below right:$\omega$] (o\i) at (\i*1.5,-.4) {};
}
\node (adots) at (12,-2.6) {$\cdots$};
\node (ldots) at (12,-1.5) {$\cdots$};
\node (odots) at (12,-.4) {$\cdots$};
\node[p0s,label=below right:$\omega$] (b) at (0,-3.9) {$$};
\foreach \i in {0,...,7}{
\path[-stealth]
(o\i) edge[] (l\i);
}
\foreach \i in {1,...,7}{
\path[-stealth]
(l\i) edge[bend left=0] (a\i);
}
\foreach \i [remember=\i as \lasti (initially 0)]in {1,...,7}{
\path[-stealth]
(o\lasti) edge (o\i);
}
\foreach \i [remember=\i as \lasti (initially 7)]in {6,...,0}{
\path[-stealth]
(a\lasti) edge (a\i)
(l\lasti) edge[fault] (l\i)
;
}
\path[-stealth]
(adots) edge (a7)
(o7) edge (odots)
(ldots) edge[fault] (l7)
(b) edge (a0)
(a0) edge[loop left] ()
(b) edge[fault,loop left] ()
(l0) edge[loop left] ()
;
\end{tikzpicture}
\caption{A one-counter reachability game with all possible resilience values (depicted as labels below vertices).}
\label{fig_reachresilvalues}
\end{figure}
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_reachoptimality_vs_resilience}}
\reachoptimalityvsresilience*
\begin{proof}
Fix a reachability optimal strategy~$\sigma_\mathrm{opt}$ for Player~$0$ in $\mathcal{G}$.
We sketch the proof ideas, but leave the straightforward details to the reader.
{\boldmath$\mathrm{val}_\mathcal{G}(v,\sigma_\mathrm{opt}) \le r_{\mathcal{G}'}(v)$:}
The statement is trivial if $r_{\mathcal{G}'}(v) = \omega +1$.
Hence, assume $r_{\mathcal{G}'}(v) \in \omega$, say $r_{\mathcal{G}'}(v) = k$.
Then, Player~$1$ has a winning strategy~$\tau$ for $(\mathcal{A}'_\mathrm{rig}, \mathrm{Safety}(F)_\mathrm{rig} \cup R_{\ge k+1})$ from $v$ due Lemma~\ref{lemma_riggedgamescharacterizeresilience}.\ref{lemma_riggedgamescharacterizeresilience_k}.
This strategy can be turned into a strategy~$\sigma$ for Player~$0$ in $\mathcal{A}$ that mimics $\tau$ while ignoring the auxiliary vertices in $\mathcal{A}'_\mathrm{rig}$ that are not in $\mathcal{A}$.
An induction shows that $F$ is reached within $k$ moves when starting in $v$ and playing according to $\sigma$.
Thus, $\mathrm{val}(v,\sigma_\mathrm{opt}) \le \mathrm{val}(v,\sigma) \le k = r_{\mathcal{G}'}(v)$.
{\boldmath$r_{\mathcal{G}'}(v) \le \mathrm{val}_\mathcal{G}(v,\sigma_\mathrm{opt})$:}
The statement is trivial, if $\mathrm{val}_\mathcal{G}(v,\sigma_\mathrm{opt}) = \omega +1$.
Hence, assume $\mathrm{val}_\mathcal{G}(v,\sigma_\mathrm{opt}) \in \omega$, say $\mathrm{val}_\mathcal{G}(v,\sigma_\mathrm{opt}) = k$.
Then, by definition, $F$ is reached within $k$ moves when starting in $v$ and playing according to $\sigma_\mathrm{opt}$.
The strategy~$\sigma_\mathrm{opt}$ for Player~$0$ in $\mathcal{G}$ can be turned into a strategy~$\tau$ for Player~$1$ in $\mathcal{A}'_\mathrm{rig}$ that simulates the moves of $\sigma_\mathrm{opt}$ and, as long as $F$ has not been visited, simulate a disturbance whenever possible (there is a unique disturbance edge at every vertex with outgoing disturbances edges).
After visiting $F$, no more disturbances are simulated.
An induction shows that $F$ is reached and at most $k$ disturbances are simulated when starting in $v$ and playing according to $\tau$.
Hence, Player~$1$ wins $(\mathcal{A}'_\mathrm{rig}, \mathrm{Safety}(F)_\mathrm{rig} \cup R_{\ge k+1})$, which implies $r_\mathcal{G}(v) \le k = \mathrm{val}_\mathcal{G}(v,\sigma_\mathrm{opt})$ by Lemma~\ref{lemma_riggedgamescharacterizeresilience}.\ref{lemma_riggedgamescharacterizeresilience_k}.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_riggedpushdowngameseffective}}
\riggedpushdowngameseffective*
\begin{proof}
Given a pushdown arena~$\mathcal{A}$ induced by a PDS~$\mathcal{P}$ with set~$Q$ of states, a partition~$\set{Q_0, Q_1}$ of $Q$, and a transition relation~$\Delta$ inducing the disturbance edges, a PDS~$\mathcal{P}'$ with set~$Q'$ of states and a partition~$\set{Q_0', Q_1'}$ of $Q'$ inducing~$\mathcal{A}_\mathrm{rig}$ can be computed in linear time.
If $\mathcal{P}$ is one-counter, then so is $\mathcal{P}'$.
Further, given a coloring~$\Omega$ of $Q$, one can determine
\begin{itemize}
\item a coloring~$\Omega'$ of $Q'$ such that $\mathrm{Parity}(\Omega') = \mathrm{Parity}(\Omega)_\mathrm{rig} $, and
\item a coloring~$\Omega''$ of $Q'$ such that $\mathrm{Parity}(\Omega'') = \mathrm{Parity}(\Omega)_\mathrm{rig}\cup \mathrm{B\ddot{u}chi}(D)$, where $D$ is the set of disturbances edges of $\mathcal{A}$.
\end{itemize}
In $\Omega'$, all vertices in $V$ inherit their colors from $\Omega$ and auxiliary vertices are colored by zero, which makes them irrelevant, while in $\Omega''$, all vertices in $V$ inherit their colors from $\Omega$, all vertices in $D$ are assigned an even color that is larger than all colors in $\Omega$'s range, and all other auxiliary vertices are colored by zero.
Hence, the games characterizing the existence of $(\omega+1)$-resilient and $\omega$-resilient strategies are pushdown (one-counter) parity games that can be efficiently constructed.
Finally, checking whether Player~$0$ wins a pushdown parity game from the initial vertex is $\textsc{ExpTime}$-complete~\cite{Walukiewicz01} while checking whether Player~$0$ wins a one-counter parity game from the initial vertex is $\textsc{PSpace}\xspace$-complete~\cite{JancarS07,Serre06}.
Furthermore, the first algorithm directly yields winning strategies for the rigged games, which can easily be turned into $(\omega+1)$-resilient or $\omega$-resilient strategies for the original game.
The lower bounds hold already
for determining the winner
of a disturbance-free pushdown (one-counter) safety game,
which is hard for $\textsc{ExpTime}$~\cite{Walukiewicz01} ($\textsc{PSpace}\xspace$~\cite{JancarS07}~\footnote{The result cited pertains to emptiness of alternating word automata over a singleton alphabet. However it is easy to see that this problem can be reduced to solving one-counter safety games.}).
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_riggedpushdowngameseffective_k}}
\riggedpushdowngameseffectivek*
\begin{proof}
Assume the input~$\mathcal{G} = (\mathcal{A}, \mathrm{Parity}(\Omega))$ is induced by a PDS~$\mathcal{P}$ with set~$Q$ of states, a partition~$\set{Q_0, Q_1}$ of $Q$, and a coloring~$\Omega$ of $Q$.
Then, we construct a PDS~$\mathcal{P}'$ with set~$Q'$ of states and a partition $\set{Q_0', Q_1'}$ of $Q'$ inducing~$\mathcal{A}_\mathrm{rig}$ as for the proof of Lemma~\ref{lemma_riggedgamescharacterizeresilience}.
Now, we turn $\mathcal{P}'$ into a PDS~$\mathcal{P}'_k$ with set~$Q' \times \set{0,\ldots, k}$ of states which uses the additional component to keep track of the number of simulated disturbances, up to $k$.
Further, we use the partition $\set{Q_0' \times \set{0,\ldots, k}, Q_1' \times \set{0,\ldots, k} }$ and define the coloring~$\Omega'$ such that $\Omega'(q,k') = \Omega(q)$ for $k' < k$ and $\Omega'(q,k) = 1$.
The resulting pushdown game is equivalent to $(\mathcal{A}_\mathrm{rig}, \mathrm{Win}_\mathrm{rig} \cup R_{\ge k})$ and the winner from the initial vertex~$((q_I, 0), \bot)$ can be determined in exponential time in $k$ and the size of $\mathcal{P}$~\cite{Walukiewicz01}, i.e., in doubly-exponential time in the size of the input, as $k$ is encoded in binary.
Due to Lemma~\ref{lemma_riggedgamescharacterizeresilience}.\ref{lemma_riggedgamescharacterizeresilience_k}, Player~$0$ wins from the initial vertex if and only if she has a $k$-resilient strategy from $v_I$ in $\mathcal{G}$, i.e., if and only if $r_\mathcal{G}(v_I) \ge k$.
Furthermore, the algorithm computes winning strategies for Player~$0$ in doubly-exponential time, if they exist at all.
These can easily be turned into $k$-resilient strategies for the original game.
If the input is one-counter, then the resulting pushdown game is one-counter as well and the winner from the initial vertex can be determined in polynomial space in $k$ and the size of $\mathcal{P}$~\cite{Serre06}, i.e., in exponential space in the input.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_ubresilienceinitialvertexsafety}}
\ubresilienceinitialvertexsafety*
To prove this result, we apply a result about winning strategies for Player~$1$ in pushdown safety games (Player~$1$ has a reachability condition in a safety game: he wins if $F$ is visited at least once).
Fix a disturbance-free pushdown safety game~$\mathcal{G} = (\mathcal{A}, \mathrm{Safety}(F))$ with initial vertex~$v_I$.
We say that a winning strategy~$\tau$ for Player~$1$ from $v_I$ \emph{bounds the stack height to} $n \in \omega$ if every play~$v_0 v_1 v_2 \cdots$ that starts in $v_I$ and is consistent with $\tau$ satisfies the following condition for all $j \in \omega$: either there is some $j' \le j$ with $v_{j'} \in F$ or $\mathrm{sh}(v_j) \le n$.
Thus, such a strategy ensures a visit to $F$ when starting in the initial vertex, and ensures that the stack height~$n$ is never exceeded before $F$ is visited for the first time.
The next proposition shows that such a strategy always exists for $n = h(\mathcal{P})$, if Player~$1$ wins from $v_I$ at all.
\begin{restatable}{lem}{2}
\label{lemma_ubstackheightreachability}
If $v_I \in \mathcal{W}_1(\mathcal{G})$, then Player~$1$ has a winning strategy~$\tau$ that bounds the stack height to $h(\mathcal{P})$, where $\mathcal{P}$ is the PDS underlying $\mathcal{G}$.
\end{restatable}
\begin{proof}
We transform $\mathcal{G}$ into a parity game as described at the end of Section~\ref{subsec_games} on Page~\pageref{cons_safety2parity}.
This transformation can be implemented on the PDS inducing $\mathcal{G}$ without increasing the number of states or the number of stack symbols.
Furthermore, the parity condition only uses two colors, say $0$ for states outside of $F$ and $1$ for states in $F$, which are sinks.
Now, the desired result follows from a result on the existence of strategies in pushdown games that bound the occurrence of undesirable colors (here, the color~$0$, which is undesirable for Player~$1$)~\cite{FridmanZ12}.
Slightly more formally, in the resulting parity game, the \emph{stair score} for the color~$0$ after a play prefix (see \cite{FridmanZ12} for definitions) is equal to the stack height of the prefix.
Now, the main result in the work cited above shows that Player~$1$ has a strategy that bounds the stair score for $0$ by $h(\mathcal{P})$, if he wins at all.
Thus, this strategy bounds the stack height to $h(\mathcal{P})$.
\end{proof}
Now, we are able to prove the upper bound~$b(\mathcal{P})$ on the resilience of the initial vertex of a pushdown safety game induced by $\mathcal{P}$ in case this value is finite.
\begin{proof}[Proof of Lemma~\ref{lemma_ubresilienceinitialvertexsafety}]
Let $ r_\mathcal{G}(v_I) \neq \omega+1 $.
As pushdown arenas are finitely branching, Lemma~\ref{lemma_resiliencevaluessafety} yields $ r_\mathcal{G}(v_I) \in \omega$, say $r_\mathcal{G}(v_I) = k$.
By definition, Player~$0$ has a $k$-resilient strategy for $\mathcal{G}$ from $v_I$, but no $(k+1)$-resilient strategy.
Hence, due to Lemma~\ref{lemma_riggedgamescharacterizeresilience}.\ref{lemma_riggedgamescharacterizeresilience_k}, Player~$1$ wins the game~$(\mathcal{A}_\mathrm{rig}, \mathrm{Safety}(F)_\mathrm{rig} \cup R_{\ge k+1})$ from $v_I$.
Thus, he also wins the safety game~$(\mathcal{A}_\mathrm{rig}, \mathrm{Safety}(F)_\mathrm{rig} )$ from $v_I$, as every winning strategy for Player~$1$ for the former game is also one for the latter.
Hence, applying Lemma~\ref{lemma_ubstackheightreachability} yields the existence of a winning strategy~$\tau$ for the latter game from $v_I$ that bounds the stack height by $h(\mathcal{P})$.
Note that we can assume $\tau$ to be positional (see Lemma~\ref{lemma_positionalstategies} on Page~\pageref{lemma_positionalstategies} for a stronger statement and note that the construction presented in its proof preserves bounds on the stack height).
Now, every play that starts in $v_I$ and is consistent with $\tau$ visits each vertex with stack height at most $h(\mathcal{P})$ at most once before reaching $F$.
There are at most $b(\mathcal{P})$ such vertices, i.e., after at most $b(\mathcal{P})-1$ moves, $F$ is reached.
Now, we show that Player~$0$ has no $b(\mathcal{P})$-resilient strategy from $v_I$ in $\mathcal{G}$.
To this end, we show for that every strategy~$\sigma$ for her, there is a play~$\rho$ that starts in $v_I$, is consistent with $\sigma$, has at most $b(\mathcal{P})-1$ many disturbances, and visits a vertex in $F$, i.e., it is losing for Player~$0$.
Let $\sigma'$ be the strategy for Player~$0$ in $\mathcal{A}_\mathrm{rig}$ obtained by transforming $\sigma$ as described in the proof of Lemma~\ref{lemma_riggedgamescharacterizeresilience}.
Now, let $\rho'$ be the unique play of $\mathcal{A}_\mathrm{rig}$ starting in $v_I$ that is consistent with $\sigma'$ and $\tau$, which visits $F$ after at most $b(\mathcal{P})-1$ many moves.
Hence, there are at most $b(\mathcal{P})-1$ many simulated disturbances in $\rho'$ before the first visit to $F$.
Now, $t(\rho')$ starts in $v$, is consistent with $\sigma$, and there are at most $b(\mathcal{P})-1$ many disturbances in $t(\rho')$ before the first visit to $F$ (which occurs).
Now, we just replace the suffix of $t(\rho')$ after the first visit to $F$ by some disturbance-free suffix so that the resulting play~$\rho$ is still consistent with $\sigma$.
We obtain a play~$\rho$ starting in $v_I$, consistent with $\sigma$, with at most $b(\mathcal{P})-1$ many disturbances that is losing for Player~$0$.
Hence, $\sigma$ is indeed not $b(\mathcal{P})$-resilient.
As we have picked $\sigma$ arbitrarily, there is no $b(\mathcal{P})$-resilient strategy from $v_I$ and therefore $r_\mathcal{G}(v_I) < b(\mathcal{P})$.
\end{proof}
\subsubsection{Proof of Theorem~\ref{theorem_determiningresiliencesafetypds}}
\determiningresiliencesafetypds*
\begin{proof}
Algorithm~\ref{algorithm_ocssafety} is correct due to
Lemma~\ref{lemma_ubresilienceinitialvertexsafety}.
The triply-exponential running time stems from the doubly-exponential bound $b(\mathcal{P})$ presented in Lemma~\ref{lemma_ubresilienceinitialvertexsafety}, which has to be plugged into Lemma~\ref{lemma_riggedpushdowngameseffective_k} to implement the check in Line~$4$.
The check in Line~$1$ runs in exponential time (Lemma~\ref{lemma_riggedpushdowngameseffective}) and the for-loop terminates after at most doubly-exponentially many iterations.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_lbresilienceinitialvertexsafety}}
\lbresilienceinitialvertexsafety*
\begin{proof}
\ref{lemma_lbresilienceinitialvertexsafety_ocs}.)
We show the game~$\mathcal{G}_2$ in Figure~\ref{fig_safetyresillowerbounds_ocs} and later explain the general case.
\begin{figure}
\centering
\def\times{1.4}
\def-.9{-.9}
\begin{tikzpicture}[]
\draw[black!30, fill = black!10, rounded corners] (-2,-2.4) rectangle (11.7,-3.9);
\draw[black!30, fill = black!10, rounded corners] (-2,-4.2) rectangle (11.7,-6.6);
\foreach \i in {1,2,...,7}{
\node[p0s] (c\i) at (\times*\i,1*-.9) {};
}
\foreach \i in {0,1,...,7}{
\node[p1s] (i\i) at (\times*\i,0*-.9) {};
\node[p0s] (d\i) at (\times*\i,2*-.9) {};
\node[p1s] (20\i) at (\times*\i,3*-.9) {};
\node[p1s] (21\i) at (\times*\i,4*-.9) {};
\node[p1s] (30\i) at (\times*\i,5*-.9) {};
\node[p1s] (31\i) at (\times*\i,6*-.9) {};
\node[p1s] (32\i) at (\times*\i,7*-.9) {};
}
\node (idots) at (11.2,0*-.9) {$\cdots$};
\node (cdots) at (11.2,1*-.9) {$\cdots$};
\node (ddots) at (11.2,2*-.9) {$\cdots$};
\node (20dots) at (11.2,3*-.9) {$\cdots$};
\node (21dots) at (11.2,4*-.9) {$\cdots$};
\node (30dots) at (11.2,5*-.9) {$\cdots$};
\node (31dots) at (11.2,6*-.9) {$\cdots$};
\node (32dots) at (11.2,7*-.9) {$\cdots$};
\node[p1s, double] (s) at (0,8*-.9) {};
\node (il) at (-1.5,0*-.9) {$i$};
\node (cl) at (-1.5,1*-.9) {$c$};
\node (dl) at (-1.5,2*-.9) {$d$};
\node (20l) at (-1.5,3*-.9) {$(2,0)$};
\node (21l) at (-1.5,4*-.9) {$(2,1)$};
\node (30l) at (-1.5,5*-.9) {$(3,0)$};
\node (31l) at (-1.5,6*-.9) {$(3,1)$};
\node (32l) at (-1.5,7*-.9) {$(3,2)$};
\node (sl) at (-1.5,8*-.9) {$s$};
\foreach \i in {1,...,7}{
\path[-stealth]
(i\i) edge (c\i)
(d\i) edge[loop below] ()
(c\i) edge (d\i)
(c\i) edge[bend left=50] (20\i)
(c\i) edge[bend right] (30\i)
;
}
\foreach \i [remember=\i as \lastx (initially 0)]in {1,...,7}{
\path[-stealth]
(i\lastx) edge (i\i)
;
}
\foreach \i [remember=\i as \lastx (initially 7)]in {6,...,0}{
\path[-stealth]
(d\lastx) edge[fault] (d\i)
(20\lastx) edge (21\i)
(21\lastx) edge (20\i)
(30\lastx) edge (31\i)
(31\lastx) edge (32\i)
(32\lastx) edge (30\i)
;
}
\path[-stealth]
(-.7,0) edge (i0)
(s) edge[loop right] ()
(210) edge[loop left] ()
(310) edge[loop left] ()
(320) edge[loop left] ()
(idots) edge (i7)
(ddots) edge[fault] (d7)
(20dots) edge (217)
(21dots) edge (207)
(30dots) edge (317)
(31dots) edge (327)
(32dots) edge (307)
;
\path[rounded corners,-stealth,draw, thick]
(d0) -- (-.8,2*-.9) -- (-.8,8*-.9) -- (s)
;
\path[rounded corners,draw, thick]
(200) -- (-.8,3*-.9) -- (-.8,3*-.9-1)
(300) -- (-.8,5*-.9) -- (-.8,5*-.9-1)
;
\end{tikzpicture}
\caption{The one-couter safety game~$\mathcal{G}_2$ for the proof of Lemma~\ref{lemma_lbresilienceinitialvertexsafety}.\ref{lemma_lbresilienceinitialvertexsafety_ocs}. Player~$0$ wins if and only if $(s,\bot)$ is never visited. Vertices in the upper gray rectangle implement a modulo-$2$ counter while vertices in the lower rectangle implement a modulo-3 counter.}
\label{fig_safetyresillowerbounds_ocs}
\end{figure}
The winning condition is defined such that Player~$0$ wins a play if and only if the state~$s$ is never reached.
Now, a play starting in the initial vertex of $\mathcal{G}_2$ proceeds as follows:
Player~$1$ either stays in the state~$i$ ad infinitum, and thereby allows Player~$0$ to win, or he eventually moves to some vertex of the form~$(c,A^n\bot)$.
Now, Player~$0$ has three choices, moving to $((2,0),A^n\bot)$, $((3,0),A^n\bot)$, or $(d,A^n\bot)$.
In the first case, there is only one continuation of the play prefix, which results in a disturbance-free play that is winning for Player~$0$ if and only if $n \bmod 2 \neq 0$.
Similarly, in the second case, there is only one continuation of the play prefix, which results in a disturbance-free play that is winning for Player~$0$ if and only if $n \bmod 3 \neq 0$.
Finally, moving to $(d,A^n\bot)$ means that Player~$0$ wins if strictly less than $n$ disturbances occur in the continuation of the play prefix, but loses if $n$ disturbances occur.
We claim that the initial vertex has resilience~$6 = \prim{2}$. A $6$-resilient strategy for Player~$0$ moves from $(c,A^n\bot)$ to $(d,A^n\bot)$ if $n$ is a multiple of $6$.
Otherwise, it moves to $((p_j,0), A^n\bot)$ for some $p_j \in \set{2,3}$ such that $n \bmod p_j \neq 6$, which always exists.
Applying the reasoning above implies that every play starting in the initial vertex, consistent with the strategy, and with at most $5$ disturbances is winning for Player~$0$.
Thus, the strategy is indeed $6$-resilient.
Now, consider an arbitrary strategy~$\sigma$ for Player~$0$.
We show that it is not $7$-resilient, which is yields the desired result.
To this end, consider the unique play prefix leading to $(c,A^6\bot)$, which is consistent with $\sigma$.
If $\sigma$ prescribes a move to some $((p_j,0),A^6\bot)$, then, as argued above, there is disturbance-free play that is consistent with $\sigma$, but losing for her.
The only other choice for $\sigma$ is to move to $(d,A^6\bot)$.
Then, as argued above as well, there is a play that is consistent with $\sigma$ with $6$ disturbances that is losing for her.
In both cases, we have shown that the strategy is indeed not $7$-resilient.
The general case is obtained by having modulo counters in $\mathcal{G}_k$ for the first $k$ prime numbers instead of only the first two as in $\mathcal{G}_2$.
Using the same reasoning as above for arbitrary $k$ instead of $k=2$ shows that the initial vertex of $\mathcal{G}_k$ has resilience~$\prim{k}$.
Finally, the number of states of the one-counter system inducing $\mathcal{G}_k$ is bounded by $\bigo(k^3)$.
\ref{lemma_lbresilienceinitialvertexsafety_pds}.)
We modify the one-counter safety game~$\mathcal{G}_k$ to obtain a pushdown safety game~$\mathcal{G}_k'$.
We use the stack alphabet~$\set{0,1}$, which allows us to interpret stack contents as binary encodings of natural numbers, with the least significant bit at the top of the stack.
In the following, we give an informal account of the structure of $\mathcal{G}_k'$ and leave the implementation by a pushdown system to the reader.
Here, we reuse the modulo-counters of $\mathcal{G}_k$ which forces that Player~$1$ to reach a stack height that is a multiple of $\prim{k}$, as he would lose otherwise.
Thus, Player~$1$ is initially forced to push a multiple of $\prim{k}$ $1$'s on the stack and then gives control to Player~$0$.
As the stack height is a multiple of $\prim{k}$, she can only go to a state~$d$ where all $0$'s are popped from the stack until the first $1$ is uncovered (note that initially there is no $0$ to pop).
If there is no such $1$, i.e., if the bottom of the stack is reached by removing $0$'s, then the play reaches a losing sink for Player~$0$.
Otherwise, if a $1$ is uncovered, then Player~$0$ only has a self-loop that leaves the stack unchanged, but there is also a disturbance edge that removes the topmost $1$ by a $0$ and hands back control to Player~$1$.
He can now push as many $1$'s as necessary to again reach a stack height that is a multiple of $\prim{k}$.
Now, if Player~$1$ never exceeds the stack height~$\prim{k}$, the stack always contains $\prim{k}$ bits when Player~$0$ gains control.
Assume now that Player~$0$ uses a strategy which moves to $d$ in that situation and uses the correct modulo counter to win in all other situations (as described in more detail above for $\mathcal{G}_k$).
Then, the stack contents reached at the positions where Player~$0$ gains control implement a binary counter with $\prim{k}$ bits that is decremented each time Player~$0$ gains control, starting with the value~$1^{\prim{k}}$.
Hence, as each decrement requires exactly one disturbance (and there are no others), the strategy described above is $(2^{\prim{k}}-1)$-resilient from the initial vertex.
On the other hand, $2^{\prim{k}}-1$ disturbances suffice to reach a stack containing only $0$'s at some configuration where Player~$0$ gains control.
Then, the unique continuation of that play is losing for her.
The only other choice for Player~$0$ is to enter a modulo counter at an \myquot{unsuitable} configuration, which also leads to a losing play with less than $2^{\prim{k}}$ disturbances.
Hence, Player~$0$ has no $2^{\prim{k}}$-resilient strategy from the initial vertex, i.e., it has indeed resilience~$2^{\prim{k}}-1$.
Finally, the number of states of the one-counter system inducing $\mathcal{G}_k$ is bounded by $\bigo(k^3)$.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_stratgraphcharacterizeswinninggamek}}
\stratgraphcharacterizeswinninggamek*
To simplify the proof, we transform $\mathcal{G}_k$ into a game~$\mathcal{G}_k'$ where all reachable vertices in $F$ are sinks of stack height zero.
To do this, we replace all outgoing (standard and disturbance) edges of vertices~$(q,A^n\bot) \in F$ with $n>0$ by an edge to $(q,A^{n-1}\bot)$ and the all outgoing (standard and disturbance) edges of vertices~$(q,\bot) \in F$ by an edge to a sink vertex~$(q_f,\bot)$, where $q_f$ is a fresh state.
Then, $\mathcal{G}_k'$ is the game played in the modified arena with winning condition~$\mathrm{Safety}(\set{q_f})_\mathrm{rig} \cup R_{\ge k}$.
Intuitively, once a vertex in $F$ is reached in the modified arena, the players no longer have strategic choices;
instead, the stack is emptied (without simulating any disturbances) and the unsafe sink vertex~$(q_f,\bot)$ is reached.
It is straightforward to verify that we have $v \in \mathcal{W}_i(\mathcal{G}_k)$ if and only $v \in \mathcal{W}_i(\mathcal{G}_k')$ for every vertex of $\mathcal{A}_\mathrm{rig}$ and $i \in \set{0,1}$ by transferring winning strategies between the games.
Hence, in the following, we assume without loss of generality, that the only vertices of $\mathcal{G}_k$ in $F$ that are reachable from the initial vertex are sinks of stack height zero.
In this situation, a play can no longer simulate a disturbance edge once a vertex in $F$ has been reached.
To prove Lemma~\ref{lemma_stratgraphcharacterizeswinninggamek}, we show that if Player~$1$ wins $\mathcal{G}_k$ with some arbitrary winning strategy, then also with a winning strategy that can be turned into a strategy graph.
To simplify our notation, given a strategy~$\tau$, let $\mathrm{maxSh}(\tau) = \sup_v \mathrm{sh}(v)$, where $v$ ranges over all vertices reachable by a play prefix starting in $v_I$ that is consistent with $\tau$, i.e., $\mathrm{maxSh}(\tau)$ is the maximal stack height visited by a play that is starting in the initial vertex and consistent with $\tau$.
Using this, we show that Player~$1$ wins $\mathcal{G}_k$ from $v_I$ if and only if he has a positional winning strategy from $v_I$ with $\mathrm{maxSh}(\tau) \le (2k)^{\size{Q}^2}$.
The latter can then be transformed into a strategy graph.
We only have to consider the implication from left to right, as the other one is trivial.
Let Player~$1$ win $\mathcal{G}_k$ from $v_I$, i.e., he has a winning strategy~$\tau$ for $\mathcal{G}_k$ from $v_I$.
We proceed in two steps:
\begin{enumerate}
\item We turn $\tau$ in a positional winning strategy~$\tau'$ from $v_I$ (Lemma~\ref{lemma_positionalstategies}).
\item We turn $\tau'$ into a positional winning strategy~$\tau''$ with $\mathrm{maxSh}(\tau'') \le (2k)^{\size{Q}^2}$ (Lemma~\ref{lemma_positionalstategiesexponentialstackheightgamek}).
\end{enumerate}
For the first step, we generalize a standard argument for turning an arbitrary, not necessarily positional, winning strategy~$\tau$ in a reachability game into a positional one:
At a vertex~$v \notin F$, consider all play prefixes that are consistent with $\tau$ and end in $v$, and mimic the move $\tau$ prescribes for a longest one (call it $\mathrm{rep}(v)$).
The resulting strategy~$\tau'$ is obviously positional and winning as every play consistent with $\tau'$ and ending in some $v \notin F$ can be shown to be at most as long as the play~$\mathrm{rep}(v)$ whose moves are mimicked to define $\tau'(v)$.
Here, we have to refine this argument to ensure that the resulting strategy~$\tau'$ still simulates at most $k-1$ disturbances during each play.
\begin{restatable}{lem}{3}
\label{lemma_positionalstategies}
If Player~$1$ wins $\mathcal{G}_k$ from $v_I$ then he has a positional winning strategy for $\mathcal{G}_k$ from $v_I$.
\end{restatable}
\begin{proof}
Assume a winning strategy $\tau$ for Player~$1$ from $v_I$.
Let us call a play prefix~$v_0 \cdots v_j$ \emph{unsettled} if it starts in $v_I$, is consistent with $\tau$, and
no strict prefix contains a vertex in the target $F$.
Notice that there must be a uniform bound~$\ell \in \omega$ such that $\size{w} < \ell$ for every unsettled $w$.
Indeed, if there was no such bound, then it is possible to arrange an infinite set of arbitrarily long play prefixes not visiting $F$ into an infinite finitely branching tree.
By König's Lemma, this tree has an infinite path which corresponds to an infinite play starting in $v_I$, consistent with $\tau$, but not containing a vertex in $F$,
which contradicts the assumption that $\tau$ is winning.
Given an unsettled prefix~$w$, let $\mathrm{val}(w) = d \cdot \ell + \size{w}$ where $d$ is the number of simulated disturbances during $w$.
Let $U(v)$ for $v \in V'$ denote the set of unsettled play prefixes ending in $v$.
Further, for every $v \in V$ with non-empty $U(v)$ let $\mathrm{rep}(v)$ be an element from $U(v)$ such that $\mathrm{val}(\mathrm{rep}(v)) \ge \mathrm{val}(w)$ for all $w \in U(v)$.
Such an element exists, as the $\mathrm{val}(w)$ for $w \in U(v)$ are bounded by $ k \cdot\ell -1$:
Each unsettled prefix is consistent with the winning strategy~$\tau$, which implies that it simulates at most $k-1$ disturbance edges, and its length is bounded by $\ell-1$, as argued above.
Based on this we define the positional strategy~$\tau'$ via $\tau'(v) = \tau(\mathrm{rep}(v))$ if $\mathrm{rep}(v)$ is defined and $\tau'(v) = v'$ for some arbitrary successor~$v'$ of $v$ if $\mathrm{rep}(v)$ is undefined (note that it suffices to define $\tau'(v)$ for $v \in V_1$ to define a positional strategy for Player~$1$).
We claim that $\tau'$ is winning from $v_I$.
To this end, let $\rho = v_0v_1v_2 \cdots$ start in $v_I$ and be consistent with $\tau'$.
We need to show that it visits a vertex in $F$ and that it simulates at most $k-1$ disturbance edges.
A simple induction shows that every length-$j$ prefix $v_0 \cdots v_{j-1}$ that does not visit $F$ must satisfy that
\begin{equation}
\mathrm{val}(v_0 \cdots v_j) \le \mathrm{val}(\mathrm{rep}(v_j))
\hfill \label{eq}
\end{equation}
The induction start $j = 0$ is trivial, as we have $v_0 = v_I$ and $v_I \in U(v_I)$, which implies $\mathrm{val}(v_I ) \le \mathrm{val}(\mathrm{rep}(v_I))$ as required.
For the induction step, consider some $j > 0$ such that $v_0 \cdots v_{j-1}$ does not visit $F$.
The induction hypothesis yields $\mathrm{val}(v_0 \cdots v_{j-1}) \le \mathrm{val}(\mathrm{rep}(v_{j-1}))$.
Let $ \mathrm{rep}(v_{j-1}) = wv_{j-1}$, which is consistent with $\tau$.
If $v_{j-1} \in V_0'$, then $wv_{j-1}v_j$ is consistent with $\tau$ as well, as it is Player~$0$'s turn at $v_{j-1}$.
Similarly, if $v_{j-1} \in V_1'$, then we have
\[v_j = \tau'(v_0 \cdots v_{j-1}) = \tau(\mathrm{rep}(v_{j-1})) = \tau(wv_{j-1}). \]
Hence, $wv_{j-1}v_j$ is again consistent with $\tau$.
Furthermore, $wv_{j-1}$ does not contain a vertex in $F$, as $v_{j-1}$ is not in $F$ (recall that vertices in $F$ are sinks).
Thus, we conclude that $wv_{j-1}v_j$ is unsettled, which implies $\mathrm{val}(wv_{j-1}v_j) \le \mathrm{val}(\mathrm{rep}(v_j))$, by our definition of $\mathrm{rep}(v_j)$.
To finish the induction step,
let $x = 1$ if $ v_j \in D$, i.e. a disturbance edge is simulated, and $x = 0$ otherwise.
Then, we have \begin{align*}
\mathrm{val}(v_0 \cdots v_j) ={}& \mathrm{val}(v_0 \cdots v_{j-1}) + x\cdot \ell + 1 \\
\le{}& \mathrm{val}(\mathrm{rep}(v_{j-1})) + x\cdot \ell + 1 \\
={}& \mathrm{val}(wv_{j-1}) + x\cdot \ell + 1 \\
={}& \mathrm{val}(wv_{j-1}v_j) \le \mathrm{val}(\mathrm{rep}(v_j)).
\end{align*}
Applying Equation~\ref{eq}, we can show that $\rho$ is indeed winning.
First, towards a contradiction, assume $\rho$ does not visit a vertex in $F$.
Then, Equation~\ref{eq} is applicable to every prefix $v_0 \cdots v_j$ and we thus obtain for every $j>0$, that
\[
j+1 = \size{v_0 \cdots v_j} \le \mathrm{val}(v_0 \cdots v_j) \le \mathrm{val}(\mathrm{rep}(v_j))
\le k \cdot\ell -1
\]
which is a contradiction as the term on the right is constant.
Second, again towards a contradiction, assume that $\rho$ simulates at least $k$ disturbance edges.
Then, let $j$ be minimal such that the prefix~$v_0 \cdots v_j$ simulates exactly $k$ disturbance edges.
As vertices in $F$ are sinks, and therefore have no outgoing edges simulating disturbance edges, Equation~\ref{eq} is applicable to $v_0 \cdots v_j$ and we obtain that
\[
k\cdot \ell \le \mathrm{val}(v_0 \cdots v_j) \le \mathrm{val}(\mathrm{rep}(v_j))
\]
which is impossible as $\mathrm{val}(\mathrm{rep}(v_j)) \le k \cdot\ell -1$.
Consequently, $\rho$ visits $F$ and simulates at most $k-1$ disturbance edges. As $\rho$ was an arbitrary play consistent with $\tau'$, this strategy is indeed winning.
\end{proof}
The second step of our construction is to bound the stack height reached by plays consistent with the winning strategy (while preserving positionality).
To this end, we generalize a classical argument for pushdown safety games:
In such games, Player~$1$, who has a reachability objective, has a positional winning strategy~$\tau$ from $v_I$ with exponentially bounded~$\mathrm{maxSh}(\tau)$, if he wins at all from $v_I$.
This is typically proven by a \myquot{hill-cutting} argument
\cite{Val1973}
showing that a winning strategy exceeding this bound can be turned into one of smaller maximal stack height by removing infixes of plays that increase the stack without reaching states that have not been reached at smaller stack height already.
Here, we again have to generalize this argument to additionally ensure that the number of simulated disturbances remains bounded by $k-1$. This is done using ``summarizations'' of paths in pushdown systems (see e.g.~\cite{RepsHS95,HMM2016}) that take the number of disturbances into account.
\begin{restatable}{lem}{4}
\label{lemma_positionalstategiesexponentialstackheightgamek}
If Player~$1$ wins $\mathcal{G}_k$ from $v_I$ then he has a positional winning strategy from $v_I$ with $\mathrm{maxSh}(\tau) \le (2k)^{\size{Q}^2}$.
\end{restatable}
\begin{proof}
By Lemma~\ref{lemma_positionalstategies}
we can pick a positional strategy~$\tau$ for Player~$1$ that is winning $\mathcal{G}_k$ from $v_I$.
We show how to turn this into a winning strategy that satisfies the claim.
Notice first that $\mathrm{maxSh}(\tau)$ must be finite.
Indeed, if it is unbounded, then for every $n \in \omega$ there is a play prefix~$w_n$ starting in $v_I$, consistent with $\tau$, and ending in a vertex of stack height~$n$.
As the stack height is increased by at most one during each move, we have $\size{w_n} \ge n$.
Furthermore, as vertices in $F$ are sinks, these play prefixes can be assumed to not contain a vertex in $F$.
The prefixes~$w_n$ can be arranged in an infinite finitely branching tree.
By König's Lemma, this tree has an infinite path, which corresponds to an infinite play starting in $v_I$, consistent with $\tau$, but not visiting a vertex in $F$.
This contradicts $\tau$ being a winning strategy.
It suffices to show that
if $\mathrm{maxSh}(\tau) > (2k)^{\size{Q}^2}$, then $\tau$ can be turned into a positional winning strategy~$\tau'$ from $v_I$ with strictly smaller maximal stack height.
For the sake of readability, we will identify a stack content~$A^n\bot$ of the one-counter system underlying $\mathcal{G}_k$ by the number~$n \in \omega$.
Hence, vertices of $\mathcal{G}_k$ are from now on denoted by $(q,n)$ with $n \in \omega$.
Let $R$ denote the set of vertices reachable from $v_I$ via play prefixes that are consistent with $\tau$.
For $(q,n) \in R$ with $n > 0$ let $H(q,n)$ be the set of vertices of the form~$(q',n-1)$ reachable from $(q,n)$ via a play prefix~$(q,n)(q_1,n_1) \cdots (q_j,n_j)(q',n-1)$ that is consistent with $\tau$ and such that $n_{j'} \ge n$ for every $j' \in \set{1, \ldots, j}$, i.e., the last vertex of the play prefix is the first time the stack height along the play prefix is strictly smaller than $n$.
We call such a play prefix a \emph{hill} from $(q,n)$ to $(q',n-1)$.
For all $n > 0$ define the partial function~$h_n \colon Q \rightarrow 2^Q$ that maps $q$ to $H(q,n)$ whenever $(q,n) \in R$, and leaves $h_n(q)$ undefined otherwise.
Similarly, define the partial function~$d_n \colon Q \times Q \rightarrow \set{0, \ldots, k-1}$ by mapping each pair~$(q,q')$ with $q' \in H(q,n)$ to the maximal number of disturbances simulated during any hill from $(q,n)$ to $(q',n-1)$.
This value is bounded by $k-1$, as each hill is part of a play that is consistent with $\tau$.
For $(q,q')$ with $q' \notin H(q,n)$, we leave $d_n(q,q')$ undefined.
There are at most $(2k)^{\size{Q}^2}$ many different pairs of such functions $h_n$ and $d_n$.
Hence, if $R$ contains a vertex~$(q,n)$ with $n > (2k)^{\size{Q}^2}$, then there are $0 < n_\ell < n_u$ such that $h_{n_\ell} = h_{n_u}$ and $d_{n_\ell} = d_{n_u}$.
Let $s = n_u -n_\ell$.
We define the positional strategy~$\tau'$ via $\tau'(q,n) = \tau(q,n)$, if $n < n_\ell$ and $\tau'(q,n) = \tau(q, n + s)$ if $n \ge n_\ell$ (recall that it suffices to define $\tau'(v)$ for every $v \in V_1'$ to define $\tau'$).
We claim that $\tau'$ is still winning for Player~$1$ from $v_I$ in $\mathcal{G}_k$.
To this end, consider an arbitrary play $\rho' = (q_0, n_0)(q_1, n_1)(q_2, n_u) \cdots$ that starts in $v_I$ and is consistent with $\tau'$.
We need to show that it visits $F$ and simulates at most $k-1$ disturbance edges.
If every $n_j$ is strictly smaller than $n_\ell$, then $\rho'$ is also consistent with $\tau$, as only the first case of the definition of $\tau$ is applied.
Hence, it is winning for Player~$1$, as $\tau$ is a winning strategy from $v_I$.
It remains to consider the case where $\rho'$ reaches stack height~$n_\ell$.
Here, we turn $\rho'$ into a play~$\rho$ starting in $v_I$ and consistent with $\tau$, which implies that $\rho$ visits $F$ and simulates at most $k-1$ disturbance edges.
Using the relation between $\rho$ and $\rho'$, we argue that the latter play is also winning.
The following remark is useful throughout our argument and follows immediately from the fact that at stack heights~$n$ greater or equal than $n_\ell$, $\tau'$ mimics the behavior of $\tau$ at stack height~$n+s$.
\begin{remark}
\label{remark_copycatstrategy}
Let $j$ and $j'$ be positions of $\rho'$ such that $n_j = n_\ell$ and $n_{j''} \ge n_\ell$ for every $j'' \in \set{j+1, \ldots, j'}$, i.e., the infix between positions~$j$ and $j'$ starts at stack height~$n_\ell$ and never reaches a smaller stack height. Then, $(q_j, n_j+s) \cdots (q_{j'+1}, n_{j'+1}+s)$ is consistent with $\tau$ (note the $+1$!).
\end{remark}
We inductively construct $\rho$ by defining a sequence~$(w_m)_{m \in \omega}$ of strictly increasing prefixes whose limit is $\rho$.
To define this sequence, we simultaneously construct a sequence~$(j_m)_{m \in \omega}$ of strictly increasing positions of $\rho'$.
During the construction, we satisfy the following invariant:
Each $w_m$ is consistent with $\tau$, ends in $(q_{j_m}, n_{j_m})$ where $n_{j_m}$ is strictly smaller than $n_\ell$, and $w_m$ simulates at least as many disturbances as $(q_0, n_0) \cdots (q_{j_m}, n_{j_m})$.
We start with $j_0 = 0$ and $w_0 = (q_0, n_0) = v_I$, which satisfies the invariant due to our choice of $n_\ell$ being greater than zero.
We define $w_m$ and $j_m$ for $m >0$, based on $w_{m-1}$ and $j_{m-1}$, as follows.
Due to the invariant, $w_{m-1}$ ends in $(q_{j_{m-1}}, n_{j_{m-1}})$ with $n_{j_{m-1}} < n_\ell$ and is consistent with $\tau$.
We consider two cases.
In the first case, if the vertex~$(q_{j_{m-1}+1}, n_{j_{m-1}+1})$, i.e., the next one after $(q_{j_{m-1}}, n_{j_{m-1}})$ in $\rho'$, satisfies $n_{j_{m-1}+1} < n_\ell$, then we define $w_{m} = w_{m-1}(q_{j_{m-1}+1}, n_{j_{m-1}+1})$ and $j_m = j_{m-1}+1$.
Note that the move from $(q_{j_{m-1}}, n_{j_{m-1}})$ to $(q_{j_{m-1}+1}, n_{j_{m-1}+1})$ is consistent with $\tau$, as it is either Player~$0$'s turn or the first case of the definition of $\tau'$ is applied (which mimics $\tau$) due to our invariant.
Hence, $w_m$ is again consistent with $\tau$.
Similarly, the requirement on the number of simulated disturbances is satisfied as the same edge is used to extend both play prefixes.
In the second case, we have $n_{j_{m-1}+1} \ge n_\ell$, which implies $n_{j_{m-1}+1} = n_\ell$, as the stack height can increase by at most one during every transition.
We claim there is some $j > j_{m-1}+1$ such that $n_j = n-1$.
Towards a contradiction, assume there is no such $j$.
Then, Remark~\ref{remark_copycatstrategy} is applicable to every pair $(j_{m-1}+1,j)$ with $j > j_{m-1}+1$.
This yields an infinite play
\[
\rho_c =
(q_{j_{m-1}+1}, n_{j_{m-1}+1}+s)
(q_{j_{m-1}+2}, n_{j_{m-1}+2}+s)
(q_{j_{m-1}+3}, n_{j_{m-1}+3}+s)\cdots
\]
that is consistent with $\tau$.
The play prefix~$w_{m-1}$ starts in $v_I$, is consistent with $\tau$, and ends in $(q_{j_{m-1}}, n_{j_{m-1}})$.
Further, the move from $(q_{j_{m-1}}, n_{j_{m-1}})$ to $(q_{j_{m-1}+1}, n_{j_{m-1}+1})$ in $\rho'$ is consistent with $\tau'$ and therefore also with $\tau$, as $n_{j_{m-1}} < n_\ell$ by out invariant.
Thus, we have shown $(q_{j_{m-1}+1}, n_{j_{m-1}+1}) \in R$, i.e., there is a play prefix~$w_c$ starting in $v_I$, consistent with $\tau$, and ending in $(q_{j_{m-1}+1}, n_{j_{m-1}+1})$.
Altogether, we can combine $w_c$ and $\rho_c$ into an infinite play starting in $v_I$ and consistent with $\tau$ that has $\rho_c$ as suffix.
Now, $\rho_c$ contains by construction no vertex of stack height zero.
As vertices in $F$ are sinks of stack height zero, the combined play can therefore not visit $F$.
This contradicts the assumption that $\tau$ is winning from $v_I$.
Thus, let $j > j_{m-1}+1$ be minimal such that $n_j = n-1$.
Applying Remark~\ref{remark_copycatstrategy} for $j_{m-1}+1$ and $j-1$ shows that
\[
w = (q_{j_{m-1}+1}, n_{j_{m-1}+1}+s) \cdots (q_{j}, n_{j}+s)
\]
is a hill from $(q_{j_{m-1}+1}, n_{j_{m-1}+1}+s) = (q_{j_{m-1}+1}, n_u)$ to $(q_{j}, n_{j}+s) = (q_{j}, n_u-1)$.
Hence, by the choice of $n_\ell$ and $n_u$ there is also a hill~$w'$ from
$(q_{j_{m-1}+1}, n_\ell)$ to $ (q_{j}, n_\ell-1)$ that has at least as many simulated disturbances as $w$.
We obtain $w_m$ from $w_{m-1}$ by appending $w'$ and define $j_m = j$.
The requirement on the stack height~$n_{j_m}$ is satisfied by our choice of $j_m = j$ while $w_m$ is consistent with $\tau$, as $w_{m-1}$, the move from $(q_{j_{m-1}}, n_{j_{m-1}})$ (the last vertex of $w_{m-1}$) to $(q_{j_{m-1}+1}, n_{j_{m-1}+1})$ (the first vertex of $w'$), and $w'$ are all consistent with $\tau$.
Finally, the requirement on the number of simulated disturbances is satisfied, as $(q_{j_{m-1}+1}, n_{j_{m-1}+1}) \cdots (q_{j}, n_{j})$ simulates the same number of disturbances as $w$, which is at most the number of disturbances simulated by $w'$.
Consider the resulting play~$\rho$, which is by construction winning for Player~$1$ and consequently simulates at most $k-1$ disturbances.
An inductive application of the invariant above shows that $\rho'$ therefore also simulates at most $k-1$ disturbances.
Furthermore, $\rho$ visits a vertex in $F$, which has stack height zero.
When such a vertex is added during the inductive construction described above, then only in the first case (when $n_{j_{m-1}+1} < n_\ell$) and only because the same vertex appears in $\rho'$, i.e., $\rho'$ visits $F$ as well.
Hence, $\rho'$ is indeed winning for Player~$1$.
To conclude, we have to show $\mathrm{maxSh}(\tau') < \mathrm{maxSh}(\tau)$.
An induction on $n$ shows that if $(q,n)$ is reachable from $v_I$ by a play prefix that is consistent with $\tau'$, then:
\begin{itemize}
\item If $n \le n_\ell$, then $(q,n)$ is reachable from $v_I$ by a play prefix that is consistent with $\tau$.
\item If $n > n_\ell$, then $(q,n+s)$ is reachable from $v_I$ by a play prefix that is consistent with $\tau$.
\end{itemize}
This implies $\mathrm{maxSh}(\tau') + s \le \mathrm{maxSh}(\tau)$, which yields the desired bound due to $s >0$.
\end{proof}
Having proved the existence of positional winning strategies with exponential maximal stack height, it is straightforward to show that these are essentially strategy graphs, which proves Lemma~\ref{lemma_stratgraphcharacterizeswinninggamek}.
\begin{proof}[Proof of Lemma~\ref{lemma_stratgraphcharacterizeswinninggamek}]
Let Player~$1$ win $\mathcal{G}_k$ from $v_I$.
Then, Lemma~\ref{lemma_positionalstategiesexponentialstackheightgamek} yields a positional winning strategy~$\tau$ for him from $v_I$ with $\mathrm{maxSh}(\tau) \le (2k)^{\size{Q}^2}$.
We turn $\tau$ into a strategy graph for $\mathcal{G}_k$ by defining
\begin{itemize}
\item $V^\circ$ to be the set of vertices visited by plays starting in $v_I$ that are consistent with $\tau$,
\item $E^\circ$ to be the set of edges traversed by these plays (ignoring the self-loops at vertices in $F$),
\item $\mu_r^\circ(v)$ to be the maximal number of disturbance edges simulated on plays starting in $v$ that are consistent with $\tau$, and
\item $\mu_d^\circ(v)$ to be the maximal length of a play prefix starting in $v$, being consistent with $\tau$, and the last vertex (but no other) being in $F$.
\end{itemize}
It is straightforward to prove that $(V^\circ, E^\circ, \mu_r^\circ, \mu_d^\circ)$ satisfies all properties required of a strategy graph for $\mathcal{G}_k$.
Conversely, assume there is a strategy graph~$(V^\circ,E^\circ,\mu_r^\circ,\mu_d^\circ)$ for $\mathcal{G}_k$.
We turn it into a positional winning strategy~$\tau$ for Player~$1$ from $v_I$.
Let $v \in V_1'$.
If $v \in V^\circ \setminus F$, then there is a unique outgoing edge~$(v,v') \in E^\circ \cap E'$ due to Property~\ref{graphproperty:strategy1} of the strategy graph definition.
Then, we define $\tau(v) = v'$.
Otherwise, i.e., if $v \notin V^\circ \setminus F$, then define $\tau(v)$ to be an arbitrary successor of $v$ in $\mathcal{A}_\mathrm{rig}$.
We claim that $\tau$ is indeed a winning strategy for Player~$1$ for $\mathcal{G}_k$ from $v_I$.
To this end, let $\rho = (v_0, 0) (v_1, 0) (v_2, 0) \cdots$ be a play starting in $v_I$ that is consistent with $\tau$.
We need to show that $\rho$ is winning for Player~$1$, i.e., that it visits $F$ and contains at most $k-1$ simulated disturbance edges.
An induction applying the definition of $\tau$ and Property~\ref{graphproperty:strategy0} of the strategy graph definition shows that if $v_0 \cdots v_{j-1}$ does not contain a vertex from $F$, then $v_0 \cdots v_{j}$ is a path through the graph~$(V^\circ, E^\circ)$.
Hence, we have $ \mu_d^\circ(v_0) > \mu_d^\circ(v_0) > \cdots > \mu_d^\circ(v_{j-1}) > \mu_d^\circ(v_j)$ by Property~\ref{graphproperty:valuesd}.
As the range of $\mu_d^\circ$ is finite, this yields an upper bound on the length of prefixes of $\rho$ that do not visit $F$, which implies that $\rho$ contains a vertex of $F$.
Hence, let $j$ be the minimal position of $\rho$ with $v_j \in F$.
As vertices in $F$ are sinks, no disturbance edges are simulated in $\rho$ after position~$j$.
Due to Property~\ref{graphproperty:valuesr}, we have $ \mu_r^\circ(v_0) \ge \mu_d^\circ(v_0) \ge \cdots \ge \mu_r^\circ(v_{j-1}) \ge \mu_r^\circ(v_j)$ with strict inequality whenever a disturbance edge is simulated, as $v_0 \cdots v_j$ is a path through $(V^\circ, E^\circ)$ as argued above.
Hence, as the range of $\mu_r^\circ$ has at most $k$ elements, there are at most $k-1$ simulated disturbances in $\rho$ before position~$j$ and none afterwards, as argued above.
Altogether, $\rho$ visits $F$ and contains at most $k-1$ simulated disturbance edges, i.e., it is indeed winning for Player~$1$ in $\mathcal{G}_k$.
\end{proof}
\subsubsection{Proof of Lemma~\ref{lemma_stratgraphexistenceinpspace}}
\stratgraphexistenceinpspace*
\begin{proof}
Notice that all defining conditions of strategy graphs are local, and can be verified
for a vertex $v=(q,n)$ if the values of $\mu_r^\circ(v')$ and $\mu_d^\circ(v')$ are known for all direct neighbors,
which have the form $(q',n')$ with $n'\in\set{n-1,n,n+1}$.
A strategy graph can therefore be guessed and verified on the fly,
keeping in memory these values for vertices in $Q \times \set{n, n+1, n+2}$ while incrementing $n$ from $0$ to $(2k)^{\size{Q}^2}$.
This requires polynomial space, both for the labeling of the vertices (as the numbers are at most exponential in the size of the input) and for the counter.
\end{proof}
\subsection{Proofs Omitted in Section~\ref{sec_arbitrary}}
\input{appendix/app_arbitrary}
\subsection{Proofs Omitted in Section~\ref{sec_riggedgames}}
\input{appendix/app_riggedgames}
\subsection{Proofs Omitted in Section~\ref{sec_pushdown}}
\input{appendix/app_pushdown}
\subsection{Proofs Omitted in Section~\ref{sec_ocs}}
\input{appendix/app_ocs}
\subsection{Proofs Omitted in Section~\ref{sec_outlook}}
\input{appendix/app_outlook}
\subsection*{Determining Finite Resilience Values for OC-Safety Games}
We consider strategies of Player 1, the reachability player, in the game where they control disturbances.
If such a reachability strategy ensures to reach the target with at most $k$ disturbances, it serves as a witness that the initial configuration has resilience strictly less than $k$.
W.l.o.g., assume that the set of target configurations is $T \subseteq Q\times\{0\}$, i.e., contains only configurations with counter value zero.
\newcommand{\Win}[2][]{\mathit{W}_{#2}^{#1}}
\newcommand{\Base}[1]{\mathit{B}_{#1}}
\newcommand{\Period}[1]{\mathit{P}_{#1}}
\newcommand{\Pat}[2][]{\mathit{R}_{#2}^{#1}}
\begin{definition}
Let $\Win[]{i}\subseteq Q\times\+{N}$ denote the winning region for the reachability player that may use $i\in\+{N}$ disturbances.
That is, $\Win{0}$ is simply the player-1 attractor set of $T$ and $\Win{i+1}$ is the attractor of $\PreD{\Win{i}}$,
the set of configurations from which player 1 can reach $\Win{i}$ in one step of a disturbance edge.
We write $\Win[j]{i}\eqby{def} \Win{i}\cap (Q\times\{j\})$ for the projection into configurations with counter $j\in\+{N}$
and $\Pat[j]{i}\eqby{def} \{p\mid (p,j) \in \Win{i}\}$.
\end{definition}
\begin{lemma}
\label{lem:OC-repetition}
Suppose $i,j,k\in\+{N}$ such that $\Pat[i]{l} = \Pat[j]{l}$ for all $l\le k$.
Then
$\Pat[i+1]{k} = \Pat[j+1]{k}$
for all $l\le k$.
\end{lemma}
\begin{proof}
Induction on $k$, where both basis and step relies on the fact that (prefixes of) reachability strategies from $(p,i+1)$ can be reused from $(p,j+1)$ and vice-versa.
\qed
\end{proof}
\begin{lemma}
$\Win{k} = \Base{k} \cup \Period{k}$,
where $\Base{k}$ is finite and $\Period{k}$ is periodic set with period
$m_k\le 2^{\card{Q}^2}$.
\end{lemma}
\newcommand{\PA}[1]{\mathit{P}^\infty_{#1}}
\begin{proof}
We recursively construct $m_k$ which is a period \emph{for all} sets $\Period{l}, l\le k$.
For $k=0$, we use that there are at most $2^{\card{Q}}$ many different sets $\Pat[l]{0}$.
The existence of a suitable $m_0\le 2^{\card{Q}}$ thus follows from Lemma~\ref{lem:OC-repetition}.
For $k>0$
first note that if $\Win{k}$ has period $m_k$ then $\PreD{\Win{k}}$ is also periodic with period $m_k$.
%
Let's denote by $\PA{k}$ the finite arena resulting from hard-coding counter values modulo $m_k$
and consider as reachability target the projection of $T_k \eqby{def} \bigcup_{l\le k}\PreD{\Period{l}}\cup \Period{l}$ into this residual space, that is, $$T^\infty_k \eqby{def} \{(p,(n\mod m_k)) \mid (p,n) \in T_k\}.$$
Notice that for sufficiently large $n$, if $(p,n\mod m_k)$ is in the winning region for this new reachability game on $\PA{k}$ then $(p,n) \in \Win{k+1}$.
%
This holds because a strategy in $\PA{k}$ can be used to reach $T^\infty_k$
in no more than $\size{\PA{k}} = \card{Q}\cdot m_k$ steps.
The same strategy will thus reach $T_k\subseteq \Win{k+1}$
from $(p,n)$
assuming that $n\ge b_k + \card{Q}\cdot m_k$, where $b_k$ denotes the height of the base set $B_k$.
%
There are two cases:
\begin{enumerate}
\item There exists some level $c\ge b_k$ such that $\Pat[c]{k+1} = \emptyset$.
Then for every configuration
$(p,n)\in\Period{k+1}$ it must hold that
$(p,n\mod m_k)$ is in the winning region of the reachability game $\PA{k}$.
To see this just pick a sufficiently high $n'>c$ such that $n'\equiv_{m_k}n$.
No strategy from $(p,n')$ in the original arena may allow to drop to counter value $c>b_k$
and therefore must reach
$T_k = \bigcup_{l\le k}\PreD{\Period{l}}\cup \Period{l}$
without leaving $Q\times\+{N}_{\ge b_k}$.
We conclude that $\Period{k+1}$ is periodic in $m_k$.
\item $\Pat[c]{k+1} \neq \emptyset$ for every level $c$.
Then there must exist
$i<j<2^{\card{Q}}$ with
$\Pat[i\cdot m_k]{k+1} = \Pat[j\cdot m_k]{k+1}$.
By Lemma~\ref{lem:OC-repetition} we derive that
$\Period{k+1}$ has period at most $m_k\cdot 2^{\card{Q}}$.
\end{enumerate}
Notice the pigeon hole principle guarantees that the second case above applies at most $\card{Q}$ times.
As the first case does not increase the bound on the period, this means that $m_k \le 2^{\card{Q}^2}$, as required. \qed
\end{proof}
\section{Introduction}
\input{content/intro}
\subsection*{Related Work}
\label{subsec_relatedwork}
\input{content/relatedwork}
\section{Preliminaries}
\label{sec_prelims}
\input{content/prelims}
\subsection{Infinite Games with Disturbances}
\label{subsec_games}
\input{content/games}
\subsection{Pushdown Games}
\label{subsec_pushdowndefs}
\input{content/pushdowndefs}
\subsection{Infinite Games without Disturbances}
\label{subsec_gameswithoutdisturbances}
\input{content/gameswithoutfaults}
\subsection*{Resilient Strategies}
\label{subsec_gameswithdisturbances}
\input{content/gameswithfaults}
\section{Resilience in Infinite Safety Games}
\label{sec_arbitrary}
\input{content/infinitearenas}
\section{Characterizing Resilience Values via Classical Games}
\label{sec_riggedgames}
\input{content/riggedgames}
\section{Resilience in Pushdown Safety Games}
\label{sec_pushdown}
\input{content/pushdownarenas}
\section{Resilience in One-counter Safety Games}
\label{sec_ocs}
\input{content/ocs}
\section{Beyond Safety: Reachability Games with Disturbances}
\label{sec_outlook}
\input{content/outlook}
\subsection{Resilience in Pushdown Reachability Games}
\label{subsec_reachability}
\input{content/outlookreach}
\subsection{Optimal Strategies in One-counter Reachability Games}
\label{subsec_optreachability}
\input{content/outlookoptimal}
\section{Conclusion}
\label{sec_conclusion}
\input{content/conclusion}
\bibliographystyle{plain}
|
1,116,691,498,511 | arxiv | \section{Introduction}
How \newer{can one} determine the relative strength of players who engage in a one-on-one competitive game? This is easy to find out for a group of two players: just let them play a match. For more players, tournaments solve this problem by ranking the players after a limited number of pairwise matches among the participants. The \textit{tournament format} defines a general structure of matches to be played and the method for deriving a ranking from the results of those matches.
\subsubsection*{Tournament Formats}
Most tournaments follow an elimination, a round-robin, or a Swiss-system format. In each round of an \textit{elimination} tournament, such as the second stage of the FIFA World Cup, only players who won their match in the previous round are paired again. The last player standing wins the tournament, and the remaining players' strength can only be estimated very roughly from the round they were eliminated in. \textit{Round-robin} tournaments are also called all-play-all tournaments, because each player plays against each other player once. The player with the highest score at the end of the tournament is declared the winner. The pool stage of the FIFA World Cup consists of round-robin tournaments.
\new{The \emph{Swiss-system} tournament format is widely used in competitive games like most e-sports, badminton, and chess, the last of which this paper focuses on.} In \new{such} tournaments, the number of rounds is predefined, but the pairing of players in each of these rounds depends on the results of previous rounds.
This format offers a convenient golden middle way between the earlier \new{mentioned} two tournament formats.
However, the
features \new{of the Swiss system} challenge organizers \new{greatly}. Firstly, unlike in elimination tournaments, the goal is to determine a whole ranking of the players and not only to declare the winner. Secondly, the final ranking of each player is greatly influenced by her assigned opponents, which is not an issue in round-robin tournaments
Therefore, a mechanism that computes suitable player pairings for Swiss-system tournaments is crucially important. However, designing such a system is a challenging task as it boils down to solving a complex combinatorial optimization problem. Interestingly, the state-of-the-art solution to this problem in chess tournaments relies on a complex set of declarative rules and not on a combinatorial optimization algorithm. In this paper we provide an algorithmic approach and we demonstrate that it outperforms the declarative state-of-the-art solution. For this, we do not try to mimic the FIDE solution but instead focus on the most important features of the Swiss system and derive a maximum weight matching formulation that enforces them.
\subsubsection*{The Swiss-System in Chess}
In Swiss-system chess tournaments, there are two well-defined and rigid \textit{absolute} and two milder \textit{quality} pairing criteria.
\begin{enumerate}
\item[(A1)] No two players play against each other more than once.
\item[(A2)] In each round before the last one, the difference of matches played with \newer{white}
and matches played with black pieces is between $-2$ and $2$ for every player.
\item[(Q1)] Opponents have equal or similar score.
\item[(Q2)] Each player has a balanced color distribution.
\end{enumerate}
Criterion (A1) ensures variety, while criterion (A2) ensures fairness, since the player with white pieces starts the game, and thus has an advantage over her opponent. These absolute criteria must be obeyed at any cost, which often enforces the relaxation of the two quality criteria.
In order to implement criterion~(Q1), players with equal score are grouped into \textit{score groups}. In each round, a chosen \textit{pairing system} allocates each player an opponent from the same score group. If a complete pairing is not possible within a score group, then one or more players are moved to another score group. Criterion~(Q2) requires that after each round of the tournament, the difference between matches played with black and white pieces is small for each player.
Adhering to these four criteria makes pairing design truly challenging. Pairings at FIDE tournaments were traditionally calculated manually by so-called arbiters, often using trial-and-error. Today, pairings are computed by decision-making software, but the FIDE pairing criteria are still written for human instead of computer execution. Over the years, more and more criteria were added to resolve ambiguities, which increased the complexity to a level at which pairing decisions are very challenging to comprehend for most players and even arbiters.
\subsection{Related Literature}
Novel algorithms that assist tournament scheduling regularly evoke interest in the AI community \citep{LJCS14,KW15,CIT16,GRSZ18,Hos18}.
\new{We first elaborate on existing work on comparing tournament formats, and then turn to approaches that utilize matchings for scheduling tournaments.}
\subsubsection{Comparing Tournament Formats}
\citet{appleton1995may} gives an overview of tournament formats and compares them with respect to how often the best player wins.
\citet{scarf2009numerical} simulate different tournament formats using team data from the UEFA champions league.
\citet{elmenreich2009robustness} compare several sorting algorithms, including one based on a Swiss-system tournament, with respect to their robustness, which is defined as the degree of similarity between the resulting ranking and the true strength order of players. They find round-robin sort, merge sort, and Swiss-system sort to be the most robust overall.
\subsubsection{Automated Matching Approaches} A tournament schedule can be seen as a set of matchings---one for each round.
\citet{glickman2005adaptive} propose an algorithm based on maximum weight perfect matchings to find the schedule. This algorithm maximizes the information gain about players' skill. The authors' approach compares favorably against random and Swiss-system pairing if at least 16 rounds are played. However, almost all real-world Swiss-system chess tournaments have less than 10 rounds \newer{according to}~\url{chess-results.com} \cite{herzog2020chess}.
\citet{kujansuu1999stable} use the stable roommates problem, see \cite{irving1985efficient}, to model a Swiss-system tournament pairing decision. Each player $p$ has a preference list, which ranks the other players by how desirable a match between player $p$ and each other player would be. The desirability depends on score difference and color balance. In comparison to the official FIDE pairing, this approach produces pairings with slightly better color balance but higher score differences between paired players, or, in other words, clearly favors criterion~(Q2) over~(Q1).
\subsubsection{Weighted Matching Models for Chess Tournaments} The two papers closest to ours focus on modeling the exact FIDE pairing criteria and computing the prescribed pairings.
\citet{olafsson1990weighted} pairs players using a maximum weight matching algorithm on a graph, where players and possible matches are represented by vertices and edges. Edge weights are set so that they model the 1985 FIDE pairing criteria. At that time, pairing criteria were more ambiguous than today, and pairing was done by hand, which sometimes took several hours. In contrast, using {\'O}lafsson's method, pairings could be calculated fast. Pairings calculated with the commercial software built by {\'O}lafsson are claimed to be preferred by experts to manually calculated pairings. However, {\'O}lafsson only provides examples and does not present any comparison based on formal criteria.
A more recent attempt to convert the FIDE pairing criteria into a weighted matching instance was undertaken by
\citet{BFP17}. Due to the extensive criterion system, only a subset of the criteria were modeled. The authors show that a pairing respecting these selected criteria can be calculated in polynomial time\new{, and leave it as a challenging open question whether the other FIDE criteria can also be integrated into a single weighted matching model}. The contribution appears to be purely theoretical, since neither a comparison with other pairing programs, nor implementation details are provided
Our work breaks the line of research that attempts to implement the declarative FIDE pairing criteria \newer{via} weighted matchings. Instead, we design new pairing rules \newer{along with a
different mechanism to compute the pairings}, and demonstrate their superiority compared to the FIDE pairing criteria and engine. This clearly differentiates our approach from the one in~\cite{olafsson1990weighted,BFP17}.
\subsection{Preliminaries and FIDE Criteria}\label{chap:background}
\textit{Players} are entities participating in a Swiss-system tournament. Each player \newer{has} an \emph{Elo rating}, which is a measure designed to capture her current playing strength from the outcome of her earlier matches \cite{elo1978rating}. In a \textit{match} two players, $a$ and $b$, play against each other. The three possible \textit{match results} are: $a$ wins and $b$ loses, $a$ and $b$ draw, $a$ loses and $b$ wins. The winner receives 1 point, the loser 0 points, while a draw is worth 0.5 points. A Swiss-system tournament consists of multiple \textit{rounds}, each of which is defined by a \textit{pairing}: a set of disjoint pairs of players, where each pair plays a match. At the end of the tournament, a strict ranking of the players is derived from the match results.
\subsubsection{Bye Allocation} In general, each player plays exactly one match per round. For an odd number of players, one of them receives a so-called `bye', which is a point rewarded without a match. This is always the player currently ranked last among those who have not yet received a bye
\subsubsection{Color Balance}
The FIDE Handbook~\cite[Chapter C.04.1]{fide2020handbook} states that `For each player the difference between the number of black and the number of white games shall not be greater than 2 or less than -2.' This criterion may only be relaxed \newer{in the last round}. This corresponds to our criterion (A2).
\newer{Also, a ban on a color that is assigned to a player three times consecutively, and further} milder criteria \newer{are phrased} to ensure a color assignment as close to an alternating white-black sequence as possible \cite[Chapters C.04.3.A.6 and C.04.3.C]{fide2020handbook}.
\subsubsection{Pairing Systems}
\label{sec:pairing_systems}
Players are always ranked by their current \newer{tournament} score. Furthermore, within each score group
the players are ranked by their Elo rank. The score groups and this ranking are the input of the \emph{pairing system}, which assigns an opponent to each player. Three main pairing systems are defined for chess tournaments. Table~\ref{tab:pairing_systems_example_pairing} shows an example pairing for each of them.
\begin{itemize}
\item \textbf{Dutch:}
Each score group is cut into an upper and a lower half. The upper half is then paired against the lower half so that the $i$th ranked player in the upper half plays against the $i$th ranked player in the lower half. Dutch is the de facto standard for major chess tournaments.
\item \textbf{Burstein:}
For each score group, the highest ranked unpaired player is paired against the lowest ranked unpaired player repeatedly until all players are paired.
\item \textbf{Monrad:}
In ascending rank order each unpaired player in a score group is paired against the next highest ranked player in that score group.
\end{itemize}
\begin{table}[h]
\centering
\begin{tabular}{ccccccc}
Dutch &&& Burstein &&& Monrad\\
1--5 &&& 1--8 &&& 1--2\\
2--6 &&& 2--7 &&& 3--4\\
3--7 &&& 3--6 &&& 5--6\\
4--8 &&& 4--5 &&& 7--8\\
\end{tabular}
\caption{Example pairing for each pairing system in a score group of 8 players. Players are referenced by rank within the score group, i.e., player 1 has the highest Elo rank.}
\label{tab:pairing_systems_example_pairing}
\end{table}
\noindent For \newer{comparison}, we propose two additional pairing systems based on randomness.
\begin{itemize}
\item \textbf{Random:} Every player within a score group is paired against a random player from her score group.
\item \textbf{Random2:} Every player from the top half of her score group is paired against a random player from the bottom half of her score group.
\end{itemize}
\subsubsection*{Floating Players}
Players \newer{who are} paired outside of their own score group are called~\emph{floaters}. To ensure that opponents are of similar strength--our criterion~(Q1)--, the FIDE criteria require to minimize the number of such floaters \new{and aim to float them to a score group of similar score}. However,
\new{floating} is unavoidable, e.g., in score groups with an odd number of players, and also in score groups where the first or second criterion eliminates too many possible matches.
\subsubsection*{The BBP Pairing Engine}
\label{sec:bbp_engine}
A \textit{pairing engine} is used to calculate the pairing for each round, based on the results of previous rounds. The BBP \newer{pairing} engine \newer{was developed by}
\citet{bierema2017bbp}. It implements the FIDE criteria strictly \cite[C.04.3 and C.04.4.2]{fide2020handbook} for the Dutch and Burstein pairing systems and outputs the unique pairing adhering to each of them. \newer{BBP uses a weighted matching algorithm, similarly as the approaches in \cite{olafsson1990weighted,BFP17}. The main difference to our algorithm is that while the weighted model of BBP was designed to follow the declarative criteria of FIDE and output the prescribed pairings, our pairing engine relies on a different weighted model, computes completely different pairs, and while doing so, it is able to reach a better ranking quality and a higher degree of fairness.} The output of Dutch BBP will serve as a base for our comparisons \newer{throughout} the paper, because Dutch is the pairing system implemented by all 8 pairing programs currently endorsed by the FIDE \cite[C.04.A.10.Annex-3]{fide2020handbook}.
\subsubsection*{Final Ranking}
The major organizing principle for the final ranking of players is obviously the final score. Players with the same final score are sorted by tiebreakers. The FIDE \cite[Chapter C.02.13]{fide2020handbook} defines 14 types of tiebreakers, and the tournament organizer lists some of them to be used at the specific tournament. If all tiebreaks fail, the tie is required to be broken by drawing of lots.
\subsection{Our Contribution}
In this paper, we present a \newer{novel} mechanism \newer{for calculating pairings in Swiss-system chess tournaments. With this, we contest the state-of-the-art mechanism endorsed by FIDE}. We compare the two systems by three measures: ranking quality, number of floaters, and color balance quality, in accordance with the FIDE tournament schedule goals.
Our main findings are summarized in the following list.
\begin{enumerate}
\setlength\itemsep{0mm}
\item We implemented \newer{the pairing systems Dutch, Burstein, Monrad, Random, and Random2} with an extensible and easy-to-understand approach that uses maximum weight matchings.
\item The pairing systems in descending order by expected \textbf{ranking quality} are: Burstein $>$ Random2 $>$ Dutch $=$ Dutch BBP $>$ Random $>$ Monrad. In particular, our implementations of Burstein and Random2 both \new{yield} higher ranking quality, while our implementation of Dutch \new{yields} similar ranking quality as the one reached by the Dutch BBP \new{pairing engine}.
\item We utilize our weighted matching model to define a \newer{novel} measure called `normalized strength difference', which we identify as the \newer{main} reason for a good ranking quality
\item The pairing systems in ascending order by expected \textbf{number of floaters} are: Burstein $<$ Random2 $=$ Dutch $=$ Monrad $<$ Dutch BBP $<$ Random. Compared to Dutch BBP, our \newer{mechanism} is fairer in terms of matching more players within their own score group.
\item All our pairing systems ensure the same \textbf{color balance quality} as Dutch BBP, with Random even reaching a better color balance. \new{Moreover, we show that our approach can easily be modified to enforce an even stronger color balance. This does not significantly affect the ranking quality---only the number of floaters increases slightly.}
\item \new{As the previous points demonstrate, o}ur implementations of Burstein and Random2 either outperform or are on a par with Dutch BBP. Our implementation of Dutch leads to pairings that perform just as well or even better than the ones prescribed by the official FIDE (Dutch) criteria and computed by Dutch BBP.
\end{enumerate}
\section{Pairings via Maximum Weight Matching}
\label{sec:mwm_engine}
Our novel \newer{mechanism} is based on computing a maximum weight matching (MWM) in an auxiliary, suitably weighted graph. The MWM engine is optimized for simplicity: score groups, color balances, and the employed pairing system are modeled by weights, so only a single \newer{computation} of a MWM is needed \new{in each round}. We \newer{now} describe the MWM engine.
\subsection{Input}
Each tournament has $n$ players $P = \{p_1,\dots,p_n\}$, a chosen pairing system (Dutch, Burstein, Monrad, Random, or Random2), and a maximum allowed color difference~$\beta$. \new{As criterion~(A2) states, FIDE aims for $\beta=2$.} If $n$ is odd, the weakest performing player who has not received a bye yet is given one, in accordance with the FIDE rules. In the MWM engine we will exclude the same player while constructing the auxiliary graph. Hence, from this point on we can assume that $n$ is even.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{example}
\caption{Example pairings of a 4-round tournament with 8 players generated via the MWM engine using the Dutch pairing system. Initially players are sorted decreasingly according to their Elo rating. Bold edges are possible matches within the same score group whereas dashed edges are other possible matches. The maximum weight matchings are shown in red. Arrows indicate the match outcomes (winner points to loser, no draws), and the color column shows the corresponding color distribution. The table for round $i+1$ is based on the table of round $i$. As score and color difference are equal, the pairing in round 1 is enforced by the Dutch pairing system. The pairing in round~2 is the outcome of optimizing first for criterion~(Q1) and then for criterion~(Q2), e.g., in $G_2$ we have $w(p_1,p_3) = w(p_4,p_6) = (0,0,-1)$ and $w(p_1,p_4) = w(p_3,p_6) = (0,-2,0)$ so the MWM picks the edges $\{p_1,p_3\}$ and $\{p_4,p_6\}$. In $G_3$ players $p_3$ and $p_4$ are paired since $w(p_3,p_4) = (0,0,0)$ whereas the weight of any other incident edge of both $p_3$ and $p_4$ has lexicographically lower weight. The matching in $G_4$ is enforced by maximizing the number of matches within score groups. If $p_1$ and $p_2$ would be paired, then, since $p_3$ and $p_4$ already played, player $p_4$ would float to a match with a player with score $1$, which implies that no match within the group with score $1$ is possible.}
\label{fig:example}
\end{figure*
Before each round of the tournament, the following input parameters are defined for each player $p_i \in P$:
\begin{itemize}
\item $Elo(p_i)$: the Elo rating of $p_i$ prior to the tournament. This remains unchanged for all rounds.
\item $s(p_i)$: the current score of $p_i$, defined as the sum of points player $p_i$ collected so far.
\item $r(p_i)$: the current rank of $p_i$, calculated from ordering all players in decreasing order according to their scores and their Elo ratings. Higher score and higher Elo rating yield better rank. Players with equal Elo rating are ordered randomly at the beginning, and their order is kept for all rounds.
\item $cd(p_i)$: the current color difference of $p_i$, defined as the number of matches played with white minus the number of matches played with black pieces.
\end{itemize}
\subsection{Graph Construction}
With these parameters as input, we construct the corresponding auxiliary weighted graph $G_r = (V,E,w)$ for round $r$ as follows. Let $V := P$ and for all pairs of players $p_i\neq p_j$, let the edge set $E$ contain the edge $\{p_i,p_j\}$ if
\begin{enumerate}
\item[(1)] $p_i$ and $p_j$ have not yet played against each other, and
\item[(2)] $|cd(p_i) + cd(p_j)| < 2\beta$.
\end{enumerate}
\new{These rules ensure criteria~(A1) and (A2). The second condition in our model will enforce $-2 \leq cd(p_i) \leq 2$ together with our color assignment rule in Section~\ref{sec:alg}.} \newer{In the appendix we additionally consider a variant where $-1 \leq cd(p_i) \leq 1$ is enforced. This implements FIDE's criterion that the color assignment should be as close to an alternating white-black sequence as possible and that no player can be assigned the same color three times in a row.
The weight of an edge $\{p_i,p_j\} \in E$ is defined as the tuple $$w(p_i,p_j) := (-|s(p_i)-s(p_j)|, -|cd(p_i) + cd(p_j)|, \pi(p_i,p_j)),$$ where the value of $\pi(p_i,p_j)$ depends on the pairing system \new{as follows.}
\begin{itemize}
\item Monrad: $\pi(p_i,p_j) :=-\left| r(p_i)-r(p_j) \right|$.
\item Burstein: $\pi(p_i,p_j) :=\left| r(p_i)-r(p_j) \right|^{1.01}$.
\item Dutch: $\pi(p_i,p_j) :=-\left|\frac{\text{sg size}}{2}- |r(p_i)-r(p_j)| \right|^{1.01}$, where sg size is set to 0 if $p_i$ and $p_j$ belong to different score groups, and it is the size of the score group of $p_i$ and $p_j$ otherwise.
\item Random: $\pi(p_i,p_j) :=$ random number in the interval $(0,1)$.
\item Random2: $\pi(p_i,p_j)$ is set to a random number in the interval $(0,1)$ if $p_i$ and $p_j$ belong to different halves of the same score group, otherwise it is set to a random number in the interval $(-1,0)$.
\end{itemize}
The exponent 1.01 in the function for Burstein rewards a larger rank difference, i.e., the Burstein pairing in Table~\ref{tab:pairing_systems_example_pairing} indeed carries a larger weight than the Dutch pairing, which has the same sum of rank differences. Similarly, the exponent for Dutch penalizes a larger distance from $\frac{\text{sg size}}{2}$. Notice that this exponent could be an arbitrary number as long as it is larger than~1.
\subsection{Algorithm}
\label{sec:alg}
The edge weights of $G_r$ are compared lexicographically and a maximum weight matching is sought for. This implies that pairing players within their score groups has the highest priority, optimizing color balance is second, and adhering to the pairing system is last. The comprehensive rules of our framework consist of our two absolute rules for \new{including an edge in }
the graph $G_r$, and this priority ordering serving as our quality rule. See Figure~\ref{fig:example} for an illustration
Before round $r$, we compute a maximum weight matching $M$ in graph $G_r$ and derive the player pairing from the edges in~$M$. If $\{p_i,p_j\} \in M$ then the players $p_i$ and $p_j$ will play against each other in round~$r$. Between them, the respective player with the lower color difference will play white. If they have the same color difference\new{, then} colors are assigned randomly.
\section{Assumptions and Experimental Setup}
In our simulations we assume that each player $p_i \in P$ has true playing strength $str(p_i)$ that is approximated by her Elo rating $Elo(p_i)$ and we treat both values as constant throughout the tournament. The probabilities of match results and optimal rankings are defined by the playing strength. More precisely, each player's playing strength is a random number drawn from a uniform distribution of values between 1400 and 2200. \new{We also justified our claims on ranking quality using other \newer{realistic} player strength distributions. We elaborate on these in the appendix. The results are in line with the results for the uniform distribution.}
\new{Elo ratings are used for computing $r(p_i)$
and for breaking ties in the final order.} The Elo rating of player $p_i$ is randomly drawn from a normal distribution with mean $str(p_i)$ and standard deviation $\frac{3000-str(p_i)}{20}$. This function mirrors the assumption that a higher Elo rating estimates the strength more accurately.
To avoid the noise introduced by byes, we assume that the number of players $n$ is even. The number of rounds is chosen to lie between $\lceil\log_2 n \rceil$ and $\frac{n}{2}$, as at least $\lceil\log_2 n \rceil$ rounds ensure that a player who wins all matches is the sole winner and at most $\frac{n}{2}$ rounds ensures that, according to Dirac's theorem \cite{dirac1952some}, a perfect matching always exists. The tiebreakers used for obtaining the final tournament ranking are based on the FIDE recommendation \cite[C.02.13.16.5]{fide2020handbook}.
\subsubsection*{Computing the Maximum Weight Matching}
First we transform each edge weight given as a tuple to a rational number. In particular, $w(p_i,p_j)$ is transformed to $10000 \cdot (-|s(p_i)-s(p_j)|) + 100 \cdot (-|cd(p_i) + cd(p_j)|)+ \pi(p_i,p_j)$. The factors 10000 and 100 ensure that each lexicographically maximum solution corresponds to a maximum weight solution with the new weights and vice versa. We compute pairings using the LEMON Graph Library \cite{dezsHo2011lemon} implementation of the maximum weight perfect matching algorithm, which is based on the blossom algorithm of
\citet{edmonds1965paths} and has the same time and space complexity \cite{Kol09}. The implementation we use has $O(nm \log{n})$ time complexity, where $n$ is the number of players and $m$ is the number of edges in the constructed graph~$G_r$.
\subsubsection{Realistic Probabilistic Model for Match Results}\label{sec:probabilistic_model}
The results of the individual matches are computed via a probabilistic model that is designed to be as realistic as possible. Match results are drawn at random from a suitably chosen probability distribution based on the players' strength and on the assigned colors for that match. For this, we use the probability distribution proposed by
\citet{milvang2016prob}, which was featured in a recent news article of the FIDE commission System of Pairings and Programs \cite{fide2020news}. Milvang's probability distribution was engineered via a Data Science approach that used real-world data from almost 4 million real chess matches from 50\,000 tournaments. It is based on Elo ratings and color information, whereas we use true strength values instead of Elo ratings to get unbiased match result probabilities.
Using Milvang's approach, the probability for a certain outcome of a match depends on the actual strengths of the involved players, not only on their strength difference. Draw probability increases with mean strength of the players. The probabilities also depend on colors, as the player playing with white pieces has an advantage. See Table~\ref{tab:example_probabilities} for some example values drawn from Milvang's distribution.
\begin{table}[h]
\centering
\begin{tabular}{rccc}
Player Strengths & Win White & Win Black & Draw\\
1200 (w) vs 1400 (b) & 26 \%& 57 \%& 17 \%\\
2200 (w) vs 2400 (b) & 14 \%& 55 \%&31 \%\\
2400 (w) vs 2200 (b) & 63 \%& 11 \%& 26 \%
\end{tabular}
\caption{Example match outcome probabilities drawn from Milvang's probability distribution \cite{milvang2016prob}.
}
\label{tab:example_probabilities}
\end{table}
\subsubsection{Measuring Ranking Quality} Ranking quality measures how similar the tournament's final ranking is to the ranking that sorts the players by their strength. One popular measure for the difference between two rankings is the Kendall $\tau$ distance \cite{kendall1945treatment}. It counts the number of discordant pairs: pairs of elements $x$ and $y$, where $x < y$ in one ranking, but $y < x$ in the other. We use its normalized variant, where $\tau \in [-1, 1]$, and $\tau = 1$ means the rankings are identical, while $\tau = -1$ means one ranking is the inverse of the other. A higher Kendall $\tau$ is better, because it indicates a larger degree of similarity between the true and the output ranking.
We also justify our claims on ranking quality using two other well-known and possibly more sophisticated similarity measures, the Spearman $\rho$ distance \cite{spearman1904proof} and normalized discounted cumulative gain (NDCG). We elaborate on these measures and their behavior for our problem in the appendix. The results are in line with the ones derived for the Kendall $\tau$ distance.
\subsubsection{Measuring Fairness
We measure fairness in terms of the two relaxable criteria of Swiss-system chess tournaments: (Q1) on the equal score of opponents and (Q2) on the color distribution balance. Adhering to (Q1) is measured by the number of float pairs, which equals the number of matches with opponents from different score groups throughout the tournament. We measure the absolute color difference of a round as the sum of color differences for all players: $acd = \sum_{p_i \in P}{|cd(p_i)|}$. Note that as $|cd(p_i)| \geq 1$ for all players after each odd round, $acd \geq n$ in those rounds.
\subsubsection{Presentation of the Data}
Data is presented in the form of \textit{violin plots} \cite{hintze1998violin}, \textit{letter value plots} \cite{HWK17}, \new{and \emph{scatter plots} \cite{FD05}}. For violin plots, kernel density estimation is used to show a smoothed probability density function of the underlying distribution. Additionally, similar to box plots, quartiles are shown by dashed lines. Letter value plots are enhanced box plots that show more quantiles. Unlike violin plots, they are suitable for discrete values, as all shown values are actual observations without smoothing.
Our plots compare the MWM implementation of the five pairing systems with the BBP implementation of Dutch.
\section{Simulation Results}
All simulations use the following parameters, unless noted otherwise:
\begin{itemize}
\item number of players $n$: 32
\item number of rounds: 7
\item strength range: between 1400 and 2200
\item maximum allowed color difference $\beta$: 2
\item sample size: 100\,000
\end{itemize}
These values were chosen to be as realistic as possible, based on parameters of more than 320\,000 real-world tournaments uploaded to the website \url{chess-results.com}.\footnote{The data was kindly provided by Heinz Herzog, author of the FIDE-endorsed tournament manager \url{Swiss-Manager} \cite{herzog2020swiss} and \url{chess-results.com} \cite{herzog2020chess}.} The experiments were run on a compute server using version 20.04.1 of the Ubuntu operating system. It is powered by 48 Intel Xeon Gold 5118 CPUs running at 2.3 GHz and 62.4 GiB of RAM. We emphasize that \new{the standard real-life challenge at a tournament, that is, }
computing a single pairing via a maximum weight matching for a tournament round can be \new{solved}
in a fraction of a second on a standard laptop.
\subsection{Ranking Quality}
\label{sec:renking_q}
The pairing system of a \new{Swiss-system} tournament has a major impact on the obtained ranking quality, as Figure~\ref{fig:ranking_quality} shows. Burstein and Random2 achieve the best ranking quality, followed by Dutch and Dutch BBP. Random has a worse ranking quality and Monrad performs by far the worst. For other strength ranges, Figure~\ref{fig:mean_strength} shows consistent results.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{color_figures/ranking_quality_pairing_systems.pdf}
\caption{Ranking quality measured by normalized Kendall~$\tau$. A higher value means a better ranking quality.}
\label{fig:ranking_quality}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth]{color_figures/mean_strength.pdf}
\caption{Ranking quality measured by normalized Kendall~$\tau$ for different strength ranges.}
\label{fig:mean_strength}
\end{figure}
\noindent Comparing Dutch to Dutch BBP shows that they behave very similarly, with slight advantage for Dutch. This is remarkable, since Dutch BBP is based on complex and rigid declarative criteria that are time-tested, while Dutch is the output of our easy-to-understand, purely \newer{matching-based} approach.
Together with the performance of Burstein and Random2 this shows that more transparent pairing systems can outperform the state-of-the-art Dutch BBP in terms of ranking quality.
We provide additional experimental results on the ranking quality in the appendix. There we present consistent results also for fewer or more players, \new{for other strength range sizes, and for different player strength distributions}. Additionally, we elaborate on how our flexible maximum weight matching model enabled us to detect the exact reason why certain pairing systems produce better rankings, which might help designing better pairing systems in the future.
\subsection{Fairness}
The highly complex pairing criteria of the FIDE were designed with a focus on two fairness goals phrased as quality criteria, (Q1): minimizing the number of float pairs and (Q2): minimizing the absolute color difference.
Criterion~(Q1) is at the heart of Swiss-system tournaments as pairing players of equal score ensures well-balanced matches. \newer{As Figure~\ref{fig:float_pairs} shows}, Burstein, Dutch, and Random2 \newer{beat} Dutch BBP in terms of the number of float pairs. \newer{In the appendix we} show consistent results for other simulation parameters.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{color_figures/float_pairs_pairing_systems.pdf}
\caption{Number of float pairs out of the $7\cdot 16 = 112$ matches of the tournament. Recall that floating is often unavoidable due to the size of the score group. A lower number indicates a better implementation of criterion~(Q1).}
\label{fig:float_pairs}
\end{figure}
\noindent Figure~\ref{fig:color_difference_6} focuses on criterion~(Q2) and shows that an absolute color difference very similar to the one guaranteed by Dutch BBP can be achieved via our MWM engine. The pairing system Random even outperforms Dutch BBP in this regard. \new{In the appendix, we provide additional experiments with different numbers of rounds and numbers of players that lead to consistent results.} Also, we report there on experiments in which an even stronger color difference constraint is enforced, and observe the impact on the obtained ranking quality and the number of float pairs. \newer{Interestingly, the obtained ranking quality is almost the same but this comes at a cost of a slightly increased number of float pairs. }
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{color_figures/6_color_difference_pairing_systems.pdf}
\caption{Absolute color difference after 6 rounds. A lower $acd$ means a better color distribution. Recall that a $acd \geq n$ for each odd round, while $acd = 0$ is possible after each even round.}
\label{fig:color_difference_6}
\end{figure}
\noindent Hence, our maximum weight matching approach with edge weights that prioritize matches within score groups and secondly optimize for color balance is on a par with the sophisticated official FIDE criteria for criterion~(Q2) and it even outperforms them for criterion~(Q1). Thus, our more transparent approach ensures the same color balance quality but achieves even fewer float pairs. \newer{Moreover, our approach also allows for a different trade-off between criteria (Q1) and (Q2) that does not affect the obtained ranking quality.}
\section{Conclusion}
The experimental results of our MWM engine with Burstein or Random2 demonstrate that it is possible to outperform the state-of-the-art FIDE pairing criteria in terms of both ranking quality and fairness, i.e., criteria (Q1) and (Q2), with a novel efficient \newer{mechanism} that is more transparent and intelligible to all participants. The direct comparison of our MWM Dutch engine versus Dutch BBP shows that even if the same pairing system is used,
MWM achieves the same ranking quality but is more powerful since it yields an improved fairness. We believe that the key to this is the direct formulation of the most important criteria as a maximum weight matching problem.
The only scenario for which we might advise against using our \newer{mechanism} is when the arbiter has no access to a computing device. In order to manually produce pairings in our framework, the arbiter would need to calculate the edge weights and then execute Edmonds' blossom algorithm. \newer{Instead,} the FIDE~\cite[Chapter C.04.3.D]{fide2020handbook} provides manually executable rules.
However, these rules include exhaustive search routines that can make the execution very slow, i.e., highly exponential in the number of players \cite{BFP17}. Therefore, the ill-fated arbiter has to choose between learning \new{to execute} Edmonds' blossom algorithm and following a cumbersome exponential-time pairing routine.
A clear advantage of our \newer{mechanism} is that it is easily extendable: as Random and Random2 already demonstrate, a new pairing system can be implemented simply by specifying how edge weights are calculated. Similarly, as we have also demonstrated, the color balance can be adjusted by simply changing the parameter~$\beta$. By thinning out the edge set in our graph, we can also reach an alternating black-white sequence for each player instead of just minimizing the color difference in each round. Also, the flexibility of the maximum weight matching approach proved to be essential for uncovering the driving force behind the achieved high ranking quality: the normalized strength difference. Hence, our approach was not only valuable for computing better pairings but also in the analysis of the obtained ranking quality.
Last but not least, the flexibility of the MWM engine likely allows \newer{to incorporate additional quality criteria like measuring fairness via the average opponent ratings. Also} quality criteria of other games and sports tournaments organized in the Swiss system \newer{can be} integrated into the model.
\pagebreak
\section{Introduction}
How \newer{can one} determine the relative strength of players who engage in a one-on-one competitive game? This is easy to find out for a group of two players: just let them play a match. For more players, tournaments solve this problem by ranking the players after a limited number of pairwise matches among the participants. The \textit{tournament format} defines a general structure of matches to be played and the method for deriving a ranking from the results of those matches.
\subsubsection*{Tournament Formats}
Most tournaments follow an elimination, a round-robin, or a Swiss-system format. In each round of an \textit{elimination} tournament, such as the second stage of the FIFA World Cup, only players who won their match in the previous round are paired again. The last player standing wins the tournament, and the remaining players' strength can only be estimated very roughly from the round they were eliminated in. \textit{Round-robin} tournaments are also called all-play-all tournaments, because each player plays against each other player once. The player with the highest score at the end of the tournament is declared the winner. The pool stage of the FIFA World Cup consists of round-robin tournaments.
\new{The \emph{Swiss-system} tournament format is widely used in competitive games like most e-sports, badminton, and chess, the last of which this paper focuses on.} In \new{such} tournaments, the number of rounds is predefined, but the pairing of players in each of these rounds depends on the results of previous rounds.
This format offers a convenient golden middle way between the earlier \new{mentioned} two tournament formats.
However, the
features \new{of the Swiss system} challenge organizers \new{greatly}. Firstly, unlike in elimination tournaments, the goal is to determine a whole ranking of the players and not only to declare the winner. Secondly, the final ranking of each player is greatly influenced by her assigned opponents, which is not an issue in round-robin tournaments
Therefore, a mechanism that computes suitable player pairings for Swiss-system tournaments is crucially important. However, designing such a system is a challenging task as it boils down to solving a complex combinatorial optimization problem. Interestingly, the state-of-the-art solution to this problem in chess tournaments relies on a complex set of declarative rules and not on a combinatorial optimization algorithm. In this paper we provide an algorithmic approach and we demonstrate that it outperforms the declarative state-of-the-art solution. For this, we do not try to mimic the FIDE solution but instead focus on the most important features of the Swiss system and derive a maximum weight matching formulation that enforces them.
\subsubsection*{The Swiss-System in Chess}
In Swiss-system chess tournaments, there are two well-defined and rigid \textit{absolute} and two milder \textit{quality} pairing criteria.
\begin{enumerate}
\item[(A1)] No two players play against each other more than once.
\item[(A2)] In each round before the last one, the difference of matches played with \newer{white}
and matches played with black pieces is between $-2$ and $2$ for every player.
\item[(Q1)] Opponents have equal or similar score.
\item[(Q2)] Each player has a balanced color distribution.
\end{enumerate}
Criterion (A1) ensures variety, while criterion (A2) ensures fairness, since the player with white pieces starts the game, and thus has an advantage over her opponent. These absolute criteria must be obeyed at any cost, which often enforces the relaxation of the two quality criteria.
In order to implement criterion~(Q1), players with equal score are grouped into \textit{score groups}. In each round, a chosen \textit{pairing system} allocates each player an opponent from the same score group. If a complete pairing is not possible within a score group, then one or more players are moved to another score group. Criterion~(Q2) requires that after each round of the tournament, the difference between matches played with black and white pieces is small for each player.
Adhering to these four criteria makes pairing design truly challenging. Pairings at FIDE tournaments were traditionally calculated manually by so-called arbiters, often using trial-and-error. Today, pairings are computed by decision-making software, but the FIDE pairing criteria are still written for human instead of computer execution. Over the years, more and more criteria were added to resolve ambiguities, which increased the complexity to a level at which pairing decisions are very challenging to comprehend for most players and even arbiters.
\subsection{Related Literature}
Novel algorithms that assist tournament scheduling regularly evoke interest in the AI community \citep{LJCS14,KW15,CIT16,GRSZ18,Hos18}.
\new{We first elaborate on existing work on comparing tournament formats, and then turn to approaches that utilize matchings for scheduling tournaments.}
\subsubsection{Comparing Tournament Formats}
\citet{appleton1995may} gives an overview of tournament formats and compares them with respect to how often the best player wins.
\citet{scarf2009numerical} simulate different tournament formats using team data from the UEFA champions league.
\citet{elmenreich2009robustness} compare several sorting algorithms, including one based on a Swiss-system tournament, with respect to their robustness, which is defined as the degree of similarity between the resulting ranking and the true strength order of players. They find round-robin sort, merge sort, and Swiss-system sort to be the most robust overall.
\subsubsection{Automated Matching Approaches} A tournament schedule can be seen as a set of matchings---one for each round.
\citet{glickman2005adaptive} propose an algorithm based on maximum weight perfect matchings to find the schedule. This algorithm maximizes the information gain about players' skill. The authors' approach compares favorably against random and Swiss-system pairing if at least 16 rounds are played. However, almost all real-world Swiss-system chess tournaments have less than 10 rounds \newer{according to}~\url{chess-results.com} \cite{herzog2020chess}.
\citet{kujansuu1999stable} use the stable roommates problem, see \cite{irving1985efficient}, to model a Swiss-system tournament pairing decision. Each player $p$ has a preference list, which ranks the other players by how desirable a match between player $p$ and each other player would be. The desirability depends on score difference and color balance. In comparison to the official FIDE pairing, this approach produces pairings with slightly better color balance but higher score differences between paired players, or, in other words, clearly favors criterion~(Q2) over~(Q1).
\subsubsection{Weighted Matching Models for Chess Tournaments} The two papers closest to ours focus on modeling the exact FIDE pairing criteria and computing the prescribed pairings.
\citet{olafsson1990weighted} pairs players using a maximum weight matching algorithm on a graph, where players and possible matches are represented by vertices and edges. Edge weights are set so that they model the 1985 FIDE pairing criteria. At that time, pairing criteria were more ambiguous than today, and pairing was done by hand, which sometimes took several hours. In contrast, using {\'O}lafsson's method, pairings could be calculated fast. Pairings calculated with the commercial software built by {\'O}lafsson are claimed to be preferred by experts to manually calculated pairings. However, {\'O}lafsson only provides examples and does not present any comparison based on formal criteria.
A more recent attempt to convert the FIDE pairing criteria into a weighted matching instance was undertaken by
\citet{BFP17}. Due to the extensive criterion system, only a subset of the criteria were modeled. The authors show that a pairing respecting these selected criteria can be calculated in polynomial time\new{, and leave it as a challenging open question whether the other FIDE criteria can also be integrated into a single weighted matching model}. The contribution appears to be purely theoretical, since neither a comparison with other pairing programs, nor implementation details are provided
Our work breaks the line of research that attempts to implement the declarative FIDE pairing criteria \newer{via} weighted matchings. Instead, we design new pairing rules \newer{along with a
different mechanism to compute the pairings}, and demonstrate their superiority compared to the FIDE pairing criteria and engine. This clearly differentiates our approach from the one in~\cite{olafsson1990weighted,BFP17}.
\subsection{Preliminaries and FIDE Criteria}\label{chap:background}
\textit{Players} are entities participating in a Swiss-system tournament. Each player \newer{has} an \emph{Elo rating}, which is a measure designed to capture her current playing strength from the outcome of her earlier matches \cite{elo1978rating}. In a \textit{match} two players, $a$ and $b$, play against each other. The three possible \textit{match results} are: $a$ wins and $b$ loses, $a$ and $b$ draw, $a$ loses and $b$ wins. The winner receives 1 point, the loser 0 points, while a draw is worth 0.5 points. A Swiss-system tournament consists of multiple \textit{rounds}, each of which is defined by a \textit{pairing}: a set of disjoint pairs of players, where each pair plays a match. At the end of the tournament, a strict ranking of the players is derived from the match results.
\subsubsection{Bye Allocation} In general, each player plays exactly one match per round. For an odd number of players, one of them receives a so-called `bye', which is a point rewarded without a match. This is always the player currently ranked last among those who have not yet received a bye
\subsubsection{Color Balance}
The FIDE Handbook~\cite[Chapter C.04.1]{fide2020handbook} states that `For each player the difference between the number of black and the number of white games shall not be greater than 2 or less than -2.' This criterion may only be relaxed \newer{in the last round}. This corresponds to our criterion (A2).
\newer{Also, a ban on a color that is assigned to a player three times consecutively, and further} milder criteria \newer{are phrased} to ensure a color assignment as close to an alternating white-black sequence as possible \cite[Chapters C.04.3.A.6 and C.04.3.C]{fide2020handbook}.
\subsubsection{Pairing Systems}
\label{sec:pairing_systems}
Players are always ranked by their current \newer{tournament} score. Furthermore, within each score group
the players are ranked by their Elo rank. The score groups and this ranking are the input of the \emph{pairing system}, which assigns an opponent to each player. Three main pairing systems are defined for chess tournaments. Table~\ref{tab:pairing_systems_example_pairing} shows an example pairing for each of them.
\begin{itemize}
\item \textbf{Dutch:}
Each score group is cut into an upper and a lower half. The upper half is then paired against the lower half so that the $i$th ranked player in the upper half plays against the $i$th ranked player in the lower half. Dutch is the de facto standard for major chess tournaments.
\item \textbf{Burstein:}
For each score group, the highest ranked unpaired player is paired against the lowest ranked unpaired player repeatedly until all players are paired.
\item \textbf{Monrad:}
In ascending rank order each unpaired player in a score group is paired against the next highest ranked player in that score group.
\end{itemize}
\begin{table}[h]
\centering
\begin{tabular}{ccccccc}
Dutch &&& Burstein &&& Monrad\\
1--5 &&& 1--8 &&& 1--2\\
2--6 &&& 2--7 &&& 3--4\\
3--7 &&& 3--6 &&& 5--6\\
4--8 &&& 4--5 &&& 7--8\\
\end{tabular}
\caption{Example pairing for each pairing system in a score group of 8 players. Players are referenced by rank within the score group, i.e., player 1 has the highest Elo rank.}
\label{tab:pairing_systems_example_pairing}
\end{table}
\noindent For \newer{comparison}, we propose two additional pairing systems based on randomness.
\begin{itemize}
\item \textbf{Random:} Every player within a score group is paired against a random player from her score group.
\item \textbf{Random2:} Every player from the top half of her score group is paired against a random player from the bottom half of her score group.
\end{itemize}
\subsubsection*{Floating Players}
Players \newer{who are} paired outside of their own score group are called~\emph{floaters}. To ensure that opponents are of similar strength--our criterion~(Q1)--, the FIDE criteria require to minimize the number of such floaters \new{and aim to float them to a score group of similar score}. However,
\new{floating} is unavoidable, e.g., in score groups with an odd number of players, and also in score groups where the first or second criterion eliminates too many possible matches.
\subsubsection*{The BBP Pairing Engine}
\label{sec:bbp_engine}
A \textit{pairing engine} is used to calculate the pairing for each round, based on the results of previous rounds. The BBP \newer{pairing} engine \newer{was developed by}
\citet{bierema2017bbp}. It implements the FIDE criteria strictly \cite[C.04.3 and C.04.4.2]{fide2020handbook} for the Dutch and Burstein pairing systems and outputs the unique pairing adhering to each of them. \newer{BBP uses a weighted matching algorithm, similarly as the approaches in \cite{olafsson1990weighted,BFP17}. The main difference to our algorithm is that while the weighted model of BBP was designed to follow the declarative criteria of FIDE and output the prescribed pairings, our pairing engine relies on a different weighted model, computes completely different pairs, and while doing so, it is able to reach a better ranking quality and a higher degree of fairness.} The output of Dutch BBP will serve as a base for our comparisons \newer{throughout} the paper, because Dutch is the pairing system implemented by all 8 pairing programs currently endorsed by the FIDE \cite[C.04.A.10.Annex-3]{fide2020handbook}.
\subsubsection*{Final Ranking}
The major organizing principle for the final ranking of players is obviously the final score. Players with the same final score are sorted by tiebreakers. The FIDE \cite[Chapter C.02.13]{fide2020handbook} defines 14 types of tiebreakers, and the tournament organizer lists some of them to be used at the specific tournament. If all tiebreaks fail, the tie is required to be broken by drawing of lots.
\subsection{Our Contribution}
In this paper, we present a \newer{novel} mechanism \newer{for calculating pairings in Swiss-system chess tournaments. With this, we contest the state-of-the-art mechanism endorsed by FIDE}. We compare the two systems by three measures: ranking quality, number of floaters, and color balance quality, in accordance with the FIDE tournament schedule goals.
Our main findings are summarized in the following list.
\begin{enumerate}
\setlength\itemsep{0mm}
\item We implemented \newer{the pairing systems Dutch, Burstein, Monrad, Random, and Random2} with an extensible and easy-to-understand approach that uses maximum weight matchings.
\item The pairing systems in descending order by expected \textbf{ranking quality} are: Burstein $>$ Random2 $>$ Dutch $=$ Dutch BBP $>$ Random $>$ Monrad. In particular, our implementations of Burstein and Random2 both \new{yield} higher ranking quality, while our implementation of Dutch \new{yields} similar ranking quality as the one reached by the Dutch BBP \new{pairing engine}.
\item We utilize our weighted matching model to define a \newer{novel} measure called `normalized strength difference', which we identify as the \newer{main} reason for a good ranking quality
\item The pairing systems in ascending order by expected \textbf{number of floaters} are: Burstein $<$ Random2 $=$ Dutch $=$ Monrad $<$ Dutch BBP $<$ Random. Compared to Dutch BBP, our \newer{mechanism} is fairer in terms of matching more players within their own score group.
\item All our pairing systems ensure the same \textbf{color balance quality} as Dutch BBP, with Random even reaching a better color balance. \new{Moreover, we show that our approach can easily be modified to enforce an even stronger color balance. This does not significantly affect the ranking quality---only the number of floaters increases slightly.}
\item \new{As the previous points demonstrate, o}ur implementations of Burstein and Random2 either outperform or are on a par with Dutch BBP. Our implementation of Dutch leads to pairings that perform just as well or even better than the ones prescribed by the official FIDE (Dutch) criteria and computed by Dutch BBP.
\end{enumerate}
\section{Pairings via Maximum Weight Matching}
\label{sec:mwm_engine}
Our novel \newer{mechanism} is based on computing a maximum weight matching (MWM) in an auxiliary, suitably weighted graph. The MWM engine is optimized for simplicity: score groups, color balances, and the employed pairing system are modeled by weights, so only a single \newer{computation} of a MWM is needed \new{in each round}. We \newer{now} describe the MWM engine.
\subsection{Input}
Each tournament has $n$ players $P = \{p_1,\dots,p_n\}$, a chosen pairing system (Dutch, Burstein, Monrad, Random, or Random2), and a maximum allowed color difference~$\beta$. \new{As criterion~(A2) states, FIDE aims for $\beta=2$.} If $n$ is odd, the weakest performing player who has not received a bye yet is given one, in accordance with the FIDE rules. In the MWM engine we will exclude the same player while constructing the auxiliary graph. Hence, from this point on we can assume that $n$ is even.
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{example}
\caption{Example pairings of a 4-round tournament with 8 players generated via the MWM engine using the Dutch pairing system. Initially players are sorted decreasingly according to their Elo rating. Bold edges are possible matches within the same score group whereas dashed edges are other possible matches. The maximum weight matchings are shown in red. Arrows indicate the match outcomes (winner points to loser, no draws), and the color column shows the corresponding color distribution. The table for round $i+1$ is based on the table of round $i$. As score and color difference are equal, the pairing in round 1 is enforced by the Dutch pairing system. The pairing in round~2 is the outcome of optimizing first for criterion~(Q1) and then for criterion~(Q2), e.g., in $G_2$ we have $w(p_1,p_3) = w(p_4,p_6) = (0,0,-1)$ and $w(p_1,p_4) = w(p_3,p_6) = (0,-2,0)$ so the MWM picks the edges $\{p_1,p_3\}$ and $\{p_4,p_6\}$. In $G_3$ players $p_3$ and $p_4$ are paired since $w(p_3,p_4) = (0,0,0)$ whereas the weight of any other incident edge of both $p_3$ and $p_4$ has lexicographically lower weight. The matching in $G_4$ is enforced by maximizing the number of matches within score groups. If $p_1$ and $p_2$ would be paired, then, since $p_3$ and $p_4$ already played, player $p_4$ would float to a match with a player with score $1$, which implies that no match within the group with score $1$ is possible.}
\label{fig:example}
\end{figure*
Before each round of the tournament, the following input parameters are defined for each player $p_i \in P$:
\begin{itemize}
\item $Elo(p_i)$: the Elo rating of $p_i$ prior to the tournament. This remains unchanged for all rounds.
\item $s(p_i)$: the current score of $p_i$, defined as the sum of points player $p_i$ collected so far.
\item $r(p_i)$: the current rank of $p_i$, calculated from ordering all players in decreasing order according to their scores and their Elo ratings. Higher score and higher Elo rating yield better rank. Players with equal Elo rating are ordered randomly at the beginning, and their order is kept for all rounds.
\item $cd(p_i)$: the current color difference of $p_i$, defined as the number of matches played with white minus the number of matches played with black pieces.
\end{itemize}
\subsection{Graph Construction}
With these parameters as input, we construct the corresponding auxiliary weighted graph $G_r = (V,E,w)$ for round $r$ as follows. Let $V := P$ and for all pairs of players $p_i\neq p_j$, let the edge set $E$ contain the edge $\{p_i,p_j\}$ if
\begin{enumerate}
\item[(1)] $p_i$ and $p_j$ have not yet played against each other, and
\item[(2)] $|cd(p_i) + cd(p_j)| < 2\beta$.
\end{enumerate}
\new{These rules ensure criteria~(A1) and (A2). The second condition in our model will enforce $-2 \leq cd(p_i) \leq 2$ together with our color assignment rule in Section~\ref{sec:alg}.} \newer{In the appendix we additionally consider a variant where $-1 \leq cd(p_i) \leq 1$ is enforced. This implements FIDE's criterion that the color assignment should be as close to an alternating white-black sequence as possible and that no player can be assigned the same color three times in a row.
The weight of an edge $\{p_i,p_j\} \in E$ is defined as the tuple $$w(p_i,p_j) := (-|s(p_i)-s(p_j)|, -|cd(p_i) + cd(p_j)|, \pi(p_i,p_j)),$$ where the value of $\pi(p_i,p_j)$ depends on the pairing system \new{as follows.}
\begin{itemize}
\item Monrad: $\pi(p_i,p_j) :=-\left| r(p_i)-r(p_j) \right|$.
\item Burstein: $\pi(p_i,p_j) :=\left| r(p_i)-r(p_j) \right|^{1.01}$.
\item Dutch: $\pi(p_i,p_j) :=-\left|\frac{\text{sg size}}{2}- |r(p_i)-r(p_j)| \right|^{1.01}$, where sg size is set to 0 if $p_i$ and $p_j$ belong to different score groups, and it is the size of the score group of $p_i$ and $p_j$ otherwise.
\item Random: $\pi(p_i,p_j) :=$ random number in the interval $(0,1)$.
\item Random2: $\pi(p_i,p_j)$ is set to a random number in the interval $(0,1)$ if $p_i$ and $p_j$ belong to different halves of the same score group, otherwise it is set to a random number in the interval $(-1,0)$.
\end{itemize}
The exponent 1.01 in the function for Burstein rewards a larger rank difference, i.e., the Burstein pairing in Table~\ref{tab:pairing_systems_example_pairing} indeed carries a larger weight than the Dutch pairing, which has the same sum of rank differences. Similarly, the exponent for Dutch penalizes a larger distance from $\frac{\text{sg size}}{2}$. Notice that this exponent could be an arbitrary number as long as it is larger than~1.
\subsection{Algorithm}
\label{sec:alg}
The edge weights of $G_r$ are compared lexicographically and a maximum weight matching is sought for. This implies that pairing players within their score groups has the highest priority, optimizing color balance is second, and adhering to the pairing system is last. The comprehensive rules of our framework consist of our two absolute rules for \new{including an edge in }
the graph $G_r$, and this priority ordering serving as our quality rule. See Figure~\ref{fig:example} for an illustration
Before round $r$, we compute a maximum weight matching $M$ in graph $G_r$ and derive the player pairing from the edges in~$M$. If $\{p_i,p_j\} \in M$ then the players $p_i$ and $p_j$ will play against each other in round~$r$. Between them, the respective player with the lower color difference will play white. If they have the same color difference\new{, then} colors are assigned randomly.
\section{Assumptions and Experimental Setup}
In our simulations we assume that each player $p_i \in P$ has true playing strength $str(p_i)$ that is approximated by her Elo rating $Elo(p_i)$ and we treat both values as constant throughout the tournament. The probabilities of match results and optimal rankings are defined by the playing strength. More precisely, each player's playing strength is a random number drawn from a uniform distribution of values between 1400 and 2200. \new{We also justified our claims on ranking quality using other \newer{realistic} player strength distributions. We elaborate on these in the appendix. The results are in line with the results for the uniform distribution.}
\new{Elo ratings are used for computing $r(p_i)$
and for breaking ties in the final order.} The Elo rating of player $p_i$ is randomly drawn from a normal distribution with mean $str(p_i)$ and standard deviation $\frac{3000-str(p_i)}{20}$. This function mirrors the assumption that a higher Elo rating estimates the strength more accurately.
To avoid the noise introduced by byes, we assume that the number of players $n$ is even. The number of rounds is chosen to lie between $\lceil\log_2 n \rceil$ and $\frac{n}{2}$, as at least $\lceil\log_2 n \rceil$ rounds ensure that a player who wins all matches is the sole winner and at most $\frac{n}{2}$ rounds ensures that, according to Dirac's theorem \cite{dirac1952some}, a perfect matching always exists. The tiebreakers used for obtaining the final tournament ranking are based on the FIDE recommendation \cite[C.02.13.16.5]{fide2020handbook}.
\subsubsection*{Computing the Maximum Weight Matching}
First we transform each edge weight given as a tuple to a rational number. In particular, $w(p_i,p_j)$ is transformed to $10000 \cdot (-|s(p_i)-s(p_j)|) + 100 \cdot (-|cd(p_i) + cd(p_j)|)+ \pi(p_i,p_j)$. The factors 10000 and 100 ensure that each lexicographically maximum solution corresponds to a maximum weight solution with the new weights and vice versa. We compute pairings using the LEMON Graph Library \cite{dezsHo2011lemon} implementation of the maximum weight perfect matching algorithm, which is based on the blossom algorithm of
\citet{edmonds1965paths} and has the same time and space complexity \cite{Kol09}. The implementation we use has $O(nm \log{n})$ time complexity, where $n$ is the number of players and $m$ is the number of edges in the constructed graph~$G_r$.
\subsubsection{Realistic Probabilistic Model for Match Results}\label{sec:probabilistic_model}
The results of the individual matches are computed via a probabilistic model that is designed to be as realistic as possible. Match results are drawn at random from a suitably chosen probability distribution based on the players' strength and on the assigned colors for that match. For this, we use the probability distribution proposed by
\citet{milvang2016prob}, which was featured in a recent news article of the FIDE commission System of Pairings and Programs \cite{fide2020news}. Milvang's probability distribution was engineered via a Data Science approach that used real-world data from almost 4 million real chess matches from 50\,000 tournaments. It is based on Elo ratings and color information, whereas we use true strength values instead of Elo ratings to get unbiased match result probabilities.
Using Milvang's approach, the probability for a certain outcome of a match depends on the actual strengths of the involved players, not only on their strength difference. Draw probability increases with mean strength of the players. The probabilities also depend on colors, as the player playing with white pieces has an advantage. See Table~\ref{tab:example_probabilities} for some example values drawn from Milvang's distribution.
\begin{table}[h]
\centering
\begin{tabular}{rccc}
Player Strengths & Win White & Win Black & Draw\\
1200 (w) vs 1400 (b) & 26 \%& 57 \%& 17 \%\\
2200 (w) vs 2400 (b) & 14 \%& 55 \%&31 \%\\
2400 (w) vs 2200 (b) & 63 \%& 11 \%& 26 \%
\end{tabular}
\caption{Example match outcome probabilities drawn from Milvang's probability distribution \cite{milvang2016prob}.
}
\label{tab:example_probabilities}
\end{table}
\subsubsection{Measuring Ranking Quality} Ranking quality measures how similar the tournament's final ranking is to the ranking that sorts the players by their strength. One popular measure for the difference between two rankings is the Kendall $\tau$ distance \cite{kendall1945treatment}. It counts the number of discordant pairs: pairs of elements $x$ and $y$, where $x < y$ in one ranking, but $y < x$ in the other. We use its normalized variant, where $\tau \in [-1, 1]$, and $\tau = 1$ means the rankings are identical, while $\tau = -1$ means one ranking is the inverse of the other. A higher Kendall $\tau$ is better, because it indicates a larger degree of similarity between the true and the output ranking.
We also justify our claims on ranking quality using two other well-known and possibly more sophisticated similarity measures, the Spearman $\rho$ distance \cite{spearman1904proof} and normalized discounted cumulative gain (NDCG). We elaborate on these measures and their behavior for our problem in the appendix. The results are in line with the ones derived for the Kendall $\tau$ distance.
\subsubsection{Measuring Fairness
We measure fairness in terms of the two relaxable criteria of Swiss-system chess tournaments: (Q1) on the equal score of opponents and (Q2) on the color distribution balance. Adhering to (Q1) is measured by the number of float pairs, which equals the number of matches with opponents from different score groups throughout the tournament. We measure the absolute color difference of a round as the sum of color differences for all players: $acd = \sum_{p_i \in P}{|cd(p_i)|}$. Note that as $|cd(p_i)| \geq 1$ for all players after each odd round, $acd \geq n$ in those rounds.
\subsubsection{Presentation of the Data}
Data is presented in the form of \textit{violin plots} \cite{hintze1998violin}, \textit{letter value plots} \cite{HWK17}, \new{and \emph{scatter plots} \cite{FD05}}. For violin plots, kernel density estimation is used to show a smoothed probability density function of the underlying distribution. Additionally, similar to box plots, quartiles are shown by dashed lines. Letter value plots are enhanced box plots that show more quantiles. Unlike violin plots, they are suitable for discrete values, as all shown values are actual observations without smoothing.
Our plots compare the MWM implementation of the five pairing systems with the BBP implementation of Dutch.
\section{Simulation Results}
All simulations use the following parameters, unless noted otherwise:
\begin{itemize}
\item number of players $n$: 32
\item number of rounds: 7
\item strength range: between 1400 and 2200
\item maximum allowed color difference $\beta$: 2
\item sample size: 100\,000
\end{itemize}
These values were chosen to be as realistic as possible, based on parameters of more than 320\,000 real-world tournaments uploaded to the website \url{chess-results.com}.\footnote{The data was kindly provided by Heinz Herzog, author of the FIDE-endorsed tournament manager \url{Swiss-Manager} \cite{herzog2020swiss} and \url{chess-results.com} \cite{herzog2020chess}.} The experiments were run on a compute server using version 20.04.1 of the Ubuntu operating system. It is powered by 48 Intel Xeon Gold 5118 CPUs running at 2.3 GHz and 62.4 GiB of RAM. We emphasize that \new{the standard real-life challenge at a tournament, that is, }
computing a single pairing via a maximum weight matching for a tournament round can be \new{solved}
in a fraction of a second on a standard laptop.
\subsection{Ranking Quality}
\label{sec:renking_q}
The pairing system of a \new{Swiss-system} tournament has a major impact on the obtained ranking quality, as Figure~\ref{fig:ranking_quality} shows. Burstein and Random2 achieve the best ranking quality, followed by Dutch and Dutch BBP. Random has a worse ranking quality and Monrad performs by far the worst. For other strength ranges, Figure~\ref{fig:mean_strength} shows consistent results.
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{color_figures/ranking_quality_pairing_systems.pdf}
\caption{Ranking quality measured by normalized Kendall~$\tau$. A higher value means a better ranking quality.}
\label{fig:ranking_quality}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth]{color_figures/mean_strength.pdf}
\caption{Ranking quality measured by normalized Kendall~$\tau$ for different strength ranges.}
\label{fig:mean_strength}
\end{figure}
\noindent Comparing Dutch to Dutch BBP shows that they behave very similarly, with slight advantage for Dutch. This is remarkable, since Dutch BBP is based on complex and rigid declarative criteria that are time-tested, while Dutch is the output of our easy-to-understand, purely \newer{matching-based} approach.
Together with the performance of Burstein and Random2 this shows that more transparent pairing systems can outperform the state-of-the-art Dutch BBP in terms of ranking quality.
We provide additional experimental results on the ranking quality in the appendix. There we present consistent results also for fewer or more players, \new{for other strength range sizes, and for different player strength distributions}. Additionally, we elaborate on how our flexible maximum weight matching model enabled us to detect the exact reason why certain pairing systems produce better rankings, which might help designing better pairing systems in the future.
\subsection{Fairness}
The highly complex pairing criteria of the FIDE were designed with a focus on two fairness goals phrased as quality criteria, (Q1): minimizing the number of float pairs and (Q2): minimizing the absolute color difference.
Criterion~(Q1) is at the heart of Swiss-system tournaments as pairing players of equal score ensures well-balanced matches. \newer{As Figure~\ref{fig:float_pairs} shows}, Burstein, Dutch, and Random2 \newer{beat} Dutch BBP in terms of the number of float pairs. \newer{In the appendix we} show consistent results for other simulation parameters.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{color_figures/float_pairs_pairing_systems.pdf}
\caption{Number of float pairs out of the $7\cdot 16 = 112$ matches of the tournament. Recall that floating is often unavoidable due to the size of the score group. A lower number indicates a better implementation of criterion~(Q1).}
\label{fig:float_pairs}
\end{figure}
\noindent Figure~\ref{fig:color_difference_6} focuses on criterion~(Q2) and shows that an absolute color difference very similar to the one guaranteed by Dutch BBP can be achieved via our MWM engine. The pairing system Random even outperforms Dutch BBP in this regard. \new{In the appendix, we provide additional experiments with different numbers of rounds and numbers of players that lead to consistent results.} Also, we report there on experiments in which an even stronger color difference constraint is enforced, and observe the impact on the obtained ranking quality and the number of float pairs. \newer{Interestingly, the obtained ranking quality is almost the same but this comes at a cost of a slightly increased number of float pairs. }
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{color_figures/6_color_difference_pairing_systems.pdf}
\caption{Absolute color difference after 6 rounds. A lower $acd$ means a better color distribution. Recall that a $acd \geq n$ for each odd round, while $acd = 0$ is possible after each even round.}
\label{fig:color_difference_6}
\end{figure}
\noindent Hence, our maximum weight matching approach with edge weights that prioritize matches within score groups and secondly optimize for color balance is on a par with the sophisticated official FIDE criteria for criterion~(Q2) and it even outperforms them for criterion~(Q1). Thus, our more transparent approach ensures the same color balance quality but achieves even fewer float pairs. \newer{Moreover, our approach also allows for a different trade-off between criteria (Q1) and (Q2) that does not affect the obtained ranking quality.}
\section{Conclusion}
The experimental results of our MWM engine with Burstein or Random2 demonstrate that it is possible to outperform the state-of-the-art FIDE pairing criteria in terms of both ranking quality and fairness, i.e., criteria (Q1) and (Q2), with a novel efficient \newer{mechanism} that is more transparent and intelligible to all participants. The direct comparison of our MWM Dutch engine versus Dutch BBP shows that even if the same pairing system is used,
MWM achieves the same ranking quality but is more powerful since it yields an improved fairness. We believe that the key to this is the direct formulation of the most important criteria as a maximum weight matching problem.
The only scenario for which we might advise against using our \newer{mechanism} is when the arbiter has no access to a computing device. In order to manually produce pairings in our framework, the arbiter would need to calculate the edge weights and then execute Edmonds' blossom algorithm. \newer{Instead,} the FIDE~\cite[Chapter C.04.3.D]{fide2020handbook} provides manually executable rules.
However, these rules include exhaustive search routines that can make the execution very slow, i.e., highly exponential in the number of players \cite{BFP17}. Therefore, the ill-fated arbiter has to choose between learning \new{to execute} Edmonds' blossom algorithm and following a cumbersome exponential-time pairing routine.
A clear advantage of our \newer{mechanism} is that it is easily extendable: as Random and Random2 already demonstrate, a new pairing system can be implemented simply by specifying how edge weights are calculated. Similarly, as we have also demonstrated, the color balance can be adjusted by simply changing the parameter~$\beta$. By thinning out the edge set in our graph, we can also reach an alternating black-white sequence for each player instead of just minimizing the color difference in each round. Also, the flexibility of the maximum weight matching approach proved to be essential for uncovering the driving force behind the achieved high ranking quality: the normalized strength difference. Hence, our approach was not only valuable for computing better pairings but also in the analysis of the obtained ranking quality.
Last but not least, the flexibility of the MWM engine likely allows \newer{to incorporate additional quality criteria like measuring fairness via the average opponent ratings. Also} quality criteria of other games and sports tournaments organized in the Swiss system \newer{can be} integrated into the model.
\pagebreak
|
1,116,691,498,512 | arxiv | \section{Introduction}
In relativistic nucleon--nucleon collisions dielectrons are emitted from various sources. Pseudoscalar and vector-mesons can decay directly or via Dalitz decays into a real or a virtual photon that internally converts into an $\rm e^{+}e^{-}$ pair. In addition, semi-leptonic decays from open heavy-flavour (HF) hadrons can produce a correlated dielectron when following the hadronisation and decay pattern
\begin{equation}
c\overline{c} \rightarrow D\overline{D} \rightarrow XY e^{+}e^{-}.
\label{eq:cc2ee}
\end{equation}
The same holds true for dielectron pairs from open-beauty hadrons. The analysis of these pairs can then shed light on the correlation of the heavy-quark pairs, especially in the low transverse momentum regime, which is not easily accessible in other analyses.
In nucleon--nucleus collisions the before mentioned sources of dielectrons can be modified. Of particular interest are possible modifications of the heavy-flavour production via cold nuclear matter effects, e.g.\ shadowing. Additional sources of dielectrons such as thermal radiation from a hot medium, i.e.\ hadron gas or QGP, possibly formed in collisions of small systems.
\section{Data analysis}
We report on the results from two data taking periods~\cite{ref-ee7,ref-ee13}. In 2010, the ALICE detector recorded $370\times10^{6}$ minimum bias (MB) pp events at $\sqrt{s} = 7$\,TeV. In another pp data taking, in 2016, a total of $455\times10^{6}$ MB were recorded at $\sqrt{s} = 13$\,TeV. In addition, a dedicated high multiplicity (HM) trigger selected 0.036\% of the highest multiplicity pp collisions and collected $79.2\times10^{6}$ HM events.
In ALICE electrons\footnote{Electrons here and in the whole document also refers to their anti-particles, positrons.} are identified in the central barrel using the Inner Tracking System (ITS), the Time Projection Chamber (TPC), and the Time-Of-Flight system (TOF) in a kinematic range of $p_{\rm T,e} > 0.2$\,GeV/$c$ and $\eta_{\rm e} < |0.8|$. The selected electrons are then combined to opposite-sign (OS) pairs. The OS invariant mass distribution contains all correlated signal pairs, but in addition a combinatorial background. The background is estimated by constructing a spectrum of same-sign (SS) pairs.
Residual differences in the acceptance for SS and OS pairs are estimated using event mixing and taken into account during the subtraction of the background. The spectrum is then corrected for tracking and particle identification inefficiencies.
\section{Results}
Dielectrons were measured as a function of invariant mass ($m_{\rm ee}$) and pair transverse momentum ($p_{\rm T,ee}$) in both data taking periods. In addition, a measurement as function of $\rm DCA_{ee}$, the distance of closest approach of the electrons to the primary vertex normalised to its resolution and summed in quadrature, was performed in the 7\,TeV data sample. The $m_{\rm ee}$ spectra integrated over $p_{\rm T,ee}$, and $\rm DCA_{ee}$ are shown in Fig. \ref{fig:mee} in comparison with an expectation of the cross sections from known hadronic sources, the hadronic cocktail.
\begin{figure}[ht!]
\centering
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.33,
trim = 0 100 10 130, clip]{./plots/2018-10-11-2018-09-03-invmassintegrated}
\end{minipage}
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.33,
trim = 0 80 10 120, clip]{./plots/2018-10-11-2018-09-25-2018-09-25-Signal_cocktail_pt0_pythia.pdf}
\end{minipage}
\caption{Dielectron cross section in pp collisions at $\sqrt{s} = 7$\,TeV (left) and $\sqrt{s} = 13$\,TeV (right) as a function of $m_{\rm ee}$ in comparison with a cocktail of known hadronic sources~\cite{ref-ee7,ref-ee13}.}
\label{fig:mee}
\end{figure}
The measured $m_{\rm ee}$ spectra are well described by the hadronic cocktail within statistical and systematic uncertainties.
At intermediate mass both spectra are described by a contribution from charm and beauty calculated with PYTHIA6~\cite{ref-pythia6} with the Perugia2011 tune~\cite{ref-perugia2011} normalised to the cross sections measured in single heavy-flavour hadron measurements~\cite{ref-ccbar,ref-bbbar}.
In addition, it can be seen that at LHC energies the mass spectra are dominated over a wide mass range by the contribution from semi-leptonic decays of correlated open heavy-flavour hadrons.
This can be used to select and further study the production of heavy-flavour quarks in high-energy collisions.
In the mass window of 1.1\,GeV/$c^{2}$ < $m_{\rm ee}$ < 2.7\,GeV/$c^{2}$, the so called intermediate mass region (IMR), the contributions from the HF hadrons can be selected without any significant contribution from other sources.
\begin{figure}[ht!]
\centering
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 100 0 130, clip]{./plots/2018-May-09-heavyflavourptee}
\end{minipage}
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 100 1 130, clip]{./plots/2018-May-09-heavyflavourdca}
\end{minipage}
\caption{Dielectron cross section in pp collisions at $\sqrt{s} = 7$\,TeV as a function of $p_{\rm T, ee}$ (left) and $\rm DCA_{ee}$ (right) in comparison with a cocktail of known hadronic sources~\cite{ref-ee7}.}
\label{fig:pteedca}
\end{figure}
In Fig. \ref{fig:pteedca}, the $p_{\rm T,ee}$ and $\rm DCA_{ee}$ spectra measured at $\sqrt{s} = 7$\,TeV in the IMR are depicted.
For both observables one can see that the contribution from charm and beauty have a different spectral shape. In the $p_{\rm T,ee}$ case the charm contribution dominates up to about 3\,GeV/$c$. Above this, the spectrum is dominated by the contribution from beauty quarks. For the $\rm DCA_{ee}$ observable, the crossing point is around 4$\sigma$. The distinct shapes of the two contributions in $p_{\rm T,ee}$ and $\rm DCA_{ee}$ are used to disentangle them with a two-component fit to either the $m_{\rm ee}$-$p_{\rm T,ee}$ or the $\rm DCA_{ee}$ distributions. The result is presented in Fig. \ref{fig:xsection} using heavy-flavor distributions obtained from PYTHIA, as described before, and POWHEG~\cite{ref-powheg}.
\begin{figure}[ht!]
\centering
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 100 10 160, clip]{./plots/2018-May-09-oneSigmaPythiaDCA0to8}
\end{minipage}
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 100 10 160, clip]{./plots/2018-May-09-oneSigmaPowhegDCA0to8}
\end{minipage}
\caption{Total $\rm c\overline{c}$ and $\rm b\overline{b}$ cross sections with systematic and statistical uncertainties, extracted from fits of the measured dielectron yield from heavy-flavour hadron decays to ($m_{\rm ee}$, $p_{\rm T,ee}$) and to $\rm DCA_{ee}$ with PYTHIA (left) and POWHEG (right) in comparison with published cross sections from independent measurements (lines)~\cite{ref-ee7}.}
\label{fig:xsection}
\end{figure}
The two approaches are consistent within uncertainties, in the case of PYTHIA, as well as POWHEG. However, we can see a significant shift in the charm cross section when using POWHEG instead of PYTHIA.
This shift can be understood since PYTHIA calculates the leading-order contributions and POWHEG in addition also includes the next-to-leading order contribution, which changes the overall correlation. In a measurement of the HF cross section in the dielectron channel we are sensitive to these different contributions. The cross sections extracted with both models are in agreement with independent measurements within uncertainties.
\begin{figure}[ht]
\centering
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 170 10 180, clip]{./plots/2018-May-10-Ratio_ptbin0}
\end{minipage}
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 170 10 180, clip]{./plots/2018-May-10-Ratio_ptbin4}
\end{minipage}
\caption{Ratio of dielectron production in high-multiplicity events over inelastic events integrated over $p_{\rm T,ee}$ (left) and for $p_{\rm T,ee} > 3$\,GeV/$c$ (right)~\cite{ref-ee13}.}
\label{fig:ratio}
\end{figure}
Similar findings can be reported in the 13\,TeV analysis. In this analysis the production of dielectrons was studied as function of $m_{\rm ee}$ and $p_{\rm T,ee}$ for a minimum bias and a high-multiplicity data sample.
In Fig. \ref{fig:ratio}, the ratio of the high-multiplicity dielectron spectrum over the inelastic one is shown as function of $m_{\rm ee}$, integrated over $p_{\rm T,ee}$ and for $p_{\rm T,ee} > 3$\,GeV/$c^2$, left and right, respectively. The ratios are compared with ratios of the expected hadronic contributions. The cocktail ratio reflects modifications measured independently at high multiplicity.
We use a measurement of the multiplicity dependence of D mesons~\cite{ref-DmesonsMult} to scale the heavy-flavour production, including the B mesons. For the light-flavour part of the cocktail a measurement of the multiplicity dependence of the $p_{\rm T}$ spectra is used, which shows a hardening with multiplicity~\cite{ref-ptMult}.
No significant deviation from the cocktail in both $p_{\rm T,ee}$ intervals is observed.
The high $p_{\rm T,ee}$ part of the spectrum, dominated by the heavy-flavour contribution from beauty quarks, can be described by a cocktail constructed from D-meson measurements. This suggests a similar scaling of charm and beauty production with multiplicity at LHC energies.
In p--Pb collisions, we can further study the modification of HF hadron production resulting e.g.\ from the modification of the parton distribution functions. These modifications would be expected for small $Q^{2}$ and $x_{\rm Bj}$ in the production process of the HF quark pair and thus at low $p_{\rm T}$. This makes dielectrons a prime probe for this sort of measurements, since standard selection criteria in the analysis preserve most of the HF cross sections.
\begin{figure}[ht!]
\centering
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 100 10 180, clip]{./plots/2018-09-27-PionImrComp_400}
\end{minipage}
\begin{minipage}{0.47\textwidth}
\includegraphics[scale=0.35,
trim = 0 100 10 180, clip]{./plots/2018-09-27-DCArms_400}
\end{minipage}
\caption{$\rm DCA_{ee}$ spectra in the $\pi$-mass region and the IMR normalised to unity (left) and the $\rm \langle DCA_{ee}\rangle$ as function of $m_{\rm ee}$ in p--Pb collisions at $\sqrt{s_{\rm NN}}=5.02$\,TeV.}
\label{fig:DCAppb}
\end{figure}
In Fig. \ref{fig:DCAppb} (left), we show the $\rm DCA_{ee}$ distributions for the mass region dominated by the $\pi^{0}$ Dalitz decays in comparison with the HF dominated IMR. The spectra are normalised to unity for direct comparison of the shapes. It is apparent that the HF dominated mass region has a much wider distribution. This will give the opportunity to disentangle not only the charm and beauty distributions, but in addition study a possible prompt source, such as thermal radiation. The sensitivity of the $\rm DCA_{ee}$ on the mixture of prompt and non-prompt contributions to the spectrum can be derived from Fig. \ref{fig:DCAppb} (right). We show the $\rm \langle DCA_{ee} \rangle$ as function a of $m_{\rm ee}$. For low masses ($< 0.5$\,GeV/$c^{2}$), we see a rather flat distribution. This is the region dominated by the Dalitz decays of $\pi^{0}$ and $\eta$ mesons, both prompt sources. With increasing masses the charm contribution becomes more significant, and with this the $\rm \langle DCA_{ee} \rangle$ rises. Significant drops in the distribution can be associated with the narrow contributions of the resonance decays of $\rho, \omega, \phi$ mesons. At masses larger than the $\phi$ the spectrum is completely dominated by the charm and beauty contributions and the $\rm \langle DCA_{ee} \rangle$ reaches a maximum. The falling off of $\rm \langle DCA_{ee} \rangle$ could be interpreted as a rising prompt contribution in the radiative tail of the $J/\psi$, at whose mass the spectrum is completely dominated by a prompt source again.
\section{Conclusion}
We presented the measurement of the dielectron production cross section as function of $m_{\rm ee}$, $p_{\rm T,ee}$ and $\rm DCA_{ee}$ in pp collisions at $\sqrt{s} = 7$\,TeV and as function of $m_{\rm ee}$, $p_{\rm T,ee}$ and multiplicity at $\sqrt{s} = 13$\,TeV. The spectra are well described with a hadronic cocktail for all observables. Cross sections for the production of HF quarks were extracted. The extracted cross sections are in agreement with previous measurements of single HF hadrons. A strong model dependence for this measurement points to a sensitivity to the heavy-quark production mechanisms in these models. The comparison of minimum bias and high-multiplicity $m_{\rm ee}$ spectra for $p_{\rm T,ee} > 3$\,GeV/$c$ suggests that the scaling of beauty production follows the previously observed modifications of charm production. We do not observe an indication of an additional source of thermal radiation in high multiplicity pp events within the precision of the data.
In p--Pb collisions at $\sqrt{s_{\rm NN}} = 5.02$\,TeV, the $\rm DCA_{ee}$ distribution shows promising possibilities for further studies of possible modifications of the HF contributions due to cold nuclear matter effects and to disentangle prompt and non-prompt sources. The latter would be another possibility to study possible thermal radiation in small collision systems.
|
1,116,691,498,513 | arxiv |
\section{Introduction}
\label{sec:introduction}
\input{introduction.tex}
\section{Overview of Polarized SILL}
\label{sec:overview-sill}
\input{overview-sill.tex}
\section{Overview of the Semantics}
\label{sec:overview-semantics}
\input{overview-semantics.tex}
\section{Background and Notation}
\label{sec:background}
\input{background.tex}
\section{Semantic Clauses}
\label{sec:semantic-clauses}
\input{semantic-clauses.tex}
\section{Illustrative Example: Flipping Bit Streams}
\label{sec:bit-streams}
\input{bit-streams-short.tex}
\section{Related and Future Work}
\label{sec:related-work}
\input{related-work.tex}
\section*{Acknowledgments}
This work is funded in part by a Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship.
The author thanks anonymous referees, Robert Atkey, Stephen Brookes, and Frank Pfenning for their comments.
\bibliographystyle{plainurl}
\subsection{Rules for Term Formation}
\label{sec:rules-term-formation}
\[
\infer[\rn{I-\{\}}]{
\jtypef{\Psi}{\tProc{a}{P}{\overline{a_i}}}{\Tproc{a:A}{\overline{a_i:A_i}}}
}{
\jtypem{\Psi}{\overline{a_i:A_i}}{P}{a}{A}
}
\]
\[
\infer[\rn{F-Var}]{\jtypef{\Psi, x: \tau}{x}{\tau}}{\mathstrut}
\quad
\infer[\rn{F-Fix}]{
\jtypef{\Psi}{\tFix{x}{M}}{\tau}
}{
\jtypef{\Psi,x:\tau}{M}{\tau}
}
\]
\[
\infer[\rn{F-Fun}]{
\jtypef{\Psi}{\lambda x : \tau.M}{\tau \to \sigma}
}{
\jtypef{\Psi, x:\tau}{M}{\sigma}
}
\quad
\infer[\rn{F-App}]{
\jtypef{\Psi}{MN}{\sigma}
}{
\jtypef{\Psi}{M}{\tau \to \sigma}
&
\jtypef{\Psi}{N}{\tau}
}
\]
\subsection{Rules for Process Formation}
\label{sec:rules-proc-form}
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{Fwd}]{
\jtypem{\Psi}{a:A}{\tFwd{b}{a}}{b}{A}
}{}
\quad
\infer[\rn{Cut}]{
\jtypem{\Psi}{\Delta_1,\Delta_2}{\tCut{a}{P}{Q}}{c}{C}
}{
\jtypem{\Psi}{\Delta_1}{P}{a}{A}
&
\jtypem{\Psi}{a:A,\Delta_2}{Q}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\Tu R$}]{
\jtypem{\Psi}{\cdot}{\tClose a}{a}{\Tu}
}{}
\quad
\infer[\rn{$\Tu L$}]{
\jtypem{\Psi}{\Delta, a : \Tu}{\tWait{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta}{P}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\Tds{} R$}]{
\jtypem{\Psi}{\Delta}{\tSendS{a}{P}}{a}{\Tds A}
}{
\jtypem{\Psi}{\Delta}{P}{a}{A}
}
\quad
\infer[\rn{$\Tds{} L$}]{
\jtypem{\Psi}{\Delta,a : \Tds A}{\tRecvS{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\Tus R$}]{
\jtypem{\Psi}{\Delta}{\tRecvS{a}{P}}{a}{\Tus{A}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{}
}
\quad
\infer[\rn{$\Tus L$}]{
\jtypem{\Psi}{\Delta,a : \Tus A}{\tSendS{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\Tplus R_k$}]{
\jtypem{\Psi}{\Delta}{\tSendL{a}{k}{P}}{a}{{\Tplus\{l:A_l\}}_{l \in L}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{A_k}\quad(k \in L)
}
\quad
\infer[\rn{$\Tplus L$}]{
\jtypem{\Psi}{\Delta,a:{\Tplus\{l : A_l\}}_{l \in L}}{\tCase{a}{\left\{l_l \Rightarrow P_l\right\}_{i\in I}}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a:A_l}{P_l}{c}{C}\quad(\forall l \in L)
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\Tamp R$}]{
\jtypem{\Psi}{\Delta}{\tCase{a}{\left\{l \Rightarrow P_l\right\}_{l \in L}}}{a}{{\Tamp\{l :A_l \}}_{l \in L}}
}{
\jtypem{\Psi}{\Delta}{P_l}{a}{A_l}\quad(\forall l \in L)
}
\quad
\infer[\rn{$\Tamp L_k$}]{
\jtypem{\Psi}{\Delta,a:{\Tamp\{l : A_l\}}_{l \in L}}{\tSendL{a}{k}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a:A_k}{P}{c}{C}
&
(k \in L)
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\Tot R^*$}]{
\jtypem{\Psi}{\Delta, b : B}{\tSendC{a}{b}{P}}{a}{B \Tot A}
}{
\jtypem{\Psi}{\Delta}{P}{a}{A}
}
\quad
\infer[\rn{$\Tot L$}]{
\jtypem{\Psi}{\Delta, a : B \Tot A}{\tRecvC{b}{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta, a : A, b : B}{P}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{${\Tlolly}R$}]{
\jtypem{\Psi}{\Delta}{\tRecvC{b}{a}{P}}{a}{B \Tlolly A}
}{
\jtypem{\Psi}{\Delta, b : B}{P}{a}{A}
}
\quad
\infer[\rn{${\Tlolly}L$}]{
\jtypem{\Psi}{\Delta, b : B, a : B \Tlolly A}{\tSendC{a}{b}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\Tand{}{} R$}]{
\jtypem{\Psi}{\Delta}{\tSendV{a}{M}{P}}{a}{\Tand{\tau}{A}}
}{
\jtypef{\Psi}{M}{\tau}
&
\jtypem{\Psi}{\Delta}{P}{a}{A}
}
\quad
\infer[\rn{$\Tand{}{} L$}]{
\jtypem{\Psi}{\Delta, a:\Tand{\tau}{A}}{\tRecvV{x}{a}{P}}{c}{C}
}{
\jtypem{\Psi,x:\tau}{\Delta, a:A}{P}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{${\Timp{}{}} R$}]{
\jtypem{\Psi}{\Delta}{\tRecvV{x}{a}{P}}{a}{\Timp{\tau}{A}}
}{
\jtypem{\Psi,x:\tau}{\Delta}{P}{a}{A}
}
\quad
\infer[\rn{${\Timp{}{}} L$}]{
\jtypem{\Psi}{\Delta,a : \Timp{\tau}{A}}{\tSendV{a}{M}{P}}{c}{C}
}{
\jtypef{\Psi}{M}{\tau}
&
\jtypem{\Psi}{\Delta, a : A}{P}{c}{C}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\rho^+R$}]{
\jtypem{\Psi}{\Delta}{\tSendU{a}{P}}{a}{\Trec{\alpha}{A}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}
&
\cdot \vdash \jisst[+]{\Trec{\alpha}{A}}
}
\,
\infer[\rn{$\rho^+L$}]{
\jtypem{\Psi}{\Delta, a : \Trec{\alpha}{A}}{\tRecvU{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta, a : [\Trec{\alpha}{A}/\alpha]A}{P}{c}{C}
&
\cdot \vdash \jisst[+]{\Trec{\alpha}{A}}
}
\end{adjustbox}
\]
\[
\begin{adjustbox}{max width=\textwidth}
\infer[\rn{$\rho^-R$}]{
\jtypem{\Psi}{\Delta}{\tRecvU{a}{P}}{a}{\Trec{\alpha}{A}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}
&
\cdot \vdash \jisst[-]{\Trec{\alpha}{A}}
}
\,
\infer[\rn{$\rho^-L$}]{
\jtypem{\Psi}{\Delta, a : \Trec{\alpha}{A}}{\tSendU{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta, a : [\Trec{\alpha}{A}/\alpha]A}{P}{c}{C}
&
\cdot \vdash \jisst[-]{\Trec{\alpha}{A}}
}
\end{adjustbox}
\]
\[
\infer[\rn{E-\{\}}]{
\jtypem{\Psi}{\overline{a_i:A_i}}{\tProc{a}{M}{\overline a_i}}{a}{A}
}{
\jtypef{\Psi}{M}{\Tproc{a:A}{\overline{a_i:A_i}}}
}
\]
\subsection{Rules for Type Formation}
\label{sec:rules-type-formation}
\[
\infer[\rn{C$\Tu$}]{\Xi\vdash\jisst[+]{\Tu}}{}
\quad
\infer[\rn{CVar}]{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}{}
\quad
\infer[\rn{C$\rho$}]{
\Xi \vdash \jisst[p]{\Trec{\alpha}{A}}
}{
\Xi, \jisst[p]{\alpha} \vdash \jisst[p]{A}
}
\]
\[
\infer[\rn{C$\Tds{}$}]{
\Xi\vdash\jisst[+]{\Tds A}
}{
\Xi\vdash\jisst[-]{A}
}
\quad
\infer[\rn{C$\Tus{}$}]{
\Xi\vdash\jisst[-]{\Tus A}
}{
\Xi\vdash\jisst[+]{A}
}
\]
\[
\infer[\rn{C$\Tplus$}]{
\Xi\vdash\jisst[+]{{\Tplus\{l : A_l\}}_{l \in L}}
}{
\Xi\vdash\jisst[+]{A_l}\quad(\forall l \in L)
}
\quad
\infer[\rn{C$\Tamp$}]{
\Xi\vdash\jisst[-]{{\Tamp\{l :A_l \}}_{l \in L}}
}{
\Xi\vdash\jisst[-]{A_l}\quad(\forall l \in L)
}
\]
\[
\infer[\rn{C$\Tot$}]{
\Xi\vdash\jisst[+]{A \Tot B}
}{
\Xi\vdash\jisst[+]{A}
&
\Xi\vdash\jisst[+]{B}
}
\quad
\infer[\rn{C$\Tlolly$}]{
\Xi\vdash\jisst[-]{A \Tlolly B}
}{
\Xi\vdash\jisst[+]{A}
&
\Xi\vdash\jisst[-]{B}
}
\]
\[
\infer[\rn{C$\Tand{}{}$}]{
\Xi\vdash\jisst[+]{\Tand{\tau}{A}}
}{
\jisft{\tau}
&
\Xi\vdash\jisst[+]{A}
}
\quad
\infer[\rn{C$\Timp{}{}$}]{
\Xi\vdash\jisst[-]{\Timp{\tau}{A}}
}{
\jisft{\tau}
&
\Xi\vdash\jisst[-]{A}
}
\]
\[
\infer[\rn{T\{\}}]{
\jisft{\Tproc{a_0:A_0}{a_1:A_1,\dotsc,a_n:A_n}}
}{
\cdot\vdash\jisst{A_i} \quad(0\leq i \leq n)
}
\quad
\infer[\rn{T$\to$}]{
\jisft{\tau \to \sigma}
}{
\jisft{\tau}
&
\jisft{\sigma}
}
\]
\subsection{Functional Programming and Value Transmission}
\label{sec:funct-progr-value}
The functional layer is the simply-typed $\lambda$-calculus with a call-by-value semantics and a fixed-point operator.
Arrow types are formed by the rule \rn{T$\to$}.
They are interpreted as strict function spaces in $\moabcs$ to enforce a call-by-value semantics:
\begin{equation}
\label[intn]{eq:73}
\sembr{\jisft{\tau \to \sigma}} = \moabcs[\sembr{\jisft\tau} \to \sembr{\jisft{\sigma}}].
\end{equation}
The typing rules \rn{F-Var}, \rn{F-Fun}, \rn{F-App}, and \rn{F-Fix} for the functional layer are standard.
The call-by-value semantics is as in \cite{stoy_1977:_denot_seman}.
We let $u$ range over $\sembr{\Psi}$.
The environment $\upd{u}{x \mapsto v} \in \sembr{\Psi, x : \tau}$ maps $x$ to $v$ and $y$ to $u(y)$ for all $y \in \Psi$.
The fixed-point operator \rn{F-Fix} is interpreted using the fixed-point operator defined in \cref{sec:background}.
\begin{align}
\sembr{\jtypef{\Psi,x:\tau}{x}{\tau}}u &= \pi^{\Psi,x}_{x}u\label[intn]{eq:39}\\
\sembr{\jtypef{\Psi}{\lambda x: \tau.M}{\tau \to \sigma}}u &= \strictfn\left(\lambda v \in \sembr{\tau}.\sembr{\jtypef{\Psi,x:\tau}{M}{\sigma}}\upd{u}{x \mapsto v}\right)\label[intn]{eq:41}\\
\sembr{\jtypef{\Psi}{MN}{\sigma}}u &= \sembr{\jtypef{\Psi}{M}{\tau \to \sigma}}u(\sembr{\jtypef{\Psi}{N}{\tau}}u)\label[intn]{eq:42}\\
\sembr{\jtypef{\Psi}{\tFix{x}{M}}{\tau}}u &= \sfix{\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}}}u \label[intn]{eq:28}
\end{align}
\begin{proposition}[restate=recsubst,name={}]
\label{cor:122}
For all $\jtypef{\Psi,x : \tau}{M}{\tau}$, we have $[\tFix{x}{M}/x]M \equiv \tFix{x}{M}$.
\end{proposition}
For simplicity, the only base types are those of quoted processes.
They are formed by \rn{T\{\}}, and its interpretation is:
\begin{equation}
\label[intn]{eq:72}
\sembr{\jisft{\Tproc{a:A}{\overline{a_i:A_i}}}} = \sembr{\overline{a_i:A_i} \vdash a : A}_\bot
\end{equation}
where we abbreviate $\moabc\left[\sembr{\Delta}^+ \times \sembr{a:A}^- \to \sembr{\Delta}^- \times \sembr{a:A}^+\right]$ as $\sembr{\Delta \vdash a : A}$.
The distinction between the two ``bottom'' elements in $\sembr{\overline{a_i:A_i} \vdash a : A}_\bot$ is semantically meaningful.
The genuine bottom element denotes the absence of a value of type $\Tproc{a:A}{\overline{a_i:A_i}}$.
The lifted bottom element $\upim{\lambda x.\bot}$ corresponds to a stuck process that produces no output.
The \rn{I-\{\}} introduction rule quotes processes.
Its denotation is:
\begin{equation}
\label[intn]{eq:191919}
\sembr{\jtypef{\Psi}{\tProc{a}{P}{\overline{a_i}}}{\Tproc{a:A}{\overline{a_i:A_i}}}} = \up \circ \sembr{\jtypem{\Psi}{\overline{a_i:A_i}}{P}{a}{A}}.
\end{equation}
Because the unit $\up$ is not strict, quoting respects the semantic distinction between quoted stuck processes $\upim{\lambda x.\bot}$ and the absence $\bot$ of a value of type $\Tproc{a:A}{\overline{a_i:A_i}}$.
The \rn{E-\{\}} elimination rule spawns quoted processes.
Its denotation is:
\begin{equation}
\label[intn]{eq:115}
\sembr{\jtypem{\Psi}{\overline{a_i:A_i}}{\tProc{a}{M}{\overline a_i}}{a}{A}} = \down \circ \sembr{\jtypef{\Psi}{M}{\Tproc{a:A}{\overline{a_i:A_i}}}}.
\end{equation}
Two cases are possible when unquoting $M$.
If $\sembr{\jtypef{\Psi}{M}{\Tproc{c:A}{\overline{a_i:A_i}}}}u$ is $\bot$, then $\down(\bot)$ is the constant function $\lambda x.\bot$.
This represents the process that never produces output.
If $\sembr{\jtypef{\Psi}{M}{\Tproc{c:A}{\overline{a_i:A_i}}}}u = \upim{p}$, then $\down(\upim{p}) = p$ is the denotation of some quoted process.
\Cref{eq:191919,eq:115} satisfy the following $\eta$-like property:
\begin{proposition}[restate=quoteunquote,name={}]
\label{prop:fscd:1}
For all $\jtypem{\Psi}{\overline{a_i:A_i}}{P}{a}{A}$, we have $P \equiv \tProc{a}{\tProc{a}{P}{\overline a_i}}{\overline a_i}$.
\end{proposition}
A communication of type $\Tand{\tau}{A}$ is a value $v \in \sembr{\tau}$ followed by a communication satisfying $A$.
We use lifting to account for the potential lack of a communication: the bottom element represents the absence of communication.
The value travels in the positive direction, so it only appears in the positive aspect.
\begin{align}
\sembr{\Xi\vdash\jisst{\Tand{\tau}{A}}}^- &= \sembr{\Xi\vdash \jisst{A}}^-\label[intn]{eq:15104}\\
\sembr{\Xi\vdash\jisst{\Tand{\tau}{A}}}^+ &= \left(\sembr{\tau}\times\sembr{\Xi\vdash \jisst{A}}^+\right)_\bot\label[intn]{eq:15106}
\end{align}
The process $\tSendV{a}{M}{P}$ sends a functional value on $a$ and continues as $P$.
To send the term $M$ on $a$, we evaluate it under the current environment $u$ to get an element $\sembr{\jtypef{\Psi}{M}{\tau}}u \in \sembr{\tau}$.
Divergence is represented by $\bot_{\sembr{\tau}}$; the other elements represent values of type $\tau$.
If $\sembr{\jtypef{\Psi}{M}{\tau}}u$ represents a value, then we pair it with the output of the continuation process $P$ on $a^+$.
Otherwise, the process transmits nothing.
This gives the clause:
\begin{equation}
\label[intn]{eq:1005}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendV{a}{M}{P}}{a}{\Tand{\tau}{A}}}u(\delta^+,a^-)\\
&= \begin{cases}
\bot & \text{if }\sembr{\jtypef{\Psi}{M}{\tau}}u = \bot\\
\left(\delta^-,\upim{\left(v,a^+\right)}\right) & \text{if }\sembr{\jtypef{\Psi}{M}{\tau}}u = v \neq \bot
\end{cases}\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta}{P}{a}{A}}u(\delta^+,a^-) = (\delta^-, a^+).
\end{aligned}
\end{equation}
The process $\tRecvV{x}{a}{P}$ blocks until it receives a communication on the channel $a$.
If a communication $\upim{(v,\alpha^+)}$ arrives on $a^+$, then the process binds $v$ to $x$ in the environment and continues as $P$ with the remaining communication $\alpha^+$ on $a^+$.
If it receives no message, then it should produce no output, \ie, it should produce $\bot$ on all channels.
This means that its denotation should be strict in the $a^+$ component.
\begin{equation}
\label[intn]{eq:1006}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, a:\Tand{\tau}{A}}{\tRecvV{x}{a}{P}}{c}{C}}u\\
&= \strictfn_{a^+}\left( \lambda \left(\delta^+,a^+ : \upim{\left(v, \alpha^+\right)},c^-\right) . \right.\\
&\qquad\quad\qquad\qquad \left. \sembr{\jtypem{\Psi,x:\tau}{\Delta, a:A}{P}{c}{C}}\upd{u}{x \mapsto v}(\delta^+, \alpha^+, c^-) \right)
\end{aligned}
\end{equation}
We abuse notation to pattern match on the component $a^+$.
By strictness, we know that it will be an element of the form $\upim{(v,\alpha^+)}$.
\Cref{eq:1006} illustrates a general principle in our semantics.
The bottom element $\bot$ represents the absence of communication.
When a process waits for input on a channel $a$ it uses (it provides), its denotation is strict in $a^+$ (resp.,~$a^-$).
The $\eta$-property for value transmission is subtle because the functional term $M$ might diverge.
The equivalence $\tCut{a}{P}{[M/x]Q} \; \equiv \; \tCut{a}{{(\tSendV{a}{M}{P})}}{{(\tRecvV{x}{a}{Q})}}$ does not hold in general.
If $x$ does not appear free in $Q$, the substitution on the left has no effect and $\tCut{a}{P}{Q}$ runs as usual.
However, if $M$ diverges, then the process on the right is stuck.
Indeed, $\tRecvV{x}{a}{Q}$ waits on $a$ but $\tSendV{a}{M}{P}$ gets stuck evaluating $M$ and sends nothing.
The two processes in the equivalence have equal denotations whenever $M$ converges.
Process equivalence requires the processes to have equal denotations under all environments $u$.
For the equivalence to hold, $M$ must then converge under every environment $u \in \sembr{\Psi}$.
This justifies the statement of \cref{prop:29}.
Its proof uses the substitution property (\cref{prop:fscd:19}) and the Knaster-Tarski-style formulation of the trace.
\begin{proposition}[restate=etaquote,name={}]%
\label{prop:29}
For all $\jtypem{\Psi}{\Delta_1}{P}{a}{A}$, all $\jtypem{\Psi,x : \tau}{a : A, \Delta_2}{Q}{c}{C}$, and all $\jtypef{\Psi}{M}{\tau}$, we have $\tCut{a}{P}{[M/x]Q} \equiv \tCut{a}{(\tSendV{a}{M}{P})}{(\tRecvV{x}{a}{Q})}$ whenever $\sembr{\jtypef{\Psi}{M}{\tau}}u \neq \bot$ for all $u \in \sembr{\Psi}$.
\end{proposition}
\subsection{Manipulating channels}
\label{sec:manip-chann}
Forwarding denotes the function that copies data from $a^+$ to $b^+$ and from~$b^-$~to~$a^-$:
\begin{equation}
\label[intn]{eq:10}
\begin{aligned}
&\sembr{\jtypem{\Psi}{a:A}{\tFwd{b}{a}}{b}{A}}u(a^+ : \alpha, b^- : \beta) = (a^- : \beta, b^+ : \alpha).
\end{aligned}
\end{equation}
Process composition ``connects'' the common channel $a$ of two communicating processes.
As motivated in \cref{sec:overview-semantics}, we use the trace operator to fix $a$'s positive and negative aspects:
\begin{equation}
\label[intn]{eq:11}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta_1, \Delta_2}{\tCut{a}{P}{Q}}{c}{C}}u\\
&= \Tr{\left(\sembr{\jtypem{\Psi}{\Delta_1}{P}{a}{A}}u \times \sembr{\jtypem{\Psi}{a : A,\Delta_2}{Q}{c}{C}}u\right)}{a^- \times a^+}.
\end{aligned}
\end{equation}
Associativity of composition follows from the trace operator axioms:
\begin{proposition}[restate=assoccut,name={}]
\label{prop:1}
For all ${\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}}$, all ${\jtypem{\Psi}{c_1:C_1,\Delta_2}{P_2}{c_2}{C_2}}$, and all ${\jtypem{\Psi}{c_2:C_2,\Delta_3}{P_3}{c_3}{C_3}}$, we have $\tCut{c_1}{P_1}{(\tCut{c_2}{P_2}{P_3})} \equiv {\tCut{c_2}{(\tCut{c_1}{P_1}{P_2})}{P_3}}$.
\end{proposition}
Processes can close channels of type $\Tu$.
The close message is the only communication possible on a channel of type $\Tu$.
The positive aspect of \rn{C$\Tu$} is the constant functor onto the two-element pointed domain $\{\ast\}_\bot = \{ \bot \sqsubseteq \ast \}$.
The element $\ast$ represents the close message, while $\bot$ represents the absence of communication.
All communication on a channel of type $\Tu$ is positive, so the negative aspect is the constant functor onto the terminal object $\{\bot\}$.
\begin{align}
\sembr{\Xi \vdash \jisst{\Tu}}^- &= \lambda \xi.\{\bot\}\label[intn]{eq:1502}\\
\sembr{\Xi \vdash \jisst{\Tu}}^+ &= \lambda \xi.\{\ast\}_\bot\label[intn]{eq:1501}
\end{align}
In our asynchronous setting, $\tClose a$ does not wait for a client before sending the close message.
We interpret \rn{$\Tu R$} as the constant function that sends the close message $\ast$ on $a^+$:
\begin{equation}
\label[intn]{eq:1509}
\sembr{\jtypem{\Psi}{\cdot}{\tClose a}{a}{\Tu}}u(a^- : \bot) = (a^+ : \ast).
\end{equation}
The process $\tWait{a}{P}$ blocks until it receives the close message, so its denotation is strict in the component $a^+$.
All other communication is handled by $P$.
We interpret \rn{$\Tu L$} by:
\begin{equation}
\label[intn]{eq:21}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a : \Tu}{\tWait{a}{P}}{c}{C}}u = \strictfn_{a^+}\left(\lambda (\delta^+,a^+,c^-).(\delta^-,\bot,c^+)\right))\\
&\text{where } \sembr{\jtypem{\Psi}{\Delta}{P}{c}{C}}u(\delta^+,c^-) = (\delta^-, c^+).
\end{aligned}
\end{equation}
\begin{proposition}[restate=etaunit,name={}]
\label{prop:9}
For all $\jtypem{\Psi}{\Delta}{P}{c}{C}$, we have $P \equiv \tCut{a}{\tClose a}{(\tWait{a}{P})}$.
\end{proposition}
Processes can communicate channels.
We cannot directly observe a channel, only the communications it carries.
For this reason, we treat communications of type $A \Tot B$ as a pair of communications: one for the sent channel and one for the continuation channel.
This is analogous to the denotation of $A \Tot B$ given by Atkey~\citeN{atkey_2017:_obser_commun_seman_class_proces}.
We account for the potential absence of communication by lifting.
\begin{align}
\sembr{\Xi\vdash\jisst{A \Tot B}}^- &= \sembr{\Xi\vdash\jisst{A}}^- \times \sembr{\Xi\vdash\jisst{B}}^-\label[intn]{eq:13}\\
\sembr{\Xi\vdash\jisst{A \Tot B}}^+ &= \left(\sembr{\Xi\vdash\jisst{A}}^+ \times \sembr{\Xi\vdash\jisst{B}}^+\right)_\bot\label[intn]{eq:14}
\end{align}
To send the channel $b$ over $a$, the process $\tSendC{a}{b}{P}$ relays the positive communication from $b^+$ to the $\sembr{B}^+$-component of $\sembr{a : B \Tot A}^+$, and the negative communication on the $\sembr{B}^-$-component of $\sembr{a : B \Tot A}^-$ to $b^-$.
The continuation $P$ handles all other communication.
\begin{equation}
\label[intn]{eq:17}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, b : B}{\tSendC{a}{b}{P}}{a}{B \Tot A}}u(\delta^+,b^+,(a^-_B,a^-_A)) = \left(\delta^-,a_B^-,\upim{\left(b^+,a_A^+\right)}\right)\\
&\text{ where }\sembr{\jtypem{\Psi}{\Delta}{P}{a}{A}}u(\delta^+,a_A^-) = (\delta^-,a_A^+).
\end{aligned}
\end{equation}
The client $\tRecvC{b}{a}{Q}$ blocks until it receives a channel on $a$.
When it receives a communication $\upim{(a_B^+, a_A^+)}$ on $\sembr{a : B \Tot A}^+$, it unpacks it into the two positive channels $\sembr{a : A, b : B}^+$ expected by $Q$.
It then repacks the negative communication $Q$ produces on $\sembr{a : A, b : B}^-$ and relays it over~$\sembr{a : B \Tot A}^-$.
\begin{equation}
\label[intn]{eq:26}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, a : B \Tot A}{\tRecvC{b}{a}{Q}}{c}{C}}u(\delta^+,a^+,c^-)\\
&= \strictfn_{a^+}\left(\lambda (\delta^+,a^+ : \upim{(a_B^+, a_A^+)},c^-) . (\delta^-,(b^-,a^-),c^+) \right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta, a : A, b : B}{Q}{c}{C}}u(\delta^+,a_A^+,a_B^+,c^-) = (\delta^-,a^-,b^-,c^+).
\end{aligned}
\end{equation}
\begin{proposition}[restate=etatensor,name={}]
\label{prop:522}
For all $\jtypem{\Psi}{\Delta_1}{P}{a}{A}$ and $\jtypem{\Psi}{a : A, \Delta_2, b : B}{Q}{c}{C}$, we have $\tCut{a}{P}{Q} \equiv \tCut{b}{\left(\tSendC{a}{b}{P}\right)}{\left(\tRecvC{b}{a}{Q}\right)}$.
\end{proposition}
\begin{example}
\label{ex:fscd:11}
The process below blocks until it receives a channel $a$ of type $\Tu$ over the channel $b$, at which point the type of $b$ becomes $\Tu$.
Then, the process waits for the close messages on $a$ and $b$ before closing $c$.
The element $\upim{(\ast,\ast)} \in \sembr{\Tu \Tot \Tu}^+ = (\sembr{\Tu}^+ \times \sembr{\Tu}^+)_\bot$ corresponds to receiving the channel $a$, the close message on $a$, and the close message on $b$.
The element $\upim{(\bot,\bot)}$ corresponds to receiving $a$ but no close messages, while the elements $\upim{(\ast,\bot)}$ and $\upim{(\bot,\ast)}$ correspond to receiving $a$ and one close message.
The element $\bot$ means that $a$ is never received.
We see that the process only closes $c$ in the first case:
\begin{align*}
&\sembr{\jtypem{\cdot}{b : \Tu \Tot \Tu}{\tRecvC{a}{b}{\tWait{a}{\tWait{b}{\tClose{c}}}}}{c}{\Tu}}\bot(b^+ : \beta, c^- : \bot)\\
&=
\begin{cases}
(b^- : (\bot,\bot), c^+ : \ast) & \text{if }\beta = \upim{(\ast, \ast)}\\
(b^- : (\bot,\bot), c^+ : \bot) & \text{otherwise.}
\end{cases}
\end{align*}
\end{example}
\subsection{Shifts in Polarity}
\label{sec:shifts-polarity}
Synchronization is encoded using ``polarity shifts''.
A communication of type $\Tds A$ is a synchronization message (the ``shift'' message) followed by a communication of type $A$.
``Downshifting'' $A$ to $\Tds A$ introduces only \textit{positive} communication (the ``shift'' message), so the negative aspect of $\Tds A$ is the same as the negative aspect of $A$.
We interpret $\sembr{\Tds A}^+$ by lifting $\sembr{A}^+$.
The element $\bot_{\sembr{\Tds A}^+}$ captures the absence of communication.
The elements $\upim{a}$ for $a \in \sembr{A}^+$ capture a $\ms{shift}$ message followed by a communication satisfying $A$.
\begin{align}
\sembr{\Xi\vdash\jisst{\Tds A}}^- &= \sembr{\Xi\vdash\jisst{A}}^-\label[intn]{eq:1506}\\
\sembr{\Xi\vdash\jisst{\Tds A}}^+ &= \sembr{\Xi\vdash\jisst{A}}^+_\bot\label[intn]{eq:1507}
\end{align}
In our asynchronous setting, the shift message is always sent.
This corresponds to lifting the output of $P$ on the $a^+$ component.
We interpret \rn{$\Tds{} R$} as:
\begin{equation}
\label[intn]{eq:59}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendS{a}{P}}{a}{\Tds A}}u\\
&= \left(\ms{id} \times \left(a^+ : \up\right)\right) \circ \sembr{\jtypem{\Psi}{\Delta}{P}{a}{A}}u.
\end{aligned}
\end{equation}
The client blocks until it receives the shift message on $a^+$.
We lower $\sembr{A}^+_\bot$ to $\sembr{A}^+$ to extract the positive communication expected by $P$.
\begin{equation}
\label[intn]{eq:56}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a : \Tds A}{\tRecvS{a}{P}}{c}{C}}u\\
&= \strictfn_{a^+}\left(\sembr{\jtypem{\Psi}{\Delta, a : A}{P}{c}{C}}u \circ \left(\ms{id} \times \left(a^+ : \down\right)\right)\right)
\end{aligned}
\end{equation}
\begin{proposition}[restate=etadshift,name={}]
\label{prop:7}
For all $\jtypem{\Psi}{\Delta_1}{P}{a}{A}$ and $\jtypem{\Psi}{a : A, \Delta_2}{Q}{c}{C}$, we have $\tCut{a}{P}{Q} \equiv \tCut{a}{(\tSendS{a}{P})}{(\tRecvS{a}{Q})}$.
\end{proposition}
\begin{example}
\label{ex:fscd:3}
Upshifts are the polar duals of downshifts.
The following process waits for its client to synchronize with it before closing the channel.
The protocol $\Tus\Tu$ has denotations $\sembr{\Tus\Tu}^- = \sembr{\Tu}^-_{\bot} = \{\bot\}_\bot$ and $\sembr{\Tus\Tu}^+ = \sembr{\Tu}^+ = \{\ast\}_\bot$.
The element $\upim{\bot} \in \sembr{\Tus\Tu}^-$ captures the synchronizing shift message.
The process closes $a$ if and only if it receives the shift message:
\[
\sembr{\jtypem{\cdot}{\cdot}{\tRecvS{a}{\tClose{a}}}{a}{\Tus{\Tu}}}\bot(a^- : \alpha) =
\begin{cases}
(a^+ : \bot) & \text{if }\alpha = \bot\\
(a^+ : \ast) & \text{if }\alpha = \upim{\bot}.
\end{cases}
\]
\end{example}
\subsection{Making Choices}
\label{sec:making-choices}
The internal choice type $\Tplus \{ l : A_l \}_{l \in L}$ prescribes a choice between session types $\{ A_l \}_{l \in L}$ ($L$ finite).
A communication of type $\Tplus \{ l : A_l \}_{l \in L}$ is a label $k \in L$ sent in the positive direction followed by a communication satisfying $A_k$.
Denotationally, this corresponds to tagging a communication $a_k \in \sembr{A_k}^+$ with the label $k$.
Tagged communications $(k, a_k)$ are the elements of the disjoint union $\biguplus_{l \in L} \sembr{A_l}^+$.
To account for the potential lack of communication, we lift this disjoint union.
This lifted disjoint union is isomorphic to the coalesced sum $\bigoplus_{l \in L} \sembr{A_l}^+_\bot$.
Coalesced sums are coproducts in $\moabcs$, and we define the interpretation using a coalesced sum to make this structure evident.
Explicitly, its elements are $\bot$ and $(k,\upim{a_k})$ for $k \in L$ and $a_k \in \sembr{A_k}^+$.
The client does not know a priori which branch it will take: it must be ready to send negative information for every possible branch.
This justifies \cref{eq:202020}.
\begin{align}
\sembr{\Xi\vdash\jisst{\Tplus\{l:A_l\}_{l \in L}}}^- &= \prod_{l \in L} \sembr{\Xi\vdash\jisst{A_l}}^-\label[intn]{eq:202020}\\
\sembr{\Xi\vdash\jisst{\Tplus\{l:A_l\}_{l \in L}}}^+ &= \bigoplus_{l \in L} \sembr{\Xi\vdash\jisst{A_l}}^+_\bot\label[intn]{eq:222222}
\end{align}
\begin{samepage}%
To interpret \rn{$\Tplus R_k$}, we extract from $a^-$ the negative information $a_k^-$ required by the continuation process $P$.
Output on $a^+$ is the label $k$ followed by the output of $P$ on $a^+$:
\begin{equation}
\label[intn]{eq:52}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendL{a}{k}{P}}{a}{\Tplus\{l:A_l\}_{l \in L}}}u\left(\delta^+, \left(a_l^-\right)_{l \in L}\right) = \left(\delta^-, \left(k, \upim{a_k^+}\right)\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta}{P}{a}{A_k}}u\left(\delta^+, a_k^-\right) = \left(\delta^-, a_k^+\right).
\end{aligned}
\end{equation}
The client $\tCase{a}{\left\{l \Rightarrow P_l\right\}_{l \in L}}$ blocks until it receives a communication $(k, \upim{a_k^+})$ on $a^+$, and then it takes the branch $P_k$.
\end{samepage}%
To transmit $P_k$'s output $a_k^-$ on $a^-$ back to the provider, we place $a_k^-$ in the $k$ component of the product sent on $a^-$.
The other branches were not taken and produced no communication, so their respective components in $a^-$ are filled with $\bot$.
\begin{equation}
\label[intn]{eq:53}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a:\Tplus\{l:A_l\}_{l \in L}}{\tCase{a}{\left\{l \Rightarrow P_l\right\}_{l \in L}}}{c}{C}}u\\
&= \strictfn_{a^+}\left(\lambda \left(\delta^+, a^+ : \left(k, \upim{a_k^+}\right), c^-\right).\left(\delta^-, a^- : \left(k : a_k^-, l \neq k : \bot\right)_{l \in L}, c^+\right)\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta,a:A_k}{P_k}{c}{C}}u(\delta^+, a_k^+, c^-) = (\delta^-, a_k^-, c^+)
\end{aligned}
\end{equation}
\begin{proposition}[restate=etaplus,name={}]
\label{prop:1253}
Let $k \in L$.
If ${\jtypem{\Psi}{\Delta_1}{P}{a}{A_k}}$, and ${\jtypem{\Psi}{a : A_l, \Delta_2}{Q_l}{c}{C}}$ for all $l \in L$, then
$\tCut{a}{P}{Q_k} \equiv \tCut{a}{\left(\tSendL{a}{k}{P}\right)}{\left(\tCase{a}{\left\{l \Rightarrow Q_l\right\}_{l \in L}}\right)}$.
\end{proposition}
\begin{example}
We build on \cref{ex:fscd:3}.
External choices $\Tamp\{l : A_l\}_{l \in L}$ are the polar duals of internal choices.
Let $A = \Tamp\{\mt{j} : \Tus\Tu, \mt{k} : \Tus\Tu\}$.
A provider of $A$ receives a label and a synchronizing shift before closing the channel.
The elements $(l,\upim{\upim{\bot}}) \in \sembr{A}^-$ correspond to receiving the label $l$ over $a$ followed by a shift, while the elements $(l,\upim{\bot})$ correspond to receiving $l$ but no shift.
The communication on $a^+$ depends on the label $l$ received: the close message is in the $l$ component of the output on $a^+$.
\begin{align*}
&\sembr{\jtypem{\cdot}{\cdot}{\tCase{a}{\left\{l \Rightarrow \tRecvS{a}{\tClose{a}}\right\}_{l \in \{\mt{j},\mt{k}\}}}}{a}{A}}\bot(a^- : \alpha)\\
&=
\begin{cases}
(a^+ : (\mt{j} : \ast, \mt{k} : \bot)) & \text{if } \alpha = (\mt{j},\upim{\upim{\bot}})\\
(a^+ : (\mt{j} : \bot, \mt{k} : \ast)) & \text{if } \alpha = (\mt{k},\upim{\upim{\bot}})\\
(a^+ : (\mt{j} : \bot, \mt{k} : \bot)) & \text{if } \alpha = (l,\upim{\bot}) \text{ for }l \in \{\mt{j},\mt{k}\}\text{ or if } \alpha = \bot
\end{cases}
\end{align*}
\end{example}
\subsection{Recursive Types}
\label{sec:recursive-types}
The substitution property (\cref{prop:11}) forces the denotation of the variable rule \rn{CVar}:
\begin{align}
\sembr{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}^q &= \pi^{\Xi,\alpha}_\alpha \quad (q \in \{{-},{+}\})\label[intn]{eq:fscd:7}
\end{align}
We interpret recursive types by parametrized solutions of recursive domain equations.
Every locally continuous functor $G : \moabcs \to \moabcs$ has a canonical fixed point $\FIX(G)$ in $\moabcs$.
Given a locally continuous functor $F : \sembr{\Xi} \times \moabcs \to \moabcs$, the mapping $D \mapsto \FIX(F(D, {-}))$ extends to a locally continuous functor $\sfix{F} : \sembr{\Xi} \to \moabcs$ \cite[Proposition~5.2.7]{abramsky_jung_1995:_domain_theor}.
The fixed-point property $F(D, \sfix{F} D) \cong \sfix{F} D$ is witnessed by a canonical natural isomorphism $\ms{Fold} : F \circ \langle \ms{id}_{\sembr{\Xi}}, \sfix F \rangle \nto \sfix{F}$ with inverse $\ms{Unfold}$.
The rule \rn{C$\rho$} denotes:
\begin{align}
\sembr{\Xi\vdash\jisst{\Trec{\alpha}{A}}}^p &= \sfix{\left(\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p\right)}\quad (p \in \{{-},{+}\})\label[intn]{eq:20051}
\end{align}
Let $\ms{Fold}^p \,{:}\, \sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p \circ \langle \ms{id}_{\sembr{\Xi}}, \!\sfix{\left(\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p\right)} \rangle \nto \sembr{\jstype{\Xi}{\Trec{\alpha}{A}}}^p$ be the canonical natural isomorphism and let $\ms{Unfold}^p$ be its inverse.
These isomorphisms capture the semantic folding and unfolding of recursive types.
Indeed, by \cref{prop:11,eq:fscd:7,eq:20051}, the domain of $\ms{Fold}^p$ is:
\[
\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p \circ \left\langle \ms{id}_{\sembr{\Xi}}, \sfix{\left(\sembr{\jstype{\Xi}{\Trec{\alpha}{A}}}^p\right)} \right\rangle = \sembr{\Xi\vdash [\Trec{\alpha}{A}/\alpha]\jisst{A}}^p.
\]
\begin{samepage}
Processes unfold recursive types by transmitting unfold messages.
Semantically, this denotes pre- and post-composition with $\Fold^p$ and $\Unfold^p$.
We interpret \rn{$\rho^+R$} and \rn{$\rho^+L$}~by:
\begin{align}
\begin{split}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendU{a}{P}}{a}{\Trec{\alpha}{A}}}u\\
&= \left(\ms{id} \times \left(a^+ : \ms{Fold}^+\right)\right)
\circ \sembr{\jtypem{\Psi}{\Delta}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}}u
\circ \left(\ms{id} \times \left(a^- : \ms{Unfold}^-\right)\right),\label[intn]{eq:fossacs:2}
\end{split}\\
\begin{split}
&\sembr{\jtypem{\Psi}{\Delta, a : \Trec{\alpha}{A}}{\tRecvU{a}{P}}{c}{C}}u\\
&= \left(\ms{id} \!\times \!\left(a^- : \ms{Fold}^-\right)\right)
\circ \sembr{\jtypem{\Psi}{\Delta, a : [\Trec{\alpha}{A}/\alpha]A}{P}{c}{C}}u
\circ \left(\ms{id} \!\times \!\left(a^+ : \ms{Unfold}^+\right)\right)\!.\label[intn]{eq:fossacs:3}
\end{split}
\end{align}
\begin{proposition}[restate=etafunfold,name={}]
\label{prop:fscd:18}
If $\jtypem{\Psi}{\Delta_1}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}$ and ${\jtypem{\Psi}{a : [\Trec{\alpha}{A}/\alpha]A, \Delta_2}{Q}{c}{C}}$, then $\tCut{a}{P}{Q} \equiv \tCut{a}{(\tSendU{a}{P})}{(\tRecvU{a}{Q})}$.
\end{proposition}
\end{samepage}
\subsection{Structural Properties}
\label{sec:struct-prop}
Our semantics respects the exchange rule because we interpret structural contexts as indexed products.
It also respects weakening (\cref{prop:extended:8,prop:extended:9}) and substitution (\cref{prop:11,prop:fscd:19}).
These propositions follow by induction on the derivations.
\begin{proposition}[restate=cohsestyp,name=Coherence of Session Types]
\label{prop:extended:8}
If $\Xi \vdash \jisst[q]{A}$, then $\sembr{\Xi,\Theta\vdash \jisst[q]{A}} = \sembr{\Xi \vdash \jisst[q]{A}}^p\pi^{\Xi,\Theta}_\Xi$ for all $p \in \{{-},{+}\}$.
\end{proposition}
\begin{proposition}[restate=cohterpro,name=Coherence of Terms and Processes]
\label{prop:extended:9}
If $\jtypef{\Psi}{M}{\tau}$, then $\sembr{\jtypef{\Phi,\Psi}{M}{\tau}} = \sembr{\jtypef{\Psi}{M}{\tau}} \circ \pi^{\Phi,\Psi}_\Psi$.
If ${\jtypem{\Psi\!}{\!\Delta\!}{\!P\!}{\!a\!}{\!A}}$, then $\sembr{\jtypem{\Phi,\Psi\!}{\!\Delta\!}{\!P\!}{\!a\!}{\!A}}\,{=}\,\sembr{\jtypem{\Psi\!}{\!\Delta\!}{\!P\!}{\!a\!}{\!A}} \circ \pi^{\Phi,\Psi}_\Psi$.
\end{proposition}
\begin{proposition}[restate=ssst,name=Semantic Substitution of Session Types]\label{prop:11}
Let $\Xi = \jisst[p_1]{\alpha_1},\dotsc,\jisst[p_n]{\alpha_n}$.
For all $p \in \{{-},{+}\}$ and all choices of types $\Theta \vdash \jisst[p_i]{A_i}$ ($1 \leq i \leq n$), if $\Xi \vdash \jisst[q]{B}$, then
\[
\sembr{\Theta \vdash \jisst[q]{[\vec A/\vec \alpha]B}}^p = \sembr{\Xi \vdash \jisst[q]{B}}^p \circ \langle \sembr{\Theta \vdash \jisst[p_i]{A_i}}^p \mid 1 \leq i \leq n \rangle.
\]
\end{proposition}
\begin{proposition}[restate=ssft,name=Semantic Substitution of Functional Terms]
\label{prop:fscd:19}
Let $\Psi = {x_1:\tau_1},\dotsc,{x_n:\tau_n}$.
For all choices of terms $\jtypef{\Phi}{M_i}{\tau_i}$ ($1 \leq i \leq n$),
if $\jtypef{\Psi}{N}{\tau}$ and $\jtypem{\Psi}{\Delta}{P}{c}{C}$, then
\begin{align*}
\sembr{\jtypef{\Phi}{[\vec{M}/\vec{x}]N}{\tau}} &= \sembr{\jtypef{\Psi}{N}{\tau}} \circ \langle \sembr{\jtypef{\Phi}{M_i}{\tau_i}} \mid 1 \leq i \leq n \rangle,\\
\sembr{\jtypem{\Phi}{\Delta}{[\vec{M}/\vec{x}]P}{c}{C}} &= \sembr{\jtypem{\Psi}{\Delta}{P}{c}{C}} \circ \langle \sembr{\jtypef{\Phi}{M_i}{\tau_i}} \mid 1 \leq i \leq n \rangle.
\end{align*}
\end{proposition}
\subsection{Semantic Weakening}
\label{sec:semantic-weakening}
We show that weakening is semantically well-behaved, \ie, that the semantic clauses are coherent~\cite[p.~218]{tennent_1995:_denot_seman}.
\cohterpro*
\begin{proof}
By induction on the derivation of $\jtypef{\Psi}{M}{\tau}$ and $\jtypem{\Psi}{\Delta}{P}{a}{A}$.
Except where stated otherwise, each of rule cases uses the same proof outline.
We refer to it below as the ``standard proof''.
We give this proof for a generic rule for forming functional terms.
The proof for rules forming processes is analogous.
Consider the rule
\[
\adjustbox{width=\linewidth,keepaspectratio}\bgroup
\infer{
\jtypef{\Psi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{\tau}
}{
\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}
&
\cdots
&
\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}
&
\jtypef{\Psi}{M_1}{\tau_1}
&
\cdots
&
\jtypef{\Psi}{M_m}{\tau_m}
&
\mathcal{J}_1
&
\cdots
&
\mathcal{J}_l
}
\egroup
\]
Assume its interpretation is given by
\begin{equation}
\label[intn]{eq:fscd:47}
\begin{aligned}
&\sembr{\jtypef{\Psi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{\tau}}\\
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Psi}}\left(
\sembr{\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}},
\dotsc,
\sembr{\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}},\right.\\
&\qquad\qquad\left.
\sembr{\jtypef{\Psi}{M_1}{\tau_1}},
\dotsc,
\sembr{\jtypef{\Psi}{M_m}{\tau_m}}
\right)
\end{aligned}
\end{equation}
where $\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}$ is a natural interpretation
\begin{equation*}
\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}} :
\left(\prod_{i = 1}^n \moabc\left( {-} , \sembr{\Delta_i \vdash c_i : C_i} \right)\right)
\times
\left(\prod_{i = 1}^m \moabc\left( {-} , \sembr{\tau_i} \right) \right)
\nto
\moabc\left( {-} , \sembr{\tau} \right).
\end{equation*}
Given any other functional context $\Phi$ disjoint from $\Psi$, we would like to show that
\begin{align*}
&\sembr{\jtypef{\Phi,\Psi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{\tau}}\\
&= \sembr{\jtypef{\Psi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{\tau}} \circ \pi^{\Phi,\Psi}_\Psi.
\end{align*}
By the induction hypothesis, we know that for $1 \leq i \leq n$ and $1 \leq j \leq m$,
\begin{align*}
\sembr{\jtypem{\Phi,\Psi}{\Delta_i}{P_i}{c_i}{C_i}} &= \sembr{\jtypem{\Psi}{\Delta_i}{P_i}{c_i}{C_i}} \circ \pi^{\Phi,\Psi}_\Psi\\
\sembr{\jtypef{\Phi,\Psi}{M_j}{\tau_j}} &= \sembr{\jtypef{\Psi}{M_j}{\tau_j}} \circ \pi^{\Phi,\Psi}_\Psi.
\end{align*}
Using these facts we get:
\begin{align*}
&\sembr{\jtypef{\Phi,\Psi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{\tau}}\\
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Phi}}\left(
\sembr{\jtypem{\Phi,\Psi}{\Delta_1}{P_1}{c_1}{C_1}},
\dotsc,
\sembr{\jtypem{\Phi,\Psi}{\Delta_n}{P_n}{c_n}{C_n}},\right.\\
&\qquad\qquad\left.
\sembr{\jtypef{\Phi,\Psi}{M_1}{\tau_1}},
\dotsc,
\sembr{\jtypef{\Phi,\Psi}{M_m}{\tau_m}}
\right)\\
\shortintertext{which by the induction hypothesis,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Phi}}\left(
\sembr{\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}} \circ \pi^{\Phi,\Psi}_\Psi,
\dotsc,
\sembr{\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}} \circ \pi^{\Phi,\Psi}_\Psi,\right.\\
&\qquad\qquad\left.
\sembr{\jtypef{\Psi}{M_1}{\tau_1}} \circ \pi^{\Phi,\Psi}_\Psi,
\dotsc,
\sembr{\jtypef{\Psi}{M_m}{\tau_m}} \circ \pi^{\Phi,\Psi}_\Psi
\right)\\
\shortintertext{which by naturality of $\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}$,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Psi}}\left(
\sembr{\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}}, \dotsc,
\sembr{\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}},\right.\\
&\qquad\qquad\left.
\sembr{\jtypef{\Psi}{M_1}{\tau_1}},
\dotsc,
\sembr{\jtypef{\Psi}{M_m}{\tau_m}}
\right) \circ \pi^{\Phi,\Psi}_\Psi\\
&= \sembr{\jtypef{\Psi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{\tau}} \circ \pi^{\Phi,\Psi}_\Psi.
\end{align*}
This is what we wanted to show.
\begin{description}
\item[Case \rn{I-\{\}}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{F-Var}.]
\[
\infer[\rn{F-Var}]{\jtypef{\Psi, x: \tau}{x}{\tau}}{\mathstrut}
\]
Recall \cref{eq:39}:
\begin{align*}
\sembr{\jtypef{\Psi,x:\tau}{x}{\tau}}u &= \pi^{\Psi,x}_{x}u.
\end{align*}
We use identities of products and projections to compute:
\begin{align*}
&\sembr{\jtypef{\Phi,\Psi,x : \tau}{x}{\tau}}\\
&= \pi^{\Phi,\Psi,x}_x\\
&= \pi^{\Psi,x}_x \circ \pi^{\Phi,\Psi,x}_{\Psi,x}\\
&= \sembr{\jtypef{\Psi,x : \tau}{x}{\tau}} \circ \pi^{\Phi,\Psi,x}_{\Psi,x}.
\end{align*}
\item[Case \rn{F-Fix}.]
\[
\infer[\rn{F-Fix}]{
\jtypef{\Psi}{\tFix{x}{M}}{\tau}
}{
\jtypef{\Psi,x:\tau}{M}{\tau}
}
\]
Recall \cref{eq:28}:
\begin{align*}
\sembr{\jtypef{\Psi}{\tFix{x}{M}}{\tau}}u &= \sfix{\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}}}u
\end{align*}
The result follows from a tweak of the standard proof and \cref{prop:fscd:2}.
Let $\eta : \moabc\left( {-} \times \sembr{x : \tau} , \sembr{\tau} \right) \nto \moabc\left( {-} , \sembr{\tau} \right)$ be the natural interpretation given by \cref{prop:fscd:2}.
Then
\begin{align*}
&\sembr{\jtypef{\Phi,\Psi}{\tFix{x}{M}}{\tau}}\\
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Phi,\Psi,x:\tau}{M}{\tau}}\right)\\
\shortintertext{which by the induction hypothesis:}
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}} \circ \pi^{\Phi,\Psi,x}_{\Psi,x}\right)\\
\shortintertext{which by an identity of projections:}
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}} \circ \left(\pi^{\Phi,\Psi}_{\Psi} \times \left(x : \ms{id}_{\sembr{\tau}}\right)\right)\right)\\
\shortintertext{which by naturality:}
&= \eta_{\sembr{\Psi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}}\right) \circ \pi^{\Phi,\Psi}_{\Psi}\\
&= \sembr{\jtypef{\Psi}{\tFix{x}{M}}{\tau}} \circ \pi^{\Phi,\Psi}_{\Psi}.
\end{align*}
\item[Case \rn{F-Fun}.]
\[
\infer[\rn{F-Fun}]{
\jtypef{\Psi}{\lambda x : \tau.M}{\tau \to \sigma}
}{
\jtypef{\Psi, x:\tau}{M}{\sigma}
}
\]
This case is analogous to the case \rn{F-Fix}.
\item[Case \rn{F-App}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{Fwd}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{Cut}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tu R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tu L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tds{} R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tds{} L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tus R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tus L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tplus R_k$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tplus L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tamp R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tamp L_k$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tot R^*$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tot L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{${\Tlolly}R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{${\Tlolly}L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tand{}{} R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tand{}{} L$}.]
\[
\infer[\rn{$\Tand{}{} L$}]{
\jtypem{\Psi}{\Delta, a:\Tand{\tau}{A}}{\tRecvV{x}{a}{P}}{c}{C}
}{
\jtypem{\Psi,x:\tau}{\Delta, a:A}{P}{c}{C}
}
\]
This case is analogous to the case \rn{F-Fix}.
\item[Case \rn{${\Timp{}{}} R$}.]
\[
\infer[\rn{${\Timp{}{}} R$}]{
\jtypem{\Psi}{\Delta}{\tRecvV{x}{a}{P}}{a}{\Timp{\tau}{A}}
}{
\jtypem{\Psi,x:\tau}{\Delta}{P}{a}{A}
}
\]
This case is analogous to the case \rn{F-Fix}.
\item[Case \rn{${\Timp{}{}} L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^+R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^+L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^-R$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^-L$}.]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{E-\{\}}.]
By the standard proof and \cref{prop:fscd:2}.\qedhere
\end{description}
\end{proof}
\subsection{Semantic Substitution}
\label{sec:semant-subst}
Our goal in this section is to show that substitution is given by composition.
\defin{Context morphisms} give us a semantic account of substitutions.
The judgment $\jcmf{\sigma}{\Phi}{\Psi}$ means the substitution $\sigma$ is a morphism of functional variable contexts from $\Phi$ to $\Psi$.
It is inductively defined by the rules
\[
\infer[\rn{S-T-Empty}]{
\jcmf{\cdot}{\Phi}{\cdot}
}{
}
\qquad
\infer[\rn{S-T-F}]{
\jcmf{\sigma,M}{\Phi}{\Psi,x:\tau}
}{
\jcmf{\sigma}{\Phi}{\Psi}
&
\jtypef{\Phi}{M}{\tau}
}
\]
Consider a morphism $\sigma = N_1,\dotsc,N_n$ with $n \geq 0$ satisfying $\jcmf{\sigma{}}{\Phi}{{x_1:\tau_1},\dotsc,{x_n:\tau_n}}$.
Given a functional term $\jtypef{x_1:\tau_1,\dotsc,x_n:\tau_n}{M}{\tau}$ or a process $\jtypem{x_1:\tau_1,\dotsc,x_n:\tau_n}{\Delta}{P}{c}{C}$, we write $\sigma M$ and $\sigma P$ for the results of the simultaneous substitutions $[N_1,\dotsc,N_n/x_1,\dotsc,x_n] M$ and $[N_1,\dotsc, N_n/x_1,\dotsc,x_n] P$, respectively.
The judgments $\jtypef{\Psi}{M}{\tau}$ and $\jtypem{\Psi}{\Delta}{P}{a}{A}$ satisfy the following syntactic substitution property:
\begin{proposition}[Syntactic Substitution of Terms]
\label{prop:fscd:3}
Let $\jcmf{\sigma}{\Phi}{\Psi}$ be arbitrary.
\begin{proplist}
\item If $\jtypef{\Psi}{N}{\tau}$, then $\jtypef{\Phi}{\sigma N}{\tau}$.
\item If $\jtypem{\Psi}{\Delta}{P}{c}{C}$, then $\jtypem{\Phi}{\Delta}{\sigma P}{c}{C}$.
\end{proplist}
\end{proposition}
\begin{proof}
By induction on the derivation of $\jtypef{\Psi}{M}{\tau}$ and $\jtypem{\Psi}{\Delta}{P}{a}{A}$.
\end{proof}
A context morphism $\jcmf{\sigma}{\Phi}{\Psi}$ is interpreted as a continuous morphism $\sembr{\jcmf{\sigma}{\Phi}{\Psi}} : \sembr{\Phi} \to \sembr{\Psi}$.
It is recursively defined on the derivation of $\jcmf{\sigma}{\Phi}{\Psi}$.
The interpretations of \rn{S-T-Empty} and \rn{S-T-F} are respectively
\begin{align}
\sembr{\jcmf{\cdot}{\Phi}{\cdot}} &= \top_{\sembr{\Phi}}\label[intn]{eq:fscd:32}\\
\sembr{\jcmf{\sigma, M}{\Phi}{\Psi,x:\tau}} &= \langle \sembr{\jcmf{\sigma}{\Phi}{\Psi}}, x : \sembr{\jtypef{\Phi}{M}{\tau}} \rangle\label[intn]{eq:fscd:33}
\end{align}
where $\top_{\sembr{\Phi}}$ is the unique morphism from $\sembr{\Phi}$ to the terminal object $\top$ of $\moabc$.
\begin{lemma}[Weakening of Context Morphisms]
\label{lemma:fscd:13}
Let $\jcmf{\sigma}{\Phi}{\Psi}$ be arbitrary and $\Gamma,\Phi$ a context.
Then $\jcmf{\sigma}{\Gamma,\Phi}{\Psi}$ and
\[
\sembr{\jcmf{\sigma}{\Gamma,\Phi}{\Psi}} = \sembr{\jcmf{\sigma}{\Phi}{\Psi}} \circ \pi^{\Gamma,\Phi}_\Phi.
\]
\end{lemma}
\begin{proof}
By induction on the derivation of $\jcmf{\sigma}{\Phi}{\Psi}$.
\begin{description}[listparindent=\parindent]
\item[Case \rn{S-T-Empty}.]
Then $\jcmf{\sigma}{\Gamma,\Phi}{\cdot}$ by \rn{S-T-Empty}.
By terminality,
\[
\sembr{\jcmf{\sigma}{\Gamma,\Phi}{\cdot}} = \top_{\sembr{\Gamma,\Phi}} = \top_{\sembr{\Phi}} \circ \pi^{\Gamma,\Phi}_\Phi = \sembr{\jcmf{\sigma}{\Phi}{\cdot}} \circ \pi^{\Gamma,\Phi}_\Phi.
\]
\item[Case \rn{S-T-F}.]
\[
\infer[\rn{S-T-F}]{
\jcmf{\sigma,M}{\Phi}{\Psi,x:\tau}
}{
\jcmf{\sigma}{\Phi}{\Psi}
&
\jtypef{\Phi}{M}{\tau}
}
\]
By the induction hypothesis, $\jcmf{\sigma}{\Gamma,\Phi}{\Psi}$ and
\[
\sembr{\jcmf{\sigma}{\Gamma,\Phi}{\Psi}} = \sembr{\jcmf{\sigma}{\Phi}{\Psi}} \circ \pi^{\Gamma,\Phi}_\Phi.
\]
By weakening, $\jtypef{\Gamma,\Phi}{M}{\tau}$, and by \cref{prop:extended:9},
\[
\sembr{\jtypef{\Gamma,\Phi}{M}{\tau}} = \sembr{\jtypef{\Phi}{M}{\tau}} \circ \pi^{\Gamma,\Phi}_\Phi.
\]
So $\jcmf{\sigma,M}{\Gamma,\Phi}{\Psi,x:\tau}$ by \rn{S-T-F}.
By \cref{eq:fscd:33},
\begin{align*}
&\sembr{\jcmf{\sigma,M}{\Gamma,\Phi}{\Psi,x:\tau}}\\
&= \langle \sembr{\jcmf{\sigma}{\Gamma,\Phi}{\Psi}}, x : \sembr{\jtypef{\Gamma,\Phi}{M}{\tau}} \rangle\\
&= \langle \sembr{\jcmf{\sigma}{\Phi}{\Psi}} \circ \pi^{\Gamma,\Phi}_\Phi, x : \sembr{\jtypef{\Phi}{M}{\tau}} \circ \pi^{\Gamma,\Phi}_\Phi \rangle\\
&= \langle \sembr{\jcmf{\sigma}{\Phi}{\Psi}}, x : \sembr{\jtypef{\Phi}{M}{\tau}} \rangle \circ \pi^{\Gamma,\Phi}_\Phi\\
&= \sembr{\jcmf{\sigma,M}{\Phi}{\Psi,x:\tau}} \circ \pi^{\Gamma,\Phi}_\Phi.
\end{align*}
\end{description}
We conclude the result by induction.
\end{proof}
\begin{proposition}[Semantic Substitution of Terms]\label{prop:extended:10}
Let $\jcmf{\sigma}{\Phi}{\Psi}$ be arbitrary.
\begin{proplist}
\item If $\jtypef{\Psi}{N}{\tau}$, then $\sembr{\jtypef{\Phi}{\sigma N}{\tau}} = \sembr{\jtypef{\Psi}{N}{\tau}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}$. \label{item:extended:2}
\item If $\jtypem{\Psi}{\Delta}{P}{c}{C}$, then $\sembr{\jtypem{\Phi}{\Delta}{\sigma P}{c}{C}} = \sembr{\jtypem{\Psi}{\Delta}{P}{c}{C}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}$. \label{item:extended:3}
\end{proplist}
\end{proposition}
\begin{proof}
By induction on the derivation of $\jtypef{\Psi}{M}{\tau}$ and $\jtypem{\Psi}{\Delta}{P}{a}{A}$.
Except where stated otherwise, each of rule cases uses the same proof outline.
We refer to it below as the ``standard proof''.
We give this proof for a generic rule.
Consider the rule
\[
\adjustbox{width=\linewidth,keepaspectratio}\bgroup
\infer{
\jtypef{\Psi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{\tau}
}{
\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}
&
\cdots
&
\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}
&
\jtypef{\Psi}{M_1}{\tau_1}
&
\cdots
&
\jtypef{\Psi}{M_m}{\tau_m}
&
\mathcal{J}_1
&
\cdots
&
\mathcal{J}_l
}
\egroup
\]
Assume its interpretation is given by
\begin{equation}
\label[intn]{eq:fscd:1}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{c}{C}}\\
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Psi}}\left(
\sembr{\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}},
\dotsc,
\sembr{\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}},\right.\\
&\qquad\qquad\left.
\sembr{\jtypef{\Psi}{M_1}{\tau_1}},
\dotsc,
\sembr{\jtypef{\Psi}{M_m}{\tau_m}}
\right)
\end{aligned}
\end{equation}
where $\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}$ is a natural interpretation
\begin{equation*}
\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}} :
\left(\prod_{i = 1}^n \moabc\left( {-} , \sembr{\Delta_i \vdash c_i : C_i} \right)\right)
\times
\left(\prod_{i = 1}^m \moabc\left( {-} , \sembr{\tau_i} \right) \right)
\nto
\moabc\left( {-} , \sembr{\tau} \right).
\end{equation*}
Given any context morphism $\jcmf{\sigma}{\Phi}{\Psi}$, we would like to show that
\begin{align*}
&\sembr{\jtypem{\Phi}{\Delta}{\sigma\left(F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)\right)}{c}{C}}\\
&= \sembr{\jtypem{\Psi}{\Delta}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{c}{C}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}.
\end{align*}
By the definition of syntactic substitution, we know that
\[
\sigma\left(F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)\right) = F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(\sigma P_1,\dotsc,\sigma P_n,\sigma M_1,\dotsc,\sigma M_m).
\]
By the induction hypothesis, we know that for $1 \leq i \leq n$ and $1 \leq j \leq m$,
\begin{align*}
\sembr{\jtypem{\Phi}{\Delta_i}{\sigma P_i}{c_i}{C_i}} &= \sembr{\jtypem{\Psi}{\Delta_i}{P_i}{c_i}{C_i}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}\\
\sembr{\jtypef{\Phi}{\sigma M_j}{\tau_j}} &= \sembr{\jtypef{\Psi}{M_j}{\tau_j}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}.
\end{align*}
Using these facts we get:
\begin{align*}
&\sembr{\jtypem{\Phi}{\Delta}{\sigma\left(F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)\right)}{c}{C}}\\
&= \sembr{\jtypem{\Phi}{\Delta}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(\sigma P_1,\dotsc,\sigma P_n,\sigma M_1,\dotsc,\sigma M_m)}{c}{C}}\\
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Phi}}\left(
\sembr{\jtypem{\Phi}{\Delta_1}{\sigma P_1}{c_1}{C_1}},
\dotsc,
\sembr{\jtypem{\Phi}{\Delta_n}{\sigma P_n}{c_n}{C_n}},\right.\\
&\qquad\qquad\left.
\sembr{\jtypef{\Phi}{\sigma M_1}{\tau_1}},
\dotsc,
\sembr{\jtypef{\Phi}{\sigma M_m}{\tau_m}}
\right)\\
\shortintertext{which by the induction hypothesis,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Phi}}\left(
\sembr{\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}},\dotsc,\right.\\
&\qquad\qquad\sembr{\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}},\\
&\qquad\qquad\left.
\sembr{\jtypef{\Psi}{M_1}{\tau_1}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}},
\dotsc,
\sembr{\jtypef{\Psi}{M_m}{\tau_m}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}
\right)\\
\shortintertext{which by naturality of $\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}$,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}}_{\sembr{\Psi}}\left(
\sembr{\jtypem{\Psi}{\Delta_1}{P_1}{c_1}{C_1}}, \dotsc,
\sembr{\jtypem{\Psi}{\Delta_n}{P_n}{c_n}{C_n}},\right.\\
&\qquad\qquad\left.
\sembr{\jtypef{\Psi}{M_1}{\tau_1}},
\dotsc,
\sembr{\jtypef{\Psi}{M_m}{\tau_m}}
\right) \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}\\
&= \sembr{\jtypem{\Psi}{\Delta}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_l}(P_1,\dotsc,P_n,M_1,\dotsc,M_m)}{c}{C}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}.
\end{align*}
\begin{description}
\item[Case \rn{I-\{\}}.]
\[
\infer[\rn{I-\{\}}]{
\jtypef{\Psi}{\tProc{a}{P}{\overline{a_i}}}{\Tproc{a:A}{\overline{a_i:A_i}}}
}{
\jtypem{\Psi}{\overline{a_i:A_i}}{P}{a}{A}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{F-Var}.]
\[
\infer[\rn{F-Var}]{\jtypef{\Psi, x: \tau}{x}{\tau}}{\mathstrut}
\]
Recall \cref{eq:39}:
\begin{align*}
\sembr{\jtypef{\Psi,x:\tau}{x}{\tau}}u &= \pi^{\Psi,x}_{x}u.
\end{align*}
Let $\jcmf{\sigma, M}{\Phi}{\Psi, x : \tau}$ be arbitrary.
Then
\begin{align*}
&\sembr{\jtypef{\Phi}{(\sigma,M) x}{\tau}}\\
&= \sembr{\jtypef{\Phi}{M}{\tau}}\\
&= \pi^{\Psi,x}_x \circ \langle \sembr{\jcmf{\sigma}{\Phi}{\Psi}}, x : \sembr{\jtypef{\Phi}{M}{\tau}} \rangle\\
&= \pi^{\Psi,x}_x \circ \sembr{\jcmf{\sigma, M}{\Phi}{\Psi, x : \tau}}\\
&= \sembr{\jtypef{\Psi,x : \tau}{x}{\tau}} \circ \sembr{\jcmf{\sigma, M}{\Phi}{\Psi, x : \tau}}.
\end{align*}
\item[Case \rn{F-Fix}.]
\[
\infer[\rn{F-Fix}]{
\jtypef{\Psi}{\tFix{x}{M}}{\tau}
}{
\jtypef{\Psi,x:\tau}{M}{\tau}
}
\]
Recall \cref{eq:28}:
\begin{align*}
\sembr{\jtypef{\Psi}{\tFix{x}{M}}{\tau}}u &= \sfix{\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}}}u
\end{align*}
The result follows from a tweak of the standard proof and \cref{prop:fscd:2}.
Let $\jcmf{\sigma}{\Phi}{\Psi}$ be arbitrary.
By \cref{lemma:fscd:13}, we can weaken it to $\jcmf{\sigma,x}{\Phi,x : \tau}{\Psi, x : \tau}$.
Let $\eta : \moabc\left( \cdot \times \sembr{x : \tau} , \sembr{\tau} \right) \nto \moabc\left( \cdot , \sembr{\tau} \right)$ be the natural interpretation given by \cref{prop:fscd:2}.
Then
\begin{align*}
&\sembr{\jtypef{\Phi}{\sigma(\tFix{x}{M})}{\tau}}\\
&= \sembr{\jtypef{\Phi}{\tFix{x}{(\sigma,x)M}}{\tau}}\\
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Phi,x:\tau}{(\sigma,x)M}{\tau}}\right)\\
\shortintertext{which by the induction hypothesis:}
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}} \circ \sembr{\jcmf{\sigma,x}{\Phi, x : \tau}{\Psi,x:\tau}}\right)\\
\shortintertext{which by \cref{eq:fscd:33}:}
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}} \circ \langle \sembr{\jcmf{\sigma}{\Phi, x : \tau}{\Psi}}, x : \sembr{\jtypef{\Phi,x : \tau}{x}{\tau}} \rangle\right)\\
\shortintertext{which by \cref{eq:39}:}
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}} \circ \langle \sembr{\jcmf{\sigma}{\Phi, x : \tau}{\Psi}}, x : \pi^{\Phi,x}_x \rangle\right)\\
\shortintertext{which by \cref{lemma:fscd:13}:}
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}} \circ \langle \sembr{\jcmf{\sigma}{\Phi}{\Psi}} \circ \pi^{\Phi, x}_\Phi, x : \pi^{\Phi,x}_x \rangle\right)\\
\shortintertext{which by a property of products and projections:}
&= \eta_{\sembr{\Phi}}\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}} \circ \left(\sembr{\jcmf{\sigma}{\Phi}{\Psi}} \times \sembr{x : \tau}\right)\right)\\
&= \left(\eta_{\sembr{\Phi}} \circ \moabc\left( \sembr{\jcmf{\sigma}{\Phi}{\Psi}} \times \sembr{x : \tau} , \sembr{\tau} \right)\right) \left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}}\right)\\
\shortintertext{which by naturality:}
&= \left(\moabc\left( \sembr{\jcmf{\sigma}{\Phi}{\Psi}} , \sembr{\tau} \right) \circ \eta_{\sembr{\Psi}}\right)\left(\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}}\right)\\
\shortintertext{which by definition of natural interpretation:}
&= \moabc\left( \sembr{\jcmf{\sigma}{\Phi}{\Psi}} , \sembr{\tau} \right)\left(\sembr{\jtypef{\Psi}{\tFix{x}{M}}{\tau}}\right)\\
&= \sembr{\jtypef{\Psi}{\tFix{x}{M}}{\tau}} \circ \sembr{\jcmf{\sigma}{\Phi}{\Psi}}.
\end{align*}
\item[Case \rn{F-Fun}.]
\[
\infer[\rn{F-Fun}]{
\jtypef{\Psi}{\lambda x : \tau.M}{\tau \to \sigma}
}{
\jtypef{\Psi, x:\tau}{M}{\sigma}
}
\]
This case is analogous to the case \rn{F-Fix}.
\item[Case \rn{F-App}.]
\[
\infer[\rn{F-App}]{
\jtypef{\Psi}{MN}{\sigma}
}{
\jtypef{\Psi}{M}{\tau \to \sigma}
&
\jtypef{\Psi}{N}{\tau}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{Fwd}.]
\[
\infer[\rn{Fwd}]{
\jtypem{\Psi}{a:A}{\tFwd{b}{a}}{b}{A}
}{}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{Cut}.]
\[
\infer[\rn{Cut}]{
\jtypem{\Psi}{\Delta_1,\Delta_2}{\tCut{a}{P}{Q}}{c}{C}
}{
\jtypem{\Psi}{\Delta_1}{P}{a}{A}
&
\jtypem{\Psi}{a:A,\Delta_2}{Q}{c}{C}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tu R$}.]
\[
\infer[\rn{$\Tu R$}]{
\jtypem{\Psi}{\cdot}{\tClose a}{a}{\Tu}
}{}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tu L$}.]
\[
\infer[\rn{$\Tu L$}]{
\jtypem{\Psi}{\Delta, a : \Tu}{\tWait{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta}{P}{c}{C}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tds{} R$}.]
\[
\infer[\rn{$\Tds{} R$}]{
\jtypem{\Psi}{\Delta}{\tSendS{a}{P}}{a}{\Tds A}
}{
\jtypem{\Psi}{\Delta}{P}{a}{A}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tds{} L$}.]
\[
\infer[\rn{$\Tds{} L$}]{
\jtypem{\Psi}{\Delta,a : \Tds A}{\tRecvS{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tus R$}.]
\[
\infer[\rn{$\Tus R$}]{
\jtypem{\Psi}{\Delta}{\tRecvS{a}{P}}{a}{\Tus{A}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tus L$}.]
\[
\infer[\rn{$\Tus L$}]{
\jtypem{\Psi}{\Delta,a : \Tus A}{\tSendS{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tplus R_k$}.]
\[
\infer[\rn{$\Tplus R_k$}]{
\jtypem{\Psi}{\Delta}{\tSendL{a}{k}{P}}{a}{{\Tplus\{l:A_l\}}_{l \in L}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{A_k}\quad(k \in L)
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tplus L$}.]
\[
\infer[\rn{$\Tplus L$}]{
\jtypem{\Psi}{\Delta,a:{\Tplus\{l : A_l\}}_{l \in L}}{\tCase{a}{\left\{l_l \Rightarrow P_l\right\}_{i\in I}}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a:A_l}{P_l}{c}{C}\quad(\forall l \in L)
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tamp R$}.]
\[
\infer[\rn{$\Tamp R$}]{
\jtypem{\Psi}{\Delta}{\tCase{a}{\left\{l \Rightarrow P_l\right\}_{l \in L}}}{a}{{\Tamp\{l :A_l \}}_{l \in L}}
}{
\jtypem{\Psi}{\Delta}{P_l}{a}{A_l}\quad(\forall l \in L)
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tamp L_k$}.]
\[
\infer[\rn{$\Tamp L_k$}]{
\jtypem{\Psi}{\Delta,a:{\Tamp\{l : A_l\}}_{l \in L}}{\tSendL{a}{k}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a:A_k}{P}{c}{C}
&
(k \in L)
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tot R^*$}.]
\[
\infer[\rn{$\Tot R^*$}]{
\jtypem{\Psi}{\Delta, b : B}{\tSendC{a}{b}{P}}{a}{B \Tot A}
}{
\jtypem{\Psi}{\Delta}{P}{a}{A}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tot L$}.]
\[
\infer[\rn{$\Tot L$}]{
\jtypem{\Psi}{\Delta, a : B \Tot A}{\tRecvC{b}{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta, a : A, b : B}{P}{c}{C}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{${\Tlolly}R$}.]
\[
\infer[\rn{${\Tlolly}R$}]{
\jtypem{\Psi}{\Delta}{\tRecvC{b}{a}{P}}{a}{B \Tlolly A}
}{
\jtypem{\Psi}{\Delta, b : B}{P}{a}{A}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{${\Tlolly}L$}.]
\[
\infer[\rn{${\Tlolly}L$}]{
\jtypem{\Psi}{\Delta, b : B, a : B \Tlolly A}{\tSendC{a}{bb}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tand{}{} R$}.]
\[
\infer[\rn{$\Tand{}{} R$}]{
\jtypem{\Psi}{\Delta}{\tSendV{a}{M}{P}}{a}{\Tand{\tau}{A}}
}{
\jtypef{\Psi}{M}{\tau}
&
\jtypem{\Psi}{\Delta}{P}{a}{A}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\Tand{}{} L$}.]
\[
\infer[\rn{$\Tand{}{} L$}]{
\jtypem{\Psi}{\Delta, a:\Tand{\tau}{A}}{\tRecvV{x}{a}{P}}{c}{C}
}{
\jtypem{\Psi,x:\tau}{\Delta, a:A}{P}{c}{C}
}
\]
This case is analogous to the case \rn{F-Fix}.
\item[Case \rn{${\Timp{}{}} R$}.]
\[
\infer[\rn{${\Timp{}{}} R$}]{
\jtypem{\Psi}{\Delta}{\tRecvV{x}{a}{P}}{a}{\Timp{\tau}{A}}
}{
\jtypem{\Psi,x:\tau}{\Delta}{P}{a}{A}
}
\]
This case is analogous to the case \rn{F-Fix}.
\item[Case \rn{${\Timp{}{}} L$}.]
\[
\infer[\rn{${\Timp{}{}} L$}]{
\jtypem{\Psi}{\Delta,a : \Timp{\tau}{A}}{\tSendV{a}{M}{P}}{c}{C}
}{
\jtypef{\Psi}{M}{\tau}
&
\jtypem{\Psi}{\Delta, a : A}{P}{c}{C}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^+R$}.]
\[
\infer[\rn{$\rho^+R$}]{
\jtypem{\Psi}{\Delta}{\tSendU{a}{P}}{a}{\Trec{\alpha}{A}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}
&
\cdot \vdash \jisst[+]{\Trec{\alpha}{A}}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^+L$}.]
\[
\infer[\rn{$\rho^+L$}]{
\jtypem{\Psi}{\Delta, a : \Trec{\alpha}{A}}{\tRecvU{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta, a : [\Trec{\alpha}{A}/\alpha]A}{P}{c}{C}
&
\cdot \vdash \jisst[+]{\Trec{\alpha}{A}}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^-R$}.]
\[
\infer[\rn{$\rho^-R$}]{
\jtypem{\Psi}{\Delta}{\tRecvU{a}{P}}{a}{\Trec{\alpha}{A}}
}{
\jtypem{\Psi}{\Delta}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}
&
\cdot \vdash \jisst[-]{\Trec{\alpha}{A}}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{$\rho^-L$}.]
\[
\infer[\rn{$\rho^-L$}]{
\jtypem{\Psi}{\Delta, a : \Trec{\alpha}{A}}{\tSendU{a}{P}}{c}{C}
}{
\jtypem{\Psi}{\Delta, a : [\Trec{\alpha}{A}/\alpha]A}{P}{c}{C}
&
\cdot \vdash \jisst[-]{\Trec{\alpha}{A}}
}
\]
By the standard proof and \cref{prop:fscd:2}.
\item[Case \rn{E-\{\}}.]
\[
\infer[\rn{E-\{\}}]{
\jtypem{\Psi}{\overline{a_i:A_i}}{\tProc{a}{M}{\overline a_i}}{a}{A}
}{
\jtypef{\Psi}{M}{\Tproc{a:A}{\overline{a_i:A_i}}}
}
\]
By the standard proof and \cref{prop:fscd:2}.\qedhere
\end{description}
\end{proof}
\subsection{Semantic Weakening}
\label{sec:semantic-weakening-types}
We show that weakening is semantically well-behaved, \ie, that the semantic clauses are coherent~\cite[p.~218]{tennent_1995:_denot_seman}.
\begin{proposition}[Coherence]
\label{prop:fscd:8}
Let $\Theta,\Xi$ be a context of type variables.
If $\jstype[p]{\Xi}{A}$, then the following diagram commutes for $q \in \{{-},{+}\}$:
\begin{equation}
\label[diagram]{eq:114}
\begin{tikzcd}[column sep=7em]
\sembr{\Theta,\Xi} \ar[dr, "{\sembr{\jstype[p]{\Theta,\Xi}{A}}^q}"] \ar[d, swap, "\pi^{\Theta,\Xi}_\Xi"] &\\
\sembr{\Xi} \ar[r, swap, "{\sembr{\jstype[p]{\Xi}{{A}}}^q}"] & \moabcs
\end{tikzcd}
\end{equation}
\end{proposition}
\begin{proof}
By induction on the derivation of $\jstype[p]{\Xi}{A}$.
Except where otherwise stated, each rule uses the same proof outline.
We refer to it below as the ``standard proof''.
Consider a type-forming rule
\[
\infer{
\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}
}{
\jstype{\Xi}{A_1}
&
\cdots
&
\jstype{\Xi}{A_n}
&
\mathcal{J}_1
&
\cdots
&
\mathcal{J}_m
}
\]
Assume its interpretation is given by
\[
\begin{aligned}
&\sembr{\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p\\
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}_{\sembr{\Xi}}\left(\sembr{\jstype{\Xi}{A_1}}^p, \dotsc, \sembr{\jstype{\Xi}{A_n}}^p\right).
\end{aligned}
\]
where $\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}$ is a natural interpretation
\[
\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}_{\sembr{\Xi}} : \left( \prod_{i = 1}^n \CFP\left( \sembr{\Xi}, \moabcs \right) \right) \to \CFP\left( \sembr{\Xi} , \moabcs \right).
\]
Given any other context of type variables $\Theta$ disjoint from $\Xi$, we would like to show that
\begin{align*}
&\sembr{\jstype{\Theta,\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p\\
&= \sembr{\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p \circ \pi^{\Theta,\Xi}_\Xi.
\end{align*}
By the induction hypothesis, we have for all $1 \leq i \leq n$,
\[
\sembr{\jstype{\Theta,\Xi}{A_i}}^p = \sembr{\jstype{\Xi}{A_i}}^p \circ \pi^{\Theta,\Xi}_\Xi.
\]
Using these facts we get:
\begin{align*}
&\sembr{\jstype{\Theta,\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p\\
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}_{\sembr{\Theta,\Xi}}\left(\sembr{\jstype{\Theta,\Xi}{A_1}}^p, \dotsc, \sembr{\jstype{\Theta,\Xi}{A_n}}^p\right)\\
\shortintertext{which by the induction hypothesis,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}_{\sembr{\Theta,\Xi}}\left(
\sembr{\jstype{\Xi}{A_1}}^p \circ \pi^{\Theta,\Xi}_\Xi,
\dotsc,
\sembr{\jstype{\Xi}{A_n}}^p \circ \pi^{\Theta,\Xi}_\Xi\right)\\
\shortintertext{which by naturality of $\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}$,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}_{\sembr{\Xi}}\left(
\sembr{\jstype{\Xi}{A_1}}^p,
\dotsc,
\sembr{\jstype{\Xi}{A_n}}^p\right) \circ \pi^{\Theta,\Xi}_\Xi\\
&=\sembr{\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p \circ \pi^{\Theta,\Xi}_\Xi.
\end{align*}
This is what we wanted to show.
\begin{description}[listparindent=\parindent]
\item[Case \rn{C$\Tu$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{CVar}.]
\[
\infer[\rn{CVar}]{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}{}
\]
Recall \cref{eq:fscd:7}:
\begin{align*}
\sembr{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}^q &= \pi^{\Xi,\alpha}_\alpha \quad (q \in \{{-},{+}\})
\end{align*}
Then
\[
\sembr{\jstype{\Theta,\Xi,\jisst{\alpha}}{\alpha}}^q = \pi^{\Theta,\Xi,\alpha}_\alpha = \pi^{\Xi,\alpha}_\alpha\pi^{\Theta,\Xi,\alpha}_{\Xi,\alpha} = \sembr{\jstype{\Xi,\jisst{\alpha}}{\alpha}}\pi^{\Theta,\Xi,\alpha}_{\Xi,\alpha}.
\]
This is what we wanted to show.
\item[Case \rn{C$\rho$}.]
\[
\infer[\rn{C$\rho$}]{
\Xi \vdash \jisst[p]{\Trec{\alpha}{A}}
}{
\Xi, \jisst[p]{\alpha} \vdash \jisst[p]{A}
}
\]
Recall \cref{eq:20051}:
\begin{align*}
\sembr{\Xi\vdash\jisst{\Trec{\alpha}{A}}}^p &= \sfix{\left(\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p\right)}\quad (p \in \{{-},{+}\})
\end{align*}
The result follows from a tweak of the standard proof and \cref{prop:fscd:4}.
Let $\eta^p : \CFP\left( {-} \times \sembr{\alpha}, \moabcs \right) \nto \CFP\left( {-}, \moabcs \right)$ be the natural interpretation given by \cref{prop:fscd:4}.
Then
\begin{align*}
&\sembr{\jstype{\Theta,\Xi}{\Trec{\alpha}{A}}}^p\\
&= \eta^p_{\sembr{\Theta,\Xi}}\left(\sembr{\jstype{\Theta,\Xi,\alpha}{A}}^p\right)\\
\shortintertext{which by the induction hypothesis:}
&= \eta^p_{\sembr{\Theta,\Xi}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p \pi^{\Theta,\Xi,\alpha}_{\Xi,\alpha}\right)\\
\shortintertext{which by a property of products and projections:}
&= \eta^p_{\sembr{\Theta,\Xi}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p \left(\pi^{\Theta,\Xi}_{\Xi} \times \sembr{\alpha}\right)\right)\\
\shortintertext{which by naturality:}
&= \eta^p_{\sembr{\Xi}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p\right)\pi^{\Theta,\Xi}_{\Xi}\\
&= \sembr{\jstype{\Xi}{\Trec{\alpha}{A}}}^p\pi^{\Theta,\Xi}_{\Xi}.
\end{align*}
\item[Case \rn{C$\Tds{}$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tus{}$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tplus$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tamp$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tot$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tlolly$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tand{}{}$}.]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Timp{}{}$}.]
By the standard proof and \cref{prop:fscd:4}.\qedhere
\end{description}
\end{proof}
\subsection{Semantic Substitution}
\label{sec:semant-subst-types}
Our goal in this section is to show that substitution is given by composition.
\defin{Context morphisms} give us a semantic account of substitutions.
To this end, we introduce explicit typing rules for type substitutions.
We use the judgment $\jcms{\sigma}{\Xi}{\Theta}$ to mean that the substitution $\sigma$ is a morphism of type variable contexts from $\Xi$ to $\Theta$.
It is inductively defined by
\[
\infer[\rn{S-S-Empty}]{
\jcms{\cdot}{\Theta}{\cdot}
}{\mathstrut}
\quad
\infer[\rn{S-S-T$^p$}]{
\jcms{\sigma,A}{\Theta}{\Xi,\jisst[p]{\alpha}}
}{
\jcms{\sigma}{\Theta}{\Xi}
&
\jstype[p]{\Theta}{A}
}
\]
These rules ensure that substitutions of types for type variables respects polarities, \ie, that we only substitute positive session types for a positive type variable and negative session types for negative type variables.
Given a morphism $\sigma = A_1,\dots,A_n$ satisfying $\jcms{\sigma}{\Xi}{\alpha_1,\dotsc,\alpha_n}$ with $n \geq 0$, and a session type $\alpha_1,\dotsc,\alpha_n \vdash B$, we write $\sigma B$ for the result of the simultaneous substitution $[A_1,\dotsc,A_n/\alpha_1,\dotsc,\alpha_n]B$.
\begin{proposition}[Syntactic Substitution of Session Types]
\label{prop:fscd:5}
Let $\jcms{\sigma}{\Theta}{\Xi}$ be arbitrary.
If $\jstype[p]{\Xi}{A}$, then $\jstype[p]{\Theta}{\sigma A}$.
\end{proposition}
Context morphisms are subject to the same classes of interpretation as session types.
A context morphism $\jcms{\sigma}{\Theta}{\Xi}$ gives rise to polarized interpretations, which are locally continuous functors $\sembr{\Theta} \to \sembr{\Xi}$.
The polarized interpretations of \rn{S-S-Empty} and \rn{S-S-T$^p$} are respectively
\begin{align*}
\sembr{\jcms{\cdot}{\Theta}{\cdot}}^q &= \top_{\sembr{\Theta}}\\
\sembr{\jcms{\sigma,A}{\Theta}{\Xi,\jisst[p]{\alpha}}}^q &= \langle \sembr{\jcms{\sigma}{\Theta}{\Xi}}^q, \alpha : \sembr{\jstype[p]{\Theta}{A}}^q \rangle
\end{align*}
where $\top_{\sembr{\Theta}}$ is the constant functor from $\sembr{\Theta}$ to the terminal category $\top$ and $q \in \{{-},{+}\}$.
\begin{lemma}[Weakening of Context Morphisms]
\label{lemma:fscd:14}
Let $\jcms{\sigma}{\Theta}{\Xi}$ be arbitrary and $\Theta,\Omega$ a context.
Then $\jcms{\sigma}{\Omega,\Theta}{\Xi}$ and where $p$ ranges over $\{{-},{+}\}$,
\[
\sembr{\jcms{\sigma}{\Omega,\Theta}{\Xi}}^p = \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p\pi^{\Omega,\Theta}_\Theta.
\]
\end{lemma}
\begin{proof}
By induction on the derivation of $\jcms{\sigma}{\Theta}{\Xi}$.
\begin{description}[listparindent=\parindent]
\item[Case \rn{S-S-Empty}.]
Then $\jcms{\sigma}{\Omega,\Theta}{\cdot}$ by \rn{S-S-Empty}.
Moreover,
\[
\sembr{\jcms{\sigma}{\Omega,\Theta}{\cdot}}^p = \top_{\CFP} = \top_{\Theta}\pi^{\Omega,\Theta}_\Theta = \sembr{\jcms{\sigma}{\Theta}{\cdot}}^p\pi^{\Omega,\Theta}_\Theta.
\]
\item[Case \rn{S-S-T${}^q$}.]
\[
\infer[\rn{S-S-T$^q$}]{
\jcms{\sigma,A}{\Theta}{\Xi,\jisst[q]{\alpha}}
}{
\jcms{\sigma}{\Theta}{\Xi}
&
\jstype[q]{\Theta}{A}
}
\]
Then by the induction hypothesis, $\jcms{\sigma}{\Omega,\Theta}{\Xi}$ and
\[
\sembr{\jcms{\sigma}{\Omega,\Theta}{\Xi}}^p = \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p \pi^{\Omega,\Theta}_\Theta.
\]
By weakening, $\jstype[q]{\Omega,\Theta}{A}$, and by \cref{prop:fscd:8}
\[
\sembr{\jstype[q]{\Omega,\Theta}{A}}^p = \sembr{\jstype[q]{\Theta}{A}}^p \pi^{\Omega,\Theta}_\Theta.
\]
By \rn{S-S-T$^q$}, $\jcms{\sigma,A}{\Omega,\Theta}{\Xi,\jisst[q]{\alpha}}$.
By the interpretation of \rn{S-S-T$^q$},
\begin{align*}
&\sembr{\jcms{\sigma,A}{\Omega,\Theta}{\Xi,\jisst[q]{\alpha}}}^p\\
&= \langle \sembr{\jcms{\sigma}{\Omega,\Theta}{\Xi}}^p , \alpha : \sembr{\jstype[q]{\Omega,\Theta}{A}}^p \rangle\\
&= \langle \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p \pi^{\Omega,\Theta}_\Theta, \alpha : \sembr{\jstype[q]{\Theta}{A}}^p \pi^{\Omega,\Theta}_\Theta \rangle\\
&= \langle \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p, \alpha : \sembr{\jstype[q]{\Theta}{A}}^p\rangle \pi^{\Omega,\Theta}_\Theta\\
&= \sembr{\jcms{\sigma,A}{\Theta}{\Xi,\jisst[q]{\alpha}}}^p\pi^{\Omega,\Theta}_\Theta .
\end{align*}
\end{description}
We conclude the result by induction.
\end{proof}
\begin{proposition}[Semantic Substitution of Session Types]\label{prop:fscd:7}
Let $\jcms{\sigma}{\Theta}{\Xi}$ be arbitrary and let $p$ range over $\{{-},{+}\}$.
If $\Xi \vdash \jisst[q]{A}$, then
\[
\sembr{\Theta \vdash \jisst[q]{\sigma A}}^p = \sembr{\Xi \vdash \jisst[q]{A}}^p \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p.
\]
\end{proposition}
\begin{proof}
By induction on the derivation of $\jstype{\Xi}{A}$.
As in \cref{prop:extended:10}, we describe the standard case.
Consider a type-forming rule
\[
\infer{
\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}
}{
\jstype{\Xi}{A_1}
&
\cdots
&
\jstype{\Xi}{A_n}
&
\mathcal{J}_1
&
\cdots
&
\mathcal{J}_m
}
\]
Assume $\sembr{\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p$ is given by a natural interpretation
\[
\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}^p_{\sembr{\Xi}} : \left( \prod_{i = 1}^n \CFP\left( \sembr{\Xi} , \moabcs \right) \right) \to \CFP\left( \sembr{\Xi}, \moabcs \right).
\]
We need to show that
\begin{align*}
&\sembr{\jstype{\Theta}{\sigma\left(F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)\right)}}^p\\
&= \sembr{\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p.
\end{align*}
By the definition of syntactic substitution, we know that
\[
\sigma\left(F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)\right) = F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(\sigma A_1,\dotsc,\sigma A_n).
\]
By the induction hypothesis, we know for $1 \leq i \leq n$ that
\[
\sembr{\jstype{\Theta}{\sigma A_i}}^p = \sembr{\jstype{\Xi}{A_i}}^p \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p.
\]
Using these facts, we get:
\begin{align*}
&\sembr{\jstype{\Theta}{\sigma\left(F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)\right)}}^p\\
&= \sembr{\jstype{\Theta}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(\sigma A_1,\dotsc,\sigma A_n)}}^p\\
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}^p_{\sembr{\Theta}}\left(\left(\sembr{\jstype{\Theta}{\sigma A_i}}^p\right)_{1 \leq i \leq n}\right)\\
\shortintertext{which by the induction hypothesis,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}^p_{\sembr{\Theta}}\left(\left( \sembr{\jstype{\Xi}{A_i}}^p \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p\right)_{1 \leq i \leq n}\right)\\
\shortintertext{which by naturality of $\sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}^p$,}
&= \sembr{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}}^p_{\sembr{\Xi}}\left(\left( \sembr{\jstype{\Xi}{A_i}}^p\right)_{1 \leq i \leq n}\right) \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p\\
&= \sembr{\jstype{\Xi}{F_{\mathcal{J}_1,\dotsc,\mathcal{J}_m}(A_1,\dotsc,A_n)}}^p \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p.
\end{align*}
\begin{description}
\item[Case \rn{C$\Tu$}.]
\[
\infer[\rn{C$\Tu$}]{\jstype[+]{\Xi}{\Tu}}{}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{CVar}.]
\[
\infer[\rn{CVar}]{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}{}
\]
Recall \cref{eq:fscd:7}:
\begin{align*}
\sembr{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}^q &= \pi^{\Xi,\alpha}_\alpha \quad (q \in \{{-},{+}\})
\end{align*}
Let $\jcms{\sigma,A}{\Theta}{\Xi,\alpha}$ be arbitrary.
\begin{align*}
&\sembr{\jstype{\Theta}{(\sigma,A)\alpha}}^q\\
&= \sembr{\jstype{\Theta}{A}}^q\\
&= \ms{id}_{\pi^{\Xi,\alpha}_\alpha} \circ \langle \sembr{\jcms{\sigma}{\Theta}{\Xi}}^q, \alpha : \sembr{\jstype{\Theta}{A}}^q \rangle\\
&= \sembr{\Xi,\jisst[p]{\alpha} \vdash \jisst[p]{\alpha}}^q \circ \sembr{\jcms{\sigma,A}{\Theta}{\Xi,\alpha}}^q.
\end{align*}
\item[Case \rn{C$\rho$}.]
\[
\infer[\rn{C$\rho$}]{
\Xi \vdash \jisst[p]{\Trec{\alpha}{A}}
}{
\Xi, \jisst[p]{\alpha} \vdash \jisst[p]{A}
}
\]
Recall \cref{eq:20051}:
\begin{align*}
\sembr{\Xi\vdash\jisst{\Trec{\alpha}{A}}}^p &= \sfix{\left(\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p\right)}\quad (p \in \{{-},{+}\})
\end{align*}
The result follows from a tweak of the standard proof and \cref{prop:fscd:4}.
Let $\jcms{\sigma}{\Theta}{\Xi}$ be arbitrary.
By \cref{lemma:fscd:14}, we can weaken it to $\jcms{\sigma,\alpha}{\Theta,\alpha}{\Xi,\alpha}$.
Let $\eta^p : \CFP\left( {-} \times \sembr{\alpha}, \moabcs \right) \nto \CFP\left( {-}, \moabcs \right)$ be the natural interpretation given by \cref{prop:fscd:4}.
Then
\begin{align*}
&\sembr{\jstype{\Theta}{\sigma\left(\Trec{\alpha}{A}\right)}}^p\\
&= \sembr{\jstype{\Theta}{\Trec{\alpha}{(\sigma,\alpha)A}}}^p\\
&= \eta^p_{\sembr{\Theta}}\left(\sembr{\jstype{\Theta,\alpha}{(\sigma,\alpha)A}}^p\right)\\
\shortintertext{which by the induction hypothesis:}
&= \eta^p_{\sembr{\Theta}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p \circ \sembr{\jcms{\sigma,\alpha}{\Theta,\alpha}{\Xi,\alpha}}^p\right)\\
\shortintertext{which by the interpretation of \rn{S-S-T$^q$}:}
&= \eta^p_{\sembr{\Theta}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p \circ \langle \sembr{\jcms{\sigma}{\Theta,\alpha}{\Xi}}^p, \alpha : \sembr{\jstype{\Theta,\alpha}{\alpha}}^p \rangle \right)\\
\shortintertext{which by \cref{eq:20051}:}
&= \eta^p_{\sembr{\Theta}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p \circ \langle \sembr{\jcms{\sigma}{\Theta,\alpha}{\Xi}}^p, \alpha : \ms{id}_{\pi^{\Theta,\alpha}_\alpha} \rangle \right)\\
\shortintertext{which by \cref{lemma:fscd:14}:}
&= \eta^p_{\sembr{\Theta}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p \circ \langle \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p\pi^{\Theta,\alpha}_\Theta, \alpha : \ms{id}_{\pi^{\Theta,\alpha}_\alpha} \rangle \right)\\
\shortintertext{which by a property of products and projections:}
&= \eta^p_{\sembr{\Theta}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p \circ \left( \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p \times \sembr{\alpha} \right) \right)\\
\shortintertext{which by naturality:}
&= \eta^p_{\sembr{\Xi}}\left(\sembr{\jstype{\Xi,\alpha}{A}}^p\right) \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p\\
&= \sembr{\jstype{\Xi}{\Trec{\alpha}{A}}}^p \circ \sembr{\jcms{\sigma}{\Theta}{\Xi}}^p.
\end{align*}
\item[Case \rn{C$\Tds{}$}.]
\[
\infer[\rn{C$\Tds{}$}]{
\jstype[+]{\Xi}{\Tds A}
}{
\jstype[-]{\Xi}{A}
}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tus{}$}.]
\[
\infer[\rn{C$\Tus{}$}]{
\jstype[-]{\Xi}{\Tus A}
}{
\jstype[+]{\Xi}{A}
}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tplus$}.]
\[
\infer[\rn{C$\Tplus$}]{
\jstype[+]{\Xi}{{\Tplus\{l : A_l\}}_{l \in L}}
}{
\jstype[+]{\Xi}{A_l}\quad(\forall l \in L)
}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tamp$}.]
\[
\infer[\rn{C$\Tamp$}]{
\jstype[-]{\Xi}{{\Tamp\{l :A_l \}}_{l \in L}}
}{
\jstype[-]{\Xi}{A_l}\quad(\forall l \in L)
}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tot$}.]
\[
\infer[\rn{C$\Tot$}]{
\jstype[+]{\Xi}{A \Tot B}
}{
\jstype[+]{\Xi}{A}
&
\jstype[+]{\Xi}{B}
}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tlolly$}.]
\[
\infer[\rn{C$\Tlolly$}]{
\jstype[-]{\Xi}{A \Tlolly B}
}{
\jstype[+]{\Xi}{A}
&
\jstype[-]{\Xi}{B}
}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Tand{}{}$}.]
\[
\infer[\rn{C$\Tand{}{}$}]{
\jstype[+]{\Xi}{\Tand{\tau}{A}}
}{
\jisft{\tau}
&
\jstype[+]{\Xi}{A}
}
\]
By the standard proof and \cref{prop:fscd:4}.
\item[Case \rn{C$\Timp{}{}$}.]
\[
\infer[\rn{C$\Timp{}{}$}]{
\jstype[-]{\Xi}{\Timp{\tau}{A}}
}{
\jisft{\tau}
&
\jstype[-]{\Xi}{A}
}
\]
By the standard proof and \cref{prop:fscd:4}.\qedhere
\end{description}
\end{proof}
\subsection{Interpretations are Well-Defined}
\label{sec:interpr-are-well}
In this section we show that our interpretations are all well-defined, \ie, that the polarized aspects are locally continuous functors.
\begin{proposition}[Functorial Interpretations are Well-Defined]
\label{prop:fscd:9}
If $\jstype{\Xi}{A}$, then the interpretations $\sembr{\jstype{\Xi}{A}}^-$ and $\sembr{\jstype{\Xi}{A}}^+$ are locally continuous functors from $\sembr{\Xi}$ to $\moabcs$.
\end{proposition}
\begin{proof}
By induction on the derivation of $\jstype{\Xi}{A}$.
We must show that each of the functor interpretations is locally continuous.
\begin{description}
\item[Case \rn{C$\Tu$}.]
\[
\infer[\rn{C$\Tu$}]{\jstype[+]{\Xi}{\Tu}}{}
\]
Recall \cref{eq:1502,eq:1501}:
\begin{align*}
\sembr{\jstype{\Xi}{\Tu}}^- &= \lambda \xi.\top_{\moabcs}\\
\sembr{\jstype{\Xi}{\Tu}}^+ &= \lambda \xi.\{\ast\}_\bot
\end{align*}
Constant functors are easily seen to be locally continuous.
\item[Case \rn{CVar}.]
\[
\infer[\rn{CVar}]{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}{}
\]
Recall \cref{eq:fscd:7}:
\begin{align*}
\sembr{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}^q &= \pi^{\Xi,\alpha}_\alpha \quad (q \in \{{-},{+}\})
\end{align*}
Projection functors are easily seen to be locally continuous.
\item[Case \rn{C$\rho$}.]
\[
\infer[\rn{C$\rho$}]{
\Xi \vdash \jisst[p]{\Trec{\alpha}{A}}
}{
\Xi, \jisst[p]{\alpha} \vdash \jisst[p]{A}
}
\]
Recall \cref{eq:20051}:
\begin{align*}
\sembr{\jstype{\Xi}{\Trec{\alpha}{A}}}^p &= \sfix{\left(\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p\right)}\quad (p \in \{{-},{+}\})
\end{align*}
By the induction hypothesis, $\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p$ is locally continuous.
By \cite[Proposition~5.2.7]{abramsky_jung_1995:_domain_theor}, we know that whenever $F : \sembr{\Xi} \times \moabcs \to \moabcs$ is locally continuous, then so is $\sfix{F} : \sembr{\Xi} \to \moabcs$.
It follows that the interpretations are locally continuous.
\item[Case \rn{C$\Tds{}$}.]
\[
\infer[\rn{C$\Tds{}$}]{
\jstype[+]{\Xi}{\Tds A}
}{
\jstype[-]{\Xi}{A}
}
\]
Recall \cref{eq:1506,eq:1507}:
\begin{align*}
\sembr{\jstype{\Xi}{\Tds A}}^- &= \sembr{\jstype{\Xi}{A}}^-\\
\sembr{\jstype{\Xi}{\Tds A}}^+ &= \sembr{\jstype{\Xi}{A}}^+_\bot
\end{align*}
By the induction hypothesis, the interpretations for $\jstype{\Xi}{A}$ are locally continuous.
The lifting functor is locally continuous.
Locally continuous functors are closed under composition.
This gives the result.
\item[Case \rn{C$\Tus{}$}.]
\[
\infer[\rn{C$\Tus{}$}]{
\jstype[-]{\Xi}{\Tus A}
}{
\jstype[+]{\Xi}{A}
}
\]
This case is analogous to the case \rn{C$\Tds{}$}.
\item[Case \rn{C$\Tplus$}.]
\[
\infer[\rn{C$\Tplus$}]{
\jstype[+]{\Xi}{{\Tplus\{l : A_l\}}_{l \in L}}
}{
\jstype[+]{\Xi}{A_l}\quad(\forall l \in L)
}
\]
Recall \cref{eq:202020,eq:222222}:
\begin{align*}
\sembr{\jstype{\Xi}{\Tplus\{l:A_l\}_{l \in L}}}^- &= \prod_{l \in L} \sembr{\jstype{\Xi}{A_l}}^-\\
\sembr{\jstype{\Xi}{\Tplus\{l:A_l\}_{l \in L}}}^+ &= \bigoplus_{l \in L} \sembr{\jstype{\Xi}{A_l}}^+_\bot
\end{align*}
By the induction hypothesis, the interpretations for $\jstype{\Xi}{A}$ are locally continuous.
The lifting, coalesced-sum, and product functors are locally continuous.
Locally continuous functors are closed under composition.
This gives the result.
\item[Case \rn{C$\Tamp$}.]
\[
\infer[\rn{C$\Tamp$}]{
\jstype[-]{\Xi}{{\Tamp\{l :A_l \}}_{l \in L}}
}{
\jstype[-]{\Xi}{A_l}\quad(\forall l \in L)
}
\]
This case is analogous to the case \rn{C$\Tplus$}.
\item[Case \rn{C$\Tot$}.]
\[
\infer[\rn{C$\Tot$}]{
\jstype[+]{\Xi}{A \Tot B}
}{
\jstype[+]{\Xi}{A}
&
\jstype[+]{\Xi}{B}
}
\]
Recall \cref{eq:13,eq:14}:
\begin{align*}
\sembr{\jstype{\Xi}{A \Tot B}}^- &= \sembr{\jstype{\Xi}{A}}^- \times \sembr{\jstype{\Xi}{B}}^-\\
\sembr{\jstype{\Xi}{A \Tot B}}^+ &= \left(\sembr{\jstype{\Xi}{A}}^+ \times \sembr{\jstype{\Xi}{B}}^+\right)_\bot
\end{align*}
By the induction hypothesis, the interpretations for $\jstype{\Xi}{A}$ and $\jstype{\Xi}{B}$ are locally continuous.
The lifting and product functors are locally continuous.
Locally continuous functors are closed under composition.
This gives the result.
\item[Case \rn{C$\Tlolly$}.]
\[
\infer[\rn{C$\Tlolly$}]{
\jstype[-]{\Xi}{A \Tlolly B}
}{
\jstype[+]{\Xi}{A}
&
\jstype[-]{\Xi}{B}
}
\]
This case is analogous to the case \rn{C$\Tot$}.
\item[Case \rn{C$\Tand{}{}$}.]
\[
\infer[\rn{C$\Tand{}{}$}]{
\jstype[+]{\Xi}{\Tand{\tau}{A}}
}{
\jisft{\tau}
&
\jstype[+]{\Xi}{A}
}
\]
Recall \cref{eq:15104,eq:15106}:
\begin{align*}
\sembr{\jstype{\Xi}{\Tand{\tau}{A}}}^- &= \sembr{\Xi\vdash \jisst{A}}^-\\
\sembr{\jstype{\Xi}{\Tand{\tau}{A}}}^+ &= \left(\sembr{\tau}\times\sembr{\Xi\vdash \jisst{A}}^+\right)_\bot
\end{align*}
By the induction hypothesis, the interpretations for $\jstype{\Xi}{A}$ are locally continuous.
The constant functor $\sembr{\tau}$, the lifting functor, and the product functors are locally continuous.
Locally continuous functors are closed under composition.
This gives the result.
\item[Case \rn{C$\Timp{}{}$}.]
\[
\infer[\rn{C$\Timp{}{}$}]{
\jstype[-]{\Xi}{\Timp{\tau}{A}}
}{
\jisft{\tau}
&
\jstype[-]{\Xi}{A}
}
\]
This case is analogous to the case \rn{C$\Tand{}{}$}.\qedhere
\end{description}
\end{proof}
\subsection{Clauses for Term Formation (\cref{sec:rules-term-formation})}
\label{sec:claus-term-form}
\begin{description}
\item[Rule \rn{I-\{\}}.] \Cref{eq:191919}:
\begin{equation*}
\sembr{\jtypef{\Psi}{\tProc{a}{P}{\overline{a_i}}}{\Tproc{a:A}{\overline{a_i:A_i}}}} = \up \circ \sembr{\jtypem{\Psi}{\overline{a_i:A_i}}{P}{a}{A}}
\end{equation*}
\item[Rule \rn{F-Var}.] \Cref{eq:39}:
\begin{align*}
\sembr{\jtypef{\Psi,x:\tau}{x}{\tau}}u &= \pi^{\Psi,x}_{x}u
\end{align*}
\item[Rule \rn{F-Fix}.] \Cref{eq:28}:
\begin{align*}
\sembr{\jtypef{\Psi}{\tFix{x}{M}}{\tau}}u &= \sfix{\sembr{\jtypef{\Psi,x:\tau}{M}{\tau}}}u
\end{align*}
\item[Rule \rn{F-Fun}.] \Cref{eq:41}:
\begin{align*}
\sembr{\jtypef{\Psi}{\lambda x: \tau.M}{\tau \to \sigma}}u &= \strictfn\left(\lambda v \in \sembr{\tau}.\sembr{\jtypef{\Psi,x:\tau}{M}{\sigma}}\upd{u}{x \mapsto v}\right)
\end{align*}
\item[Rule \rn{F-App}.] \Cref{eq:42}:
\begin{align*}
\sembr{\jtypef{\Psi}{MN}{\sigma}}u &= \sembr{\jtypef{\Psi}{M}{\tau \to \sigma}}u(\sembr{\jtypef{\Psi}{N}{\tau}}u)
\end{align*}
\end{description}
\subsection{Clauses for Process Formation (\cref{sec:rules-proc-form})}
\label{sec:claus-proc-form}
\begin{description}
\item[Rule \rn{Fwd}.] \Cref{eq:10}:
\[
\sembr{\jtypem{\Psi}{a:A}{\tFwd{b}{a}}{b}{A}}u(a^+ : \alpha, b^- : \beta) = (a^- : \beta, b^+ : \alpha)
\]
\item[Rule \rn{Cut}.] \Cref{eq:11}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta_1, \Delta_2}{\tCut{a}{P}{Q}}{c}{C}}u\\
&= \Tr{\left(\sembr{\jtypem{\Psi}{\Delta_1}{P}{a}{A}}u \times \sembr{\jtypem{\Psi}{a : A,\Delta_2}{Q}{c}{C}}u\right)}{a^- \times a^+}
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tu R$}.] \Cref{eq:1509}:
\[
\sembr{\jtypem{\Psi}{\cdot}{\tClose a}{a}{\Tu}}u(a^- : \bot) = (a^+ : \ast)
\]
\item[Rule \rn{$\Tu L$}.] \Cref{eq:21}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a : \Tu}{\tWait{a}{P}}{c}{C}}u = \strictfn_{a^+}\left(\lambda (\delta^+,a^+,c^-).(\delta^-,\bot,c^+)\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta}{P}{c}{C}}u(\delta^+,c^-) = (\delta^-, c^+)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tds{} R$}.] \Cref{eq:59}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendS{a}{P}}{a}{\Tds A}}u\\
&= \left(\ms{id} \times \left(a^+ : \up\right)\right) \circ \sembr{\jtypem{\Psi}{\Delta}{P}{a}{A}}u
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tds{} L$}.] \Cref{eq:56}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a : \Tds A}{\tRecvS{a}{P}}{c}{C}}u\\
&= \strictfn_{a^+}\left(\sembr{\jtypem{\Psi}{\Delta, a : A}{P}{c}{C}}u \circ \left(\ms{id} \times \left(a^+ : \down\right)\right)\right)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tus{} R$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:3}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tRecvS{a}{P}}{a}{\Tus{A}}}u\\
&= \strictfn_{a^-}\left(\sembr{\jtypem{\Psi}{\Delta}{P}{a}{A}}u \circ \left(\ms{id} \times \left(a^- : \down\right)\right)\right)
\end{aligned}
\end{equation}
\item[Rule \rn{$\Tus{} L$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:4}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a : \Tus A}{\tSendS{a}{P}}{c}{C}}u\\
&= \left(\ms{id} \times \left(a^- : \up\right)\right) \circ \sembr{\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}}u
\end{aligned}
\end{equation}
\item[Rule \rn{$\Tplus R_k$}.] \Cref{eq:52}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendL{a}{k}{P}}{a}{\Tplus\{l:A_l\}_{l \in L}}}u\left(\delta^+, \left(a_l^-\right)_{l \in L}\right) = \left(\delta^-, \left(k, \upim{a_k^+}\right)\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta}{P}{a}{A_k}}u\left(\delta^+, a_k^-\right) = \left(\delta^-, a_k^+\right)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tplus L$}.] \Cref{eq:53}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a:\Tplus\{l:A_l\}_{l \in L}}{\tCase{a}{\left\{l \Rightarrow P_l\right\}_{l \in L}}}{c}{C}}u\\
&= \strictfn_{a^+}\left(\lambda \left(\delta^+, a^+ : \left(k, \upim{a_k^+}\right), c^-\right).\left(\delta^-, a^- : \left(k : a_k^-, l \neq k : \bot\right)_{l \in L}, c^+\right)\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta,a:A_k}{P_k}{c}{C}}u(\delta^+, a_k^+, c^-) = (\delta^-, a_k^-, c^+)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tamp R$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:5}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tCase{a}{\left\{l \Rightarrow P_l\right\}_{l \in L}}}{a}{{\Tamp\{l :A_l \}}_{l \in L}}}u\\
&= \strictfn_{a^-}\left(\lambda \left(\delta^+, a^- : \left(k, \upim{a_k^-}\right)\right).\left(\delta^-, a^+ : \left(k : a_k^+, l \neq k : \bot\right)_{l \in L}\right)\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta}{P_k}{a}{A_k}}u(\delta^+, a_k^-) = (\delta^-, a_k^+)
\end{aligned}
\end{equation}
\item[Rule \rn{$\Tamp L_k$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:6}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a:{\Tamp\{l : A_l\}}_{l \in L}}{\tSendL{a}{k}{P}}{c}{C}}u\left(\delta^+, \left(a_l^+\right)_{l \in L}, c^-\right) = \left(\delta^-, \left(k, \upim{a_k^-}\right), c^+\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta,a:A_k}{P}{c}{C}}u\left(\delta^+, a_k^+, c^-\right) = \left(\delta^-, a_k^-, c^+\right)
\end{aligned}
\end{equation}
\item[Rule \rn{$\Tot R^*$}.] \Cref{eq:17}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, b : B}{\tSendC{a}{b}{P}}{a}{B \Tot A}}u(\delta^+,b^+,(a^-_B,a^-_A)) = \left(\delta^-,a_B^-,\upim{\left(b^+,a_A^+\right)}\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta}{P}{a}{A}}u(\delta^+,a_A^-) = (\delta^-,a_A^+)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tot L$}.] \Cref{eq:26}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, a : B \Tot A}{\tRecvC{b}{a}{Q}}{c}{C}}u(\delta^+,a^+,c^-)\\
&= \strictfn_{a^+}\left(\lambda (\delta^+,a^+ : \upim{(a_B^+, a_A^+)},c^-) . (\delta^-,(b^-,a^-),c^+) \right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta, a : A, b : B}{Q}{c}{C}}u(\delta^+,a_A^+,a_B^+,c^-) = (\delta^-,a^-,b^-,c^+)
\end{aligned}
\end{equation*}
\item[Rule \rn{${\Tlolly}R$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:12}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tRecvC{b}{a}{P}}{a}{B \Tlolly A}}u(\delta^+,a^-)\\
&= \strictfn_{a^-}\left(\lambda (\delta^+,a^- : \upim{(a_B^+, a_A^-)}) . (\delta^+,(b^-,a^+)) \right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta, b : B}{P}{a}{A}}u(\delta^+,a_B^+,a_A^-) = (\delta^-,b^-,a^+)
\end{aligned}
\end{equation}
\item[Rule \rn{${\Tlolly}L$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:28}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, b : B, a : B \Tlolly A}{\tSendC{a}{b}{P}}{c}{C}}u(\delta^+,b^+,(a^-_B,a^+_A),c^-)\\
&= \left(\delta^-,a_B^-,\upim{\left(b^+,a_A^-\right)},c^+\right)\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta,a : A}{P}{c}{C}}u(\delta^+,a_A^+,c^-) = (\delta^-,a_A^-,c^+)
\end{aligned}
\end{equation}
\item[Rule \rn{$\Tand{}{} R$}.] \Cref{eq:1005}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendV{a}{M}{P}}{a}{\Tand{\tau}{A}}}u(\delta^+,a^-)\\
&= \begin{cases}
\bot & \text{if }\sembr{\jtypef{\Psi}{M}{\tau}}u = \bot\\
\left(\delta^-,\upim{\left(v,a^+\right)}\right) & \text{if }\sembr{\jtypef{\Psi}{M}{\tau}}u = v \neq \bot
\end{cases}\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta}{P}{a}{A}}u(\delta^+,a^-) = (\delta^-, a^+)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\Tand{}{} L$}.] \Cref{eq:1006}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, a:\Tand{\tau}{A}}{\tRecvV{x}{a}{P}}{c}{C}}u\\
&= \strictfn_{a^+}\left( \lambda \left(\delta^+,a^+ : \upim{\left(v, \alpha^+\right)},c^-\right) . \right.\\
&\qquad\quad\qquad\qquad \left. \sembr{\jtypem{\Psi,x:\tau}{\Delta, a:A}{P}{c}{C}}\upd{u}{x \mapsto v}(\delta^+, \alpha^+, c^-) \right)
\end{aligned}
\end{equation*}
\item[Rule \rn{${\Timp{}{}} R$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:30}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tRecvV{x}{a}{P}}{a}{\Timp{\tau}{A}}}u\\
&= \strictfn_{a^-}\left( \lambda \left(\delta^+,a^- : \upim{\left(v, \alpha^-\right)}\right) . \sembr{\jtypem{\Psi,x:\tau}{\Delta}{P}{a}{A}}\upd{u}{x \mapsto v}(\delta^+, \alpha^-) \right)
\end{aligned}
\end{equation}
\item[Rule \rn{${\Timp{}{}} L$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:31}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta,a : \Timp{\tau}{A}}{\tSendV{a}{M}{P}}{c}{C}}u(\delta^+,a^+,c^-)\\
&= \begin{cases}
\bot & \text{if }\sembr{\jtypef{\Psi}{M}{\tau}}u = \bot\\
\left(\delta^-,\upim{\left(v,a^-\right)}, c^+\right) & \text{if }\sembr{\jtypef{\Psi}{M}{\tau}}u = v \neq \bot
\end{cases}\\
&\text{where }\sembr{\jtypem{\Psi}{\Delta, a : A}{P}{c}{C}}u(\delta^+,a^+,c^-) = (\delta^-, a^-,c^+)
\end{aligned}
\end{equation}
\item[Rule \rn{$\rho^+R$}.] \Cref{eq:fossacs:2}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tSendU{a}{P}}{a}{\Trec{\alpha}{A}}}u\\
&= \left(\ms{id} \times \left(a^+ : \ms{Fold}\right)\right)
\circ \sembr{\jtypem{\Psi}{\Delta}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}}u
\circ \left(\ms{id} \times \left(a^- : \ms{Unfold}\right)\right)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\rho^+L$}.] \Cref{eq:fossacs:3}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, a : \Trec{\alpha}{A}}{\tRecvU{a}{P}}{c}{C}}u\\
&= \left(\ms{id} \times \left(a^- : \ms{Fold}\right)\right)
\circ \sembr{\jtypem{\Psi}{\Delta, a : [\Trec{\alpha}{A}/\alpha]A}{P}{c}{C}}u
\circ \left(\ms{id} \times \left(a^+ : \ms{Unfold}\right)\right)
\end{aligned}
\end{equation*}
\item[Rule \rn{$\rho^-R$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:50}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta}{\tRecvU{a}{P}}{a}{\Trec{\alpha}{A}}}u\\
&= \left(\ms{id} \times \left(a^+ : \ms{Fold}\right)\right)
\circ \sembr{\jtypem{\Psi}{\Delta}{P}{a}{[\Trec{\alpha}{A}/\alpha]A}}u
\circ \left(\ms{id} \times \left(a^- : \ms{Unfold}\right)\right)
\end{aligned}
\end{equation}
\item[Rule \rn{$\rho^-L$}.] Omitted.
\begin{equation}
\label[intn]{eq:fscd:51}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\Delta, a : \Trec{\alpha}{A}}{\tSendU{a}{P}}{c}{C}}u\\
&= \left(\ms{id} \times \left(a^- : \ms{Fold}\right)\right)
\circ \sembr{\jtypem{\Psi}{\Delta, a : [\Trec{\alpha}{A}/\alpha]A}{P}{c}{C}}u
\circ \left(\ms{id} \times \left(a^+ : \ms{Unfold}\right)\right)
\end{aligned}
\end{equation}
\item[Rule \rn{E-\{\}}] \Cref{eq:115}:
\begin{equation*}
\begin{aligned}
&\sembr{\jtypem{\Psi}{\overline{a_i:A_i}}{\tProc{a}{M}{\overline a_i}}{a}{A}} = \down \circ \sembr{\jtypef{\Psi}{M}{\Tproc{a:A}{\overline{a_i:A_i}}}}
\end{aligned}
\end{equation*}
\end{description}
\subsection{Clauses for Type Formation (\cref{sec:rules-type-formation})}
\label{sec:claus-type-form}
\begin{description}
\item[Rule \rn{C$\Tu$}.] \Cref{eq:1502,eq:1501}:
\begin{align*}
\sembr{\Xi \vdash \jisst{\Tu}}^- &= \lambda \xi.\{\bot\}\\
\sembr{\Xi \vdash \jisst{\Tu}}^+ &= \lambda \xi.\{\ast\}_\bot
\end{align*}
\item[Rule \rn{CVar}.] \Cref{eq:fscd:7}:
\begin{align*}
\sembr{\Xi,\jisst[p]{\alpha}\vdash\jisst[p]{\alpha}}^q &= \pi^{\Xi,\alpha}_\alpha \quad (q \in \{{-},{+}\})
\end{align*}
\item[Rule \rn{C$\rho$}.] \Cref{eq:20051}:
\begin{align*}
\sembr{\Xi\vdash\jisst{\Trec{\alpha}{A}}}^p &= \sfix{\left(\sembr{\Xi,\jisst{\alpha}\vdash \jisst{A}}^p\right)}\quad (p \in \{{-},{+}\})
\end{align*}
\item[Rule \rn{C$\Tds{}$}.] \Cref{eq:1506,eq:1507}:
\begin{align*}
\sembr{\Xi\vdash\jisst{\Tds A}}^- &= \sembr{\Xi\vdash\jisst{A}}^-\\
\sembr{\Xi\vdash\jisst{\Tds A}}^+ &= \sembr{\Xi\vdash\jisst{A}}^+_\bot
\end{align*}
\item[Rule \rn{C$\Tus{}$}.] Omitted.
\begin{align}
\sembr{\Xi\vdash\jisst{\Tus A}}^- &= \sembr{\Xi\vdash\jisst{A}}^-_\bot\label[intn]{eq:fscd:14}\\
\sembr{\Xi\vdash\jisst{\Tus A}}^+ &= \sembr{\Xi\vdash\jisst{A}}^+\label[intn]{eq:fscd:15}
\end{align}
\item[Rule \rn{C$\Tplus$}.] \Cref{eq:202020,eq:222222}:
\begin{align*}
\sembr{\Xi\vdash\jisst{\Tplus\{l:A_l\}_{l \in L}}}^- &= \prod_{l \in L} \sembr{\Xi\vdash\jisst{A_l}}^-\\
\sembr{\Xi\vdash\jisst{\Tplus\{l:A_l\}_{l \in L}}}^+ &= \bigoplus_{l \in L} \sembr{\Xi\vdash\jisst{A_l}}^+_\bot
\end{align*}
\item[Rule \rn{C$\Tamp$}.] Omitted.
\begin{align}
\sembr{\Xi\vdash\jisst{\Tamp\{l:A_l\}_{l \in L}}}^- &= \bigoplus_{l \in L} \sembr{\Xi\vdash\jisst{A_l}}^-_\bot\label[intn]{eq:fscd:19}\\
\sembr{\Xi\vdash\jisst{\Tamp\{l:A_l\}_{l \in L}}}^+ &= \prod_{l \in L} \sembr{\Xi\vdash\jisst{A_l}}^+\label[intn]{eq:fscd:20}
\end{align}
\item[Rule \rn{C$\Tot$}.] \Cref{eq:13,eq:14}:
\begin{align*}
\sembr{\Xi\vdash\jisst{A \Tot B}}^- &= \sembr{\Xi\vdash\jisst{A}}^- \times \sembr{\Xi\vdash\jisst{B}}^-\\
\sembr{\Xi\vdash\jisst{A \Tot B}}^+ &= \left(\sembr{\Xi\vdash\jisst{A}}^+ \times \sembr{\Xi\vdash\jisst{B}}^+\right)_\bot
\end{align*}
\item[Rule \rn{C$\Tlolly$}.] Omitted.
\begin{align}
\sembr{\Xi\vdash\jisst{A \Tlolly B}}^- &= \left(\sembr{\Xi\vdash\jisst{A}}^+ \times \sembr{\Xi\vdash\jisst{B}}^-\right)_\bot\label[intn]{eq:1001}\\
\sembr{\Xi\vdash\jisst{A \Tlolly B}}^+ &= \sembr{\Xi\vdash\jisst{A}}^- \times \sembr{\Xi\vdash\jisst{B}}^+\label[intn]{eq:1002}
\end{align}
\item[Rule \rn{C$\Tand{}{}$}.] \Cref{eq:15104,eq:15106}:
\begin{align*}
\sembr{\Xi\vdash\jisst{\Tand{\tau}{A}}}^- &= \sembr{\Xi\vdash \jisst{A}}^-\\
\sembr{\Xi\vdash\jisst{\Tand{\tau}{A}}}^+ &= \left(\sembr{\tau}\times\sembr{\Xi\vdash \jisst{A}}^+\right)_\bot
\end{align*}
\item[Rule \rn{C$\Timp{}{}$}.] Omitted.
\begin{align}
\sembr{\Xi\vdash\jisst{\Timp{\tau}{A}}}^- &= \left(\sembr{\tau}\times\sembr{\Xi\vdash \jisst{A}}^-\right)_\bot\label[intn]{eq:fscd:24}\\
\sembr{\Xi\vdash\jisst{\Timp{\tau}{A}}}^+ &= \sembr{\Xi\vdash \jisst{A}}^+\label[intn]{eq:fscd:25}
\end{align}
\item[Rule \rn{T\{\}}.] \Cref{eq:72}:
\begin{align*}
\sembr{\jisft{\Tproc{a:A}{\overline{a_i:A_i}}}} &= \sembr{\overline{a_i:A_i} \vdash a : A}_\bot
\end{align*}
\item[Rule \rn{T$\to$}.] \Cref{eq:73}:
\begin{align*}
\sembr{\jisft{\tau \to \sigma}} &= [\sembr{\jisft\tau} \sto \sembr{\jisft{\sigma}}]
\end{align*}
\end{description}
|
1,116,691,498,514 | arxiv | \section{Introduction}
Inverse reinforcement learning (IRL)~\cite{ng2000algorithms} captures the problem of inferring a reward function that reflects the objective function of an expert, from limited observations of the expert's behavior.
Traditionally, IRL required planning algorithms as an inner step~\cite{ziebart2008maximum}, which makes IRL expensive in high-dimensional control tasks.
Later work alleviates this by using adversarial training objectives~\cite{ho2016generative,fu2018learning}, inspired by the generative adversarial network (GAN) method~\cite{goodfellow2014generative}.
Essentially, these methods iteratively train a discriminator to measure the difference between the agent's and the expert's behaviors, and optimize a policy to reduce such difference via reinforcement learning.
Combined with modern deep RL algorithms~\cite{schulman2015trust}, adversarial imitation and IRL show improved scalability to high-dimensional tasks.
Recently, adversarial imitation and IRL have been extended to multi-agent imitation~\cite{song2018multi} and multi-agent IRL~\cite{yu2019multi}, respectively, where the agents in the same environment aim to learn rewards or policies from multiple experts' demonstrations.
Both these previous works have shown strong theoretical relationship between single-agent and multi-agent learning methods, and demonstrated that the proposed methods outperform baselines.
However, there remains some questions on their empirical performances.
First of all, both~\citet{song2018multi} and \citet{yu2019multi} have focused on the performance \textit{after} convergence, but the sample efficiency of the proposed methods in terms of agent-environment interactions has not been considered rigorously.
Also, both used the multi-agent extension of ACKTR, MACK~\cite{wu2017scalable}, which is built upon the \emph{centralized-training decentralized-execution} framework~\cite{lowe2017multi,foerster2018counterfactual} and uses centralized critics to stabilize training.
If such centralized critics are not carefully designed, the resultant MARL algorithm may scale poorly with the number of agents due to the curse of dimensionality, i.e., the joint observation-action space grows exponentially with the number of agents.
In this work, we propose multi-agent discriminator-actor-attention-critic (MA-DAAC), a multi-agent algorithm capable of sample-efficient imitation and inverse reward learning, regardless of the number of agents.
MA-DAAC uses multi-agent actor-attention-critic~\cite{iqbal2019actor} that scales to large numbers of agents thanks to a shared attention-critic network~\cite{vaswani2017attention}.
We verify that MA-DAAC is more sample-efficient than the multi-agent imitation and IRL baselines, and demonstrate that the reward functions learned by MA-DAAC lead to improved empirical performances.
Finally, MA-DAAC is shown to be more robust to smaller number of experts' demonstration.
\section{Preliminaries\vspace{-0.3in}}~\label{sec:background}
\subsection{Markov Games and Notations}
In a Markov Game~\cite{littman1994markov}, multiple agents observe a shared environment state and take individual actions based on their observations.
Then, each agent gets a reward and the environment transitions to a new state.
Mathematically, a Markov Game is defined by
$\langle\mathcal{S}, \{\mathcal{A}_i\}_{i=1}^N, \{r_i\}_{i=1}^N, P_T, \nu, \gamma\rangle$, where
$N$ is the number of agents,
$\mathcal{S}$ is the state space,
$\mathcal{A}_i$ is the action space for agent $i$,
$r_i(s, a_1, ..., a_N)$ is a reward function for agent $i$,
$P_T(s'|s, a_1, ..., a_N)$ is a state transition distribution,
$\nu(s)$ is an initial state distribution,
$\gamma$ is a discount factor.
Also, the policy $\pi_i(a_i|s)$ is the probability of the $i$-th agent choosing an action $a_i$ at the state $s$.
For succinct notation, we use bars to indicate joint quantities over the agents, e.g.,
$\bar{\mathcal{A}}=\mathcal{A}_1\times\cdots\mathcal{A}_N$ is a joint action space,
$\bar{a}=(a_1, ..., a_N)$ is a joint action,
$\bar{\pi}=(\pi_1, ..., \pi_N)$ is a joint policy,
$\bar{r}(s, \bar{a})=(r_1(s, \bar{a}), r_2(s, \bar{a}), ..., r_N(s, \bar{a}))$ is a joint reward.
The value function of the $i$-th agent with respect to $\bar{\pi}$ is defined by
$
Q_i^{\bar{\pi}}(s, \bar{a})=
\mathbb{E}^{\bar{\pi}}
[
{\textstyle \sum_{t=0}^\infty}
\gamma^t
r_i(s_t, \bar{a}_t)
|
s_0=s,
\bar{a}_0=\bar{a}
]
$,
where the superscript $\bar{\pi}$ on the expectation implies that states and joint actions are sampled from $\nu, P_T$ and $\bar{\pi}$.
Additionally, the $\gamma$-discounted state occupancy measure $\rho^{\bar{\pi}}$ of the joint policy $\bar{\pi}$ is defined by
$
\rho^{\bar{\pi}}(s, \bar{a})
=
\mathbb{E}^{\bar{\pi}}
[
\sum_{t=0}^\infty
\gamma^t
\mathbb{I}\{s_t=s, \bar{a}_t=\bar{a}\}
]
$
,
where $\mathbb{I}\{\cdot\}$ is an indicator function.
In this work, we consider a partially observable Markov Game,
where each agent can only use its own observation $o_i\in\mathcal{O}_i$ from the environment's state $s\in\mathcal{S}$.
Due to the partial observability, we consider the policy $\pi_i(a_i|o_i)$ which is the probability that the $i$-th agent chooses an action $a_i$ after observing $o_i$.
Also, we consider value functions of the form $Q_i^{\bar{\pi}}(\bar{o}, \bar{a})$ that use the joint observation $\bar{o}:=(o_1, ..., o_N)$ instead of the state $s$, which is commonly done in previous works~\cite{lowe2017multi,iqbal2019actor}.
\subsection{Multi-Agent Adversarial Imitation and IRL}
In the multi-agent IRL problem, $N$ agents respectively try to mimic $N$ experts' policies $\bar{\pi}^E=(\pi_{1}^E, ..., \pi_{N}^E)$.
Here, each agent is not allowed to access to its own target expert's policy directly and should rely on a limited amount of experts' demonstration.
There are two possible objectives in this problem; (1) policy imitation -- learning policies close to those of the experts -- and (2) reward learning -- recovering reward functions that lead to expert-like behavior.
Multi-agent generative adversarial imitation learning (MA-GAIL) enables agents to learn experts' policies by optimizing the following mini-max objective~\cite{song2018multi}:
\begin{align*}
\min_{\bar{\pi}}
\max_{D_1,\ldots,D_N}
&\mathbb{E}_{s, \bar{a}\sim\rho^{\bar{\pi}}}
\left[
{\textstyle\sum_{i=1}^N}
\log D_i(s, a_i)
\right]\\
&+
\mathbb{E}_{s, \bar{a}\sim\rho^{\bar{\pi}^E}}
\left[
{\textstyle\sum_{i=1}^N}
\log
(1-D_i(s, a_i))
\right].
\end{align*}
In practice, MA-GAIL iteratively optimizes discriminators $D_1, \ldots , D_N$ and policies $\bar{\pi}$, where the discriminators are trained to classify whether state-action pairs come from agents or experts and the policies are optimized with MARL methods and reward functions $\log D_i(s, a_i)$ defined by the discriminators. Although MA-GAIL successfully imitates experts, learned rewards $\log D_i(s, a_i)$ of MA-GAIL cannot be used as reward functions due to discriminators converging to $1/2$~\cite{goodfellow2014generative, fu2018learning}.
MA-AIRL addressed this issue by modifying the following parts in MA-GAIL.
First, structured form of discriminators motivated by logistic stochastic best response equilibrium (LSBRE) was used~\cite{yu2019multi}:
\begin{align*}
D_i(s, a_i, s')&=\frac{\exp(f_i(s, a_i, s'))}{\exp(f_i(s, a_i, s'))+\pi_i(a_i|s)},\\
f_i(s, a_i, s')&=g(s, a_i)+\gamma h(s') - h(s).
\end{align*}
In addition, MA-AIRL used $\log D_i(s, a_i, s') - \log (1 - D_i(s, a_i, s'))$ as reward functions instead of the functions $\log D_i(s, a_i)$ of MA-GAIL.
It turns out that either $f_i$ or $g$ can recover the reward functions that lead to the experts' behavior.
\section{Related Works}
\subsection{Sample-Efficient Adversarial Imitation Learning}
In~\citet{kostrikov2018discriminator}, TD3~\cite{fujimoto2018addressing} was used with a discriminator.
To stabilize their algorithm, they proposed to learn the terminal-state values using a discriminator and use them in the RL inner loop of imitation learning, whereas conventional off-policy reinforcement learning algorithms implicitly consider zero terminal-state values and do not account for them.
In~\citet{sasaki2018sample}, another sample-efficient imitation learning algorithm was proposed.
In contrast with prior works, their method did not use discriminators by considering the Bellman consistency of the imitation learning reward signal.
Then, by using off-policy actor-critic~\cite{degris2012off}, it was shown that the proposed method is much more sample-efficient than GAIL.
\subsection{Scalable Multi-Agent Learning}
For large number of agents,
it has been regarded as a challenging problem for MARL to achieve coordinated multi-agent behavior.
Although existing works using centralized critics such as MADDPG~\cite{lowe2017multi} and COMA~\cite{foerster2018counterfactual} make a handful of agents coordinated, they struggle when the number of agents to manage increases.
This is mainly due to the exponential growth of the critic inputs with the increasing number of agents, which possibly increases the input variance of training as well.
MAAC~\cite{iqbal2019actor} addressed such an issue by using the attention mechanism~\cite{vaswani2017attention} and a shared critic network, and it outperformed existing algorithms for large number of agents.
Thanks to the attention mechanism, MAAC is trained to focus only on part of joint observations and actions, which leads to rapid and efficient training.
Mean-field MARL~\cite{yang2018mean} is another approach to address the scalability issue of MARL, which is based on mean-field approximation for the centralized critics.
However, its application is restricted to the situation where all agents are homogeneous, whereas MAAC can be applied to much general scenarios in which non-homogeneous agents exist.
Meanwhile, there were some multi-agent imitation learning algorithms applied to large-scale environments.
\citet{le2017coordinated} proposed multi-agent imitation learning in the index-free control setting, where agents are not allowed to know the indices of their own expert.
The proposed method trains a model that infers and assigns the role of each agent with rollout trajectories and given experts' demonstration and exploits that model to make highly coordinated behavior.
\citet{sanghvi2019modeling} uses multi-agent imitation learning to learn social group communication among agents with a single shared policy network among agents.
However, they used multi-agent behavioral cloning and focused on the environments with homogeneous agents.
In contrast with those works, \textit{our algorithm can deal with non-homogeneous agents as well}.
In addition, it has been reported in existing literature~\cite{kostrikov2018discriminator,sasaki2018sample,song2018multi} that behavioral cloning performs poorly when there are only small number of experts' demonstration due to the co-variate shift problem~\cite{ross2011reduction}.
For these reasons, we narrow down our scope to MA-GAIL and MA-AIRL in this work.
\begin{algorithm}[t]
\caption{Multi-Agent Discriminator-Actor-Attention-Critic (MA-DAAC)}
\label{algorithm:MAA2CIL}
\begin{algorithmic}[1]
\STATE {\bfseries Input:}
a buffer $\mathcal{B}_A$ for agents' rollout trajectories,
experts' demonstration $\mathcal{B}_E$,
policy networks $\bar{\pi}_{\bar{\theta}}=\{\pi_{\theta_i}\}_{i=1}^N$,
a shared attention critic $\bar{Q}_{\bar{\phi}}=\{Q_{\phi_i}\}_{i=1}^N$, and
discriminators $\bar{D}_{\bar{\omega},\bar{\psi}}=\{D_{\omega_i,\psi_i}\}_{i=1}^N$
\FOR{each iteration}
\STATE Sample rollout trajectories:
$
(\bar{o}, \bar{a}, \bar{o}')\sim \bar{\pi}_{\bar{\theta}}.
$
\STATE Add sampled trajectories to $\mathcal{B}_A$.
\FOR{each training iteration}
\STATE Sample $(\bar{o}^A, \bar{a}^A, \bar{o}'{}^A)$ from $\mathcal{B}_A$.
\STATE Sample $(\bar{o}^E, \bar{a}^E, \bar{o}'{}^E)$ from $\mathcal{B}_E$.
\STATE Update rewards by using discriminators.
\begin{align*}
r_i^A(\bar{o}^A, \bar{a}^A, \bar{o}'{}^A)&=
\log D_{\omega_i,\psi_i}(\bar{o}^A, \bar{a}^A, \bar{o}'{}^A)\\
&-
\log (1 - D_{\omega_i,\psi_i}(\bar{o}^A, \bar{a}^A, \bar{o}'{}^A))
\end{align*}
\STATE
// Policy learning via MAAC
Update $\bar{\theta}, \bar{\phi}$ with \eqref{eq:critic} and \eqref{eq:policy}.
\STATE
// Reward learning
Update $\bar{\omega},\bar{\psi}$ with \eqref{eq:disc_op}.
\ENDFOR
\ENDFOR
\STATE {\bfseries Output:} $\bar{\theta}, \bar{\omega}$
\end{algorithmic}
\end{algorithm}
\section{Our Method\vspace{-0.2in}}~\label{sec:ourapproach}
We consider multi-agent IRL problems where agents desire to learn experts' behavior as well as reward functions that lead to such behavior.
Note that agents cannot access to experts directly, but learning agents are allowed to used the limited amount of experts' demonstration.
In such setting, we introduce MA-DAAC, our multi-agent IRL method as outlined in \textbf{Algorithm~\ref{algorithm:MAA2CIL}}.
Our method iteratively trains discriminators and policies using experts' demonstration and agents' rollout trajectories -- using MARL and a reward signal modeled by the discriminators -- similar to MA-GAIL and MA-AIRL.
\textbf{MARL Algorithm.}
In our method, we use MAAC, which are off-policy MARL algorithms and shown to be sample-efficient and scalable to large number of agents~\cite{iqbal2019actor}.
We summarize it as follows.
Assuming discrete action spaces, let
\begin{align*}
\bar{\vec{Q}}_{\bar{\phi}}(\bar{o}, \bar{a})
=
(\vec{Q}_{\phi_1}(\bar{o}, \bar{a}), ..., \vec{Q}_{\phi_N}(\bar{o}, \bar{a})),
\end{align*}
denote action values, where each element of $\bar{\vec{Q}}_{\bar{\phi}}(\bar{o}, \bar{a})$ is a vector-valued action value of the corresponding agents and $\bar{\phi}=(\phi_1, ..., \phi_N)$ is a set of neural network parameters for the critic network.
In MAAC, the $i$-th agent's action value was modeled as a neural network
\begin{align*}
\vec{Q}_{\phi_i}(\bar{o}, \bar{a})
=
\vec{f}_i(
g_i^{\mbox{\scriptsize{local}}}(o_i),
g_i^{\mbox{\scriptsize{global}}}(\bar{o}, \bar{a})
)
,
\end{align*}
where $g_i^{\mbox{\scriptsize{local}}}$ is a network looking at the $i$-th agent's local observation and action, $g_i^{\mbox{\scriptsize{global}}}$ is another network that considers \emph{other} agents' observations and actions. Finally, $\vec{f}_i$ is a network that takes into account the extracted features from both of the previous neural networks.
The main idea of MAAC is to model $g_i^{\mbox{\scriptsize{global}}}$ with an attention mechanism~\cite{vaswani2017attention} and to share this network among agents, i.e., for the $i$-th agent's embedding $e_i$,
\begin{align*}
g_i^{\mbox{\scriptsize{global}}}(\bar{o}, \bar{a})
=
\sum_{j\ne i}
P_{i\rightarrow j}(o_i, a_i, o_j, a_j)
v_j(o_j, a_j),
\end{align*}
where
\begin{align*}
P_{i\rightarrow j}(o_i, a_i, o_j, a_j)
&\propto
\exp(
(W_Ke_j(o_j, a_j))^T
W_Qe_i(o_i, a_i)
),\\
v_j(o_j, a_j)
&=
\sigma(W_Ve_j(o_j, a_j))
\end{align*}
for element-wise non-linear activation $\sigma$ and shared neural network parameters $W_K$, $W_Q$ and $W_V$ among agents. The objective of the critic training is to minimize the sum of temporal difference (TD) errors:
\begin{align}
\underset{\bar{\phi}}{\mathrm{argmin}}~
\mathbb{E}_{\bar{o}, \bar{a}\sim\rho_{\mathcal{B}}}
\left[
{\textstyle \sum_{i=1}^N }
(
y_i(\bar{o}, \bar{a}, \bar{o}')
-
Q_{\phi_i}(\bar{o}, \bar{a})
)^2
\right],\label{eq:critic}
\end{align}
where $
y_i(\bar{o}, \bar{a}, \bar{o}')
=
r_i(\bar{o}, \bar{a})
+
\gamma
\mathbb{E}_{\bar{a}'\sim\bar{\pi}_{\bar{\theta}'}(\cdot|\bar{o}')}
Q_{\phi_i'}(\bar{o}', \bar{a}')
$, $\bar{o}, \bar{a}\sim\rho_{\mathcal{B}}$ implies $\bar{o}, \bar{a}$ are sampled from an experience replay buffer $\mathcal{B}$, $Q_{\phi_i}$ is the value for $a_i$ in $\vec{Q}_{\phi_i}$, $\bar{\theta}'$ is a target policy parameter, $\phi_i'$ is a target critic parameter, and $r_i$ is the $i$-th agent's reward function.
For policy updates, the policy gradient
\begin{align}
&\mathbb{E}_{
\bar{o}\sim\rho_{\mathcal{B}},
\bar{a}\sim\bar{\pi}_{\bar{\theta}}(\cdot|\bar{o})
}
\nabla_{\theta_i}
\log\pi_{\theta_i}(a_i|o_i)
A_i(\bar{o}, \bar{a}), \label{eq:policy}\\
&A_i(\bar{o}, \bar{a})
=
Q_{\phi_i}(\bar{o}, \bar{a})
-
\sum_{a_i'\in\mathcal{A}_i}
\pi_i(a_i'|o_i)
Q_{\phi_i}(\bar{o}, \bar{a}_{-i}(a_i')) \nonumber
\end{align}
was used, where $\bar{a}_{-i}(a_i')$ is the change of the $i$-th action in $\bar{a}$ to $a_i'$.
\textbf{Discriminator.}
We consider two types of discriminator models that were considered in MA-GAIL~\cite{song2018multi}; a \emph{centralized discriminator} that takes all agents' observations and actions as its input and outputs multi-head classification for each agent; a \emph{decentralized discriminator} that takes each agent's local observations and actions and outputs a single-head classification result.
It should be noted that both sample efficiency and scalability for both discriminators have not been rigorously analyzed in MA-GAIL and MA-AIRL~\cite{song2018multi,yu2019multi}.
Let $
\bar{D}(\bar{o}, \bar{a}, \bar{o}')=(D_1(\bar{o}, \bar{a}, \bar{o}'), ..., D_N(\bar{o}, \bar{a}, \bar{o}'))
$
denote the vector-valued discriminator output.
This is a general expression for both types of discriminators since one can consider a shared feature among $D_1, ..., D_N$
for the centralized discriminator, while decentralized discriminators do not share feature but ignore other agents' observations and actions, i.e., $D_i(o_i, a_i, o_i')$.
For each training iteration, we train discriminator by maximizing
\begin{align}
&\mathbb{E}_{\bar{o}^A, \bar{a}^A\sim\rho_{\bar{\pi}^A}}
\left[
{\textstyle \sum_{i=1}^N}
\log (1 - D_i(\bar{o}^A, \bar{a}^A, \bar{o}'{}^A))
\right]
\nonumber
\\
&+
\mathbb{E}_{\bar{o}^E, \bar{a}^E\sim\rho_{\bar{\pi}^E}}
\left[
{\textstyle \sum_{i=1}^N}
\log D_i(\bar{o}^E, \bar{a}^E, \bar{o}'{}^E)
\right].
\label{eq:disc_op}
\end{align}
Intuitively, discriminators are trained in a way that expert-like behavior gets higher values, whereas non-expert-like behavior results in lower values.
Also, note that the objective $\eqref{eq:disc_op}$ means the use of rollout trajectories sampled from agents' policies $\bar{\pi}^A$ in the first expectation.
In practice, however, we use the samples from the replay buffer \emph{without} off-policy correction to enhance the sample-efficiency of discriminator training via sample reuse.
As shown in our experiments, such an abuse of samples does not harm the performance, which is similar to the results in~\citet{kostrikov2018discriminator}.
Similar to MA-AIRL, we assume the discriminators are parameterized neural networks such that
\begin{align}
D_{\omega_i, \psi_i}(\bar{o}, \bar{a}, \bar{o}')&=\frac{\exp(f_{\omega_i, \psi_i}(\bar{o}, \bar{a}, \bar{o}'))}{\exp(f_{\omega_i, \psi_i}(\bar{o}, \bar{a}, \bar{o}'))+\pi_i(a_i|o_i)},\nonumber\\
f_{\omega_i, \psi_i}(\bar{o}, \bar{a}, \bar{o}')&=g_{\omega_i}(\bar{o}, \bar{a})+\gamma h_{\psi_i}(\bar{o}') - h_{\psi_i}(\bar{o}).\label{eq:reward}
\end{align}
During training, we use $\log D_i(\bar{o}, \bar{a}, \bar{o}') - \log (1 - D_i(\bar{o}, \bar{a}, \bar{o}'))$ for $i=1, ..., N$ as agents' reward functions. Especially for centralized discriminators, we use \emph{observation-only} rewards~\cite{fu2018learning} that ignore the action inputs of the discriminators and lead to slightly better performance than those using action inputs. In Section~\ref{sec:discussion}, we discuss about it in detail.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figure/small_scale.pdf}
\caption{
Imitation performance and reward learning performance in small-scale environments.
The results in the same row share their environments; \emph{Keey Away}, \emph{Cooperative Communication}, \emph{Cooperative Navigation} from top to bottom row.
For the first three columns, we report $\mathrm{NSS}$ of policies during training, where 50, 100 and 200 experts' demonstration were used from left to right column. Note that MA-DAAC converges to the best performance among all methods sample-efficiently.
For the last column, we report $\mathrm{NSS}$ of policies learned by reward functions from MA-DAAC. The result show that MA-DAAC always recover better reward functions than the baselines. Note that means and 95\% confidence intervals over 10 runs are considered.
}
\label{fig:smallscale}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figure/large_scale.pdf}
\caption{
Imitation performance and reward learning performance in \emph{Rover Tower} tasks.
The results in the same row share the number of agents; 8, 12 and 16 from top to bottom row.
For the first three columns, we report $\mathrm{NSS}$ of policies during training, where 50, 100 and 200 experts' demonstration were used from left to right column.
Regardless of the number of agents, MA-DAAC converges much faster than the baselines. Also, its convergence is not affected much by the number of agents, whereas those of baselines becomes slower for the increasing number of agents.
For the last column, we report $\mathrm{NSS}$ of policies learned by reward functions from MA-DAAC.
The policies learned by the rewards from MA-DAAC achieve higher $\mathrm{NSS}$.
Note that means and 99\% confidence intervals over 10 runs are considered.
}
\label{fig:largescale}
\end{figure*}
\section{Experiments\vspace{-0.2in}}~\label{sec:experiments}
Our experiments are designed to answer the following questions:
\emph{1. Is MA-DAAC capable of recovering multi-agent reward functions effectively?}
\newline
\emph{2. Is MA-DAAC sample-efficient in terms of the number of agent-environment interactions and available expert demonstration?}
\newline
\emph{3. Is MA-DAAC scalable to the environments with many of agents?}
We evaluate our methods from two perspectives; \emph{policy imitation} and \emph{reward learning}. We briefly summarize our experiment setup in the following sections and include more detailed information in Appendix.
\subsection{Experiment Setup}
\textbf{Tasks.}
We consider two classes of environments, which respectively cover small-scale and large-scale environments. All of them run on OpenAI Multi-agent Particle Environment (MPE)~\cite{mordatch2017emergence}.
The small-scale environments include:
\emph{Keep Away $-$} There are 2 agents ``reacher" and ``pusher", where reacher tries to reach the goal and pusher tries to push it away from the goal.
\emph{Cooperative Communication $-$} There are 2 agents, ``speaker" and ``listener". One of three landmarks is randomly chosen as a target at each episode and its location can only be seen by speaker. Speaker cannot move, whereas listener can observe speaker's message and move toward the target landmark.
\emph{Cooperative Navigation $-$} There are 3 agents and 3 landmarks, and the goal of agents is to cover as many landmarks as possible.
We also consider a large-scale environments proposed in MAAC~\cite{iqbal2019actor} to measure the scalability of MA-DAAC and the existing methods:
\emph{Rover Tower~\cite{iqbal2019actor} $-$} There are even number of agents (8, 12, 16), where half of them are ``rovers" and the others are ``towers". For each episode, rovers and towers are \emph{randomly paired}, and each tower has its own goal destination. Similar to \textit{Cooperative Communication}, towers cannot move but can communicate with rovers so that rovers can move toward corresponding goals.
\textbf{Experts.}
For experts' policies, we trained policies by using MAAC over either 50,000 episodes (\emph{Keep Away}, \emph{Cooperative Communication}, \emph{Cooperative Navigation}) or 100,000 episodes (\emph{Rover Tower}).
Then, we considered those trained policies as experts and generated trajectories from them, where the actions in those episodes are always taken with the largest probability.
Throughout our experiment, we vary the number of available demonstration from 50, 100 to 200.
\textbf{Performance Measure.}
In multi-agent IRL problems, we need to define a proper performance measure to see the gap between learned agents and experts during training.
However, episodic-score-based measure widely used in single-agent learning~\cite{ho2016generative, kostrikov2018discriminator, sasaki2018sample} cannot be directly used in our problems due to the multiple reward functions for each agent and their \emph{unnormalized} scales.
Therefore, we define the \emph{normalized score similarity} ($\mathrm{NSS}$) as follows:
\begin{align*}
\mathrm{NSS}
=
\frac{1}{N}\sum_{i=1}^N
\frac{{\mbox{score}}_{A_i} - {\mbox{score}}_{R_i}}{{\mbox{score}}_{E_i} - {\mbox{score}}_{R_i}}.
\end{align*}
Here, ${\mbox{score}}_{A_i}$ is the $i$-th agent's (episode) score during training, ${\mbox{score}}_{E_i}$ is the $i$-th expert's \emph{average} score for experts' demonstration, and ${\mbox{score}}_{R_i}$ is the \emph{average} score of the $i$-th agent when uniformly random actions were taken by all agents.
Intuitively, $\mathrm{NSS}$ gets close to 1 if every agent shows expert-like behavior since such behavior will lead to the experts' score.
One advantage of $\mathrm{NSS}$ is that we can evaluate multi-agent imitation performance for both competitive and cooperative tasks.
In our experiments, we show that it is an effective measure.
\subsection{Small-Scale Environments}
\textbf{Policy Imitation.}
The results in the small-scale environments are summarized in \emph{Figure~\ref{fig:smallscale}} (column 1-3).
For all small-scale environments, we demonstrate MA-DAAC converges faster than the baselines.
This is due to the use of MAAC -- an off-policy MARL methods -- rather than using MACK -- an on-policy MARL methods -- proposed by~\citet{song2018multi}.
Also, there is only a negligible gap between the performances of using centralized discriminators and using decentralized discriminators.
It should be noted that in \citet{song2018multi}, imitation learning with centralized discriminators leads to slightly better performance compared to its decentralized counterparts, whereas we get comparable results for both types of discriminators. We believe this small difference comes from using different MACK implementations and experts' demonstration.
\textbf{Reward Learning.}
For all imitation and IRL methods, we first train both policies and rewards over 50,000 episodes and \emph{re-train} policies with MACK and learned rewards over 50,000 episodes.
For rewards from MA-DAAC and MA-AIRL, we use $g_{\omega_i}$ in~\eqref{eq:reward}, a learned reward without potential functions $h_{\psi_i}$~\cite{fu2018learning,yu2019multi}.
For MA-GAIL, we use $\log D_i(\bar{o}, \bar{a})$ as compared by~\citet{yu2019multi}.
The results of reward learning are described in \emph{Figure~\ref{fig:smallscale}} (column 4).
Note that the policies trained by either imitation or IRL methods have comparable mean $\mathrm{NSS}$ for given the number of expert demonstration and environment.
Nevertheless, the rewards learned by either MA-DAAC or MA-AIRL with decentralized discriminators achieve the best retraining performance.
One interesting observation is that rewards from MA-DAAC with centralized discriminators lead to better performance compared to those from MA-AIRL with centralized discriminators.
We believe using the off-policy samples in MA-DAAC makes the reward training more robust since MA-AIRL is trained by on-policy samples and can easily overfit to the latest rollouts. Additional results in small-scale environments are given in Appendix.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figure/num_parameter.pdf}
\caption{The number of trainable parameters for each method with decentralized discriminator in \emph{Rover Tower} tasks. The number of parameters linearly increases for MA-DAAC, whereas it increases for MA-AIRL. Note that MA-DAAC performs better than MA-AIRL for both policy imitation and reward learning with fewer number of parameters in the environments with either 12 or 16 agents.}
\label{fig:largescale2}
\end{figure}
\subsection{Large-Scale Environments}
\textbf{Policy Imitation.}
The imitation performances in the large-scale environments are depicted in Figure~\ref{fig:largescale} (column 1-3).
In the large-scale environments, we observe that sample efficiencies of the methods with decentralized discriminators are extremely higher than those with centralized discriminators. This is in contrast with the results in the small-scale environments in the sense that both efficiencies are comparable in those environments.
This comes from the higher variance of training centralized discriminators in the large-scale environments compared to the small-scale counterparts.
Among the methods with decentralized discriminators, we find out MA-DAAC learns much faster than the baselines.
Especially, MA-DAAC robustly achieves the best $\mathrm{NSS}$ irrespective of the number of agents, whereas the convergence of the baselines becomes slower for the increasing number of agents.
This comes from the fact that MA-DAAC uses a shared attention-based critic as well as the off-policy samples via replay.
\textbf{Reward Learning.}
For the large-scale environments, we train policies and rewards over 100,000 episodes and retrain policies with MAAC and learned reward over 100,000 episodes.
Also, the same reward models as those in the small-scale environments are used.
The reward learning results are described in \emph{Figure~\ref{fig:smallscale}} (column 4).
Among all methods, learned rewards from MA-DAAC with decentralized discriminators leads to the best $\mathrm{NSS}$.
The performance of retrained policies decreases as the number of agents increases.
\textbf{Number of Learnable Parameters.}
MA-DAAC is more efficient than baselines in terms of the number of parameters. As depicted in \emph{Figure~\ref{fig:largescale2}}, the number of MA-DAAC's parameters linearly increases, while the number of the baselines's parameters exponentially increase. Such an exponential increase comes from the fact that MACK doesn't share critic networks among the agents.
Note that the number of discriminator and policy parameters linearly increases for all cases. Additional results in small-scale environments are given in Appendix.
\begin{table}[t]
\caption{Imitation learning performance relative to that of MA-GAIL in \emph{Rover Tower} tasks. Note that the gain becomes much larger for the decreasing amount of available experts' demonstration. The evaluation score after training 100,000 episodes were used.}
\label{table:largescale1}
\begin{center}
\begin{tabular}{ccccc}
\toprule
\multirow{2}{*}{\# Agents} & \multirow{2}{*}{Algorithm} &\multicolumn{3}{c}{\# Expert Traj.}\\
\cline{3-5}
& & 50 & 100 & 200 \\
\midrule
\multirow{2}{*}{8} & MA-AIRL & $\mathrm{ 21.459 }$ & $\mathrm{ 8.279 }$ & $\mathbf{ 1.590 }$ \\
& MA-DAAC & $\mathbf{ 46.684 }$ & $\mathbf{ 12.716 }$ & $\mathrm{ 0.042 }$ \\
\cline{1-5}
\multirow{2}{*}{12} & MA-AIRL & $\mathrm{ 9.285 }$ & $\mathrm{ 8.910 }$ & $\mathbf{ 2.400 }$ \\
& MA-DAAC & $\mathbf{ 44.574 }$ & $\mathbf{ 20.154 }$ & $\mathrm{ 1.406 }$ \\
\cline{1-5}
\multirow{2}{*}{16} & MA-AIRL & $\mathrm{ 6.472 }$ & $\mathrm{ 7.918 }$ & $\mathrm{ 3.538 }$ \\
& MA-DAAC & $\mathbf{ 41.167 }$ & $\mathbf{ 24.982 }$ & $\mathbf{ 8.145 }$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\begin{table}[t]
\caption{Reward learning performance relative to that of MA-GAIL in \emph{Rover Tower} tasks. MA-DAAC consistently performs better than the baselines. The evaluation scores after training 100,000 episodes were used.}
\label{table:largescale2}
\begin{center}
\begin{tabular}{ccccc}
\toprule
\multirow{2}{*}{\# Agents} & \multirow{2}{*}{Algorithm} &\multicolumn{3}{c}{\# Expert Traj.}\\
\cline{3-5}
& & 50 & 100 & 200 \\
\midrule
\multirow{2}{*}{8} & MA-AIRL & $\mathrm{ 25.700 }$ & $\mathrm{ 24.963 }$ & $\mathrm{ 47.944 }$ \\
& MA-DAAC & $\mathbf{ 54.640 }$ & $\mathbf{ 50.468 }$ & $\mathbf{ 64.144 }$ \\
\cline{1-5}
\multirow{2}{*}{12} & MA-AIRL & $\mathrm{ 21.489 }$ & $\mathrm{ 13.177 }$ & $\mathrm{ 25.360 }$ \\
& MA-DAAC & $\mathbf{ 40.887 }$ & $\mathbf{ 36.511 }$ & $\mathbf{ 44.920 }$ \\
\cline{1-5}
\multirow{2}{*}{16} & MA-AIRL & $\mathrm{ 32.875 }$ & $\mathrm{ 16.451 }$ & $\mathrm{ 26.510 }$ \\
& MA-DAAC & $\mathbf{ 56.326 }$ & $\mathbf{ 43.364 }$ & $\mathbf{ 48.945 }$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsection{Effect of Number of Experts' Demonstration}
For both small-scale and large-scale environments, we vary the number of available experts' demonstration among 50, 100, 200 and check its effect on policy imitation and reward learning performances (See \emph{Figure~\ref{fig:smallscale},~\ref{fig:largescale}}).
In the large-scale environments, the performance is highly affected by the number of experts' demonstration, whereas there's a negligible effect in the small-scale environments.
We believe this comes from the different size of joint observation-action spaces between small-scale and large-scale environments.
Specifically for the fixed amount of experts' demonstration, the \emph{effective amount} of training data -- the number of experts' demonstration relative to input dimensions -- decreases and causes discriminators to be more biased as the number of agents increases.
In the end, this leads to learning in the large-scale environments more difficult than learning in the small-scale environments.
Especially for learning with decentralized discriminators in the large-scale environments, we demonstrate that MA-DAAC performs better than the baselines with small amount of experts' demonstration (\emph{Table~\ref{table:largescale1},~\ref{table:largescale2}}).
For policy imitation (\emph{Table~\ref{table:largescale1}}), we observe the relative score of MA-DAAC becomes larger as the number of available demonstration decreases irrespective of the number of agents, whereas the relative score of MA-AIRL is much smaller than that of MA-DAAC.
This result supports that our method is much more robust than the baseline with respect to the number of experts' demonstration.
For reward learning (\emph{Table~\ref{table:largescale2}})), we again observe that the relative socre of MA-DAAC is consistently higher than that of MA-AIRL.
\section{Discussion}\label{sec:discussion}
\emph{Why do decentralized discriminators work well?}
Let's think about the sources of coordination that can lead to successful learning with decentralized discriminators.
The first one is centralized critics, which take joint observations and actions as inputs as used in many MARL algorithms.
The second one is achieved by sampling experts' joint observations and actions that happened in the same time step.
Since experts' joint trajectories include information about how to coordinate at the specific time step, decentralized discriminators can be sufficiently trained toward the coordination of experts.
These two mechanisms allow decentralized discriminators to focus on local experiences, which highly reduces the input spaces and leads to better scalability.
\emph{Why don't we use observation-only decentralized discriminators?}
In single-agent AIRL~\cite{fu2018learning}, it was shown that a discriminator model that ignores action inputs can recover a reward function that is robust to changes of environment.
In the multi-agent problems, however, we find that observation-only discriminators may fail to imitate well, depending on the task.
In \emph{Cooperative Communication}, for example, the speaker's observation $o_s$ is fixed as the color of a goal landmark in an episode, i.e., $o_{s,0}=\cdots=o_{s,T-1}$ for the length $T$ of the episode, and the speaker's action (message) $a_{s,t}$ at $t$ becomes the part of the listener's successor observation $o_{l,t+1}$ at $t+1$. In such setting, if observation-only decentralized discriminators $D_s(o_s, o_s')$ and $D_l(o_l, o_l')$ are used, the speaker cannot learn how to send a correct message since $D_s$ does not include the message information $a_s$. Due to the incorrect message from the speaker, the observation of the listener becomes noisy, which results in poor performance of listener as well. On the other hand, an observation-only centralized discriminator $\bar{D}(o_s, o_l, o_s', o_l')$ does not suffer from the above issue since the centralized discriminators of both speaker and listener can exploit the full observation transition $(o_s, o_l)\rightarrow(o_s', o_l')$ and match the transition with transitions in experts' demonstration.
Such a problem, from the partial observable nature of multi-agent problem, opens a new challenge: learning reward functions that are both scalable and robust. We leave this problem to future work.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure/discussion.pdf}
\caption{Pearson's correlation coefficient (PCC) between ground truth rewards and learned rewards (left) and NSS (right) during retraining with learned rewards. The results imply the experts' behavior can be achieved by the learned rewards having low correlation with the ground truth.}
\label{fig:discussion}
\end{figure}
\emph{Is the correlation between learned rewards and ground truth rewards meaningful?}
In the experiments of MA-AIRL~\cite{yu2019multi}, statistical correlations between ground truth rewards -- the rewards used to train experts' policies -- and learned rewards were regarded as the performance measure of reward learning. We also check statistical correlations in our implementation, but the reward recovery performance reported in~\citet{yu2019multi} cannot be reproduced.
Discrepancies in the results may come from differences between our implementation and that of~\citet{yu2019multi}.
However, we additionally find that high correlation between the learned rewards and the ground truth rewards is not a necessary condition of learned rewards leading to the experts' behavior, as depicted in \emph{Figure.~\ref{fig:discussion}}.
It is well known that IRL problems are ill-defined, therefore there can exist multiple rewards that can be matched to the experts' observed trajectories.
\section{Conclusion}\label{sec:conclusion}
We propose MA-DAAC, a multi-agent IRL method that is much more scalable and sample-efficient than existing works.
We massively and rigorously analyze the performance of MA-DAAC and compare to baselines in terms of sample efficiency and the retraining score with newly defined measure ($\mathrm{NSS}$), using various types of discriminators (decentralized and centralized).
We show that MA-DAAC with decentralized discriminators outperforms other methods.
One interesting future direction is a scalable multi-agent IRL with a centralized discriminator, so that we can efficiently interpret sequential behavior of a large number of agents by looking at the resultant centralized reward functions.
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